Properties

Label 504.2.a.c
Level $504$
Weight $2$
Character orbit 504.a
Self dual yes
Analytic conductor $4.024$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{5} - q^{7} + 4 q^{11} + 2 q^{13} + 6 q^{17} + 8 q^{19} - q^{25} - 6 q^{29} + 8 q^{31} + 2 q^{35} - 2 q^{37} - 2 q^{41} - 4 q^{43} + 8 q^{47} + q^{49} - 6 q^{53} - 8 q^{55} - 6 q^{61} - 4 q^{65} - 4 q^{67} + 8 q^{71} + 10 q^{73} - 4 q^{77} + 16 q^{79} - 8 q^{83} - 12 q^{85} + 6 q^{89} - 2 q^{91} - 16 q^{95} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 −2.00000 0 −1.00000 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 504.2.a.c 1
3.b odd 2 1 56.2.a.a 1
4.b odd 2 1 1008.2.a.d 1
7.b odd 2 1 3528.2.a.x 1
7.c even 3 2 3528.2.s.t 2
7.d odd 6 2 3528.2.s.e 2
8.b even 2 1 4032.2.a.bb 1
8.d odd 2 1 4032.2.a.bk 1
12.b even 2 1 112.2.a.b 1
15.d odd 2 1 1400.2.a.g 1
15.e even 4 2 1400.2.g.g 2
21.c even 2 1 392.2.a.d 1
21.g even 6 2 392.2.i.d 2
21.h odd 6 2 392.2.i.c 2
24.f even 2 1 448.2.a.e 1
24.h odd 2 1 448.2.a.d 1
28.d even 2 1 7056.2.a.bo 1
33.d even 2 1 6776.2.a.g 1
39.d odd 2 1 9464.2.a.c 1
48.i odd 4 2 1792.2.b.i 2
48.k even 4 2 1792.2.b.d 2
60.h even 2 1 2800.2.a.p 1
60.l odd 4 2 2800.2.g.p 2
84.h odd 2 1 784.2.a.e 1
84.j odd 6 2 784.2.i.g 2
84.n even 6 2 784.2.i.e 2
105.g even 2 1 9800.2.a.u 1
168.e odd 2 1 3136.2.a.p 1
168.i even 2 1 3136.2.a.q 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.a 1 3.b odd 2 1
112.2.a.b 1 12.b even 2 1
392.2.a.d 1 21.c even 2 1
392.2.i.c 2 21.h odd 6 2
392.2.i.d 2 21.g even 6 2
448.2.a.d 1 24.h odd 2 1
448.2.a.e 1 24.f even 2 1
504.2.a.c 1 1.a even 1 1 trivial
784.2.a.e 1 84.h odd 2 1
784.2.i.e 2 84.n even 6 2
784.2.i.g 2 84.j odd 6 2
1008.2.a.d 1 4.b odd 2 1
1400.2.a.g 1 15.d odd 2 1
1400.2.g.g 2 15.e even 4 2
1792.2.b.d 2 48.k even 4 2
1792.2.b.i 2 48.i odd 4 2
2800.2.a.p 1 60.h even 2 1
2800.2.g.p 2 60.l odd 4 2
3136.2.a.p 1 168.e odd 2 1
3136.2.a.q 1 168.i even 2 1
3528.2.a.x 1 7.b odd 2 1
3528.2.s.e 2 7.d odd 6 2
3528.2.s.t 2 7.c even 3 2
4032.2.a.bb 1 8.b even 2 1
4032.2.a.bk 1 8.d odd 2 1
6776.2.a.g 1 33.d even 2 1
7056.2.a.bo 1 28.d even 2 1
9464.2.a.c 1 39.d odd 2 1
9800.2.a.u 1 105.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(504))\):

\( T_{5} + 2 \) Copy content Toggle raw display
\( T_{11} - 4 \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 2 \) Copy content Toggle raw display
$7$ \( T + 1 \) Copy content Toggle raw display
$11$ \( T - 4 \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T - 6 \) Copy content Toggle raw display
$19$ \( T - 8 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T - 8 \) Copy content Toggle raw display
$37$ \( T + 2 \) Copy content Toggle raw display
$41$ \( T + 2 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T - 8 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 6 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T - 8 \) Copy content Toggle raw display
$73$ \( T - 10 \) Copy content Toggle raw display
$79$ \( T - 16 \) Copy content Toggle raw display
$83$ \( T + 8 \) Copy content Toggle raw display
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T + 6 \) Copy content Toggle raw display
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