Newform invariants
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below.
We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 4 x^{10} + 9 x^{8} - 16 x^{6} + 36 x^{4} - 64 x^{2} + 64\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \((\)\( -\nu^{11} - 4 \nu^{9} - \nu^{7} + 8 \nu^{5} - 12 \nu^{3} - 112 \nu \)\()/80\) |
| \(\beta_{2}\) | \(=\) | \((\)\( -3 \nu^{11} + 8 \nu^{9} - 43 \nu^{7} + 44 \nu^{5} - 76 \nu^{3} + 144 \nu \)\()/160\) |
| \(\beta_{3}\) | \(=\) | \((\)\( \nu^{11} - 8 \nu^{9} + 9 \nu^{7} - 20 \nu^{5} + 52 \nu^{3} - 80 \nu \)\()/32\) |
| \(\beta_{4}\) | \(=\) | \((\)\( -\nu^{11} + \nu^{9} - \nu^{7} - 7 \nu^{5} - 12 \nu^{3} - 12 \nu \)\()/20\) |
| \(\beta_{5}\) | \(=\) | \((\)\( \nu^{11} - 2 \nu^{10} - \nu^{9} + 2 \nu^{8} + \nu^{7} - 2 \nu^{6} - 13 \nu^{5} + 26 \nu^{4} + 32 \nu^{3} - 24 \nu^{2} - 28 \nu + 16 \)\()/40\) |
| \(\beta_{6}\) | \(=\) | \((\)\( -\nu^{11} - 2 \nu^{10} + \nu^{9} + 2 \nu^{8} - \nu^{7} - 2 \nu^{6} + 13 \nu^{5} + 26 \nu^{4} - 32 \nu^{3} - 24 \nu^{2} + 28 \nu + 16 \)\()/40\) |
| \(\beta_{7}\) | \(=\) | \((\)\( -2 \nu^{11} + 5 \nu^{10} + 2 \nu^{9} - 20 \nu^{8} - 2 \nu^{7} + 45 \nu^{6} + 26 \nu^{5} - 80 \nu^{4} - 64 \nu^{3} + 100 \nu^{2} + 56 \nu - 240 \)\()/80\) |
| \(\beta_{8}\) | \(=\) | \((\)\( \nu^{11} - 3 \nu^{10} - \nu^{9} - 2 \nu^{8} + \nu^{7} - 3 \nu^{6} - 13 \nu^{5} - 6 \nu^{4} + 32 \nu^{3} + 4 \nu^{2} - 28 \nu - 56 \)\()/40\) |
| \(\beta_{9}\) | \(=\) | \((\)\( 9 \nu^{10} - 24 \nu^{8} + 49 \nu^{6} - 132 \nu^{4} + 308 \nu^{2} - 352 \)\()/80\) |
| \(\beta_{10}\) | \(=\) | \((\)\( -3 \nu^{11} + 8 \nu^{9} - 13 \nu^{7} + 24 \nu^{5} - 46 \nu^{3} + 84 \nu \)\()/40\) |
| \(\beta_{11}\) | \(=\) | \((\)\( 2 \nu^{11} + 11 \nu^{10} - 2 \nu^{9} - 36 \nu^{8} + 2 \nu^{7} + 51 \nu^{6} - 26 \nu^{5} - 88 \nu^{4} + 64 \nu^{3} + 252 \nu^{2} - 56 \nu - 368 \)\()/80\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \((\)\(\beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - 3 \beta_{1}\)\()/4\) |
| \(\nu^{2}\) | \(=\) | \((\)\(\beta_{9} - \beta_{7} + \beta_{6} + 1\)\()/2\) |
| \(\nu^{3}\) | \(=\) | \((\)\(2 \beta_{10} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2}\)\()/2\) |
| \(\nu^{4}\) | \(=\) | \((\)\(\beta_{11} - \beta_{8} - \beta_{7} + 2 \beta_{6} + \beta_{5} - 1\)\()/2\) |
| \(\nu^{5}\) | \(=\) | \((\)\(2 \beta_{10} - 3 \beta_{4} - 2 \beta_{2} + 3 \beta_{1}\)\()/2\) |
| \(\nu^{6}\) | \(=\) | \((\)\(-3 \beta_{11} + 2 \beta_{9} - 3 \beta_{8} + 5 \beta_{7} - 2 \beta_{6} + 9 \beta_{5} + 3\)\()/2\) |
| \(\nu^{7}\) | \(=\) | \((\)\(2 \beta_{10} + 2 \beta_{6} - 2 \beta_{5} - \beta_{4} - 10 \beta_{2} - \beta_{1}\)\()/2\) |
| \(\nu^{8}\) | \(=\) | \((\)\(-9 \beta_{11} + 8 \beta_{9} - 7 \beta_{8} + \beta_{7} - 6 \beta_{6} + 11 \beta_{5} - 15\)\()/2\) |
| \(\nu^{9}\) | \(=\) | \((\)\(6 \beta_{10} - 10 \beta_{6} + 10 \beta_{5} - \beta_{4} - 10 \beta_{3} - 16 \beta_{2} + 7 \beta_{1}\)\()/2\) |
| \(\nu^{10}\) | \(=\) | \((\)\(7 \beta_{11} - 6 \beta_{9} - 17 \beta_{8} - 5 \beta_{7} - 10 \beta_{6} - 5 \beta_{5} - 27\)\()/2\) |
| \(\nu^{11}\) | \(=\) | \((\)\(-34 \beta_{10} - 6 \beta_{6} + 6 \beta_{5} - 7 \beta_{4} - 28 \beta_{3} + 14 \beta_{2} + 5 \beta_{1}\)\()/2\) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).
