LMFDB - Embedded newform 2052.3.m.a.1493.17

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			1493.17
Character	$\chi$	$=$	2052.1493
Dual form			2052.3.m.a.881.24

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.56237i q^{5} +(-4.59594 + 7.96040i) q^{7} +O(q^{10})$	$q-1.56237i q^{5} +(-4.59594 + 7.96040i) q^{7} +(18.3761 + 10.6095i) q^{11} +(0.163320 - 0.282878i) q^{13} +(-22.8775 - 13.2083i) q^{17} +(8.51862 - 16.9833i) q^{19} +(2.17715 + 1.25698i) q^{23} +22.5590 q^{25} +52.9670i q^{29} +(-4.11110 - 7.12064i) q^{31} +(12.4371 + 7.18057i) q^{35} +30.2712 q^{37} -10.9070i q^{41} +(30.7940 + 53.3368i) q^{43} -41.4547i q^{47} +(-17.7453 - 30.7358i) q^{49} +(-54.4144 + 31.4162i) q^{53} +(16.5759 - 28.7104i) q^{55} +78.8050i q^{59} -57.7876 q^{61} +(-0.441962 - 0.255167i) q^{65} +(30.3656 - 52.5947i) q^{67} +(-48.8284 - 28.1911i) q^{71} +(30.1874 - 52.2860i) q^{73} +(-168.911 + 97.5209i) q^{77} +(-34.6994 - 60.1011i) q^{79} +(-1.98665 - 1.14700i) q^{83} +(-20.6363 + 35.7431i) q^{85} +(-92.6623 + 53.4986i) q^{89} +(1.50122 + 2.60018i) q^{91} +(-26.5343 - 13.3093i) q^{95} +(13.8646 + 24.0142i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/2052\mathbb{Z}\right)^\times$.

$n$	$325$	$1027$	$1217$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{1}{6}\right)$

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
$5$		−	1.56237i		−	0.312475i	−0.987720	−	0.156237i	$-0.950064\pi$
$5$		−	1.56237i		−	0.312475i	0.987720	−	0.156237i	$-0.0499365\pi$
$6$		0			0
$7$	−4.59594	+	7.96040i	−0.656563	+	1.13720i	0.324937	+	0.945736i	$0.394657\pi$
$7$	−4.59594	+	7.96040i	−0.656563	+	1.13720i	−0.981500	+	0.191464i	$0.938676\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	18.3761	+	10.6095i	1.67056	+	0.964496i	0.967326	+	0.253535i	$0.0815931\pi$
$11$	18.3761	+	10.6095i	1.67056	+	0.964496i	0.703231	+	0.710962i	$0.251740\pi$
$12$		0			0
$13$	0.163320	−	0.282878i	0.0125631	−	0.0217599i	−0.859676	−	0.510840i	$-0.829334\pi$
$13$	0.163320	−	0.282878i	0.0125631	−	0.0217599i	0.872239	+	0.489081i	$0.162668\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−22.8775	−	13.2083i	−1.34573	−	0.776960i	−0.358092	−	0.933686i	$-0.616572\pi$
$17$	−22.8775	−	13.2083i	−1.34573	−	0.776960i	−0.987642	+	0.156727i	$0.949906\pi$
$18$		0			0
$19$	8.51862	−	16.9833i	0.448349	−	0.893859i
$19$	8.51862	−	16.9833i	0.448349	−	0.893859i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.17715	+	1.25698i	0.0946589	+	0.0546513i	0.546582	−	0.837405i	$-0.315929\pi$
$23$	2.17715	+	1.25698i	0.0946589	+	0.0546513i	−0.451923	+	0.892057i	$0.649262\pi$
$24$		0			0
$25$	22.5590			0.902360
$26$		0			0
$27$		0			0
$28$		0			0
$29$			52.9670i			1.82645i	0.407458	+	0.913224i	$0.366415\pi$
$29$			52.9670i			1.82645i	−0.407458	+	0.913224i	$0.633585\pi$
$30$		0			0
$31$	−4.11110	−	7.12064i	−0.132616	−	0.229698i	0.792068	−	0.610433i	$-0.209004\pi$
$31$	−4.11110	−	7.12064i	−0.132616	−	0.229698i	−0.924684	+	0.380735i	$0.875671\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	12.4371	+	7.18057i	0.355346	+	0.205159i
$36$		0			0
$37$	30.2712			0.818141			0.409071	−	0.912503i	$-0.365853\pi$
$37$	30.2712			0.818141			0.409071	+	0.912503i	$0.365853\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	10.9070i		−	0.266025i	−0.991114	−	0.133012i	$-0.957535\pi$
$41$		−	10.9070i		−	0.266025i	0.991114	−	0.133012i	$-0.0424650\pi$
$42$		0			0
$43$	30.7940	+	53.3368i	0.716140	+	1.24039i	0.962518	+	0.271218i	$0.0874263\pi$
$43$	30.7940	+	53.3368i	0.716140	+	1.24039i	−0.246378	+	0.969174i	$0.579240\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	41.4547i		−	0.882015i	−0.897503	−	0.441008i	$-0.854621\pi$
$47$		−	41.4547i		−	0.882015i	0.897503	−	0.441008i	$-0.145379\pi$
$48$		0			0
$49$	−17.7453	−	30.7358i	−0.362149	−	0.627261i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−54.4144	+	31.4162i	−1.02669	+	0.592758i	−0.916034	−	0.401101i	$-0.868628\pi$
$53$	−54.4144	+	31.4162i	−1.02669	+	0.592758i	−0.110653	+	0.993859i	$0.535294\pi$
$54$		0			0
$55$	16.5759	−	28.7104i	0.301381	−	0.522007i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			78.8050i			1.33568i	0.744305	+	0.667839i	$0.232781\pi$
$59$			78.8050i			1.33568i	−0.744305	+	0.667839i	$0.767219\pi$
$60$		0			0
$61$	−57.7876			−0.947337			−0.473669	−	0.880703i	$-0.657071\pi$
$61$	−57.7876			−0.947337			−0.473669	+	0.880703i	$0.657071\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.441962	−	0.255167i	−0.00679941	−	0.00392564i
$66$		0			0
$67$	30.3656	−	52.5947i	0.453218	−	0.784996i	−0.545366	−	0.838198i	$-0.683609\pi$
$67$	30.3656	−	52.5947i	0.453218	−	0.784996i	0.998584	+	0.0532020i	$0.0169427\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−48.8284	−	28.1911i	−0.687724	−	0.397058i	0.115035	−	0.993361i	$-0.463302\pi$
$71$	−48.8284	−	28.1911i	−0.687724	−	0.397058i	−0.802759	+	0.596304i	$0.796635\pi$
$72$		0			0
$73$	30.1874	−	52.2860i	0.413526	−	0.716247i	−0.581747	−	0.813370i	$-0.697631\pi$
$73$	30.1874	−	52.2860i	0.413526	−	0.716247i	0.995272	+	0.0971226i	$0.0309639\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−168.911	+	97.5209i	−2.19365	+	1.26650i
$78$		0			0
$79$	−34.6994	−	60.1011i	−0.439232	−	0.760773i	0.558398	−	0.829573i	$-0.311416\pi$
$79$	−34.6994	−	60.1011i	−0.439232	−	0.760773i	−0.997630	+	0.0688003i	$0.978083\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.98665	−	1.14700i	−0.0239356	−	0.0138192i	0.487984	−	0.872852i	$-0.337732\pi$
$83$	−1.98665	−	1.14700i	−0.0239356	−	0.0138192i	−0.511920	+	0.859033i	$0.671066\pi$
$84$		0			0
$85$	−20.6363	+	35.7431i	−0.242780	+	0.420507i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−92.6623	+	53.4986i	−1.04115	+	0.601108i	−0.920159	−	0.391545i	$-0.871941\pi$
