LMFDB - Embedded newform 2052.3.bl.a.145.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2052,3,Mod(145,2052)]

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, 2, 5]))

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

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

chi := DirichletCharacter("2052.145");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2052 = 2^{2} \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2052.bl (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			145.8
Character	$\chi$	$=$	2052.145
Dual form			2052.3.bl.a.1585.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.96221 q^{5} +(6.83689 + 11.8418i) q^{7} +O(q^{10})$	$q-5.96221 q^{5} +(6.83689 + 11.8418i) q^{7} +(0.0305186 + 0.0528598i) q^{11} +(17.8742 - 10.3197i) q^{13} +(-15.1064 - 26.1650i) q^{17} +(-16.6230 - 9.20190i) q^{19} +(-0.855542 - 1.48184i) q^{23} +10.5479 q^{25} +50.9509i q^{29} +(-2.97698 - 1.71876i) q^{31} +(-40.7629 - 70.6035i) q^{35} -18.8728i q^{37} -43.2335i q^{41} +(20.5707 - 35.6295i) q^{43} +8.12095 q^{47} +(-68.9861 + 119.487i) q^{49} +(-38.9115 - 22.4656i) q^{53} +(-0.181958 - 0.315161i) q^{55} -46.4391i q^{59} +76.9545 q^{61} +(-106.570 + 61.5280i) q^{65} +(35.6393 - 20.5764i) q^{67} +(35.0078 - 20.2118i) q^{71} +(19.3561 + 33.5258i) q^{73} +(-0.417305 + 0.722793i) q^{77} +(-21.7113 - 12.5350i) q^{79} +(-17.9661 - 31.1181i) q^{83} +(90.0673 + 156.001i) q^{85} +(35.8967 + 20.7250i) q^{89} +(244.408 + 141.109i) q^{91} +(99.1099 + 54.8636i) q^{95} +(-109.866 - 63.4310i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - q^{7}+O(q^{10}) $ 80 * q - q^7	$ 80 q - q^{7} + 6 q^{11} - 15 q^{13} + 21 q^{17} - 20 q^{19} - 24 q^{23} + 400 q^{25} + 24 q^{31} + 54 q^{35} + 76 q^{43} - 24 q^{47} - 267 q^{49} + 36 q^{53} + 14 q^{61} - 288 q^{65} - 21 q^{67} + 81 q^{71} + 55 q^{73} - 30 q^{77} - 51 q^{79} + 93 q^{83} - 216 q^{89} + 96 q^{91} + 432 q^{95} + 90 q^{97}+O(q^{100}) $ 80 * q - q^7 + 6 * q^11 - 15 * q^13 + 21 * q^17 - 20 * q^19 - 24 * q^23 + 400 * q^25 + 24 * q^31 + 54 * q^35 + 76 * q^43 - 24 * q^47 - 267 * q^49 + 36 * q^53 + 14 * q^61 - 288 * q^65 - 21 * q^67 + 81 * q^71 + 55 * q^73 - 30 * q^77 - 51 * q^79 + 93 * q^83 - 216 * q^89 + 96 * q^91 + 432 * q^95 + 90 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

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
$5$	−5.96221			−1.19244			−0.596221	−	0.802821i	$-0.703332\pi$
$5$	−5.96221			−1.19244			−0.596221	+	0.802821i	$0.703332\pi$
$6$		0			0
$7$	6.83689	+	11.8418i	0.976699	+	1.69169i	0.674212	+	0.738538i	$0.264483\pi$
$7$	6.83689	+	11.8418i	0.976699	+	1.69169i	0.302487	+	0.953154i	$0.402183\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.0305186	+	0.0528598i	0.00277442	+	0.00480544i	0.867409	−	0.497595i	$-0.165784\pi$
$11$	0.0305186	+	0.0528598i	0.00277442	+	0.00480544i	−0.864635	+	0.502401i	$0.832450\pi$
$12$		0			0
$13$	17.8742	−	10.3197i	1.37494	−	0.793821i	0.383393	−	0.923585i	$-0.374756\pi$
$13$	17.8742	−	10.3197i	1.37494	−	0.793821i	0.991545	+	0.129765i	$0.0414222\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−15.1064	−	26.1650i	−0.888610	−	1.53912i	−0.841519	−	0.540227i	$-0.818338\pi$
$17$	−15.1064	−	26.1650i	−0.888610	−	1.53912i	−0.0470906	−	0.998891i	$-0.514995\pi$
$18$		0			0
$19$	−16.6230	−	9.20190i	−0.874896	−	0.484310i
$19$	−16.6230	−	9.20190i	−0.874896	−	0.484310i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.855542	−	1.48184i	−0.0371975	−	0.0644279i	0.846827	−	0.531868i	$-0.178510\pi$
$23$	−0.855542	−	1.48184i	−0.0371975	−	0.0644279i	−0.884025	+	0.467440i	$0.845176\pi$
$24$		0			0
$25$	10.5479			0.421916
$26$		0			0
$27$		0			0
$28$		0			0
$29$			50.9509i			1.75693i	0.477810	+	0.878463i	$0.341431\pi$
$29$			50.9509i			1.75693i	−0.477810	+	0.878463i	$0.658569\pi$
$30$		0			0
$31$	−2.97698	−	1.71876i	−0.0960316	−	0.0554439i	0.451215	−	0.892415i	$-0.350991\pi$
$31$	−2.97698	−	1.71876i	−0.0960316	−	0.0554439i	−0.547247	+	0.836971i	$0.684324\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−40.7629	−	70.6035i	−1.16466	−	2.01724i
$36$		0			0
$37$		−	18.8728i		−	0.510077i	−0.966931	−	0.255038i	$-0.917912\pi$
$37$		−	18.8728i		−	0.510077i	0.966931	−	0.255038i	$-0.0820881\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	43.2335i		−	1.05448i	−0.849718	−	0.527238i	$-0.823228\pi$
$41$		−	43.2335i		−	1.05448i	0.849718	−	0.527238i	$-0.176772\pi$
$42$		0			0
$43$	20.5707	−	35.6295i	0.478389	−	0.828594i	−0.521304	−	0.853371i	$-0.674554\pi$
$43$	20.5707	−	35.6295i	0.478389	−	0.828594i	0.999693	+	0.0247770i	$0.00788756\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	8.12095			0.172786			0.0863931	−	0.996261i	$-0.472466\pi$
$47$	8.12095			0.172786			0.0863931	+	0.996261i	$0.472466\pi$
$48$		0			0
$49$	−68.9861	+	119.487i	−1.40788	+	2.43852i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−38.9115	−	22.4656i	−0.734180	−	0.423879i	0.0857695	−	0.996315i	$-0.472665\pi$
$53$	−38.9115	−	22.4656i	−0.734180	−	0.423879i	−0.819949	+	0.572436i	$0.805998\pi$
$54$		0			0
$55$	−0.181958	−	0.315161i	−0.00330833	−	0.00573020i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	46.4391i		−	0.787104i	−0.919302	−	0.393552i	$-0.871246\pi$
$59$		−	46.4391i		−	0.787104i	0.919302	−	0.393552i	$-0.128754\pi$
$60$		0			0
$61$	76.9545			1.26155			0.630775	−	0.775966i	$-0.282737\pi$
$61$	76.9545			1.26155			0.630775	+	0.775966i	$0.282737\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−106.570	+	61.5280i	−1.63953	+	0.946585i
$66$		0			0
$67$	35.6393	−	20.5764i	0.531930	−	0.307110i	−0.209872	−	0.977729i	$-0.567305\pi$
$67$	35.6393	−	20.5764i	0.531930	−	0.307110i	0.741802	+	0.670619i	$0.233971\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	35.0078	−	20.2118i	0.493068	−	0.284673i	−0.232778	−	0.972530i	$-0.574782\pi$
$71$	35.0078	−	20.2118i	0.493068	−	0.284673i	0.725846	+	0.687857i	$0.241448\pi$
$72$		0			0
