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 \(x^{12} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 318 x^{8} - 678 x^{7} + 1225 x^{6} - 1698 x^{5} + 1905 x^{4} - 1584 x^{3} + 936 x^{2} - 342 x + 57\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \((\)\( 6 \nu^{11} - 33 \nu^{10} + 127 \nu^{9} - 324 \nu^{8} + 438 \nu^{7} - 252 \nu^{6} - 1278 \nu^{5} + 3234 \nu^{4} - 5701 \nu^{3} + 5358 \nu^{2} - 3477 \nu + 1060 \)\()/218\) |
| \(\beta_{2}\) | \(=\) | \((\)\( 36 \nu^{11} - 89 \nu^{10} + 544 \nu^{9} - 745 \nu^{8} + 2301 \nu^{7} - 1512 \nu^{6} + 3777 \nu^{5} - 2069 \nu^{4} + 5579 \nu^{3} - 6002 \nu^{2} + 5080 \nu - 1706 \)\()/218\) |
| \(\beta_{3}\) | \(=\) | \((\)\( -27 \nu^{11} + 94 \nu^{10} - 408 \nu^{9} + 586 \nu^{8} - 445 \nu^{7} - 2572 \nu^{6} + 9021 \nu^{5} - 18150 \nu^{4} + 24401 \nu^{3} - 20623 \nu^{2} + 10469 \nu - 2263 \)\()/218\) |
| \(\beta_{4}\) | \(=\) | \((\)\( 26 \nu^{11} - 34 \nu^{10} + 187 \nu^{9} + 449 \nu^{8} - 1590 \nu^{7} + 6865 \nu^{6} - 12623 \nu^{5} + 20118 \nu^{4} - 19981 \nu^{3} + 12972 \nu^{2} - 4712 \nu + 524 \)\()/218\) |
| \(\beta_{5}\) | \(=\) | \((\)\( -2 \nu^{11} + 120 \nu^{10} - 551 \nu^{9} + 2615 \nu^{8} - 6686 \nu^{7} + 15780 \nu^{6} - 23990 \nu^{5} + 31404 \nu^{4} - 25822 \nu^{3} + 15545 \nu^{2} - 4618 \nu + 555 \)\()/218\) |
| \(\beta_{6}\) | \(=\) | \((\)\( -27 \nu^{11} + 203 \nu^{10} - 953 \nu^{9} + 3311 \nu^{8} - 8075 \nu^{7} + 16285 \nu^{6} - 23134 \nu^{5} + 26758 \nu^{4} - 19635 \nu^{3} + 9570 \nu^{2} - 1957 \nu - 83 \)\()/218\) |
| \(\beta_{7}\) | \(=\) | \((\)\( \nu^{10} - 5 \nu^{9} + 25 \nu^{8} - 70 \nu^{7} + 173 \nu^{6} - 295 \nu^{5} + 412 \nu^{4} - 404 \nu^{3} + 279 \nu^{2} - 116 \nu + 26 \)\()/2\) |
| \(\beta_{8}\) | \(=\) | \((\)\( 2 \nu^{11} + 98 \nu^{10} - 539 \nu^{9} + 2726 \nu^{8} - 8138 \nu^{7} + 20190 \nu^{6} - 36614 \nu^{5} + 52308 \nu^{4} - 55710 \nu^{3} + 41571 \nu^{2} - 19689 \nu + 4350 \)\()/218\) |
| \(\beta_{9}\) | \(=\) | \((\)\( 26 \nu^{11} - 252 \nu^{10} + 1277 \nu^{9} - 4892 \nu^{8} + 13234 \nu^{7} - 28887 \nu^{6} + 47327 \nu^{5} - 60760 \nu^{4} + 56973 \nu^{3} - 36296 \nu^{2} + 13927 \nu - 2201 \)\()/218\) |
| \(\beta_{10}\) | \(=\) | \((\)\( 36 \nu^{11} - 307 \nu^{10} + 1634 \nu^{9} - 6086 \nu^{8} + 17125 \nu^{7} - 37373 \nu^{6} + 64054 \nu^{5} - 84255 \nu^{4} + 84604 \nu^{3} - 58431 \nu^{2} + 25899 \nu - 5194 \)\()/218\) |
