Newform invariants
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{12}\) 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 \(x^{13} - 4 x^{12} - 11 x^{11} + 56 x^{10} + 26 x^{9} - 263 x^{8} + 50 x^{7} + 478 x^{6} - 174 x^{5} - 311 x^{4} + 84 x^{3} + 69 x^{2} - 12 x - 4\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \( \nu \) |
| \(\beta_{2}\) | \(=\) | \( \nu^{2} - 3 \) |
| \(\beta_{3}\) | \(=\) | \((\)\( -105 \nu^{12} + 402 \nu^{11} + 1471 \nu^{10} - 6122 \nu^{9} - 7424 \nu^{8} + 32643 \nu^{7} + 19248 \nu^{6} - 71846 \nu^{5} - 35248 \nu^{4} + 56201 \nu^{3} + 35530 \nu^{2} - 1463 \nu - 4612 \)\()/2162\) |
| \(\beta_{4}\) | \(=\) | \((\)\( 944 \nu^{12} - 3120 \nu^{11} - 12772 \nu^{10} + 44868 \nu^{9} + 58087 \nu^{8} - 219999 \nu^{7} - 110536 \nu^{6} + 428762 \nu^{5} + 119516 \nu^{4} - 298916 \nu^{3} - 91598 \nu^{2} + 44430 \nu + 11577 \)\()/1081\) |
| \(\beta_{5}\) | \(=\) | \((\)\( 2617 \nu^{12} - 8228 \nu^{11} - 37363 \nu^{10} + 120936 \nu^{9} + 187114 \nu^{8} - 614931 \nu^{7} - 418518 \nu^{6} + 1279188 \nu^{5} + 499382 \nu^{4} - 1003841 \nu^{3} - 321158 \nu^{2} + 182903 \nu + 44488 \)\()/2162\) |
| \(\beta_{6}\) | \(=\) | \((\)\( 1537 \nu^{12} - 5483 \nu^{11} - 19453 \nu^{10} + 78197 \nu^{9} + 75852 \nu^{8} - 378472 \nu^{7} - 93135 \nu^{6} + 723044 \nu^{5} + 43093 \nu^{4} - 495112 \nu^{3} - 63961 \nu^{2} + 80294 \nu + 9559 \)\()/1081\) |
| \(\beta_{7}\) | \(=\) | \((\)\( 1680 \nu^{12} - 6432 \nu^{11} - 19212 \nu^{10} + 89304 \nu^{9} + 55005 \nu^{8} - 412026 \nu^{7} + 21737 \nu^{6} + 714974 \nu^{5} - 146249 \nu^{4} - 398713 \nu^{3} + 5531 \nu^{2} + 50433 \nu + 7851 \)\()/1081\) |
| \(\beta_{8}\) | \(=\) | \((\)\( -4041 \nu^{12} + 14730 \nu^{11} + 49447 \nu^{10} - 207380 \nu^{9} - 177062 \nu^{8} + 981221 \nu^{7} + 151906 \nu^{6} - 1793936 \nu^{5} + 9716 \nu^{4} + 1126967 \nu^{3} + 153280 \nu^{2} - 171323 \nu - 31592 \)\()/2162\) |
| \(\beta_{9}\) | \(=\) | \((\)\( 4041 \nu^{12} - 14730 \nu^{11} - 49447 \nu^{10} + 207380 \nu^{9} + 177062 \nu^{8} - 981221 \nu^{7} - 151906 \nu^{6} + 1793936 \nu^{5} - 9716 \nu^{4} - 1124805 \nu^{3} - 153280 \nu^{2} + 160513 \nu + 31592 \)\()/2162\) |
| \(\beta_{10}\) | \(=\) | \((\)\( -2764 \nu^{12} + 10088 \nu^{11} + 33585 \nu^{10} - 141614 \nu^{9} - 117946 \nu^{8} + 666901 \nu^{7} + 90681 \nu^{6} - 1209623 \nu^{5} + 23336 \nu^{4} + 751304 \nu^{3} + 104974 \nu^{2} - 113389 \nu - 23055 \)\()/1081\) |
