Newform invariants
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) 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 \(x^{12} + x^{10} - 8 x^{6} + 16 x^{2} + 64\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \((\)\( \nu^{9} + \nu^{7} + 8 \nu \)\()/8\) |
| \(\beta_{2}\) | \(=\) | \((\)\( \nu^{8} + \nu^{6} + 4 \nu^{4} - 4 \nu^{2} \)\()/8\) |
| \(\beta_{3}\) | \(=\) | \((\)\( \nu^{11} - \nu^{9} + 2 \nu^{7} + 4 \nu^{5} + 8 \nu^{3} \)\()/64\) |
| \(\beta_{4}\) | \(=\) | \((\)\( -\nu^{8} - \nu^{6} + 4 \nu^{4} + 12 \nu^{2} \)\()/8\) |
| \(\beta_{5}\) | \(=\) | \((\)\( -\nu^{11} - 3 \nu^{9} + 2 \nu^{7} + 4 \nu^{5} + 24 \nu^{3} \)\()/32\) |
| \(\beta_{6}\) | \(=\) | \((\)\( -\nu^{8} + 3 \nu^{6} + 4 \nu^{2} - 16 \)\()/8\) |
| \(\beta_{7}\) | \(=\) | \((\)\( \nu^{10} + \nu^{8} - 4 \nu^{6} - 12 \nu^{4} + 16 \nu^{2} + 32 \)\()/16\) |
| \(\beta_{8}\) | \(=\) | \((\)\( -\nu^{11} + \nu^{9} - 2 \nu^{7} + 12 \nu^{5} - 24 \nu^{3} + 32 \nu \)\()/32\) |
| \(\beta_{9}\) | \(=\) | \((\)\( \nu^{11} - \nu^{9} - 6 \nu^{7} - 4 \nu^{5} + 40 \nu^{3} + 32 \nu \)\()/32\) |
| \(\beta_{10}\) | \(=\) | \((\)\( -\nu^{10} + \nu^{6} + 4 \nu^{4} + 4 \nu^{2} - 8 \)\()/8\) |
| \(\beta_{11}\) | \(=\) | \((\)\( 3 \nu^{11} - 3 \nu^{9} - 10 \nu^{7} - 36 \nu^{5} - 8 \nu^{3} + 128 \nu \)\()/64\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \((\)\(\beta_{11} + \beta_{8} + \beta_{5} + \beta_{3} + \beta_{1}\)\()/4\) |
| \(\nu^{2}\) | \(=\) | \((\)\(\beta_{10} + 2 \beta_{7} + \beta_{6} + \beta_{4} + \beta_{2} - 1\)\()/4\) |
| \(\nu^{3}\) | \(=\) | \((\)\(-\beta_{11} + 2 \beta_{9} - \beta_{8} + \beta_{5} - \beta_{3} + \beta_{1}\)\()/4\) |
| \(\nu^{4}\) | \(=\) | \((\)\(-\beta_{10} - 2 \beta_{7} - \beta_{6} + 3 \beta_{4} + 3 \beta_{2} + 1\)\()/4\) |
| \(\nu^{5}\) | \(=\) | \((\)\(-3 \beta_{11} + 2 \beta_{9} + 5 \beta_{8} - \beta_{5} + 13 \beta_{3} - \beta_{1}\)\()/4\) |
| \(\nu^{6}\) | \(=\) | \((\)\(\beta_{10} + 2 \beta_{7} + 9 \beta_{6} - 3 \beta_{4} + 5 \beta_{2} + 15\)\()/4\) |
| \(\nu^{7}\) | \(=\) | \((\)\(3 \beta_{11} - 10 \beta_{9} - 5 \beta_{8} + 9 \beta_{5} + 19 \beta_{3} + 9 \beta_{1}\)\()/4\) |
| \(\nu^{8}\) | \(=\) | \((\)\(7 \beta_{10} + 14 \beta_{7} - \beta_{6} - 5 \beta_{4} + 19 \beta_{2} - 23\)\()/4\) |
| \(\nu^{9}\) | \(=\) | \((\)\(-11 \beta_{11} + 10 \beta_{9} - 3 \beta_{8} - 17 \beta_{5} - 27 \beta_{3} + 15 \beta_{1}\)\()/4\) |
| \(\nu^{10}\) | \(=\) | \((\)\(-31 \beta_{10} + 2 \beta_{7} + 9 \beta_{6} + 13 \beta_{4} + 21 \beta_{2} - 17\)\()/4\) |
| \(\nu^{11}\) | \(=\) | \((\)\(3 \beta_{11} + 6 \beta_{9} - 5 \beta_{8} - 39 \beta_{5} + 147 \beta_{3} - 7 \beta_{1}\)\()/4\) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times\).
| \(n\) |
\(577\) |
\(901\) |
\(1601\) |
\(1951\) |
| \(\chi(n)\) |
\(1\) |
\(-1\) |
\(1\) |
\(1\) |
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 \( T_{7}^{6} - 24 T_{7}^{4} + 128 T_{7}^{2} - 64 \) acting on \(S_{2}^{\mathrm{new}}(2400, [\chi])\).
