LMFDB - Embedded newform 684.3.s.a.445.33

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,3,Mod(445,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 1]))

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

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

chi := DirichletCharacter("684.445");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	684.s (of order $6$, degree $2$, 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:	$18.6376500822$
Analytic rank:	$0$
Dimension:	$80$
Relative dimension:	$40$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			445.33
Character	$\chi$	$=$	684.445
Dual form			684.3.s.a.601.33

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.33228 + 1.88692i) q^{3} +(2.16848 + 3.75592i) q^{5} +(-1.29489 - 2.24282i) q^{7} +(1.87907 + 8.80165i) q^{9} +O(q^{10})$	\(q+(2.33228 + 1.88692i) q^{3} +(2.16848 + 3.75592i) q^{5} +(-1.29489 - 2.24282i) q^{7} +(1.87907 + 8.80165i) q^{9} +(-2.53449 - 4.38986i) q^{11} +21.8819i q^{13} +(-2.02961 + 12.8516i) q^{15} +(-7.69782 + 13.3330i) q^{17} +(-10.9829 - 15.5040i) q^{19} +(1.21197 - 7.67423i) q^{21} -23.4214 q^{23} +(3.09539 - 5.36138i) q^{25} +(-12.2255 + 24.0736i) q^{27} +(39.2793 + 22.6779i) q^{29} +(18.5714 + 10.7222i) q^{31} +(2.37218 - 15.0208i) q^{33} +(5.61588 - 9.72700i) q^{35} +50.5347i q^{37} +(-41.2894 + 51.0347i) q^{39} +(-26.7428 + 15.4400i) q^{41} -3.10733 q^{43} +(-28.9836 + 26.1438i) q^{45} +(34.8212 - 60.3120i) q^{47} +(21.1465 - 36.6268i) q^{49} +(-43.1118 + 16.5712i) q^{51} +(-54.5703 + 31.5062i) q^{53} +(10.9920 - 19.0387i) q^{55} +(3.63962 - 56.8837i) q^{57} +(-70.8303 + 40.8939i) q^{59} +(53.1069 - 91.9838i) q^{61} +(17.3073 - 15.6116i) q^{63} +(-82.1865 + 47.4504i) q^{65} -13.1074i q^{67} +(-54.6253 - 44.1943i) q^{69} +(81.6733 + 47.1541i) q^{71} +(-52.8222 + 91.4908i) q^{73} +(17.3358 - 6.66348i) q^{75} +(-6.56377 + 11.3688i) q^{77} -26.5901i q^{79} +(-73.9382 + 33.0778i) q^{81} +(36.6877 + 63.5449i) q^{83} -66.7703 q^{85} +(48.8189 + 127.008i) q^{87} +(107.613 - 62.1305i) q^{89} +(49.0770 - 28.3346i) q^{91} +(23.0818 + 60.0500i) q^{93} +(34.4156 - 74.8712i) q^{95} +30.5295i q^{97} +(33.8756 - 30.5565i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - q^{7} + 4 q^{9}+O(q^{10}) $ 80 * q - q^7 + 4 * q^9	$ 80 q - q^{7} + 4 q^{9} - 6 q^{11} + 33 q^{15} - 21 q^{17} - 20 q^{19} - 48 q^{23} - 200 q^{25} - 63 q^{27} - 27 q^{29} - 24 q^{31} + 27 q^{33} - 54 q^{35} - 81 q^{39} - 18 q^{41} - 152 q^{43} + 188 q^{45} - 12 q^{47} - 267 q^{49} - 126 q^{51} - 36 q^{53} + 126 q^{57} - 135 q^{59} - 7 q^{61} - 190 q^{63} - 288 q^{65} + 48 q^{69} - 81 q^{71} + 55 q^{73} + 165 q^{75} + 30 q^{77} + 28 q^{81} - 93 q^{83} + 306 q^{87} + 216 q^{89} + 96 q^{91} + 24 q^{93} + 288 q^{95} - 241 q^{99}+O(q^{100}) $ 80 * q - q^7 + 4 * q^9 - 6 * q^11 + 33 * q^15 - 21 * q^17 - 20 * q^19 - 48 * q^23 - 200 * q^25 - 63 * q^27 - 27 * q^29 - 24 * q^31 + 27 * q^33 - 54 * q^35 - 81 * q^39 - 18 * q^41 - 152 * q^43 + 188 * q^45 - 12 * q^47 - 267 * q^49 - 126 * q^51 - 36 * q^53 + 126 * q^57 - 135 * q^59 - 7 * q^61 - 190 * q^63 - 288 * q^65 + 48 * q^69 - 81 * q^71 + 55 * q^73 + 165 * q^75 + 30 * q^77 + 28 * q^81 - 93 * q^83 + 306 * q^87 + 216 * q^89 + 96 * q^91 + 24 * q^93 + 288 * q^95 - 241 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$e\left(\frac{1}{3}\right)$

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
$3$	2.33228	+	1.88692i	0.777427	+	0.628973i
$3$	2.33228	+	1.88692i	0.777427	+	0.628973i
$4$		0			0
$5$	2.16848	+	3.75592i	0.433696	+	0.751183i	0.997188	−	0.0749380i	$-0.0238759\pi$
$5$	2.16848	+	3.75592i	0.433696	+	0.751183i	−0.563492	+	0.826121i	$0.690543\pi$
$6$		0			0
$7$	−1.29489	−	2.24282i	−0.184984	−	0.320402i	0.758587	−	0.651572i	$-0.225890\pi$
$7$	−1.29489	−	2.24282i	−0.184984	−	0.320402i	−0.943571	+	0.331170i	$0.892557\pi$
$8$		0			0
$9$	1.87907	+	8.80165i	0.208785	+	0.977962i
$10$		0			0
$11$	−2.53449	−	4.38986i	−0.230408	−	0.399079i	0.727520	−	0.686086i	$-0.240673\pi$
$11$	−2.53449	−	4.38986i	−0.230408	−	0.399079i	−0.957928	+	0.287008i	$0.907339\pi$
$12$		0			0
$13$			21.8819i			1.68322i	0.540084	+	0.841611i	$0.318392\pi$
$13$			21.8819i			1.68322i	−0.540084	+	0.841611i	$0.681608\pi$
$14$		0			0
$15$	−2.02961	+	12.8516i	−0.135307	+	0.856773i
$16$		0			0
$17$	−7.69782	+	13.3330i	−0.452813	+	0.784295i	−0.998560	−	0.0536551i	$-0.982913\pi$
$17$	−7.69782	+	13.3330i	−0.452813	+	0.784295i	0.545746	+	0.837950i	$0.316246\pi$
$18$		0			0
$19$	−10.9829	−	15.5040i	−0.578049	−	0.816002i
$19$	−10.9829	−	15.5040i	−0.578049	−	0.816002i
$20$		0			0
$21$	1.21197	−	7.67423i	0.0577127	−	0.365439i
$22$		0			0
$23$	−23.4214			−1.01832			−0.509161	−	0.860671i	$-0.670044\pi$
$23$	−23.4214			−1.01832			−0.509161	+	0.860671i	$0.670044\pi$
$24$		0			0
$25$	3.09539	−	5.36138i	0.123816	−	0.214455i
$26$		0			0
$27$	−12.2255	+	24.0736i	−0.452797	+	0.891614i
$28$		0			0
$29$	39.2793	+	22.6779i	1.35446	+	0.781996i	0.988870	−	0.148781i	$-0.0475349\pi$
$29$	39.2793	+	22.6779i	1.35446	+	0.781996i	0.365587	+	0.930777i	$0.380868\pi$
$30$		0			0
$31$	18.5714	+	10.7222i	0.599078	+	0.345878i	0.768679	−	0.639635i	$-0.220914\pi$
$31$	18.5714	+	10.7222i	0.599078	+	0.345878i	−0.169601	+	0.985513i	$0.554248\pi$
$32$		0			0
$33$	2.37218	−	15.0208i	0.0718843	−	0.455175i
$34$		0			0
$35$	5.61588	−	9.72700i	0.160454	−	0.277914i
$36$		0			0
$37$			50.5347i			1.36580i	0.730510	+	0.682902i	$0.239282\pi$
$37$			50.5347i			1.36580i	−0.730510	+	0.682902i	$0.760718\pi$
$38$		0			0
$39$	−41.2894	+	51.0347i	−1.05870	+	1.30858i
$40$		0			0
$41$	−26.7428	+	15.4400i	−0.652264	+	0.376585i	−0.789323	−	0.613978i	$-0.789568\pi$
$41$	−26.7428	+	15.4400i	−0.652264	+	0.376585i	0.137059	+	0.990563i	$0.456235\pi$
$42$		0			0
$43$	−3.10733			−0.0722635			−0.0361318	−	0.999347i	$-0.511504\pi$
$43$	−3.10733			−0.0722635			−0.0361318	+	0.999347i	$0.511504\pi$
$44$		0			0
$45$	−28.9836	+	26.1438i	−0.644079	+	0.580974i
$46$		0			0
$47$	34.8212	−	60.3120i	0.740876	−	1.28323i	−0.211221	−	0.977438i	$-0.567744\pi$
$47$	34.8212	−	60.3120i	0.740876	−	1.28323i	0.952097	−	0.305796i	$-0.0989226\pi$
$48$		0			0
$49$	21.1465	−	36.6268i	0.431562	−	0.747487i
$50$		0			0
$51$	−43.1118	+	16.5712i	−0.845330	+	0.324925i
$52$		0			0
$53$	−54.5703	+	31.5062i	−1.02963	+	0.594456i	−0.916878	−	0.399167i	$-0.869299\pi$
$53$	−54.5703	+	31.5062i	−1.02963	+	0.594456i	−0.112750	+	0.993623i	$0.535966\pi$
$54$		0			0
$55$	10.9920	−	19.0387i	0.199854	−	0.346157i
$56$		0			0
$57$	3.63962	−	56.8837i	0.0638529	−	0.997959i
$58$		0			0
$59$	−70.8303	+	40.8939i	−1.20051	+	0.693116i	−0.960669	−	0.277695i	$-0.910429\pi$
$59$	−70.8303	+	40.8939i	−1.20051	+	0.693116i	−0.239844	+	0.970812i	$0.577096\pi$
$60$		0			0
$61$	53.1069	−	91.9838i	0.870605	−	1.50793i	0.00923264	−	0.999957i	$-0.497061\pi$
