Newform orbit 1710.2.l.b

• Label: 1710.2.l.b
• Level: $1710$
• Weight: $2$
• Character orbit: 1710.l
• Analytic conductor: $13.654$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1710.l (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.6544187456\)
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 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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1710\mathbb{Z}\right)^\times\).

\(n\)	\(191\)	\(1027\)	\(1351\)
\(\chi(n)\)	\(1\)	\(1\)	\(-\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1261.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−0.500000

−

0.866025i

0

−0.500000

+

0.866025i

−0.500000

−

0.866025i

0

2.00000

1.00000

0

−0.500000

+

0.866025i

1531.1

−0.500000

+

0.866025i

0

−0.500000

−

0.866025i

−0.500000

+

0.866025i

0

2.00000

1.00000

0

−0.500000

−

0.866025i

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	1710.2.l.b		2
3.b	odd	2	1		190.2.e.b	✓	2
12.b	even	2	1		1520.2.q.e		2
15.d	odd	2	1		950.2.e.a		2
15.e	even	4	2		950.2.j.a		4
19.c	even	3	1	inner	1710.2.l.b		2
57.f	even	6	1		3610.2.a.i		1
57.h	odd	6	1		190.2.e.b	✓	2
57.h	odd	6	1		3610.2.a.a		1
228.m	even	6	1		1520.2.q.e		2
285.n	odd	6	1		950.2.e.a		2
285.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	3.b	odd	2	1
190.2.e.b	✓	2	57.h	odd	6	1
950.2.e.a		2	15.d	odd	2	1
950.2.e.a		2	285.n	odd	6	1
950.2.j.a		4	15.e	even	4	2
950.2.j.a		4	285.v	even	12	2
1520.2.q.e		2	12.b	even	2	1
1520.2.q.e		2	228.m	even	6	1
1710.2.l.b		2	1.a	even	1	1	trivial
1710.2.l.b		2	19.c	even	3	1	inner
3610.2.a.a		1	57.h	odd	6	1
3610.2.a.i		1	57.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 \(S_{2}^{\mathrm{new}}(1710, [\chi])\):

\( T_{7} - 2 \)

\( T_{11} - 3 \)

Hecke characteristic polynomials

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