Properties

Label 9.6.a.a
Level 9
Weight 6
Character orbit 9.a
Self dual Yes
Analytic conductor 1.443
Analytic rank 0
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 9.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(1.44345437832\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 6q^{2} \) \(\mathstrut +\mathstrut 4q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 40q^{7} \) \(\mathstrut -\mathstrut 168q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 6q^{2} \) \(\mathstrut +\mathstrut 4q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 40q^{7} \) \(\mathstrut -\mathstrut 168q^{8} \) \(\mathstrut -\mathstrut 36q^{10} \) \(\mathstrut +\mathstrut 564q^{11} \) \(\mathstrut +\mathstrut 638q^{13} \) \(\mathstrut -\mathstrut 240q^{14} \) \(\mathstrut -\mathstrut 1136q^{16} \) \(\mathstrut -\mathstrut 882q^{17} \) \(\mathstrut -\mathstrut 556q^{19} \) \(\mathstrut -\mathstrut 24q^{20} \) \(\mathstrut +\mathstrut 3384q^{22} \) \(\mathstrut +\mathstrut 840q^{23} \) \(\mathstrut -\mathstrut 3089q^{25} \) \(\mathstrut +\mathstrut 3828q^{26} \) \(\mathstrut -\mathstrut 160q^{28} \) \(\mathstrut -\mathstrut 4638q^{29} \) \(\mathstrut +\mathstrut 4400q^{31} \) \(\mathstrut -\mathstrut 1440q^{32} \) \(\mathstrut -\mathstrut 5292q^{34} \) \(\mathstrut +\mathstrut 240q^{35} \) \(\mathstrut -\mathstrut 2410q^{37} \) \(\mathstrut -\mathstrut 3336q^{38} \) \(\mathstrut +\mathstrut 1008q^{40} \) \(\mathstrut +\mathstrut 6870q^{41} \) \(\mathstrut +\mathstrut 9644q^{43} \) \(\mathstrut +\mathstrut 2256q^{44} \) \(\mathstrut +\mathstrut 5040q^{46} \) \(\mathstrut +\mathstrut 18672q^{47} \) \(\mathstrut -\mathstrut 15207q^{49} \) \(\mathstrut -\mathstrut 18534q^{50} \) \(\mathstrut +\mathstrut 2552q^{52} \) \(\mathstrut -\mathstrut 33750q^{53} \) \(\mathstrut -\mathstrut 3384q^{55} \) \(\mathstrut +\mathstrut 6720q^{56} \) \(\mathstrut -\mathstrut 27828q^{58} \) \(\mathstrut +\mathstrut 18084q^{59} \) \(\mathstrut +\mathstrut 39758q^{61} \) \(\mathstrut +\mathstrut 26400q^{62} \) \(\mathstrut +\mathstrut 27712q^{64} \) \(\mathstrut -\mathstrut 3828q^{65} \) \(\mathstrut -\mathstrut 23068q^{67} \) \(\mathstrut -\mathstrut 3528q^{68} \) \(\mathstrut +\mathstrut 1440q^{70} \) \(\mathstrut +\mathstrut 4248q^{71} \) \(\mathstrut -\mathstrut 41110q^{73} \) \(\mathstrut -\mathstrut 14460q^{74} \) \(\mathstrut -\mathstrut 2224q^{76} \) \(\mathstrut -\mathstrut 22560q^{77} \) \(\mathstrut +\mathstrut 21920q^{79} \) \(\mathstrut +\mathstrut 6816q^{80} \) \(\mathstrut +\mathstrut 41220q^{82} \) \(\mathstrut -\mathstrut 82452q^{83} \) \(\mathstrut +\mathstrut 5292q^{85} \) \(\mathstrut +\mathstrut 57864q^{86} \) \(\mathstrut -\mathstrut 94752q^{88} \) \(\mathstrut +\mathstrut 94086q^{89} \) \(\mathstrut -\mathstrut 25520q^{91} \) \(\mathstrut +\mathstrut 3360q^{92} \) \(\mathstrut +\mathstrut 112032q^{94} \) \(\mathstrut +\mathstrut 3336q^{95} \) \(\mathstrut +\mathstrut 49442q^{97} \) \(\mathstrut -\mathstrut 91242q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 0 4.00000 −6.00000 0 −40.0000 −168.000 0 −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 \(S_{6}^{\mathrm{new}}(\Gamma_0(9))\).