LMFDB - Embedded newform 1600.2.n.k.1407.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1600,2,Mod(1343,1600)] 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("1600.1343"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1600, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 3])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.n (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,0,0,-2,0,0,0,0,0,8,0,0,0,8,0,8,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

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

Self dual:	no
Analytic conductor:	$12.7760643234$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 800)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1407.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1600.1407
Dual form			1600.2.n.k.1343.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 1.00000i) q^{3} +(-1.00000 + 1.00000i) q^{7} -1.00000i q^{9} +4.00000i q^{11} +(4.00000 - 4.00000i) q^{13} +(4.00000 + 4.00000i) q^{17} +4.00000 q^{19} -2.00000 q^{21} +(-5.00000 - 5.00000i) q^{23} +(4.00000 - 4.00000i) q^{27} +2.00000i q^{29} +8.00000i q^{31} +(-4.00000 + 4.00000i) q^{33} +8.00000 q^{39} -4.00000 q^{41} +(7.00000 + 7.00000i) q^{43} +(-3.00000 + 3.00000i) q^{47} +5.00000i q^{49} +8.00000i q^{51} +(4.00000 - 4.00000i) q^{53} +(4.00000 + 4.00000i) q^{57} +4.00000 q^{59} +8.00000 q^{61} +(1.00000 + 1.00000i) q^{63} +(-3.00000 + 3.00000i) q^{67} -10.0000i q^{69} +16.0000i q^{71} +(-4.00000 + 4.00000i) q^{73} +(-4.00000 - 4.00000i) q^{77} +8.00000 q^{79} +5.00000 q^{81} +(-5.00000 - 5.00000i) q^{83} +(-2.00000 + 2.00000i) q^{87} -10.0000i q^{89} +8.00000i q^{91} +(-8.00000 + 8.00000i) q^{93} +(-12.0000 - 12.0000i) q^{97} +4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{7} + 8 q^{13} + 8 q^{17} + 8 q^{19} - 4 q^{21} - 10 q^{23} + 8 q^{27} - 8 q^{33} + 16 q^{39} - 8 q^{41} + 14 q^{43} - 6 q^{47} + 8 q^{53} + 8 q^{57} + 8 q^{59} + 16 q^{61} + 2 q^{63}+ \cdots + 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^7 + 8 * q^13 + 8 * q^17 + 8 * q^19 - 4 * q^21 - 10 * q^23 + 8 * q^27 - 8 * q^33 + 16 * q^39 - 8 * q^41 + 14 * q^43 - 6 * q^47 + 8 * q^53 + 8 * q^57 + 8 * q^59 + 16 * q^61 + 2 * q^63 - 6 * q^67 - 8 * q^73 - 8 * q^77 + 16 * q^79 + 10 * q^81 - 10 * q^83 - 4 * q^87 - 16 * q^93 - 24 * q^97 + 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$-1$

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
$3$	1.00000	+	1.00000i	0.577350	+	0.577350i	0.934172	−	0.356822i	$-0.116140\pi$
$3$	1.00000	+	1.00000i	0.577350	+	0.577350i	−0.356822	+	0.934172i	$0.616140\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.00000	+	1.00000i	−0.377964	+	0.377964i	−0.870367	−	0.492403i	$-0.836119\pi$
$7$	−1.00000	+	1.00000i	−0.377964	+	0.377964i	0.492403	+	0.870367i	$0.336119\pi$
$8$		0			0
$9$		−	1.00000i		−	0.333333i
$10$		0			0
$11$			4.00000i			1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$			4.00000i			1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$		0			0
$13$	4.00000	−	4.00000i	1.10940	−	1.10940i	0.116171	−	0.993229i	$-0.462938\pi$
$13$	4.00000	−	4.00000i	1.10940	−	1.10940i	0.993229	−	0.116171i	$-0.0370621\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.00000	+	4.00000i	0.970143	+	0.970143i	0.999567	−	0.0294245i	$-0.00936746\pi$
