LMFDB - Embedded newform 3720.2.a.j.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3720,2,Mod(1,3720)] 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("3720.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3720.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,2,0,-4] 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:	$29.7043495519$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	3720.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} -3.41421 q^{7} +1.00000 q^{9} +2.82843 q^{11} -2.58579 q^{13} +1.00000 q^{15} -2.82843 q^{17} +2.82843 q^{19} -3.41421 q^{21} -7.65685 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.75736 q^{29} -1.00000 q^{31} +2.82843 q^{33} -3.41421 q^{35} -7.07107 q^{37} -2.58579 q^{39} +0.828427 q^{41} -1.65685 q^{43} +1.00000 q^{45} -0.343146 q^{47} +4.65685 q^{49} -2.82843 q^{51} -10.8284 q^{53} +2.82843 q^{55} +2.82843 q^{57} +2.58579 q^{59} +0.828427 q^{61} -3.41421 q^{63} -2.58579 q^{65} +5.07107 q^{67} -7.65685 q^{69} -2.58579 q^{71} -6.58579 q^{73} +1.00000 q^{75} -9.65685 q^{77} +3.17157 q^{79} +1.00000 q^{81} -2.00000 q^{83} -2.82843 q^{85} -1.75736 q^{87} +10.7279 q^{89} +8.82843 q^{91} -1.00000 q^{93} +2.82843 q^{95} -7.65685 q^{97} +2.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} - 4 q^{7} + 2 q^{9} - 8 q^{13} + 2 q^{15} - 4 q^{21} - 4 q^{23} + 2 q^{25} + 2 q^{27} - 12 q^{29} - 2 q^{31} - 4 q^{35} - 8 q^{39} - 4 q^{41} + 8 q^{43} + 2 q^{45} - 12 q^{47} - 2 q^{49}+ \cdots - 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 - 4 * q^7 + 2 * q^9 - 8 * q^13 + 2 * q^15 - 4 * q^21 - 4 * q^23 + 2 * q^25 + 2 * q^27 - 12 * q^29 - 2 * q^31 - 4 * q^35 - 8 * q^39 - 4 * q^41 + 8 * q^43 + 2 * q^45 - 12 * q^47 - 2 * q^49 - 16 * q^53 + 8 * q^59 - 4 * q^61 - 4 * q^63 - 8 * q^65 - 4 * q^67 - 4 * q^69 - 8 * q^71 - 16 * q^73 + 2 * q^75 - 8 * q^77 + 12 * q^79 + 2 * q^81 - 4 * q^83 - 12 * q^87 - 4 * q^89 + 12 * q^91 - 2 * q^93 - 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.41421	−1.29045	−0.645226	−	0.763992i	$-0.723237\pi$
$7$	−3.41421	−1.29045	−0.645226	+	0.763992i	$0.723237\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.82843	0.852803	0.426401	−	0.904534i	$-0.359781\pi$
$11$	2.82843	0.852803	0.426401	+	0.904534i	$0.359781\pi$
$12$	0	0
$13$	−2.58579	−0.717168	−0.358584	−	0.933497i	$-0.616740\pi$
$13$	−2.58579	−0.717168	−0.358584	+	0.933497i	$0.616740\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−2.82843	−0.685994	−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843	−0.685994	−0.342997	+	0.939336i	$0.611442\pi$
$18$	0	0
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	0	0
$21$	−3.41421	−0.745042
$22$	0	0
$23$	−7.65685	−1.59656	−0.798282	−	0.602284i	$-0.794258\pi$
$23$	−7.65685	−1.59656	−0.798282	+	0.602284i	$0.794258\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.75736	−0.326333	−0.163167	−	0.986599i	$-0.552171\pi$
$29$	−1.75736	−0.326333	−0.163167	+	0.986599i	$0.552171\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	2.82843	0.492366
$34$	0	0
$35$	−3.41421	−0.577107
$36$	0	0
$37$	−7.07107	−1.16248	−0.581238	−	0.813733i	$-0.697432\pi$
$37$	−7.07107	−1.16248	−0.581238	+	0.813733i	$0.697432\pi$
$38$	0	0
$39$	−2.58579	−0.414057
$40$	0	0
$41$	0.828427	0.129379	0.0646893	−	0.997905i	$-0.479394\pi$
$41$	0.828427	0.129379	0.0646893	+	0.997905i	$0.479394\pi$
$42$	0	0
$43$	−1.65685	−0.252668	−0.126334	−	0.991988i	$-0.540321\pi$
$43$	−1.65685	−0.252668	−0.126334	+	0.991988i	$0.540321\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−0.343146	−0.0500530	−0.0250265	−	0.999687i	$-0.507967\pi$
$47$	−0.343146	−0.0500530	−0.0250265	+	0.999687i	$0.507967\pi$
$48$	0	0
$49$	4.65685	0.665265
$50$	0	0
