Newform invariants
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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 \(x^{8} - 3 x^{7} + 5 x^{6} - 3 x^{5} + 4 x^{4} + 3 x^{3} + 5 x^{2} + 3 x + 1\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \( \nu \) |
| \(\beta_{2}\) | \(=\) | \((\)\( -\nu^{7} + 2 \nu^{6} - 3 \nu^{5} - 4 \nu^{3} - 7 \nu^{2} - 12 \nu - 7 \)\()/8\) |
| \(\beta_{3}\) | \(=\) | \((\)\( \nu^{7} - 7 \nu^{5} + 20 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} + 6 \nu + 9 \)\()/8\) |
| \(\beta_{4}\) | \(=\) | \((\)\( -\nu^{7} + 4 \nu^{6} - 9 \nu^{5} + 12 \nu^{4} - 16 \nu^{3} + 13 \nu^{2} - 10 \nu - 1 \)\()/8\) |
| \(\beta_{5}\) | \(=\) | \((\)\( -3 \nu^{7} + 10 \nu^{6} - 17 \nu^{5} + 8 \nu^{4} - 4 \nu^{3} - 13 \nu^{2} - 8 \nu - 5 \)\()/8\) |
| \(\beta_{6}\) | \(=\) | \((\)\( 3 \nu^{7} - 12 \nu^{6} + 23 \nu^{5} - 20 \nu^{4} + 16 \nu^{3} + \nu^{2} + 6 \nu - 1 \)\()/8\) |
| \(\beta_{7}\) | \(=\) | \((\)\( -5 \nu^{7} + 18 \nu^{6} - 35 \nu^{5} + 32 \nu^{4} - 28 \nu^{3} - 11 \nu^{2} - 12 \nu - 7 \)\()/8\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \(\beta_{1}\) |
| \(\nu^{2}\) | \(=\) | \(\beta_{6} + \beta_{5} + \beta_{4} - \beta_{2}\) |
| \(\nu^{3}\) | \(=\) | \(\beta_{7} + 3 \beta_{6} + 2 \beta_{5} + \beta_{4} - 3 \beta_{2} - 2 \beta_{1}\) |
| \(\nu^{4}\) | \(=\) | \(3 \beta_{7} + 4 \beta_{6} + \beta_{4} - \beta_{3} - 5 \beta_{2} - 4 \beta_{1}\) |
| \(\nu^{5}\) | \(=\) | \(4 \beta_{7} - 6 \beta_{5} - 4 \beta_{3} - 6 \beta_{2} - 6 \beta_{1} - 1\) |
| \(\nu^{6}\) | \(=\) | \(-16 \beta_{6} - 16 \beta_{5} - 6 \beta_{4} - 6 \beta_{3} - 7 \beta_{1} - 6\) |
| \(\nu^{7}\) | \(=\) | \(-16 \beta_{7} - 51 \beta_{6} - 29 \beta_{5} - 23 \beta_{4} + 29 \beta_{2} - 16\) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/165\mathbb{Z}\right)^\times\).
| \(n\) |
\(46\) |
\(56\) |
\(67\) |
| \(\chi(n)\) |
\(\beta_{4}\) |
\(1\) |
\(1\) |
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 \( T_{2}^{8} + 3 T_{2}^{6} + 5 T_{2}^{5} + 24 T_{2}^{4} - 85 T_{2}^{3} + 177 T_{2}^{2} - 165 T_{2} + 121 \) acting on \(S_{2}^{\mathrm{new}}(165, [\chi])\).