$61$	53.1069	−	91.9838i	0.870605	−	1.50793i	0.861372	−	0.507974i	$-0.169606\pi$
$62$		0			0
$63$	17.3073	−	15.6116i	0.274719	−	0.247803i
$64$		0			0
$65$	−82.1865	+	47.4504i	−1.26441	+	0.730007i
$66$		0			0
$67$		−	13.1074i		−	0.195632i	−0.995205	−	0.0978161i	$-0.968814\pi$
$67$		−	13.1074i		−	0.195632i	0.995205	−	0.0978161i	$-0.0311857\pi$
$68$		0			0
$69$	−54.6253	−	44.1943i	−0.791671	−	0.640497i
$70$		0			0
$71$	81.6733	+	47.1541i	1.15033	+	0.664142i	0.948967	−	0.315375i	$-0.102130\pi$
$71$	81.6733	+	47.1541i	1.15033	+	0.664142i	0.201361	+	0.979517i	$0.435464\pi$
$72$		0			0
$73$	−52.8222	+	91.4908i	−0.723592	+	1.25330i	0.235958	+	0.971763i	$0.424177\pi$
$73$	−52.8222	+	91.4908i	−0.723592	+	1.25330i	−0.959551	+	0.281536i	$0.909156\pi$
$74$		0			0
$75$	17.3358	−	6.66348i	0.231144	−	0.0888464i
$76$		0			0
$77$	−6.56377	+	11.3688i	−0.0852437	+	0.147646i
$78$		0			0
$79$		−	26.5901i		−	0.336584i	−0.985737	−	0.168292i	$-0.946175\pi$
$79$		−	26.5901i		−	0.336584i	0.985737	−	0.168292i	$-0.0538252\pi$
$80$		0			0
$81$	−73.9382	+	33.0778i	−0.912818	+	0.408368i
$82$		0			0
$83$	36.6877	+	63.5449i	0.442020	+	0.765601i	0.997839	−	0.0657020i	$-0.0209287\pi$
$83$	36.6877	+	63.5449i	0.442020	+	0.765601i	−0.555819	+	0.831303i	$0.687595\pi$
$84$		0			0
$85$	−66.7703			−0.785533
$86$		0			0
$87$	48.8189	+	127.008i	0.561137	+	1.45986i
$88$		0			0
$89$	107.613	−	62.1305i	1.20914	−	0.698095i	0.246566	−	0.969126i	$-0.420698\pi$
$89$	107.613	−	62.1305i	1.20914	−	0.698095i	0.962571	+	0.271031i	$0.0873645\pi$
$90$		0			0
$91$	49.0770	−	28.3346i	0.539308	−	0.311370i
$92$		0			0
$93$	23.0818	+	60.0500i	0.248192	+	0.645699i
$94$		0			0
$95$	34.4156	−	74.8712i	0.362270	−	0.788117i
$96$		0			0
$97$			30.5295i			0.314738i	0.987540	+	0.157369i	$0.0503011\pi$
$97$			30.5295i			0.314738i	−0.987540	+	0.157369i	$0.949699\pi$
$98$		0			0
$99$	33.8756	−	30.5565i	0.342178	−	0.308652i
$100$		0			0
$101$	15.9016	−	27.5423i	0.157441	−	0.272696i	−0.776504	−	0.630112i	$-0.783009\pi$
$101$	15.9016	−	27.5423i	0.157441	−	0.272696i	0.933945	+	0.357416i	$0.116342\pi$
$102$		0			0
$103$	−97.2013	−	56.1192i	−0.943702	−	0.544847i	−0.0525834	−	0.998617i	$-0.516746\pi$
$103$	−97.2013	−	56.1192i	−0.943702	−	0.544847i	−0.891119	+	0.453770i	$0.850079\pi$
$104$		0			0
$105$	31.4519	−	12.0894i	0.299542	−	0.115137i
$106$		0			0
$107$			109.389i			1.02233i	0.859483	+	0.511164i	$0.170785\pi$
$107$			109.389i			1.02233i	−0.859483	+	0.511164i	$0.829215\pi$
$108$		0			0
$109$	−98.1817	−	56.6853i	−0.900750	−	0.520048i	−0.0233065	−	0.999728i	$-0.507419\pi$
$109$	−98.1817	−	56.6853i	−0.900750	−	0.520048i	−0.877443	+	0.479680i	$0.840753\pi$
$110$		0			0
$111$	−95.3550	+	117.861i	−0.859054	+	1.06181i
$112$		0			0
$113$	55.4318	+	32.0036i	0.490547	+	0.283217i	0.724801	−	0.688958i	$-0.241932\pi$
$113$	55.4318	+	32.0036i	0.490547	+	0.283217i	−0.234254	+	0.972175i	$0.575265\pi$
$114$		0			0
$115$	−50.7888	−	87.9689i	−0.441642	−	0.764947i
$116$		0			0
$117$	−192.597	+	41.1175i	−1.64613	+	0.351432i
$118$		0			0
$119$	39.8713			0.335053
$120$		0			0
$121$	47.6527	−	82.5370i	0.393824	−	0.682124i
$122$		0			0
$123$	−91.5058	−	14.4512i	−0.743950	−	0.117490i
$124$		0			0
$125$	135.273			1.08219
$126$		0			0
$127$	−74.5505	+	43.0418i	−0.587012	+	0.338911i	−0.763915	−	0.645317i	$-0.776725\pi$
$127$	−74.5505	+	43.0418i	−0.587012	+	0.338911i	0.176903	+	0.984228i	$0.443392\pi$
$128$		0			0
$129$	−7.24717	−	5.86329i	−0.0561796	−	0.0454518i
$130$		0			0
$131$	87.1902	+	151.018i	0.665574	+	1.15281i	0.979129	+	0.203238i	$0.0651465\pi$
$131$	87.1902	+	151.018i	0.665574	+	1.15281i	−0.313555	+	0.949570i	$0.601520\pi$
$132$		0			0
$133$	−20.5510	+	44.7087i	−0.154519	+	0.336156i
$134$		0			0
$135$	−116.929	+	6.28506i	−0.866141	+	0.0465560i
$136$		0			0
$137$	9.01544	−	15.6152i	0.0658062	−	0.113980i	−0.831245	−	0.555906i	$-0.812371\pi$
$137$	9.01544	−	15.6152i	0.0658062	−	0.113980i	0.897051	+	0.441926i	$0.145705\pi$
$138$		0			0
$139$	163.001			1.17267			0.586335	−	0.810069i	$-0.300570\pi$
$139$	163.001			1.17267			0.586335	+	0.810069i	$0.300570\pi$
$140$		0			0
$141$	195.017	−	74.9598i	1.38310	−	0.531630i
$142$		0			0
$143$	96.0585	−	55.4594i	0.671738	−	0.387828i
$144$		0			0
$145$			196.706i			1.35659i
$146$		0			0
$147$	118.432	−	45.5223i	0.805657	−	0.309675i
$148$		0			0
$149$	64.3447	+	111.448i	0.431843	+	0.747975i	0.997032	−	0.0769870i	$-0.0245300\pi$
$149$	64.3447	+	111.448i	0.431843	+	0.747975i	−0.565189	+	0.824962i	$0.691197\pi$
$150$		0			0
$151$	202.799	−	117.086i	1.34304	−	0.775404i	0.355787	−	0.934567i	$-0.384213\pi$
$151$	202.799	−	117.086i	1.34304	−	0.775404i	0.987252	+	0.159163i	$0.0508795\pi$
$152$		0			0
$153$	−131.817	−	42.6999i	−0.861551	−	0.279085i
$154$		0			0
$155$			93.0036i			0.600024i
$156$		0			0
$157$	−98.6463	−	170.860i	−0.628320	−	1.08828i	−0.987889	−	0.155164i	$-0.950410\pi$
$157$	−98.6463	−	170.860i	−0.628320	−	1.08828i	0.359569	−	0.933119i	$-0.382924\pi$
$158$		0			0
$159$	−186.723	−	29.4886i	−1.17436	−	0.185463i
$160$		0			0
$161$	30.3281	+	52.5299i	0.188374	+	0.326273i
$162$		0			0
$163$	−28.1217			−0.172526			−0.0862628	−	0.996272i	$-0.527492\pi$
$163$	−28.1217			−0.172526			−0.0862628	+	0.996272i	$0.527492\pi$
$164$		0			0
$165$	61.5608	−	23.6625i	0.373096	−	0.143409i
$166$		0			0
$167$		−	184.761i		−	1.10636i	−0.833063	−	0.553178i	$-0.813415\pi$
$167$		−	184.761i		−	1.10636i	0.833063	−	0.553178i	$-0.186585\pi$
$168$		0			0
$169$	−309.817			−1.83324
$170$		0			0
$171$	115.824	−	125.801i	0.677331	−	0.735679i
$172$		0			0
$173$			249.241i			1.44070i	0.693612	+	0.720349i	$0.256018\pi$
$173$			249.241i			1.44070i	−0.693612	+	0.720349i	$0.743982\pi$
$174$		0			0
$175$	−16.0328			−0.0916159
$176$		0			0
$177$	−242.359	−	38.2751i	−1.36926	−	0.216243i
$178$		0			0
$179$			127.328i			0.711328i	0.934614	+	0.355664i	$0.115745\pi$
$179$			127.328i			0.711328i	−0.934614	+	0.355664i	$0.884255\pi$
$180$		0			0
$181$	300.786	−	173.659i	1.66180	−	0.959440i	0.689943	−	0.723864i	$-0.257636\pi$
$181$	300.786	−	173.659i	1.66180	−	0.959440i	0.971856	−	0.235576i	$-0.0756976\pi$
$182$		0			0
$183$	297.426	−	114.324i	1.62528	−	0.624719i
$184$		0			0
$185$	−189.804	+	109.584i	−1.02597	+	0.592343i
$186$		0			0
$187$	78.0402			0.417327
$188$		0			0
$189$	69.8233	−	3.75307i	0.369435	−	0.0198575i
$190$		0			0
$191$	−22.7702	−	39.4392i	−0.119216	−	0.206488i	0.800241	−	0.599678i	$-0.204705\pi$
$191$	−22.7702	−	39.4392i	−0.119216	−	0.206488i	−0.919457	+	0.393190i	$0.871371\pi$
$192$		0			0
$193$	237.222	−	136.960i	1.22913	−	0.709638i	0.262282	−	0.964991i	$-0.415525\pi$
$193$	237.222	−	136.960i	1.22913	−	0.709638i	0.966848	+	0.255353i	$0.0821917\pi$
$194$		0			0
$195$	−281.217	−	44.4117i	−1.44214	−	0.227752i
