Newform orbit 1456.2.cc.c

• Label: 1456.2.cc.c
• Level: $1456$
• Weight: $2$
• Character orbit: 1456.cc
• Analytic conductor: $11.626$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1456.cc (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.6262185343\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{6})\)
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:	\( 1 \)
Twist minimal:	no (minimal twist has level 91)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

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

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{11} - 5\nu^{9} - 2\nu^{8} + 15\nu^{7} + 2\nu^{6} - 30\nu^{5} + 4\nu^{4} + 60\nu^{3} - 16\nu^{2} - 80\nu ) / 32 \)
\(\beta_{3}\)	\(=\)	\( ( - 3 \nu^{11} + 4 \nu^{10} + 7 \nu^{9} - 6 \nu^{8} - 13 \nu^{7} + 30 \nu^{6} - 6 \nu^{5} - 28 \nu^{4} - 4 \nu^{3} + 16 \nu^{2} - 48 \nu + 96 ) / 32 \)
\(\beta_{4}\)	\(=\)	\( ( 3 \nu^{11} + 2 \nu^{10} - 11 \nu^{9} - 8 \nu^{8} + 21 \nu^{7} + 4 \nu^{6} - 42 \nu^{5} + 84 \nu^{3} + 24 \nu^{2} - 64 \nu - 64 ) / 32 \)
\(\beta_{5}\)	\(=\)	\( ( \nu^{11} + 3 \nu^{10} - 3 \nu^{9} - 13 \nu^{8} + 7 \nu^{7} + 23 \nu^{6} - 26 \nu^{5} - 38 \nu^{4} + 60 \nu^{3} + 68 \nu^{2} - 56 \nu - 64 ) / 16 \)
\(\beta_{6}\)	\(=\)	\( ( - 5 \nu^{11} + 4 \nu^{10} + 13 \nu^{9} - 10 \nu^{8} - 23 \nu^{7} + 42 \nu^{6} + 10 \nu^{5} - 68 \nu^{4} - 20 \nu^{3} + 48 \nu^{2} - 32 \nu + 64 ) / 32 \)
\(\beta_{7}\)	\(=\)	\( ( - 5 \nu^{11} + 2 \nu^{10} + 17 \nu^{9} - 8 \nu^{8} - 39 \nu^{7} + 44 \nu^{6} + 50 \nu^{5} - 88 \nu^{4} - 68 \nu^{3} + 120 \nu^{2} + 16 \nu - 32 ) / 32 \)
\(\beta_{8}\)	\(=\)	\( ( - \nu^{11} + \nu^{10} + 5 \nu^{9} - 3 \nu^{8} - 13 \nu^{7} + 13 \nu^{6} + 20 \nu^{5} - 34 \nu^{4} - 28 \nu^{3} + 52 \nu^{2} + 24 \nu - 32 ) / 8 \)
\(\beta_{9}\)	\(=\)	\( ( - \nu^{11} + 3 \nu^{10} + 5 \nu^{9} - 13 \nu^{8} - 13 \nu^{7} + 35 \nu^{6} + 12 \nu^{5} - 70 \nu^{4} + 8 \nu^{3} + 108 \nu^{2} - 16 \nu - 80 ) / 16 \)
\(\beta_{10}\)	\(=\)	\( ( - 3 \nu^{11} + 15 \nu^{9} - 2 \nu^{8} - 37 \nu^{7} + 18 \nu^{6} + 66 \nu^{5} - 68 \nu^{4} - 92 \nu^{3} + 96 \nu^{2} + 96 \nu - 64 ) / 16 \)
\(\beta_{11}\)	\(=\)	\( ( - \nu^{11} + 4 \nu^{10} + 9 \nu^{9} - 18 \nu^{8} - 27 \nu^{7} + 50 \nu^{6} + 34 \nu^{5} - 116 \nu^{4} - 20 \nu^{3} + 192 \nu^{2} + 16 \nu - 160 ) / 16 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times\).

\(n\)	\(561\)	\(911\)	\(1093\)	\(1249\)
\(\chi(n)\)	\(1 - \beta_{9}\)	\(1\)	\(1\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

