Newform orbit 1183.2.c.g

• Label: 1183.2.c.g
• Level: $1183$
• Weight: $2$
• Character orbit: 1183.c
• Analytic conductor: $9.446$
• Analytic rank: $0$
• Dimension: $8$
• 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:	\(8\)
Coefficient field:	8.0.11667456256.1
Defining polynomial:	\( x^{8} + 13x^{6} + 44x^{4} + 21x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 13x^{6} + 44x^{4} + 21x^{2} + 1 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{6} + 10\nu^{4} + 22\nu^{2} + 3 ) / 8 \)
\(\beta_{3}\)	\(=\)	\( ( -\nu^{6} - 10\nu^{4} - 14\nu^{2} + 21 ) / 8 \)
\(\beta_{4}\)	\(=\)	\( ( -3\nu^{6} - 38\nu^{4} - 122\nu^{2} - 33 ) / 8 \)
\(\beta_{5}\)	\(=\)	\( ( -3\nu^{7} - 38\nu^{5} - 122\nu^{3} - 41\nu ) / 8 \)
\(\beta_{6}\)	\(=\)	\( ( -5\nu^{7} - 66\nu^{5} - 230\nu^{3} - 127\nu ) / 8 \)
\(\beta_{7}\)	\(=\)	\( -\nu^{7} - 13\nu^{5} - 43\nu^{3} - 14\nu \)

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

	−	2.74108i
	−	2.22001i
	−	0.710287i
	−	0.231361i
		0.231361i
		0.710287i
		2.22001i
		2.74108i

−

2.74108i

1.36482

−5.51353

0.741082i

−

3.74108i

1.00000i

9.63087i

−1.13727

2.03137

337.2

−

2.22001i

0.549551

−2.92843

4.22001i

−

1.22001i

−

1.00000i

2.06113i

−2.69799

9.36845

337.3

−

0.710287i

2.40788

1.49549

−

1.28971i

−

1.71029i

1.00000i

−

2.48280i

2.79790

−0.916066

337.4

−

0.231361i

−3.32225

1.94647

2.23136i

0.768639i

−

1.00000i

−

0.913059i

8.03736

0.516249

337.5

0.231361i

−3.32225

1.94647

−

2.23136i

−

0.768639i

1.00000i

0.913059i

8.03736

0.516249

337.6

0.710287i

2.40788

1.49549

1.28971i

1.71029i

−

1.00000i

2.48280i

2.79790

−0.916066

337.7

2.22001i

0.549551

−2.92843

−

4.22001i

1.22001i

1.00000i

−

2.06113i

−2.69799

9.36845

337.8

2.74108i

1.36482

−5.51353

−

0.741082i

3.74108i

−

1.00000i

−

9.63087i

−1.13727

2.03137

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.g		8
13.b	even	2	1	inner	1183.2.c.g		8
13.d	odd	4	1		1183.2.a.k		4
13.d	odd	4	1		1183.2.a.l		4
13.f	odd	12	2		91.2.f.c	✓	8
39.k	even	12	2		819.2.o.h		8
52.l	even	12	2		1456.2.s.q		8
91.i	even	4	1		8281.2.a.bp		4
91.i	even	4	1		8281.2.a.bt		4
91.w	even	12	2		637.2.g.j		8
91.x	odd	12	2		637.2.h.h		8
91.ba	even	12	2		637.2.h.i		8
91.bc	even	12	2		637.2.f.i		8
91.bd	odd	12	2		637.2.g.k		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
91.2.f.c	✓	8	13.f	odd	12	2
637.2.f.i		8	91.bc	even	12	2
637.2.g.j		8	91.w	even	12	2
637.2.g.k		8	91.bd	odd	12	2
637.2.h.h		8	91.x	odd	12	2
637.2.h.i		8	91.ba	even	12	2
819.2.o.h		8	39.k	even	12	2
1183.2.a.k		4	13.d	odd	4	1
1183.2.a.l		4	13.d	odd	4	1
1183.2.c.g		8	1.a	even	1	1	trivial
1183.2.c.g		8	13.b	even	2	1	inner
1456.2.s.q		8	52.l	even	12	2
8281.2.a.bp		4	91.i	even	4	1
8281.2.a.bt		4	91.i	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 13T_{2}^{6} + 44T_{2}^{4} + 21T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1183, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^8 + 13*T^6 + 44*T^4 + 21*T^2 + 1
$3$	\( (T^{4} - T^{3} - 9 T^{2} + 16 T - 6)^{2} \)
$5$	T^8 + 25*T^6 + 140*T^4 + 217*T^2 + 81
$7$	\( (T^{2} + 1)^{4} \)
$11$	T^8 + 19*T^6 + 101*T^4 + 148*T^2 + 36