Properties

Label 1.30.a.a
Level 1
Weight 30
Character orbit 1.a
Self dual Yes
Analytic conductor 5.328
Analytic rank 1
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 30 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(5.3278042383\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{51349}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 96\sqrt{51349}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 4320 - \beta ) q^{2} \) \( + ( -2483820 + 552 \beta ) q^{3} \) \( + ( -44976128 - 8640 \beta ) q^{4} \) \( + ( -8738894250 - 116000 \beta ) q^{5} \) \( + ( -271954378368 + 4868460 \beta ) q^{6} \) \( + ( -1510156341400 - 67855536 \beta ) q^{7} \) \( + ( 1575148584960 + 544522240 \beta ) q^{8} \) \( + ( 81734784761853 - 2742137280 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \(+(4320 - \beta) q^{2}\) \(+(-2483820 + 552 \beta) q^{3}\) \(+(-44976128 - 8640 \beta) q^{4}\) \(+(-8738894250 - 116000 \beta) q^{5}\) \(+(-271954378368 + 4868460 \beta) q^{6}\) \(+(-1510156341400 - 67855536 \beta) q^{7}\) \(+(1575148584960 + 544522240 \beta) q^{8}\) \(+(81734784761853 - 2742137280 \beta) q^{9}\) \(+(17142933384000 + 8237774250 \beta) q^{10}\) \(+(-1027690259794308 - 10592091400 \beta) q^{11}\) \(+(-2145265138114560 - 3366617856 \beta) q^{12}\) \(+(8569685589666110 - 113971542048 \beta) q^{13}\) \(+(25587561674029824 + 1217020425880 \beta) q^{14}\) \(+(-8596175696253000 - 4535746506000 \beta) q^{15}\) \(+(-226734541031604224 + 5415752171520 \beta) q^{16}\) \(+(-332397963953874990 + 19090509870144 \beta) q^{17}\) \(+(1650762432440880480 - 93580817811453 \beta) q^{18}\) \(+(616084722726077540 + 142801363479240 \beta) q^{19}\) \(+(867334050906624000 + 80721277168000 \beta) q^{20}\) \(+(-13974556738124410848 - 665065363025280 \beta) q^{21}\) \(+(572898742456487040 + 981932424946308 \beta) q^{22}\) \(+(-9294215252187189960 - 351986937759248 \beta) q^{23}\) \(+(\)\(13\!\cdots\!20\)\( - 483013211258880 \beta) q^{24}\) \(+(-\)\(10\!\cdots\!25\)\( + 2027423466000000 \beta) q^{25}\) \(+(90956066298888877632 - 9062042651313470 \beta) q^{26}\) \(+(-\)\(74\!\cdots\!40\)\( + 14044608301937040 \beta) q^{27}\) \(+(\)\(34\!\cdots\!60\)\( + 16099630062340608 \beta) q^{28}\) \(+(\)\(49\!\cdots\!10\)\( - 63411615083585440 \beta) q^{29}\) \(+(\)\(21\!\cdots\!00\)\( - 10998249209667000 \beta) q^{30}\) \(+(-\)\(54\!\cdots\!28\)\( + 195576850719316800 \beta) q^{31}\) \(+(-\)\(43\!\cdots\!80\)\( - 42207561180512256 \beta) q^{32}\) \(+(-\)\(21\!\cdots\!40\)\( - 540976174945310016 \beta) q^{33}\) \(+(-\)\(10\!\cdots\!96\)\( + 414868966592897070 \beta) q^{34}\) \(+(\)\(16\!\cdots\!00\)\( + 768160488983468000 \beta) q^{35}\) \(+(\)\(75\!\cdots\!16\)\( - 582857823043558080 \beta) q^{36}\) \(+(\)\(49\!\cdots\!30\)\( - 1363639917490075296 \beta) q^{37}\) \(+(-\)\(64\!\cdots\!60\)\( + 817167504239260 \beta) q^{38}\) \(+(-\)\(51\!\cdots\!64\)\( + 5013551241065356080 \beta) q^{39}\) \(+(-\)\(43\!\cdots\!00\)\( - 4941239507988480000 \beta) q^{40}\) \(+(-\)\(53\!\cdots\!38\)\( - 2173657014743715200 \beta) q^{41}\) \(+(\)\(25\!\cdots\!60\)\( + 11101474369855201248 \beta) q^{42}\) \(+(\)\(25\!\cdots\!00\)\( - 19178453595369039048 \beta) q^{43}\) \(+(\)\(89\!\cdots\!24\)\( + 9355635103216920320 \beta) q^{44}\) \(+(-\)\(56\!\cdots\!50\)\( + 14482012676527692000 \beta) q^{45}\) \(+(\)\(12\!\cdots\!32\)\( + 7773631681067238600 \beta) q^{46}\) \(+(-\)\(22\!