LMFDB - Embedded newform 639.3.d.b.496.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [639,3,Mod(496,639)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(639, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("639.496");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 639 = 3^{2} \cdot 71 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	639.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$17.4114888926$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	7.7.294755098673.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 14x^{5} + 56x^{3} - 56x - 21 $ x^7 - 14x^5 + 56x^3 - 56*x - 21
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	no (minimal twist has level 71)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			496.5
Root			$1.64039$ of defining polynomial
Character	$\chi$	$=$	639.496

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.30911 q^{2} -2.28622 q^{4} -9.99851 q^{5} -8.22938 q^{8} +O(q^{10})$	$q+1.30911 q^{2} -2.28622 q^{4} -9.99851 q^{5} -8.22938 q^{8} -13.0892 q^{10} -1.62833 q^{16} +35.0113 q^{19} +22.8588 q^{20} +74.9703 q^{25} -10.6009 q^{29} +30.7859 q^{32} -72.3160 q^{37} +45.8338 q^{38} +82.2816 q^{40} +30.5899 q^{43} +49.0000 q^{49} +98.1447 q^{50} -13.8778 q^{58} +46.8155 q^{64} +71.0000 q^{71} -79.0966 q^{73} -94.6699 q^{74} -80.0434 q^{76} +99.0936 q^{79} +16.2808 q^{80} +9.47851 q^{83} +40.0457 q^{86} -2.37599 q^{89} -350.061 q^{95} +64.1466 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 28 q^{4}+O(q^{10}) $ 7 * q + 28 * q^4	$ 7 q + 28 q^{4} + 112 q^{16} + 217 q^{20} + 175 q^{25} - 35 q^{38} + 343 q^{49} + 448 q^{64} + 497 q^{71} - 539 q^{74} + 868 q^{80}+O(q^{100}) $ 7 * q + 28 * q^4 + 112 * q^16 + 217 * q^20 + 175 * q^25 - 35 * q^38 + 343 * q^49 + 448 * q^64 + 497 * q^71 - 539 * q^74 + 868 * q^80

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/639\mathbb{Z}\right)^\times$.

$n$	$433$	$569$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See 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 $

$2$	1.30911	0.654557	0.327279	−	0.944928i	$-0.393868\pi$
$2$	1.30911	0.654557	0.327279	+	0.944928i	$0.393868\pi$
$3$	0	0
$3$	0	0
$4$	−2.28622	−0.571555
$5$	−9.99851	−1.99970	−0.999851	−	0.0172361i	$-0.994513\pi$
$5$	−9.99851	−1.99970	−0.999851	+	0.0172361i	$0.994513\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−8.22938	−1.02867
$9$	0	0
$10$	−13.0892	−1.30892
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	−1.62833	−0.101770
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	35.0113	1.84270	0.921349	−	0.388736i	$-0.127088\pi$
$19$	35.0113	1.84270	0.921349	+	0.388736i	$0.127088\pi$
$20$	22.8588	1.14294
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	74.9703	2.99881
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−10.6009	−0.365548	−0.182774	−	0.983155i	$-0.558508\pi$
$29$	−10.6009	−0.365548	−0.182774	+	0.983155i	$0.558508\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	30.7859	0.962058
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−72.3160	−1.95449	−0.977243	−	0.212121i	$-0.931963\pi$
$37$	−72.3160	−1.95449	−0.977243	+	0.212121i	$0.931963\pi$
$38$	45.8338	1.20615
$39$	0	0
$40$	82.2816	2.05704
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	30.5899	0.711393	0.355697	−	0.934601i	$-0.384244\pi$
$43$	30.5899	0.711393	0.355697	+	0.934601i	$0.384244\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	49.0000	1.00000