$89$	−92.6623	+	53.4986i	−1.04115	+	0.601108i	−0.120991	+	0.992654i	$0.538607\pi$
$90$		0			0
$91$	1.50122	+	2.60018i	0.0164969	+	0.0285734i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−26.5343	−	13.3093i	−0.279308	−	0.140098i
$96$		0			0
$97$	13.8646	+	24.0142i	0.142934	+	0.247569i	0.928600	−	0.371082i	$-0.121013\pi$
$97$	13.8646	+	24.0142i	0.142934	+	0.247569i	−0.785666	+	0.618650i	$0.787680\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	14.4308i		−	0.142879i	−0.997445	−	0.0714394i	$-0.977241\pi$
$101$		−	14.4308i		−	0.142879i	0.997445	−	0.0714394i	$-0.0227593\pi$
$102$		0			0
$103$	83.0455	+	143.839i	0.806267	+	1.39650i	0.915432	+	0.402472i	$0.131849\pi$
$103$	83.0455	+	143.839i	0.806267	+	1.39650i	−0.109165	+	0.994024i	$0.534818\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			131.867i			1.23241i	0.787587	+	0.616203i	$0.211330\pi$
$107$			131.867i			1.23241i	−0.787587	+	0.616203i	$0.788670\pi$
$108$		0			0
$109$	−73.6078	+	127.492i	−0.675301	+	1.16965i	0.301080	+	0.953599i	$0.402653\pi$
$109$	−73.6078	+	127.492i	−0.675301	+	1.16965i	−0.976381	+	0.216056i	$0.930681\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−7.71947	+	4.45684i	−0.0683139	+	0.0394411i	−0.533768	−	0.845631i	$-0.679224\pi$
$113$	−7.71947	+	4.45684i	−0.0683139	+	0.0394411i	0.465454	+	0.885072i	$0.345891\pi$
$114$		0			0
$115$	1.96387	−	3.40153i	0.0170771	−	0.0295785i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	210.287	−	121.409i	1.76712	−	1.02025i
$120$		0			0
$121$	164.621	+	285.132i	1.36051	+	2.35647i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	74.3049i		−	0.594439i
$126$		0			0
$127$	−54.7983	−	94.9135i	−0.431483	−	0.747351i	0.565518	−	0.824736i	$-0.308676\pi$
$127$	−54.7983	−	94.9135i	−0.431483	−	0.747351i	−0.997001	+	0.0773852i	$0.975343\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			36.4161i			0.277985i	0.990293	+	0.138993i	$0.0443864\pi$
$131$			36.4161i			0.277985i	−0.990293	+	0.138993i	$0.955614\pi$
$132$		0			0
$133$	96.0429	+	145.866i	0.722127	+	1.09674i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			219.638i			1.60320i	0.597860	+	0.801600i	$0.296018\pi$
$137$			219.638i			1.60320i	−0.597860	+	0.801600i	$0.703982\pi$
$138$		0			0
$139$	−66.5128	+	115.204i	−0.478509	+	0.828803i	−0.999696	−	0.0246398i	$-0.992156\pi$
$139$	−66.5128	+	115.204i	−0.478509	+	0.828803i	0.521187	+	0.853443i	$0.325489\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	6.00237	−	3.46547i	0.0419746	−	0.0242341i
$144$		0			0
$145$	82.7542			0.570718
$146$		0			0
$147$		0			0
$148$		0			0
$149$			93.6039i			0.628214i	0.949388	+	0.314107i	$0.101705\pi$
$149$			93.6039i			0.628214i	−0.949388	+	0.314107i	$0.898295\pi$
$150$		0			0
$151$	−1.49138	+	2.58314i	−0.00987668	+	0.0171069i	−0.870922	−	0.491422i	$-0.836477\pi$
$151$	−1.49138	+	2.58314i	−0.00987668	+	0.0171069i	0.861045	+	0.508529i	$0.169811\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−11.1251	+	6.42307i	−0.0717748	+	0.0414392i
$156$		0			0
$157$	−40.0109			−0.254847			−0.127423	−	0.991848i	$-0.540671\pi$
$157$	−40.0109			−0.254847			−0.127423	+	0.991848i	$0.540671\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−20.0121	+	11.5540i	−0.124299	+	0.0717640i
$162$		0			0
$163$	−73.5546			−0.451255			−0.225628	−	0.974214i	$-0.572443\pi$
$163$	−73.5546			−0.451255			−0.225628	+	0.974214i	$0.572443\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	85.1627	+	49.1687i	0.509956	+	0.294423i	0.732816	−	0.680427i	$-0.238206\pi$
$167$	85.1627	+	49.1687i	0.509956	+	0.294423i	−0.222860	+	0.974851i	$0.571539\pi$
$168$		0			0
$169$	84.4467	+	146.266i	0.499684	+	0.865479i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	40.7061	−	23.5017i	0.235296	−	0.135848i	−0.377717	−	0.925921i	$-0.623291\pi$
$173$	40.7061	−	23.5017i	0.235296	−	0.135848i	0.613013	+	0.790073i	$0.289957\pi$
$174$		0			0
$175$	−103.680	+	179.579i	−0.592456	+	1.02616i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			250.921i			1.40179i	0.713262	+	0.700897i	$0.247217\pi$
$179$			250.921i			1.40179i	−0.713262	+	0.700897i	$0.752783\pi$
$180$		0			0
$181$	122.712	+	212.544i	0.677969	+	1.17428i	0.975591	+	0.219594i	$0.0704732\pi$
$181$	122.712	+	212.544i	0.677969	+	1.17428i	−0.297622	+	0.954684i	$0.596193\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	47.2949i		−	0.255648i
$186$		0			0
$187$	−280.266	−	485.435i	−1.49875	−	2.59591i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−240.460	−	138.830i	−1.25895	−	0.726857i	−0.286082	−	0.958205i	$-0.592353\pi$
$191$	−240.460	−	138.830i	−1.25895	−	0.726857i	−0.972871	+	0.231348i	$0.925686\pi$
$192$		0			0
$193$	229.151			1.18731			0.593654	−	0.804720i	$-0.297685\pi$
$193$	229.151			1.18731			0.593654	+	0.804720i	$0.297685\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	174.829i		−	0.887455i	−0.896162	−	0.443727i	$-0.853656\pi$
$197$		−	174.829i		−	0.887455i	0.896162	−	0.443727i	$-0.146344\pi$
$198$		0			0
$199$	163.425	+	283.060i	0.821230	+	1.42241i	0.904767	+	0.425907i	$0.140045\pi$
$199$	163.425	+	283.060i	0.821230	+	1.42241i	−0.0835370	+	0.996505i	$0.526622\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−421.638	−	243.433i	−2.07704	−	1.19918i
$204$		0			0
$205$	−17.0408			−0.0831259
$206$		0			0
$207$		0			0
$208$		0			0
$209$	336.723	−	221.710i	1.61112	−	1.06081i
$210$		0			0
$211$	60.2414			0.285504			0.142752	−	0.989758i	$-0.454405\pi$
$211$	60.2414			0.285504			0.142752	+	0.989758i	$0.454405\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	83.3320	−	48.1118i	0.387591	−	0.223776i
$216$		0			0
$217$	75.5775			0.348283
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−7.47269	+	4.31436i	−0.0338131	+	0.0195220i
