Newform invariants
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) 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 \(x^{9} - x^{8} - 11 x^{7} + 9 x^{6} + 32 x^{5} - 17 x^{4} - 27 x^{3} + 10 x^{2} + 3 x - 1\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \( \nu \) |
| \(\beta_{2}\) | \(=\) | \((\)\( 3 \nu^{8} + 4 \nu^{7} - 41 \nu^{6} - 47 \nu^{5} + 164 \nu^{4} + 141 \nu^{3} - 181 \nu^{2} - 89 \nu + 31 \)\()/13\) |
| \(\beta_{3}\) | \(=\) | \((\)\( 7 \nu^{8} - 8 \nu^{7} - 74 \nu^{6} + 68 \nu^{5} + 192 \nu^{4} - 100 \nu^{3} - 106 \nu^{2} + 22 \nu - 23 \)\()/13\) |
| \(\beta_{4}\) | \(=\) | \( -\nu^{8} + \nu^{7} + 11 \nu^{6} - 9 \nu^{5} - 32 \nu^{4} + 17 \nu^{3} + 27 \nu^{2} - 10 \nu - 3 \) |
| \(\beta_{5}\) | \(=\) | \((\)\( 16 \nu^{8} - 9 \nu^{7} - 184 \nu^{6} + 70 \nu^{5} + 580 \nu^{4} - 67 \nu^{3} - 532 \nu^{2} - 24 \nu + 70 \)\()/13\) |
| \(\beta_{6}\) | \(=\) | \((\)\( -25 \nu^{8} + 23 \nu^{7} + 268 \nu^{6} - 189 \nu^{5} - 734 \nu^{4} + 242 \nu^{3} + 555 \nu^{2} - 34 \nu - 59 \)\()/13\) |
| \(\beta_{7}\) | \(=\) | \((\)\( -27 \nu^{8} + 16 \nu^{7} + 304 \nu^{6} - 123 \nu^{5} - 917 \nu^{4} + 122 \nu^{3} + 771 \nu^{2} - 18 \nu - 71 \)\()/13\) |
| \(\beta_{8}\) | \(=\) | \((\)\( 57 \nu^{8} - 41 \nu^{7} - 636 \nu^{6} + 329 \nu^{5} + 1894 \nu^{4} - 389 \nu^{3} - 1606 \nu^{2} + 38 \nu + 160 \)\()/13\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \(\beta_{1}\) |
| \(\nu^{2}\) | \(=\) | \(\beta_{8} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} + \beta_{1} + 3\) |
| \(\nu^{3}\) | \(=\) | \(\beta_{5} + \beta_{4} - \beta_{2} + 5 \beta_{1}\) |
| \(\nu^{4}\) | \(=\) | \(8 \beta_{8} + \beta_{7} + 7 \beta_{6} - 8 \beta_{5} + 7 \beta_{4} - 2 \beta_{3} - 7 \beta_{2} + 7 \beta_{1} + 16\) |
| \(\nu^{5}\) | \(=\) | \(-\beta_{8} - 2 \beta_{7} + 9 \beta_{5} + 8 \beta_{4} - \beta_{3} - 10 \beta_{2} + 30 \beta_{1} - 1\) |
| \(\nu^{6}\) | \(=\) | \(56 \beta_{8} + 6 \beta_{7} + 47 \beta_{6} - 57 \beta_{5} + 50 \beta_{4} - 20 \beta_{3} - 51 \beta_{2} + 47 \beta_{1} + 100\) |
| \(\nu^{7}\) | \(=\) | \(-10 \beta_{8} - 24 \beta_{7} + 66 \beta_{5} + 61 \beta_{4} - 14 \beta_{3} - 81 \beta_{2} + 197 \beta_{1} - 12\) |
| \(\nu^{8}\) | \(=\) | \(386 \beta_{8} + 28 \beta_{7} + 320 \beta_{6} - 396 \beta_{5} + 358 \beta_{4} - 161 \beta_{3} - 372 \beta_{2} + 322 \beta_{1} + 663\) |
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
|
| \(2\) |
\(-1\) |
| \(5\) |
\(-1\) |
| \(151\) |
\(1\) |
This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6040))\):
