LMFDB - Embedded newform 2070.2.a.z.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2070,2,Mod(1,2070)] 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("2070.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2070, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2070.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-3,0,3,3,0,3,-3,0,-3,-3,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$16.5290332184$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1101.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 12 $ x^3 - x^2 - 9*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.11903$ of defining polynomial
Character	$\chi$	$=$	2070.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.50973 q^{7} -1.00000 q^{8} -1.00000 q^{10} -4.33763 q^{11} -3.72833 q^{13} -4.50973 q^{14} +1.00000 q^{16} -1.11903 q^{17} +4.50973 q^{19} +1.00000 q^{20} +4.33763 q^{22} +1.00000 q^{23} +1.00000 q^{25} +3.72833 q^{26} +4.50973 q^{28} +8.23805 q^{29} +1.72833 q^{31} -1.00000 q^{32} +1.11903 q^{34} +4.50973 q^{35} -0.781399 q^{37} -4.50973 q^{38} -1.00000 q^{40} -3.90043 q^{41} +8.00000 q^{43} -4.33763 q^{44} -1.00000 q^{46} +11.4567 q^{47} +13.3376 q^{49} -1.00000 q^{50} -3.72833 q^{52} +6.00000 q^{53} -4.33763 q^{55} -4.50973 q^{56} -8.23805 q^{58} +2.23805 q^{59} +3.55623 q^{61} -1.72833 q^{62} +1.00000 q^{64} -3.72833 q^{65} +2.43720 q^{67} -1.11903 q^{68} -4.50973 q^{70} -7.11903 q^{71} -9.45665 q^{73} +0.781399 q^{74} +4.50973 q^{76} -19.5615 q^{77} -14.9133 q^{79} +1.00000 q^{80} +3.90043 q^{82} -2.78140 q^{83} -1.11903 q^{85} -8.00000 q^{86} +4.33763 q^{88} +7.69471 q^{89} -16.8137 q^{91} +1.00000 q^{92} -11.4567 q^{94} +4.50973 q^{95} -0.642920 q^{97} -13.3376 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} + 3 q^{7} - 3 q^{8} - 3 q^{10} - 3 q^{11} - q^{13} - 3 q^{14} + 3 q^{16} + 7 q^{17} + 3 q^{19} + 3 q^{20} + 3 q^{22} + 3 q^{23} + 3 q^{25} + q^{26} + 3 q^{28} + 4 q^{29}+ \cdots - 30 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 + 3 * q^7 - 3 * q^8 - 3 * q^10 - 3 * q^11 - q^13 - 3 * q^14 + 3 * q^16 + 7 * q^17 + 3 * q^19 + 3 * q^20 + 3 * q^22 + 3 * q^23 + 3 * q^25 + q^26 + 3 * q^28 + 4 * q^29 - 5 * q^31 - 3 * q^32 - 7 * q^34 + 3 * q^35 - 2 * q^37 - 3 * q^38 - 3 * q^40 - q^41 + 24 * q^43 - 3 * q^44 - 3 * q^46 + 14 * q^47 + 30 * q^49 - 3 * q^50 - q^52 + 18 * q^53 - 3 * q^55 - 3 * q^56 - 4 * q^58 - 14 * q^59 + q^61 + 5 * q^62 + 3 * q^64 - q^65 + 8 * q^67 + 7 * q^68 - 3 * q^70 - 11 * q^71 - 8 * q^73 + 2 * q^74 + 3 * q^76 + 24 * q^77 - 4 * q^79 + 3 * q^80 + q^82 - 8 * q^83 + 7 * q^85 - 24 * q^86 + 3 * q^88 - 18 * q^89 + q^91 + 3 * q^92 - 14 * q^94 + 3 * q^95 - 33 * q^97 - 30 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.50973	1.70452	0.852258	−	0.523122i	$-0.175233\pi$
$7$	4.50973	1.70452	0.852258	+	0.523122i	$0.175233\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	−4.33763	−1.30784	−0.653922	−	0.756562i	$-0.726878\pi$
$11$	−4.33763	−1.30784	−0.653922	+	0.756562i	$0.726878\pi$
$12$	0	0
$13$	−3.72833	−1.03405	−0.517026	−	0.855970i	$-0.672961\pi$