$50$	98.1447	1.96289
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−13.8778	−0.239272
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	46.8155	0.731492
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	71.0000	1.00000
$71$	71.0000	1.00000
$72$	0	0
$73$	−79.0966	−1.08351	−0.541757	−	0.840535i	$-0.682241\pi$
$73$	−79.0966	−1.08351	−0.541757	+	0.840535i	$0.682241\pi$
$74$	−94.6699	−1.27932
$75$	0	0
$76$	−80.0434	−1.05320
$77$	0	0
$78$	0	0
$79$	99.0936	1.25435	0.627175	−	0.778879i	$-0.284211\pi$
$79$	99.0936	1.25435	0.627175	+	0.778879i	$0.284211\pi$
$80$	16.2808	0.203510
$81$	0	0
$82$	0	0
$83$	9.47851	0.114199	0.0570995	−	0.998368i	$-0.481815\pi$
$83$	9.47851	0.114199	0.0570995	+	0.998368i	$0.481815\pi$
$84$	0	0
$85$	0	0
$86$	40.0457	0.465648
$87$	0	0
$88$	0	0
$89$	−2.37599	−0.0266965	−0.0133483	−	0.999911i	$-0.504249\pi$
$89$	−2.37599	−0.0266965	−0.0133483	+	0.999911i	$0.504249\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−350.061	−3.68485
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	64.1466	0.654557
$99$	0	0
$100$	−171.399	−1.71399
$101$	116.806	1.15649	0.578246	−	0.815862i	$-0.303737\pi$
$101$	116.806	1.15649	0.578246	+	0.815862i	$0.303737\pi$
$102$	0	0
$103$	97.5646	0.947229	0.473615	−	0.880732i	$-0.342949\pi$
$103$	97.5646	0.947229	0.473615	+	0.880732i	$0.342949\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	70.0000	0.654206	0.327103	−	0.944989i	$-0.393928\pi$
$107$	70.0000	0.654206	0.327103	+	0.944989i	$0.393928\pi$
$108$	0	0
$109$	−194.471	−1.78414	−0.892069	−	0.451899i	$-0.850747\pi$
$109$	−194.471	−1.78414	−0.892069	+	0.451899i	$0.850747\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	24.2360	0.208931
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	121.000	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−499.629	−3.99703
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−61.8566	−0.483254
$129$	0	0
$130$	0	0
$131$	250.557	1.91265	0.956325	−	0.292305i	$-0.0944221\pi$
$131$	250.557	1.91265	0.956325	+	0.292305i	$0.0944221\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	92.9471	0.654557
$143$	0	0
$144$	0	0
$145$	105.993	0.730987
$146$	−103.546	−0.709222
$147$	0	0
$148$	165.330	1.11710
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	207.448	1.37383	0.686914	−	0.726739i	$-0.258965\pi$
$151$	207.448	1.37383	0.686914	+	0.726739i	$0.258965\pi$
$152$	−288.121	−1.89553
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−84.5877	−0.538775	−0.269388	−	0.963032i	$-0.586821\pi$
$157$	−84.5877	−0.538775	−0.269388	+	0.963032i	$0.586821\pi$
$158$	129.725	0.821044
$159$	0	0
$160$	−307.813	−1.92383
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	12.4085	0.0747497
$167$	182.054	1.09014	0.545070	−	0.838390i	$-0.316503\pi$
$167$	182.054	1.09014	0.545070	+	0.838390i	$0.316503\pi$
$168$	0	0
$169$	169.000	1.00000
$170$	0	0
$171$	0	0
$172$	−69.9352	−0.406600
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	−3.11045	−0.0174744
$179$	147.600	0.824583	0.412291	−	0.911052i	$-0.364729\pi$
$179$	147.600	0.824583	0.412291	+	0.911052i	$0.364729\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	723.053	3.90839
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	−458.270	−2.41194
$191$	186.246	0.975111	0.487555	−	0.873092i	$-0.337889\pi$
$191$	186.246	0.975111	0.487555	+	0.873092i	$0.337889\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	−112.025	−0.571555
