LMFDB - Embedded newform 369.3.l.b.325.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [369,3,Mod(55,369)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(369, base_ring=CyclotomicField(8))

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

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

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

chi := DirichletCharacter("369.55");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 369 = 3^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	369.l (of order $8$, degree $4$, 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:	$10.0545217549$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$5$ over $\Q(\zeta_{8})$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 66 x^{18} + 1853 x^{16} + 28868 x^{14} + 272678 x^{12} + 1600296 x^{10} + 5739482 x^{8} + \cdots + 776161 $ x^20 + 66x^18 + 1853x^16 + 28868x^14 + 272678x^12 + 1600296x^10 + 5739482x^8 + 11863964x^6 + 12547553x^4 + 5476470*x^2 + 776161
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 41)
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			325.5
Root			$-3.37131i$ of defining polynomial
Character	$\chi$	$=$	369.325
Dual form			369.3.l.b.109.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.38388 - 2.38388i) q^{2} -7.36576i q^{4} +(-2.78016 - 2.78016i) q^{5} +(-9.79337 - 4.05655i) q^{7} +(-8.02356 - 8.02356i) q^{8} +O(q^{10})$	\(q+(2.38388 - 2.38388i) q^{2} -7.36576i q^{4} +(-2.78016 - 2.78016i) q^{5} +(-9.79337 - 4.05655i) q^{7} +(-8.02356 - 8.02356i) q^{8} -13.2551 q^{10} +(-4.61148 + 11.1331i) q^{11} +(-2.11730 - 0.877014i) q^{13} +(-33.0165 + 13.6759i) q^{14} -8.79138 q^{16} +(2.83394 - 1.17386i) q^{17} +(8.73721 - 3.61907i) q^{19} +(-20.4780 + 20.4780i) q^{20} +(15.5467 + 37.5332i) q^{22} -20.5237i q^{23} -9.54144i q^{25} +(-7.13808 + 2.95669i) q^{26} +(-29.8795 + 72.1356i) q^{28} +(-8.67290 - 3.59243i) q^{29} -33.7758i q^{31} +(11.1367 - 11.1367i) q^{32} +(3.95744 - 9.55410i) q^{34} +(15.9493 + 38.5049i) q^{35} -60.1877 q^{37} +(12.2010 - 29.4559i) q^{38} +44.6136i q^{40} +(6.99603 - 40.3987i) q^{41} +(6.83594 - 6.83594i) q^{43} +(82.0037 + 33.9670i) q^{44} +(-48.9261 - 48.9261i) q^{46} +(-0.199100 + 0.0824700i) q^{47} +(44.8062 + 44.8062i) q^{49} +(-22.7456 - 22.7456i) q^{50} +(-6.45987 + 15.5955i) q^{52} +(6.93524 - 16.7431i) q^{53} +(43.7724 - 18.1311i) q^{55} +(46.0298 + 111.126i) q^{56} +(-29.2391 + 12.1112i) q^{58} +36.2067 q^{59} +(-81.0896 + 81.0896i) q^{61} +(-80.5174 - 80.5174i) q^{62} -88.2625i q^{64} +(3.44819 + 8.32466i) q^{65} +(49.6077 - 20.5482i) q^{67} +(-8.64634 - 20.8741i) q^{68} +(129.812 + 53.7700i) q^{70} +(13.8205 + 5.72464i) q^{71} +(47.3455 - 47.3455i) q^{73} +(-143.480 + 143.480i) q^{74} +(-26.6572 - 64.3562i) q^{76} +(90.3238 - 90.3238i) q^{77} +(14.7604 - 35.6349i) q^{79} +(24.4414 + 24.4414i) q^{80} +(-79.6280 - 112.983i) q^{82} +99.1394 q^{83} +(-11.1423 - 4.61529i) q^{85} -32.5921i q^{86} +(126.328 - 52.3266i) q^{88} +(40.3183 + 16.7004i) q^{89} +(17.1778 + 17.1778i) q^{91} -151.173 q^{92} +(-0.278032 + 0.671230i) q^{94} +(-34.3524 - 14.2292i) q^{95} +(50.8068 + 122.659i) q^{97} +213.625 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 8 q^{2} + 12 q^{5} - 4 q^{7} - 36 q^{8}+O(q^{10}) $ 20 * q + 8 * q^2 + 12 * q^5 - 4 * q^7 - 36 * q^8	$ 20 q + 8 q^{2} + 12 q^{5} - 4 q^{7} - 36 q^{8} + 16 q^{10} + 12 q^{11} - 48 q^{13} - 88 q^{14} - 36 q^{16} + 28 q^{17} + 76 q^{19} + 16 q^{20} - 116 q^{22} - 40 q^{26} + 72 q^{28} + 24 q^{29} - 176 q^{32} + 80 q^{34} - 60 q^{35} + 208 q^{37} + 380 q^{38} + 116 q^{41} - 40 q^{43} - 116 q^{44} - 176 q^{46} + 64 q^{47} + 168 q^{49} + 148 q^{50} - 184 q^{52} + 120 q^{53} + 20 q^{55} - 188 q^{56} + 36 q^{58} + 512 q^{59} - 460 q^{61} - 68 q^{62} - 432 q^{65} + 300 q^{67} - 120 q^{68} + 308 q^{70} + 108 q^{71} + 60 q^{73} - 140 q^{74} + 872 q^{76} - 112 q^{77} - 208 q^{79} + 68 q^{80} - 376 q^{82} + 120 q^{83} + 172 q^{85} + 316 q^{88} - 268 q^{89} - 800 q^{91} + 448 q^{92} - 212 q^{94} + 184 q^{95} - 120 q^{97} + 20 q^{98}+O(q^{100}) $ 20 * q + 8 * q^2 + 12 * q^5 - 4 * q^7 - 36 * q^8 + 16 * q^10 + 12 * q^11 - 48 * q^13 - 88 * q^14 - 36 * q^16 + 28 * q^17 + 76 * q^19 + 16 * q^20 - 116 * q^22 - 40 * q^26 + 72 * q^28 + 24 * q^29 - 176 * q^32 + 80 * q^34 - 60 * q^35 + 208 * q^37 + 380 * q^38 + 116 * q^41 - 40 * q^43 - 116 * q^44 - 176 * q^46 + 64 * q^47 + 168 * q^49 + 148 * q^50 - 184 * q^52 + 120 * q^53 + 20 * q^55 - 188 * q^56 + 36 * q^58 + 512 * q^59 - 460 * q^61 - 68 * q^62 - 432 * q^65 + 300 * q^67 - 120 * q^68 + 308 * q^70 + 108 * q^71 + 60 * q^73 - 140 * q^74 + 872 * q^76 - 112 * q^77 - 208 * q^79 + 68 * q^80 - 376 * q^82 + 120 * q^83 + 172 * q^85 + 316 * q^88 - 268 * q^89 - 800 * q^91 + 448 * q^92 - 212 * q^94 + 184 * q^95 - 120 * q^97 + 20 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/369\mathbb{Z}\right)^\times$.

$n$	$83$	$334$
$\chi(n)$	$1$	$e\left(\frac{7}{8}\right)$

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 $

$2$	2.38388	−	2.38388i	1.19194	−	1.19194i	0.215418	−	0.976522i	$-0.430889\pi$
$2$	2.38388	−	2.38388i	1.19194	−	1.19194i	0.976522	−	0.215418i	$-0.0691112\pi$
$3$		0			0
$3$		0			0
$4$		−	7.36576i		−	1.84144i
$5$	−2.78016	−	2.78016i	−0.556032	−	0.556032i	0.372144	−	0.928175i	$-0.378623\pi$
$5$	−2.78016	−	2.78016i	−0.556032	−	0.556032i	−0.928175	+	0.372144i	$0.878623\pi$
$6$		0			0
$7$	−9.79337	−	4.05655i	−1.39905	−	0.579506i	−0.449547	−	0.893257i	$-0.648415\pi$
$7$	−9.79337	−	4.05655i	−1.39905	−	0.579506i	−0.949506	+	0.313750i	$0.898415\pi$
$8$	−8.02356	−	8.02356i	−1.00295	−	1.00295i
$9$		0			0
$10$	−13.2551			−1.32551
$11$	−4.61148	+	11.1331i	−0.419225	+	1.01210i	0.563347	+	0.826220i	$0.309513\pi$
$11$	−4.61148	+	11.1331i	−0.419225	+	1.01210i	−0.982573	+	0.185879i	$0.940487\pi$
$12$		0			0
$13$	−2.11730	−	0.877014i	−0.162869	−	0.0674626i	0.299759	−	0.954015i	$-0.403094\pi$
$13$	−2.11730	−	0.877014i	−0.162869	−	0.0674626i	−0.462628	+	0.886552i	$0.653094\pi$
$14$	−33.0165	+	13.6759i	−2.35832	+	0.976849i
$15$		0			0
$16$	−8.79138			−0.549461
$17$	2.83394	−	1.17386i	0.166702	−	0.0690504i	−0.297772	−	0.954637i	$-0.596243\pi$
$17$	2.83394	−	1.17386i	0.166702	−	0.0690504i	0.464474	+	0.885587i	$0.346243\pi$
$18$		0			0
$19$	8.73721	−	3.61907i	0.459853	−	0.190477i	−0.140716	−	0.990050i	$-0.544941\pi$
$19$	8.73721	−	3.61907i	0.459853	−	0.190477i	0.600569	+	0.799573i	$0.294941\pi$
$20$	−20.4780	+	20.4780i	−1.02390	+	1.02390i
$21$		0			0
$22$	15.5467	+	37.5332i	0.706670	+	1.70605i
$23$		−	20.5237i		−	0.892336i	−0.894949	−	0.446168i	$-0.852788\pi$
$23$		−	20.5237i		−	0.892336i	0.894949	−	0.446168i	$-0.147212\pi$
$24$		0			0
$25$		−	9.54144i		−	0.381658i
$26$	−7.13808	+	2.95669i	−0.274541	+	0.113719i
$27$		0			0
$28$	−29.8795	+	72.1356i	−1.06713	+	2.57627i
$29$	−8.67290	−	3.59243i	−0.299066	−	0.123877i	0.228105	−	0.973637i	$-0.426747\pi$
$29$	−8.67290	−	3.59243i	−0.299066	−	0.123877i	−0.527171	+	0.849760i	$0.676747\pi$
$30$		0			0
$31$		−	33.7758i		−	1.08954i	−0.838585	−	0.544771i	$-0.816617\pi$
$31$		−	33.7758i		−	1.08954i	0.838585	−	0.544771i	$-0.183383\pi$
$32$	11.1367	−	11.1367i	0.348021	−	0.348021i
$33$		0			0
$34$	3.95744	−	9.55410i	0.116395	−	0.281003i
$35$	15.9493	+	38.5049i	0.455693	+	1.10014i
$36$		0			0
$37$	−60.1877			−1.62669			−0.813347	−	0.581778i	$-0.802357\pi$
$37$	−60.1877			−1.62669			−0.813347	+	0.581778i	$0.802357\pi$
$38$	12.2010	−	29.4559i	0.321079	−	0.775154i
$39$		0			0
$40$			44.6136i			1.11534i
$41$	6.99603	−	40.3987i	0.170635	−	0.985334i
$41$	6.99603	−	40.3987i	0.170635	−	0.985334i
$42$		0			0
$43$	6.83594	−	6.83594i	0.158975	−	0.158975i	−0.623137	−	0.782112i	$-0.714142\pi$
