Newform invariants
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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 \(x^{6}\mathstrut +\mathstrut \) \(253102\) \(x^{4}\mathstrut +\mathstrut \) \(17425276096\) \(x^{2}\mathstrut +\mathstrut \) \(250659115499520\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \( 18 \nu \) |
| \(\beta_{2}\) | \(=\) | \((\)\( 927 \nu^{5} - 22464 \nu^{4} + 158192658 \nu^{3} - 8746814592 \nu^{2} + 4583765471040 \nu - 519220416540672 \)\()/\)\(1041302528\) |
| \(\beta_{3}\) | \(=\) | \((\)\( -927 \nu^{5} + 162432 \nu^{4} - 158192658 \nu^{3} + 11341271808 \nu^{2} - 4574393748288 \nu - 624717182509056 \)\()/\)\(520651264\) |
| \(\beta_{4}\) | \(=\) | \((\)\( 129501 \nu^{5} + 252288 \nu^{4} + 36839351862 \nu^{3} + 46328530176 \nu^{2} + 2340669617528256 \nu + 1452164483653632 \)\()/\)\(520651264\) |
| \(\beta_{5}\) | \(=\) | \((\)\( 927 \nu^{5} + 2217024 \nu^{4} + 158192658 \nu^{3} + 370146519936 \nu^{2} + 4583765471040 \nu + 9641799262384128 \)\()/\)\(1041302528\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \(\beta_{1}\)\(/18\) |
| \(\nu^{2}\) | \(=\) | \((\)\(\beta_{5}\mathstrut -\mathstrut \) \(8\) \(\beta_{3}\mathstrut -\mathstrut \) \(17\) \(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut -\mathstrut \) \(27335016\)\()/324\) |
| \(\nu^{3}\) | \(=\) | \((\)\(-\)\(736\) \(\beta_{5}\mathstrut +\mathstrut \) \(103\) \(\beta_{4}\mathstrut +\mathstrut \) \(3393\) \(\beta_{3}\mathstrut -\mathstrut \) \(21256\) \(\beta_{2}\mathstrut -\mathstrut \) \(18690797\) \(\beta_{1}\)\()/2916\) |
| \(\nu^{4}\) | \(=\) | \((\)\(-\)\(250237\) \(\beta_{5}\mathstrut +\mathstrut \) \(18272248\) \(\beta_{3}\mathstrut +\mathstrut \) \(36794733\) \(\beta_{2}\mathstrut -\mathstrut \) \(18272248\) \(\beta_{1}\mathstrut +\mathstrut \) \(42588282369096\)\()/4374\) |
| \(\nu^{5}\) | \(=\) | \((\)\(103238176\) \(\beta_{5}\mathstrut -\mathstrut \) \(8788481\) \(\beta_{4}\mathstrut -\mathstrut \) \(481592855\) \(\beta_{3}\mathstrut +\mathstrut \) \(3026837432\) \(\beta_{2}\mathstrut +\mathstrut \) \(1194462231243\) \(\beta_{1}\)\()/1458\) |
Character Values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).
| \(n\) |
\(2\) |
| \(\chi(n)\) |
\(-1\) |
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 \(T_{2}^{6} \) \(\mathstrut +\mathstrut 82005048 T_{2}^{4} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!96\)\( T_{2}^{2} \) \(\mathstrut +\mathstrut \)\(85\!\cdots\!80\)\( \) acting on \(S_{25}^{\mathrm{new}}(3, [\chi])\).
