Newform orbit 8712.2.a.bm

• Label: 8712.2.a.bm
• Level: $8712$
• Weight: $2$
• Character orbit: 8712.a
• Self dual: yes
• Analytic conductor: $69.566$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 8712 = 2^{3} \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8712.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(69.5656702409\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 88)
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.

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

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

−0.618034
1.61803

0

0

0

−0.618034

0

−1.61803

0

0

0

1.2

0

0

0

1.61803

0

0.618034

0

0

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(1\)
\(3\)	\(-1\)
\(11\)	\(-1\)

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	8712.2.a.bm		2
3.b	odd	2	1		968.2.a.h		2
11.b	odd	2	1		8712.2.a.bp		2
11.d	odd	10	2		792.2.r.b		4
12.b	even	2	1		1936.2.a.u		2
24.f	even	2	1		7744.2.a.cc		2
24.h	odd	2	1		7744.2.a.cq		2
33.d	even	2	1		968.2.a.i		2
33.f	even	10	2		88.2.i.a	✓	4
33.f	even	10	2		968.2.i.d		4
33.h	odd	10	2		968.2.i.c		4
33.h	odd	10	2		968.2.i.k		4
132.d	odd	2	1		1936.2.a.t		2
132.n	odd	10	2		176.2.m.a		4
264.m	even	2	1		7744.2.a.cr		2
264.p	odd	2	1		7744.2.a.cb		2
264.r	odd	10	2		704.2.m.g		4
264.u	even	10	2		704.2.m.b		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
88.2.i.a	✓	4	33.f	even	10	2
176.2.m.a		4	132.n	odd	10	2
704.2.m.b		4	264.u	even	10	2
704.2.m.g		4	264.r	odd	10	2
792.2.r.b		4	11.d	odd	10	2
968.2.a.h		2	3.b	odd	2	1
968.2.a.i		2	33.d	even	2	1
968.2.i.c		4	33.h	odd	10	2
968.2.i.d		4	33.f	even	10	2
968.2.i.k		4	33.h	odd	10	2
1936.2.a.t		2	132.d	odd	2	1
1936.2.a.u		2	12.b	even	2	1
7744.2.a.cb		2	264.p	odd	2	1
7744.2.a.cc		2	24.f	even	2	1
7744.2.a.cq		2	24.h	odd	2	1
7744.2.a.cr		2	264.m	even	2	1
8712.2.a.bm		2	1.a	even	1	1	trivial
8712.2.a.bp		2	11.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8712))\):

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

\( T_{7}^{2} + T_{7} - 1 \)

\( T_{13}^{2} - 5T_{13} - 5 \)

\( T_{17}^{2} - 3T_{17} - 9 \)

Hecke characteristic polynomials

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