Properties

Label 56.1.h.a
Level 56
Weight 1
Character orbit 56.h
Self dual yes
Analytic conductor 0.028
Analytic rank 0
Dimension 1
Projective image \(D_{2}\)
CM/RM discs -7, -56, 8
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 56.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.0279476407074\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{2}, \sqrt{-7})\)
Artin image $D_4$
Artin field Galois closure of 4.0.392.1

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{7} - q^{8} - q^{9} + O(q^{10}) \) \( q - q^{2} + q^{4} - q^{7} - q^{8} - q^{9} + q^{14} + q^{16} + q^{18} + 2q^{23} - q^{25} - q^{28} - q^{32} - q^{36} - 2q^{46} + q^{49} + q^{50} + q^{56} + q^{63} + q^{64} - 2q^{71} + q^{72} - 2q^{79} + q^{81} + 2q^{92} - q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/56\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(29\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
0
−1.00000 0 1.00000 0 0 −1.00000 −1.00000 −1.00000 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
8.b even 2 1 RM by \(\Q(\sqrt{2}) \)
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 56.1.h.a 1
3.b odd 2 1 504.1.l.a 1
4.b odd 2 1 224.1.h.a 1
5.b even 2 1 1400.1.m.a 1
5.c odd 4 2 1400.1.c.a 2
7.b odd 2 1 CM 56.1.h.a 1
7.c even 3 2 392.1.j.a 2
7.d odd 6 2 392.1.j.a 2
8.b even 2 1 RM 56.1.h.a 1
8.d odd 2 1 224.1.h.a 1
12.b even 2 1 2016.1.l.a 1
16.e even 4 2 1792.1.c.b 1
16.f odd 4 2 1792.1.c.a 1
21.c even 2 1 504.1.l.a 1
21.g even 6 2 3528.1.bw.a 2
21.h odd 6 2 3528.1.bw.a 2
24.f even 2 1 2016.1.l.a 1
24.h odd 2 1 504.1.l.a 1
28.d even 2 1 224.1.h.a 1
28.f even 6 2 1568.1.n.a 2
28.g odd 6 2 1568.1.n.a 2
35.c odd 2 1 1400.1.m.a 1
35.f even 4 2 1400.1.c.a 2
40.f even 2 1 1400.1.m.a 1
40.i odd 4 2 1400.1.c.a 2
56.e even 2 1 224.1.h.a 1
56.h odd 2 1 CM 56.1.h.a 1
56.j odd 6 2 392.1.j.a 2
56.k odd 6 2 1568.1.n.a 2
56.m even 6 2 1568.1.n.a 2
56.p even 6 2 392.1.j.a 2
84.h odd 2 1 2016.1.l.a 1
112.j even 4 2 1792.1.c.a 1
112.l odd 4 2 1792.1.c.b 1
168.e odd 2 1 2016.1.l.a 1
168.i even 2 1 504.1.l.a 1
168.s odd 6 2 3528.1.bw.a 2
168.ba even 6 2 3528.1.bw.a 2
280.c odd 2 1 1400.1.m.a 1
280.s even 4 2 1400.1.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.1.h.a 1 1.a even 1 1 trivial
56.1.h.a 1 7.b odd 2 1 CM
56.1.h.a 1 8.b even 2 1 RM
56.1.h.a 1 56.h odd 2 1 CM
224.1.h.a 1 4.b odd 2 1
224.1.h.a 1 8.d odd 2 1
224.1.h.a 1 28.d even 2 1
224.1.h.a 1 56.e even 2 1
392.1.j.a 2 7.c even 3 2
392.1.j.a 2 7.d odd 6 2
392.1.j.a 2 56.j odd 6 2
392.1.j.a 2 56.p even 6 2
504.1.l.a 1 3.b odd 2 1
504.1.l.a 1 21.c even 2 1
504.1.l.a 1 24.h odd 2 1
504.1.l.a 1 168.i even 2 1
1400.1.c.a 2 5.c odd 4 2
1400.1.c.a 2 35.f even 4 2
1400.1.c.a 2 40.i odd 4 2
1400.1.c.a 2 280.s even 4 2
1400.1.m.a 1 5.b even 2 1
1400.1.m.a 1 35.c odd 2 1
1400.1.m.a 1 40.f even 2 1
1400.1.m.a 1 280.c odd 2 1
1568.1.n.a 2 28.f even 6 2
1568.1.n.a 2 28.g odd 6 2
1568.1.n.a 2 56.k odd 6 2
1568.1.n.a 2 56.m even 6 2
1792.1.c.a 1 16.f odd 4 2
1792.1.c.a 1 112.j even 4 2
1792.1.c.b 1 16.e even 4 2
1792.1.c.b 1 112.l odd 4 2
2016.1.l.a 1 12.b even 2 1
2016.1.l.a 1 24.f even 2 1
2016.1.l.a 1 84.h odd 2 1
2016.1.l.a 1 168.e odd 2 1
3528.1.bw.a 2 21.g even 6 2
3528.1.bw.a 2 21.h odd 6 2
3528.1.bw.a 2 168.s odd 6 2
3528.1.bw.a 2 168.ba even 6 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(56, [\chi])\).

Hecke characteristic polynomials

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