$196$		0			0
$197$	137.224			0.696569			0.348285	−	0.937389i	$-0.386764\pi$
$197$	137.224			0.696569			0.348285	+	0.937389i	$0.386764\pi$
$198$		0			0
$199$	−35.9623	−	62.2886i	−0.180715	−	0.313008i	0.761409	−	0.648272i	$-0.224508\pi$
$199$	−35.9623	−	62.2886i	−0.180715	−	0.313008i	−0.942124	+	0.335264i	$0.891175\pi$
$200$		0			0
$201$	24.7325	−	30.5700i	0.123047	−	0.152090i
$202$		0			0
$203$		−	117.461i		−	0.578628i
$204$		0			0
$205$	−115.983	−	66.9626i	−0.565769	−	0.326647i
$206$		0			0
$207$	−44.0104	−	206.147i	−0.212610	−	0.995880i
$208$		0			0
$209$	−40.2245	+	87.5084i	−0.192462	+	0.418700i
$210$		0			0
$211$	312.499	−	180.422i	1.48104	−	0.855079i	0.481271	−	0.876572i	$-0.340175\pi$
$211$	312.499	−	180.422i	1.48104	−	0.855079i	0.999769	+	0.0214931i	$0.00684199\pi$
$212$		0			0
$213$	101.509	+	264.088i	0.476568	+	1.23985i
$214$		0			0
$215$	−6.73819	−	11.6709i	−0.0313404	−	0.0542832i
$216$		0			0
$217$		−	55.5364i		−	0.255928i
$218$		0			0
$219$	−295.832	+	113.711i	−1.35083	+	0.519228i
$220$		0			0
$221$	−291.752	−	168.443i	−1.32014	−	0.762185i
$222$		0			0
$223$		−	216.541i		−	0.971036i	−0.874227	−	0.485518i	$-0.838631\pi$
$223$		−	216.541i		−	0.971036i	0.874227	−	0.485518i	$-0.161369\pi$
$224$		0			0
$225$	53.0055	+	17.1702i	0.235580	+	0.0763120i
$226$		0			0
$227$	61.4046	−	35.4520i	0.270505	−	0.156176i	−0.358612	−	0.933487i	$-0.616750\pi$
$227$	61.4046	−	35.4520i	0.270505	−	0.156176i	0.629117	+	0.777311i	$0.283417\pi$
$228$		0			0
$229$	−147.217	+	254.987i	−0.642869	+	1.11348i	0.341920	+	0.939729i	$0.388923\pi$
$229$	−147.217	+	254.987i	−0.642869	+	1.11348i	−0.984789	+	0.173753i	$0.944411\pi$
$230$		0			0
$231$	−36.7605	+	14.1299i	−0.159136	+	0.0611683i
$232$		0			0
$233$	221.160	−	383.061i	0.949185	−	1.64404i	0.202038	−	0.979378i	$-0.435243\pi$
$233$	221.160	−	383.061i	0.949185	−	1.64404i	0.747147	−	0.664659i	$-0.231423\pi$
$234$		0			0
$235$	302.036			1.28526
$236$		0			0
$237$	50.1735	−	62.0157i	0.211702	−	0.261670i
$238$		0			0
$239$	184.572	−	319.688i	0.772268	−	1.33761i	−0.164048	−	0.986452i	$-0.552455\pi$
$239$	184.572	−	319.688i	0.772268	−	1.33761i	0.936317	−	0.351156i	$-0.114211\pi$
$240$		0			0
$241$	36.5643	+	21.1104i	0.151719	+	0.0875952i	0.573938	−	0.818899i	$-0.305415\pi$
$241$	36.5643	+	21.1104i	0.151719	+	0.0875952i	−0.422218	+	0.906494i	$0.638748\pi$
$242$		0			0
$243$	−234.860	−	62.3689i	−0.966501	−	0.256662i
$244$		0			0
$245$	183.423			0.748666
$246$		0			0
$247$	339.258	−	240.327i	1.37351	−	0.972985i
$248$		0			0
$249$	−34.3382	+	217.431i	−0.137904	+	0.873218i
$250$		0			0
$251$	213.785	+	370.287i	0.851734	+	1.47525i	0.879642	+	0.475636i	$0.157782\pi$
$251$	213.785	+	370.287i	0.851734	+	1.47525i	−0.0279079	+	0.999610i	$0.508885\pi$
$252$		0			0
$253$	59.3613	+	102.817i	0.234630	+	0.406390i
$254$		0			0
$255$	−155.727	−	125.990i	−0.610694	−	0.494079i
$256$		0			0
$257$			182.247i			0.709132i	0.935031	+	0.354566i	$0.115371\pi$
$257$			182.247i			0.709132i	−0.935031	+	0.354566i	$0.884629\pi$
$258$		0			0
$259$	113.340	−	65.4369i	0.437606	−	0.252652i
$260$		0			0
$261$	−125.795	+	388.336i	−0.481972	+	1.48788i
$262$		0			0
$263$	−25.9609			−0.0987105			−0.0493552	−	0.998781i	$-0.515717\pi$
$263$	−25.9609			−0.0987105			−0.0493552	+	0.998781i	$0.515717\pi$
$264$		0			0
$265$	−236.669	−	136.641i	−0.893091	−	0.515627i
$266$		0			0
$267$	368.219	+	58.1517i	1.37910	+	0.217797i
$268$		0			0
$269$	−79.2628	−	45.7624i	−0.294657	−	0.170121i	0.345383	−	0.938462i	$-0.387749\pi$
$269$	−79.2628	−	45.7624i	−0.294657	−	0.170121i	−0.640040	+	0.768341i	$0.721082\pi$
$270$		0			0
$271$	86.2540	−	149.396i	0.318281	−	0.551278i	−0.661849	−	0.749637i	$-0.730228\pi$
$271$	86.2540	−	149.396i	0.318281	−	0.551278i	0.980129	+	0.198359i	$0.0635612\pi$
$272$		0			0
$273$	167.927	+	26.5201i	0.615116	+	0.0971432i
$274$		0			0
$275$	−31.3810			−0.114113
$276$		0			0
$277$	−76.1738	−	131.937i	−0.274996	−	0.476306i	0.695138	−	0.718876i	$-0.255343\pi$
$277$	−76.1738	−	131.937i	−0.274996	−	0.476306i	−0.970134	+	0.242569i	$0.922010\pi$
$278$		0			0
$279$	−59.4763	+	183.607i	−0.213177	+	0.658090i
$280$		0			0
$281$	123.373	+	71.2294i	0.439050	+	0.253485i	0.703194	−	0.710998i	$-0.251756\pi$
$281$	123.373	+	71.2294i	0.439050	+	0.253485i	−0.264145	+	0.964483i	$0.585090\pi$
$282$		0			0
$283$	−74.9562	−	129.828i	−0.264863	−	0.458756i	0.702665	−	0.711521i	$-0.251993\pi$
$283$	−74.9562	−	129.828i	−0.264863	−	0.458756i	−0.967528	+	0.252765i	$0.918660\pi$
$284$		0			0
$285$	221.543	−	109.681i	0.777343	−	0.384846i
$286$		0			0
$287$	69.2580	+	39.9862i	0.241317	+	0.139325i
$288$		0			0
$289$	25.9871	+	45.0109i	0.0899206	+	0.155747i
$290$		0			0
$291$	−57.6068	+	71.2035i	−0.197962	+	0.244685i
$292$		0			0
$293$	−228.709	−	132.045i	−0.780576	−	0.450666i	0.0560582	−	0.998428i	$-0.482147\pi$
$293$	−228.709	−	132.045i	−0.780576	−	0.450666i	−0.836634	+	0.547762i	$0.815480\pi$
$294$		0			0
$295$	−307.188	−	177.355i	−1.04131	−	0.601203i
$296$		0			0
$297$	136.665	−	7.34589i	0.460152	−	0.0247336i
$298$		0			0
$299$		−	512.505i		−	1.71406i
$300$		0			0
$301$	4.02365	+	6.96917i	0.0133676	+	0.0231534i
$302$		0			0
$303$	89.0571	−	34.2314i	0.293918	−	0.112975i
$304$		0			0
$305$	460.645			1.51031
$306$		0			0
$307$	−521.476	−	301.074i	−1.69862	−	0.980699i	−0.947073	−	0.321019i	$-0.895975\pi$
$307$	−521.476	−	301.074i	−1.69862	−	0.980699i	−0.751547	−	0.659680i	$-0.770692\pi$
$308$		0			0
$309$	−120.808	−	314.297i	−0.390965	−	1.01714i
$310$		0			0
$311$	−36.3171	+	62.9031i	−0.116775	+	0.202261i	−0.918488	−	0.395449i	$-0.870589\pi$
$311$	−36.3171	+	62.9031i	−0.116775	+	0.202261i	0.801713	+	0.597710i	$0.203922\pi$
$312$		0			0
$313$	−67.3754	+	116.698i	−0.215257	+	0.372836i	−0.953352	−	0.301861i	$-0.902392\pi$
$313$	−67.3754	+	116.698i	−0.215257	+	0.372836i	0.738095	+	0.674697i	$0.235725\pi$
$314$		0			0
$315$	96.1663	+	31.1514i	0.305290	+	0.0988933i
$316$		0			0
$317$	7.39553	+	4.26981i	0.0233298	+	0.0134694i	0.511620	−	0.859212i	$-0.329046\pi$
$317$	7.39553	+	4.26981i	0.0233298	+	0.0134694i	−0.488290	+	0.872682i	$0.662379\pi$
$318$		0			0
$319$		−	229.907i		−	0.720713i
$320$		0			0
$321$	−206.408	+	255.126i	−0.643017	+	0.794785i
$322$		0			0
$323$	291.260	−	27.0882i	0.901735	−	0.0838645i
$324$		0			0
$325$	117.317	+	67.7331i	0.360976	+	0.208409i
$326$		0			0
$327$	−122.027	−	317.467i	−0.373171	−	0.970847i
$328$		0			0
$329$	−180.358			−0.548201
$330$		0			0
$331$	98.2815	−	56.7428i	0.296923	−	0.171428i	−0.344137	−	0.938919i	$-0.611828\pi$
$331$	98.2815	−	56.7428i	0.296923	−	0.171428i	0.641060	+	0.767491i	$0.278495\pi$
$332$		0			0
$333$	−444.789	+	94.9581i	−1.33570	+	0.285159i
$334$		0			0
$335$	49.2302	−	28.4230i	0.146956	−	0.0848449i
$336$		0			0
$337$	−45.1249	+	26.0529i	−0.133902	+	0.0773082i	−0.565455	−	0.824779i	$-0.691299\pi$