$43$	6.83594	−	6.83594i	0.158975	−	0.158975i	0.782112	+	0.623137i	$0.214142\pi$
$44$	82.0037	+	33.9670i	1.86372	+	0.771978i
$45$		0			0
$46$	−48.9261	−	48.9261i	−1.06361	−	1.06361i
$47$	−0.199100	+	0.0824700i	−0.00423618	+	0.00175468i	−0.384801	−	0.923000i	$-0.625730\pi$
$47$	−0.199100	+	0.0824700i	−0.00423618	+	0.00175468i	0.380564	+	0.924754i	$0.375730\pi$
$48$		0			0
$49$	44.8062	+	44.8062i	0.914413	+	0.914413i
$50$	−22.7456	−	22.7456i	−0.454913	−	0.454913i
$51$		0			0
$52$	−6.45987	+	15.5955i	−0.124228	+	0.299914i
$53$	6.93524	−	16.7431i	0.130854	−	0.315908i	−0.844850	−	0.535003i	$-0.820310\pi$
$53$	6.93524	−	16.7431i	0.130854	−	0.315908i	0.975704	+	0.219095i	$0.0703104\pi$
$54$		0			0
$55$	43.7724	−	18.1311i	0.795862	−	0.329657i
$56$	46.0298	+	111.126i	0.821960	+	1.98439i
$57$		0			0
$58$	−29.2391	+	12.1112i	−0.504122	+	0.208814i
$59$	36.2067			0.613673			0.306836	−	0.951762i	$-0.400730\pi$
$59$	36.2067			0.613673			0.306836	+	0.951762i	$0.400730\pi$
$60$		0			0
$61$	−81.0896	+	81.0896i	−1.32934	+	1.32934i	−0.423391	+	0.905947i	$0.639160\pi$
$61$	−81.0896	+	81.0896i	−1.32934	+	1.32934i	−0.905947	+	0.423391i	$0.860840\pi$
$62$	−80.5174	−	80.5174i	−1.29867	−	1.29867i
$63$		0			0
$64$		−	88.2625i		−	1.37910i
$65$	3.44819	+	8.32466i	0.0530490	+	0.128072i
$66$		0			0
$67$	49.6077	−	20.5482i	0.740413	−	0.306689i	0.0195897	−	0.999808i	$-0.493764\pi$
$67$	49.6077	−	20.5482i	0.740413	−	0.306689i	0.720823	+	0.693119i	$0.243764\pi$
$68$	−8.64634	−	20.8741i	−0.127152	−	0.306972i
$69$		0			0
$70$	129.812	+	53.7700i	1.85446	+	0.768143i
$71$	13.8205	+	5.72464i	0.194655	+	0.0806287i	0.477881	−	0.878424i	$-0.341405\pi$
$71$	13.8205	+	5.72464i	0.194655	+	0.0806287i	−0.283227	+	0.959053i	$0.591405\pi$
$72$		0			0
$73$	47.3455	−	47.3455i	0.648569	−	0.648569i	−0.304078	−	0.952647i	$-0.598348\pi$
$73$	47.3455	−	47.3455i	0.648569	−	0.648569i	0.952647	+	0.304078i	$0.0983484\pi$
$74$	−143.480	+	143.480i	−1.93892	+	1.93892i
$75$		0			0
$76$	−26.6572	−	64.3562i	−0.350753	−	0.846792i
$77$	90.3238	−	90.3238i	1.17304	−	1.17304i
$78$		0			0
$79$	14.7604	−	35.6349i	0.186841	−	0.451074i	−0.802507	−	0.596643i	$-0.796501\pi$
$79$	14.7604	−	35.6349i	0.186841	−	0.451074i	0.989348	+	0.145568i	$0.0465010\pi$
$80$	24.4414	+	24.4414i	0.305518	+	0.305518i
$81$		0			0
$82$	−79.6280	−	112.983i	−0.971073	−	1.37785i
$83$	99.1394			1.19445			0.597225	−	0.802073i	$-0.296270\pi$
$83$	99.1394			1.19445			0.597225	+	0.802073i	$0.296270\pi$
$84$		0			0
$85$	−11.1423	−	4.61529i	−0.131086	−	0.0542976i
$86$		−	32.5921i		−	0.378978i
$87$		0			0
$88$	126.328	−	52.3266i	1.43554	−	0.594620i
$89$	40.3183	+	16.7004i	0.453015	+	0.187645i	0.597511	−	0.801860i	$-0.296156\pi$
$89$	40.3183	+	16.7004i	0.453015	+	0.187645i	−0.144497	+	0.989505i	$0.546156\pi$
$90$		0			0
$91$	17.1778	+	17.1778i	0.188767	+	0.188767i
$92$	−151.173			−1.64318
$93$		0			0
$94$	−0.278032	+	0.671230i	−0.00295779	+	0.00714074i
$95$	−34.3524	−	14.2292i	−0.361604	−	0.149781i
$96$		0			0
$97$	50.8068	+	122.659i	0.523782	+	1.26452i	0.935538	+	0.353227i	$0.114916\pi$
$97$	50.8068	+	122.659i	0.523782	+	1.26452i	−0.411756	+	0.911294i	$0.635084\pi$
$98$	213.625			2.17985
$99$		0			0
$100$	−70.2800			−0.702800
$101$	−137.221	+	56.8388i	−1.35862	+	0.562761i	−0.938680	−	0.344789i	$-0.887950\pi$
$101$	−137.221	+	56.8388i	−1.35862	+	0.562761i	−0.419944	+	0.907550i	$0.637950\pi$
$102$		0			0
$103$	127.979	+	127.979i	1.24252	+	1.24252i	0.958953	+	0.283565i	$0.0915170\pi$
$103$	127.979	+	127.979i	1.24252	+	1.24252i	0.283565	+	0.958953i	$0.408483\pi$
$104$	9.95150	+	24.0251i	0.0956875	+	0.231010i
$105$		0			0
$106$	−23.3809	−	56.4464i	−0.220574	−	0.532513i
$107$			69.5523i			0.650022i	0.945710	+	0.325011i	$0.105368\pi$
$107$			69.5523i			0.650022i	−0.945710	+	0.325011i	$0.894632\pi$
$108$		0			0
$109$	10.0089	+	24.1636i	0.0918247	+	0.221684i	0.963119	−	0.269077i	$-0.0867186\pi$
$109$	10.0089	+	24.1636i	0.0918247	+	0.221684i	−0.871294	+	0.490762i	$0.836719\pi$
$110$	61.1257	−	147.571i	0.555688	−	1.34155i
$111$		0			0
$112$	86.0972	+	35.6626i	0.768725	+	0.318416i
$113$		−	183.205i		−	1.62129i	−0.585541	−	0.810643i	$-0.699118\pi$
$113$		−	183.205i		−	1.62129i	0.585541	−	0.810643i	$-0.300882\pi$
$114$		0			0
$115$	−57.0592	+	57.0592i	−0.496167	+	0.496167i
$116$	−26.4610	+	63.8825i	−0.228112	+	0.550711i
$117$		0			0
$118$	86.3124	−	86.3124i	0.731461	−	0.731461i
$119$	−32.5156			−0.273240
$120$		0			0
$121$	−17.1201	−	17.1201i	−0.141489	−	0.141489i
$122$			386.616i			3.16898i
$123$		0			0
$124$	−248.784			−2.00633
$125$	−96.0307	+	96.0307i	−0.768245	+	0.768245i
$126$		0			0
$127$		−	147.141i		−	1.15859i	−0.815119	−	0.579294i	$-0.803328\pi$
$127$		−	147.141i		−	1.15859i	0.815119	−	0.579294i	$-0.196672\pi$
$128$	−165.860	−	165.860i	−1.29578	−	1.29578i
$129$		0			0
$130$	28.0650	+	11.6249i	0.215885	+	0.0894225i
$131$	−94.0344	−	94.0344i	−0.717820	−	0.717820i	0.250338	−	0.968158i	$-0.419458\pi$
$131$	−94.0344	−	94.0344i	−0.717820	−	0.717820i	−0.968158	+	0.250338i	$0.919458\pi$
$132$		0			0
$133$	−100.248			−0.753741
$134$	69.2743	−	167.243i	0.516973	−	1.24808i
$135$		0			0
$136$	−32.1568	−	13.3198i	−0.236447	−	0.0979396i
$137$	103.079	−	42.6969i	0.752405	−	0.311656i	0.0266823	−	0.999644i	$-0.491506\pi$
$137$	103.079	−	42.6969i	0.752405	−	0.311656i	0.725722	+	0.687988i	$0.241506\pi$
$138$		0			0
$139$	106.826			0.768535			0.384268	−	0.923222i	$-0.374454\pi$
$139$	106.826			0.768535			0.384268	+	0.923222i	$0.374454\pi$
$140$	283.618	−	117.478i	2.02584	−	0.839132i
$141$		0			0
$142$	46.5932	−	19.2995i	0.328121	−	0.135912i
$143$	19.5278	−	19.5278i	0.136558	−	0.136558i
$144$		0			0
$145$	14.1245	+	34.0996i	0.0974104	+	0.235170i
$146$		−	225.732i		−	1.54611i
$147$		0			0
$148$			443.328i			2.99546i
$149$	88.6747	−	36.7303i	0.595132	−	0.246512i	−0.0647247	−	0.997903i	$-0.520617\pi$
$149$	88.6747	−	36.7303i	0.595132	−	0.246512i	0.659857	+	0.751391i	$0.270617\pi$
$150$		0			0
$151$	−15.5155	+	37.4578i	−0.102752	+	0.248065i	−0.966892	−	0.255187i	$-0.917863\pi$
$151$	−15.5155	+	37.4578i	−0.102752	+	0.248065i	0.864140	+	0.503252i	$0.167863\pi$
$152$	−99.1414	−	41.0657i	−0.652246	−	0.270169i
$153$		0			0
$154$		−	430.642i		−	2.79638i
$155$	−93.9021	+	93.9021i	−0.605820	+	0.605820i
$156$		0			0
$157$	7.16318	−	17.2934i	0.0456253	−	0.110149i	−0.899424	−	0.437078i	$-0.856014\pi$
$157$	7.16318	−	17.2934i	0.0456253	−	0.110149i	0.945049	+	0.326928i	$0.106014\pi$
$158$	−49.7621	−	120.136i	−0.314950	−	0.760357i
$159$		0			0
$160$	−61.9234			−0.387022
$161$	−83.2555	+	200.996i	−0.517115	+	1.24843i
$162$		0			0
$163$		−	135.937i		−	0.833972i	−0.908913	−	0.416986i	$-0.863086\pi$
$163$		−	135.937i		−	0.833972i	0.908913	−	0.416986i	$-0.136914\pi$
$164$	−297.567	−	51.5311i	−1.81443	−	0.314214i
$165$		0			0
$166$	236.336	−	236.336i	1.42371	−	1.42371i
$167$	−218.746	−	90.6077i	−1.30986	−	0.542561i	−0.385015	−	0.922910i	$-0.625804\pi$
$167$	−218.746	−	90.6077i	−1.30986	−	0.542561i	−0.924843	+	0.380350i	$0.875804\pi$
$168$		0			0
$169$	−115.787	−	115.787i	−0.685132	−	0.685132i
$170$	−37.5642	+	15.5596i	−0.220966	+	0.0915271i
$171$		0			0
$172$	−50.3519	−	50.3519i	−0.292743	−	0.292743i
$173$	36.3303	+	36.3303i	0.210002	+	0.210002i	0.804268	−	0.594267i	$-0.202558\pi$
$173$	36.3303	+	36.3303i	0.210002	+	0.210002i	−0.594267	+	0.804268i	$0.702558\pi$
$174$		0			0
$175$	−38.7053	+	93.4428i	−0.221173	+	0.533959i
$176$	40.5412	−	97.8752i	0.230348	−	0.556109i
$177$		0			0
$178$	135.926	−	56.3023i	0.763627	−	0.316305i
$179$	−27.8175	−	67.1573i	−0.155405	−	0.375181i	0.826932	−	0.562302i	$-0.190084\pi$
$179$	−27.8175	−	67.1573i	−0.155405	−	0.375181i	−0.982337	+	0.187122i	$0.940084\pi$
