Newform orbit 1152.3.b.f

• Label: 1152.3.b.f
• Level: $1152$
• Weight: $3$
• Character orbit: 1152.b
• Analytic conductor: $31.390$
• Analytic rank: $0$
• Dimension: $4$
• CM discriminant: -24
• Inner twists: $8$

Newspace parameters

Level:	$N$	$=$	$1152 = 2^{7} \cdot 3^{2}$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	1152.b (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$31.3897264543$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
Defining polynomial:	$x^{4} + 9$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{7}$
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^5 + b2 * q^7 + b3 * q^11 + 21 * q^25 + 25*b1 * q^29 - 5*b2 * q^31 - b3 * q^35 - 47 * q^49 + 47*b1 * q^53 + 4*b2 * q^55 - 6*b3 * q^59 + 50 * q^73 + 96*b1 * q^77 + 15*b2 * q^79 + 5*b3 * q^83 - 190 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 84 q^{25} - 188 q^{49} + 200 q^{73} - 760 q^{97}+O(q^{100})$

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

$\beta_{1}$	$=$	$( 2\nu^{2} ) / 3$
$\beta_{2}$	$=$	$( 4\nu^{3} + 12\nu ) / 3$
$\beta_{3}$	$=$	$( -8\nu^{3} + 24\nu ) / 3$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$1$	$-1$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

703.1

1.22474	−	1.22474i
−1.22474	+	1.22474i
−1.22474	−	1.22474i
1.22474	+	1.22474i

0

0

0

−

2.00000i

0

−

9.79796i

0

0

0

703.2

0

0

0

−

2.00000i

0

9.79796i

0

0

0

703.3

0

0

0

2.00000i

0

−

9.79796i

0

0

0

703.4

0

0

0

2.00000i

0

9.79796i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
24.h	odd	2	1	CM by $\Q(\sqrt{-6})$
3.b	odd	2	1	inner
4.b	odd	2	1	inner
8.b	even	2	1	inner
8.d	odd	2	1	inner
12.b	even	2	1	inner
24.f	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1152.3.b.f	✓	4
3.b	odd	2	1	inner	1152.3.b.f	✓	4
4.b	odd	2	1	inner	1152.3.b.f	✓	4
8.b	even	2	1	inner	1152.3.b.f	✓	4
8.d	odd	2	1	inner	1152.3.b.f	✓	4
12.b	even	2	1	inner	1152.3.b.f	✓	4
16.e	even	4	1		2304.3.g.i		2
16.e	even	4	1		2304.3.g.l		2
16.f	odd	4	1		2304.3.g.i		2
16.f	odd	4	1		2304.3.g.l		2
24.f	even	2	1	inner	1152.3.b.f	✓	4
24.h	odd	2	1	CM	1152.3.b.f	✓	4
48.i	odd	4	1		2304.3.g.i		2
48.i	odd	4	1		2304.3.g.l		2
48.k	even	4	1		2304.3.g.i		2
48.k	even	4	1		2304.3.g.l		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1152.3.b.f	✓	4	1.a	even	1	1	trivial
1152.3.b.f	✓	4	3.b	odd	2	1	inner
1152.3.b.f	✓	4	4.b	odd	2	1	inner
1152.3.b.f	✓	4	8.b	even	2	1	inner
1152.3.b.f	✓	4	8.d	odd	2	1	inner
1152.3.b.f	✓	4	12.b	even	2	1	inner
1152.3.b.f	✓	4	24.f	even	2	1	inner
1152.3.b.f	✓	4	24.h	odd	2	1	CM
2304.3.g.i		2	16.e	even	4	1
2304.3.g.i		2	16.f	odd	4	1
2304.3.g.i		2	48.i	odd	4	1
2304.3.g.i		2	48.k	even	4	1
2304.3.g.l		2	16.e	even	4	1
2304.3.g.l		2	16.f	odd	4	1
2304.3.g.l		2	48.i	odd	4	1
2304.3.g.l		2	48.k	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(1152, [\chi])$:

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

$T_{7}^{2} + 96$

$T_{17}$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4}$
$3$	$T^{4}$
$5$	$(T^{2} + 4)^{2}$
$7$	$(T^{2} + 96)^{2}$
$11$	$(T^{2} - 384)^{2}$