Newform orbit 117.2.g.a

• Label: 117.2.g.a
• Level: $117$
• Weight: $2$
• Character orbit: 117.g
• Analytic conductor: $0.934$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	117.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.934249703649\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{3}]$

$q$-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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\)	\(28\)	\(92\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

55.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

0

1.00000

−

1.73205i

0

0

0.500000

−

0.866025i

0

0

0

100.1

0

0

1.00000

+

1.73205i

0

0

0.500000

+

0.866025i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
13.c	even	3	1	inner
39.i	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	117.2.g.a	✓	2
3.b	odd	2	1	CM	117.2.g.a	✓	2
4.b	odd	2	1		1872.2.t.g		2
12.b	even	2	1		1872.2.t.g		2
13.c	even	3	1	inner	117.2.g.a	✓	2
13.c	even	3	1		1521.2.a.b		1
13.e	even	6	1		1521.2.a.c		1
13.f	odd	12	2		1521.2.b.e		2
39.h	odd	6	1		1521.2.a.c		1
39.i	odd	6	1	inner	117.2.g.a	✓	2
39.i	odd	6	1		1521.2.a.b		1
39.k	even	12	2		1521.2.b.e		2
52.j	odd	6	1		1872.2.t.g		2
156.p	even	6	1		1872.2.t.g		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
117.2.g.a	✓	2	1.a	even	1	1	trivial
117.2.g.a	✓	2	3.b	odd	2	1	CM
117.2.g.a	✓	2	13.c	even	3	1	inner
117.2.g.a	✓	2	39.i	odd	6	1	inner
1521.2.a.b		1	13.c	even	3	1
1521.2.a.b		1	39.i	odd	6	1
1521.2.a.c		1	13.e	even	6	1
1521.2.a.c		1	39.h	odd	6	1
1521.2.b.e		2	13.f	odd	12	2
1521.2.b.e		2	39.k	even	12	2
1872.2.t.g		2	4.b	odd	2	1
1872.2.t.g		2	12.b	even	2	1
1872.2.t.g		2	52.j	odd	6	1
1872.2.t.g		2	156.p	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(117, [\chi])\).

Hecke characteristic polynomials

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