Newform orbit 91.2.g.a

• Label: 91.2.g.a
• Level: $91$
• Weight: $2$
• Character orbit: 91.g
• Analytic conductor: $0.727$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	91.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.726638658394\)
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:	$\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\)	\(=\)	\( q - \zeta_{6} q^{2} - 3 q^{3} + ( - \zeta_{6} + 1) q^{4} + (3 \zeta_{6} - 3) q^{5} + 3 \zeta_{6} q^{6} + (\zeta_{6} - 3) q^{7} - 3 q^{8} + 6 q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 6 q^{3} + q^{4} - 3 q^{5} + 3 q^{6} - 5 q^{7} - 6 q^{8} + 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(-1 + \zeta_{6}\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

9.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−0.500000

−

0.866025i

−3.00000

0.500000

−

0.866025i

−1.50000

+

2.59808i

1.50000

+

2.59808i

−2.50000

+

0.866025i

−3.00000

6.00000

3.00000

81.1

−0.500000

+

0.866025i

−3.00000

0.500000

+

0.866025i

−1.50000

−

2.59808i

1.50000

−

2.59808i

−2.50000

−

0.866025i

−3.00000

6.00000

3.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
91.g	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	91.2.g.a	✓	2
3.b	odd	2	1		819.2.n.c		2
7.b	odd	2	1		637.2.g.a		2
7.c	even	3	1		91.2.h.a	yes	2
7.c	even	3	1		637.2.f.b		2
7.d	odd	6	1		637.2.f.a		2
7.d	odd	6	1		637.2.h.a		2
13.c	even	3	1		91.2.h.a	yes	2
13.c	even	3	1		1183.2.e.a		2
13.e	even	6	1		1183.2.e.c		2
21.h	odd	6	1		819.2.s.a		2
39.i	odd	6	1		819.2.s.a		2
91.g	even	3	1	inner	91.2.g.a	✓	2
91.g	even	3	1		8281.2.a.i		1
91.h	even	3	1		637.2.f.b		2
91.h	even	3	1		1183.2.e.a		2
91.k	even	6	1		1183.2.e.c		2
91.m	odd	6	1		637.2.g.a		2
91.m	odd	6	1		8281.2.a.j		1
91.n	odd	6	1		637.2.h.a		2
91.p	odd	6	1		8281.2.a.g		1
91.u	even	6	1		8281.2.a.c		1
91.v	odd	6	1		637.2.f.a		2
273.bm	odd	6	1		819.2.n.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
91.2.g.a	✓	2	1.a	even	1	1	trivial
91.2.g.a	✓	2	91.g	even	3	1	inner
91.2.h.a	yes	2	7.c	even	3	1
91.2.h.a	yes	2	13.c	even	3	1
637.2.f.a		2	7.d	odd	6	1
637.2.f.a		2	91.v	odd	6	1
637.2.f.b		2	7.c	even	3	1
637.2.f.b		2	91.h	even	3	1
637.2.g.a		2	7.b	odd	2	1
637.2.g.a		2	91.m	odd	6	1
637.2.h.a		2	7.d	odd	6	1
637.2.h.a		2	91.n	odd	6	1
819.2.n.c		2	3.b	odd	2	1
819.2.n.c		2	273.bm	odd	6	1
819.2.s.a		2	21.h	odd	6	1
819.2.s.a		2	39.i	odd	6	1
1183.2.e.a		2	13.c	even	3	1
1183.2.e.a		2	91.h	even	3	1
1183.2.e.c		2	13.e	even	6	1
1183.2.e.c		2	91.k	even	6	1
8281.2.a.c		1	91.u	even	6	1
8281.2.a.g		1	91.p	odd	6	1
8281.2.a.i		1	91.g	even	3	1
8281.2.a.j		1	91.m	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(91, [\chi])\).

Hecke characteristic polynomials

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