LMFDB - Embedded newform 2052.3.bl.a.1585.2

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			1585.2
Character	$\chi$	$=$	2052.1585
Dual form			2052.3.bl.a.145.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.37687 q^{5} +(-4.50755 + 7.80731i) q^{7} +O(q^{10})$	$q-8.37687 q^{5} +(-4.50755 + 7.80731i) q^{7} +(1.48869 - 2.57849i) q^{11} +(-4.82268 - 2.78438i) q^{13} +(13.0392 - 22.5845i) q^{17} +(-10.0502 - 16.1243i) q^{19} +(9.46832 - 16.3996i) q^{23} +45.1719 q^{25} -15.7214i q^{29} +(16.0762 - 9.28158i) q^{31} +(37.7592 - 65.4008i) q^{35} -32.1993i q^{37} +72.6702i q^{41} +(31.5902 + 54.7158i) q^{43} -40.7377 q^{47} +(-16.1361 - 27.9485i) q^{49} +(-73.4198 + 42.3890i) q^{53} +(-12.4706 + 21.5997i) q^{55} -15.5284i q^{59} +16.6558 q^{61} +(40.3990 + 23.3244i) q^{65} +(54.7201 + 31.5927i) q^{67} +(-84.2994 - 48.6703i) q^{71} +(20.4220 - 35.3719i) q^{73} +(13.4207 + 23.2454i) q^{77} +(-93.3133 + 53.8744i) q^{79} +(57.3231 - 99.2866i) q^{83} +(-109.227 + 189.188i) q^{85} +(-2.70957 + 1.56437i) q^{89} +(43.4770 - 25.1015i) q^{91} +(84.1890 + 135.071i) q^{95} +(-104.467 + 60.3140i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/2052\mathbb{Z}\right)^\times$.

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

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
$5$	−8.37687			−1.67537			−0.837687	−	0.546151i	$-0.816093\pi$
$5$	−8.37687			−1.67537			−0.837687	+	0.546151i	$0.816093\pi$
$6$		0			0
$7$	−4.50755	+	7.80731i	−0.643936	+	1.11533i	0.340610	+	0.940205i	$0.389366\pi$
$7$	−4.50755	+	7.80731i	−0.643936	+	1.11533i	−0.984546	+	0.175125i	$0.943967\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.48869	−	2.57849i	0.135336	−	0.234408i	−0.790390	−	0.612604i	$-0.790122\pi$
$11$	1.48869	−	2.57849i	0.135336	−	0.234408i	0.925726	+	0.378196i	$0.123455\pi$
$12$		0			0
$13$	−4.82268	−	2.78438i	−0.370976	−	0.214183i	0.302909	−	0.953019i	$-0.402042\pi$
$13$	−4.82268	−	2.78438i	−0.370976	−	0.214183i	−0.673885	+	0.738837i	$0.735376\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	13.0392	−	22.5845i	0.767010	−	1.32850i	−0.172167	−	0.985068i	$-0.555077\pi$
$17$	13.0392	−	22.5845i	0.767010	−	1.32850i	0.939177	−	0.343433i	$-0.111590\pi$
$18$		0			0
$19$	−10.0502	−	16.1243i	−0.528957	−	0.848649i
$19$	−10.0502	−	16.1243i	−0.528957	−	0.848649i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	9.46832	−	16.3996i	0.411666	−	0.713027i	−0.583406	−	0.812181i	$-0.698280\pi$
$23$	9.46832	−	16.3996i	0.411666	−	0.713027i	0.995072	+	0.0991539i	$0.0316136\pi$
$24$		0			0
$25$	45.1719			1.80688
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	15.7214i		−	0.542119i	−0.962563	−	0.271059i	$-0.912626\pi$
$29$		−	15.7214i		−	0.542119i	0.962563	−	0.271059i	$-0.0873740\pi$
$30$		0			0
$31$	16.0762	−	9.28158i	0.518586	−	0.299406i	−0.217770	−	0.976000i	$-0.569878\pi$
$31$	16.0762	−	9.28158i	0.518586	−	0.299406i	0.736356	+	0.676594i	$0.236545\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	37.7592	−	65.4008i	1.07883	−	1.86859i
$36$		0			0
$37$		−	32.1993i		−	0.870251i	−0.900370	−	0.435126i	$-0.856704\pi$
$37$		−	32.1993i		−	0.870251i	0.900370	−	0.435126i	$-0.143296\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			72.6702i			1.77244i	0.463262	+	0.886222i	$0.346679\pi$
$41$			72.6702i			1.77244i	−0.463262	+	0.886222i	$0.653321\pi$
$42$		0			0
$43$	31.5902	+	54.7158i	0.734655	+	1.27246i	0.954875	+	0.297009i	$0.0959892\pi$
$43$	31.5902	+	54.7158i	0.734655	+	1.27246i	−0.220220	+	0.975450i	$0.570677\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−40.7377			−0.866759			−0.433379	−	0.901212i	$-0.642679\pi$
$47$	−40.7377			−0.866759			−0.433379	+	0.901212i	$0.642679\pi$
$48$		0			0
$49$	−16.1361	−	27.9485i	−0.329307	−	0.570377i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−73.4198	+	42.3890i	−1.38528	+	0.799792i	−0.992779	−	0.119960i	$-0.961723\pi$
$53$	−73.4198	+	42.3890i	−1.38528	+	0.799792i	−0.392501	+	0.919752i	$0.628390\pi$
$54$		0			0
$55$	−12.4706	+	21.5997i	−0.226738	+	0.392721i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	15.5284i		−	0.263193i	−0.991303	−	0.131596i	$-0.957990\pi$
$59$		−	15.5284i		−	0.263193i	0.991303	−	0.131596i	$-0.0420103\pi$
$60$		0			0
$61$	16.6558			0.273046			0.136523	−	0.990637i	$-0.456407\pi$
$61$	16.6558			0.273046			0.136523	+	0.990637i	$0.456407\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	40.3990	+	23.3244i	0.621523	+	0.358836i
$66$		0			0
$67$	54.7201	+	31.5927i	0.816718	+	0.471533i	0.849284	−	0.527937i	$-0.177034\pi$
$67$	54.7201	+	31.5927i	0.816718	+	0.471533i	−0.0325651	+	0.999470i	$0.510368\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−84.2994	−	48.6703i	−1.18732	−	0.685497i	−0.229620	−	0.973280i	$-0.573748\pi$
$71$	−84.2994	−	48.6703i	−1.18732	−	0.685497i	−0.957695	+	0.287784i	$0.907082\pi$
$72$		0			0
$73$	20.4220	−	35.3719i	0.279753	−	0.484547i	−0.691570	−	0.722309i	$-0.743081\pi$
$73$	20.4220	−	35.3719i	0.279753	−	0.484547i	0.971323	+	0.237763i	$0.0764140\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	13.4207	+	23.2454i	0.174295	+	0.301888i
$78$		0			0
$79$	−93.3133	+	53.8744i	−1.18118	+	0.681955i	−0.956287	−	0.292428i	$-0.905537\pi$
$79$	−93.3133	+	53.8744i	−1.18118	+	0.681955i	−0.224893	+	0.974383i	$0.572203\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	57.3231	−	99.2866i	0.690640	−	1.19622i	−0.280988	−	0.959711i	$-0.590662\pi$
$83$	57.3231	−	99.2866i	0.690640	−	1.19622i	0.971628	−	0.236512i	$-0.0760044\pi$
$84$		0			0
$85$	−109.227	+	189.188i	−1.28503	+	2.22574i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.70957	+	1.56437i	−0.0304446	+	0.0175772i	−0.515145	−	0.857103i	$-0.672262\pi$
