LMFDB - Embedded newform 415.4.b.a.84.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [415,4,Mod(84,415)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("415.84"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(415, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 415 = 5 \cdot 83 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	415.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$24.4857926524$
Analytic rank:	$0$
Dimension:	$124$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			84.1
Character	$\chi$	$=$	415.84
Dual form			415.4.b.a.84.124

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-5.59070i q^{2} +9.32458i q^{3} -23.2559 q^{4} +(-3.95795 + 10.4563i) q^{5} +52.1309 q^{6} +14.9769i q^{7} +85.2909i q^{8} -59.9478 q^{9} +(58.4581 + 22.1277i) q^{10} -14.4939 q^{11} -216.851i q^{12} +73.8958i q^{13} +83.7312 q^{14} +(-97.5008 - 36.9062i) q^{15} +290.789 q^{16} +61.2669i q^{17} +335.150i q^{18} +47.8919 q^{19} +(92.0455 - 243.171i) q^{20} -139.653 q^{21} +81.0309i q^{22} -157.044i q^{23} -795.302 q^{24} +(-93.6693 - 82.7711i) q^{25} +413.129 q^{26} -307.224i q^{27} -348.301i q^{28} -100.730 q^{29} +(-206.331 + 545.097i) q^{30} -16.7128 q^{31} -943.383i q^{32} -135.149i q^{33} +342.525 q^{34} +(-156.603 - 59.2777i) q^{35} +1394.14 q^{36} +72.9466i q^{37} -267.749i q^{38} -689.048 q^{39} +(-891.829 - 337.577i) q^{40} +5.60615 q^{41} +780.758i q^{42} +189.430i q^{43} +337.068 q^{44} +(237.270 - 626.834i) q^{45} -877.983 q^{46} +427.438i q^{47} +2711.48i q^{48} +118.693 q^{49} +(-462.748 + 523.677i) q^{50} -571.288 q^{51} -1718.51i q^{52} +424.295i q^{53} -1717.60 q^{54} +(57.3660 - 151.553i) q^{55} -1277.39 q^{56} +446.572i q^{57} +563.150i q^{58} +223.761 q^{59} +(2267.47 + 858.285i) q^{60} +575.383 q^{61} +93.4361i q^{62} -897.831i q^{63} -2947.86 q^{64} +(-772.679 - 292.476i) q^{65} -755.579 q^{66} -1042.02i q^{67} -1424.82i q^{68} +1464.37 q^{69} +(-331.404 + 875.520i) q^{70} -437.616 q^{71} -5113.00i q^{72} -91.6976i q^{73} +407.822 q^{74} +(771.806 - 873.427i) q^{75} -1113.77 q^{76} -217.073i q^{77} +3852.26i q^{78} -24.3050 q^{79} +(-1150.93 + 3040.58i) q^{80} +1246.15 q^{81} -31.3423i q^{82} +83.0000i q^{83} +3247.76 q^{84} +(-640.627 - 242.491i) q^{85} +1059.05 q^{86} -939.264i q^{87} -1236.20i q^{88} +726.778 q^{89} +(-3504.44 - 1326.51i) q^{90} -1106.73 q^{91} +3652.19i q^{92} -155.840i q^{93} +2389.68 q^{94} +(-189.553 + 500.773i) q^{95} +8796.65 q^{96} +887.385i q^{97} -663.576i q^{98} +868.877 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 124 q - 504 q^{4} + 6 q^{5} - 1088 q^{9} + 62 q^{10} + 60 q^{11} + 124 q^{14} - 120 q^{15} + 2080 q^{16} + 124 q^{19} - 206 q^{20} + 24 q^{21} + 368 q^{24} - 430 q^{25} - 176 q^{26} + 208 q^{29} - 52 q^{30}+ \cdots + 2668 q^{99}+O(q^{100}) $ 124 * q - 504 * q^4 + 6 * q^5 - 1088 * q^9 + 62 * q^10 + 60 * q^11 + 124 * q^14 - 120 * q^15 + 2080 * q^16 + 124 * q^19 - 206 * q^20 + 24 * q^21 + 368 * q^24 - 430 * q^25 - 176 * q^26 + 208 * q^29 - 52 * q^30 + 372 * q^31 + 496 * q^34 + 320 * q^35 + 4592 * q^36 - 592 * q^39 - 430 * q^40 - 384 * q^41 - 636 * q^44 - 118 * q^45 - 1216 * q^46 - 6504 * q^49 + 440 * q^50 - 696 * q^51 + 8 * q^54 - 684 * q^55 - 1328 * q^56 - 1068 * q^59 + 4698 * q^60 + 952 * q^61 - 9180 * q^64 - 16 * q^65 - 284 * q^66 + 3496 * q^69 - 1296 * q^70 - 2976 * q^71 - 1732 * q^74 + 1796 * q^75 + 804 * q^76 + 1080 * q^79 - 1758 * q^80 + 8700 * q^81 + 6752 * q^84 + 1036 * q^85 - 2692 * q^86 + 832 * q^89 - 5586 * q^90 + 1464 * q^91 + 2620 * q^94 + 1240 * q^95 + 5892 * q^96 + 2668 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/415\mathbb{Z}\right)^\times$.

$n$	$167$	$251$
$\chi(n)$	$-1$	$1$

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 $

$2$		−	5.59070i		−	1.97661i	−0.152493	−	0.988305i	$-0.548730\pi$
$2$		−	5.59070i		−	1.97661i	0.152493	−	0.988305i	$-0.451270\pi$
$3$			9.32458i			1.79452i	0.441506	+	0.897258i	$0.354444\pi$
$3$			9.32458i			1.79452i	−0.441506	+	0.897258i	$0.645556\pi$
$4$	−23.2559			−2.90698
$5$	−3.95795	+	10.4563i	−0.354009	+	0.935242i
$5$	−3.95795	+	10.4563i	−0.354009	+	0.935242i
$6$	52.1309			3.54706
$7$			14.9769i			0.808676i	0.914610	+	0.404338i	$0.132498\pi$
$7$			14.9769i			0.808676i	−0.914610	+	0.404338i	$0.867502\pi$
$8$			85.2909i			3.76936i
$9$	−59.9478			−2.22029
$10$	58.4581	+	22.1277i	1.84861	+	0.699738i
$11$	−14.4939			−0.397279			−0.198640	−	0.980073i	$-0.563652\pi$
$11$	−14.4939			−0.397279			−0.198640	+	0.980073i	$0.563652\pi$
$12$		−	216.851i		−	5.21663i
$13$			73.8958i			1.57654i	0.615330	+	0.788270i	$0.289023\pi$
$13$			73.8958i			1.57654i	−0.615330	+	0.788270i	$0.710977\pi$
$14$	83.7312			1.59844
$15$	−97.5008	−	36.9062i	−1.67831	−	0.635276i
$16$	290.789			4.54357
$17$			61.2669i			0.874083i	0.899441	+	0.437042i	$0.143974\pi$
$17$			61.2669i			0.874083i	−0.899441	+	0.437042i	$0.856026\pi$
$18$			335.150i			4.38864i
$19$	47.8919			0.578271			0.289136	−	0.957288i	$-0.406632\pi$
$19$	47.8919			0.578271			0.289136	+	0.957288i	$0.406632\pi$
$20$	92.0455	−	243.171i	1.02910	−	2.71873i
$21$	−139.653			−1.45118
$22$			81.0309i			0.785266i
$23$		−	157.044i		−	1.42373i	−0.702315	−	0.711867i	$-0.747850\pi$
$23$		−	157.044i		−	1.42373i	0.702315	−	0.711867i	$-0.252150\pi$
$24$	−795.302			−6.76418
$25$	−93.6693	−	82.7711i	−0.749355	−	0.662169i
$26$	413.129			3.11620
$27$		−	307.224i		−	2.18983i
$28$		−	348.301i		−	2.35081i
$29$	−100.730			−0.645002			−0.322501	−	0.946569i	$-0.604524\pi$
$29$	−100.730			−0.645002			−0.322501	+	0.946569i	$0.604524\pi$
$30$	−206.331	+	545.097i	−1.25569	+	3.31736i
$31$	−16.7128			−0.0968292			−0.0484146	−	0.998827i	$-0.515417\pi$
$31$	−16.7128			−0.0968292			−0.0484146	+	0.998827i	$0.515417\pi$
$32$		−	943.383i		−	5.21150i
$33$		−	135.149i		−	0.712924i
$34$	342.525			1.72772
$35$	−156.603	−	59.2777i	−0.756308	−	0.286279i
$36$	1394.14			6.45434
$37$			72.9466i			0.324118i	0.986781	+	0.162059i	$0.0518134\pi$
$37$			72.9466i			0.324118i	−0.986781	+	0.162059i	$0.948187\pi$
$38$		−	267.749i		−	1.14302i
$39$	−689.048			−2.82913
$40$	−891.829	−	337.577i	−3.52526	−	1.33439i
$41$	5.60615			0.0213545			0.0106772	−	0.999943i	$-0.496601\pi$
$41$	5.60615			0.0213545			0.0106772	+	0.999943i	$0.496601\pi$
$42$			780.758i			2.86842i
$43$			189.430i			0.671809i	0.941896	+	0.335905i	$0.109042\pi$
$43$			189.430i			0.671809i	−0.941896	+	0.335905i	$0.890958\pi$
$44$	337.068			1.15488
$45$	237.270	−	626.834i	0.786003	−	2.07651i
$46$	−877.983			−2.81416
$47$			427.438i			1.32656i	0.748372	+	0.663280i	$0.230836\pi$
$47$			427.438i			1.32656i	−0.748372	+	0.663280i	$0.769164\pi$
$48$			2711.48i			8.15351i
$49$	118.693			0.346043
$50$	−462.748	+	523.677i	−1.30885	+	1.48118i
$51$	−571.288			−1.56856
$52$		−	1718.51i		−	4.58298i
$53$			424.295i			1.09965i	0.835280	+	0.549825i	$0.185306\pi$
$53$			424.295i			1.09965i	−0.835280	+	0.549825i	$0.814694\pi$
$54$	−1717.60			−4.32844
$55$	57.3660	−	151.553i	0.140641	−	0.371552i
$56$	−1277.39			−3.04819
$57$			446.572i			1.03772i