$73$	19.3561	+	33.5258i	0.265152	+	0.459257i	0.967603	−	0.252475i	$-0.0812445\pi$
$73$	19.3561	+	33.5258i	0.265152	+	0.459257i	−0.702451	+	0.711732i	$0.747911\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.417305	+	0.722793i	−0.00541954	+	0.00938693i
$78$		0			0
$79$	−21.7113	−	12.5350i	−0.274826	−	0.158671i	0.356253	−	0.934390i	$-0.384054\pi$
$79$	−21.7113	−	12.5350i	−0.274826	−	0.158671i	−0.631079	+	0.775719i	$0.717388\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−17.9661	−	31.1181i	−0.216459	−	0.374917i	0.737264	−	0.675605i	$-0.236117\pi$
$83$	−17.9661	−	31.1181i	−0.216459	−	0.374917i	−0.953723	+	0.300687i	$0.902784\pi$
$84$		0			0
$85$	90.0673	+	156.001i	1.05962	+	1.83531i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	35.8967	+	20.7250i	0.403334	+	0.232865i	0.687921	−	0.725785i	$-0.258523\pi$
$89$	35.8967	+	20.7250i	0.403334	+	0.232865i	−0.284588	+	0.958650i	$0.591857\pi$
$90$		0			0
$91$	244.408	+	141.109i	2.68580	+	1.55065i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	99.1099	+	54.8636i	1.04326	+	0.577512i
$96$		0			0
$97$	−109.866	−	63.4310i	−1.13264	−	0.653928i	−0.188040	−	0.982161i	$-0.560213\pi$
$97$	−109.866	−	63.4310i	−1.13264	−	0.653928i	−0.944597	+	0.328233i	$0.893547\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	184.431			1.82605			0.913026	−	0.407900i	$-0.133739\pi$
$101$	184.431			1.82605			0.913026	+	0.407900i	$0.133739\pi$
$102$		0			0
$103$	145.759	+	84.1540i	1.41514	+	0.817030i	0.995866	−	0.0908319i	$-0.0289526\pi$
$103$	145.759	+	84.1540i	1.41514	+	0.817030i	0.419270	+	0.907861i	$0.362286\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	84.5050i		−	0.789767i	−0.918731	−	0.394883i	$-0.870785\pi$
$107$		−	84.5050i		−	0.789767i	0.918731	−	0.394883i	$-0.129215\pi$
$108$		0			0
$109$	−96.1522	+	55.5135i	−0.882130	+	0.509298i	−0.871360	−	0.490644i	$-0.836762\pi$
$109$	−96.1522	+	55.5135i	−0.882130	+	0.509298i	−0.0107698	+	0.999942i	$0.503428\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−46.3810	−	26.7781i	−0.410452	−	0.236974i	0.280532	−	0.959845i	$-0.409489\pi$
$113$	−46.3810	−	26.7781i	−0.410452	−	0.236974i	−0.690984	+	0.722870i	$0.742822\pi$
$114$		0			0
$115$	5.10092	+	8.83505i	0.0443558	+	0.0768265i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	206.561	−	357.774i	1.73581	−	3.00651i
$120$		0			0
$121$	60.4981	−	104.786i	0.499985	−	0.865999i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	86.1664			0.689331
$126$		0			0
$127$	−78.0863	−	45.0832i	−0.614853	−	0.354985i	0.160009	−	0.987115i	$-0.448848\pi$
$127$	−78.0863	−	45.0832i	−0.614853	−	0.354985i	−0.774862	+	0.632130i	$0.782181\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	29.6584			0.226400			0.113200	−	0.993572i	$-0.463890\pi$
$131$	29.6584			0.226400			0.113200	+	0.993572i	$0.463890\pi$
$132$		0			0
$133$	−4.68242	−	259.760i	−0.0352062	−	1.95308i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−180.800			−1.31971			−0.659854	−	0.751394i	$-0.729382\pi$
$137$	−180.800			−1.31971			−0.659854	+	0.751394i	$0.729382\pi$
$138$		0			0
$139$	−43.5844	−	75.4905i	−0.313557	−	0.543097i	0.665573	−	0.746333i	$-0.268187\pi$
$139$	−43.5844	−	75.4905i	−0.313557	−	0.543097i	−0.979130	+	0.203236i	$0.934854\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.09099	+	0.629884i	0.00762931	+	0.00440478i
$144$		0			0
$145$		−	303.780i		−	2.09503i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	46.7893			0.314022			0.157011	−	0.987597i	$-0.449814\pi$
$149$	46.7893			0.314022			0.157011	+	0.987597i	$0.449814\pi$
$150$		0			0
$151$	155.846	−	89.9778i	1.03209	−	0.595880i	0.114510	−	0.993422i	$-0.463470\pi$
$151$	155.846	−	89.9778i	1.03209	−	0.595880i	0.917584	+	0.397543i	$0.130137\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	17.7494	+	10.2476i	0.114512	+	0.0661135i
$156$		0			0
$157$	−79.7566			−0.508004			−0.254002	−	0.967204i	$-0.581747\pi$
$157$	−79.7566			−0.508004			−0.254002	+	0.967204i	$0.581747\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	11.6985	−	20.2624i	0.0726615	−	0.125853i
$162$		0			0
$163$	−46.4827			−0.285170			−0.142585	−	0.989783i	$-0.545541\pi$
$163$	−46.4827			−0.285170			−0.142585	+	0.989783i	$0.545541\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	79.9901	−	46.1823i	0.478983	−	0.276541i	−0.241010	−	0.970523i	$-0.577479\pi$
$167$	79.9901	−	46.1823i	0.478983	−	0.276541i	0.719993	+	0.693982i	$0.244145\pi$
$168$		0			0
$169$	128.491	−	222.553i	0.760303	−	1.31688i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	75.6930	+	43.7014i	0.437532	+	0.252609i	0.702550	−	0.711634i	$-0.252045\pi$
$173$	75.6930	+	43.7014i	0.437532	+	0.252609i	−0.265018	+	0.964243i	$0.585378\pi$
$174$		0			0
$175$	72.1149	+	124.907i	0.412085	+	0.713752i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	155.213i		−	0.867109i	−0.901127	−	0.433555i	$-0.857259\pi$
$179$		−	155.213i		−	0.867109i	0.901127	−	0.433555i	$-0.142741\pi$
$180$		0			0
$181$	58.5550	+	33.8067i	0.323508	+	0.186778i	0.652955	−	0.757397i	$-0.273529\pi$
$181$	58.5550	+	33.8067i	0.323508	+	0.186778i	−0.329447	+	0.944174i	$0.606862\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			112.524i			0.608236i
$186$		0			0
$187$	0.922051	−	1.59704i	0.00493075	−	0.00854032i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	73.8255	+	127.869i	0.386521	+	0.669474i	0.991979	−	0.126403i	$-0.0403434\pi$
$191$	73.8255	+	127.869i	0.386521	+	0.669474i	−0.605458	+	0.795877i	$0.707010\pi$
$192$		0			0
$193$		−	105.302i		−	0.545607i	−0.962070	−	0.272803i	$-0.912049\pi$
$193$		−	105.302i		−	0.545607i	0.962070	−	0.272803i	$-0.0879509\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	289.808			1.47111			0.735553	−	0.677467i	$-0.236922\pi$
$197$	289.808			1.47111			0.735553	+	0.677467i	$0.236922\pi$
$198$		0			0
$199$	184.816	−	320.111i	0.928725	−	1.60860i	0.143268	−	0.989684i	$-0.454239\pi$