| $p$ |
$F_p(T)$ |
| $2$ |
\( 121 - 165 T + 177 T^{2} - 85 T^{3} + 24 T^{4} + 5 T^{5} + 3 T^{6} + T^{8} \) |
| $3$ |
\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| $5$ |
\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| $7$ |
\( 1681 - 164 T + 1027 T^{2} + 253 T^{3} + 180 T^{4} + 7 T^{5} - 3 T^{6} - T^{7} + T^{8} \) |
| $11$ |
\( 14641 + 3993 T + 968 T^{2} + 11 T^{3} - 85 T^{4} + T^{5} + 8 T^{6} + 3 T^{7} + T^{8} \) |
| $13$ |
\( 1 - 3 T + 11 T^{2} - 39 T^{3} + 94 T^{4} - 87 T^{5} + 39 T^{6} - 6 T^{7} + T^{8} \) |
| $17$ |
\( ( 625 + 125 T + 25 T^{2} + 5 T^{3} + T^{4} )^{2} \) |
| $19$ |
\( 961 - 3813 T + 38671 T^{2} - 2379 T^{3} + 874 T^{4} + 123 T^{5} + 9 T^{6} - 6 T^{7} + T^{8} \) |
| $23$ |
\( ( -1 - 5 T - T^{2} + 5 T^{3} + T^{4} )^{2} \) |
| $29$ |
\( 290521 + 18865 T + 34153 T^{2} - 5285 T^{3} + 844 T^{4} + 95 T^{5} + 17 T^{6} + T^{8} \) |
| $31$ |
\( 19321 + 4309 T + 2829 T^{2} + 253 T^{3} + 174 T^{4} + T^{5} + 11 T^{6} - 3 T^{7} + T^{8} \) |
| $37$ |
\( 185761 + 227999 T + 136645 T^{2} + 49391 T^{3} + 12234 T^{4} + 2111 T^{5} + 255 T^{6} + 19 T^{7} + T^{8} \) |
| $41$ |
\( 4289041 + 2091710 T + 708673 T^{2} + 150785 T^{3} + 23634 T^{4} + 2855 T^{5} + 327 T^{6} + 25 T^{7} + T^{8} \) |
| $43$ |
\( ( 1861 - 63 T - 92 T^{2} + 2 T^{3} + T^{4} )^{2} \) |
| $47$ |
\( 121 - 990 T + 3837 T^{2} - 6815 T^{3} + 10194 T^{4} - 665 T^{5} + 123 T^{6} - 15 T^{7} + T^{8} \) |
| $53$ |
\( 1615441 - 557969 T + 343174 T^{2} - 35863 T^{3} + 7329 T^{4} + 379 T^{5} - 14 T^{6} - 7 T^{7} + T^{8} \) |
| $59$ |
\( 5285401 - 3253085 T + 1296507 T^{2} - 331435 T^{3} + 58754 T^{4} - 7285 T^{5} + 633 T^{6} - 35 T^{7} + T^{8} \) |
| $61$ |
\( 3575881 + 1760521 T + 459192 T^{2} + 41503 T^{3} + 8805 T^{4} - 1043 T^{5} + 242 T^{6} - 21 T^{7} + T^{8} \) |
| $67$ |
\( ( -3379 - 1768 T - 136 T^{2} + 13 T^{3} + T^{4} )^{2} \) |
| $71$ |
\( 5527201 - 2879975 T + 1218443 T^{2} - 269975 T^{3} + 37544 T^{4} - 3625 T^{5} + 347 T^{6} - 25 T^{7} + T^{8} \) |
| $73$ |
\( 84621601 - 18664771 T + 3462030 T^{2} - 293209 T^{3} + 19089 T^{4} + 141 T^{5} + 30 T^{6} - T^{7} + T^{8} \) |
| $79$ |
\( 1437601 - 1402830 T + 669978 T^{2} - 183930 T^{3} + 37084 T^{4} - 4980 T^{5} + 487 T^{6} - 30 T^{7} + T^{8} \) |
| $83$ |
\( 149352841 - 29978113 T + 2466101 T^{2} + 85701 T^{3} + 18074 T^{4} - 297 T^{5} + 359 T^{6} - 11 T^{7} + T^{8} \) |
| $89$ |
\( ( -271 + 132 T + 45 T^{2} - 16 T^{3} + T^{4} )^{2} \) |
| $97$ |
\( 625 - 1875 T + 13000 T^{2} - 3375 T^{3} + 2675 T^{4} + 225 T^{5} - 20 T^{6} - 5 T^{7} + T^{8} \) |
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