LMFDB - Embedded newform 69.6.c.b.68.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,6,Mod(68,69)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("69.68");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	69.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$11.0664835671$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			68.9
Character	$\chi$	$=$	69.68
Dual form			69.6.c.b.68.23

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.41836i q^{2} +(3.03549 - 15.2901i) q^{3} +2.64141 q^{4} -41.1824 q^{5} +(-82.8470 - 16.4474i) q^{6} +67.8449i q^{7} -187.700i q^{8} +(-224.572 - 92.8256i) q^{9} +O(q^{10})$	\(q-5.41836i q^{2} +(3.03549 - 15.2901i) q^{3} +2.64141 q^{4} -41.1824 q^{5} +(-82.8470 - 16.4474i) q^{6} +67.8449i q^{7} -187.700i q^{8} +(-224.572 - 92.8256i) q^{9} +223.141i q^{10} -718.835 q^{11} +(8.01796 - 40.3872i) q^{12} -17.0158 q^{13} +367.608 q^{14} +(-125.009 + 629.681i) q^{15} -932.498 q^{16} +1853.18 q^{17} +(-502.962 + 1216.81i) q^{18} -1139.13i q^{19} -108.779 q^{20} +(1037.35 + 205.943i) q^{21} +3894.90i q^{22} +(-1493.34 + 2050.92i) q^{23} +(-2869.94 - 569.760i) q^{24} -1429.01 q^{25} +92.1977i q^{26} +(-2100.99 + 3151.94i) q^{27} +179.206i q^{28} -1744.71i q^{29} +(3411.84 + 677.342i) q^{30} +3913.17 q^{31} -953.777i q^{32} +(-2182.02 + 10991.0i) q^{33} -10041.2i q^{34} -2794.01i q^{35} +(-593.185 - 245.190i) q^{36} -9229.25i q^{37} -6172.22 q^{38} +(-51.6513 + 260.173i) q^{39} +7729.91i q^{40} -5011.22i q^{41} +(1115.87 - 5620.75i) q^{42} -5505.60i q^{43} -1898.73 q^{44} +(9248.39 + 3822.78i) q^{45} +(11112.6 + 8091.43i) q^{46} -20518.7i q^{47} +(-2830.59 + 14257.9i) q^{48} +12204.1 q^{49} +7742.90i q^{50} +(5625.31 - 28335.2i) q^{51} -44.9456 q^{52} -36672.8 q^{53} +(17078.3 + 11383.9i) q^{54} +29603.3 q^{55} +12734.5 q^{56} +(-17417.4 - 3457.82i) q^{57} -9453.46 q^{58} -30068.0i q^{59} +(-330.199 + 1663.24i) q^{60} -30394.4i q^{61} -21202.9i q^{62} +(6297.75 - 15236.0i) q^{63} -35007.8 q^{64} +700.751 q^{65} +(59553.3 + 11822.9i) q^{66} +42629.9i q^{67} +4895.00 q^{68} +(26825.7 + 29058.7i) q^{69} -15139.0 q^{70} +42884.6i q^{71} +(-17423.3 + 42152.0i) q^{72} +10487.2 q^{73} -50007.4 q^{74} +(-4337.75 + 21849.7i) q^{75} -3008.91i q^{76} -48769.3i q^{77} +(1409.71 + 279.865i) q^{78} -37112.3i q^{79} +38402.5 q^{80} +(41815.8 + 41692.0i) q^{81} -27152.6 q^{82} +65765.4 q^{83} +(2740.07 + 543.978i) q^{84} -76318.4 q^{85} -29831.3 q^{86} +(-26676.7 - 5296.05i) q^{87} +134925. i q^{88} -107918. q^{89} +(20713.2 - 50111.1i) q^{90} -1154.44i q^{91} +(-3944.50 + 5417.32i) q^{92} +(11878.4 - 59832.6i) q^{93} -111177. q^{94} +46912.1i q^{95} +(-14583.3 - 2895.18i) q^{96} +69228.0i q^{97} -66126.0i