Newspace parameters
| Level: | \( N \) | \(=\) | \( 25 = 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 34 \) |
| Character orbit: | \([\chi]\) | \(=\) | 25.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(172.457072203\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
| Defining polynomial: | \(x^{16} - 26286285043 x^{14} + 277176279803774573548 x^{12} - 1496006322727341104924267746816 x^{10} + 4376228902975645752284567792282218577920 x^{8} - 6785210496827316795155770011788917433647451078656 x^{6} + 5093319366938751307797338793282286802764384084972313509888 x^{4} - 1608229292592013531797229664939538769264311330074914543142623510528 x^{2} + 163532475457876517394407745867705361011086521692855999713211555842344615936\) |
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | multiple of \( 2^{85}\cdot 3^{26}\cdot 5^{66}\cdot 7^{4}\cdot 11^{8} \) |
| Twist minimal: | no (minimal twist has level 5) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) 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 \(x^{16} - 26286285043 x^{14} + 277176279803774573548 x^{12} - 1496006322727341104924267746816 x^{10} + 4376228902975645752284567792282218577920 x^{8} - 6785210496827316795155770011788917433647451078656 x^{6} + 5093319366938751307797338793282286802764384084972313509888 x^{4} - 1608229292592013531797229664939538769264311330074914543142623510528 x^{2} + 163532475457876517394407745867705361011086521692855999713211555842344615936\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \( 2 \nu \) |
| \(\beta_{2}\) | \(=\) | \( 4 \nu^{2} - 13143142522 \) |
| \(\beta_{3}\) | \(=\) | \((\)\(-\)\(23\!\cdots\!27\)\( \nu^{14} + \)\(48\!\cdots\!85\)\( \nu^{12} - \)\(37\!\cdots\!60\)\( \nu^{10} + \)\(14\!\cdots\!00\)\( \nu^{8} - \)\(26\!\cdots\!00\)\( \nu^{6} + \)\(22\!\cdots\!32\)\( \nu^{4} - \)\(85\!\cdots\!00\)\( \nu^{2} + \)\(10\!\cdots\!20\)\(\)\()/ \)\(17\!\cdots\!40\)\( \) |
| \(\beta_{4}\) | \(=\) | \((\)\(\)\(39\!\cdots\!63\)\( \nu^{14} - \)\(40\!\cdots\!33\)\( \nu^{12} - \)\(15\!\cdots\!28\)\( \nu^{10} + \)\(32\!\cdots\!20\)\( \nu^{8} - \)\(13\!\cdots\!00\)\( \nu^{6} + \)\(23\!\cdots\!12\)\( \nu^{4} - \)\(14\!\cdots\!92\)\( \nu^{2} + \)\(22\!\cdots\!08\)\(\)\()/ \)\(57\!\cdots\!80\)\( \) |
| \(\beta_{5}\) | \(=\) | \((\)\(-\)\(77\!\cdots\!41\)\( \nu^{14} + \)\(66\!\cdots\!15\)\( \nu^{12} + \)\(68\!\cdots\!84\)\( \nu^{10} - \)\(10\!\cdots\!20\)\( \nu^{8} + \)\(47\!\cdots\!80\)\( \nu^{6} - \)\(87\!\cdots\!24\)\( \nu^{4} + \)\(62\!\cdots\!40\)\( \nu^{2} - \)\(11\!\cdots\!84\)\(\)\()/ \)\(42\!\cdots\!60\)\( \) |
| \(\beta_{6}\) | \(=\) | \((\)\(-\)\(40\!\cdots\!83\)\( \nu^{14} + \)\(77\!\cdots\!25\)\( \nu^{12} - \)\(52\!\cdots\!28\)\( \nu^{10} + \)\(14\!\cdots\!20\)\( \nu^{8} - \)\(99\!\cdots\!40\)\( \nu^{6} - \)\(32\!\cdots\!12\)\( \nu^{4} + \)\(66\!\cdots\!80\)\( \nu^{2} - \)\(25\!\cdots\!72\)\(\)\()/ \)\(57\!\cdots\!80\)\( \) |
