LMFDB - Embedded newform 2052.3.m.a.881.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2052,3,Mod(881,2052)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2052, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5, 2]))

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

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

chi := DirichletCharacter("2052.881");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2052 = 2^{2} \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2052.m (of order $6$, degree $2$, not 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:	no
Analytic conductor:	$55.9129502467$
Analytic rank:	$0$
Dimension:	$80$
Relative dimension:	$40$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 684)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			881.14
Character	$\chi$	$=$	2052.881
Dual form			2052.3.m.a.1493.27

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.92593i q^{5} +(-2.71227 - 4.69779i) q^{7} +O(q^{10})$	$q-2.92593i q^{5} +(-2.71227 - 4.69779i) q^{7} +(-9.00863 + 5.20114i) q^{11} +(12.2388 + 21.1983i) q^{13} +(19.6840 - 11.3645i) q^{17} +(-5.11536 - 18.2984i) q^{19} +(-25.4052 + 14.6677i) q^{23} +16.4389 q^{25} -30.7594i q^{29} +(-8.67884 + 15.0322i) q^{31} +(-13.7454 + 7.93593i) q^{35} +47.8168 q^{37} +76.3790i q^{41} +(3.12848 - 5.41869i) q^{43} -17.6934i q^{47} +(9.78715 - 16.9518i) q^{49} +(-12.9368 - 7.46908i) q^{53} +(15.2182 + 26.3586i) q^{55} +21.7645i q^{59} +112.033 q^{61} +(62.0248 - 35.8100i) q^{65} +(-10.5822 - 18.3289i) q^{67} +(43.5246 - 25.1289i) q^{71} +(-6.91001 - 11.9685i) q^{73} +(48.8677 + 28.2138i) q^{77} +(-0.652435 + 1.13005i) q^{79} +(-66.2341 + 38.2403i) q^{83} +(-33.2519 - 57.5940i) q^{85} +(-123.051 - 71.0438i) q^{89} +(66.3902 - 114.991i) q^{91} +(-53.5400 + 14.9672i) q^{95} +(92.4918 - 160.201i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q + q^{7}+O(q^{10}) $ 80 * q + q^7	$ 80 q + q^{7} - 18 q^{11} - 5 q^{13} + 9 q^{17} + 20 q^{19} - 72 q^{23} - 400 q^{25} - 8 q^{31} + 22 q^{37} - 44 q^{43} - 267 q^{49} + 36 q^{53} - 14 q^{61} + 144 q^{65} - 77 q^{67} + 135 q^{71} + 43 q^{73} - 216 q^{77} - 17 q^{79} + 171 q^{83} - 216 q^{89} + 122 q^{91} + 288 q^{95} - 8 q^{97}+O(q^{100}) $ 80 * q + q^7 - 18 * q^11 - 5 * q^13 + 9 * q^17 + 20 * q^19 - 72 * q^23 - 400 * q^25 - 8 * q^31 + 22 * q^37 - 44 * q^43 - 267 * q^49 + 36 * q^53 - 14 * q^61 + 144 * q^65 - 77 * q^67 + 135 * q^71 + 43 * q^73 - 216 * q^77 - 17 * q^79 + 171 * q^83 - 216 * q^89 + 122 * q^91 + 288 * q^95 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2052\mathbb{Z}\right)^\times$.

$n$	$325$	$1027$	$1217$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{5}{6}\right)$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		−	2.92593i		−	0.585186i	−0.956237	−	0.292593i	$-0.905482\pi$
$5$		−	2.92593i		−	0.585186i	0.956237	−	0.292593i	$-0.0945182\pi$
$6$		0			0
$7$	−2.71227	−	4.69779i	−0.387468	−	0.671114i	0.604641	−	0.796498i	$-0.293317\pi$
$7$	−2.71227	−	4.69779i	−0.387468	−	0.671114i	−0.992108	+	0.125385i	$0.959983\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−9.00863	+	5.20114i	−0.818966	+	0.472830i	−0.850060	−	0.526686i	$-0.823434\pi$
$11$	−9.00863	+	5.20114i	−0.818966	+	0.472830i	0.0310935	+	0.999516i	$0.490101\pi$
$12$		0			0
$13$	12.2388	+	21.1983i	0.941450	+	1.63064i	0.762708	+	0.646743i	$0.223869\pi$
$13$	12.2388	+	21.1983i	0.941450	+	1.63064i	0.178742	+	0.983896i	$0.442797\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	19.6840	−	11.3645i	1.15788	−	0.668503i	0.207086	−	0.978323i	$-0.433602\pi$
$17$	19.6840	−	11.3645i	1.15788	−	0.668503i	0.950795	+	0.309820i	$0.100269\pi$
$18$		0			0
$19$	−5.11536	−	18.2984i	−0.269230	−	0.963076i
$19$	−5.11536	−	18.2984i	−0.269230	−	0.963076i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−25.4052	+	14.6677i	−1.10457	+	0.637727i	−0.937419	−	0.348204i	$-0.886792\pi$
$23$	−25.4052	+	14.6677i	−1.10457	+	0.637727i	−0.167156	+	0.985930i	$0.553458\pi$
$24$		0			0
$25$	16.4389			0.657557
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	30.7594i		−	1.06067i	−0.847789	−	0.530334i	$-0.822067\pi$
$29$		−	30.7594i		−	1.06067i	0.847789	−	0.530334i	$-0.177933\pi$
$30$		0			0
$31$	−8.67884	+	15.0322i	−0.279962	+	0.484909i	−0.971375	−	0.237551i	$-0.923655\pi$
$31$	−8.67884	+	15.0322i	−0.279962	+	0.484909i	0.691413	+	0.722460i	$0.256989\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−13.7454	+	7.93593i	−0.392726	+	0.226741i
$36$		0			0
$37$	47.8168			1.29234			0.646172	−	0.763192i	$-0.276369\pi$
$37$	47.8168			1.29234			0.646172	+	0.763192i	$0.276369\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			76.3790i			1.86290i	0.363866	+	0.931451i	$0.381456\pi$
$41$			76.3790i			1.86290i	−0.363866	+	0.931451i	$0.618544\pi$
$42$		0			0
$43$	3.12848	−	5.41869i	0.0727554	−	0.126016i	−0.827353	−	0.561683i	$-0.810154\pi$
$43$	3.12848	−	5.41869i	0.0727554	−	0.126016i	0.900108	+	0.435667i	$0.143487\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	17.6934i		−	0.376456i	−0.982125	−	0.188228i	$-0.939726\pi$
$47$		−	17.6934i		−	0.376456i	0.982125	−	0.188228i	$-0.0602744\pi$
$48$		0			0
$49$	9.78715	−	16.9518i	0.199738	−	0.345956i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−12.9368	−	7.46908i	−0.244091	−	0.140926i	0.372964	−	0.927846i	$-0.378341\pi$
$53$	−12.9368	−	7.46908i	−0.244091	−	0.140926i	−0.617056	+	0.786920i	$0.711675\pi$
$54$		0			0
$55$	15.2182	+	26.3586i	0.276694	+	0.479248i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			21.7645i			0.368889i	0.982843	+	0.184445i	$0.0590487\pi$
$59$			21.7645i			0.368889i	−0.982843	+	0.184445i	$0.940951\pi$
$60$		0			0
$61$	112.033			1.83661			0.918305	−	0.395873i	$-0.129558\pi$
$61$	112.033			1.83661			0.918305	+	0.395873i	$0.129558\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	62.0248	−	35.8100i	0.954228	−	0.550924i
$66$		0			0
$67$	−10.5822	−	18.3289i	−0.157943	−	0.273566i	0.776183	−	0.630507i	$-0.217153\pi$
