Properties

Label 3.20.a.a
Level 3
Weight 20
Character orbit 3.a
Self dual Yes
Analytic conductor 6.865
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 20 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(6.86450089669\)
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

\(f(q)\) \(=\) \( q - 1104q^{2} + 19683q^{3} + 694528q^{4} + 3516270q^{5} - 21730032q^{6} - 195590584q^{7} - 187944960q^{8} + 387420489q^{9} + O(q^{10}) \) \( q - 1104q^{2} + 19683q^{3} + 694528q^{4} + 3516270q^{5} - 21730032q^{6} - 195590584q^{7} - 187944960q^{8} + 387420489q^{9} - 3881962080q^{10} - 2746857948q^{11} + 13670394624q^{12} - 44400445258q^{13} + 215932004736q^{14} + 69210742410q^{15} - 156641460224q^{16} - 785982517614q^{17} - 427712219856q^{18} + 315410465180q^{19} + 2442147970560q^{20} - 3849809464872q^{21} + 3032531174592q^{22} + 4900560535752q^{23} - 3699320647680q^{24} - 6709331615225q^{25} + 49018091564832q^{26} + 7625597484987q^{27} - 135843137124352q^{28} + 12188520672150q^{29} - 76408659620640q^{30} - 42713658601168q^{31} + 271469459275776q^{32} - 54066404990484q^{33} + 867724699445856q^{34} - 687749302801680q^{35} + 269074377384192q^{36} - 423452395388194q^{37} - 348213153558720q^{38} - 873933964013214q^{39} - 660865224499200q^{40} - 1113920690896038q^{41} + 4250189649218688q^{42} + 1136100238138052q^{43} - 1907769756908544q^{44} + 1362275042856030q^{45} - 5410218831470208q^{46} + 1531372040448816q^{47} - 3083173861588992q^{48} + 26856781364087913q^{49} + 7407102103208400q^{50} - 15470493894196362q^{51} - 30837352444148224q^{52} - 18059320314853218q^{53} - 8418659623425648q^{54} - 9658694196813960q^{55} + 36760264486256640q^{56} + 6208224186137940q^{57} - 13456126822053600q^{58} + 92700438637662420q^{59} + 48068798504532480q^{60} + 21352962331944422q^{61} + 47155879095689472q^{62} - 75775799697075576q^{63} - 217577045142536192q^{64} - 156123953647347660q^{65} + 59689311109494336q^{66} + 268065007707894476q^{67} - 545886865993416192q^{68} + 96457733025206616q^{69} + 759275230293054720q^{70} - 113273531338221288q^{71} - 72813728308285440q^{72} - 545956267317696358q^{73} + 467491444508566176q^{74} - 132059774182473675q^{75} + 219061399560535040q^{76} + 537259550214361632q^{77} + 964823096270588256q^{78} - 1807609924990106560q^{79} - 550793667341844480q^{80} + 150094635296999121q^{81} + 1229768442749225952q^{82} + 1469958731688321372q^{83} - 2673800468018620416q^{84} - 2763726747210579780q^{85} - 1254254662904409408q^{86} + 239906652389928450q^{87} + 516258107162542080q^{88} + 2974040568798940170q^{89} - 1503951647313057120q^{90} + 8684309017872250672q^{91} + 3403576507774765056q^{92} - 840732942246789744q^{93} - 1690634732655492864q^{94} + 1109068356398478600q^{95} + 5343333366925099008q^{96} - 6925686051327380254q^{97} - 29649886625953055952q^{98} - 1064189049427696572q^{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
−1104.00 19683.0 694528. 3.51627e6 −2.17300e7 −1.95591e8 −1.87945e8 3.87420e8 −3.88196e9
\(n\): 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 \( T_{2} + 1104 \) acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(3))\).