$89$	−2.70957	+	1.56437i	−0.0304446	+	0.0175772i	0.484700	+	0.874680i	$0.338929\pi$
$90$		0			0
$91$	43.4770	−	25.1015i	0.477769	−	0.275840i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	84.1890	+	135.071i	0.886200	+	1.42180i
$96$		0			0
$97$	−104.467	+	60.3140i	−1.07698	+	0.621794i	−0.930080	−	0.367358i	$-0.880262\pi$
$97$	−104.467	+	60.3140i	−1.07698	+	0.621794i	−0.146898	+	0.989152i	$0.546929\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−17.6335			−0.174589			−0.0872944	−	0.996183i	$-0.527822\pi$
$101$	−17.6335			−0.174589			−0.0872944	+	0.996183i	$0.527822\pi$
$102$		0			0
$103$	−100.731	+	58.1573i	−0.977976	+	0.564634i	−0.901658	−	0.432449i	$-0.857649\pi$
$103$	−100.731	+	58.1573i	−0.977976	+	0.564634i	−0.0763172	+	0.997084i	$0.524316\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	5.14531i		−	0.0480870i	−0.999711	−	0.0240435i	$-0.992346\pi$
$107$		−	5.14531i		−	0.0480870i	0.999711	−	0.0240435i	$-0.00765403\pi$
$108$		0			0
$109$	−9.07991	−	5.24229i	−0.0833019	−	0.0480944i	0.457770	−	0.889070i	$-0.348648\pi$
$109$	−9.07991	−	5.24229i	−0.0833019	−	0.0480944i	−0.541072	+	0.840976i	$0.681981\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−177.083	+	102.239i	−1.56711	+	0.904772i	−0.570607	+	0.821223i	$0.693292\pi$
$113$	−177.083	+	102.239i	−1.56711	+	0.904772i	−0.996504	+	0.0835482i	$0.973375\pi$
$114$		0			0
$115$	−79.3149	+	137.377i	−0.689695	+	1.19459i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	117.550	+	203.602i	0.987811	+	1.71094i
$120$		0			0
$121$	56.0676	+	97.1119i	0.463369	+	0.802578i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−168.978			−1.35182
$126$		0			0
$127$	36.6506	−	21.1602i	0.288587	−	0.166616i	−0.348717	−	0.937228i	$-0.613383\pi$
$127$	36.6506	−	21.1602i	0.288587	−	0.166616i	0.637305	+	0.770612i	$0.280049\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	217.081			1.65710			0.828552	−	0.559912i	$-0.189165\pi$
$131$	217.081			1.65710			0.828552	+	0.559912i	$0.189165\pi$
$132$		0			0
$133$	171.189	−	5.78357i	1.28714	−	0.0434855i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−58.9349			−0.430181			−0.215091	−	0.976594i	$-0.569005\pi$
$137$	−58.9349			−0.430181			−0.215091	+	0.976594i	$0.569005\pi$
$138$		0			0
$139$	−1.08302	+	1.87585i	−0.00779155	+	0.0134954i	−0.869895	−	0.493237i	$-0.835813\pi$
$139$	−1.08302	+	1.87585i	−0.00779155	+	0.0134954i	0.862103	+	0.506732i	$0.169147\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−14.3590	+	8.29016i	−0.100412	+	0.0579731i
$144$		0			0
$145$			131.697i			0.908252i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−142.301			−0.955043			−0.477522	−	0.878620i	$-0.658465\pi$
$149$	−142.301			−0.955043			−0.477522	+	0.878620i	$0.658465\pi$
$150$		0			0
$151$	247.779	+	143.055i	1.64092	+	0.947387i	0.980506	+	0.196487i	$0.0629535\pi$
$151$	247.779	+	143.055i	1.64092	+	0.947387i	0.660416	+	0.750900i	$0.270380\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−134.668	+	77.7506i	−0.868825	+	0.501617i
$156$		0			0
$157$	174.920			1.11414			0.557070	−	0.830465i	$-0.311925\pi$
$157$	174.920			1.11414			0.557070	+	0.830465i	$0.311925\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	85.3579	+	147.844i	0.530173	+	0.918287i
$162$		0			0
$163$	280.563			1.72124			0.860622	−	0.509244i	$-0.170075\pi$
$163$	280.563			1.72124			0.860622	+	0.509244i	$0.170075\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.92518	−	2.84355i	−0.0294921	−	0.0170273i	0.485181	−	0.874413i	$-0.338754\pi$
$167$	−4.92518	−	2.84355i	−0.0294921	−	0.0170273i	−0.514674	+	0.857386i	$0.672087\pi$
$168$		0			0
$169$	−68.9945	−	119.502i	−0.408251	−	0.707112i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−0.896616	+	0.517661i	−0.00518275	+	0.00299226i	−0.502589	−	0.864525i	$-0.667619\pi$
$173$	−0.896616	+	0.517661i	−0.00518275	+	0.00299226i	0.497406	+	0.867518i	$0.334286\pi$
$174$		0			0
$175$	−203.615	+	352.671i	−1.16351	+	2.01527i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			185.084i			1.03399i	0.855988	+	0.516996i	$0.172950\pi$
$179$			185.084i			1.03399i	−0.855988	+	0.516996i	$0.827050\pi$
$180$		0			0
$181$	77.7940	−	44.9144i	0.429801	−	0.248146i	−0.269461	−	0.963011i	$-0.586845\pi$
$181$	77.7940	−	44.9144i	0.429801	−	0.248146i	0.699262	+	0.714866i	$0.253512\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			269.729i			1.45800i
$186$		0			0
$187$	−38.8226	−	67.2427i	−0.207608	−	0.359587i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−107.195	+	185.668i	−0.561233	+	0.972083i	0.436157	+	0.899871i	$0.356339\pi$
$191$	−107.195	+	185.668i	−0.561233	+	0.972083i	−0.997389	+	0.0722127i	$0.976994\pi$
$192$		0			0
$193$			8.48799i			0.0439792i	0.999758	+	0.0219896i	$0.00700008\pi$
$193$			8.48799i			0.0439792i	−0.999758	+	0.0219896i	$0.993000\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	182.684			0.927331			0.463666	−	0.886010i	$-0.346534\pi$
$197$	182.684			0.927331			0.463666	+	0.886010i	$0.346534\pi$
$198$		0			0
$199$	−117.295	−	203.161i	−0.589422	−	1.02091i	−0.994308	−	0.106541i	$-0.966022\pi$
$199$	−117.295	−	203.161i	−0.589422	−	1.02091i	0.404886	−	0.914367i	$-0.367311\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	122.742	+	70.8653i	0.604641	+	0.349090i
$204$		0			0
$205$		−	608.749i		−	2.96951i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−56.5380	+	1.91012i	−0.270517	+	0.00913932i
$210$		0			0
$211$		−	66.7424i		−	0.316314i	−0.987414	−	0.158157i	$-0.949445\pi$
$211$		−	66.7424i		−	0.316314i	0.987414	−	0.158157i	$-0.0505553\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−264.627	−	458.347i	−1.23082	−	2.13185i
$216$		0			0
$217$			167.349i			0.771193i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−125.768	+	72.6120i	−0.569084	+	0.328561i
