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 \(x^{8} + 14 x^{6} + 61 x^{4} + 84 x^{2} + 4\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \((\)\( \nu^{5} + 9 \nu^{3} + 16 \nu \)\()/2\) |
| \(\beta_{2}\) | \(=\) | \((\)\( \nu^{6} + 11 \nu^{4} + 30 \nu^{2} + 4 \nu + 12 \)\()/4\) |
| \(\beta_{3}\) | \(=\) | \((\)\( -\nu^{6} - 9 \nu^{4} - 16 \nu^{2} + 2 \)\()/2\) |
| \(\beta_{4}\) | \(=\) | \((\)\( -\nu^{6} - 11 \nu^{4} - 30 \nu^{2} + 4 \nu - 12 \)\()/4\) |
| \(\beta_{5}\) | \(=\) | \((\)\( \nu^{5} + 11 \nu^{3} + 26 \nu \)\()/2\) |
| \(\beta_{6}\) | \(=\) | \((\)\( -\nu^{6} - 11 \nu^{4} - 26 \nu^{2} + 4 \)\()/2\) |
| \(\beta_{7}\) | \(=\) | \((\)\( \nu^{7} + 13 \nu^{5} + 50 \nu^{3} + 54 \nu \)\()/2\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \((\)\(\beta_{4} + \beta_{2}\)\()/2\) |
| \(\nu^{2}\) | \(=\) | \((\)\(\beta_{6} - \beta_{4} + \beta_{2} - 8\)\()/2\) |
| \(\nu^{3}\) | \(=\) | \((\)\(2 \beta_{5} - 5 \beta_{4} - 5 \beta_{2} - 2 \beta_{1}\)\()/2\) |
| \(\nu^{4}\) | \(=\) | \((\)\(-7 \beta_{6} + 5 \beta_{4} + 2 \beta_{3} - 5 \beta_{2} + 42\)\()/2\) |
| \(\nu^{5}\) | \(=\) | \((\)\(-18 \beta_{5} + 29 \beta_{4} + 29 \beta_{2} + 22 \beta_{1}\)\()/2\) |
| \(\nu^{6}\) | \(=\) | \((\)\(47 \beta_{6} - 29 \beta_{4} - 22 \beta_{3} + 29 \beta_{2} - 246\)\()/2\) |
| \(\nu^{7}\) | \(=\) | \((\)\(4 \beta_{7} + 134 \beta_{5} - 181 \beta_{4} - 181 \beta_{2} - 186 \beta_{1}\)\()/2\) |
Character Values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).
| \(n\) |
\(127\) |
\(421\) |
\(449\) |
\(577\) |
| \(\chi(n)\) |
\(-1\) |
\(1\) |
\(1\) |
\(-1\) |
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 \( T_{19}^{4} - 4 T_{19}^{3} - 44 T_{19}^{2} + 96 T_{19} + 64 \) acting on \(S_{2}^{\mathrm{new}}(672, [\chi])\).
