Properties

Label 126.4.a.d
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: no (minimal twist has level 14)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{2} + 4q^{4} + 12q^{5} + 7q^{7} - 8q^{8} + O(q^{10}) \) \( q - 2q^{2} + 4q^{4} + 12q^{5} + 7q^{7} - 8q^{8} - 24q^{10} - 48q^{11} + 56q^{13} - 14q^{14} + 16q^{16} + 114q^{17} + 2q^{19} + 48q^{20} + 96q^{22} + 120q^{23} + 19q^{25} - 112q^{26} + 28q^{28} + 54q^{29} + 236q^{31} - 32q^{32} - 228q^{34} + 84q^{35} + 146q^{37} - 4q^{38} - 96q^{40} - 126q^{41} - 376q^{43} - 192q^{44} - 240q^{46} + 12q^{47} + 49q^{49} - 38q^{50} + 224q^{52} - 174q^{53} - 576q^{55} - 56q^{56} - 108q^{58} - 138q^{59} + 380q^{61} - 472q^{62} + 64q^{64} + 672q^{65} - 484q^{67} + 456q^{68} - 168q^{70} - 576q^{71} - 1150q^{73} - 292q^{74} + 8q^{76} - 336q^{77} + 776q^{79} + 192q^{80} + 252q^{82} - 378q^{83} + 1368q^{85} + 752q^{86} + 384q^{88} + 390q^{89} + 392q^{91} + 480q^{92} - 24q^{94} + 24q^{95} - 1330q^{97} - 98q^{98} + O(q^{100}) \)

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 12.0000 0 7.00000 −8.00000 0 −24.0000
\(n\): 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.d 1
3.b odd 2 1 14.4.a.b 1
4.b odd 2 1 1008.4.a.r 1
7.b odd 2 1 882.4.a.b 1
7.c even 3 2 882.4.g.p 2
7.d odd 6 2 882.4.g.v 2
12.b even 2 1 112.4.a.e 1
15.d odd 2 1 350.4.a.f 1
15.e even 4 2 350.4.c.g 2
21.c even 2 1 98.4.a.e 1
21.g even 6 2 98.4.c.b 2
21.h odd 6 2 98.4.c.c 2
24.f even 2 1 448.4.a.g 1
24.h odd 2 1 448.4.a.k 1
33.d even 2 1 1694.4.a.b 1
39.d odd 2 1 2366.4.a.c 1
84.h odd 2 1 784.4.a.h 1
105.g even 2 1 2450.4.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.b 1 3.b odd 2 1
98.4.a.e 1 21.c even 2 1
98.4.c.b 2 21.g even 6 2
98.4.c.c 2 21.h odd 6 2
112.4.a.e 1 12.b even 2 1
126.4.a.d 1 1.a even 1 1 trivial
350.4.a.f 1 15.d odd 2 1
350.4.c.g 2 15.e even 4 2
448.4.a.g 1 24.f even 2 1
448.4.a.k 1 24.h odd 2 1
784.4.a.h 1 84.h odd 2 1
882.4.a.b 1 7.b odd 2 1
882.4.g.p 2 7.c even 3 2
882.4.g.v 2 7.d odd 6 2
1008.4.a.r 1 4.b odd 2 1
1694.4.a.b 1 33.d even 2 1
2366.4.a.c 1 39.d odd 2 1
2450.4.a.i 1 105.g even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 12 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(126))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( T \)
$5$ \( -12 + T \)
$7$ \( -7 + T \)
$11$ \( 48 + T \)
$13$ \( -56 + T \)
$17$ \( -114 + T \)
$19$ \( -2 + T \)
$23$ \( -120 + T \)
$29$ \( -54 + T \)
$31$ \( -236 + T \)
$37$ \( -146 + T \)
$41$ \( 126 + T \)
$43$ \( 376 + T \)
$47$ \( -12 + T \)
$53$ \( 174 + T \)
$59$ \( 138 + T \)
$61$ \( -380 + T \)
$67$ \( 484 + T \)
$71$ \( 576 + T \)
$73$ \( 1150 + T \)
$79$ \( -776 + T \)
$83$ \( 378 + T \)
$89$ \( -390 + T \)
$97$ \( 1330 + T \)
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