$337$	−45.1249	+	26.0529i	−0.133902	+	0.0773082i	0.431553	+	0.902088i	$0.357966\pi$
$338$		0			0
$339$	68.8944	+	179.237i	0.203228	+	0.528722i
$340$		0			0
$341$		−	108.701i		−	0.318772i
$342$		0			0
$343$	−236.429			−0.689297
$344$		0			0
$345$	47.5364	−	301.003i	0.137787	−	0.872471i
$346$		0			0
$347$	103.374	+	179.049i	0.297908	+	0.515991i	0.975657	−	0.219302i	$-0.0703779\pi$
$347$	103.374	+	179.049i	0.297908	+	0.515991i	−0.677749	+	0.735293i	$0.737045\pi$
$348$		0			0
$349$	161.919	+	280.451i	0.463950	+	0.803585i	0.999153	−	0.0411380i	$-0.0130983\pi$
$349$	161.919	+	280.451i	0.463950	+	0.803585i	−0.535203	+	0.844723i	$0.679765\pi$
$350$		0			0
$351$	−526.775	−	267.517i	−1.50078	−	0.762157i
$352$		0			0
$353$	−197.960	−	342.876i	−0.560792	−	0.971321i	−0.997428	−	0.0716821i	$-0.977163\pi$
$353$	−197.960	−	342.876i	−0.560792	−	0.971321i	0.436635	−	0.899639i	$-0.356170\pi$
$354$		0			0
$355$			409.011i			1.15214i
$356$		0			0
$357$	92.9911	+	75.2340i	0.260479	+	0.210740i
$358$		0			0
$359$	−107.272	+	185.800i	−0.298808	+	0.517550i	−0.975863	−	0.218382i	$-0.929922\pi$
$359$	−107.272	+	185.800i	−0.298808	+	0.517550i	0.677056	+	0.735932i	$0.263256\pi$
$360$		0			0
$361$	−119.751	+	340.560i	−0.331719	+	0.943378i
$362$		0			0
$363$	266.880	−	102.582i	0.735207	−	0.282596i
$364$		0			0
$365$	−458.176			−1.25528
$366$		0			0
$367$	263.539	−	456.463i	0.718089	−	1.24377i	−0.243667	−	0.969859i	$-0.578350\pi$
$367$	263.539	−	456.463i	0.718089	−	1.24377i	0.961756	−	0.273908i	$-0.0883164\pi$
$368$		0			0
$369$	−186.149	−	206.368i	−0.504469	−	0.559264i
$370$		0			0
$371$	141.325	+	81.5941i	0.380930	+	0.219930i
$372$		0			0
$373$	600.925	+	346.944i	1.61106	+	0.930145i	0.989125	+	0.147076i	$0.0469861\pi$
$373$	600.925	+	346.944i	1.61106	+	0.930145i	0.621934	+	0.783070i	$0.286347\pi$
$374$		0			0
$375$	315.495	+	255.250i	0.841320	+	0.680666i
$376$		0			0
$377$	−496.235	+	859.504i	−1.31627	+	2.27985i
$378$		0			0
$379$		−	229.436i		−	0.605373i	−0.953090	−	0.302687i	$-0.902116\pi$
$379$		−	229.436i		−	0.605373i	0.953090	−	0.302687i	$-0.0978835\pi$
$380$		0			0
$381$	−255.089	−	40.2854i	−0.669525	−	0.105736i
$382$		0			0
$383$	−325.033	+	187.658i	−0.848651	+	0.489969i	−0.860196	−	0.509964i	$-0.829659\pi$
$383$	−325.033	+	187.658i	−0.848651	+	0.489969i	0.0115443	+	0.999933i	$0.496325\pi$
$384$		0			0
$385$	−56.9336			−0.147879
$386$		0			0
$387$	−5.83888	−	27.3497i	−0.0150875	−	0.0706710i
$388$		0			0
$389$	172.066	−	298.027i	0.442330	−	0.766137i	−0.555532	−	0.831495i	$-0.687485\pi$
$389$	172.066	−	298.027i	0.442330	−	0.766137i	0.997862	+	0.0653576i	$0.0208188\pi$
$390$		0			0
$391$	180.294	−	312.278i	0.461110	−	0.798665i
$392$		0			0
$393$	−81.6066	+	516.737i	−0.207650	+	1.31485i
$394$		0			0
$395$	99.8704	−	57.6602i	0.252836	−	0.145975i
$396$		0			0
$397$	135.167	−	234.117i	0.340472	−	0.589714i	−0.644049	−	0.764984i	$-0.722747\pi$
$397$	135.167	−	234.117i	0.340472	−	0.589714i	0.984520	+	0.175270i	$0.0560799\pi$
$398$		0			0
$399$	−132.292	+	65.4951i	−0.331560	+	0.164148i
$400$		0			0
$401$	−428.105	+	247.167i	−1.06759	+	0.616376i	−0.927523	−	0.373766i	$-0.878066\pi$
$401$	−428.105	+	247.167i	−1.06759	+	0.616376i	−0.140071	+	0.990141i	$0.544733\pi$
$402$		0			0
$403$	−234.622	+	406.378i	−0.582190	+	1.00838i
$404$		0			0
$405$	−284.571	−	205.977i	−0.702644	−	0.508586i
$406$		0			0
$407$	221.841	−	128.080i	0.545063	−	0.314692i
$408$		0			0
$409$			667.431i			1.63186i	0.578150	+	0.815930i	$0.303775\pi$
$409$			667.431i			1.63186i	−0.578150	+	0.815930i	$0.696225\pi$
$410$		0			0
$411$	50.4912	−	19.4076i	0.122850	−	0.0472205i
$412$		0			0
$413$	183.435	+	105.906i	0.444152	+	0.256431i
$414$		0			0
$415$	−159.113	+	275.592i	−0.383405	+	0.664076i
$416$		0			0
$417$	380.164	+	307.570i	0.911665	+	0.737578i
$418$		0			0
$419$	−27.4769	+	47.5913i	−0.0655773	+	0.113583i	−0.896950	−	0.442132i	$-0.854222\pi$
$419$	−27.4769	+	47.5913i	−0.0655773	+	0.113583i	0.831373	+	0.555715i	$0.187556\pi$
$420$		0			0
$421$			535.684i			1.27241i	0.771521	+	0.636204i	$0.219496\pi$
$421$			535.684i			1.27241i	−0.771521	+	0.636204i	$0.780504\pi$
$422$		0			0
$423$	596.277	+	193.154i	1.40964	+	0.456628i
$424$		0			0
$425$	47.6556	+	82.5419i	0.112131	+	0.194216i
$426$		0			0
$427$	−275.070			−0.644193
$428$		0			0
$429$	328.683	+	51.9078i	0.766161	+	0.120997i
$430$		0			0
$431$	−65.6385	+	37.8964i	−0.152294	+	0.0879267i	−0.574210	−	0.818708i	$-0.694691\pi$
$431$	−65.6385	+	37.8964i	−0.152294	+	0.0879267i	0.421917	+	0.906635i	$0.361357\pi$
$432$		0			0
$433$	238.225	−	137.539i	0.550173	−	0.317642i	−0.199019	−	0.979996i	$-0.563776\pi$
$433$	238.225	−	137.539i	0.550173	−	0.317642i	0.749192	+	0.662353i	$0.230442\pi$
$434$		0			0
$435$	−371.169	+	458.774i	−0.853262	+	1.05465i
$436$		0			0
$437$	257.236	+	363.126i	0.588640	+	0.830953i
$438$		0			0
$439$			76.3724i			0.173969i	0.996210	+	0.0869845i	$0.0277230\pi$
$439$			76.3724i			0.173969i	−0.996210	+	0.0869845i	$0.972277\pi$
$440$		0			0
$441$	362.113	+	117.300i	0.821117	+	0.265987i
$442$		0			0
$443$	−293.656	+	508.626i	−0.662879	+	1.14814i	0.316976	+	0.948434i	$0.397333\pi$
$443$	−293.656	+	508.626i	−0.662879	+	1.14814i	−0.979856	+	0.199707i	$0.936001\pi$
$444$		0			0
$445$	466.714	+	269.457i	1.04880	+	0.605522i
$446$		0			0
$447$	−60.2241	+	381.342i	−0.134729	+	0.853114i
$448$		0			0
$449$			29.7194i			0.0661903i	0.999452	+	0.0330951i	$0.0105364\pi$
$449$			29.7194i			0.0661903i	−0.999452	+	0.0330951i	$0.989464\pi$
$450$		0			0
$451$	135.559	+	78.2649i	0.300574	+	0.173536i
$452$		0			0
$453$	693.916	+	109.588i	1.53182	+	0.241916i
$454$		0			0
$455$	212.845	+	122.886i	0.467791	+	0.270079i
$456$		0			0
$457$	−186.684	−	323.347i	−0.408500	−	0.707543i	0.586222	−	0.810150i	$-0.300615\pi$
$457$	−186.684	−	323.347i	−0.408500	−	0.707543i	−0.994722	+	0.102608i	$0.967281\pi$
$458$		0			0
$459$	−226.864	−	348.317i	−0.494256	−	0.758861i
$460$		0			0
$461$	−313.694			−0.680464			−0.340232	−	0.940342i	$-0.610506\pi$
$461$	−313.694			−0.680464			−0.340232	+	0.940342i	$0.610506\pi$
$462$		0			0
$463$	−370.575	+	641.855i	−0.800379	+	1.38630i	0.118988	+	0.992896i	$0.462035\pi$
$463$	−370.575	+	641.855i	−0.800379	+	1.38630i	−0.919367	+	0.393401i	$0.871298\pi$
$464$		0			0
$465$	−175.490	+	216.911i	−0.377399	+	0.466474i
$466$		0			0
$467$	−249.203			−0.533626			−0.266813	−	0.963748i	$-0.585971\pi$
$467$	−249.203			−0.533626			−0.266813	+	0.963748i	$0.585971\pi$
$468$		0			0
$469$	−29.3974	+	16.9726i	−0.0626810	+	0.0361889i
$470$		0			0
$471$	92.3290	−	584.632i	0.196028	−	1.24126i
$472$		0			0
$473$	7.87550	+	13.6408i	0.0166501	+	0.0288388i
$474$		0			0
$475$	−117.120	+	10.8925i	−0.246567	+	0.0229316i
$476$		0			0
$477$	−379.848	−	421.107i	−0.796327	−	0.882824i
