Newform orbit 1014.2.e.b

• Label: 1014.2.e.b
• Level: $1014$
• Weight: $2$
• Character orbit: 1014.e
• Analytic conductor: $8.097$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1014.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.09683076496\)
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 78)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).

\(n\)	\(677\)	\(847\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}\)

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

−0.500000

+

0.866025i

−0.500000

+

0.866025i

−0.500000

−

0.866025i

2.00000

−0.500000

−

0.866025i

−1.00000

−

1.73205i

1.00000

−0.500000

−

0.866025i

−1.00000

+

1.73205i

991.1

−0.500000

−

0.866025i

−0.500000

−

0.866025i

−0.500000

+

0.866025i

2.00000

−0.500000

+

0.866025i

−1.00000

+

1.73205i

1.00000

−0.500000

+

0.866025i

−1.00000

−

1.73205i

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	1014.2.e.b		2
13.b	even	2	1		1014.2.e.e		2
13.c	even	3	1		1014.2.a.g		1
13.c	even	3	1	inner	1014.2.e.b		2
13.d	odd	4	2		1014.2.i.c		4
13.e	even	6	1		1014.2.a.b		1
13.e	even	6	1		1014.2.e.e		2
13.f	odd	12	2		78.2.b.a	✓	2
13.f	odd	12	2		1014.2.i.c		4
39.h	odd	6	1		3042.2.a.n		1
39.i	odd	6	1		3042.2.a.c		1
39.k	even	12	2		234.2.b.a		2
52.i	odd	6	1		8112.2.a.g		1
52.j	odd	6	1		8112.2.a.j		1
52.l	even	12	2		624.2.c.a		2
65.o	even	12	2		1950.2.f.g		2
65.s	odd	12	2		1950.2.b.c		2
65.t	even	12	2		1950.2.f.d		2
91.bc	even	12	2		3822.2.c.d		2
104.u	even	12	2		2496.2.c.m		2
104.x	odd	12	2		2496.2.c.f		2
156.v	odd	12	2		1872.2.c.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
78.2.b.a	✓	2	13.f	odd	12	2
234.2.b.a		2	39.k	even	12	2
624.2.c.a		2	52.l	even	12	2
1014.2.a.b		1	13.e	even	6	1
1014.2.a.g		1	13.c	even	3	1
1014.2.e.b		2	1.a	even	1	1	trivial
1014.2.e.b		2	13.c	even	3	1	inner
1014.2.e.e		2	13.b	even	2	1
1014.2.e.e		2	13.e	even	6	1
1014.2.i.c		4	13.d	odd	4	2
1014.2.i.c		4	13.f	odd	12	2
1872.2.c.b		2	156.v	odd	12	2
1950.2.b.c		2	65.s	odd	12	2
1950.2.f.d		2	65.t	even	12	2
1950.2.f.g		2	65.o	even	12	2
2496.2.c.f		2	104.x	odd	12	2
2496.2.c.m		2	104.u	even	12	2
3042.2.a.c		1	39.i	odd	6	1
3042.2.a.n		1	39.h	odd	6	1
3822.2.c.d		2	91.bc	even	12	2
8112.2.a.g		1	52.i	odd	6	1
8112.2.a.j		1	52.j	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1014, [\chi])\):

\( T_{5} - 2 \)

\( T_{7}^{2} + 2T_{7} + 4 \)

Hecke characteristic polynomials

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