$199$	184.816	−	320.111i	0.928725	−	1.60860i	0.785457	−	0.618916i	$-0.212428\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−603.352	+	348.345i	−2.97218	+	1.71599i
$204$		0			0
$205$			257.767i			1.25740i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.0209015	−	1.15952i	−0.000100007	−	0.00554794i
$210$		0			0
$211$		−	31.6844i		−	0.150163i	−0.997177	−	0.0750814i	$-0.976078\pi$
$211$		−	31.6844i		−	0.150163i	0.997177	−	0.0750814i	$-0.0239217\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−122.647	+	212.431i	−0.570451	+	0.988050i
$216$		0			0
$217$		−	47.0039i		−	0.216608i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−540.028	−	311.785i	−2.44357	−	1.41079i
$222$		0			0
$223$	4.10878	+	2.37220i	0.0184250	+	0.0106377i	0.509184	−	0.860658i	$-0.329947\pi$
$223$	4.10878	+	2.37220i	0.0184250	+	0.0106377i	−0.490759	+	0.871295i	$0.663281\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	215.175	−	124.231i	0.947908	−	0.547275i	0.0554775	−	0.998460i	$-0.482332\pi$
$227$	215.175	−	124.231i	0.947908	−	0.547275i	0.892430	+	0.451185i	$0.148999\pi$
$228$		0			0
$229$	21.6534	−	37.5047i	0.0945562	−	0.163776i	−0.814867	−	0.579648i	$-0.803190\pi$
$229$	21.6534	−	37.5047i	0.0945562	−	0.163776i	0.909423	+	0.415872i	$0.136523\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−74.3349	−	128.752i	−0.319034	−	0.552583i	0.661253	−	0.750163i	$-0.270025\pi$
$233$	−74.3349	−	128.752i	−0.319034	−	0.552583i	−0.980287	+	0.197580i	$0.936692\pi$
$234$		0			0
$235$	−48.4188			−0.206037
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−111.083	+	192.401i	−0.464781	+	0.805023i	−0.999192	−	0.0402012i	$-0.987200\pi$
$239$	−111.083	+	192.401i	−0.464781	+	0.805023i	0.534411	+	0.845225i	$0.320533\pi$
$240$		0			0
$241$		−	103.268i		−	0.428500i	−0.976779	−	0.214250i	$-0.931269\pi$
$241$		−	103.268i		−	0.428500i	0.976779	−	0.214250i	$-0.0687307\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	411.310	−	712.409i	1.67881	−	2.90779i
$246$		0			0
$247$	−392.084	+	7.06769i	−1.58738	+	0.0286141i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	65.3474	−	113.185i	0.260348	−	0.450937i	−0.705986	−	0.708226i	$-0.749496\pi$
$251$	65.3474	−	113.185i	0.260348	−	0.450937i	0.966334	+	0.257289i	$0.0828292\pi$
$252$		0			0
$253$	0.0522199	−	0.0904476i	0.000206403	−	0.000357500i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	294.809	−	170.208i	1.14712	−	0.662287i	0.198933	−	0.980013i	$-0.436252\pi$
$257$	294.809	−	170.208i	1.14712	−	0.662287i	0.948183	+	0.317726i	$0.102919\pi$
$258$		0			0
$259$	223.489	−	129.031i	0.862892	−	0.498191i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	199.254	−	345.118i	0.757619	−	1.31223i	−0.186443	−	0.982466i	$-0.559696\pi$
$263$	199.254	−	345.118i	0.757619	−	1.31223i	0.944062	−	0.329768i	$-0.106971\pi$
$264$		0			0
$265$	231.999	+	133.944i	0.875466	+	0.505451i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−413.044	+	238.471i	−1.53548	+	0.886509i	−0.536383	+	0.843975i	$0.680210\pi$
$269$	−413.044	+	238.471i	−1.53548	+	0.886509i	−0.999095	+	0.0425337i	$0.986457\pi$
$270$		0			0
$271$	45.3497	+	78.5479i	0.167342	+	0.289845i	0.937484	−	0.348027i	$-0.113148\pi$
$271$	45.3497	+	78.5479i	0.167342	+	0.289845i	−0.770143	+	0.637872i	$0.779815\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.321907	+	0.557560i	0.00117057	+	0.00202749i
$276$		0			0
$277$	140.593	+	243.515i	0.507557	+	0.879115i	0.999962	+	0.00874854i	$0.00278478\pi$
$277$	140.593	+	243.515i	0.507557	+	0.879115i	−0.492404	+	0.870367i	$0.663882\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	66.6712i		−	0.237264i	−0.992938	−	0.118632i	$-0.962149\pi$
$281$		−	66.6712i		−	0.237264i	0.992938	−	0.118632i	$-0.0378509\pi$
$282$		0			0
$283$	88.0549			0.311148			0.155574	−	0.987824i	$-0.450277\pi$
$283$	88.0549			0.311148			0.155574	+	0.987824i	$0.450277\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	511.964	−	295.583i	1.78385	−	1.02991i
$288$		0			0
$289$	−311.905	+	540.235i	−1.07926	+	1.86932i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	141.493	+	81.6909i	0.482910	+	0.278808i	0.721629	−	0.692280i	$-0.243394\pi$
$293$	141.493	+	81.6909i	0.482910	+	0.278808i	−0.238718	+	0.971089i	$0.576727\pi$
$294$		0			0
$295$			276.880i			0.938575i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−30.5842	−	17.6578i	−0.102288	−	0.0590563i
$300$		0			0
$301$	562.559			1.86897
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−458.819			−1.50432
$306$		0			0
$307$	199.159	−	114.984i	0.648725	−	0.374542i	−0.139242	−	0.990258i	$-0.544467\pi$
$307$	199.159	−	114.984i	0.648725	−	0.374542i	0.787968	+	0.615717i	$0.211133\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	28.8869	−	50.0336i	0.0928840	−	0.160880i	−0.815840	−	0.578278i	$-0.803725\pi$
$311$	28.8869	−	50.0336i	0.0928840	−	0.160880i	0.908724	+	0.417399i	$0.137058\pi$
$312$		0			0
$313$	84.4539			0.269821			0.134910	−	0.990858i	$-0.456925\pi$
$313$	84.4539			0.269821			0.134910	+	0.990858i	$0.456925\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			459.347i			1.44905i	0.689251	+	0.724523i	$0.257940\pi$
$317$			459.347i			1.44905i	−0.689251	+	0.724523i	$0.742060\pi$
$318$		0			0
$319$	−2.69325	+	1.55495i	−0.00844280	+	0.00487445i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	10.3460	+	573.949i	0.0320309	+	1.77693i
$324$		0			0
$325$	188.535	−	108.851i	0.580108	−	0.334926i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	55.5220	+	96.1670i	0.168760	+	0.292301i
$330$		0			0
$331$	33.9831	−	19.6202i	0.102668	−	0.0592755i	−0.447787	−	0.894140i	$-0.647788\pi$
$331$	33.9831	−	19.6202i	0.102668	−	0.0592755i	0.550455	+	0.834865i	$0.314454\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−212.489	+	122.680i	−0.634295	+	0.366210i
$336$		0			0
$337$		−	531.335i		−	1.57666i	−0.615252	−	0.788330i	$-0.710946\pi$
$337$		−	531.335i		−	1.57666i	0.615252	−	0.788330i	$-0.289054\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	0.209817i		−	0.000615298i
