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 \(x^{8} - 14 x^{6} - 4 x^{5} + 60 x^{4} + 31 x^{3} - 75 x^{2} - 60 x - 11\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \( \nu \) |
| \(\beta_{2}\) | \(=\) | \( \nu^{2} - 4 \) |
| \(\beta_{3}\) | \(=\) | \((\)\( -\nu^{7} + 2 \nu^{6} + 10 \nu^{5} - 13 \nu^{4} - 34 \nu^{3} + 10 \nu^{2} + 46 \nu + 19 \)\()/3\) |
| \(\beta_{4}\) | \(=\) | \((\)\( 2 \nu^{7} - 4 \nu^{6} - 17 \nu^{5} + 26 \nu^{4} + 38 \nu^{3} - 29 \nu^{2} - 26 \nu - 8 \)\()/3\) |
| \(\beta_{5}\) | \(=\) | \((\)\( \nu^{7} + \nu^{6} - 16 \nu^{5} - 11 \nu^{4} + 73 \nu^{3} + 35 \nu^{2} - 94 \nu - 40 \)\()/3\) |
| \(\beta_{6}\) | \(=\) | \((\)\( 2 \nu^{7} - 4 \nu^{6} - 20 \nu^{5} + 29 \nu^{4} + 65 \nu^{3} - 41 \nu^{2} - 80 \nu - 17 \)\()/3\) |
| \(\beta_{7}\) | \(=\) | \((\)\( -2 \nu^{7} + \nu^{6} + 26 \nu^{5} - 5 \nu^{4} - 101 \nu^{3} - 7 \nu^{2} + 113 \nu + 44 \)\()/3\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \(\beta_{1}\) |
| \(\nu^{2}\) | \(=\) | \(\beta_{2} + 4\) |
| \(\nu^{3}\) | \(=\) | \(\beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + 5 \beta_{1} + 2\) |
| \(\nu^{4}\) | \(=\) | \(\beta_{7} + 2 \beta_{6} + \beta_{5} + 3 \beta_{3} + 8 \beta_{2} + \beta_{1} + 23\) |
| \(\nu^{5}\) | \(=\) | \(10 \beta_{7} + 10 \beta_{6} + 10 \beta_{5} + \beta_{4} + 12 \beta_{3} + 13 \beta_{2} + 28 \beta_{1} + 22\) |
| \(\nu^{6}\) | \(=\) | \(15 \beta_{7} + 23 \beta_{6} + 16 \beta_{5} + 2 \beta_{4} + 36 \beta_{3} + 62 \beta_{2} + 15 \beta_{1} + 149\) |
| \(\nu^{7}\) | \(=\) | \(83 \beta_{7} + 86 \beta_{6} + 85 \beta_{5} + 14 \beta_{4} + 116 \beta_{3} + 126 \beta_{2} + 173 \beta_{1} + 210\) |
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 does not admit any (nontrivial) inner twists.
| \( p \) |
Sign
|
| \(2\) |
\(-1\) |
| \(7\) |
\(1\) |
| \(41\) |
\(-1\) |
This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8036))\):
| \( T_{3}^{8} - 14 T_{3}^{6} + 4 T_{3}^{5} + 60 T_{3}^{4} - 31 T_{3}^{3} - 75 T_{3}^{2} + 60 T_{3} - 11 \) |
| \( T_{5}^{8} - 18 T_{5}^{6} + 12 T_{5}^{5} + 98 T_{5}^{4} - 117 T_{5}^{3} - 115 T_{5}^{2} + 196 T_{5} - 51 \) |
| \(T_{11}^{8} + \cdots\) |