$17$	4.00000	+	4.00000i	0.970143	+	0.970143i	−0.0294245	+	0.999567i	$0.509367\pi$
$18$		0			0
$19$	4.00000			0.917663			0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000			0.917663			0.458831	+	0.888523i	$0.348268\pi$
$20$		0			0
$21$	−2.00000			−0.436436
$22$		0			0
$23$	−5.00000	−	5.00000i	−1.04257	−	1.04257i	−0.999053	−	0.0435195i	$-0.986143\pi$
$23$	−5.00000	−	5.00000i	−1.04257	−	1.04257i	−0.0435195	−	0.999053i	$-0.513857\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	4.00000	−	4.00000i	0.769800	−	0.769800i
$28$		0			0
$29$			2.00000i			0.371391i	0.982607	+	0.185695i	$0.0594537\pi$
$29$			2.00000i			0.371391i	−0.982607	+	0.185695i	$0.940546\pi$
$30$		0			0
$31$			8.00000i			1.43684i	0.695608	+	0.718421i	$0.255135\pi$
$31$			8.00000i			1.43684i	−0.695608	+	0.718421i	$0.744865\pi$
$32$		0			0
$33$	−4.00000	+	4.00000i	−0.696311	+	0.696311i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$37$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$38$		0			0
$39$	8.00000			1.28103
$40$		0			0
$41$	−4.00000			−0.624695			−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000			−0.624695			−0.312348	+	0.949968i	$0.601115\pi$
$42$		0			0
$43$	7.00000	+	7.00000i	1.06749	+	1.06749i	0.997551	+	0.0699387i	$0.0222804\pi$
$43$	7.00000	+	7.00000i	1.06749	+	1.06749i	0.0699387	+	0.997551i	$0.477720\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.00000	+	3.00000i	−0.437595	+	0.437595i	−0.891202	−	0.453607i	$-0.850137\pi$
$47$	−3.00000	+	3.00000i	−0.437595	+	0.437595i	0.453607	+	0.891202i	$0.350137\pi$
$48$		0			0
$49$			5.00000i			0.714286i
$50$		0			0
$51$			8.00000i			1.12022i
$52$		0			0
$53$	4.00000	−	4.00000i	0.549442	−	0.549442i	−0.376837	−	0.926279i	$-0.622988\pi$
$53$	4.00000	−	4.00000i	0.549442	−	0.549442i	0.926279	+	0.376837i	$0.122988\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	4.00000	+	4.00000i	0.529813	+	0.529813i
$58$		0			0
$59$	4.00000			0.520756			0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000			0.520756			0.260378	+	0.965507i	$0.416153\pi$
$60$		0			0
$61$	8.00000			1.02430			0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000			1.02430			0.512148	+	0.858898i	$0.328850\pi$
$62$		0			0
$63$	1.00000	+	1.00000i	0.125988	+	0.125988i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−3.00000	+	3.00000i	−0.366508	+	0.366508i	−0.866202	−	0.499694i	$-0.833446\pi$
$67$	−3.00000	+	3.00000i	−0.366508	+	0.366508i	0.499694	+	0.866202i	$0.333446\pi$
$68$		0			0
$69$		−	10.0000i		−	1.20386i
$70$		0			0
$71$			16.0000i			1.89885i	0.313993	+	0.949425i	$0.398333\pi$
$71$			16.0000i			1.89885i	−0.313993	+	0.949425i	$0.601667\pi$
$72$		0			0
$73$	−4.00000	+	4.00000i	−0.468165	+	0.468165i	−0.901319	−	0.433155i	$-0.857400\pi$
$73$	−4.00000	+	4.00000i	−0.468165	+	0.468165i	0.433155	+	0.901319i	$0.357400\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.00000	−	4.00000i	−0.455842	−	0.455842i
$78$		0			0
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$		0			0
$81$	5.00000			0.555556
$82$		0			0
$83$	−5.00000	−	5.00000i	−0.548821	−	0.548821i	0.377279	−	0.926100i	$-0.376860\pi$
$83$	−5.00000	−	5.00000i	−0.548821	−	0.548821i	−0.926100	+	0.377279i	$0.876860\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−2.00000	+	2.00000i	−0.214423	+	0.214423i
