Newform orbit 975.2.i.f

• Label: 975.2.i.f
• Level: $975$
• Weight: $2$
• Character orbit: 975.i
• Analytic conductor: $7.785$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(7.78541419707\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\)	\(301\)	\(326\)	\(352\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(1\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

451.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0.500000

−

0.866025i

−0.500000

+

0.866025i

0.500000

+

0.866025i

0

0.500000

+

0.866025i

1.00000

+

1.73205i

3.00000

−0.500000

−

0.866025i

0

601.1

0.500000

+

0.866025i

−0.500000

−

0.866025i

0.500000

−

0.866025i

0

0.500000

−

0.866025i

1.00000

−

1.73205i

3.00000

−0.500000

+

0.866025i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	975.2.i.f		2
5.b	even	2	1		39.2.e.a	✓	2
5.c	odd	4	2		975.2.bb.d		4
13.c	even	3	1	inner	975.2.i.f		2
15.d	odd	2	1		117.2.g.b		2
20.d	odd	2	1		624.2.q.c		2
60.h	even	2	1		1872.2.t.j		2
65.d	even	2	1		507.2.e.c		2
65.g	odd	4	2		507.2.j.d		4
65.l	even	6	1		507.2.a.b		1
65.l	even	6	1		507.2.e.c		2
65.n	even	6	1		39.2.e.a	✓	2
65.n	even	6	1		507.2.a.c		1
65.q	odd	12	2		975.2.bb.d		4
65.s	odd	12	2		507.2.b.b		2
65.s	odd	12	2		507.2.j.d		4
195.x	odd	6	1		117.2.g.b		2
195.x	odd	6	1		1521.2.a.a		1
195.y	odd	6	1		1521.2.a.d		1
195.bh	even	12	2		1521.2.b.c		2
260.v	odd	6	1		624.2.q.c		2
260.v	odd	6	1		8112.2.a.w		1
260.w	odd	6	1		8112.2.a.bc		1
780.br	even	6	1		1872.2.t.j		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
39.2.e.a	✓	2	5.b	even	2	1
39.2.e.a	✓	2	65.n	even	6	1
117.2.g.b		2	15.d	odd	2	1
117.2.g.b		2	195.x	odd	6	1
507.2.a.b		1	65.l	even	6	1
507.2.a.c		1	65.n	even	6	1
507.2.b.b		2	65.s	odd	12	2
507.2.e.c		2	65.d	even	2	1
507.2.e.c		2	65.l	even	6	1
507.2.j.d		4	65.g	odd	4	2
507.2.j.d		4	65.s	odd	12	2
624.2.q.c		2	20.d	odd	2	1
624.2.q.c		2	260.v	odd	6	1
975.2.i.f		2	1.a	even	1	1	trivial
975.2.i.f		2	13.c	even	3	1	inner
975.2.bb.d		4	5.c	odd	4	2
975.2.bb.d		4	65.q	odd	12	2
1521.2.a.a		1	195.x	odd	6	1
1521.2.a.d		1	195.y	odd	6	1
1521.2.b.c		2	195.bh	even	12	2
1872.2.t.j		2	60.h	even	2	1
1872.2.t.j		2	780.br	even	6	1
8112.2.a.w		1	260.v	odd	6	1
8112.2.a.bc		1	260.w	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\):

\( T_{2}^{2} - T_{2} + 1 \)

\( T_{7}^{2} - 2T_{7} + 4 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - T + 1 \)
$3$	\( T^{2} + T + 1 \)
$5$	\( T^{2} \)
$7$	\( T^{2} - 2T + 4 \)
$11$	\( T^{2} - 2T + 4 \)