Newform orbit 1152.4.a.r

• Label: 1152.4.a.r
• Level: $1152$
• Weight: $4$
• Character orbit: 1152.a
• Self dual: yes
• Analytic conductor: $67.970$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$1152 = 2^{7} \cdot 3^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1152.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$67.9702003266$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{3})$
Defining polynomial:	$x^{2} - 3$
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	no (minimal twist has level 128)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = 4\sqrt{3}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + (2*b - 2) * q^5 + (-2*b + 4) * q^7 + (-b + 46) * q^11 + (6*b + 50) * q^13 + (12*b - 46) * q^17 + (7*b - 2) * q^19 + (-18*b + 4) * q^23 + (-8*b + 71) * q^25 + (10*b - 42) * q^29 + (-16*b + 192) * q^31 + (12*b - 200) * q^35 + (14*b - 86) * q^37 + (8*b + 150) * q^41 + (5*b - 150) * q^43 + (-44*b - 8) * q^47 + (-16*b - 135) * q^49 + (-14*b + 6) * q^53 + (94*b - 188) * q^55 + (-33*b - 322) * q^59 + (70*b + 146) * q^61 + (88*b + 476) * q^65 + (83*b + 86) * q^67 + (42*b + 204) * q^71 + (52*b + 206) * q^73 + (-96*b + 280) * q^77 + (-36*b + 200) * q^79 + (13*b + 474) * q^83 + (-116*b + 1244) * q^85 + (-116*b - 286) * q^89 + (-76*b - 376) * q^91 + (-18*b + 676) * q^95 + (-92*b + 1102) * q^97
$\operatorname{Tr}(f)(q)$	$=$	2 * q - 4 * q^5 + 8 * q^7 + 92 * q^11 + 100 * q^13 - 92 * q^17 - 4 * q^19 + 8 * q^23 + 142 * q^25 - 84 * q^29 + 384 * q^31 - 400 * q^35 - 172 * q^37 + 300 * q^41 - 300 * q^43 - 16 * q^47 - 270 * q^49 + 12 * q^53 - 376 * q^55 - 644 * q^59 + 292 * q^61 + 952 * q^65 + 172 * q^67 + 408 * q^71 + 412 * q^73 + 560 * q^77 + 400 * q^79 + 948 * q^83 + 2488 * q^85 - 572 * q^89 - 752 * q^91 + 1352 * q^95 + 2204 * 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

−15.8564

0

17.8564

0

0

0

1.2

0

0

0

11.8564

0

−9.85641

0

0

0

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$-1$

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	1152.4.a.r		2
3.b	odd	2	1		128.4.a.f	yes	2
4.b	odd	2	1		1152.4.a.q		2
8.b	even	2	1		1152.4.a.t		2
8.d	odd	2	1		1152.4.a.s		2
12.b	even	2	1		128.4.a.h	yes	2
24.f	even	2	1		128.4.a.e	✓	2
24.h	odd	2	1		128.4.a.g	yes	2
48.i	odd	4	2		256.4.b.h		4
48.k	even	4	2		256.4.b.i		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
128.4.a.e	✓	2	24.f	even	2	1
128.4.a.f	yes	2	3.b	odd	2	1
128.4.a.g	yes	2	24.h	odd	2	1
128.4.a.h	yes	2	12.b	even	2	1
256.4.b.h		4	48.i	odd	4	2
256.4.b.i		4	48.k	even	4	2
1152.4.a.q		2	4.b	odd	2	1
1152.4.a.r		2	1.a	even	1	1	trivial
1152.4.a.s		2	8.d	odd	2	1
1152.4.a.t		2	8.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(1152))$:

$T_{5}^{2} + 4T_{5} - 188$

$T_{7}^{2} - 8T_{7} - 176$

$T_{13}^{2} - 100T_{13} + 772$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2}$
$5$	$T^{2} + 4T - 188$
$7$	$T^{2} - 8T - 176$
$11$	$T^{2} - 92T + 2068$