Properties

Label 7744.2.a.v
Level $7744$
Weight $2$
Character orbit 7744.a
Self dual yes
Analytic conductor $61.836$
Analytic rank $1$
Dimension $1$
CM discriminant -4
Inner twists $2$

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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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 32)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{5} - 3q^{9} + O(q^{10}) \) \( q + 2q^{5} - 3q^{9} + 6q^{13} - 2q^{17} - q^{25} - 10q^{29} + 2q^{37} - 10q^{41} - 6q^{45} - 7q^{49} - 14q^{53} - 10q^{61} + 12q^{65} + 6q^{73} + 9q^{81} - 4q^{85} + 10q^{89} + 18q^{97} + O(q^{100}) \)

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 0 0 2.00000 0 0 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7744.2.a.v 1
4.b odd 2 1 CM 7744.2.a.v 1
8.b even 2 1 3872.2.a.f 1
8.d odd 2 1 3872.2.a.f 1
11.b odd 2 1 64.2.a.a 1
33.d even 2 1 576.2.a.c 1
44.c even 2 1 64.2.a.a 1
55.d odd 2 1 1600.2.a.n 1
55.e even 4 2 1600.2.c.l 2
77.b even 2 1 3136.2.a.m 1
88.b odd 2 1 32.2.a.a 1
88.g even 2 1 32.2.a.a 1
132.d odd 2 1 576.2.a.c 1
176.i even 4 2 256.2.b.b 2
176.l odd 4 2 256.2.b.b 2
220.g even 2 1 1600.2.a.n 1
220.i odd 4 2 1600.2.c.l 2
264.m even 2 1 288.2.a.d 1
264.p odd 2 1 288.2.a.d 1
308.g odd 2 1 3136.2.a.m 1
352.o odd 8 4 1024.2.e.j 4
352.q even 8 4 1024.2.e.j 4
440.c even 2 1 800.2.a.d 1
440.o odd 2 1 800.2.a.d 1
440.t even 4 2 800.2.c.e 2
440.w odd 4 2 800.2.c.e 2
528.s odd 4 2 2304.2.d.j 2
528.x even 4 2 2304.2.d.j 2
616.g odd 2 1 1568.2.a.e 1
616.o even 2 1 1568.2.a.e 1
616.s even 6 2 1568.2.i.f 2
616.y even 6 2 1568.2.i.g 2
616.z odd 6 2 1568.2.i.f 2
616.bg odd 6 2 1568.2.i.g 2
792.s odd 6 2 2592.2.i.e 2
792.w even 6 2 2592.2.i.e 2
792.z even 6 2 2592.2.i.t 2
792.be odd 6 2 2592.2.i.t 2
1144.h odd 2 1 5408.2.a.g 1
1144.o even 2 1 5408.2.a.g 1
1320.b odd 2 1 7200.2.a.v 1
1320.u even 2 1 7200.2.a.v 1
1320.bn odd 4 2 7200.2.f.m 2
1320.bt even 4 2 7200.2.f.m 2
1496.g even 2 1 9248.2.a.f 1
1496.p odd 2 1 9248.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.a.a 1 88.b odd 2 1
32.2.a.a 1 88.g even 2 1
64.2.a.a 1 11.b odd 2 1
64.2.a.a 1 44.c even 2 1
256.2.b.b 2 176.i even 4 2
256.2.b.b 2 176.l odd 4 2
288.2.a.d 1 264.m even 2 1
288.2.a.d 1 264.p odd 2 1
576.2.a.c 1 33.d even 2 1
576.2.a.c 1 132.d odd 2 1
800.2.a.d 1 440.c even 2 1
800.2.a.d 1 440.o odd 2 1
800.2.c.e 2 440.t even 4 2
800.2.c.e 2 440.w odd 4 2
1024.2.e.j 4 352.o odd 8 4
1024.2.e.j 4 352.q even 8 4
1568.2.a.e 1 616.g odd 2 1
1568.2.a.e 1 616.o even 2 1
1568.2.i.f 2 616.s even 6 2
1568.2.i.f 2 616.z odd 6 2
1568.2.i.g 2 616.y even 6 2
1568.2.i.g 2 616.bg odd 6 2
1600.2.a.n 1 55.d odd 2 1
1600.2.a.n 1 220.g even 2 1
1600.2.c.l 2 55.e even 4 2
1600.2.c.l 2 220.i odd 4 2
2304.2.d.j 2 528.s odd 4 2
2304.2.d.j 2 528.x even 4 2
2592.2.i.e 2 792.s odd 6 2
2592.2.i.e 2 792.w even 6 2
2592.2.i.t 2 792.z even 6 2
2592.2.i.t 2 792.be odd 6 2
3136.2.a.m 1 77.b even 2 1
3136.2.a.m 1 308.g odd 2 1
3872.2.a.f 1 8.b even 2 1
3872.2.a.f 1 8.d odd 2 1
5408.2.a.g 1 1144.h odd 2 1
5408.2.a.g 1 1144.o even 2 1
7200.2.a.v 1 1320.b odd 2 1
7200.2.a.v 1 1320.u even 2 1
7200.2.f.m 2 1320.bn odd 4 2
7200.2.f.m 2 1320.bt even 4 2
7744.2.a.v 1 1.a even 1 1 trivial
7744.2.a.v 1 4.b odd 2 1 CM
9248.2.a.f 1 1496.g even 2 1
9248.2.a.f 1 1496.p odd 2 1

Hecke kernels

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

\( T_{3} \)
\( T_{5} - 2 \)
\( T_{7} \)
\( T_{13} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -2 + T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( -6 + T \)
$17$ \( 2 + T \)
$19$ \( T \)
$23$ \( T \)
$29$ \( 10 + T \)
$31$ \( T \)
$37$ \( -2 + T \)
$41$ \( 10 + T \)
$43$ \( T \)
$47$ \( T \)
$53$ \( 14 + T \)
$59$ \( T \)
$61$ \( 10 + T \)
$67$ \( T \)
$71$ \( T \)
$73$ \( -6 + T \)
$79$ \( T \)
$83$ \( T \)
$89$ \( -10 + T \)
$97$ \( -18 + T \)
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