LMFDB - Embedded newform 75.4.e.c.68.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [75,4,Mod(32,75)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("75.32"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	75.e (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,6,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$4.42514325043$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.28356903014400.8
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 209x^{4} + 1600 $ x^8 + 209*x^4 + 1600
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			68.1
Root			$-2.66260 + 2.66260i$ of defining polynomial
Character	$\chi$	$=$	75.68
Dual form			75.4.e.c.32.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-2.66260 - 2.66260i) q^{2} +(2.80471 + 4.37420i) q^{3} +6.17891i q^{4} +(4.17891 - 19.1146i) q^{6} +(-9.35782 + 9.35782i) q^{7} +(-4.84884 + 4.84884i) q^{8} +(-11.2672 + 24.5367i) q^{9} +34.1375i q^{11} +(-27.0278 + 17.3301i) q^{12} +(-2.82109 - 2.82109i) q^{13} +49.8323 q^{14} +75.2524 q^{16} +(64.2384 + 64.2384i) q^{17} +(95.3316 - 35.3316i) q^{18} +19.0735i q^{19} +(-67.1789 - 14.6869i) q^{21} +(90.8945 - 90.8945i) q^{22} +(-51.4018 + 51.4018i) q^{23} +(-34.8094 - 7.61018i) q^{24} +15.0229i q^{26} +(-138.930 + 19.5337i) q^{27} +(-57.8211 - 57.8211i) q^{28} -50.5042 q^{29} -93.3673 q^{31} +(-161.576 - 161.576i) q^{32} +(-149.324 + 95.7458i) q^{33} -342.083i q^{34} +(-151.610 - 69.6188i) q^{36} +(161.537 - 161.537i) q^{37} +(50.7850 - 50.7850i) q^{38} +(4.42765 - 20.2524i) q^{39} -88.7935i q^{41} +(139.765 + 217.976i) q^{42} +(-176.399 - 176.399i) q^{43} -210.932 q^{44} +273.725 q^{46} +(38.2843 + 38.2843i) q^{47} +(211.061 + 329.169i) q^{48} +167.863i q^{49} +(-100.821 + 461.162i) q^{51} +(17.4313 - 17.4313i) q^{52} +(344.569 - 344.569i) q^{53} +(421.925 + 317.904i) q^{54} -90.7492i q^{56} +(-83.4310 + 53.4956i) q^{57} +(134.473 + 134.473i) q^{58} +421.133 q^{59} +2.00000 q^{61} +(248.600 + 248.600i) q^{62} +(-124.174 - 335.046i) q^{63} +258.409i q^{64} +(652.524 + 142.657i) q^{66} +(-430.987 + 430.987i) q^{67} +(-396.923 + 396.923i) q^{68} +(-369.009 - 80.6742i) q^{69} -733.866i q^{71} +(-64.3420 - 173.607i) q^{72} +(348.073 + 348.073i) q^{73} -860.216 q^{74} -117.853 q^{76} +(-319.452 - 319.452i) q^{77} +(-65.7131 + 42.1349i) q^{78} +588.019i q^{79} +(-475.102 - 552.919i) q^{81} +(-236.422 + 236.422i) q^{82} +(-217.997 + 217.997i) q^{83} +(90.7492 - 415.092i) q^{84} +939.362i q^{86} +(-141.650 - 220.915i) q^{87} +(-165.527 - 165.527i) q^{88} +1272.00 q^{89} +52.7985 q^{91} +(-317.607 - 317.607i) q^{92} +(-261.868 - 408.407i) q^{93} -203.872i q^{94} +(253.591 - 1159.94i) q^{96} +(432.111 - 432.111i) q^{97} +(446.951 - 446.951i) q^{98} +(-837.622 - 384.633i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{3} - 12 q^{6} + 16 q^{7} - 132 q^{12} - 68 q^{13} + 284 q^{16} + 240 q^{18} - 492 q^{21} + 500 q^{22} - 702 q^{27} - 508 q^{28} + 616 q^{31} + 240 q^{33} - 804 q^{36} + 1156 q^{37} - 540 q^{42}+ \cdots - 1904 q^{97}+O(q^{100}) $ 8 * q + 6 * q^3 - 12 * q^6 + 16 * q^7 - 132 * q^12 - 68 * q^13 + 284 * q^16 + 240 * q^18 - 492 * q^21 + 500 * q^22 - 702 * q^27 - 508 * q^28 + 616 * q^31 + 240 * q^33 - 804 * q^36 + 1156 * q^37 - 540 * q^42 - 548 * q^43 + 736 * q^46 + 1116 * q^48 - 852 * q^51 - 224 * q^52 - 684 * q^57 - 60 * q^58 + 16 * q^61 - 1428 * q^63 + 2040 * q^66 - 404 * q^67 + 1800 * q^72 + 2512 * q^73 - 1488 * q^76 + 360 * q^78 + 288 * q^81 - 2800 * q^82 + 1680 * q^87 - 2460 * q^88 - 1304 * q^91 - 3408 * q^93 + 4164 * q^96 - 1904 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	−2.66260	−	2.66260i	−0.941372	−	0.941372i	0.0570018	−	0.998374i	$-0.481846\pi$
$2$	−2.66260	−	2.66260i	−0.941372	−	0.941372i	−0.998374	+	0.0570018i	$0.981846\pi$
$3$	2.80471	+	4.37420i	0.539767	+	0.841814i
$3$	2.80471	+	4.37420i	0.539767	+	0.841814i
$4$			6.17891i			0.772364i
$5$		0			0
$5$		0			0
$6$	4.17891	−	19.1146i	0.284339	−	1.30058i
$7$	−9.35782	+	9.35782i	−0.505275	+	0.505275i	−0.913072	−	0.407798i	$-0.866297\pi$
$7$	−9.35782	+	9.35782i	−0.505275	+	0.505275i	0.407798	+	0.913072i	$0.366297\pi$
$8$	−4.84884	+	4.84884i	−0.214291	+	0.214291i
$9$	−11.2672	+	24.5367i	−0.417303	+	0.908768i
$10$		0			0
$11$			34.1375i			0.935712i	0.883805	+	0.467856i	$0.154973\pi$
$11$			34.1375i			0.935712i	−0.883805	+	0.467856i	$0.845027\pi$
$12$	−27.0278	+	17.3301i	−0.650187	+	0.416897i
$13$	−2.82109	−	2.82109i	−0.0601869	−	0.0601869i	0.676373	−	0.736560i	$-0.263551\pi$
$13$	−2.82109	−	2.82109i	−0.0601869	−	0.0601869i	−0.736560	+	0.676373i	$0.763551\pi$
$14$	49.8323			0.951303
$15$		0			0
$16$	75.2524			1.17582
$17$	64.2384	+	64.2384i	0.916477	+	0.916477i	0.996771	−	0.0802942i	$-0.0255860\pi$
$17$	64.2384	+	64.2384i	0.916477	+	0.916477i	−0.0802942	+	0.996771i	$0.525586\pi$
$18$	95.3316	−	35.3316i	1.24833	−	0.462652i
$19$			19.0735i			0.230303i	0.993348	+	0.115151i	$0.0367353\pi$
$19$			19.0735i			0.230303i	−0.993348	+	0.115151i	$0.963265\pi$
$20$		0			0
$21$	−67.1789	−	14.6869i	−0.698078	−	0.152617i
$22$	90.8945	−	90.8945i	0.880854	−	0.880854i
$23$	−51.4018	+	51.4018i	−0.466001	+	0.466001i	−0.900616	−	0.434616i	$-0.856884\pi$
$23$	−51.4018	+	51.4018i	−0.466001	+	0.466001i	0.434616	+	0.900616i	$0.356884\pi$
$24$	−34.8094	−	7.61018i	−0.296060	−	0.0647259i
$25$		0			0
$26$			15.0229i			0.113317i
$27$	−138.930	+	19.5337i	−0.990260	+	0.139232i
$28$	−57.8211	−	57.8211i	−0.390256	−	0.390256i
$29$	−50.5042			−0.323393			−0.161697	−	0.986841i	$-0.551697\pi$
$29$	−50.5042			−0.323393			−0.161697	+	0.986841i	$0.551697\pi$
$30$		0			0
$31$	−93.3673			−0.540944			−0.270472	−	0.962728i	$-0.587180\pi$
$31$	−93.3673			−0.540944			−0.270472	+	0.962728i	$0.587180\pi$
$32$	−161.576	−	161.576i	−0.892592	−	0.892592i
$33$	−149.324	+	95.7458i	−0.787696	+	0.505067i
$34$		−	342.083i		−	1.72549i
$35$		0			0
$36$	−151.610	−	69.6188i	−0.701899	−	0.322309i
$37$	161.537	−	161.537i	0.717743	−	0.717743i	−0.250400	−	0.968142i	$-0.580562\pi$
$37$	161.537	−	161.537i	0.717743	−	0.717743i	0.968142	+	0.250400i	$0.0805621\pi$
$38$	50.7850	−	50.7850i	0.216800	−	0.216800i
$39$	4.42765	−	20.2524i	0.0181793	−	0.0831531i
$40$		0			0
$41$		−	88.7935i		−	0.338225i	−0.985597	−	0.169112i	$-0.945910\pi$
$41$		−	88.7935i		−	0.338225i	0.985597	−	0.169112i	$-0.0540901\pi$
$42$	139.765	+	217.976i	0.513482	+	0.800820i
$43$	−176.399	−	176.399i	−0.625596	−	0.625596i	0.321361	−	0.946957i	$-0.395860\pi$
$43$	−176.399	−	176.399i	−0.625596	−	0.625596i	−0.946957	+	0.321361i	$0.895860\pi$
$44$	−210.932			−0.722710
$45$		0			0
$46$	273.725			0.877360
$47$	38.2843	+	38.2843i	0.118816	+	0.118816i	0.764015	−	0.645199i	$-0.223225\pi$
$47$	38.2843	+	38.2843i	0.118816	+	0.118816i	−0.645199	+	0.764015i	$0.723225\pi$
$48$	211.061	+	329.169i	0.634668	+	0.989820i
$49$			167.863i			0.489395i
$50$		0			0
$51$	−100.821	+	461.162i	−0.276819	+	1.26619i
$52$	17.4313	−	17.4313i	0.0464862	−	0.0464862i
$53$	344.569	−	344.569i	0.893022	−	0.893022i	−0.101784	−	0.994806i	$-0.532455\pi$
$53$	344.569	−	344.569i	0.893022	−	0.893022i	0.994806	+	0.101784i	$0.0324551\pi$
$54$	421.925	+	317.904i	1.06327	+	0.801134i
$55$		0			0
$56$		−	90.7492i		−	0.216551i
$57$	−83.4310	+	53.4956i	−0.193872	+	0.124310i
$58$	134.473	+	134.473i	0.304433	+	0.304433i
$59$	421.133			0.929268			0.464634	−	0.885503i	$-0.346186\pi$
$59$	421.133			0.929268			0.464634	+	0.885503i	$0.346186\pi$
$60$		0			0
$61$	2.00000			0.00419793			0.00209897	−	0.999998i	$-0.499332\pi$
$61$	2.00000			0.00419793			0.00209897	+	0.999998i	$0.499332\pi$
$62$	248.600	+	248.600i	0.509229	+	0.509229i
$63$	−124.174	−	335.046i	−0.248325	−	0.670030i
$64$			258.409i			0.504704i
$65$		0			0
$66$	652.524	+	142.657i	1.21697	+	0.266059i
