Properties

Label 2.40.a.a
Level 2
Weight 40
Character orbit 2.a
Self dual Yes
Analytic conductor 19.268
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 40 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(19.2679102779\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 524288q^{2} \) \(\mathstrut -\mathstrut 735458292q^{3} \) \(\mathstrut +\mathstrut 274877906944q^{4} \) \(\mathstrut -\mathstrut 16226178983250q^{5} \) \(\mathstrut -\mathstrut 385591956996096q^{6} \) \(\mathstrut +\mathstrut 16050065775887864q^{7} \) \(\mathstrut +\mathstrut 144115188075855872q^{8} \) \(\mathstrut -\mathstrut 3511656253747419003q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 524288q^{2} \) \(\mathstrut -\mathstrut 735458292q^{3} \) \(\mathstrut +\mathstrut 274877906944q^{4} \) \(\mathstrut -\mathstrut 16226178983250q^{5} \) \(\mathstrut -\mathstrut 385591956996096q^{6} \) \(\mathstrut +\mathstrut 16050065775887864q^{7} \) \(\mathstrut +\mathstrut 144115188075855872q^{8} \) \(\mathstrut -\mathstrut 3511656253747419003q^{9} \) \(\mathstrut -\mathstrut 8507190926770176000q^{10} \) \(\mathstrut -\mathstrut 167429747630019631548q^{11} \) \(\mathstrut -\mathstrut 202161235949569179648q^{12} \) \(\mathstrut -\mathstrut 1323691058888421756442q^{13} \) \(\mathstrut +\mathstrut 8414856885508696440832q^{14} \) \(\mathstrut +\mathstrut 11933677880707341609000q^{15} \) \(\mathstrut +\mathstrut 75557863725914323419136q^{16} \) \(\mathstrut -\mathstrut 496480799590583480551566q^{17} \) \(\mathstrut -\mathstrut 1841119233964726814244864q^{18} \) \(\mathstrut -\mathstrut 11499782498758130928946180q^{19} \) \(\mathstrut -\mathstrut 4460218116614482034688000q^{20} \) \(\mathstrut -\mathstrut 11804153962022143240968288q^{21} \) \(\mathstrut -\mathstrut 87781407525447732585037824q^{22} \) \(\mathstrut -\mathstrut 666241934758249389046846872q^{23} \) \(\mathstrut -\mathstrut 105990710073527726059290624q^{24} \) \(\mathstrut -\mathstrut 1555700519149392472049515625q^{25} \) \(\mathstrut -\mathstrut 693995337882492865841463296q^{26} \) \(\mathstrut +\mathstrut 5563162001547330308271078840q^{27} \) \(\mathstrut +\mathstrut 4411808486789583439570927616q^{28} \) \(\mathstrut +\mathstrut 44018303948798095210174459350q^{29} \) \(\mathstrut +\mathstrut 6256684108720290717499392000q^{30} \) \(\mathstrut +\mathstrut 15831006217138043611820986592q^{31} \) \(\mathstrut +\mathstrut 39614081257132168796771975168q^{32} \) \(\mathstrut +\mathstrut 123137596221965286144761396016q^{33} \) \(\mathstrut -\mathstrut 260298925455747831851419435008q^{34} \) \(\mathstrut -\mathstrut 260431239972491763445534278000q^{35} \) \(\mathstrut -\mathstrut 965276720936898691986811256832q^{36} \) \(\mathstrut -\mathstrut 2586566128509039416001203660146q^{37} \) \(\mathstrut -\mathstrut 6029197966708902948475334819840q^{38} \) \(\mathstrut +\mathstrut 973519565305750083568473317064q^{39} \) \(\mathstrut -\mathstrut 2338438835923573557002502144000q^{40} \) \(\mathstrut +\mathstrut 51237316431927477788465354103642q^{41} \) \(\mathstrut -\mathstrut 6188776272440665435520781778944q^{42} \) \(\mathstrut +\mathstrut 78798404752245159214502811925028q^{43} \) \(\mathstrut -\mathstrut 46022738588701940821544310669312q^{44} \) \(\mathstrut +\mathstrut 56980762900954799280410268699750q^{45} \) \(\mathstrut -\mathstrut 349302651490533055684593252827136q^{46} \) \(\mathstrut +\mathstrut 242751106429257069824806165133904q^{47} \) \(\mathstrut -\mathstrut 55569657403029704440173362675712q^{48} \) \(\mathstrut -\mathstrut 651939068719534238995907517406647q^{49} \) \(\mathstrut -\mathstrut 815635113783796680385896448000000q^{50} \) \(\mathstrut +\mathstrut 365140920877684825889869948285272q^{51} \) \(\mathstrut -\mathstrut 