Newform orbit 700.2.g.c

• Label: 700.2.g.c
• Level: $700$
• Weight: $2$
• Character orbit: 700.g
• Analytic conductor: $5.590$
• Analytic rank: $0$
• Dimension: $4$
• CM discriminant: -7
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	700.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.58952814149\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{-7})\)
Defining polynomial:	\( x^{4} - x^{3} - x^{2} - 2x + 4 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - x^{2} - 2x + 4 \) :

\(\beta_{1}\)	\(=\)	\( ( \nu^{3} + \nu^{2} + \nu - 4 ) / 2 \)
\(\beta_{2}\)	\(=\)	\( ( -\nu^{3} - \nu^{2} + 3\nu + 2 ) / 2 \)
\(\beta_{3}\)	\(=\)	\( -\nu^{3} + 3 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(351\)	\(477\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

251.1

1.39564	−	0.228425i
1.39564	+	0.228425i
−0.895644	−	1.09445i
−0.895644	+	1.09445i

−1.39564

−

0.228425i

0

1.89564

+

0.637600i

0

0

−

2.64575i

−2.50000

−

1.32288i

−3.00000

0

251.2

−1.39564

+

0.228425i

0

1.89564

−

0.637600i

0

0

2.64575i

−2.50000

+

1.32288i

−3.00000

0

251.3

0.895644

−

1.09445i

0

−0.395644

−

1.96048i

0

0

−

2.64575i

−2.50000

−

1.32288i

−3.00000

0

251.4

0.895644

+

1.09445i

0

−0.395644

+

1.96048i

0

0

2.64575i

−2.50000

+

1.32288i

−3.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by \(\Q(\sqrt{-7}) \)
4.b	odd	2	1	inner
28.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	700.2.g.c	✓	4
4.b	odd	2	1	inner	700.2.g.c	✓	4
5.b	even	2	1		700.2.g.e	yes	4
5.c	odd	4	2		700.2.c.h		8
7.b	odd	2	1	CM	700.2.g.c	✓	4
20.d	odd	2	1		700.2.g.e	yes	4
20.e	even	4	2		700.2.c.h		8
28.d	even	2	1	inner	700.2.g.c	✓	4
35.c	odd	2	1		700.2.g.e	yes	4
35.f	even	4	2		700.2.c.h		8
140.c	even	2	1		700.2.g.e	yes	4
140.j	odd	4	2		700.2.c.h		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
700.2.c.h		8	5.c	odd	4	2
700.2.c.h		8	20.e	even	4	2
700.2.c.h		8	35.f	even	4	2
700.2.c.h		8	140.j	odd	4	2
700.2.g.c	✓	4	1.a	even	1	1	trivial
700.2.g.c	✓	4	4.b	odd	2	1	inner
700.2.g.c	✓	4	7.b	odd	2	1	CM
700.2.g.c	✓	4	28.d	even	2	1	inner
700.2.g.e	yes	4	5.b	even	2	1
700.2.g.e	yes	4	20.d	odd	2	1
700.2.g.e	yes	4	35.c	odd	2	1
700.2.g.e	yes	4	140.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(700, [\chi])\):

\( T_{3} \)

\( T_{11}^{4} + 38T_{11}^{2} + 25 \)

\( T_{19} \)

\( T_{37}^{2} - 6T_{37} - 75 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^4 + T^3 - T^2 + 2*T + 4
$3$	\( T^{4} \)
$5$	\( T^{4} \)
$7$	\( (T^{2} + 7)^{2} \)
$11$	\( T^{4} + 38T^{2} + 25 \)