Newform orbit 676.2.e.c

• Label: 676.2.e.c
• Level: $676$
• Weight: $2$
• Character orbit: 676.e
• Analytic conductor: $5.398$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 676 = 2^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	676.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.39788717664\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 52)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/676\mathbb{Z}\right)^\times\).

\(n\)	\(339\)	\(509\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

529.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

0

0

2.00000

0

1.00000

+

1.73205i

0

1.50000

+

2.59808i

0

653.1

0

0

0

2.00000

0

1.00000

−

1.73205i

0

1.50000

−

2.59808i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	676.2.e.c		2
13.b	even	2	1		676.2.e.b		2
13.c	even	3	1		52.2.a.a	✓	1
13.c	even	3	1	inner	676.2.e.c		2
13.d	odd	4	2		676.2.h.c		4
13.e	even	6	1		676.2.a.c		1
13.e	even	6	1		676.2.e.b		2
13.f	odd	12	2		676.2.d.c		2
13.f	odd	12	2		676.2.h.c		4
39.h	odd	6	1		6084.2.a.m		1
39.i	odd	6	1		468.2.a.b		1
39.k	even	12	2		6084.2.b.m		2
52.i	odd	6	1		2704.2.a.g		1
52.j	odd	6	1		208.2.a.c		1
52.l	even	12	2		2704.2.f.f		2
65.n	even	6	1		1300.2.a.d		1
65.q	odd	12	2		1300.2.c.c		2
91.g	even	3	1		2548.2.j.e		2
91.h	even	3	1		2548.2.j.e		2
91.m	odd	6	1		2548.2.j.f		2
91.n	odd	6	1		2548.2.a.e		1
91.v	odd	6	1		2548.2.j.f		2
104.n	odd	6	1		832.2.a.f		1
104.r	even	6	1		832.2.a.e		1
117.f	even	3	1		4212.2.i.d		2
117.h	even	3	1		4212.2.i.d		2
117.k	odd	6	1		4212.2.i.i		2
117.u	odd	6	1		4212.2.i.i		2
143.k	odd	6	1		6292.2.a.g		1
156.p	even	6	1		1872.2.a.f		1
208.bg	odd	12	2		3328.2.b.e		2
208.bj	even	12	2		3328.2.b.q		2
260.v	odd	6	1		5200.2.a.q		1
312.bh	odd	6	1		7488.2.a.bn		1
312.bn	even	6	1		7488.2.a.bw		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
52.2.a.a	✓	1	13.c	even	3	1
208.2.a.c		1	52.j	odd	6	1
468.2.a.b		1	39.i	odd	6	1
676.2.a.c		1	13.e	even	6	1
676.2.d.c		2	13.f	odd	12	2
676.2.e.b		2	13.b	even	2	1
676.2.e.b		2	13.e	even	6	1
676.2.e.c		2	1.a	even	1	1	trivial
676.2.e.c		2	13.c	even	3	1	inner
676.2.h.c		4	13.d	odd	4	2
676.2.h.c		4	13.f	odd	12	2
832.2.a.e		1	104.r	even	6	1
832.2.a.f		1	104.n	odd	6	1
1300.2.a.d		1	65.n	even	6	1
1300.2.c.c		2	65.q	odd	12	2
1872.2.a.f		1	156.p	even	6	1
2548.2.a.e		1	91.n	odd	6	1
2548.2.j.e		2	91.g	even	3	1
2548.2.j.e		2	91.h	even	3	1
2548.2.j.f		2	91.m	odd	6	1
2548.2.j.f		2	91.v	odd	6	1
2704.2.a.g		1	52.i	odd	6	1
2704.2.f.f		2	52.l	even	12	2
3328.2.b.e		2	208.bg	odd	12	2
3328.2.b.q		2	208.bj	even	12	2
4212.2.i.d		2	117.f	even	3	1
4212.2.i.d		2	117.h	even	3	1
4212.2.i.i		2	117.k	odd	6	1
4212.2.i.i		2	117.u	odd	6	1
5200.2.a.q		1	260.v	odd	6	1
6084.2.a.m		1	39.h	odd	6	1
6084.2.b.m		2	39.k	even	12	2
6292.2.a.g		1	143.k	odd	6	1
7488.2.a.bn		1	312.bh	odd	6	1
7488.2.a.bw		1	312.bn	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(676, [\chi])\):

\( T_{3} \)

\( T_{5} - 2 \)

Hecke characteristic polynomials

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