| \(\beta_{7}\) | \(=\) | \((\)\(-\)\(19\!\cdots\!57\)\( \nu^{14} + \)\(58\!\cdots\!35\)\( \nu^{12} - \)\(67\!\cdots\!32\)\( \nu^{10} + \)\(38\!\cdots\!20\)\( \nu^{8} - \)\(10\!\cdots\!00\)\( \nu^{6} + \)\(13\!\cdots\!92\)\( \nu^{4} - \)\(42\!\cdots\!20\)\( \nu^{2} + \)\(11\!\cdots\!92\)\(\)\()/ \)\(28\!\cdots\!40\)\( \) |
| \(\beta_{8}\) | \(=\) | \((\)\(\)\(13\!\cdots\!33\)\( \nu^{14} - \)\(26\!\cdots\!83\)\( \nu^{12} + \)\(20\!\cdots\!96\)\( \nu^{10} - \)\(73\!\cdots\!00\)\( \nu^{8} + \)\(13\!\cdots\!20\)\( \nu^{6} - \)\(11\!\cdots\!88\)\( \nu^{4} + \)\(45\!\cdots\!28\)\( \nu^{2} - \)\(52\!\cdots\!56\)\(\)\()/ \)\(17\!\cdots\!40\)\( \) |
| \(\beta_{9}\) | \(=\) | \((\)\(-\)\(37\!\cdots\!75\)\( \nu^{15} + \)\(83\!\cdots\!69\)\( \nu^{13} - \)\(73\!\cdots\!20\)\( \nu^{11} + \)\(32\!\cdots\!20\)\( \nu^{9} - \)\(76\!\cdots\!00\)\( \nu^{7} + \)\(91\!\cdots\!00\)\( \nu^{5} - \)\(48\!\cdots\!04\)\( \nu^{3} + \)\(74\!\cdots\!40\)\( \nu\)\()/ \)\(22\!\cdots\!40\)\( \) |
| \(\beta_{10}\) | \(=\) | \((\)\(-\)\(29\!\cdots\!35\)\( \nu^{15} + \)\(60\!\cdots\!73\)\( \nu^{13} - \)\(48\!\cdots\!32\)\( \nu^{11} + \)\(18\!\cdots\!40\)\( \nu^{9} - \)\(38\!\cdots\!60\)\( \nu^{7} + \)\(40\!\cdots\!00\)\( \nu^{5} - \)\(19\!\cdots\!28\)\( \nu^{3} + \)\(29\!\cdots\!72\)\( \nu\)\()/ \)\(33\!\cdots\!60\)\( \) |
| \(\beta_{11}\) | \(=\) | \((\)\(\)\(25\!\cdots\!45\)\( \nu^{15} - \)\(54\!\cdots\!55\)\( \nu^{13} + \)\(47\!\cdots\!84\)\( \nu^{11} - \)\(20\!\cdots\!00\)\( \nu^{9} + \)\(46\!\cdots\!20\)\( \nu^{7} - \)\(54\!\cdots\!00\)\( \nu^{5} + \)\(29\!\cdots\!60\)\( \nu^{3} - \)\(71\!\cdots\!84\)\( \nu\)\()/ \)\(66\!\cdots\!20\)\( \) |
| \(\beta_{12}\) | \(=\) | \((\)\(\)\(33\!\cdots\!81\)\( \nu^{15} - \)\(10\!\cdots\!71\)\( \nu^{13} + \)\(12\!\cdots\!64\)\( \nu^{11} - \)\(80\!\cdots\!80\)\( \nu^{9} + \)\(30\!\cdots\!00\)\( \nu^{7} - \)\(62\!\cdots\!36\)\( \nu^{5} + \)\(59\!\cdots\!76\)\( \nu^{3} - \)\(14\!\cdots\!24\)\( \nu\)\()/ \)\(33\!\cdots\!60\)\( \) |
| \(\beta_{13}\) | \(=\) | \((\)\(-\)\(39\!\cdots\!53\)\( \nu^{15} + \)\(10\!\cdots\!39\)\( \nu^{13} - \)\(12\!\cdots\!24\)\( \nu^{11} + \)\(68\!\cdots\!20\)\( \nu^{9} - \)\(21\!\cdots\!80\)\( \nu^{7} + \)\(38\!\cdots\!28\)\( \nu^{5} - \)\(36\!\cdots\!84\)\( \nu^{3} + \)\(13\!\cdots\!44\)\( \nu\)\()/ \)\(22\!\cdots\!40\)\( \) |
| \(\beta_{14}\) | \(=\) | \((\)\(-\)\(24\!\cdots\!05\)\( \nu^{15} + \)\(26\!\cdots\!47\)\( \nu^{13} + \)\(54\!\cdots\!56\)\( \nu^{11} - \)\(54\!\cdots\!40\)\( \nu^{9} + \)\(20\!\cdots\!40\)\( \nu^{7} - \)\(34\!\cdots\!60\)\( \nu^{5} + \)\(22\!\cdots\!88\)\( \nu^{3} - \)\(38\!\cdots\!76\)\( \nu\)\()/ \)\(11\!\cdots\!20\)\( \) |