$180$		0			0
$181$	−80.9954	+	33.5494i	−0.447488	+	0.185356i	−0.595036	−	0.803699i	$-0.702862\pi$
$181$	−80.9954	+	33.5494i	−0.447488	+	0.185356i	0.147548	+	0.989055i	$0.452862\pi$
$182$	81.8998			0.449999
$183$		0			0
$184$	−164.674	+	164.674i	−0.894965	+	0.894965i
$185$	167.331	+	167.331i	0.904494	+	0.904494i
$186$		0			0
$187$			36.9637i			0.197667i
$188$	0.607454	+	1.46652i	0.00323114	+	0.00780066i
$189$		0			0
$190$	−115.813	+	47.9712i	−0.609541	+	0.252480i
$191$	118.697	+	286.560i	0.621451	+	1.50032i	0.850000	+	0.526783i	$0.176602\pi$
$191$	118.697	+	286.560i	0.621451	+	1.50032i	−0.228549	+	0.973532i	$0.573398\pi$
$192$		0			0
$193$	271.296	+	112.375i	1.40568	+	0.582252i	0.951219	−	0.308515i	$-0.0998322\pi$
$193$	271.296	+	112.375i	1.40568	+	0.582252i	0.454460	+	0.890767i	$0.349832\pi$
$194$	413.520	+	171.286i	2.13155	+	0.882916i
$195$		0			0
$196$	330.032	−	330.032i	1.68384	−	1.68384i
$197$	−5.26587	+	5.26587i	−0.0267303	+	0.0267303i	−0.720346	−	0.693615i	$-0.756017\pi$
$197$	−5.26587	+	5.26587i	−0.0267303	+	0.0267303i	0.693615	+	0.720346i	$0.256017\pi$
$198$		0			0
$199$	−25.2153	−	60.8750i	−0.126710	−	0.305905i	0.847776	−	0.530355i	$-0.177941\pi$
$199$	−25.2153	−	60.8750i	−0.126710	−	0.305905i	−0.974486	+	0.224450i	$0.927941\pi$
$200$	−76.5564	+	76.5564i	−0.382782	+	0.382782i
$201$		0			0
$202$	−191.622	+	462.615i	−0.948621	+	2.29017i
$203$	70.3641	+	70.3641i	0.346621	+	0.346621i
$204$		0			0
$205$	−131.765	+	92.8647i	−0.642755	+	0.452999i
$206$	610.174			2.96201
$207$		0			0
$208$	18.6140	+	7.71016i	0.0894902	+	0.0370681i
$209$			113.961i			0.545270i
$210$		0			0
$211$	188.285	−	77.9903i	0.892347	−	0.369622i	0.111074	−	0.993812i	$-0.464571\pi$
$211$	188.285	−	77.9903i	0.892347	−	0.369622i	0.781273	+	0.624190i	$0.214571\pi$
$212$	−123.326	−	51.0833i	−0.581726	−	0.240959i
$213$		0			0
$214$	165.804	+	165.804i	0.774787	+	0.774787i
$215$	−38.0100			−0.176791
$216$		0			0
$217$	−137.013	+	330.779i	−0.631397	+	1.52433i
$218$	81.4631	+	33.7431i	0.373684	+	0.154785i
$219$		0			0
$220$	−133.549	−	322.417i	−0.607043	−	1.46553i
$221$	−7.02978			−0.0318090
$222$		0			0
$223$	−32.4398			−0.145470			−0.0727350	−	0.997351i	$-0.523173\pi$
$223$	−32.4398			−0.145470			−0.0727350	+	0.997351i	$0.523173\pi$
$224$	−154.242	+	63.8891i	−0.688580	+	0.285219i
$225$		0			0
$226$	−436.739	−	436.739i	−1.93248	−	1.93248i
$227$	51.2534	+	123.737i	0.225786	+	0.545095i	0.995656	−	0.0931051i	$-0.0296793\pi$
$227$	51.2534	+	123.737i	0.225786	+	0.545095i	−0.769871	+	0.638200i	$0.779679\pi$
$228$		0			0
$229$	43.8595	+	105.886i	0.191526	+	0.462385i	0.990248	−	0.139315i	$-0.0444901\pi$
$229$	43.8595	+	105.886i	0.191526	+	0.462385i	−0.798722	+	0.601700i	$0.794490\pi$
$230$			272.045i			1.18280i
$231$		0			0
$232$	40.7635	+	98.4117i	0.175705	+	0.424188i
$233$	55.5861	−	134.197i	0.238567	−	0.575951i	−0.758568	−	0.651594i	$-0.774101\pi$
$233$	55.5861	−	134.197i	0.238567	−	0.575951i	0.997135	+	0.0756424i	$0.0241008\pi$
$234$		0			0
$235$	0.782810	+	0.324250i	0.00333111	+	0.00137979i
$236$		−	266.690i		−	1.13004i
$237$		0			0
$238$	−77.5133	+	77.5133i	−0.325686	+	0.325686i
$239$	−106.730	+	257.669i	−0.446568	+	1.07811i	0.527031	+	0.849846i	$0.323305\pi$
$239$	−106.730	+	257.669i	−0.446568	+	1.07811i	−0.973599	+	0.228265i	$0.926695\pi$
$240$		0			0
$241$	−28.2681	+	28.2681i	−0.117295	+	0.117295i	−0.763318	−	0.646023i	$-0.776431\pi$
$241$	−28.2681	+	28.2681i	−0.117295	+	0.117295i	0.646023	+	0.763318i	$0.276431\pi$
$242$	−81.6246			−0.337292
$243$		0			0
$244$	597.287	+	597.287i	2.44790	+	2.44790i
$245$		−	249.137i		−	1.01689i
$246$		0			0
$247$	−21.6732			−0.0877459
$248$	−271.002	+	271.002i	−1.09275	+	1.09275i
$249$		0			0
$250$			457.851i			1.83140i
$251$	−90.3600	−	90.3600i	−0.360000	−	0.360000i	0.503813	−	0.863813i	$-0.331930\pi$
$251$	−90.3600	−	90.3600i	−0.360000	−	0.360000i	−0.863813	+	0.503813i	$0.831930\pi$
$252$		0			0
$253$	228.493	+	94.6448i	0.903133	+	0.374090i
$254$	−350.766	−	350.766i	−1.38097	−	1.38097i
$255$		0			0
$256$	−437.732			−1.70989
$257$	−84.5632	+	204.154i	−0.329040	+	0.794372i	0.669625	+	0.742700i	$0.266455\pi$
$257$	−84.5632	+	204.154i	−0.329040	+	0.794372i	−0.998664	+	0.0516720i	$0.983545\pi$
$258$		0			0
$259$	589.440	+	244.154i	2.27583	+	0.942680i
$260$	61.3175	−	25.3985i	0.235836	−	0.0976866i
$261$		0			0
$262$	−448.334			−1.71120
$263$	435.904	−	180.557i	1.65743	−	0.686530i	0.659553	−	0.751658i	$-0.270745\pi$
$263$	435.904	−	180.557i	1.65743	−	0.686530i	0.997877	+	0.0651277i	$0.0207455\pi$
$264$		0			0
$265$	−65.8296	+	27.2675i	−0.248414	+	0.102896i
$266$	−238.978	+	238.978i	−0.898414	+	0.898414i
$267$		0			0
$268$	−151.353	−	365.398i	−0.564749	−	1.36343i
$269$		−	38.3082i		−	0.142410i	−0.997462	−	0.0712048i	$-0.977316\pi$
$269$		−	38.3082i		−	0.142410i	0.997462	−	0.0712048i	$-0.0226844\pi$
$270$		0			0
$271$			66.2839i			0.244590i	0.992494	+	0.122295i	$0.0390254\pi$
$271$			66.2839i			0.244590i	−0.992494	+	0.122295i	$0.960975\pi$
$272$	−24.9142	+	10.3198i	−0.0915964	+	0.0379405i
$273$		0			0
$274$	143.945	−	347.513i	0.525345	−	1.26830i
$275$	106.226	+	44.0002i	0.386276	+	0.160001i
$276$		0			0
$277$		−	330.119i		−	1.19176i	−0.803072	−	0.595882i	$-0.796802\pi$
$277$		−	330.119i		−	1.19176i	0.803072	−	0.595882i	$-0.203198\pi$
$278$	254.661	−	254.661i	0.916047	−	0.916047i
$279$		0			0
$280$	180.977	−	436.917i	0.646346	−	1.56042i
$281$	−168.265	−	406.227i	−0.598807	−	1.44565i	−0.874798	−	0.484488i	$-0.839006\pi$
$281$	−168.265	−	406.227i	−0.598807	−	1.44565i	0.275991	−	0.961160i	$-0.410994\pi$
$282$		0			0
$283$	−127.048			−0.448933			−0.224467	−	0.974482i	$-0.572064\pi$
$283$	−127.048			−0.448933			−0.224467	+	0.974482i	$0.572064\pi$
$284$	42.1663	−	101.798i	0.148473	−	0.358445i
$285$		0			0
$286$		−	93.1036i		−	0.325537i
$287$	−232.394	+	367.260i	−0.809735	+	1.27965i
$288$		0			0
$289$	−197.701	+	197.701i	−0.684085	+	0.684085i
$290$	114.960	+	47.6182i	0.396415	+	0.164201i
$291$		0			0
$292$	−348.736	−	348.736i	−1.19430	−	1.19430i
$293$	−393.156	+	162.851i	−1.34183	+	0.555804i	−0.934006	−	0.357257i	$-0.883712\pi$
$293$	−393.156	+	162.851i	−1.34183	+	0.555804i	−0.407823	+	0.913061i	$0.633712\pi$
$294$		0			0
$295$	−100.660	−	100.660i	−0.341221	−	0.341221i
$296$	482.920	+	482.920i	1.63149	+	1.63149i
$297$		0			0
$298$	123.829	−	298.950i	0.415534	−	1.00319i
$299$	−17.9996	+	43.4549i	−0.0601993	+	0.145334i
$300$		0			0
$301$	−94.6771	+	39.2165i	−0.314542	+	0.130288i
$302$	52.3077	+	126.282i	0.173204	+	0.418152i
$303$		0			0
$304$	−76.8121	+	31.8166i	−0.252671	+	0.104660i
$305$	450.884			1.47831
$306$		0			0
$307$	−327.066	+	327.066i	−1.06536	+	1.06536i	−0.0676533	+	0.997709i	$0.521551\pi$
$307$	−327.066	+	327.066i	−1.06536	+	1.06536i	−0.997709	+	0.0676533i	$0.978449\pi$
$308$	−665.303	−	665.303i	−2.16008	−	2.16008i
$309$		0			0
$310$			447.702i			1.44420i
$311$	−43.4851	−	104.982i	−0.139823	−	0.337564i	0.838420	−	0.545025i	$-0.183480\pi$
$311$	−43.4851	−	104.982i	−0.139823	−	0.337564i	−0.978243	+	0.207461i	$0.933480\pi$
$312$		0			0
$313$	353.499	−	146.424i	1.12939	−	0.467808i	0.261816	−	0.965118i	$-0.415679\pi$
$313$	353.499	−	146.424i	1.12939	−	0.467808i	0.867573	+	0.497309i	$0.165679\pi$
$314$	−24.1493	−	58.3016i	−0.0769087	−	0.185674i
$315$		0			0
$316$	−262.478	−	108.722i	−0.830626	−	0.344057i
$317$	−346.574	−	143.556i	−1.09329	−	0.452857i	−0.238139	−	0.971231i	$-0.576537\pi$
$317$	−346.574	−	143.556i	−1.09329	−	0.452857i	−0.855154	+	0.518374i	$0.826537\pi$
$318$		0			0
$319$	79.9898	−	79.9898i	0.250752	−	0.250752i
$320$	−245.384	+	245.384i	−0.766824	+	0.766824i
$321$		0			0
$322$	280.680	+	677.622i	0.871678	+	2.10442i