$58$			563.150i			1.27492i
$59$	223.761			0.493749			0.246875	−	0.969047i	$-0.420596\pi$
$59$	223.761			0.493749			0.246875	+	0.969047i	$0.420596\pi$
$60$	2267.47	+	858.285i	4.87881	+	1.84674i
$61$	575.383			1.20771			0.603854	−	0.797095i	$-0.293631\pi$
$61$	575.383			1.20771			0.603854	+	0.797095i	$0.293631\pi$
$62$			93.4361i			0.191394i
$63$		−	897.831i		−	1.79549i
$64$	−2947.86			−5.75753
$65$	−772.679	−	292.476i	−1.47445	−	0.558110i
$66$	−755.579			−1.40917
$67$		−	1042.02i		−	1.90004i	−0.312189	−	0.950020i	$-0.601062\pi$
$67$		−	1042.02i		−	1.90004i	0.312189	−	0.950020i	$-0.398938\pi$
$68$		−	1424.82i		−	2.54095i
$69$	1464.37			2.55491
$70$	−331.404	+	875.520i	−0.565861	+	1.49492i
$71$	−437.616			−0.731485			−0.365742	−	0.930716i	$-0.619185\pi$
$71$	−437.616			−0.731485			−0.365742	+	0.930716i	$0.619185\pi$
$72$		−	5113.00i		−	8.36907i
$73$		−	91.6976i		−	0.147019i	−0.997295	−	0.0735095i	$-0.976580\pi$
$73$		−	91.6976i		−	0.147019i	0.997295	−	0.0735095i	$-0.0234199\pi$
$74$	407.822			0.640654
$75$	771.806	−	873.427i	1.18827	−	1.34473i
$76$	−1113.77			−1.68103
$77$		−	217.073i		−	0.321270i
$78$			3852.26i			5.59208i
$79$	−24.3050			−0.0346143			−0.0173072	−	0.999850i	$-0.505509\pi$
$79$	−24.3050			−0.0346143			−0.0173072	+	0.999850i	$0.505509\pi$
$80$	−1150.93	+	3040.58i	−1.60847	+	4.24934i
$81$	1246.15			1.70939
$82$		−	31.3423i		−	0.0422095i
$83$			83.0000i			0.109764i
$83$			83.0000i			0.109764i
$84$	3247.76			4.21856
$85$	−640.627	−	242.491i	−0.817479	−	0.309434i
$86$	1059.05			1.32790
$87$		−	939.264i		−	1.15747i
$88$		−	1236.20i		−	1.49749i
$89$	726.778			0.865600			0.432800	−	0.901490i	$-0.357526\pi$
$89$	726.778			0.865600			0.432800	+	0.901490i	$0.357526\pi$
$90$	−3504.44	−	1326.51i	−4.10444	−	1.55362i
$91$	−1106.73			−1.27491
$92$			3652.19i			4.13877i
$93$		−	155.840i		−	0.173762i
$94$	2389.68			2.62209
$95$	−189.553	+	500.773i	−0.204713	+	0.540823i
$96$	8796.65			9.35213
$97$			887.385i			0.928869i	0.885607	+	0.464434i	$0.153742\pi$
$97$			887.385i			0.928869i	−0.885607	+	0.464434i	$0.846258\pi$
$98$		−	663.576i		−	0.683992i
$99$	868.877			0.882075
$100$	2178.36	+	1924.91i	2.17836	+	1.92491i
$101$	−1506.11			−1.48380			−0.741898	−	0.670513i	$-0.766074\pi$
$101$	−1506.11			−1.48380			−0.741898	+	0.670513i	$0.766074\pi$
$102$			3193.90i			3.10042i
$103$		−	1471.86i		−	1.40803i	−0.710187	−	0.704013i	$-0.751390\pi$
$103$		−	1471.86i		−	1.40803i	0.710187	−	0.704013i	$-0.248610\pi$
$104$	−6302.64			−5.94255
$105$	552.740	−	1460.26i	0.513732	−	1.35721i
$106$	2372.11			2.17358
$107$			745.458i			0.673515i	0.941591	+	0.336757i	$0.109330\pi$
$107$			745.458i			0.673515i	−0.941591	+	0.336757i	$0.890670\pi$
$108$			7144.77i			6.36580i
$109$	1675.06			1.47194			0.735970	−	0.677014i	$-0.236726\pi$
$109$	1675.06			1.47194			0.735970	+	0.677014i	$0.236726\pi$
$110$	−847.285	−	320.716i	−0.734413	−	0.277991i
$111$	−680.197			−0.581634
$112$			4355.11i			3.67428i
$113$		−	1422.34i		−	1.18409i	−0.805903	−	0.592047i	$-0.798320\pi$
$113$		−	1422.34i		−	1.18409i	0.805903	−	0.592047i	$-0.201680\pi$
$114$	2496.65			2.05116
$115$	1642.10	+	621.570i	1.33154	+	0.504015i
$116$	2342.56			1.87501
$117$		−	4429.89i		−	3.50037i
$118$		−	1250.98i		−	0.975949i
$119$	−917.588			−0.706850
$120$	3147.76	−	8315.93i	2.39458	−	6.32615i
$121$	−1120.93			−0.842169
$122$		−	3216.79i		−	2.38717i
$123$			52.2750i			0.0383210i
$124$	388.671			0.281481
$125$	1236.22	−	651.833i	0.884567	−	0.466414i
$126$	−5019.50			−3.54899
$127$			43.6856i			0.0305234i	0.999884	+	0.0152617i	$0.00485814\pi$
$127$			43.6856i			0.0305234i	−0.999884	+	0.0152617i	$0.995142\pi$
$128$			8933.51i			6.16889i
$129$	−1766.35			−1.20557
$130$	−1635.14	+	4319.81i	−1.10317	+	2.91440i
$131$	1635.48			1.09079			0.545393	−	0.838181i	$-0.316381\pi$
$131$	1635.48			1.09079			0.545393	+	0.838181i	$0.316381\pi$
$132$			3143.02i			2.07246i
$133$			717.271i			0.467634i
$134$	−5825.60			−3.75564
$135$	3212.44	+	1215.98i	2.04802	+	0.775220i
$136$	−5225.51			−3.29474
$137$			1252.41i			0.781027i	0.920597	+	0.390513i	$0.127703\pi$
$137$			1252.41i			0.781027i	−0.920597	+	0.390513i	$0.872297\pi$
$138$		−	8186.83i		−	5.05006i
$139$	1005.60			0.613624			0.306812	−	0.951770i	$-0.400738\pi$
$139$	1005.60			0.613624			0.306812	+	0.951770i	$0.400738\pi$
$140$	3641.94	+	1378.55i	2.19857	+	0.832208i
$141$	−3985.68			−2.38053
$142$			2446.58i			1.44586i
$143$		−	1071.04i		−	0.626326i
$144$	−17432.1			−10.0880
$145$	398.683	−	1053.26i	0.228337	−	0.603233i
$146$	−512.653			−0.290599
$147$			1106.76i			0.620980i
$148$		−	1696.44i		−	0.942205i
$149$	−2038.93			−1.12104			−0.560522	−	0.828140i	$-0.689399\pi$
$149$	−2038.93			−1.12104			−0.560522	+	0.828140i	$0.689399\pi$
$150$	−4883.07	−	4314.93i	−2.65800	−	2.34875i
$151$	1709.73			0.921428			0.460714	−	0.887549i	$-0.347593\pi$
$151$	1709.73			0.921428			0.460714	+	0.887549i	$0.347593\pi$
$152$			4084.74i			2.17971i
$153$		−	3672.82i		−	1.94072i
$154$	−1213.59			−0.635025
$155$	66.1483	−	174.754i	0.0342785	−	0.0905588i
$156$	16024.4			8.22422
$157$		−	1866.01i		−	0.948558i	−0.880375	−	0.474279i	$-0.842709\pi$
$157$		−	1866.01i		−	0.948558i	0.880375	−	0.474279i	$-0.157291\pi$
$158$			135.882i			0.0684190i
$159$	−3956.38			−1.97334
$160$	9864.31	+	3733.86i	4.87402	+	1.84492i
$161$	2352.03			1.15134
$162$		−	6966.84i		−	3.37880i
$163$			3457.90i			1.66162i	0.556559	+	0.830808i	$0.312121\pi$
$163$			3457.90i			1.66162i	−0.556559	+	0.830808i	$0.687879\pi$
$164$	−130.376			−0.0620772
$165$	1413.17	+	534.914i	0.666756	+	0.252382i
$166$	464.028			0.216961
$167$			1539.61i			0.713407i	0.934218	+	0.356703i	$0.116099\pi$
$167$			1539.61i			0.713407i	−0.934218	+	0.356703i	$0.883901\pi$
$168$		−	11911.1i		−	5.47003i
$169$	−3263.59			−1.48548
$170$	−1355.69	+	3581.55i	−0.611629	+	1.61584i
$171$	−2871.01			−1.28393
$172$		−	4405.36i		−	1.95294i
$173$		−	626.204i		−	0.275199i	−0.990488	−	0.137600i	$-0.956061\pi$
$173$		−	626.204i		−	0.275199i	0.990488	−	0.137600i	$-0.0439387\pi$
$174$	−5251.14			−2.28786
$175$	1239.65	−	1402.87i	0.535480	−	0.605985i
$176$	−4214.66			−1.80507
$177$			2086.48i			0.886041i
$178$		−	4063.20i		−	1.71095i
$179$	−524.737			−0.219110			−0.109555	−	0.993981i	$-0.534943\pi$
$179$	−524.737			−0.219110			−0.109555	+	0.993981i	$0.534943\pi$
$180$	−5517.92	+	14577.6i	−2.28490	+	6.03637i
$181$	−118.464			−0.0486483			−0.0243242	−	0.999704i	$-0.507743\pi$
$181$	−118.464			−0.0486483			−0.0243242	+	0.999704i	$0.507743\pi$
$182$			6187.39i			2.52000i
$183$			5365.20i			2.16725i
$184$	13394.4			5.36657
$185$	−762.753	−	288.719i	−0.303128	−	0.114741i
$186$	−871.253			−0.343459
$187$		−	887.996i		−	0.347255i
$188$		−	9940.45i		−	3.85629i
$189$	4601.27			1.77086
$190$	2799.67	+	1059.74i	1.06900	+	0.404638i
$191$	−1026.64			−0.388928			−0.194464	−	0.980910i	$-0.562297\pi$
$191$	−1026.64			−0.388928			−0.194464	+	0.980910i	$0.562297\pi$
$192$		−	27487.5i		−	10.3320i
$193$			2005.04i			0.747801i	0.927469	+	0.373901i	$0.121980\pi$
