LMFDB - Embedded newform 476.2.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [476,2,Mod(1,476)] 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("476.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(476, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 476 = 2^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	476.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-3,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$3.80087913621$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	476.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.30278 q^{3} +0.302776 q^{5} +1.00000 q^{7} +7.90833 q^{9} -2.60555 q^{11} +0.605551 q^{13} -1.00000 q^{15} -1.00000 q^{17} -6.00000 q^{19} -3.30278 q^{21} +4.60555 q^{23} -4.90833 q^{25} -16.2111 q^{27} -1.39445 q^{29} -10.3028 q^{31} +8.60555 q^{33} +0.302776 q^{35} -7.21110 q^{37} -2.00000 q^{39} -8.51388 q^{41} -0.697224 q^{43} +2.39445 q^{45} +10.0000 q^{47} +1.00000 q^{49} +3.30278 q^{51} +4.30278 q^{53} -0.788897 q^{55} +19.8167 q^{57} -9.21110 q^{59} -0.697224 q^{61} +7.90833 q^{63} +0.183346 q^{65} +6.30278 q^{67} -15.2111 q^{69} -2.00000 q^{71} +4.51388 q^{73} +16.2111 q^{75} -2.60555 q^{77} -6.00000 q^{79} +29.8167 q^{81} -5.21110 q^{83} -0.302776 q^{85} +4.60555 q^{87} -2.00000 q^{89} +0.605551 q^{91} +34.0278 q^{93} -1.81665 q^{95} -12.9083 q^{97} -20.6056 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9} + 2 q^{11} - 6 q^{13} - 2 q^{15} - 2 q^{17} - 12 q^{19} - 3 q^{21} + 2 q^{23} + q^{25} - 18 q^{27} - 10 q^{29} - 17 q^{31} + 10 q^{33} - 3 q^{35} - 4 q^{39}+ \cdots - 34 q^{99}+O(q^{100}) $ 2 * q - 3 * q^3 - 3 * q^5 + 2 * q^7 + 5 * q^9 + 2 * q^11 - 6 * q^13 - 2 * q^15 - 2 * q^17 - 12 * q^19 - 3 * q^21 + 2 * q^23 + q^25 - 18 * q^27 - 10 * q^29 - 17 * q^31 + 10 * q^33 - 3 * q^35 - 4 * q^39 + q^41 - 5 * q^43 + 12 * q^45 + 20 * q^47 + 2 * q^49 + 3 * q^51 + 5 * q^53 - 16 * q^55 + 18 * q^57 - 4 * q^59 - 5 * q^61 + 5 * q^63 + 22 * q^65 + 9 * q^67 - 16 * q^69 - 4 * q^71 - 9 * q^73 + 18 * q^75 + 2 * q^77 - 12 * q^79 + 38 * q^81 + 4 * q^83 + 3 * q^85 + 2 * q^87 - 4 * q^89 - 6 * q^91 + 32 * q^93 + 18 * q^95 - 15 * q^97 - 34 * q^99

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
$3$	−3.30278	−1.90686	−0.953429	−	0.301617i	$-0.902474\pi$
$3$	−3.30278	−1.90686	−0.953429	+	0.301617i	$0.902474\pi$
$4$	0	0
$5$	0.302776	0.135405	0.0677027	−	0.997706i	$-0.478433\pi$
$5$	0.302776	0.135405	0.0677027	+	0.997706i	$0.478433\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	7.90833	2.63611
$10$	0	0
$11$	−2.60555	−0.785603	−0.392802	−	0.919623i	$-0.628494\pi$