$88$		0			0
$89$		−	10.0000i		−	1.06000i	−0.847998	−	0.529999i	$-0.822192\pi$
$89$		−	10.0000i		−	1.06000i	0.847998	−	0.529999i	$-0.177808\pi$
$90$		0			0
$91$			8.00000i			0.838628i
$92$		0			0
$93$	−8.00000	+	8.00000i	−0.829561	+	0.829561i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−12.0000	−	12.0000i	−1.21842	−	1.21842i	−0.968187	−	0.250229i	$-0.919494\pi$
$97$	−12.0000	−	12.0000i	−1.21842	−	1.21842i	−0.250229	−	0.968187i	$-0.580506\pi$
$98$		0			0
$99$	4.00000			0.402015

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	1600.2.n.k.1407.1		2
4.3	odd	2		1600.2.n.f.1407.1		2
5.2	odd	4		1600.2.n.i.1343.1		2
5.3	odd	4		1600.2.n.f.1343.1		2
5.4	even	2		1600.2.n.d.1407.1		2
8.3	odd	2		800.2.n.g.607.1	yes	2
8.5	even	2		800.2.n.b.607.1	yes	2
20.3	even	4	inner	1600.2.n.k.1343.1		2
20.7	even	4		1600.2.n.d.1343.1		2
20.19	odd	2		1600.2.n.i.1407.1		2
40.3	even	4		800.2.n.b.543.1	✓	2
40.13	odd	4		800.2.n.g.543.1	yes	2
40.19	odd	2		800.2.n.d.607.1	yes	2
40.27	even	4		800.2.n.i.543.1	yes	2
40.29	even	2		800.2.n.i.607.1	yes	2
40.37	odd	4		800.2.n.d.543.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
800.2.n.b.543.1	✓	2	40.3	even	4
800.2.n.b.607.1	yes	2	8.5	even	2
800.2.n.d.543.1	yes	2	40.37	odd	4
800.2.n.d.607.1	yes	2	40.19	odd	2
800.2.n.g.543.1	yes	2	40.13	odd	4
800.2.n.g.607.1	yes	2	8.3	odd	2
800.2.n.i.543.1	yes	2	40.27	even	4
800.2.n.i.607.1	yes	2	40.29	even	2
1600.2.n.d.1343.1		2	20.7	even	4
1600.2.n.d.1407.1		2	5.4	even	2
1600.2.n.f.1343.1		2	5.3	odd	4
1600.2.n.f.1407.1		2	4.3	odd	2
1600.2.n.i.1343.1		2	5.2	odd	4
1600.2.n.i.1407.1		2	20.19	odd	2
1600.2.n.k.1343.1		2	20.3	even	4	inner
1600.2.n.k.1407.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.n (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.00000i) q^{3} +(-1.00000 + 1.00000i) q^{7} -1.00000i q^{9} +4.00000i q^{11} +(4.00000 - 4.00000i) q^{13} +(4.00000 + 4.00000i) q^{17} +4.00000 q^{19} -2.00000 q^{21} +(-5.00000 - 5.00000i) q^{23} +(4.00000 - 4.00000i) q^{27} +2.00000i q^{29} +8.00000i q^{31} +(-4.00000 + 4.00000i) q^{33} +8.00000 q^{39} -4.00000 q^{41} +(7.00000 + 7.00000i) q^{43} +(-3.00000 + 3.00000i) q^{47} +5.00000i q^{49} +8.00000i q^{51} +(4.00000 - 4.00000i) q^{53} +(4.00000 + 4.00000i) q^{57} +4.00000 q^{59} +8.00000 q^{61} +(1.00000 + 1.00000i) q^{63} +(-3.00000 + 3.00000i) q^{67} -10.0000i q^{69} +16.0000i q^{71} +(-4.00000 + 4.00000i) q^{73} +(-4.00000 - 4.00000i) q^{77} +8.00000 q^{79} +5.00000 q^{81} +(-5.00000 - 5.00000i) q^{83} +(-2.00000 + 2.00000i) q^{87} -10.0000i q^{89} +8.00000i q^{91} +(-8.00000 + 8.00000i) q^{93} +(-12.0000 - 12.0000i) q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{7} + 8 q^{13} + 8 q^{17} + 8 q^{19} - 4 q^{21} - 10 q^{23} + 8 q^{27} - 8 q^{33} + 16 q^{39} - 8 q^{41} + 14 q^{43} - 6 q^{47} + 8 q^{53} + 8 q^{57} + 8 q^{59} + 16 q^{61} + 2 q^{63}+ \cdots + 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^7 + 8 * q^13 + 8 * q^17 + 8 * q^19 - 4 * q^21 - 10 * q^23 + 8 * q^27 - 8 * q^33 + 16 * q^39 - 8 * q^41 + 14 * q^43 - 6 * q^47 + 8 * q^53 + 8 * q^57 + 8 * q^59 + 16 * q^61 + 2 * q^63 - 6 * q^67 - 8 * q^73 - 8 * q^77 + 16 * q^79 + 10 * q^81 - 10 * q^83 - 4 * q^87 - 16 * q^93 - 24 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more