Properties

Label 126.4.a.h
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

\(f(q)\) \(=\) \( q + 2q^{2} + 4q^{4} + 14q^{5} - 7q^{7} + 8q^{8} + O(q^{10}) \) \( q + 2q^{2} + 4q^{4} + 14q^{5} - 7q^{7} + 8q^{8} + 28q^{10} + 28q^{11} + 18q^{13} - 14q^{14} + 16q^{16} - 74q^{17} + 80q^{19} + 56q^{20} + 56q^{22} + 112q^{23} + 71q^{25} + 36q^{26} - 28q^{28} - 190q^{29} + 72q^{31} + 32q^{32} - 148q^{34} - 98q^{35} - 346q^{37} + 160q^{38} + 112q^{40} - 162q^{41} - 412q^{43} + 112q^{44} + 224q^{46} - 24q^{47} + 49q^{49} + 142q^{50} + 72q^{52} - 318q^{53} + 392q^{55} - 56q^{56} - 380q^{58} + 200q^{59} - 198q^{61} + 144q^{62} + 64q^{64} + 252q^{65} - 716q^{67} - 296q^{68} - 196q^{70} - 392q^{71} + 538q^{73} - 692q^{74} + 320q^{76} - 196q^{77} + 240q^{79} + 224q^{80} - 324q^{82} + 1072q^{83} - 1036q^{85} - 824q^{86} + 224q^{88} - 810q^{89} - 126q^{91} + 448q^{92} - 48q^{94} + 1120q^{95} + 1354q^{97} + 98q^{98} + O(q^{100}) \)

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 14.0000 0 −7.00000 8.00000 0 28.0000
\(n\): 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.h 1
3.b odd 2 1 14.4.a.a 1
4.b odd 2 1 1008.4.a.s 1
7.b odd 2 1 882.4.a.i 1
7.c even 3 2 882.4.g.b 2
7.d odd 6 2 882.4.g.k 2
12.b even 2 1 112.4.a.a 1
15.d odd 2 1 350.4.a.l 1
15.e even 4 2 350.4.c.b 2
21.c even 2 1 98.4.a.a 1
21.g even 6 2 98.4.c.f 2
21.h odd 6 2 98.4.c.d 2
24.f even 2 1 448.4.a.o 1
24.h odd 2 1 448.4.a.b 1
33.d even 2 1 1694.4.a.g 1
39.d odd 2 1 2366.4.a.h 1
84.h odd 2 1 784.4.a.s 1
105.g even 2 1 2450.4.a.bo 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.a 1 3.b odd 2 1
98.4.a.a 1 21.c even 2 1
98.4.c.d 2 21.h odd 6 2
98.4.c.f 2 21.g even 6 2
112.4.a.a 1 12.b even 2 1
126.4.a.h 1 1.a even 1 1 trivial
350.4.a.l 1 15.d odd 2 1
350.4.c.b 2 15.e even 4 2
448.4.a.b 1 24.h odd 2 1
448.4.a.o 1 24.f even 2 1
784.4.a.s 1 84.h odd 2 1
882.4.a.i 1 7.b odd 2 1
882.4.g.b 2 7.c even 3 2
882.4.g.k 2 7.d odd 6 2
1008.4.a.s 1 4.b odd 2 1
1694.4.a.g 1 33.d even 2 1
2366.4.a.h 1 39.d odd 2 1
2450.4.a.bo 1 105.g even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 14 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(126))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( T \)
$5$ \( -14 + T \)
$7$ \( 7 + T \)
$11$ \( -28 + T \)
$13$ \( -18 + T \)
$17$ \( 74 + T \)
$19$ \( -80 + T \)
$23$ \( -112 + T \)
$29$ \( 190 + T \)
$31$ \( -72 + T \)
$37$ \( 346 + T \)
$41$ \( 162 + T \)
$43$ \( 412 + T \)
$47$ \( 24 + T \)
$53$ \( 318 + T \)
$59$ \( -200 + T \)
$61$ \( 198 + T \)
$67$ \( 716 + T \)
$71$ \( 392 + T \)
$73$ \( -538 + T \)
$79$ \( -240 + T \)
$83$ \( -1072 + T \)
$89$ \( 810 + T \)
$97$ \( -1354 + T \)
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