Newform orbit 86.2.a.a

• Label: 86.2.a.a
• Level: $86$
• Weight: $2$
• Character orbit: 86.a
• Self dual: yes
• Analytic conductor: $0.687$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 86 = 2 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	86.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.686713457383\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{21}) \)
Defining polynomial:	\( x^{2} - x - 5 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{21})\). We also show the integral \(q\)-expansion of the trace form.

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

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

2.79129
−1.79129

−1.00000

−2.79129

1.00000

3.79129

2.79129

2.00000

−1.00000

4.79129

−3.79129

1.2

−1.00000

1.79129

1.00000

−0.791288

−1.79129

2.00000

−1.00000

0.208712

0.791288

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(1\)
\(43\)	\(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	86.2.a.a	✓	2
3.b	odd	2	1		774.2.a.l		2
4.b	odd	2	1		688.2.a.g		2
5.b	even	2	1		2150.2.a.z		2
5.c	odd	4	2		2150.2.b.p		4
7.b	odd	2	1		4214.2.a.i		2
8.b	even	2	1		2752.2.a.n		2
8.d	odd	2	1		2752.2.a.i		2
12.b	even	2	1		6192.2.a.be		2
43.b	odd	2	1		3698.2.a.f		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
86.2.a.a	✓	2	1.a	even	1	1	trivial
688.2.a.g		2	4.b	odd	2	1
774.2.a.l		2	3.b	odd	2	1
2150.2.a.z		2	5.b	even	2	1
2150.2.b.p		4	5.c	odd	4	2
2752.2.a.i		2	8.d	odd	2	1
2752.2.a.n		2	8.b	even	2	1
3698.2.a.f		2	43.b	odd	2	1
4214.2.a.i		2	7.b	odd	2	1
6192.2.a.be		2	12.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} - 5 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(86))\).

Hecke characteristic polynomials

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