LMFDB - Embedded newform 1296.2.s.j.431.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1296,2,Mod(431,1296)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1296.431"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1296, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1296.s (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$10.3486121020$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			431.3
Root			$0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	1296.431
Dual form			1296.2.s.j.863.3

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.776457 - 0.448288i) q^{5} +(1.09808 + 0.633975i) q^{7} +(2.12132 - 3.67423i) q^{11} +(-0.500000 - 0.866025i) q^{13} -0.896575i q^{17} -4.73205i q^{19} +(-3.67423 - 6.36396i) q^{23} +(-2.09808 + 3.63397i) q^{25} +(7.14042 + 4.12252i) q^{29} +(-5.19615 + 3.00000i) q^{31} +1.13681 q^{35} +1.19615 q^{37} +(6.36396 - 3.67423i) q^{41} +(7.09808 + 4.09808i) q^{43} +(5.79555 - 10.0382i) q^{47} +(-2.69615 - 4.66987i) q^{49} +7.34847i q^{53} -3.80385i q^{55} +(-5.79555 - 10.0382i) q^{59} +(1.59808 - 2.76795i) q^{61} +(-0.776457 - 0.448288i) q^{65} +(-9.29423 + 5.36603i) q^{67} +1.13681 q^{71} +9.19615 q^{73} +(4.65874 - 2.68973i) q^{77} +(4.09808 + 2.36603i) q^{79} +(4.24264 - 7.34847i) q^{83} +(-0.401924 - 0.696152i) q^{85} -11.8313i q^{89} -1.26795i q^{91} +(-2.12132 - 3.67423i) q^{95} +(-7.19615 + 12.4641i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{7} - 4 q^{13} + 4 q^{25} - 32 q^{37} + 36 q^{43} + 20 q^{49} - 8 q^{61} - 12 q^{67} + 32 q^{73} + 12 q^{79} - 24 q^{85} - 16 q^{97}+O(q^{100}) $ 8 * q - 12 * q^7 - 4 * q^13 + 4 * q^25 - 32 * q^37 + 36 * q^43 + 20 * q^49 - 8 * q^61 - 12 * q^67 + 32 * q^73 + 12 * q^79 - 24 * q^85 - 16 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$. You can download additional coefficients here.

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.776457	−	0.448288i	0.347242	−	0.200480i	−0.316228	−	0.948683i	$-0.602416\pi$
$5$	0.776457	−	0.448288i	0.347242	−	0.200480i	0.663470	+	0.748203i	$0.269083\pi$
$6$		0			0
$7$	1.09808	+	0.633975i	0.415034	+	0.239620i	0.692950	−	0.720985i	$-0.256311\pi$
$7$	1.09808	+	0.633975i	0.415034	+	0.239620i	−0.277916	+	0.960605i	$0.589644\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.12132	−	3.67423i	0.639602	−	1.10782i	−0.345918	−	0.938265i	$-0.612432\pi$
$11$	2.12132	−	3.67423i	0.639602	−	1.10782i	0.985520	−	0.169559i	$-0.0542342\pi$
$12$		0			0
$13$	−0.500000	−	0.866025i	−0.138675	−	0.240192i	0.788320	−	0.615265i	$-0.210951\pi$
$13$	−0.500000	−	0.866025i	−0.138675	−	0.240192i	−0.926995	+	0.375073i	$0.877618\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	0.896575i		−	0.217451i	−0.994072	−	0.108726i	$-0.965323\pi$
$17$		−	0.896575i		−	0.217451i	0.994072	−	0.108726i	$-0.0346770\pi$
$18$		0			0
$19$		−	4.73205i		−	1.08561i	−0.839860	−	0.542803i	$-0.817363\pi$
$19$		−	4.73205i		−	1.08561i	0.839860	−	0.542803i	$-0.182637\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.67423	−	6.36396i	−0.766131	−	1.32698i	−0.939647	−	0.342147i	$-0.888846\pi$
