Newform orbit 1078.2.e.c

• Label: 1078.2.e.c
• Level: $1078$
• Weight: $2$
• Character orbit: 1078.e
• Analytic conductor: $8.608$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1078.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.60787333789\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 154)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1078\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(981\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

67.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−0.500000

−

0.866025i

0

−0.500000

+

0.866025i

1.00000

+

1.73205i

0

0

1.00000

1.50000

+

2.59808i

1.00000

−

1.73205i

177.1

−0.500000

+

0.866025i

0

−0.500000

−

0.866025i

1.00000

−

1.73205i

0

0

1.00000

1.50000

−

2.59808i

1.00000

+

1.73205i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1078.2.e.c		2
7.b	odd	2	1		1078.2.e.b		2
7.c	even	3	1		1078.2.a.j		1
7.c	even	3	1	inner	1078.2.e.c		2
7.d	odd	6	1		154.2.a.c	✓	1
7.d	odd	6	1		1078.2.e.b		2
21.g	even	6	1		1386.2.a.b		1
21.h	odd	6	1		9702.2.a.v		1
28.f	even	6	1		1232.2.a.h		1
28.g	odd	6	1		8624.2.a.o		1
35.i	odd	6	1		3850.2.a.f		1
35.k	even	12	2		3850.2.c.l		2
56.j	odd	6	1		4928.2.a.n		1
56.m	even	6	1		4928.2.a.o		1
77.i	even	6	1		1694.2.a.c		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
154.2.a.c	✓	1	7.d	odd	6	1
1078.2.a.j		1	7.c	even	3	1
1078.2.e.b		2	7.b	odd	2	1
1078.2.e.b		2	7.d	odd	6	1
1078.2.e.c		2	1.a	even	1	1	trivial
1078.2.e.c		2	7.c	even	3	1	inner
1232.2.a.h		1	28.f	even	6	1
1386.2.a.b		1	21.g	even	6	1
1694.2.a.c		1	77.i	even	6	1
3850.2.a.f		1	35.i	odd	6	1
3850.2.c.l		2	35.k	even	12	2
4928.2.a.n		1	56.j	odd	6	1
4928.2.a.o		1	56.m	even	6	1
8624.2.a.o		1	28.g	odd	6	1
9702.2.a.v		1	21.h	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1078, [\chi])\):

\( T_{3} \)

\( T_{5}^{2} - 2T_{5} + 4 \)

\( T_{13} + 2 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + T + 1 \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 2T + 4 \)
$7$	\( T^{2} \)
$11$	\( T^{2} - T + 1 \)