Newform invariants
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) 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 \(x^{14} - 2 x^{13} - 42 x^{12} + 70 x^{11} + 680 x^{10} - 882 x^{9} - 5559 x^{8} + 5066 x^{7} + 24455 x^{6} - 12990 x^{5} - 55580 x^{4} + 9808 x^{3} + 53551 x^{2} + 6282 x - 7083\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \( \nu \) |
| \(\beta_{2}\) | \(=\) | \((\)\(-6885747582 \nu^{13} + 33914318185 \nu^{12} + 202430140259 \nu^{11} - 1106373672733 \nu^{10} - 1870674412587 \nu^{9} + 12477302034149 \nu^{8} + 6907753319984 \nu^{7} - 63433606982404 \nu^{6} - 11816541353639 \nu^{5} + 149813081971647 \nu^{4} + 23331243759189 \nu^{3} - 143402427852725 \nu^{2} - 36206262335215 \nu + 19597736159910\)\()/ 837106466514 \) |
| \(\beta_{3}\) | \(=\) | \((\)\(-11004140892 \nu^{13} + 49436090470 \nu^{12} + 340112131925 \nu^{11} - 1615645352599 \nu^{10} - 3518878903680 \nu^{9} + 18377007078962 \nu^{8} + 16571607563132 \nu^{7} - 95500280471449 \nu^{6} - 40945399675328 \nu^{5} + 234011215846473 \nu^{4} + 63297600417105 \nu^{3} - 232233141936539 \nu^{2} - 56298435608683 \nu + 34191931183770\)\()/ 837106466514 \) |
| \(\beta_{4}\) | \(=\) | \((\)\(-19118330939 \nu^{13} + 86554061461 \nu^{12} + 595251271041 \nu^{11} - 2872804185614 \nu^{10} - 6118651043878 \nu^{9} + 33227082074766 \nu^{8} + 26870136857865 \nu^{7} - 173410067107969 \nu^{6} - 54385976946928 \nu^{5} + 418064433687984 \nu^{4} + 69177996434839 \nu^{3} - 401075285306924 \nu^{2} - 68967721612466 \nu + 53337950263866\)\()/ 837106466514 \) |
| \(\beta_{5}\) | \(=\) | \((\)\(-23639242991 \nu^{13} + 109271241761 \nu^{12} + 703785403742 \nu^{11} - 3523804015461 \nu^{10} - 6629286791539 \nu^{9} + 38941941028796 \nu^{8} + 24638483686895 \nu^{7} - 191223255135815 \nu^{6} - 34950317548215 \nu^{5} + 426625795784475 \nu^{4} + 35993911720081 \nu^{3} - 377615249540662 \nu^{2} - 56296105207467 \nu + 49744016665776\)\()/ 837106466514 \) |
| \(\beta_{6}\) | \(=\) | \((\)\(-25609954388 \nu^{13} + 109910586859 \nu^{12} + 842563500174 \nu^{11} - 3739110618812 \nu^{10} - 9624887605363 \nu^{9} + 45107127742866 \nu^{8} + 50850635857116 \nu^{7} - 250882871913973 \nu^{6} - 135245303862964 \nu^{5} + 659898648295785 \nu^{4} + 189622681806508 \nu^{3} - 696858166252685 \nu^{2} - 135681262744841 \nu + 105376014149448\)\()/ 837106466514 \) |
| \(\beta_{7}\) | \(=\) | \((\)\(31764401845 \nu^{13} - 134861372299 \nu^{12} - 1026991737247 \nu^{11} + 4499074114704 \nu^{10} + 11379461904392 \nu^{9} - 52596458855725 \nu^{8} - 57437914514824 \nu^{7} + 279816457466485 \nu^{6} + 143968104585861 \nu^{5} - 694429136478564 \nu^{4} - 192408481665842 \nu^{3} + 688027259578091 \nu^{2} + 132289761077346 \nu - 101151393317133\)\()/ 837106466514 \) |