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	−273.900	−1.37638	−0.688190	−	0.725531i	$-0.741594\pi$
$199$	−273.900	−1.37638	−0.688190	+	0.725531i	$0.741594\pi$
$200$	−616.959	−3.08480
$201$	0	0
$202$	152.912	0.756991
$203$	0	0
$204$	0	0
$205$	0	0
$206$	127.723	0.620016
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	91.6380	0.428215
$215$	−305.854	−1.42257
$216$	0	0
$217$	0	0
$218$	−254.585	−1.16782
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	84.7777	0.380169	0.190085	−	0.981768i	$-0.439124\pi$
$223$	84.7777	0.380169	0.190085	+	0.981768i	$0.439124\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	−449.339	−1.96218	−0.981089	−	0.193557i	$-0.937998\pi$
$229$	−449.339	−1.96218	−0.981089	+	0.193557i	$0.937998\pi$
$230$	0	0
$231$	0	0
$232$	87.2387	0.376029
$233$	−370.064	−1.58826	−0.794129	−	0.607749i	$-0.792073\pi$
$233$	−370.064	−1.58826	−0.794129	+	0.607749i	$0.792073\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	158.403	0.654557
$243$	0	0
$244$	0	0
$245$	−489.927	−1.99970
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	−654.071	−2.61628
$251$	501.625	1.99851	0.999253	−	0.0386396i	$-0.0123024\pi$
$251$	501.625	1.99851	0.999253	+	0.0386396i	$0.0123024\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−268.239	−1.04781
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	328.008	1.25194
$263$	−23.8604	−0.0907239	−0.0453619	−	0.998971i	$-0.514444\pi$
$263$	−23.8604	−0.0907239	−0.0453619	+	0.998971i	$0.514444\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	−391.318	−1.44398	−0.721989	−	0.691904i	$-0.756772\pi$
$271$	−391.318	−1.44398	−0.721989	+	0.691904i	$0.756772\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	527.158	1.90310	0.951549	−	0.307499i	$-0.0994920\pi$
$277$	527.158	1.90310	0.951549	+	0.307499i	$0.0994920\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	−162.322	−0.571555
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	289.000	1.00000
$290$	138.757	0.478473
$291$	0	0
$292$	180.832	0.619288
$293$	550.000	1.87713	0.938567	−	0.345098i	$-0.112154\pi$
$293$	550.000	1.87713	0.938567	+	0.345098i	$0.112154\pi$
$294$	0	0
$295$	0	0
$296$	595.116	2.01053
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	271.573	0.899249
$303$	0	0
$304$	−57.0097	−0.187532
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−414.554	−1.33297	−0.666485	−	0.745518i	$-0.732202\pi$
$311$	−414.554	−1.33297	−0.666485	+	0.745518i	$0.732202\pi$
$312$	0	0
$313$	−451.859	−1.44364	−0.721819	−	0.692082i	$-0.756694\pi$
$313$	−451.859	−1.44364	−0.721819	+	0.692082i	$0.756694\pi$
$314$	−110.735	−0.352659
$315$	0	0
$316$	−226.550	−0.716929
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	−468.086	−1.46277
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	−21.6699	−0.0652709
$333$	0	0
$334$	238.329	0.713560
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	221.240	0.654557
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	−251.736	−0.731791
$345$	0	0
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	−709.895	−1.99970
$356$	5.43204	0.0152585
$357$	0	0
$358$	193.226	0.539736
$359$	−194.918	−0.542948	−0.271474	−	0.962446i	$-0.587511\pi$
$359$	−194.918	−0.542948	−0.271474	+	0.962446i	$0.587511\pi$
$360$	0	0
$361$	864.789	2.39554
$362$	0	0
$363$	0	0
$364$	0	0
$365$	790.848	2.16671
$366$	0	0
