Newform orbit 840.2.g.b

• Label: 840.2.g.b
• Level: $840$
• Weight: $2$
• Character orbit: 840.g
• Analytic conductor: $6.707$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	840.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.70743376979\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	12.0.3058043990573056.1
Defining polynomial:	\( x^{12} + x^{10} - 8x^{7} - 16x^{5} + 16x^{2} + 64 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{6} \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + x^{10} - 8x^{7} - 16x^{5} + 16x^{2} + 64 \) :

\(\beta_{1}\)	\(=\)	\( ( \nu^{7} + \nu^{5} - 8 ) / 4 \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{10} + \nu^{8} - 8\nu^{3} - 16 ) / 16 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{10} + \nu^{8} - 8\nu^{5} + 16\nu + 16 ) / 16 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{11} + \nu^{9} - 8\nu^{4} - 16\nu ) / 32 \)
\(\beta_{5}\)	\(=\)	\( ( -\nu^{11} - \nu^{9} + 8\nu^{6} + 16\nu^{4} - 16\nu ) / 32 \)
\(\beta_{6}\)	\(=\)	\( ( -\nu^{7} - \nu^{5} + 8\nu^{2} + 8 ) / 4 \)
\(\beta_{7}\)	\(=\)	\( ( -\nu^{10} - \nu^{8} + 8\nu^{5} + 16\nu^{3} - 16 ) / 16 \)
\(\beta_{8}\)	\(=\)	\( ( \nu^{8} + \nu^{6} - 4\nu^{3} - 4\nu ) / 4 \)
\(\beta_{9}\)	\(=\)	\( ( \nu^{11} + 5\nu^{9} + 4\nu^{7} - 8\nu^{6} - 16\nu^{4} - 32\nu^{2} + 16\nu ) / 32 \)
\(\beta_{10}\)	\(=\)	\( ( \nu^{8} + \nu^{6} - 4\nu^{3} - 12\nu ) / 4 \)
\(\beta_{11}\)	\(=\)	\( ( \nu^{11} - 3\nu^{9} - 12\nu^{7} - 8\nu^{6} - 8\nu^{5} + 16\nu^{4} + 64\nu^{2} + 16\nu + 64 ) / 32 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/840\mathbb{Z}\right)^\times\).

\(n\)	\(241\)	\(281\)	\(337\)	\(421\)	\(631\)
\(\chi(n)\)	\(1\)	\(1\)	\(1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

421.1

−0.210663	−	1.39844i
−0.210663	+	1.39844i
0.723626	+	1.21506i
0.723626	−	1.21506i
−1.23149	−	0.695292i
−1.23149	+	0.695292i
1.41133	−	0.0902148i
1.41133	+	0.0902148i
−1.11568	+	0.869059i
−1.11568	−	0.869059i
0.422872	−	1.34951i
0.422872	+	1.34951i

−1.39844

−

0.210663i

−

1.00000i

1.91124

+

0.589197i

−

1.00000i

−0.210663

+

1.39844i

−1.00000

−2.54863

−

1.22658i

−1.00000

−0.210663

+

1.39844i

421.2

−1.39844

+

0.210663i

1.00000i

1.91124

−

0.589197i

1.00000i

−0.210663

−

1.39844i

−1.00000

−2.54863

+

1.22658i

−1.00000

−0.210663

−

1.39844i

421.3

−1.21506

−

0.723626i

1.00000i

0.952732

+

1.75849i

1.00000i

0.723626

−

1.21506i

−1.00000

0.114868

−

2.82609i

−1.00000

0.723626

−

1.21506i

421.4

−1.21506

+

0.723626i

−

1.00000i

0.952732

−

1.75849i

−

1.00000i

0.723626

+

1.21506i

−1.00000

0.114868

+

2.82609i

−1.00000

0.723626

+

1.21506i

421.5

−0.695292

−

1.23149i

−

1.00000i

−1.03314

+

1.71249i

−

1.00000i

−1.23149

+

0.695292i

−1.00000

2.82725

+

0.0816198i

−1.00000

−1.23149

+

0.695292i

421.6

−0.695292

+

1.23149i

1.00000i

−1.03314

−

1.71249i

1.00000i

−1.23149

−

0.695292i

−1.00000

2.82725

−

0.0816198i

−1.00000

−1.23149

−

0.695292i

421.7

0.0902148

−

1.41133i

1.00000i

−1.98372

−

0.254646i

1.00000i

1.41133

+

0.0902148i

−1.00000

−0.538352

+

2.77672i

−1.00000

1.41133

+

0.0902148i

421.8

0.0902148

+

1.41133i

−

1.00000i

−1.98372

+

0.254646i

−

1.00000i

1.41133

−

0.0902148i

−1.00000

−0.538352

−

2.77672i

−1.00000

1.41133

−

0.0902148i

421.9

0.869059

−

1.11568i

−

1.00000i

−0.489471

−

1.93918i

−

1.00000i

−1.11568

−

0.869059i

−1.00000

−2.58888

−

1.13917i

−1.00000

−1.11568

−

0.869059i

421.10

0.869059

+

1.11568i

1.00000i

−0.489471

+

1.93918i

1.00000i

−1.11568

+

0.869059i

−1.00000

−2.58888

+

1.13917i

−1.00000

−1.11568

+

0.869059i

421.11

1.34951

−

0.422872i

1.00000i

1.64236

−

1.14134i

1.00000i

0.422872

+

1.34951i

−1.00000

1.73374

−

2.23476i

−1.00000

0.422872

+

1.34951i

421.12

1.34951

+

0.422872i

−

1.00000i

1.64236

+

1.14134i

−

1.00000i

0.422872

−

1.34951i

−1.00000

1.73374

+

2.23476i

−1.00000

0.422872

−

1.34951i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	840.2.g.b	✓	12
4.b	odd	2	1		3360.2.g.c		12
8.b	even	2	1	inner	840.2.g.b	✓	12
8.d	odd	2	1		3360.2.g.c		12

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
840.2.g.b	✓	12	1.a	even	1	1	trivial
840.2.g.b	✓	12	8.b	even	2	1	inner
3360.2.g.c		12	4.b	odd	2	1
3360.2.g.c		12	8.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{12} + 64T_{11}^{10} + 1232T_{11}^{8} + 8320T_{11}^{6} + 12864T_{11}^{4} + 6656T_{11}^{2} + 1024 \) acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^12 + 2*T^11 + T^10 + 16*T^2 + 64*T + 64
$3$	\( (T^{2} + 1)^{6} \)
$5$	\( (T^{2} + 1)^{6} \)
$7$	\( (T + 1)^{12} \)
$11$	T^12 + 64*T^10 + 1232*T^8 + 8320*T^6 + 12864*T^4 + 6656*T^2 + 1024