Newform orbit 117.2.r.a

• Label: 117.2.r.a
• Level: $117$
• Weight: $2$
• Character orbit: 117.r
• Analytic conductor: $0.934$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	117.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.934249703649\)
Analytic rank:	\(1\)
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + (\zeta_{6} - 2) q^{2} + (\zeta_{6} - 2) q^{3} + ( - \zeta_{6} + 1) q^{4} + (\zeta_{6} - 2) q^{5} + ( - 3 \zeta_{6} + 3) q^{6} + ( - 2 \zeta_{6} + 1) q^{7} + ( - 2 \zeta_{6} + 1) q^{8} + ( - 3 \zeta_{6} + 3) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} - 3 q^{3} + q^{4} - 3 q^{5} + 3 q^{6} + 3 q^{9}+O(q^{10}) \)

Character values

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

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

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

43.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−1.50000

+

0.866025i

−1.50000

+

0.866025i

0.500000

−

0.866025i

−1.50000

+

0.866025i

1.50000

−

2.59808i

−

1.73205i

−

1.73205i

1.50000

−

2.59808i

1.50000

−

2.59808i

49.1

−1.50000

−

0.866025i

−1.50000

−

0.866025i

0.500000

+

0.866025i

−1.50000

−

0.866025i

1.50000

+

2.59808i

1.73205i

1.73205i

1.50000

+

2.59808i

1.50000

+

2.59808i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
117.r	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	117.2.r.a	yes	2
3.b	odd	2	1		351.2.r.a		2
9.c	even	3	1		117.2.l.a	✓	2
9.d	odd	6	1		351.2.l.a		2
13.e	even	6	1		117.2.l.a	✓	2
39.h	odd	6	1		351.2.l.a		2
117.m	odd	6	1		351.2.r.a		2
117.r	even	6	1	inner	117.2.r.a	yes	2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
117.2.l.a	✓	2	9.c	even	3	1
117.2.l.a	✓	2	13.e	even	6	1
117.2.r.a	yes	2	1.a	even	1	1	trivial
117.2.r.a	yes	2	117.r	even	6	1	inner
351.2.l.a		2	9.d	odd	6	1
351.2.l.a		2	39.h	odd	6	1
351.2.r.a		2	3.b	odd	2	1
351.2.r.a		2	117.m	odd	6	1

Hecke kernels

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

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 3T + 3 \)
$3$	\( T^{2} + 3T + 3 \)
$5$	\( T^{2} + 3T + 3 \)
$7$	\( T^{2} + 3 \)
$11$	\( T^{2} + 6T + 12 \)