$342$		0			0
$343$	−1216.59			−3.54690
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−424.279			−1.22271			−0.611353	−	0.791358i	$-0.709375\pi$
$347$	−424.279			−1.22271			−0.611353	+	0.791358i	$0.709375\pi$
$348$		0			0
$349$	328.765	+	569.438i	0.942020	+	1.63163i	0.761611	+	0.648034i	$0.224409\pi$
$349$	328.765	+	569.438i	0.942020	+	1.63163i	0.180409	+	0.983592i	$0.442258\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−10.2091	−	17.6827i	−0.0289210	−	0.0500926i	0.851203	−	0.524837i	$-0.175874\pi$
$353$	−10.2091	−	17.6827i	−0.0289210	−	0.0500926i	−0.880124	+	0.474745i	$0.842540\pi$
$354$		0			0
$355$	−208.724	+	120.507i	−0.587955	+	0.339456i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	177.673	+	307.739i	0.494911	+	0.857211i	0.999983	−	0.00586638i	$-0.00186734\pi$
$359$	177.673	+	307.739i	0.494911	+	0.857211i	−0.505072	+	0.863077i	$0.668534\pi$
$360$		0			0
$361$	191.650	+	305.927i	0.530887	+	0.847443i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−115.405	−	199.887i	−0.316178	−	0.547637i
$366$		0			0
$367$	−540.288			−1.47217			−0.736087	−	0.676887i	$-0.763329\pi$
$367$	−540.288			−1.47217			−0.736087	+	0.676887i	$0.763329\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	614.379i		−	1.65601i
$372$		0			0
$373$	−317.781	−	183.471i	−0.851961	−	0.491880i	0.00935134	−	0.999956i	$-0.497023\pi$
$373$	−317.781	−	183.471i	−0.851961	−	0.491880i	−0.861312	+	0.508077i	$0.830357\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	525.796	+	910.705i	1.39468	+	2.41566i
$378$		0			0
$379$		−	495.103i		−	1.30634i	−0.757211	−	0.653170i	$-0.773439\pi$
$379$		−	495.103i		−	1.30634i	0.757211	−	0.653170i	$-0.226561\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			602.533i			1.57319i	0.617467	+	0.786597i	$0.288159\pi$
$383$			602.533i			1.57319i	−0.617467	+	0.786597i	$0.711841\pi$
$384$		0			0
$385$	2.48806	−	4.30944i	0.00646249	−	0.0111934i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	591.183			1.51975			0.759876	−	0.650068i	$-0.225260\pi$
$389$	591.183			1.51975			0.759876	+	0.650068i	$0.225260\pi$
$390$		0			0
$391$	−25.8483	+	44.7705i	−0.0661081	+	0.114503i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	129.447	+	74.7363i	0.327714	+	0.189206i
$396$		0			0
$397$	111.011	+	192.276i	0.279624	+	0.484322i	0.971291	−	0.237894i	$-0.0764570\pi$
$397$	111.011	+	192.276i	0.279624	+	0.484322i	−0.691668	+	0.722216i	$0.743124\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	293.337i		−	0.731513i	−0.930711	−	0.365757i	$-0.880810\pi$
$401$		−	293.337i		−	0.731513i	0.930711	−	0.365757i	$-0.119190\pi$
$402$		0			0
$403$	−70.9481			−0.176050
$404$		0			0
$405$		0			0
$406$		0			0
$407$	0.997614	−	0.575973i	0.00245114	−	0.00141517i
$408$		0			0
$409$	−49.7824	+	28.7419i	−0.121717	+	0.0702735i	−0.559623	−	0.828748i	$-0.689054\pi$
$409$	−49.7824	+	28.7419i	−0.121717	+	0.0702735i	0.437905	+	0.899021i	$0.355721\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	549.925	−	317.499i	1.33154	−	0.768763i
$414$		0			0
$415$	107.117	+	185.533i	0.258114	+	0.447067i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−198.879	+	344.468i	−0.474650	+	0.822119i	−0.999579	−	0.0290278i	$-0.990759\pi$
$419$	−198.879	+	344.468i	−0.474650	+	0.822119i	0.524928	+	0.851147i	$0.324092\pi$
$420$		0			0
$421$	−121.075	−	69.9029i	−0.287590	−	0.166040i	0.349265	−	0.937024i	$-0.386431\pi$
$421$	−121.075	−	69.9029i	−0.287590	−	0.166040i	−0.636854	+	0.770984i	$0.719765\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−159.341	−	275.986i	−0.374919	−	0.649379i
$426$		0			0
$427$	526.130	+	911.283i	1.23215	+	2.13415i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−644.642	−	372.184i	−1.49569	−	0.863536i	−0.495701	−	0.868493i	$-0.665089\pi$
$431$	−644.642	−	372.184i	−1.49569	−	0.863536i	−0.999988	+	0.00495690i	$0.998422\pi$
$432$		0			0
$433$	475.819	+	274.714i	1.09889	+	0.634444i	0.935929	−	0.352188i	$-0.114562\pi$
$433$	475.819	+	274.714i	1.09889	+	0.634444i	0.162961	+	0.986633i	$0.447896\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.585940	+	32.5053i	0.00134082	+	0.0743829i
$438$		0			0
$439$	−631.963	−	364.864i	−1.43955	−	0.831125i	−0.441733	−	0.897147i	$-0.645636\pi$
$439$	−631.963	−	364.864i	−1.43955	−	0.831125i	−0.997818	+	0.0660215i	$0.978969\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	119.035			0.268703			0.134351	−	0.990934i	$-0.457105\pi$
$443$	119.035			0.268703			0.134351	+	0.990934i	$0.457105\pi$
$444$		0			0
$445$	−214.024	−	123.567i	−0.480952	−	0.277678i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	141.672i		−	0.315529i	−0.987477	−	0.157764i	$-0.949571\pi$
$449$		−	141.672i		−	0.315529i	0.987477	−	0.157764i	$-0.0504286\pi$
$450$		0			0
$451$	2.28531	−	1.31943i	0.00506722	−	0.00292556i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1457.21	−	841.320i	−3.20266	−	1.84906i
$456$		0			0
$457$	18.8326	+	32.6191i	0.0412093	+	0.0713766i	0.885894	−	0.463887i	$-0.153546\pi$
$457$	18.8326	+	32.6191i	0.0412093	+	0.0713766i	−0.844685	+	0.535264i	$0.820212\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−247.430	+	428.561i	−0.536724	+	0.929633i	0.462354	+	0.886695i	$0.347005\pi$
$461$	−247.430	+	428.561i	−0.536724	+	0.929633i	−0.999078	+	0.0429373i	$0.986328\pi$
$462$		0			0
$463$	265.993	−	460.714i	0.574500	−	0.995063i	−0.421596	−	0.906784i	$-0.638530\pi$
$463$	265.993	−	460.714i	0.574500	−	0.995063i	0.996096	−	0.0882791i	$-0.0281367\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	472.877			1.01258			0.506292	−	0.862362i	$-0.331016\pi$
$467$	472.877			1.01258			0.506292	+	0.862362i	$0.331016\pi$
$468$		0			0
$469$	487.324	+	281.356i	1.03907	+	0.599907i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2.51116			0.00530901
$474$		0			0
$475$	−175.338	−	97.0607i	−0.369133	−	0.204338i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−534.382			−1.11562			−0.557810	−	0.829969i	$-0.688358\pi$