$222$		0			0
$223$	−199.752	+	115.327i	−0.895749	+	0.517161i	−0.875819	−	0.482641i	$-0.839678\pi$
$223$	−199.752	+	115.327i	−0.895749	+	0.517161i	−0.0199303	+	0.999801i	$0.506344\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−76.8934	−	44.3944i	−0.338737	−	0.195570i	0.320976	−	0.947087i	$-0.395989\pi$
$227$	−76.8934	−	44.3944i	−0.338737	−	0.195570i	−0.659713	+	0.751517i	$0.729322\pi$
$228$		0			0
$229$	67.4283	+	116.789i	0.294447	+	0.509997i	0.974856	−	0.222836i	$-0.0715313\pi$
$229$	67.4283	+	116.789i	0.294447	+	0.509997i	−0.680409	+	0.732832i	$0.738198\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−33.1778	+	57.4656i	−0.142394	+	0.246633i	−0.928398	−	0.371588i	$-0.878813\pi$
$233$	−33.1778	+	57.4656i	−0.142394	+	0.246633i	0.786004	+	0.618222i	$0.212147\pi$
$234$		0			0
$235$	341.254			1.45215
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−157.173	−	272.231i	−0.657626	−	1.13904i	−0.981228	−	0.192849i	$-0.938227\pi$
$239$	−157.173	−	272.231i	−0.657626	−	1.13904i	0.323602	−	0.946193i	$-0.395106\pi$
$240$		0			0
$241$			50.7114i			0.210421i	0.994450	+	0.105210i	$0.0335516\pi$
$241$			50.7114i			0.210421i	−0.994450	+	0.105210i	$0.966448\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	135.170	+	234.121i	0.551713	+	0.955595i
$246$		0			0
$247$	3.57259	+	105.746i	0.0144639	+	0.428121i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−10.9851	−	19.0267i	−0.0437653	−	0.0758038i	0.843313	−	0.537423i	$-0.180602\pi$
$251$	−10.9851	−	19.0267i	−0.0437653	−	0.0758038i	−0.887078	+	0.461619i	$0.847269\pi$
$252$		0			0
$253$	−28.1908	−	48.8279i	−0.111426	−	0.192996i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	355.279	+	205.121i	1.38241	+	0.798135i	0.992444	−	0.122695i	$-0.0391537\pi$
$257$	355.279	+	205.121i	1.38241	+	0.798135i	0.389965	+	0.920830i	$0.372487\pi$
$258$		0			0
$259$	251.390	+	145.140i	0.970618	+	0.560386i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	127.108	+	220.157i	0.483300	+	0.837100i	0.999816	−	0.0191774i	$-0.00610471\pi$
$263$	127.108	+	220.157i	0.483300	+	0.837100i	−0.516516	+	0.856277i	$0.672771\pi$
$264$		0			0
$265$	615.028	−	355.087i	2.32086	−	1.33995i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.296857	+	0.171391i	0.00110356	+	0.000637140i	0.500552	−	0.865707i	$-0.333131\pi$
$269$	0.296857	+	0.171391i	0.00110356	+	0.000637140i	−0.499448	+	0.866344i	$0.666464\pi$
$270$		0			0
$271$	177.957	−	308.230i	0.656666	−	1.13738i	−0.324807	−	0.945780i	$-0.605299\pi$
$271$	177.957	−	308.230i	0.656666	−	1.13738i	0.981473	−	0.191599i	$-0.0613673\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	67.2471	−	116.475i	0.244535	−	0.423547i
$276$		0			0
$277$	−146.001	+	252.882i	−0.527081	+	0.912931i	0.472421	+	0.881373i	$0.343380\pi$
$277$	−146.001	+	252.882i	−0.527081	+	0.912931i	−0.999502	+	0.0315577i	$0.989953\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			338.871i			1.20594i	0.797762	+	0.602972i	$0.206017\pi$
$281$			338.871i			1.20594i	−0.797762	+	0.602972i	$0.793983\pi$
$282$		0			0
$283$	−76.8677			−0.271617			−0.135809	−	0.990735i	$-0.543363\pi$
$283$	−76.8677			−0.271617			−0.135809	+	0.990735i	$0.543363\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−567.359	−	327.565i	−1.97686	−	1.14134i
$288$		0			0
$289$	−195.540	−	338.685i	−0.676610	−	1.17192i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−13.8270	+	7.98304i	−0.0471913	+	0.0272459i	−0.523410	−	0.852081i	$-0.675340\pi$
$293$	−13.8270	+	7.98304i	−0.0471913	+	0.0272459i	0.476219	+	0.879327i	$0.342007\pi$
$294$		0			0
$295$			130.079i			0.440946i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−91.3254	+	52.7268i	−0.305436	+	0.176344i
$300$		0			0
$301$	−569.577			−1.89228
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−139.524			−0.457454
$306$		0			0
$307$	392.315	+	226.503i	1.27790	+	0.737795i	0.976461	−	0.215692i	$-0.0692006\pi$
$307$	392.315	+	226.503i	1.27790	+	0.737795i	0.301436	+	0.953486i	$0.402534\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	179.030	+	310.089i	0.575659	+	0.997071i	0.995970	+	0.0896903i	$0.0285877\pi$
$311$	179.030	+	310.089i	0.575659	+	0.997071i	−0.420311	+	0.907380i	$0.638079\pi$
$312$		0			0
$313$	579.223			1.85055			0.925277	−	0.379293i	$-0.123833\pi$
$313$	579.223			1.85055			0.925277	+	0.379293i	$0.123833\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	318.874i		−	1.00591i	−0.864312	−	0.502956i	$-0.832246\pi$
$317$		−	318.874i		−	1.00591i	0.864312	−	0.502956i	$-0.167754\pi$
$318$		0			0
$319$	−40.5376	−	23.4044i	−0.127077	−	0.0733680i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−495.206	+	16.7304i	−1.53315	+	0.0517968i
$324$		0			0
$325$	−217.850	−	125.776i	−0.670308	−	0.387002i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	183.627	−	318.052i	0.558137	−	0.966722i
$330$		0			0
$331$	−454.593	−	262.459i	−1.37339	−	0.792928i	−0.382038	−	0.924147i	$-0.624778\pi$
$331$	−454.593	−	262.459i	−1.37339	−	0.792928i	−0.991353	+	0.131219i	$0.958111\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−458.383	−	264.648i	−1.36831	−	0.789993i
$336$		0			0
$337$		−	211.397i		−	0.627290i	−0.949540	−	0.313645i	$-0.898450\pi$
$337$		−	211.397i		−	0.627290i	0.949540	−	0.313645i	$-0.101550\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	55.2696i		−	0.162081i
$342$		0			0
$343$	−150.804			−0.439661
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−274.738			−0.791751			−0.395875	−	0.918304i	$-0.629559\pi$
$347$	−274.738			−0.791751			−0.395875	+	0.918304i	$0.629559\pi$
$348$		0			0
$349$	−50.0768	+	86.7355i	−0.143486	+	0.248526i	−0.928807	−	0.370563i	$-0.879165\pi$
