Newform orbit 91.2.f.a

• Label: 91.2.f.a
• 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:	\(\Q(\sqrt{-3}, \sqrt{5})\)
Defining polynomial:	\( x^{4} - x^{3} + 2x^{2} + x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
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 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.

\(f(q)\)	\(=\)	\( q + ( - \beta_{3} - \beta_1 - 1) q^{2} + (\beta_{3} + \beta_1 + 1) q^{3} + (3 \beta_{2} + 3 \beta_1) q^{4} + ( - \beta_{2} + 1) q^{5} + ( - 2 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{6} + \beta_{3} q^{7} + ( - 4 \beta_{2} + 1) q^{8} + ( - \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{2} + 3 q^{3} - 3 q^{4} + 6 q^{5} + 7 q^{6} - 2 q^{7} + 12 q^{8} - q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 2x^{2} + x + 1 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{3} + 1 ) / 2 \)
\(\beta_{3}\)	\(=\)	\( ( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2 \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(\beta_{3}\)	\(1\)

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

0.809017	−	1.40126i
−0.309017	+	0.535233i
0.809017	+	1.40126i
−0.309017	−	0.535233i

−1.30902

+

2.26728i

1.30902

−

2.26728i

−2.42705

−

4.20378i

2.61803

3.42705

+

5.93583i

−0.500000

−

0.866025i

7.47214

−1.92705

−

3.33775i

−3.42705

+

5.93583i

22.2

−0.190983

+

0.330792i

0.190983

−

0.330792i

0.927051

+

1.60570i

0.381966

0.0729490

+

0.126351i

−0.500000

−

0.866025i

−1.47214

1.42705

+

2.47172i

−0.0729490

+

0.126351i

29.1

−1.30902

−

2.26728i

1.30902

+

2.26728i

−2.42705

+

4.20378i

2.61803

3.42705

−

5.93583i

−0.500000

+

0.866025i

7.47214

−1.92705

+

3.33775i

−3.42705

−

5.93583i

29.2

−0.190983

−

0.330792i

0.190983

+

0.330792i

0.927051

−

1.60570i

0.381966

0.0729490

−

0.126351i

−0.500000

+

0.866025i

−1.47214

1.42705

−

2.47172i

−0.0729490

−

0.126351i

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

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 3T_{2}^{3} + 8T_{2}^{2} + 3T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(91, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} + 3 T^{3} + 8 T^{2} + 3 T + 1 \)
$3$	\( T^{4} - 3 T^{3} + 8 T^{2} - 3 T + 1 \)
$5$	\( (T^{2} - 3 T + 1)^{2} \)
$7$	\( (T^{2} + T + 1)^{2} \)
$11$	\( T^{4} - 3 T^{3} + 18 T^{2} + 27 T + 81 \)