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 \(x^{8} - 2 x^{7} + 13 x^{6} - 2 x^{5} + 91 x^{4} - 50 x^{3} + 190 x^{2} + 100 x + 100\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \( \nu \) |
| \(\beta_{2}\) | \(=\) | \((\)\( \nu^{7} + 20 \nu^{6} - 51 \nu^{5} + 304 \nu^{4} - 193 \nu^{3} + 1752 \nu^{2} - 2510 \nu + 2630 \)\()/630\) |
| \(\beta_{3}\) | \(=\) | \((\)\( -87 \nu^{7} + 24 \nu^{6} - 841 \nu^{5} - 1276 \nu^{4} - 10117 \nu^{3} - 4640 \nu^{2} - 2900 \nu - 13700 \)\()/21630\) |
| \(\beta_{4}\) | \(=\) | \((\)\( 137 \nu^{7} - 361 \nu^{6} + 1805 \nu^{5} - 1115 \nu^{4} + 11191 \nu^{3} - 16967 \nu^{2} + 21390 \nu - 10830 \)\()/21630\) |
| \(\beta_{5}\) | \(=\) | \((\)\( 472 \nu^{7} - 1249 \nu^{6} + 6966 \nu^{5} - 8699 \nu^{4} + 48092 \nu^{3} - 74565 \nu^{2} + 78220 \nu - 131200 \)\()/64890\) |
| \(\beta_{6}\) | \(=\) | \((\)\( 661 \nu^{7} - 2047 \nu^{6} + 8793 \nu^{5} - 5927 \nu^{4} + 44711 \nu^{3} - 64485 \nu^{2} + 84520 \nu + 126050 \)\()/64890\) |
| \(\beta_{7}\) | \(=\) | \((\)\( -977 \nu^{7} + 1985 \nu^{6} - 15693 \nu^{5} + 4657 \nu^{4} - 114889 \nu^{3} + 25521 \nu^{2} - 381050 \nu - 55810 \)\()/64890\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \(\beta_{1}\) |
| \(\nu^{2}\) | \(=\) | \(2 \beta_{6} + \beta_{5} - 5 \beta_{4} - \beta_{3} + \beta_{1} - 5\) |
| \(\nu^{3}\) | \(=\) | \(\beta_{7} + \beta_{6} - 3 \beta_{5} - \beta_{4} - 9 \beta_{3} - 2 \beta_{2} - 5\) |
| \(\nu^{4}\) | \(=\) | \(-2 \beta_{7} - 11 \beta_{6} - 24 \beta_{5} + 41 \beta_{4} - 2 \beta_{2} - 17 \beta_{1}\) |
| \(\nu^{5}\) | \(=\) | \(-26 \beta_{7} - 42 \beta_{6} - 8 \beta_{5} + 72 \beta_{4} + 95 \beta_{3} + 13 \beta_{2} - 95 \beta_{1} + 85\) |
| \(\nu^{6}\) | \(=\) | \(-34 \beta_{7} - 121 \beta_{6} + 189 \beta_{5} + 34 \beta_{4} + 243 \beta_{3} + 68 \beta_{2} + 475\) |
| \(\nu^{7}\) | \(=\) | \(155 \beta_{7} + 311 \beta_{6} + 777 \beta_{5} - 905 \beta_{4} + 155 \beta_{2} + 1081 \beta_{1}\) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).
| \(n\) |
\(22\) |
\(31\) |
\(71\) |
| \(\chi(n)\) |
\(1\) |
\(-\beta_{4}\) |
\(1\) |
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 \(T_{2}^{8} - \cdots\) acting on \(S_{3}^{\mathrm{new}}(105, [\chi])\).