$478$		0			0
$479$	288.598	−	499.866i	0.602501	−	1.04356i	−0.389940	−	0.920840i	$-0.627504\pi$
$479$	288.598	−	499.866i	0.602501	−	1.04356i	0.992441	−	0.122722i	$-0.0391624\pi$
$480$		0			0
$481$	−1105.80			−2.29895
$482$		0			0
$483$	−28.3859	+	179.741i	−0.0587701	+	0.372135i
$484$		0			0
$485$	−114.666	+	66.2027i	−0.236426	+	0.136500i
$486$		0			0
$487$			255.758i			0.525170i	0.964909	+	0.262585i	$0.0845751\pi$
$487$			255.758i			0.525170i	−0.964909	+	0.262585i	$0.915425\pi$
$488$		0			0
$489$	−65.5876	−	53.0633i	−0.134126	−	0.108514i
$490$		0			0
$491$	423.273	+	733.130i	0.862063	+	1.49314i	0.869934	+	0.493168i	$0.164161\pi$
$491$	423.273	+	733.130i	0.862063	+	1.49314i	−0.00787085	+	0.999969i	$0.502505\pi$
$492$		0			0
$493$	−604.730	+	349.141i	−1.22663	+	0.708196i
$494$		0			0
$495$	188.226	+	60.9727i	0.380255	+	0.123177i
$496$		0			0
$497$		−	244.237i		−	0.491423i
$498$		0			0
$499$	−217.974	−	377.542i	−0.436822	−	0.756598i	0.560620	−	0.828073i	$-0.310563\pi$
$499$	−217.974	−	377.542i	−0.436822	−	0.756598i	−0.997442	+	0.0714751i	$0.977229\pi$
$500$		0			0
$501$	348.630	−	430.915i	0.695868	−	0.860111i
$502$		0			0
$503$	58.6957	+	101.664i	0.116691	+	0.202115i	0.918455	−	0.395527i	$-0.129438\pi$
$503$	58.6957	+	101.664i	0.116691	+	0.202115i	−0.801763	+	0.597642i	$0.796105\pi$
$504$		0			0
$505$	137.929			0.273126
$506$		0			0
$507$	−722.580	−	584.600i	−1.42521	−	1.15306i
$508$		0			0
$509$		−	144.358i		−	0.283611i	−0.989895	−	0.141806i	$-0.954709\pi$
$509$		−	144.358i		−	0.283611i	0.989895	−	0.141806i	$-0.0452908\pi$
$510$		0			0
$511$	273.596			0.535413
$512$		0			0
$513$	507.510	−	74.8535i	0.989297	−	0.145913i
$514$		0			0
$515$		−	486.774i		−	0.945191i
$516$		0			0
$517$	−353.015			−0.682815
$518$		0			0
$519$	−470.297	+	581.299i	−0.906161	+	1.12004i
$520$		0			0
$521$			325.337i			0.624447i	0.950009	+	0.312224i	$0.101074\pi$
$521$			325.337i			0.624447i	−0.950009	+	0.312224i	$0.898926\pi$
$522$		0			0
$523$	−264.213	+	152.544i	−0.505188	+	0.291670i	−0.730853	−	0.682534i	$-0.760878\pi$
$523$	−264.213	+	152.544i	−0.505188	+	0.291670i	0.225665	+	0.974205i	$0.427544\pi$
$524$		0			0
$525$	−37.3929	−	30.2526i	−0.0712246	−	0.0576239i
$526$		0			0
$527$	−285.919	+	165.075i	−0.542541	+	0.313236i
$528$		0			0
$529$	19.5623			0.0369798
$530$		0			0
$531$	−493.028	−	546.581i	−0.928490	−	1.02934i
$532$		0			0
$533$	−337.856	−	585.184i	−0.633876	−	1.09791i
$534$		0			0
$535$	−410.856	+	237.208i	−0.767956	+	0.443379i
$536$		0			0
$537$	−240.257	+	296.964i	−0.447406	+	0.553006i
$538$		0			0
$539$	−214.382			−0.397741
$540$		0			0
$541$	−181.001	−	313.502i	−0.334567	−	0.579487i	0.648835	−	0.760929i	$-0.275257\pi$
$541$	−181.001	−	313.502i	−0.334567	−	0.579487i	−0.983402	+	0.181443i	$0.941923\pi$
$542$		0			0
$543$	1029.20	+	162.538i	1.89539	+	0.299333i
$544$		0			0
$545$		−	491.683i		−	0.902171i
$546$		0			0
$547$	574.565	+	331.725i	1.05039	+	0.606444i	0.922759	−	0.385377i	$-0.125929\pi$
$547$	574.565	+	331.725i	1.05039	+	0.606444i	0.127633	+	0.991821i	$0.459262\pi$
$548$		0			0
$549$	909.401	+	294.585i	1.65647	+	0.536584i
$550$		0			0
$551$	−79.8023	−	858.057i	−0.144832	−	1.55727i
$552$		0			0
$553$	−59.6368	+	34.4313i	−0.107842	+	0.0622628i
$554$		0			0
$555$	−649.452	−	102.566i	−1.17018	−	0.184803i
$556$		0			0
$557$	−439.853	−	761.848i	−0.789683	−	1.36777i	−0.926161	−	0.377127i	$-0.876912\pi$
$557$	−439.853	−	761.848i	−0.789683	−	1.36777i	0.136479	−	0.990643i	$-0.456421\pi$
$558$		0			0
$559$		−	67.9943i		−	0.121636i
$560$		0			0
$561$	182.012	+	147.256i	0.324441	+	0.262488i
$562$		0			0
$563$	−9.20040	−	5.31185i	−0.0163417	−	0.00943491i	0.491807	−	0.870704i	$-0.336337\pi$
$563$	−9.20040	−	5.31185i	−0.0163417	−	0.00943491i	−0.508149	+	0.861269i	$0.669670\pi$
$564$		0			0
$565$			277.596i			0.491321i
$566$		0			0
$567$	169.929	+	122.998i	0.299699	+	0.216927i
$568$		0			0
$569$	−88.8009	+	51.2692i	−0.156065	+	0.0901041i	−0.575999	−	0.817451i	$-0.695387\pi$
$569$	−88.8009	+	51.2692i	−0.156065	+	0.0901041i	0.419934	+	0.907555i	$0.362053\pi$
$570$		0			0
$571$	−170.885	+	295.981i	−0.299273	+	0.518355i	−0.975970	−	0.217906i	$-0.930077\pi$
$571$	−170.885	+	295.981i	−0.299273	+	0.518355i	0.676697	+	0.736261i	$0.263411\pi$
$572$		0			0
$573$	21.3120	−	134.949i	0.0371938	−	0.235513i
$574$		0			0
$575$	−72.4985	+	125.571i	−0.126084	+	0.218384i
$576$		0			0
$577$	−113.080			−0.195980			−0.0979900	−	0.995187i	$-0.531241\pi$
$577$	−113.080			−0.195980			−0.0979900	+	0.995187i	$0.531241\pi$
$578$		0			0
$579$	811.701	+	128.189i	1.40190	+	0.221398i
$580$		0			0
$581$	95.0130	−	164.567i	0.163533	−	0.283248i
$582$		0			0
$583$	276.616	+	159.704i	0.474469	+	0.273935i
$584$		0			0
$585$	−572.076	−	634.215i	−0.977908	−	1.08413i
$586$		0			0
$587$	−463.608			−0.789792			−0.394896	−	0.918726i	$-0.629219\pi$
$587$	−463.608			−0.789792			−0.394896	+	0.918726i	$0.629219\pi$
$588$		0			0
$589$	−37.7309	−	405.694i	−0.0640593	−	0.688784i
$590$		0			0
$591$	320.045	+	258.931i	0.541532	+	0.438123i
$592$		0			0
$593$	−212.766	−	368.522i	−0.358796	−	0.621454i	0.628964	−	0.777435i	$-0.283479\pi$
$593$	−212.766	−	368.522i	−0.358796	−	0.621454i	−0.987760	+	0.155981i	$0.950146\pi$
$594$		0			0
$595$	86.4602	+	149.753i	0.145311	+	0.251686i
$596$		0			0
$597$	33.6593	−	213.133i	0.0563808	−	0.357006i
$598$		0			0
$599$			558.072i			0.931672i	0.884871	+	0.465836i	$0.154246\pi$
$599$			558.072i			0.931672i	−0.884871	+	0.465836i	$0.845754\pi$
$600$		0			0
$601$	43.1112	−	24.8902i	0.0717324	−	0.0414147i	−0.463705	−	0.885990i	$-0.653480\pi$
$601$	43.1112	−	24.8902i	0.0717324	−	0.0414147i	0.535437	+	0.844575i	$0.320147\pi$
$602$		0			0
$603$	115.366	−	24.6296i	0.191321	−	0.0408451i
$604$		0			0
$605$	413.336			0.683200
$606$		0			0
$607$	666.122	+	384.586i	1.09740	+	0.633585i	0.935537	−	0.353228i	$-0.114916\pi$
$607$	666.122	+	384.586i	1.09740	+	0.633585i	0.161864	+	0.986813i	$0.448249\pi$
$608$		0			0
$609$	221.640	−	273.953i	0.363942	−	0.449841i
$610$		0			0
$611$	1319.74	+	761.953i	2.15997	+	1.24706i
$612$		0			0
$613$	4.67669	−	8.10026i	0.00762918	−	0.0132141i	−0.862186	−	0.506593i	$-0.830905\pi$
$613$	4.67669	−	8.10026i	0.00762918	−	0.0132141i	0.869815	+	0.493378i	$0.164238\pi$
$614$		0			0
$615$	−144.151	−	375.025i	−0.234392	−	0.609797i
$616$		0			0
$617$	−499.671			−0.809839			−0.404920	−	0.914352i	$-0.632701\pi$
$617$	−499.671			−0.809839			−0.404920	+	0.914352i	$0.632701\pi$
$618$		0			0
$619$	−0.402372	−	0.696929i	−0.000650036	−	0.00112590i	0.865700	−	0.500563i	$-0.166874\pi$
$619$	−0.402372	−	0.696929i	−0.000650036	−	0.00112590i	−0.866350	+	0.499437i	$0.833540\pi$
$620$		0			0
$621$	286.339	−	563.837i	0.461093	−	0.907950i
$622$		0			0
$623$	−278.694	−	160.904i	−0.447342	−	0.258273i