$479$	−534.382			−1.11562			−0.557810	+	0.829969i	$0.688358\pi$
$480$		0			0
$481$	−194.761	−	337.337i	−0.404909	−	0.701324i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	655.042	+	378.189i	1.35060	+	0.779771i
$486$		0			0
$487$		−	519.550i		−	1.06684i	−0.845851	−	0.533419i	$-0.820907\pi$
$487$		−	519.550i		−	1.06684i	0.845851	−	0.533419i	$-0.179093\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	329.798			0.671685			0.335843	−	0.941918i	$-0.390979\pi$
$491$	329.798			0.671685			0.335843	+	0.941918i	$0.390979\pi$
$492$		0			0
$493$	1333.13	−	769.682i	2.70412	−	1.56122i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	478.689	+	276.371i	0.963158	+	0.556079i
$498$		0			0
$499$	502.951			1.00792			0.503959	−	0.863727i	$-0.331876\pi$
$499$	502.951			1.00792			0.503959	+	0.863727i	$0.331876\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−182.006	+	315.244i	−0.361842	+	0.626729i	−0.988264	−	0.152755i	$-0.951185\pi$
$503$	−182.006	+	315.244i	−0.361842	+	0.626729i	0.626422	+	0.779484i	$0.284519\pi$
$504$		0			0
$505$	−1099.62			−2.17746
$506$		0			0
$507$		0			0
$508$		0			0
$509$	600.445	−	346.667i	1.17966	−	0.681075i	0.223721	−	0.974653i	$-0.428179\pi$
$509$	600.445	−	346.667i	1.17966	−	0.681075i	0.955935	+	0.293578i	$0.0948461\pi$
$510$		0			0
$511$	−264.671	+	458.424i	−0.517947	+	0.897111i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−869.046	−	501.744i	−1.68747	−	0.974260i
$516$		0			0
$517$	0.247840	+	0.429272i	0.000479381	+	0.000830313i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	456.825i		−	0.876823i	−0.898774	−	0.438411i	$-0.855541\pi$
$521$		−	456.825i		−	0.876823i	0.898774	−	0.438411i	$-0.144459\pi$
$522$		0			0
$523$	−177.780	−	102.641i	−0.339924	−	0.196255i	0.320315	−	0.947311i	$-0.396211\pi$
$523$	−177.780	−	102.641i	−0.339924	−	0.196255i	−0.660238	+	0.751056i	$0.729545\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			103.857i			0.197072i
$528$		0			0
$529$	263.036	−	455.592i	0.497233	−	0.861232i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−446.156	−	772.764i	−0.837065	−	1.44984i
$534$		0			0
$535$			503.836i			0.941750i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−8.42145			−0.0156242
$540$		0			0
$541$	101.226	−	175.329i	0.187109	−	0.324082i	−0.757176	−	0.653211i	$-0.773422\pi$
$541$	101.226	−	175.329i	0.187109	−	0.324082i	0.944285	+	0.329128i	$0.106755\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	573.279	−	330.983i	1.05189	−	0.607308i
$546$		0			0
$547$		−	36.2433i		−	0.0662583i	−0.999451	−	0.0331291i	$-0.989453\pi$
$547$		−	36.2433i		−	0.0662583i	0.999451	−	0.0331291i	$-0.0105473\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	468.845	−	846.958i	0.850898	−	1.53713i
$552$		0			0
$553$		−	342.802i		−	0.619895i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−220.256	+	381.495i	−0.395433	+	0.684910i	−0.993156	−	0.116793i	$-0.962739\pi$
$557$	−220.256	+	381.495i	−0.395433	+	0.684910i	0.597724	+	0.801702i	$0.296072\pi$
$558$		0			0
$559$		−	849.132i		−	1.51902i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−489.577	−	282.657i	−0.869586	−	0.502056i	−0.00237548	−	0.999997i	$-0.500756\pi$
$563$	−489.577	−	282.657i	−0.869586	−	0.502056i	−0.867211	+	0.497941i	$0.834089\pi$
$564$		0			0
$565$	276.533	+	159.657i	0.489440	+	0.282578i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−457.777	+	264.298i	−0.804529	+	0.464495i	−0.845052	−	0.534684i	$-0.820431\pi$
$569$	−457.777	+	264.298i	−0.804529	+	0.464495i	0.0405235	+	0.999179i	$0.487097\pi$
$570$		0			0
$571$	−254.349	+	440.545i	−0.445444	+	0.771532i	−0.998083	−	0.0618886i	$-0.980288\pi$
$571$	−254.349	+	440.545i	−0.445444	+	0.771532i	0.552639	+	0.833421i	$0.313621\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−9.02418	−	15.6303i	−0.0156942	−	0.0271832i
$576$		0			0
$577$	696.388			1.20691			0.603456	−	0.797396i	$-0.293790\pi$
$577$	696.388			1.20691			0.603456	+	0.797396i	$0.293790\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	245.664	−	425.503i	0.422830	−	0.732362i
$582$		0			0
$583$		−	2.74247i		−	0.00470407i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−511.342	+	885.670i	−0.871110	+	1.50881i	−0.0102613	+	0.999947i	$0.503266\pi$
$587$	−511.342	+	885.670i	−0.871110	+	1.50881i	−0.860849	+	0.508860i	$0.830067\pi$
$588$		0			0
$589$	33.6706	+	55.9648i	0.0571656	+	0.0950167i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	10.7225	−	18.5719i	0.0180817	−	0.0313185i	−0.856843	−	0.515578i	$-0.827577\pi$
$593$	10.7225	−	18.5719i	0.0180817	−	0.0313185i	0.874925	+	0.484259i	$0.160911\pi$
$594$		0			0
$595$	−1231.56	+	2133.12i	−2.06985	+	3.58508i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−406.542	+	234.717i	−0.678702	+	0.391849i	−0.799366	−	0.600845i	$-0.794831\pi$
$599$	−406.542	+	234.717i	−0.678702	+	0.391849i	0.120664	+	0.992693i	$0.461498\pi$
$600$		0			0
$601$	−446.289	+	257.665i	−0.742577	+	0.428727i	−0.823006	−	0.568033i	$-0.807705\pi$
$601$	−446.289	+	257.665i	−0.742577	+	0.428727i	0.0804283	+	0.996760i	$0.474371\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−360.702	+	624.755i	−0.596202	+	1.03265i
$606$		0			0
$607$	−676.578	−	390.622i	−1.11463	−	0.643529i	−0.174602	−	0.984639i	$-0.555864\pi$
$607$	−676.578	−	390.622i	−1.11463	−	0.643529i	−0.940023	+	0.341110i	$0.889197\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	145.155	−	83.8055i	0.237570	−	0.137161i
$612$		0			0
$613$	−442.760	−	766.882i	−0.722283	−	1.25103i	−0.960082	−	0.279717i	$-0.909759\pi$
$613$	−442.760	−	766.882i	−0.722283	−	1.25103i	0.237799	−	0.971314i	$-0.423574\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−281.038	−	486.773i	−0.455492	−	0.788935i	0.543225	−	0.839587i	$-0.317203\pi$
$617$	−281.038	−	486.773i	−0.455492	−	0.788935i	−0.998716	+	0.0506526i	$0.983870\pi$
$618$		0			0
$619$	−276.310	−	478.582i	−0.446381	−	0.773154i	0.551767	−	0.833998i	$-0.313954\pi$