$193$			2005.04i			0.747801i	−0.927469	+	0.373901i	$0.878020\pi$
$194$	4961.10			1.83601
$195$	2727.21	−	7204.90i	1.00154	−	2.64592i
$196$	−2760.31			−1.00594
$197$		−	445.231i		−	0.161022i	−0.996754	−	0.0805112i	$-0.974345\pi$
$197$		−	445.231i		−	0.161022i	0.996754	−	0.0805112i	$-0.0256553\pi$
$198$		−	4857.62i		−	1.74352i
$199$	−390.091			−0.138959			−0.0694795	−	0.997583i	$-0.522134\pi$
$199$	−390.091			−0.138959			−0.0694795	+	0.997583i	$0.522134\pi$
$200$	7059.62	−	7989.14i	2.49595	−	2.82459i
$201$	9716.38			3.40965
$202$			8420.19i			2.93288i
$203$		−	1508.62i		−	0.521598i
$204$	13285.8			4.55977
$205$	−22.1889	+	58.6197i	−0.00755969	+	0.0199716i
$206$	−8228.72			−2.78312
$207$			9414.42i			3.16110i
$208$			21488.1i			7.16312i
$209$	−694.140			−0.229735
$210$	−8163.86	−	3090.20i	−2.68267	−	1.01545i
$211$	1950.45			0.636373			0.318187	−	0.948028i	$-0.396926\pi$
$211$	1950.45			0.636373			0.318187	+	0.948028i	$0.396926\pi$
$212$		−	9867.36i		−	3.19667i
$213$		−	4080.58i		−	1.31266i
$214$	4167.63			1.33128
$215$	−1980.74	−	749.753i	−0.628304	−	0.237827i
$216$	26203.5			8.25426
$217$		−	250.306i		−	0.0783035i
$218$		−	9364.74i		−	2.90945i
$219$	855.041			0.263828
$220$	−1334.10	+	3524.49i	−0.408840	+	1.08010i
$221$	−4527.37			−1.37803
$222$			3802.77i			1.14966i
$223$			3313.38i			0.994980i	0.867470	+	0.497490i	$0.165745\pi$
$223$			3313.38i			0.994980i	−0.867470	+	0.497490i	$0.834255\pi$
$224$	14128.9			4.21442
$225$	5615.27	+	4961.95i	1.66378	+	1.47021i
$226$	−7951.88			−2.34049
$227$		−	320.243i		−	0.0936356i	−0.998903	−	0.0468178i	$-0.985092\pi$
$227$		−	320.243i		−	0.0936356i	0.998903	−	0.0468178i	$-0.0149080\pi$
$228$		−	10385.4i		−	3.01663i
$229$	−655.779			−0.189236			−0.0946182	−	0.995514i	$-0.530163\pi$
$229$	−655.779			−0.189236			−0.0946182	+	0.995514i	$0.530163\pi$
$230$	3475.01	−	9180.48i	0.996241	−	2.63192i
$231$	2024.12			0.576524
$232$		−	8591.34i		−	2.43125i
$233$		−	1364.10i		−	0.383543i	−0.981440	−	0.191771i	$-0.938577\pi$
$233$		−	1364.10i		−	0.383543i	0.981440	−	0.191771i	$-0.0614232\pi$
$234$	−24766.2			−6.91887
$235$	−4469.43	−	1691.78i	−1.24065	−	0.469615i
$236$	−5203.76			−1.43532
$237$		−	226.634i		−	0.0621160i
$238$			5129.95i			1.39717i
$239$	−6912.12			−1.87074			−0.935371	−	0.353667i	$-0.884935\pi$
$239$	−6912.12			−1.87074			−0.935371	+	0.353667i	$0.884935\pi$
$240$	−28352.1	−	10731.9i	−7.62551	−	2.88642i
$241$	−2021.80			−0.540397			−0.270199	−	0.962805i	$-0.587089\pi$
$241$	−2021.80			−0.540397			−0.270199	+	0.962805i	$0.587089\pi$
$242$			6266.76i			1.66464i
$243$			3324.75i			0.877707i
$244$	−13381.0			−3.51079
$245$	−469.780	+	1241.09i	−0.122503	+	0.323634i
$246$	292.254			0.0757456
$247$			3539.01i			0.911668i
$248$		−	1425.45i		−	0.364984i
$249$	−773.940			−0.196974
$250$	−3644.20	−	6911.33i	−0.921918	−	1.74844i
$251$	1192.74			0.299939			0.149970	−	0.988691i	$-0.452082\pi$
$251$	1192.74			0.299939			0.149970	+	0.988691i	$0.452082\pi$
$252$			20879.9i			5.21947i
$253$			2276.17i			0.565620i
$254$	244.233			0.0603328
$255$	2261.13	−	5973.58i	0.555284	−	1.46698i
$256$	26361.7			6.43595
$257$		−	109.586i		−	0.0265983i	−0.999912	−	0.0132992i	$-0.995767\pi$
$257$		−	109.586i		−	0.0265983i	0.999912	−	0.0132992i	$-0.00423338\pi$
$258$			9875.15i			2.38295i
$259$	−1092.51			−0.262106
$260$	17969.3	+	6801.78i	4.28619	+	1.62242i
$261$	6038.54			1.43209
$262$		−	9143.49i		−	2.15606i
$263$		−	5009.05i		−	1.17442i	−0.809436	−	0.587208i	$-0.800227\pi$
$263$		−	5009.05i		−	1.17442i	0.809436	−	0.587208i	$-0.199773\pi$
$264$	11527.0			2.68727
$265$	−4436.57	−	1679.34i	−1.02844	−	0.389287i
$266$	4010.05			0.924330
$267$			6776.90i			1.55333i
$268$			24233.0i			5.52339i
$269$	−6954.12			−1.57621			−0.788105	−	0.615541i	$-0.788937\pi$
$269$	−6954.12			−1.57621			−0.788105	+	0.615541i	$0.788937\pi$
$270$	6798.16	−	17959.8i	1.53231	−	4.04813i
$271$	−2676.62			−0.599975			−0.299987	−	0.953943i	$-0.596982\pi$
$271$	−2676.62			−0.599975			−0.299987	+	0.953943i	$0.596982\pi$
$272$			17815.7i			3.97146i
$273$		−	10319.8i		−	2.28785i
$274$	7001.85			1.54379
$275$	1357.63	+	1199.68i	0.297703	+	0.263066i
$276$	−34055.1			−7.42709
$277$		−	3135.52i		−	0.680127i	−0.940402	−	0.340064i	$-0.889551\pi$
$277$		−	3135.52i		−	0.680127i	0.940402	−	0.340064i	$-0.110449\pi$
$278$		−	5622.00i		−	1.21290i
$279$	1001.90			0.214989
$280$	5055.85	−	13356.8i	1.07909	−	2.85080i
$281$	4159.43			0.883028			0.441514	−	0.897254i	$-0.354441\pi$
$281$	4159.43			0.883028			0.441514	+	0.897254i	$0.354441\pi$
$282$			22282.7i			4.70538i
$283$			2191.06i			0.460229i	0.973164	+	0.230114i	$0.0739100\pi$
$283$			2191.06i			0.460229i	−0.973164	+	0.230114i	$0.926090\pi$
$284$	10177.1			2.12641
$285$	−4669.50	−	1767.51i	−0.970517	−	0.367362i
$286$	−5987.85			−1.23800
$287$			83.9627i			0.0172689i
$288$			56553.7i			11.5710i
$289$	1159.36			0.235979
$290$	−5888.48	−	2228.92i	−1.19236	−	0.451333i
$291$	−8274.49			−1.66687
$292$			2132.51i			0.427382i
$293$		−	6157.10i		−	1.22765i	−0.789442	−	0.613825i	$-0.789630\pi$
$293$		−	6157.10i		−	1.22765i	0.789442	−	0.613825i	$-0.210370\pi$
$294$	6187.56			1.22744
$295$	−885.634	+	2339.72i	−0.174792	+	0.461775i
$296$	−6221.68			−1.22172
$297$			4452.88i			0.869973i
$298$			11399.0i			2.21587i
$299$	11604.9			2.24457
$300$	−17949.0	+	20312.3i	−3.45429	+	3.90911i
$301$	−2837.07			−0.543276
$302$		−	9558.56i		−	1.82130i
$303$		−	14043.8i		−	2.66270i
$304$	13926.4			2.62742
$305$	−2277.33	+	6016.39i	−0.427540	+	1.12950i
$306$	−20533.6			−3.83604
$307$			5530.05i			1.02807i	0.857770	+	0.514033i	$0.171849\pi$
$307$			5530.05i			1.02807i	−0.857770	+	0.514033i	$0.828151\pi$
$308$			5048.23i			0.933927i
$309$	13724.5			2.52673
$310$	−976.998	−	369.815i	−0.178999	−	0.0677551i
$311$	−99.9673			−0.0182271			−0.00911355	−	0.999958i	$-0.502901\pi$
$311$	−99.9673			−0.0182271			−0.00911355	+	0.999958i	$0.502901\pi$
$312$		−	58769.5i		−	10.6640i
$313$			29.8755i			0.00539509i	0.999996	+	0.00269754i	$0.000858656\pi$
$313$			29.8755i			0.00539509i	−0.999996	+	0.00269754i	$0.999141\pi$
$314$	−10432.3			−1.87493
$315$	9388.01	+	3553.57i	1.67922	+	0.635622i
$316$	565.235			0.100623
$317$			10947.0i			1.93957i	0.243967	+	0.969784i	$0.421551\pi$
$317$			10947.0i			1.93957i	−0.243967	+	0.969784i	$0.578449\pi$
$318$			22118.9i			3.90052i
$319$	1459.97			0.256246
$320$	11667.5	−	30823.7i	2.03822	−	5.38469i
$321$	−6951.08			−1.20863
$322$		−	13149.5i		−	2.27575i
$323$			2934.19i			0.505457i
$324$	−28980.3			−4.96918
$325$	6116.44	−	6921.77i	1.04394	−	1.18139i
$326$	19332.0			3.28437
$327$			15619.2i			2.64142i
$328$			478.154i			0.0804928i
$329$	−6401.70			−1.07276
$330$	2990.54	−	7900.58i	0.498860	−	1.31792i
$331$	6033.58			1.00192			0.500960	−	0.865471i	$-0.332980\pi$
$331$	6033.58			1.00192			0.500960	+	0.865471i	$0.332980\pi$
$332$		−	1930.24i		−	0.319083i
$333$		−	4372.99i		−	0.719635i
$334$	8607.51			1.41013
$335$	10895.7	+	4124.25i	1.77700	+	0.672632i
$336$	−40609.5			−6.59355