225.1

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

0

−1.41289

+

2.44719i

0

0.518957i

0

0.866025

−

0.500000i

0

−2.49250

−

4.31714i

0

225.2

0

−0.583963

+

1.01145i

0

1.81487i

0

0.866025

−

0.500000i

0

0.817975

+

1.41677i

0

225.3

0

−0.291146

+

0.504280i

0

−

1.68817i

0

−0.866025

+

0.500000i

0

1.33047

+

2.30444i

0

225.4

0

−0.172975

+

0.299601i

0

3.25812i

0

−0.866025

+

0.500000i

0

1.44016

+

2.49443i

0

225.5

0

1.13082

−

1.95864i

0

−

3.60178i

0

0.866025

−

0.500000i

0

−1.05753

−

1.83169i

0

225.6

0

1.33015

−

2.30388i

0

3.16209i

0

−0.866025

+

0.500000i

0

−2.03858

−

3.53092i

0

673.1

0

−1.41289

−

2.44719i

0

−

0.518957i

0

0.866025

+

0.500000i

0

−2.49250

+

4.31714i

0

673.2

0

−0.583963

−

1.01145i

0

−

1.81487i

0

0.866025

+

0.500000i

0

0.817975

−

1.41677i

0

673.3

0

−0.291146

−

0.504280i

0

1.68817i

0

−0.866025

−

0.500000i

0

1.33047

−

2.30444i

0

673.4

0

−0.172975

−

0.299601i

0

−

3.25812i

0

−0.866025

−

0.500000i

0

1.44016

−

2.49443i

0

673.5

0

1.13082

+

1.95864i

0

3.60178i

0

0.866025

+

0.500000i

0

−1.05753

+

1.83169i

0

673.6

0

1.33015

+

2.30388i

0

−

3.16209i

0

−0.866025

−

0.500000i

0

−2.03858

+

3.53092i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.e	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1456.2.cc.c		12
4.b	odd	2	1		91.2.q.a	✓	12
12.b	even	2	1		819.2.ct.a		12
13.e	even	6	1	inner	1456.2.cc.c		12
28.d	even	2	1		637.2.q.h		12
28.f	even	6	1		637.2.k.g		12
28.f	even	6	1		637.2.u.i		12
28.g	odd	6	1		637.2.k.h		12
28.g	odd	6	1		637.2.u.h		12
52.i	odd	6	1		91.2.q.a	✓	12
52.i	odd	6	1		1183.2.c.i		12
52.j	odd	6	1		1183.2.c.i		12
52.l	even	12	1		1183.2.a.m		6
52.l	even	12	1		1183.2.a.p		6
156.r	even	6	1		819.2.ct.a		12
364.s	odd	6	1		637.2.k.h		12
364.w	even	6	1		637.2.u.i		12
364.bc	even	6	1		637.2.q.h		12
364.bk	odd	6	1		637.2.u.h		12
364.bp	even	6	1		637.2.k.g		12
364.bv	odd	12	1		8281.2.a.by		6
364.bv	odd	12	1		8281.2.a.ch		6

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
91.2.q.a	✓	12	4.b	odd	2	1
91.2.q.a	✓	12	52.i	odd	6	1
637.2.k.g		12	28.f	even	6	1
637.2.k.g		12	364.bp	even	6	1
637.2.k.h		12	28.g	odd	6	1
637.2.k.h		12	364.s	odd	6	1
637.2.q.h		12	28.d	even	2	1
637.2.q.h		12	364.bc	even	6	1
637.2.u.h		12	28.g	odd	6	1
637.2.u.h		12	364.bk	odd	6	1
637.2.u.i		12	28.f	even	6	1
637.2.u.i		12	364.w	even	6	1
819.2.ct.a		12	12.b	even	2	1
819.2.ct.a		12	156.r	even	6	1
1183.2.a.m		6	52.l	even	12	1
1183.2.a.p		6	52.l	even	12	1
1183.2.c.i		12	52.i	odd	6	1
1183.2.c.i		12	52.j	odd	6	1
1456.2.cc.c		12	1.a	even	1	1	trivial
1456.2.cc.c		12	13.e	even	6	1	inner
8281.2.a.by		6	364.bv	odd	12	1
8281.2.a.ch		6	364.bv	odd	12	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 11 T_{3}^{10} + 4 T_{3}^{9} + 96 T_{3}^{8} + 42 T_{3}^{7} + 287 T_{3}^{6} + 390 T_{3}^{5} + 709 T_{3}^{4} + 516 T_{3}^{3} + 300 T_{3}^{2} + 80 T_{3} + 16 \) acting on \(S_{2}^{\mathrm{new}}(1456, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{12} \)
$3$	T^12 + 11*T^10 + 4*T^9 + 96*T^8 + 42*T^7 + 287*T^6 + 390*T^5 + 709*T^4 + 516*T^3 + 300*T^2 + 80*T + 16
$5$	T^12 + 40*T^10 + 600*T^8 + 4146*T^6 + 13040*T^4 + 16148*T^2 + 3481
$7$	\( (T^{4} - T^{2} + 1)^{3} \)
$11$	T^12 + 6*T^11 - 7*T^10 - 114*T^9 + 248*T^8 + 522*T^7 - 771*T^6 - 1494*T^5 + 2025*T^4 + 2544*T^3 - 80*T^2 - 768*T + 256