Properties

Label 7744.2.a.bc
Level $7744$
Weight $2$
Character orbit 7744.a
Self dual yes
Analytic conductor $61.836$
Analytic rank $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 44)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} + 3 q^{5} + 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + 3 q^{5} + 2 q^{7} - 2 q^{9} - 4 q^{13} + 3 q^{15} - 6 q^{17} - 8 q^{19} + 2 q^{21} + 3 q^{23} + 4 q^{25} - 5 q^{27} - 5 q^{31} + 6 q^{35} + q^{37} - 4 q^{39} + 10 q^{43} - 6 q^{45} - 3 q^{49} - 6 q^{51} + 6 q^{53} - 8 q^{57} + 3 q^{59} - 4 q^{61} - 4 q^{63} - 12 q^{65} - q^{67} + 3 q^{69} - 15 q^{71} + 4 q^{73} + 4 q^{75} + 2 q^{79} + q^{81} - 6 q^{83} - 18 q^{85} - 9 q^{89} - 8 q^{91} - 5 q^{93} - 24 q^{95} - 7 q^{97}+O(q^{100}) \) 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 3.00000 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.bc 1
4.b odd 2 1 7744.2.a.m 1
8.b even 2 1 1936.2.a.c 1
8.d odd 2 1 484.2.a.a 1
11.b odd 2 1 704.2.a.i 1
24.f even 2 1 4356.2.a.j 1
33.d even 2 1 6336.2.a.i 1
44.c even 2 1 704.2.a.f 1
88.b odd 2 1 176.2.a.a 1
88.g even 2 1 44.2.a.a 1
88.k even 10 4 484.2.e.a 4
88.l odd 10 4 484.2.e.b 4
132.d odd 2 1 6336.2.a.j 1
176.i even 4 2 2816.2.c.e 2
176.l odd 4 2 2816.2.c.k 2
264.m even 2 1 1584.2.a.p 1
264.p odd 2 1 396.2.a.c 1
440.c even 2 1 1100.2.a.b 1
440.o odd 2 1 4400.2.a.v 1
440.t even 4 2 4400.2.b.k 2
440.w odd 4 2 1100.2.b.c 2
616.g odd 2 1 2156.2.a.a 1
616.o even 2 1 8624.2.a.w 1
616.y even 6 2 2156.2.i.b 2
616.z odd 6 2 2156.2.i.c 2
792.s odd 6 2 3564.2.i.a 2
792.z even 6 2 3564.2.i.j 2
1144.o even 2 1 7436.2.a.d 1
1320.b odd 2 1 9900.2.a.h 1
1320.bt even 4 2 9900.2.c.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.2.a.a 1 88.g even 2 1
176.2.a.a 1 88.b odd 2 1
396.2.a.c 1 264.p odd 2 1
484.2.a.a 1 8.d odd 2 1
484.2.e.a 4 88.k even 10 4
484.2.e.b 4 88.l odd 10 4
704.2.a.f 1 44.c even 2 1
704.2.a.i 1 11.b odd 2 1
1100.2.a.b 1 440.c even 2 1
1100.2.b.c 2 440.w odd 4 2
1584.2.a.p 1 264.m even 2 1
1936.2.a.c 1 8.b even 2 1
2156.2.a.a 1 616.g odd 2 1
2156.2.i.b 2 616.y even 6 2
2156.2.i.c 2 616.z odd 6 2
2816.2.c.e 2 176.i even 4 2
2816.2.c.k 2 176.l odd 4 2
3564.2.i.a 2 792.s odd 6 2
3564.2.i.j 2 792.z even 6 2
4356.2.a.j 1 24.f even 2 1
4400.2.a.v 1 440.o odd 2 1
4400.2.b.k 2 440.t even 4 2
6336.2.a.i 1 33.d even 2 1
6336.2.a.j 1 132.d odd 2 1
7436.2.a.d 1 1144.o even 2 1
7744.2.a.m 1 4.b odd 2 1
7744.2.a.bc 1 1.a even 1 1 trivial
8624.2.a.w 1 616.o even 2 1
9900.2.a.h 1 1320.b odd 2 1
9900.2.c.g 2 1320.bt even 4 2

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
\( T_{5} - 3 \) 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
$5$ \( T - 3 \) 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
$17$ \( T + 6 \) Copy content Toggle raw display
$19$ \( T + 8 \) Copy content Toggle raw display
$23$ \( T - 3 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 5 \) Copy content Toggle raw display
$37$ \( T - 1 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T - 10 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T - 3 \) Copy content Toggle raw display
$61$ \( T + 4 \) Copy content Toggle raw display
$67$ \( T + 1 \) Copy content Toggle raw display
$71$ \( T + 15 \) Copy content Toggle raw display
$73$ \( T - 4 \) Copy content Toggle raw display
$79$ \( T - 2 \) Copy content Toggle raw display
$83$ \( T + 6 \) Copy content Toggle raw display
$89$ \( T + 9 \) Copy content Toggle raw display
$97$ \( T + 7 \) Copy content Toggle raw display
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