Properties

Label 12.12.a.b
Level 12
Weight 12
Character orbit 12.a
Self dual Yes
Analytic conductor 9.220
Analytic rank 0
Dimension 1
CM No
Inner twists 1

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Newspace parameters

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

Newform invariants

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

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 243q^{3} \) \(\mathstrut +\mathstrut 2862q^{5} \) \(\mathstrut +\mathstrut 9128q^{7} \) \(\mathstrut +\mathstrut 59049q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 243q^{3} \) \(\mathstrut +\mathstrut 2862q^{5} \) \(\mathstrut +\mathstrut 9128q^{7} \) \(\mathstrut +\mathstrut 59049q^{9} \) \(\mathstrut +\mathstrut 668196q^{11} \) \(\mathstrut +\mathstrut 2052950q^{13} \) \(\mathstrut +\mathstrut 695466q^{15} \) \(\mathstrut +\mathstrut 1604178q^{17} \) \(\mathstrut -\mathstrut 230500q^{19} \) \(\mathstrut +\mathstrut 2218104q^{21} \) \(\mathstrut -\mathstrut 43012728q^{23} \) \(\mathstrut -\mathstrut 40637081q^{25} \) \(\mathstrut +\mathstrut 14348907q^{27} \) \(\mathstrut -\mathstrut 141745194q^{29} \) \(\mathstrut +\mathstrut 233221904q^{31} \) \(\mathstrut +\mathstrut 162371628q^{33} \) \(\mathstrut +\mathstrut 26124336q^{35} \) \(\mathstrut +\mathstrut 278269694q^{37} \) \(\mathstrut +\mathstrut 498866850q^{39} \) \(\mathstrut -\mathstrut 1181577510q^{41} \) \(\mathstrut +\mathstrut 856975172q^{43} \) \(\mathstrut +\mathstrut 168998238q^{45} \) \(\mathstrut -\mathstrut 1664054928q^{47} \) \(\mathstrut -\mathstrut 1894006359q^{49} \) \(\mathstrut +\mathstrut 389815254q^{51} \) \(\mathstrut -\mathstrut 3851181666q^{53} \) \(\mathstrut +\mathstrut 1912376952q^{55} \) \(\mathstrut -\mathstrut 56011500q^{57} \) \(\mathstrut +\mathstrut 10339000596q^{59} \) \(\mathstrut +\mathstrut 185948102q^{61} \) \(\mathstrut +\mathstrut 538999272q^{63} \) \(\mathstrut +\mathstrut 5875542900q^{65} \) \(\mathstrut +\mathstrut 2915010572q^{67} \) \(\mathstrut -\mathstrut 10452092904q^{69} \) \(\mathstrut +\mathstrut 12662314200q^{71} \) \(\mathstrut -\mathstrut 15201270694q^{73} \) \(\mathstrut -\mathstrut 9874810683q^{75} \) \(\mathstrut +\mathstrut 6099293088q^{77} \) \(\mathstrut -\mathstrut 36644027488q^{79} \) \(\mathstrut +\mathstrut 3486784401q^{81} \) \(\mathstrut -\mathstrut 9217637028q^{83} \) \(\mathstrut +\mathstrut 4591157436q^{85} \) \(\mathstrut -\mathstrut 34444082142q^{87} \) \(\mathstrut +\mathstrut 30573828810q^{89} \) \(\mathstrut +\mathstrut 18739327600q^{91} \) \(\mathstrut +\mathstrut 56672922672q^{93} \) \(\mathstrut -\mathstrut 659691000q^{95} \) \(\mathstrut +\mathstrut 145701815906q^{97} \) \(\mathstrut +\mathstrut 39456305604q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 2862.00 0 9128.00 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 \(T_{5} \) \(\mathstrut -\mathstrut 2862 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(12))\).