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 \(x^{12} - 23 x^{10} + 198 x^{8} - 719 x^{6} + 886 x^{4} + 585 x^{2} + 81\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \((\)\( 17 \nu^{10} - 260 \nu^{8} + 1311 \nu^{6} - 2275 \nu^{4} - 1388 \nu^{2} + 33786 \)\()/7875\) |
| \(\beta_{2}\) | \(=\) | \((\)\( -19 \nu^{10} + 445 \nu^{8} - 3627 \nu^{6} + 9800 \nu^{4} + 5566 \nu^{2} - 24327 \)\()/7875\) |
| \(\beta_{3}\) | \(=\) | \((\)\( -18 \nu^{10} + 415 \nu^{8} - 3594 \nu^{6} + 13350 \nu^{4} - 18398 \nu^{2} - 5994 \)\()/1125\) |
| \(\beta_{4}\) | \(=\) | \((\)\( -368 \nu^{10} + 8665 \nu^{8} - 77019 \nu^{6} + 295225 \nu^{4} - 411623 \nu^{2} - 133794 \)\()/7875\) |
| \(\beta_{5}\) | \(=\) | \((\)\( 181 \nu^{11} - 4055 \nu^{9} + 33723 \nu^{7} - 115325 \nu^{5} + 124141 \nu^{3} + 132273 \nu \)\()/23625\) |
| \(\beta_{6}\) | \(=\) | \((\)\( 106 \nu^{10} - 2455 \nu^{8} + 21423 \nu^{6} - 80150 \nu^{4} + 111416 \nu^{2} + 29448 \)\()/1575\) |
| \(\beta_{7}\) | \(=\) | \((\)\( -88 \nu^{11} + 2015 \nu^{9} - 17154 \nu^{7} + 60725 \nu^{5} - 67168 \nu^{3} - 70929 \nu \)\()/6750\) |
| \(\beta_{8}\) | \(=\) | \((\)\( -316 \nu^{11} + 7355 \nu^{9} - 64053 \nu^{7} + 233450 \nu^{5} - 264751 \nu^{3} - 277353 \nu \)\()/23625\) |
| \(\beta_{9}\) | \(=\) | \((\)\( -1528 \nu^{11} + 35465 \nu^{9} - 309924 \nu^{7} + 1163225 \nu^{5} - 1604758 \nu^{3} - 514449 \nu \)\()/23625\) |
| \(\beta_{10}\) | \(=\) | \((\)\( 2818 \nu^{11} - 65540 \nu^{9} + 575244 \nu^{7} - 2181725 \nu^{5} + 3102073 \nu^{3} + 800019 \nu \)\()/23625\) |
| \(\beta_{11}\) | \(=\) | \((\)\( -811 \nu^{11} + 18830 \nu^{9} - 164763 \nu^{7} + 620450 \nu^{5} - 860821 \nu^{3} - 275688 \nu \)\()/6750\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \((\)\(4 \beta_{11} + 4 \beta_{10} + \beta_{9} - 2 \beta_{8} - 2 \beta_{7} + 2 \beta_{5}\)\()/6\) |
| \(\nu^{2}\) | \(=\) | \((\)\(\beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_{1} + 11\)\()/3\) |
| \(\nu^{3}\) | \(=\) | \((\)\(26 \beta_{11} + 20 \beta_{10} - 7 \beta_{9} - 10 \beta_{8} - 4 \beta_{7} + 13 \beta_{5}\)\()/6\) |
| \(\nu^{4}\) | \(=\) | \((\)\(14 \beta_{6} + 8 \beta_{4} + 35 \beta_{3} - \beta_{2} - 5 \beta_{1} + 79\)\()/3\) |
| \(\nu^{5}\) | \(=\) | \((\)\(154 \beta_{11} + 94 \beta_{10} - 92 \beta_{9} - 92 \beta_{8} + 58 \beta_{7} + 113 \beta_{5}\)\()/6\) |
| \(\nu^{6}\) | \(=\) | \((\)\(151 \beta_{6} + 46 \beta_{4} + 505 \beta_{3} - 26 \beta_{2} + 2 \beta_{1} + 560\)\()/3\) |
| \(\nu^{7}\) | \(=\) | \((\)\(692 \beta_{11} + 350 \beta_{10} - 565 \beta_{9} - 868 \beta_{8} + 1166 \beta_{7} + 1102 \beta_{5}\)\()/6\) |
| \(\nu^{8}\) | \(=\) | \((\)\(1430 \beta_{6} + 242 \beta_{4} + 5399 \beta_{3} - 190 \beta_{2} + 460 \beta_{1} + 3580\)\()/3\) |
| \(\nu^{9}\) | \(=\) | \((\)\(262 \beta_{11} - 218 \beta_{10} - 947 \beta_{9} - 7154 \beta_{8} + 14116 \beta_{7} + 11039 \beta_{5}\)\()/6\) |
| \(\nu^{10}\) | \(=\) | \((\)\(12181 \beta_{6} + 1306 \beta_{4} + 48394 \beta_{3} - 953 \beta_{2} + 7520 \beta_{1} + 17075\)\()/3\) |
| \(\nu^{11}\) | \(=\) | \((\)\(-45694 \beta_{11} - 26842 \beta_{10} + 29504 \beta_{9} - 48850 \beta_{8} + 140162 \beta_{7} + 104395 \beta_{5}\)\()/6\) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).