$67$	−430.987	+	430.987i	−0.785872	+	0.785872i	−0.980815	−	0.194943i	$-0.937548\pi$
$67$	−430.987	+	430.987i	−0.785872	+	0.785872i	0.194943	+	0.980815i	$0.437548\pi$
$68$	−396.923	+	396.923i	−0.707853	+	0.707853i
$69$	−369.009	−	80.6742i	−0.643818	−	0.140754i
$70$		0			0
$71$		−	733.866i		−	1.22667i	−0.789822	−	0.613337i	$-0.789827\pi$
$71$		−	733.866i		−	1.22667i	0.789822	−	0.613337i	$-0.210173\pi$
$72$	−64.3420	−	173.607i	−0.105316	−	0.284164i
$73$	348.073	+	348.073i	0.558067	+	0.558067i	0.928757	−	0.370690i	$-0.120879\pi$
$73$	348.073	+	348.073i	0.558067	+	0.558067i	−0.370690	+	0.928757i	$0.620879\pi$
$74$	−860.216			−1.35133
$75$		0			0
$76$	−117.853			−0.177877
$77$	−319.452	−	319.452i	−0.472792	−	0.472792i
$78$	−65.7131	+	42.1349i	−0.0953915	+	0.0611646i
$79$			588.019i			0.837434i	0.908117	+	0.418717i	$0.137520\pi$
$79$			588.019i			0.837434i	−0.908117	+	0.418717i	$0.862480\pi$
$80$		0			0
$81$	−475.102	−	552.919i	−0.651717	−	0.758462i
$82$	−236.422	+	236.422i	−0.318395	+	0.318395i
$83$	−217.997	+	217.997i	−0.288293	+	0.288293i	−0.836405	−	0.548112i	$-0.815347\pi$
$83$	−217.997	+	217.997i	−0.288293	+	0.288293i	0.548112	+	0.836405i	$0.315347\pi$
$84$	90.7492	−	415.092i	0.117876	−	0.539170i
$85$		0			0
$86$			939.362i			1.17784i
$87$	−141.650	−	220.915i	−0.174557	−	0.272237i
$88$	−165.527	−	165.527i	−0.200514	−	0.200514i
$89$	1272.00			1.51497			0.757483	−	0.652855i	$-0.226429\pi$
$89$	1272.00			1.51497			0.757483	+	0.652855i	$0.226429\pi$
$90$		0			0
$91$	52.7985			0.0608219
$92$	−317.607	−	317.607i	−0.359922	−	0.359922i
$93$	−261.868	−	408.407i	−0.291984	−	0.455374i
$94$		−	203.872i		−	0.223700i
$95$		0			0
$96$	253.591	−	1159.94i	0.269605	−	1.23319i
$97$	432.111	−	432.111i	0.452312	−	0.452312i	−0.443809	−	0.896121i	$-0.646373\pi$
$97$	432.111	−	432.111i	0.452312	−	0.452312i	0.896121	+	0.443809i	$0.146373\pi$
$98$	446.951	−	446.951i	0.460703	−	0.460703i
$99$	−837.622	−	384.633i	−0.850345	−	0.390475i
$100$		0			0
$101$			1662.30i			1.63767i	0.574029	+	0.818835i	$0.305380\pi$
$101$			1662.30i			1.63767i	−0.574029	+	0.818835i	$0.694620\pi$
$102$	1496.34	−	959.444i	1.45254	−	0.931364i
$103$	774.495	+	774.495i	0.740906	+	0.740906i	0.972752	−	0.231847i	$-0.0744768\pi$
$103$	774.495	+	774.495i	0.740906	+	0.740906i	−0.231847	+	0.972752i	$0.574477\pi$
$104$	27.3581			0.0257950
$105$		0			0
$106$	−1834.90			−1.68133
$107$	1170.26	+	1170.26i	1.05732	+	1.05732i	0.998254	+	0.0590633i	$0.0188114\pi$
$107$	1170.26	+	1170.26i	1.05732	+	1.05732i	0.0590633	+	0.998254i	$0.481189\pi$
$108$	−120.697	−	858.433i	−0.107537	−	0.764841i
$109$		−	1264.60i		−	1.11125i	−0.831432	−	0.555627i	$-0.812478\pi$
$109$		−	1264.60i		−	1.11125i	0.831432	−	0.555627i	$-0.187522\pi$
$110$		0			0
$111$	1159.66	+	253.529i	0.991620	+	0.216792i
$112$	−704.198	+	704.198i	−0.594111	+	0.594111i
$113$	−381.173	+	381.173i	−0.317325	+	0.317325i	−0.847739	−	0.530414i	$-0.822037\pi$
$113$	−381.173	+	381.173i	−0.317325	+	0.317325i	0.530414	+	0.847739i	$0.322037\pi$
$114$	364.581	+	79.7062i	0.299528	+	0.0654839i
$115$		0			0
$116$		−	312.061i		−	0.249777i
$117$	101.006	−	37.4346i	0.0798121	−	0.0295798i
$118$	−1121.31	−	1121.31i	−0.874787	−	0.874787i
$119$	−1202.26			−0.926145
$120$		0			0
$121$	165.633			0.124442
$122$	−5.32521	−	5.32521i	−0.00395182	−	0.00395182i
$123$	388.400	−	249.040i	0.284722	−	0.182563i
$124$		−	576.908i		−	0.417805i
$125$		0			0
$126$	−561.469	+	1222.72i	−0.396981	+	0.864513i
$127$	439.588	−	439.588i	0.307142	−	0.307142i	−0.536658	−	0.843800i	$-0.680313\pi$
$127$	439.588	−	439.588i	0.307142	−	0.307142i	0.843800	+	0.536658i	$0.180313\pi$
$128$	−604.571	+	604.571i	−0.417477	+	0.417477i
$129$	276.856	−	1266.35i	0.188959	−	0.864312i
$130$		0			0
$131$		−	1399.28i		−	0.933247i	−0.884456	−	0.466623i	$-0.845470\pi$
$131$		−	1399.28i		−	0.933247i	0.884456	−	0.466623i	$-0.154530\pi$
$132$	−591.605	−	922.659i	−0.390095	−	0.608388i
$133$	−178.486	−	178.486i	−0.116366	−	0.116366i
$134$	2295.09			1.47960
$135$		0			0
$136$	−622.964			−0.392785
$137$	−1092.28	−	1092.28i	−0.681169	−	0.681169i	0.279095	−	0.960264i	$-0.409966\pi$
$137$	−1092.28	−	1092.28i	−0.681169	−	0.681169i	−0.960264	+	0.279095i	$0.909966\pi$
$138$	767.720	+	1197.33i	0.473570	+	0.738574i
$139$			2498.43i			1.52456i	0.647245	+	0.762282i	$0.275921\pi$
$139$			2498.43i			1.52456i	−0.647245	+	0.762282i	$0.724079\pi$
$140$		0			0
$141$	−60.0866	+	274.840i	−0.0358880	+	0.164154i
$142$	−1953.99	+	1953.99i	−1.15476	+	1.15476i
$143$	96.3049	−	96.3049i	0.0563177	−	0.0563177i
$144$	−847.881	+	1846.45i	−0.490672	+	1.06855i
$145$		0			0
$146$		−	1853.56i		−	1.05070i
$147$	−734.264	+	470.806i	−0.411980	+	0.264159i
$148$	998.121	+	998.121i	0.554358	+	0.554358i
$149$	−3570.40			−1.96308			−0.981538	−	0.191270i	$-0.938739\pi$
$149$	−3570.40			−1.96308			−0.981538	+	0.191270i	$0.938739\pi$
$150$		0			0
$151$	2687.14			1.44819			0.724094	−	0.689701i	$-0.242258\pi$
$151$	2687.14			1.44819			0.724094	+	0.689701i	$0.242258\pi$
$152$	−92.4842	−	92.4842i	−0.0493517	−	0.0493517i
$153$	−2299.99	+	852.415i	−1.21531	+	0.450416i
$154$			1701.15i			0.890146i
$155$		0			0
$156$	125.137	+	27.3581i	0.0642245	+	0.0140410i
$157$	1810.48	−	1810.48i	0.920333	−	0.920333i	−0.0767201	−	0.997053i	$-0.524445\pi$
$157$	1810.48	−	1810.48i	0.920333	−	0.920333i	0.997053	+	0.0767201i	$0.0244448\pi$
$158$	1565.66	−	1565.66i	0.788337	−	0.788337i
$159$	2473.63	+	540.795i	1.23378	+	0.269735i
$160$		0			0
$161$		−	962.017i		−	0.470916i
$162$	−207.196	+	2737.21i	−0.100487	+	1.32750i
$163$	2679.06	+	2679.06i	1.28736	+	1.28736i	0.936382	+	0.350982i	$0.114152\pi$
$163$	2679.06	+	2679.06i	1.28736	+	1.28736i	0.350982	+	0.936382i	$0.385848\pi$
$164$	548.647			0.261232
$165$		0			0
$166$	1160.88			0.542782
$167$	−139.543	−	139.543i	−0.0646597	−	0.0646597i	0.674037	−	0.738697i	$-0.264559\pi$
$167$	−139.543	−	139.543i	−0.0646597	−	0.0646597i	−0.738697	+	0.674037i	$0.764559\pi$
$168$	396.955	−	254.525i	0.182296	−	0.116887i
$169$		−	2181.08i		−	0.992755i
$170$		0			0
$171$	−468.000	−	214.904i	−0.209292	−	0.0961059i
$172$	1089.95	−	1089.95i	0.483188	−	0.483188i
$173$	881.613	−	881.613i	0.387444	−	0.387444i	−0.486331	−	0.873775i	$-0.661665\pi$
$173$	881.613	−	881.613i	0.387444	−	0.387444i	0.873775	+	0.486331i	$0.161665\pi$
$174$	−211.053	+	965.367i	−0.0919532	+	0.420599i
$175$		0			0
$176$			2568.93i			1.10023i
$177$	1181.16	+	1842.12i	0.501588	+	0.782271i
$178$	−3386.84	−	3386.84i	−1.42615	−	1.42615i
$179$	2512.87			1.04928			0.524638	−	0.851325i	$-0.324201\pi$
$179$	2512.87			1.04928			0.524638	+	0.851325i	$0.324201\pi$
$180$		0			0
$181$	269.796			0.110795			0.0553973	−	0.998464i	$-0.482357\pi$
$181$	269.796			0.110795			0.0553973	+	0.998464i	$0.482357\pi$
$182$	−140.581	−	140.581i	−0.0572560	−	0.0572560i
$183$	5.60943	+	8.74839i	0.00226591	+	0.00353388i
$184$		−	498.478i		−	0.199719i
$185$		0			0
$186$	−390.173	+	1784.68i	−0.153811	+	0.703542i
$187$	−2192.94	+	2192.94i	−0.857559	+	0.857559i
$188$	−236.555	+	236.555i	−0.0917690	+	0.0917690i
$189$	1117.29	−	1482.87i	0.430003	−	0.570703i
$190$		0			0
$191$			2420.22i			0.916864i	0.888729	+	0.458432i	$0.151589\pi$
$191$			2420.22i			0.916864i	−0.888729	+	0.458432i	$0.848411\pi$
$192$	−1130.33	+	724.762i	−0.424867	+	0.272423i
$193$	−1965.28	−	1965.28i	−0.732973	−	0.732973i	0.238234	−	0.971208i	$-0.423431\pi$
$193$	−1965.28	−	1965.28i	−0.732973	−	0.732973i	−0.971208	+	0.238234i	$0.923431\pi$
$194$	−2301.08			−0.851588
$195$		0			0
$196$	−1037.21			−0.377991
$197$	832.602	+	832.602i	0.301119	+	0.301119i	0.841452	−	0.540333i	$-0.181702\pi$
$197$	832.602	+	832.602i	0.301119	+	0.301119i	−0.540333	+	0.841452i	$0.681702\pi$
$198$	1206.13	+	3254.38i	0.432909	+	1.16807i