$367$	681.979	1.85825	0.929127	−	0.369761i	$-0.120560\pi$
$367$	681.979	1.85825	0.929127	+	0.369761i	$0.120560\pi$
$368$	0	0
$369$	0	0
$370$	946.559	2.55827
$371$	0	0
$372$	0	0
$373$	−175.324	−0.470038	−0.235019	−	0.971991i	$-0.575515\pi$
$373$	−175.324	−0.470038	−0.235019	+	0.971991i	$0.575515\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	642.266	1.69463	0.847317	−	0.531087i	$-0.178216\pi$
$379$	642.266	1.69463	0.847317	+	0.531087i	$0.178216\pi$
$380$	800.315	2.10609
$381$	0	0
$382$	243.818	0.638266
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	−403.240	−1.02867
$393$	0	0
$394$	0	0
$395$	−990.789	−2.50833
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	−358.566	−0.900919
$399$	0	0
$400$	−122.076	−0.305190
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	−267.044	−0.660999
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−817.912	−1.99978	−0.999892	−	0.0146943i	$-0.995323\pi$
$409$	−817.912	−1.99978	−0.999892	+	0.0146943i	$0.995323\pi$
$410$	0	0
$411$	0	0
$412$	−223.054	−0.541393
$413$	0	0
$414$	0	0
$415$	−94.7710	−0.228364
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−830.889	−1.98303	−0.991514	−	0.130001i	$-0.958502\pi$
$419$	−830.889	−1.98303	−0.991514	+	0.130001i	$0.958502\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	−160.035	−0.373914
$429$	0	0
$430$	−400.397	−0.931157
$431$	810.536	1.88059	0.940297	−	0.340355i	$-0.110547\pi$
$431$	810.536	1.88059	0.940297	+	0.340355i	$0.110547\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	444.603	1.01973
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	23.7564	0.0533852
$446$	110.984	0.248843
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	−588.236	−1.28436
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	684.252	1.47787	0.738933	−	0.673779i	$-0.235330\pi$
$463$	684.252	1.47787	0.738933	+	0.673779i	$0.235330\pi$
$464$	17.2617	0.0372019
$465$	0	0
$466$	−484.456	−1.03961
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	2624.81	5.52591
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−276.633	−0.571555
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	−641.371	−1.30892
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	43.5005	0.0871753	0.0435876	−	0.999050i	$-0.486121\pi$
$499$	43.5005	0.0871753	0.0435876	+	0.999050i	$0.486121\pi$
$500$	1142.26	2.28452
$501$	0	0
$502$	656.685	1.30814
$503$	927.795	1.84452	0.922261	−	0.386567i	$-0.126339\pi$
$503$	927.795	1.84452	0.922261	+	0.386567i	$0.126339\pi$
$504$	0	0
$505$	−1167.88	−2.31264
$506$	0	0
$507$	0	0
$508$	0	0
$509$	118.000	0.231827	0.115914	−	0.993259i	$-0.463020\pi$
$509$	118.000	0.231827	0.115914	+	0.993259i	$0.463020\pi$
$510$	0	0
$511$	0	0
$512$	−103.730	−0.202597
$513$	0	0
$514$	0	0
$515$	−975.501	−1.89418
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	997.771	1.91511	0.957554	−	0.288254i	$-0.0930748\pi$
$521$	997.771	1.91511	0.957554	+	0.288254i	$0.0930748\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−572.829	−1.09318
$525$	0	0
$526$	−31.2360	−0.0593840
$527$	0	0
$528$	0	0
$529$	529.000	1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−699.896	−1.30822
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	−512.280	−0.945166
$543$	0	0
$544$	0	0
$545$	1944.42	3.56775
$546$	0	0
$547$	−1082.75	−1.97944	−0.989718	−	0.143034i	$-0.954314\pi$
$547$	−1082.75	−1.97944	−0.989718	+	0.143034i	$0.954314\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−371.151	−0.673594
