Properties

Label 7644.2.a.k
Level $7644$
Weight $2$
Character orbit 7644.a
Self dual yes
Analytic conductor $61.038$
Analytic rank $0$
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: \(0\)
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

\(f(q)\) \(=\) \( q + q^{3} + 4q^{5} + q^{9} + O(q^{10}) \) \( q + q^{3} + 4q^{5} + q^{9} - 4q^{11} - q^{13} + 4q^{15} - 2q^{17} + 2q^{19} + 11q^{25} + q^{27} - 6q^{29} + 10q^{31} - 4q^{33} + 10q^{37} - q^{39} - 8q^{41} + 4q^{43} + 4q^{45} + 4q^{47} - 2q^{51} - 10q^{53} - 16q^{55} + 2q^{57} + 8q^{59} + 14q^{61} - 4q^{65} + 2q^{67} + 16q^{71} + 10q^{73} + 11q^{75} - 16q^{79} + q^{81} - 8q^{85} - 6q^{87} + 4q^{89} + 10q^{93} + 8q^{95} + 2q^{97} - 4q^{99} + O(q^{100}) \)

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 4.00000 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.k 1
7.b odd 2 1 156.2.a.a 1
21.c even 2 1 468.2.a.d 1
28.d even 2 1 624.2.a.e 1
35.c odd 2 1 3900.2.a.m 1
35.f even 4 2 3900.2.h.b 2
56.e even 2 1 2496.2.a.o 1
56.h odd 2 1 2496.2.a.bc 1
63.l odd 6 2 4212.2.i.l 2
63.o even 6 2 4212.2.i.b 2
84.h odd 2 1 1872.2.a.s 1
91.b odd 2 1 2028.2.a.c 1
91.i even 4 2 2028.2.b.a 2
91.n odd 6 2 2028.2.i.e 2
91.t odd 6 2 2028.2.i.g 2
91.bc even 12 4 2028.2.q.h 4
168.e odd 2 1 7488.2.a.d 1
168.i even 2 1 7488.2.a.c 1
273.g even 2 1 6084.2.a.b 1
273.o odd 4 2 6084.2.b.j 2
364.h even 2 1 8112.2.a.bi 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.a.a 1 7.b odd 2 1
468.2.a.d 1 21.c even 2 1
624.2.a.e 1 28.d even 2 1
1872.2.a.s 1 84.h odd 2 1
2028.2.a.c 1 91.b odd 2 1
2028.2.b.a 2 91.i even 4 2
2028.2.i.e 2 91.n odd 6 2
2028.2.i.g 2 91.t odd 6 2
2028.2.q.h 4 91.bc even 12 4
2496.2.a.o 1 56.e even 2 1
2496.2.a.bc 1 56.h odd 2 1
3900.2.a.m 1 35.c odd 2 1
3900.2.h.b 2 35.f even 4 2
4212.2.i.b 2 63.o even 6 2
4212.2.i.l 2 63.l odd 6 2
6084.2.a.b 1 273.g even 2 1
6084.2.b.j 2 273.o odd 4 2
7488.2.a.c 1 168.i even 2 1
7488.2.a.d 1 168.e odd 2 1
7644.2.a.k 1 1.a even 1 1 trivial
8112.2.a.bi 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7644))\):

\( T_{5} - 4 \)
\( T_{11} + 4 \)
\( T_{17} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -1 + T \)
$5$ \( -4 + T \)
$7$ \( T \)
$11$ \( 4 + T \)
$13$ \( 1 + T \)
$17$ \( 2 + T \)
$19$ \( -2 + T \)
$23$ \( T \)
$29$ \( 6 + T \)
$31$ \( -10 + T \)
$37$ \( -10 + T \)
$41$ \( 8 + T \)
$43$ \( -4 + T \)
$47$ \( -4 + T \)
$53$ \( 10 + T \)
$59$ \( -8 + T \)
$61$ \( -14 + T \)
$67$ \( -2 + T \)
$71$ \( -16 + T \)
$73$ \( -10 + T \)
$79$ \( 16 + T \)
$83$ \( T \)
$89$ \( -4 + T \)
$97$ \( -2 + T \)
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