q^{98} +(161430. + 66726.3i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 408 q^{4} - 528 q^{6} - 444 q^{9}+O(q^{10}) $ 32 * q - 408 * q^4 - 528 * q^6 - 444 * q^9	$ 32 q - 408 q^{4} - 528 q^{6} - 444 q^{9} - 2484 q^{12} + 520 q^{13} + 4936 q^{16} + 7188 q^{18} + 18660 q^{24} + 36032 q^{25} - 22032 q^{27} + 6544 q^{31} - 33912 q^{36} - 63912 q^{39} + 54328 q^{46} + 88284 q^{48} - 207664 q^{49} + 46296 q^{52} - 38628 q^{54} - 139296 q^{55} - 184144 q^{58} + 486584 q^{64} - 113580 q^{69} + 37176 q^{70} - 15504 q^{72} - 93896 q^{73} + 249840 q^{75} + 368028 q^{78} - 339372 q^{81} - 23512 q^{82} + 259584 q^{85} + 509928 q^{87} + 82740 q^{93} - 562000 q^{94} + 1404 q^{96}+O(q^{100}) $ 32 * q - 408 * q^4 - 528 * q^6 - 444 * q^9 - 2484 * q^12 + 520 * q^13 + 4936 * q^16 + 7188 * q^18 + 18660 * q^24 + 36032 * q^25 - 22032 * q^27 + 6544 * q^31 - 33912 * q^36 - 63912 * q^39 + 54328 * q^46 + 88284 * q^48 - 207664 * q^49 + 46296 * q^52 - 38628 * q^54 - 139296 * q^55 - 184144 * q^58 + 486584 * q^64 - 113580 * q^69 + 37176 * q^70 - 15504 * q^72 - 93896 * q^73 + 249840 * q^75 + 368028 * q^78 - 339372 * q^81 - 23512 * q^82 + 259584 * q^85 + 509928 * q^87 + 82740 * q^93 - 562000 * q^94 + 1404 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/69\mathbb{Z}\right)^\times$.

$n$	$28$	$47$
$\chi(n)$	$-1$	$-1$

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 $

$2$		−	5.41836i		−	0.957839i	−0.877859	−	0.478920i	$-0.841029\pi$
$2$		−	5.41836i		−	0.957839i	0.877859	−	0.478920i	$-0.158971\pi$
$3$	3.03549	−	15.2901i	0.194727	−	0.980858i
$3$	3.03549	−	15.2901i	0.194727	−	0.980858i
$4$	2.64141			0.0825439
$5$	−41.1824			−0.736693			−0.368346	−	0.929689i	$-0.620076\pi$
$5$	−41.1824			−0.736693			−0.368346	+	0.929689i	$0.620076\pi$
$6$	−82.8470	−	16.4474i	−0.939504	−	0.186517i
$7$			67.8449i			0.523326i	0.965159	+	0.261663i	$0.0842709\pi$
$7$			67.8449i			0.523326i	−0.965159	+	0.261663i	$0.915729\pi$
$8$		−	187.700i		−	1.03690i
$9$	−224.572	−	92.8256i	−0.924163	−	0.381998i
$10$			223.141i			0.705633i
$11$	−718.835			−1.79121			−0.895607	−	0.444847i	$-0.853258\pi$
$11$	−718.835			−1.79121			−0.895607	+	0.444847i	$0.853258\pi$
$12$	8.01796	−	40.3872i	0.0160735	−	0.0809638i
$13$	−17.0158			−0.0279251			−0.0139625	−	0.999903i	$-0.504445\pi$
$13$	−17.0158			−0.0279251			−0.0139625	+	0.999903i	$0.504445\pi$
$14$	367.608			0.501262
$15$	−125.009	+	629.681i	−0.143454	+	0.722591i
$16$	−932.498			−0.910643
$17$	1853.18			1.55523			0.777616	−	0.628739i	$-0.216429\pi$
$17$	1853.18			1.55523			0.777616	+	0.628739i	$0.216429\pi$
$18$	−502.962	+	1216.81i	−0.365893	+	0.885200i
$19$		−	1139.13i		−	0.723918i	−0.932194	−	0.361959i	$-0.882108\pi$
