Newspace parameters
| Level: | \( N \) | \(=\) | \( 91 = 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 91.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(28.4270373191\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
| Defining polynomial: | \(x^{10} - 3 x^{9} - 816 x^{8} + 2298 x^{7} + 213848 x^{6} - 507132 x^{5} - 19919976 x^{4} + 24331248 x^{3} + 727257184 x^{2} - 56397312 x - 7335224320\) |
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) 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 \(x^{10} - 3 x^{9} - 816 x^{8} + 2298 x^{7} + 213848 x^{6} - 507132 x^{5} - 19919976 x^{4} + 24331248 x^{3} + 727257184 x^{2} - 56397312 x - 7335224320\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \( \nu \) |
| \(\beta_{2}\) | \(=\) | \( \nu^{2} - 164 \) |
| \(\beta_{3}\) | \(=\) | \((\)\(2710730315 \nu^{9} - 10591077029 \nu^{8} - 2080232699020 \nu^{7} + 8256593138470 \nu^{6} + 475250169039696 \nu^{5} - 1887482839380516 \nu^{4} - 29505722844621864 \nu^{3} + 108035987258967312 \nu^{2} + 459226885383275744 \nu - 2026471443331588352\)\()/ 6604053619701888 \) |
| \(\beta_{4}\) | \(=\) | \((\)\(-3194242487 \nu^{9} - 1907850873301 \nu^{8} + 3542382315514 \nu^{7} + 1446371109921230 \nu^{6} - 1352801131155108 \nu^{5} - 324395969902423524 \nu^{4} + 161974354745208864 \nu^{3} + 19185289940564666448 \nu^{2} + 7458647501074047040 \nu - 239545848385365834496\)\()/ 6604053619701888 \) |
| \(\beta_{5}\) | \(=\) | \((\)\(8702011532 \nu^{9} + 219416775281 \nu^{8} - 6637241612701 \nu^{7} - 166817327091610 \nu^{6} + 1504914496875870 \nu^{5} + 37830391473107604 \nu^{4} - 87075056020249188 \nu^{3} - 2345441329966057200 \nu^{2} - 656398435870468144 \nu + 29275385637077899904\)\()/ 1100675603283648 \) |
| \(\beta_{6}\) | \(=\) | \((\)\(-77448321017 \nu^{9} - 118851619051 \nu^{8} + 58643694893350 \nu^{7} + 91889448816290 \nu^{6} - 13112355777656796 \nu^{5} - 22934572829074620 \nu^{4} + 763336811734808160 \nu^{3} + 2040479023768700784 \nu^{2} - 8015617409167559744 \nu - 21333548537638031104\)\()/ 6604053619701888 \) |
| \(\beta_{7}\) | \(=\) | \((\)\(27147386411 \nu^{9} + 204615595672 \nu^{8} - 20771638781257 \nu^{7} - 155247362966420 \nu^{6} + 4755818968563294 \nu^{5} + 35348820970636656 \nu^{4} - 297274827829562508 \nu^{3} - 2268088978451838432 \nu^{2} + 2762106303633476912 \nu + 25656397174687265824\)\()/ 1651013404925472 \) |
| \(\beta_{8}\) | \(=\) | \((\)\(-19136930095 \nu^{9} - 96570802877 \nu^{8} + 14586243188762 \nu^{7} + 73109374683374 \nu^{6} - 3310768376771748 \nu^{5} - 16795865975028196 \nu^{4} + 201942480187656800 \nu^{3} + 1138764405671422352 \nu^{2} - 2024655463680966592 \nu - 12870878638627250048\)\()/ 733783735522432 \) |