$619$	−276.310	−	478.582i	−0.446381	−	0.773154i	−0.998147	+	0.0608448i	$0.980621\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			566.777i			0.909755i
$624$		0			0
$625$	−777.439			−1.24390
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−493.808	+	285.100i	−0.785068	+	0.453259i
$630$		0			0
$631$	365.132	−	632.427i	0.578656	−	1.00226i	−0.416978	−	0.908916i	$-0.636911\pi$
$631$	365.132	−	632.427i	0.578656	−	1.00226i	0.995634	−	0.0933444i	$-0.0297558\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	465.567	+	268.795i	0.733176	+	0.423299i
$636$		0			0
$637$			2847.66i			4.47042i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	604.563	+	349.045i	0.943156	+	0.544531i	0.890948	−	0.454105i	$-0.150041\pi$
$641$	604.563	+	349.045i	0.943156	+	0.544531i	0.0522078	+	0.998636i	$0.483374\pi$
$642$		0			0
$643$	−389.541			−0.605817			−0.302909	−	0.953020i	$-0.597958\pi$
$643$	−389.541			−0.605817			−0.302909	+	0.953020i	$0.597958\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	535.718			0.828002			0.414001	−	0.910276i	$-0.364131\pi$
$647$	535.718			0.828002			0.414001	+	0.910276i	$0.364131\pi$
$648$		0			0
$649$	2.45476	−	1.41726i	0.00378238	−	0.00218376i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−387.113	+	670.499i	−0.592822	+	1.02680i	0.401028	+	0.916066i	$0.368653\pi$
$653$	−387.113	+	670.499i	−0.592822	+	1.02680i	−0.993850	+	0.110733i	$0.964680\pi$
$654$		0			0
$655$	−176.829			−0.269969
$656$		0			0
$657$		0			0
$658$		0			0
$659$			137.735i			0.209007i	0.994525	+	0.104503i	$0.0333253\pi$
$659$			137.735i			0.209007i	−0.994525	+	0.104503i	$0.966675\pi$
$660$		0			0
$661$	955.488	−	551.651i	1.44552	−	0.834571i	0.447309	−	0.894380i	$-0.352382\pi$
$661$	955.488	−	551.651i	1.44552	−	0.834571i	0.998210	+	0.0598090i	$0.0190492\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	27.9176	+	1548.74i	0.0419813	+	2.32893i
$666$		0			0
$667$	75.5011	−	43.5906i	0.113195	−	0.0653532i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	2.34855	+	4.06780i	0.00350007	+	0.00606230i
$672$		0			0
$673$	−793.377	+	458.056i	−1.17887	+	0.680618i	−0.955752	−	0.294174i	$-0.904955\pi$
$673$	−793.377	+	458.056i	−1.17887	+	0.680618i	−0.223114	+	0.974792i	$0.571622\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	518.834	−	299.549i	0.766372	−	0.442465i	−0.0652071	−	0.997872i	$-0.520771\pi$
$677$	518.834	−	299.549i	0.766372	−	0.442465i	0.831579	+	0.555407i	$0.187437\pi$
$678$		0			0
$679$		−	1734.68i		−	2.55476i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			884.634i			1.29522i	0.761973	+	0.647609i	$0.224231\pi$
$683$			884.634i			1.29522i	−0.761973	+	0.647609i	$0.775769\pi$
$684$		0			0
$685$	1077.97			1.57367
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−927.349			−1.34594
$690$		0			0
$691$	341.903	+	592.194i	0.494795	+	0.857010i	0.999982	−	0.00600019i	$-0.00190993\pi$
$691$	341.903	+	592.194i	0.494795	+	0.857010i	−0.505187	+	0.863010i	$0.668577\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	259.859	+	450.090i	0.373898	+	0.647611i
$696$		0			0
$697$	−1131.20	+	653.101i	−1.62296	+	0.937018i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	424.494	+	735.244i	0.605554	+	1.04885i	0.991964	+	0.126523i	$0.0403819\pi$
$701$	424.494	+	735.244i	0.605554	+	1.04885i	−0.386409	+	0.922327i	$0.626285\pi$
$702$		0			0
$703$	−173.666	+	313.724i	−0.247035	+	0.446264i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1260.94	+	2184.01i	1.78350	+	3.08912i
$708$		0			0
$709$	−1310.98			−1.84905			−0.924526	−	0.381119i	$-0.875539\pi$
$709$	−1310.98			−1.84905			−0.924526	+	0.381119i	$0.875539\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			5.88188i			0.00824949i
$714$		0			0
$715$	−6.50471	−	3.75550i	−0.00909750	−	0.00525245i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−266.689	−	461.919i	−0.370916	−	0.642446i	0.618790	−	0.785556i	$-0.287623\pi$
$719$	−266.689	−	461.919i	−0.370916	−	0.642446i	−0.989707	+	0.143110i	$0.954290\pi$
$720$		0			0
$721$			2301.41i			3.19197i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			537.425i			0.741276i
$726$		0			0
$727$	260.552	−	451.289i	0.358393	−	0.620755i	−0.629299	−	0.777163i	$-0.716658\pi$
$727$	260.552	−	451.289i	0.358393	−	0.620755i	0.987693	+	0.156408i	$0.0499914\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1243.00			−1.70040
$732$		0			0
$733$	−50.8982	+	88.1583i	−0.0694382	+	0.120271i	−0.898654	−	0.438658i	$-0.855454\pi$
$733$	−50.8982	+	88.1583i	−0.0694382	+	0.120271i	0.829216	+	0.558928i	$0.188787\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2.17532	+	1.25592i	0.00295159	+	0.00170410i
$738$		0			0
$739$	−190.049	−	329.174i	−0.257170	−	0.445432i	0.708313	−	0.705899i	$-0.249457\pi$
$739$	−190.049	−	329.174i	−0.257170	−	0.445432i	−0.965483	+	0.260467i	$0.916123\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	549.604i		−	0.739709i	−0.929090	−	0.369855i	$-0.879407\pi$
$743$		−	549.604i		−	0.739709i	0.929090	−	0.369855i	$-0.120593\pi$
$744$		0			0
$745$	−278.967			−0.374453
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1000.70	−	577.752i	1.33604	−	0.771364i
$750$		0			0
$751$	−1102.86	+	636.737i	−1.46852	+	0.847852i	−0.999378	−	0.0352683i	$-0.988771\pi$
$751$	−1102.86	+	636.737i	−1.46852	+	0.847852i	−0.469146	+	0.883121i	$0.655438\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−929.187	+	536.466i	−1.23071	+	0.710551i
$756$		0			0
$757$	−265.936	−	460.615i	−0.351303	−	0.608474i	0.635175	−	0.772368i	$-0.280928\pi$
$757$	−265.936	−	460.615i	−0.351303	−	0.608474i	−0.986478	+	0.163894i	$0.947595\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−323.880	+	560.976i	−0.425598	+	0.737157i	−0.996476	−	0.0838780i	$-0.973269\pi$
$761$	−323.880	+	560.976i	−0.425598	+	0.737157i	0.570878	+	0.821035i	$0.306603\pi$
$762$		0			0
$763$	−1314.76	−	759.079i	−1.72315	−	0.994861i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−479.237	−	830.062i	−0.624819	−	1.08222i
