Newform orbit 56.3.h.c

• Label: 56.3.h.c
• Level: $56$
• Weight: $3$
• Character orbit: 56.h
• Self dual: yes
• Analytic conductor: $1.526$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -56
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	yes
Analytic conductor:	\(1.52588948042\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + 2 q^{2} + \beta q^{3} + 4 q^{4} - 3 \beta q^{5} + 2 \beta q^{6} - 7 q^{7} + 8 q^{8} - q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 8 q^{4} - 14 q^{7} + 16 q^{8} - 2 q^{9}+O(q^{10}) \)

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

−1.41421
1.41421

2.00000

−2.82843

4.00000

8.48528

−5.65685

−7.00000

8.00000

−1.00000

16.9706

13.2

2.00000

2.82843

4.00000

−8.48528

5.65685

−7.00000

8.00000

−1.00000

−16.9706

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
56.h	odd	2	1	CM by \(\Q(\sqrt{-14}) \)
7.b	odd	2	1	inner
8.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	56.3.h.c	✓	2
3.b	odd	2	1		504.3.l.a		2
4.b	odd	2	1		224.3.h.c		2
7.b	odd	2	1	inner	56.3.h.c	✓	2
7.c	even	3	2		392.3.j.a		4
7.d	odd	6	2		392.3.j.a		4
8.b	even	2	1	inner	56.3.h.c	✓	2
8.d	odd	2	1		224.3.h.c		2
12.b	even	2	1		2016.3.l.c		2
21.c	even	2	1		504.3.l.a		2
24.f	even	2	1		2016.3.l.c		2
24.h	odd	2	1		504.3.l.a		2
28.d	even	2	1		224.3.h.c		2
56.e	even	2	1		224.3.h.c		2
56.h	odd	2	1	CM	56.3.h.c	✓	2
56.j	odd	6	2		392.3.j.a		4
56.p	even	6	2		392.3.j.a		4
84.h	odd	2	1		2016.3.l.c		2
168.e	odd	2	1		2016.3.l.c		2
168.i	even	2	1		504.3.l.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
56.3.h.c	✓	2	1.a	even	1	1	trivial
56.3.h.c	✓	2	7.b	odd	2	1	inner
56.3.h.c	✓	2	8.b	even	2	1	inner
56.3.h.c	✓	2	56.h	odd	2	1	CM
224.3.h.c		2	4.b	odd	2	1
224.3.h.c		2	8.d	odd	2	1
224.3.h.c		2	28.d	even	2	1
224.3.h.c		2	56.e	even	2	1
392.3.j.a		4	7.c	even	3	2
392.3.j.a		4	7.d	odd	6	2
392.3.j.a		4	56.j	odd	6	2
392.3.j.a		4	56.p	even	6	2
504.3.l.a		2	3.b	odd	2	1
504.3.l.a		2	21.c	even	2	1
504.3.l.a		2	24.h	odd	2	1
504.3.l.a		2	168.i	even	2	1
2016.3.l.c		2	12.b	even	2	1
2016.3.l.c		2	24.f	even	2	1
2016.3.l.c		2	84.h	odd	2	1
2016.3.l.c		2	168.e	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 8 \) acting on \(S_{3}^{\mathrm{new}}(56, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T - 2)^{2} \)
$3$	\( T^{2} - 8 \)
$5$	\( T^{2} - 72 \)
$7$	\( (T + 7)^{2} \)
$11$	\( T^{2} \)