$552$	0	0
$553$	0	0
$554$	690.110	1.24569
$555$	0	0
$556$	0	0
$557$	1109.95	1.99272	0.996362	−	0.0852214i	$-0.0271598\pi$
$557$	1109.95	1.99272	0.996362	+	0.0852214i	$0.0271598\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	−584.286	−1.02867
$569$	894.814	1.57261	0.786304	−	0.617839i	$-0.211992\pi$
$569$	894.814	1.57261	0.786304	+	0.617839i	$0.211992\pi$
$570$	0	0
$571$	−724.074	−1.26808	−0.634040	−	0.773300i	$-0.718605\pi$
$571$	−724.074	−1.26808	−0.634040	+	0.773300i	$0.718605\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	914.811	1.58546	0.792731	−	0.609572i	$-0.208659\pi$
$577$	914.811	1.58546	0.792731	+	0.609572i	$0.208659\pi$
$578$	378.334	0.654557
$579$	0	0
$580$	−242.324	−0.417799
$581$	0	0
$582$	0	0
$583$	0	0
$584$	650.916	1.11458
$585$	0	0
$586$	720.013	1.22869
$587$	240.310	0.409387	0.204694	−	0.978826i	$-0.434380\pi$
$587$	240.310	0.409387	0.204694	+	0.978826i	$0.434380\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	117.754	0.198909
$593$	−981.896	−1.65581	−0.827906	−	0.560867i	$-0.810468\pi$
$593$	−981.896	−1.65581	−0.827906	+	0.560867i	$0.810468\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	−474.271	−0.785218
$605$	−1209.82	−1.99970
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	1077.85	1.77278
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	785.049	1.28067	0.640334	−	0.768097i	$-0.278796\pi$
$613$	785.049	1.28067	0.640334	+	0.768097i	$0.278796\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	826.713	1.33989	0.669946	−	0.742410i	$-0.266317\pi$
$617$	826.713	1.33989	0.669946	+	0.742410i	$0.266317\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	−542.699	−0.872506
$623$	0	0
$624$	0	0
$625$	3121.29	4.99406
$626$	−591.535	−0.944944
$627$	0	0
$628$	193.386	0.307940
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	−815.479	−1.29031
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	618.474	0.966365
$641$	−485.186	−0.756920	−0.378460	−	0.925618i	$-0.623546\pi$
$641$	−485.186	−0.756920	−0.378460	+	0.925618i	$0.623546\pi$
$642$	0	0
$643$	−1270.00	−1.97512	−0.987558	−	0.157253i	$-0.949736\pi$
$643$	−1270.00	−1.97512	−0.987558	+	0.157253i	$0.949736\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1010.00	−1.56105	−0.780526	−	0.625124i	$-0.785048\pi$
$647$	−1010.00	−1.56105	−0.780526	+	0.625124i	$0.785048\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	−2505.20	−3.82473
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−797.989	−1.21091	−0.605454	−	0.795880i	$-0.707009\pi$
$659$	−797.989	−1.21091	−0.605454	+	0.795880i	$0.707009\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	−78.0023	−0.117473
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−416.214	−0.623075
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	−386.371	−0.571555
$677$	−1319.77	−1.94944	−0.974720	−	0.223429i	$-0.928275\pi$
$677$	−1319.77	−1.94944	−0.974720	+	0.223429i	$0.928275\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−49.8103	−0.0723987
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	−2531.88	−3.60153
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	−929.333	−1.30892
$711$	0	0
$712$	19.5529	0.0274620
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−337.447	−0.471294
$717$	0	0
$718$	−255.170	−0.355390
$719$	−506.292	−0.704162	−0.352081	−	0.935970i	$-0.614526\pi$
$719$	−506.292	−0.704162	−0.352081	+	0.935970i	$0.614526\pi$
