Properties

Label 12.12.a.a
Level 12
Weight 12
Character orbit 12.a
Self dual Yes
Analytic conductor 9.220
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

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

Newform invariants

Self dual: Yes
Analytic conductor: \(9.22011816672\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 243q^{3} \) \(\mathstrut +\mathstrut 9990q^{5} \) \(\mathstrut -\mathstrut 86128q^{7} \) \(\mathstrut +\mathstrut 59049q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 243q^{3} \) \(\mathstrut +\mathstrut 9990q^{5} \) \(\mathstrut -\mathstrut 86128q^{7} \) \(\mathstrut +\mathstrut 59049q^{9} \) \(\mathstrut -\mathstrut 806004q^{11} \) \(\mathstrut -\mathstrut 960250q^{13} \) \(\mathstrut -\mathstrut 2427570q^{15} \) \(\mathstrut -\mathstrut 4306878q^{17} \) \(\mathstrut +\mathstrut 401300q^{19} \) \(\mathstrut +\mathstrut 20929104q^{21} \) \(\mathstrut +\mathstrut 17751528q^{23} \) \(\mathstrut +\mathstrut 50971975q^{25} \) \(\mathstrut -\mathstrut 14348907q^{27} \) \(\mathstrut -\mathstrut 84704994q^{29} \) \(\mathstrut +\mathstrut 140930504q^{31} \) \(\mathstrut +\mathstrut 195858972q^{33} \) \(\mathstrut -\mathstrut 860418720q^{35} \) \(\mathstrut -\mathstrut 413506594q^{37} \) \(\mathstrut +\mathstrut 233340750q^{39} \) \(\mathstrut +\mathstrut 150094890q^{41} \) \(\mathstrut +\mathstrut 706702028q^{43} \) \(\mathstrut +\mathstrut 589899510q^{45} \) \(\mathstrut -\mathstrut 2475725472q^{47} \) \(\mathstrut +\mathstrut 5440705641q^{49} \) \(\mathstrut +\mathstrut 1046571354q^{51} \) \(\mathstrut +\mathstrut 1600124166q^{53} \) \(\mathstrut -\mathstrut 8051979960q^{55} \) \(\mathstrut -\mathstrut 97515900q^{57} \) \(\mathstrut +\mathstrut 3945492396q^{59} \) \(\mathstrut -\mathstrut 885973498q^{61} \) \(\mathstrut -\mathstrut 5085772272q^{63} \) \(\mathstrut -\mathstrut 9592897500q^{65} \) \(\mathstrut -\mathstrut 4881597772q^{67} \) \(\mathstrut -\mathstrut 4313621304q^{69} \) \(\mathstrut +\mathstrut 12631469400q^{71} \) \(\mathstrut +\mathstrut 1423335194q^{73} \) \(\mathstrut -\mathstrut 12386189925q^{75} \) \(\mathstrut +\mathstrut 69419512512q^{77} \) \(\mathstrut +\mathstrut 667407512q^{79} \) \(\mathstrut +\mathstrut 3486784401q^{81} \) \(\mathstrut +\mathstrut 5716071828q^{83} \) \(\mathstrut -\mathstrut 43025711220q^{85} \) \(\mathstrut +\mathstrut 20583313542q^{87} \) \(\mathstrut -\mathstrut 85738736790q^{89} \) \(\mathstrut +\mathstrut 82704412000q^{91} \) \(\mathstrut -\mathstrut 34246112472q^{93} \) \(\mathstrut +\mathstrut 4008987000q^{95} \) \(\mathstrut -\mathstrut 52302647806q^{97} \) \(\mathstrut -\mathstrut 47593730196q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
0 −243.000 0 9990.00 0 −86128.0 0 59049.0 0
\(n\): 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
\(2\) \(-1\)
\(3\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{5} \) \(\mathstrut -\mathstrut 9990 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(12))\).