Newform orbit 8410.2.a.p

• Label: 8410.2.a.p
• Level: $8410$
• Weight: $2$
• Character orbit: 8410.a
• Self dual: yes
• Analytic conductor: $67.154$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 8410 = 2 \cdot 5 \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8410.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(67.1541880999\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 290)
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.

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

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

−0.618034
1.61803

−1.00000

0.381966

1.00000

−1.00000

−0.381966

−0.618034

−1.00000

−2.85410

1.00000

1.2

−1.00000

2.61803

1.00000

−1.00000

−2.61803

1.61803

−1.00000

3.85410

1.00000

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(1\)
\(5\)	\(1\)
\(29\)	\(-1\)

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	8410.2.a.p		2
29.b	even	2	1		8410.2.a.q		2
29.c	odd	4	2		290.2.c.c	✓	4
87.f	even	4	2		2610.2.f.d		4
116.e	even	4	2		2320.2.g.g		4
145.e	even	4	2		1450.2.d.h		4
145.f	odd	4	2		1450.2.c.d		4
145.j	even	4	2		1450.2.d.e		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
290.2.c.c	✓	4	29.c	odd	4	2
1450.2.c.d		4	145.f	odd	4	2
1450.2.d.e		4	145.j	even	4	2
1450.2.d.h		4	145.e	even	4	2
2320.2.g.g		4	116.e	even	4	2
2610.2.f.d		4	87.f	even	4	2
8410.2.a.p		2	1.a	even	1	1	trivial
8410.2.a.q		2	29.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8410))\):

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

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

\( T_{11}^{2} - 20 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 1)^{2} \)
$3$	\( T^{2} - 3T + 1 \)
$5$	\( (T + 1)^{2} \)
$7$	\( T^{2} - T - 1 \)
$11$	\( T^{2} - 20 \)