$67$	−10.5822	−	18.3289i	−0.157943	−	0.273566i	−0.934127	+	0.356941i	$0.883820\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	43.5246	−	25.1289i	0.613023	−	0.353929i	−0.161125	−	0.986934i	$-0.551512\pi$
$71$	43.5246	−	25.1289i	0.613023	−	0.353929i	0.774147	+	0.633005i	$0.218179\pi$
$72$		0			0
$73$	−6.91001	−	11.9685i	−0.0946577	−	0.163952i	0.814808	−	0.579731i	$-0.196842\pi$
$73$	−6.91001	−	11.9685i	−0.0946577	−	0.163952i	−0.909466	+	0.415779i	$0.863509\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	48.8677	+	28.2138i	0.634646	+	0.366413i
$78$		0			0
$79$	−0.652435	+	1.13005i	−0.00825868	+	0.0143044i	−0.870125	−	0.492831i	$-0.835962\pi$
$79$	−0.652435	+	1.13005i	−0.00825868	+	0.0143044i	0.861867	+	0.507135i	$0.169296\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−66.2341	+	38.2403i	−0.798002	+	0.460726i	−0.842772	−	0.538271i	$-0.819078\pi$
$83$	−66.2341	+	38.2403i	−0.798002	+	0.460726i	0.0447703	+	0.998997i	$0.485744\pi$
$84$		0			0
$85$	−33.2519	−	57.5940i	−0.391199	−	0.677576i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−123.051	−	71.0438i	−1.38260	−	0.798244i	−0.390133	−	0.920758i	$-0.627571\pi$
$89$	−123.051	−	71.0438i	−1.38260	−	0.798244i	−0.992467	+	0.122514i	$0.960904\pi$
$90$		0			0
$91$	66.3902	−	114.991i	0.729563	−	1.26364i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−53.5400	+	14.9672i	−0.563579	+	0.157550i
$96$		0			0
$97$	92.4918	−	160.201i	0.953524	−	1.65155i	0.215814	−	0.976434i	$-0.430759\pi$
$97$	92.4918	−	160.201i	0.953524	−	1.65155i	0.737710	−	0.675118i	$-0.235907\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	59.1628i		−	0.585771i	−0.956148	−	0.292885i	$-0.905385\pi$
$101$		−	59.1628i		−	0.585771i	0.956148	−	0.292885i	$-0.0946154\pi$
$102$		0			0
$103$	84.6622	−	146.639i	0.821963	−	1.42368i	−0.0822557	−	0.996611i	$-0.526212\pi$
$103$	84.6622	−	146.639i	0.821963	−	1.42368i	0.904218	−	0.427070i	$-0.140454\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	131.165i		−	1.22584i	−0.790144	−	0.612921i	$-0.789994\pi$
$107$		−	131.165i		−	1.22584i	0.790144	−	0.612921i	$-0.210006\pi$
$108$		0			0
$109$	−7.82050	−	13.5455i	−0.0717477	−	0.124271i	0.827920	−	0.560847i	$-0.189524\pi$
$109$	−7.82050	−	13.5455i	−0.0717477	−	0.124271i	−0.899667	+	0.436576i	$0.856191\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	20.4851	+	11.8271i	0.181284	+	0.104665i	0.587896	−	0.808937i	$-0.299956\pi$
$113$	20.4851	+	11.8271i	0.181284	+	0.104665i	−0.406612	+	0.913601i	$0.633290\pi$
$114$		0			0
$115$	42.9167	+	74.3339i	0.373189	+	0.646382i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−106.777	−	61.6475i	−0.897283	−	0.518046i
$120$		0			0
$121$	−6.39638	+	11.0789i	−0.0528627	+	0.0915608i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	121.247i		−	0.969980i
$126$		0			0
$127$	−41.1013	+	71.1895i	−0.323632	+	0.560547i	−0.981235	−	0.192818i	$-0.938237\pi$
$127$	−41.1013	+	71.1895i	−0.323632	+	0.560547i	0.657603	+	0.753365i	$0.271571\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			79.3492i			0.605719i	0.953035	+	0.302860i	$0.0979413\pi$
$131$			79.3492i			0.605719i	−0.953035	+	0.302860i	$0.902059\pi$
$132$		0			0
$133$	−72.0881	+	73.6613i	−0.542016	+	0.553844i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	46.5484i		−	0.339769i	−0.985464	−	0.169885i	$-0.945661\pi$
$137$		−	46.5484i		−	0.339769i	0.985464	−	0.169885i	$-0.0543395\pi$
$138$		0			0
$139$	−36.9779	−	64.0477i	−0.266028	−	0.460774i	0.701804	−	0.712370i	$-0.252378\pi$
$139$	−36.9779	−	64.0477i	−0.266028	−	0.460774i	−0.967833	+	0.251595i	$0.919045\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−220.511	−	127.312i	−1.54203	−	0.890292i
$144$		0			0
$145$	−89.9998			−0.620689
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	197.372i		−	1.32464i	−0.749220	−	0.662321i	$-0.769572\pi$
$149$		−	197.372i		−	1.32464i	0.749220	−	0.662321i	$-0.230428\pi$
$150$		0			0
$151$	104.491	+	180.983i	0.691992	+	1.19856i	0.971184	+	0.238330i	$0.0765998\pi$
$151$	104.491	+	180.983i	0.691992	+	1.19856i	−0.279193	+	0.960235i	$0.590067\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	43.9832	+	25.3937i	0.283762	+	0.163830i
$156$		0			0
$157$	−43.4243			−0.276588			−0.138294	−	0.990391i	$-0.544162\pi$
$157$	−43.4243			−0.276588			−0.138294	+	0.990391i	$0.544162\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	137.812	+	79.5657i	0.855974	+	0.494197i
$162$		0			0
$163$	268.887			1.64962			0.824808	−	0.565413i	$-0.191283\pi$
$163$	268.887			1.64962			0.824808	+	0.565413i	$0.191283\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	278.934	−	161.043i	1.67026	−	0.964327i	0.702775	−	0.711412i	$-0.251944\pi$
$167$	278.934	−	161.043i	1.67026	−	0.964327i	0.967488	−	0.252915i	$-0.0813894\pi$
$168$		0			0
$169$	−215.079	+	372.527i	−1.27266	+	2.20430i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−30.7529	−	17.7552i	−0.177762	−	0.102631i	0.408479	−	0.912768i	$-0.366059\pi$
$173$	−30.7529	−	17.7552i	−0.177762	−	0.102631i	−0.586241	+	0.810137i	$0.699393\pi$
$174$		0			0
$175$	−44.5869	−	77.2267i	−0.254782	−	0.441295i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	214.059i		−	1.19586i	−0.801549	−	0.597930i	$-0.795990\pi$
$179$		−	214.059i		−	1.19586i	0.801549	−	0.597930i	$-0.204010\pi$
$180$		0			0
$181$	40.8818	−	70.8094i	0.225866	−	0.391212i	−0.730713	−	0.682685i	$-0.760812\pi$
$181$	40.8818	−	70.8094i	0.225866	−	0.391212i	0.956579	+	0.291473i	$0.0941454\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	139.909i		−	0.756262i
$186$		0			0
$187$	−118.217	+	204.758i	−0.632177	+	1.09496i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	284.070	−	164.008i	1.48728	−	0.858681i	0.487384	−	0.873188i	$-0.337951\pi$
$191$	284.070	−	164.008i	1.48728	−	0.858681i	0.999895	+	0.0145071i	$0.00461792\pi$
$192$		0			0
$193$	219.666			1.13817			0.569083	−	0.822280i	$-0.307298\pi$
