Properties

Label 2.28.a.a
Level 2
Weight 28
Character orbit 2.a
Self dual Yes
Analytic conductor 9.237
Analytic rank 0
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 28 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(9.23711149676\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q - 8192q^{2} + 3984828q^{3} + 67108864q^{4} - 2851889250q^{5} - 32643710976q^{6} + 368721063704q^{7} - 549755813888q^{8} + 8253256704597q^{9} + O(q^{10}) \) \( q - 8192q^{2} + 3984828q^{3} + 67108864q^{4} - 2851889250q^{5} - 32643710976q^{6} + 368721063704q^{7} - 549755813888q^{8} + 8253256704597q^{9} + 23362676736000q^{10} + 59896911912852q^{11} + 267417280315392q^{12} + 1902611126010998q^{13} - 3020562953863168q^{14} - 11364288136299000q^{15} + 4503599627370496q^{16} - 3449864157282126q^{17} - 67610678924058624q^{18} + 158487413654686700q^{19} - 191387047821312000q^{20} + 1469290018837482912q^{21} - 490675502390083584q^{22} - 1257122371521270072q^{23} - 2190682360343691264q^{24} + 682691697341734375q^{25} - 15586190344282095616q^{26} + 2501114032760077080q^{27} + 24744451718047072256q^{28} + 52884218157232223910q^{29} + 93096248412561408000q^{30} - 236582716743417120928q^{31} - 36893488147419103232q^{32} + 238678891703866209456q^{33} + 28261287176455176192q^{34} - 1051551637826002782000q^{35} + 553866681745888247808q^{36} - 700609078644771513346q^{37} - 1298328892659193446400q^{38} + 7581578088040153138344q^{39} + 1567842695752187904000q^{40} - 297530762921478223878q^{41} - 12036423834316660015104q^{42} + 162258398551243950068q^{43} + 4019613715579564720128q^{44} - 23537374073330609882250q^{45} + 10298346467502244429824q^{46} - 8701760862057607485936q^{47} + 17946069895935518834688q^{48} + 70242860455474946060073q^{49} - 5592610384623488000000q^{50} - 13747115290134219584328q^{51} + 127682071300358927286272q^{52} - 204136777825800152405202q^{53} - 20489126156370551439360q^{54} - 170819359192459555641000q^{55} - 202706548474241615921152q^{56} + 631545083578777893387600q^{57} - 433227515144046378270720q^{58} - 46341379034584555074780q^{59} - 762644466995703054336000q^{60} + 536423474575698953006342q^{61} + 1938085615562073054642176q^{62} + 3043149591141175546647288q^{63} + 302231454903657293676544q^{64} - 5426036217201160577971500q^{65} - 1955257480838071987863552q^{66} - 2854848488828581020392836q^{67} - 231516464549520803364864q^{68} - 5009416425464359578467616q^{69} + 8614311017070614790144000q^{70} + 11069760379179718709565912q^{71} - 4537275856862316526043136q^{72} - 4933635680015518303566502q^{73} + 5739389572257968237330432q^{74} + 2720408990934868706062500q^{75} + 10635910288664112712908800q^{76} + 22085253073091578788323808q^{77} - 62108287697224934509314048q^{78} + 31411864518460750828112240q^{79} - 12843767363601923309568000q^{80} - 52969504340591706370842999q^{81} + 2437372009852749610008576q^{82} - 62818814968751953942411092q^{83} + 98602384050722078843731968q^{84} + 9838630504113204356545500q^{85} - 1329220800931790438957056q^{86} + 210734513271047368338837480q^{87} - 32928675558027794187288576q^{88} - 370405798981885975118175990q^{89} + 192818168408724356155392000q^{90} + 701532798197840364962616592q^{91} - 84364054261778386369118208q^{92} - 942741433995237359153280384q^{93} + 71284824981975920524787712q^{94} - 451988551262104211847975000q^{95} - 147014204587503770293764096q^{96} + 512277538175158422436165154q^{97} - 575429512851250758124118016q^{98} + 494344589829401689171780644q^{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
−8192.00 3.98483e6 6.71089e7 −2.85189e9 −3.26437e10 3.68721e11 −5.49756e11 8.25326e12 2.33627e13
\(n\): 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 \( T_{3} - 3984828 \) acting on \(S_{28}^{\mathrm{new}}(\Gamma_0(2))\).