Properties

Label 12.18.a.a
Level 12
Weight 18
Character orbit 12.a
Self dual Yes
Analytic conductor 21.987
Analytic rank 0
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: \(0\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 6561q^{3} \) \(\mathstrut -\mathstrut 1608930q^{5} \) \(\mathstrut -\mathstrut 9417184q^{7} \) \(\mathstrut +\mathstrut 43046721q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 6561q^{3} \) \(\mathstrut -\mathstrut 1608930q^{5} \) \(\mathstrut -\mathstrut 9417184q^{7} \) \(\mathstrut +\mathstrut 43046721q^{9} \) \(\mathstrut -\mathstrut 186910524q^{11} \) \(\mathstrut -\mathstrut 2625442690q^{13} \) \(\mathstrut +\mathstrut 10556189730q^{15} \) \(\mathstrut +\mathstrut 43782311106q^{17} \) \(\mathstrut -\mathstrut 96594985540q^{19} \) \(\mathstrut +\mathstrut 61786144224q^{21} \) \(\mathstrut +\mathstrut 290867937336q^{23} \) \(\mathstrut +\mathstrut 1825716291775q^{25} \) \(\mathstrut -\mathstrut 282429536481q^{27} \) \(\mathstrut +\mathstrut 1398617429094q^{29} \) \(\mathstrut +\mathstrut 7647898359464q^{31} \) \(\mathstrut +\mathstrut 1226319947964q^{33} \) \(\mathstrut +\mathstrut 15151589853120q^{35} \) \(\mathstrut -\mathstrut 33369516616762q^{37} \) \(\mathstrut +\mathstrut 17225529489090q^{39} \) \(\mathstrut -\mathstrut 12032733393990q^{41} \) \(\mathstrut -\mathstrut 755092495804q^{43} \) \(\mathstrut -\mathstrut 69259160818530q^{45} \) \(\mathstrut -\mathstrut 280540358127936q^{47} \) \(\mathstrut -\mathstrut 143947159497351q^{49} \) \(\mathstrut -\mathstrut 287255743166466q^{51} \) \(\mathstrut +\mathstrut 460570203615582q^{53} \) \(\mathstrut +\mathstrut 300725949379320q^{55} \) \(\mathstrut +\mathstrut 633759700127940q^{57} \) \(\mathstrut +\mathstrut 1078467799153284q^{59} \) \(\mathstrut -\mathstrut 1980778975313218q^{61} \) \(\mathstrut -\mathstrut 405378892253664q^{63} \) \(\mathstrut +\mathstrut 4224153507221700q^{65} \) \(\mathstrut +\mathstrut 4850190377589884q^{67} \) \(\mathstrut -\mathstrut 1908384536861496q^{69} \) \(\mathstrut +\mathstrut 2707574704052040q^{71} \) \(\mathstrut -\mathstrut 5002264428090742q^{73} \) \(\mathstrut -\mathstrut 11978524590335775q^{75} \) \(\mathstrut +\mathstrut 1760170796044416q^{77} \) \(\mathstrut -\mathstrut 9774477292907752q^{79} \) \(\mathstrut +\mathstrut 1853020188851841q^{81} \) \(\mathstrut +\mathstrut 17112919183614396q^{83} \) \(\mathstrut -\mathstrut 70442673807776580q^{85} \) \(\mathstrut -\mathstrut 9176328952285734q^{87} \) \(\mathstrut +\mathstrut 34698182155846650q^{89} \) \(\mathstrut +\mathstrut 24724276893184960q^{91} \) \(\mathstrut -\mathstrut 50177861136443304q^{93} \) \(\mathstrut +\mathstrut 155414570084872200q^{95} \) \(\mathstrut +\mathstrut 68616916871806082q^{97} \) \(\mathstrut -\mathstrut 8045885178591804q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −1.60893e6 0 −9.41718e6 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 \(T_{5} \) \(\mathstrut +\mathstrut 1608930 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(12))\).