$199$			1540.54i			0.548775i	0.961619	+	0.274387i	$0.0884750\pi$
$199$			1540.54i			0.548775i	−0.961619	+	0.274387i	$0.911525\pi$
$200$		0			0
$201$	−3094.02	−	676.426i	−1.08575	−	0.237370i
$202$	4426.04	−	4426.04i	1.54166	−	1.54166i
$203$	472.609	−	472.609i	0.163402	−	0.163402i
$204$	−2849.48	−	622.964i	−0.977957	−	0.213805i
$205$		0			0
$206$		−	4124.35i		−	1.39494i
$207$	−682.079	−	1840.38i	−0.229023	−	0.617949i
$208$	−212.294	−	212.294i	−0.0707689	−	0.0707689i
$209$	−651.119			−0.215497
$210$		0			0
$211$	−10.9380			−0.00356874			−0.00178437	−	0.999998i	$-0.500568\pi$
$211$	−10.9380			−0.00356874			−0.00178437	+	0.999998i	$0.500568\pi$
$212$	2129.06	+	2129.06i	0.689738	+	0.689738i
$213$	3210.07	−	2058.28i	1.03263	−	0.662118i
$214$		−	6231.86i		−	1.99066i
$215$		0			0
$216$	578.932	−	768.364i	0.182367	−	0.242039i
$217$	873.714	−	873.714i	0.273325	−	0.273325i
$218$	−3367.13	+	3367.13i	−1.04610	+	1.04610i
$219$	−546.295	+	2498.79i	−0.168563	+	0.771016i
$220$		0			0
$221$		−	362.445i		−	0.110320i
$222$	−2412.66	−	3762.75i	−0.729401	−	1.13757i
$223$	−831.512	−	831.512i	−0.249696	−	0.249696i	0.571150	−	0.820846i	$-0.306497\pi$
$223$	−831.512	−	831.512i	−0.249696	−	0.249696i	−0.820846	+	0.571150i	$0.806497\pi$
$224$	3024.00			0.902008
$225$		0			0
$226$	2029.83			0.597443
$227$	−3441.18	−	3441.18i	−1.00616	−	1.00616i	−0.999981	−	0.00618314i	$-0.998032\pi$
$227$	−3441.18	−	3441.18i	−1.00616	−	1.00616i	−0.00618314	−	0.999981i	$-0.501968\pi$
$228$	−330.544	−	515.512i	−0.0960124	−	0.149740i
$229$		−	1680.38i		−	0.484903i	−0.970164	−	0.242451i	$-0.922049\pi$
$229$		−	1680.38i		−	0.484903i	0.970164	−	0.242451i	$-0.0779515\pi$
$230$		0			0
$231$	501.375	−	2293.32i	0.142805	−	0.653200i
$232$	244.887	−	244.887i	0.0693001	−	0.0693001i
$233$	−2106.74	+	2106.74i	−0.592348	+	0.592348i	−0.938265	−	0.345917i	$-0.887568\pi$
$233$	−2106.74	+	2106.74i	−0.592348	+	0.592348i	0.345917	+	0.938265i	$0.387568\pi$
$234$	−368.613	−	169.265i	−0.102978	−	0.0472873i
$235$		0			0
$236$			2602.14i			0.717733i
$237$	−2572.11	+	1649.22i	−0.704964	+	0.452019i
$238$	3201.15	+	3201.15i	0.871847	+	0.871847i
$239$	−261.125			−0.0706728			−0.0353364	−	0.999375i	$-0.511250\pi$
$239$	−261.125			−0.0706728			−0.0353364	+	0.999375i	$0.511250\pi$
$240$		0			0
$241$	−6001.45			−1.60410			−0.802048	−	0.597259i	$-0.796256\pi$
$241$	−6001.45			−1.60410			−0.802048	+	0.597259i	$0.796256\pi$
$242$	−441.014	−	441.014i	−0.117147	−	0.117147i
$243$	1086.05	−	3628.97i	0.286709	−	0.958018i
$244$			12.3578i			0.00324233i
$245$		0			0
$246$	−1697.25	−	371.060i	−0.439889	−	0.0961704i
$247$	53.8080	−	53.8080i	0.0138612	−	0.0138612i
$248$	452.723	−	452.723i	0.115919	−	0.115919i
$249$	−1564.98	−	342.143i	−0.398300	−	0.0870781i
$250$		0			0
$251$			3044.59i			0.765630i	0.923825	+	0.382815i	$0.125045\pi$
$251$			3044.59i			0.765630i	−0.923825	+	0.382815i	$0.874955\pi$
$252$	2070.22	−	767.260i	0.517506	−	0.191797i
$253$	−1754.73	−	1754.73i	−0.436042	−	0.436042i
$254$	−2340.89			−0.578271
$255$		0			0
$256$	5286.74			1.29071
$257$	946.317	+	946.317i	0.229687	+	0.229687i	0.812562	−	0.582875i	$-0.198072\pi$
$257$	946.317	+	946.317i	0.229687	+	0.229687i	−0.582875	+	0.812562i	$0.698072\pi$
$258$	−4108.95	+	2634.64i	−0.991521	+	0.635758i
$259$			3023.26i			0.725314i
$260$		0			0
$261$	569.040	−	1239.21i	0.134953	−	0.293889i
$262$	−3725.72	+	3725.72i	−0.878532	+	0.878532i
$263$	−67.0257	+	67.0257i	−0.0157148	+	0.0157148i	−0.714921	−	0.699206i	$-0.753537\pi$
$263$	−67.0257	+	67.0257i	−0.0157148	+	0.0157148i	0.699206	+	0.714921i	$0.253537\pi$
$264$	259.792	−	1188.31i	0.0605648	−	0.277027i
$265$		0			0
$266$			950.474i			0.219088i
$267$	3567.60	+	5563.99i	0.817729	+	1.27532i
$268$	−2663.03	−	2663.03i	−0.606979	−	0.606979i
$269$	2658.15			0.602492			0.301246	−	0.953546i	$-0.402597\pi$
$269$	2658.15			0.602492			0.301246	+	0.953546i	$0.402597\pi$
$270$		0			0
$271$	145.673			0.0326530			0.0163265	−	0.999867i	$-0.494803\pi$
$271$	145.673			0.0326530			0.0163265	+	0.999867i	$0.494803\pi$
$272$	4834.09	+	4834.09i	1.07761	+	1.07761i
$273$	148.085	+	230.951i	0.0328296	+	0.0512007i
$274$			5816.64i			1.28247i
$275$		0			0
$276$	498.478	−	2280.07i	0.108713	−	0.497261i
$277$	1074.57	−	1074.57i	0.233085	−	0.233085i	−0.580894	−	0.813979i	$-0.697297\pi$
$277$	1074.57	−	1074.57i	0.233085	−	0.233085i	0.813979	+	0.580894i	$0.197297\pi$
$278$	6652.34	−	6652.34i	1.43518	−	1.43518i
$279$	1051.98	−	2290.93i	0.225737	−	0.491592i
$280$		0			0
$281$			2020.29i			0.428898i	0.976735	+	0.214449i	$0.0687956\pi$
$281$			2020.29i			0.428898i	−0.976735	+	0.214449i	$0.931204\pi$
$282$	891.776	−	571.802i	0.188314	−	0.120746i
$283$	2400.34	+	2400.34i	0.504189	+	0.504189i	0.912737	−	0.408548i	$-0.133965\pi$
$283$	2400.34	+	2400.34i	0.504189	+	0.504189i	−0.408548	+	0.912737i	$0.633965\pi$
$284$	4534.49			0.947438
$285$		0			0
$286$	−512.844			−0.106032
$287$	830.913	+	830.913i	0.170896	+	0.170896i
$288$	5785.06	−	2144.05i	1.18364	−	0.438678i
$289$			3340.15i			0.679860i
$290$		0			0
$291$	3102.09	+	678.191i	0.624906	+	0.136619i
$292$	−2150.71	+	2150.71i	−0.431031	+	0.431031i
$293$	−2533.13	+	2533.13i	−0.505075	+	0.505075i	−0.913011	−	0.407936i	$-0.866249\pi$
$293$	−2533.13	+	2533.13i	−0.505075	+	0.505075i	0.407936	+	0.913011i	$0.366249\pi$
$294$	3208.62	+	701.482i	0.636499	+	0.139154i
$295$		0			0
$296$			1566.53i			0.307611i
$297$	−666.830	−	4742.71i	−0.130281	−	0.926598i
$298$	9506.54	+	9506.54i	1.84798	+	1.84798i
$299$	290.018			0.0560943
$300$		0			0
$301$	3301.42			0.632196
$302$	−7154.79	−	7154.79i	−1.36328	−	1.36328i
$303$	−7271.21	+	4662.27i	−1.37861	+	0.883961i
$304$			1435.32i			0.270794i
$305$		0			0
$306$	8393.59	+	3854.31i	1.56807	+	0.720052i
$307$	−3159.93	+	3159.93i	−0.587449	+	0.587449i	−0.936940	−	0.349491i	$-0.886355\pi$
$307$	−3159.93	+	3159.93i	−0.587449	+	0.587449i	0.349491	+	0.936940i	$0.386355\pi$
$308$	1973.87	−	1973.87i	0.365167	−	0.365167i
$309$	−1215.56	+	5560.03i	−0.223788	+	1.02362i
$310$		0			0
$311$		−	7206.19i		−	1.31391i	−0.753931	−	0.656954i	$-0.771845\pi$
$311$		−	7206.19i		−	1.31391i	0.753931	−	0.656954i	$-0.228155\pi$
$312$	76.7315	+	119.670i	0.0139233	+	0.0217146i
$313$	−2029.31	−	2029.31i	−0.366464	−	0.366464i	0.499722	−	0.866186i	$-0.333436\pi$
$313$	−2029.31	−	2029.31i	−0.366464	−	0.366464i	−0.866186	+	0.499722i	$0.833436\pi$
$314$	−9641.19			−1.73275
$315$		0			0
$316$	−3633.31			−0.646804
$317$	689.223	+	689.223i	0.122116	+	0.122116i	0.765524	−	0.643408i	$-0.222480\pi$
$317$	689.223	+	689.223i	0.122116	+	0.122116i	−0.643408	+	0.765524i	$0.722480\pi$
$318$	−5146.37	−	8026.21i	−0.907528	−	1.41537i
$319$		−	1724.09i		−	0.302603i
$320$		0			0
$321$	−1836.70	+	8401.16i	−0.319360	+	1.46077i
$322$	−2561.47	+	2561.47i	−0.443308	+	0.443308i
$323$	−1225.25	+	1225.25i	−0.211067	+	0.211067i
$324$	3416.44	−	2935.61i	0.585809	−	0.503363i
$325$		0			0
$326$		−	14266.6i		−	2.42378i
$327$	5531.60	−	3546.84i	0.935469	−	0.599818i
$328$	430.546	+	430.546i	0.0724784	+	0.0724784i
$329$	−716.516			−0.120069
$330$		0			0
$331$	−8226.53			−1.36608			−0.683038	−	0.730383i	$-0.739342\pi$
$331$	−8226.53			−1.36608			−0.683038	+	0.730383i	$0.739342\pi$
$332$	−1346.99	−	1346.99i	−0.222667	−	0.222667i
$333$	2143.52	+	5783.64i	0.352745	+	0.951777i
$334$			743.096i			0.121738i
$335$		0			0
$336$	−5055.37	−	1105.23i	−0.820813	−	0.179449i
$337$	−1777.34	+	1777.34i	−0.287294	+	0.287294i	−0.836009	−	0.548715i	$-0.815117\pi$
$337$	−1777.34	+	1777.34i	−0.287294	+	0.287294i	0.548715	+	0.836009i	$0.315117\pi$
$338$	−5807.36	+	5807.36i	−0.934552	+	0.934552i
$339$	−2736.41	−	598.245i	−0.438411	−	0.0958472i
$340$		0			0
$341$		−	3187.32i		−	0.506168i
