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 \(x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \( \nu \) |
| \(\beta_{2}\) | \(=\) | \((\)\( -\nu^{6} + 5 \nu^{4} - 5 \nu^{2} - 12 \)\()/20\) |
| \(\beta_{3}\) | \(=\) | \((\)\( -\nu^{7} + 5 \nu^{5} - 5 \nu^{3} - 12 \nu \)\()/40\) |
| \(\beta_{4}\) | \(=\) | \((\)\( 3 \nu^{6} + 5 \nu^{4} + 15 \nu^{2} + 36 \)\()/20\) |
| \(\beta_{5}\) | \(=\) | \((\)\( -3 \nu^{7} - 5 \nu^{5} + 5 \nu^{3} - 16 \nu \)\()/40\) |
| \(\beta_{6}\) | \(=\) | \((\)\( -\nu^{6} - 7 \)\()/5\) |
| \(\beta_{7}\) | \(=\) | \((\)\( \nu^{7} + 3 \nu^{5} + 5 \nu^{3} + 12 \nu \)\()/8\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \(\beta_{1}\) |
| \(\nu^{2}\) | \(=\) | \(\beta_{6} + \beta_{4} - \beta_{2} - 1\) |
| \(\nu^{3}\) | \(=\) | \(\beta_{7} + 2 \beta_{5} - \beta_{3} - \beta_{1}\) |
| \(\nu^{4}\) | \(=\) | \(\beta_{4} + 3 \beta_{2}\) |
| \(\nu^{5}\) | \(=\) | \(\beta_{7} + 5 \beta_{3}\) |
| \(\nu^{6}\) | \(=\) | \(-5 \beta_{6} - 7\) |
| \(\nu^{7}\) | \(=\) | \(-10 \beta_{5} - 10 \beta_{3} - 7 \beta_{1}\) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/65\mathbb{Z}\right)^\times\).
| \(n\) |
\(27\) |
\(41\) |
| \(\chi(n)\) |
\(-1\) |
\(1 - \beta_{4}\) |
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 is the entire newspace \(S_{2}^{\mathrm{new}}(65, [\chi])\).
| $p$ |
$F_p(T)$ |
| $2$ |
\( 1 + 5 T^{2} + 24 T^{4} + 5 T^{6} + T^{8} \) |
| $3$ |
\( ( 1 - T^{2} + T^{4} )^{2} \) |
| $5$ |
\( 625 - 34 T^{4} + T^{8} \) |
| $7$ |
\( ( 9 + 3 T^{2} + T^{4} )^{2} \) |
| $11$ |
\( ( 49 - 7 T^{2} + T^{4} )^{2} \) |
| $13$ |
\( ( 169 - 22 T^{2} + T^{4} )^{2} \) |
| $17$ |
\( ( 441 - 21 T^{2} + T^{4} )^{2} \) |
| $19$ |
\( ( 3 + 3 T + T^{2} )^{4} \) |
| $23$ |
\( ( 441 - 21 T^{2} + T^{4} )^{2} \) |
| $29$ |
\( ( 441 + 21 T^{2} + T^{4} )^{2} \) |
| $31$ |
\( ( 3600 + 132 T^{2} + T^{4} )^{2} \) |
| $37$ |
\( ( 3969 + 63 T^{2} + T^{4} )^{2} \) |
| $41$ |
\( ( 49 - 7 T^{2} + T^{4} )^{2} \) |
| $43$ |
\( 50625 - 25650 T^{2} + 12771 T^{4} - 114 T^{6} + T^{8} \) |
| $47$ |
\( ( 256 - 80 T^{2} + T^{4} )^{2} \) |
| $53$ |
\( ( 144 + 60 T^{2} + T^{4} )^{2} \) |
| $59$ |
\( ( 2209 + 1410 T + 347 T^{2} + 30 T^{3} + T^{4} )^{2} \) |
| $61$ |
\( ( 225 - 180 T + 129 T^{2} - 12 T^{3} + T^{4} )^{2} \) |
| $67$ |
\( 50625 + 49950 T^{2} + 49059 T^{4} + 222 T^{6} + T^{8} \) |
| $71$ |
\( ( 625 + 150 T - 13 T^{2} - 6 T^{3} + T^{4} )^{2} \) |
| $73$ |
\( T^{8} \) |
| $79$ |
\( ( -6 + T )^{8} \) |
| $83$ |
\( ( 4624 - 164 T^{2} + T^{4} )^{2} \) |
| $89$ |
\( ( 1681 - 984 T + 233 T^{2} - 24 T^{3} + T^{4} )^{2} \) |
| $97$ |
\( 6765201 + 390150 T^{2} + 19899 T^{4} + 150 T^{6} + T^{8} \) |
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