Properties

Label 504.4.a.b
Level $504$
Weight $4$
Character orbit 504.a
Self dual yes
Analytic conductor $29.737$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 504.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.7369626429\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 4q^{5} - 7q^{7} + O(q^{10}) \) \( q - 4q^{5} - 7q^{7} + 26q^{11} + 2q^{13} + 36q^{17} - 76q^{19} + 114q^{23} - 109q^{25} - 6q^{29} - 256q^{31} + 28q^{35} - 86q^{37} - 160q^{41} - 220q^{43} - 308q^{47} + 49q^{49} - 258q^{53} - 104q^{55} - 264q^{59} + 606q^{61} - 8q^{65} - 520q^{67} + 286q^{71} - 530q^{73} - 182q^{77} - 44q^{79} - 1012q^{83} - 144q^{85} - 768q^{89} - 14q^{91} + 304q^{95} + 222q^{97} + O(q^{100}) \)

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
0 0 0 −4.00000 0 −7.00000 0 0 0
\(n\): 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 504.4.a.b 1
3.b odd 2 1 168.4.a.c 1
4.b odd 2 1 1008.4.a.i 1
12.b even 2 1 336.4.a.j 1
21.c even 2 1 1176.4.a.j 1
24.f even 2 1 1344.4.a.e 1
24.h odd 2 1 1344.4.a.s 1
84.h odd 2 1 2352.4.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.a.c 1 3.b odd 2 1
336.4.a.j 1 12.b even 2 1
504.4.a.b 1 1.a even 1 1 trivial
1008.4.a.i 1 4.b odd 2 1
1176.4.a.j 1 21.c even 2 1
1344.4.a.e 1 24.f even 2 1
1344.4.a.s 1 24.h odd 2 1
2352.4.a.h 1 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(504))\):

\( T_{5} + 4 \)
\( T_{11} - 26 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 4 + T \)
$7$ \( 7 + T \)
$11$ \( -26 + T \)
$13$ \( -2 + T \)
$17$ \( -36 + T \)
$19$ \( 76 + T \)
$23$ \( -114 + T \)
$29$ \( 6 + T \)
$31$ \( 256 + T \)
$37$ \( 86 + T \)
$41$ \( 160 + T \)
$43$ \( 220 + T \)
$47$ \( 308 + T \)
$53$ \( 258 + T \)
$59$ \( 264 + T \)
$61$ \( -606 + T \)
$67$ \( 520 + T \)
$71$ \( -286 + T \)
$73$ \( 530 + T \)
$79$ \( 44 + T \)
$83$ \( 1012 + T \)
$89$ \( 768 + T \)
$97$ \( -222 + T \)
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