$13$	−3.72833	−1.03405	−0.517026	+	0.855970i	$0.672961\pi$
$14$	−4.50973	−1.20527
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.11903	−0.271404	−0.135702	−	0.990750i	$-0.543329\pi$
$17$	−1.11903	−0.271404	−0.135702	+	0.990750i	$0.543329\pi$
$18$	0	0
$19$	4.50973	1.03460	0.517301	−	0.855803i	$-0.326937\pi$
$19$	4.50973	1.03460	0.517301	+	0.855803i	$0.326937\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	4.33763	0.924785
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	3.72833	0.731185
$27$	0	0
$28$	4.50973	0.852258
$29$	8.23805	1.52977	0.764884	−	0.644168i	$-0.222796\pi$
$29$	8.23805	1.52977	0.764884	+	0.644168i	$0.222796\pi$
$30$	0	0
$31$	1.72833	0.310417	0.155208	−	0.987882i	$-0.450395\pi$
$31$	1.72833	0.310417	0.155208	+	0.987882i	$0.450395\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	1.11903	0.191911
$35$	4.50973	0.762283
$36$	0	0
$37$	−0.781399	−0.128461	−0.0642306	−	0.997935i	$-0.520459\pi$
$37$	−0.781399	−0.128461	−0.0642306	+	0.997935i	$0.520459\pi$
$38$	−4.50973	−0.731574
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−3.90043	−0.609144	−0.304572	−	0.952489i	$-0.598513\pi$
$41$	−3.90043	−0.609144	−0.304572	+	0.952489i	$0.598513\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	−4.33763	−0.653922
$45$	0	0
$46$	−1.00000	−0.147442
$47$	11.4567	1.67112	0.835562	−	0.549396i	$-0.185142\pi$
$47$	11.4567	1.67112	0.835562	+	0.549396i	$0.185142\pi$
$48$	0	0
$49$	13.3376	1.90538
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−3.72833	−0.517026
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	−4.33763	−0.584886
$56$	−4.50973	−0.602637
$57$	0	0
$58$	−8.23805	−1.08171
$59$	2.23805	0.291370	0.145685	−	0.989331i	$-0.453461\pi$
$59$	2.23805	0.291370	0.145685	+	0.989331i	$0.453461\pi$
$60$	0	0
$61$	3.55623	0.455329	0.227664	−	0.973740i	$-0.426891\pi$
$61$	3.55623	0.455329	0.227664	+	0.973740i	$0.426891\pi$
$62$	−1.72833	−0.219498
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.72833	−0.462442
$66$	0	0
$67$	2.43720	0.297752	0.148876	−	0.988856i	$-0.452435\pi$
$67$	2.43720	0.297752	0.148876	+	0.988856i	$0.452435\pi$
$68$	−1.11903	−0.135702
$69$	0	0
$70$	−4.50973	−0.539015
$71$	−7.11903	−0.844873	−0.422437	−	0.906393i	$-0.638825\pi$
$71$	−7.11903	−0.844873	−0.422437	+	0.906393i	$0.638825\pi$
$72$	0	0
$73$	−9.45665	−1.10682	−0.553409	−	0.832910i	$-0.686673\pi$
$73$	−9.45665	−1.10682	−0.553409	+	0.832910i	$0.686673\pi$
$74$	0.781399	0.0908357
$75$	0	0
$76$	4.50973	0.517301
$77$	−19.5615	−2.22924
$78$	0	0
$79$	−14.9133	−1.67788	−0.838939	−	0.544225i	$-0.816824\pi$
$79$	−14.9133	−1.67788	−0.838939	+	0.544225i	$0.816824\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	3.90043	0.430730
$83$	−2.78140	−0.305298	−0.152649	−	0.988280i	$-0.548780\pi$
$83$	−2.78140	−0.305298	−0.152649	+	0.988280i	$0.548780\pi$
$84$	0	0
$85$	−1.11903	−0.121375
$86$	−8.00000	−0.862662
$87$	0	0
$88$	4.33763	0.462393
$89$	7.69471	0.815637	0.407819	−	0.913063i	$-0.366290\pi$
$89$	7.69471	0.815637	0.407819	+	0.913063i	$0.366290\pi$
$90$	0	0
$91$	−16.8137	−1.76256
$92$	1.00000	0.104257
$93$	0	0
$94$	−11.4567	−1.18166