| \(\beta_{9}\) | \(=\) | \((\)\(131832796597 \nu^{9} + 712892783321 \nu^{8} - 100446640439600 \nu^{7} - 544641329504038 \nu^{6} + 22756691833994232 \nu^{5} + 126705574187831268 \nu^{4} - 1375386805042063368 \nu^{3} - 8777582157342320688 \nu^{2} + 13016678190206610976 \nu + 107398239353653408256\)\()/ 3302026809850944 \) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \(\beta_{1}\) |
| \(\nu^{2}\) | \(=\) | \(\beta_{2} + 164\) |
| \(\nu^{3}\) | \(=\) | \(\beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} + 291 \beta_{1} - 39\) |
| \(\nu^{4}\) | \(=\) | \(3 \beta_{9} - 7 \beta_{8} - 8 \beta_{7} + 14 \beta_{6} + \beta_{5} + 2 \beta_{4} - 33 \beta_{3} + 383 \beta_{2} - 132 \beta_{1} + 47819\) |
| \(\nu^{5}\) | \(=\) | \(-9 \beta_{9} - 49 \beta_{8} + 446 \beta_{7} + 888 \beta_{6} + 387 \beta_{5} + 428 \beta_{4} - 1683 \beta_{3} - 733 \beta_{2} + 96370 \beta_{1} - 34217\) |
| \(\nu^{6}\) | \(=\) | \(1077 \beta_{9} - 4935 \beta_{8} - 5062 \beta_{7} + 7492 \beta_{6} + 659 \beta_{5} + 942 \beta_{4} - 13063 \beta_{3} + 134191 \beta_{2} - 62784 \beta_{1} + 15912781\) |
| \(\nu^{7}\) | \(=\) | \(-12079 \beta_{9} - 34639 \beta_{8} + 172474 \beta_{7} + 325824 \beta_{6} + 136231 \beta_{5} + 160342 \beta_{4} - 1061051 \beta_{3} - 238425 \beta_{2} + 32895336 \beta_{1} - 14490419\) |
| \(\nu^{8}\) | \(=\) | \(303237 \beta_{9} - 2550843 \beta_{8} - 2497490 \beta_{7} + 3250664 \beta_{6} + 313483 \beta_{5} + 351318 \beta_{4} - 4270631 \beta_{3} + 46699871 \beta_{2} - 22807272 \beta_{1} + 5459291269\) |
| \(\nu^{9}\) | \(=\) | \(-7698359 \beta_{9} - 17800439 \beta_{8} + 65138942 \beta_{7} + 115752360 \beta_{6} + 47493695 \beta_{5} + 58790582 \beta_{4} - 527514763 \beta_{3} - 75670077 \beta_{2} + 11356612544 \beta_{1} - 5176466335\) |
Embeddings
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.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−19.2053 | 20.2953 | 240.842 | −234.137 | −389.776 | −343.000 | −2167.15 | −1775.10 | 4496.66 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −17.3557 | −89.1399 | 173.221 | −9.67118 | 1547.09 | −343.000 | −784.843 | 5758.93 | 167.850 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −9.01767 | −31.1400 | −46.6816 | 514.920 | 280.810 | −343.000 | 1575.22 | −1217.30 | −4643.38 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −8.95987 | −21.0756 | −47.7208 | −55.0053 | 188.835 | −343.000 | 1574.43 | −1742.82 | 492.840 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −3.66228 | 42.3449 | −114.588 | −251.753 | −155.079 | −343.000 | 888.424 | −393.907 | 921.992 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 4.83483 | 62.4820 | −104.624 | 314.530 | 302.090 | −343.000 | −1124.70 | 1717.00 | 1520.70 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 6.50047 | −52.7234 | −85.7438 | 295.835 | −342.727 | −343.000 | −1389.44 | 592.761 | 1923.06 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 6.95530 | −32.1557 | −79.6238 | −381.581 | −223.652 | −343.000 | −1444.09 | −1153.01 | −2654.01 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 17.8475 | 86.1795 | 190.532 | 250.298 | 1538.09 | −343.000 | 1116.04 | 5239.91 | 4467.19 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 19.0627 | −86.0671 | 235.388 | −217.435 | −1640.67 | −343.000 | 2047.10 | 5220.54 | −4144.91 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(1\) |
| \(13\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 91.8.a.d | ✓ | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 91.8.a.d | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{10} + \cdots\) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(91))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ | \( -7335224320 + 56397312 T + 727257184 T^{2} - 24331248 T^{3} - 19919976 T^{4} + 507132 T^{5} + 213848 T^{6} - 2298 T^{7} - 816 T^{8} + 3 T^{9} + T^{10} \) |
| $3$ | \( 39502613236151808 + 2052856949954400 T - 109539306717408 T^{2} - 6995852893656 T^{3} + 22654161816 T^{4} + 5376507204 T^{5} + 32490260 T^{6} - 1352210 T^{7} - 11958 T^{8} + 101 T^{9} + T^{10} \) |
| $5$ | \( \)\(31\!\cdots\!00\)\( + \)\(38\!\cdots\!50\)\( T + 68426832004052315775 T^{2} - 8753316903155360 T^{3} - 3166169499742990 T^{4} - 3318493729288 T^{5} + 53404092472 T^{6} + 53550872 T^{7} - 388682 T^{8} - 226 T^{9} + T^{10} \) |
| $7$ | \( ( 343 + T )^{10} \) |
| $11$ | \( -\)\(10\!\cdots\!20\)\( + \)\(89\!\cdots\!28\)\( T + \)\(43\!\cdots\!12\)\( T^{2} - \)\(28\!\cdots\!12\)\( T^{3} - \)\(10\!\cdots\!40\)\( T^{4} + 13920095146632385708 T^{5} + 6461146520665052 T^{6} - 144647825502 T^{7} - 138374754 T^{8} - 451 T^{9} + T^{10} \) |
| $13$ | \( ( 2197 + T )^{10} \) |
| $17$ | \( -\)\(90\!\cdots\!44\)\( + \)\(17\!\cdots\!84\)\( T + \)\(49\!\cdots\!84\)\( T^{2} - \)\(19\!\cdots\!04\)\( T^{3} - \)\(53\!\cdots\!44\)\( T^{4} + \)\(72\!\cdots\!00\)\( T^{5} + 1934207046484823300 T^{6} - 14283851697540 T^{7} - 2446710978 T^{8} + 8654 T^{9} + T^{10} \) |
| $19$ | \( \)\(12\!\cdots\!44\)\( - \)\(53\!\cdots\!98\)\( T + \)\(55\!\cdots\!11\)\( T^{2} + \)\(41\!\cdots\!08\)\( T^{3} - \)\(66\!\cdots\!14\)\( T^{4} - \)\(13\!\cdots\!92\)\( T^{5} + 2304389011404901120 T^{6} + 23931800477192 T^{7} - 2779703538 T^{8} - 10130 T^{9} + T^{10} \) |