\cdots\!60\)\( - 16406150082376208416 \beta) q^{47}\) \(+(\)\(19\!\cdots\!40\)\( - \)\(13\!\cdots\!48\)\( \beta) q^{48}\) \(+(\)\(12\!\cdots\!57\)\( + \)\(20\!\cdots\!00\)\( \beta) q^{49}\) \(+(-\)\(14\!\cdots\!00\)\( + \)\(11\!\cdots\!25\)\( \beta) q^{50}\) \(+(\)\(58\!\cdots\!92\)\( - \)\(23\!\cdots\!60\)\( \beta) q^{51}\) \(+(\)\(80\!\cdots\!00\)\( - 68916084831206960256 \beta) q^{52}\) \(+(-\)\(80\!\cdots\!70\)\( - \)\(29\!\cdots\!48\)\( \beta) q^{53}\) \(+(-\)\(98\!\cdots\!60\)\( + \)\(80\!\cdots\!40\)\( \beta) q^{54}\) \(+(\)\(95\!\cdots\!00\)\( + \)\(21\!\cdots\!00\)\( \beta) q^{55}\) \(+(-\)\(19\!\cdots\!60\)\( - \)\(92\!\cdots\!60\)\( \beta) q^{56}\) \(+(\)\(35\!\cdots\!20\)\( - 14614115692211094720 \beta) q^{57}\) \(+(\)\(32\!\cdots\!60\)\( - \)\(77\!\cdots\!10\)\( \beta) q^{58}\) \(+(-\)\(41\!\cdots\!80\)\( + \)\(21\!\cdots\!20\)\( \beta) q^{59}\) \(+(\)\(18\!\cdots\!00\)\( + \)\(27\!\cdots\!00\)\( \beta) q^{60}\) \(+(-\)\(79\!\cdots\!58\)\( - \)\(24\!\cdots\!00\)\( \beta) q^{61}\) \(+(-\)\(94\!\cdots\!60\)\( + \)\(13\!\cdots\!28\)\( \beta) q^{62}\) \(+(-\)\(35\!\cdots\!80\)\( - \)\(14\!\cdots\!08\)\( \beta) q^{63}\) \(+(\)\(12\!\cdots\!92\)\( + \)\(12\!\cdots\!20\)\( \beta) q^{64}\) \(+(-\)\(68\!\cdots\!00\)\( + 1901725065631664000 \beta) q^{65}\) \(+(\)\(25\!\cdots\!44\)\( - \)\(21\!\cdots\!80\)\( \beta) q^{66}\) \(+(\)\(12\!\cdots\!60\)\( + \)\(53\!\cdots\!44\)\( \beta) q^{67}\) \(+(-\)\(63\!\cdots\!20\)\( + \)\(20\!\cdots\!68\)\( \beta) q^{68}\) \(+(-\)\(68\!\cdots\!64\)\( - \)\(42\!\cdots\!60\)\( \beta) q^{69}\) \(+(-\)\(29\!\cdots\!00\)\( - \)\(13\!\cdots\!00\)\( \beta) q^{70}\) \(+(-\)\(94\!\cdots\!68\)\( - \)\(12\!\cdots\!00\)\( \beta) q^{71}\) \(+(-\)\(57\!\cdots\!20\)\( + \)\(40\!\cdots\!20\)\( \beta) q^{72}\) \(+(\)\(49\!\cdots\!90\)\( + \)\(43\!\cdots\!52\)\( \beta) q^{73}\) \(+(\)\(85\!\cdots\!64\)\( - \)\(55\!\cdots\!50\)\( \beta) q^{74}\) \(+(\)\(78\!\cdots\!00\)\( - \)\(62\!\cdots\!00\)\( \beta) q^{75}\) \(+(-\)\(61\!\cdots\!20\)\( - \)\(11\!\cdots\!20\)\( \beta) q^{76}\) \(+(\)\(18\!\cdots\!00\)\( + \)\(85\!\cdots\!88\)\( \beta) q^{77}\) \(+(-\)\(25\!\cdots\!00\)\( + \)\(72\!\cdots\!64\)\( \beta) q^{78}\) \(+(-\)\(36\!\cdots\!40\)\( - \)\(15\!\cdots\!40\)\( \beta) q^{79}\) \(+(\)\(16\!\cdots\!00\)\( - \)\(21\!\cdots\!00\)\( \beta) q^{80}\) \(+(-\)\(80\!\cdots\!79\)\( - \)\(26\!\cdots\!40\)\( \beta) q^{81}\) \(+(\)\(79\!\cdots\!40\)\( + \)\(43\!\cdots\!38\)\( \beta) q^{82}\) \(+(\)\(11\!\cdots\!20\)\( + \)\(25\!\cdots\!52\)\( \beta) q^{83}\) \(+(\)\(33\!\cdots\!44\)\( + \)\(15\!\cdots\!60\)\( \beta) q^{84}\) \(+(\)\(18\!\cdots\!00\)\( - \)\(12\!\cdots\!00\)\( \beta) q^{85}\) \(+(\)\(10\!\cdots\!32\)\( - \)\(33\!\cdots\!60\)\( \beta) q^{86}\) \(+(-\)\(17\!\cdots\!20\)\( + \)\(43\!\cdots\!20\)\( \beta) q^{87}\) \(+(-\)\(43\!\cdots\!80\)\( - \)\(57\!\cdots\!20\)\( \beta) q^{88}\) \(+(\)\(29\!\cdots\!30\)\( + \)\(16\!\cdots\!80\)\( \beta) q^{89}\) \(+(-\)\(92\!\cdots\!00\)\( + \)\(62\!\cdots\!50\)\( \beta) q^{90}\) \(+(-\)\(92\!\cdots\!48\)\( - \)\(40\!\cdots\!60\)\( \beta) q^{91}\) \(+(\)\(18\!\cdots\!60\)\( + \)\(96\!\cdots\!44\)\( \beta) q^{92}\) \(+(\)\(52\!\cdots\!60\)\( - \)\(78\!\cdots\!56\)\( \beta) q^{93}\) \(+(-\)\(20\!\cdots\!56\)\( + \)\(21\!\cdots\!40\)\( \beta) q^{94}\) \(+(-\)\(13\!\cdots\!00\)\( - \)\(13\!\cdots\!00\)\( \beta) q^{95}\) \(+(-\)\(12\!