$193$	219.666			1.13817			0.569083	+	0.822280i	$0.307298\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			161.830i			0.821470i	0.911755	+	0.410735i	$0.134728\pi$
$197$			161.830i			0.821470i	−0.911755	+	0.410735i	$0.865272\pi$
$198$		0			0
$199$	37.0576	−	64.1857i	0.186219	−	0.322541i	−0.757767	−	0.652525i	$-0.773710\pi$
$199$	37.0576	−	64.1857i	0.186219	−	0.322541i	0.943987	+	0.329984i	$0.107043\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−144.501	+	83.4278i	−0.711829	+	0.410975i
$204$		0			0
$205$	223.480			1.09014
$206$		0			0
$207$		0			0
$208$		0			0
$209$	141.255	+	138.238i	0.675862	+	0.661427i
$210$		0			0
$211$	−124.478			−0.589941			−0.294970	−	0.955506i	$-0.595310\pi$
$211$	−124.478			−0.589941			−0.294970	+	0.955506i	$0.595310\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−15.8547	−	9.15373i	−0.0737429	−	0.0425755i
$216$		0			0
$217$	94.1575			0.433906
$218$		0			0
$219$		0			0
$220$		0			0
$221$	481.818	+	278.178i	2.18017	+	1.25872i
$222$		0			0
$223$	160.216	−	277.502i	0.718456	−	1.24440i	−0.243155	−	0.969987i	$-0.578182\pi$
$223$	160.216	−	277.502i	0.718456	−	1.24440i	0.961611	−	0.274415i	$-0.0884842\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	137.264	−	79.2492i	0.604685	−	0.349115i	−0.166197	−	0.986093i	$-0.553149\pi$
$227$	137.264	−	79.2492i	0.604685	−	0.349115i	0.770883	+	0.636977i	$0.219816\pi$
$228$		0			0
$229$	−124.132	+	215.003i	−0.542061	+	0.938876i	0.456725	+	0.889608i	$0.349022\pi$
$229$	−124.132	+	215.003i	−0.542061	+	0.938876i	−0.998786	+	0.0492684i	$0.984311\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	277.122	−	159.996i	1.18936	−	0.686680i	0.231202	−	0.972906i	$-0.425734\pi$
$233$	277.122	−	159.996i	1.18936	−	0.686680i	0.958162	+	0.286226i	$0.0924008\pi$
$234$		0			0
$235$	−51.7698			−0.220297
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−13.0028	−	7.50716i	−0.0544049	−	0.0314107i	0.472551	−	0.881303i	$-0.343333\pi$
$239$	−13.0028	−	7.50716i	−0.0544049	−	0.0314107i	−0.526956	+	0.849893i	$0.676667\pi$
$240$		0			0
$241$	−45.7719			−0.189925			−0.0949625	−	0.995481i	$-0.530273\pi$
$241$	−45.7719			−0.189925			−0.0949625	+	0.995481i	$0.530273\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−49.5999	−	28.6365i	−0.202449	−	0.116884i
$246$		0			0
$247$	325.290	−	332.389i	1.31696	−	1.34570i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	306.214	+	176.793i	1.21997	+	0.704353i	0.964912	−	0.262572i	$-0.0845707\pi$
$251$	306.214	+	176.793i	1.21997	+	0.704353i	0.255062	+	0.966925i	$0.417904\pi$
$252$		0			0
$253$	152.578	−	264.272i	0.603073	−	1.04455i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−136.692	+	78.9192i	−0.531876	+	0.307079i	−0.741780	−	0.670643i	$-0.766018\pi$
$257$	−136.692	+	78.9192i	−0.531876	+	0.307079i	0.209904	+	0.977722i	$0.432685\pi$
$258$		0			0
$259$	−129.692	−	224.633i	−0.500742	−	0.867310i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−242.974	−	140.281i	−0.923856	−	0.533389i	−0.0389930	−	0.999239i	$-0.512415\pi$
$263$	−242.974	−	140.281i	−0.923856	−	0.533389i	−0.884863	+	0.465851i	$0.845748\pi$
$264$		0			0
$265$	−21.8540	+	37.8523i	−0.0824680	+	0.142839i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−293.137	+	169.243i	−1.08973	+	0.629156i	−0.933505	−	0.358566i	$-0.883266\pi$
$269$	−293.137	+	169.243i	−1.08973	+	0.629156i	−0.156225	+	0.987721i	$0.549933\pi$
$270$		0			0
$271$	144.466	+	250.222i	0.533084	+	0.923329i	0.999253	+	0.0386333i	$0.0123004\pi$
$271$	144.466	+	250.222i	0.533084	+	0.923329i	−0.466169	+	0.884696i	$0.654366\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−148.092	+	85.5011i	−0.538517	+	0.310913i
$276$		0			0
$277$	−254.832	−	441.382i	−0.919971	−	1.59344i	−0.799456	−	0.600725i	$-0.794879\pi$
$277$	−254.832	−	441.382i	−0.919971	−	1.59344i	−0.120515	−	0.992712i	$-0.538454\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	404.740i		−	1.44036i	−0.693790	−	0.720178i	$-0.744060\pi$
$281$		−	404.740i		−	1.44036i	0.693790	−	0.720178i	$-0.255940\pi$
$282$		0			0
$283$	−369.422			−1.30538			−0.652689	−	0.757626i	$-0.726359\pi$
$283$	−369.422			−1.30538			−0.652689	+	0.757626i	$0.726359\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	358.813	−	207.161i	1.25022	−	0.721814i
$288$		0			0
$289$	113.806	−	197.118i	0.393792	−	0.682068i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−115.170	−	66.4934i	−0.393071	−	0.226940i	0.290419	−	0.956900i	$-0.406205\pi$
$293$	−115.170	−	66.4934i	−0.393071	−	0.226940i	−0.683490	+	0.729960i	$0.739539\pi$
$294$		0			0
$295$	63.6814			0.215869
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−621.861	−	359.032i	−2.07980	−	1.20078i
$300$		0			0
$301$	−33.9412			−0.112761
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	327.802i		−	1.07476i
$306$		0			0
$307$	−49.3725	−	85.5156i	−0.160822	−	0.278552i	0.774341	−	0.632768i	$-0.218081\pi$
$307$	−49.3725	−	85.5156i	−0.160822	−	0.278552i	−0.935164	+	0.354215i	$0.884748\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−102.641	−	59.2596i	−0.330034	−	0.190545i	0.325822	−	0.945431i	$-0.394359\pi$
$311$	−102.641	−	59.2596i	−0.330034	−	0.190545i	−0.655856	+	0.754886i	$0.727692\pi$
$312$		0			0
$313$	−39.0892			−0.124886			−0.0624429	−	0.998049i	$-0.519889\pi$
$313$	−39.0892			−0.124886			−0.0624429	+	0.998049i	$0.519889\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			563.219i			1.77672i	0.459152	+	0.888358i	$0.348153\pi$
$317$			563.219i			1.77672i	−0.459152	+	0.888358i	$0.651847\pi$
$318$		0			0
$319$	159.984	+	277.100i	0.501516	+	0.868652i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−308.644	−	302.052i	−0.955555	−	0.935147i
$324$		0			0
$325$	201.193	+	348.477i	0.619057	+	1.07224i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−83.1202	+	47.9895i	−0.252645	+	0.145865i
$330$		0			0
$331$	275.829	+	477.751i	0.833322	+	1.44336i	0.895390	+	0.445284i	$0.146897\pi$
$331$	275.829	+	477.751i	0.833322	+	1.44336i	−0.0620679	+	0.998072i	$0.519770\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−53.6292	+	30.9628i	−0.160087	+	0.0924263i
