Newform orbit 700.2.g.a

• Label: 700.2.g.a
• Level: $700$
• Weight: $2$
• Character orbit: 700.g
• Analytic conductor: $5.590$
• Analytic rank: $0$
• Dimension: $2$
• 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:	\(2\)
Coefficient field:	\(\Q(\sqrt{-7}) \)
Defining polynomial:	\( x^{2} - x + 2 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{-7})\). We also show the integral \(q\)-expansion of the trace form.

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

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

0.500000	−	1.32288i
0.500000	+	1.32288i

0.500000

−

1.32288i

0

−1.50000

−

1.32288i

0

0

2.64575i

−2.50000

+

1.32288i

−3.00000

0

251.2

0.500000

+

1.32288i

0

−1.50000

+

1.32288i

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.a		2
4.b	odd	2	1	inner	700.2.g.a		2
5.b	even	2	1		28.2.d.a	✓	2
5.c	odd	4	2		700.2.c.d		4
7.b	odd	2	1	CM	700.2.g.a		2
15.d	odd	2	1		252.2.b.a		2
20.d	odd	2	1		28.2.d.a	✓	2
20.e	even	4	2		700.2.c.d		4
28.d	even	2	1	inner	700.2.g.a		2
35.c	odd	2	1		28.2.d.a	✓	2
35.f	even	4	2		700.2.c.d		4
35.i	odd	6	2		196.2.f.b		4
35.j	even	6	2		196.2.f.b		4
40.e	odd	2	1		448.2.f.b		2
40.f	even	2	1		448.2.f.b		2
60.h	even	2	1		252.2.b.a		2
80.k	odd	4	2		1792.2.e.b		4
80.q	even	4	2		1792.2.e.b		4
105.g	even	2	1		252.2.b.a		2
120.i	odd	2	1		4032.2.b.e		2
120.m	even	2	1		4032.2.b.e		2
140.c	even	2	1		28.2.d.a	✓	2
140.j	odd	4	2		700.2.c.d		4
140.p	odd	6	2		196.2.f.b		4
140.s	even	6	2		196.2.f.b		4
280.c	odd	2	1		448.2.f.b		2
280.n	even	2	1		448.2.f.b		2
420.o	odd	2	1		252.2.b.a		2
560.be	even	4	2		1792.2.e.b		4
560.bf	odd	4	2		1792.2.e.b		4
840.b	odd	2	1		4032.2.b.e		2
840.u	even	2	1		4032.2.b.e		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
28.2.d.a	✓	2	5.b	even	2	1
28.2.d.a	✓	2	20.d	odd	2	1
28.2.d.a	✓	2	35.c	odd	2	1
28.2.d.a	✓	2	140.c	even	2	1
196.2.f.b		4	35.i	odd	6	2
196.2.f.b		4	35.j	even	6	2
196.2.f.b		4	140.p	odd	6	2
196.2.f.b		4	140.s	even	6	2
252.2.b.a		2	15.d	odd	2	1
252.2.b.a		2	60.h	even	2	1
252.2.b.a		2	105.g	even	2	1
252.2.b.a		2	420.o	odd	2	1
448.2.f.b		2	40.e	odd	2	1
448.2.f.b		2	40.f	even	2	1
448.2.f.b		2	280.c	odd	2	1
448.2.f.b		2	280.n	even	2	1
700.2.c.d		4	5.c	odd	4	2
700.2.c.d		4	20.e	even	4	2
700.2.c.d		4	35.f	even	4	2
700.2.c.d		4	140.j	odd	4	2
700.2.g.a		2	1.a	even	1	1	trivial
700.2.g.a		2	4.b	odd	2	1	inner
700.2.g.a		2	7.b	odd	2	1	CM
700.2.g.a		2	28.d	even	2	1	inner
1792.2.e.b		4	80.k	odd	4	2
1792.2.e.b		4	80.q	even	4	2
1792.2.e.b		4	560.be	even	4	2
1792.2.e.b		4	560.bf	odd	4	2
4032.2.b.e		2	120.i	odd	2	1
4032.2.b.e		2	120.m	even	2	1
4032.2.b.e		2	840.b	odd	2	1
4032.2.b.e		2	840.u	even	2	1

Hecke kernels

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

\( T_{3} \)

\( T_{11}^{2} + 28 \)

\( T_{19} \)

\( T_{37} + 6 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - T + 2 \)
$3$	\( T^{2} \)
$5$	\( T^{2} \)
$7$	\( T^{2} + 7 \)
$11$	\( T^{2} + 28 \)