$342$	673.895	+	1818.30i	0.106550	+	0.287493i
$343$	−4780.56	−	4780.56i	−0.752554	−	0.752554i
$344$	1710.66			0.268119
$345$		0			0
$346$	−4694.77			−0.729458
$347$	1715.22	+	1715.22i	0.265354	+	0.265354i	0.827225	−	0.561871i	$-0.189918\pi$
$347$	1715.22	+	1715.22i	0.265354	+	0.265354i	−0.561871	+	0.827225i	$0.689918\pi$
$348$	1365.02	−	875.242i	0.210266	−	0.134821i
$349$		−	8603.96i		−	1.31965i	−0.751417	−	0.659827i	$-0.770629\pi$
$349$		−	8603.96i		−	1.31965i	0.751417	−	0.659827i	$-0.229371\pi$
$350$		0			0
$351$	447.039	+	336.827i	0.0679806	+	0.0512208i
$352$	5515.81	−	5515.81i	0.835209	−	0.835209i
$353$	5425.13	−	5425.13i	0.817990	−	0.817990i	−0.167827	−	0.985816i	$-0.553675\pi$
$353$	5425.13	−	5425.13i	0.817990	−	0.817990i	0.985816	+	0.167827i	$0.0536750\pi$
$354$	1759.87	−	8049.77i	0.264227	−	1.20859i
$355$		0			0
$356$			7859.58i			1.17010i
$357$	−3372.00	−	5258.93i	−0.499903	−	0.779642i
$358$	−6690.76	−	6690.76i	−0.987759	−	0.987759i
$359$	−11418.9			−1.67874			−0.839370	−	0.543560i	$-0.817076\pi$
$359$	−11418.9			−1.67874			−0.839370	+	0.543560i	$0.817076\pi$
$360$		0			0
$361$	6495.20			0.946961
$362$	−718.361	−	718.361i	−0.104299	−	0.104299i
$363$	464.552	+	724.510i	0.0671699	+	0.104757i
$364$			326.237i			0.0469766i
$365$		0			0
$366$	8.35782	−	38.2292i	0.00119363	−	0.00545976i
$367$	−6554.73	+	6554.73i	−0.932299	+	0.932299i	−0.997849	−	0.0655499i	$-0.979120\pi$
$367$	−6554.73	+	6554.73i	−0.932299	+	0.932299i	0.0655499	+	0.997849i	$0.479120\pi$
$368$	−3868.11	+	3868.11i	−0.547932	+	0.547932i
$369$	2178.70	+	1000.45i	0.307368	+	0.141142i
$370$		0			0
$371$			6448.83i			0.902443i
$372$	2523.51	−	1618.06i	0.351714	−	0.225518i
$373$	5967.46	+	5967.46i	0.828374	+	0.828374i	0.987292	−	0.158918i	$-0.0508005\pi$
$373$	5967.46	+	5967.46i	0.828374	+	0.828374i	−0.158918	+	0.987292i	$0.550801\pi$
$374$	11677.8			1.61456
$375$		0			0
$376$	−371.270			−0.0509222
$377$	142.477	+	142.477i	0.0194640	+	0.0194640i
$378$	−6923.18	+	973.408i	−0.942037	+	0.132452i
$379$		−	1680.48i		−	0.227758i	−0.993495	−	0.113879i	$-0.963672\pi$
$379$		−	1680.48i		−	0.227758i	0.993495	−	0.113879i	$-0.0363276\pi$
$380$		0			0
$381$	3155.76	+	689.925i	0.424342	+	0.0927715i
$382$	6444.09	−	6444.09i	0.863110	−	0.863110i
$383$	7493.42	−	7493.42i	0.999728	−	0.999728i	−0.000271480	−	1.00000i	$-0.500086\pi$
$383$	7493.42	−	7493.42i	0.999728	−	0.999728i	1.00000		0.000271480i	$8.64148e-5\pi$
$384$	−4340.16	−	948.864i	−0.576779	−	0.126098i
$385$		0			0
$386$			10465.5i			1.38000i
$387$	6315.78	−	2340.74i	0.829584	−	0.307459i
$388$	2669.98	+	2669.98i	0.349349	+	0.349349i
$389$	7966.97			1.03841			0.519205	−	0.854650i	$-0.326228\pi$
$389$	7966.97			1.03841			0.519205	+	0.854650i	$0.326228\pi$
$390$		0			0
$391$	−6603.94			−0.854158
$392$	−813.939	−	813.939i	−0.104873	−	0.104873i
$393$	6120.71	−	3924.57i	0.785620	−	0.503736i
$394$		−	4433.78i		−	0.566930i
$395$		0			0
$396$	2376.61	−	5175.59i	0.301589	−	0.656776i
$397$	8188.88	−	8188.88i	1.03523	−	1.03523i	0.0358786	−	0.999356i	$-0.488577\pi$
$397$	8188.88	−	8188.88i	1.03523	−	1.03523i	0.999356	−	0.0358786i	$-0.0114230\pi$
$398$	4101.85	−	4101.85i	0.516601	−	0.516601i
$399$	280.130	−	1281.33i	0.0351480	−	0.160769i
$400$		0			0
$401$		−	5167.66i		−	0.643542i	−0.946817	−	0.321771i	$-0.895722\pi$
$401$		−	5167.66i		−	0.643542i	0.946817	−	0.321771i	$-0.104278\pi$
$402$	6437.08	+	10039.2i	0.798638	+	1.24555i
$403$	263.398	+	263.398i	0.0325577	+	0.0325577i
$404$	−10271.2			−1.26488
$405$		0			0
$406$	−2516.74			−0.307645
$407$	5514.46	+	5514.46i	0.671601	+	0.671601i
$408$	−1747.24	−	2724.97i	−0.212012	−	0.330652i
$409$			8514.82i			1.02941i	0.857366	+	0.514707i	$0.172099\pi$
$409$			8514.82i			1.02941i	−0.857366	+	0.514707i	$0.827901\pi$
$410$		0			0
$411$	1714.32	−	7841.41i	0.205745	−	0.941090i
$412$	−4785.54	+	4785.54i	−0.572249	+	0.572249i
$413$	−3940.88	+	3940.88i	−0.469535	+	0.469535i
$414$	−3084.11	+	6716.32i	−0.366125	+	0.797316i
$415$		0			0
$416$			911.644i			0.107445i
$417$	−10928.6	+	7007.39i	−1.28340	+	0.822910i
$418$	1733.67	+	1733.67i	0.202863	+	0.202863i
$419$	11939.7			1.39211			0.696053	−	0.717990i	$-0.254938\pi$
$419$	11939.7			1.39211			0.696053	+	0.717990i	$0.254938\pi$
$420$		0			0
$421$	10873.3			1.25875			0.629373	−	0.777103i	$-0.283312\pi$
$421$	10873.3			1.25875			0.629373	+	0.777103i	$0.283312\pi$
$422$	29.1236	+	29.1236i	0.00335951	+	0.00335951i
$423$	−1370.73	+	508.016i	−0.157558	+	0.0583938i
$424$			3341.52i			0.382733i
$425$		0			0
$426$	−14027.5	−	3066.76i	−1.59539	−	0.348791i
$427$	−18.7156	+	18.7156i	−0.00212111	+	0.00212111i
$428$	−7230.91	+	7230.91i	−0.816634	+	0.816634i
$429$	691.364	+	151.149i	0.0778074	+	0.0170106i
$430$		0			0
$431$			7603.48i			0.849760i	0.905250	+	0.424880i	$0.139684\pi$
$431$			7603.48i			0.849760i	−0.905250	+	0.424880i	$0.860316\pi$
$432$	−10454.8	+	1469.95i	−1.16437	+	0.163711i
$433$	−4681.19	−	4681.19i	−0.519547	−	0.519547i	0.397887	−	0.917434i	$-0.369743\pi$
$433$	−4681.19	−	4681.19i	−0.519547	−	0.519547i	−0.917434	+	0.397887i	$0.869743\pi$
$434$	−4652.70			−0.514601
$435$		0			0
$436$	7813.84			0.858292
$437$	−980.409	−	980.409i	−0.107321	−	0.107321i
$438$	8107.85	−	5198.71i	0.884493	−	0.567133i
$439$		−	8608.08i		−	0.935857i	−0.883766	−	0.467929i	$-0.845000\pi$
$439$		−	8608.08i		−	0.935857i	0.883766	−	0.467929i	$-0.155000\pi$
$440$		0			0
$441$	−4118.80	−	1891.34i	−0.444746	−	0.204226i
$442$	−965.047	+	965.047i	−0.103852	+	0.103852i
$443$	−6466.81	+	6466.81i	−0.693561	+	0.693561i	−0.963014	−	0.269453i	$-0.913157\pi$
$443$	−6466.81	+	6466.81i	−0.693561	+	0.693561i	0.269453	+	0.963014i	$0.413157\pi$
$444$	−1566.53	+	7165.42i	−0.167442	+	0.765891i
$445$		0			0
$446$			4427.97i			0.470113i
$447$	−10013.9	−	15617.6i	−1.05960	−	1.65254i
$448$	−2418.14	−	2418.14i	−0.255014	−	0.255014i
$449$	356.370			0.0374569			0.0187284	−	0.999825i	$-0.494038\pi$
$449$	356.370			0.0374569			0.0187284	+	0.999825i	$0.494038\pi$
$450$		0			0
$451$	3031.19			0.316481
$452$	−2355.24	−	2355.24i	−0.245091	−	0.245091i
$453$	7536.66	+	11754.1i	0.781685	+	1.21911i
$454$			18325.0i			1.89435i
$455$		0			0
$456$	145.152	−	663.935i	0.0149065	−	0.0681834i
$457$	−1512.80	+	1512.80i	−0.154849	+	0.154849i	−0.780280	−	0.625431i	$-0.784923\pi$
$457$	−1512.80	+	1512.80i	−0.154849	+	0.154849i	0.625431	+	0.780280i	$0.284923\pi$
$458$	−4474.19	+	4474.19i	−0.456474	+	0.456474i
$459$	−10179.4	−	7669.81i	−1.03515	−	0.779948i
$460$		0			0
$461$		−	13307.9i		−	1.34449i	−0.740327	−	0.672246i	$-0.765330\pi$
$461$		−	13307.9i		−	1.34449i	0.740327	−	0.672246i	$-0.234670\pi$
$462$	−7441.16	+	4771.23i	−0.749338	+	0.480472i
$463$	1237.43	+	1237.43i	0.124208	+	0.124208i	0.766478	−	0.642270i	$-0.222007\pi$
$463$	1237.43	+	1237.43i	0.124208	+	0.124208i	−0.642270	+	0.766478i	$0.722007\pi$
$464$	−3800.56			−0.380251
$465$		0			0
$466$	11218.8			1.11524
$467$	−8201.87	−	8201.87i	−0.812713	−	0.812713i	0.172327	−	0.985040i	$-0.444872\pi$
$467$	−8201.87	−	8201.87i	−0.812713	−	0.812713i	−0.985040	+	0.172327i	$0.944872\pi$
$468$	231.305	+	624.107i	0.0228463	+	0.0616439i
$469$		−	8066.19i		−	0.794162i
$470$		0			0
$471$	12997.3	+	2841.52i	1.27151	+	0.277984i
$472$	−2042.01	+	2042.01i	−0.199133	+	0.199133i
$473$	6021.83	−	6021.83i	0.585378	−	0.585378i
$474$	11239.7	+	2457.28i	1.08915	+	0.238115i
$475$		0			0
$476$		−	7428.67i		−	0.715321i
$477$	4572.28	+	12336.9i	0.438889	+	1.18421i
$478$	695.273	+	695.273i	0.0665294	+	0.0665294i
$479$	11419.1			1.08926			0.544629	−	0.838677i	$-0.316671\pi$
$479$	11419.1			1.08926			0.544629	+	0.838677i	$0.316671\pi$
$480$		0			0
$481$	−911.420			−0.0863974
$482$	15979.5	+	15979.5i	1.51005	+	1.51005i
$483$	4208.05	−	2698.18i	0.396424	−	0.254185i
