Newform orbit 961.2.d.h

• Label: 961.2.d.h
• Level: $961$
• Weight: $2$
• Character orbit: 961.d
• Analytic conductor: $7.674$
• Analytic rank: $0$
• Dimension: $8$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 961 = 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	961.d (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.67362363425\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{5})\)
Coefficient field:	8.0.64000000.2
Defining polynomial:	\( x^{8} + 2x^{6} + 4x^{4} + 8x^{2} + 16 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (\beta_{6} + 2 \beta_{4} + \beta_{2}) q^{2} + (\beta_{7} + \beta_{5} + \beta_1) q^{3} + ( - 3 \beta_{4} - 3 \beta_{2}) q^{4} + ( - 2 \beta_{6} - 2 \beta_{4} - 1) q^{5} + ( - \beta_{7} - \beta_{5} - \beta_{3}) q^{6} + (\beta_{6} + \beta_{4} + \beta_{2} + 1) q^{7} + (4 \beta_{4} + 5 \beta_{2} + 4) q^{8} + ( - 2 \beta_{4} - \beta_{2} - 2) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 12 q^{4} + 2 q^{7} + 14 q^{8} - 10 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 2x^{6} + 4x^{4} + 8x^{2} + 16 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{2} ) / 2 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{3} ) / 2 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{4} ) / 4 \)
\(\beta_{5}\)	\(=\)	\( ( \nu^{5} ) / 4 \)
\(\beta_{6}\)	\(=\)	\( ( \nu^{6} ) / 8 \)
\(\beta_{7}\)	\(=\)	\( ( \nu^{7} ) / 8 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/961\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(\beta_{2}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

374.1

1.14412	−	0.831254i
−1.14412	+	0.831254i
1.14412	+	0.831254i
−1.14412	−	0.831254i
−0.437016	+	1.34500i
0.437016	−	1.34500i
−0.437016	−	1.34500i
0.437016	+	1.34500i

−2.11803

−

1.53884i

−0.707107

+

0.513743i

1.50000

+

4.61653i

2.23607

2.28825

−0.309017

−

0.951057i

2.30902

−

7.10642i

−0.690983

+

2.12663i

−4.73607

−

3.44095i

374.2

−2.11803

−

1.53884i

0.707107

−

0.513743i

1.50000

+

4.61653i

2.23607

−2.28825

−0.309017

−

0.951057i

2.30902

−

7.10642i

−0.690983

+

2.12663i

−4.73607

−

3.44095i

388.1

−2.11803

+

1.53884i

−0.707107

−

0.513743i

1.50000

−

4.61653i

2.23607

2.28825

−0.309017

+

0.951057i

2.30902

+

7.10642i

−0.690983

−

2.12663i

−4.73607

+

3.44095i

388.2

−2.11803

+

1.53884i

0.707107

+

0.513743i

1.50000

−

4.61653i

2.23607

−2.28825

−0.309017

+

0.951057i

2.30902

+

7.10642i

−0.690983

−

2.12663i

−4.73607

+

3.44095i

531.1

0.118034

+

0.363271i

−0.707107

+

2.17625i

1.50000

−

1.08981i

−2.23607

−0.874032

0.809017

−

0.587785i

1.19098

+

0.865300i

−1.80902

−

1.31433i

−0.263932

−

0.812299i

531.2

0.118034

+

0.363271i

0.707107

−

2.17625i

1.50000

−

1.08981i

−2.23607

0.874032

0.809017

−

0.587785i

1.19098

+

0.865300i

−1.80902

−

1.31433i

−0.263932

−

0.812299i

628.1

0.118034

−

0.363271i

−0.707107

−

2.17625i

1.50000

+

1.08981i

−2.23607

−0.874032

0.809017

+

0.587785i

1.19098

−

0.865300i

−1.80902

+

1.31433i

−0.263932

+

0.812299i

628.2

0.118034

−

0.363271i

0.707107

+

2.17625i

1.50000

+

1.08981i

−2.23607

0.874032

0.809017

+

0.587785i

1.19098

−

0.865300i

−1.80902

+

1.31433i

−0.263932

+

0.812299i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
31.b	odd	2	1	inner
31.d	even	5	1	inner
31.f	odd	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	961.2.d.h		8
31.b	odd	2	1	inner	961.2.d.h		8
31.c	even	3	2		961.2.g.i		16
31.d	even	5	1		961.2.a.h	✓	4
31.d	even	5	1	inner	961.2.d.h		8
31.d	even	5	2		961.2.d.j		8
31.e	odd	6	2		961.2.g.i		16
31.f	odd	10	1		961.2.a.h	✓	4
31.f	odd	10	1	inner	961.2.d.h		8
31.f	odd	10	2		961.2.d.j		8
31.g	even	15	2		961.2.c.h		8
31.g	even	15	2		961.2.g.i		16
31.g	even	15	4		961.2.g.p		16
31.h	odd	30	2		961.2.c.h		8
31.h	odd	30	2		961.2.g.i		16
31.h	odd	30	4		961.2.g.p		16
93.k	even	10	1		8649.2.a.r		4
93.l	odd	10	1		8649.2.a.r		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
961.2.a.h	✓	4	31.d	even	5	1
961.2.a.h	✓	4	31.f	odd	10	1
961.2.c.h		8	31.g	even	15	2
961.2.c.h		8	31.h	odd	30	2
961.2.d.h		8	1.a	even	1	1	trivial
961.2.d.h		8	31.b	odd	2	1	inner
961.2.d.h		8	31.d	even	5	1	inner
961.2.d.h		8	31.f	odd	10	1	inner
961.2.d.j		8	31.d	even	5	2
961.2.d.j		8	31.f	odd	10	2
961.2.g.i		16	31.c	even	3	2
961.2.g.i		16	31.e	odd	6	2
961.2.g.i		16	31.g	even	15	2
961.2.g.i		16	31.h	odd	30	2
961.2.g.p		16	31.g	even	15	4
961.2.g.p		16	31.h	odd	30	4
8649.2.a.r		4	93.k	even	10	1
8649.2.a.r		4	93.l	odd	10	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(961, [\chi])\):

\( T_{2}^{4} + 4T_{2}^{3} + 6T_{2}^{2} - T_{2} + 1 \)

\( T_{3}^{8} + 8T_{3}^{6} + 24T_{3}^{4} - 8T_{3}^{2} + 16 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	(T^4 + 4*T^3 + 6*T^2 - T + 1)^2
$3$	T^8 + 8*T^6 + 24*T^4 - 8*T^2 + 16
$5$	\( (T^{2} - 5)^{4} \)
$7$	(T^4 - T^3 + T^2 - T + 1)^2
$11$	T^8 + 18*T^6 + 324*T^4 + 5832*T^2 + 104976