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 \(x^{8} - 2 x^{7} - 2 x^{5} + 9 x^{4} - 4 x^{3} - 16 x + 16\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \((\)\( -\nu^{7} + 2 \nu^{4} + 3 \nu^{3} - 6 \nu^{2} - 4 \nu \)\()/8\) |
| \(\beta_{2}\) | \(=\) | \((\)\( -5 \nu^{7} + 2 \nu^{6} + 4 \nu^{5} + 18 \nu^{4} - 21 \nu^{3} - 12 \nu^{2} - 20 \nu + 56 \)\()/8\) |
| \(\beta_{3}\) | \(=\) | \((\)\( 5 \nu^{7} - 4 \nu^{6} - 4 \nu^{5} - 18 \nu^{4} + 25 \nu^{3} + 10 \nu^{2} + 24 \nu - 64 \)\()/8\) |
| \(\beta_{4}\) | \(=\) | \((\)\( 3 \nu^{7} - 2 \nu^{6} - 2 \nu^{5} - 10 \nu^{4} + 15 \nu^{3} + 8 \nu^{2} + 10 \nu - 36 \)\()/4\) |
| \(\beta_{5}\) | \(=\) | \((\)\( -3 \nu^{7} + 2 \nu^{6} + 4 \nu^{5} + 10 \nu^{4} - 15 \nu^{3} - 12 \nu^{2} - 8 \nu + 36 \)\()/4\) |
| \(\beta_{6}\) | \(=\) | \((\)\( -4 \nu^{7} + 3 \nu^{6} + 4 \nu^{5} + 12 \nu^{4} - 18 \nu^{3} - 9 \nu^{2} - 10 \nu + 44 \)\()/4\) |
| \(\beta_{7}\) | \(=\) | \((\)\( 7 \nu^{7} - 4 \nu^{6} - 6 \nu^{5} - 22 \nu^{4} + 31 \nu^{3} + 18 \nu^{2} + 26 \nu - 80 \)\()/4\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \((\)\(\beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + 1\)\()/2\) |
| \(\nu^{2}\) | \(=\) | \((\)\(\beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} - \beta_{1}\)\()/2\) |
| \(\nu^{3}\) | \(=\) | \((\)\(\beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_{1} + 3\)\()/2\) |
| \(\nu^{4}\) | \(=\) | \((\)\(4 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{3} + 5 \beta_{2} + \beta_{1}\)\()/2\) |
| \(\nu^{5}\) | \(=\) | \((\)\(-\beta_{7} + \beta_{6} + 2 \beta_{5} + 7 \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_{1} - 1\)\()/2\) |
| \(\nu^{6}\) | \(=\) | \((\)\(4 \beta_{7} + 3 \beta_{6} + \beta_{5} - \beta_{4} - 8 \beta_{3} - 5 \beta_{2} + 5 \beta_{1}\)\()/2\) |
| \(\nu^{7}\) | \(=\) | \((\)\(7 \beta_{7} - 5 \beta_{6} + 8 \beta_{5} - \beta_{4} - 11 \beta_{3} + 3 \beta_{2} - 2 \beta_{1} + 5\)\()/2\) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).
| \(n\) |
\(151\) |
\(301\) |
\(401\) |
\(577\) |
| \(\chi(n)\) |
\(1\) |
\(-1\) |
\(1\) |
\(1\) |
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 \( T_{7}^{4} - 4 T_{7}^{3} - 6 T_{7}^{2} + 12 T_{7} + 1 \) acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\).