$349$	−50.0768	+	86.7355i	−0.143486	+	0.248526i	0.785321	+	0.619089i	$0.212498\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−285.912	+	495.215i	−0.809950	+	1.40287i	0.102948	+	0.994687i	$0.467172\pi$
$353$	−285.912	+	495.215i	−0.809950	+	1.40287i	−0.912898	+	0.408187i	$0.866161\pi$
$354$		0			0
$355$	706.165	+	407.704i	1.98920	+	1.14846i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−30.1543	+	52.2287i	−0.0839951	+	0.145484i	−0.904963	−	0.425491i	$-0.860101\pi$
$359$	−30.1543	+	52.2287i	−0.0839951	+	0.145484i	0.820967	+	0.570975i	$0.193435\pi$
$360$		0			0
$361$	−158.988	+	324.105i	−0.440410	+	0.897797i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−171.072	+	296.306i	−0.468691	+	0.811797i
$366$		0			0
$367$	−524.596			−1.42942			−0.714709	−	0.699422i	$-0.753441\pi$
$367$	−524.596			−1.42942			−0.714709	+	0.699422i	$0.753441\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	764.282i		−	2.06006i
$372$		0			0
$373$	525.452	−	303.370i	1.40872	−	0.813324i	0.413454	−	0.910525i	$-0.364322\pi$
$373$	525.452	−	303.370i	1.40872	−	0.813324i	0.995265	+	0.0972011i	$0.0309890\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−43.7744	+	75.8195i	−0.116113	+	0.201113i
$378$		0			0
$379$			488.868i			1.28989i	0.764229	+	0.644945i	$0.223120\pi$
$379$			488.868i			1.28989i	−0.764229	+	0.644945i	$0.776880\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			177.453i			0.463323i	0.972796	+	0.231662i	$0.0744162\pi$
$383$			177.453i			0.463323i	−0.972796	+	0.231662i	$0.925584\pi$
$384$		0			0
$385$	−112.424	−	194.723i	−0.292009	−	0.505775i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	670.487			1.72362			0.861808	−	0.507234i	$-0.169332\pi$
$389$	670.487			1.72362			0.861808	+	0.507234i	$0.169332\pi$
$390$		0			0
$391$	−246.918	−	427.675i	−0.631504	−	1.09380i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	781.673	−	451.299i	1.97892	−	1.14253i
$396$		0			0
$397$	−143.711	+	248.915i	−0.361992	+	0.626989i	−0.988289	−	0.152596i	$-0.951237\pi$
$397$	−143.711	+	248.915i	−0.361992	+	0.626989i	0.626296	+	0.779585i	$0.284570\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			134.473i			0.335344i	0.985843	+	0.167672i	$0.0536250\pi$
$401$			134.473i			0.335344i	−0.985843	+	0.167672i	$0.946375\pi$
$402$		0			0
$403$	−103.374			−0.256510
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−83.0256	−	47.9348i	−0.203994	−	0.117776i
$408$		0			0
$409$	518.835	+	299.550i	1.26855	+	0.732395i	0.974713	−	0.223461i	$-0.0717356\pi$
$409$	518.835	+	299.550i	1.26855	+	0.732395i	0.293833	+	0.955857i	$0.405069\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	121.235	+	69.9949i	0.293547	+	0.169479i
$414$		0			0
$415$	−480.188	+	831.711i	−1.15708	+	2.00412i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	161.894	+	280.408i	0.386381	+	0.669232i	0.991960	−	0.126553i	$-0.0403915\pi$
$419$	161.894	+	280.408i	0.386381	+	0.669232i	−0.605578	+	0.795786i	$0.707058\pi$
$420$		0			0
$421$	−564.730	+	326.047i	−1.34140	+	0.774459i	−0.987013	−	0.160639i	$-0.948645\pi$
$421$	−564.730	+	326.047i	−1.34140	+	0.774459i	−0.354389	+	0.935098i	$0.615311\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	589.005	−	1020.19i	1.38589	−	2.40044i
$426$		0			0
$427$	−75.0769	+	130.037i	−0.175824	+	0.304536i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−99.9113	+	57.6838i	−0.231813	+	0.133837i	−0.611408	−	0.791315i	$-0.709397\pi$
$431$	−99.9113	+	57.6838i	−0.231813	+	0.133837i	0.379595	+	0.925153i	$0.376063\pi$
$432$		0			0
$433$	−60.5946	+	34.9843i	−0.139941	+	0.0807952i	−0.568336	−	0.822797i	$-0.692413\pi$
$433$	−60.5946	+	34.9843i	−0.139941	+	0.0807952i	0.428395	+	0.903592i	$0.359079\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−359.591	+	12.1487i	−0.822863	+	0.0278001i
$438$		0			0
$439$	388.864	−	224.511i	0.885795	−	0.511414i	0.0132301	−	0.999912i	$-0.495789\pi$
$439$	388.864	−	224.511i	0.885795	−	0.511414i	0.872565	+	0.488499i	$0.162455\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−192.761			−0.435127			−0.217564	−	0.976046i	$-0.569811\pi$
$443$	−192.761			−0.435127			−0.217564	+	0.976046i	$0.569811\pi$
$444$		0			0
$445$	22.6977	−	13.1045i	0.0510061	−	0.0294484i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			264.055i			0.588096i	0.955791	+	0.294048i	$0.0950025\pi$
$449$			264.055i			0.588096i	−0.955791	+	0.294048i	$0.904997\pi$
$450$		0			0
$451$	187.379	+	108.183i	0.415475	+	0.239875i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−364.201	+	210.272i	−0.800442	+	0.462135i
$456$		0			0
$457$	−225.519	+	390.610i	−0.493476	+	0.854726i	−0.999972	−	0.00751646i	$-0.997607\pi$
$457$	−225.519	+	390.610i	−0.493476	+	0.854726i	0.506495	+	0.862243i	$0.330941\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	299.891	+	519.426i	0.650522	+	1.12674i	0.982996	+	0.183625i	$0.0587834\pi$
$461$	299.891	+	519.426i	0.650522	+	1.12674i	−0.332474	+	0.943112i	$0.607883\pi$
$462$		0			0
$463$	−39.3790	−	68.2064i	−0.0850518	−	0.147314i	0.820361	−	0.571845i	$-0.193772\pi$
$463$	−39.3790	−	68.2064i	−0.0850518	−	0.147314i	−0.905413	+	0.424531i	$0.860439\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−570.793			−1.22225			−0.611127	−	0.791532i	$-0.709284\pi$
$467$	−570.793			−1.22225			−0.611127	+	0.791532i	$0.709284\pi$
$468$		0			0
$469$	−493.308	+	284.811i	−1.05183	+	0.607274i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	188.112			0.397700
$474$		0			0
$475$	−453.986	−	728.367i	−0.955760	−	1.53340i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	924.676			1.93043			0.965215	−	0.261457i	$-0.0842030\pi$
$479$	924.676			1.93043			0.965215	+	0.261457i	$0.0842030\pi$
$480$		0			0