$51$	−2.82843	−0.396059
$52$	0	0
$53$	−10.8284	−1.48740	−0.743699	−	0.668514i	$-0.766931\pi$
$53$	−10.8284	−1.48740	−0.743699	+	0.668514i	$0.766931\pi$
$54$	0	0
$55$	2.82843	0.381385
$56$	0	0
$57$	2.82843	0.374634
$58$	0	0
$59$	2.58579	0.336641	0.168320	−	0.985732i	$-0.446166\pi$
$59$	2.58579	0.336641	0.168320	+	0.985732i	$0.446166\pi$
$60$	0	0
$61$	0.828427	0.106069	0.0530346	−	0.998593i	$-0.483111\pi$
$61$	0.828427	0.106069	0.0530346	+	0.998593i	$0.483111\pi$
$62$	0	0
$63$	−3.41421	−0.430150
$64$	0	0
$65$	−2.58579	−0.320727
$66$	0	0
$67$	5.07107	0.619530	0.309765	−	0.950813i	$-0.399750\pi$
$67$	5.07107	0.619530	0.309765	+	0.950813i	$0.399750\pi$
$68$	0	0
$69$	−7.65685	−0.921777
$70$	0	0
$71$	−2.58579	−0.306876	−0.153438	−	0.988158i	$-0.549035\pi$
$71$	−2.58579	−0.306876	−0.153438	+	0.988158i	$0.549035\pi$
$72$	0	0
$73$	−6.58579	−0.770808	−0.385404	−	0.922748i	$-0.625938\pi$
$73$	−6.58579	−0.770808	−0.385404	+	0.922748i	$0.625938\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−9.65685	−1.10050
$78$	0	0
$79$	3.17157	0.356830	0.178415	−	0.983955i	$-0.442903\pi$
$79$	3.17157	0.356830	0.178415	+	0.983955i	$0.442903\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.00000	−0.219529	−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000	−0.219529	−0.109764	+	0.993958i	$0.535010\pi$
$84$	0	0
$85$	−2.82843	−0.306786
$86$	0	0
$87$	−1.75736	−0.188409
$88$	0	0
$89$	10.7279	1.13716	0.568579	−	0.822629i	$-0.307493\pi$
$89$	10.7279	1.13716	0.568579	+	0.822629i	$0.307493\pi$
$90$	0	0
$91$	8.82843	0.925471
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	2.82843	0.290191
$96$	0	0
$97$	−7.65685	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$97$	−7.65685	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$98$	0	0
$99$	2.82843	0.284268

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	3720.2.a.j.1.1	✓	2
4.3	odd	2		7440.2.a.bh.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3720.2.a.j.1.1	✓	2	1.1	even	1	trivial
7440.2.a.bh.1.2		2	4.3	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:	\( N \)	\(=\)	\( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3720.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} -3.41421 q^{7} +1.00000 q^{9} +2.82843 q^{11} -2.58579 q^{13} +1.00000 q^{15} -2.82843 q^{17} +2.82843 q^{19} -3.41421 q^{21} -7.65685 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.75736 q^{29} -1.00000 q^{31} +2.82843 q^{33} -3.41421 q^{35} -7.07107 q^{37} -2.58579 q^{39} +0.828427 q^{41} -1.65685 q^{43} +1.00000 q^{45} -0.343146 q^{47} +4.65685 q^{49} -2.82843 q^{51} -10.8284 q^{53} +2.82843 q^{55} +2.82843 q^{57} +2.58579 q^{59} +0.828427 q^{61} -3.41421 q^{63} -2.58579 q^{65} +5.07107 q^{67} -7.65685 q^{69} -2.58579 q^{71} -6.58579 q^{73} +1.00000 q^{75} -9.65685 q^{77} +3.17157 q^{79} +1.00000 q^{81} -2.00000 q^{83} -2.82843 q^{85} -1.75736 q^{87} +10.7279 q^{89} +8.82843 q^{91} -1.00000 q^{93} +2.82843 q^{95} -7.65685 q^{97} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} - 4 q^{7} + 2 q^{9} - 8 q^{13} + 2 q^{15} - 4 q^{21} - 4 q^{23} + 2 q^{25} + 2 q^{27} - 12 q^{29} - 2 q^{31} - 4 q^{35} - 8 q^{39} - 4 q^{41} + 8 q^{43} + 2 q^{45} - 12 q^{47} - 2 q^{49}+ \cdots - 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 - 4 * q^7 + 2 * q^9 - 8 * q^13 + 2 * q^15 - 4 * q^21 - 4 * q^23 + 2 * q^25 + 2 * q^27 - 12 * q^29 - 2 * q^31 - 4 * q^35 - 8 * q^39 - 4 * q^41 + 8 * q^43 + 2 * q^45 - 12 * q^47 - 2 * q^49 - 16 * q^53 + 8 * q^59 - 4 * q^61 - 4 * q^63 - 8 * q^65 - 4 * q^67 - 4 * q^69 - 8 * q^71 - 16 * q^73 + 2 * q^75 - 8 * q^77 + 12 * q^79 + 2 * q^81 - 4 * q^83 - 12 * q^87 - 4 * q^89 + 12 * q^91 - 2 * q^93 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more