$624$		0			0
$625$	215.952	+	374.040i	0.345524	+	0.598464i
$626$		0			0
$627$	−258.936	+	128.194i	−0.412976	+	0.204456i
$628$		0			0
$629$	−673.781	−	389.007i	−1.07119	−	0.618454i
$630$		0			0
$631$	−303.378	−	525.466i	−0.480789	−	0.832751i	0.518968	−	0.854794i	$-0.326316\pi$
$631$	−303.378	−	525.466i	−0.480789	−	0.832751i	−0.999757	+	0.0220423i	$0.992983\pi$
$632$		0			0
$633$	1069.28	+	168.868i	1.68922	+	0.266773i
$634$		0			0
$635$	−323.323	−	186.670i	−0.509169	−	0.293969i
$636$		0			0
$637$	801.465	+	462.726i	1.25819	+	0.726414i
$638$		0			0
$639$	−261.565	+	807.466i	−0.409334	+	1.26364i
$640$		0			0
$641$		−	922.538i		−	1.43922i	−0.694380	−	0.719609i	$-0.744321\pi$
$641$		−	922.538i		−	1.43922i	0.694380	−	0.719609i	$-0.255679\pi$
$642$		0			0
$643$	51.3500	+	88.9409i	0.0798601	+	0.138322i	0.903189	−	0.429242i	$-0.141219\pi$
$643$	51.3500	+	88.9409i	0.0798601	+	0.138322i	−0.823329	+	0.567564i	$0.807886\pi$
$644$		0			0
$645$	6.30668	−	39.9342i	0.00977779	−	0.0619135i
$646$		0			0
$647$	−504.674			−0.780022			−0.390011	−	0.920810i	$-0.627529\pi$
$647$	−504.674			−0.780022			−0.390011	+	0.920810i	$0.627529\pi$
$648$		0			0
$649$	359.037	+	207.290i	0.553216	+	0.319399i
$650$		0			0
$651$	104.793	−	129.526i	0.160972	−	0.198965i
$652$		0			0
$653$	43.1142	−	74.6761i	0.0660249	−	0.114358i	−0.831123	−	0.556088i	$-0.812302\pi$
$653$	43.1142	−	74.6761i	0.0660249	−	0.114358i	0.897148	+	0.441730i	$0.145635\pi$
$654$		0			0
$655$	−378.140	+	654.958i	−0.577313	+	0.999936i
$656$		0			0
$657$	−904.527	−	293.006i	−1.37675	−	0.445975i
$658$		0			0
$659$	839.278	+	484.557i	1.27356	+	0.735292i	0.975657	−	0.219303i	$-0.0703784\pi$
$659$	839.278	+	484.557i	1.27356	+	0.735292i	0.297906	+	0.954595i	$0.403712\pi$
$660$		0			0
$661$			373.018i			0.564323i	0.959367	+	0.282162i	$0.0910515\pi$
$661$			373.018i			0.564323i	−0.959367	+	0.282162i	$0.908949\pi$
$662$		0			0
$663$	−362.608	−	943.368i	−0.546921	−	1.42288i
$664$		0			0
$665$	−212.487	+	19.7620i	−0.319529	+	0.0297173i
$666$		0			0
$667$	−919.976	−	531.148i	−1.37927	−	0.796324i
$668$		0			0
$669$	408.595	−	505.034i	0.610756	−	0.754909i
$670$		0			0
$671$	−538.395			−0.802378
$672$		0			0
$673$	−862.461	+	497.942i	−1.28152	+	0.739885i	−0.977126	−	0.212662i	$-0.931787\pi$
$673$	−862.461	+	497.942i	−1.28152	+	0.739885i	−0.304392	+	0.952547i	$0.598453\pi$
$674$		0			0
$675$	91.2248	+	140.063i	0.135148	+	0.207500i
$676$		0			0
$677$	−519.176	+	299.746i	−0.766878	+	0.442757i	−0.831760	−	0.555136i	$-0.812666\pi$
$677$	−519.176	+	299.746i	−0.766878	+	0.442757i	0.0648820	+	0.997893i	$0.479333\pi$
$678$		0			0
$679$	68.4721	−	39.5324i	0.100843	−	0.0582215i
$680$		0			0
$681$	210.108	+	33.1816i	0.308528	+	0.0487249i
$682$		0			0
$683$		−	215.711i		−	0.315829i	−0.987453	−	0.157915i	$-0.949523\pi$
$683$		−	215.711i		−	0.315829i	0.987453	−	0.157915i	$-0.0504771\pi$
$684$		0			0
$685$	78.1992			0.114159
$686$		0			0
$687$	−824.492	+	316.915i	−1.20013	+	0.461303i
$688$		0			0
$689$	−689.415	−	1194.10i	−1.00060	−	1.73309i
$690$		0			0
$691$	−43.0756	−	74.6090i	−0.0623380	−	0.107973i	0.833172	−	0.553014i	$-0.186522\pi$
$691$	−43.0756	−	74.6090i	−0.0623380	−	0.107973i	−0.895510	+	0.445041i	$0.853189\pi$
$692$		0			0
$693$	−112.398	−	36.4093i	−0.162190	−	0.0525387i
$694$		0			0
$695$	353.465	+	612.219i	0.508582	+	0.880890i
$696$		0			0
$697$		−	475.417i		−	0.682090i
$698$		0			0
$699$	1238.61	−	476.093i	1.77198	−	0.681106i
$700$		0			0
$701$	275.942	−	477.945i	0.393640	−	0.681805i	−0.599287	−	0.800535i	$-0.704549\pi$
$701$	275.942	−	477.945i	0.393640	−	0.681805i	0.992927	+	0.118730i	$0.0378823\pi$
$702$		0			0
$703$	783.493	−	555.019i	1.11450	−	0.789501i
$704$		0			0
$705$	704.432	+	569.918i	0.999195	+	0.808394i
$706$		0			0
$707$	−82.3631			−0.116497
$708$		0			0
$709$	660.256	−	1143.60i	0.931250	−	1.61297i	0.150061	−	0.988677i	$-0.452053\pi$
$709$	660.256	−	1143.60i	0.931250	−	1.61297i	0.781189	−	0.624295i	$-0.214614\pi$
$710$		0			0
$711$	234.037	−	49.9646i	0.329166	−	0.0702738i
$712$		0			0
$713$	−434.969	−	251.129i	−0.610055	−	0.352215i
$714$		0			0
$715$	416.602	+	240.525i	0.582660	+	0.336399i
$716$		0			0
$717$	1033.70	−	397.330i	1.44170	−	0.554156i
$718$		0			0
$719$	−621.273	+	1076.08i	−0.864079	+	1.49663i	0.00387879	+	0.999992i	$0.498765\pi$
$719$	−621.273	+	1076.08i	−0.864079	+	1.49663i	−0.867958	+	0.496637i	$0.834568\pi$
$720$		0			0
$721$			290.673i			0.403152i
$722$		0			0
$723$	45.4446	+	118.229i	0.0628556	+	0.163526i
$724$		0			0
$725$	243.170	−	140.394i	0.335406	−	0.193647i
$726$		0			0
$727$	390.792			0.537540			0.268770	−	0.963204i	$-0.413383\pi$
$727$	390.792			0.537540			0.268770	+	0.963204i	$0.413383\pi$
$728$		0			0
$729$	−430.074	−	588.623i	−0.589950	−	0.807439i
$730$		0			0
$731$	23.9197	−	41.4301i	0.0327219	−	0.0566760i
$732$		0			0
$733$	474.561	−	821.964i	0.647423	−	1.12137i	−0.336314	−	0.941750i	$-0.609180\pi$
$733$	474.561	−	821.964i	0.647423	−	1.12137i	0.983736	−	0.179619i	$-0.0574865\pi$
$734$		0			0
$735$	427.794	+	346.105i	0.582033	+	0.470891i
$736$		0			0
$737$	−57.5395	+	33.2205i	−0.0780726	+	0.0450753i
$738$		0			0
$739$	−77.2261	+	133.759i	−0.104501	+	0.181001i	−0.913534	−	0.406762i	$-0.866658\pi$
$739$	−77.2261	+	133.759i	−0.104501	+	0.181001i	0.809033	+	0.587763i	$0.199991\pi$
$740$		0			0
$741$	1244.72	+	79.6417i	1.67979	+	0.107479i
$742$		0			0
$743$	−2.39045	+	1.38013i	−0.00321730	+	0.00185751i	−0.501608	−	0.865095i	$-0.667258\pi$
$743$	−2.39045	+	1.38013i	−0.00321730	+	0.00185751i	0.498390	+	0.866953i	$0.333925\pi$
$744$		0			0
$745$	−279.060	+	483.346i	−0.374577	+	0.648787i
$746$		0			0
$747$	−490.362	+	442.317i	−0.656441	+	0.592125i
$748$		0			0
$749$	245.339	−	141.647i	0.327556	−	0.189115i
$750$		0			0
$751$		−	259.268i		−	0.345231i	−0.984989	−	0.172615i	$-0.944778\pi$
$751$		−	259.268i		−	0.345231i	0.984989	−	0.172615i	$-0.0552218\pi$
$752$		0			0
$753$	−200.095	+	1267.01i	−0.265730	+	1.68261i
$754$		0			0
$755$	879.531	+	507.797i	1.16494	+	0.672579i
$756$		0			0
$757$	−396.040	+	685.962i	−0.523171	+	0.906158i	0.476466	+	0.879193i	$0.341918\pi$
$757$	−396.040	+	685.962i	−0.523171	+	0.906158i	−0.999636	+	0.0269652i	$0.991416\pi$
$758$		0			0
$759$	−55.5598	+	351.808i	−0.0732014	+	0.463515i
$760$		0			0
$761$	−730.012	+	1264.42i	−0.959280	+	1.66152i	−0.235026	+	0.971989i	$0.575518\pi$
$761$	−730.012	+	1264.42i	−0.959280	+	1.66152i	−0.724254	+	0.689533i	$0.757816\pi$
$762$		0			0
$763$			293.605i			0.384803i
$764$		0			0
$765$	−125.466	−	587.689i	−0.164008	−	0.768221i
$766$		0			0
$767$	−894.835	−	1549.90i	−1.16667	−	2.02073i
$768$		0			0
$769$	−597.342			−0.776778			−0.388389	−	0.921495i	$-0.626968\pi$
$769$	−597.342			−0.776778			−0.388389	+	0.921495i	$0.626968\pi$
$770$		0			0
