Properties

Label 126.4.a.a
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 42)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{2} + 4q^{4} - 18q^{5} + 7q^{7} - 8q^{8} + O(q^{10}) \) \( q - 2q^{2} + 4q^{4} - 18q^{5} + 7q^{7} - 8q^{8} + 36q^{10} + 72q^{11} - 34q^{13} - 14q^{14} + 16q^{16} - 6q^{17} + 92q^{19} - 72q^{20} - 144q^{22} + 180q^{23} + 199q^{25} + 68q^{26} + 28q^{28} + 114q^{29} + 56q^{31} - 32q^{32} + 12q^{34} - 126q^{35} - 34q^{37} - 184q^{38} + 144q^{40} - 6q^{41} + 164q^{43} + 288q^{44} - 360q^{46} - 168q^{47} + 49q^{49} - 398q^{50} - 136q^{52} - 654q^{53} - 1296q^{55} - 56q^{56} - 228q^{58} + 492q^{59} - 250q^{61} - 112q^{62} + 64q^{64} + 612q^{65} - 124q^{67} - 24q^{68} + 252q^{70} - 36q^{71} + 1010q^{73} + 68q^{74} + 368q^{76} + 504q^{77} + 56q^{79} - 288q^{80} + 12q^{82} - 228q^{83} + 108q^{85} - 328q^{86} - 576q^{88} - 390q^{89} - 238q^{91} + 720q^{92} + 336q^{94} - 1656q^{95} - 70q^{97} - 98q^{98} + O(q^{100}) \)

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 −18.0000 0 7.00000 −8.00000 0 36.0000
\(n\): 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.a 1
3.b odd 2 1 42.4.a.a 1
4.b odd 2 1 1008.4.a.b 1
7.b odd 2 1 882.4.a.g 1
7.c even 3 2 882.4.g.w 2
7.d odd 6 2 882.4.g.o 2
12.b even 2 1 336.4.a.l 1
15.d odd 2 1 1050.4.a.g 1
15.e even 4 2 1050.4.g.a 2
21.c even 2 1 294.4.a.i 1
21.g even 6 2 294.4.e.b 2
21.h odd 6 2 294.4.e.c 2
24.f even 2 1 1344.4.a.a 1
24.h odd 2 1 1344.4.a.o 1
84.h odd 2 1 2352.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.a.a 1 3.b odd 2 1
126.4.a.a 1 1.a even 1 1 trivial
294.4.a.i 1 21.c even 2 1
294.4.e.b 2 21.g even 6 2
294.4.e.c 2 21.h odd 6 2
336.4.a.l 1 12.b even 2 1
882.4.a.g 1 7.b odd 2 1
882.4.g.o 2 7.d odd 6 2
882.4.g.w 2 7.c even 3 2
1008.4.a.b 1 4.b odd 2 1
1050.4.a.g 1 15.d odd 2 1
1050.4.g.a 2 15.e even 4 2
1344.4.a.a 1 24.f even 2 1
1344.4.a.o 1 24.h odd 2 1
2352.4.a.a 1 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 18 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(126))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( T \)
$5$ \( 18 + T \)
$7$ \( -7 + T \)
$11$ \( -72 + T \)
$13$ \( 34 + T \)
$17$ \( 6 + T \)
$19$ \( -92 + T \)
$23$ \( -180 + T \)
$29$ \( -114 + T \)
$31$ \( -56 + T \)
$37$ \( 34 + T \)
$41$ \( 6 + T \)
$43$ \( -164 + T \)
$47$ \( 168 + T \)
$53$ \( 654 + T \)
$59$ \( -492 + T \)
$61$ \( 250 + T \)
$67$ \( 124 + T \)
$71$ \( 36 + T \)
$73$ \( -1010 + T \)
$79$ \( -56 + T \)
$83$ \( 228 + T \)
$89$ \( 390 + T \)
$97$ \( 70 + T \)
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