Properties

Label 2.30.a.a
Level 2
Weight 30
Character orbit 2.a
Self dual Yes
Analytic conductor 10.656
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 30 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(10.6556084766\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 16384q^{2} - 2792556q^{3} + 268435456q^{4} + 6651856470q^{5} + 45753237504q^{6} + 1432518476648q^{7} - 4398046511104q^{8} - 60832008351747q^{9} + O(q^{10}) \) \( q - 16384q^{2} - 2792556q^{3} + 268435456q^{4} + 6651856470q^{5} + 45753237504q^{6} + 1432518476648q^{7} - 4398046511104q^{8} - 60832008351747q^{9} - 108984016404480q^{10} - 777022624156548q^{11} - 749621043265536q^{12} - 27095750590848706q^{13} - 23470382721400832q^{14} - 18575681696437320q^{15} + 72057594037927936q^{16} + 623720384075229138q^{17} + 996671624835022848q^{18} + 23397560477447780q^{19} + 1785594124771000320q^{20} - 4000388067074232288q^{21} + 12730738674180882432q^{22} - 101961500467007428296q^{23} + 12281791172862541824q^{24} - 142017320425614842225q^{25} + 443936777680465199104q^{26} + 361530962007289406280q^{27} + 384538750507431231488q^{28} - 193842816899629681650q^{29} + 304343968914429050880q^{30} - 4411693069206519719008q^{31} - 1180591620717411303424q^{32} + 2169879191224113056688q^{33} - 10219034772688554196992q^{34} + 9528907297285542712560q^{35} - 16329467901297014341632q^{36} + 68464702455096345074678q^{37} - 383345630862504427520q^{38} + 75666400886978099032536q^{39} - 29255174140248069242880q^{40} - 130597382880843284243958q^{41} + 65542358090944221806592q^{42} - 655176150269911693888996q^{43} - 208580422437779577765888q^{44} - 404645788337662317753090q^{45} + 1670537223651449705201664q^{46} + 1883486986380896623798128q^{47} - 201224866576179885244416q^{48} - 1167796569875273205521703q^{49} + 2326811777853273575014400q^{50} - 1741774100871585580696728q^{51} - 7273460165516741822119936q^{52} + 6529902016158550020796614q^{53} - 5923323281527429632491520q^{54} - 5168642969832112106665560q^{55} - 6300282888313753296699392q^{56} - 65338997896659662725680q^{57} + 3175920712083532704153600q^{58} + 44894091442793669254526700q^{59} - 4986371586694005569617920q^{60} + 35309137728183249168305582q^{61} + 72281179245879619076227072q^{62} - 87142975935483025789504056q^{63} + 19342813113834066795298816q^{64} - 180237043877243287797227820q^{65} - 35551300669015868320776192q^{66} - 383658355413958878684363052q^{67} + 167428665715729271963516928q^{68} + 284733199898144395932564576q^{69} - 156121617158726331802583040q^{70} + 728985379862346402417775272q^{71} + 267542002094850282973298688q^{72} + 724286660588086686263964074q^{73} - 1121725685024298517703524352q^{74} + 396591320258473281344477100q^{75} + 6280734816051272540487680q^{76} - 1113099265877769586834291104q^{77} - 1239718312132249174549069824q^{78} - 2013411690690004127061099760q^{79} + 479316773113824366475345920q^{80} + 3165328231904882846785848921q^{81} + 2139707521119736369053007872q^{82} - 7236735323128970857830291036q^{83} - 1073845994962030130079203328q^{84} + 4148898472281697908337822860q^{85} + 10734406046022233192677310464q^{86} + 541316921389962265269797400q^{87} + 3417381641220580602116308992q^{88} + 13578154221906201595278362490q^{89} + 6629716596124259414066626560q^{90} - 38815163360036734248562017488q^{91} - 27370081872305351970024062976q^{92} + 12319899950571081880434104448q^{93} - 30859050784864610284308529152q^{94} + 155637214044127304480136600q^{95} + 3296868213984131239844511744q^{96} + 22518383238623495861116621538q^{97} + 19133179000836476199267581952q^{98} + 47267846762187498167577289356q^{99} + 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
0
−16384.0 −2.79256e6 2.68435e8 6.65186e9 4.57532e10 1.43252e12 −4.39805e12 −6.08320e13 −1.08984e14
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{3} + 2792556 \) acting on \(S_{30}^{\mathrm{new}}(\Gamma_0(2))\).