| $23$ | \( \)\(40\!\cdots\!12\)\( + \)\(98\!\cdots\!63\)\( T - \)\(16\!\cdots\!79\)\( T^{2} - \)\(70\!\cdots\!20\)\( T^{3} - \)\(36\!\cdots\!92\)\( T^{4} + \)\(19\!\cdots\!26\)\( T^{5} + \)\(21\!\cdots\!94\)\( T^{6} - 1802551003101324 T^{7} - 26598140036 T^{8} + 52155 T^{9} + T^{10} \) |
| $29$ | \( \)\(97\!\cdots\!12\)\( + \)\(67\!\cdots\!42\)\( T - \)\(79\!\cdots\!83\)\( T^{2} - \)\(34\!\cdots\!92\)\( T^{3} + \)\(54\!\cdots\!44\)\( T^{4} + \)\(57\!\cdots\!88\)\( T^{5} - \)\(50\!\cdots\!66\)\( T^{6} + 25653743750267328 T^{7} + 32572021804 T^{8} - 520154 T^{9} + T^{10} \) |
| $31$ | \( -\)\(12\!\cdots\!20\)\( + \)\(11\!\cdots\!49\)\( T - \)\(96\!\cdots\!13\)\( T^{2} - \)\(38\!\cdots\!86\)\( T^{3} + \)\(51\!\cdots\!38\)\( T^{4} + \)\(22\!\cdots\!20\)\( T^{5} - \)\(57\!\cdots\!68\)\( T^{6} + 15636436463998842 T^{7} + 115972438466 T^{8} - 692605 T^{9} + T^{10} \) |
| $37$ | \( \)\(31\!\cdots\!80\)\( - \)\(16\!\cdots\!00\)\( T + \)\(21\!\cdots\!56\)\( T^{2} + \)\(15\!\cdots\!36\)\( T^{3} - \)\(33\!\cdots\!12\)\( T^{4} - \)\(48\!\cdots\!12\)\( T^{5} + \)\(77\!\cdots\!32\)\( T^{6} + 13052517093030974 T^{7} - 507451621388 T^{8} + 20511 T^{9} + T^{10} \) |
| $41$ | \( \)\(73\!\cdots\!40\)\( - \)\(70\!\cdots\!20\)\( T + \)\(43\!\cdots\!88\)\( T^{2} + \)\(13\!\cdots\!60\)\( T^{3} - \)\(26\!\cdots\!84\)\( T^{4} - \)\(92\!\cdots\!28\)\( T^{5} + \)\(23\!\cdots\!36\)\( T^{6} + 291300210818398600 T^{7} - 820326494378 T^{8} - 355049 T^{9} + T^{10} \) |
| $43$ | \( -\)\(50\!\cdots\!12\)\( + \)\(47\!\cdots\!92\)\( T + \)\(16\!\cdots\!21\)\( T^{2} - \)\(26\!\cdots\!68\)\( T^{3} + \)\(10\!\cdots\!68\)\( T^{4} + \)\(23\!\cdots\!48\)\( T^{5} - \)\(26\!\cdots\!18\)\( T^{6} + 574610072326096284 T^{7} + 21156175684 T^{8} - 1256772 T^{9} + T^{10} \) |
| $47$ | \( \)\(68\!\cdots\!72\)\( + \)\(12\!\cdots\!05\)\( T - \)\(11\!\cdots\!61\)\( T^{2} - \)\(11\!\cdots\!46\)\( T^{3} + \)\(23\!\cdots\!90\)\( T^{4} - \)\(39\!\cdots\!12\)\( T^{5} - \)\(66\!\cdots\!32\)\( T^{6} + 2196811713209202814 T^{7} - 1082581165810 T^{8} - 1260721 T^{9} + T^{10} \) |
| $53$ | \( -\)\(10\!\cdots\!00\)\( - \)\(27\!\cdots\!90\)\( T + \)\(17\!\cdots\!81\)\( T^{2} + \)\(48\!\cdots\!40\)\( T^{3} - \)\(29\!\cdots\!36\)\( T^{4} - \)\(25\!\cdots\!08\)\( T^{5} + \)\(17\!\cdots\!14\)\( T^{6} + 4425635112957776172 T^{7} - 3809172816676 T^{8} - 928854 T^{9} + T^{10} \) |
| $59$ | \( \)\(58\!\cdots\!00\)\( + \)\(16\!\cdots\!08\)\( T - \)\(13\!\cdots\!04\)\( T^{2} + \)\(18\!\cdots\!00\)\( T^{3} + \)\(33\!\cdots\!40\)\( T^{4} - \)\(59\!\cdots\!60\)\( T^{5} + \)\(26\!\cdots\!36\)\( T^{6} + 26572368044049123144 T^{7} - 6748019155212 T^{8} - 3144446 T^{9} + T^{10} \) |