$23$	−3.67423	−	6.36396i	−0.766131	−	1.32698i	0.173516	−	0.984831i	$-0.444487\pi$
$24$		0			0
$25$	−2.09808	+	3.63397i	−0.419615	+	0.726795i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	7.14042	+	4.12252i	1.32594	+	0.765533i	0.984669	−	0.174431i	$-0.0558086\pi$
$29$	7.14042	+	4.12252i	1.32594	+	0.765533i	0.341273	+	0.939964i	$0.389142\pi$
$30$		0			0
$31$	−5.19615	+	3.00000i	−0.933257	+	0.538816i	−0.887840	−	0.460152i	$-0.847795\pi$
$31$	−5.19615	+	3.00000i	−0.933257	+	0.538816i	−0.0454165	+	0.998968i	$0.514461\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.13681			0.192156
$36$		0			0
$37$	1.19615			0.196646			0.0983231	−	0.995155i	$-0.468652\pi$
$37$	1.19615			0.196646			0.0983231	+	0.995155i	$0.468652\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	6.36396	−	3.67423i	0.993884	−	0.573819i	0.0874508	−	0.996169i	$-0.472128\pi$
$41$	6.36396	−	3.67423i	0.993884	−	0.573819i	0.906433	+	0.422350i	$0.138795\pi$
$42$		0			0
$43$	7.09808	+	4.09808i	1.08245	+	0.624951i	0.931555	−	0.363600i	$-0.118452\pi$
$43$	7.09808	+	4.09808i	1.08245	+	0.624951i	0.150891	+	0.988550i	$0.451786\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.79555	−	10.0382i	0.845369	−	1.46422i	−0.0399322	−	0.999202i	$-0.512714\pi$
$47$	5.79555	−	10.0382i	0.845369	−	1.46422i	0.885301	−	0.465019i	$-0.153952\pi$
$48$		0			0
$49$	−2.69615	−	4.66987i	−0.385165	−	0.667125i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			7.34847i			1.00939i	0.863298	+	0.504695i	$0.168395\pi$
$53$			7.34847i			1.00939i	−0.863298	+	0.504695i	$0.831605\pi$
$54$		0			0
$55$		−	3.80385i		−	0.512911i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.79555	−	10.0382i	−0.754517	−	1.30686i	−0.945614	−	0.325291i	$-0.894538\pi$
$59$	−5.79555	−	10.0382i	−0.754517	−	1.30686i	0.191097	−	0.981571i	$-0.438795\pi$
$60$		0			0
$61$	1.59808	−	2.76795i	0.204613	−	0.354400i	−0.745397	−	0.666621i	$-0.767740\pi$
$61$	1.59808	−	2.76795i	0.204613	−	0.354400i	0.950009	+	0.312222i	$0.101073\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.776457	−	0.448288i	−0.0963077	−	0.0556033i
$66$		0			0
$67$	−9.29423	+	5.36603i	−1.13547	+	0.655564i	−0.945305	−	0.326188i	$-0.894236\pi$
$67$	−9.29423	+	5.36603i	−1.13547	+	0.655564i	−0.190166	+	0.981752i	$0.560903\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	1.13681			0.134915			0.0674574	−	0.997722i	$-0.478511\pi$
$71$	1.13681			0.134915			0.0674574	+	0.997722i	$0.478511\pi$
$72$		0			0
$73$	9.19615			1.07633			0.538164	−	0.842840i	$-0.319118\pi$
$73$	9.19615			1.07633			0.538164	+	0.842840i	$0.319118\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4.65874	−	2.68973i	0.530913	−	0.306523i
$78$		0			0
$79$	4.09808	+	2.36603i	0.461070	+	0.266199i	0.712494	−	0.701678i	$-0.247566\pi$
$79$	4.09808	+	2.36603i	0.461070	+	0.266199i	−0.251424	+	0.967877i	$0.580899\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4.24264	−	7.34847i	0.465690	−	0.806599i	−0.533542	−	0.845774i	$-0.679139\pi$
$83$	4.24264	−	7.34847i	0.465690	−	0.806599i	0.999232	+	0.0391742i	$0.0124727\pi$
$84$		0			0