$19$		−	1139.13i		−	0.723918i	0.932194	−	0.361959i	$-0.117892\pi$
$20$	−108.779			−0.0608095
$21$	1037.35	+	205.943i	0.513308	+	0.101906i
$22$			3894.90i			1.71569i
$23$	−1493.34	+	2050.92i	−0.588624	+	0.808407i
$23$	−1493.34	+	2050.92i	−0.588624	+	0.808407i
$24$	−2869.94	−	569.760i	−1.01705	−	0.201913i
$25$	−1429.01			−0.457284
$26$			92.1977i			0.0267477i
$27$	−2100.99	+	3151.94i	−0.554645	+	0.832087i
$28$			179.206i			0.0431974i
$29$		−	1744.71i		−	0.385237i	−0.981274	−	0.192619i	$-0.938302\pi$
$29$		−	1744.71i		−	0.385237i	0.981274	−	0.192619i	$-0.0616980\pi$
$30$	3411.84	+	677.342i	0.692126	+	0.137406i
$31$	3913.17			0.731348			0.365674	−	0.930743i	$-0.380838\pi$
$31$	3913.17			0.731348			0.365674	+	0.930743i	$0.380838\pi$
$32$		−	953.777i		−	0.164654i
$33$	−2182.02	+	10991.0i	−0.348797	+	1.75693i
$34$		−	10041.2i		−	1.48966i
$35$		−	2794.01i		−	0.385530i
$36$	−593.185	−	245.190i	−0.0762840	−	0.0315316i
$37$		−	9229.25i		−	1.10831i	−0.832413	−	0.554156i	$-0.813041\pi$
$37$		−	9229.25i		−	1.10831i	0.832413	−	0.554156i	$-0.186959\pi$
$38$	−6172.22			−0.693398
$39$	−51.6513	+	260.173i	−0.00543776	+	0.0273905i
$40$			7729.91i			0.763879i
$41$		−	5011.22i		−	0.465569i	−0.972528	−	0.232784i	$-0.925216\pi$
$41$		−	5011.22i		−	0.465569i	0.972528	−	0.232784i	$-0.0747836\pi$
$42$	1115.87	−	5620.75i	0.0976091	−	0.491667i
$43$		−	5505.60i		−	0.454081i	−0.973885	−	0.227041i	$-0.927095\pi$
$43$		−	5505.60i		−	0.454081i	0.973885	−	0.227041i	$-0.0729050\pi$
$44$	−1898.73			−0.147854
$45$	9248.39	+	3822.78i	0.680824	+	0.281416i
$46$	11112.6	+	8091.43i	0.774324	+	0.563807i
$47$		−	20518.7i		−	1.35489i	−0.735573	−	0.677446i	$-0.763087\pi$
$47$		−	20518.7i		−	1.35489i	0.735573	−	0.677446i	$-0.236913\pi$
$48$	−2830.59	+	14257.9i	−0.177327	+	0.893211i
$49$	12204.1			0.726130
$50$			7742.90i			0.438004i
$51$	5625.31	−	28335.2i	0.302846	−	1.52546i
$52$	−44.9456			−0.00230504
$53$	−36672.8			−1.79331			−0.896653	−	0.442734i	$-0.854009\pi$
$53$	−36672.8			−1.79331			−0.896653	+	0.442734i	$0.854009\pi$
$54$	17078.3	+	11383.9i	0.797005	+	0.531261i
$55$	29603.3			1.31957
$56$	12734.5			0.542638
$57$	−17417.4	−	3457.82i	−0.710061	−	0.140966i
$58$	−9453.46			−0.368995
$59$		−	30068.0i		−	1.12454i	−0.826955	−	0.562269i	$-0.809929\pi$
$59$		−	30068.0i		−	1.12454i	0.826955	−	0.562269i	$-0.190071\pi$
$60$	−330.199	+	1663.24i	−0.0118412	+	0.0596455i
$61$		−	30394.4i		−	1.04585i	−0.852379	−	0.522924i	$-0.824841\pi$
$61$		−	30394.4i		−	1.04585i	0.852379	−	0.522924i	$-0.175159\pi$
$62$		−	21202.9i		−	0.700514i
$63$	6297.75	−	15236.0i	0.199910	−	0.483638i
$64$	−35007.8			−1.06835
$65$	700.751			0.0205722
$66$	59553.3	+	11822.9i	1.68285	+	0.334092i