| $p$ |
$F_p(T)$ |
| $2$
| 1
|
| $3$
| \( ( 1 + T^{2} )^{6} \)
|
| $5$
| 1
|
| $7$
| \( ( 1 + 18 T^{2} + 191 T^{4} + 1532 T^{6} + 9359 T^{8} + 43218 T^{10} + 117649 T^{12} )^{2} \)
|
| $11$
| \( ( 1 - 34 T^{2} + 503 T^{4} - 5436 T^{6} + 60863 T^{8} - 497794 T^{10} + 1771561 T^{12} )^{2} \)
|
| $13$
| \( ( 1 - 30 T^{2} + 743 T^{4} - 10436 T^{6} + 125567 T^{8} - 856830 T^{10} + 4826809 T^{12} )^{2} \)
|
| $17$
| \( ( 1 + 66 T^{2} + 2255 T^{4} + 47324 T^{6} + 651695 T^{8} + 5512386 T^{10} + 24137569 T^{12} )^{2} \)
|
| $19$
| \( ( 1 - 54 T^{2} + 1367 T^{4} - 25652 T^{6} + 493487 T^{8} - 7037334 T^{10} + 47045881 T^{12} )^{2} \)
|
| $23$
| \( ( 1 + 46 T^{2} + 1775 T^{4} + 40932 T^{6} + 938975 T^{8} + 12872686 T^{10} + 148035889 T^{12} )^{2} \)
|
| $29$
| \( ( 1 - 66 T^{2} + 3207 T^{4} - 111228 T^{6} + 2697087 T^{8} - 46680546 T^{10} + 594823321 T^{12} )^{2} \)
|
| $31$
| \( ( 1 - 8 T + 89 T^{2} - 432 T^{3} + 2759 T^{4} - 7688 T^{5} + 29791 T^{6} )^{4} \)
|
| $37$
| \( ( 1 - 158 T^{2} + 11191 T^{4} - 496772 T^{6} + 15320479 T^{8} - 296117438 T^{10} + 2565726409 T^{12} )^{2} \)
|
| $41$
| \( ( 1 + 2 T + 23 T^{2} + 220 T^{3} + 943 T^{4} + 3362 T^{5} + 68921 T^{6} )^{4} \)
|
| $43$
| \( ( 1 - 130 T^{2} + 9815 T^{4} - 518268 T^{6} + 18147935 T^{8} - 444444130 T^{10} + 6321363049 T^{12} )^{2} \)
|
| $47$
| \( ( 1 + 222 T^{2} + 22367 T^{4} + 1328324 T^{6} + 49408703 T^{8} + 1083289182 T^{10} + 10779215329 T^{12} )^{2} \)
|
| $53$
| \( ( 1 - 238 T^{2} + 26391 T^{4} - 1758052 T^{6} + 74132319 T^{8} - 1877934478 T^{10} + 22164361129 T^{12} )^{2} \)
|
| $59$
| \( ( 1 - 178 T^{2} + 20567 T^{4} - 1418652 T^{6} + 71593727 T^{8} - 2156890258 T^{10} + 42180533641 T^{12} )^{2} \)
|
| $61$
| \( ( 1 - 190 T^{2} + 20039 T^{4} - 1419204 T^{6} + 74565119 T^{8} - 2630709790 T^{10} + 51520374361 T^{12} )^{2} \)
|
| $67$
| \( ( 1 - 274 T^{2} + 37127 T^{4} - 3112476 T^{6} + 166663103 T^{8} - 5521407154 T^{10} + 90458382169 T^{12} )^{2} \)
|
| $71$
| \( ( 1 + 8 T + 133 T^{2} + 1008 T^{3} + 9443 T^{4} + 40328 T^{5} + 357911 T^{6} )^{4} \)
|
| $73$
| \( ( 1 + 54 T^{2} + 2367 T^{4} + 531700 T^{6} + 12613743 T^{8} + 1533505014 T^{10} + 151334226289 T^{12} )^{2} \)
|
| $79$
| \( ( 1 - 8 T + 233 T^{2} - 1200 T^{3} + 18407 T^{4} - 49928 T^{5} + 493039 T^{6} )^{4} \)
|
| $83$
| \( ( 1 - 306 T^{2} + 50855 T^{4} - 5168732 T^{6} + 350340095 T^{8} - 14522246226 T^{10} + 326940373369 T^{12} )^{2} \)
|
| $89$
| \( ( 1 - 10 T + 103 T^{2} - 396 T^{3} + 9167 T^{4} - 79210 T^{5} + 704969 T^{6} )^{4} \)
|
| $97$
| \( ( 1 + 246 T^{2} + 39183 T^{4} + 4535476 T^{6} + 368672847 T^{8} + 21778203126 T^{10} + 832972004929 T^{12} )^{2} \)
|
|
show more
|
|
show less
|