363853427707736419646289108533248q^{52} \) \(\mathstrut +\mathstrut 695517143534500492850281534063998q^{53} \) \(\mathstrut +\mathstrut 2916699079467246712662827382865920q^{54} \) \(\mathstrut +\mathstrut 2716745052165076042209066263571000q^{55} \) \(\mathstrut +\mathstrut 2313058247921937122365762497937408q^{56} \) \(\mathstrut +\mathstrut 8457610394908147094115130902724560q^{57} \) \(\mathstrut +\mathstrut 23078268540707455741551946943692800q^{58} \) \(\mathstrut -\mathstrut 20181524948439110783048183071512300q^{59} \) \(\mathstrut +\mathstrut 3280304397992743779696321232896000q^{60} \) \(\mathstrut -\mathstrut 128433361186908960737422356033926218q^{61} \) \(\mathstrut +\mathstrut 8300006587570870609154401418346496q^{62} \) \(\mathstrut -\mathstrut 56362313854954038402712120750679592q^{63} \) \(\mathstrut +\mathstrut 20769187434139310514121985316880384q^{64} \) \(\mathstrut +\mathstrut 21478448040051247211141230697596500q^{65} \) \(\mathstrut +\mathstrut 64559564048021735942264662794436608q^{66} \) \(\mathstrut +\mathstrut 456917726301331948233545546175061964q^{67} \) \(\mathstrut -\mathstrut 136471603029343119265716992741474304q^{68} \) \(\mathstrut +\mathstrut 489993155396077528578437508466662624q^{69} \) \(\mathstrut -\mathstrut 136540973942697761673332275544064000q^{70} \) \(\mathstrut -\mathstrut 99458563668407119053976138500473928q^{71} \) \(\mathstrut -\mathstrut 506083001466564741424381300221935616q^{72} \) \(\mathstrut +\mathstrut 812887758504563318241014461088181818q^{73} \) \(\mathstrut -\mathstrut 1356105582383747257336439064570626048q^{74} \) \(\mathstrut +\mathstrut 1144152846677125480331194500989812500q^{75} \) \(\mathstrut -\mathstrut 3161036143569877309050236342024273920q^{76} \) \(\mathstrut -\mathstrut 2687258462302120296336444819844733472q^{77} \) \(\mathstrut +\mathstrut 510404625855021099813947738456850432q^{78} \) \(\mathstrut -\mathstrut 8572745601192170985075907678235922640q^{79} \) \(\mathstrut -\mathstrut 1226015420408698533053727844073472000q^{80} \) \(\mathstrut +\mathstrut 10139707022978115662060222115898060521q^{81} \) \(\mathstrut +\mathstrut 26863110157462393474758923572290256896q^{82} \) \(\mathstrut +\mathstrut 7125890269767880907477351827779977148q^{83} \) \(\mathstrut -\mathstrut 3244701134325371599858319637318991872q^{84} \) \(\mathstrut +\mathstrut 8055986315903880876730455347075269500q^{85} \) \(\mathstrut +\mathstrut 41313058030745110034253250258549080064q^{86} \) \(\mathstrut -\mathstrut 32373626638919902556128288895574430200q^{87} \) \(\mathstrut -\mathstrut 24129169569193363149445823552192249856q^{88} \) \(\mathstrut -\mathstrut 130969521476936175444631870523462811990q^{89} \) \(\mathstrut +\mathstrut 29874330219815789805127738956054528000q^{90} \) \(\mathstrut -\mathstrut 21245328562113825215144039666911619888q^{91} \) \(\mathstrut -\mathstrut 183135188544668594698764027338233479168q^{92} \) \(\mathstrut -\mathstrut 11643044793097726682971373808707220864q^{93} \) \(\mathstrut +\mathstrut 127271492087582330624307974705724260352q^{94} \) \(\mathstrut +\mathstrut 186597529093295353304318304986371485000q^{95} \) \(\mathstrut -\mathstrut 29134504540519637681529611970523693056q^{96} \) \(\mathstrut +\mathstrut 761042994574827986180495148895603139234q^{97} \) \(\mathstrut -\mathstrut 341803830460827167094686360486096142336q^{98} \) \(\mathstrut +\mathstrut 587955720328310544684461155526033506644q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
0
524288. −7.35458e8 2.74878e11 −1.62262e13 −3.85592e14 1.60501e16 1.44115e17 −3.51166e18 −8.50719e18
\(n\): 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 \(T_{3} \) \(\mathstrut +\mathstrut 735458292 \) acting on \(S_{40}^{\mathrm{new}}(\Gamma_0(2))\).