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 \(x^{8} - 2 x^{7} + 15 x^{6} - 2 x^{5} + 133 x^{4} - 84 x^{3} + 276 x^{2} + 144 x + 144\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \((\)\( -473 \nu^{7} + 198 \nu^{6} - 5547 \nu^{5} - 7826 \nu^{4} - 75285 \nu^{3} - 29928 \nu^{2} + 140724 \nu - 256788 \)\()/159300\) |
| \(\beta_{2}\) | \(=\) | \((\)\( 677 \nu^{7} - 1827 \nu^{6} + 10353 \nu^{5} - 6901 \nu^{4} + 82215 \nu^{3} - 132153 \nu^{2} + 156924 \nu + 78912 \)\()/159300\) |
| \(\beta_{3}\) | \(=\) | \((\)\( -473 \nu^{7} + 198 \nu^{6} - 5547 \nu^{5} - 7826 \nu^{4} - 75285 \nu^{3} - 29928 \nu^{2} - 98226 \nu - 177138 \)\()/79650\) |
| \(\beta_{4}\) | \(=\) | \((\)\( 1013 \nu^{7} - 2688 \nu^{6} + 16707 \nu^{5} - 19444 \nu^{4} + 143085 \nu^{3} - 232782 \nu^{2} + 247356 \nu - 401472 \)\()/159300\) |
| \(\beta_{5}\) | \(=\) | \((\)\( 644 \nu^{7} - 2019 \nu^{6} + 9966 \nu^{5} - 7447 \nu^{4} + 68730 \nu^{3} - 107691 \nu^{2} + 129078 \nu + 194364 \)\()/79650\) |
| \(\beta_{6}\) | \(=\) | \((\)\( -2407 \nu^{7} + 7182 \nu^{6} - 45123 \nu^{5} + 47066 \nu^{4} - 389565 \nu^{3} + 475248 \nu^{2} - 1245834 \nu + 209808 \)\()/159300\) |
| \(\beta_{7}\) | \(=\) | \((\)\( 5753 \nu^{7} - 4878 \nu^{6} + 67467 \nu^{5} + 95186 \nu^{4} + 657585 \nu^{3} + 364008 \nu^{2} + 544536 \nu + 2015568 \)\()/318600\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \((\)\(-\beta_{3} + 2 \beta_{1} + 1\)\()/3\) |
| \(\nu^{2}\) | \(=\) | \((\)\(6 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - 18 \beta_{2} + \beta_{1} - 1\)\()/3\) |
| \(\nu^{3}\) | \(=\) | \((\)\(-6 \beta_{7} + 6 \beta_{5} - 6 \beta_{4} - 13 \beta_{3} - 7 \beta_{1} - 32\)\()/3\) |
| \(\nu^{4}\) | \(=\) | \((\)\(-12 \beta_{6} - 45 \beta_{5} - 90 \beta_{4} + 25 \beta_{3} + 186 \beta_{2} - 38 \beta_{1} - 199\)\()/3\) |
| \(\nu^{5}\) | \(=\) | \((\)\(90 \beta_{7} - 90 \beta_{6} - 240 \beta_{5} - 120 \beta_{4} + 338 \beta_{3} + 288 \beta_{2} - 169 \beta_{1} + 169\)\()/3\) |
| \(\nu^{6}\) | \(=\) | \((\)\(240 \beta_{7} - 627 \beta_{5} + 627 \beta_{4} + 445 \beta_{3} + 205 \beta_{1} + 2912\)\()/3\) |
| \(\nu^{7}\) | \(=\) | \((\)\(1254 \beta_{6} + 1962 \beta_{5} + 3924 \beta_{4} - 2293 \beta_{3} - 5316 \beta_{2} + 3332 \beta_{1} + 6355\)\()/3\) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).
| \(n\) |
\(55\) |
\(109\) |
\(137\) |
| \(\chi(n)\) |
\(1\) |
\(1\) |
\(\beta_{2}\) |
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 subspace can be constructed as the kernel of the linear operator \(T_{5}^{8} - \cdots\) acting on \(S_{3}^{\mathrm{new}}(216, [\chi])\).