$222$		0			0
$223$	−130.050	−	225.254i	−0.583185	−	1.01011i	−0.995099	−	0.0988836i	$-0.968473\pi$
$223$	−130.050	−	225.254i	−0.583185	−	1.01011i	0.411914	−	0.911223i	$-0.364860\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	143.205	+	82.6796i	0.630860	+	0.364227i	0.781085	−	0.624425i	$-0.214667\pi$
$227$	143.205	+	82.6796i	0.630860	+	0.364227i	−0.150225	+	0.988652i	$0.548000\pi$
$228$		0			0
$229$	−166.723	−	288.773i	−0.728048	−	1.26102i	−0.957707	−	0.287745i	$-0.907095\pi$
$229$	−166.723	−	288.773i	−0.728048	−	1.26102i	0.229659	−	0.973271i	$-0.426239\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	46.3651	+	26.7689i	0.198992	+	0.114888i	0.596185	−	0.802847i	$-0.296682\pi$
$233$	46.3651	+	26.7689i	0.198992	+	0.114888i	−0.397193	+	0.917735i	$0.630016\pi$
$234$		0			0
$235$	−64.7677			−0.275607
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−364.835	+	210.637i	−1.52651	+	0.881328i	−0.527000	+	0.849865i	$0.676683\pi$
$239$	−364.835	+	210.637i	−1.52651	+	0.881328i	−0.999505	+	0.0314631i	$0.989983\pi$
$240$		0			0
$241$	141.455			0.586949			0.293475	−	0.955967i	$-0.405188\pi$
$241$	141.455			0.586949			0.293475	+	0.955967i	$0.405188\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−48.0208	+	27.7248i	−0.196003	+	0.113162i
$246$		0			0
$247$	−3.41295	−	5.18345i	−0.0138176	−	0.0209856i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−111.588	+	64.4253i	−0.444573	+	0.256675i	−0.705536	−	0.708674i	$-0.749294\pi$
$251$	−111.588	+	64.4253i	−0.444573	+	0.256675i	0.260962	+	0.965349i	$0.415960\pi$
$252$		0			0
$253$	26.6718	+	46.1969i	0.105422	+	0.182596i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	4.49375	+	2.59447i	0.0174854	+	0.0100952i	0.508717	−	0.860934i	$-0.330120\pi$
$257$	4.49375	+	2.59447i	0.0174854	+	0.0100952i	−0.491232	+	0.871029i	$0.663453\pi$
$258$		0			0
$259$	−139.125	+	240.971i	−0.537161	+	0.930390i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	18.3290	−	10.5822i	0.0696919	−	0.0402366i	−0.464749	−	0.885442i	$-0.653855\pi$
$263$	18.3290	−	10.5822i	0.0696919	−	0.0402366i	0.534441	+	0.845206i	$0.320522\pi$
$264$		0			0
$265$	49.0838	+	85.0156i	0.185222	+	0.320813i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	180.342	+	104.121i	0.670418	+	0.387066i	0.796235	−	0.604988i	$-0.206822\pi$
$269$	180.342	+	104.121i	0.670418	+	0.387066i	−0.125817	+	0.992053i	$0.540155\pi$
$270$		0			0
$271$	150.784	−	261.165i	0.556398	−	0.963710i	−0.441395	−	0.897313i	$-0.645516\pi$
$271$	150.784	−	261.165i	0.556398	−	0.963710i	0.997793	−	0.0663970i	$-0.0211504\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	414.547	+	239.339i	1.50744	+	0.870323i
$276$		0			0
$277$	−106.417	+	184.320i	−0.384177	+	0.665414i	−0.991655	−	0.128923i	$-0.958848\pi$
$277$	−106.417	+	184.320i	−0.384177	+	0.665414i	0.607478	+	0.794337i	$0.292181\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			242.264i			0.862150i	0.902316	+	0.431075i	$0.141866\pi$
$281$			242.264i			0.862150i	−0.902316	+	0.431075i	$0.858134\pi$
$282$		0			0
$283$	336.324			1.18843			0.594213	−	0.804308i	$-0.297464\pi$
$283$	336.324			1.18843			0.594213	+	0.804308i	$0.297464\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	86.8242	+	50.1280i	0.302523	+	0.174662i
$288$		0			0
$289$	204.419	+	354.064i	0.707333	+	1.22514i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−179.353	+	103.550i	−0.612127	+	0.353412i	−0.773797	−	0.633433i	$-0.781645\pi$
$293$	−179.353	+	103.550i	−0.612127	+	0.353412i	0.161670	+	0.986845i	$0.448312\pi$
$294$		0			0
$295$	123.123			0.417366
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.711145	−	0.410580i	0.00237841	−	0.00137318i
$300$		0			0
$301$	−566.110			−1.88076
$302$		0			0
$303$		0			0
$304$		0			0
$305$			90.2858i			0.296019i
$306$		0			0
$307$	137.863	−	238.786i	0.449066	−	0.777806i	−0.549259	−	0.835652i	$-0.685090\pi$
$307$	137.863	−	238.786i	0.449066	−	0.777806i	0.998325	+	0.0578463i	$0.0184233\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−237.877	+	137.339i	−0.764879	+	0.441603i	−0.831045	−	0.556205i	$-0.812257\pi$
$311$	−237.877	+	137.339i	−0.764879	+	0.441603i	0.0661657	+	0.997809i	$0.478923\pi$
$312$		0			0
$313$	−485.034			−1.54963			−0.774815	−	0.632189i	$-0.782157\pi$
$313$	−485.034			−1.54963			−0.774815	+	0.632189i	$0.782157\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	406.534i		−	1.28244i	−0.767357	−	0.641220i	$-0.778429\pi$
$317$		−	406.534i		−	1.28244i	0.767357	−	0.641220i	$-0.221571\pi$
$318$		0			0
$319$	−561.951	+	973.328i	−1.76160	+	3.05118i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−419.206	+	276.019i	−1.29785	+	0.854547i
$324$		0			0
$325$	3.68433	−	6.38145i	0.0113364	−	0.0196352i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	329.996	+	190.523i	1.00303	+	0.579098i
$330$		0			0
$331$	−114.803	+	198.845i	−0.346838	+	0.600741i	−0.985686	−	0.168592i	$-0.946078\pi$
$331$	−114.803	+	198.845i	−0.346838	+	0.600741i	0.638848	+	0.769333i	$0.279411\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−82.1726	−	47.4424i	−0.245291	−	0.141619i
$336$		0			0
$337$	444.001			1.31751			0.658755	−	0.752358i	$-0.271083\pi$
$337$	444.001			1.31751			0.658755	+	0.752358i	$0.271083\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	174.466i		−	0.511631i
$342$		0			0
$343$	−124.176			−0.362030
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	53.3296i		−	0.153688i	−0.997043	−	0.0768439i	$-0.975516\pi$
$347$		−	53.3296i		−	0.153688i	0.997043	−	0.0768439i	$-0.0244843\pi$
$348$		0			0
$349$	−174.035	+	301.437i	−0.498666	+	0.863715i	−0.999999	−	0.00153938i	$-0.999510\pi$