$67$			42629.9i			1.16019i	0.814550	+	0.580093i	$0.196984\pi$
$67$			42629.9i			1.16019i	−0.814550	+	0.580093i	$0.803016\pi$
$68$	4895.00			0.128375
$69$	26825.7	+	29058.7i	0.678311	+	0.734775i
$70$	−15139.0			−0.369276
$71$			42884.6i			1.00961i	0.863232	+	0.504807i	$0.168436\pi$
$71$			42884.6i			1.00961i	−0.863232	+	0.504807i	$0.831564\pi$
$72$	−17423.3	+	42152.0i	−0.396095	+	0.958267i
$73$	10487.2			0.230330			0.115165	−	0.993346i	$-0.463260\pi$
$73$	10487.2			0.230330			0.115165	+	0.993346i	$0.463260\pi$
$74$	−50007.4			−1.06158
$75$	−4337.75	+	21849.7i	−0.0890454	+	0.448530i
$76$		−	3008.91i		−	0.0597551i
$77$		−	48769.3i		−	0.937388i
$78$	1409.71	+	279.865i	0.0262357	+	0.00520850i
$79$		−	37112.3i		−	0.669037i	−0.942389	−	0.334519i	$-0.891426\pi$
$79$		−	37112.3i		−	0.669037i	0.942389	−	0.334519i	$-0.108574\pi$
$80$	38402.5			0.670864
$81$	41815.8	+	41692.0i	0.708154	+	0.706058i
$82$	−27152.6			−0.445940
$83$	65765.4			1.04786			0.523929	−	0.851762i	$-0.324466\pi$
$83$	65765.4			1.04786			0.523929	+	0.851762i	$0.324466\pi$
$84$	2740.07	+	543.978i	0.0423705	+	0.00841168i
$85$	−76318.4			−1.14573
$86$	−29831.3			−0.434937
$87$	−26676.7	−	5296.05i	−0.377863	−	0.0750160i
$88$			134925.i			1.85731i
$89$	−107918.			−1.44418			−0.722088	−	0.691801i	$-0.756818\pi$
$89$	−107918.			−1.44418			−0.722088	+	0.691801i	$0.756818\pi$
$90$	20713.2	−	50111.1i	0.269551	−	0.652120i
$91$		−	1154.44i		−	0.0146139i
$92$	−3944.50	+	5417.32i	−0.0485873	+	0.0667291i
$93$	11878.4	−	59832.6i	0.142413	−	0.717349i
$94$	−111177.			−1.29777
$95$			46912.1i			0.533306i
$96$	−14583.3	−	2895.18i	−0.161502	−	0.0320625i
$97$			69228.0i			0.747055i	0.927619	+	0.373528i	$0.121852\pi$
$97$			69228.0i			0.747055i	−0.927619	+	0.373528i	$0.878148\pi$
$98$		−	66126.0i		−	0.695516i
$99$	161430.	+	66726.3i	1.65537	+	0.684241i
$100$	−3774.60			−0.0377460
$101$		−	18385.9i		−	0.179342i	−0.995971	−	0.0896709i	$-0.971418\pi$
$101$		−	18385.9i		−	0.179342i	0.995971	−	0.0896709i	$-0.0285815\pi$
$102$	−153530.	−	30479.9i	−1.46115	−	0.290077i
$103$		−	3247.41i		−	0.0301608i	−0.999886	−	0.0150804i	$-0.995200\pi$
$103$		−	3247.41i		−	0.0301608i	0.999886	−	0.0150804i	$-0.00480043\pi$
$104$			3193.86i			0.0289556i
$105$	−42720.6	−	8481.20i	−0.378150	−	0.0750731i
$106$			198706.i			1.71770i
$107$	34846.8			0.294241			0.147121	−	0.989119i	$-0.452999\pi$
$107$	34846.8			0.294241			0.147121	+	0.989119i	$0.452999\pi$
$108$	−5549.58	+	8325.55i	−0.0457826	+	0.0686837i
$109$		−	227445.i		−	1.83362i	−0.399318	−	0.916812i	$-0.630753\pi$
$109$		−	227445.i		−	1.83362i	0.399318	−	0.916812i	$-0.369247\pi$
$110$		−	160401.i		−	1.26394i
$111$	−141116.	−	28015.3i	−1.08710	−	0.215818i
$112$		−	63265.2i		−	0.476563i
