Newform orbit 85.2.d.a

• Label: 85.2.d.a
• Level: $85$
• Weight: $2$
• Character orbit: 85.d
• Analytic conductor: $0.679$
• Analytic rank: $0$
• Dimension: $6$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 85 = 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	85.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.678728417181\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	6.0.350464.1
Defining polynomial:	\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2 \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) :

\(\beta_{1}\)	\(=\)	\( ( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23 \)
\(\beta_{2}\)	\(=\)	\( ( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23 \)
\(\beta_{3}\)	\(=\)	\( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \)
\(\beta_{4}\)	\(=\)	\( ( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23 \)
\(\beta_{5}\)	\(=\)	\( ( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/85\mathbb{Z}\right)^\times\).

\(n\)	\(52\)	\(71\)
\(\chi(n)\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

16.1

−0.854638	+	0.854638i
−0.854638	−	0.854638i
1.45161	+	1.45161i
1.45161	−	1.45161i
0.403032	−	0.403032i
0.403032	+	0.403032i

−2.17009

−

0.539189i

2.70928

1.00000i

1.17009i

−

4.87936i

−1.53919

2.70928

−

2.17009i

16.2

−2.17009

0.539189i

2.70928

−

1.00000i

−

1.17009i

4.87936i

−1.53919

2.70928

2.17009i

16.3

−0.311108

−

2.21432i

−1.90321

−

1.00000i

0.688892i

−

1.59210i

1.21432

−1.90321

0.311108i

16.4

−0.311108

2.21432i

−1.90321

1.00000i

−

0.688892i

1.59210i

1.21432

−1.90321

−

0.311108i

16.5

1.48119

−

1.67513i

0.193937

1.00000i

−

2.48119i

1.28726i

−2.67513

0.193937

1.48119i

16.6

1.48119

1.67513i

0.193937

−

1.00000i

2.48119i

−

1.28726i

−2.67513

0.193937

−

1.48119i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	85.2.d.a	✓	6
3.b	odd	2	1		765.2.g.b		6
4.b	odd	2	1		1360.2.c.f		6
5.b	even	2	1		425.2.d.c		6
5.c	odd	4	1		425.2.c.a		6
5.c	odd	4	1		425.2.c.b		6
17.b	even	2	1	inner	85.2.d.a	✓	6
17.c	even	4	1		1445.2.a.j		3
17.c	even	4	1		1445.2.a.k		3
51.c	odd	2	1		765.2.g.b		6
68.d	odd	2	1		1360.2.c.f		6
85.c	even	2	1		425.2.d.c		6
85.g	odd	4	1		425.2.c.a		6
85.g	odd	4	1		425.2.c.b		6
85.j	even	4	1		7225.2.a.q		3
85.j	even	4	1		7225.2.a.r		3

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
85.2.d.a	✓	6	1.a	even	1	1	trivial
85.2.d.a	✓	6	17.b	even	2	1	inner
425.2.c.a		6	5.c	odd	4	1
425.2.c.a		6	85.g	odd	4	1
425.2.c.b		6	5.c	odd	4	1
425.2.c.b		6	85.g	odd	4	1
425.2.d.c		6	5.b	even	2	1
425.2.d.c		6	85.c	even	2	1
765.2.g.b		6	3.b	odd	2	1
765.2.g.b		6	51.c	odd	2	1
1360.2.c.f		6	4.b	odd	2	1
1360.2.c.f		6	68.d	odd	2	1
1445.2.a.j		3	17.c	even	4	1
1445.2.a.k		3	17.c	even	4	1
7225.2.a.q		3	85.j	even	4	1
7225.2.a.r		3	85.j	even	4	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(85, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{3} + T^{2} - 3 T - 1)^{2} \)
$3$	T^6 + 8*T^4 + 16*T^2 + 4
$5$	\( (T^{2} + 1)^{3} \)
$7$	T^6 + 28*T^4 + 104*T^2 + 100
$11$	T^6 + 12*T^4 + 20*T^2 + 4