$337$		−	4423.63i		−	0.715046i	−0.933904	−	0.357523i	$-0.883621\pi$
$337$		−	4423.63i		−	0.715046i	0.933904	−	0.357523i	$-0.116379\pi$
$338$			18245.8i			2.93621i
$339$	13262.7			2.12488
$340$	14898.3	+	5639.34i	2.37640	+	0.899519i
$341$	242.233			0.0384682
$342$			16051.0i			2.53783i
$343$			6914.72i			1.08851i
$344$	−16156.7			−2.53229
$345$	−5795.88	+	15311.9i	−0.904463	+	2.38946i
$346$	−3500.92			−0.543961
$347$		−	5121.98i		−	0.792398i	−0.918165	−	0.396199i	$-0.870329\pi$
$347$		−	5121.98i		−	0.792398i	0.918165	−	0.396199i	$-0.129671\pi$
$348$			21843.4i			3.36474i
$349$	−9409.22			−1.44316			−0.721582	−	0.692329i	$-0.756584\pi$
$349$	−9409.22			−1.44316			−0.721582	+	0.692329i	$0.756584\pi$
$350$	−7843.05	−	6930.52i	−1.19780	−	1.05843i
$351$	22702.6			3.45235
$352$			13673.3i			2.07042i
$353$			12038.5i			1.81514i	0.419900	+	0.907570i	$0.362065\pi$
$353$			12038.5i			1.81514i	−0.419900	+	0.907570i	$0.637935\pi$
$354$	11664.9			1.75136
$355$	1732.06	−	4575.85i	0.258952	−	0.684115i
$356$	−16901.9			−2.51628
$357$		−	8556.12i		−	1.26845i
$358$			2933.65i			0.433095i
$359$	6561.58			0.964643			0.482321	−	0.875994i	$-0.339794\pi$
$359$	6561.58			0.964643			0.482321	+	0.875994i	$0.339794\pi$
$360$	53463.2	+	20237.0i	7.82711	+	2.96273i
$361$	−4565.37			−0.665602
$362$			662.295i			0.0961587i
$363$		−	10452.2i		−	1.51129i
$364$	25738.0			3.70614
$365$	958.819	+	362.934i	0.137498	+	0.0520461i
$366$	29995.2			4.28381
$367$			11976.8i			1.70350i	0.523945	+	0.851752i	$0.324460\pi$
$367$			11976.8i			1.70350i	−0.523945	+	0.851752i	$0.675540\pi$
$368$		−	45666.5i		−	6.46883i
$369$	−336.077			−0.0474131
$370$	−1614.14	+	4264.32i	−0.226797	+	0.599166i
$371$	−6354.62			−0.889261
$372$			3624.19i			0.505122i
$373$		−	6115.44i		−	0.848915i	−0.905448	−	0.424458i	$-0.860465\pi$
$373$		−	6115.44i		−	0.848915i	0.905448	−	0.424458i	$-0.139535\pi$
$374$	−4964.51			−0.686388
$375$	6078.07	+	11527.2i	0.836987	+	1.58737i
$376$	−36456.6			−5.00028
$377$		−	7443.52i		−	1.01687i
$378$		−	25724.3i		−	3.50030i
$379$	−2016.81			−0.273342			−0.136671	−	0.990617i	$-0.543640\pi$
$379$	−2016.81			−0.273342			−0.136671	+	0.990617i	$0.543640\pi$
$380$	4408.23	−	11645.9i	0.595099	−	1.57216i
$381$	−407.350			−0.0547747
$382$			5739.65i			0.768760i
$383$		−	2376.30i		−	0.317033i	−0.987356	−	0.158516i	$-0.949329\pi$
$383$		−	2376.30i		−	0.317033i	0.987356	−	0.158516i	$-0.0506710\pi$
$384$	−83301.2			−11.0702
$385$	2269.79	+	859.164i	0.300465	+	0.113733i
$386$	11209.5			1.47811
$387$		−	11355.9i		−	1.49161i
$388$		−	20636.9i		−	2.70021i
$389$	3720.23			0.484893			0.242446	−	0.970165i	$-0.422050\pi$
$389$	3720.23			0.484893			0.242446	+	0.970165i	$0.422050\pi$
$390$	−40280.4	−	15247.0i	−5.22994	−	1.97965i
$391$	9621.58			1.24446
$392$			10123.4i			1.30436i
$393$			15250.2i			1.95743i
$394$	−2489.15			−0.318278
$395$	96.1980	−	254.141i	0.0122538	−	0.0323728i
$396$	−20206.5			−2.56418
$397$		−	2374.06i		−	0.300128i	−0.988676	−	0.150064i	$-0.952052\pi$
$397$		−	2374.06i		−	0.300128i	0.988676	−	0.150064i	$-0.0479480\pi$
$398$			2180.88i			0.274667i
$399$	−6688.25			−0.839177
$400$	−27238.0	−	24068.9i	−3.40475	−	3.00861i
$401$	−15666.2			−1.95095			−0.975477	−	0.220100i	$-0.929362\pi$
$401$	−15666.2			−1.95095			−0.975477	+	0.220100i	$0.929362\pi$
$402$		−	54321.3i		−	6.73955i
$403$		−	1235.01i		−	0.152655i
$404$	35025.8			4.31337
$405$	−4932.19	+	13030.1i	−0.605142	+	1.59870i
$406$	−8434.23			−1.03100
$407$		−	1057.28i		−	0.128765i
$408$		−	48725.7i		−	5.91246i
$409$	−6198.90			−0.749428			−0.374714	−	0.927140i	$-0.622259\pi$
$409$	−6198.90			−0.749428			−0.374714	+	0.927140i	$0.622259\pi$
$410$	327.725	+	124.051i	0.0394761	+	0.0149426i
$411$	−11678.2			−1.40157
$412$			34229.4i			4.09311i
$413$			3351.24i			0.399283i
$414$	52633.2			6.24826
$415$	−867.875	−	328.509i	−0.102656	−	0.0388576i
$416$	69712.1			8.21614
$417$			9376.79i			1.10116i
$418$			3880.72i			0.454097i
$419$	4579.20			0.533910			0.266955	−	0.963709i	$-0.413982\pi$
$419$	4579.20			0.533910			0.266955	+	0.963709i	$0.413982\pi$
$420$	−12854.4	+	33959.6i	−1.49341	+	3.94538i
$421$	8991.46			1.04090			0.520448	−	0.853894i	$-0.325765\pi$
$421$	8991.46			1.04090			0.520448	+	0.853894i	$0.325765\pi$
$422$		−	10904.4i		−	1.25786i
$423$		−	25624.0i		−	2.94535i
$424$	−36188.5			−4.14498
$425$	5071.13	−	5738.83i	0.578791	−	0.654998i
$426$	−22813.3			−2.59462
$427$			8617.44i			0.976645i
$428$		−	17336.3i		−	1.95790i
$429$	9986.98			1.12395
$430$	−4191.64	+	11073.7i	−0.470091	+	1.24191i
$431$	−10252.5			−1.14582			−0.572909	−	0.819619i	$-0.694185\pi$
$431$	−10252.5			−1.14582			−0.572909	+	0.819619i	$0.694185\pi$
$432$		−	89337.4i		−	9.94964i
$433$			16744.8i			1.85844i	0.369527	+	0.929220i	$0.379520\pi$
$433$			16744.8i			1.85844i	−0.369527	+	0.929220i	$0.620480\pi$
$434$	−1399.38			−0.154775
$435$	9821.25	+	3717.56i	1.08251	+	0.409754i
$436$	−38954.9			−4.27891
$437$		−	7521.12i		−	0.823304i
$438$		−	4780.28i		−	0.521485i
$439$	2067.29			0.224752			0.112376	−	0.993666i	$-0.464154\pi$
$439$	2067.29			0.224752			0.112376	+	0.993666i	$0.464154\pi$
$440$	12926.1	+	4892.80i	1.40051	+	0.530125i
$441$	−7115.38			−0.768316
$442$			25311.1i			2.72382i
$443$		−	11792.3i		−	1.26472i	−0.774674	−	0.632360i	$-0.782086\pi$
$443$		−	11792.3i		−	1.26472i	0.774674	−	0.632360i	$-0.217914\pi$
$444$	15818.6			1.69080
$445$	−2876.55	+	7599.43i	−0.306430	+	0.809545i
$446$	18524.1			1.96669
$447$		−	19012.2i		−	2.01173i
$448$		−	44149.7i		−	4.65598i
$449$	6250.42			0.656961			0.328481	−	0.944511i	$-0.393463\pi$
$449$	6250.42			0.656961			0.328481	+	0.944511i	$0.393463\pi$
$450$	27740.7	−	31393.3i	2.90602	−	3.28865i
$451$	−81.2550			−0.00848370
$452$			33077.8i			3.44214i
$453$			15942.5i			1.65352i
$454$	−1790.38			−0.185081
$455$	4380.38	−	11572.3i	0.451330	−	1.19235i
$456$	−38088.5			−3.91153
$457$		−	11685.9i		−	1.19616i	−0.801438	−	0.598078i	$-0.795931\pi$
$457$		−	11685.9i		−	1.19616i	0.801438	−	0.598078i	$-0.204069\pi$
$458$			3666.26i			0.374046i
$459$	18822.7			1.91409
$460$	−38188.4	−	14455.2i	−3.87075	−	1.46516i
$461$	12521.7			1.26506			0.632532	−	0.774534i	$-0.282016\pi$
$461$	12521.7			1.26506			0.632532	+	0.774534i	$0.282016\pi$
$462$		−	11316.2i		−	1.13956i
$463$			8658.25i			0.869078i	0.900653	+	0.434539i	$0.143089\pi$
$463$			8658.25i			0.869078i	−0.900653	+	0.434539i	$0.856911\pi$
$464$	−29291.1			−2.93061
$465$	1629.51	+	616.805i	0.162509	+	0.0615133i
$466$	−7626.29			−0.758114
$467$			8094.52i			0.802077i	0.916061	+	0.401038i	$0.131351\pi$
$467$			8094.52i			0.802077i	−0.916061	+	0.401038i	$0.868649\pi$
$468$			103021.i			10.1755i
$469$	15606.2			1.53652
$470$	−9458.21	+	24987.2i	−0.928244	+	2.45229i
$471$	17399.7			1.70220
$472$			19084.8i			1.86112i
$473$		−	2745.58i		−	0.266896i
$474$	−1267.04			−0.122779
$475$	−4486.00	−	3964.06i	−0.433330	−	0.382913i
$476$	21339.3			2.05480
$477$		−	25435.6i		−	2.44154i
$478$			38643.5i			3.69773i
$479$	−7469.82			−0.712536			−0.356268	−	0.934384i	$-0.615951\pi$
