Properties

Label 7644.2.a.a
Level $7644$
Weight $2$
Character orbit 7644.a
Self dual yes
Analytic conductor $61.038$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7644 = 2^{2} \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7644.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.0376473051\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 156)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} + q^{9} + O(q^{10}) \) \( q - q^{3} + q^{9} - q^{13} + 6q^{17} - 2q^{19} - 5q^{25} - q^{27} - 6q^{29} - 2q^{31} + 2q^{37} + q^{39} + 12q^{41} - 4q^{43} - 6q^{51} + 6q^{53} + 2q^{57} - 12q^{59} - 2q^{61} - 10q^{67} + 12q^{71} - 14q^{73} + 5q^{75} + 8q^{79} + q^{81} - 12q^{83} + 6q^{87} + 2q^{93} + 10q^{97} + O(q^{100}) \)

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 −1.00000 0 0 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)
\(13\) \(1\)

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 7644.2.a.a 1
7.b odd 2 1 156.2.a.b 1
21.c even 2 1 468.2.a.c 1
28.d even 2 1 624.2.a.b 1
35.c odd 2 1 3900.2.a.a 1
35.f even 4 2 3900.2.h.e 2
56.e even 2 1 2496.2.a.v 1
56.h odd 2 1 2496.2.a.h 1
63.l odd 6 2 4212.2.i.f 2
63.o even 6 2 4212.2.i.g 2
84.h odd 2 1 1872.2.a.i 1
91.b odd 2 1 2028.2.a.e 1
91.i even 4 2 2028.2.b.d 2
91.n odd 6 2 2028.2.i.b 2
91.t odd 6 2 2028.2.i.c 2
91.bc even 12 4 2028.2.q.d 4
168.e odd 2 1 7488.2.a.bb 1
168.i even 2 1 7488.2.a.bf 1
273.g even 2 1 6084.2.a.h 1
273.o odd 4 2 6084.2.b.a 2
364.h even 2 1 8112.2.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.a.b 1 7.b odd 2 1
468.2.a.c 1 21.c even 2 1
624.2.a.b 1 28.d even 2 1
1872.2.a.i 1 84.h odd 2 1
2028.2.a.e 1 91.b odd 2 1
2028.2.b.d 2 91.i even 4 2
2028.2.i.b 2 91.n odd 6 2
2028.2.i.c 2 91.t odd 6 2
2028.2.q.d 4 91.bc even 12 4
2496.2.a.h 1 56.h odd 2 1
2496.2.a.v 1 56.e even 2 1
3900.2.a.a 1 35.c odd 2 1
3900.2.h.e 2 35.f even 4 2
4212.2.i.f 2 63.l odd 6 2
4212.2.i.g 2 63.o even 6 2
6084.2.a.h 1 273.g even 2 1
6084.2.b.a 2 273.o odd 4 2
7488.2.a.bb 1 168.e odd 2 1
7488.2.a.bf 1 168.i even 2 1
7644.2.a.a 1 1.a even 1 1 trivial
8112.2.a.i 1 364.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7644))\):

\( T_{5} \)
\( T_{11} \)
\( T_{17} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 1 + T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( 1 + T \)
$17$ \( -6 + T \)
$19$ \( 2 + T \)
$23$ \( T \)
$29$ \( 6 + T \)
$31$ \( 2 + T \)
$37$ \( -2 + T \)
$41$ \( -12 + T \)
$43$ \( 4 + T \)
$47$ \( T \)
$53$ \( -6 + T \)
$59$ \( 12 + T \)
$61$ \( 2 + T \)
$67$ \( 10 + T \)
$71$ \( -12 + T \)
$73$ \( 14 + T \)
$79$ \( -8 + T \)
$83$ \( 12 + T \)
$89$ \( T \)
$97$ \( -10 + T \)
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