Newform orbit 840.2.u.c

• Label: 840.2.u.c
• Level: $840$
• Weight: $2$
• Character orbit: 840.u
• Analytic conductor: $6.707$
• Analytic rank: $0$
• Dimension: $8$
• CM discriminant: -56
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.70743376979\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	8.0.2517630976.5
Defining polynomial:	\( x^{8} - 2x^{6} + 11x^{4} + 4x^{2} + 4 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\)	\(=\)	\( q - \beta_1 q^{2} - \beta_{2} q^{3} + 2 q^{4} - \beta_{6} q^{5} + (\beta_{7} + \beta_{3}) q^{6} - \beta_{4} q^{7} - 2 \beta_1 q^{8} + ( - \beta_{4} + \beta_1) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{6} + 11x^{4} + 4x^{2} + 4 \) :

\(\beta_{1}\)	\(=\)	\( ( \nu^{6} + 2\nu^{4} + \nu^{2} + 26 ) / 18 \)
\(\beta_{2}\)	\(=\)	\( ( -\nu^{7} + 4\nu^{5} - 19\nu^{3} + 22\nu ) / 12 \)
\(\beta_{3}\)	\(=\)	\( ( 7\nu^{7} - 4\nu^{5} + 61\nu^{3} + 110\nu ) / 36 \)
\(\beta_{4}\)	\(=\)	\( ( -4\nu^{6} + 10\nu^{4} - 40\nu^{2} - 5 ) / 9 \)
\(\beta_{5}\)	\(=\)	\( ( \nu^{6} - 2\nu^{4} + 13\nu^{2} + 2 ) / 2 \)
\(\beta_{6}\)	\(=\)	\( ( 13\nu^{7} - 28\nu^{5} + 139\nu^{3} + 86\nu ) / 36 \)
\(\beta_{7}\)	\(=\)	\( ( -13\nu^{7} + 28\nu^{5} - 139\nu^{3} - 14\nu ) / 36 \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

629.1

0.435132	+	0.629640i
0.435132	−	0.629640i
−0.435132	+	0.629640i
−0.435132	−	0.629640i
1.52009	−	1.05050i
1.52009	+	1.05050i
−1.52009	−	1.05050i
−1.52009	+	1.05050i

−1.41421

−1.48563

−

0.890446i

2.00000

0.615370

−

2.14973i

2.10100

+

1.25928i

2.64575i

−2.82843

1.41421

+

2.64575i

−0.870264

+

3.04017i

629.2

−1.41421

−1.48563

+

0.890446i

2.00000

0.615370

+

2.14973i

2.10100

−

1.25928i

−

2.64575i

−2.82843

1.41421

−

2.64575i

−0.870264

−

3.04017i

629.3

−1.41421

1.48563

−

0.890446i

2.00000

−0.615370

−

2.14973i

−2.10100

+

1.25928i

−

2.64575i

−2.82843

1.41421

−

2.64575i

0.870264

+

3.04017i

629.4

−1.41421

1.48563

+

0.890446i

2.00000

−0.615370

+

2.14973i

−2.10100

−

1.25928i

2.64575i

−2.82843

1.41421

+

2.64575i

0.870264

−

3.04017i

629.5

1.41421

−0.890446

−

1.48563i

2.00000

−2.14973

+

0.615370i

−1.25928

−

2.10100i

2.64575i

2.82843

−1.41421

+

2.64575i

−3.04017

+

0.870264i

629.6

1.41421

−0.890446

+

1.48563i

2.00000

−2.14973

−

0.615370i

−1.25928

+

2.10100i

−

2.64575i

2.82843

−1.41421

−

2.64575i

−3.04017

−

0.870264i

629.7

1.41421

0.890446

−

1.48563i

2.00000

2.14973

+

0.615370i

1.25928

−

2.10100i

−

2.64575i

2.82843

−1.41421

−

2.64575i

3.04017

+

0.870264i

629.8

1.41421

0.890446

+

1.48563i

2.00000

2.14973

−

0.615370i

1.25928

+

2.10100i

2.64575i

2.82843

−1.41421

+

2.64575i

3.04017

−

0.870264i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
56.h	odd	2	1	CM by \(\Q(\sqrt{-14}) \)
7.b	odd	2	1	inner
8.b	even	2	1	inner
15.d	odd	2	1	inner
105.g	even	2	1	inner
120.i	odd	2	1	inner
840.u	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.u.c	✓	8
3.b	odd	2	1		840.2.u.d	yes	8
5.b	even	2	1		840.2.u.d	yes	8
7.b	odd	2	1	inner	840.2.u.c	✓	8
8.b	even	2	1	inner	840.2.u.c	✓	8
15.d	odd	2	1	inner	840.2.u.c	✓	8
21.c	even	2	1		840.2.u.d	yes	8
24.h	odd	2	1		840.2.u.d	yes	8
35.c	odd	2	1		840.2.u.d	yes	8
40.f	even	2	1		840.2.u.d	yes	8
56.h	odd	2	1	CM	840.2.u.c	✓	8
105.g	even	2	1	inner	840.2.u.c	✓	8
120.i	odd	2	1	inner	840.2.u.c	✓	8
168.i	even	2	1		840.2.u.d	yes	8
280.c	odd	2	1		840.2.u.d	yes	8
840.u	even	2	1	inner	840.2.u.c	✓	8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
840.2.u.c	✓	8	1.a	even	1	1	trivial
840.2.u.c	✓	8	7.b	odd	2	1	inner
840.2.u.c	✓	8	8.b	even	2	1	inner
840.2.u.c	✓	8	15.d	odd	2	1	inner
840.2.u.c	✓	8	56.h	odd	2	1	CM
840.2.u.c	✓	8	105.g	even	2	1	inner
840.2.u.c	✓	8	120.i	odd	2	1	inner
840.2.u.c	✓	8	840.u	even	2	1	inner
840.2.u.d	yes	8	3.b	odd	2	1
840.2.u.d	yes	8	5.b	even	2	1
840.2.u.d	yes	8	21.c	even	2	1
840.2.u.d	yes	8	24.h	odd	2	1
840.2.u.d	yes	8	35.c	odd	2	1
840.2.u.d	yes	8	40.f	even	2	1
840.2.u.d	yes	8	168.i	even	2	1
840.2.u.d	yes	8	280.c	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\):

\( T_{11} \)

\( T_{23} + 6 \)

\( T_{73} \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} - 2)^{4} \)
$3$	\( T^{8} + 10T^{4} + 81 \)
$5$	\( T^{8} - 22T^{4} + 625 \)
$7$	\( (T^{2} + 7)^{4} \)
$11$	\( T^{8} \)