LMFDB - Embedded newform 1664.2.a.v.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1664,2,Mod(1,1664)] 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("1664.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1664 = 2^{7} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1664.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,2,0,2,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$13.2871068963$
Analytic rank:	$0$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	1664.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214 q^{3} -1.82843 q^{5} +2.41421 q^{7} -2.82843 q^{9} -4.82843 q^{11} -1.00000 q^{13} -0.757359 q^{15} +5.00000 q^{17} +4.82843 q^{19} +1.00000 q^{21} +4.00000 q^{23} -1.65685 q^{25} -2.41421 q^{27} +3.65685 q^{29} +6.00000 q^{31} -2.00000 q^{33} -4.41421 q^{35} +8.65685 q^{37} -0.414214 q^{39} +8.00000 q^{41} -0.757359 q^{43} +5.17157 q^{45} +6.07107 q^{47} -1.17157 q^{49} +2.07107 q^{51} +8.00000 q^{53} +8.82843 q^{55} +2.00000 q^{57} -14.4853 q^{59} +9.65685 q^{61} -6.82843 q^{63} +1.82843 q^{65} -8.82843 q^{67} +1.65685 q^{69} -9.72792 q^{71} +13.3137 q^{73} -0.686292 q^{75} -11.6569 q^{77} -7.31371 q^{79} +7.48528 q^{81} +6.34315 q^{83} -9.14214 q^{85} +1.51472 q^{87} -7.65685 q^{89} -2.41421 q^{91} +2.48528 q^{93} -8.82843 q^{95} +14.0000 q^{97} +13.6569 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} - 4 q^{11} - 2 q^{13} - 10 q^{15} + 10 q^{17} + 4 q^{19} + 2 q^{21} + 8 q^{23} + 8 q^{25} - 2 q^{27} - 4 q^{29} + 12 q^{31} - 4 q^{33} - 6 q^{35} + 6 q^{37} + 2 q^{39}+ \cdots + 16 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 - 4 * q^11 - 2 * q^13 - 10 * q^15 + 10 * q^17 + 4 * q^19 + 2 * q^21 + 8 * q^23 + 8 * q^25 - 2 * q^27 - 4 * q^29 + 12 * q^31 - 4 * q^33 - 6 * q^35 + 6 * q^37 + 2 * q^39 + 16 * q^41 - 10 * q^43 + 16 * q^45 - 2 * q^47 - 8 * q^49 - 10 * q^51 + 16 * q^53 + 12 * q^55 + 4 * q^57 - 12 * q^59 + 8 * q^61 - 8 * q^63 - 2 * q^65 - 12 * q^67 - 8 * q^69 + 6 * q^71 + 4 * q^73 - 24 * q^75 - 12 * q^77 + 8 * q^79 - 2 * q^81 + 24 * q^83 + 10 * q^85 + 20 * q^87 - 4 * q^89 - 2 * q^91 - 12 * q^93 - 12 * q^95 + 28 * q^97 + 16 * 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
$3$	0.414214	0.239146	0.119573	−	0.992825i	$-0.461847\pi$
$3$	0.414214	0.239146	0.119573	+	0.992825i	$0.461847\pi$
$4$	0	0
$5$	−1.82843	−0.817697	−0.408849	−	0.912602i	$-0.634070\pi$
$5$	−1.82843	−0.817697	−0.408849	+	0.912602i	$0.634070\pi$
$6$	0	0
$7$	2.41421	0.912487	0.456243	−	0.889855i	$-0.349195\pi$
$7$	2.41421	0.912487	0.456243	+	0.889855i	$0.349195\pi$
$8$	0	0
$9$	−2.82843	−0.942809
$10$	0	0
$11$	−4.82843	−1.45583	−0.727913	−	0.685670i	$-0.759509\pi$
