Properties

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

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Newspace parameters

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

Newform invariants

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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 27q^{3} \) \(\mathstrut -\mathstrut 378q^{5} \) \(\mathstrut -\mathstrut 832q^{7} \) \(\mathstrut +\mathstrut 729q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 27q^{3} \) \(\mathstrut -\mathstrut 378q^{5} \) \(\mathstrut -\mathstrut 832q^{7} \) \(\mathstrut +\mathstrut 729q^{9} \) \(\mathstrut -\mathstrut 2484q^{11} \) \(\mathstrut +\mathstrut 14870q^{13} \) \(\mathstrut +\mathstrut 10206q^{15} \) \(\mathstrut -\mathstrut 22302q^{17} \) \(\mathstrut -\mathstrut 16300q^{19} \) \(\mathstrut +\mathstrut 22464q^{21} \) \(\mathstrut -\mathstrut 115128q^{23} \) \(\mathstrut +\mathstrut 64759q^{25} \) \(\mathstrut -\mathstrut 19683q^{27} \) \(\mathstrut +\mathstrut 157086q^{29} \) \(\mathstrut -\mathstrut 16456q^{31} \) \(\mathstrut +\mathstrut 67068q^{33} \) \(\mathstrut +\mathstrut 314496q^{35} \) \(\mathstrut -\mathstrut 149266q^{37} \) \(\mathstrut -\mathstrut 401490q^{39} \) \(\mathstrut -\mathstrut 241110q^{41} \) \(\mathstrut -\mathstrut 443188q^{43} \) \(\mathstrut -\mathstrut 275562q^{45} \) \(\mathstrut +\mathstrut 922752q^{47} \) \(\mathstrut -\mathstrut 131319q^{49} \) \(\mathstrut +\mathstrut 602154q^{51} \) \(\mathstrut -\mathstrut 697626q^{53} \) \(\mathstrut +\mathstrut 938952q^{55} \) \(\mathstrut +\mathstrut 440100q^{57} \) \(\mathstrut +\mathstrut 870156q^{59} \) \(\mathstrut +\mathstrut 2067062q^{61} \) \(\mathstrut -\mathstrut 606528q^{63} \) \(\mathstrut -\mathstrut 5620860q^{65} \) \(\mathstrut -\mathstrut 1680748q^{67} \) \(\mathstrut +\mathstrut 3108456q^{69} \) \(\mathstrut -\mathstrut 1070280q^{71} \) \(\mathstrut -\mathstrut 2403334q^{73} \) \(\mathstrut -\mathstrut 1748493q^{75} \) \(\mathstrut +\mathstrut 2066688q^{77} \) \(\mathstrut +\mathstrut 2301512q^{79} \) \(\mathstrut +\mathstrut 531441q^{81} \) \(\mathstrut +\mathstrut 4708692q^{83} \) \(\mathstrut +\mathstrut 8430156q^{85} \) \(\mathstrut -\mathstrut 4241322q^{87} \) \(\mathstrut +\mathstrut 4143690q^{89} \) \(\mathstrut -\mathstrut 12371840q^{91} \) \(\mathstrut +\mathstrut 444312q^{93} \) \(\mathstrut +\mathstrut 6161400q^{95} \) \(\mathstrut -\mathstrut 1622974q^{97} \) \(\mathstrut -\mathstrut 1810836q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −378.000 0 −832.000 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 \(T_{5} \) \(\mathstrut +\mathstrut 378 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(12))\).