# symmetric_square

## 1640 days ago by jimlb

Note the code that is implemented computes $L(j,\textrm{Sym}^2,f)$ directly, but not the algebraic part.  This means we need to calculate $\langle f, f \rangle$ first using the formula $\langle f, f \rangle = \frac{(k-1)! L(k,\textrm{Sym}^2 f)}{2^{2k-1}\pi^{k+1}}$.

R=RealField(750);R
 \newcommand{\Bold}{\mathbf{#1}}\Bold{R}
def Cusp_Forms(N,k): S=CuspForms(N,k) return S
S=Cusp_Forms(1,16);S
 \newcommand{\Bold}{\mathbf{#1}}\verb|Cuspidal|\phantom{\verb!x!}\verb|subspace|\phantom{\verb!x!}\verb|of|\phantom{\verb!x!}\verb|dimension|\phantom{\verb!x!}\verb|1|\phantom{\verb!x!}\verb|of|\phantom{\verb!x!}\verb|Modular|\phantom{\verb!x!}\verb|Forms|\phantom{\verb!x!}\verb|space|\phantom{\verb!x!}\verb|of|\phantom{\verb!x!}\verb|dimension|\phantom{\verb!x!}\verb|2|\phantom{\verb!x!}\verb|for|\phantom{\verb!x!}\verb|Modular|\phantom{\verb!x!}\verb|Group|\phantom{\verb!x!}\verb|SL(2,Z)|\phantom{\verb!x!}\verb|of|\phantom{\verb!x!}\verb|weight|\phantom{\verb!x!}\verb|16|\phantom{\verb!x!}\verb|over|\phantom{\verb!x!}\verb|Rational|\phantom{\verb!x!}\verb|Field|
f=S.basis();f
 \newcommand{\Bold}{\mathbf{#1}}q + 216q^{2} - 3348q^{3} + 13888q^{4} + 52110q^{5} + O(q^{6})
L=f.symsquare_lseries(prec=750);L
 \newcommand{\Bold}{\mathbf{#1}}\verb|Dokchitser|\phantom{\verb!x!}\verb|L-series|\phantom{\verb!x!}\verb|of|\phantom{\verb!x!}\verb|conductor|\phantom{\verb!x!}\verb|1|\phantom{\verb!x!}\verb|and|\phantom{\verb!x!}\verb|weight|\phantom{\verb!x!}\verb|31|

Now that we have $\langle f, f \rangle$ we can calculate the algebraic values $L_{\textrm{alg}}(r+k-1,\textrm{Sym}^2 f) = \frac{L(r+k-1,\textrm{Sym}^2 f)}{\langle f, f \rangle \pi^{2r+k-1}}$.

def symsquare_lseries_alg(L,r,f): A=L(r+f.weight()-1)/(Petersson_Product(L, f.weight()) * pi^(2*r+f.weight()-1)) return A

We start by checking that we recover the Loeffler-Zerbes example.

 verbose -1 (370: dokchitser.py, __call__) Warning: Loss of 2 decimal digits due to cancellation \newcommand{\Bold}{\mathbf{#1}}2.16906134759063332432422387083870700932303748188526319309632394974464706450176214523975094722363528137860620368361765490603762578448552476645009519432477453094069562978621956326456951558774955045251445650004274252008735056327 \times 10^{-6} verbose -1 (370: dokchitser.py, __call__) Warning: Loss of 2 decimal digits due to cancellation

The above value for $\langle f, f \rangle$ agrees with what Loeffler-Zerbes obtain for their example.

psi = DirichletGroup(7).0^2;psi
 \newcommand{\Bold}{\mathbf{#1}}\hbox{Dirichlet character modulo } 7 \hbox{ of conductor } 7 \hbox{ mapping } 3 \mapsto \zeta_{6} - 1

We want to calculate $L_{\textrm{alg}}(j,\textrm{Sym}^2 f \otimes \psi) = \frac{(j-1)! (j-16)! G(\psi^{-1})^2}{2^{2j+1} \pi^{2j-15} \langle f, f \rangle} L(j, \textrm{Sym}^2 f \otimes \psi).$

G=DirichletGroup(7); for j in [0..5]: G[j];
 \newcommand{\Bold}{\mathbf{#1}}\hbox{Dirichlet character modulo } 7 \hbox{ of conductor } 1 \hbox{ mapping } 3 \mapsto 1 \newcommand{\Bold}{\mathbf{#1}}\hbox{Dirichlet character modulo } 7 \hbox{ of conductor } 7 \hbox{ mapping } 3 \mapsto \zeta_{6} \newcommand{\Bold}{\mathbf{#1}}\hbox{Dirichlet character modulo } 7 \hbox{ of conductor } 7 \hbox{ mapping } 3 \mapsto \zeta_{6} - 1 \newcommand{\Bold}{\mathbf{#1}}\hbox{Dirichlet character modulo } 7 \hbox{ of conductor } 7 \hbox{ mapping } 3 \mapsto -1 \newcommand{\Bold}{\mathbf{#1}}\hbox{Dirichlet character modulo } 7 \hbox{ of conductor } 7 \hbox{ mapping } 3 \mapsto -\zeta_{6} \newcommand{\Bold}{\mathbf{#1}}\hbox{Dirichlet character modulo } 7 \hbox{ of conductor } 7 \hbox{ mapping } 3 \mapsto -\zeta_{6} + 1
chi=G;
for j in [0..5]: chi(j)*psi(j)
 \newcommand{\Bold}{\mathbf{#1}}0 \newcommand{\Bold}{\mathbf{#1}}1 \newcommand{\Bold}{\mathbf{#1}}1 \newcommand{\Bold}{\mathbf{#1}}1 \newcommand{\Bold}{\mathbf{#1}}1 \newcommand{\Bold}{\mathbf{#1}}1
psi(3)*chi(3);
 \newcommand{\Bold}{\mathbf{#1}}\zeta_{6}

