Properties

Label 2.36.a.a
Level 2
Weight 36
Character orbit 2.a
Self dual Yes
Analytic conductor 15.519
Analytic rank 0
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 36 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(15.5190261267\)
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

\(f(q)\) \(=\) \( q - 131072q^{2} + 36494748q^{3} + 17179869184q^{4} + 389070858750q^{5} - 4783439609856q^{6} - 129689369490856q^{7} - 2251799813685248q^{8} - 48699678467416203q^{9} + O(q^{10}) \) \( q - 131072q^{2} + 36494748q^{3} + 17179869184q^{4} + 389070858750q^{5} - 4783439609856q^{6} - 129689369490856q^{7} - 2251799813685248q^{8} - 48699678467416203q^{9} - 50996295598080000q^{10} + 469638570722172852q^{11} + 626974996543045632q^{12} + 28541547044626383638q^{13} + 16998645037905477632q^{14} + 14199042944224845000q^{15} + 295147905179352825856q^{16} + 5390298709064366404914q^{17} + 6383164256081176559616q^{18} + 34188850221443340757580q^{19} + 6684186456631541760000q^{20} - 4732980857847678024288q^{21} - 61556466741696640057344q^{22} - 170205212851012933201272q^{23} - 82178866746890077077504q^{24} - 2759006912544907909765625q^{25} - 3740997654233269356199936q^{26} - 3603171123788009898640680q^{27} - 2228046402408346764181504q^{28} + 45455583968151176507388870q^{29} - 1861096956785438883840000q^{30} + 193288220828538443977926752q^{31} - 38685626227668133590597632q^{32} + 17139341289585876246181296q^{33} - 706517232394484633424887808q^{34} - 50458354358553394192590000q^{35} - 836654105373071974021988352q^{36} + 1499846546966350893154927454q^{37} - 4481200976225021559777525760q^{38} + 1041616566923784625020133224q^{39} - 876109687243609441566720000q^{40} + 15121824929562972920630265402q^{41} + 620361266999810853999476736q^{42} + 37564365389338064172981957908q^{43} + 8068329208767662005596192768q^{44} - 18947625722166505994874326250q^{45} + 22309137658807967180557123584q^{46} - 310291642984003302036824689776q^{47} + 10771348422248376182702604288q^{48} - 361999359706729010622050013207q^{49} + 361628554041086169548800000000q^{50} + 196717593032029367727002391672q^{51} + 490340044535663081055838011392q^{52} + 652810129063511369533379325198q^{53} + 472274845537142033434631208960q^{54} + 182722682012998399193576655000q^{55} + 292034498056466827074798092288q^{56} + 1247713473241318917226011189840q^{57} - 5957954301873511007176473968640q^{58} - 1257024650427227659122067488060q^{59} + 243937700319781045382676480000q^{60} + 166767144602738383087145451302q^{61} - 25334673680438190929074815238144q^{62} + 6315830594846623801247954739768q^{63} + 5070602400912917605986812821504q^{64} + 11104684218706311654943609132500q^{65} - 2246487741508599971339474829312q^{66} + 112725124573795147642205351427164q^{67} + 92604626684409889872266894770176q^{68} - 6211596351284078541921254919456q^{69} + 6613677422484310483611156480000q^{70} - 397712012261633827759413397231848q^{71} + 109661926899459289779010057273344q^{72} - 333895359134508153342394992690022q^{73} - 196587886603973544267602651250688q^{74} - 100689262003584452850103223437500q^{75} + 587359974355766025883159856414720q^{76} - 60907130125545381755213265441312q^{77} - 136526766659834298370638901936128q^{78} + 1382904481345969270449030874945520q^{79} + 114833448926394376725033123840000q^{80} + 2305023337385798466719650984903881q^{81} - 1982047837167677986652850146770944q^{82} - 828639202304461825856973692415732q^{83} - 81311991988199208255419414740992q^{84} + 2097208147654689446184540199897500q^{85} - 4923636500311318747281091186917376q^{86} + 1658890082110517212540676948654760q^{87} - 1057532046051594994397504178487296q^{88} - 13863918224809647470926585526996790q^{89} + 2483503198655808273760167690240000q^{90} - 3701535240511200150901272589014128q^{91} - 2924103291215277874289983302402048q^{92} + 7054004910505861721286554376698496q^{93} + 40670546229199280804570685738319872q^{94} + 13301885315332088253020526171825000q^{95} - 1411822180400939163019195749236736q^{96} + 12254634389030627568202089383184674q^{97} + 47447980075480384880253339331067904q^{98} - 22871247390066722862046658391520956q^{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
−131072. 3.64947e7 1.71799e10 3.89071e11 −4.78344e12 −1.29689e14 −2.25180e15 −4.86997e16 −5.09963e16
\(n\): 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 \( T_{3} - 36494748 \) acting on \(S_{36}^{\mathrm{new}}(\Gamma_0(2))\).