$11$	−2.60555	−0.785603	−0.392802	+	0.919623i	$0.628494\pi$
$12$	0	0
$13$	0.605551	0.167950	0.0839749	−	0.996468i	$-0.473238\pi$
$13$	0.605551	0.167950	0.0839749	+	0.996468i	$0.473238\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	−3.30278	−0.720725
$22$	0	0
$23$	4.60555	0.960324	0.480162	−	0.877180i	$-0.340578\pi$
$23$	4.60555	0.960324	0.480162	+	0.877180i	$0.340578\pi$
$24$	0	0
$25$	−4.90833	−0.981665
$26$	0	0
$27$	−16.2111	−3.11983
$28$	0	0
$29$	−1.39445	−0.258943	−0.129471	−	0.991583i	$-0.541328\pi$
$29$	−1.39445	−0.258943	−0.129471	+	0.991583i	$0.541328\pi$
$30$	0	0
$31$	−10.3028	−1.85043	−0.925217	−	0.379439i	$-0.876117\pi$
$31$	−10.3028	−1.85043	−0.925217	+	0.379439i	$0.876117\pi$
$32$	0	0
$33$	8.60555	1.49803
$34$	0	0
$35$	0.302776	0.0511784
$36$	0	0
$37$	−7.21110	−1.18550	−0.592749	−	0.805387i	$-0.701957\pi$
$37$	−7.21110	−1.18550	−0.592749	+	0.805387i	$0.701957\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−8.51388	−1.32964	−0.664822	−	0.747002i	$-0.731493\pi$
$41$	−8.51388	−1.32964	−0.664822	+	0.747002i	$0.731493\pi$
$42$	0	0
$43$	−0.697224	−0.106326	−0.0531629	−	0.998586i	$-0.516930\pi$
$43$	−0.697224	−0.106326	−0.0531629	+	0.998586i	$0.516930\pi$
$44$	0	0
$45$	2.39445	0.356943
$46$	0	0
$47$	10.0000	1.45865	0.729325	−	0.684167i	$-0.239834\pi$
$47$	10.0000	1.45865	0.729325	+	0.684167i	$0.239834\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.30278	0.462481
$52$	0	0
$53$	4.30278	0.591032	0.295516	−	0.955338i	$-0.404508\pi$
$53$	4.30278	0.591032	0.295516	+	0.955338i	$0.404508\pi$
$54$	0	0
$55$	−0.788897	−0.106375
$56$	0	0
$57$	19.8167	2.62478
$58$	0	0
$59$	−9.21110	−1.19918	−0.599592	−	0.800306i	$-0.704670\pi$
$59$	−9.21110	−1.19918	−0.599592	+	0.800306i	$0.704670\pi$
$60$	0	0
$61$	−0.697224	−0.0892704	−0.0446352	−	0.999003i	$-0.514213\pi$
$61$	−0.697224	−0.0892704	−0.0446352	+	0.999003i	$0.514213\pi$
$62$	0	0
$63$	7.90833	0.996356
$64$	0	0
$65$	0.183346	0.0227413
$66$	0	0
$67$	6.30278	0.770007	0.385003	−	0.922915i	$-0.374200\pi$
$67$	6.30278	0.770007	0.385003	+	0.922915i	$0.374200\pi$
$68$	0	0
$69$	−15.2111	−1.83120
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	4.51388	0.528309	0.264155	−	0.964480i	$-0.414907\pi$
$73$	4.51388	0.528309	0.264155	+	0.964480i	$0.414907\pi$
$74$	0	0
$75$	16.2111	1.87190
$76$	0	0
$77$	−2.60555	−0.296930
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	0	0
$81$	29.8167	3.31296
$82$	0	0
$83$	−5.21110	−0.571993	−0.285996	−	0.958231i	$-0.592325\pi$
$83$	−5.21110	−0.571993	−0.285996	+	0.958231i	$0.592325\pi$
$84$	0	0
$85$	−0.302776	−0.0328406
$86$	0	0