$720$	0	0
$721$	0	0
$722$	1132.11	1.56802
$723$	0	0
$724$	0	0
$725$	−794.752	−1.09621
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0	0
$730$	1035.31	1.41823
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	892.789	1.21633
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−1078.00	−1.45873	−0.729364	−	0.684126i	$-0.760184\pi$
$739$	−1078.00	−1.45873	−0.729364	+	0.684126i	$0.760184\pi$
$740$	−1653.06	−2.23386
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	−229.519	−0.307666
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−2074.17	−2.74725
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	840.800	1.10924
$759$	0	0
$760$	2880.78	3.79050
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	−425.799	−0.557329
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−79.7879	−0.101770
$785$	845.751	1.07739
$786$	0	0
$787$	702.800	0.893012	0.446506	−	0.894781i	$-0.352668\pi$
$787$	702.800	0.893012	0.446506	+	0.894781i	$0.352668\pi$
$788$	0	0
$789$	0	0
$790$	−1297.06	−1.64184
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	626.194	0.786676
$797$	−1315.14	−1.65012	−0.825059	−	0.565046i	$-0.808858\pi$
$797$	−1315.14	−1.65012	−0.825059	+	0.565046i	$0.808858\pi$
$798$	0	0
$799$	0	0
$800$	2308.02	2.88503
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−961.239	−1.18965
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	−1141.51	−1.40753	−0.703766	−	0.710432i	$-0.748500\pi$
$811$	−1141.51	−1.40753	−0.703766	+	0.710432i	$0.748500\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1070.99	1.31088
$818$	−1070.74	−1.30897
$819$	0	0
$820$	0	0
$821$	1352.45	1.64732	0.823660	−	0.567085i	$-0.191929\pi$
$821$	1352.45	1.64732	0.823660	+	0.567085i	$0.191929\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	−802.896	−0.974389
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	−208.544	−0.251561	−0.125781	−	0.992058i	$-0.540144\pi$
$829$	−208.544	−0.251561	−0.125781	+	0.992058i	$0.540144\pi$
$830$	−124.066	−0.149477
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1820.26	−2.17996
$836$	0	0
$837$	0	0
$838$	−1087.73	−1.29801
$839$	718.797	0.856731	0.428366	−	0.903606i	$-0.359090\pi$
$839$	718.797	0.856731	0.428366	+	0.903606i	$0.359090\pi$
$840$	0	0
$841$	−728.621	−0.866375
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1689.75	−1.99970
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	1334.47	1.56444	0.782219	−	0.623003i	$-0.214088\pi$
$853$	1334.47	1.56444	0.782219	+	0.623003i	$0.214088\pi$
$854$	0	0
$855$	0	0
$856$	−576.057	−0.672963
$857$	1151.07	1.34314	0.671570	−	0.740941i	$-0.265620\pi$
$857$	1151.07	1.34314	0.671570	+	0.740941i	$0.265620\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	699.248	0.813080
$861$	0	0
$862$	1061.08	1.23096
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	1600.38	1.83529
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	818.584	0.933391	0.466695	−	0.884418i	$-0.345444\pi$
$877$	818.584	0.933391	0.466695	+	0.884418i	$0.345444\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	312.734	0.354976	0.177488	−	0.984123i	$-0.443203\pi$
$881$	312.734	0.354976	0.177488	+	0.984123i	$0.443203\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	31.0998	0.0349436
$891$	0	0
$892$	−193.820	−0.217288
$893$	0	0
$894$	0	0
$895$	−1475.78	−1.64892
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	1027.29	1.12149
