Newform orbit 1152.4.d.a

• Label: 1152.4.d.a
• Level: $1152$
• Weight: $4$
• Character orbit: 1152.d
• Analytic conductor: $67.970$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$1152 = 2^{7} \cdot 3^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1152.d (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$67.9702003266$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1})$
Defining polynomial:	$x^{2} + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	no (minimal twist has level 128)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = 4i$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + 3*b * q^5 - 32 * q^7 - 2*b * q^11 - 5*b * q^13 + 98 * q^17 - 22*b * q^19 - 32 * q^23 - 19 * q^25 - 43*b * q^29 - 256 * q^31 - 96*b * q^35 - 23*b * q^37 + 102 * q^41 + 74*b * q^43 + 320 * q^47 + 681 * q^49 + 19*b * q^53 + 96 * q^55 + 102*b * q^59 + 159*b * q^61 + 240 * q^65 + 138*b * q^67 + 416 * q^71 - 138 * q^73 + 64*b * q^77 - 64 * q^79 - 98*b * q^83 + 294*b * q^85 - 582 * q^89 + 160*b * q^91 + 1056 * q^95 + 238 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 64 q^{7} + 196 q^{17} - 64 q^{23} - 38 q^{25} - 512 q^{31} + 204 q^{41} + 640 q^{47} + 1362 q^{49} + 192 q^{55} + 480 q^{65} + 832 q^{71} - 276 q^{73} - 128 q^{79} - 1164 q^{89} + 2112 q^{95} + 476 q^{97}+O(q^{100})$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

577.1

	−	1.00000i
		1.00000i

0

0

0

−

12.0000i

0

−32.0000

0

0

0

577.2

0

0

0

12.0000i

0

−32.0000

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1152.4.d.a		2
3.b	odd	2	1		128.4.b.a	✓	2
4.b	odd	2	1		1152.4.d.h		2
8.b	even	2	1	inner	1152.4.d.a		2
8.d	odd	2	1		1152.4.d.h		2
12.b	even	2	1		128.4.b.d	yes	2
16.e	even	4	1		2304.4.a.d		1
16.e	even	4	1		2304.4.a.n		1
16.f	odd	4	1		2304.4.a.c		1
16.f	odd	4	1		2304.4.a.m		1
24.f	even	2	1		128.4.b.d	yes	2
24.h	odd	2	1		128.4.b.a	✓	2
48.i	odd	4	1		256.4.a.b		1
48.i	odd	4	1		256.4.a.g		1
48.k	even	4	1		256.4.a.c		1
48.k	even	4	1		256.4.a.f		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
128.4.b.a	✓	2	3.b	odd	2	1
128.4.b.a	✓	2	24.h	odd	2	1
128.4.b.d	yes	2	12.b	even	2	1
128.4.b.d	yes	2	24.f	even	2	1
256.4.a.b		1	48.i	odd	4	1
256.4.a.c		1	48.k	even	4	1
256.4.a.f		1	48.k	even	4	1
256.4.a.g		1	48.i	odd	4	1
1152.4.d.a		2	1.a	even	1	1	trivial
1152.4.d.a		2	8.b	even	2	1	inner
1152.4.d.h		2	4.b	odd	2	1
1152.4.d.h		2	8.d	odd	2	1
2304.4.a.c		1	16.f	odd	4	1
2304.4.a.d		1	16.e	even	4	1
2304.4.a.m		1	16.f	odd	4	1
2304.4.a.n		1	16.e	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(1152, [\chi])$:

$T_{5}^{2} + 144$

$T_{7} + 32$

$T_{17} - 98$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2}$
$5$	$T^{2} + 144$
$7$	$(T + 32)^{2}$
$11$	$T^{2} + 64$