$349$	−174.035	+	301.437i	−0.498666	+	0.863715i	0.501333	+	0.865255i	$0.332843\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	155.310	+	89.6686i	0.439973	+	0.254019i	0.703586	−	0.710610i	$-0.251581\pi$
$353$	155.310	+	89.6686i	0.439973	+	0.254019i	−0.263613	+	0.964628i	$0.584914\pi$
$354$		0			0
$355$	−44.0450	+	76.2882i	−0.124070	+	0.214896i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−192.832	−	111.332i	−0.537138	−	0.310117i	0.206780	−	0.978387i	$-0.433701\pi$
$359$	−192.832	−	111.332i	−0.537138	−	0.310117i	−0.743918	+	0.668271i	$0.767035\pi$
$360$		0			0
$361$	−215.866	−	289.349i	−0.597967	−	0.801521i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−81.6903	−	47.1639i	−0.223809	−	0.129216i
$366$		0			0
$367$	466.936			1.27231			0.636153	−	0.771563i	$-0.280525\pi$
$367$	466.936			1.27231			0.636153	+	0.771563i	$0.280525\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	577.547i		−	1.55673i
$372$		0			0
$373$	−29.7770	−	51.5752i	−0.0798310	−	0.138271i	0.823346	−	0.567540i	$-0.192105\pi$
$373$	−29.7770	−	51.5752i	−0.0798310	−	0.138271i	−0.903177	+	0.429269i	$0.858771\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	14.9832	+	8.65056i	0.0397433	+	0.0229458i
$378$		0			0
$379$	61.2909			0.161717			0.0808587	−	0.996726i	$-0.474234\pi$
$379$	61.2909			0.161717			0.0808587	+	0.996726i	$0.474234\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	202.153i		−	0.527814i	−0.964548	−	0.263907i	$-0.914989\pi$
$383$		−	202.153i		−	0.527814i	0.964548	−	0.263907i	$-0.0850112\pi$
$384$		0			0
$385$	152.364	+	263.902i	0.395751	+	0.685460i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			773.464i			1.98834i	0.107826	+	0.994170i	$0.465611\pi$
$389$			773.464i			1.98834i	−0.107826	+	0.994170i	$0.534389\pi$
$390$		0			0
$391$	−33.2052	−	57.5131i	−0.0849237	−	0.147092i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−93.9003	+	54.2133i	−0.237722	+	0.137249i
$396$		0			0
$397$	111.155	−	192.525i	0.279986	−	0.484951i	−0.691395	−	0.722477i	$-0.743003\pi$
$397$	111.155	−	192.525i	0.279986	−	0.484951i	0.971381	+	0.237527i	$0.0763367\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	495.758i		−	1.23630i	−0.786058	−	0.618152i	$-0.787881\pi$
$401$		−	495.758i		−	1.23630i	0.786058	−	0.618152i	$-0.212119\pi$
$402$		0			0
$403$	−2.68570			−0.00666427
$404$		0			0
$405$		0			0
$406$		0			0
$407$	556.268	+	321.161i	1.36675	+	0.789094i
$408$		0			0
$409$	403.387	−	698.686i	0.986275	−	1.70828i	0.350149	−	0.936694i	$-0.386131\pi$
$409$	403.387	−	698.686i	0.986275	−	1.70828i	0.636126	−	0.771585i	$-0.280536\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−627.320	−	362.183i	−1.51893	−	0.876957i
$414$		0			0
$415$	−1.79203	+	3.10389i	−0.00431815	+	0.00747926i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−235.914	+	136.205i	−0.563041	+	0.325072i	−0.754365	−	0.656455i	$-0.772055\pi$
$419$	−235.914	+	136.205i	−0.563041	+	0.325072i	0.191324	+	0.981527i	$0.438722\pi$
$420$		0			0
$421$	400.333	+	693.396i	0.950909	+	1.64702i	0.743465	+	0.668775i	$0.233181\pi$
$421$	400.333	+	693.396i	0.950909	+	1.64702i	0.207444	+	0.978247i	$0.433486\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−516.093	−	297.966i	−1.21434	−	0.701097i
$426$		0			0
$427$	265.588	−	460.012i	0.621987	−	1.07731i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−76.3653	+	44.0895i	−0.177182	+	0.102296i	−0.585968	−	0.810334i	$-0.699286\pi$
$431$	−76.3653	+	44.0895i	−0.177182	+	0.102296i	0.408786	+	0.912630i	$0.365952\pi$
$432$		0			0
$433$	−405.267	−	701.944i	−0.935952	−	1.62112i	−0.772927	−	0.634495i	$-0.781208\pi$
$433$	−405.267	−	701.944i	−0.935952	−	1.62112i	−0.163026	−	0.986622i	$-0.552125\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	39.8940	−	26.2675i	0.0912907	−	0.0601088i
$438$		0			0
$439$	4.18943	+	7.25630i	0.00954312	+	0.0165292i	0.870758	−	0.491713i	$-0.163629\pi$
$439$	4.18943	+	7.25630i	0.00954312	+	0.0165292i	−0.861214	+	0.508242i	$0.830296\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	93.4493i		−	0.210947i	−0.994422	−	0.105473i	$-0.966364\pi$
$443$		−	93.4493i		−	0.210947i	0.994422	−	0.105473i	$-0.0336358\pi$
$444$		0			0
$445$	83.5848	+	144.773i	0.187831	+	0.325333i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	122.697i		−	0.273268i	−0.990622	−	0.136634i	$-0.956372\pi$
$449$		−	122.697i		−	0.273268i	0.990622	−	0.136634i	$-0.0436284\pi$
$450$		0			0
$451$	115.717	−	200.429i	0.256580	−	0.444409i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	4.06246	−	2.34546i	0.00892848	−	0.00515486i
$456$		0			0
$457$	213.401	−	369.622i	0.466962	−	0.808801i	−0.532326	−	0.846539i	$-0.678682\pi$
$457$	213.401	−	369.622i	0.466962	−	0.808801i	0.999288	+	0.0377382i	$0.0120153\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−144.545	+	83.4531i	−0.313546	+	0.181026i	−0.648512	−	0.761204i	$-0.724609\pi$
$461$	−144.545	+	83.4531i	−0.313546	+	0.181026i	0.334966	+	0.942230i	$0.391275\pi$
$462$		0			0
$463$	−422.303	−	731.450i	−0.912101	−	1.57980i	−0.811092	−	0.584919i	$-0.801126\pi$
$463$	−422.303	−	731.450i	−0.912101	−	1.57980i	−0.101009	−	0.994886i	$-0.532207\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	612.701i		−	1.31199i	−0.754764	−	0.655997i	$-0.772248\pi$
$467$		−	612.701i		−	1.31199i	0.754764	−	0.655997i	$-0.227752\pi$
$468$		0			0
$469$	279.117	+	483.444i	0.595132	+	1.03080i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			1306.83i			2.76286i
$474$		0			0
$475$	192.172	−	383.126i	0.404572	−	0.806582i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			138.380i			0.288892i	0.989513	+	0.144446i	$0.0461401\pi$
$479$			138.380i			0.288892i	−0.989513	+	0.144446i	$0.953860\pi$
$480$		0			0
