Newform orbit 832.4.e.a

• Label: 832.4.e.a
• Level: $832$
• Weight: $4$
• Character orbit: 832.e
• Analytic conductor: $49.090$
• Analytic rank: $0$
• Dimension: $4$
• CM discriminant: -104
• Inner twists: $8$

Newspace parameters

Level:	$N$	$=$	$832 = 2^{6} \cdot 13$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	832.e (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$49.0895891248$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
Defining polynomial:	$x^{4} + 7x^{2} + 9$
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2^{4}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + b1 * q^3 - 6*b3 * q^5 + 5*b2 * q^7 + 23 * q^9 + 13*b3 * q^13 - 6*b2 * q^15 + 114 * q^17 - 20*b3 * q^21 + 343 * q^25 + 50*b1 * q^27 + 41*b2 * q^31 - 390*b1 * q^35 + 82*b3 * q^37 + 13*b2 * q^39 + 109*b1 * q^43 - 138*b3 * q^45 - 39*b2 * q^47 - 957 * q^49 + 114*b1 * q^51 + 115*b2 * q^63 - 1014 * q^65 + 141*b2 * q^71 + 343*b1 * q^75 + 421 * q^81 - 684*b3 * q^85 + 845*b1 * q^91 - 164*b3 * q^93
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 92 q^{9} + 456 q^{17} + 1372 q^{25} - 3828 q^{49} - 4056 q^{65} + 1684 q^{81}+O(q^{100})$

Basis of coefficient ring in terms of a root $\nu$ of $x^{4} + 7x^{2} + 9$ :

$\beta_{1}$	$=$	$( 2\nu^{3} + 8\nu ) / 3$
$\beta_{2}$	$=$	$( 2\nu^{3} + 20\nu ) / 3$
$\beta_{3}$	$=$	$2\nu^{2} + 7$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/832\mathbb{Z}\right)^\times$.

$n$	$261$	$703$	$769$
$\chi(n)$	$-1$	$1$	$-1$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

545.1

	−	1.30278i
		2.30278i
		1.30278i
	−	2.30278i

0

−

2.00000i

0

−21.6333

0

−

36.0555i

0

23.0000

0

545.2

0

−

2.00000i

0

21.6333

0

36.0555i

0

23.0000

0

545.3

0

2.00000i

0

−21.6333

0

36.0555i

0

23.0000

0

545.4

0

2.00000i

0

21.6333

0

−

36.0555i

0

23.0000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
104.h	odd	2	1	CM by $\Q(\sqrt{-26})$
4.b	odd	2	1	inner
8.b	even	2	1	inner
8.d	odd	2	1	inner
13.b	even	2	1	inner
52.b	odd	2	1	inner
104.e	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	832.4.e.a	✓	4
4.b	odd	2	1	inner	832.4.e.a	✓	4
8.b	even	2	1	inner	832.4.e.a	✓	4
8.d	odd	2	1	inner	832.4.e.a	✓	4
13.b	even	2	1	inner	832.4.e.a	✓	4
52.b	odd	2	1	inner	832.4.e.a	✓	4
104.e	even	2	1	inner	832.4.e.a	✓	4
104.h	odd	2	1	CM	832.4.e.a	✓	4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
832.4.e.a	✓	4	1.a	even	1	1	trivial
832.4.e.a	✓	4	4.b	odd	2	1	inner
832.4.e.a	✓	4	8.b	even	2	1	inner
832.4.e.a	✓	4	8.d	odd	2	1	inner
832.4.e.a	✓	4	13.b	even	2	1	inner
832.4.e.a	✓	4	52.b	odd	2	1	inner
832.4.e.a	✓	4	104.e	even	2	1	inner
832.4.e.a	✓	4	104.h	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 $S_{4}^{\mathrm{new}}(832, [\chi])$:

$T_{3}^{2} + 4$

$T_{5}^{2} - 468$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4}$
$3$	$(T^{2} + 4)^{2}$
$5$	$(T^{2} - 468)^{2}$
$7$	$(T^{2} + 1300)^{2}$
$11$	$T^{4}$