Newform invariants
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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 \(x^{6} - 2 x^{5} - 4725 x^{4} + 92430 x^{3} + 1610577 x^{2} - 16081740 x - 24661341\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \((\)\( -4393 \nu^{5} - 288577 \nu^{4} + 17990538 \nu^{3} + 820304928 \nu^{2} - 23681399193 \nu - 121791601863 \)\()/ 828150480 \) |
| \(\beta_{2}\) | \(=\) | \((\)\( -23423 \nu^{5} - 142247 \nu^{4} + 103907598 \nu^{3} - 1434178872 \nu^{2} - 30899430423 \nu + 62101268487 \)\()/ 3312601920 \) |
| \(\beta_{3}\) | \(=\) | \((\)\( -23423 \nu^{5} - 142247 \nu^{4} + 103907598 \nu^{3} - 1434178872 \nu^{2} - 17649022743 \nu + 58788666567 \)\()/ 3312601920 \) |
| \(\beta_{4}\) | \(=\) | \((\)\( 5543 \nu^{5} - 64113 \nu^{4} - 28829198 \nu^{3} + 639322632 \nu^{2} + 11916019503 \nu - 53106861087 \)\()/ 552100320 \) |
| \(\beta_{5}\) | \(=\) | \((\)\( -6041 \nu^{5} + 3471 \nu^{4} + 28357426 \nu^{3} - 539262904 \nu^{2} - 9893470161 \nu + 97030518849 \)\()/ 184033440 \) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \((\)\(\beta_{3} - \beta_{2} + 1\)\()/4\) |
| \(\nu^{2}\) | \(=\) | \((\)\(-36 \beta_{5} - 39 \beta_{4} - 29 \beta_{3} + 143 \beta_{2} - 3 \beta_{1} + 12622\)\()/8\) |
| \(\nu^{3}\) | \(=\) | \((\)\(1044 \beta_{5} - 69 \beta_{4} + 6971 \beta_{3} - 12485 \beta_{2} + 759 \beta_{1} - 335116\)\()/8\) |
| \(\nu^{4}\) | \(=\) | \((\)\(-77850 \beta_{5} - 80907 \beta_{4} - 148531 \beta_{3} + 410857 \beta_{2} - 21057 \beta_{1} + 25100360\)\()/4\) |
| \(\nu^{5}\) | \(=\) | \((\)\(3890574 \beta_{5} + 1532274 \beta_{4} + 15932809 \beta_{3} - 33812113 \beta_{2} + 1903236 \beta_{1} - 1272876977\)\()/4\) |
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.
| \( p \) |
Sign
|
| \(2\) |
\(1\) |
| \(3\) |
\(-1\) |
| \(19\) |
\(1\) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 65 T_{5}^{5} - 9557 T_{5}^{4} - 445577 T_{5}^{3} + 29529208 T_{5}^{2} + 602356212 T_{5} - 26557430352 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(912))\).
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{6} \) |
| $3$ |
\( ( -9 + T )^{6} \) |
| $5$ |
\( -26557430352 + 602356212 T + 29529208 T^{2} - 445577 T^{3} - 9557 T^{4} + 65 T^{5} + T^{6} \) |
| $7$ |
\( -143821013760 - 47041668096 T + 616968276 T^{2} + 7618369 T^{3} - 58601 T^{4} - 149 T^{5} + T^{6} \) |
| $11$ |
\( 1637954874112 + 267853720384 T + 4031719924 T^{2} - 17133601 T^{3} - 251657 T^{4} - 203 T^{5} + T^{6} \) |
| $13$ |
\( -6708606782009472 + 15213557591328 T + 264801371792 T^{2} - 343769584 T^{3} - 1307040 T^{4} + 298 T^{5} + T^{6} \) |
| $17$ |
\( 112483698699536520 - 1696792136443948 T + 4545676438234 T^{2} + 4944257071 T^{3} - 4786951 T^{4} - 1319 T^{5} + T^{6} \) |
