Properties

Label 9.12.a.b
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

\(f(q)\) \(=\) \( q + 24q^{2} - 1472q^{4} - 4830q^{5} - 16744q^{7} - 84480q^{8} + O(q^{10}) \) \( q + 24q^{2} - 1472q^{4} - 4830q^{5} - 16744q^{7} - 84480q^{8} - 115920q^{10} - 534612q^{11} - 577738q^{13} - 401856q^{14} + 987136q^{16} + 6905934q^{17} + 10661420q^{19} + 7109760q^{20} - 12830688q^{22} - 18643272q^{23} - 25499225q^{25} - 13865712q^{26} + 24647168q^{28} - 128406630q^{29} - 52843168q^{31} + 196706304q^{32} + 165742416q^{34} + 80873520q^{35} - 182213314q^{37} + 255874080q^{38} + 408038400q^{40} - 308120442q^{41} - 17125708q^{43} + 786948864q^{44} - 447438528q^{46} - 2687348496q^{47} - 1696965207q^{49} - 611981400q^{50} + 850430336q^{52} + 1596055698q^{53} + 2582175960q^{55} + 1414533120q^{56} - 3081759120q^{58} + 5189203740q^{59} + 6956478662q^{61} - 1268236032q^{62} + 2699296768q^{64} + 2790474540q^{65} - 15481826884q^{67} - 10165534848q^{68} + 1940964480q^{70} - 9791485272q^{71} + 1463791322q^{73} - 4373119536q^{74} - 15693610240q^{76} + 8951543328q^{77} + 38116845680q^{79} - 4767866880q^{80} - 7394890608q^{82} + 29335099668q^{83} - 33355661220q^{85} - 411016992q^{86} + 45164021760q^{88} + 24992917110q^{89} + 9673645072q^{91} + 27442896384q^{92} - 64496363904q^{94} - 51494658600q^{95} + 75013568546q^{97} - 40727164968q^{98} + O(q^{100}) \)

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
24.0000 0 −1472.00 −4830.00 0 −16744.0 −84480.0 0 −115920.
\(n\): 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 \( T_{2} - 24 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(9))\).