Newform orbit 117.4.g.b

• Label: 117.4.g.b
• Level: $117$
• Weight: $4$
• Character orbit: 117.g
• Analytic conductor: $6.903$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	117.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.90322347067\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 39)
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.

\(f(q)\)	\(=\)	\( q + ( - 3 \zeta_{6} + 3) q^{2} - \zeta_{6} q^{4} + 9 q^{5} - 2 \zeta_{6} q^{7} + 21 q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} - q^{4} + 18 q^{5} - 2 q^{7} + 42 q^{8}+O(q^{10}) \)

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

1.50000

+

2.59808i

0

−0.500000

+

0.866025i

9.00000

0

−1.00000

+

1.73205i

21.0000

0

13.5000

+

23.3827i

100.1

1.50000

−

2.59808i

0

−0.500000

−

0.866025i

9.00000

0

−1.00000

−

1.73205i

21.0000

0

13.5000

−

23.3827i

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	117.4.g.b		2
3.b	odd	2	1		39.4.e.a	✓	2
12.b	even	2	1		624.4.q.b		2
13.c	even	3	1	inner	117.4.g.b		2
13.c	even	3	1		1521.4.a.c		1
13.e	even	6	1		1521.4.a.j		1
39.h	odd	6	1		507.4.a.a		1
39.i	odd	6	1		39.4.e.a	✓	2
39.i	odd	6	1		507.4.a.e		1
39.k	even	12	2		507.4.b.c		2
156.p	even	6	1		624.4.q.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
39.4.e.a	✓	2	3.b	odd	2	1
39.4.e.a	✓	2	39.i	odd	6	1
117.4.g.b		2	1.a	even	1	1	trivial
117.4.g.b		2	13.c	even	3	1	inner
507.4.a.a		1	39.h	odd	6	1
507.4.a.e		1	39.i	odd	6	1
507.4.b.c		2	39.k	even	12	2
624.4.q.b		2	12.b	even	2	1
624.4.q.b		2	156.p	even	6	1
1521.4.a.c		1	13.c	even	3	1
1521.4.a.j		1	13.e	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 3T_{2} + 9 \) acting on \(S_{4}^{\mathrm{new}}(117, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 3T + 9 \)
$3$	\( T^{2} \)
$5$	\( (T - 9)^{2} \)
$7$	\( T^{2} + 2T + 4 \)
$11$	\( T^{2} - 30T + 900 \)