Properties

Label 3.24.a.a
Level 3
Weight 24
Character orbit 3.a
Self dual Yes
Analytic conductor 10.056
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 24 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(10.0561211204\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 1128q^{2} \) \(\mathstrut +\mathstrut 177147q^{3} \) \(\mathstrut -\mathstrut 7116224q^{4} \) \(\mathstrut -\mathstrut 48863730q^{5} \) \(\mathstrut +\mathstrut 199821816q^{6} \) \(\mathstrut -\mathstrut 1723688680q^{7} \) \(\mathstrut -\mathstrut 17489450496q^{8} \) \(\mathstrut +\mathstrut 31381059609q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 1128q^{2} \) \(\mathstrut +\mathstrut 177147q^{3} \) \(\mathstrut -\mathstrut 7116224q^{4} \) \(\mathstrut -\mathstrut 48863730q^{5} \) \(\mathstrut +\mathstrut 199821816q^{6} \) \(\mathstrut -\mathstrut 1723688680q^{7} \) \(\mathstrut -\mathstrut 17489450496q^{8} \) \(\mathstrut +\mathstrut 31381059609q^{9} \) \(\mathstrut -\mathstrut 55118287440q^{10} \) \(\mathstrut -\mathstrut 1428263180124q^{11} \) \(\mathstrut -\mathstrut 1260617732928q^{12} \) \(\mathstrut -\mathstrut 8220964044826q^{13} \) \(\mathstrut -\mathstrut 1944320831040q^{14} \) \(\mathstrut -\mathstrut 8656063178310q^{15} \) \(\mathstrut +\mathstrut 39967113416704q^{16} \) \(\mathstrut -\mathstrut 5989210330446q^{17} \) \(\mathstrut +\mathstrut 35397835238952q^{18} \) \(\mathstrut +\mathstrut 680005481275676q^{19} \) \(\mathstrut +\mathstrut 347725248155520q^{20} \) \(\mathstrut -\mathstrut 305346278595960q^{21} \) \(\mathstrut -\mathstrut 1611080867179872q^{22} \) \(\mathstrut +\mathstrut 15440648191080q^{23} \) \(\mathstrut -\mathstrut 3098203687014912q^{24} \) \(\mathstrut -\mathstrut 9533264845565225q^{25} \) \(\mathstrut -\mathstrut 9273247442563728q^{26} \) \(\mathstrut +\mathstrut 5559060566555523q^{27} \) \(\mathstrut +\mathstrut 12266154753144320q^{28} \) \(\mathstrut +\mathstrut 115094192813324022q^{29} \) \(\mathstrut -\mathstrut 9764039265133680q^{30} \) \(\mathstrut -\mathstrut 90829724501108800q^{31} \) \(\mathstrut +\mathstrut 191795048280391680q^{32} \) \(\mathstrut -\mathstrut 253012537569426228q^{33} \) \(\mathstrut -\mathstrut 6755829252743088q^{34} \) \(\mathstrut +\mathstrut 84225858263576400q^{35} \) \(\mathstrut -\mathstrut 223314649534996416q^{36} \) \(\mathstrut -\mathstrut 1297873386623227570q^{37} \) \(\mathstrut +\mathstrut 767046182878962528q^{38} \) \(\mathstrut -\mathstrut 1456319117648791422q^{39} \) \(\mathstrut +\mathstrut 854599786884910080q^{40} \) \(\mathstrut +\mathstrut 5214036225478655130q^{41} \) \(\mathstrut -\mathstrut 344430602256242880q^{42} \) \(\mathstrut -\mathstrut 2410434516296794108q^{43} \) \(\mathstrut +\mathstrut 10163840720714731776q^{44} \) \(\mathstrut -\mathstrut 1533395623848081570q^{45} \) \(\mathstrut +\mathstrut 17417051159538240q^{46} \) \(\mathstrut -\mathstrut 23132669525900803824q^{47} \) \(\mathstrut +\mathstrut 7080054240428863488q^{48} \) \(\mathstrut -\mathstrut 24397644674520773943q^{49} \) \(\mathstrut -\mathstrut 10753522745797573800q^{50} \) \(\mathstrut -\mathstrut 