$479$	−7469.82			−0.712536			−0.356268	+	0.934384i	$0.615951\pi$
$480$	−34816.7	+	91980.6i	−3.31074	+	8.74650i
$481$	−5390.45			−0.510984
$482$			11303.3i			1.06815i
$483$			21931.6i			2.06610i
$484$	26068.1			2.44817
$485$	−9278.78	−	3512.22i	−0.868717	−	0.328828i
$486$	18587.7			1.73488
$487$			1877.21i			0.174670i	0.996179	+	0.0873350i	$0.0278351\pi$
$487$			1877.21i			0.174670i	−0.996179	+	0.0873350i	$0.972165\pi$
$488$			49074.9i			4.55229i
$489$	−32243.4			−2.98180
$490$	6938.56	+	2626.40i	0.639698	+	0.242140i
$491$	−11626.0			−1.06858			−0.534290	−	0.845301i	$-0.679421\pi$
$491$	−11626.0			−1.06858			−0.534290	+	0.845301i	$0.679421\pi$
$492$		−	1215.70i		−	0.111398i
$493$		−	6171.41i		−	0.563786i
$494$	19785.5			1.80201
$495$	−3438.97	+	9085.25i	−0.312263	+	0.824953i
$496$	−4859.89			−0.439951
$497$		−	6554.12i		−	0.591534i
$498$			4326.86i			0.389340i
$499$	10254.8			0.919979			0.459989	−	0.887924i	$-0.347853\pi$
$499$	10254.8			0.919979			0.459989	+	0.887924i	$0.347853\pi$
$500$	−28749.4	+	15158.9i	−2.57142	+	1.35586i
$501$	−14356.3			−1.28022
$502$		−	6668.22i		−	0.592863i
$503$			12156.4i			1.07759i	0.842437	+	0.538794i	$0.181120\pi$
$503$			12156.4i			1.07759i	−0.842437	+	0.538794i	$0.818880\pi$
$504$	76576.9			6.76787
$505$	5961.09	−	15748.3i	0.525278	−	1.38771i
$506$	12725.4			1.11801
$507$		−	30431.6i		−	2.66571i
$508$		−	1015.95i		−	0.0887310i
$509$	−8695.86			−0.757244			−0.378622	−	0.925551i	$-0.623602\pi$
$509$	−8695.86			−0.757244			−0.378622	+	0.925551i	$0.623602\pi$
$510$	−33396.4	−	12641.3i	−2.89965	−	1.09758i
$511$	1373.34			0.118891
$512$		−	75911.9i		−	6.55247i
$513$		−	14713.6i		−	1.26631i
$514$	−612.661			−0.0525745
$515$	15390.2	+	5825.54i	1.31684	+	0.498455i
$516$	41078.1			3.50458
$517$		−	6195.24i		−	0.527015i
$518$			6107.91i			0.518081i
$519$	5839.09			0.493849
$520$	24945.5	−	65902.5i	2.10372	−	5.55772i
$521$	−2862.77			−0.240730			−0.120365	−	0.992730i	$-0.538406\pi$
$521$	−2862.77			−0.240730			−0.120365	+	0.992730i	$0.538406\pi$
$522$		−	33759.6i		−	2.83069i
$523$		−	10391.1i		−	0.868778i	−0.900725	−	0.434389i	$-0.856964\pi$
$523$		−	10391.1i		−	0.868778i	0.900725	−	0.434389i	$-0.143036\pi$
$524$	−38034.6			−3.17090
$525$	13081.2	+	11559.2i	1.08745	+	0.960928i
$526$	−28004.1			−2.32136
$527$		−	1023.94i		−	0.0846368i
$528$		−	39299.9i		−	3.23922i
$529$	−12495.7			−1.02702
$530$	−9388.67	+	24803.5i	−0.769467	+	2.03282i
$531$	−13414.0			−1.09627
$532$		−	16680.8i		−	1.35940i
$533$			414.271i			0.0336662i
$534$	37887.6			3.07033
$535$	−7794.74	−	2950.48i	−0.629899	−	0.238431i
$536$	88874.6			7.16194
$537$		−	4892.96i		−	0.393197i
$538$			38878.4i			3.11555i
$539$	−1720.32			−0.137476
$540$	−74708.0	−	28278.6i	−5.95356	−	2.25355i
$541$	13489.4			1.07200			0.536001	−	0.844217i	$-0.319934\pi$
$541$	13489.4			1.07200			0.536001	+	0.844217i	$0.319934\pi$
$542$			14964.2i			1.18592i
$543$		−	1104.63i		−	0.0873002i
$544$	57798.2			4.55529
$545$	−6629.79	+	17515.0i	−0.521081	+	1.37662i
$546$	−57694.8			−4.52218
$547$			16705.8i			1.30583i	0.757432	+	0.652914i	$0.226454\pi$
$547$			16705.8i			1.30583i	−0.757432	+	0.652914i	$0.773546\pi$
$548$		−	29125.9i		−	2.27043i
$549$	−34492.9			−2.68146
$550$	6707.02	−	7590.11i	0.519979	−	0.588443i
$551$	−4824.14			−0.372986
$552$			124897.i			9.63039i
$553$		−	364.014i		−	0.0279918i
$554$	−17529.7			−1.34435
$555$	2692.18	−	7112.36i	0.205904	−	0.543969i
$556$	−23386.1			−1.78380
$557$		−	20236.0i		−	1.53937i	−0.638425	−	0.769684i	$-0.720414\pi$
$557$		−	20236.0i		−	1.53937i	0.638425	−	0.769684i	$-0.279586\pi$
$558$		−	5601.29i		−	0.424949i
$559$	−13998.1			−1.05913
$560$	−45538.4	−	17237.3i	−3.43634	−	1.30073i
$561$	8280.19			0.623155
$562$		−	23254.1i		−	1.74540i
$563$		−	14552.9i		−	1.08940i	−0.838631	−	0.544700i	$-0.816643\pi$
$563$		−	14552.9i		−	1.08940i	0.838631	−	0.544700i	$-0.183357\pi$
$564$	92690.5			6.92017
$565$	14872.5	+	5629.55i	1.10741	+	0.419180i
$566$	12249.5			0.909692
$567$			18663.4i			1.38235i
$568$		−	37324.6i		−	2.75723i
$569$	−21936.3			−1.61620			−0.808100	−	0.589045i	$-0.799504\pi$
$569$	−21936.3			−1.61620			−0.808100	+	0.589045i	$0.799504\pi$
$570$	−9881.59	+	26105.7i	−0.726130	+	1.91833i
$571$	−16923.7			−1.24034			−0.620169	−	0.784468i	$-0.712936\pi$
$571$	−16923.7			−1.24034			−0.620169	+	0.784468i	$0.712936\pi$
$572$			24907.9i			1.82072i
$573$		−	9573.02i		−	0.697938i
$574$	469.410			0.0341338
$575$	−12998.7	+	14710.2i	−0.942752	+	1.06688i
$576$	176718.			12.7834
$577$		−	4444.56i		−	0.320675i	−0.987062	−	0.160338i	$-0.948742\pi$
$577$		−	4444.56i		−	0.320675i	0.987062	−	0.160338i	$-0.0512583\pi$
$578$		−	6481.65i		−	0.466438i
$579$	−18696.1			−1.34194
$580$	−9271.73	+	24494.6i	−0.663772	+	1.75359i
$581$	−1243.08			−0.0887637
$582$			46260.2i			3.29475i
$583$		−	6149.69i		−	0.436868i
$584$	7820.97			0.554168
$585$	46320.4	+	17533.3i	3.27370	+	1.23917i
$586$	−34422.5			−2.42658
$587$		−	11466.1i		−	0.806227i	−0.915150	−	0.403113i	$-0.867928\pi$
$587$		−	11466.1i		−	0.806227i	0.915150	−	0.403113i	$-0.132072\pi$
$588$		−	25738.7i		−	1.80518i
$589$	−800.407			−0.0559936
$590$	13080.6	+	4951.31i	0.912748	+	0.345495i
$591$	4151.59			0.288957
$592$			21212.0i			1.47265i
$593$		−	8509.90i		−	0.589308i	−0.955604	−	0.294654i	$-0.904796\pi$
$593$		−	8509.90i		−	0.589308i	0.955604	−	0.294654i	$-0.0952045\pi$
$594$	24894.7			1.71960
$595$	3631.76	−	9594.59i	0.250232	−	0.661076i
$596$	47417.1			3.25886
$597$		−	3637.44i		−	0.249364i
$598$		−	64879.3i		−	4.43664i
$599$	13153.8			0.897245			0.448622	−	0.893721i	$-0.351915\pi$
$599$	13153.8			0.897245			0.448622	+	0.893721i	$0.351915\pi$
$600$	74495.4	+	65828.0i	5.06877	+	4.47903i
$601$	−2039.26			−0.138408			−0.0692040	−	0.997603i	$-0.522046\pi$
$601$	−2039.26			−0.138408			−0.0692040	+	0.997603i	$0.522046\pi$
$602$			15861.2i			1.07384i
$603$			62466.7i			4.21864i
$604$	−39761.2			−2.67858
$605$	4436.57	−	11720.8i	0.298136	−	0.787632i
$606$	−78514.7			−5.26311
$607$			4875.90i			0.326041i	0.986623	+	0.163020i	$0.0521236\pi$
$607$			4875.90i			0.326041i	−0.986623	+	0.163020i	$0.947876\pi$
$608$		−	45180.4i		−	3.01366i
$609$	14067.2			0.936016
$610$	33635.8	+	12731.9i	2.23258	+	0.845080i
$611$	−31585.9			−2.09137
$612$			85414.6i			5.64163i
$613$			13804.3i			0.909545i	0.890608	+	0.454772i	$0.150279\pi$
$613$			13804.3i			0.909545i	−0.890608	+	0.454772i	$0.849721\pi$
$614$	30916.8			2.03209
$615$	−546.605	−	206.902i	−0.0358394	−	0.0135660i
$616$	18514.4			1.21098
$617$		−	11865.1i		−	0.774181i	−0.922042	−	0.387091i	$-0.873480\pi$
$617$		−	11865.1i		−	0.774181i	0.922042	−	0.387091i	$-0.126520\pi$
$618$		−	76729.4i		−	4.99435i
$619$	2097.46			0.136194			0.0680971	−	0.997679i	$-0.478307\pi$
$619$	2097.46			0.136194			0.0680971	+	0.997679i	$0.478307\pi$
$620$	−1538.34	+	4064.06i	−0.0996469	+	0.263253i
$621$	−48247.7			−3.11773
$622$			558.887i			0.0360278i
$623$			10884.9i			0.699989i
$624$	−200367.			−12.8543
$625$	1922.89	+	15506.2i	0.123065	+	0.992399i
$626$	167.025			0.0106640
