Properties

Label 12.18.a.b
Level 12
Weight 18
Character orbit 12.a
Self dual Yes
Analytic conductor 21.987
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 18 \)
Character orbit: \([\chi]\) = 12.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(21.9866504813\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 6561q^{3} \) \(\mathstrut +\mathstrut 130950q^{5} \) \(\mathstrut -\mathstrut 14846776q^{7} \) \(\mathstrut +\mathstrut 43046721q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 6561q^{3} \) \(\mathstrut +\mathstrut 130950q^{5} \) \(\mathstrut -\mathstrut 14846776q^{7} \) \(\mathstrut +\mathstrut 43046721q^{9} \) \(\mathstrut -\mathstrut 845469684q^{11} \) \(\mathstrut +\mathstrut 1751414990q^{13} \) \(\mathstrut +\mathstrut 859162950q^{15} \) \(\mathstrut -\mathstrut 47147886q^{17} \) \(\mathstrut -\mathstrut 56973573100q^{19} \) \(\mathstrut -\mathstrut 97409697336q^{21} \) \(\mathstrut -\mathstrut 371395374696q^{23} \) \(\mathstrut -\mathstrut 745791550625q^{25} \) \(\mathstrut +\mathstrut 282429536481q^{27} \) \(\mathstrut -\mathstrut 3681168479586q^{29} \) \(\mathstrut -\mathstrut 5479889229856q^{31} \) \(\mathstrut -\mathstrut 5547126596724q^{33} \) \(\mathstrut -\mathstrut 1944185317200q^{35} \) \(\mathstrut -\mathstrut 5446958938138q^{37} \) \(\mathstrut +\mathstrut 11491033749390q^{39} \) \(\mathstrut +\mathstrut 29773337634090q^{41} \) \(\mathstrut +\mathstrut 98485895466284q^{43} \) \(\mathstrut +\mathstrut 5636968114950q^{45} \) \(\mathstrut +\mathstrut 107861800207536q^{47} \) \(\mathstrut -\mathstrut 12203756393031q^{49} \) \(\mathstrut -\mathstrut 309337280046q^{51} \) \(\mathstrut +\mathstrut 626472886328118q^{53} \) \(\mathstrut -\mathstrut 110714255119800q^{55} \) \(\mathstrut -\mathstrut 373803613109100q^{57} \) \(\mathstrut -\mathstrut 1260971066668356q^{59} \) \(\mathstrut -\mathstrut 956343149707138q^{61} \) \(\mathstrut -\mathstrut 639105024221496q^{63} \) \(\mathstrut +\mathstrut 229347792940500q^{65} \) \(\mathstrut -\mathstrut 5519389511567164q^{67} \) \(\mathstrut -\mathstrut 2436725053380456q^{69} \) \(\mathstrut +\mathstrut 9303053873586120q^{71} \) \(\mathstrut +\mathstrut 3692590926453962q^{73} \) \(\mathstrut -\mathstrut 4893138363650625q^{75} \) \(\mathstrut +\mathstrut 12552499013138784q^{77} \) \(\mathstrut -\mathstrut 2597720120860912q^{79} \) \(\mathstrut +\mathstrut 1853020188851841q^{81} \) \(\mathstrut +\mathstrut 26266742515599444q^{83} \) \(\mathstrut -\mathstrut 6174015671700q^{85} \) \(\mathstrut -\mathstrut 24152146394563746q^{87} \) \(\mathstrut +\mathstrut 63717157489864410q^{89} \) \(\mathstrut -\mathstrut 26002866039572240q^{91} \) \(\mathstrut -\mathstrut 35953553237085216q^{93} \) \(\mathstrut -\mathstrut 7460689397445000q^{95} \) \(\mathstrut -\mathstrut 78558896505972382q^{97} \) \(\mathstrut -\mathstrut 36394697601106164q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 6561.00 0 130950. 0 −1.48468e7 0 4.30467e7 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 \(T_{5} \) \(\mathstrut -\mathstrut 130950 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(12))\).