The following matrices are the ones we will need to use.
This worksheet is to calculate the generators as given in the Aoki-Ibukiyama paper. You probably want less terms below so they are easier to read if you want to do comparisons. This just shows how to do the easiest ones; one still needs $\theta_4$ and $\chi_{14}$. Note that $\theta_4$ and $\chi_{14}$ are cusp forms ($\chi_{14}$ from the Aoki-Ibukiyama paper and $\theta_{4}$ from an earlier paper on level 3), but we will still need them for the expansion.
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Quadratic form in 2 variables over Integer Ring with coefficients: [ 2 1 ] [ * 2 ] Quadratic form in 2 variables over Integer Ring with coefficients: [ 2 1 ] [ * 2 ] |
{(0, 0, 0): 1, (0, 0, 1): 0, (0, 0, 2): 4, (0, 0, 3): 2, (0, 0, 4): 0, (0, 0, 5): 2, (0, 0, 6): 0, (0, 0, 7): 0, (0, 0, 8): 8, (0, 0, 9): 0, (0, 0, 10): 0, (0, 0, 11): 0, (0, 0, 12): 6, (0, 0, 13): 0, (0, 0, 14): 0, (0, 0, 15): 0, (0, 0, 16): 0, (0, 0, 17): 4, (0, 0, 18): 4, (0, 0, 19): 0, (0, 0, 20): 6, (0, 0, 21): 0, (0, 0, 22): 0, (0, 0, 23): 4, (0, 0, 24): 0, (0, 0, 25): 0, (2, 1, 2): 4, (2, 2, 8): 8, (3, 0, 5): 4} {(0, 0, 0): 1, (0, 0, 1): 0, (0, 0, 2): 4, (0, 0, 3): 2, (0, 0, 4): 0, (0, 0, 5): 2, (0, 0, 6): 0, (0, 0, 7): 0, (0, 0, 8): 8, (0, 0, 9): 0, (0, 0, 10): 0, (0, 0, 11): 0, (0, 0, 12): 6, (0, 0, 13): 0, (0, 0, 14): 0, (0, 0, 15): 0, (0, 0, 16): 0, (0, 0, 17): 4, (0, 0, 18): 4, (0, 0, 19): 0, (0, 0, 20): 6, (0, 0, 21): 0, (0, 0, 22): 0, (0, 0, 23): 4, (0, 0, 24): 0, (0, 0, 25): 0, (2, 1, 2): 4, (2, 2, 8): 8, (3, 0, 5): 4} |
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Since $\chi_{14}$ involves derivatives, it is probably best to encode these as functions with exponentials rather than formal $q$-series.
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Quadratic form in 6 variables over Integer Ring with coefficients: [ 2 -1 0 0 0 0 ] [ * 2 -1 0 0 0 ] [ * * 2 -1 0 -1 ] [ * * * 2 -1 0 ] [ * * * * 2 0 ] [ * * * * * 2 ] Quadratic form in 6 variables over Integer Ring with coefficients: [ 2 -1 0 0 0 0 ] [ * 2 -1 0 0 0 ] [ * * 2 -1 0 -1 ] [ * * * 2 -1 0 ] [ * * * * 2 0 ] [ * * * * * 2 ] |
Quadratic form in 6 variables over Integer Ring with coefficients: [ 4 5 6 4 2 3 ] [ * 10 12 8 4 6 ] [ * * 18 12 6 9 ] [ * * * 10 5 6 ] [ * * * * 4 3 ] [ * * * * * 6 ] Quadratic form in 6 variables over Integer Ring with coefficients: [ 4 5 6 4 2 3 ] [ * 10 12 8 4 6 ] [ * * 18 12 6 9 ] [ * * * 10 5 6 ] [ * * * * 4 3 ] [ * * * * * 6 ] |
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WARNING: Output truncated! full_output.txt ... ^C Traceback (click to the left of this block for traceback) ... __SAGE__ WARNING: Output truncated! full_output.txt ... ^C Traceback (most recent call last): for y3 in [-2..2]: File "", line 1, in <module> File "/tmp/tmpjK1XTI/___code___.py", line 3, in <module> exec compile(u'for x1 in (ellipsis_range(-_sage_const_2 ,Ellipsis,_sage_const_2 )):\n for x2 in (ellipsis_range(-_sage_const_2 ,Ellipsis,_sage_const_2 )):\n for x3 in (ellipsis_range(-_sage_const_2 ,Ellipsis,_sage_const_2 )):\n for x4 in (ellipsis_range(-_sage_const_2 ,Ellipsis,_sage_const_2 )):\n for y1 in (ellipsis_range(-_sage_const_2 ,Ellipsis,_sage_const_2 )):\n for y2 in (ellipsis_range(-_sage_const_2 ,Ellipsis,_sage_const_2 )):\n for y3 in (ellipsis_range(-_sage_const_2 ,Ellipsis,_sage_const_2 )):\n for y4 in (ellipsis_range(-_sage_const_2 ,Ellipsis,_sage_const_2 )):\n (c**_sage_const_2 -d**_sage_const_2 ) File "", line 9, in <module> File "/usr/local/sage-6.10/local/lib/python2.7/site-packages/sage/repl/rich_output/display_manager.py", line 746, in displayhook return self._backend.displayhook(plain_text, rich_output) File "/usr/local/sage-6.10/local/lib/python2.7/site-packages/sage/repl/rich_output/backend_base.py", line 538, in displayhook return self.display_immediately(plain_text, rich_output) File "/usr/local/sage-6.10/local/lib/python2.7/site-packages/sage/repl/rich_output/backend_sagenb.py", line 368, in display_immediately print(rich_output.mathjax()) File "sage/ext/interrupt/interrupt.pyx", line 203, in sage.ext.interrupt.interrupt.sage_python_check_interrupt (/usr/local/sage-6.10/src/build/cythonized/sage/ext/interrupt/interrupt.c:1891) File "sage/ext/interrupt/interrupt.pyx", line 88, in sage.ext.interrupt.interrupt.sig_raise_exception (/usr/local/sage-6.10/src/build/cythonized/sage/ext/interrupt/interrupt.c:925) KeyboardInterrupt __SAGE__ |
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