| $p$ |
$F_p(T)$ |
| $2$
| \( 1 - 2 T - 3 T^{2} + 14 T^{3} - 13 T^{4} - 18 T^{5} + 70 T^{6} - 44 T^{7} - 156 T^{8} - 176 T^{9} + 1120 T^{10} - 1152 T^{11} - 3328 T^{12} + 14336 T^{13} - 12288 T^{14} - 32768 T^{15} + 65536 T^{16} \)
|
| $3$
| \( ( 1 + 3 T + 3 T^{2} )^{4} \)
|
| $5$
| \( ( 1 - 5 T^{2} + 25 T^{4} )^{2} \)
|
| $7$
| \( 1 + 16 T + 109 T^{2} + 784 T^{3} + 6664 T^{4} + 38416 T^{5} + 261709 T^{6} + 1882384 T^{7} + 5764801 T^{8} \)
|
| $11$
| \( 1 - 20 T - 147 T^{2} + 2960 T^{3} + 57131 T^{4} - 519480 T^{5} - 8882912 T^{6} + 8003440 T^{7} + 1602642534 T^{8} + 968416240 T^{9} - 130054714592 T^{10} - 920290508280 T^{11} + 12246537230411 T^{12} + 76774776818960 T^{13} - 461348971377987 T^{14} - 7594996671664820 T^{15} + 45949729863572161 T^{16} \)
|
| $13$
| \( 1 - 188 T^{2} + 39826 T^{4} - 8798048 T^{6} + 2113175419 T^{8} - 251281048928 T^{10} + 32487291694546 T^{12} - 4380040003026428 T^{14} + 665416609183179841 T^{16} \)
|
| $17$
| \( 1 + 18 T + 766 T^{2} + 11844 T^{3} + 262438 T^{4} + 1906254 T^{5} + 22625848 T^{6} - 382636314 T^{7} - 5169040877 T^{8} - 110581894746 T^{9} + 1889733450808 T^{10} + 46012337456526 T^{11} + 1830703831301158 T^{12} + 23877431756917956 T^{13} + 446288633717996926 T^{14} + 3030800878069216722 T^{15} + 48661191875666868481 T^{16} \)
|
| $19$
| \( 1 + 598 T^{2} + 183481 T^{4} - 682560 T^{5} - 48973562 T^{6} - 501474240 T^{7} - 32269961996 T^{8} - 181032200640 T^{9} - 6382283573402 T^{10} - 32111636535360 T^{11} + 3116161130325721 T^{12} + 1323562321601564278 T^{14} + \)\(28\!\cdots\!81\)\( T^{16} \)
|
| $23$
| \( 1 - 62 T + 1497 T^{2} + 6014 T^{3} - 1196893 T^{4} + 31086552 T^{5} - 143771420 T^{6} - 13896561704 T^{7} + 513552019554 T^{8} - 7351281141416 T^{9} - 40233137944220 T^{10} + 4601925361264728 T^{11} - 93729870105931933 T^{12} + 249139038438885086 T^{13} + 32806192774734420537 T^{14} - \)\(71\!\cdots\!58\)\( T^{15} + \)\(61\!\cdots\!61\)\( T^{16} \)
|
| $29$
| \( ( 1 + 50 T + 1234 T^{2} - 15850 T^{3} - 1164374 T^{4} - 13329850 T^{5} + 872784754 T^{6} + 29741166050 T^{7} + 500246412961 T^{8} )^{2} \)
|
| $31$
| \( 1 + 126 T + 9883 T^{2} + 578466 T^{3} + 27206317 T^{4} + 1079090100 T^{5} + 38160094402 T^{6} + 1243487527488 T^{7} + 38998740329170 T^{8} + 1194991513915968 T^{9} + 35241648542229442 T^{10} + 957696435880658100 T^{11} + 23204023931078714797 T^{12} + \)\(47\!\cdots\!66\)\( T^{13} + \)\(77\!\cdots\!63\)\( T^{14} + \)\(95\!\cdots\!46\)\( T^{15} + \)\(72\!\cdots\!81\)\( T^{16} \)
