Properties

Label 3.6.a.a
Level 3
Weight 6
Character orbit 3.a
Self dual Yes
Analytic conductor 0.481
Analytic rank 0
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(0.481151459439\)
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

\(f(q)\) \(=\) \( q - 6q^{2} + 9q^{3} + 4q^{4} + 6q^{5} - 54q^{6} - 40q^{7} + 168q^{8} + 81q^{9} + O(q^{10}) \) \( q - 6q^{2} + 9q^{3} + 4q^{4} + 6q^{5} - 54q^{6} - 40q^{7} + 168q^{8} + 81q^{9} - 36q^{10} - 564q^{11} + 36q^{12} + 638q^{13} + 240q^{14} + 54q^{15} - 1136q^{16} + 882q^{17} - 486q^{18} - 556q^{19} + 24q^{20} - 360q^{21} + 3384q^{22} - 840q^{23} + 1512q^{24} - 3089q^{25} - 3828q^{26} + 729q^{27} - 160q^{28} + 4638q^{29} - 324q^{30} + 4400q^{31} + 1440q^{32} - 5076q^{33} - 5292q^{34} - 240q^{35} + 324q^{36} - 2410q^{37} + 3336q^{38} + 5742q^{39} + 1008q^{40} - 6870q^{41} + 2160q^{42} + 9644q^{43} - 2256q^{44} + 486q^{45} + 5040q^{46} - 18672q^{47} - 10224q^{48} - 15207q^{49} + 18534q^{50} + 7938q^{51} + 2552q^{52} + 33750q^{53} - 4374q^{54} - 3384q^{55} - 6720q^{56} - 5004q^{57} - 27828q^{58} - 18084q^{59} + 216q^{60} + 39758q^{61} - 26400q^{62} - 3240q^{63} + 27712q^{64} + 3828q^{65} + 30456q^{66} - 23068q^{67} + 3528q^{68} - 7560q^{69} + 1440q^{70} - 4248q^{71} + 13608q^{72} - 41110q^{73} + 14460q^{74} - 27801q^{75} - 2224q^{76} + 22560q^{77} - 34452q^{78} + 21920q^{79} - 6816q^{80} + 6561q^{81} + 41220q^{82} + 82452q^{83} - 1440q^{84} + 5292q^{85} - 57864q^{86} + 41742q^{87} - 94752q^{88} - 94086q^{89} - 2916q^{90} - 25520q^{91} - 3360q^{92} + 39600q^{93} + 112032q^{94} - 3336q^{95} + 12960q^{96} + 49442q^{97} + 91242q^{98} - 45684q^{99} + O(q^{100}) \)

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
−6.00000 9.00000 4.00000 6.00000 −54.0000 −40.0000 168.000 81.0000 −36.0000
\(n\): 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\).