Properties

Label 126.4.a.f
Level $126$
Weight $4$
Character orbit 126.a
Self dual yes
Analytic conductor $7.434$
Analytic rank $1$
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: \(1\)
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

\(f(q)\) \(=\) \( q + 2q^{2} + 4q^{4} - 22q^{5} - 7q^{7} + 8q^{8} + O(q^{10}) \) \( q + 2q^{2} + 4q^{4} - 22q^{5} - 7q^{7} + 8q^{8} - 44q^{10} - 26q^{11} - 54q^{13} - 14q^{14} + 16q^{16} - 74q^{17} + 116q^{19} - 88q^{20} - 52q^{22} + 58q^{23} + 359q^{25} - 108q^{26} - 28q^{28} - 208q^{29} - 252q^{31} + 32q^{32} - 148q^{34} + 154q^{35} + 50q^{37} + 232q^{38} - 176q^{40} - 126q^{41} + 164q^{43} - 104q^{44} + 116q^{46} + 444q^{47} + 49q^{49} + 718q^{50} - 216q^{52} - 12q^{53} + 572q^{55} - 56q^{56} - 416q^{58} - 124q^{59} - 162q^{61} - 504q^{62} + 64q^{64} + 1188q^{65} - 860q^{67} - 296q^{68} + 308q^{70} + 238q^{71} - 146q^{73} + 100q^{74} + 464q^{76} + 182q^{77} - 984q^{79} - 352q^{80} - 252q^{82} - 656q^{83} + 1628q^{85} + 328q^{86} - 208q^{88} + 954q^{89} + 378q^{91} + 232q^{92} + 888q^{94} - 2552q^{95} + 526q^{97} + 98q^{98} + O(q^{100}) \)

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 −22.0000 0 −7.00000 8.00000 0 −44.0000
\(n\): 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.f yes 1
3.b odd 2 1 126.4.a.e 1
4.b odd 2 1 1008.4.a.a 1
7.b odd 2 1 882.4.a.s 1
7.c even 3 2 882.4.g.m 2
7.d odd 6 2 882.4.g.a 2
12.b even 2 1 1008.4.a.w 1
21.c even 2 1 882.4.a.a 1
21.g even 6 2 882.4.g.x 2
21.h odd 6 2 882.4.g.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.a.e 1 3.b odd 2 1
126.4.a.f yes 1 1.a even 1 1 trivial
882.4.a.a 1 21.c even 2 1
882.4.a.s 1 7.b odd 2 1
882.4.g.a 2 7.d odd 6 2
882.4.g.m 2 7.c even 3 2
882.4.g.n 2 21.h odd 6 2
882.4.g.x 2 21.g even 6 2
1008.4.a.a 1 4.b odd 2 1
1008.4.a.w 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 22 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(126))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( T \)
$5$ \( 22 + T \)
$7$ \( 7 + T \)
$11$ \( 26 + T \)
$13$ \( 54 + T \)
$17$ \( 74 + T \)
$19$ \( -116 + T \)
$23$ \( -58 + T \)
$29$ \( 208 + T \)
$31$ \( 252 + T \)
$37$ \( -50 + T \)
$41$ \( 126 + T \)
$43$ \( -164 + T \)
$47$ \( -444 + T \)
$53$ \( 12 + T \)
$59$ \( 124 + T \)
$61$ \( 162 + T \)
$67$ \( 860 + T \)
$71$ \( -238 + T \)
$73$ \( 146 + T \)
$79$ \( 984 + T \)
$83$ \( 656 + T \)
$89$ \( -954 + T \)
$97$ \( -526 + T \)
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