$484$			1023.43i			0.0961147i
$485$		0			0
$486$	−12554.2	+	6770.77i	−1.17175	+	0.631952i
$487$	6066.93	−	6066.93i	0.564515	−	0.564515i	−0.366071	−	0.930587i	$-0.619297\pi$
$487$	6066.93	−	6066.93i	0.564515	−	0.564515i	0.930587	+	0.366071i	$0.119297\pi$
$488$	−9.69769	+	9.69769i	−0.000899577	+	0.000899577i
$489$	−4204.74	+	19232.7i	−0.388845	+	1.77860i
$490$		0			0
$491$		−	8978.88i		−	0.825277i	−0.910895	−	0.412639i	$-0.864607\pi$
$491$		−	8978.88i		−	0.825277i	0.910895	−	0.412639i	$-0.135393\pi$
$492$	1538.80	+	2399.89i	0.141005	+	0.219909i
$493$	−3244.31	−	3244.31i	−0.296382	−	0.296382i
$494$	−286.538			−0.0260971
$495$		0			0
$496$	−7026.11			−0.636051
$497$	6867.38	+	6867.38i	0.619807	+	0.619807i
$498$	3255.94	+	5077.92i	0.292976	+	0.456922i
$499$		−	7674.34i		−	0.688478i	−0.938882	−	0.344239i	$-0.888137\pi$
$499$		−	7674.34i		−	0.688478i	0.938882	−	0.344239i	$-0.111863\pi$
$500$		0			0
$501$	219.010	−	1001.77i	0.0195303	−	0.0893326i
$502$	8106.54	−	8106.54i	0.720743	−	0.720743i
$503$	4044.23	−	4044.23i	0.358496	−	0.358496i	−0.504762	−	0.863258i	$-0.668420\pi$
$503$	4044.23	−	4044.23i	0.358496	−	0.358496i	0.863258	+	0.504762i	$0.168420\pi$
$504$	2226.69	+	1022.49i	0.196795	+	0.0903674i
$505$		0			0
$506$			9344.28i			0.820957i
$507$	9540.48	−	6117.31i	0.835715	−	0.535857i
$508$	2716.17	+	2716.17i	0.237226	+	0.237226i
$509$	−12532.5			−1.09134			−0.545672	−	0.837999i	$-0.683725\pi$
$509$	−12532.5			−1.09134			−0.545672	+	0.837999i	$0.683725\pi$
$510$		0			0
$511$	−6514.42			−0.563955
$512$	−9239.91	−	9239.91i	−0.797559	−	0.797559i
$513$	−372.574	−	2649.87i	−0.0320654	−	0.228059i
$514$		−	5039.33i		−	0.432443i
$515$		0			0
$516$	7824.69	+	1710.66i	0.667563	+	0.145945i
$517$	−1306.93	+	1306.93i	−0.111177	+	0.111177i
$518$	8049.75	−	8049.75i	0.682791	−	0.682791i
$519$	6329.02	+	1383.68i	0.535285	+	0.117026i
$520$		0			0
$521$			19201.8i			1.61468i	0.590089	+	0.807338i	$0.299093\pi$
$521$			19201.8i			1.61468i	−0.590089	+	0.807338i	$0.700907\pi$
$522$	−4814.65	+	1784.39i	−0.403700	+	0.149618i
$523$	−5472.69	−	5472.69i	−0.457560	−	0.457560i	0.440294	−	0.897854i	$-0.354874\pi$
$523$	−5472.69	−	5472.69i	−0.457560	−	0.457560i	−0.897854	+	0.440294i	$0.854874\pi$
$524$	8646.00			0.720806
$525$		0			0
$526$	356.926			0.0295869
$527$	−5997.77	−	5997.77i	−0.495762	−	0.495762i
$528$	−11237.0	+	7205.10i	−0.926187	+	0.593867i
$529$			6882.71i			0.565687i
$530$		0			0
$531$	−4744.97	+	10333.2i	−0.387786	+	0.844488i
$532$	1102.85	−	1102.85i	0.0898769	−	0.0898769i
$533$	−250.495	+	250.495i	−0.0203567	+	0.0203567i
$534$	5315.58	−	24313.8i	0.430763	−	1.97034i
$535$		0			0
$536$		−	4179.58i		−	0.336810i
$537$	7047.87	+	10991.8i	0.566365	+	0.883295i
$538$	−7077.60	−	7077.60i	−0.567169	−	0.567169i
$539$	−5730.40			−0.457933
$540$		0			0
$541$	12778.2			1.01548			0.507741	−	0.861510i	$-0.330481\pi$
$541$	12778.2			1.01548			0.507741	+	0.861510i	$0.330481\pi$
$542$	−387.868	−	387.868i	−0.0307387	−	0.0307387i
$543$	756.702	+	1180.14i	0.0598033	+	0.0932684i
$544$		−	20758.8i		−	1.63608i
$545$		0			0
$546$	220.640	−	1009.22i	0.0172940	−	0.0791038i
$547$	2414.12	−	2414.12i	0.188702	−	0.188702i	−0.606433	−	0.795135i	$-0.707400\pi$
$547$	2414.12	−	2414.12i	0.188702	−	0.188702i	0.795135	+	0.606433i	$0.207400\pi$
$548$	6749.12	−	6749.12i	0.526110	−	0.526110i
$549$	−22.5343	+	49.0735i	−0.00175181	+	0.00381494i
$550$		0			0
$551$		−	963.290i		−	0.0744783i
$552$	2180.44	−	1398.09i	0.168126	−	0.107802i
$553$	−5502.57	−	5502.57i	−0.423134	−	0.423134i
$554$	−5722.30			−0.438840
$555$		0			0
$556$	−15437.6			−1.17752
$557$	−5573.05	−	5573.05i	−0.423946	−	0.423946i	0.462614	−	0.886560i	$-0.346912\pi$
$557$	−5573.05	−	5573.05i	−0.423946	−	0.423946i	−0.886560	+	0.462614i	$0.846912\pi$
$558$	−8900.85	+	3298.81i	−0.675274	+	0.250268i
$559$			995.277i			0.0753054i
$560$		0			0
$561$	−15742.9	−	3441.78i	−1.18479	−	0.259023i
$562$	5379.23	−	5379.23i	0.403753	−	0.403753i
$563$	−3488.75	+	3488.75i	−0.261160	+	0.261160i	−0.825525	−	0.564365i	$-0.809121\pi$
$563$	−3488.75	+	3488.75i	−0.261160	+	0.261160i	0.564365	+	0.825525i	$0.309121\pi$
$564$	−1698.21	−	371.270i	−0.126786	−	0.0277186i
$565$		0			0
$566$		−	12782.3i		−	0.949260i
$567$	9620.03	+	728.199i	0.712528	+	0.0539356i
$568$	3558.40	+	3558.40i	0.262865	+	0.262865i
$569$	−4924.15			−0.362796			−0.181398	−	0.983410i	$-0.558062\pi$
$569$	−4924.15			−0.362796			−0.181398	+	0.983410i	$0.558062\pi$
$570$		0			0
$571$	−5642.12			−0.413512			−0.206756	−	0.978393i	$-0.566291\pi$
$571$	−5642.12			−0.413512			−0.206756	+	0.978393i	$0.566291\pi$
$572$	595.059	+	595.059i	0.0434977	+	0.0434977i
$573$	−10586.5	+	6788.02i	−0.771829	+	0.494893i
$574$		−	4424.78i		−	0.321754i
$575$		0			0
$576$	−6340.50	−	2911.53i	−0.458659	−	0.210614i
$577$	−8505.39	+	8505.39i	−0.613663	+	0.613663i	−0.943899	−	0.330235i	$-0.892872\pi$
$577$	−8505.39	+	8505.39i	−0.613663	+	0.613663i	0.330235	+	0.943899i	$0.392872\pi$
$578$	8893.50	−	8893.50i	0.640002	−	0.640002i
$579$	3084.47	−	14108.6i	0.221392	−	1.01266i
$580$		0			0
$581$		−	4079.96i		−	0.291334i
$582$	−6453.87	−	10065.4i	−0.459659	−	0.716879i
$583$	11762.7	+	11762.7i	0.835612	+	0.835612i
$584$	−3375.51			−0.239177
$585$		0			0
$586$	13489.4			0.950927
$587$	1464.72	+	1464.72i	0.102990	+	0.102990i	0.756724	−	0.653734i	$-0.226798\pi$
$587$	1464.72	+	1464.72i	0.102990	+	0.102990i	−0.653734	+	0.756724i	$0.726798\pi$
$588$	−2909.07	−	4536.95i	−0.204027	−	0.318198i
$589$		−	1780.84i		−	0.124581i
$590$		0			0
$591$	−1306.75	+	5977.17i	−0.0909521	+	0.416020i
$592$	12156.0	−	12156.0i	0.843935	−	0.843935i
$593$	−18086.4	+	18086.4i	−1.25248	+	1.25248i	−0.297871	+	0.954606i	$0.596277\pi$
$593$	−18086.4	+	18086.4i	−1.25248	+	1.25248i	−0.954606	+	0.297871i	$0.903723\pi$
$594$	−10852.4	+	14403.4i	−0.749631	+	0.994917i
$595$		0			0
$596$		−	22061.1i		−	1.51621i
$597$	−6738.63	+	4320.78i	−0.461966	+	0.296211i
$598$	−772.204	−	772.204i	−0.0528056	−	0.0528056i
$599$	−21899.3			−1.49379			−0.746897	−	0.664940i	$-0.768457\pi$
$599$	−21899.3			−1.49379			−0.746897	+	0.664940i	$0.768457\pi$
$600$		0			0
$601$	−12431.8			−0.843766			−0.421883	−	0.906650i	$-0.638631\pi$
$601$	−12431.8			−0.843766			−0.421883	+	0.906650i	$0.638631\pi$
$602$	−8790.38	−	8790.38i	−0.595132	−	0.595132i
$603$	−5719.00	−	15431.0i	−0.386229	−	1.04212i
$604$			16603.6i			1.11853i
$605$		0			0
$606$	31774.1	+	6946.59i	2.12993	+	0.465653i
$607$	−8237.79	+	8237.79i	−0.550843	+	0.550843i	−0.926684	−	0.375841i	$-0.877354\pi$
$607$	−8237.79	+	8237.79i	−0.550843	+	0.550843i	0.375841	+	0.926684i	$0.377354\pi$
$608$	3081.82	−	3081.82i	0.205566	−	0.205566i
$609$	3392.82	+	741.752i	0.225754	+	0.0493552i
$610$		0			0
$611$		−	216.007i		−	0.0143023i
$612$	−5267.00	−	14211.4i	−0.347885	−	0.938663i
$613$	8913.02	+	8913.02i	0.587265	+	0.587265i	0.936890	−	0.349625i	$-0.113691\pi$
$613$	8913.02	+	8913.02i	0.587265	+	0.587265i	−0.349625	+	0.936890i	$0.613691\pi$
$614$	16827.3			1.10602
$615$		0			0
$616$	3097.95			0.202630
$617$	−1378.18	−	1378.18i	−0.0899244	−	0.0899244i	0.660714	−	0.750638i	$-0.270254\pi$
$617$	−1378.18	−	1378.18i	−0.0899244	−	0.0899244i	−0.750638	+	0.660714i	$0.770254\pi$
$618$	18040.7	−	11567.6i	1.17428	−	0.752941i
$619$			18926.2i			1.22893i	0.788945	+	0.614464i	$0.210628\pi$
$619$			18926.2i			1.22893i	−0.788945	+	0.614464i	$0.789372\pi$
$620$		0			0
$621$	6137.16	−	8145.29i	0.396580	−	0.526344i
$622$	−19187.2	+	19187.2i	−1.23688	+	1.23688i
$623$	−11903.2	+	11903.2i	−0.765474	+	0.765474i
$624$	333.191	−	1524.04i	0.0213755	−	0.0977730i
$625$		0			0
$626$			10806.5i			0.689959i
$627$	−1826.20	−	2848.12i	−0.116318	−	0.181408i
$628$	11186.8	+	11186.8i	0.710831	+	0.710831i
$629$	20753.7			1.31559