$113$	29482.7			0.217206			0.108603	−	0.994085i	$-0.465362\pi$
$113$	29482.7			0.217206			0.108603	+	0.994085i	$0.465362\pi$
$114$	−18735.7	+	94373.6i	−0.135023	+	0.680124i
$115$	61499.1	−	84462.0i	0.433635	−	0.595548i
$116$		−	4608.48i		−	0.0317990i
$117$	3821.27	+	1579.50i	0.0258073	+	0.0106673i
$118$	−162919.			−1.07713
$119$			125729.i			0.813894i
$120$	118191.	+	23464.1i	0.749256	+	0.148748i
$121$	355672.			2.20845
$122$	−164688.			−1.00175
$123$	−76621.8	−	15211.5i	−0.456657	−	0.0906587i
$124$	10336.3			0.0603684
$125$	187545.			1.07357
$126$	−82554.3	−	34123.4i	−0.463248	−	0.191481i
$127$	218296.			1.20098			0.600491	−	0.799631i	$-0.294972\pi$
$127$	218296.			1.20098			0.600491	+	0.799631i	$0.294972\pi$
$128$			159164.i			0.858658i
$129$	−84180.9	−	16712.2i	−0.445389	−	0.0884218i
$130$		−	3796.92i		−	0.0197049i
$131$			366952.i			1.86823i	0.356971	+	0.934116i	$0.383810\pi$
$131$			366952.i			1.86823i	−0.356971	+	0.934116i	$0.616190\pi$
$132$	−5763.59	+	29031.7i	−0.0287911	+	0.145023i
$133$	77284.2			0.378845
$134$	230984.			1.11127
$135$	86523.9	−	129804.i	0.408603	−	0.612992i
$136$		−	347841.i		−	1.61263i
$137$	126931.			0.577786			0.288893	−	0.957361i	$-0.406713\pi$
$137$	126931.			0.577786			0.288893	+	0.957361i	$0.406713\pi$
$138$	157451.	−	145351.i	0.703796	−	0.649713i
$139$	147020.			0.645416			0.322708	−	0.946499i	$-0.395407\pi$
$139$	147020.			0.645416			0.322708	+	0.946499i	$0.395407\pi$
$140$		−	7380.13i		−	0.0318232i
$141$	−313732.	−	62284.2i	−1.32896	−	0.263834i
$142$	232364.			0.967048
$143$	12231.6			0.0500198
$144$	209413.	+	86559.7i	0.841582	+	0.347864i
$145$			71851.3i			0.283801i
$146$		−	56823.2i		−	0.220619i
$147$	37045.3	−	186601.i	0.141397	−	0.712230i
$148$		−	24378.2i		−	0.0914844i
$149$	193942.			0.715660			0.357830	−	0.933787i	$-0.383517\pi$
$149$	193942.			0.715660			0.357830	+	0.933787i	$0.383517\pi$
$150$	118389.	+	23503.5i	0.429620	+	0.0852912i
$151$	−22344.9			−0.0797510			−0.0398755	−	0.999205i	$-0.512696\pi$
$151$	−22344.9			−0.0797510			−0.0398755	+	0.999205i	$0.512696\pi$
$152$	−213814.			−0.750633
$153$	−416172.	−	172023.i	−1.43729	−	0.594097i
$154$	−264249.			−0.897867
$155$	−161154.			−0.538779
$156$	−136.432	+	687.221i	−0.000448854	+	0.00226092i
$157$		−	139190.i		−	0.450670i	−0.974281	−	0.225335i	$-0.927652\pi$
$157$		−	139190.i		−	0.450670i	0.974281	−	0.225335i	$-0.0723476\pi$
$158$	−201088.			−0.640830
$159$	−111320.	+	560729.i	−0.349205	+	1.75898i
$160$			39278.8i			0.121299i
$161$	−139145.	−	101315.i	−0.423060	−	0.308042i
$162$	225902.	−	226573.i	0.676290	−	0.678298i
$163$	−128054.			−0.377507			−0.188753	−	0.982025i	$-0.560445\pi$
$163$	−128054.			−0.377507			−0.188753	+	0.982025i	$0.560445\pi$