| \(\beta_{11}\) | \(=\) | \((\)\(-5977 \nu^{12} + 21092 \nu^{11} + 75787 \nu^{10} - 299544 \nu^{9} - 297124 \nu^{8} + 1438983 \nu^{7} + 375044 \nu^{6} - 2709136 \nu^{5} - 204722 \nu^{4} + 1799105 \nu^{3} + 294962 \nu^{2} - 275121 \nu - 47218\)\()/2162\) |
| \(\beta_{12}\) | \(=\) | \((\)\(6745 \nu^{12} - 23044 \nu^{11} - 88523 \nu^{10} + 329744 \nu^{9} + 376628 \nu^{8} - 1604297 \nu^{7} - 611328 \nu^{6} + 3091452 \nu^{5} + 548460 \nu^{4} - 2142659 \nu^{3} - 502024 \nu^{2} + 347655 \nu + 78110\)\()/2162\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \(\beta_{1}\) |
| \(\nu^{2}\) | \(=\) | \(\beta_{2} + 3\) |
| \(\nu^{3}\) | \(=\) | \(\beta_{9} + \beta_{8} + 5 \beta_{1}\) |
| \(\nu^{4}\) | \(=\) | \(\beta_{11} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{5} + 7 \beta_{2} + \beta_{1} + 15\) |
| \(\nu^{5}\) | \(=\) | \(2 \beta_{12} + 2 \beta_{11} + 9 \beta_{9} + 9 \beta_{8} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 30 \beta_{1} - 1\) |
| \(\nu^{6}\) | \(=\) | \(\beta_{12} + 11 \beta_{11} + 12 \beta_{9} + 12 \beta_{8} + 9 \beta_{7} + 2 \beta_{6} + 8 \beta_{5} + \beta_{4} + 2 \beta_{3} + 44 \beta_{2} + 14 \beta_{1} + 82\) |
| \(\nu^{7}\) | \(=\) | \(25 \beta_{12} + 26 \beta_{11} + 66 \beta_{9} + 70 \beta_{8} + 3 \beta_{7} + 14 \beta_{6} - 13 \beta_{5} - 8 \beta_{4} + 10 \beta_{3} + 14 \beta_{2} + 191 \beta_{1} - 6\) |
| \(\nu^{8}\) | \(=\) | \(17 \beta_{12} + 98 \beta_{11} - 4 \beta_{10} + 104 \beta_{9} + 112 \beta_{8} + 67 \beta_{7} + 32 \beta_{6} + 49 \beta_{5} + 17 \beta_{4} + 24 \beta_{3} + 274 \beta_{2} + 141 \beta_{1} + 471\) |
| \(\nu^{9}\) | \(=\) | \(229 \beta_{12} + 251 \beta_{11} - 8 \beta_{10} + 454 \beta_{9} + 529 \beta_{8} + 49 \beta_{7} + 147 \beta_{6} - 121 \beta_{5} - 41 \beta_{4} + 76 \beta_{3} + 140 \beta_{2} + 1259 \beta_{1} - 11\) |
| \(\nu^{10}\) | \(=\) | \(196 \beta_{12} + 811 \beta_{11} - 75 \beta_{10} + 797 \beta_{9} + 963 \beta_{8} + 488 \beta_{7} + 357 \beta_{6} + 266 \beta_{5} + 196 \beta_{4} + 207 \beta_{3} + 1727 \beta_{2} + 1254 \beta_{1} + 2806\) |
| \(\nu^{11}\) | \(=\) | \(1867 \beta_{12} + 2161 \beta_{11} - 166 \beta_{10} + 3042 \beta_{9} + 3979 \beta_{8} + 550 \beta_{7} + 1364 \beta_{6} - 995 \beta_{5} - 100 \beta_{4} + 534 \beta_{3} + 1226 \beta_{2} + 8497 \beta_{1} + 202\) |
| \(\nu^{12}\) | \(=\) | \(1927 \beta_{12} + 6467 \beta_{11} - 937 \beta_{10} + 5748 \beta_{9} + 7966 \beta_{8} + 3595 \beta_{7} + 3425 \beta_{6} + 1281 \beta_{5} + 1932 \beta_{4} + 1587 \beta_{3} + 11075 \beta_{2} + 10469 \beta_{1} + 17210\) |
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):
