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 \(x^{8} - 3 x^{7} + 4 x^{6} - 7 x^{5} + 11 x^{4} + 5 x^{3} - 10 x^{2} - 25 x + 25\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \((\)\( -64 \nu^{7} + 206 \nu^{6} - 318 \nu^{5} + 399 \nu^{4} - 732 \nu^{3} - 126 \nu^{2} + 2175 \nu + 235 \)\()/1355\) |
| \(\beta_{2}\) | \(=\) | \((\)\( 406 \nu^{7} - 714 \nu^{6} + 747 \nu^{5} - 1896 \nu^{4} + 2103 \nu^{3} + 4949 \nu^{2} + 1065 \nu - 7800 \)\()/1355\) |
| \(\beta_{3}\) | \(=\) | \((\)\( -420 \nu^{7} + 776 \nu^{6} - 698 \nu^{5} + 1924 \nu^{4} - 2297 \nu^{3} - 5129 \nu^{2} - 1055 \nu + 10265 \)\()/1355\) |
| \(\beta_{4}\) | \(=\) | \((\)\( 504 \nu^{7} - 877 \nu^{6} + 946 \nu^{5} - 2363 \nu^{4} + 2919 \nu^{3} + 5125 \nu^{2} + 3705 \nu - 11505 \)\()/1355\) |
| \(\beta_{5}\) | \(=\) | \((\)\( -613 \nu^{7} + 1321 \nu^{6} - 1513 \nu^{5} + 3394 \nu^{4} - 4352 \nu^{3} - 6178 \nu^{2} - 530 \nu + 12985 \)\()/1355\) |
| \(\beta_{6}\) | \(=\) | \((\)\( -714 \nu^{7} + 1265 \nu^{6} - 1295 \nu^{5} + 3325 \nu^{4} - 3390 \nu^{3} - 8367 \nu^{2} - 2200 \nu + 14605 \)\()/1355\) |
| \(\beta_{7}\) | \(=\) | \((\)\( -1020 \nu^{7} + 1962 \nu^{6} - 2121 \nu^{5} + 5563 \nu^{4} - 6314 \nu^{3} - 10443 \nu^{2} - 3530 \nu + 21445 \)\()/1355\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \((\)\(\beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} + 3 \beta_{1} + 3\)\()/5\) |
| \(\nu^{2}\) | \(=\) | \((\)\(2 \beta_{7} + 3 \beta_{6} - 4 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + 11 \beta_{2} + 6 \beta_{1} - 4\)\()/5\) |
| \(\nu^{3}\) | \(=\) | \((\)\(-\beta_{7} + 16 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} - 6 \beta_{3} + 17 \beta_{2} - 3 \beta_{1} + 2\)\()/5\) |
| \(\nu^{4}\) | \(=\) | \((\)\(13 \beta_{7} - 3 \beta_{6} - 6 \beta_{5} + 13 \beta_{4} - 7 \beta_{3} - 6 \beta_{2} - 6 \beta_{1} + 14\)\()/5\) |
| \(\nu^{5}\) | \(=\) | \((\)\(16 \beta_{7} - 6 \beta_{6} - 22 \beta_{5} + 6 \beta_{4} + 51 \beta_{3} + 43 \beta_{2} + 8 \beta_{1} - 67\)\()/5\) |
| \(\nu^{6}\) | \(=\) | \((\)\(-23 \beta_{7} + 63 \beta_{6} + 46 \beta_{5} + 12 \beta_{4} + 52 \beta_{3} + 156 \beta_{2} - 34 \beta_{1} - 144\)\()/5\) |
| \(\nu^{7}\) | \(=\) | \((\)\(-31 \beta_{7} - 9 \beta_{6} + 137 \beta_{5} + 84 \beta_{4} - 31 \beta_{3} - 33 \beta_{2} - 168 \beta_{1} + 62\)\()/5\) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).
| \(n\) |
\(101\) |
\(177\) |
\(351\) |
| \(\chi(n)\) |
\(1\) |
\(-\beta_{3}\) |
\(1\) |
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 \(T_{3}^{8} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(400, [\chi])\).