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{8} \) |
| $3$ |
\( T^{8} \) |
| $5$ |
\( 16 - 528 T + 5612 T^{2} + 6468 T^{3} + 2661 T^{4} + 294 T^{5} - 37 T^{6} - 6 T^{7} + T^{8} \) |
| $7$ |
\( 11664 + 3888 T + 4860 T^{2} + 108 T^{3} + 1197 T^{4} + 126 T^{5} + 69 T^{6} - 6 T^{7} + T^{8} \) |
| $11$ |
\( 105616729 + 58455576 T + 11688824 T^{2} + 500544 T^{3} - 50235 T^{4} - 3168 T^{5} + 344 T^{6} + 36 T^{7} + T^{8} \) |
| $13$ |
\( 2611456 - 6270080 T + 14487184 T^{2} - 1407128 T^{3} + 179137 T^{4} - 2846 T^{5} + 547 T^{6} - 14 T^{7} + T^{8} \) |
| $17$ |
\( 7020428944 + 120824648 T^{2} + 687753 T^{4} + 1454 T^{6} + T^{8} \) |
| $19$ |
\( ( 226348 - 4004 T - 1179 T^{2} - 2 T^{3} + T^{4} )^{2} \) |
| $23$ |
\( 11198718976 - 3042651648 T + 228044192 T^{2} + 12909648 T^{3} - 670143 T^{4} - 45798 T^{5} + 3917 T^{6} - 102 T^{7} + T^{8} \) |
| $29$ |
\( 106450807824 - 9055894608 T - 37821492 T^{2} + 25063668 T^{3} + 86949 T^{4} - 102942 T^{5} + 5235 T^{6} - 114 T^{7} + T^{8} \) |
| $31$ |
\( 152712134656 - 12067409920 T + 1224387712 T^{2} - 17678560 T^{3} + 1633465 T^{4} + 27110 T^{5} + 3193 T^{6} + 50 T^{7} + T^{8} \) |
| $37$ |
\( ( 206496 + 27360 T - 276 T^{2} - 60 T^{3} + T^{4} )^{2} \) |
| $41$ |
\( 1919025613521 + 435967071768 T + 44376688026 T^{2} + 2581267824 T^{3} + 93582171 T^{4} + 2165328 T^{5} + 31434 T^{6} + 264 T^{7} + T^{8} \) |
| $43$ |
\( 1352729498761 + 211873953592 T + 28244463112 T^{2} + 838981528 T^{3} + 24309277 T^{4} + 245392 T^{5} + 5032 T^{6} + 28 T^{7} + T^{8} \) |
| $47$ |
\( 4615347568896 + 475641590400 T + 16526225232 T^{2} + 19261800 T^{3} - 8914095 T^{4} - 13050 T^{5} + 7413 T^{6} + 150 T^{7} + T^{8} \) |
| $53$ |
\( 78435844096 + 2590797824 T^{2} + 12396816 T^{4} + 7016 T^{6} + T^{8} \) |
| $59$ |
\( 127589696809 - 43823785536 T + 6086181872 T^{2} - 367082496 T^{3} + 4892493 T^{4} + 323136 T^{5} + 896 T^{6} - 108 T^{7} + T^{8} \) |
| $61$ |
\( 133593174016 + 46766967808 T + 13427579584 T^{2} + 1020419248 T^{3} + 63457201 T^{4} + 368674 T^{5} + 8251 T^{6} - 14 T^{7} + T^{8} \) |
| $67$ |
\( 17391015625 - 4879375000 T + 1606375000 T^{2} + 61325000 T^{3} + 3848125 T^{4} + 38000 T^{5} + 2200 T^{6} + 20 T^{7} + T^{8} \) |
| $71$ |
\( 114698616545536 + 406952754944 T^{2} + 206774880 T^{4} + 26864 T^{6} + T^{8} \) |
| $73$ |
\( ( 2961976 - 487192 T - 9831 T^{2} + 38 T^{3} + T^{4} )^{2} \) |
| $79$ |
\( 103529078405776 - 749891898800 T + 123470983324 T^{2} + 1384089748 T^{3} + 122492077 T^{4} + 449026 T^{5} + 12277 T^{6} - 26 T^{7} + T^{8} \) |
| $83$ |
\( 1085363908864 + 263110694016 T + 28654548944 T^{2} + 1792361544 T^{3} + 70034865 T^{4} + 1745862 T^{5} + 27269 T^{6} + 246 T^{7} + T^{8} \) |
| $89$ |
\( 309931236458496 + 1618052474880 T^{2} + 407712528 T^{4} + 34920 T^{6} + T^{8} \) |
| $97$ |
\( 3435006304129 + 837036947756 T + 171307640890 T^{2} + 7083794672 T^{3} + 202097299 T^{4} + 3255536 T^{5} + 38074 T^{6} + 236 T^{7} + T^{8} \) |
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