$481$	4.94389	−	8.56308i	0.0102784	−	0.0178027i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	37.5191	−	21.6616i	0.0773589	−	0.0446632i
$486$		0			0
$487$	−767.796			−1.57658			−0.788291	−	0.615302i	$-0.789034\pi$
$487$	−767.796			−1.57658			−0.788291	+	0.615302i	$0.789034\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	20.6719i		−	0.0421017i	−0.999778	−	0.0210508i	$-0.993299\pi$
$491$		−	20.6719i		−	0.0421017i	0.999778	−	0.0210508i	$-0.00670118\pi$
$492$		0			0
$493$	699.605	−	1211.75i	1.41908	−	2.45791i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	448.825	−	259.129i	0.903068	−	0.521387i
$498$		0			0
$499$	−821.607			−1.64651			−0.823253	−	0.567674i	$-0.807843\pi$
$499$	−821.607			−1.64651			−0.823253	+	0.567674i	$0.807843\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	331.574	−	191.434i	0.659192	−	0.380585i	−0.132777	−	0.991146i	$-0.542389\pi$
$503$	331.574	−	191.434i	0.659192	−	0.380585i	0.791969	+	0.610561i	$0.209056\pi$
$504$		0			0
$505$	−22.5462			−0.0446460
$506$		0			0
$507$		0			0
$508$		0			0
$509$	483.713	+	279.272i	0.950321	+	0.548668i	0.893180	−	0.449698i	$-0.148469\pi$
$509$	483.713	+	279.272i	0.950321	+	0.548668i	0.0571400	+	0.998366i	$0.481802\pi$
$510$		0			0
$511$	277.479	+	480.607i	0.543011	+	0.940523i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	224.730	−	129.748i	0.436369	−	0.251938i
$516$		0			0
$517$	439.812	−	761.777i	0.850701	−	1.47346i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	1013.96i		−	1.94618i	−0.230417	−	0.973092i	$-0.574009\pi$
$521$		−	1013.96i		−	1.94618i	0.230417	−	0.973092i	$-0.425991\pi$
$522$		0			0
$523$	400.647	+	693.941i	0.766056	+	1.32685i	0.939687	+	0.342036i	$0.111117\pi$
$523$	400.647	+	693.941i	0.766056	+	1.32685i	−0.173631	+	0.984811i	$0.555550\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			217.203i			0.412150i
$528$		0			0
$529$	−261.340	−	452.654i	−0.494026	−	0.855679i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3.08536	−	1.78133i	−0.00578866	−	0.00334209i
$534$		0			0
$535$	206.026			0.385096
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	753.073i		−	1.39717i
$540$		0			0
$541$	−12.7450	−	22.0750i	−0.0235582	−	0.0408040i	0.854006	−	0.520263i	$-0.174166\pi$
$541$	−12.7450	−	22.0750i	−0.0235582	−	0.0408040i	−0.877564	+	0.479459i	$0.840833\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	199.191	+	115.003i	0.365487	+	0.211014i
$546$		0			0
$547$	7.32208			0.0133859			0.00669295	−	0.999978i	$-0.497870\pi$
$547$	7.32208			0.0133859			0.00669295	+	0.999978i	$0.497870\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	899.555	+	451.206i	1.63259	+	0.818885i
$552$		0			0
$553$	637.905			1.15353
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−235.341	+	135.874i	−0.422514	+	0.243939i	−0.696153	−	0.717894i	$-0.745106\pi$
$557$	−235.341	+	135.874i	−0.422514	+	0.243939i	0.273638	+	0.961833i	$0.411773\pi$
$558$		0			0
$559$	20.1171			0.0359877
$560$		0			0
$561$		0			0
$562$		0			0
$563$	125.587	−	72.5078i	0.223068	−	0.128788i	−0.384302	−	0.923207i	$-0.625558\pi$
$563$	125.587	−	72.5078i	0.223068	−	0.128788i	0.607370	+	0.794419i	$0.292225\pi$
$564$		0			0
$565$	6.96324	+	12.0607i	0.0123243	+	0.0213464i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	944.345	+	545.218i	1.65966	+	0.958204i	0.972872	+	0.231344i	$0.0743122\pi$
$569$	944.345	+	545.218i	1.65966	+	0.958204i	0.686785	+	0.726860i	$0.259021\pi$
$570$		0			0
$571$	−234.969	−	406.979i	−0.411505	−	0.712748i	0.583549	−	0.812078i	$-0.301663\pi$
$571$	−234.969	−	406.979i	−0.411505	−	0.712748i	−0.995055	+	0.0993297i	$0.968330\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	49.1144	+	28.3562i	0.0854163	+	0.0493151i
$576$		0			0
$577$	−43.8582			−0.0760107			−0.0380053	−	0.999278i	$-0.512100\pi$
$577$	−43.8582			−0.0760107			−0.0380053	+	0.999278i	$0.512100\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	18.2611	−	10.5430i	0.0314304	−	0.0181464i
$582$		0			0
$583$	−1333.23			−2.28685
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−373.819	+	215.824i	−0.636829	+	0.367674i	−0.783392	−	0.621528i	$-0.786512\pi$
$587$	−373.819	+	215.824i	−0.636829	+	0.367674i	0.146563	+	0.989201i	$0.453179\pi$
$588$		0			0
$589$	−155.953	+	9.16212i	−0.264776	+	0.0155554i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−839.777	+	484.845i	−1.41615	+	0.817614i	−0.995958	−	0.0898213i	$-0.971370\pi$
$593$	−839.777	+	484.845i	−1.41615	+	0.817614i	−0.420191	+	0.907435i	$0.638037\pi$
$594$		0			0
$595$	−189.686	−	328.547i	−0.318801	−	0.552179i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	721.606	+	416.619i	1.20468	+	0.695525i	0.961593	−	0.274479i	$-0.0885052\pi$
$599$	721.606	+	416.619i	1.20468	+	0.695525i	0.243091	+	0.970003i	$0.421839\pi$
$600$		0			0
$601$	−39.4076	+	68.2560i	−0.0655701	+	0.113571i	−0.896947	−	0.442139i	$-0.854220\pi$
$601$	−39.4076	+	68.2560i	−0.0655701	+	0.113571i	0.831377	+	0.555709i	$0.187553\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	445.483	−	257.200i	0.736336	−	0.425124i
$606$		0			0
$607$	−237.075	−	410.626i	−0.390568	−	0.676484i	0.601956	−	0.798529i	$-0.294388\pi$
$607$	−237.075	−	410.626i	−0.390568	−	0.676484i	−0.992525	+	0.122045i	$0.961055\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−11.7266	−	6.77038i	−0.0191925	−	0.0110808i
$612$		0			0
$613$	197.970	−	342.894i	0.322953	−	0.559370i	−0.658143	−	0.752893i	$-0.728658\pi$
$613$	197.970	−	342.894i	0.322953	−	0.559370i	0.981096	+	0.193522i	$0.0619912\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	657.897	+	379.837i	1.06628	+	0.615619i	0.927164	−	0.374657i	$-0.122239\pi$
$617$	657.897	+	379.837i	1.06628	+	0.615619i	0.139120	+	0.990276i	$0.455573\pi$
$618$		0			0
