Newform orbit 1520.2.q.e

• Label: 1520.2.q.e
• Level: $1520$
• Weight: $2$
• Character orbit: 1520.q
• Analytic conductor: $12.137$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.1372611072\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 190)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\)	\(191\)	\(401\)	\(1141\)	\(1217\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}\)	\(1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

881.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

−0.500000

−

0.866025i

0

0.500000

+

0.866025i

0

−2.00000

0

1.00000

−

1.73205i

0

961.1

0

−0.500000

+

0.866025i

0

0.500000

−

0.866025i

0

−2.00000

0

1.00000

+

1.73205i

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1520.2.q.e		2
4.b	odd	2	1		190.2.e.b	✓	2
12.b	even	2	1		1710.2.l.b		2
19.c	even	3	1	inner	1520.2.q.e		2
20.d	odd	2	1		950.2.e.a		2
20.e	even	4	2		950.2.j.a		4
76.f	even	6	1		3610.2.a.i		1
76.g	odd	6	1		190.2.e.b	✓	2
76.g	odd	6	1		3610.2.a.a		1
228.m	even	6	1		1710.2.l.b		2
380.p	odd	6	1		950.2.e.a		2
380.v	even	12	2		950.2.j.a		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
190.2.e.b	✓	2	4.b	odd	2	1
190.2.e.b	✓	2	76.g	odd	6	1
950.2.e.a		2	20.d	odd	2	1
950.2.e.a		2	380.p	odd	6	1
950.2.j.a		4	20.e	even	4	2
950.2.j.a		4	380.v	even	12	2
1520.2.q.e		2	1.a	even	1	1	trivial
1520.2.q.e		2	19.c	even	3	1	inner
1710.2.l.b		2	12.b	even	2	1
1710.2.l.b		2	228.m	even	6	1
3610.2.a.a		1	76.g	odd	6	1
3610.2.a.i		1	76.f	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1520, [\chi])\):

\( T_{3}^{2} + T_{3} + 1 \)

\( T_{7} + 2 \)

Hecke characteristic polynomials

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