$164$		−	13236.7i		−	0.0384299i
$165$	89860.6	−	452636.i	0.256956	−	1.29431i
$166$		−	356340.i		−	1.00368i
$167$		−	183367.i		−	0.508781i	−0.967102	−	0.254391i	$-0.918125\pi$
$167$		−	183367.i		−	0.508781i	0.967102	−	0.254391i	$-0.0818749\pi$
$168$	38655.3	−	194711.i	0.105666	−	0.532251i
$169$	−371003.			−0.999220
$170$			413520.i			1.09742i
$171$	−105741.	+	255816.i	−0.276536	+	0.669019i
$172$		−	14542.5i		−	0.0374816i
$173$		−	343548.i		−	0.872716i	−0.899773	−	0.436358i	$-0.856268\pi$
$173$		−	343548.i		−	0.872716i	0.899773	−	0.436358i	$-0.143732\pi$
$174$	−28695.9	+	144544.i	−0.0718532	+	0.361932i
$175$		−	96951.2i		−	0.239308i
$176$	670312.			1.63116
$177$	−459741.	−	91271.0i	−1.10301	−	0.218978i
$178$			584740.i			1.38329i
$179$			185482.i			0.432684i	0.976318	+	0.216342i	$0.0694125\pi$
$179$			185482.i			0.432684i	−0.976318	+	0.216342i	$0.930587\pi$
$180$	24428.8	+	10097.5i	0.0561979	+	0.0232291i
$181$			667226.i			1.51383i	0.653514	+	0.756915i	$0.273294\pi$
$181$			667226.i			1.51383i	−0.653514	+	0.756915i	$0.726706\pi$
$182$	−6255.15			−0.0139978
$183$	−464732.	−	92261.8i	−1.02583	−	0.203655i
$184$	384958.	+	280298.i	0.838240	+	0.610346i
$185$			380082.i			0.816485i
$186$	−324194.	−	64361.3i	−0.687105	−	0.136409i
$187$	−1.33213e6			−2.78575
$188$		−	54198.1i		−	0.111838i
$189$	−213843.	−	142542.i	−0.435453	−	0.290260i
$190$	254187.			0.510821
$191$	280522.			0.556396			0.278198	−	0.960524i	$-0.410263\pi$
$191$	280522.			0.556396			0.278198	+	0.960524i	$0.410263\pi$
$192$	−106266.	+	535272.i	−0.208037	+	1.04790i
$193$	561191.			1.08447			0.542235	−	0.840227i	$-0.317578\pi$
$193$	561191.			1.08447			0.542235	+	0.840227i	$0.317578\pi$
$194$	375102.			0.715559
$195$	2127.12	−	10714.5i	0.00400596	−	0.0201784i
$196$	32235.9			0.0599376
$197$			111595.i			0.204870i	0.994740	+	0.102435i	$0.0326633\pi$
$197$			111595.i			0.204870i	−0.994740	+	0.102435i	$0.967337\pi$
$198$	361547.	−	874685.i	0.655393	−	1.58558i
$199$			60669.5i			0.108602i	0.998525	+	0.0543010i	$0.0172930\pi$
$199$			60669.5i			0.108602i	−0.998525	+	0.0543010i	$0.982707\pi$
$200$			268225.i			0.474159i
$201$	651814.	+	129403.i	1.13798	+	0.225919i
$202$	−99621.4			−0.171781
$203$	118370.			0.201605
$204$	14858.7	−	74844.8i	0.0249981	−	0.125918i
$205$			206374.i			0.342981i
$206$	−17595.6			−0.0288892
$207$	525739.	−	321960.i	0.852795	−	0.522246i
$208$	15867.2			0.0254298
$209$			818847.i			1.29669i
$210$	−45954.2	+	231476.i	−0.0719080	+	0.362207i
$211$	−722950.			−1.11790			−0.558949	−	0.829202i	$-0.688795\pi$
$211$	−722950.			−1.11790			−0.558949	+	0.829202i	$0.688795\pi$
$212$	−96867.7			−0.148026
$213$	655708.	+	130176.i	0.990288	+	0.196599i