$630$		0			0
$631$	−26118.8			−1.64782			−0.823909	−	0.566723i	$-0.808211\pi$
$631$	−26118.8			−1.64782			−0.823909	+	0.566723i	$0.808211\pi$
$632$	−2851.21	−	2851.21i	−0.179454	−	0.179454i
$633$	−30.6780	−	47.8450i	−0.00192629	−	0.00300421i
$634$		−	3670.26i		−	0.229912i
$635$		0			0
$636$	−3341.52	+	15284.3i	−0.208333	+	0.952929i
$637$	473.556	−	473.556i	0.0294552	−	0.0294552i
$638$	−4590.56	+	4590.56i	−0.284862	+	0.284862i
$639$	18006.7	+	8268.59i	1.11476	+	0.511894i
$640$		0			0
$641$			15846.5i			0.976442i	0.872720	+	0.488221i	$0.162354\pi$
$641$			15846.5i			0.976442i	−0.872720	+	0.488221i	$0.837646\pi$
$642$	27259.4	−	17478.6i	1.67577	−	1.07449i
$643$	−89.9404	−	89.9404i	−0.00551618	−	0.00551618i	0.704343	−	0.709859i	$-0.251242\pi$
$643$	−89.9404	−	89.9404i	−0.00551618	−	0.00551618i	−0.709859	+	0.704343i	$0.751242\pi$
$644$	5944.21			0.363719
$645$		0			0
$646$	6524.70			0.397385
$647$	20057.0	+	20057.0i	1.21874	+	1.21874i	0.968074	+	0.250665i	$0.0806492\pi$
$647$	20057.0	+	20057.0i	1.21874	+	1.21874i	0.250665	+	0.968074i	$0.419351\pi$
$648$	4984.71	+	377.323i	0.302188	+	0.0228745i
$649$			14376.4i			0.869527i
$650$		0			0
$651$	6272.31	+	1371.28i	0.377621	+	0.0825570i
$652$	−16553.7	+	16553.7i	−0.994313	+	0.994313i
$653$	20478.4	−	20478.4i	1.22723	−	1.22723i	0.262225	−	0.965007i	$-0.415544\pi$
$653$	20478.4	−	20478.4i	1.22723	−	1.22723i	0.965007	−	0.262225i	$-0.0844562\pi$
$654$	−24172.3	−	5284.64i	−1.44528	−	0.315972i
$655$		0			0
$656$		−	6681.92i		−	0.397691i
$657$	−12462.4	+	4618.78i	−0.740036	+	0.274271i
$658$	1907.80	+	1907.80i	0.113030	+	0.113030i
$659$	1169.87			0.0691529			0.0345765	−	0.999402i	$-0.488992\pi$
$659$	1169.87			0.0691529			0.0345765	+	0.999402i	$0.488992\pi$
$660$		0			0
$661$	19622.6			1.15466			0.577329	−	0.816511i	$-0.304095\pi$
$661$	19622.6			1.15466			0.577329	+	0.816511i	$0.304095\pi$
$662$	21904.0	+	21904.0i	1.28599	+	1.28599i
$663$	1585.41	−	1016.55i	0.0928688	−	0.0595471i
$664$		−	2114.07i		−	0.123557i
$665$		0			0
$666$	9692.20	−	21106.9i	0.563912	−	1.22804i
$667$	2596.01	−	2596.01i	0.150701	−	0.150701i
$668$	862.224	−	862.224i	0.0499408	−	0.0499408i
$669$	1305.04	−	5969.35i	0.0754199	−	0.344975i
$670$		0			0
$671$			68.2750i			0.00392806i
$672$	8481.46	+	13227.6i	0.486874	+	0.759323i
$673$	−15256.3	−	15256.3i	−0.873830	−	0.873830i	0.119057	−	0.992887i	$-0.462013\pi$
$673$	−15256.3	−	15256.3i	−0.873830	−	0.873830i	−0.992887	+	0.119057i	$0.962013\pi$
$674$	9464.72			0.540901
$675$		0			0
$676$	13476.7			0.766768
$677$	16.1429	+	16.1429i	0.000916428	+	0.000916428i	0.707565	−	0.706648i	$-0.249794\pi$
$677$	16.1429	+	16.1429i	0.000916428	+	0.000916428i	−0.706648	+	0.707565i	$0.749794\pi$
$678$	5693.08	+	8878.86i	0.322480	+	0.502936i
$679$			8087.23i			0.457083i
$680$		0			0
$681$	5400.87	−	24703.9i	0.303909	−	1.39010i
$682$	−8486.57	+	8486.57i	−0.476492	+	0.476492i
$683$	3894.05	−	3894.05i	0.218157	−	0.218157i	−0.589564	−	0.807722i	$-0.700700\pi$
$683$	3894.05	−	3894.05i	0.218157	−	0.218157i	0.807722	+	0.589564i	$0.200700\pi$
$684$	1327.87	−	2891.73i	0.0742287	−	0.161649i
$685$		0			0
$686$			25457.5i			1.41687i
$687$	7350.31	−	4712.99i	0.408198	−	0.261735i
$688$	−13274.5	−	13274.5i	−0.735587	−	0.735587i
$689$	−1944.12			−0.107497
$690$		0			0
$691$	−1041.11			−0.0573163			−0.0286581	−	0.999589i	$-0.509123\pi$
$691$	−1041.11			−0.0573163			−0.0286581	+	0.999589i	$0.509123\pi$
$692$	5447.40	+	5447.40i	0.299247	+	0.299247i
$693$	11437.6	−	4238.99i	0.626955	−	0.232361i
$694$		−	9133.91i		−	0.499594i
$695$		0			0
$696$	1758.02	+	384.346i	0.0957438	+	0.0209319i
$697$	5703.96	−	5703.96i	0.309975	−	0.309975i
$698$	−22908.9	+	22908.9i	−1.24229	+	1.24229i
$699$	−15124.1	−	3306.49i	−0.818378	−	0.178917i
$700$		0			0
$701$			29885.3i			1.61020i	0.593138	+	0.805101i	$0.297889\pi$
$701$			29885.3i			1.61020i	−0.593138	+	0.805101i	$0.702111\pi$
$702$	−293.452	−	2087.12i	−0.0157773	−	0.112213i
$703$	3081.06	+	3081.06i	0.165298	+	0.165298i
$704$	−8821.42			−0.472258
$705$		0			0
$706$	−28889.9			−1.54007
$707$	−15555.5	−	15555.5i	−0.827473	−	0.827473i
$708$	−11382.3	+	7298.26i	−0.604198	+	0.387409i
$709$			12115.0i			0.641734i	0.947124	+	0.320867i	$0.103974\pi$
$709$			12115.0i			0.641734i	−0.947124	+	0.320867i	$0.896026\pi$
$710$		0			0
$711$	−14428.1	−	6625.31i	−0.761033	−	0.349463i
$712$	−6167.74	+	6167.74i	−0.324643	+	0.324643i
$713$	4799.24	−	4799.24i	0.252080	−	0.252080i
$714$	−5024.15	+	22980.8i	−0.263339	+	1.20453i
$715$		0			0
$716$			15526.8i			0.810422i
$717$	−732.381	−	1142.21i	−0.0381468	−	0.0594933i
$718$	30404.1	+	30404.1i	1.58032	+	1.58032i
$719$	−6371.24			−0.330469			−0.165234	−	0.986254i	$-0.552838\pi$
$719$	−6371.24			−0.330469			−0.165234	+	0.986254i	$0.552838\pi$
$720$		0			0
$721$	−14495.2			−0.748722
$722$	−17294.1	−	17294.1i	−0.891443	−	0.891443i
$723$	−16832.3	−	26251.5i	−0.865839	−	1.35035i
$724$			1667.05i			0.0855737i
$725$		0			0
$726$	692.164	−	3166.00i	0.0353838	−	0.161848i
$727$	−15771.9	+	15771.9i	−0.804604	+	0.804604i	−0.983811	−	0.179207i	$-0.942647\pi$
$727$	−15771.9	+	15771.9i	−0.804604	+	0.804604i	0.179207	+	0.983811i	$0.442647\pi$
$728$	−256.012	+	256.012i	−0.0130336	+	0.0130336i
$729$	18919.9	−	5427.61i	0.961229	−	0.275751i
$730$		0			0
$731$		−	22663.2i		−	1.14669i
$732$	−54.0555	+	34.6601i	−0.00272944	+	0.00175010i
$733$	5626.05	+	5626.05i	0.283496	+	0.283496i	0.834502	−	0.551005i	$-0.185756\pi$
$733$	5626.05	+	5626.05i	0.283496	+	0.283496i	−0.551005	+	0.834502i	$0.685756\pi$
$734$	34905.3			1.75528
$735$		0			0
$736$	16610.6			0.831897
$737$	−14712.8	−	14712.8i	−0.735350	−	0.735350i
$738$	−3137.21	−	8464.82i	−0.156480	−	0.422215i
$739$		−	30340.1i		−	1.51026i	−0.655577	−	0.755129i	$-0.727574\pi$
$739$		−	30340.1i		−	1.51026i	0.655577	−	0.755129i	$-0.272426\pi$
$740$		0			0
$741$	386.282	+	84.4506i	0.0191504	+	0.00418674i
$742$	17170.7	−	17170.7i	0.849535	−	0.849535i
$743$	2368.77	−	2368.77i	0.116961	−	0.116961i	−0.646204	−	0.763165i	$-0.723645\pi$
$743$	2368.77	−	2368.77i	0.116961	−	0.116961i	0.763165	+	0.646204i	$0.223645\pi$
$744$	3250.06	+	710.541i	0.160152	+	0.0350130i
$745$		0			0
$746$		−	31778.0i		−	1.55962i
$747$	−2892.73	−	7805.16i	−0.141686	−	0.382297i
$748$	−13550.0	−	13550.0i	−0.662347	−	0.662347i
$749$	−21902.1			−1.06847
$750$		0			0
$751$	10606.2			0.515346			0.257673	−	0.966232i	$-0.417044\pi$
$751$	10606.2			0.515346			0.257673	+	0.966232i	$0.417044\pi$
$752$	2880.99	+	2880.99i	0.139706	+	0.139706i
$753$	−13317.6	+	8539.21i	−0.644518	+	0.413262i
$754$		−	758.720i		−	0.0366458i
$755$		0			0
$756$	9162.52	+	6903.60i	0.440790	+	0.332119i
$757$	18470.4	−	18470.4i	0.886812	−	0.886812i	−0.107404	−	0.994215i	$-0.534254\pi$
$757$	18470.4	−	18470.4i	0.886812	−	0.886812i	0.994215	+	0.107404i	$0.0342538\pi$
$758$	−4474.44	+	4474.44i	−0.214405	+	0.214405i
$759$	2754.01	−	12597.0i	0.131705	−	0.602428i
$760$		0			0
$761$		−	13568.0i		−	0.646307i	−0.946347	−	0.323153i	$-0.895257\pi$
$761$		−	13568.0i		−	0.646307i	0.946347	−	0.323153i	$-0.104743\pi$
$762$	−6565.54	−	10239.5i	−0.312132	−	0.486797i
$763$	11833.9	+	11833.9i	0.561488	+	0.561488i
$764$	−14954.3			−0.708152
$765$		0			0
$766$	−39904.0			−1.88223
$767$	−1188.05	−	1188.05i	−0.0559298	−	0.0559298i
$768$	14827.8	+	23125.2i	0.696682	+	1.08654i
$769$		−	11029.1i		−	0.517190i	−0.965986	−	0.258595i	$-0.916741\pi$
$769$		−	11029.1i		−	0.517190i	0.965986	−	0.258595i	$-0.0832594\pi$
$770$		0			0
$771$	−1485.23	+	6793.52i	−0.0693764	+	0.317332i
$772$	12143.3	−	12143.3i	0.566122	−	0.566122i
$773$	7090.94	−	7090.94i	0.329940	−	0.329940i	−0.522624	−	0.852563i	$-0.675047\pi$
$773$	7090.94	−	7090.94i	0.329940	−	0.329940i	0.852563	+	0.522624i	$0.175047\pi$