| \(n\) |
\(101\) |
\(127\) |
| \(\chi(n)\) |
\(\beta_{3}\) |
\(-1\) |
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 \( T_{2}^{12} - 35 T_{2}^{10} + 930 T_{2}^{8} - 10307 T_{2}^{6} + 86710 T_{2}^{4} - 2655 T_{2}^{2} + 81 \) acting on \(S_{4}^{\mathrm{new}}(225, [\chi])\).
| $p$ |
$F_p(T)$ |
| $2$ |
\( 81 - 2655 T^{2} + 86710 T^{4} - 10307 T^{6} + 930 T^{8} - 35 T^{10} + T^{12} \) |
| $3$ |
\( 387420489 + 30292137 T^{2} + 1640250 T^{4} + 77517 T^{6} + 2250 T^{8} + 57 T^{10} + T^{12} \) |
| $5$ |
\( T^{12} \) |
| $7$ |
\( 811313702977761 - 14156533177983 T^{2} + 205458430878 T^{4} - 668166075 T^{6} + 1631674 T^{8} - 1459 T^{10} + T^{12} \) |
| $11$ |
\( ( 1896428304 - 122631168 T + 7320184 T^{2} - 126520 T^{3} + 3012 T^{4} + 14 T^{5} + T^{6} )^{2} \) |
| $13$ |
\( 32259361226118390016 - 68392067190914304 T^{2} + 108054911793792 T^{4} - 66957956704 T^{6} + 30260592 T^{8} - 6504 T^{10} + T^{12} \) |
| $17$ |
\( ( 24437192976 + 27666832 T^{2} + 9716 T^{4} + T^{6} )^{2} \) |
| $19$ |
\( ( -57316 + 7292 T - 164 T^{2} + T^{3} )^{4} \) |
| $23$ |
\( 14036574567470305761 - 165939452573967927 T^{2} + 1815958493546022 T^{4} - 1715751132507 T^{6} + 1469463282 T^{8} - 38907 T^{10} + T^{12} \) |
| $29$ |
\( ( 11463342489 - 2926248177 T + 782851006 T^{2} + 9370019 T^{3} + 84894 T^{4} + 335 T^{5} + T^{6} )^{2} \) |
| $31$ |
\( ( 97238137683600 - 137382616080 T + 3665151504 T^{2} - 14817816 T^{3} + 137836 T^{4} - 352 T^{5} + T^{6} )^{2} \) |
| $37$ |
\( ( 11124119360656 + 3292053600 T^{2} + 112188 T^{4} + T^{6} )^{2} \) |
| $41$ |
\( ( 52247959475625 + 322359380175 T + 3340579834 T^{2} + 6116911 T^{3} + 79566 T^{4} + 187 T^{5} + T^{6} )^{2} \) |
| $43$ |
\( \)\(56\!\cdots\!76\)\( - \)\(44\!\cdots\!32\)\( T^{2} + \)\(26\!\cdots\!32\)\( T^{4} - 5925591427156192 T^{6} + 98606441232 T^{8} - 342492 T^{10} + T^{12} \) |
| $47$ |
\( \)\(38\!\cdots\!21\)\( - \)\(35\!\cdots\!19\)\( T^{2} + 31737062392410542434 T^{4} - 1439749703256995 T^{6} + 57564148998 T^{8} - 251663 T^{10} + T^{12} \) |
| $53$ |
\( ( 10565984287296 + 6553907440 T^{2} + 320300 T^{4} + T^{6} )^{2} \) |
| $59$ |
\( ( 16224618881802816 + 45173097261024 T + 163730383744 T^{2} + 149067880 T^{3} + 443448 T^{4} + 298 T^{5} + T^{6} )^{2} \) |
| $61$ |
\( ( 3092861048569009 - 32773423076579 T + 267254918066 T^{2} - 736785779 T^{3} + 1481414 T^{4} - 1439 T^{5} + T^{6} )^{2} \) |
| $67$ |
\( \)\(18\!\cdots\!61\)\( - \)\(18\!\cdots\!07\)\( T^{2} + \)\(12\!\cdots\!02\)\( T^{4} - 435051095147093547 T^{6} + 1092386569462 T^{8} - 1231447 T^{10} + T^{12} \) |
| $71$ |
\( ( 223775052 - 685460 T - 70 T^{2} + T^{3} )^{4} \) |
| $73$ |
\( ( 18205496100929536 + 238382112768 T^{2} + 881280 T^{4} + T^{6} )^{2} \) |
| $79$ |
\( ( 19133137244437056 + 53658650075616 T + 203324256864 T^{2} + 128458200 T^{3} + 533848 T^{4} + 382 T^{5} + T^{6} )^{2} \) |
| $83$ |
\( \)\(83\!\cdots\!81\)\( - \)\(26\!\cdots\!71\)\( T^{2} + \)\(58\!\cdots\!74\)\( T^{4} - 7018863830915788035 T^{6} + 6138727687818 T^{8} - 3012507 T^{10} + T^{12} \) |
| $89$ |
\( ( -125506395 + 238491 T + 1719 T^{2} + T^{3} )^{4} \) |
| $97$ |
\( \)\(81\!\cdots\!16\)\( - \)\(40\!\cdots\!20\)\( T^{2} + \)\(13\!\cdots\!40\)\( T^{4} - 2460481517295808 T^{6} + 32055240480 T^{8} - 214860 T^{10} + T^{12} \) |
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