Properties

Label 3.10.a.a
Level 3
Weight 10
Character orbit 3.a
Self dual Yes
Analytic conductor 1.545
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(1.54510750849\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 36q^{2} \) \(\mathstrut -\mathstrut 81q^{3} \) \(\mathstrut +\mathstrut 784q^{4} \) \(\mathstrut -\mathstrut 1314q^{5} \) \(\mathstrut +\mathstrut 2916q^{6} \) \(\mathstrut -\mathstrut 4480q^{7} \) \(\mathstrut -\mathstrut 9792q^{8} \) \(\mathstrut +\mathstrut 6561q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 36q^{2} \) \(\mathstrut -\mathstrut 81q^{3} \) \(\mathstrut +\mathstrut 784q^{4} \) \(\mathstrut -\mathstrut 1314q^{5} \) \(\mathstrut +\mathstrut 2916q^{6} \) \(\mathstrut -\mathstrut 4480q^{7} \) \(\mathstrut -\mathstrut 9792q^{8} \) \(\mathstrut +\mathstrut 6561q^{9} \) \(\mathstrut +\mathstrut 47304q^{10} \) \(\mathstrut +\mathstrut 1476q^{11} \) \(\mathstrut -\mathstrut 63504q^{12} \) \(\mathstrut -\mathstrut 151522q^{13} \) \(\mathstrut +\mathstrut 161280q^{14} \) \(\mathstrut +\mathstrut 106434q^{15} \) \(\mathstrut -\mathstrut 48896q^{16} \) \(\mathstrut +\mathstrut 108162q^{17} \) \(\mathstrut -\mathstrut 236196q^{18} \) \(\mathstrut +\mathstrut 593084q^{19} \) \(\mathstrut -\mathstrut 1030176q^{20} \) \(\mathstrut +\mathstrut 362880q^{21} \) \(\mathstrut -\mathstrut 53136q^{22} \) \(\mathstrut -\mathstrut 969480q^{23} \) \(\mathstrut +\mathstrut 793152q^{24} \) \(\mathstrut -\mathstrut 226529q^{25} \) \(\mathstrut +\mathstrut 5454792q^{26} \) \(\mathstrut -\mathstrut 531441q^{27} \) \(\mathstrut -\mathstrut 3512320q^{28} \) \(\mathstrut -\mathstrut 6642522q^{29} \) \(\mathstrut -\mathstrut 3831624q^{30} \) \(\mathstrut +\mathstrut 7070600q^{31} \) \(\mathstrut +\mathstrut 6773760q^{32} \) \(\mathstrut -\mathstrut 119556q^{33} \) \(\mathstrut -\mathstrut 3893832q^{34} \) \(\mathstrut +\mathstrut 5886720q^{35} \) \(\mathstrut +\mathstrut 5143824q^{36} \) \(\mathstrut -\mathstrut 7472410q^{37} \) \(\mathstrut -\mathstrut 21351024q^{38} \) \(\mathstrut +\mathstrut 12273282q^{39} \) \(\mathstrut +\mathstrut 12866688q^{40} \) \(\mathstrut -\mathstrut 4350150q^{41} \) \(\mathstrut -\mathstrut 13063680q^{42} \) \(\mathstrut -\mathstrut 4358716q^{43} \) \(\mathstrut +\mathstrut 1157184q^{44} \) \(\mathstrut -\mathstrut 8621154q^{45} \) \(\mathstrut +\mathstrut 34901280q^{46} \) \(\mathstrut +\mathstrut 28309248q^{47} \) \(\mathstrut +\mathstrut 3960576q^{48} \) \(\mathstrut -\mathstrut 20283207q^{49} \) \(\mathstrut +\mathstrut 8155044q^{50} \) \(\mathstrut -\mathstrut 8761122q^{51} \) \(\mathstrut -\mathstrut 118793248q^{52} \) \(\mathstrut +\mathstrut 16111710q^{53} \) \(\mathstrut +\mathstrut 19131876q^{54} \) \(\mathstrut -\mathstrut 1939464q^{55} \) \(\mathstrut +\mathstrut 43868160q^{56} \) \(\mathstrut -\mathstrut 48039804q^{57} \) \(\mathstrut +\mathstrut 239130792q^{58} \) \(\mathstrut -\mathstrut 86075964q^{59} \) \(\mathstrut +\mathstrut 83444256q^{60} \) \(\mathstrut +\mathstrut 32213918q^{61} \) \(\mathstrut -\mathstrut 254541600q^{62} \) \(\mathstrut -\mathstrut 29393280q^{63} \) \(\mathstrut -\mathstrut 218820608q^{64} \) \(\mathstrut +\mathstrut 199099908q^{65} \) \(\mathstrut +\mathstrut 4304016q^{66} \) \(\mathstrut +\mathstrut 99531452q^{67} \) \(\mathstrut +\mathstrut 84799008q^{68} \) \(\mathstrut +\mathstrut 78527880q^{69} \) \(\mathstrut -\mathstrut 211921920q^{70} \) \(\mathstrut -\mathstrut 44170488q^{71} \) \(\mathstrut -\mathstrut 64245312q^{72} \) \(\mathstrut -\mathstrut 23560630q^{73} \) \(\mathstrut +\mathstrut 269006760q^{74} \) \(\mathstrut +\mathstrut 18348849q^{75} \) \(\mathstrut +\mathstrut 464977856q^{76} \) \(\mathstrut -\mathstrut 6612480q^{77} \) \(\mathstrut -\mathstrut 441838152q^{78} \) \(\mathstrut -\mathstrut 401754760q^{79} \) \(\mathstrut +\mathstrut 64249344q^{80} \) \(\mathstrut +\mathstrut 43046721q^{81} \) \(\mathstrut +\mathstrut 156605400q^{82} \) \(\mathstrut -\mathstrut 744528708q^{83} \) \(\mathstrut +\mathstrut 284497920q^{84} \) \(\mathstrut -\mathstrut 142124868q^{85} \) \(\mathstrut +\mathstrut 156913776q^{86} \) \(\mathstrut +\mathstrut 538044282q^{87} \) \(\mathstrut -\mathstrut 14452992q^{88} \) \(\mathstrut +\mathstrut 769871034q^{89} \) \(\mathstrut +\mathstrut 310361544q^{90} \) \(\mathstrut +\mathstrut 678818560q^{91} \) \(\mathstrut -\mathstrut 760072320q^{92} \) \(\mathstrut -\mathstrut 572718600q^{93} \) \(\mathstrut -\mathstrut 1019132928q^{94} \) \(\mathstrut -\mathstrut 779312376q^{95} \) \(\mathstrut -\mathstrut 548674560q^{96} \) \(\mathstrut +\mathstrut 907130882q^{97} \) \(\mathstrut +\mathstrut 730195452q^{98} \) \(\mathstrut +\mathstrut 9684036q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−36.0000 −81.0000 784.000 −1314.00 2916.00 −4480.00 −9792.00 6561.00 47304.0
\(n\): 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 \(T_{2} \) \(\mathstrut +\mathstrut 36 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(3))\).