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 \(x^{8} - x^{7} - \)\(45\!\cdots\!12\)\( x^{6} + \)\(33\!\cdots\!60\)\( x^{5} + \)\(59\!\cdots\!16\)\( x^{4} - \)\(12\!\cdots\!20\)\( x^{3} - \)\(20\!\cdots\!88\)\( x^{2} + \)\(46\!\cdots\!68\)\( x + \)\(12\!\cdots\!76\)\(\):
\(1\) | \(=\) | \(\beta_0\) |
\(\nu\) | \(=\) | \((\)\(\beta_{1} + 3\)\()/24\) |
\(\nu^{2}\) | \(=\) | \((\)\(\beta_{3} - 62091 \beta_{2} - 26761022262152 \beta_{1} + 65733123937758428360847400200\)\()/576\) |
\(\nu^{3}\) | \(=\) | \((\)\(\beta_{7} + 295 \beta_{6} + 160002 \beta_{5} - 2004356999 \beta_{4} + 51851570017067 \beta_{3} + 3276903991940428836 \beta_{2} + 112986517961279136356684832712 \beta_{1} - 1759085591639276897686588409592076569213504\)\()/13824\) |
\(\nu^{4}\) | \(=\) | \((\)\(3926523411123 \beta_{7} + 6959296706126917 \beta_{6} + 10823831194312423526 \beta_{5} - 64844235599751336866149 \beta_{4} + 19321453573813038341957137905 \beta_{3} - 1394410694473382624668011668515956 \beta_{2} - 298710901026801058381966898320637168298536 \beta_{1} + 928369598546200014171379235704963029862290826589385463872\)\()/41472\) |
\(\nu^{5}\) | \(=\) | \((\)\(2621067272289636342635857733 \beta_{7} + 781805958213198104016871956611 \beta_{6} + 626720668829570339117854559945610 \beta_{5} - 6015663869305187979638343702232227939 \beta_{4} + 205850449531108259889513399496641754951863 \beta_{3} - 41887807299957782555913661895122244965673155244 \beta_{2} + 229515919805726057201751531896339304137050604018359815144 \beta_{1} - 2454400080858624658714224866616934605524010680291289642149797035267648\)\()/124416\) |
\(\nu^{6}\) | \(=\) | \((\)\(2456698967573154102027892526992622040587 \beta_{7} + 2287336922187280642817740919038569124333229 \beta_{6} + 4173765778599191392471236993532318130284922646 \beta_{5} - 22719176458145430265227683260813234396944017761485 \beta_{4} + 4820608307462255271214311913264579460921736185013012921 \beta_{3} - 421824107166754602266817224392645212904577134947783234195476 \beta_{2} + 14896020318557401322538306359430870429741697951273163415269325560472 \beta_{1} + 209538866711613197149097879153829515723664322115869508558284660196534474158205620800\)\()/41472\) |
\(\nu^{7}\) | \(=\) | \((\)\(\)\(22\!\cdots\!91\)\( \beta_{7} + \)\(67\!\cdots\!33\)\( \beta_{6} + \)\(63\!\cdots\!18\)\( \beta_{5} - \)\(57\!\cdots\!61\)\( \beta_{4} + \)\(23\!\cdots\!85\)\( \beta_{3} - \)\(59\!\cdots\!32\)\( \beta_{2} + \)\(18\!\cdots\!88\)\( \beta_{1} + \)\(40\!\cdots\!88\)\(\)\()/41472\) |
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.
This newform subspace is the entire newspace \(S_{96}^{\mathrm{new}}(\Gamma_0(1))\).