LMFDB - Embedded newform 760.2.a.j.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [760,2,Mod(1,760)] 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("760.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(760, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 760 = 2^{3} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	760.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-1,0,3,0,-1] 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:	yes
Analytic conductor:	$6.06863055362$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.11491$ of defining polynomial
Character	$\chi$	$=$	760.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.642074 q^{3} +1.00000 q^{5} +3.58774 q^{7} -2.58774 q^{9} +1.35793 q^{13} -0.642074 q^{15} +5.58774 q^{17} +1.00000 q^{19} -2.30359 q^{21} -4.87189 q^{23} +1.00000 q^{25} +3.58774 q^{27} +9.58774 q^{29} -7.17548 q^{31} +3.58774 q^{35} -0.945668 q^{37} -0.871889 q^{39} +10.4596 q^{41} -2.71585 q^{43} -2.58774 q^{45} -5.89134 q^{47} +5.87189 q^{49} -3.58774 q^{51} +9.81756 q^{53} -0.642074 q^{57} +10.1560 q^{59} +3.28415 q^{61} -9.28415 q^{63} +1.35793 q^{65} +10.3859 q^{67} +3.12811 q^{69} +14.3510 q^{71} -4.15604 q^{73} -0.642074 q^{75} +1.28415 q^{79} +5.45963 q^{81} -11.1755 q^{83} +5.58774 q^{85} -6.15604 q^{87} -6.45963 q^{89} +4.87189 q^{91} +4.60719 q^{93} +1.00000 q^{95} -13.4053 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} + 3 q^{5} - q^{7} + 4 q^{9} + 5 q^{13} - q^{15} + 5 q^{17} + 3 q^{19} + 3 q^{21} - q^{23} + 3 q^{25} - q^{27} + 17 q^{29} + 2 q^{31} - q^{35} + 8 q^{37} + 11 q^{39} + 6 q^{41} - 10 q^{43}+ \cdots - 4 q^{97}+O(q^{100}) $ 3 * q - q^3 + 3 * q^5 - q^7 + 4 * q^9 + 5 * q^13 - q^15 + 5 * q^17 + 3 * q^19 + 3 * q^21 - q^23 + 3 * q^25 - q^27 + 17 * q^29 + 2 * q^31 - q^35 + 8 * q^37 + 11 * q^39 + 6 * q^41 - 10 * q^43 + 4 * q^45 + 4 * q^47 + 4 * q^49 + q^51 + 5 * q^53 - q^57 + 15 * q^59 + 8 * q^61 - 26 * q^63 + 5 * q^65 + 3 * q^67 + 23 * q^69 - 4 * q^71 + 3 * q^73 - q^75 + 2 * q^79 - 9 * q^81 - 10 * q^83 + 5 * q^85 - 3 * q^87 + 6 * q^89 + q^91 - 6 * q^93 + 3 * q^95 - 4 * q^97

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
$3$	−0.642074	−0.370701	−0.185351	−	0.982672i	$-0.559342\pi$
$3$	−0.642074	−0.370701	−0.185351	+	0.982672i	$0.559342\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.58774	1.35604	0.678019	−	0.735044i	$-0.262839\pi$
$7$	3.58774	1.35604	0.678019	+	0.735044i	$0.262839\pi$
$8$	0	0
$9$	−2.58774	−0.862580
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	1.35793	0.376621	0.188311	−	0.982110i	$-0.439699\pi$
$13$	1.35793	0.376621	0.188311	+	0.982110i	$0.439699\pi$
$14$	0	0
$15$	−0.642074	−0.165783