| \(\beta_{11}\) | \(=\) | \((\)\( 91 \nu^{11} - 664 \nu^{10} + 3325 \nu^{9} - 11563 \nu^{8} + 30405 \nu^{7} - 62791 \nu^{6} + 99754 \nu^{5} - 123498 \nu^{4} + 112914 \nu^{3} - 71337 \nu^{2} + 28743 \nu - 5469 \)\()/218\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \((\)\(2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_{1} + 2\)\()/3\) |
| \(\nu^{2}\) | \(=\) | \((\)\(2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + 4 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 5 \beta_{3} - \beta_{2} - \beta_{1} - 7\)\()/3\) |
| \(\nu^{3}\) | \(=\) | \((\)\(-7 \beta_{11} + 2 \beta_{10} + 5 \beta_{9} - 5 \beta_{8} + \beta_{7} - 13 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} - 4 \beta_{1} - 10\)\()/3\) |
| \(\nu^{4}\) | \(=\) | \((\)\(-16 \beta_{11} + 11 \beta_{10} + 8 \beta_{9} - 8 \beta_{8} - 20 \beta_{7} - 7 \beta_{6} + 14 \beta_{5} - 2 \beta_{4} - 31 \beta_{3} - \beta_{2} - 7 \beta_{1} + 32\)\()/3\) |
| \(\nu^{5}\) | \(=\) | \((\)\(23 \beta_{11} + 14 \beta_{10} - 34 \beta_{9} + 25 \beta_{8} - 26 \beta_{7} + 65 \beta_{6} - 10 \beta_{5} + 4 \beta_{4} - 10 \beta_{3} - 16 \beta_{2} + 29 \beta_{1} + 77\)\()/3\) |
| \(\nu^{6}\) | \(=\) | \((\)\(110 \beta_{11} - 49 \beta_{10} - 88 \beta_{9} + 67 \beta_{8} + 85 \beta_{7} + 101 \beta_{6} - 94 \beta_{5} - 17 \beta_{4} + 161 \beta_{3} + 17 \beta_{2} + 104 \beta_{1} - 121\)\()/3\) |
| \(\nu^{7}\) | \(=\) | \((\)\(-25 \beta_{11} - 187 \beta_{10} + 158 \beta_{9} - 101 \beta_{8} + 229 \beta_{7} - 271 \beta_{6} - 46 \beta_{5} - 56 \beta_{4} + 212 \beta_{3} + 149 \beta_{2} - 82 \beta_{1} - 535\)\()/3\) |
| \(\nu^{8}\) | \(=\) | \((\)\(-652 \beta_{11} + 41 \beta_{10} + 761 \beta_{9} - 509 \beta_{8} - 263 \beta_{7} - 826 \beta_{6} + 515 \beta_{5} + 151 \beta_{4} - 724 \beta_{3} - 19 \beta_{2} - 829 \beta_{1} + 257\)\()/3\) |
| \(\nu^{9}\) | \(=\) | \((\)\(-478 \beta_{11} + 1304 \beta_{10} - 268 \beta_{9} + 160 \beta_{8} - 1571 \beta_{7} + 821 \beta_{6} + 812 \beta_{5} + 640 \beta_{4} - 1951 \beta_{3} - 1108 \beta_{2} - 406 \beta_{1} + 3359\)\()/3\) |
| \(\nu^{10}\) | \(=\) | \((\)\(3353 \beta_{11} + 1451 \beta_{10} - 5224 \beta_{9} + 3352 \beta_{8} + 25 \beta_{7} + 5630 \beta_{6} - 2308 \beta_{5} - 521 \beta_{4} + 2375 \beta_{3} - 913 \beta_{2} + 4949 \beta_{1} + 1451\)\()/3\) |
| \(\nu^{11}\) | \(=\) | \((\)\(6041 \beta_{11} - 6547 \beta_{10} - 3685 \beta_{9} + 2323 \beta_{8} + 9241 \beta_{7} + 326 \beta_{6} - 7273 \beta_{5} - 5168 \beta_{4} + 14021 \beta_{3} + 6587 \beta_{2} + 7973 \beta_{1} - 18736\)\()/3\) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).