$85$	−0.401924	−	0.696152i	−0.0435948	−	0.0755083i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	11.8313i		−	1.25412i	−0.778971	−	0.627060i	$-0.784258\pi$
$89$		−	11.8313i		−	1.25412i	0.778971	−	0.627060i	$-0.215742\pi$
$90$		0			0
$91$		−	1.26795i		−	0.132917i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.12132	−	3.67423i	−0.217643	−	0.376969i
$96$		0			0
$97$	−7.19615	+	12.4641i	−0.730659	+	1.26554i	0.225944	+	0.974140i	$0.427454\pi$
$97$	−7.19615	+	12.4641i	−0.730659	+	1.26554i	−0.956602	+	0.291397i	$0.905880\pi$
$98$		0			0
$99$		0			0

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.2.s.j.431.3		8
3.2	odd	2	inner	1296.2.s.j.431.2		8
4.3	odd	2		1296.2.s.l.431.3		8
9.2	odd	6		1296.2.c.g.1295.3	✓	8
9.4	even	3		1296.2.s.l.863.2		8
9.5	odd	6		1296.2.s.l.863.3		8
9.7	even	3		1296.2.c.g.1295.5	yes	8
12.11	even	2		1296.2.s.l.431.2		8
36.7	odd	6		1296.2.c.g.1295.6	yes	8
36.11	even	6		1296.2.c.g.1295.4	yes	8
36.23	even	6	inner	1296.2.s.j.863.3		8
36.31	odd	6	inner	1296.2.s.j.863.2		8
72.11	even	6		5184.2.c.h.5183.6		8
72.29	odd	6		5184.2.c.h.5183.5		8
72.43	odd	6		5184.2.c.h.5183.4		8
72.61	even	6		5184.2.c.h.5183.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1296.2.c.g.1295.3	✓	8	9.2	odd	6
1296.2.c.g.1295.4	yes	8	36.11	even	6
1296.2.c.g.1295.5	yes	8	9.7	even	3
1296.2.c.g.1295.6	yes	8	36.7	odd	6
1296.2.s.j.431.2		8	3.2	odd	2	inner
1296.2.s.j.431.3		8	1.1	even	1	trivial
1296.2.s.j.863.2		8	36.31	odd	6	inner
1296.2.s.j.863.3		8	36.23	even	6	inner
1296.2.s.l.431.2		8	12.11	even	2
1296.2.s.l.431.3		8	4.3	odd	2
1296.2.s.l.863.2		8	9.4	even	3
1296.2.s.l.863.3		8	9.5	odd	6
5184.2.c.h.5183.3		8	72.61	even	6
5184.2.c.h.5183.4		8	72.43	odd	6
5184.2.c.h.5183.5		8	72.29	odd	6
5184.2.c.h.5183.6		8	72.11	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1296.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.776457 - 0.448288i) q^{5} +(1.09808 + 0.633975i) q^{7} +(2.12132 - 3.67423i) q^{11} +(-0.500000 - 0.866025i) q^{13} -0.896575i q^{17} -4.73205i q^{19} +(-3.67423 - 6.36396i) q^{23} +(-2.09808 + 3.63397i) q^{25} +(7.14042 + 4.12252i) q^{29} +(-5.19615 + 3.00000i) q^{31} +1.13681 q^{35} +1.19615 q^{37} +(6.36396 - 3.67423i) q^{41} +(7.09808 + 4.09808i) q^{43} +(5.79555 - 10.0382i) q^{47} +(-2.69615 - 4.66987i) q^{49} +7.34847i q^{53} -3.80385i q^{55} +(-5.79555 - 10.0382i) q^{59} +(1.59808 - 2.76795i) q^{61} +(-0.776457 - 0.448288i) q^{65} +(-9.29423 + 5.36603i) q^{67} +1.13681 q^{71} +9.19615 q^{73} +(4.65874 - 2.68973i) q^{77} +(4.09808 + 2.36603i) q^{79} +(4.24264 - 7.34847i) q^{83} +(-0.401924 - 0.696152i) q^{85} -11.8313i q^{89} -1.26795i q^{91} +(-2.12132 - 3.67423i) q^{95} +(-7.19615 + 12.4641i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{7} - 4 q^{13} + 4 q^{25} - 32 q^{37} + 36 q^{43} + 20 q^{49} - 8 q^{61} - 12 q^{67} + 32 q^{73} + 12 q^{79} - 24 q^{85} - 16 q^{97}+O(q^{100}) \) 8 * q - 12 * q^7 - 4 * q^13 + 4 * q^25 - 32 * q^37 + 36 * q^43 + 20 * q^49 - 8 * q^61 - 12 * q^67 + 32 * q^73 + 12 * q^79 - 24 * q^85 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more