$481$	−89.6550	+	155.287i	−0.186393	+	0.322842i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	875.105	−	505.242i	1.80434	−	1.04174i
$486$		0			0
$487$			236.919i			0.486486i	0.969965	+	0.243243i	$0.0782113\pi$
$487$			236.919i			0.486486i	−0.969965	+	0.243243i	$0.921789\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−552.296			−1.12484			−0.562420	−	0.826852i	$-0.690129\pi$
$491$	−552.296			−1.12484			−0.562420	+	0.826852i	$0.690129\pi$
$492$		0			0
$493$	−355.061	−	204.995i	−0.720205	−	0.415811i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	759.968	−	438.768i	1.52911	−	0.882832i
$498$		0			0
$499$	−370.123			−0.741730			−0.370865	−	0.928687i	$-0.620939\pi$
$499$	−370.123			−0.741730			−0.370865	+	0.928687i	$0.620939\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	410.372	+	710.785i	0.815848	+	1.41309i	0.908717	+	0.417412i	$0.137063\pi$
$503$	410.372	+	710.785i	0.815848	+	1.41309i	−0.0928687	+	0.995678i	$0.529604\pi$
$504$		0			0
$505$	147.713			0.292502
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−32.7472	−	18.9066i	−0.0643363	−	0.0371446i	0.467487	−	0.884000i	$-0.345160\pi$
$509$	−32.7472	−	18.9066i	−0.0643363	−	0.0371446i	−0.531823	+	0.846855i	$0.678493\pi$
$510$		0			0
$511$	184.106	+	318.882i	0.360286	+	0.624034i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	843.814	−	487.177i	1.63847	−	0.945974i
$516$		0			0
$517$	−60.6458	+	105.042i	−0.117303	+	0.203175i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	190.170i		−	0.365009i	−0.983205	−	0.182504i	$-0.941580\pi$
$521$		−	190.170i		−	0.365009i	0.983205	−	0.182504i	$-0.0584204\pi$
$522$		0			0
$523$	86.4225	−	49.8961i	0.165244	−	0.0954035i	−0.415097	−	0.909777i	$-0.636253\pi$
$523$	86.4225	−	49.8961i	0.165244	−	0.0954035i	0.580341	+	0.814373i	$0.302919\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	484.097i		−	0.918589i
$528$		0			0
$529$	85.2017	+	147.574i	0.161062	+	0.278967i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	202.341	−	350.465i	0.379627	−	0.657533i
$534$		0			0
$535$			43.1016i			0.0805638i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−96.0865			−0.178268
$540$		0			0
$541$	151.059	+	261.642i	0.279222	+	0.483627i	0.971192	−	0.238300i	$-0.0765901\pi$
$541$	151.059	+	261.642i	0.279222	+	0.483627i	−0.691970	+	0.721927i	$0.743257\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	76.0612	+	43.9140i	0.139562	+	0.0805761i
$546$		0			0
$547$		−	362.251i		−	0.662250i	−0.943587	−	0.331125i	$-0.892572\pi$
$547$		−	362.251i		−	0.662250i	0.943587	−	0.331125i	$-0.107428\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−253.498	+	158.003i	−0.460069	+	0.286757i
$552$		0			0
$553$		−	971.368i		−	1.75654i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−198.669	−	344.104i	−0.356676	−	0.617781i	0.630727	−	0.776005i	$-0.282757\pi$
$557$	−198.669	−	344.104i	−0.356676	−	0.617781i	−0.987403	+	0.158223i	$0.949423\pi$
$558$		0			0
$559$		−	351.836i		−	0.629402i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	250.075	−	144.381i	0.444182	−	0.256449i	−0.261188	−	0.965288i	$-0.584114\pi$
$563$	250.075	−	144.381i	0.444182	−	0.256449i	0.705370	+	0.708839i	$0.250781\pi$
$564$		0			0
$565$	1483.41	−	856.444i	2.62550	−	1.51583i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	506.239	+	292.277i	0.889700	+	0.513668i	0.873844	−	0.486206i	$-0.161620\pi$
$569$	506.239	+	292.277i	0.889700	+	0.513668i	0.0158554	+	0.999874i	$0.494953\pi$
$570$		0			0
$571$	−523.855	−	907.343i	−0.917434	−	1.58904i	−0.803298	−	0.595577i	$-0.796924\pi$
$571$	−523.855	−	907.343i	−0.917434	−	1.58904i	−0.114135	−	0.993465i	$-0.536410\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	427.703	−	740.803i	0.743831	−	1.28835i
$576$		0			0
$577$	514.036			0.890877			0.445438	−	0.895313i	$-0.353048\pi$
$577$	514.036			0.890877			0.445438	+	0.895313i	$0.353048\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	516.774	+	895.079i	0.889456	+	1.54058i
$582$		0			0
$583$			252.416i			0.432961i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−253.000	−	438.209i	−0.431005	−	0.746523i	0.565955	−	0.824436i	$-0.308508\pi$
$587$	−253.000	−	438.209i	−0.431005	−	0.746523i	−0.996960	+	0.0779134i	$0.975174\pi$
$588$		0			0
$589$	−311.228	−	165.936i	−0.528400	−	0.281725i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	110.508	+	191.405i	0.186353	+	0.322774i	0.944032	−	0.329855i	$-0.107000\pi$
$593$	110.508	+	191.405i	0.186353	+	0.322774i	−0.757678	+	0.652628i	$0.773666\pi$
$594$		0			0
$595$	−984.697	−	1705.55i	−1.65495	−	2.86646i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−129.840	−	74.9630i	−0.216761	−	0.125147i	0.387689	−	0.921790i	$-0.373274\pi$
$599$	−129.840	−	74.9630i	−0.216761	−	0.125147i	−0.604449	+	0.796643i	$0.706607\pi$
$600$		0			0
$601$	933.044	+	538.693i	1.55249	+	0.896328i	0.997939	+	0.0641750i	$0.0204416\pi$
$601$	933.044	+	538.693i	1.55249	+	0.896328i	0.554547	+	0.832153i	$0.312892\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−469.671	−	813.494i	−0.776316	−	1.34462i
$606$		0			0
$607$	−990.463	+	571.844i	−1.63174	+	0.942083i	−0.648178	+	0.761489i	$0.724469\pi$
$607$	−990.463	+	571.844i	−1.63174	+	0.942083i	−0.983558	+	0.180594i	$0.942198\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	196.465	+	113.429i	0.321546	+	0.185645i
$612$		0			0
$613$	−357.940	+	619.969i	−0.583914	+	1.01137i	0.411095	+	0.911592i	$0.365146\pi$
$613$	−357.940	+	619.969i	−0.583914	+	1.01137i	−0.995010	+	0.0997770i	$0.968187\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−573.579	+	993.468i	−0.929625	+	1.61016i	−0.145677	+	0.989332i	$0.546536\pi$
$617$	−573.579	+	993.468i	−0.929625	+	1.61016i	−0.783948	+	0.620826i	$0.786797\pi$
$618$		0			0