$323$	20.5124	−	20.5124i	0.0635060	−	0.0635060i
$324$		0			0
$325$	−8.36798	+	20.2021i	−0.0257476	+	0.0621602i
$326$	−324.058	−	324.058i	−0.994044	−	0.994044i
$327$		0			0
$328$	−380.275	+	268.009i	−1.15937	+	0.817099i
$329$	2.28441			0.00694348
$330$		0			0
$331$	−284.193	−	117.717i	−0.858589	−	0.355639i	−0.0904333	−	0.995903i	$-0.528825\pi$
$331$	−284.193	−	117.717i	−0.858589	−	0.355639i	−0.768155	+	0.640263i	$0.778825\pi$
$332$		−	730.237i		−	2.19951i
$333$		0			0
$334$	−737.462	+	305.467i	−2.20797	+	0.914571i
$335$	−195.044	−	80.7900i	−0.582222	−	0.241164i
$336$		0			0
$337$	−440.163	−	440.163i	−1.30612	−	1.30612i	−0.924192	−	0.381928i	$-0.875260\pi$
$337$	−440.163	−	440.163i	−1.30612	−	1.30612i	−0.381928	−	0.924192i	$-0.624740\pi$
$338$	−552.046			−1.63327
$339$		0			0
$340$	−33.9951	+	82.0715i	−0.0999857	+	0.241387i
$341$	376.029	+	155.756i	1.10272	+	0.456764i
$342$		0			0
$343$	−58.2747	−	140.688i	−0.169897	−	0.410168i
$344$	−109.697			−0.318887
$345$		0			0
$346$	173.214			0.500618
$347$	363.094	−	150.398i	1.04638	−	0.433425i	0.207782	−	0.978175i	$-0.433376\pi$
$347$	363.094	−	150.398i	1.04638	−	0.433425i	0.838598	+	0.544751i	$0.183376\pi$
$348$		0			0
$349$	−65.9189	−	65.9189i	−0.188879	−	0.188879i	0.606332	−	0.795212i	$-0.292640\pi$
$349$	−65.9189	−	65.9189i	−0.188879	−	0.188879i	−0.795212	+	0.606332i	$0.792640\pi$
$350$	130.488	+	315.025i	0.372822	+	0.900072i
$351$		0			0
$352$	72.6291	+	175.342i	0.206333	+	0.498131i
$353$			548.467i			1.55373i	0.629667	+	0.776865i	$0.283191\pi$
$353$			548.467i			1.55373i	−0.629667	+	0.776865i	$0.716809\pi$
$354$		0			0
$355$	−22.5078	−	54.3385i	−0.0634022	−	0.153066i
$356$	123.011	−	296.975i	0.345537	−	0.834199i
$357$		0			0
$358$	−226.408	−	93.7814i	−0.632426	−	0.261959i
$359$		−	204.799i		−	0.570470i	−0.958458	−	0.285235i	$-0.907928\pi$
$359$		−	204.799i		−	0.570470i	0.958458	−	0.285235i	$-0.0920717\pi$
$360$		0			0
$361$	−192.024	+	192.024i	−0.531924	+	0.531924i
$362$	−113.105	+	273.061i	−0.312446	+	0.754312i
$363$		0			0
$364$	126.528	−	126.528i	0.347604	−	0.347604i
$365$	−263.256			−0.721249
$366$		0			0
$367$	−387.267	−	387.267i	−1.05522	−	1.05522i	−0.998383	−	0.0568391i	$-0.981898\pi$
$367$	−387.267	−	387.267i	−1.05522	−	1.05522i	−0.0568391	−	0.998383i	$-0.518102\pi$
$368$			180.432i			0.490304i
$369$		0			0
$370$	797.795			2.15620
$371$	−135.839	+	135.839i	−0.366142	+	0.366142i
$372$		0			0
$373$		−	14.5539i		−	0.0390186i	−0.999810	−	0.0195093i	$-0.993790\pi$
$373$		−	14.5539i		−	0.0390186i	0.999810	−	0.0195093i	$-0.00621040\pi$
$374$	88.1171	+	88.1171i	0.235607	+	0.235607i
$375$		0			0
$376$	2.25920	+	0.935790i	0.00600850	+	0.00248880i
$377$	15.2125	+	15.2125i	0.0403515	+	0.0403515i
$378$		0			0
$379$	589.750			1.55607			0.778034	−	0.628222i	$-0.216217\pi$
$379$	589.750			1.55607			0.778034	+	0.628222i	$0.216217\pi$
$380$	−104.809	+	253.032i	−0.275813	+	0.665872i
$381$		0			0
$382$	966.085	+	400.165i	2.52902	+	1.04755i
$383$	363.597	−	150.607i	0.949340	−	0.393229i	0.146357	−	0.989232i	$-0.453245\pi$
$383$	363.597	−	150.607i	0.949340	−	0.393229i	0.802983	+	0.596002i	$0.203245\pi$
$384$		0			0
$385$	−502.229			−1.30449
$386$	914.625	−	378.850i	2.36949	−	0.981476i
$387$		0			0
$388$	903.473	−	374.231i	2.32854	−	0.964512i
$389$	−353.165	+	353.165i	−0.907879	+	0.907879i	−0.996101	−	0.0882215i	$-0.971882\pi$
$389$	−353.165	+	353.165i	−0.907879	+	0.907879i	0.0882215	+	0.996101i	$0.471882\pi$
$390$		0			0
$391$	−24.0919	−	58.1630i	−0.0616161	−	0.148755i
$392$		−	719.012i		−	1.83421i
$393$		0			0
$394$			25.1064i			0.0637219i
$395$	−140.107	+	58.0342i	−0.354701	+	0.146922i
$396$		0			0
$397$	−70.4174	+	170.003i	−0.177374	+	0.428218i	−0.987414	−	0.158156i	$-0.949445\pi$
$397$	−70.4174	+	170.003i	−0.177374	+	0.428218i	0.810040	+	0.586374i	$0.199445\pi$
$398$	−205.229	−	85.0086i	−0.515651	−	0.213589i
$399$		0			0
$400$			83.8824i			0.209706i
$401$	95.8849	−	95.8849i	0.239114	−	0.239114i	−0.577369	−	0.816483i	$-0.695921\pi$
$401$	95.8849	−	95.8849i	0.239114	−	0.239114i	0.816483	+	0.577369i	$0.195921\pi$
$402$		0			0
$403$	−29.6218	+	71.5134i	−0.0735033	+	0.177453i
$404$	418.661	+	1010.74i	1.03629	+	2.50182i
$405$		0			0
$406$	335.479			0.826302
$407$	277.554	−	670.075i	0.681952	−	1.64638i
$408$		0			0
$409$			108.593i			0.265509i	0.991149	+	0.132755i	$0.0423822\pi$
$409$			108.593i			0.265509i	−0.991149	+	0.132755i	$0.957618\pi$
$410$	−92.7332	+	535.490i	−0.226179	+	1.30607i
$411$		0			0
$412$	942.665	−	942.665i	2.28802	−	2.28802i
$413$	−354.585	−	146.874i	−0.858560	−	0.355627i
$414$		0			0
$415$	−275.623	−	275.623i	−0.664152	−	0.664152i
$416$	−33.3467	+	13.8126i	−0.0801603	+	0.0332035i
$417$		0			0
$418$	271.670	+	271.670i	0.649929	+	0.649929i
$419$	201.190	+	201.190i	0.480167	+	0.480167i	0.905185	−	0.425018i	$-0.139732\pi$
$419$	201.190	+	201.190i	0.480167	+	0.480167i	−0.425018	+	0.905185i	$0.639732\pi$
$420$		0			0
$421$	26.2486	−	63.3697i	0.0623482	−	0.150522i	−0.889635	−	0.456673i	$-0.849041\pi$
$421$	26.2486	−	63.3697i	0.0623482	−	0.150522i	0.951983	+	0.306151i	$0.0990411\pi$
$422$	262.930	−	634.769i	0.623056	−	1.50419i
$423$		0			0
$424$	−189.985	+	78.6944i	−0.448078	+	0.185600i
$425$	−11.2003	−	27.0399i	−0.0263536	−	0.0636232i
$426$		0			0
$427$	1123.08	−	465.197i	2.63017	−	1.08945i
$428$	512.306			1.19698
$429$		0			0
$430$	−90.6112	+	90.6112i	−0.210724	+	0.210724i
$431$	−75.2035	−	75.2035i	−0.174486	−	0.174486i	0.614461	−	0.788947i	$-0.289373\pi$
$431$	−75.2035	−	75.2035i	−0.174486	−	0.174486i	−0.788947	+	0.614461i	$0.789373\pi$
$432$		0			0
$433$		−	725.948i		−	1.67655i	−0.545245	−	0.838277i	$-0.683563\pi$
$433$		−	725.948i		−	1.67655i	0.545245	−	0.838277i	$-0.316437\pi$
$434$	461.914	+	1115.16i	1.06432	+	2.56949i
$435$		0			0
$436$	177.983	−	73.7231i	0.408219	−	0.169090i
$437$	−74.2768	−	179.320i	−0.169970	−	0.410344i
$438$		0			0
$439$	368.226	+	152.524i	0.838784	+	0.347436i	0.760374	−	0.649485i	$-0.225016\pi$
$439$	368.226	+	152.524i	0.838784	+	0.347436i	0.0784103	+	0.996921i	$0.475016\pi$
$440$	−496.687	−	205.734i	−1.12883	−	0.467578i
$441$		0			0
$442$	−16.7582	+	16.7582i	−0.0379144	+	0.0379144i
$443$	256.839	−	256.839i	0.579772	−	0.579772i	−0.355069	−	0.934840i	$-0.615542\pi$
$443$	256.839	−	256.839i	0.579772	−	0.579772i	0.934840	+	0.355069i	$0.115542\pi$
$444$		0			0
$445$	−65.6615	−	158.521i	−0.147554	−	0.356227i
$446$	−77.3326	+	77.3326i	−0.173391	+	0.173391i
$447$		0			0
$448$	−358.041	+	864.387i	−0.799198	+	1.92944i
$449$	326.223	+	326.223i	0.726555	+	0.726555i	0.969932	−	0.243377i	$-0.0782551\pi$
$449$	326.223	+	326.223i	0.726555	+	0.726555i	−0.243377	+	0.969932i	$0.578255\pi$
$450$		0			0
$451$	417.501	+	264.185i	0.925722	+	0.585777i
$452$	−1349.45			−2.98550
$453$		0			0
$454$	417.155	+	172.791i	0.918843	+	0.380597i
$455$		−	95.5142i		−	0.209921i
$456$		0			0
$457$	−454.419	+	188.227i	−0.994353	+	0.411875i	−0.819723	−	0.572760i	$-0.805873\pi$
$457$	−454.419	+	188.227i	−0.994353	+	0.411875i	−0.174630	+	0.984634i	$0.555873\pi$
$458$	356.976	+	147.864i	0.779423	+	0.322848i
$459$		0			0
$460$	420.285	+	420.285i	0.913662	+	0.913662i
$461$	37.6854			0.0817471			0.0408735	−	0.999164i	$-0.486986\pi$
$461$	37.6854			0.0817471			0.0408735	+	0.999164i	$0.486986\pi$
$462$		0			0
$463$	306.369	−	739.640i	0.661704	−	1.59750i	−0.133427	−	0.991059i	$-0.542598\pi$
$463$	306.369	−	739.640i	0.661704	−	1.59750i	0.795132	−	0.606437i	$-0.207402\pi$
$464$	76.2468	+	31.5824i	0.164325	+	0.0680656i
$465$		0			0
$466$	−187.398	−	452.419i	−0.402142	−	0.970856i
$467$	258.503			0.553539			0.276769	−	0.960936i	$-0.410736\pi$
$467$	258.503			0.553539			0.276769	+	0.960936i	$0.410736\pi$
$468$		0			0
