Newform orbit 1183.2.c.e

• Label: 1183.2.c.e
• Level: $1183$
• Weight: $2$
• Character orbit: 1183.c
• Analytic conductor: $9.446$
• Analytic rank: $0$
• Dimension: $4$
• 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:	\(4\)
Coefficient field:	\(\Q(\zeta_{12})\)
Defining polynomial:	\( x^{4} - x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring

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

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

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

−

1.73205i

−0.732051

−1.00000

−

1.73205i

1.26795i

−

1.00000i

−

1.73205i

−2.46410

−3.00000

337.2

−

1.73205i

2.73205

−1.00000

−

1.73205i

−

4.73205i

1.00000i

−

1.73205i

4.46410

−3.00000

337.3

1.73205i

−0.732051

−1.00000

1.73205i

−

1.26795i

1.00000i

1.73205i

−2.46410

−3.00000

337.4

1.73205i

2.73205

−1.00000

1.73205i

4.73205i

−

1.00000i

1.73205i

4.46410

−3.00000

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

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(1183, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} + 3)^{2} \)
$3$	\( (T^{2} - 2 T - 2)^{2} \)
$5$	\( (T^{2} + 3)^{2} \)
$7$	\( (T^{2} + 1)^{2} \)
$11$	\( T^{4} + 24T^{2} + 36 \)