Newform orbit 1183.2.c.i

• Label: 1183.2.c.i
• Level: $1183$
• Weight: $2$
• Character orbit: 1183.c
• Analytic conductor: $9.446$
• Analytic rank: $0$
• Dimension: $12$
• 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:	\(12\)
Coefficient field:	12.0.58891012706304.1
Defining polynomial:	\( x^{12} - 5x^{10} - 2x^{9} + 15x^{8} + 2x^{7} - 30x^{6} + 4x^{5} + 60x^{4} - 16x^{3} - 80x^{2} + 64 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + ( - \beta_{7} + \beta_1) q^{2} - \beta_{2} q^{3} + (\beta_{5} - \beta_{4} - 1) q^{4} + ( - \beta_{8} - \beta_{7} + \beta_{6}) q^{5} + (\beta_{11} + \beta_{10} - \beta_{8} - \beta_{7}) q^{6} - \beta_{7} q^{7} + (\beta_{10} + 2 \beta_{8} + 2 \beta_{7}) q^{8} + (2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2}) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{4} + 8 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5x^{10} - 2x^{9} + 15x^{8} + 2x^{7} - 30x^{6} + 4x^{5} + 60x^{4} - 16x^{3} - 80x^{2} + 64 \) :

\(\beta_{1}\)	\(=\)	\( ( \nu^{11} - 5\nu^{9} - 2\nu^{8} + 15\nu^{7} + 2\nu^{6} - 30\nu^{5} + 4\nu^{4} + 60\nu^{3} - 16\nu^{2} - 48\nu ) / 32 \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{10} - 3\nu^{9} - \nu^{8} + 9\nu^{7} + \nu^{6} - 23\nu^{5} + 16\nu^{4} + 26\nu^{3} - 32\nu^{2} - 28\nu + 48 ) / 8 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{11} + 6 \nu^{10} - 5 \nu^{9} - 24 \nu^{8} + 3 \nu^{7} + 52 \nu^{6} - 34 \nu^{5} - 88 \nu^{4} + 100 \nu^{3} + 168 \nu^{2} - 112 \nu - 160 ) / 32 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{11} + 4 \nu^{10} - 17 \nu^{9} - 6 \nu^{8} + 51 \nu^{7} + 6 \nu^{6} - 122 \nu^{5} + 68 \nu^{4} + 164 \nu^{3} - 144 \nu^{2} - 224 \nu + 192 ) / 32 \)
\(\beta_{5}\)	\(=\)	\( ( - \nu^{11} - 2 \nu^{10} + 17 \nu^{9} - 4 \nu^{8} - 55 \nu^{7} + 24 \nu^{6} + 126 \nu^{5} - 128 \nu^{4} - 156 \nu^{3} + 232 \nu^{2} + 192 \nu - 288 ) / 32 \)
\(\beta_{6}\)	\(=\)	\( ( \nu^{11} - 8 \nu^{10} - 5 \nu^{9} + 30 \nu^{8} + 15 \nu^{7} - 78 \nu^{6} + 18 \nu^{5} + 124 \nu^{4} - 68 \nu^{3} - 192 \nu^{2} + 112 \nu + 96 ) / 32 \)
\(\beta_{7}\)	\(=\)	\( ( - 5 \nu^{11} + 2 \nu^{10} + 17 \nu^{9} - 8 \nu^{8} - 39 \nu^{7} + 44 \nu^{6} + 50 \nu^{5} - 88 \nu^{4} - 68 \nu^{3} + 120 \nu^{2} + 16 \nu - 32 ) / 32 \)
\(\beta_{8}\)	\(=\)	\( ( - \nu^{11} + 5 \nu^{10} + 7 \nu^{9} - 19 \nu^{8} - 19 \nu^{7} + 49 \nu^{6} + 14 \nu^{5} - 106 \nu^{4} + 4 \nu^{3} + 172 \nu^{2} - 8 \nu - 128 ) / 16 \)
\(\beta_{9}\)	\(=\)	\( ( - 5 \nu^{11} + 9 \nu^{9} + 18 \nu^{8} - 11 \nu^{7} - 18 \nu^{6} + 6 \nu^{5} + 68 \nu^{4} - 76 \nu^{3} - 160 \nu^{2} + 16 \nu + 288 ) / 32 \)
\(\beta_{10}\)	\(=\)	\( ( - \nu^{11} + 3 \nu^{10} + 5 \nu^{9} - 13 \nu^{8} - 13 \nu^{7} + 35 \nu^{6} + 12 \nu^{5} - 70 \nu^{4} + 8 \nu^{3} + 108 \nu^{2} - 16 \nu - 88 ) / 8 \)
\(\beta_{11}\)	\(=\)	\( ( - 3 \nu^{11} + \nu^{10} + 12 \nu^{9} - 3 \nu^{8} - 28 \nu^{7} + 19 \nu^{6} + 43 \nu^{5} - 52 \nu^{4} - 66 \nu^{3} + 64 \nu^{2} + 52 \nu - 16 ) / 8 \)