1060970642407517562q^{51} \) \(\mathstrut +\mathstrut 58502221638927857024q^{52} \) \(\mathstrut -\mathstrut 44512631945276522850q^{53} \) \(\mathstrut +\mathstrut 6270620319074629944q^{54} \) \(\mathstrut +\mathstrut 69790266402520502520q^{55} \) \(\mathstrut +\mathstrut 30146367839375585280q^{56} \) \(\mathstrut +\mathstrut 120460930991542176372q^{57} \) \(\mathstrut +\mathstrut 129826249493429496816q^{58} \) \(\mathstrut -\mathstrut 323974479000840790476q^{59} \) \(\mathstrut +\mathstrut 61598484535005901440q^{60} \) \(\mathstrut -\mathstrut 199406203121599312522q^{61} \) \(\mathstrut -\mathstrut 102455929237250726400q^{62} \) \(\mathstrut -\mathstrut 54091177214438526120q^{63} \) \(\mathstrut -\mathstrut 118923632883988692992q^{64} \) \(\mathstrut +\mathstrut 401706967426085560980q^{65} \) \(\mathstrut -\mathstrut 285398142378312785184q^{66} \) \(\mathstrut -\mathstrut 646392500721161158996q^{67} \) \(\mathstrut +\mathstrut 42620562294567755904q^{68} \) \(\mathstrut +\mathstrut 2735264505105248760q^{69} \) \(\mathstrut +\mathstrut 95006768121314179200q^{70} \) \(\mathstrut +\mathstrut 3551461551813260928312q^{71} \) \(\mathstrut -\mathstrut 548837488543630616064q^{72} \) \(\mathstrut +\mathstrut 3353187900182300778170q^{73} \) \(\mathstrut -\mathstrut 1464001180111000698960q^{74} \) \(\mathstrut -\mathstrut 1688789267597342913075q^{75} \) \(\mathstrut -\mathstrut 4839071325985516167424q^{76} \) \(\mathstrut +\mathstrut 2461881075640539796320q^{77} \) \(\mathstrut -\mathstrut 1642727964707836724016q^{78} \) \(\mathstrut -\mathstrut 6872134095241809038320q^{79} \) \(\mathstrut -\mathstrut 1952942238873201745920q^{80} \) \(\mathstrut +\mathstrut 984770902183611232881q^{81} \) \(\mathstrut +\mathstrut 5881432862339922986640q^{82} \) \(\mathstrut -\mathstrut 1169769717495414820644q^{83} \) \(\mathstrut +\mathstrut 2172912516055256855040q^{84} \) \(\mathstrut +\mathstrut 292655156500124123580q^{85} \) \(\mathstrut -\mathstrut 2718970134382783753824q^{86} \) \(\mathstrut +\mathstrut 20388590974301910525234q^{87} \) \(\mathstrut +\mathstrut 24979538184038229141504q^{88} \) \(\mathstrut -\mathstrut 23457212631337905637974q^{89} \) \(\mathstrut -\mathstrut 1729670263700636010960q^{90} \) \(\mathstrut +\mathstrut 14170382662753588769680q^{91} \) \(\mathstrut -\mathstrut 109879111232920081920q^{92} \) \(\mathstrut -\mathstrut 16090213206197920593600q^{93} \) \(\mathstrut -\mathstrut 26093651225216106713472q^{94} \) \(\mathstrut -\mathstrut 33227604235574687631480q^{95} \) \(\mathstrut +\mathstrut 33975917417726544936960q^{96} \) \(\mathstrut -\mathstrut 30603881563463466110686q^{97} \) \(\mathstrut -\mathstrut 27520543192859433007704q^{98} \) \(\mathstrut -\mathstrut 44820411992811148011516q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
1128.00 177147. −7.11622e6 −4.88637e7 1.99822e8 −1.72369e9 −1.74895e10 3.13811e10 −5.51183e10
\(n\): 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
\(3\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2} \) \(\mathstrut -\mathstrut 1128 \) acting on \(S_{24}^{\mathrm{new}}(\Gamma_0(3))\).