Newform orbit 108.2.e.a

• Label: 108.2.e.a
• Level: $108$
• Weight: $2$
• Character orbit: 108.e
• Analytic conductor: $0.862$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	108.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.862384341830\)
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:	no (minimal twist has level 36)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(55\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

37.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

0

0

1.50000

−

2.59808i

0

0.500000

+

0.866025i

0

0

0

73.1

0

0

0

1.50000

+

2.59808i

0

0.500000

−

0.866025i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	108.2.e.a		2
3.b	odd	2	1		36.2.e.a	✓	2
4.b	odd	2	1		432.2.i.c		2
5.b	even	2	1		2700.2.i.b		2
5.c	odd	4	2		2700.2.s.b		4
7.b	odd	2	1		5292.2.j.a		2
7.c	even	3	1		5292.2.i.c		2
7.c	even	3	1		5292.2.l.a		2
7.d	odd	6	1		5292.2.i.a		2
7.d	odd	6	1		5292.2.l.c		2
8.b	even	2	1		1728.2.i.d		2
8.d	odd	2	1		1728.2.i.c		2
9.c	even	3	1	inner	108.2.e.a		2
9.c	even	3	1		324.2.a.a		1
9.d	odd	6	1		36.2.e.a	✓	2
9.d	odd	6	1		324.2.a.c		1
12.b	even	2	1		144.2.i.a		2
15.d	odd	2	1		900.2.i.b		2
15.e	even	4	2		900.2.s.b		4
21.c	even	2	1		1764.2.j.b		2
21.g	even	6	1		1764.2.i.c		2
21.g	even	6	1		1764.2.l.a		2
21.h	odd	6	1		1764.2.i.a		2
21.h	odd	6	1		1764.2.l.c		2
24.f	even	2	1		576.2.i.e		2
24.h	odd	2	1		576.2.i.f		2
36.f	odd	6	1		432.2.i.c		2
36.f	odd	6	1		1296.2.a.b		1
36.h	even	6	1		144.2.i.a		2
36.h	even	6	1		1296.2.a.k		1
45.h	odd	6	1		900.2.i.b		2
45.h	odd	6	1		8100.2.a.j		1
45.j	even	6	1		2700.2.i.b		2
45.j	even	6	1		8100.2.a.g		1
45.k	odd	12	2		2700.2.s.b		4
45.k	odd	12	2		8100.2.d.c		2
45.l	even	12	2		900.2.s.b		4
45.l	even	12	2		8100.2.d.h		2
63.g	even	3	1		5292.2.i.c		2
63.h	even	3	1		5292.2.l.a		2
63.i	even	6	1		1764.2.l.a		2
63.j	odd	6	1		1764.2.l.c		2
63.k	odd	6	1		5292.2.i.a		2
63.l	odd	6	1		5292.2.j.a		2
63.n	odd	6	1		1764.2.i.a		2
63.o	even	6	1		1764.2.j.b		2
63.s	even	6	1		1764.2.i.c		2
63.t	odd	6	1		5292.2.l.c		2
72.j	odd	6	1		576.2.i.f		2
72.j	odd	6	1		5184.2.a.e		1
72.l	even	6	1		576.2.i.e		2
72.l	even	6	1		5184.2.a.f		1
72.n	even	6	1		1728.2.i.d		2
72.n	even	6	1		5184.2.a.ba		1
72.p	odd	6	1		1728.2.i.c		2
72.p	odd	6	1		5184.2.a.bb		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
36.2.e.a	✓	2	3.b	odd	2	1
36.2.e.a	✓	2	9.d	odd	6	1
108.2.e.a		2	1.a	even	1	1	trivial
108.2.e.a		2	9.c	even	3	1	inner
144.2.i.a		2	12.b	even	2	1
144.2.i.a		2	36.h	even	6	1
324.2.a.a		1	9.c	even	3	1
324.2.a.c		1	9.d	odd	6	1
432.2.i.c		2	4.b	odd	2	1
432.2.i.c		2	36.f	odd	6	1
576.2.i.e		2	24.f	even	2	1
576.2.i.e		2	72.l	even	6	1
576.2.i.f		2	24.h	odd	2	1
576.2.i.f		2	72.j	odd	6	1
900.2.i.b		2	15.d	odd	2	1
900.2.i.b		2	45.h	odd	6	1
900.2.s.b		4	15.e	even	4	2
900.2.s.b		4	45.l	even	12	2
1296.2.a.b		1	36.f	odd	6	1
1296.2.a.k		1	36.h	even	6	1
1728.2.i.c		2	8.d	odd	2	1
1728.2.i.c		2	72.p	odd	6	1
1728.2.i.d		2	8.b	even	2	1
1728.2.i.d		2	72.n	even	6	1
1764.2.i.a		2	21.h	odd	6	1
1764.2.i.a		2	63.n	odd	6	1
1764.2.i.c		2	21.g	even	6	1
1764.2.i.c		2	63.s	even	6	1
1764.2.j.b		2	21.c	even	2	1
1764.2.j.b		2	63.o	even	6	1
1764.2.l.a		2	21.g	even	6	1
1764.2.l.a		2	63.i	even	6	1
1764.2.l.c		2	21.h	odd	6	1
1764.2.l.c		2	63.j	odd	6	1
2700.2.i.b		2	5.b	even	2	1
2700.2.i.b		2	45.j	even	6	1
2700.2.s.b		4	5.c	odd	4	2
2700.2.s.b		4	45.k	odd	12	2
5184.2.a.e		1	72.j	odd	6	1
5184.2.a.f		1	72.l	even	6	1
5184.2.a.ba		1	72.n	even	6	1
5184.2.a.bb		1	72.p	odd	6	1
5292.2.i.a		2	7.d	odd	6	1
5292.2.i.a		2	63.k	odd	6	1
5292.2.i.c		2	7.c	even	3	1
5292.2.i.c		2	63.g	even	3	1
5292.2.j.a		2	7.b	odd	2	1
5292.2.j.a		2	63.l	odd	6	1
5292.2.l.a		2	7.c	even	3	1
5292.2.l.a		2	63.h	even	3	1
5292.2.l.c		2	7.d	odd	6	1
5292.2.l.c		2	63.t	odd	6	1
8100.2.a.g		1	45.j	even	6	1
8100.2.a.j		1	45.h	odd	6	1
8100.2.d.c		2	45.k	odd	12	2
8100.2.d.h		2	45.l	even	12	2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(108, [\chi])\).

Hecke characteristic polynomials

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