|
| $37$
| \( 1 + 80 T + 1194 T^{2} + 28960 T^{3} + 3461705 T^{4} - 28416960 T^{5} - 6540374054 T^{6} - 198858748720 T^{7} - 6603314864556 T^{8} - 272237626997680 T^{9} - 12257713977418694 T^{10} - 72910144735496640 T^{11} + 12159167688035595305 T^{12} + \)\(13\!\cdots\!40\)\( T^{13} + \)\(78\!\cdots\!14\)\( T^{14} + \)\(72\!\cdots\!20\)\( T^{15} + \)\(12\!\cdots\!41\)\( T^{16} \)
|
| $41$
| \( 1 - 10106 T^{2} + 48877645 T^{4} - 146585251874 T^{6} + 296639674915264 T^{8} - 414214887920726114 T^{10} + \)\(39\!\cdots\!45\)\( T^{12} - \)\(22\!\cdots\!86\)\( T^{14} + \)\(63\!\cdots\!41\)\( T^{16} \)
|
| $43$
| \( ( 1 - 176 T + 17017 T^{2} - 1139948 T^{3} + 56853640 T^{4} - 2107763852 T^{5} + 58177736617 T^{6} - 1112559896624 T^{7} + 11688200277601 T^{8} )^{2} \)
|
| $47$
| \( 1 + 72 T + 10951 T^{2} + 664056 T^{3} + 65519473 T^{4} + 3342900456 T^{5} + 246192812578 T^{6} + 10750018584384 T^{7} + 651041931981118 T^{8} + 23746791052904256 T^{9} + 1201342389873427618 T^{10} + 36033843838636290024 T^{11} + \)\(15\!\cdots\!53\)\( T^{12} + \)\(34\!\cdots\!44\)\( T^{13} + \)\(12\!\cdots\!91\)\( T^{14} + \)\(18\!\cdots\!68\)\( T^{15} + \)\(56\!\cdots\!21\)\( T^{16} \)
|
| $53$
| \( 1 + 76 T - 3069 T^{2} - 443764 T^{3} + 2229785 T^{4} + 1396117872 T^{5} + 34651196266 T^{6} - 1991894657480 T^{7} - 169369280357850 T^{8} - 5595232092861320 T^{9} + 273414605764143946 T^{10} + 30944060693658997488 T^{11} + \)\(13\!\cdots\!85\)\( T^{12} - \)\(77\!\cdots\!36\)\( T^{13} - \)\(15\!\cdots\!29\)\( T^{14} + \)\(10\!\cdots\!44\)\( T^{15} + \)\(38\!\cdots\!21\)\( T^{16} \)
|
| $59$
| \( 1 + 54 T + 7198 T^{2} + 336204 T^{3} + 19932742 T^{4} - 202333950 T^{5} - 35478676088 T^{6} - 7641841019598 T^{7} - 385856896323245 T^{8} - 26601248589220638 T^{9} - 429907925960363768 T^{10} - 8534553984691411950 T^{11} + \)\(29\!\cdots\!82\)\( T^{12} + \)\(17\!\cdots\!04\)\( T^{13} + \)\(12\!\cdots\!38\)\( T^{14} + \)\(33\!\cdots\!94\)\( T^{15} + \)\(21\!\cdots\!41\)\( T^{16} \)
|
| $61$
| \( 1 + 396 T + 83164 T^{2} + 12233232 T^{3} + 1413738778 T^{4} + 136250283708 T^{5} + 11318984386192 T^{6} + 825650586150588 T^{7} + 53403008176121923 T^{8} + 3072245831066337948 T^{9} + \)\(15\!\cdots\!72\)\( T^{10} + \)\(70\!\cdots\!88\)\( T^{11} + \)\(27\!\cdots\!18\)\( T^{12} + \)\(87\!\cdots\!32\)\( T^{13} + \)\(22\!\cdots\!44\)\( T^{14} + \)\(39\!\cdots\!36\)\( T^{15} + \)\(36\!\cdots\!61\)\( T^{16} \)