\cdots\!08\)\( - \)\(23\!\cdots\!40\)\( \beta) q^{96}\) \(+(\)\(54\!\cdots\!90\)\( + \)\(18\!\cdots\!84\)\( \beta) q^{97}\) \(+(-\)\(91\!\cdots\!60\)\( - \)\(35\!\cdots\!57\)\( \beta) q^{98}\) \(+(-\)\(70\!\cdots\!24\)\( + \)\(19\!\cdots\!40\)\( \beta) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 8640q^{2} \) \(\mathstrut -\mathstrut 4967640q^{3} \) \(\mathstrut -\mathstrut 89952256q^{4} \) \(\mathstrut -\mathstrut 17477788500q^{5} \) \(\mathstrut -\mathstrut 543908756736q^{6} \) \(\mathstrut -\mathstrut 3020312682800q^{7} \) \(\mathstrut +\mathstrut 3150297169920q^{8} \) \(\mathstrut +\mathstrut 163469569523706q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 8640q^{2} \) \(\mathstrut -\mathstrut 4967640q^{3} \) \(\mathstrut -\mathstrut 89952256q^{4} \) \(\mathstrut -\mathstrut 17477788500q^{5} \) \(\mathstrut -\mathstrut 543908756736q^{6} \) \(\mathstrut -\mathstrut 3020312682800q^{7} \) \(\mathstrut +\mathstrut 3150297169920q^{8} \) \(\mathstrut +\mathstrut 163469569523706q^{9} \) \(\mathstrut +\mathstrut 34285866768000q^{10} \) \(\mathstrut -\mathstrut 2055380519588616q^{11} \) \(\mathstrut -\mathstrut 4290530276229120q^{12} \) \(\mathstrut +\mathstrut 17139371179332220q^{13} \) \(\mathstrut +\mathstrut 51175123348059648q^{14} \) \(\mathstrut -\mathstrut 17192351392506000q^{15} \) \(\mathstrut -\mathstrut 453469082063208448q^{16} \) \(\mathstrut -\mathstrut 664795927907749980q^{17} \) \(\mathstrut +\mathstrut 3301524864881760960q^{18} \) \(\mathstrut +\mathstrut 1232169445452155080q^{19} \) \(\mathstrut +\mathstrut 1734668101813248000q^{20} \) \(\mathstrut -\mathstrut 27949113476248821696q^{21} \) \(\mathstrut +\mathstrut 1145797484912974080q^{22} \) \(\mathstrut -\mathstrut 18588430504374379920q^{23} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!40\)\(q^{24} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!50\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!64\)\(q^{26} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!80\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(69\!\cdots\!20\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(99\!\cdots\!20\)\(q^{29} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!00\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!56\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(87\!\cdots\!60\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!80\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!92\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!00\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!32\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(98\!\cdots\!60\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!20\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!28\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(87\!\cdots\!00\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!76\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!20\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(51\!