$627$		−	6472.56i		−	0.412263i
$628$			43395.6i			2.75744i
$629$	−4469.22			−0.283306
$630$	19866.9	−	52485.5i	1.25638	−	3.31916i
$631$	6324.92			0.399035			0.199517	−	0.979894i	$-0.436063\pi$
$631$	6324.92			0.399035			0.199517	+	0.979894i	$0.436063\pi$
$632$		−	2073.00i		−	0.130474i
$633$			18187.2i			1.14198i
$634$	61201.1			3.83377
$635$	−456.791	−	172.905i	−0.0285468	−	0.0108056i
$636$	92009.0			5.73647
$637$			8770.91i			0.545551i
$638$		−	8162.23i		−	0.506498i
$639$	26234.1			1.62411
$640$	−93411.6	−	35358.3i	−5.76940	−	2.18385i
$641$	−6497.63			−0.400376			−0.200188	−	0.979758i	$-0.564155\pi$
$641$	−6497.63			−0.400376			−0.200188	+	0.979758i	$0.564155\pi$
$642$			38861.4i			2.38900i
$643$		−	3718.06i		−	0.228034i	−0.993479	−	0.114017i	$-0.963628\pi$
$643$		−	3718.06i		−	0.228034i	0.993479	−	0.114017i	$-0.0363718\pi$
$644$	−54698.4			−3.34692
$645$	6991.14	−	18469.6i	0.426784	−	1.12750i
$646$	16404.2			0.999091
$647$			2281.41i			0.138627i	0.997595	+	0.0693135i	$0.0220809\pi$
$647$			2281.41i			0.138627i	−0.997595	+	0.0693135i	$0.977919\pi$
$648$			106285.i			6.44333i
$649$	−3243.17			−0.196156
$650$	−38697.5	−	34195.1i	−2.33514	−	2.06345i
$651$	2334.00			0.140517
$652$		−	80416.4i		−	4.83029i
$653$			1299.82i			0.0778955i	0.999241	+	0.0389477i	$0.0124006\pi$
$653$			1299.82i			0.0778955i	−0.999241	+	0.0389477i	$0.987599\pi$
$654$	87322.3			5.22106
$655$	−6473.16	+	17101.1i	−0.386148	+	1.02015i
$656$	1630.21			0.0970257
$657$			5497.07i			0.326425i
$658$			35789.9i			2.12042i
$659$	−22697.8			−1.34170			−0.670849	−	0.741594i	$-0.734070\pi$
$659$	−22697.8			−1.34170			−0.670849	+	0.741594i	$0.734070\pi$
$660$	−32864.4	−	12439.9i	−1.93825	−	0.733670i
$661$	−12366.6			−0.727691			−0.363846	−	0.931459i	$-0.618536\pi$
$661$	−12366.6			−0.727691			−0.363846	+	0.931459i	$0.618536\pi$
$662$		−	33731.9i		−	1.98040i
$663$		−	42215.8i		−	2.47289i
$664$	−7079.15			−0.413741
$665$	−7500.02	−	2838.92i	−0.437351	−	0.165547i
$666$	−24448.1			−1.42244
$667$			15819.0i			0.918312i
$668$		−	35805.1i		−	2.07386i
$669$	−30895.9			−1.78551
$670$	23057.4	−	60914.4i	1.32953	−	3.51243i
$671$	−8339.53			−0.479798
$672$			131746.i			7.56284i
$673$			6518.10i			0.373335i	0.982423	+	0.186667i	$0.0597686\pi$
$673$			6518.10i			0.373335i	−0.982423	+	0.186667i	$0.940231\pi$
$674$	−24731.2			−1.41337
$675$	−25429.3	+	28777.5i	−1.45004	+	1.64096i
$676$	75897.7			4.31826
$677$			6231.91i			0.353784i	0.984230	+	0.176892i	$0.0566043\pi$
$677$			6231.91i			0.353784i	−0.984230	+	0.176892i	$0.943396\pi$
$678$		−	74147.9i		−	4.20005i
$679$	−13290.3			−0.751154
$680$	20682.3	−	54639.6i	1.16637	−	3.08137i
$681$	2986.13			0.168031
$682$		−	1354.25i		−	0.0760367i
$683$		−	3235.02i		−	0.181236i	−0.995886	−	0.0906182i	$-0.971116\pi$
$683$		−	3235.02i		−	0.181236i	0.995886	−	0.0906182i	$-0.0288843\pi$
$684$	66767.9			3.73236
$685$	−13095.6	−	4956.97i	−0.730449	−	0.276491i
$686$	38658.1			2.15156
$687$		−	6114.87i		−	0.339588i
$688$			55084.1i			3.05241i
$689$	−31353.7			−1.73364
$690$	85604.1	+	32403.0i	4.72303	+	1.78777i
$691$	18916.8			1.04143			0.520714	−	0.853731i	$-0.325666\pi$
$691$	18916.8			1.04143			0.520714	+	0.853731i	$0.325666\pi$
$692$			14562.9i			0.799999i
$693$			13013.1i			0.713313i
$694$	−28635.4			−1.56626
$695$	−3980.10	+	10514.9i	−0.217229	+	0.573887i
$696$	80110.7			4.36291
$697$			343.472i			0.0186656i
$698$			52604.1i			2.85257i
$699$	12719.7			0.688273
$700$	−28829.2	+	32625.1i	−1.55663	+	1.76159i
$701$	11149.4			0.600725			0.300363	−	0.953825i	$-0.402892\pi$
$701$	11149.4			0.600725			0.300363	+	0.953825i	$0.402892\pi$
$702$		−	126923.i		−	6.82395i
$703$			3493.55i			0.187428i
$704$	42725.9			2.28735
$705$	15775.1	−	41675.6i	0.842731	−	2.22637i
$706$	67303.6			3.58782
$707$		−	22556.8i		−	1.19991i
$708$		−	48522.8i		−	2.57571i
$709$	−16230.7			−0.859740			−0.429870	−	0.902891i	$-0.641441\pi$
$709$	−16230.7			−0.859740			−0.429870	+	0.902891i	$0.641441\pi$
$710$	−25582.2	−	9683.41i	−1.35223	−	0.511848i
$711$	1457.03			0.0768538
$712$			61987.6i			3.26276i
$713$			2624.64i			0.137859i
$714$	−47834.7			−2.50724
$715$	11199.1	+	4239.11i	0.585767	+	0.221725i
$716$	12203.2			0.636950
$717$		−	64452.6i		−	3.35708i
$718$		−	36683.8i		−	1.90672i
$719$	−32203.3			−1.67035			−0.835174	−	0.549986i	$-0.814633\pi$
$719$	−32203.3			−1.67035			−0.835174	+	0.549986i	$0.814633\pi$
$720$	68995.4	−	182276.i	3.57126	−	9.43476i
$721$	22043.9			1.13864
$722$			25523.6i			1.31564i
$723$		−	18852.5i		−	0.969752i
$724$	2754.98			0.141420
$725$	9435.30	+	8337.52i	0.483336	+	0.427101i
$726$	−58434.9			−2.98722
$727$			3275.73i			0.167112i	0.996503	+	0.0835558i	$0.0266277\pi$
$727$			3275.73i			0.167112i	−0.996503	+	0.0835558i	$0.973372\pi$
$728$		−	94394.0i		−	4.80560i
$729$	2644.09			0.134334
$730$	2029.05	−	5360.47i	0.102875	−	0.271780i
$731$	−11605.8			−0.587217
$732$		−	124772.i		−	6.30017i
$733$		−	1802.01i		−	0.0908032i	−0.998969	−	0.0454016i	$-0.985543\pi$
$733$		−	1802.01i		−	0.0908032i	0.998969	−	0.0454016i	$-0.0144567\pi$
$734$	66958.9			3.36716
$735$	−11572.7	−	4380.50i	−0.580767	−	0.219833i
$736$	−148152.			−7.41979
$737$			15102.9i			0.754847i
$738$			1878.90i			0.0937173i
$739$	−4396.87			−0.218865			−0.109433	−	0.993994i	$-0.534903\pi$
$739$	−4396.87			−0.218865			−0.109433	+	0.993994i	$0.534903\pi$
$740$	17738.5	+	6714.41i	0.881189	+	0.333549i
$741$	−32999.8			−1.63600
$742$			35526.8i			1.75772i
$743$		−	30706.4i		−	1.51616i	−0.652159	−	0.758082i	$-0.726137\pi$
$743$		−	30706.4i		−	1.51616i	0.652159	−	0.758082i	$-0.273863\pi$
$744$	13291.7			0.654971
$745$	8069.97	−	21319.7i	0.396860	−	1.04845i
$746$	−34189.5			−1.67797
$747$		−	4975.67i		−	0.243708i
$748$			20651.1i			1.00946i
$749$	−11164.6			−0.544655
$750$	64445.2	−	33980.6i	3.13761	−	1.65440i
$751$	28809.6			1.39984			0.699918	−	0.714224i	$-0.253220\pi$
$751$	28809.6			1.39984			0.699918	+	0.714224i	$0.253220\pi$
$752$			124294.i			6.02732i
$753$			11121.8i			0.538246i
$754$	−41614.4			−2.00996
$755$	−6767.01	+	17877.5i	−0.326194	+	0.861758i
$756$	−107006.			−5.14787
$757$		−	10295.6i		−	0.494320i	−0.968975	−	0.247160i	$-0.920503\pi$
$757$		−	10295.6i		−	0.494320i	0.968975	−	0.247160i	$-0.0794974\pi$
$758$			11275.4i			0.540290i
$759$	−21224.4			−1.01501
$760$	−42711.4	−	16167.2i	−2.03856	−	0.771639i
$761$	−31242.6			−1.48823			−0.744116	−	0.668050i	$-0.767129\pi$
$761$	−31242.6			−1.48823			−0.744116	+	0.668050i	$0.767129\pi$
$762$			2277.37i			0.108268i
$763$			25087.2i			1.19032i
$764$	23875.5			1.13061
$765$	38404.2	+	14536.8i	1.81504	+	0.687032i
$766$	−13285.2			−0.626650
$767$			16535.0i			0.778415i
$768$			245811.i			11.5494i
$769$	26662.3			1.25028			0.625141	−	0.780512i	$-0.285042\pi$
$769$	26662.3			1.25028			0.625141	+	0.780512i	$0.285042\pi$
$770$	4803.33	−	12689.7i	0.224805	−	0.593902i
$771$	1021.84			0.0477312
$772$		−	46628.9i		−	2.17385i
$773$			32789.3i			1.52568i	0.646588	+	0.762839i	$0.276195\pi$
$773$			32789.3i			1.52568i	−0.646588	+	0.762839i	$0.723805\pi$