|
| $67$
| \( 1 - 184 T + 6231 T^{2} + 140176 T^{3} + 74370665 T^{4} - 7237038408 T^{5} + 24063966106 T^{6} - 15184872524680 T^{7} + 3087140085953070 T^{8} - 68164892763288520 T^{9} + 484915892741904826 T^{10} - \)\(65\!\cdots\!52\)\( T^{11} + \)\(30\!\cdots\!65\)\( T^{12} + \)\(25\!\cdots\!24\)\( T^{13} + \)\(50\!\cdots\!91\)\( T^{14} - \)\(67\!\cdots\!36\)\( T^{15} + \)\(16\!\cdots\!81\)\( T^{16} \)
|
| $71$
| \( ( 1 - 82 T + 12166 T^{2} - 846262 T^{3} + 94594474 T^{4} - 4266006742 T^{5} + 309158511046 T^{6} - 10504223281522 T^{7} + 645753531245761 T^{8} )^{2} \)
|
| $73$
| \( 1 - 348 T + 68263 T^{2} - 9707460 T^{3} + 1072498525 T^{4} - 96253557984 T^{5} + 7434307414846 T^{6} - 526544361727584 T^{7} + 37365046682274814 T^{8} - 2805954903646295136 T^{9} + \)\(21\!\cdots\!86\)\( T^{10} - \)\(14\!\cdots\!76\)\( T^{11} + \)\(86\!\cdots\!25\)\( T^{12} - \)\(41\!\cdots\!40\)\( T^{13} + \)\(15\!\cdots\!23\)\( T^{14} - \)\(42\!\cdots\!32\)\( T^{15} + \)\(65\!\cdots\!61\)\( T^{16} \)
|
| $79$
| \( 1 + 206 T + 5583 T^{2} - 659438 T^{3} + 124066817 T^{4} + 19494076044 T^{5} + 428008398310 T^{6} + 33090623674568 T^{7} + 7605703397631354 T^{8} + 206518582352978888 T^{9} + 16670961782854763110 T^{10} + \)\(47\!\cdots\!24\)\( T^{11} + \)\(18\!\cdots\!37\)\( T^{12} - \)\(62\!\cdots\!38\)\( T^{13} + \)\(32\!\cdots\!03\)\( T^{14} + \)\(75\!\cdots\!86\)\( T^{15} + \)\(23\!\cdots\!21\)\( T^{16} \)
|
| $83$
| \( 1 - 20672 T^{2} + 223804480 T^{4} - 2182268545136 T^{6} + 17948924233578718 T^{8} - \)\(10\!\cdots\!56\)\( T^{10} + \)\(50\!\cdots\!80\)\( T^{12} - \)\(22\!\cdots\!92\)\( T^{14} + \)\(50\!\cdots\!81\)\( T^{16} \)
|
| $89$
| \( 1 - 282 T + 59686 T^{2} - 9356196 T^{3} + 1240796086 T^{4} - 138656838366 T^{5} + 14271061565800 T^{6} - 1337157406377822 T^{7} + 121622616146107507 T^{8} - 10591623815918728062 T^{9} + \)\(89\!\cdots\!00\)\( T^{10} - \)\(68\!\cdots\!26\)\( T^{11} + \)\(48\!\cdots\!66\)\( T^{12} - \)\(29\!\cdots\!96\)\( T^{13} + \)\(14\!\cdots\!06\)\( T^{14} - \)\(55\!\cdots\!62\)\( T^{15} + \)\(15\!\cdots\!61\)\( T^{16} \)
|
| $97$
| \( 1 - 44576 T^{2} + 925514428 T^{4} - 12414040936928 T^{6} + 128325632901816454 T^{8} - \)\(10\!\cdots\!68\)\( T^{10} + \)\(72\!\cdots\!08\)\( T^{12} - \)\(30\!\cdots\!16\)\( T^{14} + \)\(61\!\cdots\!21\)\( T^{16} \)
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