$619$	194.222	−	336.402i	0.313767	−	0.543460i	−0.665408	−	0.746480i	$-0.731742\pi$
$619$	194.222	−	336.402i	0.313767	−	0.543460i	0.979175	+	0.203020i	$0.0650757\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	28.2059i		−	0.0452744i
$624$		0			0
$625$	286.206			0.457930
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−727.206	−	419.852i	−1.15613	−	0.667492i
$630$		0			0
$631$	−398.613	−	690.418i	−0.631717	−	1.09417i	−0.987201	−	0.159483i	$-0.949017\pi$
$631$	−398.613	−	690.418i	−0.631717	−	1.09417i	0.355484	−	0.934682i	$-0.384316\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−307.017	+	177.256i	−0.483492	+	0.279144i
$636$		0			0
$637$			179.716i			0.282128i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	666.281	−	384.678i	1.03944	−	0.600121i	0.119765	−	0.992802i	$-0.461786\pi$
$641$	666.281	−	384.678i	1.03944	−	0.600121i	0.919675	+	0.392681i	$0.128452\pi$
$642$		0			0
$643$	−318.941			−0.496021			−0.248010	−	0.968757i	$-0.579777\pi$
$643$	−318.941			−0.496021			−0.248010	+	0.968757i	$0.579777\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−585.103			−0.904333			−0.452166	−	0.891934i	$-0.649349\pi$
$647$	−585.103			−0.904333			−0.452166	+	0.891934i	$0.649349\pi$
$648$		0			0
$649$	−40.0397	−	23.1169i	−0.0616945	−	0.0356193i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	89.3580	+	154.773i	0.136842	+	0.237018i	0.926300	−	0.376788i	$-0.122971\pi$
$653$	89.3580	+	154.773i	0.136842	+	0.237018i	−0.789457	+	0.613805i	$0.789638\pi$
$654$		0			0
$655$	−1818.46			−2.77627
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	136.738i		−	0.207493i	−0.994604	−	0.103747i	$-0.966917\pi$
$659$		−	136.738i		−	0.207493i	0.994604	−	0.103747i	$-0.0330831\pi$
$660$		0			0
$661$	−548.250	−	316.532i	−0.829425	−	0.478869i	0.0242310	−	0.999706i	$-0.492286\pi$
$661$	−548.250	−	316.532i	−0.829425	−	0.478869i	−0.853656	+	0.520838i	$0.825620\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1434.03	+	48.4482i	−2.15644	+	0.0728545i
$666$		0			0
$667$	−257.826	−	148.856i	−0.386545	−	0.223172i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	24.7954	−	42.9468i	0.0369528	−	0.0640042i
$672$		0			0
$673$	293.528	+	169.468i	0.436149	+	0.251811i	0.701963	−	0.712214i	$-0.252307\pi$
$673$	293.528	+	169.468i	0.436149	+	0.251811i	−0.265814	+	0.964024i	$0.585641\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	90.2123	+	52.0841i	0.133253	+	0.0769337i	0.565145	−	0.824992i	$-0.308820\pi$
$677$	90.2123	+	52.0841i	0.133253	+	0.0769337i	−0.431892	+	0.901926i	$0.642154\pi$
$678$		0			0
$679$		−	1087.47i		−	1.60158i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			517.968i			0.758371i	0.925321	+	0.379186i	$0.123796\pi$
$683$			517.968i			0.758371i	−0.925321	+	0.379186i	$0.876204\pi$
$684$		0			0
$685$	493.690			0.720715
$686$		0			0
$687$		0			0
$688$		0			0
$689$	472.107			0.685206
$690$		0			0
$691$	−44.7261	+	77.4679i	−0.0647266	+	0.112110i	−0.896573	−	0.442897i	$-0.853951\pi$
$691$	−44.7261	+	77.4679i	−0.0647266	+	0.112110i	0.831846	+	0.555006i	$0.187284\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	9.07236	−	15.7138i	0.0130538	−	0.0226098i
$696$		0			0
$697$	1641.22	+	947.559i	2.35469	+	1.35948i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−291.669	+	505.186i	−0.416076	+	0.720665i	−0.995541	−	0.0943325i	$-0.969928\pi$
$701$	−291.669	+	505.186i	−0.416076	+	0.720665i	0.579465	+	0.814997i	$0.303262\pi$
$702$		0			0
$703$	−519.192	+	323.609i	−0.738538	+	0.460325i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	79.4838	−	137.670i	0.112424	−	0.194724i
$708$		0			0
$709$	852.625			1.20257			0.601287	−	0.799033i	$-0.294655\pi$
$709$	852.625			1.20257			0.601287	+	0.799033i	$0.294655\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	351.524i		−	0.493021i
$714$		0			0
$715$	120.283	−	69.4456i	0.168228	−	0.0971267i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	443.865	−	768.797i	0.617337	−	1.06926i	−0.372633	−	0.927979i	$-0.621545\pi$
$719$	443.865	−	768.797i	0.617337	−	1.06926i	0.989970	−	0.141280i	$-0.0451216\pi$
$720$		0			0
$721$		−	1048.59i		−	1.45435i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	710.168i		−	0.979543i
$726$		0			0
$727$	165.516	+	286.683i	0.227671	+	0.394337i	0.957117	−	0.289701i	$-0.0935557\pi$
$727$	165.516	+	286.683i	0.227671	+	0.394337i	−0.729447	+	0.684038i	$0.760222\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1647.64			2.25395
$732$		0			0
$733$	−411.381	−	712.533i	−0.561229	−	0.972078i	−0.997390	−	0.0722087i	$-0.976995\pi$
$733$	−411.381	−	712.533i	−0.561229	−	0.972078i	0.436160	−	0.899869i	$-0.356338\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	162.923	−	94.0635i	0.221062	−	0.127630i
$738$		0			0
$739$	351.505	−	608.824i	0.475649	−	0.823849i	−0.523962	−	0.851742i	$-0.675546\pi$
$739$	351.505	−	608.824i	0.475649	−	0.823849i	0.999611	+	0.0278931i	$0.00887980\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			1072.12i			1.44296i	0.692433	+	0.721482i	$0.256539\pi$
$743$			1072.12i			1.44296i	−0.692433	+	0.721482i	$0.743461\pi$
$744$		0			0
$745$	1192.04			1.60005
$746$		0			0
$747$		0			0
$748$		0			0
$749$	40.1711	+	23.1928i	0.0536329	+	0.0309650i
$750$		0			0
$751$	104.721	+	60.4606i	0.139442	+	0.0805068i	0.568098	−	0.822961i	$-0.307679\pi$
$751$	104.721	+	60.4606i	0.139442	+	0.0805068i	−0.428656	+	0.903468i	$0.641013\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−2075.62	−	1198.36i	−2.74916	−	1.58723i
$756$		0			0
$757$	−500.057	+	866.125i	−0.660578	+	1.14415i	0.319887	+	0.947456i	$0.396355\pi$
$757$	−500.057	+	866.125i	−0.660578	+	1.14415i	−0.980464	+	0.196698i	$0.936978\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−710.262	−	1230.21i	−0.933327	−	1.61657i	−0.777591	−	0.628771i	$-0.783558\pi$