$16$	0	0
$17$	5.58774	1.35523	0.677613	−	0.735419i	$-0.263014\pi$
$17$	5.58774	1.35523	0.677613	+	0.735419i	$0.263014\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−2.30359	−0.502685
$22$	0	0
$23$	−4.87189	−1.01586	−0.507930	−	0.861399i	$-0.669589\pi$
$23$	−4.87189	−1.01586	−0.507930	+	0.861399i	$0.669589\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.58774	0.690461
$28$	0	0
$29$	9.58774	1.78040	0.890199	−	0.455571i	$-0.150565\pi$
$29$	9.58774	1.78040	0.890199	+	0.455571i	$0.150565\pi$
$30$	0	0
$31$	−7.17548	−1.28875	−0.644377	−	0.764708i	$-0.722883\pi$
$31$	−7.17548	−1.28875	−0.644377	+	0.764708i	$0.722883\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.58774	0.606439
$36$	0	0
$37$	−0.945668	−0.155467	−0.0777334	−	0.996974i	$-0.524768\pi$
$37$	−0.945668	−0.155467	−0.0777334	+	0.996974i	$0.524768\pi$
$38$	0	0
$39$	−0.871889	−0.139614
$40$	0	0
$41$	10.4596	1.63352	0.816760	−	0.576978i	$-0.195768\pi$
$41$	10.4596	1.63352	0.816760	+	0.576978i	$0.195768\pi$
$42$	0	0
$43$	−2.71585	−0.414164	−0.207082	−	0.978324i	$-0.566397\pi$
$43$	−2.71585	−0.414164	−0.207082	+	0.978324i	$0.566397\pi$
$44$	0	0
$45$	−2.58774	−0.385758
$46$	0	0
$47$	−5.89134	−0.859340	−0.429670	−	0.902986i	$-0.641370\pi$
$47$	−5.89134	−0.859340	−0.429670	+	0.902986i	$0.641370\pi$
$48$	0	0
$49$	5.87189	0.838841
$50$	0	0
$51$	−3.58774	−0.502384
$52$	0	0
$53$	9.81756	1.34855	0.674273	−	0.738483i	$-0.264457\pi$
$53$	9.81756	1.34855	0.674273	+	0.738483i	$0.264457\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.642074	−0.0850447
$58$	0	0
$59$	10.1560	1.32220	0.661102	−	0.750296i	$-0.270089\pi$
$59$	10.1560	1.32220	0.661102	+	0.750296i	$0.270089\pi$
$60$	0	0
$61$	3.28415	0.420492	0.210246	−	0.977649i	$-0.432574\pi$
$61$	3.28415	0.420492	0.210246	+	0.977649i	$0.432574\pi$
$62$	0	0
$63$	−9.28415	−1.16969
$64$	0	0
$65$	1.35793	0.168430
$66$	0	0
$67$	10.3859	1.26883	0.634417	−	0.772991i	$-0.281240\pi$
$67$	10.3859	1.26883	0.634417	+	0.772991i	$0.281240\pi$
$68$	0	0
$69$	3.12811	0.376580
$70$	0	0
$71$	14.3510	1.70315	0.851573	−	0.524236i	$-0.175649\pi$
$71$	14.3510	1.70315	0.851573	+	0.524236i	$0.175649\pi$
$72$	0	0
$73$	−4.15604	−0.486427	−0.243214	−	0.969973i	$-0.578202\pi$
$73$	−4.15604	−0.486427	−0.243214	+	0.969973i	$0.578202\pi$
$74$	0	0
$75$	−0.642074	−0.0741403
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1.28415	0.144478	0.0722389	−	0.997387i	$-0.476986\pi$
$79$	1.28415	0.144478	0.0722389	+	0.997387i	$0.476986\pi$
$80$	0	0
$81$	5.45963	0.606626
$82$	0	0
$83$	−11.1755	−1.22667	−0.613334	−	0.789823i	$-0.710172\pi$
$83$	−11.1755	−1.22667	−0.613334	+	0.789823i	$0.710172\pi$
$84$	0	0
$85$	5.58774	0.606076
$86$	0	0
