Properties

Label 12.8.a.b
Level 12
Weight 8
Character orbit 12.a
Self dual Yes
Analytic conductor 3.749
Analytic rank 0
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 12.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(3.74862030581\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 27q^{3} \) \(\mathstrut +\mathstrut 270q^{5} \) \(\mathstrut +\mathstrut 1112q^{7} \) \(\mathstrut +\mathstrut 729q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 27q^{3} \) \(\mathstrut +\mathstrut 270q^{5} \) \(\mathstrut +\mathstrut 1112q^{7} \) \(\mathstrut +\mathstrut 729q^{9} \) \(\mathstrut -\mathstrut 5724q^{11} \) \(\mathstrut -\mathstrut 4570q^{13} \) \(\mathstrut +\mathstrut 7290q^{15} \) \(\mathstrut -\mathstrut 36558q^{17} \) \(\mathstrut +\mathstrut 51740q^{19} \) \(\mathstrut +\mathstrut 30024q^{21} \) \(\mathstrut +\mathstrut 22248q^{23} \) \(\mathstrut -\mathstrut 5225q^{25} \) \(\mathstrut +\mathstrut 19683q^{27} \) \(\mathstrut -\mathstrut 157194q^{29} \) \(\mathstrut -\mathstrut 103936q^{31} \) \(\mathstrut -\mathstrut 154548q^{33} \) \(\mathstrut +\mathstrut 300240q^{35} \) \(\mathstrut -\mathstrut 94834q^{37} \) \(\mathstrut -\mathstrut 123390q^{39} \) \(\mathstrut +\mathstrut 659610q^{41} \) \(\mathstrut -\mathstrut 75772q^{43} \) \(\mathstrut +\mathstrut 196830q^{45} \) \(\mathstrut +\mathstrut 405648q^{47} \) \(\mathstrut +\mathstrut 413001q^{49} \) \(\mathstrut -\mathstrut 987066q^{51} \) \(\mathstrut -\mathstrut 1346274q^{53} \) \(\mathstrut -\mathstrut 1545480q^{55} \) \(\mathstrut +\mathstrut 1396980q^{57} \) \(\mathstrut -\mathstrut 1303884q^{59} \) \(\mathstrut +\mathstrut 1833782q^{61} \) \(\mathstrut +\mathstrut 810648q^{63} \) \(\mathstrut -\mathstrut 1233900q^{65} \) \(\mathstrut +\mathstrut 1369388q^{67} \) \(\mathstrut +\mathstrut 600696q^{69} \) \(\mathstrut +\mathstrut 2714040q^{71} \) \(\mathstrut +\mathstrut 2868794q^{73} \) \(\mathstrut -\mathstrut 141075q^{75} \) \(\mathstrut -\mathstrut 6365088q^{77} \) \(\mathstrut -\mathstrut 1129648q^{79} \) \(\mathstrut +\mathstrut 531441q^{81} \) \(\mathstrut +\mathstrut 5912028q^{83} \) \(\mathstrut -\mathstrut 9870660q^{85} \) \(\mathstrut -\mathstrut 4244238q^{87} \) \(\mathstrut -\mathstrut 897750q^{89} \) \(\mathstrut -\mathstrut 5081840q^{91} \) \(\mathstrut -\mathstrut 2806272q^{93} \) \(\mathstrut +\mathstrut 13969800q^{95} \) \(\mathstrut +\mathstrut 13719074q^{97} \) \(\mathstrut -\mathstrut 4172796q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 27.0000 0 270.000 0 1112.00 0 729.000 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 \(T_{5} \) \(\mathstrut -\mathstrut 270 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(12))\).