$771$	−343.885	+	425.051i	−0.446025	+	0.551298i
$772$		0			0
$773$	160.868	−	92.8774i	0.208109	−	0.120152i	−0.392323	−	0.919827i	$-0.628329\pi$
$773$	160.868	−	92.8774i	0.208109	−	0.120152i	0.600432	+	0.799676i	$0.294995\pi$
$774$		0			0
$775$	114.972	−	66.3790i	0.148351	−	0.0856503i
$776$		0			0
$777$	387.815	+	61.2464i	0.499118	+	0.0788242i
$778$		0			0
$779$	533.097	+	245.046i	0.684335	+	0.314565i
$780$		0			0
$781$		−	478.046i		−	0.612095i
$782$		0			0
$783$	−1026.15	+	668.343i	−1.31053	+	0.853568i
$784$		0			0
$785$	427.825	−	741.014i	0.545000	−	0.943967i
$786$		0			0
$787$	−490.403	−	283.134i	−0.623129	−	0.359764i	0.154957	−	0.987921i	$-0.450476\pi$
$787$	−490.403	−	283.134i	−0.623129	−	0.359764i	−0.778086	+	0.628157i	$0.783809\pi$
$788$		0			0
$789$	−60.5480	−	48.9861i	−0.0767402	−	0.0620863i
$790$		0			0
$791$		−	165.764i		−	0.209563i
$792$		0			0
$793$	2012.78	+	1162.08i	2.53818	+	1.46542i
$794$		0			0
$795$	−294.148	−	765.261i	−0.369998	−	0.962593i
$796$		0			0
$797$	−989.785	−	571.452i	−1.24189	−	0.717004i	−0.272410	−	0.962181i	$-0.587821\pi$
$797$	−989.785	−	571.452i	−1.24189	−	0.717004i	−0.969478	+	0.245177i	$0.921154\pi$
$798$		0			0
$799$	536.094	+	928.542i	0.670957	+	1.16213i
$800$		0			0
$801$	749.063	+	830.426i	0.935160	+	1.03674i
$802$		0			0
$803$	535.510			0.666886
$804$		0			0
$805$	−131.532	+	227.820i	−0.163394	+	0.283006i
$806$		0			0
$807$	−98.5131	−	256.293i	−0.122073	−	0.317588i
$808$		0			0
$809$	1034.93			1.27927			0.639634	−	0.768680i	$-0.279086\pi$
$809$	1034.93			1.27927			0.639634	+	0.768680i	$0.279086\pi$
$810$		0			0
$811$	−346.249	+	199.907i	−0.426940	+	0.246494i	−0.698042	−	0.716057i	$-0.745945\pi$
$811$	−346.249	+	199.907i	−0.426940	+	0.246494i	0.271102	+	0.962551i	$0.412612\pi$
$812$		0			0
$813$	483.068	−	185.680i	0.594179	−	0.228388i
$814$		0			0
$815$	−60.9812	−	105.623i	−0.0748236	−	0.129598i
$816$		0			0
$817$	34.1276	+	48.1762i	0.0417719	+	0.0589672i
$818$		0			0
$819$	341.611	+	378.716i	0.417107	+	0.462413i
$820$		0			0
$821$	802.388	−	1389.78i	0.977331	−	1.69279i	0.305312	−	0.952252i	$-0.401239\pi$
$821$	802.388	−	1389.78i	0.977331	−	1.69279i	0.672019	−	0.740534i	$-0.265427\pi$
$822$		0			0
$823$	312.370			0.379550			0.189775	−	0.981828i	$-0.439224\pi$
$823$	312.370			0.379550			0.189775	+	0.981828i	$0.439224\pi$
$824$		0			0
$825$	−73.1892	−	59.2134i	−0.0887142	−	0.0717738i
$826$		0			0
$827$	284.328	−	164.157i	0.343807	−	0.198497i	−0.318147	−	0.948041i	$-0.603061\pi$
$827$	284.328	−	164.157i	0.343807	−	0.198497i	0.661954	+	0.749544i	$0.269727\pi$
$828$		0			0
$829$		−	104.537i		−	0.126100i	−0.998010	−	0.0630500i	$-0.979917\pi$
$829$		−	104.537i		−	0.126100i	0.998010	−	0.0630500i	$-0.0200828\pi$
$830$		0			0
$831$	71.2957	−	451.448i	0.0857951	−	0.543258i
$832$		0			0
$833$	325.564	+	563.894i	0.390834	+	0.676944i
$834$		0			0
$835$	693.948	−	400.651i	0.831076	−	0.479822i
$836$		0			0
$837$	−485.167	+	315.996i	−0.579650	+	0.377534i
$838$		0			0
$839$		−	871.633i		−	1.03889i	−0.854503	−	0.519447i	$-0.826138\pi$
$839$		−	871.633i		−	1.03889i	0.854503	−	0.519447i	$-0.173862\pi$
$840$		0			0
$841$	608.074	+	1053.21i	0.723036	+	1.25234i
$842$		0			0
$843$	153.336	+	398.922i	0.181893	+	0.473217i
$844$		0			0
$845$	−671.832	−	1163.65i	−0.795067	−	1.37710i
$846$		0			0
$847$	−246.820			−0.291405
$848$		0			0
$849$	70.1561	−	444.232i	0.0826338	−	0.523241i
$850$		0			0
$851$		−	1183.59i		−	1.39083i
$852$		0			0
$853$	−165.121			−0.193577			−0.0967883	−	0.995305i	$-0.530857\pi$
$853$	−165.121			−0.193577			−0.0967883	+	0.995305i	$0.530857\pi$
$854$		0			0
$855$	723.659	+	162.227i	0.846385	+	0.189739i
$856$		0			0
$857$		−	629.371i		−	0.734389i	−0.930144	−	0.367194i	$-0.880318\pi$
$857$		−	629.371i		−	0.734389i	0.930144	−	0.367194i	$-0.119682\pi$
$858$		0			0
$859$	−1517.99			−1.76716			−0.883579	−	0.468283i	$-0.844873\pi$
$859$	−1517.99			−1.76716			−0.883579	+	0.468283i	$0.844873\pi$
$860$		0			0
$861$	86.0785	+	223.943i	0.0999751	+	0.260097i
$862$		0			0
$863$			1216.77i			1.40993i	0.709243	+	0.704964i	$0.249037\pi$
$863$			1216.77i			1.40993i	−0.709243	+	0.704964i	$0.750963\pi$
$864$		0			0
$865$	−936.128	+	540.474i	−1.08223	+	0.624825i
$866$		0			0
$867$	−24.3229	+	154.014i	−0.0280541	+	0.177640i
$868$		0			0
$869$	−116.727	+	67.3924i	−0.134324	+	0.0775517i
$870$		0			0
$871$	286.814			0.329293
$872$		0			0
$873$	−268.710	+	57.3670i	−0.307801	+	0.0657125i
$874$		0			0
$875$	−175.164	−	303.393i	−0.200187	−	0.346734i
$876$		0			0
$877$	161.207	−	93.0727i	0.183816	−	0.106126i	−0.405268	−	0.914198i	$-0.632822\pi$
$877$	161.207	−	93.0727i	0.183816	−	0.106126i	0.589084	+	0.808072i	$0.299489\pi$
$878$		0			0
$879$	−284.255	−	739.522i	−0.323384	−	0.841321i
$880$		0			0
$881$	186.559			0.211758			0.105879	−	0.994379i	$-0.466234\pi$
$881$	186.559			0.211758			0.105879	+	0.994379i	$0.466234\pi$
$882$		0			0
$883$	−461.914	−	800.059i	−0.523119	−	0.906069i	−0.999638	−	0.0269051i	$-0.991435\pi$
$883$	−461.914	−	800.059i	−0.523119	−	0.906069i	0.476519	−	0.879164i	$-0.341899\pi$
$884$		0			0
$885$	−381.794	−	993.281i	−0.431405	−	1.12235i
$886$		0			0
$887$		−	1168.62i		−	1.31750i	−0.752364	−	0.658748i	$-0.771086\pi$
$887$		−	1168.62i		−	1.31750i	0.752364	−	0.658748i	$-0.228914\pi$
$888$		0			0
$889$	193.069	+	111.469i	0.217176	+	0.125387i
$890$		0			0
$891$	332.603	+	240.743i	0.373291	+	0.270195i
$892$		0			0
$893$	−1317.52	+	122.534i	−1.47538	+	0.137216i
$894$		0			0
$895$	−478.232	+	276.108i	−0.534338	+	0.308500i
$896$		0			0
$897$	967.055	−	1195.30i	1.07810	−	1.33256i
$898$		0			0
$899$	486.315	+	842.322i	0.540951	+	0.936954i
$900$		0			0
$901$		−	970.116i		−	1.07671i
$902$		0			0
$903$	−3.76598	+	23.8464i	−0.00417052	+	0.0264079i
$904$		0			0
$905$	1304.49	+	753.150i	1.44143	+	0.832210i
$906$		0			0
$907$		−	1238.15i		−	1.36510i	−0.730837	−	0.682552i	$-0.760870\pi$
$907$		−	1238.15i		−	1.36510i	0.730837	−	0.682552i	$-0.239130\pi$
$908$		0			0
$909$	272.298	+	88.2062i	0.299558	+	0.0970366i
$910$		0			0
$911$	−846.741	+	488.866i	−0.929463	+	0.536626i	−0.886642	−	0.462457i	$-0.846968\pi$
$911$	−846.741	+	488.866i	−0.929463	+	0.536626i	−0.0428212	+	0.999083i	$0.513635\pi$
$912$		0			0
$913$	185.969	−	322.108i	0.203690	−	0.352801i
$914$		0			0
$915$	1074.35	+	869.200i	1.17416	+	0.949945i
$916$		0			0
$917$	225.803	−	391.103i	0.246241	−	0.426503i
$918$		0			0
$919$	683.630			0.743884			0.371942	−	0.928256i	$-0.378692\pi$
$919$	683.630			0.743884			0.371942	+	0.928256i	$0.378692\pi$
$920$		0			0
$921$	−648.126	−	1686.17i	−0.703719	−	1.83081i
$922$		0			0
$923$	−1031.82	+	1787.17i	−1.11790	+	1.93626i
$924$		0			0
$925$	270.936	+	156.425i	0.292904	+	0.169108i
$926$		0			0
$927$	311.294	−	960.984i	0.335808	−	1.03666i
$928$		0			0