$469$	−569.181			−1.21360
$470$	2.63910	−	1.09315i	0.00561510	−	0.00232585i
$471$		0			0
$472$	−290.507	−	290.507i	−0.615480	−	0.615480i
$473$	44.5813	+	107.629i	0.0942523	+	0.227545i
$474$		0			0
$475$	−34.5311	−	83.3656i	−0.0726971	−	0.175506i
$476$			239.502i			0.503156i
$477$		0			0
$478$	359.820	+	868.681i	0.752761	+	1.81733i
$479$	60.1434	−	145.199i	0.125560	−	0.303130i	−0.848582	−	0.529064i	$-0.822543\pi$
$479$	60.1434	−	145.199i	0.125560	−	0.303130i	0.974143	+	0.225934i	$0.0725433\pi$
$480$		0			0
$481$	127.435	+	52.7854i	0.264938	+	0.109741i
$482$			134.776i			0.279617i
$483$		0			0
$484$	−126.103	+	126.103i	−0.260543	+	0.260543i
$485$	199.759	−	482.261i	0.411874	−	0.994353i
$486$		0			0
$487$	−73.4726	+	73.4726i	−0.150868	+	0.150868i	−0.778506	−	0.627638i	$-0.784022\pi$
$487$	−73.4726	+	73.4726i	−0.150868	+	0.150868i	0.627638	+	0.778506i	$0.284022\pi$
$488$	1301.26			2.66651
$489$		0			0
$490$	−593.912	−	593.912i	−1.21207	−	1.21207i
$491$		−	637.590i		−	1.29855i	−0.760552	−	0.649277i	$-0.775072\pi$
$491$		−	637.590i		−	1.29855i	0.760552	−	0.649277i	$-0.224928\pi$
$492$		0			0
$493$	−28.7955			−0.0584087
$494$	−51.6664	+	51.6664i	−0.104588	+	0.104588i
$495$		0			0
$496$			296.936i			0.598661i
$497$	−112.127	−	112.127i	−0.225607	−	0.225607i
$498$		0			0
$499$	−401.985	−	166.508i	−0.805581	−	0.333683i	−0.0583919	−	0.998294i	$-0.518597\pi$
$499$	−401.985	−	166.508i	−0.805581	−	0.333683i	−0.747190	+	0.664611i	$0.768597\pi$
$500$	707.339	+	707.339i	1.41468	+	1.41468i
$501$		0			0
$502$	−430.814			−0.858196
$503$	−116.671	+	281.668i	−0.231949	+	0.559975i	−0.996407	−	0.0846991i	$-0.973007\pi$
$503$	−116.671	+	281.668i	−0.231949	+	0.559975i	0.764457	+	0.644675i	$0.223007\pi$
$504$		0			0
$505$	539.517	+	223.475i	1.06835	+	0.442525i
$506$	770.321	−	319.077i	1.52237	−	0.630587i
$507$		0			0
$508$	−1083.80			−2.13347
$509$	−52.4461	+	21.7239i	−0.103038	+	0.0426796i	−0.433607	−	0.901102i	$-0.642759\pi$
$509$	−52.4461	+	21.7239i	−0.103038	+	0.0426796i	0.330569	+	0.943782i	$0.392759\pi$
$510$		0			0
$511$	−655.731	+	271.613i	−1.28323	+	0.531532i
$512$	−380.059	+	380.059i	−0.742304	+	0.742304i
$513$		0			0
$514$	285.089	+	688.266i	0.554648	+	1.33904i
$515$		−	711.606i		−	1.38176i
$516$		0			0
$517$		−	2.59691i		−	0.00502304i
$518$	1987.19	−	823.120i	3.83627	−	1.58904i
$519$		0			0
$520$	39.1267	−	94.4602i	0.0752436	−	0.181654i
$521$	206.145	+	85.3882i	0.395673	+	0.163893i	0.571643	−	0.820503i	$-0.306306\pi$
$521$	206.145	+	85.3882i	0.395673	+	0.163893i	−0.175970	+	0.984395i	$0.556306\pi$
$522$		0			0
$523$		−	940.733i		−	1.79872i	−0.437204	−	0.899362i	$-0.644031\pi$
$523$		−	940.733i		−	1.79872i	0.437204	−	0.899362i	$-0.355969\pi$
$524$	−692.635	+	692.635i	−1.32182	+	1.32182i
$525$		0			0
$526$	608.716	−	1469.57i	1.15725	−	2.79386i
$527$	−39.6479	−	95.7186i	−0.0752333	−	0.181629i
$528$		0			0
$529$	107.776			0.203736
$530$	−91.9274	+	221.932i	−0.173448	+	0.418740i
$531$		0			0
$532$			738.400i			1.38797i
$533$	−50.2429	+	79.4005i	−0.0942643	+	0.148969i
$534$		0			0
$535$	193.366	−	193.366i	0.361433	−	0.361433i
$536$	−562.900	−	233.161i	−1.05019	−	0.435001i
$537$		0			0
$538$	−91.3221	−	91.3221i	−0.169744	−	0.169744i
$539$	−705.455	+	292.209i	−1.30882	+	0.542132i
$540$		0			0
$541$	295.995	+	295.995i	0.547126	+	0.547126i	0.925608	−	0.378483i	$-0.123554\pi$
$541$	295.995	+	295.995i	0.547126	+	0.547126i	−0.378483	+	0.925608i	$0.623554\pi$
$542$	158.013	+	158.013i	0.291537	+	0.291537i
$543$		0			0
$544$	18.4878	−	44.6335i	0.0339849	−	0.0820469i
$545$	39.3523	−	95.0049i	0.0722061	−	0.174321i
$546$		0			0
$547$	−1.25534	+	0.519979i	−0.00229496	+	0.000950602i	−0.383831	−	0.923404i	$-0.625395\pi$
$547$	−1.25534	+	0.519979i	−0.00229496	+	0.000950602i	0.381536	+	0.924354i	$0.375395\pi$
$548$	−314.495	−	759.258i	−0.573896	−	1.38551i
$549$		0			0
$550$	358.120	−	148.338i	0.651128	−	0.269706i
$551$	−88.7782			−0.161122
$552$		0			0
$553$	−289.109	+	289.109i	−0.522801	+	0.522801i
$554$	−786.963	−	786.963i	−1.42051	−	1.42051i
$555$		0			0
$556$		−	786.857i		−	1.41521i
$557$	−199.594	−	481.864i	−0.358338	−	0.865105i	−0.995534	−	0.0944032i	$-0.969906\pi$
$557$	−199.594	−	481.864i	−0.358338	−	0.865105i	0.637196	−	0.770702i	$-0.280094\pi$
$558$		0			0
$559$	−20.4689	+	8.47851i	−0.0366170	+	0.0151673i
$560$	−140.216	−	338.511i	−0.250386	−	0.604485i
$561$		0			0
$562$	−1369.52	−	567.273i	−2.43687	−	1.00938i
$563$	507.071	+	210.036i	0.900658	+	0.373065i	0.784473	−	0.620163i	$-0.212933\pi$
$563$	507.071	+	210.036i	0.900658	+	0.373065i	0.116185	+	0.993228i	$0.462933\pi$
$564$		0			0
$565$	−509.340	+	509.340i	−0.901486	+	0.901486i
$566$	−302.867	+	302.867i	−0.535101	+	0.535101i
$567$		0			0
$568$	−64.9576	−	156.822i	−0.114362	−	0.276094i
$569$	−59.4295	+	59.4295i	−0.104446	+	0.104446i	−0.757399	−	0.652953i	$-0.773530\pi$
$569$	−59.4295	+	59.4295i	−0.104446	+	0.104446i	0.652953	+	0.757399i	$0.273530\pi$
$570$		0			0
$571$	−34.3134	+	82.8400i	−0.0600936	+	0.145079i	−0.951074	−	0.308963i	$-0.900018\pi$
$571$	−34.3134	+	82.8400i	−0.0600936	+	0.145079i	0.890981	+	0.454041i	$0.150018\pi$
$572$	−143.837	−	143.837i	−0.251463	−	0.251463i
$573$		0			0
$574$	321.504	+	1429.50i	0.560111	+	2.49042i
$575$	−195.826			−0.340567
$576$		0			0
$577$	526.602	+	218.126i	0.912655	+	0.378034i	0.789072	−	0.614300i	$-0.210562\pi$
$577$	526.602	+	218.126i	0.912655	+	0.378034i	0.123583	+	0.992334i	$0.460562\pi$
$578$			942.589i			1.63078i
$579$		0			0
$580$	251.169	−	104.038i	0.433051	−	0.179375i
$581$	−970.909	−	402.164i	−1.67110	−	0.692192i
$582$		0			0
$583$	154.421	+	154.421i	0.264874	+	0.264874i
$584$	−759.760			−1.30096
$585$		0			0
$586$	−549.020	+	1325.45i	−0.936895	+	2.26186i
$587$	723.205	+	299.561i	1.23204	+	0.510326i	0.901217	−	0.433369i	$-0.142675\pi$
$587$	723.205	+	299.561i	1.23204	+	0.510326i	0.330819	+	0.943694i	$0.392675\pi$
$588$		0			0
$589$	−122.237	−	295.106i	−0.207533	−	0.501029i
$590$	−479.924			−0.813431
$591$		0			0
$592$	529.133			0.893805
$593$	−1047.87	+	434.042i	−1.76706	+	0.731942i	−0.771676	+	0.636015i	$0.780581\pi$
$593$	−1047.87	+	434.042i	−1.76706	+	0.731942i	−0.995388	+	0.0959268i	$0.969419\pi$
$594$		0			0
$595$	90.3985	+	90.3985i	0.151930	+	0.151930i
$596$	−270.546	−	653.156i	−0.453937	−	1.09590i
$597$		0			0
$598$	60.6823	+	146.500i	0.101475	+	0.244983i
$599$		−	275.740i		−	0.460334i	−0.973151	−	0.230167i	$-0.926073\pi$
$599$		−	275.740i		−	0.460334i	0.973151	−	0.230167i	$-0.0739272\pi$
$600$		0			0
$601$	−148.724	−	359.051i	−0.247460	−	0.597422i	0.750527	−	0.660840i	$-0.229800\pi$
$601$	−148.724	−	359.051i	−0.247460	−	0.597422i	−0.997987	+	0.0634178i	$0.979800\pi$
$602$	−132.211	+	319.186i	−0.219620	+	0.530210i
$603$		0			0
$604$	275.905	+	114.284i	0.456797	+	0.189211i
$605$			95.1933i			0.157344i
$606$		0			0
$607$	434.438	−	434.438i	0.715713	−	0.715713i	−0.252011	−	0.967724i	$-0.581092\pi$
$607$	434.438	−	434.438i	0.715713	−	0.715713i	0.967724	+	0.252011i	$0.0810921\pi$
$608$	56.9990	−	137.608i	0.0937484	−	0.226329i
$609$		0			0
$610$	1074.85	−	1074.85i	1.76205	−	1.76205i
$611$	0.493882			0.000808317
$612$		0			0
$613$	554.163	+	554.163i	0.904017	+	0.904017i	0.995781	−	0.0917637i	$-0.0292504\pi$
$613$	554.163	+	554.163i	0.904017	+	0.904017i	−0.0917637	+	0.995781i	$0.529250\pi$
$614$			1559.37i			2.53969i
$615$		0			0
$616$	−1449.44			−2.35298
$617$	550.758	−	550.758i	0.892639	−	0.892639i	−0.102132	−	0.994771i	$-0.532566\pi$
$617$	550.758	−	550.758i	0.892639	−	0.892639i	0.994771	+	0.102132i	$0.0325663\pi$
$618$		0			0
$619$			590.942i			0.954671i	0.878721	+	0.477336i	$0.158397\pi$
$619$			590.942i			0.954671i	−0.878721	+	0.477336i	$0.841603\pi$