$87$	4.60555	0.493767
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	0.605551	0.0634790
$92$	0	0
$93$	34.0278	3.52851
$94$	0	0
$95$	−1.81665	−0.186385
$96$	0	0
$97$	−12.9083	−1.31064	−0.655321	−	0.755350i	$-0.727467\pi$
$97$	−12.9083	−1.31064	−0.655321	+	0.755350i	$0.727467\pi$
$98$	0	0
$99$	−20.6056	−2.07094

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	476.2.a.a.1.1	✓	2
3.2	odd	2		4284.2.a.p.1.1		2
4.3	odd	2		1904.2.a.l.1.2		2
7.6	odd	2		3332.2.a.n.1.2		2
8.3	odd	2		7616.2.a.m.1.1		2
8.5	even	2		7616.2.a.z.1.2		2
17.16	even	2		8092.2.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
476.2.a.a.1.1	✓	2	1.1	even	1	trivial
1904.2.a.l.1.2		2	4.3	odd	2
3332.2.a.n.1.2		2	7.6	odd	2
4284.2.a.p.1.1		2	3.2	odd	2
7616.2.a.m.1.1		2	8.3	odd	2
7616.2.a.z.1.2		2	8.5	even	2
8092.2.a.n.1.2		2	17.16	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 476 = 2^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	476.a (trivial)

\(f(q)\)	\(=\)	\(q-3.30278 q^{3} +0.302776 q^{5} +1.00000 q^{7} +7.90833 q^{9} -2.60555 q^{11} +0.605551 q^{13} -1.00000 q^{15} -1.00000 q^{17} -6.00000 q^{19} -3.30278 q^{21} +4.60555 q^{23} -4.90833 q^{25} -16.2111 q^{27} -1.39445 q^{29} -10.3028 q^{31} +8.60555 q^{33} +0.302776 q^{35} -7.21110 q^{37} -2.00000 q^{39} -8.51388 q^{41} -0.697224 q^{43} +2.39445 q^{45} +10.0000 q^{47} +1.00000 q^{49} +3.30278 q^{51} +4.30278 q^{53} -0.788897 q^{55} +19.8167 q^{57} -9.21110 q^{59} -0.697224 q^{61} +7.90833 q^{63} +0.183346 q^{65} +6.30278 q^{67} -15.2111 q^{69} -2.00000 q^{71} +4.51388 q^{73} +16.2111 q^{75} -2.60555 q^{77} -6.00000 q^{79} +29.8167 q^{81} -5.21110 q^{83} -0.302776 q^{85} +4.60555 q^{87} -2.00000 q^{89} +0.605551 q^{91} +34.0278 q^{93} -1.81665 q^{95} -12.9083 q^{97} -20.6056 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9} + 2 q^{11} - 6 q^{13} - 2 q^{15} - 2 q^{17} - 12 q^{19} - 3 q^{21} + 2 q^{23} + q^{25} - 18 q^{27} - 10 q^{29} - 17 q^{31} + 10 q^{33} - 3 q^{35} - 4 q^{39}+ \cdots - 34 q^{99}+O(q^{100}) \) 2 * q - 3 * q^3 - 3 * q^5 + 2 * q^7 + 5 * q^9 + 2 * q^11 - 6 * q^13 - 2 * q^15 - 2 * q^17 - 12 * q^19 - 3 * q^21 + 2 * q^23 + q^25 - 18 * q^27 - 10 * q^29 - 17 * q^31 + 10 * q^33 - 3 * q^35 - 4 * q^39 + q^41 - 5 * q^43 + 12 * q^45 + 20 * q^47 + 2 * q^49 + 3 * q^51 + 5 * q^53 - 16 * q^55 + 18 * q^57 - 4 * q^59 - 5 * q^61 + 5 * q^63 + 22 * q^65 + 9 * q^67 - 16 * q^69 - 4 * q^71 - 9 * q^73 + 18 * q^75 + 2 * q^77 - 12 * q^79 + 38 * q^81 + 4 * q^83 + 3 * q^85 + 2 * q^87 - 4 * q^89 - 6 * q^91 + 32 * q^93 + 18 * q^95 - 15 * q^97 - 34 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more