Newform invariants
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) 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 \(x^{9} - 3 x^{8} - 15 x^{7} + 45 x^{6} + 64 x^{5} - 201 x^{4} - 53 x^{3} + 252 x^{2} - 69 x - 26\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \( \nu \) |
| \(\beta_{2}\) | \(=\) | \((\)\( 25 \nu^{8} - 54 \nu^{7} - 419 \nu^{6} + 771 \nu^{5} + 2232 \nu^{4} - 3127 \nu^{3} - 3917 \nu^{2} + 2939 \nu + 773 \)\()/17\) |
| \(\beta_{3}\) | \(=\) | \((\)\( -25 \nu^{8} + 54 \nu^{7} + 419 \nu^{6} - 771 \nu^{5} - 2232 \nu^{4} + 3127 \nu^{3} + 3934 \nu^{2} - 2956 \nu - 841 \)\()/17\) |
| \(\beta_{4}\) | \(=\) | \((\)\( -50 \nu^{8} + 108 \nu^{7} + 838 \nu^{6} - 1542 \nu^{5} - 4464 \nu^{4} + 6271 \nu^{3} + 7834 \nu^{2} - 6014 \nu - 1563 \)\()/17\) |
| \(\beta_{5}\) | \(=\) | \((\)\( -110 \nu^{8} + 241 \nu^{7} + 1847 \nu^{6} - 3457 \nu^{5} - 9865 \nu^{4} + 14126 \nu^{3} + 17347 \nu^{2} - 13564 \nu - 3374 \)\()/17\) |
| \(\beta_{6}\) | \(=\) | \((\)\( -123 \nu^{8} + 265 \nu^{7} + 2071 \nu^{6} - 3794 \nu^{5} - 11095 \nu^{4} + 15461 \nu^{3} + 19596 \nu^{2} - 14789 \nu - 3965 \)\()/17\) |
| \(\beta_{7}\) | \(=\) | \((\)\( 287 \nu^{8} - 624 \nu^{7} - 4821 \nu^{6} + 8932 \nu^{5} + 25758 \nu^{4} - 36393 \nu^{3} - 45333 \nu^{2} + 34825 \nu + 9008 \)\()/17\) |
| \(\beta_{8}\) | \(=\) | \((\)\( -383 \nu^{8} + 830 \nu^{7} + 6432 \nu^{6} - 11877 \nu^{5} - 34335 \nu^{4} + 48383 \nu^{3} + 60309 \nu^{2} - 46310 \nu - 11960 \)\()/17\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \(\beta_{1}\) |
| \(\nu^{2}\) | \(=\) | \(\beta_{3} + \beta_{2} + \beta_{1} + 4\) |
| \(\nu^{3}\) | \(=\) | \(\beta_{4} + 2 \beta_{2} + 8 \beta_{1} + 1\) |
| \(\nu^{4}\) | \(=\) | \(\beta_{8} - \beta_{6} - \beta_{5} - 2 \beta_{4} + 8 \beta_{3} + 10 \beta_{2} + 11 \beta_{1} + 29\) |
| \(\nu^{5}\) | \(=\) | \(-3 \beta_{7} - 2 \beta_{6} - 4 \beta_{5} + 8 \beta_{4} + 23 \beta_{2} + 68 \beta_{1} + 19\) |
| \(\nu^{6}\) | \(=\) | \(12 \beta_{8} - 4 \beta_{7} - 13 \beta_{6} - 17 \beta_{5} - 28 \beta_{4} + 62 \beta_{3} + 97 \beta_{2} + 116 \beta_{1} + 238\) |
| \(\nu^{7}\) | \(=\) | \(\beta_{8} - 53 \beta_{7} - 38 \beta_{6} - 67 \beta_{5} + 46 \beta_{4} + 9 \beta_{3} + 243 \beta_{2} + 608 \beta_{1} + 252\) |
| \(\nu^{8}\) | \(=\) | \(114 \beta_{8} - 89 \beta_{7} - 149 \beta_{6} - 217 \beta_{5} - 313 \beta_{4} + 501 \beta_{3} + 956 \beta_{2} + 1218 \beta_{1} + 2079\) |
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
|
| \(2\) |
\(-1\) |
| \(7\) |
\(-1\) |
| \(11\) |
\(-1\) |
| \(13\) |
\(1\) |
This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):
