Newform invariants
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) 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 \(x^{3} - x^{2} - 36 x - 24\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \( 4 \nu - 1 \) |
| \(\beta_{2}\) | \(=\) | \( 8 \nu^{2} - 16 \nu - 189 \) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \((\)\(\beta_{1} + 1\)\()/4\) |
| \(\nu^{2}\) | \(=\) | \((\)\(\beta_{2} + 4 \beta_{1} + 193\)\()/8\) |
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
|
| \(2\) |
\(1\) |
| \(5\) |
\(-1\) |
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 10 T_{3}^{2} - 548 T_{3} - 408 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(320))\).
| $p$ |
$F_p(T)$ |
| $2$
| 1
|
| $3$
| \( 1 - 10 T + 181 T^{2} - 5268 T^{3} + 43983 T^{4} - 590490 T^{5} + 14348907 T^{6} \)
|
| $5$
| \( ( 1 - 25 T )^{3} \)
|
| $7$
| \( 1 - 6 T + 27153 T^{2} - 1137532 T^{3} + 456360471 T^{4} - 1694851494 T^{5} + 4747561509943 T^{6} \)
|
| $11$
| \( 1 + 396 T + 327873 T^{2} + 67617992 T^{3} + 52804274523 T^{4} + 10271220141996 T^{5} + 4177248169415651 T^{6} \)
|
| $13$
| \( 1 - 354 T + 845379 T^{2} - 291738444 T^{3} + 313883305047 T^{4} - 48801906114546 T^{5} + 51185893014090757 T^{6} \)
|
| $17$
| \( 1 - 1158 T + 1914111 T^{2} - 544550612 T^{3} + 2717763902127 T^{4} - 2334520936719942 T^{5} + 2862423051509815793 T^{6} \)
|
| $19$
| \( 1 - 3192 T + 9330057 T^{2} - 15933739216 T^{3} + 23102144807643 T^{4} - 19570363494900792 T^{5} + 15181127029874798299 T^{6} \)
|
| $23$
| \( 1 + 6126 T + 29722593 T^{2} + 84141222540 T^{3} + 191304803397399 T^{4} + 253778807694813774 T^{5} + \)\(26\!\cdots\!07\)\( T^{6} \)
|
| $29$
| \( 1 + 426 T - 939693 T^{2} - 141773503364 T^{3} - 19274183137257 T^{4} + 179221281385885626 T^{5} + \)\(86\!\cdots\!49\)\( T^{6} \)
|
| $31$
| \( 1 + 3276 T + 2292813 T^{2} - 41639420248 T^{3} + 65641289591763 T^{4} + 2685102268149104076 T^{5} + \)\(23\!\cdots\!51\)\( T^{6} \)
|
| $37$
| \( 1 - 11562 T + 90623691 T^{2} - 525372493468 T^{3} + 6284205331885287 T^{4} - 55596852513895170138 T^{5} + \)\(33\!\cdots\!93\)\( T^{6} \)
|
| $41$
| \( 1 - 12450 T + 384843783 T^{2} - 2886023186300 T^{3} + 44586538676848383 T^{4} - \)\(16\!\cdots\!50\)\( T^{5} + \)\(15\!\cdots\!01\)\( T^{6} \)
|
| $43$
| \( 1 - 26346 T + 640605069 T^{2} - 8098987233524 T^{3} + 94174353771597567 T^{4} - \)\(56\!\cdots\!54\)\( T^{5} + \)\(31\!\cdots\!07\)\( T^{6} \)
|
| $47$
| \( 1 + 36762 T + 1119958377 T^{2} + 18490559326820 T^{3} + 256856861812773639 T^{4} + \)\(19\!\cdots\!38\)\( T^{5} + \)\(12\!\cdots\!43\)\( T^{6} \)
|
| $53$
| \( 1 - 21162 T + 395789499 T^{2} + 881498066468 T^{3} + 165517384658528007 T^{4} - \)\(37\!\cdots\!38\)\( T^{5} + \)\(73\!\cdots\!57\)\( T^{6} \)
|
| $59$
| \( 1 - 35040 T + 1757220897 T^{2} - 50989349593920 T^{3} + 1256279917975876203 T^{4} - \)\(17\!\cdots\!40\)\( T^{5} + \)\(36\!\cdots\!99\)\( T^{6} \)
|
| $61$
| \( 1 - 24138 T + 393086643 T^{2} + 6904061162564 T^{3} + 331999524650307543 T^{4} - \)\(17\!\cdots\!38\)\( T^{5} + \)\(60\!\cdots\!01\)\( T^{6} \)
|
| $67$
| \( 1 + 9570 T + 659335509 T^{2} + 83080838420484 T^{3} + 890185424637524463 T^{4} + \)\(17\!\cdots\!30\)\( T^{5} + \)\(24\!\cdots\!43\)\( T^{6} \)
|
| $71$
| \( 1 + 88092 T + 7289446053 T^{2} + 329541325840584 T^{3} + 13151832521353701603 T^{4} + \)\(28\!\cdots\!92\)\( T^{5} + \)\(58\!\cdots\!51\)\( T^{6} \)
|
| $73$
| \( 1 - 66750 T + 3875077479 T^{2} - 111360074258500 T^{3} + 8033313042388954047 T^{4} - \)\(28\!\cdots\!50\)\( T^{5} + \)\(89\!\cdots\!57\)\( T^{6} \)
|
| $79$
| \( 1 - 92952 T + 11164448877 T^{2} - 573846024396496 T^{3} + 34353638858281213923 T^{4} - \)\(88\!\cdots\!52\)\( T^{5} + \)\(29\!\cdots\!99\)\( T^{6} \)
|
| $83$
| \( 1 - 30258 T + 4293702405 T^{2} - 426012342532708 T^{3} + 16913068282241846415 T^{4} - \)\(46\!\cdots\!42\)\( T^{5} + \)\(61\!\cdots\!07\)\( T^{6} \)
|
| $89$
| \( 1 - 172686 T + 26445328791 T^{2} - 2103593815517412 T^{3} + \)\(14\!\cdots\!59\)\( T^{4} - \)\(53\!\cdots\!86\)\( T^{5} + \)\(17\!\cdots\!49\)\( T^{6} \)
|
| $97$
| \( 1 - 170910 T + 30283966671 T^{2} - 2852314667192740 T^{3} + \)\(26\!\cdots\!47\)\( T^{4} - \)\(12\!\cdots\!90\)\( T^{5} + \)\(63\!\cdots\!93\)\( T^{6} \)
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