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

Related objects

Downloads

Learn more about

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

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

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 \( T_{2} + 78 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(9))\).