| \(n\) |
\(20\) |
\(40\) |
| \(\chi(n)\) |
\(1\) |
\(-\beta_{3} + \beta_{9}\) |
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 \(T_{2}^{12} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(57, [\chi])\).
| $p$ |
$F_p(T)$ |
| $2$ |
\( 64 - 96 T + 96 T^{2} - 64 T^{3} - 36 T^{4} + 18 T^{5} + 99 T^{6} - 9 T^{7} - 9 T^{8} + 8 T^{9} + 6 T^{10} + 3 T^{11} + T^{12} \) |
| $3$ |
\( ( 1 + T^{3} + T^{6} )^{2} \) |
| $5$ |
\( 18496 - 27744 T + 32640 T^{2} - 33184 T^{3} + 22608 T^{4} - 7380 T^{5} - 243 T^{6} + 927 T^{7} - 198 T^{8} - 28 T^{9} + 24 T^{10} - 6 T^{11} + T^{12} \) |
| $7$ |
\( 145161 + 37719 T + 74952 T^{2} + 31077 T^{3} + 29763 T^{4} + 11232 T^{5} + 6405 T^{6} + 2232 T^{7} + 963 T^{8} + 261 T^{9} + 66 T^{10} + 9 T^{11} + T^{12} \) |
| $11$ |
\( 207936 + 426816 T + 572400 T^{2} + 467424 T^{3} + 286236 T^{4} + 121014 T^{5} + 40749 T^{6} + 10026 T^{7} + 2241 T^{8} + 396 T^{9} + 75 T^{10} + 9 T^{11} + T^{12} \) |
| $13$ |
\( 3249 - 4104 T + 8586 T^{2} - 5463 T^{3} + 1602 T^{4} + 63 T^{5} - 264 T^{6} + 72 T^{7} + 63 T^{8} - 21 T^{9} - 6 T^{10} + 3 T^{11} + T^{12} \) |
| $17$ |
\( 41062464 - 24914304 T + 21580992 T^{2} - 12634272 T^{3} + 4728456 T^{4} - 1138374 T^{5} + 160947 T^{6} - 2403 T^{7} - 3780 T^{8} + 570 T^{9} + 18 T^{10} - 12 T^{11} + T^{12} \) |
| $19$ |
\( 47045881 + 22284891 T + 9383112 T^{2} + 3690142 T^{3} + 1153395 T^{4} + 308655 T^{5} + 78033 T^{6} + 16245 T^{7} + 3195 T^{8} + 538 T^{9} + 72 T^{10} + 9 T^{11} + T^{12} \) |
| $23$ |
\( 2483776 - 2893536 T + 2369952 T^{2} - 1393360 T^{3} + 686808 T^{4} - 228132 T^{5} + 55245 T^{6} - 6147 T^{7} - 765 T^{8} + 416 T^{9} - 18 T^{10} - 9 T^{11} + T^{12} \) |
| $29$ |
\( 87616 - 195360 T + 213696 T^{2} - 163880 T^{3} + 104076 T^{4} - 51300 T^{5} + 17217 T^{6} - 3780 T^{7} + 846 T^{8} - 125 T^{9} - 6 T^{10} + T^{12} \) |
| $31$ |
\( 185761 + 77580 T + 168165 T^{2} - 48080 T^{3} + 85509 T^{4} - 9810 T^{5} + 10578 T^{6} - 540 T^{7} + 981 T^{8} - 20 T^{9} + 36 T^{10} + T^{12} \) |
| $37$ |
\( ( 2467 - 1146 T - 2415 T^{2} - 862 T^{3} - 84 T^{4} + 6 T^{5} + T^{6} )^{2} \) |
| $41$ |
\( 87616 + 3786432 T + 69992640 T^{2} + 34553384 T^{3} + 10604664 T^{4} + 2290680 T^{5} + 280881 T^{6} + 24885 T^{7} + 6084 T^{8} + 1349 T^{9} + 183 T^{10} + 18 T^{11} + T^{12} \) |
| $43$ |