$95$	4.50973	0.462688
$96$	0	0
$97$	−0.642920	−0.0652786	−0.0326393	−	0.999467i	$-0.510391\pi$
$97$	−0.642920	−0.0652786	−0.0326393	+	0.999467i	$0.510391\pi$
$98$	−13.3376	−1.34730
$99$	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	2070.2.a.z.1.3		3
3.2	odd	2		230.2.a.d.1.1	✓	3
12.11	even	2		1840.2.a.r.1.3		3
15.2	even	4		1150.2.b.j.599.6		6
15.8	even	4		1150.2.b.j.599.1		6
15.14	odd	2		1150.2.a.q.1.3		3
24.5	odd	2		7360.2.a.bz.1.3		3
24.11	even	2		7360.2.a.ce.1.1		3
60.59	even	2		9200.2.a.cf.1.1		3
69.68	even	2		5290.2.a.r.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.2.a.d.1.1	✓	3	3.2	odd	2
1150.2.a.q.1.3		3	15.14	odd	2
1150.2.b.j.599.1		6	15.8	even	4
1150.2.b.j.599.6		6	15.2	even	4
1840.2.a.r.1.3		3	12.11	even	2
2070.2.a.z.1.3		3	1.1	even	1	trivial
5290.2.a.r.1.1		3	69.68	even	2
7360.2.a.bz.1.3		3	24.5	odd	2
7360.2.a.ce.1.1		3	24.11	even	2
9200.2.a.cf.1.1		3	60.59	even	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:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2070.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.50973 q^{7} -1.00000 q^{8} -1.00000 q^{10} -4.33763 q^{11} -3.72833 q^{13} -4.50973 q^{14} +1.00000 q^{16} -1.11903 q^{17} +4.50973 q^{19} +1.00000 q^{20} +4.33763 q^{22} +1.00000 q^{23} +1.00000 q^{25} +3.72833 q^{26} +4.50973 q^{28} +8.23805 q^{29} +1.72833 q^{31} -1.00000 q^{32} +1.11903 q^{34} +4.50973 q^{35} -0.781399 q^{37} -4.50973 q^{38} -1.00000 q^{40} -3.90043 q^{41} +8.00000 q^{43} -4.33763 q^{44} -1.00000 q^{46} +11.4567 q^{47} +13.3376 q^{49} -1.00000 q^{50} -3.72833 q^{52} +6.00000 q^{53} -4.33763 q^{55} -4.50973 q^{56} -8.23805 q^{58} +2.23805 q^{59} +3.55623 q^{61} -1.72833 q^{62} +1.00000 q^{64} -3.72833 q^{65} +2.43720 q^{67} -1.11903 q^{68} -4.50973 q^{70} -7.11903 q^{71} -9.45665 q^{73} +0.781399 q^{74} +4.50973 q^{76} -19.5615 q^{77} -14.9133 q^{79} +1.00000 q^{80} +3.90043 q^{82} -2.78140 q^{83} -1.11903 q^{85} -8.00000 q^{86} +4.33763 q^{88} +7.69471 q^{89} -16.8137 q^{91} +1.00000 q^{92} -11.4567 q^{94} +4.50973 q^{95} -0.642920 q^{97} -13.3376 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} + 3 q^{7} - 3 q^{8} - 3 q^{10} - 3 q^{11} - q^{13} - 3 q^{14} + 3 q^{16} + 7 q^{17} + 3 q^{19} + 3 q^{20} + 3 q^{22} + 3 q^{23} + 3 q^{25} + q^{26} + 3 q^{28} + 4 q^{29}+ \cdots - 30 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 + 3 * q^7 - 3 * q^8 - 3 * q^10 - 3 * q^11 - q^13 - 3 * q^14 + 3 * q^16 + 7 * q^17 + 3 * q^19 + 3 * q^20 + 3 * q^22 + 3 * q^23 + 3 * q^25 + q^26 + 3 * q^28 + 4 * q^29 - 5 * q^31 - 3 * q^32 - 7 * q^34 + 3 * q^35 - 2 * q^37 - 3 * q^38 - 3 * q^40 - q^41 + 24 * q^43 - 3 * q^44 - 3 * q^46 + 14 * q^47 + 30 * q^49 - 3 * q^50 - q^52 + 18 * q^53 - 3 * q^55 - 3 * q^56 - 4 * q^58 - 14 * q^59 + q^61 + 5 * q^62 + 3 * q^64 - q^65 + 8 * q^67 + 7 * q^68 - 3 * q^70 - 11 * q^71 - 8 * q^73 + 2 * q^74 + 3 * q^76 + 24 * q^77 - 4 * q^79 + 3 * q^80 + q^82 - 8 * q^83 + 7 * q^85 - 24 * q^86 + 3 * q^88 - 18 * q^89 + q^91 + 3 * q^92 - 14 * q^94 + 3 * q^95 - 33 * q^97 - 30 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more