| \(\beta_{15}\) | \(=\) | \((\)\(\)\(23\!\cdots\!37\)\( \nu^{15} - \)\(70\!\cdots\!03\)\( \nu^{13} + \)\(87\!\cdots\!48\)\( \nu^{11} - \)\(54\!\cdots\!80\)\( \nu^{9} + \)\(18\!\cdots\!60\)\( \nu^{7} - \)\(31\!\cdots\!72\)\( \nu^{5} + \)\(23\!\cdots\!28\)\( \nu^{3} - \)\(49\!\cdots\!88\)\( \nu\)\()/ \)\(66\!\cdots\!20\)\( \) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \(\beta_{1}\)\(/2\) |
| \(\nu^{2}\) | \(=\) | \((\)\(\beta_{2} + 13143142522\)\()/4\) |
| \(\nu^{3}\) | \(=\) | \((\)\(\beta_{11} + \beta_{10} + 1699587 \beta_{9} + 20788434710 \beta_{1}\)\()/8\) |
| \(\nu^{4}\) | \(=\) | \((\)\(\beta_{7} - 359 \beta_{6} + 171 \beta_{5} - 945 \beta_{4} + 69225 \beta_{3} + 27411674260 \beta_{2} + 273232443522298566816\)\()/16\) |
| \(\nu^{5}\) | \(=\) | \((\)\(10660 \beta_{15} - 164886 \beta_{14} + 371924 \beta_{13} + 16214668 \beta_{12} + 8171737257 \beta_{11} + 12249239047 \beta_{10} + 11871079481315179 \beta_{9} + 120700582524832670760 \beta_{1}\)\()/8\) |
| \(\nu^{6}\) | \(=\) | \((\)\(-2840318544 \beta_{8} + 10382932155 \beta_{7} - 3358385182845 \beta_{6} + 2428074969321 \beta_{5} - 5753102295867 \beta_{4} - 1048809109597917 \beta_{3} + 174914235554907059148 \beta_{2} + 1586434433353082161402564961600\)\()/16\) |
| \(\nu^{7}\) | \(=\) | \((\)\(136499115736620 \beta_{15} - 2262561078032226 \beta_{14} + 3589556647778748 \beta_{13} + 151880834833615716 \beta_{12} + 56412146521554039423 \beta_{11} + 109049727225684170937 \beta_{10} + 70927051967169655700039373 \beta_{9} + 730927623461536879752318017072 \beta_{1}\)\()/8\) |
| \(\nu^{8}\) | \(=\) | \((\)\(-42691027079367431664 \beta_{8} + 76825092896935810965 \beta_{7} - 24794474173663719316659 \beta_{6} + 23064061721645823913095 \beta_{5} - 15723412210689612002517 \beta_{4} - 19980035675571457222652307 \beta_{3} + 1105908827300269014731724189236 \beta_{2} + 9606981528873716449358067115346890688576\)\()/16\) |
| \(\nu^{9}\) | \(=\) | \((\)\(1285956383284095994631316 \beta_{15} - 21581235824005445096340414 \beta_{14} + 26844994590558708833320452 \beta_{13} + 995808552908047073449661148 \beta_{12} + 370513904015703366576893949377 \beta_{11} + 856109094579464109234386829863 \beta_{10} + 405251060054157320877267801104266995 \beta_{9} + 4515521212752738369496992352516297277104 \beta_{1}\)\()/8\) |
| \(\nu^{10}\) | \(=\) | \((\)\(-435597954259987487783700103440 \beta_{8} + 504012570392205859971069629195 \beta_{7} - 170279745747183611733858702570925 \beta_{6} + 192952894812687188095946491595865 \beta_{5} + 91732231719854797901186880862005 \beta_{4} - 214976503847936690621817920333784525 \beta_{3} + 6983035218668743874777697541147412172620 \beta_{2} + 59349827811894327294518702110279884467978794892736\)\()/16\) |
