Properties

Label 9.12.a.a
Level 9
Weight 12
Character orbit 9.a
Self dual Yes
Analytic conductor 6.915
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 9.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(6.91508862504\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 78q^{2} \) \(\mathstrut +\mathstrut 4036q^{4} \) \(\mathstrut +\mathstrut 5370q^{5} \) \(\mathstrut -\mathstrut 27760q^{7} \) \(\mathstrut -\mathstrut 155064q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 78q^{2} \) \(\mathstrut +\mathstrut 4036q^{4} \) \(\mathstrut +\mathstrut 5370q^{5} \) \(\mathstrut -\mathstrut 27760q^{7} \) \(\mathstrut -\mathstrut 155064q^{8} \) \(\mathstrut -\mathstrut 418860q^{10} \) \(\mathstrut -\mathstrut 637836q^{11} \) \(\mathstrut +\mathstrut 766214q^{13} \) \(\mathstrut +\mathstrut 2165280q^{14} \) \(\mathstrut +\mathstrut 3829264q^{16} \) \(\mathstrut -\mathstrut 3084354q^{17} \) \(\mathstrut -\mathstrut 19511404q^{19} \) \(\mathstrut +\mathstrut 21673320q^{20} \) \(\mathstrut +\mathstrut 49751208q^{22} \) \(\mathstrut -\mathstrut 15312360q^{23} \) \(\mathstrut -\mathstrut 19991225q^{25} \) \(\mathstrut -\mathstrut 59764692q^{26} \) \(\mathstrut -\mathstrut 112039360q^{28} \) \(\mathstrut -\mathstrut 10751262q^{29} \) \(\mathstrut -\mathstrut 50937400q^{31} \) \(\mathstrut +\mathstrut 18888480q^{32} \) \(\mathstrut +\mathstrut 240579612q^{34} \) \(\mathstrut -\mathstrut 149071200q^{35} \) \(\mathstrut +\mathstrut 664740830q^{37} \) \(\mathstrut +\mathstrut 1521889512q^{38} \) \(\mathstrut -\mathstrut 832693680q^{40} \) \(\mathstrut -\mathstrut 898833450q^{41} \) \(\mathstrut -\mathstrut 957947188q^{43} \) \(\mathstrut -\mathstrut 2574306096q^{44} \) \(\mathstrut +\mathstrut 1194364080q^{46} \) \(\mathstrut +\mathstrut 1555741344q^{47} \) \(\mathstrut -\mathstrut 1206709143q^{49} \) \(\mathstrut +\mathstrut 1559315550q^{50} \) \(\mathstrut +\mathstrut 3092439704q^{52} \) \(\mathstrut -\mathstrut 3792417030q^{53} \) \(\mathstrut -\mathstrut 3425179320q^{55} \) \(\mathstrut +\mathstrut 4304576640q^{56} \) \(\mathstrut +\mathstrut 838598436q^{58} \) \(\mathstrut -\mathstrut 555306924q^{59} \) \(\mathstrut +\mathstrut 4950420998q^{61} \) \(\mathstrut +\mathstrut 3973117200q^{62} \) \(\mathstrut -\mathstrut 9315634112q^{64} \) \(\mathstrut +\mathstrut 4114569180q^{65} \) \(\mathstrut +\mathstrut 5292399284q^{67} \) \(\mathstrut -\mathstrut 12448452744q^{68} \) \(\mathstrut +\mathstrut 11627553600q^{70} \) \(\mathstrut +\mathstrut 14831086248q^{71} \) \(\mathstrut +\mathstrut 13971005210q^{73} \) \(\mathstrut -\mathstrut 51849784740q^{74} \) \(\mathstrut -\mathstrut 78748026544q^{76} \) \(\mathstrut +\mathstrut 17706327360q^{77} \) \(\mathstrut +\mathstrut 3720542360q^{79} \) \(\mathstrut +\mathstrut 20563147680q^{80} \) \(\mathstrut +\mathstrut 70109009100q^{82} \) \(\mathstrut -\mathstrut 8768454036q^{83} \) \(\mathstrut -\mathstrut 16562980980q^{85} \) \(\mathstrut +\mathstrut 74719880664q^{86} \) \(\mathstrut +\mathstrut 98905401504q^{88} \) \(\mathstrut +\mathstrut 25472769174q^{89} \) \(\mathstrut -\mathstrut 21270100640q^{91} \) \(\mathstrut -\mathstrut 61800684960q^{92} \) \(\mathstrut -\mathstrut 121347824832q^{94} \) \(\mathstrut -\mathstrut 104776239480q^{95} \) \(\mathstrut -\mathstrut 39092494846q^{97} \) \(\mathstrut +\mathstrut 94123313154q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−78.0000 0 4036.00 5370.00 0 −27760.0 −155064. 0 −418860.
\(n\): 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 \(T_{2} \) \(\mathstrut +\mathstrut 78 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(9))\).