$619$	267.269	−	462.924i	0.431776	−	0.747857i	−0.565251	−	0.824919i	$-0.691221\pi$
$619$	267.269	−	462.924i	0.431776	−	0.747857i	0.997026	+	0.0770618i	$0.0245539\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	983.506i		−	1.57866i
$624$		0			0
$625$	447.883			0.716613
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−692.529	−	399.832i	−1.10100	−	0.635663i
$630$		0			0
$631$	−316.906	−	548.898i	−0.502229	−	0.869886i	−0.999997	−	0.00257536i	$-0.999180\pi$
$631$	−316.906	−	548.898i	−0.502229	−	0.869886i	0.497768	−	0.867310i	$-0.334153\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−148.290	+	85.6154i	−0.233528	+	0.134827i
$636$		0			0
$637$	−11.5927			−0.0181988
$638$		0			0
$639$		0			0
$640$		0			0
$641$	7.78120	−	4.49248i	0.0121392	−	0.00700855i	−0.493918	−	0.869508i	$-0.664436\pi$
$641$	7.78120	−	4.49248i	0.0121392	−	0.00700855i	0.506057	+	0.862500i	$0.331102\pi$
$642$		0			0
$643$	396.064			0.615963			0.307982	−	0.951392i	$-0.400346\pi$
$643$	396.064			0.615963			0.307982	+	0.951392i	$0.400346\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	749.207i		−	1.15797i	−0.815338	−	0.578985i	$-0.803449\pi$
$647$		−	749.207i		−	1.15797i	0.815338	−	0.578985i	$-0.196551\pi$
$648$		0			0
$649$	−836.079	+	1448.13i	−1.28826	+	2.23133i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−411.871	+	237.794i	−0.630737	+	0.364156i	−0.781037	−	0.624484i	$-0.785309\pi$
$653$	−411.871	+	237.794i	−0.630737	+	0.364156i	0.150301	+	0.988640i	$0.451976\pi$
$654$		0			0
$655$	56.8955			0.0868633
$656$		0			0
$657$		0			0
$658$		0			0
$659$			69.8374i			0.105975i	0.998595	+	0.0529874i	$0.0168743\pi$
$659$			69.8374i			0.105975i	−0.998595	+	0.0529874i	$0.983126\pi$
$660$		0			0
$661$	−185.179	+	320.740i	−0.280150	+	0.485234i	−0.971421	−	0.237361i	$-0.923718\pi$
$661$	−185.179	+	320.740i	−0.280150	+	0.485234i	0.691272	+	0.722595i	$0.257051\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	227.897	−	150.055i	0.342702	−	0.225646i
$666$		0			0
$667$	−66.5785	+	115.317i	−0.0998178	+	0.172889i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1061.91	−	613.095i	−1.58258	−	0.913704i
$672$		0			0
$673$	96.0890	−	166.431i	0.142777	−	0.247297i	−0.785764	−	0.618526i	$-0.787730\pi$
$673$	96.0890	−	166.431i	0.142777	−	0.247297i	0.928541	+	0.371229i	$0.121063\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	610.979	+	352.749i	0.902480	+	0.521047i	0.878004	−	0.478653i	$-0.158875\pi$
$677$	610.979	+	352.749i	0.902480	+	0.521047i	0.0244761	+	0.999700i	$0.492208\pi$
$678$		0			0
$679$	−254.883			−0.375380
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	563.037i		−	0.824358i	−0.911103	−	0.412179i	$-0.864768\pi$
$683$		−	563.037i		−	0.824358i	0.911103	−	0.412179i	$-0.135232\pi$
$684$		0			0
$685$	343.157			0.500959
$686$		0			0
$687$		0			0
$688$		0			0
$689$			20.5235i			0.0297874i
$690$		0			0
$691$	−533.521	+	924.086i	−0.772100	+	1.33732i	0.164310	+	0.986409i	$0.447460\pi$
$691$	−533.521	+	924.086i	−0.772100	+	1.33732i	−0.936410	+	0.350908i	$0.885873\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	179.991	+	103.918i	0.258980	+	0.149522i
$696$		0			0
$697$	−144.063	+	249.525i	−0.206690	+	0.357998i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	19.1885	+	11.0785i	0.0273730	+	0.0158038i	0.513624	−	0.858015i	$-0.328303\pi$
$701$	19.1885	+	11.0785i	0.0273730	+	0.0158038i	−0.486251	+	0.873819i	$0.661636\pi$
$702$		0			0
$703$	257.869	−	514.106i	0.366812	−	0.731303i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	114.875	+	66.3229i	0.162482	+	0.0938089i
$708$		0			0
$709$	499.221			0.704120			0.352060	−	0.935977i	$-0.385481\pi$
$709$	499.221			0.704120			0.352060	+	0.935977i	$0.385481\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	20.6703i		−	0.0289906i
$714$		0			0
$715$	−5.41436	−	9.37795i	−0.00757253	−	0.0131160i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−802.059	−	463.069i	−1.11552	−	0.644046i	−0.175267	−	0.984521i	$-0.556079\pi$
$719$	−802.059	−	463.069i	−1.11552	−	0.644046i	−0.940254	+	0.340475i	$0.889412\pi$
$720$		0			0
$721$	−1526.69			−2.11746
$722$		0			0
$723$		0			0
$724$		0			0
$725$			1194.88i			1.64811i
$726$		0			0
$727$	164.277	+	284.536i	0.225965	+	0.391384i	0.956609	−	0.291376i	$-0.0941130\pi$
$727$	164.277	+	284.536i	0.225965	+	0.391384i	−0.730643	+	0.682759i	$0.760780\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	1626.95i		−	2.22565i
$732$		0			0
$733$	−208.671	−	361.428i	−0.284680	−	0.493080i	0.687851	−	0.725852i	$-0.258554\pi$
$733$	−208.671	−	361.428i	−0.284680	−	0.493080i	−0.972532	+	0.232771i	$0.925221\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1116.00	−	644.325i	1.51425	−	0.874253i
$738$		0			0
$739$	293.870	−	508.998i	0.397659	−	0.688766i	−0.595778	−	0.803149i	$-0.703156\pi$
$739$	293.870	−	508.998i	0.397659	−	0.688766i	0.993437	+	0.114384i	$0.0364894\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	308.182i		−	0.414781i	−0.978258	−	0.207390i	$-0.933503\pi$
$743$		−	308.182i		−	0.414781i	0.978258	−	0.207390i	$-0.0664970\pi$
$744$		0			0
$745$	146.244			0.196301
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1049.72	−	606.055i	−1.40149	−	0.809152i
$750$		0			0
$751$	67.4421	−	116.813i	0.0898031	−	0.155543i	−0.817625	−	0.575751i	$-0.804710\pi$
$751$	67.4421	−	116.813i	0.0898031	−	0.155543i	0.907428	+	0.420208i	$0.138043\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4.03584	+	2.33009i	0.00534548	+	0.00308621i
$756$		0			0
$757$	−323.700	+	560.665i	−0.427609	+	0.740641i	−0.996660	−	0.0816616i	$-0.973977\pi$
$757$	−323.700	+	560.665i	−0.427609	+	0.740641i	0.569051	+	0.822302i	$0.307311\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	913.138	−	527.201i	1.19992	−	0.692773i	0.239382	−	0.970926i	$-0.423055\pi$
