Properties

Label 624.2.a.i
Level $624$
Weight $2$
Character orbit 624.a
Self dual yes
Analytic conductor $4.983$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 624.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.98266508613\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} + 2 q^{5} + 4 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + 2 q^{5} + 4 q^{7} + q^{9} - 4 q^{11} + q^{13} + 2 q^{15} + 2 q^{17} + 4 q^{21} - q^{25} + q^{27} - 10 q^{29} - 4 q^{31} - 4 q^{33} + 8 q^{35} - 2 q^{37} + q^{39} + 6 q^{41} + 12 q^{43} + 2 q^{45} + 9 q^{49} + 2 q^{51} + 6 q^{53} - 8 q^{55} - 12 q^{59} - 2 q^{61} + 4 q^{63} + 2 q^{65} + 8 q^{67} + 2 q^{73} - q^{75} - 16 q^{77} - 8 q^{79} + q^{81} - 4 q^{83} + 4 q^{85} - 10 q^{87} - 2 q^{89} + 4 q^{91} - 4 q^{93} + 10 q^{97} - 4 q^{99}+O(q^{100}) \) 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 1.00000 0 2.00000 0 4.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(13\) \(-1\)

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 624.2.a.i 1
3.b odd 2 1 1872.2.a.h 1
4.b odd 2 1 39.2.a.a 1
8.b even 2 1 2496.2.a.e 1
8.d odd 2 1 2496.2.a.q 1
12.b even 2 1 117.2.a.a 1
13.b even 2 1 8112.2.a.s 1
20.d odd 2 1 975.2.a.f 1
20.e even 4 2 975.2.c.f 2
24.f even 2 1 7488.2.a.bl 1
24.h odd 2 1 7488.2.a.by 1
28.d even 2 1 1911.2.a.f 1
36.f odd 6 2 1053.2.e.b 2
36.h even 6 2 1053.2.e.d 2
44.c even 2 1 4719.2.a.c 1
52.b odd 2 1 507.2.a.a 1
52.f even 4 2 507.2.b.a 2
52.i odd 6 2 507.2.e.b 2
52.j odd 6 2 507.2.e.a 2
52.l even 12 4 507.2.j.e 4
60.h even 2 1 2925.2.a.p 1
60.l odd 4 2 2925.2.c.e 2
84.h odd 2 1 5733.2.a.e 1
156.h even 2 1 1521.2.a.e 1
156.l odd 4 2 1521.2.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.a 1 4.b odd 2 1
117.2.a.a 1 12.b even 2 1
507.2.a.a 1 52.b odd 2 1
507.2.b.a 2 52.f even 4 2
507.2.e.a 2 52.j odd 6 2
507.2.e.b 2 52.i odd 6 2
507.2.j.e 4 52.l even 12 4
624.2.a.i 1 1.a even 1 1 trivial
975.2.a.f 1 20.d odd 2 1
975.2.c.f 2 20.e even 4 2
1053.2.e.b 2 36.f odd 6 2
1053.2.e.d 2 36.h even 6 2
1521.2.a.e 1 156.h even 2 1
1521.2.b.b 2 156.l odd 4 2
1872.2.a.h 1 3.b odd 2 1
1911.2.a.f 1 28.d even 2 1
2496.2.a.e 1 8.b even 2 1
2496.2.a.q 1 8.d odd 2 1
2925.2.a.p 1 60.h even 2 1
2925.2.c.e 2 60.l odd 4 2
4719.2.a.c 1 44.c even 2 1
5733.2.a.e 1 84.h odd 2 1
7488.2.a.bl 1 24.f even 2 1
7488.2.a.by 1 24.h odd 2 1
8112.2.a.s 1 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(624))\):

\( T_{5} - 2 \) Copy content Toggle raw display
\( T_{7} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T - 2 \) Copy content Toggle raw display
$7$ \( T - 4 \) Copy content Toggle raw display
$11$ \( T + 4 \) Copy content Toggle raw display
$13$ \( T - 1 \) Copy content Toggle raw display
$17$ \( T - 2 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 10 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T + 2 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T - 12 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T + 12 \) Copy content Toggle raw display
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T - 8 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 2 \) Copy content Toggle raw display
$79$ \( T + 8 \) Copy content Toggle raw display
$83$ \( T + 4 \) Copy content Toggle raw display
$89$ \( T + 2 \) Copy content Toggle raw display
$97$ \( T - 10 \) Copy content Toggle raw display
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