$768$		0			0
$769$	72.8899	+	126.249i	0.0947853	+	0.164173i	0.909519	−	0.415662i	$-0.136450\pi$
$769$	72.8899	+	126.249i	0.0947853	+	0.164173i	−0.814734	+	0.579835i	$0.803117\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−499.283	−	288.261i	−0.645903	−	0.372912i	0.140982	−	0.990012i	$-0.454974\pi$
$773$	−499.283	−	288.261i	−0.645903	−	0.372912i	−0.786885	+	0.617100i	$0.788307\pi$
$774$		0			0
$775$	−31.4009	−	18.1293i	−0.0405173	−	0.0233927i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−397.830	+	718.672i	−0.510694	+	0.922557i
$780$		0			0
$781$	2.13678	+	1.23367i	0.00273596	+	0.00157960i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	475.525			0.605765
$786$		0			0
$787$	622.039	+	359.134i	0.790392	+	0.456333i	0.840101	−	0.542431i	$-0.182496\pi$
$787$	622.039	+	359.134i	0.790392	+	0.456333i	−0.0497083	+	0.998764i	$0.515829\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	732.316i		−	0.925810i
$792$		0			0
$793$	1375.50	−	794.145i	1.73455	−	1.00144i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1164.66	−	672.418i	−1.46131	−	0.843687i	−0.462236	−	0.886757i	$-0.652953\pi$
$797$	−1164.66	−	672.418i	−1.46131	−	0.843687i	−0.999072	+	0.0430699i	$0.986286\pi$
$798$		0			0
$799$	−122.678	−	212.485i	−0.153539	−	0.265938i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.18144	+	2.04632i	−0.00147129	+	0.00254834i
$804$		0			0
$805$	−69.7488	+	120.809i	−0.0866445	+	0.150073i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−280.641			−0.346899			−0.173449	−	0.984843i	$-0.555491\pi$
$809$	−280.641			−0.346899			−0.173449	+	0.984843i	$0.555491\pi$
$810$		0			0
$811$	−107.506	−	62.0685i	−0.132559	−	0.0765333i	0.432254	−	0.901752i	$-0.357718\pi$
$811$	−107.506	−	62.0685i	−0.132559	−	0.0765333i	−0.564813	+	0.825219i	$0.691052\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	277.140			0.340049
$816$		0			0
$817$	−669.807	+	402.981i	−0.819837	+	0.493245i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−407.988			−0.496940			−0.248470	−	0.968640i	$-0.579928\pi$
$821$	−407.988			−0.496940			−0.248470	+	0.968640i	$0.579928\pi$
$822$		0			0
$823$	−672.867	−	1165.44i	−0.817578	−	1.41609i	−0.907462	−	0.420134i	$-0.861983\pi$
$823$	−672.867	−	1165.44i	−0.817578	−	1.41609i	0.0898838	−	0.995952i	$-0.471350\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	925.108	+	534.111i	1.11863	+	0.645842i	0.941051	−	0.338264i	$-0.109840\pi$
$827$	925.108	+	534.111i	1.11863	+	0.645842i	0.177580	+	0.984106i	$0.443173\pi$
$828$		0			0
$829$		−	463.384i		−	0.558968i	−0.960150	−	0.279484i	$-0.909837\pi$
$829$		−	463.384i		−	0.558968i	0.960150	−	0.279484i	$-0.0901634\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	4168.52			5.00423
$834$		0			0
$835$	−476.918	+	275.349i	−0.571159	+	0.329759i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−104.327	−	60.2334i	−0.124347	−	0.0717919i	0.436536	−	0.899687i	$-0.356205\pi$
$839$	−104.327	−	60.2334i	−0.124347	−	0.0717919i	−0.560883	+	0.827895i	$0.689538\pi$
$840$		0			0
$841$	−1754.99			−2.08679
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−766.091	+	1326.91i	−0.906616	+	1.57031i
$846$		0			0
$847$	1654.48			1.95334
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−27.9666	+	16.1465i	−0.0328632	+	0.0189736i
$852$		0			0
$853$	−411.634	+	712.971i	−0.482572	+	0.835840i	−0.999800	−	0.0200083i	$-0.993631\pi$
$853$	−411.634	+	712.971i	−0.482572	+	0.835840i	0.517228	+	0.855848i	$0.326964\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−242.598	−	140.064i	−0.283078	−	0.163435i	0.351738	−	0.936098i	$-0.385591\pi$
$857$	−242.598	−	140.064i	−0.283078	−	0.163435i	−0.634816	+	0.772663i	$0.718924\pi$
$858$		0			0
$859$	306.566	+	530.988i	0.356887	+	0.618147i	0.987439	−	0.158000i	$-0.0505046\pi$
$859$	306.566	+	530.988i	0.356887	+	0.618147i	−0.630552	+	0.776147i	$0.717171\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			825.559i			0.956615i	0.878192	+	0.478308i	$0.158750\pi$
$863$			825.559i			0.956615i	−0.878192	+	0.478308i	$0.841250\pi$
$864$		0			0
$865$	−451.297	−	260.557i	−0.521731	−	0.301222i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	1.53020i		−	0.00176088i
$870$		0			0
$871$	424.682	−	735.571i	0.487580	−	0.844513i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	589.110	+	1020.37i	0.673269	+	1.16614i
$876$		0			0
$877$			602.546i			0.687053i	0.939143	+	0.343527i	$0.111622\pi$
$877$			602.546i			0.687053i	−0.939143	+	0.343527i	$0.888378\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	649.481			0.737209			0.368605	−	0.929586i	$-0.379836\pi$
$881$	649.481			0.737209			0.368605	+	0.929586i	$0.379836\pi$
$882$		0			0
$883$	−24.3325	+	42.1452i	−0.0275567	+	0.0477295i	−0.879475	−	0.475945i	$-0.842106\pi$
$883$	−24.3325	+	42.1452i	−0.0275567	+	0.0477295i	0.851918	+	0.523675i	$0.175439\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1307.46	+	754.862i	−1.47402	+	0.851028i	−0.999572	−	0.0292530i	$-0.990687\pi$
$887$	−1307.46	+	754.862i	−1.47402	+	0.851028i	−0.474452	+	0.880281i	$0.657354\pi$
$888$		0			0
$889$		−	1232.91i		−	1.38686i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−134.995	−	74.7281i	−0.151170	−	0.0836821i
$894$		0			0
$895$			925.409i			1.03398i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	87.5723	−	151.680i	0.0974108	−	0.168720i
$900$		0			0
$901$			1357.49i			1.50665i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−349.117	−	201.563i	−0.385764	−	0.222721i
$906$		0			0
$907$	−220.691	−	127.416i	−0.243320	−	0.140481i	0.373382	−	0.927678i	$-0.378198\pi$
$907$	−220.691	−	127.416i	−0.243320	−	0.140481i	−0.616702	+	0.787197i	$0.711531\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1421.30	−	820.588i	1.56015	−	0.900755i	0.562912	−	0.826517i	$-0.309681\pi$
$911$	1421.30	−	820.588i	1.56015	−	0.900755i	0.997241	−	0.0742381i	$-0.0236525\pi$
$912$		0			0
