Newform orbit 95.2.b.b

• Label: 95.2.b.b
• Level: $95$
• Weight: $2$
• Character orbit: 95.b
• Analytic conductor: $0.759$
• Analytic rank: $0$
• Dimension: $6$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 95 = 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	95.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.758578819202\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	6.0.16516096.1
Defining polynomial:	\( x^{6} + 9x^{4} + 13x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 9x^{4} + 13x^{2} + 1 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{4} + 8\nu^{2} + 5 ) / 2 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{5} + \nu^{4} + 10\nu^{3} + 6\nu^{2} + 19\nu - 1 ) / 4 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{5} - \nu^{4} + 10\nu^{3} - 6\nu^{2} + 19\nu + 1 ) / 4 \)
\(\beta_{5}\)	\(=\)	\( ( -\nu^{5} - 8\nu^{3} - 7\nu ) / 2 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).

\(n\)	\(21\)	\(77\)
\(\chi(n)\)	\(1\)	\(-1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

39.1

		1.30397i
	−	2.68667i
	−	0.285442i
		0.285442i
		2.68667i
	−	1.30397i

−

2.41987i

−

0.537080i

−3.85577

−2.07772

+

0.826491i

−1.29966

−

3.18676i

4.49073i

2.71155

2.00000

+

5.02781i

39.2

−

1.82254i

2.31446i

−1.32164

1.94827

+

1.09737i

4.21819

−

1.45033i

−

1.23634i

−2.35673

2.00000

−

3.55080i

39.3

−

0.906968i

−

3.21789i

1.17741

−0.370556

+

2.20515i

−2.91852

2.59637i

−

2.88181i

−7.35482

2.00000

+

0.336083i

39.4

0.906968i

3.21789i

1.17741

−0.370556

−

2.20515i

−2.91852

−

2.59637i

2.88181i

−7.35482

2.00000

−

0.336083i

39.5

1.82254i

−

2.31446i

−1.32164

1.94827

−

1.09737i

4.21819

1.45033i

1.23634i

−2.35673

2.00000

+

3.55080i

39.6

2.41987i

0.537080i

−3.85577

−2.07772

−

0.826491i

−1.29966

3.18676i

−

4.49073i

2.71155

2.00000

−

5.02781i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	95.2.b.b	✓	6
3.b	odd	2	1		855.2.c.d		6
4.b	odd	2	1		1520.2.d.h		6
5.b	even	2	1	inner	95.2.b.b	✓	6
5.c	odd	4	2		475.2.a.j		6
15.d	odd	2	1		855.2.c.d		6
15.e	even	4	2		4275.2.a.br		6
19.b	odd	2	1		1805.2.b.e		6
20.d	odd	2	1		1520.2.d.h		6
20.e	even	4	2		7600.2.a.ck		6
95.d	odd	2	1		1805.2.b.e		6
95.g	even	4	2		9025.2.a.bx		6

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
95.2.b.b	✓	6	1.a	even	1	1	trivial
95.2.b.b	✓	6	5.b	even	2	1	inner
475.2.a.j		6	5.c	odd	4	2
855.2.c.d		6	3.b	odd	2	1
855.2.c.d		6	15.d	odd	2	1
1520.2.d.h		6	4.b	odd	2	1
1520.2.d.h		6	20.d	odd	2	1
1805.2.b.e		6	19.b	odd	2	1
1805.2.b.e		6	95.d	odd	2	1
4275.2.a.br		6	15.e	even	4	2
7600.2.a.ck		6	20.e	even	4	2
9025.2.a.bx		6	95.g	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 10T_{2}^{4} + 27T_{2}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(95, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{6} + 10 T^{4} + 27 T^{2} + 16 \)
$3$	\( T^{6} + 16 T^{4} + 60 T^{2} + 16 \)
$5$	T^6 + T^5 - T^4 - 2*T^3 - 5*T^2 + 25*T + 125
$7$	T^6 + 19*T^4 + 104*T^2 + 144
$11$	\( (T^{3} - T^{2} - 16 T + 12)^{2} \)