$620$	691.660	+	691.660i	1.11558	+	1.11558i
$621$		0			0
$622$	−353.928	−	146.602i	−0.569017	−	0.235694i
$623$	−327.106	−	327.106i	−0.525050	−	0.525050i
$624$		0			0
$625$	295.425			0.472680
$626$	493.641	−	1191.76i	0.788564	−	1.90376i
$627$		0			0
$628$	−127.379	−	52.7623i	−0.202833	−	0.0840163i
$629$	−170.568	+	70.6517i	−0.271174	+	0.112324i
$630$		0			0
$631$	330.673			0.524047			0.262023	−	0.965062i	$-0.415610\pi$
$631$	330.673			0.524047			0.262023	+	0.965062i	$0.415610\pi$
$632$	−404.350	+	167.487i	−0.639794	+	0.265012i
$633$		0			0
$634$	−1168.41	+	483.971i	−1.84292	+	0.763361i
$635$	−409.074	+	409.074i	−0.644212	+	0.644212i
$636$		0			0
$637$	−55.5725	−	134.164i	−0.0872410	−	0.210618i
$638$		−	381.372i		−	0.597762i
$639$		0			0
$640$			922.236i			1.44099i
$641$	−616.324	+	255.290i	−0.961504	+	0.398268i	−0.807543	−	0.589809i	$-0.799203\pi$
$641$	−616.324	+	255.290i	−0.961504	+	0.398268i	−0.153961	+	0.988077i	$0.549203\pi$
$642$		0			0
$643$	31.2194	−	75.3704i	0.0485528	−	0.117217i	−0.897742	−	0.440521i	$-0.854794\pi$
$643$	31.2194	−	75.3704i	0.0485528	−	0.117217i	0.946295	+	0.323304i	$0.104794\pi$
$644$	1480.49	+	613.240i	2.29890	+	0.952236i
$645$		0			0
$646$		−	97.7984i		−	0.151391i
$647$	756.871	−	756.871i	1.16982	−	1.16982i	0.187564	−	0.982252i	$-0.439941\pi$
$647$	756.871	−	756.871i	1.16982	−	1.16982i	0.982252	−	0.187564i	$-0.0600592\pi$
$648$		0			0
$649$	−166.966	+	403.092i	−0.257267	+	0.621098i
$650$	28.2111	+	68.1076i	0.0434017	+	0.104781i
$651$		0			0
$652$	−1001.28			−1.53571
$653$	−104.193	+	251.545i	−0.159561	+	0.385215i	−0.983360	−	0.181668i	$-0.941851\pi$
$653$	−104.193	+	251.545i	−0.159561	+	0.385215i	0.823799	+	0.566882i	$0.191851\pi$
$654$		0			0
$655$			522.861i			0.798261i
$656$	−61.5047	+	355.160i	−0.0937572	+	0.541403i
$657$		0			0
$658$	5.44575	−	5.44575i	0.00827621	−	0.00827621i
$659$	−114.144	−	47.2801i	−0.173208	−	0.0717452i	0.294394	−	0.955684i	$-0.404882\pi$
$659$	−114.144	−	47.2801i	−0.173208	−	0.0717452i	−0.467603	+	0.883939i	$0.654882\pi$
$660$		0			0
$661$	−49.1306	−	49.1306i	−0.0743277	−	0.0743277i	0.668966	−	0.743293i	$-0.266737\pi$
$661$	−49.1306	−	49.1306i	−0.0743277	−	0.0743277i	−0.743293	+	0.668966i	$0.766737\pi$
$662$	−958.103	+	396.859i	−1.44729	+	0.599486i
$663$		0			0
$664$	−795.451	−	795.451i	−1.19797	−	1.19797i
$665$	278.704	+	278.704i	0.419104	+	0.419104i
$666$		0			0
$667$	−73.7302	+	178.000i	−0.110540	+	0.266867i
$668$	−667.394	+	1611.23i	−0.999093	+	2.41202i
$669$		0			0
$670$	−657.556	+	272.368i	−0.981426	+	0.406520i
$671$	−528.835	−	1276.72i	−0.788130	−	1.90271i
$672$		0			0
$673$	−897.045	+	371.568i	−1.33290	+	0.552107i	−0.931482	−	0.363786i	$-0.881484\pi$
$673$	−897.045	+	371.568i	−1.33290	+	0.552107i	−0.401422	+	0.915893i	$0.631484\pi$
$674$	−2098.59			−3.11363
$675$		0			0
$676$	−852.861	+	852.861i	−1.26163	+	1.26163i
$677$	739.405	+	739.405i	1.09218	+	1.09218i	0.995296	+	0.0968826i	$0.0308871\pi$
$677$	739.405	+	739.405i	1.09218	+	1.09218i	0.0968826	+	0.995296i	$0.469113\pi$
$678$		0			0
$679$		−	1407.34i		−	2.07267i
$680$	52.3699	+	126.432i	0.0770145	+	0.185930i
$681$		0			0
$682$	1267.71	−	525.104i	1.85882	−	0.769947i
$683$	−364.618	−	880.266i	−0.533848	−	1.28882i	−0.928957	−	0.370189i	$-0.879293\pi$
$683$	−364.618	−	880.266i	−0.533848	−	1.28882i	0.395109	−	0.918634i	$-0.370707\pi$
$684$		0			0
$685$	−405.281	−	167.873i	−0.591651	−	0.245070i
$686$	−474.302	−	196.462i	−0.691402	−	0.286388i
$687$		0			0
$688$	−60.0973	+	60.0973i	−0.0873507	+	0.0873507i
$689$	−29.3679	+	29.3679i	−0.0426240	+	0.0426240i
$690$		0			0
$691$	−84.8429	−	204.829i	−0.122783	−	0.296424i	0.850522	−	0.525939i	$-0.176286\pi$
$691$	−84.8429	−	204.829i	−0.122783	−	0.296424i	−0.973305	+	0.229515i	$0.926286\pi$
$692$	267.600	−	267.600i	0.386705	−	0.386705i
$693$		0			0
$694$	507.040	−	1224.10i	0.730605	−	1.76384i
$695$	−296.994	−	296.994i	−0.427330	−	0.427330i
$696$		0			0
$697$	−27.5960	−	122.700i	−0.0395925	−	0.176040i
$698$	−314.285			−0.450266
$699$		0			0
$700$	688.278	+	285.094i	0.983254	+	0.407277i
$701$		−	284.354i		−	0.405641i	−0.979216	−	0.202820i	$-0.934989\pi$
$701$		−	284.354i		−	0.405641i	0.979216	−	0.202820i	$-0.0650107\pi$
$702$		0			0
$703$	−525.872	+	217.824i	−0.748040	+	0.309849i
$704$	982.635	+	407.021i	1.39579	+	0.578154i
$705$		0			0
$706$	1307.48	+	1307.48i	1.85195	+	1.85195i
$707$	1574.43			2.22691
$708$		0			0
$709$	−144.375	+	348.553i	−0.203632	+	0.491612i	−0.992396	−	0.123084i	$-0.960721\pi$
$709$	−144.375	+	348.553i	−0.203632	+	0.491612i	0.788764	+	0.614696i	$0.210721\pi$
$710$	−183.192	−	75.8807i	−0.258017	−	0.106874i
$711$		0			0
$712$	−189.500	−	457.493i	−0.266151	−	0.642547i
$713$	−693.206			−0.972238
$714$		0			0
$715$	−108.580			−0.151861
$716$	−494.665	+	204.897i	−0.690872	+	0.286169i
$717$		0			0
$718$	−488.216	−	488.216i	−0.679966	−	0.679966i
$719$	161.288	+	389.383i	0.224322	+	0.541562i	0.995468	−	0.0950965i	$-0.0303160\pi$
$719$	161.288	+	389.383i	0.224322	+	0.541562i	−0.771146	+	0.636659i	$0.780316\pi$
$720$		0			0
$721$	−734.195	−	1772.50i	−1.01830	−	2.45839i
$722$			915.526i			1.26804i
$723$		0			0
$724$	247.117	+	596.592i	0.341321	+	0.824023i
$725$	−34.2770	+	82.7520i	−0.0472786	+	0.114141i
$726$		0			0
$727$	759.674	+	314.667i	1.04494	+	0.432830i	0.838084	−	0.545541i	$-0.183676\pi$
$727$	759.674	+	314.667i	1.04494	+	0.432830i	0.206859	+	0.978371i	$0.433676\pi$
$728$		−	275.655i		−	0.378647i
$729$		0			0
$730$	−627.571	+	627.571i	−0.859686	+	0.859686i
$731$	11.3482	−	27.3970i	0.0155242	−	0.0374788i
$732$		0			0
$733$	−122.432	+	122.432i	−0.167028	+	0.167028i	−0.785672	−	0.618643i	$-0.787683\pi$
$733$	−122.432	+	122.432i	−0.167028	+	0.167028i	0.618643	+	0.785672i	$0.287683\pi$
$734$	−1846.39			−2.51552
$735$		0			0
$736$	−228.566	−	228.566i	−0.310552	−	0.310552i
$737$			647.044i			0.877943i
$738$		0			0
$739$	−1046.70			−1.41637			−0.708185	−	0.706027i	$-0.750486\pi$
$739$	−1046.70			−1.41637			−0.708185	+	0.706027i	$0.750486\pi$
$740$	1232.52	−	1232.52i	1.66557	−	1.66557i
$741$		0			0
$742$			647.646i			0.872838i
$743$	187.656	+	187.656i	0.252566	+	0.252566i	0.822022	−	0.569456i	$-0.192846\pi$
$743$	187.656	+	187.656i	0.252566	+	0.252566i	−0.569456	+	0.822022i	$0.692846\pi$
$744$		0			0
$745$	−348.646	−	144.414i	−0.467981	−	0.193844i
$746$	−34.6948	−	34.6948i	−0.0465078	−	0.0465078i
$747$		0			0
$748$	272.266			0.363992
$749$	282.142	−	681.151i	0.376692	−	0.909414i
$750$		0			0
$751$	638.232	+	264.364i	0.849843	+	0.352017i	0.764727	−	0.644354i	$-0.222874\pi$
$751$	638.232	+	264.364i	0.849843	+	0.352017i	0.0851162	+	0.996371i	$0.472874\pi$
$752$	1.75037	−	0.725025i	0.00232761	−	0.000964129i
$753$		0			0
$754$	72.5296			0.0961931
$755$	147.274	−	61.0030i	0.195065	−	0.0807987i
$756$		0			0
$757$	721.384	−	298.807i	0.952951	−	0.394725i	0.148612	−	0.988896i	$-0.452520\pi$
$757$	721.384	−	298.807i	0.952951	−	0.394725i	0.804339	+	0.594170i	$0.202520\pi$
$758$	1405.89	−	1405.89i	1.85474	−	1.85474i
$759$		0			0
$760$	161.460	+	389.798i	0.212447	+	0.512892i
$761$			1084.11i			1.42459i	0.701882	+	0.712294i	$0.252344\pi$
$761$			1084.11i			1.42459i	−0.701882	+	0.712294i	$0.747656\pi$
$762$		0			0
$763$		−	277.245i		−	0.363361i
$764$	2110.73	−	874.295i	2.76274	−	1.14436i
$765$		0			0
$766$	507.743	−	1225.80i	0.662850	−	1.60026i
$767$	−76.6604	−	31.7538i	−0.0999483	−	0.0414000i
$768$		0			0
$769$			122.901i			0.159820i	0.996802	+	0.0799098i	$0.0254632\pi$
$769$			122.901i			0.159820i	−0.996802	+	0.0799098i	$0.974537\pi$
$770$	−1197.25	+	1197.25i	−1.55487	+	1.55487i
$771$		0			0
$772$	827.724	−	1998.30i	1.07218	−	2.58847i
$773$	15.6507	+	37.7841i	0.0202467	+	0.0488799i	0.933680	−	0.358110i	$-0.116579\pi$
