Properties

Label 126.4.a.g
Level $126$
Weight $4$
Character orbit 126.a
Self dual yes
Analytic conductor $7.434$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 126.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.43424066072\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{2} + 4q^{4} + 6q^{5} + 7q^{7} + 8q^{8} + O(q^{10}) \) \( q + 2q^{2} + 4q^{4} + 6q^{5} + 7q^{7} + 8q^{8} + 12q^{10} + 30q^{11} + 2q^{13} + 14q^{14} + 16q^{16} + 66q^{17} - 52q^{19} + 24q^{20} + 60q^{22} + 114q^{23} - 89q^{25} + 4q^{26} + 28q^{28} + 72q^{29} - 196q^{31} + 32q^{32} + 132q^{34} + 42q^{35} - 286q^{37} - 104q^{38} + 48q^{40} - 378q^{41} + 164q^{43} + 120q^{44} + 228q^{46} - 228q^{47} + 49q^{49} - 178q^{50} + 8q^{52} - 348q^{53} + 180q^{55} + 56q^{56} + 144q^{58} - 348q^{59} - 106q^{61} - 392q^{62} + 64q^{64} + 12q^{65} + 596q^{67} + 264q^{68} + 84q^{70} + 630q^{71} - 1042q^{73} - 572q^{74} - 208q^{76} + 210q^{77} - 88q^{79} + 96q^{80} - 756q^{82} - 1440q^{83} + 396q^{85} + 328q^{86} + 240q^{88} + 1374q^{89} + 14q^{91} + 456q^{92} - 456q^{94} - 312q^{95} - 34q^{97} + 98q^{98} + O(q^{100}) \)

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
2.00000 0 4.00000 6.00000 0 7.00000 8.00000 0 12.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.4.a.g yes 1
3.b odd 2 1 126.4.a.b 1
4.b odd 2 1 1008.4.a.n 1
7.b odd 2 1 882.4.a.m 1
7.c even 3 2 882.4.g.e 2
7.d odd 6 2 882.4.g.h 2
12.b even 2 1 1008.4.a.g 1
21.c even 2 1 882.4.a.e 1
21.g even 6 2 882.4.g.q 2
21.h odd 6 2 882.4.g.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.a.b 1 3.b odd 2 1
126.4.a.g yes 1 1.a even 1 1 trivial
882.4.a.e 1 21.c even 2 1
882.4.a.m 1 7.b odd 2 1
882.4.g.e 2 7.c even 3 2
882.4.g.h 2 7.d odd 6 2
882.4.g.q 2 21.g even 6 2
882.4.g.t 2 21.h odd 6 2
1008.4.a.g 1 12.b even 2 1
1008.4.a.n 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 6 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(126))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( T \)
$5$ \( -6 + T \)
$7$ \( -7 + T \)
$11$ \( -30 + T \)
$13$ \( -2 + T \)
$17$ \( -66 + T \)
$19$ \( 52 + T \)
$23$ \( -114 + T \)
$29$ \( -72 + T \)
$31$ \( 196 + T \)
$37$ \( 286 + T \)
$41$ \( 378 + T \)
$43$ \( -164 + T \)
$47$ \( 228 + T \)
$53$ \( 348 + T \)
$59$ \( 348 + T \)
$61$ \( 106 + T \)
$67$ \( -596 + T \)
$71$ \( -630 + T \)
$73$ \( 1042 + T \)
$79$ \( 88 + T \)
$83$ \( 1440 + T \)
$89$ \( -1374 + T \)
$97$ \( 34 + T \)
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