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 \(x^{8} - 4 x^{7} + 16 x^{6} - 34 x^{5} + 63 x^{4} - 74 x^{3} + 70 x^{2} - 38 x + 13\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \( \nu \) |
| \(\beta_{2}\) | \(=\) | \((\)\( -\nu^{7} - 15 \nu^{6} + 32 \nu^{5} - 172 \nu^{4} + 221 \nu^{3} - 426 \nu^{2} + 235 \nu - 159 \)\()/37\) |
| \(\beta_{3}\) | \(=\) | \((\)\( -3 \nu^{7} - 8 \nu^{6} + 22 \nu^{5} - 146 \nu^{4} + 256 \nu^{3} - 390 \nu^{2} + 298 \nu - 70 \)\()/37\) |
| \(\beta_{4}\) | \(=\) | \((\)\( -3 \nu^{7} - 8 \nu^{6} + 22 \nu^{5} - 146 \nu^{4} + 256 \nu^{3} - 427 \nu^{2} + 335 \nu - 181 \)\()/37\) |
| \(\beta_{5}\) | \(=\) | \((\)\( 3 \nu^{7} - 29 \nu^{6} + 89 \nu^{5} - 261 \nu^{4} + 373 \nu^{3} - 498 \nu^{2} + 294 \nu - 152 \)\()/37\) |
| \(\beta_{6}\) | \(=\) | \((\)\( -8 \nu^{7} + 28 \nu^{6} - 114 \nu^{5} + 215 \nu^{4} - 378 \nu^{3} + 366 \nu^{2} - 266 \nu + 97 \)\()/37\) |
| \(\beta_{7}\) | \(=\) | \((\)\( 17 \nu^{7} - 41 \nu^{6} + 159 \nu^{5} - 184 \nu^{4} + 276 \nu^{3} - 84 \nu^{2} + 38 \nu + 39 \)\()/37\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \(\beta_{1}\) |
| \(\nu^{2}\) | \(=\) | \(-\beta_{4} + \beta_{3} + \beta_{1} - 3\) |
| \(\nu^{3}\) | \(=\) | \(\beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{3} - 2 \beta_{1} - 4\) |
| \(\nu^{4}\) | \(=\) | \(2 \beta_{7} + 3 \beta_{6} + 6 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 6 \beta_{1} + 7\) |
| \(\nu^{5}\) | \(=\) | \(-4 \beta_{7} - 3 \beta_{6} + 7 \beta_{5} + 6 \beta_{4} - 12 \beta_{3} - 5 \beta_{2} + \beta_{1} + 26\) |
| \(\nu^{6}\) | \(=\) | \(-17 \beta_{7} - 25 \beta_{6} + 3 \beta_{5} - 24 \beta_{4} - 5 \beta_{3} + 7 \beta_{2} + 27 \beta_{1} - 1\) |
| \(\nu^{7}\) | \(=\) | \(4 \beta_{7} - 16 \beta_{6} - 42 \beta_{5} - 54 \beta_{4} + 51 \beta_{3} + 42 \beta_{2} + 26 \beta_{1} - 122\) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/39\mathbb{Z}\right)^\times\).
| \(n\) |
\(14\) |
\(28\) |
| \(\chi(n)\) |
\(-1\) |
\(\beta_{5}\) |
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 \( T_{2}^{8} + 6 T_{2}^{6} - T_{2}^{4} - 78 T_{2}^{2} + 169 \) acting on \(S_{2}^{\mathrm{new}}(39, [\chi])\).
| $p$ |
$F_p(T)$ |
| $2$ |
\( 169 - 78 T^{2} - T^{4} + 6 T^{6} + T^{8} \) |
| $3$ |
\( 81 + 54 T - 12 T^{3} - 5 T^{4} - 4 T^{5} + 2 T^{7} + T^{8} \) |
| $5$ |
\( 169 + 38 T^{4} + T^{8} \) |
| $7$ |
\( ( 4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4} )^{2} \) |
| $11$ |
\( 2704 - 1248 T^{2} + 140 T^{4} + 24 T^{6} + T^{8} \) |
| $13$ |
\( ( 169 - 52 T + 3 T^{2} - 4 T^{3} + T^{4} )^{2} \) |
| $17$ |
\( 13689 + 3510 T^{2} + 783 T^{4} + 30 T^{6} + T^{8} \) |
| $19$ |
\( ( 16 + 16 T + 20 T^{2} + 8 T^{3} + T^{4} )^{2} \) |
| $23$ |
\( T^{8} \) |
| $29$ |
\( 2474329 - 128986 T^{2} + 5151 T^{4} - 82 T^{6} + T^{8} \) |
| $31$ |
\( ( 484 + 88 T + 8 T^{2} - 4 T^{3} + T^{4} )^{2} \) |
| $37$ |
\( ( 1369 + 592 T + 113 T^{2} + 14 T^{3} + T^{4} )^{2} \) |
| $41$ |
\( 169 + 702 T^{2} + 959 T^{4} - 54 T^{6} + T^{8} \) |
| $43$ |
\( ( 324 - 324 T + 126 T^{2} - 18 T^{3} + T^{4} )^{2} \) |
| $47$ |
\( 11075584 + 9728 T^{4} + T^{8} \) |
| $53$ |
\( ( 13 + 22 T^{2} + T^{4} )^{2} \) |
| $59$ |
\( 43264 - 4992 T^{2} - 16 T^{4} + 24 T^{6} + T^{8} \) |
| $61$ |
\( ( 49 - 7 T + T^{2} )^{4} \) |
| $67$ |
\( ( 2704 + 832 T + 164 T^{2} + 20 T^{3} + T^{4} )^{2} \) |
| $71$ |
\( 43264 + 4992 T^{2} - 16 T^{4} - 24 T^{6} + T^{8} \) |
| $73$ |
\( ( 121 + 154 T + 98 T^{2} + 14 T^{3} + T^{4} )^{2} \) |
| $79$ |
\( ( -2 + T )^{8} \) |
| $83$ |
\( 2704 + 296 T^{4} + T^{8} \) |
| $89$ |
\( 77228944 + 210912 T^{2} - 8596 T^{4} - 24 T^{6} + T^{8} \) |
| $97$ |
\( ( 484 - 572 T + 194 T^{2} - 10 T^{3} + T^{4} )^{2} \) |
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