$761$	913.138	−	527.201i	1.19992	−	0.692773i	0.960537	+	0.278152i	$0.0897219\pi$
$762$		0			0
$763$	−676.594	−	1171.89i	−0.886754	−	1.53590i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	22.2922	+	12.8704i	0.0290642	+	0.0167802i
$768$		0			0
$769$	137.478	−	238.119i	0.178775	−	0.309647i	−0.762686	−	0.646769i	$-0.776120\pi$
$769$	137.478	−	238.119i	0.178775	−	0.309647i	0.941461	+	0.337121i	$0.109453\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	568.785	−	328.388i	0.735815	−	0.424823i	−0.0847310	−	0.996404i	$-0.527003\pi$
$773$	568.785	−	328.388i	0.735815	−	0.424823i	0.820546	+	0.571581i	$0.193670\pi$
$774$		0			0
$775$	−92.7423	−	160.634i	−0.119667	−	0.207270i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−185.237	−	92.9127i	−0.237788	−	0.119272i
$780$		0			0
$781$	−598.185	−	1036.09i	−0.765922	−	1.32662i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			62.5120i			0.0796331i
$786$		0			0
$787$	266.659	+	461.866i	0.338829	+	0.586869i	0.984213	−	0.176989i	$-0.0566358\pi$
$787$	266.659	+	461.866i	0.338829	+	0.586869i	−0.645384	+	0.763859i	$0.723302\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	81.9335i		−	0.103582i
$792$		0			0
$793$	−9.43786	+	16.3469i	−0.0119015	+	0.0206139i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	9.48311	−	5.47507i	0.0118985	−	0.00686960i	−0.494039	−	0.869440i	$-0.664480\pi$
$797$	9.48311	−	5.47507i	0.0118985	−	0.00686960i	0.505937	+	0.862570i	$0.331147\pi$
$798$		0			0
$799$	−547.547	+	948.379i	−0.685290	+	1.18696i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1109.45	−	640.543i	1.38164	−	0.797688i
$804$		0			0
$805$	18.0517	+	31.2664i	0.0224244	+	0.0388403i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			200.785i			0.248189i	0.992270	+	0.124095i	$0.0396027\pi$
$809$			200.785i			0.248189i	−0.992270	+	0.124095i	$0.960397\pi$
$810$		0			0
$811$	252.299	+	436.995i	0.311096	+	0.538835i	0.978600	−	0.205772i	$-0.0659705\pi$
$811$	252.299	+	436.995i	0.311096	+	0.538835i	−0.667504	+	0.744606i	$0.732637\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			114.920i			0.141006i
$816$		0			0
$817$	1168.16	−	68.6284i	1.42982	−	0.0840005i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	758.721i		−	0.924142i	−0.886843	−	0.462071i	$-0.847106\pi$
$821$		−	758.721i		−	0.924142i	0.886843	−	0.462071i	$-0.152894\pi$
$822$		0			0
$823$	−397.225	+	688.013i	−0.482654	+	0.835982i	−0.999802	−	0.0199144i	$-0.993661\pi$
$823$	−397.225	+	688.013i	−0.482654	+	0.835982i	0.517147	+	0.855896i	$0.326994\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	154.964	−	89.4686i	0.187381	−	0.108184i	−0.403375	−	0.915035i	$-0.632163\pi$
$827$	154.964	−	89.4686i	0.187381	−	0.108184i	0.590756	+	0.806850i	$0.298830\pi$
$828$		0			0
$829$	742.995			0.896255			0.448127	−	0.893970i	$-0.352091\pi$
$829$	742.995			0.896255			0.448127	+	0.893970i	$0.352091\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			937.543i			1.12550i
$834$		0			0
$835$	76.8198	−	133.056i	0.0919998	−	0.159348i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−741.832	+	428.297i	−0.884185	+	0.510485i	−0.872036	−	0.489442i	$-0.837201\pi$
$839$	−741.832	+	428.297i	−0.884185	+	0.510485i	−0.0121493	+	0.999926i	$0.503867\pi$
$840$		0			0
$841$	−1964.50			−2.33591
$842$		0			0
$843$		0			0
$844$		0			0
$845$	228.522	−	131.937i	0.270440	−	0.156139i
$846$		0			0
$847$	−3026.36			−3.57303
$848$		0			0
$849$		0			0
$850$		0			0
$851$	65.9051	+	38.0503i	0.0774443	+	0.0447125i
$852$		0			0
$853$	376.897	+	652.805i	0.441849	+	0.765305i	0.997827	−	0.0658925i	$-0.0209895\pi$
$853$	376.897	+	652.805i	0.441849	+	0.765305i	−0.555978	+	0.831197i	$0.687656\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−908.400	+	524.465i	−1.05998	+	0.611978i	−0.925426	−	0.378929i	$-0.876292\pi$
$857$	−908.400	+	524.465i	−1.05998	+	0.611978i	−0.134551	+	0.990907i	$0.542959\pi$
$858$		0			0
$859$	318.090	−	550.949i	0.370303	−	0.641384i	−0.619309	−	0.785147i	$-0.712587\pi$
$859$	318.090	−	550.949i	0.370303	−	0.641384i	0.989612	+	0.143763i	$0.0459205\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1603.33i		−	1.85786i	−0.370255	−	0.928930i	$-0.620730\pi$
$863$		−	1603.33i		−	1.85786i	0.370255	−	0.928930i	$-0.379270\pi$
$864$		0			0
$865$	−36.7184	−	63.5981i	−0.0424490	−	0.0735239i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	1472.57i		−	1.69455i
$870$		0			0
$871$	−9.91861	−	17.1795i	−0.0113876	−	0.0197239i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	591.497	+	341.501i	0.675996	+	0.390287i
$876$		0			0
$877$	−310.652			−0.354221			−0.177111	−	0.984191i	$-0.556675\pi$
$877$	−310.652			−0.354221			−0.177111	+	0.984191i	$0.556675\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1186.34i			1.34659i	0.739375	+	0.673293i	$0.235121\pi$
$881$			1186.34i			1.34659i	−0.739375	+	0.673293i	$0.764879\pi$
$882$		0			0
$883$	160.019	+	277.162i	0.181223	+	0.313887i	0.942297	−	0.334778i	$-0.108661\pi$
$883$	160.019	+	277.162i	0.181223	+	0.313887i	−0.761075	+	0.648664i	$0.775328\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−263.073	−	151.885i	−0.296587	−	0.171235i	0.344322	−	0.938852i	$-0.388109\pi$
$887$	−263.073	−	151.885i	−0.296587	−	0.171235i	−0.640909	+	0.767617i	$0.721442\pi$
$888$		0			0
$889$	1007.40			1.13318
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−704.039	−	353.137i	−0.788397	−	0.395450i
$894$		0			0
$895$	392.032			0.438025
$896$		0			0
$897$		0			0
$898$		0			0
$899$	377.159	−	217.753i	0.419531	−	0.242217i
$900$		0			0
$901$	1659.82			1.84220
$902$		0			0
$903$		0			0
$904$		0			0
$905$	332.073	−	191.723i	0.366932	−	0.211848i