| $61$ | \( \)\(74\!\cdots\!00\)\( - \)\(99\!\cdots\!00\)\( T - \)\(98\!\cdots\!60\)\( T^{2} + \)\(19\!\cdots\!72\)\( T^{3} + \)\(50\!\cdots\!00\)\( T^{4} - \)\(86\!\cdots\!76\)\( T^{5} - \)\(53\!\cdots\!28\)\( T^{6} + 26618871183221541200 T^{7} + 6727194303282 T^{8} - 6322923 T^{9} + T^{10} \) |
| $67$ | \( \)\(65\!\cdots\!80\)\( + \)\(94\!\cdots\!80\)\( T + \)\(31\!\cdots\!56\)\( T^{2} + \)\(25\!\cdots\!48\)\( T^{3} - \)\(19\!\cdots\!00\)\( T^{4} - \)\(22\!\cdots\!24\)\( T^{5} + \)\(71\!\cdots\!00\)\( T^{6} + 54188162940512110872 T^{7} - 13812128658620 T^{8} - 3944507 T^{9} + T^{10} \) |
| $71$ | \( \)\(18\!\cdots\!24\)\( - \)\(16\!\cdots\!52\)\( T - \)\(45\!\cdots\!72\)\( T^{2} + \)\(41\!\cdots\!68\)\( T^{3} + \)\(32\!\cdots\!32\)\( T^{4} - \)\(65\!\cdots\!96\)\( T^{5} + \)\(75\!\cdots\!40\)\( T^{6} + \)\(34\!\cdots\!60\)\( T^{7} - 52251307367118 T^{8} - 6032248 T^{9} + T^{10} \) |
| $73$ | \( -\)\(50\!\cdots\!06\)\( - \)\(19\!\cdots\!23\)\( T + \)\(81\!\cdots\!19\)\( T^{2} + \)\(11\!\cdots\!34\)\( T^{3} - \)\(46\!\cdots\!42\)\( T^{4} - \)\(34\!\cdots\!28\)\( T^{5} + \)\(12\!\cdots\!52\)\( T^{6} + \)\(14\!\cdots\!42\)\( T^{7} - 67238801026160 T^{8} - 1248533 T^{9} + T^{10} \) |
| $79$ | \( \)\(31\!\cdots\!48\)\( + \)\(25\!\cdots\!49\)\( T - \)\(54\!\cdots\!43\)\( T^{2} - \)\(25\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!12\)\( T^{4} + \)\(52\!\cdots\!54\)\( T^{5} - \)\(13\!\cdots\!50\)\( T^{6} - \)\(46\!\cdots\!08\)\( T^{7} + 26226167934992 T^{8} + 14947605 T^{9} + T^{10} \) |
| $83$ | \( \)\(14\!\cdots\!56\)\( - \)\(37\!\cdots\!16\)\( T - \)\(31\!\cdots\!21\)\( T^{2} + \)\(48\!\cdots\!16\)\( T^{3} + \)\(46\!\cdots\!02\)\( T^{4} + \)\(77\!\cdots\!92\)\( T^{5} - \)\(24\!\cdots\!56\)\( T^{6} - \)\(73\!\cdots\!36\)\( T^{7} + 923666600758 T^{8} + 14177784 T^{9} + T^{10} \) |
| $89$ | \( -\)\(24\!\cdots\!00\)\( + \)\(47\!\cdots\!20\)\( T - \)\(32\!\cdots\!85\)\( T^{2} + \)\(89\!\cdots\!76\)\( T^{3} - \)\(27\!\cdots\!50\)\( T^{4} - \)\(27\!\cdots\!24\)\( T^{5} + \)\(29\!\cdots\!56\)\( T^{6} + \)\(24\!\cdots\!76\)\( T^{7} - 326948334079878 T^{8} - 6734836 T^{9} + T^{10} \) |
| $97$ | \( \)\(60\!\cdots\!54\)\( + \)\(83\!\cdots\!67\)\( T + \)\(17\!\cdots\!67\)\( T^{2} - \)\(14\!\cdots\!06\)\( T^{3} - \)\(12\!\cdots\!22\)\( T^{4} + \)\(82\!\cdots\!60\)\( T^{5} + \)\(81\!\cdots\!44\)\( T^{6} - \)\(17\!\cdots\!90\)\( T^{7} - 175877533217180 T^{8} + 12365397 T^{9} + T^{10} \) |
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