$11$	−4.82843	−1.45583	−0.727913	+	0.685670i	$0.759509\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−0.757359	−0.195549
$16$	0	0
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	0	0
$19$	4.82843	1.10772	0.553859	−	0.832611i	$-0.313155\pi$
$19$	4.82843	1.10772	0.553859	+	0.832611i	$0.313155\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	−1.65685	−0.331371
$26$	0	0
$27$	−2.41421	−0.464616
$28$	0	0
$29$	3.65685	0.679061	0.339530	−	0.940595i	$-0.389732\pi$
$29$	3.65685	0.679061	0.339530	+	0.940595i	$0.389732\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	−4.41421	−0.746138
$36$	0	0
$37$	8.65685	1.42318	0.711589	−	0.702596i	$-0.247976\pi$
$37$	8.65685	1.42318	0.711589	+	0.702596i	$0.247976\pi$
$38$	0	0
$39$	−0.414214	−0.0663273
$40$	0	0
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	0	0
$43$	−0.757359	−0.115496	−0.0577481	−	0.998331i	$-0.518392\pi$
$43$	−0.757359	−0.115496	−0.0577481	+	0.998331i	$0.518392\pi$
$44$	0	0
$45$	5.17157	0.770933
$46$	0	0
$47$	6.07107	0.885556	0.442778	−	0.896631i	$-0.353993\pi$
$47$	6.07107	0.885556	0.442778	+	0.896631i	$0.353993\pi$
$48$	0	0
$49$	−1.17157	−0.167368
$50$	0	0
$51$	2.07107	0.290008
$52$	0	0
$53$	8.00000	1.09888	0.549442	−	0.835532i	$-0.314840\pi$
$53$	8.00000	1.09888	0.549442	+	0.835532i	$0.314840\pi$
$54$	0	0
$55$	8.82843	1.19042
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	−14.4853	−1.88582	−0.942912	−	0.333043i	$-0.891924\pi$
$59$	−14.4853	−1.88582	−0.942912	+	0.333043i	$0.891924\pi$
$60$	0	0
$61$	9.65685	1.23643	0.618217	−	0.786008i	$-0.287855\pi$
$61$	9.65685	1.23643	0.618217	+	0.786008i	$0.287855\pi$
$62$	0	0
$63$	−6.82843	−0.860301
$64$	0	0
$65$	1.82843	0.226788
$66$	0	0
$67$	−8.82843	−1.07856	−0.539282	−	0.842125i	$-0.681304\pi$
$67$	−8.82843	−1.07856	−0.539282	+	0.842125i	$0.681304\pi$
$68$	0	0
$69$	1.65685	0.199462
$70$	0	0
$71$	−9.72792	−1.15449	−0.577246	−	0.816570i	$-0.695873\pi$
$71$	−9.72792	−1.15449	−0.577246	+	0.816570i	$0.695873\pi$
$72$	0	0
$73$	13.3137	1.55825	0.779126	−	0.626868i	$-0.215663\pi$
$73$	13.3137	1.55825	0.779126	+	0.626868i	$0.215663\pi$
$74$	0	0
$75$	−0.686292	−0.0792461
$76$	0	0
$77$	−11.6569	−1.32842
$78$	0	0
$79$	−7.31371	−0.822856	−0.411428	−	0.911442i	$-0.634970\pi$
$79$	−7.31371	−0.822856	−0.411428	+	0.911442i	$0.634970\pi$
$80$	0	0
$81$	7.48528	0.831698
$82$	0	0
$83$	6.34315	0.696251	0.348125	−	0.937448i	$-0.386818\pi$
$83$	6.34315	0.696251	0.348125	+	0.937448i	$0.386818\pi$
$84$	0	0
$85$	−9.14214	−0.991604
$86$	0	0
$87$	1.51472	0.162395
$88$	0	0
$89$	−7.65685	−0.811625	−0.405812	−	0.913956i	$-0.633011\pi$
$89$	−7.65685	−0.811625	−0.405812	+	0.913956i	$0.633011\pi$