Hecke Characteristic Polynomials
| $p$ |
$F_p(T)$ |
| $2$
| \( 1 + 2 T - 4 T^{3} - 6 T^{4} - 8 T^{5} + 16 T^{7} + 16 T^{8} \)
|
| $3$
| \( ( 1 + T^{2} )^{4} \)
|
| $5$
| 1
|
| $7$
| \( ( 1 - 4 T + 22 T^{2} - 72 T^{3} + 211 T^{4} - 504 T^{5} + 1078 T^{6} - 1372 T^{7} + 2401 T^{8} )^{2} \)
|
| $11$
| \( 1 - 56 T^{2} + 1612 T^{4} - 29896 T^{6} + 388998 T^{8} - 3617416 T^{10} + 23601292 T^{12} - 99207416 T^{14} + 214358881 T^{16} \)
|
| $13$
| \( 1 - 60 T^{2} + 1802 T^{4} - 36176 T^{6} + 538099 T^{8} - 6113744 T^{10} + 51466922 T^{12} - 289608540 T^{14} + 815730721 T^{16} \)
|
| $17$
| \( ( 1 + 28 T^{2} + 104 T^{3} + 350 T^{4} + 1768 T^{5} + 8092 T^{6} + 83521 T^{8} )^{2} \)
|
| $19$
| \( 1 - 36 T^{2} + 1546 T^{4} - 35120 T^{6} + 832243 T^{8} - 12678320 T^{10} + 201476266 T^{12} - 1693651716 T^{14} + 16983563041 T^{16} \)
|
| $23$
| \( ( 1 - 4 T + 36 T^{2} - 124 T^{3} + 510 T^{4} - 2852 T^{5} + 19044 T^{6} - 48668 T^{7} + 279841 T^{8} )^{2} \)
|
| $29$
| \( 1 - 88 T^{2} + 4780 T^{4} - 171048 T^{6} + 5385990 T^{8} - 143851368 T^{10} + 3380803180 T^{12} - 52344452248 T^{14} + 500246412961 T^{16} \)
|
| $31$
| \( ( 1 - 4 T + 54 T^{2} - 168 T^{3} + 2099 T^{4} - 5208 T^{5} + 51894 T^{6} - 119164 T^{7} + 923521 T^{8} )^{2} \)
|
| $37$
| \( 1 - 168 T^{2} + 14140 T^{4} - 808664 T^{6} + 34400998 T^{8} - 1107061016 T^{10} + 26500636540 T^{12} - 431042036712 T^{14} + 3512479453921 T^{16} \)
|
| $41$
| \( ( 1 + 100 T^{2} + 56 T^{3} + 5166 T^{4} + 2296 T^{5} + 168100 T^{6} + 2825761 T^{8} )^{2} \)
|
| $43$
| \( 1 - 100 T^{2} + 7434 T^{4} - 377968 T^{6} + 18442035 T^{8} - 698862832 T^{10} + 25415366634 T^{12} - 632136304900 T^{14} + 11688200277601 T^{16} \)
|
| $47$
| \( ( 1 + 116 T^{2} - 256 T^{3} + 6310 T^{4} - 12032 T^{5} + 256244 T^{6} + 4879681 T^{8} )^{2} \)
|
| $53$
| \( 1 - 168 T^{2} + 16780 T^{4} - 1266264 T^{6} + 74218758 T^{8} - 3556935576 T^{10} + 132402271180 T^{12} - 3723612669672 T^{14} + 62259690411361 T^{16} \)
|
| $59$
| \( 1 - 40 T^{2} + 2364 T^{4} - 112984 T^{6} + 20250598 T^{8} - 393297304 T^{10} + 28645441404 T^{12} - 1687221345640 T^{14} + 146830437604321 T^{16} \)
|
| $61$
| \( 1 - 252 T^{2} + 32650 T^{4} - 2942672 T^{6} + 202734451 T^{8} - 10949682512 T^{10} + 452066708650 T^{12} - 12983134338972 T^{14} + 191707312997281 T^{16} \)
|
| $67$
| \( 1 - 164 T^{2} + 20746 T^{4} - 1788592 T^{6} + 137741171 T^{8} - 8028989488 T^{10} + 418055156266 T^{12} - 14835174675716 T^{14} + 406067677556641 T^{16} \)
|
| $71$
| \( ( 1 + 20 T + 380 T^{2} + 4188 T^{3} + 43342 T^{4} + 297348 T^{5} + 1915580 T^{6} + 7158220 T^{7} + 25411681 T^{8} )^{2} \)
|
| $73$
| \( ( 1 + 8 T + 124 T^{2} + 888 T^{3} + 7014 T^{4} + 64824 T^{5} + 660796 T^{6} + 3112136 T^{7} + 28398241 T^{8} )^{2} \)
|
| $79$
| \( ( 1 + 8 T + 132 T^{2} + 1032 T^{3} + 16454 T^{4} + 81528 T^{5} + 823812 T^{6} + 3944312 T^{7} + 38950081 T^{8} )^{2} \)
|
| $83$
| \( 1 - 296 T^{2} + 53884 T^{4} - 6923736 T^{6} + 654380710 T^{8} - 47697617304 T^{10} + 2557244168764 T^{12} - 96774350517224 T^{14} + 2252292232139041 T^{16} \)
|
| $89$
| \( ( 1 + 132 T^{2} + 64 T^{3} + 18534 T^{4} + 5696 T^{5} + 1045572 T^{6} + 62742241 T^{8} )^{2} \)
|
| $97$
| \( ( 1 + 4 T + 186 T^{2} + 816 T^{3} + 26147 T^{4} + 79152 T^{5} + 1750074 T^{6} + 3650692 T^{7} + 88529281 T^{8} )^{2} \)
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