Newform orbit 1183.2.c.f

• Label: 1183.2.c.f
• Level: $1183$
• Weight: $2$
• Character orbit: 1183.c
• Analytic conductor: $9.446$
• Analytic rank: $0$
• Dimension: $6$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1183 = 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1183.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.44630255912\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	6.0.399424.1
Defining polynomial:	\( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 91)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta_{5} q^{2} + ( - \beta_{3} - \beta_1 - 1) q^{3} + ( - \beta_1 - 1) q^{4} + (\beta_{5} - \beta_{2}) q^{5} + (2 \beta_{5} - 2 \beta_{2}) q^{6} - \beta_{2} q^{7} + ( - \beta_{4} + \beta_{2}) q^{8} + (2 \beta_{3} + 3) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} - 6 q^{4} + 14 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) :

\(\beta_{1}\)	\(=\)	\( ( -\nu^{4} + 2\nu^{3} - \nu^{2} + 2\nu - 2 ) / 2 \)
\(\beta_{2}\)	\(=\)	\( ( -\nu^{5} - 3\nu^{3} + 4\nu^{2} - 2\nu + 8 ) / 4 \)
\(\beta_{3}\)	\(=\)	\( ( -\nu^{5} + 2\nu^{4} - 3\nu^{3} + 6\nu^{2} - 2\nu + 4 ) / 4 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{5} - 2\nu^{4} + 3\nu^{3} - 6\nu^{2} + 10\nu - 8 ) / 4 \)
\(\beta_{5}\)	\(=\)	\( ( 3\nu^{5} - 2\nu^{4} + 5\nu^{3} - 6\nu^{2} + 2\nu - 12 ) / 4 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).

\(n\)	\(339\)	\(1016\)
\(\chi(n)\)	\(1\)	\(-1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

337.1

−0.671462	−	1.24464i
1.40680	+	0.144584i
0.264658	+	1.38923i
0.264658	−	1.38923i
1.40680	−	0.144584i
−0.671462	+	1.24464i

−

2.34292i

−1.14637

−3.48929

1.34292i

2.68585i

−

1.00000i

3.48929i

−1.68585

3.14637

337.2

−

1.81361i

−3.10278

−1.28917

2.81361i

5.62721i

1.00000i

−

1.28917i

6.62721

5.10278

337.3

−

0.470683i

2.24914

1.77846

−

0.529317i

−

1.05863i

−

1.00000i

−

1.77846i

2.05863

−0.249141

337.4

0.470683i

2.24914

1.77846

0.529317i

1.05863i

1.00000i

1.77846i

2.05863

−0.249141

337.5

1.81361i

−3.10278

−1.28917

−

2.81361i

−

5.62721i

−

1.00000i

1.28917i

6.62721

5.10278

337.6

2.34292i

−1.14637

−3.48929

−

1.34292i

−

2.68585i

1.00000i

−

3.48929i

−1.68585

3.14637

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1183.2.c.f		6
13.b	even	2	1	inner	1183.2.c.f		6
13.d	odd	4	1		91.2.a.d	✓	3
13.d	odd	4	1		1183.2.a.i		3
39.f	even	4	1		819.2.a.i		3
52.f	even	4	1		1456.2.a.t		3
65.g	odd	4	1		2275.2.a.m		3
91.i	even	4	1		637.2.a.j		3
91.i	even	4	1		8281.2.a.bg		3
91.z	odd	12	2		637.2.e.j		6
91.bb	even	12	2		637.2.e.i		6
104.j	odd	4	1		5824.2.a.by		3
104.m	even	4	1		5824.2.a.bs		3
273.o	odd	4	1		5733.2.a.x		3

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
91.2.a.d	✓	3	13.d	odd	4	1
637.2.a.j		3	91.i	even	4	1
637.2.e.i		6	91.bb	even	12	2
637.2.e.j		6	91.z	odd	12	2
819.2.a.i		3	39.f	even	4	1
1183.2.a.i		3	13.d	odd	4	1
1183.2.c.f		6	1.a	even	1	1	trivial
1183.2.c.f		6	13.b	even	2	1	inner
1456.2.a.t		3	52.f	even	4	1
2275.2.a.m		3	65.g	odd	4	1
5733.2.a.x		3	273.o	odd	4	1
5824.2.a.bs		3	104.m	even	4	1
5824.2.a.by		3	104.j	odd	4	1
8281.2.a.bg		3	91.i	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 9T_{2}^{4} + 20T_{2}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(1183, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{6} + 9 T^{4} + 20 T^{2} + 4 \)
$3$	\( (T^{3} + 2 T^{2} - 6 T - 8)^{2} \)
$5$	\( T^{6} + 10 T^{4} + 17 T^{2} + 4 \)
$7$	\( (T^{2} + 1)^{3} \)
$11$	\( T^{6} + 16 T^{4} + 68 T^{2} + 64 \)