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

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

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

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
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 \(T_{2} \) \(\mathstrut -\mathstrut 24 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(9))\).