$774$	−23048.9	−	10584.0i	−1.07038	−	0.491515i
$775$		0			0
$776$			4190.48i			0.193852i
$777$	−13224.3	+	8479.38i	−0.610580	+	0.391501i
$778$	−21212.9	−	21212.9i	−0.977530	−	0.977530i
$779$	1693.60			0.0778940
$780$		0			0
$781$	25052.3			1.14781
$782$	17583.7	+	17583.7i	0.804080	+	0.804080i
$783$	7016.53	−	986.533i	0.320243	−	0.0450266i
$784$			12632.1i			0.575440i
$785$		0			0
$786$	−26746.6	−	5847.45i	−1.21376	−	0.265358i
$787$	17915.0	−	17915.0i	0.811437	−	0.811437i	−0.173412	−	0.984849i	$-0.555479\pi$
$787$	17915.0	−	17915.0i	0.811437	−	0.811437i	0.984849	+	0.173412i	$0.0554793\pi$
$788$	−5144.57	+	5144.57i	−0.232573	+	0.232573i
$789$	−481.172	−	105.196i	−0.0217112	−	0.00474660i
$790$		0			0
$791$		−	7133.90i		−	0.320673i
$792$	5926.52	−	2196.47i	0.265896	−	0.0985458i
$793$	−5.64218	−	5.64218i	−0.000252661	−	0.000252661i
$794$	−43607.5			−1.94908
$795$		0			0
$796$	−9518.87			−0.423854
$797$	−6043.79	−	6043.79i	−0.268610	−	0.268610i	0.559930	−	0.828540i	$-0.310828\pi$
$797$	−6043.79	−	6043.79i	−0.268610	−	0.268610i	−0.828540	+	0.559930i	$0.810828\pi$
$798$	−4157.56	+	2665.81i	−0.184431	+	0.118256i
$799$			4918.65i			0.217784i
$800$		0			0
$801$	−14331.9	+	31210.8i	−0.632199	+	1.37675i
$802$	−13759.4	+	13759.4i	−0.605813	+	0.605813i
$803$	−11882.3	+	11882.3i	−0.522191	+	0.522191i
$804$	4179.58	−	19117.6i	0.183336	−	0.838591i
$805$		0			0
$806$		−	1402.65i		−	0.0612979i
$807$	7455.35	+	11627.3i	0.325205	+	0.507186i
$808$	−8060.22	−	8060.22i	−0.350937	−	0.350937i
$809$	35063.4			1.52381			0.761905	−	0.647689i	$-0.224264\pi$
$809$	35063.4			1.52381			0.761905	+	0.647689i	$0.224264\pi$
$810$		0			0
$811$	24621.3			1.06605			0.533027	−	0.846098i	$-0.321054\pi$
$811$	24621.3			1.06605			0.533027	+	0.846098i	$0.321054\pi$
$812$	2920.21	+	2920.21i	0.126206	+	0.126206i
$813$	408.570	+	637.200i	0.0176250	+	0.0274878i
$814$		−	29365.6i		−	1.26445i
$815$		0			0
$816$	−7587.02	+	34703.5i	−0.325489	+	1.48881i
$817$	3364.54	−	3364.54i	0.144076	−	0.144076i
$818$	22671.6	−	22671.6i	0.969063	−	0.969063i
$819$	−594.890	+	1295.50i	−0.0253811	+	0.0552729i
$820$		0			0
$821$		−	14268.3i		−	0.606538i	−0.952905	−	0.303269i	$-0.901922\pi$
$821$		−	14268.3i		−	0.606538i	0.952905	−	0.303269i	$-0.0980781\pi$
$822$	−25443.1	+	16314.0i	−1.07960	+	0.692234i
$823$	−13764.1	−	13764.1i	−0.582972	−	0.582972i	0.352747	−	0.935719i	$-0.385248\pi$
$823$	−13764.1	−	13764.1i	−0.582972	−	0.582972i	−0.935719	+	0.352747i	$0.885248\pi$
$824$	−7510.81			−0.317538
$825$		0			0
$826$	20986.0			0.884015
$827$	27442.5	+	27442.5i	1.15389	+	1.15389i	0.985765	+	0.168128i	$0.0537721\pi$
$827$	27442.5	+	27442.5i	1.15389	+	1.15389i	0.168128	+	0.985765i	$0.446228\pi$
$828$	11371.6	−	4214.50i	0.477282	−	0.176889i
$829$		−	12176.9i		−	0.510159i	−0.966920	−	0.255080i	$-0.917898\pi$
$829$		−	12176.9i		−	0.510159i	0.966920	−	0.255080i	$-0.0821016\pi$
$830$		0			0
$831$	7714.23	+	1686.52i	0.322026	+	0.0704027i
$832$	728.995	−	728.995i	0.0303766	−	0.0303766i
$833$	−10783.2	+	10783.2i	−0.448519	+	0.448519i
$834$	47756.5	+	10440.7i	1.98282	+	0.433492i
$835$		0			0
$836$		−	4023.21i		−	0.166442i
$837$	12971.5	−	1823.80i	0.535675	−	0.0753165i
$838$	−31790.7	−	31790.7i	−1.31049	−	1.31049i
$839$	13942.3			0.573710			0.286855	−	0.957974i	$-0.407390\pi$
$839$	13942.3			0.573710			0.286855	+	0.957974i	$0.407390\pi$
$840$		0			0
$841$	−21838.3			−0.895417
$842$	−28951.3	−	28951.3i	−1.18495	−	1.18495i
$843$	−8837.14	+	5666.33i	−0.361053	+	0.231505i
$844$		−	67.5849i		−	0.00275636i
$845$		0			0
$846$	5002.35	+	2297.06i	0.203291	+	0.0933505i
$847$	−1549.96	+	1549.96i	−0.0628776	+	0.0628776i
$848$	25929.6	−	25929.6i	1.05003	−	1.05003i
$849$	−3767.30	+	17231.8i	−0.152289	+	0.696579i
$850$		0			0
$851$			16606.6i			0.668937i
$852$	12717.9	+	19834.7i	0.511396	+	0.797567i
$853$	32654.2	+	32654.2i	1.31074	+	1.31074i	0.920871	+	0.389866i	$0.127479\pi$
$853$	32654.2	+	32654.2i	1.31074	+	1.31074i	0.389866	+	0.920871i	$0.372521\pi$
$854$	99.6646			0.00399350
$855$		0			0
$856$	−11348.8			−0.453147
$857$	10358.9	+	10358.9i	0.412898	+	0.412898i	0.882747	−	0.469849i	$-0.155692\pi$
$857$	10358.9	+	10358.9i	0.412898	+	0.412898i	−0.469849	+	0.882747i	$0.655692\pi$
$858$	−1438.38	−	2243.28i	−0.0572325	−	0.0892590i
$859$			14100.5i			0.560072i	0.959990	+	0.280036i	$0.0903464\pi$
$859$			14100.5i			0.560072i	−0.959990	+	0.280036i	$0.909654\pi$
$860$		0			0
$861$	−1304.10	+	5965.05i	−0.0516187	+	0.236107i
$862$	20245.0	−	20245.0i	0.799941	−	0.799941i
$863$	16830.6	−	16830.6i	0.663872	−	0.663872i	−0.292419	−	0.956290i	$-0.594460\pi$
$863$	16830.6	−	16830.6i	0.663872	−	0.663872i	0.956290	+	0.292419i	$0.0944601\pi$
$864$	25603.9	+	19291.6i	1.00818	+	0.759621i
$865$		0			0
$866$			24928.3i			0.978174i
$867$	−14610.5	+	9368.17i	−0.572316	+	0.366966i
$868$	5398.60	+	5398.60i	0.211106	+	0.211106i
$869$	−20073.5			−0.783597
$870$		0			0
$871$	2431.71			0.0945984
$872$	6131.84	+	6131.84i	0.238131	+	0.238131i
$873$	5733.92	+	15471.3i	0.222295	+	0.599797i
$874$			5220.88i			0.202058i
$875$		0			0
$876$	−15439.8	−	3375.51i	−0.595504	−	0.130192i
$877$	−9832.57	+	9832.57i	−0.378589	+	0.378589i	−0.870593	−	0.492004i	$-0.836264\pi$
$877$	−9832.57	+	9832.57i	−0.378589	+	0.378589i	0.492004	+	0.870593i	$0.336264\pi$
$878$	−22919.9	+	22919.9i	−0.880990	+	0.880990i
$879$	−18185.1	−	3975.70i	−0.697802	−	0.152556i
$880$		0			0
$881$		−	41729.4i		−	1.59580i	−0.602789	−	0.797900i	$-0.705944\pi$
$881$		−	41729.4i		−	1.59580i	0.602789	−	0.797900i	$-0.294056\pi$
$882$	5930.84	+	16002.6i	0.226419	+	0.610925i
$883$	11757.3	+	11757.3i	0.448090	+	0.448090i	0.894719	−	0.446629i	$-0.147376\pi$
$883$	11757.3	+	11757.3i	0.448090	+	0.448090i	−0.446629	+	0.894719i	$0.647376\pi$
$884$	2239.51			0.0852070
$885$		0			0
$886$	34437.1			1.30580
$887$	−24303.5	−	24303.5i	−0.919989	−	0.919989i	0.0770388	−	0.997028i	$-0.475453\pi$
$887$	−24303.5	−	24303.5i	−0.919989	−	0.919989i	−0.997028	+	0.0770388i	$0.975453\pi$
$888$	−6852.32	+	4393.67i	−0.258951	+	0.166038i
$889$			8227.16i			0.310383i
$890$		0			0
$891$	18875.3	−	16218.8i	0.709702	−	0.609820i
$892$	5137.84	−	5137.84i	0.192856	−	0.192856i
$893$	−730.214	+	730.214i	−0.0273636	+	0.0273636i
$894$	−14920.4	+	68246.6i	−0.558178	+	2.55314i
$895$		0			0
$896$		−	11314.9i		−	0.421881i
$897$	813.418	+	1268.60i	0.0302779	+	0.0472210i
$898$	−948.871	−	948.871i	−0.0352609	−	0.0352609i
$899$	4715.44			0.174937
$900$		0			0
$901$	44269.1			1.63687
$902$	−8070.84	−	8070.84i	−0.297927	−	0.297927i
$903$	9259.55	+	14441.1i	0.341239	+	0.532191i
$904$		−	3696.50i		−	0.136000i
$905$		0			0
$906$	11229.3	−	51363.6i	0.411776	−	1.88349i
$907$	−1017.75	+	1017.75i	−0.0372589	+	0.0372589i	−0.725491	−	0.688232i	$-0.758387\pi$
$907$	−1017.75	+	1017.75i	−0.0372589	+	0.0372589i	0.688232	+	0.725491i	$0.258387\pi$
$908$	21262.7	−	21262.7i	0.777124	−	0.777124i
$909$	−40787.3	−	18729.4i	−1.48826	−	0.683404i
$910$		0			0
$911$			33165.2i			1.20616i	0.797681	+	0.603080i	$0.206060\pi$
$911$			33165.2i			1.20616i	−0.797681	+	0.603080i	$0.793940\pi$
$912$	−6278.38	+	4025.67i	−0.227958	+	0.146166i
$913$	−7441.88	−	7441.88i	−0.269759	−	0.269759i
$914$	8056.00			0.291541
$915$		0			0
$916$	10382.9			0.374521
$917$	13094.2	+	13094.2i	0.471546	+	0.471546i
$918$	6682.13	+	47525.4i	0.240243	+	1.70869i
$919$			2162.18i			0.0776103i	0.999247	+	0.0388051i	$0.0123552\pi$
$919$			2162.18i			0.0776103i	−0.999247	+	0.0388051i	$0.987645\pi$
$920$		0			0
$921$	−22684.9	−	4959.46i	−0.811609	−	0.177437i
$922$	−35433.7	+	35433.7i	−1.26567	+	1.26567i
$923$	−2070.30	+	2070.30i	−0.0738297	+	0.0738297i
$924$	14170.2	+	3097.95i	0.504508	+	0.110298i
$925$		0			0
$926$		−	6589.59i		−	0.233852i