This shows that $\chi$ is the inverse of $\psi$, which is necessary for the algebraic value calculation.

def symsquare_lseries_twisted_alg(L1,L2,j,chi,f): A=factorial(j-1)*factorial(j-16)*(chi.gauss_sum_numerical(prec=200))^2*L1(j)/(Petersson_Product(L2, f.weight()) * pi^(2*j-15)*2^(2*j+1)) return A
L2=L;L2
 \newcommand{\Bold}{\mathbf{#1}}\verb|Dokchitser|\phantom{\verb!x!}\verb|L-series|\phantom{\verb!x!}\verb|of|\phantom{\verb!x!}\verb|conductor|\phantom{\verb!x!}\verb|1|\phantom{\verb!x!}\verb|and|\phantom{\verb!x!}\verb|weight|\phantom{\verb!x!}\verb|31|
L1=f.symsquare_lseries(psi,prec=750);L1
 \newcommand{\Bold}{\mathbf{#1}}\verb|Dokchitser|\phantom{\verb!x!}\verb|L-series|\phantom{\verb!x!}\verb|of|\phantom{\verb!x!}\verb|conductor|\phantom{\verb!x!}\verb|343|\phantom{\verb!x!}\verb|and|\phantom{\verb!x!}\verb|weight|\phantom{\verb!x!}\verb|31|
C=ComplexField(750);C
 \newcommand{\Bold}{\mathbf{#1}}\Bold{C}
C(symsquare_lseries_twisted_alg(L1,L2,22,chi,f))
 \newcommand{\Bold}{\mathbf{#1}}7.83232263839492282800562582562036679173789562759831209434487606978887368056857835690152519442422507761952682200825732806913796773508071478680133729929954138335748594473892529566277707080741610960418593544926951510942279972071 + 10.2324170229589773827189418103229826867469652172945636976268926605408548808005901353710778160406368949479494505149603965468861115297347341088372261990113866226433566955120496088318822542228940850419984831376043340893331659396i

Note the above value agrees with Loeffler-Zerbes.

A=Rational(R(C(symsquare_lseries_twisted_alg(L1,L2,22,chi,f)).real()))
B=Rational(R(C(symsquare_lseries_twisted_alg(L1,L2,22,chi,f)).imag()/sqrt(3)))
lcm(A.denominator(),B.denominator());
 \newcommand{\Bold}{\mathbf{#1}}127958462727044286231596423387725836394844382483979872287245622166898512512567987625226089935319228440295708112910183261959396391123567073351783532786833033178703720686976532270890900116294664668406548044349158836433854060436

I am not sure how they get their value in ${\bf Q}(\sqrt{-3})$.  I tried taking the real part of our value and asking SAGE to convert it to a fraction and then divide the imaginary part by $\sqrt{3}$ and do the same thing, but that didn't work.

continued_fraction(7.83232263839494)
 \newcommand{\Bold}{\mathbf{#1}}7+ \frac{\displaystyle 1}{\displaystyle 1+ \frac{\displaystyle 1}{\displaystyle 4+ \frac{\displaystyle 1}{\displaystyle 1+ \frac{\displaystyle 1}{\displaystyle 26+ \frac{\displaystyle 1}{\displaystyle 1+ \frac{\displaystyle 1}{\displaystyle 1+ \frac{\displaystyle 1}{\displaystyle 1+ \frac{\displaystyle 1}{\displaystyle 6+ \frac{\displaystyle 1}{\displaystyle 4+ \frac{\displaystyle 1}{\displaystyle 1+ \frac{\displaystyle 1}{\displaystyle 3+ \frac{\displaystyle 1}{\displaystyle 1+ \frac{\displaystyle 1}{\displaystyle 2+ \frac{\displaystyle 1}{\displaystyle 12+ \frac{\displaystyle 1}{\displaystyle 3+ \frac{\displaystyle 1}{\displaystyle 3}}}}}}}}}}}}}}}}

We will now try this with $\psi = \chi_{K}$ for $K = {\bf Q}(\sqrt{-3})$.  This is a quadratic character of conductor $3$, so it is constructed a follows. Note that $\psi$ is its own inverse so it is less work than the above calculation.

G=DirichletGroup(3); for j in [0..1]: G[j];
 \newcommand{\Bold}{\mathbf{#1}}\hbox{Dirichlet character modulo } 3 \hbox{ of conductor } 1 \hbox{ mapping } 2 \mapsto 1 \newcommand{\Bold}{\mathbf{#1}}\hbox{Dirichlet character modulo } 3 \hbox{ of conductor } 3 \hbox{ mapping } 2 \mapsto -1
psi=G;psi
 \newcommand{\Bold}{\mathbf{#1}}\hbox{Dirichlet character modulo } 3 \hbox{ of conductor } 3 \hbox{ mapping } 2 \mapsto -1
L1=f.symsquare_lseries(psi);L1
 \newcommand{\Bold}{\mathbf{#1}}\verb|Dokchitser|\phantom{\verb!x!}\verb|L-series|\phantom{\verb!x!}\verb|of|\phantom{\verb!x!}\verb|conductor|\phantom{\verb!x!}\verb|27|\phantom{\verb!x!}\verb|and|\phantom{\verb!x!}\verb|weight|\phantom{\verb!x!}\verb|31|
Rational(CC(symsquare_lseries_twisted_alg(L1,L2,21,chi,f)).real())
 \newcommand{\Bold}{\mathbf{#1}}\frac{136976467}{56317906}