$913$	1.09660	−	1.89937i	0.00120109	−	0.00208036i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	202.771	+	351.210i	0.221124	+	0.382999i
$918$		0			0
$919$	970.835			1.05640			0.528202	−	0.849119i	$-0.322866\pi$
$919$	970.835			1.05640			0.528202	+	0.849119i	$0.322866\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	417.158	−	722.538i	0.451959	−	0.782815i
$924$		0			0
$925$		−	199.069i		−	0.215210i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−188.836	+	327.073i	−0.203268	+	0.352070i	−0.949579	−	0.313527i	$-0.898490\pi$
$929$	−188.836	+	327.073i	−0.203268	+	0.352070i	0.746312	+	0.665597i	$0.231823\pi$
$930$		0			0
$931$	2246.27	−	1351.44i	2.41275	−	1.45160i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−5.49746	+	9.52188i	−0.00587963	+	0.0101838i
$936$		0			0
$937$	472.636	−	818.629i	0.504414	−	0.873671i	−0.495573	−	0.868566i	$-0.665042\pi$
$937$	472.636	−	818.629i	0.504414	−	0.873671i	0.999987	−	0.00510436i	$-0.00162477\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−270.948	+	156.432i	−0.287936	+	0.166240i	−0.637011	−	0.770855i	$-0.719829\pi$
$941$	−270.948	+	156.432i	−0.287936	+	0.166240i	0.349075	+	0.937095i	$0.386496\pi$
$942$		0			0
$943$	−64.0653	+	36.9881i	−0.0679377	+	0.0392239i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	141.229	−	244.616i	0.149133	−	0.258306i	−0.781774	−	0.623562i	$-0.785685\pi$
$947$	141.229	−	244.616i	0.149133	−	0.258306i	0.930907	+	0.365256i	$0.119018\pi$
$948$		0			0
$949$	691.949	+	399.497i	0.729135	+	0.420966i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1014.92	−	585.963i	1.06497	−	0.614862i	0.138169	−	0.990409i	$-0.455878\pi$
$953$	1014.92	−	585.963i	1.06497	−	0.614862i	0.926803	+	0.375547i	$0.122545\pi$
$954$		0			0
$955$	−440.163	−	762.384i	−0.460903	−	0.798308i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1236.11	−	2141.01i	−1.28896	−	2.23254i
$960$		0			0
$961$	−474.592	−	822.017i	−0.493852	−	0.855377i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			627.833i			0.650604i
$966$		0			0
$967$	−1057.37			−1.09345			−0.546725	−	0.837312i	$-0.684126\pi$
$967$	−1057.37			−1.09345			−0.546725	+	0.837312i	$0.684126\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	402.603	−	232.443i	0.414628	−	0.239385i	−0.278148	−	0.960538i	$-0.589721\pi$
$971$	402.603	−	232.443i	0.414628	−	0.239385i	0.692776	+	0.721153i	$0.256387\pi$
$972$		0			0
$973$	595.964	−	1032.24i	0.612502	−	1.06088i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−374.179	−	216.032i	−0.382988	−	0.221118i	0.296130	−	0.955148i	$-0.404304\pi$
$977$	−374.179	−	216.032i	−0.382988	−	0.221118i	−0.679117	+	0.734030i	$0.737637\pi$
$978$		0			0
$979$			2.52999i			0.00258426i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−556.873	−	321.511i	−0.566504	−	0.327071i	0.189248	−	0.981929i	$-0.439395\pi$
$983$	−556.873	−	321.511i	−0.566504	−	0.327071i	−0.755752	+	0.654858i	$0.772728\pi$
$984$		0			0
$985$	−1727.89			−1.75421
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−70.3965			−0.0711795
$990$		0			0
$991$	138.794	−	80.1330i	0.140055	−	0.0808607i	−0.428335	−	0.903620i	$-0.640900\pi$
$991$	138.794	−	80.1330i	0.140055	−	0.0808607i	0.568390	+	0.822759i	$0.307566\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1101.91	+	1908.57i	−1.10745	+	1.91816i
$996$		0			0
$997$	234.440			0.235146			0.117573	−	0.993064i	$-0.462489\pi$
$997$	234.440			0.235146			0.117573	+	0.993064i	$0.462489\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2052.3.bl.a.145.8		80
3.2	odd	2		684.3.bl.a.373.2	yes	80
9.2	odd	6		684.3.s.a.601.13	yes	80
9.7	even	3		2052.3.s.a.829.33		80
19.8	odd	6		2052.3.s.a.901.33		80
57.8	even	6		684.3.s.a.445.13	✓	80
171.65	even	6		684.3.bl.a.673.2	yes	80
171.160	odd	6	inner	2052.3.bl.a.1585.8		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.s.a.445.13	✓	80	57.8	even	6
684.3.s.a.601.13	yes	80	9.2	odd	6
684.3.bl.a.373.2	yes	80	3.2	odd	2
684.3.bl.a.673.2	yes	80	171.65	even	6
2052.3.s.a.829.33		80	9.7	even	3
2052.3.s.a.901.33		80	19.8	odd	6
2052.3.bl.a.145.8		80	1.1	even	1	trivial
2052.3.bl.a.1585.8		80	171.160	odd	6	inner

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.bl (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-5.96221 q^{5} +(6.83689 + 11.8418i) q^{7} +O(q^{10})\)	\(q-5.96221 q^{5} +(6.83689 + 11.8418i) q^{7} +(0.0305186 + 0.0528598i) q^{11} +(17.8742 - 10.3197i) q^{13} +(-15.1064 - 26.1650i) q^{17} +(-16.6230 - 9.20190i) q^{19} +(-0.855542 - 1.48184i) q^{23} +10.5479 q^{25} +50.9509i q^{29} +(-2.97698 - 1.71876i) q^{31} +(-40.7629 - 70.6035i) q^{35} -18.8728i q^{37} -43.2335i q^{41} +(20.5707 - 35.6295i) q^{43} +8.12095 q^{47} +(-68.9861 + 119.487i) q^{49} +(-38.9115 - 22.4656i) q^{53} +(-0.181958 - 0.315161i) q^{55} -46.4391i q^{59} +76.9545 q^{61} +(-106.570 + 61.5280i) q^{65} +(35.6393 - 20.5764i) q^{67} +(35.0078 - 20.2118i) q^{71} +(19.3561 + 33.5258i) q^{73} +(-0.417305 + 0.722793i) q^{77} +(-21.7113 - 12.5350i) q^{79} +(-17.9661 - 31.1181i) q^{83} +(90.0673 + 156.001i) q^{85} +(35.8967 + 20.7250i) q^{89} +(244.408 + 141.109i) q^{91} +(99.1099 + 54.8636i) q^{95} +(-109.866 - 63.4310i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - q^{7}+O(q^{10}) \) 80 * q - q^7	\( 80 q - q^{7} + 6 q^{11} - 15 q^{13} + 21 q^{17} - 20 q^{19} - 24 q^{23} + 400 q^{25} + 24 q^{31} + 54 q^{35} + 76 q^{43} - 24 q^{47} - 267 q^{49} + 36 q^{53} + 14 q^{61} - 288 q^{65} - 21 q^{67} + 81 q^{71} + 55 q^{73} - 30 q^{77} - 51 q^{79} + 93 q^{83} - 216 q^{89} + 96 q^{91} + 432 q^{95} + 90 q^{97}+O(q^{100}) \) 80 * q - q^7 + 6 * q^11 - 15 * q^13 + 21 * q^17 - 20 * q^19 - 24 * q^23 + 400 * q^25 + 24 * q^31 + 54 * q^35 + 76 * q^43 - 24 * q^47 - 267 * q^49 + 36 * q^53 + 14 * q^61 - 288 * q^65 - 21 * q^67 + 81 * q^71 + 55 * q^73 - 30 * q^77 - 51 * q^79 + 93 * q^83 - 216 * q^89 + 96 * q^91 + 432 * q^95 + 90 * q^97

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