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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 1104q^{2} \) \(\mathstrut +\mathstrut 19683q^{3} \) \(\mathstrut +\mathstrut 694528q^{4} \) \(\mathstrut +\mathstrut 3516270q^{5} \) \(\mathstrut -\mathstrut 21730032q^{6} \) \(\mathstrut -\mathstrut 195590584q^{7} \) \(\mathstrut -\mathstrut 187944960q^{8} \) \(\mathstrut +\mathstrut 387420489q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 1104q^{2} \) \(\mathstrut +\mathstrut 19683q^{3} \) \(\mathstrut +\mathstrut 694528q^{4} \) \(\mathstrut +\mathstrut 3516270q^{5} \) \(\mathstrut -\mathstrut 21730032q^{6} \) \(\mathstrut -\mathstrut 195590584q^{7} \) \(\mathstrut -\mathstrut 187944960q^{8} \) \(\mathstrut +\mathstrut 387420489q^{9} \) \(\mathstrut -\mathstrut 3881962080q^{10} \) \(\mathstrut -\mathstrut 2746857948q^{11} \) \(\mathstrut +\mathstrut 13670394624q^{12} \) \(\mathstrut -\mathstrut 44400445258q^{13} \) \(\mathstrut +\mathstrut 215932004736q^{14} \) \(\mathstrut +\mathstrut 69210742410q^{15} \) \(\mathstrut -\mathstrut 156641460224q^{16} \) \(\mathstrut -\mathstrut 785982517614q^{17} \) \(\mathstrut -\mathstrut 427712219856q^{18} \) \(\mathstrut +\mathstrut 315410465180q^{19} \) \(\mathstrut +\mathstrut 2442147970560q^{20} \) \(\mathstrut -\mathstrut 3849809464872q^{21} \) \(\mathstrut +\mathstrut 3032531174592q^{22} \) \(\mathstrut +\mathstrut 4900560535752q^{23} \) \(\mathstrut -\mathstrut 3699320647680q^{24} \) \(\mathstrut -\mathstrut 6709331615225q^{25} \) \(\mathstrut +\mathstrut 49018091564832q^{26} \) \(\mathstrut +\mathstrut 7625597484987q^{27} \) \(\mathstrut -\mathstrut 135843137124352q^{28} \) \(\mathstrut +\mathstrut 12188520672150q^{29} \) \(\mathstrut -\mathstrut 76408659620640q^{30} \) \(\mathstrut -\mathstrut 42713658601168q^{31} \) \(\mathstrut +\mathstrut 271469459275776q^{32} \) \(\mathstrut -\mathstrut 54066404990484q^{33} \) \(\mathstrut +\mathstrut 867724699445856q^{34} \) \(\mathstrut -\mathstrut 687749302801680q^{35} \) \(\mathstrut +\mathstrut 269074377384192q^{36} \) \(\mathstrut -\mathstrut 423452395388194q^{37} \) \(\mathstrut -\mathstrut 348213153558720q^{38} \) \(\mathstrut -\mathstrut 873933964013214q^{39} \) \(\mathstrut -\mathstrut 660865224499200q^{40} \) \(\mathstrut -\mathstrut 1113920690896038q^{41} \) \(\mathstrut +\mathstrut 4250189649218688q^{42} \) \(\mathstrut +\mathstrut 1136100238138052q^{43} \) \(\mathstrut -\mathstrut 1907769756908544q^{44} \) \(\mathstrut +\mathstrut 1362275042856030q^{45} \) \(\mathstrut -\mathstrut 5410218831470208q^{46} \) \(\mathstrut +\mathstrut 1531372040448816q^{47} \) \(\mathstrut -\mathstrut 3083173861588992q^{48} \) \(\mathstrut +\mathstrut 26856781364087913q^{49} \) \(\mathstrut +\mathstrut 7407102103208400q^{50} \) \(\mathstrut -\mathstrut 15470493894196362q^{51} \) \(\mathstrut -\mathstrut 30837352444148224q^{52} \) \(\mathstrut -\mathstrut 18059320314853218q^{53} \) \(\mathstrut -\mathstrut 8418659623425648q^{54} \) \(\mathstrut -\mathstrut 9658694196813960q^{55} \) \(\mathstrut +\mathstrut 36760264486256640q^{56} \) \(\mathstrut +\mathstrut 6208224186137940q^{57} \) \(\mathstrut -\mathstrut 13456126822053600q^{58} \) \(\mathstrut +\mathstrut 92700438637662420q^{59} \) \(\mathstrut +\mathstrut 48068798504532480q^{60} \) \(\mathstrut +\mathstrut 21352962331944422q^{61} \) \(\mathstrut +\mathstrut 47155879095689472q^{62} \) \(\mathstrut -\mathstrut 75775799697075576q^{63} \) \(\mathstrut -\mathstrut 217577045142536192q^{64} \) \(\mathstrut -\mathstrut 156123953647347660q^{65} \) \(\mathstrut +\mathstrut 59689311109494336q^{66} \) \(\mathstrut +\mathstrut 268065007707894476q^{67} \) \(\mathstrut -\mathstrut 545886865993416192q^{68} \) \(\mathstrut +\mathstrut 96457733025206616q^{69} \) \(\mathstrut +\mathstrut 759275230293054720q^{70} \) \(\mathstrut -\mathstrut 113273531338221288q^{71} \) \(\mathstrut -\mathstrut 72813728308285440q^{72} \) \(\mathstrut -\mathstrut 545956267317696358q^{73} \) \(\mathstrut +\mathstrut 467491444508566176q^{74} \) \(\mathstrut -\mathstrut 132059774182473675q^{75} \) \(\mathstrut +\mathstrut 219061399560535040q^{76} \) \(\mathstrut +\mathstrut 537259550214361632q^{77} \) \(\mathstrut +\mathstrut 964823096270588256q^{78} \) \(\mathstrut -\mathstrut 1807609924990106560q^{79} \) \(\mathstrut -\mathstrut 550793667341844480q^{80} \) \(\mathstrut +\mathstrut 150094635296999121q^{81} \) \(\mathstrut +\mathstrut 1229768442749225952q^{82} \) \(\mathstrut +\mathstrut 1469958731688321372q^{83} \) \(\mathstrut -\mathstrut 2673800468018620416q^{84} \) \(\mathstrut -\mathstrut 2763726747210579780q^{85} \) \(\mathstrut -\mathstrut 1254254662904409408q^{86} \) \(\mathstrut +\mathstrut 239906652389928450q^{87} \) \(\mathstrut +\mathstrut 516258107162542080q^{88} \) \(\mathstrut +\mathstrut 2974040568798940170q^{89} \) \(\mathstrut -\mathstrut 1503951647313057120q^{90} \) \(\mathstrut +\mathstrut 8684309017872250672q^{91} \) \(\mathstrut +\mathstrut 3403576507774765056q^{92} \) \(\mathstrut -\mathstrut 840732942246789744q^{93} \) \(\mathstrut -\mathstrut 1690634732655492864q^{94} \) \(\mathstrut +\mathstrut 1109068356398478600q^{95} \) \(\mathstrut +\mathstrut 5343333366925099008q^{96} \) \(\mathstrut -\mathstrut 6925686051327380254q^{97} \) \(\mathstrut -\mathstrut 29649886625953055952q^{98} \) \(\mathstrut -\mathstrut 1064189049427696572q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−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 \(T_{2} \) \(\mathstrut +\mathstrut 1104 \) acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(3))\).