$927$	−27729.9	+	10277.2i	−0.982493	+	0.364129i
$928$	8160.29	+	8160.29i	0.288658	+	0.288658i
$929$	−17695.1			−0.624929			−0.312464	−	0.949930i	$-0.601154\pi$
$929$	−17695.1			−0.624929			−0.312464	+	0.949930i	$0.601154\pi$
$930$		0			0
$931$	−3201.72			−0.112709
$932$	−13017.4	−	13017.4i	−0.457508	−	0.457508i
$933$	31521.3	−	20211.3i	1.10607	−	0.709204i
$934$			43676.6i			1.53013i
$935$		0			0
$936$	−308.248	+	671.277i	−0.0107643	+	0.0234417i
$937$	30208.0	−	30208.0i	1.05320	−	1.05320i	0.0547017	−	0.998503i	$-0.482579\pi$
$937$	30208.0	−	30208.0i	1.05320	−	1.05320i	0.998503	−	0.0547017i	$-0.0174208\pi$
$938$	−21477.1	+	21477.1i	−0.747602	+	0.747602i
$939$	3184.96	−	14568.2i	0.110689	−	0.506300i
$940$		0			0
$941$			1499.59i			0.0519503i	0.999663	+	0.0259752i	$0.00826908\pi$
$941$			1499.59i			0.0519503i	−0.999663	+	0.0259752i	$0.991731\pi$
$942$	−27040.8	−	42172.4i	−0.935282	−	1.45865i
$943$	4564.14	+	4564.14i	0.157613	+	0.157613i
$944$	31691.2			1.09265
$945$		0			0
$946$	−32067.5			−1.10212
$947$	−13763.8	−	13763.8i	−0.472294	−	0.472294i	0.430362	−	0.902656i	$-0.358386\pi$
$947$	−13763.8	−	13763.8i	−0.472294	−	0.472294i	−0.902656	+	0.430362i	$0.858386\pi$
$948$	−10190.4	−	15892.8i	−0.349123	−	0.544488i
$949$		−	1963.89i		−	0.0671767i
$950$		0			0
$951$	−1081.72	+	4947.87i	−0.0368847	+	0.168713i
$952$	5829.59	−	5829.59i	0.198464	−	0.198464i
$953$	−14570.7	+	14570.7i	−0.495268	+	0.495268i	−0.909961	−	0.414693i	$-0.863889\pi$
$953$	−14570.7	+	14570.7i	−0.495268	+	0.495268i	0.414693	+	0.909961i	$0.363889\pi$
$954$	20674.1	−	45022.4i	0.701625	−	1.52794i
$955$		0			0
$956$		−	1613.47i		−	0.0545851i
$957$	7541.49	−	4835.57i	0.254735	−	0.163335i
$958$	−30404.7	−	30404.7i	−1.02540	−	1.02540i
$959$	20442.8			0.688355
$960$		0			0
$961$	−21073.6			−0.707380
$962$	2426.75	+	2426.75i	0.0813322	+	0.0813322i
$963$	−41899.7	+	15528.8i	−1.40208	+	0.519635i
$964$		−	37082.4i		−	1.23895i
$965$		0			0
$966$	−18388.6	−	4020.18i	−0.612466	−	0.133900i
$967$	−28409.1	+	28409.1i	−0.944752	+	0.944752i	−0.998552	−	0.0538000i	$-0.982867\pi$
$967$	−28409.1	+	28409.1i	−0.944752	+	0.944752i	0.0538000	+	0.998552i	$0.482867\pi$
$968$	−803.127	+	803.127i	−0.0266668	+	0.0266668i
$969$	−8795.95	−	1923.01i	−0.291606	−	0.0637522i
$970$		0			0
$971$			18059.4i			0.596864i	0.954431	+	0.298432i	$0.0964637\pi$
$971$			18059.4i			0.596864i	−0.954431	+	0.298432i	$0.903536\pi$
$972$	22423.1	+	6710.61i	0.739938	+	0.221443i
$973$	−23379.9	−	23379.9i	−0.770323	−	0.770323i
$974$	−32307.7			−1.06284
$975$		0			0
$976$	150.505			0.00493600
$977$	31166.3	+	31166.3i	1.02057	+	1.02057i	0.999784	+	0.0207872i	$0.00661725\pi$
$977$	31166.3	+	31166.3i	1.02057	+	1.02057i	0.0207872	+	0.999784i	$0.493383\pi$
$978$	62404.7	−	40013.6i	2.04037	−	1.30828i
$979$			43422.9i			1.41757i
$980$		0			0
$981$	31029.1	+	14248.5i	1.00987	+	0.463729i
$982$	−23907.2	+	23907.2i	−0.776893	+	0.776893i
$983$	−13839.4	+	13839.4i	−0.449042	+	0.449042i	−0.895036	−	0.445994i	$-0.852850\pi$
$983$	−13839.4	+	13839.4i	−0.449042	+	0.449042i	0.445994	+	0.895036i	$0.352850\pi$
$984$	−675.734	+	3090.85i	−0.0218919	+	0.100135i
$985$		0			0
$986$			17276.6i			0.558012i
$987$	−2009.62	−	3134.18i	−0.0648095	−	0.101076i
$988$	332.474	+	332.474i	0.0107059	+	0.0107059i
$989$	18134.5			0.583056
$990$		0			0
$991$	17820.9			0.571242			0.285621	−	0.958343i	$-0.407800\pi$
$991$	17820.9			0.571242			0.285621	+	0.958343i	$0.407800\pi$
$992$	15085.9	+	15085.9i	0.482842	+	0.482842i
$993$	−23073.1	−	35984.5i	−0.737363	−	1.14998i
$994$		−	36570.2i		−	1.16694i
$995$		0			0
$996$	2114.07	−	9669.89i	0.0672559	−	0.307633i
$997$	36985.0	−	36985.0i	1.17485	−	1.17485i	0.193814	−	0.981038i	$-0.437914\pi$
$997$	36985.0	−	36985.0i	1.17485	−	1.17485i	0.981038	−	0.193814i	$-0.0620857\pi$
$998$	−20433.7	+	20433.7i	−0.648114	+	0.648114i
$999$	−19286.8	+	25597.6i	−0.610819	+	0.810684i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.4.e.c.68.1		8
3.2	odd	2	inner	75.4.e.c.68.4		8
5.2	odd	4	inner	75.4.e.c.32.4		8
5.3	odd	4		15.4.e.a.2.1	✓	8
5.4	even	2		15.4.e.a.8.4	yes	8
15.2	even	4	inner	75.4.e.c.32.1		8
15.8	even	4		15.4.e.a.2.4	yes	8
15.14	odd	2		15.4.e.a.8.1	yes	8
20.3	even	4		240.4.v.c.17.4		8
20.19	odd	2		240.4.v.c.113.3		8
60.23	odd	4		240.4.v.c.17.3		8
60.59	even	2		240.4.v.c.113.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.e.a.2.1	✓	8	5.3	odd	4
15.4.e.a.2.4	yes	8	15.8	even	4
15.4.e.a.8.1	yes	8	15.14	odd	2
15.4.e.a.8.4	yes	8	5.4	even	2
75.4.e.c.32.1		8	15.2	even	4	inner
75.4.e.c.32.4		8	5.2	odd	4	inner
75.4.e.c.68.1		8	1.1	even	1	trivial
75.4.e.c.68.4		8	3.2	odd	2	inner
240.4.v.c.17.3		8	60.23	odd	4
240.4.v.c.17.4		8	20.3	even	4
240.4.v.c.113.3		8	20.19	odd	2
240.4.v.c.113.4		8	60.59	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	75.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.66260 - 2.66260i) q^{2} +(2.80471 + 4.37420i) q^{3} +6.17891i q^{4} +(4.17891 - 19.1146i) q^{6} +(-9.35782 + 9.35782i) q^{7} +(-4.84884 + 4.84884i) q^{8} +(-11.2672 + 24.5367i) q^{9} +34.1375i q^{11} +(-27.0278 + 17.3301i) q^{12} +(-2.82109 - 2.82109i) q^{13} +49.8323 q^{14} +75.2524 q^{16} +(64.2384 + 64.2384i) q^{17} +(95.3316 - 35.3316i) q^{18} +19.0735i q^{19} +(-67.1789 - 14.6869i) q^{21} +(90.8945 - 90.8945i) q^{22} +(-51.4018 + 51.4018i) q^{23} +(-34.8094 - 7.61018i) q^{24} +15.0229i q^{26} +(-138.930 + 19.5337i) q^{27} +(-57.8211 - 57.8211i) q^{28} -50.5042 q^{29} -93.3673 q^{31} +(-161.576 - 161.576i) q^{32} +(-149.324 + 95.7458i) q^{33} -342.083i q^{34} +(-151.610 - 69.6188i) q^{36} +(161.537 - 161.537i) q^{37} +(50.7850 - 50.7850i) q^{38} +(4.42765 - 20.2524i) q^{39} -88.7935i q^{41} +(139.765 + 217.976i) q^{42} +(-176.399 - 176.399i) q^{43} -210.932 q^{44} +273.725 q^{46} +(38.2843 + 38.2843i) q^{47} +(211.061 + 329.169i) q^{48} +167.863i q^{49} +(-100.821 + 461.162i) q^{51} +(17.4313 - 17.4313i) q^{52} +(344.569 - 344.569i) q^{53} +(421.925 + 317.904i) q^{54} -90.7492i q^{56} +(-83.4310 + 53.4956i) q^{57} +(134.473 + 134.473i) q^{58} +421.133 q^{59} +2.00000 q^{61} +(248.600 + 248.600i) q^{62} +(-124.174 - 335.046i) q^{63} +258.409i q^{64} +(652.524 + 142.657i) q^{66} +(-430.987 + 430.987i) q^{67} +(-396.923 + 396.923i) q^{68} +(-369.009 - 80.6742i) q^{69} -733.866i q^{71} +(-64.3420 - 173.607i) q^{72} +(348.073 + 348.073i) q^{73} -860.216 q^{74} -117.853 q^{76} +(-319.452 - 319.452i) q^{77} +(-65.7131 + 42.1349i) q^{78} +588.019i q^{79} +(-475.102 - 552.919i) q^{81} +(-236.422 + 236.422i) q^{82} +(-217.997 + 217.997i) q^{83} +(90.7492 - 415.092i) q^{84} +939.362i q^{86} +(-141.650 - 220.915i) q^{87} +(-165.527 - 165.527i) q^{88} +1272.00 q^{89} +52.7985 q^{91} +(-317.607 - 317.607i) q^{92} +(-261.868 - 408.407i) q^{93} -203.872i q^{94} +(253.591 - 1159.94i) q^{96} +(432.111 - 432.111i) q^{97} +(446.951 - 446.951i) q^{98} +(-837.622 - 384.633i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{3} - 12 q^{6} + 16 q^{7} - 132 q^{12} - 68 q^{13} + 284 q^{16} + 240 q^{18} - 492 q^{21} + 500 q^{22} - 702 q^{27} - 508 q^{28} + 616 q^{31} + 240 q^{33} - 804 q^{36} + 1156 q^{37} - 540 q^{42}+ \cdots - 1904 q^{97}+O(q^{100}) \) 8 * q + 6 * q^3 - 12 * q^6 + 16 * q^7 - 132 * q^12 - 68 * q^13 + 284 * q^16 + 240 * q^18 - 492 * q^21 + 500 * q^22 - 702 * q^27 - 508 * q^28 + 616 * q^31 + 240 * q^33 - 804 * q^36 + 1156 * q^37 - 540 * q^42 - 548 * q^43 + 736 * q^46 + 1116 * q^48 - 852 * q^51 - 224 * q^52 - 684 * q^57 - 60 * q^58 + 16 * q^61 - 1428 * q^63 + 2040 * q^66 - 404 * q^67 + 1800 * q^72 + 2512 * q^73 - 1488 * q^76 + 360 * q^78 + 288 * q^81 - 2800 * q^82 + 1680 * q^87 - 2460 * q^88 - 1304 * q^91 - 3408 * q^93 + 4164 * q^96 - 1904 * q^97

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