$774$	−63487.4			−2.94833
$775$	1565.48	+	1383.34i	0.0725594	+	0.0641173i
$776$	−75685.9			−3.50124
$777$		−	10187.2i		−	0.470354i
$778$		−	20798.7i		−	0.958444i
$779$	268.489			0.0123487
$780$	−63423.7	+	167556.i	−2.91145	+	7.69164i
$781$	6342.75			0.290604
$782$		−	53791.3i		−	2.45981i
$783$			30946.7i			1.41244i
$784$	34514.5			1.57227
$785$	19511.6	+	7385.56i	0.887131	+	0.335799i
$786$	85259.2			3.86908
$787$		−	4972.86i		−	0.225239i	−0.993638	−	0.112620i	$-0.964076\pi$
$787$		−	4972.86i		−	0.225239i	0.993638	−	0.112620i	$-0.0359241\pi$
$788$			10354.2i			0.468089i
$789$	46707.3			2.10751
$790$	−1420.83	−	537.814i	−0.0639883	−	0.0242210i
$791$	21302.2			0.957548
$792$			74107.3i			3.32486i
$793$			42518.4i			1.90400i
$794$	−13272.7			−0.593236
$795$	15659.1	−	41369.2i	0.698581	−	1.84555i
$796$	9071.91			0.403951
$797$		−	23540.4i		−	1.04623i	−0.852263	−	0.523114i	$-0.824770\pi$
$797$		−	23540.4i		−	1.04623i	0.852263	−	0.523114i	$-0.175230\pi$
$798$			37392.0i			1.65872i
$799$	−26187.8			−1.15952
$800$	−78084.8	+	88366.0i	−3.45089	+	3.90526i
$801$	−43568.8			−1.92188
$802$			87585.0i			3.85627i
$803$			1329.05i			0.0584076i
$804$	−225963.			−9.91181
$805$	−9309.19	+	24593.5i	−0.407585	+	1.07678i
$806$	−6904.54			−0.301740
$807$		−	64844.3i		−	2.82853i
$808$		−	128457.i		−	5.59296i
$809$	10364.5			0.450430			0.225215	−	0.974309i	$-0.427692\pi$
$809$	10364.5			0.450430			0.225215	+	0.974309i	$0.427692\pi$
$810$	72847.5	+	27574.4i	3.16000	+	1.19613i
$811$	18812.0			0.814524			0.407262	−	0.913311i	$-0.366484\pi$
$811$	18812.0			0.814524			0.407262	+	0.913311i	$0.366484\pi$
$812$			35084.3i			1.51628i
$813$		−	24958.4i		−	1.07666i
$814$	−5910.93			−0.254518
$815$	−36156.9	−	13686.2i	−1.55401	−	0.588228i
$816$	−166124.			−7.12685
$817$			9072.16i			0.388488i
$818$			34656.2i			1.48133i
$819$	66346.0			2.83067
$820$	516.021	−	1363.25i	0.0219759	−	0.0580572i
$821$	−34162.8			−1.45224			−0.726121	−	0.687567i	$-0.758679\pi$
$821$	−34162.8			−1.45224			−0.726121	+	0.687567i	$0.758679\pi$
$822$			65289.3i			2.77035i
$823$			12257.0i			0.519139i	0.965724	+	0.259570i	$0.0835807\pi$
$823$			12257.0i			0.519139i	−0.965724	+	0.259570i	$0.916419\pi$
$824$	125536.			5.30736
$825$	−11186.5	+	12659.4i	−0.472076	+	0.534233i
$826$	18735.8			0.789226
$827$			1780.17i			0.0748519i	0.999299	+	0.0374259i	$0.0119158\pi$
$827$			1780.17i			0.0748519i	−0.999299	+	0.0374259i	$0.988084\pi$
$828$		−	218941.i		−	9.18927i
$829$	39515.9			1.65554			0.827770	−	0.561067i	$-0.189609\pi$
$829$	39515.9			1.65554			0.827770	+	0.561067i	$0.189609\pi$
$830$	−1836.60	+	4852.02i	−0.0768063	+	0.202911i
$831$	29237.4			1.22050
$832$		−	217834.i		−	9.07698i
$833$			7271.95i			0.302471i
$834$	52422.8			2.17656
$835$	−16098.7	−	6093.71i	−0.667208	−	0.252553i
$836$	16142.8			0.667836
$837$			5134.58i			0.212039i
$838$		−	25600.9i		−	1.05533i
$839$	45133.6			1.85719			0.928596	−	0.371092i	$-0.121017\pi$
$839$	45133.6			1.85719			0.928596	+	0.371092i	$0.121017\pi$
$840$	124547.	+	47143.7i	5.11580	+	1.93644i
$841$	−14242.5			−0.583972
$842$		−	50268.5i		−	2.05744i
$843$			38785.0i			1.58461i
$844$	−45359.5			−1.84993
$845$	12917.1	−	34125.2i	0.525873	−	1.38928i
$846$	−143256.			−5.82180
$847$		−	16788.0i		−	0.681042i
$848$			123380.i			4.99634i
$849$	−20430.7			−0.825888
$850$	−32084.1	−	28351.1i	−1.29468	−	1.14404i
$851$	11455.8			0.461457
$852$			94897.5i			3.81589i
$853$		−	16109.2i		−	0.646623i	−0.946293	−	0.323311i	$-0.895204\pi$
$853$		−	16109.2i		−	0.646623i	0.946293	−	0.323311i	$-0.104796\pi$
$854$	48177.5			1.93044
$855$	11363.3	−	30020.2i	0.454523	−	1.20078i
$856$	−63580.8			−2.53872
$857$		−	42833.2i		−	1.70730i	−0.520849	−	0.853649i	$-0.674385\pi$
$857$		−	42833.2i		−	1.70730i	0.520849	−	0.853649i	$-0.325615\pi$
$858$		−	55834.2i		−	2.22162i
$859$	25795.8			1.02461			0.512305	−	0.858803i	$-0.328792\pi$
$859$	25795.8			1.02461			0.512305	+	0.858803i	$0.328792\pi$
$860$	46063.8	+	17436.2i	1.82647	+	0.691359i
$861$	−782.917			−0.0309893
$862$			57318.9i			2.26484i
$863$			5062.08i			0.199670i	0.995004	+	0.0998351i	$0.0318315\pi$
$863$			5062.08i			0.199670i	−0.995004	+	0.0998351i	$0.968168\pi$
$864$	−289830.			−11.4123
$865$	6547.79	+	2478.48i	0.257378	+	0.0974230i
$866$	93615.2			3.67341
$867$			10810.6i			0.423468i
$868$			5821.08i			0.227627i
$869$	352.275			0.0137515
$870$	20783.7	−	54907.6i	0.809924	−	2.13970i
$871$	77000.7			2.99549
$872$			142867.i			5.54828i
$873$		−	53196.8i		−	2.06236i
$874$	−42048.3			−1.62735
$875$	9762.43	+	18514.7i	0.377178	+	0.715328i
$876$	−19884.7			−0.766944
$877$		−	4371.78i		−	0.168329i	−0.996452	−	0.0841645i	$-0.973178\pi$
$877$		−	4371.78i		−	0.168329i	0.996452	−	0.0841645i	$-0.0268221\pi$
$878$		−	11557.6i		−	0.444248i
$879$	57412.4			2.20304
$880$	16681.4	−	44069.8i	0.639011	−	1.68817i
$881$	22159.0			0.847394			0.423697	−	0.905804i	$-0.360732\pi$
$881$	22159.0			0.847394			0.423697	+	0.905804i	$0.360732\pi$
$882$			39779.9i			1.51866i
$883$		−	21446.6i		−	0.817366i	−0.912676	−	0.408683i	$-0.865988\pi$
$883$		−	21446.6i		−	0.817366i	0.912676	−	0.408683i	$-0.134012\pi$
$884$	105288.			4.00590
$885$	−21816.9	−	8258.16i	−0.828663	−	0.313667i
$886$	−65927.4			−2.49986
$887$		−	24427.4i		−	0.924680i	−0.886703	−	0.462340i	$-0.847010\pi$
$887$		−	24427.4i		−	0.924680i	0.886703	−	0.462340i	$-0.152990\pi$
$888$		−	58014.6i		−	2.19239i
$889$	−654.275			−0.0246835
$890$	42486.1	+	16081.9i	1.60015	+	0.605693i
$891$	−18061.5			−0.679107
$892$		−	77055.6i		−	2.89239i
$893$			20470.8i			0.767111i
$894$	−106291.			−3.97641
$895$	2076.88	−	5486.82i	0.0775671	−	0.204921i
$896$	−133796.			−4.98863
$897$			108211.i			4.02792i
$898$		−	34944.2i		−	1.29856i
$899$	1683.48			0.0624551
$900$	−130588.	−	115394.i	−4.83659	−	4.27387i
$901$	−25995.3			−0.961186
$902$			454.272i			0.0167690i
$903$		−	26454.5i		−	0.974917i
$904$	121313.			4.46328
$905$	468.873	−	1238.70i	0.0172220	−	0.0454979i
$906$	89129.6			3.26836
$907$		−	20142.9i		−	0.737413i	−0.929546	−	0.368706i	$-0.879801\pi$
$907$		−	20142.9i		−	0.737413i	0.929546	−	0.368706i	$-0.120199\pi$
$908$			7447.53i			0.272197i
$909$	90287.9			3.29445
$910$	−64697.3	−	24489.3i	−2.35681	−	0.892103i
$911$	10580.0			0.384775			0.192388	−	0.981319i	$-0.438377\pi$
$911$	10580.0			0.384775			0.192388	+	0.981319i	$0.438377\pi$
$912$			129858.i			4.71494i
$913$		−	1202.99i		−	0.0436071i
$914$	−65332.3			−2.36433
$915$	−56100.3	−	21235.2i	−2.02691	−	0.767228i
$916$	15250.7			0.550107
$917$			24494.5i			0.882092i
$918$		−	105232.i		−	3.78341i
$919$	15173.3			0.544636			0.272318	−	0.962207i	$-0.412210\pi$
$919$	15173.3			0.544636			0.272318	+	0.962207i	$0.412210\pi$
$920$	−53014.3	+	140056.i	−1.89982	+	5.01904i
$921$	−51565.4			−1.84488
$922$		−	70005.1i		−	2.50054i
$923$		−	32338.0i		−	1.15321i
$924$	−47072.6			−1.67595
$925$	6037.87	−	6832.86i	0.214621	−	0.242879i
$926$	48405.6			1.71783
$927$			88234.8i			3.12622i
$928$			95026.8i			3.36143i