| $19$ |
\( ( 361 + T )^{6} \) |
| $23$ |
\( 10912197866748678144 + 45742188138549760 T + 26100108896512 T^{2} - 26840454464 T^{3} - 16647152 T^{4} + 1234 T^{5} + T^{6} \) |
| $29$ |
\( \)\(86\!\cdots\!48\)\( - 5798286151586715072 T + 247015746026736 T^{2} + 470208142688 T^{3} - 63158820 T^{4} - 7356 T^{5} + T^{6} \) |
| $31$ |
\( 1775258940705673216 + 21145538634270528 T - 151177840715920 T^{2} - 402954089040 T^{3} - 89208160 T^{4} + 1632 T^{5} + T^{6} \) |
| $37$ |
\( \)\(44\!\cdots\!00\)\( - 21855154949371187200 T - 3937790841698560 T^{2} + 2549548661904 T^{3} - 152187540 T^{4} - 14204 T^{5} + T^{6} \) |
| $41$ |
\( \)\(13\!\cdots\!32\)\( - \)\(13\!\cdots\!52\)\( T + 4295040987543424 T^{2} + 3521404106744 T^{3} - 218995972 T^{4} - 14734 T^{5} + T^{6} \) |
| $43$ |
\( -\)\(77\!\cdots\!60\)\( - 220901908593157840 T + 2310347035035788 T^{2} + 13046528417 T^{3} - 121424401 T^{4} - 4693 T^{5} + T^{6} \) |
| $47$ |
\( \)\(60\!\cdots\!28\)\( - \)\(19\!\cdots\!76\)\( T + 109163360832510938 T^{2} + 12059614729919 T^{3} - 875485039 T^{4} - 10955 T^{5} + T^{6} \) |
| $53$ |
\( \)\(85\!\cdots\!12\)\( - \)\(16\!\cdots\!40\)\( T - 27461508012574992 T^{2} + 76411039032672 T^{3} - 1362911364 T^{4} - 47500 T^{5} + T^{6} \) |
| $59$ |
\( -\)\(49\!\cdots\!12\)\( + \)\(19\!\cdots\!56\)\( T - 2508318902055701248 T^{2} + 107702950243712 T^{3} - 523832560 T^{4} - 61744 T^{5} + T^{6} \) |
| $61$ |
\( \)\(42\!\cdots\!00\)\( + \)\(31\!\cdots\!64\)\( T - 2680188236475503230 T^{2} - 214064110340925 T^{3} - 1397314919 T^{4} + 81581 T^{5} + T^{6} \) |
| $67$ |
\( \)\(85\!\cdots\!76\)\( - \)\(19\!\cdots\!56\)\( T + 405388370893277440 T^{2} + 112565725121280 T^{3} - 2579706560 T^{4} - 45756 T^{5} + T^{6} \) |
| $71$ |
\( \)\(11\!\cdots\!80\)\( - \)\(24\!\cdots\!80\)\( T + 12054624853039017984 T^{2} + 91249375845248 T^{3} - 7810260288 T^{4} - 10416 T^{5} + T^{6} \) |
| $73$ |
\( \)\(21\!\cdots\!08\)\( + \)\(23\!\cdots\!72\)\( T + 3166952202416264678 T^{2} - 249447316672487 T^{3} - 4851012187 T^{4} + 54615 T^{5} + T^{6} \) |
| $79$ |
\( \)\(13\!\cdots\!88\)\( + \)\(79\!\cdots\!52\)\( T - 565134851002588160 T^{2} - 91150843755968 T^{3} + 6759148176 T^{4} - 145594 T^{5} + T^{6} \) |
| $83$ |
\( \)\(12\!\cdots\!00\)\( + \)\(18\!\cdots\!40\)\( T - 42414970839760802304 T^{2} + 896709308291392 T^{3} + 1059380944 T^{4} - 160548 T^{5} + T^{6} \) |
| $89$ |
\( \)\(37\!\cdots\!84\)\( + \)\(30\!\cdots\!08\)\( T + 28929446906634701616 T^{2} - 1821987516761856 T^{3} - 18458090156 T^{4} + 97728 T^{5} + T^{6} \) |
| $97$ |
\( -\)\(53\!\cdots\!24\)\( - \)\(18\!\cdots\!28\)\( T + 55323449017729409200 T^{2} - 3353906992256 T^{3} - 14254958316 T^{4} + 760 T^{5} + T^{6} \) |
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