| \(\nu^{11}\) | \(=\) | \((\)\(10824024885239361938904064750136300 \beta_{15} - 179383935512029617559390314805343490 \beta_{14} + 185451869889771912903597551458338940 \beta_{13} + 5546135297841507326171932376632828580 \beta_{12} + 2387271776097740250934266927043208804895 \beta_{11} + 6322351216945087886365980174812970628985 \beta_{10} + 2271487433513329961689128191638580429756094125 \beta_{9} + 28184581556298296235184373414621834400296875560144 \beta_{1}\)\()/8\) |
| \(\nu^{12}\) | \(=\) | \((\)\(-3795371731956497608802273238972309166320 \beta_{8} + 3133050498262966763141349212257903137045 \beta_{7} - 1136977301752459513097751193999965326310195 \beta_{6} + 1520358054382431846847595666145506076071175 \beta_{5} + 2004263603050295589409807290094948847837355 \beta_{4} - 1909659920193502742384199090074158841641945875 \beta_{3} + 44111844391491637011147400663770371867494622427444 \beta_{2} + 370443439117400007061879612015861946285617800367874118910528\)\()/16\) |
| \(\nu^{13}\) | \(=\) | \((\)\(85857633759031760845010351228834443860534420 \beta_{15} - 1394745061941545882985146620951088071862378430 \beta_{14} + 1242326920214686755310007269372683110897409540 \beta_{13} + 27143798172317056165803141598174855765958320860 \beta_{12} + 15261437208015334006080882433939842203146470116481 \beta_{11} + 45175768197474572953077747819210402869860072658151 \beta_{10} + 12563871981596298436539785894926681829666141699918052147 \beta_{9} + 176925407400280176500970829161051137281968718940209951305520 \beta_{1}\)\()/8\) |
| \(\nu^{14}\) | \(=\) | \((\)\(-30471108549707188574009628893058269172133319139600 \beta_{8} + 18942248543119431499429482381852975952028382047691 \beta_{7} - 7501377810073741345301720240460856076386079102073069 \beta_{6} + 11549348949902495794281058976076650003033498826091161 \beta_{5} + 22484594157256890276437490449356601690592750663948405 \beta_{4} - 15473808450817914385027938899439943700318987579919256845 \beta_{3} + 278923885031473985484620348809447589373640948600998090991180 \beta_{2} + 2325408207216806167099603488648725719588214373058343073710618215216576\)\()/16\) |
| \(\nu^{15}\) | \(=\) | \((\)\(\)\(65\!\cdots\!60\)\( \beta_{15} - \)\(10\!\cdots\!06\)\( \beta_{14} + \)\(82\!\cdots\!84\)\( \beta_{13} + \)\(11\!\cdots\!68\)\( \beta_{12} + \)\(97\!\cdots\!47\)\( \beta_{11} + \)\(31\!\cdots\!77\)\( \beta_{10} + \)\(68\!\cdots\!49\)\( \beta_{9} + \)\(11\!\cdots\!20\)\( \beta_{1}\)\()/8\) |
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 |
|