\( 587917009 - 142281396 T + 857427150 T^{2} + 171228403 T^{3} - 4778694 T^{4} - 5593545 T^{5} + 402930 T^{6} + 57456 T^{7} - 8451 T^{8} + 343 T^{9} + 90 T^{10} - 15 T^{11} + T^{12} \) |
| $47$ |
\( 46656 + 139968 T + 128304 T^{2} + 40824 T^{3} + 56376 T^{4} + 11178 T^{5} + 459 T^{6} + 2997 T^{7} + 1674 T^{8} + 378 T^{9} + 63 T^{10} + 9 T^{11} + T^{12} \) |
| $53$ |
\( 1871424 + 4875552 T + 6262272 T^{2} + 5620320 T^{3} + 4036392 T^{4} + 2274480 T^{5} + 947457 T^{6} + 274239 T^{7} + 53028 T^{8} + 6531 T^{9} + 531 T^{10} + 30 T^{11} + T^{12} \) |
| $59$ |
\( 1617984 + 10761120 T + 26204256 T^{2} + 27039384 T^{3} + 16121844 T^{4} + 6036804 T^{5} + 1604913 T^{6} + 320535 T^{7} + 49104 T^{8} + 5730 T^{9} + 498 T^{10} + 30 T^{11} + T^{12} \) |
| $61$ |
\( 5329 - 75993 T + 348843 T^{2} + 4160081 T^{3} + 8575632 T^{4} + 1329138 T^{5} + 229773 T^{6} - 2970 T^{7} + 12024 T^{8} - 2761 T^{9} + 285 T^{10} - 21 T^{11} + T^{12} \) |
| $67$ |
\( 1254293056 - 2060786208 T + 857960448 T^{2} + 80549632 T^{3} + 39637476 T^{4} + 3938382 T^{5} + 1276461 T^{6} - 40950 T^{7} + 12474 T^{8} - 890 T^{9} - 66 T^{10} - 3 T^{11} + T^{12} \) |
| $71$ |
\( 241118784 + 122422752 T + 148443840 T^{2} + 33201216 T^{3} + 1499076 T^{4} - 346050 T^{5} - 108615 T^{6} - 33228 T^{7} - 396 T^{8} + 1869 T^{9} + 390 T^{10} + 30 T^{11} + T^{12} \) |
| $73$ |
\( 1056705049 + 401981562 T + 88609584 T^{2} + 10395746 T^{3} + 2441322 T^{4} - 662904 T^{5} - 162576 T^{6} + 12366 T^{7} + 4338 T^{8} + 908 T^{9} + 261 T^{10} + 24 T^{11} + T^{12} \) |
| $79$ |
\( 4606201161 + 3190318083 T + 828240876 T^{2} - 36650988 T^{3} - 27313038 T^{4} - 2208708 T^{5} - 72495 T^{6} + 144558 T^{7} + 30888 T^{8} - 5376 T^{9} + 513 T^{10} - 24 T^{11} + T^{12} \) |
| $83$ |
\( 75809912896 - 13632436032 T + 8648150160 T^{2} + 1113766400 T^{3} + 430576812 T^{4} + 28134126 T^{5} + 5809521 T^{6} + 184824 T^{7} + 53667 T^{8} + 830 T^{9} + 285 T^{10} - 3 T^{11} + T^{12} \) |
| $89$ |
\( 21486869056 - 9197852832 T + 6328237152 T^{2} - 1314646264 T^{3} - 14903244 T^{4} + 9430164 T^{5} + 3951117 T^{6} - 341937 T^{7} + 15147 T^{8} - 1036 T^{9} - 18 T^{10} - 3 T^{11} + T^{12} \) |
| $97$ |
\( 128881 + 818520 T - 1154538 T^{2} - 10936207 T^{3} + 22476942 T^{4} + 604305 T^{5} + 3824802 T^{6} + 1031418 T^{7} + 141561 T^{8} + 10529 T^{9} + 606 T^{10} + 27 T^{11} + T^{12} \) |
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