$214$		−	188812.i		−	0.281836i
$215$			226734.i			0.334518i
$216$	591618.	+	394356.i	0.862793	+	0.575113i
$217$			265489.i			0.382734i
$218$	−1.23238e6			−1.75632
$219$	31833.7	−	160349.i	0.0448515	−	0.225921i
$220$	78194.4			0.108923
$221$	−31533.4			−0.0434300
$222$	−151797.	+	764615.i	−0.206719	+	1.04126i
$223$	−868784.			−1.16990			−0.584951	−	0.811068i	$-0.698886\pi$
$223$	−868784.			−1.16990			−0.584951	+	0.811068i	$0.698886\pi$
$224$	64708.9			0.0861676
$225$	320915.	+	132649.i	0.422605	+	0.174682i
$226$		−	159748.i		−	0.208048i
$227$	783005.			1.00856			0.504278	−	0.863541i	$-0.331759\pi$
$227$	783005.			1.00856			0.504278	+	0.863541i	$0.331759\pi$
$228$	−46006.3	−	9133.51i	−0.0586112	−	0.0116359i
$229$		−	375685.i		−	0.473408i	−0.971582	−	0.236704i	$-0.923933\pi$
$229$		−	375685.i		−	0.473408i	0.971582	−	0.236704i	$-0.0760672\pi$
$230$	−457645.	−	333224.i	−0.570439	−	0.415353i
$231$	−745685.	−	148039.i	−0.919444	−	0.182535i
$232$	−327481.			−0.399454
$233$		−	1.40334e6i		−	1.69345i	−0.532028	−	0.846727i	$-0.678570\pi$
$233$		−	1.40334e6i		−	1.69345i	0.532028	−	0.846727i	$-0.321430\pi$
$234$	8558.31	−	20705.0i	0.0102176	−	0.0247193i
$235$			845008.i			0.998139i
$236$		−	79421.6i		−	0.0928237i
$237$	−567449.	−	112654.i	−0.656230	−	0.130279i
$238$	681244.			0.779579
$239$			262619.i			0.297394i	0.988883	+	0.148697i	$0.0475078\pi$
$239$			262619.i			0.297394i	−0.988883	+	0.148697i	$0.952492\pi$
$240$	116570.	−	587176.i	0.130635	−	0.658022i
$241$			54510.1i			0.0604552i	0.999543	+	0.0302276i	$0.00962322\pi$
$241$			54510.1i			0.0604552i	−0.999543	+	0.0302276i	$0.990377\pi$
$242$		−	1.92716e6i		−	2.11534i
$243$	764404.	−	512810.i	0.830439	−	0.557110i
$244$		−	80283.9i		−	0.0863284i
$245$	−502593.			−0.534935
$246$	−82421.4	+	415164.i	−0.0868365	+	0.437404i
$247$			19383.2i			0.0202155i
$248$		−	734500.i		−	0.758337i
$249$	199630.	−	1.00556e6i	0.204046	−	1.02780i
$250$		−	1.01619e6i		−	1.02831i
$251$	−720999.			−0.722354			−0.361177	−	0.932497i	$-0.617625\pi$
$251$	−720999.			−0.722354			−0.361177	+	0.932497i	$0.617625\pi$
$252$	16634.9	−	40244.6i	0.0165013	−	0.0399214i
$253$	1.07346e6	−	1.47428e6i	1.05435	−	1.44803i
$254$		−	1.18281e6i		−	1.15035i
$255$	−231664.	+	1.16691e6i	−0.223104	+	1.12380i
$256$	−257843.			−0.245898
$257$			728645.i			0.688150i	0.938942	+	0.344075i	$0.111807\pi$
$257$			728645.i			0.688150i	−0.938942	+	0.344075i	$0.888193\pi$
$258$	−90552.7	+	456122.i	−0.0846939	+	0.426611i
$259$	626158.			0.580008
$260$	1850.97			0.00169811
$261$	−161954.	+	391812.i	−0.147160	+	0.356022i
$262$	1.98828e6			1.78947
$263$	−932289.			−0.831116			−0.415558	−	0.909567i	$-0.636414\pi$
$263$	−932289.			−0.831116			−0.415558	+	0.909567i	$0.636414\pi$