$336$		0			0
$337$	188.575			0.559569			0.279785	−	0.960063i	$-0.409737\pi$
$337$	188.575			0.559569			0.279785	+	0.960063i	$0.409737\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	180.559i		−	0.529499i
$342$		0			0
$343$	−371.984			−1.08450
$344$		0			0
$345$		0			0
$346$		0			0
$347$			667.568i			1.92383i	0.273353	+	0.961914i	$0.411867\pi$
$347$			667.568i			1.92383i	−0.273353	+	0.961914i	$0.588133\pi$
$348$		0			0
$349$	41.5364	+	71.9431i	0.119015	+	0.206141i	0.919378	−	0.393376i	$-0.128693\pi$
$349$	41.5364	+	71.9431i	0.119015	+	0.206141i	−0.800362	+	0.599517i	$0.795360\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	220.283	−	127.181i	0.624032	−	0.360285i	−0.154405	−	0.988008i	$-0.549346\pi$
$353$	220.283	−	127.181i	0.624032	−	0.360285i	0.778437	+	0.627722i	$0.216013\pi$
$354$		0			0
$355$	−73.5256	−	127.350i	−0.207114	−	0.358732i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−165.360	+	95.4709i	−0.460614	+	0.265936i	−0.712302	−	0.701873i	$-0.752347\pi$
$359$	−165.360	+	95.4709i	−0.460614	+	0.265936i	0.251688	+	0.967808i	$0.419014\pi$
$360$		0			0
$361$	−308.666	+	187.206i	−0.855031	+	0.518577i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−35.0190	+	20.2182i	−0.0959424	+	0.0553924i
$366$		0			0
$367$	169.085			0.460723			0.230362	−	0.973105i	$-0.426009\pi$
$367$	169.085			0.460723			0.230362	+	0.973105i	$0.426009\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			81.0328i			0.218417i
$372$		0			0
$373$	188.822	−	327.049i	0.506225	−	0.876808i	−0.493749	−	0.869605i	$-0.664374\pi$
$373$	188.822	−	327.049i	0.506225	−	0.876808i	0.999974	−	0.00720342i	$-0.00229294\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	652.047	−	376.459i	1.72957	−	0.998566i
$378$		0			0
$379$	673.545			1.77716			0.888581	−	0.458719i	$-0.151692\pi$
$379$	673.545			1.77716			0.888581	+	0.458719i	$0.151692\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	52.6715i		−	0.137524i	−0.997633	−	0.0687618i	$-0.978095\pi$
$383$		−	52.6715i		−	0.137524i	0.997633	−	0.0687618i	$-0.0219048\pi$
$384$		0			0
$385$	82.5517	−	142.984i	0.214420	−	0.371386i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			88.3266i			0.227061i	0.993535	+	0.113530i	$0.0362159\pi$
$389$			88.3266i			0.227061i	−0.993535	+	0.113530i	$0.963784\pi$
$390$		0			0
$391$	−333.384	+	577.438i	−0.852644	+	1.47682i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3.30645	+	1.90898i	0.00837077	+	0.00483287i
$396$		0			0
$397$	50.6046	+	87.6498i	0.127468	+	0.220780i	0.922695	−	0.385531i	$-0.125982\pi$
$397$	50.6046	+	87.6498i	0.127468	+	0.220780i	−0.795227	+	0.606312i	$0.792648\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			39.8934i			0.0994847i	0.998762	+	0.0497423i	$0.0158400\pi$
$401$			39.8934i			0.0994847i	−0.998762	+	0.0497423i	$0.984160\pi$
$402$		0			0
$403$	−424.876			−1.05428
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−430.763	+	248.701i	−1.05839	+	0.611060i
$408$		0			0
$409$	−68.0284	−	117.829i	−0.166329	−	0.288090i	0.770798	−	0.637080i	$-0.219858\pi$
$409$	−68.0284	−	117.829i	−0.166329	−	0.288090i	−0.937126	+	0.348990i	$0.886525\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	102.245	−	59.0312i	0.247567	−	0.142933i
$414$		0			0
$415$	111.888	+	193.797i	0.269611	+	0.466980i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	434.800	+	251.032i	1.03771	+	0.599122i	0.919184	−	0.393830i	$-0.128850\pi$
$419$	434.800	+	251.032i	1.03771	+	0.599122i	0.118525	+	0.992951i	$0.462183\pi$
$420$		0			0
$421$	88.0829	−	152.564i	0.209223	−	0.362385i	−0.742247	−	0.670126i	$-0.766240\pi$
$421$	88.0829	−	152.564i	0.209223	−	0.362385i	0.951470	+	0.307741i	$0.0995732\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	323.583	−	186.821i	0.761373	−	0.439579i
$426$		0			0
$427$	−303.865	−	526.309i	−0.711627	−	1.23257i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−168.235	−	97.1304i	−0.390336	−	0.225361i	0.291970	−	0.956428i	$-0.405689\pi$
$431$	−168.235	−	97.1304i	−0.390336	−	0.225361i	−0.682306	+	0.731067i	$0.739023\pi$
$432$		0			0
$433$	306.183	−	530.324i	0.707120	−	1.22477i	−0.258802	−	0.965931i	$-0.583328\pi$
$433$	306.183	−	530.324i	0.707120	−	1.22477i	0.965921	−	0.258837i	$-0.0833391\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	398.353	+	389.845i	0.911563	+	0.892095i
$438$		0			0
$439$	85.8886	−	148.763i	0.195646	−	0.338869i	−0.751466	−	0.659772i	$-0.770653\pi$
$439$	85.8886	−	148.763i	0.195646	−	0.338869i	0.947112	+	0.320903i	$0.103986\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			230.077i			0.519361i	0.965695	+	0.259680i	$0.0836172\pi$
$443$			230.077i			0.519361i	−0.965695	+	0.259680i	$0.916383\pi$
$444$		0			0
$445$	−207.869	+	360.040i	−0.467122	+	0.809079i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			443.639i			0.988061i	0.869445	+	0.494031i	$0.164477\pi$
$449$			443.639i			0.988061i	−0.869445	+	0.494031i	$0.835523\pi$
$450$		0			0
$451$	−397.257	−	688.070i	−0.880837	−	1.52565i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−336.456	−	194.253i	−0.739465	−	0.426930i
$456$		0			0
$457$	372.974	+	646.009i	0.816135	+	1.41359i	0.908510	+	0.417863i	$0.137221\pi$
$457$	372.974	+	646.009i	0.816135	+	1.41359i	−0.0923753	+	0.995724i	$0.529446\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−217.202	−	125.402i	−0.471155	−	0.272021i	0.245568	−	0.969379i	$-0.421025\pi$
$461$	−217.202	−	125.402i	−0.471155	−	0.272021i	−0.716723	+	0.697358i	$0.754359\pi$
$462$		0			0
$463$	83.5338	−	144.685i	0.180419	−	0.312494i	−0.761605	−	0.648042i	$-0.775588\pi$
$463$	83.5338	−	144.685i	0.180419	−	0.312494i	0.942023	+	0.335548i	$0.108921\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			438.133i			0.938186i	0.883149	+	0.469093i	$0.155419\pi$
$467$			438.133i			0.938186i	−0.883149	+	0.469093i	$0.844581\pi$
$468$		0			0
$469$	−57.4037	+	99.4261i	−0.122396	+	0.211996i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			65.0866i			0.137604i
