Newform orbit 1170.2.i.c

• Label: 1170.2.i.c
• Level: $1170$
• Weight: $2$
• Character orbit: 1170.i
• Analytic conductor: $9.342$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1170.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.34249703649\)
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 390)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times\).

\(n\)	\(911\)	\(937\)	\(1081\)
\(\chi(n)\)	\(1\)	\(1\)	\(-\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

451.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−0.500000

+

0.866025i

0

−0.500000

−

0.866025i

1.00000

0

−1.50000

−

2.59808i

1.00000

0

−0.500000

+

0.866025i

991.1

−0.500000

−

0.866025i

0

−0.500000

+

0.866025i

1.00000

0

−1.50000

+

2.59808i

1.00000

0

−0.500000

−

0.866025i

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	1170.2.i.c		2
3.b	odd	2	1		390.2.i.e	✓	2
13.c	even	3	1	inner	1170.2.i.c		2
15.d	odd	2	1		1950.2.i.f		2
15.e	even	4	2		1950.2.z.d		4
39.h	odd	6	1		5070.2.a.r		1
39.i	odd	6	1		390.2.i.e	✓	2
39.i	odd	6	1		5070.2.a.b		1
39.k	even	12	2		5070.2.b.b		2
195.x	odd	6	1		1950.2.i.f		2
195.bl	even	12	2		1950.2.z.d		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
390.2.i.e	✓	2	3.b	odd	2	1
390.2.i.e	✓	2	39.i	odd	6	1
1170.2.i.c		2	1.a	even	1	1	trivial
1170.2.i.c		2	13.c	even	3	1	inner
1950.2.i.f		2	15.d	odd	2	1
1950.2.i.f		2	195.x	odd	6	1
1950.2.z.d		4	15.e	even	4	2
1950.2.z.d		4	195.bl	even	12	2
5070.2.a.b		1	39.i	odd	6	1
5070.2.a.r		1	39.h	odd	6	1
5070.2.b.b		2	39.k	even	12	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1170, [\chi])\):

\( T_{7}^{2} + 3T_{7} + 9 \)

\( T_{11}^{2} + 3T_{11} + 9 \)

\( T_{29}^{2} + 4T_{29} + 16 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + T + 1 \)
$3$	\( T^{2} \)
$5$	\( (T - 1)^{2} \)
$7$	\( T^{2} + 3T + 9 \)
$11$	\( T^{2} + 3T + 9 \)