Newform orbit 975.2.bc.h

• Label: 975.2.bc.h
• Level: $975$
• Weight: $2$
• Character orbit: 975.bc
• Analytic conductor: $7.785$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.bc (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.78541419707\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{12})\)
Defining polynomial:	\( x^{4} - x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 195)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (z^3 - z^2 - z + 2) * q^2 + z^2 * q^3 + (4*z^3 - 2*z^2 - 2*z + 2) * q^4 + (z^2 - z + 1) * q^6 + (2*z^2 + z + 2) * q^7 + (6*z^3 - 4*z^2 + 2) * q^8 + (z^2 - 1) * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{2} + 2 q^{3} + 4 q^{4} + 6 q^{6} + 12 q^{7} - 2 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(301\)	\(326\)	\(352\)
\(\chi(n)\)	\(1 - \zeta_{12}^{2}\)	\(1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

751.1

0.866025	+	0.500000i
−0.866025	−	0.500000i
0.866025	−	0.500000i
−0.866025	+	0.500000i

0.633975

−

0.366025i

0.500000

+

0.866025i

−0.732051

+

1.26795i

0

0.633975

+

0.366025i

3.86603

+

2.23205i

2.53590i

−0.500000

+

0.866025i

0

751.2

2.36603

−

1.36603i

0.500000

+

0.866025i

2.73205

−

4.73205i

0

2.36603

+

1.36603i

2.13397

+

1.23205i

−

9.46410i

−0.500000

+

0.866025i

0

901.1

0.633975

+

0.366025i

0.500000

−

0.866025i

−0.732051

−

1.26795i

0

0.633975

−

0.366025i

3.86603

−

2.23205i

−

2.53590i

−0.500000

−

0.866025i

0

901.2

2.36603

+

1.36603i

0.500000

−

0.866025i

2.73205

+

4.73205i

0

2.36603

−

1.36603i

2.13397

−

1.23205i

9.46410i

−0.500000

−

0.866025i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.e	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	975.2.bc.h		4
5.b	even	2	1		195.2.bb.a	✓	4
5.c	odd	4	1		975.2.w.a		4
5.c	odd	4	1		975.2.w.f		4
13.e	even	6	1	inner	975.2.bc.h		4
15.d	odd	2	1		585.2.bu.a		4
65.l	even	6	1		195.2.bb.a	✓	4
65.r	odd	12	1		975.2.w.a		4
65.r	odd	12	1		975.2.w.f		4
65.s	odd	12	1		2535.2.a.n		2
65.s	odd	12	1		2535.2.a.s		2
195.y	odd	6	1		585.2.bu.a		4
195.bh	even	12	1		7605.2.a.y		2
195.bh	even	12	1		7605.2.a.bk		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
195.2.bb.a	✓	4	5.b	even	2	1
195.2.bb.a	✓	4	65.l	even	6	1
585.2.bu.a		4	15.d	odd	2	1
585.2.bu.a		4	195.y	odd	6	1
975.2.w.a		4	5.c	odd	4	1
975.2.w.a		4	65.r	odd	12	1
975.2.w.f		4	5.c	odd	4	1
975.2.w.f		4	65.r	odd	12	1
975.2.bc.h		4	1.a	even	1	1	trivial
975.2.bc.h		4	13.e	even	6	1	inner
2535.2.a.n		2	65.s	odd	12	1
2535.2.a.s		2	65.s	odd	12	1
7605.2.a.y		2	195.bh	even	12	1
7605.2.a.bk		2	195.bh	even	12	1

Hecke kernels

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

\( T_{2}^{4} - 6T_{2}^{3} + 14T_{2}^{2} - 12T_{2} + 4 \)

\( T_{7}^{4} - 12T_{7}^{3} + 59T_{7}^{2} - 132T_{7} + 121 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^4 - 6*T^3 + 14*T^2 - 12*T + 4
$3$	\( (T^{2} - T + 1)^{2} \)
$5$	\( T^{4} \)
$7$	T^4 - 12*T^3 + 59*T^2 - 132*T + 121
$11$	\( (T^{2} + 6 T + 12)^{2} \)