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

−1.12906	−	0.851598i
−1.30089	−	0.554694i
0.759479	−	1.19298i
1.40744	−	0.138282i
−1.08105	−	0.911778i
1.34408	+	0.439820i
1.34408	−	0.439820i
−1.08105	+	0.911778i
1.40744	+	0.138282i
0.759479	+	1.19298i
−1.30089	+	0.554694i
−1.12906	+	0.851598i

−

2.70320i

−0.345949

−5.30727

−

3.25812i

0.935168i

−

1.00000i

8.94020i

−2.88032

−8.80735

337.2

−

2.10939i

2.26165

−2.44952

−

3.60178i

−

4.77070i

−

1.00000i

0.948212i

2.11505

−7.59755

337.3

−

1.38595i

−2.82577

0.0791355

−

0.518957i

3.91639i

1.00000i

−

2.88158i

4.98500

−0.719250

337.4

−

1.27656i

−1.16793

0.370384

1.81487i

1.49093i

−

1.00000i

−

3.02595i

−1.63595

2.31680

337.5

−

0.823556i

2.66029

1.32176

3.16209i

−

2.19090i

1.00000i

−

2.73565i

4.07715

2.60416

337.6

−

0.120360i

−0.582292

1.98551

1.68817i

0.0700846i

−

1.00000i

−

0.479696i

−2.66094

0.203187

337.7

0.120360i

−0.582292

1.98551

−

1.68817i

−

0.0700846i

1.00000i

0.479696i

−2.66094

0.203187

337.8

0.823556i

2.66029

1.32176

−

3.16209i

2.19090i

−

1.00000i

2.73565i

4.07715

2.60416

337.9

1.27656i

−1.16793

0.370384

−

1.81487i

−

1.49093i

1.00000i

3.02595i

−1.63595

2.31680

337.10

1.38595i

−2.82577

0.0791355

0.518957i

−

3.91639i

−

1.00000i

2.88158i

4.98500

−0.719250

337.11

2.10939i

2.26165

−2.44952

3.60178i

4.77070i

1.00000i

−

0.948212i

2.11505

−7.59755

337.12

2.70320i

−0.345949

−5.30727

3.25812i

−

0.935168i

1.00000i

−

8.94020i

−2.88032

−8.80735

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.i		12
13.b	even	2	1	inner	1183.2.c.i		12
13.c	even	3	1		91.2.q.a	✓	12
13.d	odd	4	1		1183.2.a.m		6
13.d	odd	4	1		1183.2.a.p		6
13.e	even	6	1		91.2.q.a	✓	12
39.h	odd	6	1		819.2.ct.a		12
39.i	odd	6	1		819.2.ct.a		12
52.i	odd	6	1		1456.2.cc.c		12
52.j	odd	6	1		1456.2.cc.c		12
91.g	even	3	1		637.2.u.h		12
91.h	even	3	1		637.2.k.h		12
91.i	even	4	1		8281.2.a.by		6
91.i	even	4	1		8281.2.a.ch		6
91.k	even	6	1		637.2.k.h		12
91.l	odd	6	1		637.2.k.g		12
91.m	odd	6	1		637.2.u.i		12
91.n	odd	6	1		637.2.q.h		12
91.p	odd	6	1		637.2.u.i		12
91.t	odd	6	1		637.2.q.h		12
91.u	even	6	1		637.2.u.h		12
91.v	odd	6	1		637.2.k.g		12

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
91.2.q.a	✓	12	13.c	even	3	1
91.2.q.a	✓	12	13.e	even	6	1
637.2.k.g		12	91.l	odd	6	1
637.2.k.g		12	91.v	odd	6	1
637.2.k.h		12	91.h	even	3	1
637.2.k.h		12	91.k	even	6	1
637.2.q.h		12	91.n	odd	6	1
637.2.q.h		12	91.t	odd	6	1
637.2.u.h		12	91.g	even	3	1
637.2.u.h		12	91.u	even	6	1
637.2.u.i		12	91.m	odd	6	1
637.2.u.i		12	91.p	odd	6	1
819.2.ct.a		12	39.h	odd	6	1
819.2.ct.a		12	39.i	odd	6	1
1183.2.a.m		6	13.d	odd	4	1
1183.2.a.p		6	13.d	odd	4	1
1183.2.c.i		12	1.a	even	1	1	trivial
1183.2.c.i		12	13.b	even	2	1	inner
1456.2.cc.c		12	52.i	odd	6	1
1456.2.cc.c		12	52.j	odd	6	1
8281.2.a.by		6	91.i	even	4	1
8281.2.a.ch		6	91.i	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 16T_{2}^{10} + 88T_{2}^{8} + 206T_{2}^{6} + 208T_{2}^{4} + 72T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1183, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^12 + 16*T^10 + 88*T^8 + 206*T^6 + 208*T^4 + 72*T^2 + 1
$3$	(T^6 - 11*T^4 - 2*T^3 + 25*T^2 + 20*T + 4)^2
$5$	T^12 + 40*T^10 + 600*T^8 + 4146*T^6 + 13040*T^4 + 16148*T^2 + 3481
$7$	\( (T^{2} + 1)^{6} \)
$11$	T^12 + 50*T^10 + 587*T^8 + 2538*T^6 + 4377*T^4 + 2464*T^2 + 256