Properties

Label 7056.2.a.bb
Level 7056
Weight 2
Character orbit 7056.a
Self dual yes
Analytic conductor 56.342
Analytic rank 1
Dimension 1
CM discriminant -3
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7056.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.3424436662\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 36)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q + O(q^{10}) \) \( q - 2q^{13} + 8q^{19} - 5q^{25} - 4q^{31} - 10q^{37} - 8q^{43} - 14q^{61} + 16q^{67} + 10q^{73} + 4q^{79} - 14q^{97} + O(q^{100}) \)

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7056.2.a.bb 1
3.b odd 2 1 CM 7056.2.a.bb 1
4.b odd 2 1 1764.2.a.e 1
7.b odd 2 1 144.2.a.a 1
12.b even 2 1 1764.2.a.e 1
21.c even 2 1 144.2.a.a 1
28.d even 2 1 36.2.a.a 1
28.f even 6 2 1764.2.k.h 2
28.g odd 6 2 1764.2.k.g 2
35.c odd 2 1 3600.2.a.e 1
35.f even 4 2 3600.2.f.m 2
56.e even 2 1 576.2.a.e 1
56.h odd 2 1 576.2.a.f 1
63.l odd 6 2 1296.2.i.h 2
63.o even 6 2 1296.2.i.h 2
84.h odd 2 1 36.2.a.a 1
84.j odd 6 2 1764.2.k.h 2
84.n even 6 2 1764.2.k.g 2
105.g even 2 1 3600.2.a.e 1
105.k odd 4 2 3600.2.f.m 2
112.j even 4 2 2304.2.d.q 2
112.l odd 4 2 2304.2.d.a 2
140.c even 2 1 900.2.a.g 1
140.j odd 4 2 900.2.d.b 2
168.e odd 2 1 576.2.a.e 1
168.i even 2 1 576.2.a.f 1
252.s odd 6 2 324.2.e.c 2
252.bi even 6 2 324.2.e.c 2
308.g odd 2 1 4356.2.a.g 1
336.v odd 4 2 2304.2.d.q 2
336.y even 4 2 2304.2.d.a 2
364.h even 2 1 6084.2.a.i 1
364.p odd 4 2 6084.2.b.f 2
420.o odd 2 1 900.2.a.g 1
420.w even 4 2 900.2.d.b 2
924.n even 2 1 4356.2.a.g 1
1092.d odd 2 1 6084.2.a.i 1
1092.u even 4 2 6084.2.b.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.a.a 1 28.d even 2 1
36.2.a.a 1 84.h odd 2 1
144.2.a.a 1 7.b odd 2 1
144.2.a.a 1 21.c even 2 1
324.2.e.c 2 252.s odd 6 2
324.2.e.c 2 252.bi even 6 2
576.2.a.e 1 56.e even 2 1
576.2.a.e 1 168.e odd 2 1
576.2.a.f 1 56.h odd 2 1
576.2.a.f 1 168.i even 2 1
900.2.a.g 1 140.c even 2 1
900.2.a.g 1 420.o odd 2 1
900.2.d.b 2 140.j odd 4 2
900.2.d.b 2 420.w even 4 2
1296.2.i.h 2 63.l odd 6 2
1296.2.i.h 2 63.o even 6 2
1764.2.a.e 1 4.b odd 2 1
1764.2.a.e 1 12.b even 2 1
1764.2.k.g 2 28.g odd 6 2
1764.2.k.g 2 84.n even 6 2
1764.2.k.h 2 28.f even 6 2
1764.2.k.h 2 84.j odd 6 2
2304.2.d.a 2 112.l odd 4 2
2304.2.d.a 2 336.y even 4 2
2304.2.d.q 2 112.j even 4 2
2304.2.d.q 2 336.v odd 4 2
3600.2.a.e 1 35.c odd 2 1
3600.2.a.e 1 105.g even 2 1
3600.2.f.m 2 35.f even 4 2
3600.2.f.m 2 105.k odd 4 2
4356.2.a.g 1 308.g odd 2 1
4356.2.a.g 1 924.n even 2 1
6084.2.a.i 1 364.h even 2 1
6084.2.a.i 1 1092.d odd 2 1
6084.2.b.f 2 364.p odd 4 2
6084.2.b.f 2 1092.u even 4 2
7056.2.a.bb 1 1.a even 1 1 trivial
7056.2.a.bb 1 3.b odd 2 1 CM

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7056))\):

\( T_{5} \)
\( T_{11} \)
\( T_{13} + 2 \)
\( T_{17} \)
\( T_{23} \)

Hecke Characteristic Polynomials

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