$773$	15.6507	+	37.7841i	0.0202467	+	0.0488799i	−0.913433	+	0.406989i	$0.866579\pi$
$774$		0			0
$775$	−322.270			−0.415832
$776$	576.507	−	1391.81i	0.742921	−	1.79357i
$777$		0			0
$778$			1683.81i			2.16427i
$779$	−85.0800	−	378.291i	−0.109217	−	0.485611i
$780$		0			0
$781$	−127.466	+	127.466i	−0.163208	+	0.163208i
$782$	−196.086	−	81.2214i	−0.250749	−	0.103864i
$783$		0			0
$784$	−393.909	−	393.909i	−0.502434	−	0.502434i
$785$	−67.9933	+	28.1637i	−0.0866156	+	0.0358774i
$786$		0			0
$787$	707.611	+	707.611i	0.899124	+	0.899124i	0.995359	−	0.0962346i	$-0.0306799\pi$
$787$	707.611	+	707.611i	0.899124	+	0.899124i	−0.0962346	+	0.995359i	$0.530680\pi$
$788$	38.7872	+	38.7872i	0.0492223	+	0.0492223i
$789$		0			0
$790$	−195.652	+	472.345i	−0.247660	+	0.597904i
$791$	−743.181	+	1794.20i	−0.939546	+	2.26826i
$792$		0			0
$793$	242.808	−	100.574i	0.306189	−	0.126828i
$794$	237.399	+	573.133i	0.298992	+	0.721829i
$795$		0			0
$796$	−448.391	+	185.730i	−0.563305	+	0.233329i
$797$	671.933			0.843078			0.421539	−	0.906810i	$-0.361490\pi$
$797$	671.933			0.843078			0.421539	+	0.906810i	$0.361490\pi$
$798$		0			0
$799$	−0.467430	+	0.467430i	−0.000585019	+	0.000585019i
$800$	−106.260	−	106.260i	−0.132825	−	0.132825i
$801$		0			0
$802$		−	457.156i		−	0.570020i
$803$	308.769	+	745.435i	0.384520	+	0.928313i
$804$		0			0
$805$	790.265	−	327.339i	0.981696	−	0.406632i
$806$	99.8645	+	241.094i	0.123901	+	0.299124i
$807$		0			0
$808$	1557.05	+	644.952i	1.92704	+	0.798208i
$809$	706.243	+	292.535i	0.872983	+	0.361601i	0.773771	−	0.633465i	$-0.218368\pi$
$809$	706.243	+	292.535i	0.872983	+	0.361601i	0.0992114	+	0.995066i	$0.468368\pi$
$810$		0			0
$811$	−770.751	+	770.751i	−0.950371	+	0.950371i	−0.998825	−	0.0484543i	$-0.984570\pi$
$811$	−770.751	+	770.751i	−0.950371	+	0.950371i	0.0484543	+	0.998825i	$0.484570\pi$
$812$	518.285	−	518.285i	0.638282	−	0.638282i
$813$		0			0
$814$	−935.723	−	2259.03i	−1.14954	−	2.77523i
$815$	−377.927	+	377.927i	−0.463715	+	0.463715i
$816$		0			0
$817$	34.9873	−	84.4667i	0.0428241	−	0.103386i
$818$	258.873	+	258.873i	0.316471	+	0.316471i
$819$		0			0
$820$	684.019	+	970.548i	0.834170	+	1.18360i
$821$	−623.500			−0.759440			−0.379720	−	0.925102i	$-0.623980\pi$
$821$	−623.500			−0.759440			−0.379720	+	0.925102i	$0.623980\pi$
$822$		0			0
$823$	558.248	+	231.234i	0.678308	+	0.280964i	0.695119	−	0.718894i	$-0.255351\pi$
$823$	558.248	+	231.234i	0.678308	+	0.280964i	−0.0168113	+	0.999859i	$0.505351\pi$
$824$		−	2053.70i		−	2.49236i
$825$		0			0
$826$	−1195.42	+	495.159i	−1.44724	+	0.599466i
$827$	929.687	+	385.089i	1.12417	+	0.465646i	0.865795	−	0.500399i	$-0.166813\pi$
$827$	929.687	+	385.089i	1.12417	+	0.465646i	0.258374	+	0.966045i	$0.416813\pi$
$828$		0			0
$829$	600.020	+	600.020i	0.723788	+	0.723788i	0.969375	−	0.245587i	$-0.0789806\pi$
$829$	600.020	+	600.020i	0.723788	+	0.723788i	−0.245587	+	0.969375i	$0.578981\pi$
$830$	−1314.10			−1.58326
$831$		0			0
$832$	−77.4074	+	186.878i	−0.0930378	+	0.224613i
$833$	179.574	+	74.3821i	0.215575	+	0.0892942i
$834$		0			0
$835$	356.245	+	860.053i	0.426641	+	1.03000i
$836$	839.412			1.00408
$837$		0			0
$838$	959.226			1.14466
$839$	927.665	−	384.251i	1.10568	−	0.457987i	0.246231	−	0.969211i	$-0.420808\pi$
$839$	927.665	−	384.251i	1.10568	−	0.457987i	0.859448	+	0.511224i	$0.170808\pi$
$840$		0			0
$841$	−532.363	−	532.363i	−0.633012	−	0.633012i
$842$	−88.4923	−	213.639i	−0.105098	−	0.253728i
$843$		0			0
$844$	−574.458	−	1386.86i	−0.680637	−	1.64320i
$845$			643.814i			0.761910i
$846$		0			0
$847$	98.2151	+	237.112i	0.115956	+	0.279944i
$848$	−60.9703	+	147.195i	−0.0718989	+	0.173579i
$849$		0			0
$850$	−91.1599	−	37.7597i	−0.107247	−	0.0444231i
$851$			1235.28i			1.45156i
$852$		0			0
$853$	694.593	−	694.593i	0.814295	−	0.814295i	−0.170980	−	0.985275i	$-0.554693\pi$
$853$	694.593	−	694.593i	0.814295	−	0.814295i	0.985275	+	0.170980i	$0.0546933\pi$
$854$	1568.32	−	3786.27i	1.83645	−	4.43357i
$855$		0			0
$856$	558.058	−	558.058i	0.651936	−	0.651936i
$857$	−290.882			−0.339419			−0.169710	−	0.985494i	$-0.554283\pi$
$857$	−290.882			−0.339419			−0.169710	+	0.985494i	$0.554283\pi$
$858$		0			0
$859$	−67.0806	−	67.0806i	−0.0780915	−	0.0780915i	0.666982	−	0.745074i	$-0.267586\pi$
$859$	−67.0806	−	67.0806i	−0.0780915	−	0.0780915i	−0.745074	+	0.666982i	$0.767586\pi$
$860$			279.972i			0.325549i
$861$		0			0
$862$	−358.552			−0.415954
$863$	−1122.84	+	1122.84i	−1.30109	+	1.30109i	−0.373431	+	0.927658i	$0.621819\pi$
$863$	−1122.84	+	1122.84i	−1.30109	+	1.30109i	−0.927658	+	0.373431i	$0.878181\pi$
$864$		0			0
$865$		−	202.008i		−	0.233535i
$866$	−1730.57	−	1730.57i	−1.99835	−	1.99835i
$867$		0			0
$868$	2436.44	+	1009.21i	2.80696	+	1.16268i
$869$	328.659	+	328.659i	0.378204	+	0.378204i
$870$		0			0
$871$	−123.055			−0.141280
$872$	113.571	−	274.185i	0.130242	−	0.314433i
$873$		0			0
$874$	−604.545	−	250.411i	−0.691699	−	0.286511i
$875$	1330.02	−	550.911i	1.52002	−	0.629612i
$876$		0			0
$877$	883.264			1.00714			0.503571	−	0.863954i	$-0.332019\pi$
$877$	883.264			1.00714			0.503571	+	0.863954i	$0.332019\pi$
$878$	1241.41	−	514.208i	1.41390	−	0.585658i
$879$		0			0
$880$	−384.820	+	159.398i	−0.437295	+	0.181134i
$881$	189.394	−	189.394i	0.214976	−	0.214976i	−0.591401	−	0.806377i	$-0.701425\pi$
$881$	189.394	−	189.394i	0.214976	−	0.214976i	0.806377	+	0.591401i	$0.201425\pi$
$882$		0			0
$883$	495.140	+	1195.37i	0.560748	+	1.35377i	0.909169	+	0.416427i	$0.136718\pi$
$883$	495.140	+	1195.37i	0.560748	+	1.35377i	−0.348421	+	0.937338i	$0.613282\pi$
$884$			51.7797i			0.0585743i
$885$		0			0
$886$		−	1224.55i		−	1.38211i
$887$	−152.020	+	62.9686i	−0.171386	+	0.0709905i	−0.466727	−	0.884402i	$-0.654567\pi$
$887$	−152.020	+	62.9686i	−0.171386	+	0.0709905i	0.295341	+	0.955392i	$0.404567\pi$
$888$		0			0
$889$	−596.883	+	1441.00i	−0.671409	+	1.62093i
$890$	−534.424	−	221.366i	−0.600476	−	0.248725i
$891$		0			0
$892$			238.944i			0.267874i
$893$	−1.44112	+	1.44112i	−0.00161379	+	0.00161379i
$894$		0			0
$895$	−109.371	+	264.045i	−0.122202	+	0.295022i
$896$	951.512	+	2297.15i	1.06195	+	2.56379i
$897$		0			0
$898$	1555.35			1.73202
$899$	−121.337	+	292.934i	−0.134969	+	0.325845i
$900$		0			0
$901$		−	55.5900i		−	0.0616981i
$902$	1625.06	−	365.485i	1.80161	−	0.405194i
$903$		0			0
$904$	−1469.96	+	1469.96i	−1.62606	+	1.62606i
$905$	318.452	+	131.907i	0.351881	+	0.145754i
$906$		0			0
$907$	−181.264	−	181.264i	−0.199850	−	0.199850i	0.600086	−	0.799936i	$-0.295133\pi$
$907$	−181.264	−	181.264i	−0.199850	−	0.199850i	−0.799936	+	0.600086i	$0.795133\pi$
$908$	911.414	−	377.520i	1.00376	−	0.415771i
$909$		0			0
$910$	−227.694	−	227.694i	−0.250213	−	0.250213i
$911$	395.125	+	395.125i	0.433727	+	0.433727i	0.889894	−	0.456167i	$-0.150778\pi$
$911$	395.125	+	395.125i	0.433727	+	0.433727i	−0.456167	+	0.889894i	$0.650778\pi$
$912$		0			0
$913$	−457.179	+	1103.73i	−0.500744	+	1.20890i
$914$	−634.571	+	1531.99i	−0.694279	+	1.67614i
$915$		0			0
$916$	779.933	−	323.059i	0.851455	−	0.352684i
$917$	539.459	+	1302.37i	0.588287	+	1.42025i
$918$		0			0
$919$	−250.905	+	103.928i	−0.273020	+	0.113089i	−0.514993	−	0.857194i	$-0.672205\pi$
$919$	−250.905	+	103.928i	−0.273020	+	0.113089i	0.241973	+	0.970283i	$0.422205\pi$
$920$	915.637			0.995257
$921$		0			0
$922$	89.8375	−	89.8375i	0.0974376	−	0.0974376i
$923$	−24.2415	−	24.2415i	−0.0262638	−	0.0262638i
$924$		0			0
$925$			574.278i			0.620841i
$926$	−1032.87	−	2493.56i	−1.11541	−	2.69283i
$927$		0			0
$928$	−136.595	+	56.5795i	−0.147193	+	0.0609693i
$929$	−17.5523	−	42.3751i	−0.0188938	−	0.0456137i	0.914151	−	0.405374i	$-0.132859\pi$