$929$	−235.518			−0.253517			−0.126759	−	0.991934i	$-0.540457\pi$
$929$	−235.518			−0.253517			−0.126759	+	0.991934i	$0.540457\pi$
$930$		0			0
$931$	−800.115	+	74.4135i	−0.859414	+	0.0799286i
$932$		0			0
$933$	−203.395	+	78.1802i	−0.218001	+	0.0837944i
$934$		0			0
$935$	169.229	+	293.112i	0.180993	+	0.313489i
$936$		0			0
$937$	486.033	+	841.834i	0.518712	+	0.898435i	0.999764	+	0.0217431i	$0.00692160\pi$
$937$	486.033	+	841.834i	0.518712	+	0.898435i	−0.481052	+	0.876692i	$0.659745\pi$
$938$		0			0
$939$	−377.337	+	145.039i	−0.401850	+	0.154462i
$940$		0			0
$941$		−	760.350i		−	0.808023i	−0.914754	−	0.404012i	$-0.867616\pi$
$941$		−	760.350i		−	0.808023i	0.914754	−	0.404012i	$-0.132384\pi$
$942$		0			0
$943$	626.355	−	361.626i	0.664215	−	0.383485i
$944$		0			0
$945$	165.507	+	254.112i	0.175139	+	0.268901i
$946$		0			0
$947$	−194.296			−0.205170			−0.102585	−	0.994724i	$-0.532711\pi$
$947$	−194.296			−0.205170			−0.102585	+	0.994724i	$0.532711\pi$
$948$		0			0
$949$	−2001.99	−	1155.85i	−2.10958	−	1.21797i
$950$		0			0
$951$	9.19166	+	23.9132i	0.00966526	+	0.0251453i
$952$		0			0
$953$	−151.666	−	87.5645i	−0.159146	−	0.0918830i	0.418312	−	0.908303i	$-0.362622\pi$
$953$	−151.666	−	87.5645i	−0.159146	−	0.0918830i	−0.577458	+	0.816420i	$0.695955\pi$
$954$		0			0
$955$	98.7535	−	171.046i	0.103407	−	0.179106i
$956$		0			0
$957$	433.817	−	536.209i	0.453309	−	0.560302i
$958$		0			0
$959$	−46.6960			−0.0486924
$960$		0			0
$961$	−250.568	−	433.997i	−0.260737	−	0.451609i
$962$		0			0
$963$	−962.805	+	205.549i	−0.999797	+	0.213447i
$964$		0			0
$965$	1028.82	+	593.991i	1.06614	+	0.615534i
$966$		0			0
$967$	−59.9885	−	103.903i	−0.0620357	−	0.107449i	0.833340	−	0.552761i	$-0.186426\pi$
$967$	−59.9885	−	103.903i	−0.0620357	−	0.107449i	−0.895375	+	0.445312i	$0.853093\pi$
$968$		0			0
$969$	730.414	+	486.408i	0.753781	+	0.501969i
$970$		0			0
$971$	−1464.72	−	845.656i	−1.50847	−	0.870913i	−0.999951	−	0.00985923i	$-0.996862\pi$
$971$	−1464.72	−	845.656i	−1.50847	−	0.870913i	−0.508514	−	0.861054i	$-0.669805\pi$
$972$		0			0
$973$	−211.069	−	365.581i	−0.216926	−	0.375726i
$974$		0			0
$975$	145.810	+	379.341i	0.149548	+	0.389067i
$976$		0			0
$977$	−166.168	−	95.9370i	−0.170080	−	0.0981955i	0.412544	−	0.910938i	$-0.364640\pi$
$977$	−166.168	−	95.9370i	−0.170080	−	0.0981955i	−0.582623	+	0.812742i	$0.697974\pi$
$978$		0			0
$979$	−545.489	−	314.938i	−0.557190	−	0.321694i
$980$		0			0
$981$	314.434	−	970.677i	0.320524	−	0.989477i
$982$		0			0
$983$			922.690i			0.938647i	0.883026	+	0.469323i	$0.155502\pi$
$983$			922.690i			0.938647i	−0.883026	+	0.469323i	$0.844498\pi$
$984$		0			0
$985$	297.568	+	515.402i	0.302099	+	0.523251i
$986$		0			0
$987$	−420.646	−	340.322i	−0.426187	−	0.344804i
$988$		0			0
$989$	72.7781			0.0735876
$990$		0			0
$991$	1002.18	+	578.611i	1.01129	+	0.583866i	0.911568	−	0.411150i	$-0.134873\pi$
$991$	1002.18	+	578.611i	1.01129	+	0.583866i	0.0997176	+	0.995016i	$0.468206\pi$
$992$		0			0
$993$	336.289	+	53.1091i	0.338660	+	0.0534834i
$994$		0			0
$995$	155.967	−	270.143i	0.156751	−	0.271501i
$996$		0			0
$997$	−853.144	+	1477.69i	−0.855711	+	1.48213i	0.0202734	+	0.999794i	$0.493546\pi$
$997$	−853.144	+	1477.69i	−0.855711	+	1.48213i	−0.875984	+	0.482340i	$0.839787\pi$
$998$		0			0
$999$	−1216.55	−	617.813i	−1.21777	−	0.618431i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.3.s.a.445.33	✓	80
3.2	odd	2		2052.3.s.a.901.10		80
9.2	odd	6		2052.3.bl.a.1585.31		80
9.7	even	3		684.3.bl.a.673.36	yes	80
19.12	odd	6		684.3.bl.a.373.36	yes	80
57.50	even	6		2052.3.bl.a.145.31		80
171.88	odd	6	inner	684.3.s.a.601.33	yes	80
171.164	even	6		2052.3.s.a.829.10		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.s.a.445.33	✓	80	1.1	even	1	trivial
684.3.s.a.601.33	yes	80	171.88	odd	6	inner
684.3.bl.a.373.36	yes	80	19.12	odd	6
684.3.bl.a.673.36	yes	80	9.7	even	3
2052.3.s.a.829.10		80	171.164	even	6
2052.3.s.a.901.10		80	3.2	odd	2
2052.3.bl.a.145.31		80	57.50	even	6
2052.3.bl.a.1585.31		80	9.2	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	684.s (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.33228 + 1.88692i) q^{3} +(2.16848 + 3.75592i) q^{5} +(-1.29489 - 2.24282i) q^{7} +(1.87907 + 8.80165i) q^{9} +O(q^{10})\)	\(q+(2.33228 + 1.88692i) q^{3} +(2.16848 + 3.75592i) q^{5} +(-1.29489 - 2.24282i) q^{7} +(1.87907 + 8.80165i) q^{9} +(-2.53449 - 4.38986i) q^{11} +21.8819i q^{13} +(-2.02961 + 12.8516i) q^{15} +(-7.69782 + 13.3330i) q^{17} +(-10.9829 - 15.5040i) q^{19} +(1.21197 - 7.67423i) q^{21} -23.4214 q^{23} +(3.09539 - 5.36138i) q^{25} +(-12.2255 + 24.0736i) q^{27} +(39.2793 + 22.6779i) q^{29} +(18.5714 + 10.7222i) q^{31} +(2.37218 - 15.0208i) q^{33} +(5.61588 - 9.72700i) q^{35} +50.5347i q^{37} +(-41.2894 + 51.0347i) q^{39} +(-26.7428 + 15.4400i) q^{41} -3.10733 q^{43} +(-28.9836 + 26.1438i) q^{45} +(34.8212 - 60.3120i) q^{47} +(21.1465 - 36.6268i) q^{49} +(-43.1118 + 16.5712i) q^{51} +(-54.5703 + 31.5062i) q^{53} +(10.9920 - 19.0387i) q^{55} +(3.63962 - 56.8837i) q^{57} +(-70.8303 + 40.8939i) q^{59} +(53.1069 - 91.9838i) q^{61} +(17.3073 - 15.6116i) q^{63} +(-82.1865 + 47.4504i) q^{65} -13.1074i q^{67} +(-54.6253 - 44.1943i) q^{69} +(81.6733 + 47.1541i) q^{71} +(-52.8222 + 91.4908i) q^{73} +(17.3358 - 6.66348i) q^{75} +(-6.56377 + 11.3688i) q^{77} -26.5901i q^{79} +(-73.9382 + 33.0778i) q^{81} +(36.6877 + 63.5449i) q^{83} -66.7703 q^{85} +(48.8189 + 127.008i) q^{87} +(107.613 - 62.1305i) q^{89} +(49.0770 - 28.3346i) q^{91} +(23.0818 + 60.0500i) q^{93} +(34.4156 - 74.8712i) q^{95} +30.5295i q^{97} +(33.8756 - 30.5565i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - q^{7} + 4 q^{9}+O(q^{10}) \) 80 * q - q^7 + 4 * q^9	\( 80 q - q^{7} + 4 q^{9} - 6 q^{11} + 33 q^{15} - 21 q^{17} - 20 q^{19} - 48 q^{23} - 200 q^{25} - 63 q^{27} - 27 q^{29} - 24 q^{31} + 27 q^{33} - 54 q^{35} - 81 q^{39} - 18 q^{41} - 152 q^{43} + 188 q^{45} - 12 q^{47} - 267 q^{49} - 126 q^{51} - 36 q^{53} + 126 q^{57} - 135 q^{59} - 7 q^{61} - 190 q^{63} - 288 q^{65} + 48 q^{69} - 81 q^{71} + 55 q^{73} + 165 q^{75} + 30 q^{77} + 28 q^{81} - 93 q^{83} + 306 q^{87} + 216 q^{89} + 96 q^{91} + 24 q^{93} + 288 q^{95} - 241 q^{99}+O(q^{100}) \) 80 * q - q^7 + 4 * q^9 - 6 * q^11 + 33 * q^15 - 21 * q^17 - 20 * q^19 - 48 * q^23 - 200 * q^25 - 63 * q^27 - 27 * q^29 - 24 * q^31 + 27 * q^33 - 54 * q^35 - 81 * q^39 - 18 * q^41 - 152 * q^43 + 188 * q^45 - 12 * q^47 - 267 * q^49 - 126 * q^51 - 36 * q^53 + 126 * q^57 - 135 * q^59 - 7 * q^61 - 190 * q^63 - 288 * q^65 + 48 * q^69 - 81 * q^71 + 55 * q^73 + 165 * q^75 + 30 * q^77 + 28 * q^81 - 93 * q^83 + 306 * q^87 + 216 * q^89 + 96 * q^91 + 24 * q^93 + 288 * q^95 - 241 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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