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 \(x^{8} - x^{7} - 11 x^{6} + 10 x^{5} + 37 x^{4} - 33 x^{3} - 36 x^{2} + 33 x - 4\):
\(\beta_{0}\) | \(=\) | \( 1 \) |
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} - 3 \) |
\(\beta_{3}\) | \(=\) | \( \nu^{3} - \nu^{2} - 4 \nu + 3 \) |
\(\beta_{4}\) | \(=\) | \( \nu^{5} - \nu^{4} - 7 \nu^{3} + 5 \nu^{2} + 10 \nu - 5 \) |
\(\beta_{5}\) | \(=\) | \( -\nu^{6} + \nu^{5} + 8 \nu^{4} - 6 \nu^{3} - 15 \nu^{2} + 9 \nu + 2 \) |
\(\beta_{6}\) | \(=\) | \( \nu^{6} - 2 \nu^{5} - 6 \nu^{4} + 12 \nu^{3} + 5 \nu^{2} - 15 \nu + 5 \) |
\(\beta_{7}\) | \(=\) | \( \nu^{7} - 10 \nu^{5} - \nu^{4} + 28 \nu^{3} + \nu^{2} - 21 \nu + 4 \) |
\(1\) | \(=\) | \(\beta_0\) |
\(\nu\) | \(=\) | \(\beta_{1}\) |
\(\nu^{2}\) | \(=\) | \(\beta_{2} + 3\) |
\(\nu^{3}\) | \(=\) | \(\beta_{3} + \beta_{2} + 4 \beta_{1}\) |
\(\nu^{4}\) | \(=\) | \(\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 6 \beta_{2} + 13\) |
\(\nu^{5}\) | \(=\) | \(\beta_{6} + \beta_{5} + 2 \beta_{4} + 8 \beta_{3} + 8 \beta_{2} + 18 \beta_{1} + 3\) |
\(\nu^{6}\) | \(=\) | \(9 \beta_{6} + 8 \beta_{5} + 10 \beta_{4} + 10 \beta_{3} + 35 \beta_{2} + 3 \beta_{1} + 64\) |
\(\nu^{7}\) | \(=\) | \(\beta_{7} + 11 \beta_{6} + 11 \beta_{5} + 21 \beta_{4} + 53 \beta_{3} + 57 \beta_{2} + 89 \beta_{1} + 36\) |
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
|
\(13\) |
\(-1\) |
\(31\) |
\(1\) |
This newform can be constructed as the kernel of the linear operator \(T_{2}^{8} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(403))\).