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

$q$-expansion

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

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 3.24.a.a 1
3.b odd 2 1 9.24.a.a 1
4.b odd 2 1 48.24.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.24.a.a 1 1.a even 1 1 trivial
9.24.a.a 1 3.b odd 2 1
48.24.a.a 1 4.b odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1128 + T \)
$3$ \( -177147 + T \)
$5$ \( 48863730 + T \)
$7$ \( 1723688680 + T \)
$11$ \( 1428263180124 + T \)
$13$ \( 8220964044826 + T \)
$17$ \( 5989210330446 + T \)
$19$ \( -680005481275676 + T \)
$23$ \( -15440648191080 + T \)
$29$ \( -115094192813324022 + T \)
$31$ \( 90829724501108800 + T \)
$37$ \( 1297873386623227570 + T \)
$41$ \( -5214036225478655130 + T \)
$43$ \( 2410434516296794108 + T \)
$47$ \( 23132669525900803824 + T \)
$53$ \( 44512631945276522850 + T \)
$59$ \( \)\(32\!\cdots\!76\)\( + T \)
$61$ \( \)\(19\!\cdots\!22\)\( + T \)
$67$ \( \)\(64\!\cdots\!96\)\( + T \)
$71$ \( -\)\(35\!\cdots\!12\)\( + T \)
$73$ \( -\)\(33\!\cdots\!70\)\( + T \)
$79$ \( \)\(68\!\cdots\!20\)\( + T \)
$83$ \( \)\(11\!\cdots\!44\)\( + T \)
$89$ \( \)\(23\!\cdots\!74\)\( + T \)
$97$ \( \)\(30\!\cdots\!86\)\( + T \)
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