| \(\beta_{8}\) | \(=\) | \((\)\(37435189105 \nu^{13} - 162933719250 \nu^{12} - 1191384212828 \nu^{11} + 5420978081009 \nu^{10} + 12830895743357 \nu^{9} - 63127448399537 \nu^{8} - 61998638358803 \nu^{7} + 334462797242739 \nu^{6} + 148254476159677 \nu^{5} - 827159607477825 \nu^{4} - 203381316447602 \nu^{3} + 817249872781809 \nu^{2} + 162909865954943 \nu - 113370111470034\)\()/ 837106466514 \) |
| \(\beta_{9}\) | \(=\) | \((\)\(37752739446 \nu^{13} - 161274827528 \nu^{12} - 1202197738381 \nu^{11} + 5328164864492 \nu^{10} + 12968704565682 \nu^{9} - 61372297713001 \nu^{8} - 62896893877735 \nu^{7} + 320445662317190 \nu^{6} + 150472617351805 \nu^{5} - 778793817550197 \nu^{4} - 201531065820906 \nu^{3} + 758481182157460 \nu^{2} + 153671579418602 \nu - 109196500860438\)\()/ 837106466514 \) |
| \(\beta_{10}\) | \(=\) | \((\)\(-38350226953 \nu^{13} + 161631948769 \nu^{12} + 1243644226300 \nu^{11} - 5401712538693 \nu^{10} - 13841584361987 \nu^{9} + 63333380868157 \nu^{8} + 70296111977863 \nu^{7} - 338261095735141 \nu^{6} - 178318841583876 \nu^{5} + 844077466448136 \nu^{4} + 244176912750701 \nu^{3} - 842622003635849 \nu^{2} - 170600783335470 \nu + 123665771390928\)\()/ 837106466514 \) |
| \(\beta_{11}\) | \(=\) | \((\)\(-38926641911 \nu^{13} + 173061237061 \nu^{12} + 1207022732481 \nu^{11} - 5680374898316 \nu^{10} - 12367234402564 \nu^{9} + 64701425405268 \nu^{8} + 54689127249561 \nu^{7} - 332421010230685 \nu^{6} - 113117298126910 \nu^{5} + 790930436132310 \nu^{4} + 141244421821633 \nu^{3} - 754127045359874 \nu^{2} - 128265388501436 \nu + 106272852513750\)\()/ 837106466514 \) |
| \(\beta_{12}\) | \(=\) | \((\)\(-48334159531 \nu^{13} + 220483599101 \nu^{12} + 1494088890444 \nu^{11} - 7282180176589 \nu^{10} - 15243485778272 \nu^{9} + 83843136303042 \nu^{8} + 67191645924951 \nu^{7} - 438307828557386 \nu^{6} - 140131255986980 \nu^{5} + 1069606585109949 \nu^{4} + 184820997223490 \nu^{3} - 1049938391405821 \nu^{2} - 184286450571781 \nu + 150684116159928\)\()/ 837106466514 \) |
| \(\beta_{13}\) | \(=\) | \((\)\(-102355805795 \nu^{13} + 456007662555 \nu^{12} + 3186087569260 \nu^{11} - 15003756789322 \nu^{10} - 32990628819313 \nu^{9} + 171744662126800 \nu^{8} + 149960354490979 \nu^{7} - 890755378335540 \nu^{6} - 330103734664163 \nu^{5} + 2151287002908579 \nu^{4} + 441253708065913 \nu^{3} - 2084788718214012 \nu^{2} - 392875051320628 \nu + 295489137970695\)\()/ 837106466514 \) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \(\beta_{1}\) |
| \(\nu^{2}\) | \(=\) | \(-\beta_{12} - \beta_{10} - \beta_{8} + \beta_{6} + \beta_{5} + 7\) |
| \(\nu^{3}\) | \(=\) | \(-\beta_{13} + 2 \beta_{11} - 2 \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{5} + \beta_{3} + \beta_{2} + 9 \beta_{1} + 3\) |
