Newform orbit 483.2.h.b

• Label: 483.2.h.b
• Level: $483$
• Weight: $2$
• Character orbit: 483.h
• Analytic conductor: $3.857$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	483.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.85677441763\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial:	\( x^{12} + 14x^{10} + 67x^{8} + 141x^{6} + 129x^{4} + 39x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	yes
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 14x^{10} + 67x^{8} + 141x^{6} + 129x^{4} + 39x^{2} + 1 \) :

\(\beta_{1}\)	\(=\)	\( ( 3\nu^{10} + 31\nu^{8} + 78\nu^{6} + 39\nu^{4} - 8\nu^{2} + 25 ) / 14 \)
\(\beta_{2}\)	\(=\)	\( ( 2\nu^{10} + 23\nu^{8} + 80\nu^{6} + 124\nu^{4} + 95\nu^{2} + 12 ) / 7 \)
\(\beta_{3}\)	\(=\)	\( ( -\nu^{11} - 15\nu^{9} - 82\nu^{7} - 209\nu^{5} - 212\nu^{3} - 27\nu ) / 14 \)
\(\beta_{4}\)	\(=\)	\( ( 2\nu^{11} + 23\nu^{9} + 73\nu^{7} + 54\nu^{5} - 66\nu^{3} - 58\nu ) / 7 \)
\(\beta_{5}\)	\(=\)	\( ( -3\nu^{11} - 45\nu^{9} - 232\nu^{7} - 501\nu^{5} - 426\nu^{3} - 95\nu ) / 14 \)
\(\beta_{6}\)	\(=\)	\( -\nu^{9} - 12\nu^{7} - 43\nu^{5} - 55\nu^{3} - 19\nu \)
\(\beta_{7}\)	\(=\)	\( ( -6\nu^{10} - 69\nu^{8} - 226\nu^{6} - 239\nu^{4} - 26\nu^{2} + 20 ) / 7 \)
\(\beta_{8}\)	\(=\)	\( ( -5\nu^{10} - 61\nu^{8} - 228\nu^{6} - 324\nu^{4} - 143\nu^{2} - 2 ) / 7 \)
\(\beta_{9}\)	\(=\)	(21*v^11 + 13*v^10 + 259*v^9 + 167*v^8 + 994*v^7 + 674*v^6 + 1519*v^5 + 1023*v^4 + 896*v^3 + 488*v^2 + 203*v + 43) / 28
\(\beta_{10}\)	\(=\)	\( ( -9\nu^{11} - 121\nu^{9} - 542\nu^{7} - 1041\nu^{5} - 830\nu^{3} - 173\nu ) / 14 \)
\(\beta_{11}\)	\(=\)	(-21*v^11 + 13*v^10 - 259*v^9 + 167*v^8 - 994*v^7 + 674*v^6 - 1519*v^5 + 1023*v^4 - 896*v^3 + 488*v^2 - 203*v + 43) / 28

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

160.1

		1.64421i
	−	1.28628i
	−	1.64421i
		1.28628i
	−	1.51343i
	−	0.167898i
		1.51343i
		0.167898i
		2.61062i
		0.712783i
	−	2.61062i
	−	0.712783i

−2.11491

−

1.00000i

2.47283

−1.98969

2.11491i

2.62846

−

0.302029i

−1.00000

−1.00000

4.20801

160.2

−2.11491

−

1.00000i

2.47283

1.98969

2.11491i

−2.62846

+

0.302029i

−1.00000

−1.00000

−4.20801

160.3

−2.11491

1.00000i

2.47283

−1.98969

−

2.11491i

2.62846

+

0.302029i

−1.00000

−1.00000

4.20801

160.4

−2.11491

1.00000i

2.47283

1.98969

−

2.11491i

−2.62846

−

0.302029i

−1.00000

−1.00000

−4.20801

160.5

0.254102

−

1.00000i

−1.93543

−0.458377

−

0.254102i

1.07291

+

2.41844i

−1.00000

−1.00000

−0.116474

160.6

0.254102

−

1.00000i

−1.93543

0.458377

−

0.254102i

−1.07291

−

2.41844i

−1.00000

−1.00000

0.116474

160.7

0.254102

1.00000i

−1.93543

−0.458377

0.254102i

1.07291

−

2.41844i

−1.00000

−1.00000

−0.116474

160.8

0.254102

1.00000i

−1.93543

0.458377

0.254102i

−1.07291

+

2.41844i

−1.00000

−1.00000

0.116474

160.9

1.86081

−

1.00000i

1.46260

−4.10256

−

1.86081i

−0.663393

−

2.56123i

−1.00000

−1.00000

−7.63407

160.10

1.86081

−

1.00000i

1.46260

4.10256

−

1.86081i

0.663393

+

2.56123i

−1.00000

−1.00000

7.63407

160.11

1.86081

1.00000i

1.46260

−4.10256

1.86081i

−0.663393

+

2.56123i

−1.00000

−1.00000

−7.63407

160.12

1.86081

1.00000i

1.46260

4.10256

1.86081i

0.663393

−

2.56123i

−1.00000

−1.00000

7.63407

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
23.b	odd	2	1	inner
161.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	483.2.h.b	✓	12
3.b	odd	2	1		1449.2.h.h		12
7.b	odd	2	1	inner	483.2.h.b	✓	12
21.c	even	2	1		1449.2.h.h		12
23.b	odd	2	1	inner	483.2.h.b	✓	12
69.c	even	2	1		1449.2.h.h		12
161.c	even	2	1	inner	483.2.h.b	✓	12
483.c	odd	2	1		1449.2.h.h		12

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
483.2.h.b	✓	12	1.a	even	1	1	trivial
483.2.h.b	✓	12	7.b	odd	2	1	inner
483.2.h.b	✓	12	23.b	odd	2	1	inner
483.2.h.b	✓	12	161.c	even	2	1	inner
1449.2.h.h		12	3.b	odd	2	1
1449.2.h.h		12	21.c	even	2	1
1449.2.h.h		12	69.c	even	2	1
1449.2.h.h		12	483.c	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 4T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(483, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{3} - 4 T + 1)^{4} \)
$3$	\( (T^{2} + 1)^{6} \)
$5$	(T^6 - 21*T^4 + 71*T^2 - 14)^2
$7$	T^12 + 8*T^10 - 33*T^8 - 784*T^6 - 1617*T^4 + 19208*T^2 + 117649
$11$	(T^6 + 14*T^4 + 53*T^2 + 56)^2