$761$	−710.262	−	1230.21i	−0.933327	−	1.61657i	−0.155736	−	0.987799i	$-0.549775\pi$
$762$		0			0
$763$	81.8563	−	47.2598i	0.107282	−	0.0619394i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−43.2368	+	74.8884i	−0.0563713	+	0.0976380i
$768$		0			0
$769$	−226.326	+	392.008i	−0.294312	+	0.509763i	−0.974825	−	0.222973i	$-0.928424\pi$
$769$	−226.326	+	392.008i	−0.294312	+	0.509763i	0.680513	+	0.732736i	$0.261757\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	158.416	−	91.4618i	0.204937	−	0.118321i	−0.394019	−	0.919102i	$-0.628916\pi$
$773$	158.416	−	91.4618i	0.204937	−	0.118321i	0.598956	+	0.800782i	$0.295582\pi$
$774$		0			0
$775$	726.192	−	419.267i	0.937022	−	0.540990i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1171.76	−	730.348i	1.50418	−	0.937545i
$780$		0			0
$781$	−250.992	+	144.910i	−0.321372	+	0.185544i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1465.28			−1.86660
$786$		0			0
$787$	−355.648	+	205.334i	−0.451904	+	0.260907i	−0.708634	−	0.705576i	$-0.750688\pi$
$787$	−355.648	+	205.334i	−0.451904	+	0.260907i	0.256730	+	0.966483i	$0.417355\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	1843.39i		−	2.33046i
$792$		0			0
$793$	−80.3257	−	46.3760i	−0.101293	−	0.0584818i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1016.85	−	587.077i	1.27584	−	0.736609i	0.299762	−	0.954014i	$-0.403093\pi$
$797$	1016.85	−	587.077i	1.27584	−	0.736609i	0.976081	+	0.217405i	$0.0697594\pi$
$798$		0			0
$799$	−531.186	+	920.040i	−0.664813	+	1.15149i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−60.8041	−	105.316i	−0.0757211	−	0.131153i
$804$		0			0
$805$	−715.032	−	1238.47i	−0.888239	−	1.53847i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−500.592			−0.618778			−0.309389	−	0.950936i	$-0.600125\pi$
$809$	−500.592			−0.618778			−0.309389	+	0.950936i	$0.600125\pi$
$810$		0			0
$811$	671.823	−	387.877i	0.828389	−	0.478270i	−0.0249120	−	0.999690i	$-0.507931\pi$
$811$	671.823	−	387.877i	0.828389	−	0.478270i	0.853301	+	0.521419i	$0.174597\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2350.24			−2.88373
$816$		0			0
$817$	564.768	−	1059.27i	0.691271	−	1.29654i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1449.68			1.76575			0.882875	−	0.469608i	$-0.155605\pi$
$821$	1449.68			1.76575			0.882875	+	0.469608i	$0.155605\pi$
$822$		0			0
$823$	−11.0069	+	19.0645i	−0.0133741	+	0.0231647i	−0.872635	−	0.488373i	$-0.837591\pi$
$823$	−11.0069	+	19.0645i	−0.0133741	+	0.0231647i	0.859261	+	0.511538i	$0.170924\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	417.259	−	240.905i	0.504546	−	0.291300i	−0.226043	−	0.974117i	$-0.572579\pi$
$827$	417.259	−	240.905i	0.504546	−	0.291300i	0.730589	+	0.682818i	$0.239246\pi$
$828$		0			0
$829$		−	1578.05i		−	1.90355i	−0.306791	−	0.951777i	$-0.599255\pi$
$829$		−	1578.05i		−	1.90355i	0.306791	−	0.951777i	$-0.400745\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−841.604			−1.01033
$834$		0			0
$835$	41.2576	+	23.8201i	0.0494103	+	0.0285270i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−808.937	+	467.040i	−0.964168	+	0.556663i	−0.897453	−	0.441110i	$-0.854585\pi$
$839$	−808.937	+	467.040i	−0.964168	+	0.556663i	−0.0667145	+	0.997772i	$0.521252\pi$
$840$		0			0
$841$	593.836			0.706107
$842$		0			0
$843$		0			0
$844$		0			0
$845$	577.958	+	1001.05i	0.683974	+	1.18468i
$846$		0			0
$847$	−1010.91			−1.19352
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−528.056	−	304.873i	−0.620513	−	0.358253i
$852$		0			0
$853$	−146.573	−	253.872i	−0.171832	−	0.297622i	0.767228	−	0.641374i	$-0.221635\pi$
$853$	−146.573	−	253.872i	−0.171832	−	0.297622i	−0.939061	+	0.343752i	$0.888302\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−693.932	+	400.642i	−0.809723	+	0.467494i	−0.846860	−	0.531817i	$-0.821510\pi$
$857$	−693.932	+	400.642i	−0.809723	+	0.467494i	0.0371370	+	0.999310i	$0.488176\pi$
$858$		0			0
$859$	425.123	−	736.335i	0.494905	−	0.857201i	−0.505078	−	0.863074i	$-0.668536\pi$
$859$	425.123	−	736.335i	0.494905	−	0.857201i	0.999983	+	0.00587314i	$0.00186949\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	57.6973i		−	0.0668566i	−0.999441	−	0.0334283i	$-0.989357\pi$
$863$		−	57.6973i		−	0.0668566i	0.999441	−	0.0334283i	$-0.0106425\pi$
$864$		0			0
$865$	7.51083	−	4.33638i	0.00868304	−	0.00501316i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			320.810i			0.369171i
$870$		0			0
$871$	−175.932	−	304.723i	−0.201988	−	0.349854i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	761.676	−	1319.26i	0.870487	−	1.50773i
$876$		0			0
$877$			1590.67i			1.81376i	0.421392	+	0.906879i	$0.361542\pi$
$877$			1590.67i			1.81376i	−0.421392	+	0.906879i	$0.638458\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1154.00			1.30987			0.654936	−	0.755684i	$-0.272696\pi$
$881$	1154.00			1.30987			0.654936	+	0.755684i	$0.272696\pi$
$882$		0			0
$883$	184.480	+	319.529i	0.208924	+	0.361867i	0.951376	−	0.308032i	$-0.0996705\pi$
$883$	184.480	+	319.529i	0.208924	+	0.361867i	−0.742452	+	0.669899i	$0.766337\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	934.738	+	539.671i	1.05382	+	0.608423i	0.923716	−	0.383077i	$-0.125136\pi$
$887$	934.738	+	539.671i	1.05382	+	0.608423i	0.130103	+	0.991500i	$0.458469\pi$
$888$		0			0
$889$			381.523i			0.429160i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	409.421	+	656.868i	0.458478	+	0.735574i
$894$		0			0
$895$		−	1550.43i		−	1.73232i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−145.920	−	252.741i	−0.162314	−	0.281135i
$900$		0			0
$901$			2210.87i			2.45379i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−651.670	+	376.242i	−0.720078	+	0.415737i