| \(\nu^{4}\) | \(=\) | \(-6 \beta_{13} - 9 \beta_{12} - 2 \beta_{11} - 15 \beta_{10} - 4 \beta_{9} - 18 \beta_{8} - 2 \beta_{7} + 15 \beta_{6} + 21 \beta_{5} - 5 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_{1} + 84\) |
| \(\nu^{5}\) | \(=\) | \(-24 \beta_{13} - 3 \beta_{12} + 51 \beta_{11} - 45 \beta_{10} - 14 \beta_{9} - 29 \beta_{8} - 26 \beta_{7} + 3 \beta_{6} - 17 \beta_{5} - 9 \beta_{4} + 20 \beta_{3} + 26 \beta_{2} + 113 \beta_{1} + 70\) |
| \(\nu^{6}\) | \(=\) | \(-147 \beta_{13} - 92 \beta_{12} - 37 \beta_{11} - 251 \beta_{10} - 91 \beta_{9} - 330 \beta_{8} - 61 \beta_{7} + 229 \beta_{6} + 372 \beta_{5} - 132 \beta_{4} + 44 \beta_{3} + 31 \beta_{2} + 59 \beta_{1} + 1249\) |
| \(\nu^{7}\) | \(=\) | \(-495 \beta_{13} - 61 \beta_{12} + 968 \beta_{11} - 869 \beta_{10} - 225 \beta_{9} - 659 \beta_{8} - 525 \beta_{7} + 97 \beta_{6} - 208 \beta_{5} - 291 \beta_{4} + 341 \beta_{3} + 532 \beta_{2} + 1698 \beta_{1} + 1468\) |
| \(\nu^{8}\) | \(=\) | \(-2920 \beta_{13} - 1131 \beta_{12} - 437 \beta_{11} - 4441 \beta_{10} - 1716 \beta_{9} - 6010 \beta_{8} - 1374 \beta_{7} + 3705 \beta_{6} + 6387 \beta_{5} - 2754 \beta_{4} + 804 \beta_{3} + 783 \beta_{2} + 1410 \beta_{1} + 20427\) |
| \(\nu^{9}\) | \(=\) | \(-9734 \beta_{13} - 1052 \beta_{12} + 17068 \beta_{11} - 16349 \beta_{10} - 4100 \beta_{9} - 13751 \beta_{8} - 9918 \beta_{7} + 2531 \beta_{6} - 1703 \beta_{5} - 6883 \beta_{4} + 5735 \beta_{3} + 10034 \beta_{2} + 27763 \beta_{1} + 30197\) |
| \(\nu^{10}\) | \(=\) | \(-54851 \beta_{13} - 16214 \beta_{12} - 2601 \beta_{11} - 80429 \beta_{10} - 31101 \beta_{9} - 108802 \beta_{8} - 28167 \beta_{7} + 62247 \beta_{6} + 109206 \beta_{5} - 53104 \beta_{4} + 14470 \beta_{3} + 17745 \beta_{2} + 31378 \beta_{1} + 348706\) |
| \(\nu^{11}\) | \(=\) | \(-187187 \beta_{13} - 18966 \beta_{12} + 295104 \beta_{11} - 305918 \beta_{10} - 78259 \beta_{9} - 275524 \beta_{8} - 183285 \beta_{7} + 60180 \beta_{6} + 5121 \beta_{5} - 145199 \beta_{4} + 98091 \beta_{3} + 183650 \beta_{2} + 472087 \beta_{1} + 613563\) |
| \(\nu^{12}\) | \(=\) | \(-1012139 \beta_{13} - 256618 \beta_{12} + 47933 \beta_{11} - 1470566 \beta_{10} - 559263 \beta_{9} - 1965398 \beta_{8} - 555341 \beta_{7} + 1068724 \beta_{6} + 1872765 \beta_{5} - 990846 \beta_{4} + 264711 \beta_{3} + 376610 \beta_{2} + 671115 \beta_{1} + 6080293\) |
| \(\nu^{13}\) | \(=\) | \(-3556677 \beta_{13} - 363417 \beta_{12} + 5088500 \beta_{11} - 5713758 \beta_{10} - 1507447 \beta_{9} - 5398364 \beta_{8} - 3360057 \beta_{7} + 1345486 \beta_{6} + 690113 \beta_{5} - 2905992 \beta_{4} + 1710688 \beta_{3} + 3324128 \beta_{2} + 8189616 \beta_{1} + 12308542\) |
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 does not admit any (nontrivial) inner twists.
| \( p \) |
Sign
|
| \(2\) |
\(1\) |
| \(3\) |
\(-1\) |
| \(149\) |
\(1\) |
This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8046))\):
