Newform orbit 5408.2.a.bc

• Label: 5408.2.a.bc
• Level: $5408$
• Weight: $2$
• Character orbit: 5408.a
• Self dual: yes
• Analytic conductor: $43.183$
• Analytic rank: $1$
• Dimension: $2$
• CM discriminant: -4
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$5408 = 2^{5} \cdot 13^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	5408.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$43.1830974131$
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:	no (minimal twist has level 416)
Fricke sign:	$+1$
Sato-Tate group:	$N(\mathrm{U}(1))$

$q$-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 + 2) * q^5 - 3 * q^9 + (4*b - 1) * q^17 + (-4*b + 2) * q^25 + (2*b - 5) * q^29 + (b - 6) * q^37 + (-5*b + 4) * q^41 + (3*b - 6) * q^45 - 7 * q^49 + (2*b - 7) * q^53 + (6*b + 5) * q^61 + (3*b + 8) * q^73 + 9 * q^81 + (9*b - 14) * q^85 - 16 * q^89 - 8 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 4 q^{5} - 6 q^{9} - 2 q^{17} + 4 q^{25} - 10 q^{29} - 12 q^{37} + 8 q^{41} - 12 q^{45} - 14 q^{49} - 14 q^{53} + 10 q^{61} + 16 q^{73} + 18 q^{81} - 28 q^{85} - 32 q^{89} - 16 q^{97}+O(q^{100})$

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

0.267949

0

0

0

−3.00000

0

1.2

0

0

0

3.73205

0

0

0

−3.00000

0

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$13$	$-1$

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1})$

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	5408.2.a.bc		2
4.b	odd	2	1	CM	5408.2.a.bc		2
13.b	even	2	1		5408.2.a.r		2
13.f	odd	12	2		416.2.w.b	✓	4
52.b	odd	2	1		5408.2.a.r		2
52.l	even	12	2		416.2.w.b	✓	4
104.u	even	12	2		832.2.w.f		4
104.x	odd	12	2		832.2.w.f		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
416.2.w.b	✓	4	13.f	odd	12	2
416.2.w.b	✓	4	52.l	even	12	2
832.2.w.f		4	104.u	even	12	2
832.2.w.f		4	104.x	odd	12	2
5408.2.a.r		2	13.b	even	2	1
5408.2.a.r		2	52.b	odd	2	1
5408.2.a.bc		2	1.a	even	1	1	trivial
5408.2.a.bc		2	4.b	odd	2	1	CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(5408))$:

$T_{3}$

$T_{5}^{2} - 4T_{5} + 1$

$T_{7}$

$T_{37}^{2} + 12T_{37} + 33$

Hecke characteristic polynomials

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