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

Related objects

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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: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2.40.a.a 1
3.b odd 2 1 18.40.a.a 1
4.b odd 2 1 16.40.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.40.a.a 1 1.a even 1 1 trivial
16.40.a.a 1 4.b odd 2 1
18.40.a.a 1 3.b odd 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 735458292 \) acting on \(S_{40}^{\mathrm{new}}(\Gamma_0(2))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 524288 T \)
$3$ \( 1 + 735458292 T + 4052555153018976267 T^{2} \)
$5$ \( 1 + 16226178983250 T + \)\(18\!\cdots\!25\)\( T^{2} \)
$7$ \( 1 - 16050065775887864 T + \)\(90\!\cdots\!43\)\( T^{2} \)
$11$ \( 1 + \)\(16\!\cdots\!48\)\( T + \)\(41\!\cdots\!91\)\( T^{2} \)
$13$ \( 1 + \)\(13\!\cdots\!42\)\( T + \)\(27\!\cdots\!77\)\( T^{2} \)
$17$ \( 1 + \)\(49\!\cdots\!66\)\( T + \)\(97\!\cdots\!53\)\( T^{2} \)
$19$ \( 1 + \)\(11\!\cdots\!80\)\( T + \)\(74\!\cdots\!79\)\( T^{2} \)
$23$ \( 1 + \)\(66\!\cdots\!72\)\( T + \)\(12\!\cdots\!87\)\( T^{2} \)
$29$ \( 1 - \)\(44\!\cdots\!50\)\( T + \)\(10\!\cdots\!69\)\( T^{2} \)
$31$ \( 1 - \)\(15\!\cdots\!92\)\( T + \)\(14\!\cdots\!71\)\( T^{2} \)
$37$ \( 1 + \)\(25\!\cdots\!46\)\( T + \)\(14\!\cdots\!73\)\( T^{2} \)
$41$ \( 1 - \)\(51\!\cdots\!42\)\( T + \)\(79\!\cdots\!61\)\( T^{2} \)
$43$ \( 1 - \)\(78\!\cdots\!28\)\( T + \)\(50\!\cdots\!07\)\( T^{2} \)
$47$ \( 1 - \)\(24\!\cdots\!04\)\( T + \)\(16\!\cdots\!83\)\( T^{2} \)
$53$ \( 1 - \)\(69\!\cdots\!98\)\( T + \)\(17\!\cdots\!17\)\( T^{2} \)
$59$ \( 1 + \)\(20\!\cdots\!00\)\( T + \)\(11\!\cdots\!39\)\( T^{2} \)
$61$ \( 1 + \)\(12\!\cdots\!18\)\( T + \)\(42\!\cdots\!41\)\( T^{2} \)
$67$ \( 1 - \)\(45\!\cdots\!64\)\( T + \)\(16\!\cdots\!03\)\( T^{2} \)
$71$ \( 1 + \)\(99\!\cdots\!28\)\( T + \)\(15\!\cdots\!31\)\( T^{2} \)
$73$ \( 1 - \)\(81\!\cdots\!18\)\( T + \)\(46\!\cdots\!37\)\( T^{2} \)
$79$ \( 1 + \)\(85\!\cdots\!40\)\( T + \)\(10\!\cdots\!19\)\( T^{2} \)
$83$ \( 1 - \)\(71\!\cdots\!48\)\( T + \)\(69\!\cdots\!47\)\( T^{2} \)
$89$ \( 1 + \)\(13\!\cdots\!90\)\( T + \)\(10\!\cdots\!09\)\( T^{2} \)
$97$ \( 1 - \)\(76\!\cdots\!34\)\( T + \)\(30\!\cdots\!33\)\( T^{2} \)
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