$90$	0	0
$91$	−2.41421	−0.253078
$92$	0	0
$93$	2.48528	0.257712
$94$	0	0
$95$	−8.82843	−0.905778
$96$	0	0
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	0	0
$99$	13.6569	1.37257

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	1664.2.a.v.1.2	yes	2
4.3	odd	2		1664.2.a.x.1.1	yes	2
8.3	odd	2		1664.2.a.u.1.2	✓	2
8.5	even	2		1664.2.a.w.1.1	yes	2
16.3	odd	4		3328.2.b.z.1665.2		4
16.5	even	4		3328.2.b.v.1665.2		4
16.11	odd	4		3328.2.b.z.1665.3		4
16.13	even	4		3328.2.b.v.1665.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1664.2.a.u.1.2	✓	2	8.3	odd	2
1664.2.a.v.1.2	yes	2	1.1	even	1	trivial
1664.2.a.w.1.1	yes	2	8.5	even	2
1664.2.a.x.1.1	yes	2	4.3	odd	2
3328.2.b.v.1665.2		4	16.5	even	4
3328.2.b.v.1665.3		4	16.13	even	4
3328.2.b.z.1665.2		4	16.3	odd	4
3328.2.b.z.1665.3		4	16.11	odd	4

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:	\( N \)	\(=\)	\( 1664 = 2^{7} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1664.a (trivial)

\(f(q)\)	\(=\)	\(q+0.414214 q^{3} -1.82843 q^{5} +2.41421 q^{7} -2.82843 q^{9} -4.82843 q^{11} -1.00000 q^{13} -0.757359 q^{15} +5.00000 q^{17} +4.82843 q^{19} +1.00000 q^{21} +4.00000 q^{23} -1.65685 q^{25} -2.41421 q^{27} +3.65685 q^{29} +6.00000 q^{31} -2.00000 q^{33} -4.41421 q^{35} +8.65685 q^{37} -0.414214 q^{39} +8.00000 q^{41} -0.757359 q^{43} +5.17157 q^{45} +6.07107 q^{47} -1.17157 q^{49} +2.07107 q^{51} +8.00000 q^{53} +8.82843 q^{55} +2.00000 q^{57} -14.4853 q^{59} +9.65685 q^{61} -6.82843 q^{63} +1.82843 q^{65} -8.82843 q^{67} +1.65685 q^{69} -9.72792 q^{71} +13.3137 q^{73} -0.686292 q^{75} -11.6569 q^{77} -7.31371 q^{79} +7.48528 q^{81} +6.34315 q^{83} -9.14214 q^{85} +1.51472 q^{87} -7.65685 q^{89} -2.41421 q^{91} +2.48528 q^{93} -8.82843 q^{95} +14.0000 q^{97} +13.6569 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} - 4 q^{11} - 2 q^{13} - 10 q^{15} + 10 q^{17} + 4 q^{19} + 2 q^{21} + 8 q^{23} + 8 q^{25} - 2 q^{27} - 4 q^{29} + 12 q^{31} - 4 q^{33} - 6 q^{35} + 6 q^{37} + 2 q^{39}+ \cdots + 16 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 - 4 * q^11 - 2 * q^13 - 10 * q^15 + 10 * q^17 + 4 * q^19 + 2 * q^21 + 8 * q^23 + 8 * q^25 - 2 * q^27 - 4 * q^29 + 12 * q^31 - 4 * q^33 - 6 * q^35 + 6 * q^37 + 2 * q^39 + 16 * q^41 - 10 * q^43 + 16 * q^45 - 2 * q^47 - 8 * q^49 - 10 * q^51 + 16 * q^53 + 12 * q^55 + 4 * q^57 - 12 * q^59 + 8 * q^61 - 8 * q^63 - 2 * q^65 - 12 * q^67 - 8 * q^69 + 6 * q^71 + 4 * q^73 - 24 * q^75 - 12 * q^77 + 8 * q^79 - 2 * q^81 + 24 * q^83 + 10 * q^85 + 20 * q^87 - 4 * q^89 - 2 * q^91 - 12 * q^93 - 12 * q^95 + 28 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more