Newform orbit 975.1.g.c

• Label: 975.1.g.c
• Level: $975$
• Weight: $1$
• Character orbit: 975.g
• Self dual: yes
• Analytic conductor: $0.487$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{4}$
• CM discriminant: -39
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	975.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.486588387317\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 195)
Projective image:	\(D_{4}\)
Projective field:	Galois closure of 4.2.12675.1
Artin image:	$D_8$
Artin field:	Galois closure of 8.0.2780578125.1

$q$-expansion

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).

\(n\)	\(301\)	\(326\)	\(352\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

701.1

1.41421
−1.41421

−1.41421

1.00000

1.00000

0

−1.41421

0

0

1.00000

0

701.2

1.41421

1.00000

1.00000

0

1.41421

0

0

1.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
39.d	odd	2	1	CM by \(\Q(\sqrt{-39}) \)
3.b	odd	2	1	inner
13.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	975.1.g.c		2
3.b	odd	2	1	inner	975.1.g.c		2
5.b	even	2	1		975.1.g.b		2
5.c	odd	4	2		195.1.e.a	✓	4
13.b	even	2	1	inner	975.1.g.c		2
15.d	odd	2	1		975.1.g.b		2
15.e	even	4	2		195.1.e.a	✓	4
20.e	even	4	2		3120.1.be.e		4
39.d	odd	2	1	CM	975.1.g.c		2
60.l	odd	4	2		3120.1.be.e		4
65.d	even	2	1		975.1.g.b		2
65.f	even	4	2		2535.1.f.e		4
65.h	odd	4	2		195.1.e.a	✓	4
65.k	even	4	2		2535.1.f.e		4
65.o	even	12	4		2535.1.x.e		8
65.q	odd	12	4		2535.1.y.a		8
65.r	odd	12	4		2535.1.y.a		8
65.t	even	12	4		2535.1.x.e		8
195.e	odd	2	1		975.1.g.b		2
195.j	odd	4	2		2535.1.f.e		4
195.s	even	4	2		195.1.e.a	✓	4
195.u	odd	4	2		2535.1.f.e		4
195.bc	odd	12	4		2535.1.x.e		8
195.bf	even	12	4		2535.1.y.a		8
195.bl	even	12	4		2535.1.y.a		8
195.bn	odd	12	4		2535.1.x.e		8
260.p	even	4	2		3120.1.be.e		4
780.w	odd	4	2		3120.1.be.e		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
195.1.e.a	✓	4	5.c	odd	4	2
195.1.e.a	✓	4	15.e	even	4	2
195.1.e.a	✓	4	65.h	odd	4	2
195.1.e.a	✓	4	195.s	even	4	2
975.1.g.b		2	5.b	even	2	1
975.1.g.b		2	15.d	odd	2	1
975.1.g.b		2	65.d	even	2	1
975.1.g.b		2	195.e	odd	2	1
975.1.g.c		2	1.a	even	1	1	trivial
975.1.g.c		2	3.b	odd	2	1	inner
975.1.g.c		2	13.b	even	2	1	inner
975.1.g.c		2	39.d	odd	2	1	CM
2535.1.f.e		4	65.f	even	4	2
2535.1.f.e		4	65.k	even	4	2
2535.1.f.e		4	195.j	odd	4	2
2535.1.f.e		4	195.u	odd	4	2
2535.1.x.e		8	65.o	even	12	4
2535.1.x.e		8	65.t	even	12	4
2535.1.x.e		8	195.bc	odd	12	4
2535.1.x.e		8	195.bn	odd	12	4
2535.1.y.a		8	65.q	odd	12	4
2535.1.y.a		8	65.r	odd	12	4
2535.1.y.a		8	195.bf	even	12	4
2535.1.y.a		8	195.bl	even	12	4
3120.1.be.e		4	20.e	even	4	2
3120.1.be.e		4	60.l	odd	4	2
3120.1.be.e		4	260.p	even	4	2
3120.1.be.e		4	780.w	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(975, [\chi])\):

\( T_{2}^{2} - 2 \)

\( T_{127} + 2 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 2 \)
$3$	\( (T - 1)^{2} \)
$5$	\( T^{2} \)
$7$	\( T^{2} \)
$11$	\( T^{2} - 2 \)