Newform orbit 1014.2.b.e

• Label: 1014.2.b.e
• Level: $1014$
• Weight: $2$
• Character orbit: 1014.b
• Analytic conductor: $8.097$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1014.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.09683076496\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\zeta_{12})\)
Defining polynomial:	\( x^{4} - x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 78)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + q^{3} - q^{4} + (\beta_{2} + 2 \beta_1) q^{5} + \beta_1 q^{6} + (\beta_{2} + \beta_1) q^{7} - \beta_1 q^{8} + q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) \)

Basis of coefficient ring

\(\beta_{1}\)	\(=\)	\( \zeta_{12}^{3} \)
\(\beta_{2}\)	\(=\)	\( 2\zeta_{12}^{2} - 1 \)
\(\beta_{3}\)	\(=\)	\( -\zeta_{12}^{3} + 2\zeta_{12} \)

Character values

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

\(n\)	\(677\)	\(847\)
\(\chi(n)\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

337.1

0.866025	−	0.500000i
−0.866025	−	0.500000i
−0.866025	+	0.500000i
0.866025	+	0.500000i

−

1.00000i

1.00000

−1.00000

−

3.73205i

−

1.00000i

−

2.73205i

1.00000i

1.00000

−3.73205

337.2

−

1.00000i

1.00000

−1.00000

−

0.267949i

−

1.00000i

0.732051i

1.00000i

1.00000

−0.267949

337.3

1.00000i

1.00000

−1.00000

0.267949i

1.00000i

−

0.732051i

−

1.00000i

1.00000

−0.267949

337.4

1.00000i

1.00000

−1.00000

3.73205i

1.00000i

2.73205i

−

1.00000i

1.00000

−3.73205

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1014.2.b.e		4
3.b	odd	2	1		3042.2.b.i		4
13.b	even	2	1	inner	1014.2.b.e		4
13.c	even	3	1		78.2.i.a	✓	4
13.c	even	3	1		1014.2.i.a		4
13.d	odd	4	1		1014.2.a.i		2
13.d	odd	4	1		1014.2.a.k		2
13.e	even	6	1		78.2.i.a	✓	4
13.e	even	6	1		1014.2.i.a		4
13.f	odd	12	2		1014.2.e.g		4
13.f	odd	12	2		1014.2.e.i		4
39.d	odd	2	1		3042.2.b.i		4
39.f	even	4	1		3042.2.a.p		2
39.f	even	4	1		3042.2.a.y		2
39.h	odd	6	1		234.2.l.c		4
39.i	odd	6	1		234.2.l.c		4
52.f	even	4	1		8112.2.a.bj		2
52.f	even	4	1		8112.2.a.bp		2
52.i	odd	6	1		624.2.bv.e		4
52.j	odd	6	1		624.2.bv.e		4
65.l	even	6	1		1950.2.bc.d		4
65.n	even	6	1		1950.2.bc.d		4
65.q	odd	12	1		1950.2.y.b		4
65.q	odd	12	1		1950.2.y.g		4
65.r	odd	12	1		1950.2.y.b		4
65.r	odd	12	1		1950.2.y.g		4
156.p	even	6	1		1872.2.by.h		4
156.r	even	6	1		1872.2.by.h		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
78.2.i.a	✓	4	13.c	even	3	1
78.2.i.a	✓	4	13.e	even	6	1
234.2.l.c		4	39.h	odd	6	1
234.2.l.c		4	39.i	odd	6	1
624.2.bv.e		4	52.i	odd	6	1
624.2.bv.e		4	52.j	odd	6	1
1014.2.a.i		2	13.d	odd	4	1
1014.2.a.k		2	13.d	odd	4	1
1014.2.b.e		4	1.a	even	1	1	trivial
1014.2.b.e		4	13.b	even	2	1	inner
1014.2.e.g		4	13.f	odd	12	2
1014.2.e.i		4	13.f	odd	12	2
1014.2.i.a		4	13.c	even	3	1
1014.2.i.a		4	13.e	even	6	1
1872.2.by.h		4	156.p	even	6	1
1872.2.by.h		4	156.r	even	6	1
1950.2.y.b		4	65.q	odd	12	1
1950.2.y.b		4	65.r	odd	12	1
1950.2.y.g		4	65.q	odd	12	1
1950.2.y.g		4	65.r	odd	12	1
1950.2.bc.d		4	65.l	even	6	1
1950.2.bc.d		4	65.n	even	6	1
3042.2.a.p		2	39.f	even	4	1
3042.2.a.y		2	39.f	even	4	1
3042.2.b.i		4	3.b	odd	2	1
3042.2.b.i		4	39.d	odd	2	1
8112.2.a.bj		2	52.f	even	4	1
8112.2.a.bp		2	52.f	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 14T_{5}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1014, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} + 1)^{2} \)
$3$	\( (T - 1)^{4} \)
$5$	\( T^{4} + 14T^{2} + 1 \)
$7$	\( T^{4} + 8T^{2} + 4 \)
$11$	\( T^{4} + 24T^{2} + 36 \)