| \(n\) |
\(326\) |
\(352\) |
\(496\) |
| \(\chi(n)\) |
\(1\) |
\(-1\) |
\(-1\) |
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 9 T_{2}^{4} + 17 T_{2}^{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\).
| $p$ |
$F_p(T)$ |
| $2$ |
\( ( -1 + 17 T^{2} - 9 T^{4} + T^{6} )^{2} \) |
| $3$ |
\( T^{12} \) |
| $5$ |
\( 15625 - 1250 T^{2} + 675 T^{4} - 164 T^{6} + 27 T^{8} - 2 T^{10} + T^{12} \) |
| $7$ |
\( ( -256 + 208 T^{2} - 32 T^{4} + T^{6} )^{2} \) |
| $11$ |
\( ( 64 + 84 T^{2} + 20 T^{4} + T^{6} )^{2} \) |
| $13$ |
\( 4826809 + 742586 T^{2} + 57967 T^{4} + 3500 T^{6} + 343 T^{8} + 26 T^{10} + T^{12} \) |
| $17$ |
\( ( 16384 + 2192 T^{2} + 88 T^{4} + T^{6} )^{2} \) |
| $19$ |
\( ( 6400 + 1232 T^{2} + 64 T^{4} + T^{6} )^{2} \) |
| $23$ |
\( ( 16 + T^{2} )^{6} \) |
| $29$ |
\( ( 104 - 28 T - 6 T^{2} + T^{3} )^{4} \) |
| $31$ |
\( ( 43264 + 6224 T^{2} + 160 T^{4} + T^{6} )^{2} \) |
| $37$ |
\( ( -256 + 2304 T^{2} - 132 T^{4} + T^{6} )^{2} \) |
| $41$ |
\( ( 1024 + 1460 T^{2} + 76 T^{4} + T^{6} )^{2} \) |
| $43$ |
\( ( 256 + 656 T^{2} + 72 T^{4} + T^{6} )^{2} \) |
| $47$ |
\( ( -64 + 84 T^{2} - 20 T^{4} + T^{6} )^{2} \) |
| $53$ |
\( ( 25600 + 3216 T^{2} + 104 T^{4} + T^{6} )^{2} \) |
| $59$ |
\( ( 141376 + 8500 T^{2} + 164 T^{4} + T^{6} )^{2} \) |
| $61$ |
\( ( -256 - 96 T - 2 T^{2} + T^{3} )^{4} \) |
| $67$ |
\( ( -1024 + 10064 T^{2} - 240 T^{4} + T^{6} )^{2} \) |
| $71$ |
\( ( 3356224 + 70484 T^{2} + 468 T^{4} + T^{6} )^{2} \) |
| $73$ |
\( ( -1024 + 512 T^{2} - 52 T^{4} + T^{6} )^{2} \) |
| $79$ |
\( ( 32 + 52 T + 16 T^{2} + T^{3} )^{4} \) |
| $83$ |
\( ( -12544 + 9428 T^{2} - 228 T^{4} + T^{6} )^{2} \) |
| $89$ |
\( ( 16384 + 5044 T^{2} + 140 T^{4} + T^{6} )^{2} \) |
| $97$ |
\( ( -640000 + 31232 T^{2} - 340 T^{4} + T^{6} )^{2} \) |
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