$474$		0			0
$475$	−84.0911	−	300.807i	−0.177034	−	0.633277i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	23.1763i		−	0.0483849i	−0.999707	−	0.0241924i	$-0.992299\pi$
$479$		−	23.1763i		−	0.0483849i	0.999707	−	0.0241924i	$-0.00770144\pi$
$480$		0			0
$481$	585.222	+	1013.63i	1.21668	+	2.10735i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−468.736	−	270.625i	−0.966466	−	0.557989i
$486$		0			0
$487$	−412.683			−0.847399			−0.423700	−	0.905803i	$-0.639269\pi$
$487$	−412.683			−0.847399			−0.423700	+	0.905803i	$0.639269\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	252.243i		−	0.513732i	−0.966447	−	0.256866i	$-0.917310\pi$
$491$		−	252.243i		−	0.513732i	0.966447	−	0.256866i	$-0.0826900\pi$
$492$		0			0
$493$	−349.566	−	605.467i	−0.709060	−	1.22813i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−236.101	−	136.313i	−0.475053	−	0.274272i
$498$		0			0
$499$	278.727			0.558572			0.279286	−	0.960208i	$-0.409902\pi$
$499$	278.727			0.558572			0.279286	+	0.960208i	$0.409902\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−397.715	−	229.621i	−0.790685	−	0.456502i	0.0495184	−	0.998773i	$-0.484231\pi$
$503$	−397.715	−	229.621i	−0.790685	−	0.456502i	−0.840204	+	0.542271i	$0.817565\pi$
$504$		0			0
$505$	−173.106			−0.342785
$506$		0			0
$507$		0			0
$508$		0			0
$509$	101.021	−	58.3245i	0.198469	−	0.114586i	−0.397472	−	0.917614i	$-0.630112\pi$
$509$	101.021	−	58.3245i	0.198469	−	0.114586i	0.595941	+	0.803028i	$0.296779\pi$
$510$		0			0
$511$	−37.4837	+	64.9236i	−0.0733536	+	0.127052i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−429.056	−	247.716i	−0.833119	−	0.481001i
$516$		0			0
$517$	92.0260	+	159.394i	0.178000	+	0.308305i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	280.254i		−	0.537916i	−0.963152	−	0.268958i	$-0.913321\pi$
$521$		−	280.254i		−	0.537916i	0.963152	−	0.268958i	$-0.0866793\pi$
$522$		0			0
$523$	51.0370	−	88.3987i	0.0975851	−	0.169022i	−0.813099	−	0.582125i	$-0.802222\pi$
$523$	51.0370	−	88.3987i	0.0975851	−	0.169022i	0.910685	+	0.413102i	$0.135555\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			394.524i			0.748623i
$528$		0			0
$529$	165.784	−	287.145i	0.313390	−	0.542808i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1619.11	+	934.791i	−3.03772	+	1.75383i
$534$		0			0
$535$	−383.780			−0.717346
$536$		0			0
$537$		0			0
$538$		0			0
$539$			203.617i			0.377768i
$540$		0			0
$541$	−376.134	+	651.483i	−0.695257	+	1.20422i	0.274837	+	0.961491i	$0.411376\pi$
$541$	−376.134	+	651.483i	−0.695257	+	1.20422i	−0.970094	+	0.242729i	$0.921957\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−39.6332	+	22.8822i	−0.0727215	+	0.0419858i
$546$		0			0
$547$	−655.818			−1.19894			−0.599468	−	0.800399i	$-0.704621\pi$
$547$	−655.818			−1.19894			−0.599468	+	0.800399i	$0.704621\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−562.849	+	157.345i	−1.02150	+	0.285563i
$552$		0			0
$553$	7.07833			0.0127999
$554$		0			0
$555$		0			0
$556$		0			0
$557$	497.328	+	287.133i	0.892870	+	0.515498i	0.874880	−	0.484340i	$-0.160940\pi$
$557$	497.328	+	287.133i	0.892870	+	0.515498i	0.0179895	+	0.999838i	$0.494273\pi$
$558$		0			0
$559$	153.156			0.273982
$560$		0			0
$561$		0			0
$562$		0			0
$563$	773.883	+	446.801i	1.37457	+	0.793608i	0.991499	−	0.130111i	$-0.0415334\pi$
$563$	773.883	+	446.801i	1.37457	+	0.793608i	0.383070	+	0.923719i	$0.374867\pi$
$564$		0			0
$565$	34.6053	−	59.9381i	0.0612483	−	0.106085i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	337.331	−	194.758i	0.592848	−	0.342281i	−0.173375	−	0.984856i	$-0.555467\pi$
$569$	337.331	−	194.758i	0.592848	−	0.342281i	0.766223	+	0.642575i	$0.222134\pi$
$570$		0			0
$571$	61.1358	−	105.890i	0.107068	−	0.185447i	−0.807513	−	0.589849i	$-0.799187\pi$
$571$	61.1358	−	105.890i	0.107068	−	0.185447i	0.914581	+	0.404402i	$0.132520\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−417.634	+	241.121i	−0.726321	+	0.419342i
$576$		0			0
$577$	−927.013			−1.60661			−0.803304	−	0.595569i	$-0.796926\pi$
$577$	−927.013			−1.60661			−0.803304	+	0.595569i	$0.796926\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	359.290	+	207.436i	0.618399	+	0.357033i
$582$		0			0
$583$	155.391			0.266537
$584$		0			0
$585$		0			0
$586$		0			0
$587$	459.586	+	265.342i	0.782940	+	0.452031i	0.837471	−	0.546481i	$-0.184033\pi$
$587$	459.586	+	265.342i	0.782940	+	0.452031i	−0.0545310	+	0.998512i	$0.517366\pi$
$588$		0			0
$589$	319.461	+	81.9141i	0.542379	+	0.139073i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	486.494	+	280.878i	0.820395	+	0.473655i	0.850553	−	0.525890i	$-0.176268\pi$
$593$	486.494	+	280.878i	0.820395	+	0.473655i	−0.0301577	+	0.999545i	$0.509601\pi$
$594$		0			0
$595$	−180.376	+	312.421i	−0.303154	+	0.525078i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−553.860	+	319.771i	−0.924641	+	0.533842i	−0.885113	−	0.465377i	$-0.845919\pi$
$599$	−553.860	+	319.771i	−0.924641	+	0.533842i	−0.0395283	+	0.999218i	$0.512586\pi$
$600$		0			0
$601$	−248.581	−	430.555i	−0.413612	−	0.716397i	0.581670	−	0.813425i	$-0.302400\pi$
$601$	−248.581	−	430.555i	−0.413612	−	0.716397i	−0.995282	+	0.0970281i	$0.969066\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	32.4160	+	18.7154i	0.0535801	+	0.0309345i
$606$		0			0
$607$	55.0624	−	95.3708i	0.0907123	−	0.157118i	−0.817099	−	0.576498i	$-0.804419\pi$
$607$	55.0624	−	95.3708i	0.0907123	−	0.157118i	0.907811	+	0.419379i	$0.137752\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	375.071	−	216.547i	0.613864	−	0.354415i
$612$		0			0
$613$	−51.6325	−	89.4302i	−0.0842293	−	0.145889i	0.820833	−	0.571168i	$-0.193509\pi$
$613$	−51.6325	−	89.4302i	−0.0842293	−	0.145889i	−0.905063	+	0.425279i	$0.860176\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	94.7517	−	54.7049i	0.153568	−	0.0886627i	−0.421247	−	0.906946i	$-0.638407\pi$
$617$	94.7517	−	54.7049i	0.153568	−	0.0886627i	0.574815	+	0.818283i	$0.305074\pi$
