Properties

Label 7744.2.a.x
Level $7744$
Weight $2$
Character orbit 7744.a
Self dual yes
Analytic conductor $61.836$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7744 = 2^{6} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7744.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.8361513253\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} - q^{5} + 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - q^{5} + 2 q^{7} - 2 q^{9} + 4 q^{13} - q^{15} + 2 q^{17} + 2 q^{21} - q^{23} - 4 q^{25} - 5 q^{27} + 7 q^{31} - 2 q^{35} - 3 q^{37} + 4 q^{39} + 8 q^{41} - 6 q^{43} + 2 q^{45} + 8 q^{47} - 3 q^{49} + 2 q^{51} + 6 q^{53} - 5 q^{59} + 12 q^{61} - 4 q^{63} - 4 q^{65} + 7 q^{67} - q^{69} - 3 q^{71} - 4 q^{73} - 4 q^{75} + 10 q^{79} + q^{81} - 6 q^{83} - 2 q^{85} + 15 q^{89} + 8 q^{91} + 7 q^{93} - 7 q^{97}+O(q^{100}) \) 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 1.00000 0 −1.00000 0 2.00000 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(11\) \(-1\)

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 7744.2.a.x 1
4.b odd 2 1 7744.2.a.k 1
8.b even 2 1 121.2.a.d 1
8.d odd 2 1 1936.2.a.i 1
11.b odd 2 1 704.2.a.h 1
24.h odd 2 1 1089.2.a.b 1
33.d even 2 1 6336.2.a.br 1
40.f even 2 1 3025.2.a.a 1
44.c even 2 1 704.2.a.c 1
56.h odd 2 1 5929.2.a.h 1
88.b odd 2 1 11.2.a.a 1
88.g even 2 1 176.2.a.b 1
88.o even 10 4 121.2.c.a 4
88.p odd 10 4 121.2.c.e 4
132.d odd 2 1 6336.2.a.bu 1
176.i even 4 2 2816.2.c.f 2
176.l odd 4 2 2816.2.c.j 2
264.m even 2 1 99.2.a.d 1
264.p odd 2 1 1584.2.a.g 1
440.c even 2 1 4400.2.a.i 1
440.o odd 2 1 275.2.a.b 1
440.t even 4 2 275.2.b.a 2
440.w odd 4 2 4400.2.b.h 2
616.g odd 2 1 8624.2.a.j 1
616.o even 2 1 539.2.a.a 1
616.s even 6 2 539.2.e.g 2
616.bg odd 6 2 539.2.e.h 2
792.w even 6 2 891.2.e.b 2
792.be odd 6 2 891.2.e.k 2
1144.h odd 2 1 1859.2.a.b 1
1320.u even 2 1 2475.2.a.a 1
1320.bn odd 4 2 2475.2.c.a 2
1496.p odd 2 1 3179.2.a.a 1
1672.j even 2 1 3971.2.a.b 1
1848.b odd 2 1 4851.2.a.t 1
2024.i even 2 1 5819.2.a.a 1
2552.k odd 2 1 9251.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.2.a.a 1 88.b odd 2 1
99.2.a.d 1 264.m even 2 1
121.2.a.d 1 8.b even 2 1
121.2.c.a 4 88.o even 10 4
121.2.c.e 4 88.p odd 10 4
176.2.a.b 1 88.g even 2 1
275.2.a.b 1 440.o odd 2 1
275.2.b.a 2 440.t even 4 2
539.2.a.a 1 616.o even 2 1
539.2.e.g 2 616.s even 6 2
539.2.e.h 2 616.bg odd 6 2
704.2.a.c 1 44.c even 2 1
704.2.a.h 1 11.b odd 2 1
891.2.e.b 2 792.w even 6 2
891.2.e.k 2 792.be odd 6 2
1089.2.a.b 1 24.h odd 2 1
1584.2.a.g 1 264.p odd 2 1
1859.2.a.b 1 1144.h odd 2 1
1936.2.a.i 1 8.d odd 2 1
2475.2.a.a 1 1320.u even 2 1
2475.2.c.a 2 1320.bn odd 4 2
2816.2.c.f 2 176.i even 4 2
2816.2.c.j 2 176.l odd 4 2
3025.2.a.a 1 40.f even 2 1
3179.2.a.a 1 1496.p odd 2 1
3971.2.a.b 1 1672.j even 2 1
4400.2.a.i 1 440.c even 2 1
4400.2.b.h 2 440.w odd 4 2
4851.2.a.t 1 1848.b odd 2 1
5819.2.a.a 1 2024.i even 2 1
5929.2.a.h 1 56.h odd 2 1
6336.2.a.br 1 33.d even 2 1
6336.2.a.bu 1 132.d odd 2 1
7744.2.a.k 1 4.b odd 2 1
7744.2.a.x 1 1.a even 1 1 trivial
8624.2.a.j 1 616.g odd 2 1
9251.2.a.d 1 2552.k odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7744))\):

\( T_{3} - 1 \) Copy content Toggle raw display
\( T_{5} + 1 \) Copy content Toggle raw display
\( T_{7} - 2 \) Copy content Toggle raw display
\( T_{13} - 4 \) 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
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T - 2 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 4 \) Copy content Toggle raw display
$17$ \( T - 2 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T + 1 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T - 7 \) Copy content Toggle raw display
$37$ \( T + 3 \) Copy content Toggle raw display
$41$ \( T - 8 \) Copy content Toggle raw display
$43$ \( T + 6 \) Copy content Toggle raw display
$47$ \( T - 8 \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T + 5 \) Copy content Toggle raw display
$61$ \( T - 12 \) Copy content Toggle raw display
$67$ \( T - 7 \) Copy content Toggle raw display
$71$ \( T + 3 \) Copy content Toggle raw display
$73$ \( T + 4 \) Copy content Toggle raw display
$79$ \( T - 10 \) Copy content Toggle raw display
$83$ \( T + 6 \) Copy content Toggle raw display
$89$ \( T - 15 \) Copy content Toggle raw display
$97$ \( T + 7 \) Copy content Toggle raw display
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