Newform orbit 1960.2.q.h

• Label: 1960.2.q.h
• Level: $1960$
• Weight: $2$
• Character orbit: 1960.q
• Analytic conductor: $15.651$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1960.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(15.6506787962\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 40)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1960\mathbb{Z}\right)^\times\).

\(n\)	\(981\)	\(1081\)	\(1177\)	\(1471\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}\)	\(1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

361.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

0

0

−0.500000

−

0.866025i

0

0

0

1.50000

+

2.59808i

0

961.1

0

0

0

−0.500000

+

0.866025i

0

0

0

1.50000

−

2.59808i

0

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	1960.2.q.h		2
7.b	odd	2	1		1960.2.q.i		2
7.c	even	3	1		40.2.a.a	✓	1
7.c	even	3	1	inner	1960.2.q.h		2
7.d	odd	6	1		1960.2.a.g		1
7.d	odd	6	1		1960.2.q.i		2
21.h	odd	6	1		360.2.a.a		1
28.f	even	6	1		3920.2.a.s		1
28.g	odd	6	1		80.2.a.a		1
35.i	odd	6	1		9800.2.a.x		1
35.j	even	6	1		200.2.a.c		1
35.l	odd	12	2		200.2.c.b		2
56.k	odd	6	1		320.2.a.d		1
56.p	even	6	1		320.2.a.c		1
63.g	even	3	1		3240.2.q.k		2
63.h	even	3	1		3240.2.q.k		2
63.j	odd	6	1		3240.2.q.x		2
63.n	odd	6	1		3240.2.q.x		2
77.h	odd	6	1		4840.2.a.f		1
84.n	even	6	1		720.2.a.e		1
91.r	even	6	1		6760.2.a.i		1
105.o	odd	6	1		1800.2.a.v		1
105.x	even	12	2		1800.2.f.a		2
112.u	odd	12	2		1280.2.d.a		2
112.w	even	12	2		1280.2.d.j		2
140.p	odd	6	1		400.2.a.e		1
140.w	even	12	2		400.2.c.d		2
168.s	odd	6	1		2880.2.a.t		1
168.v	even	6	1		2880.2.a.bg		1
280.bf	even	6	1		1600.2.a.o		1
280.bi	odd	6	1		1600.2.a.k		1
280.br	even	12	2		1600.2.c.m		2
280.bt	odd	12	2		1600.2.c.k		2
308.n	even	6	1		9680.2.a.q		1
420.ba	even	6	1		3600.2.a.h		1
420.bp	odd	12	2		3600.2.f.t		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
40.2.a.a	✓	1	7.c	even	3	1
80.2.a.a		1	28.g	odd	6	1
200.2.a.c		1	35.j	even	6	1
200.2.c.b		2	35.l	odd	12	2
320.2.a.c		1	56.p	even	6	1
320.2.a.d		1	56.k	odd	6	1
360.2.a.a		1	21.h	odd	6	1
400.2.a.e		1	140.p	odd	6	1
400.2.c.d		2	140.w	even	12	2
720.2.a.e		1	84.n	even	6	1
1280.2.d.a		2	112.u	odd	12	2
1280.2.d.j		2	112.w	even	12	2
1600.2.a.k		1	280.bi	odd	6	1
1600.2.a.o		1	280.bf	even	6	1
1600.2.c.k		2	280.bt	odd	12	2
1600.2.c.m		2	280.br	even	12	2
1800.2.a.v		1	105.o	odd	6	1
1800.2.f.a		2	105.x	even	12	2
1960.2.a.g		1	7.d	odd	6	1
1960.2.q.h		2	1.a	even	1	1	trivial
1960.2.q.h		2	7.c	even	3	1	inner
1960.2.q.i		2	7.b	odd	2	1
1960.2.q.i		2	7.d	odd	6	1
2880.2.a.t		1	168.s	odd	6	1
2880.2.a.bg		1	168.v	even	6	1
3240.2.q.k		2	63.g	even	3	1
3240.2.q.k		2	63.h	even	3	1
3240.2.q.x		2	63.j	odd	6	1
3240.2.q.x		2	63.n	odd	6	1
3600.2.a.h		1	420.ba	even	6	1
3600.2.f.t		2	420.bp	odd	12	2
3920.2.a.s		1	28.f	even	6	1
4840.2.a.f		1	77.h	odd	6	1
6760.2.a.i		1	91.r	even	6	1
9680.2.a.q		1	308.n	even	6	1
9800.2.a.x		1	35.i	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1960, [\chi])\):

\( T_{3} \)

\( T_{11}^{2} + 4T_{11} + 16 \)

\( T_{13} + 2 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} + T + 1 \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 4T + 16 \)