Properties

Label 56.2.a.b
Level $56$
Weight $2$
Character orbit 56.a
Self dual yes
Analytic conductor $0.447$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 56.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.447162251319\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{3} - 4q^{5} + q^{7} + q^{9} + O(q^{10}) \) \( q + 2q^{3} - 4q^{5} + q^{7} + q^{9} - 8q^{15} - 2q^{17} - 2q^{19} + 2q^{21} + 8q^{23} + 11q^{25} - 4q^{27} + 2q^{29} + 4q^{31} - 4q^{35} - 6q^{37} - 2q^{41} + 8q^{43} - 4q^{45} - 4q^{47} + q^{49} - 4q^{51} - 10q^{53} - 4q^{57} + 6q^{59} + 4q^{61} + q^{63} - 12q^{67} + 16q^{69} - 14q^{73} + 22q^{75} - 8q^{79} - 11q^{81} + 6q^{83} + 8q^{85} + 4q^{87} + 10q^{89} + 8q^{93} + 8q^{95} - 2q^{97} + O(q^{100}) \)

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 2.00000 0 −4.00000 0 1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 56.2.a.b 1
3.b odd 2 1 504.2.a.h 1
4.b odd 2 1 112.2.a.a 1
5.b even 2 1 1400.2.a.a 1
5.c odd 4 2 1400.2.g.b 2
7.b odd 2 1 392.2.a.b 1
7.c even 3 2 392.2.i.a 2
7.d odd 6 2 392.2.i.e 2
8.b even 2 1 448.2.a.c 1
8.d odd 2 1 448.2.a.h 1
11.b odd 2 1 6776.2.a.h 1
12.b even 2 1 1008.2.a.m 1
13.b even 2 1 9464.2.a.h 1
16.e even 4 2 1792.2.b.a 2
16.f odd 4 2 1792.2.b.h 2
20.d odd 2 1 2800.2.a.bd 1
20.e even 4 2 2800.2.g.g 2
21.c even 2 1 3528.2.a.b 1
21.g even 6 2 3528.2.s.ba 2
21.h odd 6 2 3528.2.s.a 2
24.f even 2 1 4032.2.a.a 1
24.h odd 2 1 4032.2.a.d 1
28.d even 2 1 784.2.a.i 1
28.f even 6 2 784.2.i.b 2
28.g odd 6 2 784.2.i.j 2
35.c odd 2 1 9800.2.a.bj 1
56.e even 2 1 3136.2.a.c 1
56.h odd 2 1 3136.2.a.w 1
84.h odd 2 1 7056.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.b 1 1.a even 1 1 trivial
112.2.a.a 1 4.b odd 2 1
392.2.a.b 1 7.b odd 2 1
392.2.i.a 2 7.c even 3 2
392.2.i.e 2 7.d odd 6 2
448.2.a.c 1 8.b even 2 1
448.2.a.h 1 8.d odd 2 1
504.2.a.h 1 3.b odd 2 1
784.2.a.i 1 28.d even 2 1
784.2.i.b 2 28.f even 6 2
784.2.i.j 2 28.g odd 6 2
1008.2.a.m 1 12.b even 2 1
1400.2.a.a 1 5.b even 2 1
1400.2.g.b 2 5.c odd 4 2
1792.2.b.a 2 16.e even 4 2
1792.2.b.h 2 16.f odd 4 2
2800.2.a.bd 1 20.d odd 2 1
2800.2.g.g 2 20.e even 4 2
3136.2.a.c 1 56.e even 2 1
3136.2.a.w 1 56.h odd 2 1
3528.2.a.b 1 21.c even 2 1
3528.2.s.a 2 21.h odd 6 2
3528.2.s.ba 2 21.g even 6 2
4032.2.a.a 1 24.f even 2 1
4032.2.a.d 1 24.h odd 2 1
6776.2.a.h 1 11.b odd 2 1
7056.2.a.c 1 84.h odd 2 1
9464.2.a.h 1 13.b even 2 1
9800.2.a.bj 1 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(56))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 2 T + 3 T^{2} \)
$5$ \( 1 + 4 T + 5 T^{2} \)
$7$ \( 1 - T \)
$11$ \( 1 + 11 T^{2} \)
$13$ \( 1 + 13 T^{2} \)
$17$ \( 1 + 2 T + 17 T^{2} \)
$19$ \( 1 + 2 T + 19 T^{2} \)
$23$ \( 1 - 8 T + 23 T^{2} \)
$29$ \( 1 - 2 T + 29 T^{2} \)
$31$ \( 1 - 4 T + 31 T^{2} \)
$37$ \( 1 + 6 T + 37 T^{2} \)
$41$ \( 1 + 2 T + 41 T^{2} \)
$43$ \( 1 - 8 T + 43 T^{2} \)
$47$ \( 1 + 4 T + 47 T^{2} \)
$53$ \( 1 + 10 T + 53 T^{2} \)
$59$ \( 1 - 6 T + 59 T^{2} \)
$61$ \( 1 - 4 T + 61 T^{2} \)
$67$ \( 1 + 12 T + 67 T^{2} \)
$71$ \( 1 + 71 T^{2} \)
$73$ \( 1 + 14 T + 73 T^{2} \)
$79$ \( 1 + 8 T + 79 T^{2} \)
$83$ \( 1 - 6 T + 83 T^{2} \)
$89$ \( 1 - 10 T + 89 T^{2} \)
$97$ \( 1 + 2 T + 97 T^{2} \)
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