Newform orbit 56.2.e.a

• Label: 56.2.e.a
• Level: $56$
• Weight: $2$
• Character orbit: 56.e
• Analytic conductor: $0.447$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -7
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(0.447162251319\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-7}) \)
Defining polynomial:	\( x^{2} - x + 2 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{-7})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta q^{2} + (\beta - 2) q^{4} + ( - 2 \beta + 1) q^{7} + ( - \beta - 2) q^{8} + 3 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - 3 q^{4} - 5 q^{8} + 6 q^{9}+O(q^{10}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

27.1

0.500000	−	1.32288i
0.500000	+	1.32288i

0.500000

−

1.32288i

0

−1.50000

−

1.32288i

0

0

2.64575i

−2.50000

+

1.32288i

3.00000

0

27.2

0.500000

+

1.32288i

0

−1.50000

+

1.32288i

0

0

−

2.64575i

−2.50000

−

1.32288i

3.00000

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	56.2.e.a	✓	2
3.b	odd	2	1		504.2.p.a		2
4.b	odd	2	1		224.2.e.a		2
7.b	odd	2	1	CM	56.2.e.a	✓	2
7.c	even	3	2		392.2.m.a		4
7.d	odd	6	2		392.2.m.a		4
8.b	even	2	1		224.2.e.a		2
8.d	odd	2	1	inner	56.2.e.a	✓	2
12.b	even	2	1		2016.2.p.a		2
16.e	even	4	2		1792.2.f.d		4
16.f	odd	4	2		1792.2.f.d		4
21.c	even	2	1		504.2.p.a		2
24.f	even	2	1		504.2.p.a		2
24.h	odd	2	1		2016.2.p.a		2
28.d	even	2	1		224.2.e.a		2
28.f	even	6	2		1568.2.q.a		4
28.g	odd	6	2		1568.2.q.a		4
56.e	even	2	1	inner	56.2.e.a	✓	2
56.h	odd	2	1		224.2.e.a		2
56.j	odd	6	2		1568.2.q.a		4
56.k	odd	6	2		392.2.m.a		4
56.m	even	6	2		392.2.m.a		4
56.p	even	6	2		1568.2.q.a		4
84.h	odd	2	1		2016.2.p.a		2
112.j	even	4	2		1792.2.f.d		4
112.l	odd	4	2		1792.2.f.d		4
168.e	odd	2	1		504.2.p.a		2
168.i	even	2	1		2016.2.p.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
56.2.e.a	✓	2	1.a	even	1	1	trivial
56.2.e.a	✓	2	7.b	odd	2	1	CM
56.2.e.a	✓	2	8.d	odd	2	1	inner
56.2.e.a	✓	2	56.e	even	2	1	inner
224.2.e.a		2	4.b	odd	2	1
224.2.e.a		2	8.b	even	2	1
224.2.e.a		2	28.d	even	2	1
224.2.e.a		2	56.h	odd	2	1
392.2.m.a		4	7.c	even	3	2
392.2.m.a		4	7.d	odd	6	2
392.2.m.a		4	56.k	odd	6	2
392.2.m.a		4	56.m	even	6	2
504.2.p.a		2	3.b	odd	2	1
504.2.p.a		2	21.c	even	2	1
504.2.p.a		2	24.f	even	2	1
504.2.p.a		2	168.e	odd	2	1
1568.2.q.a		4	28.f	even	6	2
1568.2.q.a		4	28.g	odd	6	2
1568.2.q.a		4	56.j	odd	6	2
1568.2.q.a		4	56.p	even	6	2
1792.2.f.d		4	16.e	even	4	2
1792.2.f.d		4	16.f	odd	4	2
1792.2.f.d		4	112.j	even	4	2
1792.2.f.d		4	112.l	odd	4	2
2016.2.p.a		2	12.b	even	2	1
2016.2.p.a		2	24.h	odd	2	1
2016.2.p.a		2	84.h	odd	2	1
2016.2.p.a		2	168.i	even	2	1

Hecke kernels

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

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - T + 2 \)
$3$	\( T^{2} \)
$5$	\( T^{2} \)
$7$	\( T^{2} + 7 \)
$11$	\( (T + 4)^{2} \)