Properties

Label 3.8.a.a
Level 3
Weight 8
Character orbit 3.a
Self dual Yes
Analytic conductor 0.937
Analytic rank 0
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(0.937155076452\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 27q^{3} \) \(\mathstrut -\mathstrut 92q^{4} \) \(\mathstrut +\mathstrut 390q^{5} \) \(\mathstrut -\mathstrut 162q^{6} \) \(\mathstrut -\mathstrut 64q^{7} \) \(\mathstrut -\mathstrut 1320q^{8} \) \(\mathstrut +\mathstrut 729q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 27q^{3} \) \(\mathstrut -\mathstrut 92q^{4} \) \(\mathstrut +\mathstrut 390q^{5} \) \(\mathstrut -\mathstrut 162q^{6} \) \(\mathstrut -\mathstrut 64q^{7} \) \(\mathstrut -\mathstrut 1320q^{8} \) \(\mathstrut +\mathstrut 729q^{9} \) \(\mathstrut +\mathstrut 2340q^{10} \) \(\mathstrut -\mathstrut 948q^{11} \) \(\mathstrut +\mathstrut 2484q^{12} \) \(\mathstrut -\mathstrut 5098q^{13} \) \(\mathstrut -\mathstrut 384q^{14} \) \(\mathstrut -\mathstrut 10530q^{15} \) \(\mathstrut +\mathstrut 3856q^{16} \) \(\mathstrut +\mathstrut 28386q^{17} \) \(\mathstrut +\mathstrut 4374q^{18} \) \(\mathstrut -\mathstrut 8620q^{19} \) \(\mathstrut -\mathstrut 35880q^{20} \) \(\mathstrut +\mathstrut 1728q^{21} \) \(\mathstrut -\mathstrut 5688q^{22} \) \(\mathstrut -\mathstrut 15288q^{23} \) \(\mathstrut +\mathstrut 35640q^{24} \) \(\mathstrut +\mathstrut 73975q^{25} \) \(\mathstrut -\mathstrut 30588q^{26} \) \(\mathstrut -\mathstrut 19683q^{27} \) \(\mathstrut +\mathstrut 5888q^{28} \) \(\mathstrut +\mathstrut 36510q^{29} \) \(\mathstrut -\mathstrut 63180q^{30} \) \(\mathstrut -\mathstrut 276808q^{31} \) \(\mathstrut +\mathstrut 192096q^{32} \) \(\mathstrut +\mathstrut 25596q^{33} \) \(\mathstrut +\mathstrut 170316q^{34} \) \(\mathstrut -\mathstrut 24960q^{35} \) \(\mathstrut -\mathstrut 67068q^{36} \) \(\mathstrut +\mathstrut 268526q^{37} \) \(\mathstrut -\mathstrut 51720q^{38} \) \(\mathstrut +\mathstrut 137646q^{39} \) \(\mathstrut -\mathstrut 514800q^{40} \) \(\mathstrut -\mathstrut 629718q^{41} \) \(\mathstrut +\mathstrut 10368q^{42} \) \(\mathstrut +\mathstrut 685772q^{43} \) \(\mathstrut +\mathstrut 87216q^{44} \) \(\mathstrut +\mathstrut 284310q^{45} \) \(\mathstrut -\mathstrut 91728q^{46} \) \(\mathstrut +\mathstrut 583296q^{47} \) \(\mathstrut -\mathstrut 104112q^{48} \) \(\mathstrut -\mathstrut 819447q^{49} \) \(\mathstrut +\mathstrut 443850q^{50} \) \(\mathstrut -\mathstrut 766422q^{51} \) \(\mathstrut +\mathstrut 469016q^{52} \) \(\mathstrut -\mathstrut 428058q^{53} \) \(\mathstrut -\mathstrut 118098q^{54} \) \(\mathstrut -\mathstrut 369720q^{55} \) \(\mathstrut +\mathstrut 84480q^{56} \) \(\mathstrut +\mathstrut 232740q^{57} \) \(\mathstrut +\mathstrut 219060q^{58} \) \(\mathstrut +\mathstrut 1306380q^{59} \) \(\mathstrut +\mathstrut 968760q^{60} \) \(\mathstrut +\mathstrut 300662q^{61} \) \(\mathstrut -\mathstrut 1660848q^{62} \) \(\mathstrut -\mathstrut 46656q^{63} \) \(\mathstrut +\mathstrut 659008q^{64} \) \(\mathstrut -\mathstrut 1988220q^{65} \) \(\mathstrut +\mathstrut 153576q^{66} \) \(\mathstrut -\mathstrut 507244q^{67} \) \(\mathstrut -\mathstrut 2611512q^{68} \) \(\mathstrut +\mathstrut 412776q^{69} \) \(\mathstrut -\mathstrut 149760q^{70} \) \(\mathstrut +\mathstrut 5560632q^{71} \) \(\mathstrut -\mathstrut 962280q^{72} \) \(\mathstrut +\mathstrut 1369082q^{73} \) \(\mathstrut +\mathstrut 1611156q^{74} \) \(\mathstrut -\mathstrut 1997325q^{75} \) \(\mathstrut +\mathstrut 793040q^{76} \) \(\mathstrut +\mathstrut 60672q^{77} \) \(\mathstrut +\mathstrut 825876q^{78} \) \(\mathstrut -\mathstrut 6913720q^{79} \) \(\mathstrut +\mathstrut 1503840q^{80} \) \(\mathstrut +\mathstrut 531441q^{81} \) \(\mathstrut -\mathstrut 3778308q^{82} \) \(\mathstrut -\mathstrut 4376748q^{83} \) \(\mathstrut -\mathstrut 158976q^{84} \) \(\mathstrut +\mathstrut 11070540q^{85} \) \(\mathstrut +\mathstrut 4114632q^{86} \) \(\mathstrut -\mathstrut 985770q^{87} \) \(\mathstrut +\mathstrut 1251360q^{88} \) \(\mathstrut -\mathstrut 8528310q^{89} \) \(\mathstrut +\mathstrut 1705860q^{90} \) \(\mathstrut +\mathstrut 326272q^{91} \) \(\mathstrut +\mathstrut 1406496q^{92} \) \(\mathstrut +\mathstrut 7473816q^{93} \) \(\mathstrut +\mathstrut 3499776q^{94} \) \(\mathstrut -\mathstrut 3361800q^{95} \) \(\mathstrut -\mathstrut 5186592q^{96} \) \(\mathstrut -\mathstrut 8826814q^{97} \) \(\mathstrut -\mathstrut 4916682q^{98} \) \(\mathstrut -\mathstrut 691092q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
6.00000 −27.0000 −92.0000 390.000 −162.000 −64.0000 −1320.00 729.000 2340.00
\(n\): 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

There are no other newforms in \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\).