$618$		0			0
$619$	−280.995	−	486.697i	−0.453950	−	0.786264i	0.544677	−	0.838646i	$-0.316652\pi$
$619$	−280.995	−	486.697i	−0.453950	−	0.786264i	−0.998627	+	0.0523816i	$0.983319\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			770.760i			1.23718i
$624$		0			0
$625$	56.2113			0.0899381
$626$		0			0
$627$		0			0
$628$		0			0
$629$	941.224	−	543.416i	1.49638	−	0.863936i
$630$		0			0
$631$	−310.083	+	537.079i	−0.491415	+	0.851155i	−0.999951	−	0.00988537i	$-0.996853\pi$
$631$	−310.083	+	537.079i	−0.491415	+	0.851155i	0.508537	+	0.861040i	$0.330187\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	208.296	+	120.259i	0.328024	+	0.189385i
$636$		0			0
$637$	479.134			0.752172
$638$		0			0
$639$		0			0
$640$		0			0
$641$	130.729	+	75.4762i	0.203945	+	0.117748i	0.598494	−	0.801127i	$-0.295766\pi$
$641$	130.729	+	75.4762i	0.203945	+	0.117748i	−0.394549	+	0.918875i	$0.629099\pi$
$642$		0			0
$643$	−47.4649			−0.0738179			−0.0369089	−	0.999319i	$-0.511751\pi$
$643$	−47.4649			−0.0738179			−0.0369089	+	0.999319i	$0.511751\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			36.8284i			0.0569218i	0.999595	+	0.0284609i	$0.00906061\pi$
$647$			36.8284i			0.0569218i	−0.999595	+	0.0284609i	$0.990939\pi$
$648$		0			0
$649$	−113.200	−	196.068i	−0.174422	−	0.302108i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	336.741	+	194.418i	0.515684	+	0.297730i	0.735167	−	0.677886i	$-0.237104\pi$
$653$	336.741	+	194.418i	0.515684	+	0.297730i	−0.219483	+	0.975616i	$0.570437\pi$
$654$		0			0
$655$	232.170			0.354459
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	32.0112i		−	0.0485755i	−0.999705	−	0.0242877i	$-0.992268\pi$
$659$		−	32.0112i		−	0.0485755i	0.999705	−	0.0242877i	$-0.00773179\pi$
$660$		0			0
$661$	−24.6120	−	42.6292i	−0.0372345	−	0.0644920i	0.846808	−	0.531899i	$-0.178522\pi$
$661$	−24.6120	−	42.6292i	−0.0372345	−	0.0644920i	−0.884042	+	0.467407i	$0.845188\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	215.528	+	210.925i	0.324102	+	0.317180i
$666$		0			0
$667$	451.170	+	781.449i	0.676416	+	1.17159i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1009.27	+	582.700i	−1.50412	+	0.868405i
$672$		0			0
$673$	386.780	+	669.922i	0.574710	+	0.995426i	0.996073	+	0.0885345i	$0.0282184\pi$
$673$	386.780	+	669.922i	0.574710	+	0.995426i	−0.421363	+	0.906892i	$0.638448\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1071.11	+	618.403i	−1.58214	+	0.913446i	−0.587588	+	0.809160i	$0.699922\pi$
$677$	−1071.11	+	618.403i	−1.58214	+	0.913446i	−0.994547	+	0.104286i	$0.966744\pi$
$678$		0			0
$679$	−1003.45			−1.47784
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	234.238i		−	0.342955i	−0.985188	−	0.171478i	$-0.945146\pi$
$683$		−	234.238i		−	0.342955i	0.985188	−	0.171478i	$-0.0548541\pi$
$684$		0			0
$685$	−136.197			−0.198828
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	365.652i		−	0.530700i
$690$		0			0
$691$	284.313	+	492.445i	0.411452	+	0.712655i	0.995049	−	0.0993883i	$-0.0316886\pi$
$691$	284.313	+	492.445i	0.411452	+	0.712655i	−0.583597	+	0.812043i	$0.698355\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−187.399	+	108.195i	−0.269639	+	0.155676i
$696$		0			0
$697$	868.013	+	1503.44i	1.24536	+	2.15702i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1031.29	−	595.416i	1.47117	−	0.849381i	0.471696	−	0.881761i	$-0.343642\pi$
$701$	1031.29	−	595.416i	1.47117	−	0.849381i	0.999476	+	0.0323797i	$0.0103086\pi$
$702$		0			0
$703$	−244.600	−	874.972i	−0.347937	−	1.24463i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−277.935	+	160.466i	−0.393119	+	0.226967i
$708$		0			0
$709$	−1199.06			−1.69120			−0.845599	−	0.533818i	$-0.820757\pi$
$709$	−1199.06			−1.69120			−0.845599	+	0.533818i	$0.820757\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	509.195i		−	0.714158i
$714$		0			0
$715$	−372.506	+	645.199i	−0.520987	+	0.902376i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−911.986	+	526.535i	−1.26841	+	0.732316i	−0.974687	−	0.223574i	$-0.928227\pi$
$719$	−911.986	+	526.535i	−1.26841	+	0.732316i	−0.293722	+	0.955891i	$0.594894\pi$
$720$		0			0
$721$	−918.508			−1.27394
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	505.651i		−	0.697450i
$726$		0			0
$727$	−438.485	+	759.479i	−0.603143	+	1.04468i	0.389198	+	0.921154i	$0.372752\pi$
$727$	−438.485	+	759.479i	−0.603143	+	1.04468i	−0.992342	+	0.123521i	$0.960581\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	142.215i		−	0.194549i
$732$		0			0
$733$	517.161	−	895.748i	0.705540	−	1.22203i	−0.260957	−	0.965351i	$-0.584038\pi$
$733$	517.161	−	895.748i	0.705540	−	1.22203i	0.966496	−	0.256680i	$-0.0826287\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	190.662	+	110.079i	0.258701	+	0.149361i
$738$		0			0
$739$	−349.864	−	605.982i	−0.473429	−	0.820003i	0.526108	−	0.850418i	$-0.323651\pi$
$739$	−349.864	−	605.982i	−0.473429	−	0.820003i	−0.999537	+	0.0304143i	$0.990317\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			885.269i			1.19148i	0.803178	+	0.595739i	$0.203141\pi$
$743$			885.269i			1.19148i	−0.803178	+	0.595739i	$0.796859\pi$
$744$		0			0
$745$	−577.496			−0.775162
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−616.187	+	355.756i	−0.822679	+	0.474974i
$750$		0			0
$751$	−647.242	−	1121.06i	−0.861840	−	1.49275i	−0.870151	−	0.492785i	$-0.835979\pi$
$751$	−647.242	−	1121.06i	−0.861840	−	1.49275i	0.00831164	−	0.999965i	$-0.497354\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	529.545	−	305.733i	0.701384	−	0.404944i
$756$		0			0
$757$	−80.2906	−	139.067i	−0.106064	−	0.183708i	0.808108	−	0.589034i	$-0.200492\pi$
$757$	−80.2906	−	139.067i	−0.106064	−	0.183708i	−0.914172	+	0.405325i	$0.867158\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	396.859	+	229.127i	0.521497	+	0.301087i	0.737547	−	0.675296i	$-0.235984\pi$
$761$	396.859	+	229.127i	0.521497	+	0.301087i	−0.216050	+	0.976382i	$0.569317\pi$
$762$		0			0
