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 \(x^{8} - 4 x^{7} + 14 x^{5} + 105 x^{4} - 238 x^{3} - 426 x^{2} + 548 x + 3140\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \((\)\( 107 \nu^{7} + 1001 \nu^{6} - 4543 \nu^{5} - 13153 \nu^{4} + 51884 \nu^{3} + 72002 \nu^{2} - 258088 \nu - 983740 \)\()/137550\) |
| \(\beta_{2}\) | \(=\) | \((\)\( 39 \nu^{7} - 595 \nu^{6} + 1995 \nu^{5} - 749 \nu^{4} - 406 \nu^{3} - 76426 \nu^{2} + 98312 \nu + 158560 \)\()/45850\) |
| \(\beta_{3}\) | \(=\) | \((\)\( 73 \nu^{7} + 1120 \nu^{6} - 4025 \nu^{5} + 5887 \nu^{4} + 4648 \nu^{3} + 61978 \nu^{2} - 65216 \nu + 73420 \)\()/68775\) |
| \(\beta_{4}\) | \(=\) | \((\)\( 214 \nu^{7} - 749 \nu^{6} - 833 \nu^{5} + 3955 \nu^{4} + 29491 \nu^{3} - 48566 \nu^{2} - 285092 \nu + 150790 \)\()/137550\) |
| \(\beta_{5}\) | \(=\) | \((\)\( -78 \nu^{7} + 273 \nu^{6} - 1239 \nu^{5} + 2415 \nu^{4} - 5607 \nu^{3} + 6132 \nu^{2} - 46236 \nu + 22170 \)\()/45850\) |
| \(\beta_{6}\) | \(=\) | \((\)\( -146 \nu^{7} + 511 \nu^{6} - 203 \nu^{5} - 770 \nu^{4} - 17549 \nu^{3} + 27349 \nu^{2} - 18122 \nu + 4465 \)\()/68775\) |
| \(\beta_{7}\) | \(=\) | \((\)\( -24 \nu^{7} + 84 \nu^{6} + 42 \nu^{5} - 315 \nu^{4} - 1302 \nu^{3} + 2310 \nu^{2} + 8628 \nu - 4253 \)\()/917\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \((\)\(-3 \beta_{6} + \beta_{5} - 3 \beta_{4} + 3\)\()/6\) |
| \(\nu^{2}\) | \(=\) | \((\)\(-2 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_{1} + 6\)\()/3\) |
| \(\nu^{3}\) | \(=\) | \((\)\(3 \beta_{7} - 36 \beta_{6} + 2 \beta_{5} + 12 \beta_{4} - 3 \beta_{3} - 9 \beta_{2} - 3 \beta_{1} + 15\)\()/6\) |
| \(\nu^{4}\) | \(=\) | \(\beta_{7} - 10 \beta_{6} + 8 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} - 8 \beta_{1} - 49\) |
| \(\nu^{5}\) | \(=\) | \((\)\(25 \beta_{7} + 52 \beta_{6} - 254 \beta_{5} + 272 \beta_{4} + 35 \beta_{3} - 45 \beta_{2} - 115 \beta_{1} - 767\)\()/6\) |
| \(\nu^{6}\) | \(=\) | \((\)\(30 \beta_{7} + 248 \beta_{6} - 296 \beta_{5} + 304 \beta_{4} + 229 \beta_{3} + 132 \beta_{2} - 26 \beta_{1} - 954\)\()/3\) |
| \(\nu^{7}\) | \(=\) | \((\)\(-217 \beta_{7} + 3104 \beta_{6} - 2458 \beta_{5} + 52 \beta_{4} + 1477 \beta_{3} + 1071 \beta_{2} + 217 \beta_{1} - 3805\)\()/6\) |
Character Values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7524\mathbb{Z}\right)^\times\).
| \(n\) |
\(2377\) |
\(3763\) |
\(4105\) |
\(6689\) |
| \(\chi(n)\) |
\(-1\) |
\(1\) |
\(-1\) |
\(1\) |
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 can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{2}^{\mathrm{new}}(7524, [\chi])\).
