Newform invariants
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) 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 \(x^{10} - 4 x^{9} - 11 x^{8} + 56 x^{7} + 26 x^{6} - 266 x^{5} + 52 x^{4} + 526 x^{3} - 255 x^{2} - 372 x + 239\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \( \nu \) |
| \(\beta_{2}\) | \(=\) | \( \nu^{2} - 4 \) |
| \(\beta_{3}\) | \(=\) | \((\)\( \nu^{8} - 4 \nu^{7} - 8 \nu^{6} + 46 \nu^{5} - 2 \nu^{4} - 152 \nu^{3} + 90 \nu^{2} + 154 \nu - 121 \)\()/2\) |
| \(\beta_{4}\) | \(=\) | \((\)\( \nu^{9} - 4 \nu^{8} - 8 \nu^{7} + 46 \nu^{6} - 4 \nu^{5} - 148 \nu^{4} + 110 \nu^{3} + 120 \nu^{2} - 163 \nu + 52 \)\()/2\) |
| \(\beta_{5}\) | \(=\) | \((\)\( -\nu^{9} + 5 \nu^{8} + 4 \nu^{7} - 54 \nu^{6} + 50 \nu^{5} + 148 \nu^{4} - 262 \nu^{3} - 46 \nu^{2} + 315 \nu - 153 \)\()/2\) |
| \(\beta_{6}\) | \(=\) | \( -\nu^{8} + 4 \nu^{7} + 9 \nu^{6} - 47 \nu^{5} - 9 \nu^{4} + 160 \nu^{3} - 59 \nu^{2} - 165 \nu + 102 \) |
| \(\beta_{7}\) | \(=\) | \( \nu^{8} - 4 \nu^{7} - 9 \nu^{6} + 48 \nu^{5} + 8 \nu^{4} - 169 \nu^{3} + 67 \nu^{2} + 180 \nu - 114 \) |
| \(\beta_{8}\) | \(=\) | \( -\nu^{8} + 4 \nu^{7} + 9 \nu^{6} - 48 \nu^{5} - 8 \nu^{4} + 170 \nu^{3} - 68 \nu^{2} - 185 \nu + 117 \) |
| \(\beta_{9}\) | \(=\) | \((\)\( -2 \nu^{9} + 9 \nu^{8} + 14 \nu^{7} - 104 \nu^{6} + 28 \nu^{5} + 342 \nu^{4} - 268 \nu^{3} - 318 \nu^{2} + 348 \nu - 57 \)\()/2\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \(\beta_{1}\) |
| \(\nu^{2}\) | \(=\) | \(\beta_{2} + 4\) |
| \(\nu^{3}\) | \(=\) | \(\beta_{8} + \beta_{7} + \beta_{2} + 5 \beta_{1} + 1\) |
| \(\nu^{4}\) | \(=\) | \(\beta_{5} + \beta_{4} - \beta_{3} + 8 \beta_{2} + \beta_{1} + 22\) |
| \(\nu^{5}\) | \(=\) | \(9 \beta_{8} + 10 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 9 \beta_{2} + 31 \beta_{1} + 11\) |
| \(\nu^{6}\) | \(=\) | \(\beta_{8} + 2 \beta_{7} + 2 \beta_{6} + 12 \beta_{5} + 12 \beta_{4} - 10 \beta_{3} + 58 \beta_{2} + 13 \beta_{1} + 140\) |
| \(\nu^{7}\) | \(=\) | \(\beta_{9} + 67 \beta_{8} + 82 \beta_{7} + 17 \beta_{6} + 13 \beta_{5} + 15 \beta_{4} - 10 \beta_{3} + 73 \beta_{2} + 211 \beta_{1} + 95\) |
| \(\nu^{8}\) | \(=\) | \(4 \beta_{9} + 14 \beta_{8} + 36 \beta_{7} + 38 \beta_{6} + 104 \beta_{5} + 112 \beta_{4} - 74 \beta_{3} + 420 \beta_{2} + 130 \beta_{1} + 951\) |
| \(\nu^{9}\) | \(=\) | \(24 \beta_{9} + 472 \beta_{8} + 638 \beta_{7} + 200 \beta_{6} + 120 \beta_{5} + 170 \beta_{4} - 68 \beta_{3} + 586 \beta_{2} + 1495 \beta_{1} + 782\) |
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
|
| \(3\) |
\(1\) |
| \(7\) |
\(-1\) |
| \(41\) |
\(1\) |
This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):