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{8} \) |
| $3$ |
\( 256 + 192 T + 128 T^{2} + 24 T^{3} + 25 T^{4} - 6 T^{5} + 8 T^{6} - 3 T^{7} + T^{8} \) |
| $5$ |
\( 625 - 375 T^{2} + 105 T^{4} - 15 T^{6} + T^{8} \) |
| $7$ |
\( ( 16 - 12 T - 11 T^{2} + 2 T^{3} + T^{4} )^{2} \) |
| $11$ |
\( 256 + 64 T + 352 T^{2} - 128 T^{3} + 105 T^{4} - 7 T^{5} - 3 T^{6} + T^{7} + T^{8} \) |
| $13$ |
\( 39601 + 36218 T + 21988 T^{2} + 8426 T^{3} + 2325 T^{4} + 476 T^{5} + 93 T^{6} + 13 T^{7} + T^{8} \) |
| $17$ |
\( 11881 + 14279 T + 13302 T^{2} + 6657 T^{3} + 2025 T^{4} + 383 T^{5} + 72 T^{6} + 11 T^{7} + T^{8} \) |
| $19$ |
\( 6400 + 11200 T + 12000 T^{2} + 8000 T^{3} + 3585 T^{4} + 965 T^{5} + 190 T^{6} + 20 T^{7} + T^{8} \) |
| $23$ |
\( 256 + 192 T - 192 T^{2} - 296 T^{3} + 705 T^{4} + 49 T^{5} + 63 T^{6} - 3 T^{7} + T^{8} \) |
| $29$ |
\( 25 - 300 T + 10150 T^{2} + 5700 T^{3} + 1735 T^{4} + 390 T^{5} + 105 T^{6} + 15 T^{7} + T^{8} \) |
| $31$ |
\( 1597696 - 257856 T + 56432 T^{2} - 228 T^{3} + 805 T^{4} - 237 T^{5} + 87 T^{6} - 9 T^{7} + T^{8} \) |
| $37$ |
\( 5041 - 4686 T + 3162 T^{2} - 1878 T^{3} + 2380 T^{4} + 348 T^{5} + 47 T^{6} + 6 T^{7} + T^{8} \) |
| $41$ |
\( 7921 - 11214 T + 9162 T^{2} - 4232 T^{3} + 1425 T^{4} - 128 T^{5} + 27 T^{6} + 9 T^{7} + T^{8} \) |
| $43$ |
\( ( 176 - 64 T - 39 T^{2} + 6 T^{3} + T^{4} )^{2} \) |
| $47$ |
\( 256 + 256 T + 32 T^{2} - 132 T^{3} + 125 T^{4} - 33 T^{5} + 17 T^{6} - T^{7} + T^{8} \) |
| $53$ |
\( 65536 + 73728 T + 232448 T^{2} - 12304 T^{3} - 1695 T^{4} + 206 T^{5} + 108 T^{6} - 7 T^{7} + T^{8} \) |
| $59$ |
\( 102400 + 281600 T + 320000 T^{2} + 105600 T^{3} + 17360 T^{4} + 1320 T^{5} + 260 T^{6} + 10 T^{7} + T^{8} \) |
| $61$ |
\( 2920681 - 984384 T + 570837 T^{2} - 56572 T^{3} + 9180 T^{4} + 332 T^{5} - 18 T^{6} - 6 T^{7} + T^{8} \) |
| $67$ |
\( 891136 - 1204544 T + 730272 T^{2} - 193072 T^{3} + 40505 T^{4} - 3883 T^{5} + 237 T^{6} - 11 T^{7} + T^{8} \) |
| $71$ |
\( 4096 - 19456 T + 32512 T^{2} + 21432 T^{3} + 5225 T^{4} - 297 T^{5} + 217 T^{6} - 9 T^{7} + T^{8} \) |
| $73$ |
\( 1175056 + 988608 T + 471808 T^{2} + 141726 T^{3} + 27875 T^{4} + 3231 T^{5} + 208 T^{6} + 8 T^{7} + T^{8} \) |
| $79$ |
\( 102400 - 25600 T + 25600 T^{2} + 3200 T^{3} - 240 T^{4} - 320 T^{5} + 120 T^{6} - 10 T^{7} + T^{8} \) |
| $83$ |
\( 13424896 + 6111552 T + 1867728 T^{2} + 352404 T^{3} + 46405 T^{4} + 4359 T^{5} + 403 T^{6} + 27 T^{7} + T^{8} \) |
| $89$ |
\( 9610000 + 2015000 T + 1025500 T^{2} + 30750 T^{3} + 8525 T^{4} - 750 T^{5} + 70 T^{6} + 15 T^{7} + T^{8} \) |
| $97$ |
\( ( 1 + 7 T + 124 T^{2} + 18 T^{3} + T^{4} )^{2} \) |
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