Newform orbit 8450.2.a.bg

• Label: 8450.2.a.bg
• Level: $8450$
• Weight: $2$
• Character orbit: 8450.a
• Self dual: yes
• Analytic conductor: $67.474$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8450.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(67.4735897080\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{3}) \)
Defining polynomial:	\( x^{2} - 3 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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

−1.73205
1.73205

1.00000

−2.73205

1.00000

0

−2.73205

4.46410

1.00000

4.46410

0

1.2

1.00000

0.732051

1.00000

0

0.732051

−2.46410

1.00000

−2.46410

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(-1\)
\(5\)	\(1\)
\(13\)	\(1\)

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	8450.2.a.bg		2
5.b	even	2	1		1690.2.a.l		2
13.b	even	2	1		8450.2.a.z		2
13.c	even	3	2		650.2.e.f		4
65.d	even	2	1		1690.2.a.o		2
65.g	odd	4	2		1690.2.d.h		4
65.l	even	6	2		1690.2.e.k		4
65.n	even	6	2		130.2.e.d	✓	4
65.q	odd	12	2		650.2.o.a		4
65.q	odd	12	2		650.2.o.e		4
65.s	odd	12	2		1690.2.l.c		4
65.s	odd	12	2		1690.2.l.d		4
195.x	odd	6	2		1170.2.i.n		4
260.v	odd	6	2		1040.2.q.p		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
130.2.e.d	✓	4	65.n	even	6	2
650.2.e.f		4	13.c	even	3	2
650.2.o.a		4	65.q	odd	12	2
650.2.o.e		4	65.q	odd	12	2
1040.2.q.p		4	260.v	odd	6	2
1170.2.i.n		4	195.x	odd	6	2
1690.2.a.l		2	5.b	even	2	1
1690.2.a.o		2	65.d	even	2	1
1690.2.d.h		4	65.g	odd	4	2
1690.2.e.k		4	65.l	even	6	2
1690.2.l.c		4	65.s	odd	12	2
1690.2.l.d		4	65.s	odd	12	2
8450.2.a.z		2	13.b	even	2	1
8450.2.a.bg		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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8450))\):

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

\( T_{7}^{2} - 2T_{7} - 11 \)

\( T_{11}^{2} - 3 \)

\( T_{17}^{2} + 6T_{17} + 6 \)

\( T_{31}^{2} - 10T_{31} + 22 \)

Hecke characteristic polynomials

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