Properties

Label 7056.2.a.bi
Level $7056$
Weight $2$
Character orbit 7056.a
Self dual yes
Analytic conductor $56.342$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{5} + O(q^{10}) \) \( q + q^{5} - q^{11} + 2q^{13} - 3q^{17} - 5q^{19} - 3q^{23} - 4q^{25} + 6q^{29} + q^{31} - 5q^{37} + 10q^{41} + 4q^{43} + q^{47} + 9q^{53} - q^{55} + 3q^{59} + 3q^{61} + 2q^{65} - 11q^{67} + 16q^{71} + 7q^{73} + 11q^{79} - 4q^{83} - 3q^{85} + 9q^{89} - 5q^{95} + 6q^{97} + O(q^{100}) \)

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 1.00000 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 7056.2.a.bi 1
3.b odd 2 1 784.2.a.a 1
4.b odd 2 1 3528.2.a.r 1
7.b odd 2 1 7056.2.a.s 1
7.c even 3 2 1008.2.s.e 2
12.b even 2 1 392.2.a.f 1
21.c even 2 1 784.2.a.j 1
21.g even 6 2 784.2.i.a 2
21.h odd 6 2 112.2.i.c 2
24.f even 2 1 3136.2.a.b 1
24.h odd 2 1 3136.2.a.bc 1
28.d even 2 1 3528.2.a.k 1
28.f even 6 2 3528.2.s.o 2
28.g odd 6 2 504.2.s.e 2
60.h even 2 1 9800.2.a.b 1
84.h odd 2 1 392.2.a.a 1
84.j odd 6 2 392.2.i.f 2
84.n even 6 2 56.2.i.a 2
168.e odd 2 1 3136.2.a.bb 1
168.i even 2 1 3136.2.a.a 1
168.s odd 6 2 448.2.i.a 2
168.v even 6 2 448.2.i.f 2
420.o odd 2 1 9800.2.a.bp 1
420.ba even 6 2 1400.2.q.g 2
420.bp odd 12 4 1400.2.bh.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.i.a 2 84.n even 6 2
112.2.i.c 2 21.h odd 6 2
392.2.a.a 1 84.h odd 2 1
392.2.a.f 1 12.b even 2 1
392.2.i.f 2 84.j odd 6 2
448.2.i.a 2 168.s odd 6 2
448.2.i.f 2 168.v even 6 2
504.2.s.e 2 28.g odd 6 2
784.2.a.a 1 3.b odd 2 1
784.2.a.j 1 21.c even 2 1
784.2.i.a 2 21.g even 6 2
1008.2.s.e 2 7.c even 3 2
1400.2.q.g 2 420.ba even 6 2
1400.2.bh.f 4 420.bp odd 12 4
3136.2.a.a 1 168.i even 2 1
3136.2.a.b 1 24.f even 2 1
3136.2.a.bb 1 168.e odd 2 1
3136.2.a.bc 1 24.h odd 2 1
3528.2.a.k 1 28.d even 2 1
3528.2.a.r 1 4.b odd 2 1
3528.2.s.o 2 28.f even 6 2
7056.2.a.s 1 7.b odd 2 1
7056.2.a.bi 1 1.a even 1 1 trivial
9800.2.a.b 1 60.h even 2 1
9800.2.a.bp 1 420.o odd 2 1

Hecke kernels

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

\( T_{5} - 1 \)
\( T_{11} + 1 \)
\( T_{13} - 2 \)
\( T_{17} + 3 \)
\( T_{23} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - T + 5 T^{2} \)
$7$ 1
$11$ \( 1 + T + 11 T^{2} \)
$13$ \( 1 - 2 T + 13 T^{2} \)
$17$ \( 1 + 3 T + 17 T^{2} \)
$19$ \( 1 + 5 T + 19 T^{2} \)
$23$ \( 1 + 3 T + 23 T^{2} \)
$29$ \( 1 - 6 T + 29 T^{2} \)
$31$ \( 1 - T + 31 T^{2} \)
$37$ \( 1 + 5 T + 37 T^{2} \)
$41$ \( 1 - 10 T + 41 T^{2} \)
$43$ \( 1 - 4 T + 43 T^{2} \)
$47$ \( 1 - T + 47 T^{2} \)
$53$ \( 1 - 9 T + 53 T^{2} \)
$59$ \( 1 - 3 T + 59 T^{2} \)
$61$ \( 1 - 3 T + 61 T^{2} \)
$67$ \( 1 + 11 T + 67 T^{2} \)
$71$ \( 1 - 16 T + 71 T^{2} \)
$73$ \( 1 - 7 T + 73 T^{2} \)
$79$ \( 1 - 11 T + 79 T^{2} \)
$83$ \( 1 + 4 T + 83 T^{2} \)
$89$ \( 1 - 9 T + 89 T^{2} \)
$97$ \( 1 - 6 T + 97 T^{2} \)
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