$763$	−42.4226	+	73.4782i	−0.0555998	+	0.0963017i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−461.370	+	266.372i	−0.601526	+	0.347291i
$768$		0			0
$769$	−87.8550	−	152.169i	−0.114246	−	0.197880i	0.803232	−	0.595666i	$-0.203112\pi$
$769$	−87.8550	−	152.169i	−0.114246	−	0.197880i	−0.917478	+	0.397786i	$0.869778\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	936.670	+	540.787i	1.21173	+	0.699595i	0.963137	−	0.269012i	$-0.0866971\pi$
$773$	936.670	+	540.787i	1.21173	+	0.699595i	0.248597	+	0.968607i	$0.420030\pi$
$774$		0			0
$775$	−142.671	+	247.113i	−0.184091	+	0.318855i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1397.62	−	390.706i	1.79412	−	0.501548i
$780$		0			0
$781$	−261.398	+	452.755i	−0.334697	+	0.579711i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			127.056i			0.161855i
$786$		0			0
$787$	143.087	−	247.834i	0.181813	−	0.314910i	−0.760685	−	0.649121i	$-0.775137\pi$
$787$	143.087	−	247.834i	0.181813	−	0.314910i	0.942498	+	0.334212i	$0.108470\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	128.313i		−	0.162216i
$792$		0			0
$793$	1371.16	+	2374.91i	1.72908	+	2.99485i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−207.738	−	119.938i	−0.260650	−	0.150487i	0.363981	−	0.931406i	$-0.381417\pi$
$797$	−207.738	−	119.938i	−0.260650	−	0.150487i	−0.624631	+	0.780920i	$0.714751\pi$
$798$		0			0
$799$	−201.078	−	348.277i	−0.251662	−	0.435892i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	124.499	+	71.8798i	0.155043	+	0.0895141i
$804$		0			0
$805$	232.804	−	403.228i	0.289197	−	0.500904i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			69.4467i			0.0858426i	0.999078	+	0.0429213i	$0.0136665\pi$
$809$			69.4467i			0.0858426i	−0.999078	+	0.0429213i	$0.986334\pi$
$810$		0			0
$811$	−5.39295	+	9.34086i	−0.00664975	+	0.0115177i	−0.869331	−	0.494230i	$-0.835450\pi$
$811$	−5.39295	+	9.34086i	−0.00664975	+	0.0115177i	0.862681	+	0.505748i	$0.168783\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	786.746i		−	0.965333i
$816$		0			0
$817$	−115.157	−	29.5278i	−0.140951	−	0.0361417i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	442.322i		−	0.538760i	−0.963034	−	0.269380i	$-0.913181\pi$
$821$		−	442.322i		−	0.538760i	0.963034	−	0.269380i	$-0.0868188\pi$
$822$		0			0
$823$	10.8790	+	18.8429i	0.0132187	+	0.0228954i	0.872559	−	0.488509i	$-0.162459\pi$
$823$	10.8790	+	18.8429i	0.0132187	+	0.0228954i	−0.859340	+	0.511404i	$0.829126\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−556.307	−	321.184i	−0.672681	−	0.388372i	0.124411	−	0.992231i	$-0.460296\pi$
$827$	−556.307	−	321.184i	−0.672681	−	0.388372i	−0.797092	+	0.603858i	$0.793629\pi$
$828$		0			0
$829$	−540.967			−0.652554			−0.326277	−	0.945274i	$-0.605794\pi$
$829$	−540.967			−0.652554			−0.326277	+	0.945274i	$0.605794\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	444.906i		−	0.534101i
$834$		0			0
$835$	−471.200	−	816.142i	−0.564311	−	0.977416i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	951.147	+	549.145i	1.13367	+	0.654523i	0.944855	−	0.327490i	$-0.106203\pi$
$839$	951.147	+	549.145i	1.13367	+	0.654523i	0.188812	+	0.982013i	$0.439536\pi$
$840$		0			0
$841$	−105.139			−0.125017
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1089.99	+	629.306i	1.28993	+	0.744741i
$846$		0			0
$847$	69.3949			0.0819303
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1214.80	+	701.362i	−1.42749	+	0.824163i
$852$		0			0
$853$	136.027	−	235.605i	0.159469	−	0.276208i	−0.775208	−	0.631705i	$-0.782355\pi$
$853$	136.027	−	235.605i	0.159469	−	0.276208i	0.934677	+	0.355497i	$0.115689\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−903.884	−	521.857i	−1.05471	−	0.608935i	−0.130743	−	0.991416i	$-0.541736\pi$
$857$	−903.884	−	521.857i	−1.05471	−	0.608935i	−0.923963	+	0.382481i	$0.875070\pi$
$858$		0			0
$859$	539.347	+	934.177i	0.627878	+	1.08752i	0.987977	+	0.154603i	$0.0494097\pi$
$859$	539.347	+	934.177i	0.627878	+	1.08752i	−0.360099	+	0.932914i	$0.617257\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	613.165i		−	0.710504i	−0.934771	−	0.355252i	$-0.884395\pi$
$863$		−	613.165i		−	0.710504i	0.934771	−	0.355252i	$-0.115605\pi$
$864$		0			0
$865$	−51.9505	+	89.9809i	−0.0600583	+	0.104024i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	13.5736i		−	0.0156198i
$870$		0			0
$871$	259.028	−	448.650i	0.297392	−	0.515097i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−569.596	+	328.856i	−0.650967	+	0.375836i
$876$		0			0
$877$	345.425			0.393871			0.196935	−	0.980416i	$-0.436901\pi$
$877$	345.425			0.393871			0.196935	+	0.980416i	$0.436901\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	452.169i		−	0.513245i	−0.966512	−	0.256623i	$-0.917390\pi$
$881$		−	452.169i		−	0.513245i	0.966512	−	0.256623i	$-0.0826098\pi$
$882$		0			0
$883$	40.2017	−	69.6314i	0.0455285	−	0.0788577i	−0.842363	−	0.538910i	$-0.818836\pi$
$883$	40.2017	−	69.6314i	0.0455285	−	0.0788577i	0.887892	+	0.460053i	$0.152169\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1252.29	−	723.012i	1.41183	−	0.815121i	0.416270	−	0.909241i	$-0.363337\pi$
$887$	1252.29	−	723.012i	1.41183	−	0.815121i	0.995561	+	0.0941197i	$0.0300036\pi$
$888$		0			0
$889$	445.911			0.501588
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−323.763	+	90.5084i	−0.362556	+	0.101353i
$894$		0			0
$895$	−626.321			−0.699801
$896$		0			0
$897$		0			0
$898$		0			0
$899$	462.381	+	266.956i	0.514328	+	0.296947i
$900$		0			0
$901$	−339.531			−0.376838
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−207.183	−	119.617i	−0.228932	−	0.132174i
$906$		0			0
$907$	−837.710	+	1450.96i	−0.923606	+	1.59973i	−0.129817	+	0.991538i	$0.541439\pi$
$907$	−837.710	+	1450.96i	−0.923606	+	1.59973i	−0.793788	+	0.608194i	$0.791894\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	51.2037	−	29.5624i	0.0562060	−	0.0324505i	−0.471634	−	0.881795i	$-0.656336\pi$
$911$	51.2037	−	29.5624i	0.0562060	−	0.0324505i	0.527840	+	0.849344i	$0.323002\pi$
$912$		0			0