$929$	−17.5523	−	42.3751i	−0.0188938	−	0.0456137i	−0.933045	+	0.359760i	$0.882859\pi$
$930$		0			0
$931$	553.638	+	229.324i	0.594671	+	0.246321i
$932$	−988.460	−	409.434i	−1.06058	−	0.439306i
$933$		0			0
$934$	616.239	−	616.239i	0.659785	−	0.659785i
$935$	102.765	−	102.765i	0.109909	−	0.109909i
$936$		0			0
$937$	−471.850	−	1139.15i	−0.503575	−	1.21574i	−0.947524	−	0.319686i	$-0.896423\pi$
$937$	−471.850	−	1139.15i	−0.503575	−	1.21574i	0.443949	−	0.896052i	$-0.353577\pi$
$938$	−1356.86	+	1356.86i	−1.44654	+	1.44654i
$939$		0			0
$940$	2.38835	−	5.76599i	0.00254080	−	0.00613403i
$941$	−784.033	−	784.033i	−0.833192	−	0.833192i	0.154760	−	0.987952i	$-0.450539\pi$
$941$	−784.033	−	784.033i	−0.833192	−	0.833192i	−0.987952	+	0.154760i	$0.950539\pi$
$942$		0			0
$943$	−829.132	−	143.585i	−0.879250	−	0.152264i
$944$	−318.307			−0.337189
$945$		0			0
$946$	362.851	+	150.298i	0.383563	+	0.158877i
$947$			945.504i			0.998420i	0.866481	+	0.499210i	$0.166376\pi$
$947$			945.504i			0.998420i	−0.866481	+	0.499210i	$0.833624\pi$
$948$		0			0
$949$	−141.767	+	58.7219i	−0.149386	+	0.0618777i
$950$	−281.051	−	116.415i	−0.295844	−	0.122542i
$951$		0			0
$952$	260.891	+	260.891i	0.274045	+	0.274045i
$953$	−410.140			−0.430368			−0.215184	−	0.976574i	$-0.569035\pi$
$953$	−410.140			−0.430368			−0.215184	+	0.976574i	$0.569035\pi$
$954$		0			0
$955$	466.686	−	1126.68i	0.488676	−	1.17977i
$956$	1897.92	+	786.146i	1.98528	+	0.822328i
$957$		0			0
$958$	−202.762	−	489.512i	−0.211652	−	0.510973i
$959$	−1182.70			−1.23326
$960$		0			0
$961$	−179.805			−0.187102
$962$	429.625	−	177.956i	0.446595	−	0.184986i
$963$		0			0
$964$	208.216	+	208.216i	0.215992	+	0.215992i
$965$	−441.827	−	1066.67i	−0.457852	−	1.10535i
$966$		0			0
$967$	10.4433	+	25.2123i	0.0107997	+	0.0260727i	0.929187	−	0.369609i	$-0.120508\pi$
$967$	10.4433	+	25.2123i	0.0107997	+	0.0260727i	−0.918388	+	0.395682i	$0.870508\pi$
$968$			274.729i			0.283811i
$969$		0			0
$970$	−673.451	−	1625.85i	−0.694279	−	1.67614i
$971$	−250.318	+	604.320i	−0.257794	+	0.622369i	−0.998792	−	0.0491378i	$-0.984353\pi$
$971$	−250.318	+	604.320i	−0.257794	+	0.622369i	0.740998	+	0.671507i	$0.234353\pi$
$972$		0			0
$973$	−1046.19	−	433.346i	−1.07522	−	0.445371i
$974$			350.300i			0.359651i
$975$		0			0
$976$	712.889	−	712.889i	0.730420	−	0.730420i
$977$	243.204	−	587.145i	0.248929	−	0.600967i	−0.749185	−	0.662361i	$-0.769554\pi$
$977$	243.204	−	587.145i	0.248929	−	0.600967i	0.998114	+	0.0613936i	$0.0195545\pi$
$978$		0			0
$979$	−371.854	+	371.854i	−0.379830	+	0.379830i
$980$	−1835.08			−1.87253
$981$		0			0
$982$	−1519.94	−	1519.94i	−1.54780	−	1.54780i
$983$			94.7811i			0.0964202i	0.998837	+	0.0482101i	$0.0153517\pi$
$983$			94.7811i			0.0964202i	−0.998837	+	0.0482101i	$0.984648\pi$
$984$		0			0
$985$	29.2799			0.0297258
$986$	−68.6450	+	68.6450i	−0.0696196	+	0.0696196i
$987$		0			0
$988$			159.640i			0.161579i
$989$	−140.299	−	140.299i	−0.141859	−	0.141859i
$990$		0			0
$991$	−159.074	−	65.8906i	−0.160519	−	0.0664890i	0.300977	−	0.953631i	$-0.402687\pi$
$991$	−159.074	−	65.8906i	−0.160519	−	0.0664890i	−0.461496	+	0.887142i	$0.652687\pi$
$992$	−376.150	−	376.150i	−0.379184	−	0.379184i
$993$		0			0
$994$	−534.594			−0.537821
$995$	−99.1398	+	239.345i	−0.0996380	+	0.240547i
$996$		0			0
$997$	933.834	+	386.807i	0.936644	+	0.387971i	0.798195	−	0.602399i	$-0.205788\pi$
$997$	933.834	+	386.807i	0.936644	+	0.387971i	0.138449	+	0.990370i	$0.455788\pi$
$998$	−1355.22	+	561.350i	−1.35793	+	0.562475i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	369.3.l.b.325.5		20
3.2	odd	2		41.3.e.b.38.1	yes	20
41.27	odd	8	inner	369.3.l.b.109.5		20
123.68	even	8		41.3.e.b.27.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.3.e.b.27.1	✓	20	123.68	even	8
41.3.e.b.38.1	yes	20	3.2	odd	2
369.3.l.b.109.5		20	41.27	odd	8	inner
369.3.l.b.325.5		20	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:	\( N \)	\(=\)	\( 369 = 3^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	369.l (of order \(8\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(2.38388 - 2.38388i) q^{2} -7.36576i q^{4} +(-2.78016 - 2.78016i) q^{5} +(-9.79337 - 4.05655i) q^{7} +(-8.02356 - 8.02356i) q^{8} +O(q^{10})\)	\(q+(2.38388 - 2.38388i) q^{2} -7.36576i q^{4} +(-2.78016 - 2.78016i) q^{5} +(-9.79337 - 4.05655i) q^{7} +(-8.02356 - 8.02356i) q^{8} -13.2551 q^{10} +(-4.61148 + 11.1331i) q^{11} +(-2.11730 - 0.877014i) q^{13} +(-33.0165 + 13.6759i) q^{14} -8.79138 q^{16} +(2.83394 - 1.17386i) q^{17} +(8.73721 - 3.61907i) q^{19} +(-20.4780 + 20.4780i) q^{20} +(15.5467 + 37.5332i) q^{22} -20.5237i q^{23} -9.54144i q^{25} +(-7.13808 + 2.95669i) q^{26} +(-29.8795 + 72.1356i) q^{28} +(-8.67290 - 3.59243i) q^{29} -33.7758i q^{31} +(11.1367 - 11.1367i) q^{32} +(3.95744 - 9.55410i) q^{34} +(15.9493 + 38.5049i) q^{35} -60.1877 q^{37} +(12.2010 - 29.4559i) q^{38} +44.6136i q^{40} +(6.99603 - 40.3987i) q^{41} +(6.83594 - 6.83594i) q^{43} +(82.0037 + 33.9670i) q^{44} +(-48.9261 - 48.9261i) q^{46} +(-0.199100 + 0.0824700i) q^{47} +(44.8062 + 44.8062i) q^{49} +(-22.7456 - 22.7456i) q^{50} +(-6.45987 + 15.5955i) q^{52} +(6.93524 - 16.7431i) q^{53} +(43.7724 - 18.1311i) q^{55} +(46.0298 + 111.126i) q^{56} +(-29.2391 + 12.1112i) q^{58} +36.2067 q^{59} +(-81.0896 + 81.0896i) q^{61} +(-80.5174 - 80.5174i) q^{62} -88.2625i q^{64} +(3.44819 + 8.32466i) q^{65} +(49.6077 - 20.5482i) q^{67} +(-8.64634 - 20.8741i) q^{68} +(129.812 + 53.7700i) q^{70} +(13.8205 + 5.72464i) q^{71} +(47.3455 - 47.3455i) q^{73} +(-143.480 + 143.480i) q^{74} +(-26.6572 - 64.3562i) q^{76} +(90.3238 - 90.3238i) q^{77} +(14.7604 - 35.6349i) q^{79} +(24.4414 + 24.4414i) q^{80} +(-79.6280 - 112.983i) q^{82} +99.1394 q^{83} +(-11.1423 - 4.61529i) q^{85} -32.5921i q^{86} +(126.328 - 52.3266i) q^{88} +(40.3183 + 16.7004i) q^{89} +(17.1778 + 17.1778i) q^{91} -151.173 q^{92} +(-0.278032 + 0.671230i) q^{94} +(-34.3524 - 14.2292i) q^{95} +(50.8068 + 122.659i) q^{97} +213.625 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 8 q^{2} + 12 q^{5} - 4 q^{7} - 36 q^{8}+O(q^{10}) \) 20 * q + 8 * q^2 + 12 * q^5 - 4 * q^7 - 36 * q^8	\( 20 q + 8 q^{2} + 12 q^{5} - 4 q^{7} - 36 q^{8} + 16 q^{10} + 12 q^{11} - 48 q^{13} - 88 q^{14} - 36 q^{16} + 28 q^{17} + 76 q^{19} + 16 q^{20} - 116 q^{22} - 40 q^{26} + 72 q^{28} + 24 q^{29} - 176 q^{32} + 80 q^{34} - 60 q^{35} + 208 q^{37} + 380 q^{38} + 116 q^{41} - 40 q^{43} - 116 q^{44} - 176 q^{46} + 64 q^{47} + 168 q^{49} + 148 q^{50} - 184 q^{52} + 120 q^{53} + 20 q^{55} - 188 q^{56} + 36 q^{58} + 512 q^{59} - 460 q^{61} - 68 q^{62} - 432 q^{65} + 300 q^{67} - 120 q^{68} + 308 q^{70} + 108 q^{71} + 60 q^{73} - 140 q^{74} + 872 q^{76} - 112 q^{77} - 208 q^{79} + 68 q^{80} - 376 q^{82} + 120 q^{83} + 172 q^{85} + 316 q^{88} - 268 q^{89} - 800 q^{91} + 448 q^{92} - 212 q^{94} + 184 q^{95} - 120 q^{97} + 20 q^{98}+O(q^{100}) \) 20 * q + 8 * q^2 + 12 * q^5 - 4 * q^7 - 36 * q^8 + 16 * q^10 + 12 * q^11 - 48 * q^13 - 88 * q^14 - 36 * q^16 + 28 * q^17 + 76 * q^19 + 16 * q^20 - 116 * q^22 - 40 * q^26 + 72 * q^28 + 24 * q^29 - 176 * q^32 + 80 * q^34 - 60 * q^35 + 208 * q^37 + 380 * q^38 + 116 * q^41 - 40 * q^43 - 116 * q^44 - 176 * q^46 + 64 * q^47 + 168 * q^49 + 148 * q^50 - 184 * q^52 + 120 * q^53 + 20 * q^55 - 188 * q^56 + 36 * q^58 + 512 * q^59 - 460 * q^61 - 68 * q^62 - 432 * q^65 + 300 * q^67 - 120 * q^68 + 308 * q^70 + 108 * q^71 + 60 * q^73 - 140 * q^74 + 872 * q^76 - 112 * q^77 - 208 * q^79 + 68 * q^80 - 376 * q^82 + 120 * q^83 + 172 * q^85 + 316 * q^88 - 268 * q^89 - 800 * q^91 + 448 * q^92 - 212 * q^94 + 184 * q^95 - 120 * q^97 + 20 * q^98

Introduction

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