$906$		0			0
$907$	−424.741	−	735.674i	−0.468293	−	0.811107i	0.531051	−	0.847340i	$-0.321797\pi$
$907$	−424.741	−	735.674i	−0.468293	−	0.811107i	−0.999343	+	0.0362335i	$0.988464\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	342.761	+	197.893i	0.376247	+	0.217226i	0.676184	−	0.736733i	$-0.263632\pi$
$911$	342.761	+	197.893i	0.376247	+	0.217226i	−0.299937	+	0.953959i	$0.596966\pi$
$912$		0			0
$913$	−24.3380	−	42.1546i	−0.0266572	−	0.0461716i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−289.886	−	167.366i	−0.316125	−	0.182515i
$918$		0			0
$919$	1265.47			1.37700			0.688502	−	0.725235i	$-0.258269\pi$
$919$	1265.47			1.37700			0.688502	+	0.725235i	$0.258269\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−15.9493	+	9.20834i	−0.0172799	+	0.00997653i
$924$		0			0
$925$	682.888			0.738258
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−732.006	+	422.624i	−0.787950	+	0.454923i	−0.839240	−	0.543760i	$-0.817000\pi$
$929$	−732.006	+	422.624i	−0.787950	+	0.454923i	0.0512902	+	0.998684i	$0.483667\pi$
$930$		0			0
$931$	−673.161	+	39.5477i	−0.723052	+	0.0424787i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−758.431	+	437.880i	−0.811156	+	0.468321i
$936$		0			0
$937$	119.492	+	206.966i	0.127526	+	0.220882i	0.922718	−	0.385477i	$-0.125963\pi$
$937$	119.492	+	206.966i	0.127526	+	0.220882i	−0.795191	+	0.606358i	$0.792630\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1439.49	+	831.093i	1.52975	+	0.883201i	0.999372	+	0.0354402i	$0.0112833\pi$
$941$	1439.49	+	831.093i	1.52975	+	0.883201i	0.530378	+	0.847761i	$0.322050\pi$
$942$		0			0
$943$	13.7099	−	23.7462i	0.0145386	−	0.0251816i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1181.93	−	682.387i	1.24808	−	0.720577i	0.277351	−	0.960769i	$-0.410543\pi$
$947$	1181.93	−	682.387i	1.24808	−	0.720577i	0.970726	+	0.240191i	$0.0772101\pi$
$948$		0			0
$949$	−9.86040	−	17.0787i	−0.0103903	−	0.0179965i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	797.996	+	460.723i	0.837352	+	0.483445i	0.856363	−	0.516374i	$-0.172719\pi$
$953$	797.996	+	460.723i	0.837352	+	0.483445i	−0.0190116	+	0.999819i	$0.506052\pi$
$954$		0			0
$955$	−216.904	+	375.688i	−0.227124	+	0.393391i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1748.41	−	1009.45i	−1.82316	−	1.05260i
$960$		0			0
$961$	446.698	−	773.703i	0.464826	−	0.805102i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	358.019i		−	0.371004i
$966$		0			0
$967$	189.011			0.195461			0.0977307	−	0.995213i	$-0.468842\pi$
$967$	189.011			0.195461			0.0977307	+	0.995213i	$0.468842\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	393.888	+	227.411i	0.405652	+	0.234203i	0.688920	−	0.724838i	$-0.258085\pi$
$971$	393.888	+	227.411i	0.405652	+	0.234203i	−0.283268	+	0.959041i	$0.591419\pi$
$972$		0			0
$973$	−611.378	−	1058.94i	−0.628343	−	1.08832i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−536.712	+	309.871i	−0.549347	+	0.317166i	−0.748859	−	0.662730i	$-0.769398\pi$
$977$	−536.712	+	309.871i	−0.549347	+	0.317166i	0.199511	+	0.979896i	$0.436065\pi$
$978$		0			0
$979$	−2270.37			−2.31907
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−377.628	+	218.024i	−0.384159	+	0.221794i	−0.679626	−	0.733559i	$-0.737858\pi$
$983$	−377.628	+	218.024i	−0.384159	+	0.221794i	0.295467	+	0.955353i	$0.404525\pi$
$984$		0			0
$985$	−273.148			−0.277307
$986$		0			0
$987$		0			0
$988$		0			0
$989$			154.830i			0.156552i
$990$		0			0
$991$	278.088	−	481.663i	0.280614	−	0.486038i	−0.690922	−	0.722929i	$-0.742795\pi$
$991$	278.088	−	481.663i	0.280614	−	0.486038i	0.971536	+	0.236892i	$0.0761286\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	442.245	−	255.330i	0.444468	−	0.256613i
$996$		0			0
$997$	1624.38			1.62927			0.814633	−	0.579977i	$-0.196939\pi$
$997$	1624.38			1.62927			0.814633	+	0.579977i	$0.196939\pi$
$998$		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.1493.17		80
3.2	odd	2		684.3.m.a.353.14	✓	80
9.4	even	3		684.3.be.a.581.40	yes	80
9.5	odd	6		2052.3.be.a.125.24		80
19.7	even	3		2052.3.be.a.197.24		80
57.26	odd	6		684.3.be.a.425.40	yes	80
171.121	even	3		684.3.m.a.653.14	yes	80
171.140	odd	6	inner	2052.3.m.a.881.24		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.m.a.353.14	✓	80	3.2	odd	2
684.3.m.a.653.14	yes	80	171.121	even	3
684.3.be.a.425.40	yes	80	57.26	odd	6
684.3.be.a.581.40	yes	80	9.4	even	3
2052.3.m.a.881.24		80	171.140	odd	6	inner
2052.3.m.a.1493.17		80	1.1	even	1	trivial
2052.3.be.a.125.24		80	9.5	odd	6
2052.3.be.a.197.24		80	19.7	even	3

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)

\(f(q)\)	\(=\)	\(q-1.56237i q^{5} +(-4.59594 + 7.96040i) q^{7} +O(q^{10})\)	\(q-1.56237i q^{5} +(-4.59594 + 7.96040i) q^{7} +(18.3761 + 10.6095i) q^{11} +(0.163320 - 0.282878i) q^{13} +(-22.8775 - 13.2083i) q^{17} +(8.51862 - 16.9833i) q^{19} +(2.17715 + 1.25698i) q^{23} +22.5590 q^{25} +52.9670i q^{29} +(-4.11110 - 7.12064i) q^{31} +(12.4371 + 7.18057i) q^{35} +30.2712 q^{37} -10.9070i q^{41} +(30.7940 + 53.3368i) q^{43} -41.4547i q^{47} +(-17.7453 - 30.7358i) q^{49} +(-54.4144 + 31.4162i) q^{53} +(16.5759 - 28.7104i) q^{55} +78.8050i q^{59} -57.7876 q^{61} +(-0.441962 - 0.255167i) q^{65} +(30.3656 - 52.5947i) q^{67} +(-48.8284 - 28.1911i) q^{71} +(30.1874 - 52.2860i) q^{73} +(-168.911 + 97.5209i) q^{77} +(-34.6994 - 60.1011i) q^{79} +(-1.98665 - 1.14700i) q^{83} +(-20.6363 + 35.7431i) q^{85} +(-92.6623 + 53.4986i) q^{89} +(1.50122 + 2.60018i) q^{91} +(-26.5343 - 13.3093i) q^{95} +(13.8646 + 24.0142i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + q^{7}+O(q^{10}) \) 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

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