−162009. | 1.50582e7 | 1.76568e10 | 0 | −2.43956e12 | −1.33679e14 | −1.46891e15 | −5.33231e15 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −156524. | −1.41317e8 | 1.59098e10 | 0 | 2.21195e13 | −8.25005e13 | −1.14573e15 | 1.44114e16 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −156068. | 6.96239e7 | 1.57674e10 | 0 | −1.08661e13 | 1.55193e14 | −1.12018e15 | −7.11573e14 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −114942. | −6.13071e7 | 4.62174e9 | 0 | 7.04677e12 | 1.47106e14 | 4.56113e14 | −1.80050e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −102456. | 8.77372e7 | 1.90737e9 | 0 | −8.98924e12 | −1.11815e14 | 6.84671e14 | 2.13876e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −62697.7 | −3.24591e7 | −4.65894e9 | 0 | 2.03511e12 | −2.03055e13 | 8.30673e14 | −4.50547e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −40404.6 | 1.28550e8 | −6.95740e9 | 0 | −5.19400e12 | 4.50276e13 | 6.28184e14 | 1.09659e16 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −27727.3 | −6.45007e7 | −7.82113e9 | 0 | 1.78843e12 | −3.09117e13 | 4.55035e14 | −1.39871e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 27727.3 | 6.45007e7 | −7.82113e9 | 0 | 1.78843e12 | 3.09117e13 | −4.55035e14 | −1.39871e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 40404.6 | −1.28550e8 | −6.95740e9 | 0 | −5.19400e12 | −4.50276e13 | −6.28184e14 | 1.09659e16 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 62697.7 | 3.24591e7 | −4.65894e9 | 0 | 2.03511e12 | 2.03055e13 | −8.30673e14 | −4.50547e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 102456. | −8.77372e7 | 1.90737e9 | 0 | −8.98924e12 | 1.11815e14 | −6.84671e14 | 2.13876e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 114942. | 6.13071e7 | 4.62174e9 | 0 | 7.04677e12 | −1.47106e14 | −4.56113e14 | −1.80050e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 156068. | −6.96239e7 | 1.57674e10 | 0 | −1.08661e13 | −1.55193e14 | 1.12018e15 | −7.11573e14 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 156524. | 1.41317e8 | 1.59098e10 | 0 | 2.21195e13 | 8.25005e13 | 1.14573e15 | 1.44114e16 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 162009. | −1.50582e7 | 1.76568e10 | 0 | −2.43956e12 | 1.33679e14 | 1.46891e15 | −5.33231e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 25.34.a.f | 16 | |
| 5.b | even | 2 | 1 | inner | 25.34.a.f | 16 | |
| 5.c | odd | 4 | 2 | 5.34.b.a | ✓ | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.34.b.a | ✓ | 16 | 5.c | odd | 4 | 2 | |
| 25.34.a.f | 16 | 1.a | even | 1 | 1 | trivial | |
| 25.34.a.f | 16 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \(44\!\cdots\!68\)\( T_{2}^{12} - \)\(95\!\cdots\!24\)\( T_{2}^{10} + \)\(11\!\cdots\!20\)\( T_{2}^{8} - \)\(69\!\cdots\!44\)\( T_{2}^{6} + \)\(20\!\cdots\!48\)\( T_{2}^{4} - \)\(26\!\cdots\!52\)\( T_{2}^{2} + \)\(10\!\cdots\!96\)\( \)">\(T_{2}^{16} - \cdots\) acting on \(S_{34}^{\mathrm{new}}(\Gamma_0(25))\).