Newform orbit 8424.2.a.n

• Label: 8424.2.a.n
• Level: $8424$
• Weight: $2$
• Character orbit: 8424.a
• Self dual: yes
• Analytic conductor: $67.266$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$8424 = 2^{3} \cdot 3^{4} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	8424.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$67.2659786627$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{3})$
Defining polynomial:	$x^{2} - 3$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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 $\beta = \sqrt{3}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + (b + 1) * q^5 - b * q^7 + (b - 2) * q^11 - q^13 + (-2*b - 4) * q^17 + (2*b - 1) * q^19 + (-b + 1) * q^23 + (2*b - 1) * q^25 + 9 * q^29 + (-2*b + 2) * q^31 + (-b - 3) * q^35 + (-2*b + 6) * q^37 + (-3*b - 3) * q^41 + (-4*b - 2) * q^43 + 6*b * q^47 - 4 * q^49 + (4*b + 1) * q^53 + (-b + 1) * q^55 + (-3*b - 8) * q^59 + (-3*b + 10) * q^61 + (-b - 1) * q^65 + (2*b - 4) * q^67 + (4*b + 5) * q^71 + (2*b - 2) * q^73 + (2*b - 3) * q^77 + (-5*b + 1) * q^79 + (-5*b - 4) * q^83 + (-6*b - 10) * q^85 + (5*b - 9) * q^89 + b * q^91 + (b + 5) * q^95 + (-3*b - 1) * q^97
$\operatorname{Tr}(f)(q)$	$=$	2 * q + 2 * q^5 - 4 * q^11 - 2 * q^13 - 8 * q^17 - 2 * q^19 + 2 * q^23 - 2 * q^25 + 18 * q^29 + 4 * q^31 - 6 * q^35 + 12 * q^37 - 6 * q^41 - 4 * q^43 - 8 * q^49 + 2 * q^53 + 2 * q^55 - 16 * q^59 + 20 * q^61 - 2 * q^65 - 8 * q^67 + 10 * q^71 - 4 * q^73 - 6 * q^77 + 2 * q^79 - 8 * q^83 - 20 * q^85 - 18 * q^89 + 10 * q^95 - 2 * q^97

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

0

0

0

−0.732051

0

1.73205

0

0

0

1.2

0

0

0

2.73205

0

−1.73205

0

0

0

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$-1$
$13$	$+1$

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	8424.2.a.n	yes	2
3.b	odd	2	1		8424.2.a.j	✓	2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
8424.2.a.j	✓	2	3.b	odd	2	1
8424.2.a.n	yes	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 $S_{2}^{\mathrm{new}}(\Gamma_0(8424))$:

$T_{5}^{2} - 2T_{5} - 2$

$T_{7}^{2} - 3$

Hecke characteristic polynomials

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