\cdots\!00\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!48\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!00\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!64\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!20\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!80\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!14\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!00\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!84\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!00\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!40\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!20\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!00\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!20\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(71\!\cdots\!40\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(64\!\cdots\!20\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(83\!\cdots\!60\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!00\)\(q^{60} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!16\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!20\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(70\!\cdots\!60\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!84\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!00\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(51\!\cdots\!88\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!20\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!40\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!28\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(58\!\cdots\!00\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!36\)\(q^{71} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!40\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(99\!\cdots\!80\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!28\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!40\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!00\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(51\!\cdots\!00\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(72\!\cdots\!80\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!00\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!58\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!80\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!40\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(66\!\cdots\!88\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!00\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!64\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!40\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(86\!\cdots\!60\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(58\!\cdots\!60\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!00\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!96\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!20\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!20\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!12\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!00\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!16\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!80\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!20\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!48\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
113.802
−112.802
−17433.9 9.52434e6 −2.32930e8 −1.12623e10 −1.66046e11 −2.98628e12 1.34206e13 2.20826e13 1.96347e14
1.2 26073.9 −1.44920e7 1.42978e8 −6.21544e9 −3.77862e11 −3.40335e10 −1.02703e13 1.41387e14 −1.62061e14
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Hecke kernels

There are no other newforms in \(S_{30}^{\mathrm{new}}(\Gamma_0(1))\).