$87$	−6.15604	−0.659996
$88$	0	0
$89$	−6.45963	−0.684719	−0.342360	−	0.939569i	$-0.611226\pi$
$89$	−6.45963	−0.684719	−0.342360	+	0.939569i	$0.611226\pi$
$90$	0	0
$91$	4.87189	0.510713
$92$	0	0
$93$	4.60719	0.477743
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−13.4053	−1.36110	−0.680551	−	0.732701i	$-0.738259\pi$
$97$	−13.4053	−1.36110	−0.680551	+	0.732701i	$0.738259\pi$
$98$	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	760.2.a.j.1.2	✓	3
3.2	odd	2		6840.2.a.bg.1.3		3
4.3	odd	2		1520.2.a.s.1.2		3
5.2	odd	4		3800.2.d.l.3649.4		6
5.3	odd	4		3800.2.d.l.3649.3		6
5.4	even	2		3800.2.a.x.1.2		3
8.3	odd	2		6080.2.a.bq.1.2		3
8.5	even	2		6080.2.a.bv.1.2		3
20.19	odd	2		7600.2.a.bq.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.j.1.2	✓	3	1.1	even	1	trivial
1520.2.a.s.1.2		3	4.3	odd	2
3800.2.a.x.1.2		3	5.4	even	2
3800.2.d.l.3649.3		6	5.3	odd	4
3800.2.d.l.3649.4		6	5.2	odd	4
6080.2.a.bq.1.2		3	8.3	odd	2
6080.2.a.bv.1.2		3	8.5	even	2
6840.2.a.bg.1.3		3	3.2	odd	2
7600.2.a.bq.1.2		3	20.19	odd	2

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:	\( N \)	\(=\)	\( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	760.a (trivial)

\(f(q)\)	\(=\)	\(q-0.642074 q^{3} +1.00000 q^{5} +3.58774 q^{7} -2.58774 q^{9} +1.35793 q^{13} -0.642074 q^{15} +5.58774 q^{17} +1.00000 q^{19} -2.30359 q^{21} -4.87189 q^{23} +1.00000 q^{25} +3.58774 q^{27} +9.58774 q^{29} -7.17548 q^{31} +3.58774 q^{35} -0.945668 q^{37} -0.871889 q^{39} +10.4596 q^{41} -2.71585 q^{43} -2.58774 q^{45} -5.89134 q^{47} +5.87189 q^{49} -3.58774 q^{51} +9.81756 q^{53} -0.642074 q^{57} +10.1560 q^{59} +3.28415 q^{61} -9.28415 q^{63} +1.35793 q^{65} +10.3859 q^{67} +3.12811 q^{69} +14.3510 q^{71} -4.15604 q^{73} -0.642074 q^{75} +1.28415 q^{79} +5.45963 q^{81} -11.1755 q^{83} +5.58774 q^{85} -6.15604 q^{87} -6.45963 q^{89} +4.87189 q^{91} +4.60719 q^{93} +1.00000 q^{95} -13.4053 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} + 3 q^{5} - q^{7} + 4 q^{9} + 5 q^{13} - q^{15} + 5 q^{17} + 3 q^{19} + 3 q^{21} - q^{23} + 3 q^{25} - q^{27} + 17 q^{29} + 2 q^{31} - q^{35} + 8 q^{37} + 11 q^{39} + 6 q^{41} - 10 q^{43}+ \cdots - 4 q^{97}+O(q^{100}) \) 3 * q - q^3 + 3 * q^5 - q^7 + 4 * q^9 + 5 * q^13 - q^15 + 5 * q^17 + 3 * q^19 + 3 * q^21 - q^23 + 3 * q^25 - q^27 + 17 * q^29 + 2 * q^31 - q^35 + 8 * q^37 + 11 * q^39 + 6 * q^41 - 10 * q^43 + 4 * q^45 + 4 * q^47 + 4 * q^49 + q^51 + 5 * q^53 - q^57 + 15 * q^59 + 8 * q^61 - 26 * q^63 + 5 * q^65 + 3 * q^67 + 23 * q^69 - 4 * q^71 + 3 * q^73 - q^75 + 2 * q^79 - 9 * q^81 - 10 * q^83 + 5 * q^85 - 3 * q^87 + 6 * q^89 + q^91 - 6 * q^93 + 3 * q^95 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more