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−5421.55	−5.86114
$926$	895.764	0.967347
$927$	0	0
$928$	−326.357	−0.351678
$929$	−1802.01	−1.93974	−0.969868	−	0.243632i	$-0.921661\pi$
$929$	−1802.01	−1.93974	−0.969868	+	0.243632i	$0.921661\pi$
$930$	0	0
$931$	1715.55	1.84270
$932$	846.048	0.907777
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−479.909	−0.509999	−0.255000	−	0.966941i	$-0.582075\pi$
$941$	−479.909	−0.509999	−0.255000	+	0.966941i	$0.582075\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1360.29	−1.43642	−0.718208	−	0.695828i	$-0.755038\pi$
$947$	−1360.29	−1.43642	−0.718208	+	0.695828i	$0.755038\pi$
$948$	0	0
$949$	0	0
$950$	3436.17	3.61702
$951$	0	0
$952$	0	0
$953$	106.015	0.111243	0.0556215	−	0.998452i	$-0.482286\pi$
$953$	106.015	0.111243	0.0556215	+	0.998452i	$0.482286\pi$
$954$	0	0
$955$	−1862.18	−1.94993
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	961.000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−995.755	−1.02867
$969$	0	0
$970$	0	0
$971$	−1658.00	−1.70752	−0.853759	−	0.520668i	$-0.825683\pi$
$971$	−1658.00	−1.70752	−0.853759	+	0.520668i	$0.825683\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1278.65	−1.30875	−0.654374	−	0.756171i	$-0.727068\pi$
$977$	−1278.65	−1.30875	−0.654374	+	0.756171i	$0.727068\pi$
$978$	0	0
$979$	0	0
$980$	1120.08	1.14294
$981$	0	0
$982$	0	0
$983$	1554.03	1.58091	0.790454	−	0.612521i	$-0.209845\pi$
$983$	1554.03	1.58091	0.790454	+	0.612521i	$0.209845\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	2738.59	2.75235
$996$	0	0
$997$	−1621.44	−1.62632	−0.813158	−	0.582043i	$-0.802253\pi$
$997$	−1621.44	−1.62632	−0.813158	+	0.582043i	$0.802253\pi$
$998$	56.9471	0.0570612
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	639.3.d.b.496.5		7
3.2	odd	2		71.3.b.b.70.3	✓	7
12.11	even	2		1136.3.h.b.993.6		7
71.70	odd	2	CM	639.3.d.b.496.5		7
213.212	even	2		71.3.b.b.70.3	✓	7
852.851	odd	2		1136.3.h.b.993.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
71.3.b.b.70.3	✓	7	3.2	odd	2
71.3.b.b.70.3	✓	7	213.212	even	2
639.3.d.b.496.5		7	1.1	even	1	trivial
639.3.d.b.496.5		7	71.70	odd	2	CM
1136.3.h.b.993.6		7	12.11	even	2
1136.3.h.b.993.6		7	852.851	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 639 = 3^{2} \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	639.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.30911 q^{2} -2.28622 q^{4} -9.99851 q^{5} -8.22938 q^{8} +O(q^{10})\)	\(q+1.30911 q^{2} -2.28622 q^{4} -9.99851 q^{5} -8.22938 q^{8} -13.0892 q^{10} -1.62833 q^{16} +35.0113 q^{19} +22.8588 q^{20} +74.9703 q^{25} -10.6009 q^{29} +30.7859 q^{32} -72.3160 q^{37} +45.8338 q^{38} +82.2816 q^{40} +30.5899 q^{43} +49.0000 q^{49} +98.1447 q^{50} -13.8778 q^{58} +46.8155 q^{64} +71.0000 q^{71} -79.0966 q^{73} -94.6699 q^{74} -80.0434 q^{76} +99.0936 q^{79} +16.2808 q^{80} +9.47851 q^{83} +40.0457 q^{86} -2.37599 q^{89} -350.061 q^{95} +64.1466 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 28 q^{4}+O(q^{10}) \) 7 * q + 28 * q^4	\( 7 q + 28 q^{4} + 112 q^{16} + 217 q^{20} + 175 q^{25} - 35 q^{38} + 343 q^{49} + 448 q^{64} + 497 q^{71} - 539 q^{74} + 868 q^{80}+O(q^{100}) \) 7 * q + 28 * q^4 + 112 * q^16 + 217 * q^20 + 175 * q^25 - 35 * q^38 + 343 * q^49 + 448 * q^64 + 497 * q^71 - 539 * q^74 + 868 * q^80

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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