$913$	397.786	−	688.985i	0.435691	−	0.754639i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	372.766	−	215.217i	0.406506	−	0.234697i
$918$		0			0
$919$	−811.809			−0.883361			−0.441680	−	0.897172i	$-0.645617\pi$
$919$	−811.809			−0.883361			−0.441680	+	0.897172i	$0.645617\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1065.38	+	615.099i	1.15426	+	0.666412i
$924$		0			0
$925$	786.056			0.849790
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1274.74	−	735.971i	−1.37216	−	0.792218i	−0.380963	−	0.924590i	$-0.624407\pi$
$929$	−1274.74	−	735.971i	−1.37216	−	0.792218i	−0.991200	+	0.132372i	$0.957741\pi$
$930$		0			0
$931$	−360.257	−	92.3748i	−0.386957	−	0.0992210i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	599.108	+	345.895i	0.640757	+	0.369941i
$936$		0			0
$937$	−292.962	+	507.425i	−0.312660	+	0.541543i	−0.978937	−	0.204161i	$-0.934553\pi$
$937$	−292.962	+	507.425i	−0.312660	+	0.541543i	0.666277	+	0.745704i	$0.267887\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−555.457	+	320.693i	−0.590284	+	0.340801i	−0.765210	−	0.643781i	$-0.777365\pi$
$941$	−555.457	+	320.693i	−0.590284	+	0.340801i	0.174926	+	0.984582i	$0.444031\pi$
$942$		0			0
$943$	−1120.31	−	1940.43i	−1.18802	−	2.05771i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1031.13	+	595.321i	1.08883	+	0.628639i	0.933266	−	0.359186i	$-0.116946\pi$
$947$	1031.13	+	595.321i	1.08883	+	0.628639i	0.155569	+	0.987825i	$0.450279\pi$
$948$		0			0
$949$	169.141	−	292.961i	0.178231	−	0.308705i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−834.102	+	481.569i	−0.875238	+	0.505319i	−0.869085	−	0.494662i	$-0.835292\pi$
$953$	−834.102	+	481.569i	−0.875238	+	0.505319i	−0.00615290	+	0.999981i	$0.501959\pi$
$954$		0			0
$955$	−479.876	−	831.170i	−0.502488	−	0.870335i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−218.675	+	126.252i	−0.228024	+	0.131650i
$960$		0			0
$961$	329.856	+	571.327i	0.343242	+	0.594513i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	642.728i		−	0.666039i
$966$		0			0
$967$	−1355.54			−1.40180			−0.700899	−	0.713260i	$-0.747218\pi$
$967$	−1355.54			−1.40180			−0.700899	+	0.713260i	$0.747218\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1057.02	−	610.270i	1.08859	−	0.628497i	0.155388	−	0.987853i	$-0.450337\pi$
$971$	1057.02	−	610.270i	1.08859	−	0.628497i	0.933200	+	0.359357i	$0.117004\pi$
$972$		0			0
$973$	−200.588	+	347.429i	−0.206155	+	0.357070i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−659.911	−	381.000i	−0.675446	−	0.389969i	0.122691	−	0.992445i	$-0.460848\pi$
$977$	−659.911	−	381.000i	−0.675446	−	0.389969i	−0.798137	+	0.602476i	$0.794181\pi$
$978$		0			0
$979$	1478.03			1.50974
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−956.119	−	552.016i	−0.972654	−	0.561562i	−0.0726098	−	0.997360i	$-0.523133\pi$
$983$	−956.119	−	552.016i	−0.972654	−	0.561562i	−0.900044	+	0.435798i	$0.856466\pi$
$984$		0			0
$985$	473.503			0.480713
$986$		0			0
$987$		0			0
$988$		0			0
$989$			183.551i			0.185592i
$990$		0			0
$991$	−308.321	−	534.027i	−0.311121	−	0.538877i	0.667485	−	0.744624i	$-0.267371\pi$
$991$	−308.321	−	534.027i	−0.311121	−	0.538877i	−0.978605	+	0.205747i	$0.934038\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−187.803	−	108.428i	−0.188747	−	0.108973i
$996$		0			0
$997$	580.521			0.582268			0.291134	−	0.956682i	$-0.405967\pi$
$997$	580.521			0.582268			0.291134	+	0.956682i	$0.405967\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2052.3.m.a.881.14		80
3.2	odd	2		684.3.m.a.653.16	yes	80
9.2	odd	6		2052.3.be.a.197.14		80
9.7	even	3		684.3.be.a.425.12	yes	80
19.11	even	3		2052.3.be.a.125.14		80
57.11	odd	6		684.3.be.a.581.12	yes	80
171.11	odd	6	inner	2052.3.m.a.1493.27		80
171.106	even	3		684.3.m.a.353.16	✓	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.m.a.353.16	✓	80	171.106	even	3
684.3.m.a.653.16	yes	80	3.2	odd	2
684.3.be.a.425.12	yes	80	9.7	even	3
684.3.be.a.581.12	yes	80	57.11	odd	6
2052.3.m.a.881.14		80	1.1	even	1	trivial
2052.3.m.a.1493.27		80	171.11	odd	6	inner
2052.3.be.a.125.14		80	19.11	even	3
2052.3.be.a.197.14		80	9.2	odd	6

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:	\( N \)	\(=\)	\( 2052 = 2^{2} \cdot 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2052.m (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-2.92593i q^{5} +(-2.71227 - 4.69779i) q^{7} +O(q^{10})\)	\(q-2.92593i q^{5} +(-2.71227 - 4.69779i) q^{7} +(-9.00863 + 5.20114i) q^{11} +(12.2388 + 21.1983i) q^{13} +(19.6840 - 11.3645i) q^{17} +(-5.11536 - 18.2984i) q^{19} +(-25.4052 + 14.6677i) q^{23} +16.4389 q^{25} -30.7594i q^{29} +(-8.67884 + 15.0322i) q^{31} +(-13.7454 + 7.93593i) q^{35} +47.8168 q^{37} +76.3790i q^{41} +(3.12848 - 5.41869i) q^{43} -17.6934i q^{47} +(9.78715 - 16.9518i) q^{49} +(-12.9368 - 7.46908i) q^{53} +(15.2182 + 26.3586i) q^{55} +21.7645i q^{59} +112.033 q^{61} +(62.0248 - 35.8100i) q^{65} +(-10.5822 - 18.3289i) q^{67} +(43.5246 - 25.1289i) q^{71} +(-6.91001 - 11.9685i) q^{73} +(48.8677 + 28.2138i) q^{77} +(-0.652435 + 1.13005i) q^{79} +(-66.2341 + 38.2403i) q^{83} +(-33.2519 - 57.5940i) q^{85} +(-123.051 - 71.0438i) q^{89} +(66.3902 - 114.991i) q^{91} +(-53.5400 + 14.9672i) q^{95} +(92.4918 - 160.201i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + q^{7}+O(q^{10}) \) 80 * q + q^7	\( 80 q + q^{7} - 18 q^{11} - 5 q^{13} + 9 q^{17} + 20 q^{19} - 72 q^{23} - 400 q^{25} - 8 q^{31} + 22 q^{37} - 44 q^{43} - 267 q^{49} + 36 q^{53} - 14 q^{61} + 144 q^{65} - 77 q^{67} + 135 q^{71} + 43 q^{73} - 216 q^{77} - 17 q^{79} + 171 q^{83} - 216 q^{89} + 122 q^{91} + 288 q^{95} - 8 q^{97}+O(q^{100}) \) 80 * q + q^7 - 18 * q^11 - 5 * q^13 + 9 * q^17 + 20 * q^19 - 72 * q^23 - 400 * q^25 - 8 * q^31 + 22 * q^37 - 44 * q^43 - 267 * q^49 + 36 * q^53 - 14 * q^61 + 144 * q^65 - 77 * q^67 + 135 * q^71 + 43 * q^73 - 216 * q^77 - 17 * q^79 + 171 * q^83 - 216 * q^89 + 122 * q^91 + 288 * q^95 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more