Newform orbit 8649.2.a.g

• Label: 8649.2.a.g
• Level: $8649$
• Weight: $2$
• Character orbit: 8649.a
• Self dual: yes
• Analytic conductor: $69.063$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 8649 = 3^{2} \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8649.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(69.0626127082\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 31)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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

1.61803
−0.618034

−1.61803

0

0.618034

0.381966

0

3.00000

2.23607

0

−0.618034

1.2

0.618034

0

−1.61803

2.61803

0

3.00000

−2.23607

0

1.61803

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\(-1\)
\(31\)	\(1\)

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	8649.2.a.g		2
3.b	odd	2	1		961.2.a.d		2
31.b	odd	2	1		8649.2.a.f		2
31.d	even	5	2		279.2.i.a		4
93.c	even	2	1		961.2.a.e		2
93.g	even	6	2		961.2.c.d		4
93.h	odd	6	2		961.2.c.f		4
93.k	even	10	2		961.2.d.b		4
93.k	even	10	2		961.2.d.e		4
93.l	odd	10	2		31.2.d.a	✓	4
93.l	odd	10	2		961.2.d.f		4
93.o	odd	30	4		961.2.g.b		8
93.o	odd	30	4		961.2.g.f		8
93.p	even	30	4		961.2.g.c		8
93.p	even	30	4		961.2.g.g		8
372.t	even	10	2		496.2.n.b		4
465.x	odd	10	2		775.2.k.c		4
465.bj	even	20	4		775.2.bf.a		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
31.2.d.a	✓	4	93.l	odd	10	2
279.2.i.a		4	31.d	even	5	2
496.2.n.b		4	372.t	even	10	2
775.2.k.c		4	465.x	odd	10	2
775.2.bf.a		8	465.bj	even	20	4
961.2.a.d		2	3.b	odd	2	1
961.2.a.e		2	93.c	even	2	1
961.2.c.d		4	93.g	even	6	2
961.2.c.f		4	93.h	odd	6	2
961.2.d.b		4	93.k	even	10	2
961.2.d.e		4	93.k	even	10	2
961.2.d.f		4	93.l	odd	10	2
961.2.g.b		8	93.o	odd	30	4
961.2.g.c		8	93.p	even	30	4
961.2.g.f		8	93.o	odd	30	4
961.2.g.g		8	93.p	even	30	4
8649.2.a.f		2	31.b	odd	2	1
8649.2.a.g		2	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8649))\):

\( T_{2}^{2} + T_{2} - 1 \)

\( T_{5}^{2} - 3T_{5} + 1 \)

\( T_{7} - 3 \)

\( T_{11}^{2} - 6T_{11} + 4 \)

\( T_{13}^{2} - 3T_{13} - 9 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + T - 1 \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 3T + 1 \)
$7$	\( (T - 3)^{2} \)
$11$	\( T^{2} - 6T + 4 \)