$906$		0			0
$907$	1004.34	−	579.858i	1.10733	−	0.639315i	0.169190	−	0.985583i	$-0.445885\pi$
$907$	1004.34	−	579.858i	1.10733	−	0.639315i	0.938135	+	0.346269i	$0.112552\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	228.498	+	131.923i	0.250821	+	0.144812i	0.620140	−	0.784491i	$-0.287076\pi$
$911$	228.498	+	131.923i	0.250821	+	0.144812i	−0.369319	+	0.929303i	$0.620409\pi$
$912$		0			0
$913$	−170.673	−	295.614i	−0.186936	−	0.323783i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−978.503	+	1694.82i	−1.06707	+	1.84822i
$918$		0			0
$919$	197.600			0.215016			0.107508	−	0.994204i	$-0.465713\pi$
$919$	197.600			0.215016			0.107508	+	0.994204i	$0.465713\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	271.033	+	469.442i	0.293643	+	0.508605i
$924$		0			0
$925$		−	1454.51i		−	1.57244i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	8.13637	+	14.0926i	0.00875820	+	0.0151696i	0.870371	−	0.492396i	$-0.163879\pi$
$929$	8.13637	+	14.0926i	0.00875820	+	0.0151696i	−0.861613	+	0.507566i	$0.830545\pi$
$930$		0			0
$931$	−288.480	+	541.070i	−0.309861	+	0.581171i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	325.212	+	563.284i	0.347820	+	0.602442i
$936$		0			0
$937$	136.677	+	236.732i	0.145867	+	0.252649i	0.929696	−	0.368328i	$-0.120070\pi$
$937$	136.677	+	236.732i	0.145867	+	0.252649i	−0.783829	+	0.620976i	$0.786736\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1043.64	+	602.546i	1.10908	+	0.640325i	0.938590	−	0.345035i	$-0.112133\pi$
$941$	1043.64	+	602.546i	1.10908	+	0.640325i	0.170486	+	0.985360i	$0.445466\pi$
$942$		0			0
$943$	1191.76	+	688.065i	1.26380	+	0.729655i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	514.871	+	891.783i	0.543686	+	0.941692i	0.998688	+	0.0512018i	$0.0163052\pi$
$947$	514.871	+	891.783i	0.543686	+	0.941692i	−0.455002	+	0.890490i	$0.650361\pi$
$948$		0			0
$949$	−196.977	+	113.725i	−0.207563	+	0.119837i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	618.547	+	357.118i	0.649052	+	0.374730i	0.788093	−	0.615556i	$-0.211069\pi$
$953$	618.547	+	357.118i	0.649052	+	0.374730i	−0.139041	+	0.990287i	$0.544402\pi$
$954$		0			0
$955$	897.962	−	1555.32i	0.940275	−	1.62860i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	265.652	−	460.123i	0.277009	−	0.479794i
$960$		0			0
$961$	−308.205	+	533.826i	−0.320712	+	0.555490i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	71.1028i		−	0.0736816i
$966$		0			0
$967$	−1091.59			−1.12885			−0.564423	−	0.825486i	$-0.690901\pi$
$967$	−1091.59			−1.12885			−0.564423	+	0.825486i	$0.690901\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−997.067	−	575.657i	−1.02685	−	0.592850i	−0.110766	−	0.993847i	$-0.535330\pi$
$971$	−997.067	−	575.657i	−1.02685	−	0.592850i	−0.916079	+	0.400997i	$0.868664\pi$
$972$		0			0
$973$	−9.76358	−	16.9110i	−0.0100345	−	0.0173803i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−742.979	+	428.959i	−0.760470	+	0.439057i	−0.829464	−	0.558560i	$-0.811354\pi$
$977$	−742.979	+	428.959i	−0.760470	+	0.439057i	0.0689947	+	0.997617i	$0.478021\pi$
$978$		0			0
$979$			9.31546i			0.00951528i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−759.807	+	438.675i	−0.772947	+	0.446261i	−0.833925	−	0.551878i	$-0.813911\pi$
$983$	−759.807	+	438.675i	−0.772947	+	0.446261i	0.0609777	+	0.998139i	$0.480578\pi$
$984$		0			0
$985$	−1530.32			−1.55363
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1196.42			1.20973
$990$		0			0
$991$	−452.536	−	261.272i	−0.456646	−	0.263645i	0.253987	−	0.967208i	$-0.418258\pi$
$991$	−452.536	−	261.272i	−0.456646	−	0.263645i	−0.710633	+	0.703563i	$0.751591\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	982.564	+	1701.85i	0.987502	+	1.71040i
$996$		0			0
$997$	−1154.18			−1.15766			−0.578829	−	0.815449i	$-0.696490\pi$
$997$	−1154.18			−1.15766			−0.578829	+	0.815449i	$0.696490\pi$
$998$		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.1585.2		80
3.2	odd	2		684.3.bl.a.673.29	yes	80
9.4	even	3		2052.3.s.a.901.39		80
9.5	odd	6		684.3.s.a.445.40	✓	80
19.12	odd	6		2052.3.s.a.829.39		80
57.50	even	6		684.3.s.a.601.40	yes	80
171.31	odd	6	inner	2052.3.bl.a.145.2		80
171.50	even	6		684.3.bl.a.373.29	yes	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.s.a.445.40	✓	80	9.5	odd	6
684.3.s.a.601.40	yes	80	57.50	even	6
684.3.bl.a.373.29	yes	80	171.50	even	6
684.3.bl.a.673.29	yes	80	3.2	odd	2
2052.3.s.a.829.39		80	19.12	odd	6
2052.3.s.a.901.39		80	9.4	even	3
2052.3.bl.a.145.2		80	171.31	odd	6	inner
2052.3.bl.a.1585.2		80	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-8.37687 q^{5} +(-4.50755 + 7.80731i) q^{7} +O(q^{10})\)	\(q-8.37687 q^{5} +(-4.50755 + 7.80731i) q^{7} +(1.48869 - 2.57849i) q^{11} +(-4.82268 - 2.78438i) q^{13} +(13.0392 - 22.5845i) q^{17} +(-10.0502 - 16.1243i) q^{19} +(9.46832 - 16.3996i) q^{23} +45.1719 q^{25} -15.7214i q^{29} +(16.0762 - 9.28158i) q^{31} +(37.7592 - 65.4008i) q^{35} -32.1993i q^{37} +72.6702i q^{41} +(31.5902 + 54.7158i) q^{43} -40.7377 q^{47} +(-16.1361 - 27.9485i) q^{49} +(-73.4198 + 42.3890i) q^{53} +(-12.4706 + 21.5997i) q^{55} -15.5284i q^{59} +16.6558 q^{61} +(40.3990 + 23.3244i) q^{65} +(54.7201 + 31.5927i) q^{67} +(-84.2994 - 48.6703i) q^{71} +(20.4220 - 35.3719i) q^{73} +(13.4207 + 23.2454i) q^{77} +(-93.3133 + 53.8744i) q^{79} +(57.3231 - 99.2866i) q^{83} +(-109.227 + 189.188i) q^{85} +(-2.70957 + 1.56437i) q^{89} +(43.4770 - 25.1015i) q^{91} +(84.1890 + 135.071i) q^{95} +(-104.467 + 60.3140i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - q^{7}+O(q^{10}) \) 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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