Newform orbit 91.2.f.b

• Label: 91.2.f.b
• Level: $91$
• Weight: $2$
• Character orbit: 91.f
• Analytic conductor: $0.727$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(0.726638658394\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\zeta_{12})\)
Defining polynomial:	\( x^{4} - x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta_{2} q^{2} + ( - \beta_{2} - \beta_1) q^{3} + (\beta_1 - 1) q^{4} + \beta_{3} q^{5} + ( - \beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{6} + (\beta_1 - 1) q^{7} - \beta_{3} q^{8} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 2 q^{4} - 6 q^{6} - 2 q^{7} - 2 q^{9}+O(q^{10}) \)

Basis of coefficient ring

\(\beta_{1}\)	\(=\)	\( \zeta_{12}^{2} \)
\(\beta_{2}\)	\(=\)	\( \zeta_{12}^{3} + \zeta_{12} \)
\(\beta_{3}\)	\(=\)	\( -\zeta_{12}^{3} + 2\zeta_{12} \)

Character values

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

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

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

22.1

0.866025	−	0.500000i
−0.866025	+	0.500000i
0.866025	+	0.500000i
−0.866025	−	0.500000i

−0.866025

+

1.50000i

−1.36603

+

2.36603i

−0.500000

−

0.866025i

1.73205

−2.36603

−

4.09808i

−0.500000

−

0.866025i

−1.73205

−2.23205

−

3.86603i

−1.50000

+

2.59808i

22.2

0.866025

−

1.50000i

0.366025

−

0.633975i

−0.500000

−

0.866025i

−1.73205

−0.633975

−

1.09808i

−0.500000

−

0.866025i

1.73205

1.23205

+

2.13397i

−1.50000

+

2.59808i

29.1

−0.866025

−

1.50000i

−1.36603

−

2.36603i

−0.500000

+

0.866025i

1.73205

−2.36603

+

4.09808i

−0.500000

+

0.866025i

−1.73205

−2.23205

+

3.86603i

−1.50000

−

2.59808i

29.2

0.866025

+

1.50000i

0.366025

+

0.633975i

−0.500000

+

0.866025i

−1.73205

−0.633975

+

1.09808i

−0.500000

+

0.866025i

1.73205

1.23205

−

2.13397i

−1.50000

−

2.59808i

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	91.2.f.b	✓	4
3.b	odd	2	1		819.2.o.b		4
4.b	odd	2	1		1456.2.s.o		4
7.b	odd	2	1		637.2.f.d		4
7.c	even	3	1		637.2.g.e		4
7.c	even	3	1		637.2.h.d		4
7.d	odd	6	1		637.2.g.d		4
7.d	odd	6	1		637.2.h.e		4
13.c	even	3	1	inner	91.2.f.b	✓	4
13.c	even	3	1		1183.2.a.f		2
13.e	even	6	1		1183.2.a.e		2
13.f	odd	12	2		1183.2.c.e		4
39.i	odd	6	1		819.2.o.b		4
52.j	odd	6	1		1456.2.s.o		4
91.g	even	3	1		637.2.h.d		4
91.h	even	3	1		637.2.g.e		4
91.m	odd	6	1		637.2.h.e		4
91.n	odd	6	1		637.2.f.d		4
91.n	odd	6	1		8281.2.a.r		2
91.t	odd	6	1		8281.2.a.t		2
91.v	odd	6	1		637.2.g.d		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
91.2.f.b	✓	4	1.a	even	1	1	trivial
91.2.f.b	✓	4	13.c	even	3	1	inner
637.2.f.d		4	7.b	odd	2	1
637.2.f.d		4	91.n	odd	6	1
637.2.g.d		4	7.d	odd	6	1
637.2.g.d		4	91.v	odd	6	1
637.2.g.e		4	7.c	even	3	1
637.2.g.e		4	91.h	even	3	1
637.2.h.d		4	7.c	even	3	1
637.2.h.d		4	91.g	even	3	1
637.2.h.e		4	7.d	odd	6	1
637.2.h.e		4	91.m	odd	6	1
819.2.o.b		4	3.b	odd	2	1
819.2.o.b		4	39.i	odd	6	1
1183.2.a.e		2	13.e	even	6	1
1183.2.a.f		2	13.c	even	3	1
1183.2.c.e		4	13.f	odd	12	2
1456.2.s.o		4	4.b	odd	2	1
1456.2.s.o		4	52.j	odd	6	1
8281.2.a.r		2	91.n	odd	6	1
8281.2.a.t		2	91.t	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 3T_{2}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(91, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} + 3T^{2} + 9 \)
$3$	\( T^{4} + 2 T^{3} + 6 T^{2} - 4 T + 4 \)
$5$	\( (T^{2} - 3)^{2} \)
$7$	\( (T^{2} + T + 1)^{2} \)
$11$	\( T^{4} + 6 T^{3} + 30 T^{2} + 36 T + 36 \)