$929$	17986.8			0.635230			0.317615	−	0.948220i	$-0.397118\pi$
$929$	17986.8			0.635230			0.317615	+	0.948220i	$0.397118\pi$
$930$	3448.37	−	9110.10i	0.121588	−	0.321217i
$931$	5684.43			0.200107
$932$			31723.4i			1.11495i
$933$		−	932.153i		−	0.0327088i
$934$	45254.0			1.58539
$935$	9285.17	+	3514.64i	0.324767	+	0.122932i
$936$	377830.			13.1942
$937$		−	270.976i		−	0.00944760i	−0.999989	−	0.00472380i	$-0.998496\pi$
$937$		−	270.976i		−	0.00944760i	0.999989	−	0.00472380i	$-0.00150364\pi$
$938$		−	87249.4i		−	3.03709i
$939$	−278.576			−0.00968157
$940$	103941.	+	39343.8i	3.60656	+	1.36516i
$941$	−16688.5			−0.578138			−0.289069	−	0.957308i	$-0.593346\pi$
$941$	−16688.5			−0.578138			−0.289069	+	0.957308i	$0.593346\pi$
$942$		−	97276.6i		−	3.36459i
$943$		−	880.411i		−	0.0304031i
$944$	65067.1			2.24338
$945$	−18211.6	+	48112.3i	−0.626902	+	1.65618i
$946$	−15349.7			−0.527549
$947$		−	42192.2i		−	1.44780i	−0.689907	−	0.723898i	$-0.742349\pi$
$947$		−	42192.2i		−	1.44780i	0.689907	−	0.723898i	$-0.257651\pi$
$948$			5270.58i			0.180570i
$949$	6776.07			0.231781
$950$	−22161.9	+	25079.9i	−0.756870	+	0.856524i
$951$	−102076.			−3.48059
$952$		−	78261.9i		−	2.66437i
$953$			23078.1i			0.784443i	0.919871	+	0.392221i	$0.128293\pi$
$953$			23078.1i			0.784443i	−0.919871	+	0.392221i	$0.871707\pi$
$954$	−142203.			−4.82597
$955$	4063.40	−	10734.9i	0.137684	−	0.363742i
$956$	160747.			5.43822
$957$			13613.6i			0.459838i
$958$			41761.5i			1.40841i
$959$	−18757.2			−0.631598
$960$	287418.	+	108794.i	9.66291	+	3.65762i
$961$	−29511.7			−0.990624
$962$			30136.4i			1.01002i
$963$		−	44688.5i		−	1.49540i
$964$	47018.8			1.57093
$965$	−20965.3	−	7935.82i	−0.699375	−	0.264729i
$966$	122613.			4.08387
$967$			7258.01i			0.241367i	0.992691	+	0.120683i	$0.0385086\pi$
$967$			7258.01i			0.241367i	−0.992691	+	0.120683i	$0.961491\pi$
$968$		−	95604.9i		−	3.17444i
$969$	−27360.1			−0.907051
$970$	−19635.8	+	51874.8i	−0.649965	+	1.71711i
$971$	−27971.8			−0.924468			−0.462234	−	0.886758i	$-0.652952\pi$
$971$	−27971.8			−0.924468			−0.462234	+	0.886758i	$0.652952\pi$
$972$		−	77320.0i		−	2.55148i
$973$			15060.7i			0.496223i
$974$	10494.9			0.345254
$975$	64542.6	+	57033.2i	2.12002	+	1.87336i
$976$	167315.			5.48731
$977$		−	44187.3i		−	1.44696i	−0.690347	−	0.723478i	$-0.742542\pi$
$977$		−	44187.3i		−	1.44696i	0.690347	−	0.723478i	$-0.257458\pi$
$978$			180263.i			5.89385i
$979$	−10533.8			−0.343885
$980$	10925.1	−	28862.6i	0.356113	−	0.940800i
$981$	−100416.			−3.26813
$982$			64997.3i			2.11216i
$983$		−	18022.9i		−	0.584784i	−0.956299	−	0.292392i	$-0.905549\pi$
$983$		−	18022.9i		−	0.584784i	0.956299	−	0.292392i	$-0.0944512\pi$
$984$	−4458.59			−0.144446
$985$	4655.48	+	1762.20i	0.150595	+	0.0570034i
$986$	−34502.5			−1.11438
$987$		−	59693.1i		−	1.92508i
$988$		−	82302.8i		−	2.65020i
$989$	29748.8			0.956477
$990$	50792.9	+	19226.2i	1.63061	+	0.617221i
$991$	−23876.6			−0.765353			−0.382677	−	0.923882i	$-0.624998\pi$
$991$	−23876.6			−0.765353			−0.382677	+	0.923882i	$0.624998\pi$
$992$			15766.6i			0.504626i
$993$			56260.6i			1.79796i
$994$	−36642.1			−1.16923
$995$	1543.96	−	4078.92i	0.0491928	−	0.129960i
$996$	17998.7			0.572600
$997$		−	14998.8i		−	0.476446i	−0.971210	−	0.238223i	$-0.923435\pi$
$997$		−	14998.8i		−	0.476446i	0.971210	−	0.238223i	$-0.0765649\pi$
$998$		−	57331.7i		−	1.81844i
$999$	22411.0			0.709762

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	415.4.b.a.84.1	✓	124
5.2	odd	4		2075.4.a.n.1.62		62
5.3	odd	4		2075.4.a.m.1.1		62
5.4	even	2	inner	415.4.b.a.84.124	yes	124

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
415.4.b.a.84.1	✓	124	1.1	even	1	trivial
415.4.b.a.84.124	yes	124	5.4	even	2	inner
2075.4.a.m.1.1		62	5.3	odd	4
2075.4.a.n.1.62		62	5.2	odd	4

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:	\( N \)	\(=\)	\( 415 = 5 \cdot 83 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	415.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.59070i q^{2} +9.32458i q^{3} -23.2559 q^{4} +(-3.95795 + 10.4563i) q^{5} +52.1309 q^{6} +14.9769i q^{7} +85.2909i q^{8} -59.9478 q^{9} +(58.4581 + 22.1277i) q^{10} -14.4939 q^{11} -216.851i q^{12} +73.8958i q^{13} +83.7312 q^{14} +(-97.5008 - 36.9062i) q^{15} +290.789 q^{16} +61.2669i q^{17} +335.150i q^{18} +47.8919 q^{19} +(92.0455 - 243.171i) q^{20} -139.653 q^{21} +81.0309i q^{22} -157.044i q^{23} -795.302 q^{24} +(-93.6693 - 82.7711i) q^{25} +413.129 q^{26} -307.224i q^{27} -348.301i q^{28} -100.730 q^{29} +(-206.331 + 545.097i) q^{30} -16.7128 q^{31} -943.383i q^{32} -135.149i q^{33} +342.525 q^{34} +(-156.603 - 59.2777i) q^{35} +1394.14 q^{36} +72.9466i q^{37} -267.749i q^{38} -689.048 q^{39} +(-891.829 - 337.577i) q^{40} +5.60615 q^{41} +780.758i q^{42} +189.430i q^{43} +337.068 q^{44} +(237.270 - 626.834i) q^{45} -877.983 q^{46} +427.438i q^{47} +2711.48i q^{48} +118.693 q^{49} +(-462.748 + 523.677i) q^{50} -571.288 q^{51} -1718.51i q^{52} +424.295i q^{53} -1717.60 q^{54} +(57.3660 - 151.553i) q^{55} -1277.39 q^{56} +446.572i q^{57} +563.150i q^{58} +223.761 q^{59} +(2267.47 + 858.285i) q^{60} +575.383 q^{61} +93.4361i q^{62} -897.831i q^{63} -2947.86 q^{64} +(-772.679 - 292.476i) q^{65} -755.579 q^{66} -1042.02i q^{67} -1424.82i q^{68} +1464.37 q^{69} +(-331.404 + 875.520i) q^{70} -437.616 q^{71} -5113.00i q^{72} -91.6976i q^{73} +407.822 q^{74} +(771.806 - 873.427i) q^{75} -1113.77 q^{76} -217.073i q^{77} +3852.26i q^{78} -24.3050 q^{79} +(-1150.93 + 3040.58i) q^{80} +1246.15 q^{81} -31.3423i q^{82} +83.0000i q^{83} +3247.76 q^{84} +(-640.627 - 242.491i) q^{85} +1059.05 q^{86} -939.264i q^{87} -1236.20i q^{88} +726.778 q^{89} +(-3504.44 - 1326.51i) q^{90} -1106.73 q^{91} +3652.19i q^{92} -155.840i q^{93} +2389.68 q^{94} +(-189.553 + 500.773i) q^{95} +8796.65 q^{96} +887.385i q^{97} -663.576i q^{98} +868.877 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 124 q - 504 q^{4} + 6 q^{5} - 1088 q^{9} + 62 q^{10} + 60 q^{11} + 124 q^{14} - 120 q^{15} + 2080 q^{16} + 124 q^{19} - 206 q^{20} + 24 q^{21} + 368 q^{24} - 430 q^{25} - 176 q^{26} + 208 q^{29} - 52 q^{30}+ \cdots + 2668 q^{99}+O(q^{100}) \) 124 * q - 504 * q^4 + 6 * q^5 - 1088 * q^9 + 62 * q^10 + 60 * q^11 + 124 * q^14 - 120 * q^15 + 2080 * q^16 + 124 * q^19 - 206 * q^20 + 24 * q^21 + 368 * q^24 - 430 * q^25 - 176 * q^26 + 208 * q^29 - 52 * q^30 + 372 * q^31 + 496 * q^34 + 320 * q^35 + 4592 * q^36 - 592 * q^39 - 430 * q^40 - 384 * q^41 - 636 * q^44 - 118 * q^45 - 1216 * q^46 - 6504 * q^49 + 440 * q^50 - 696 * q^51 + 8 * q^54 - 684 * q^55 - 1328 * q^56 - 1068 * q^59 + 4698 * q^60 + 952 * q^61 - 9180 * q^64 - 16 * q^65 - 284 * q^66 + 3496 * q^69 - 1296 * q^70 - 2976 * q^71 - 1732 * q^74 + 1796 * q^75 + 804 * q^76 + 1080 * q^79 - 1758 * q^80 + 8700 * q^81 + 6752 * q^84 + 1036 * q^85 - 2692 * q^86 + 832 * q^89 - 5586 * q^90 + 1464 * q^91 + 2620 * q^94 + 1240 * q^95 + 5892 * q^96 + 2668 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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