$264$	2.06301e6	+	409563.i	1.82176	+	0.361669i
$265$	1.51027e6			1.32112
$266$		−	418754.i		−	0.362873i
$267$	−327585.	+	1.65008e6i	−0.281220	+	1.41653i
$268$			112603.i			0.0957663i
$269$			1.71866e6i			1.44814i	0.689727	+	0.724070i	$0.257731\pi$
$269$			1.71866e6i			1.44814i	−0.689727	+	0.724070i	$0.742269\pi$
$270$	−703327.	−	468818.i	−0.587148	−	0.391376i
$271$	−254210.			−0.210266			−0.105133	−	0.994458i	$-0.533527\pi$
$271$	−254210.			−0.210266			−0.105133	+	0.994458i	$0.533527\pi$
$272$	−1.72809e6			−1.41626
$273$	−17651.4	−	3504.28i	−0.0143342	−	0.00284572i
$274$		−	687759.i		−	0.553426i
$275$	1.02722e6			0.819093
$276$	70857.7	+	76755.9i	0.0559905	+	0.0606512i
$277$	−1.61758e6			−1.26668			−0.633341	−	0.773873i	$-0.718317\pi$
$277$	−1.61758e6			−1.26668			−0.633341	+	0.773873i	$0.718317\pi$
$278$		−	796607.i		−	0.618204i
$279$	−878786.	−	363242.i	−0.675885	−	0.279374i
$280$	−524435.			−0.399758
$281$	2.17180e6			1.64080			0.820398	−	0.571793i	$-0.193752\pi$
$281$	2.17180e6			1.64080			0.820398	+	0.571793i	$0.193752\pi$
$282$	−337478.	+	1.69991e6i	−0.252710	+	1.27293i
$283$		−	836177.i		−	0.620629i	−0.950634	−	0.310314i	$-0.899566\pi$
$283$		−	836177.i		−	0.620629i	0.950634	−	0.310314i	$-0.100434\pi$
$284$			113276.i			0.0833375i
$285$	717289.	+	142401.i	0.523097	+	0.103849i
$286$		−	66274.9i		−	0.0479109i
$287$	339986.			0.243644
$288$	−88534.9	+	214191.i	−0.0628975	+	0.152167i
$289$	2.01442e6			1.41875
$290$	389316.			0.271836
$291$	1.05850e6	+	210141.i	0.732755	+	0.145472i
$292$	27700.8			0.0190124
$293$	−762605.			−0.518956			−0.259478	−	0.965749i	$-0.583550\pi$
$293$	−762605.			−0.518956			−0.259478	+	0.965749i	$0.583550\pi$
$294$	−1.01107e6	−	200725.i	−0.682202	−	0.135436i
$295$			1.23827e6i			0.828438i
$296$	−1.73233e6			−1.14921
$297$	1.51027e6	−	2.26572e6i	0.993488	−	1.49045i
$298$		−	1.05085e6i		−	0.685487i
$299$	25410.3	−	34898.1i	0.0164374	−	0.0225748i
$300$	−11457.8	+	57713.8i	−0.00735015	+	0.0370234i
$301$	373527.			0.237632
$302$			121073.i			0.0763887i
$303$	−281121.	−	55810.2i	−0.175909	−	0.0349226i
$304$			1.06224e6i			0.659231i
$305$			1.25171e6i			0.770469i
$306$	−932080.	+	2.25497e6i	−0.569049	+	1.37669i
$307$	−2.20009e6			−1.33228			−0.666139	−	0.745827i	$-0.732054\pi$
$307$	−2.20009e6			−1.33228			−0.666139	+	0.745827i	$0.732054\pi$
$308$		−	128819.i		−	0.0773757i
$309$	−49653.0	−	9857.47i	−0.0295835	−	0.00587312i
$310$			873188.i			0.516064i
$311$			1.36064e6i			0.797702i	0.917016	+	0.398851i	$0.130591\pi$
$311$			1.36064e6i			0.797702i	−0.917016	+	0.398851i	$0.869409\pi$
$312$	48834.3	+	9694.93i	0.0284013	+	0.00563843i
$313$		−	173231.i		−	0.0999460i	−0.998751	−	0.0499730i	$-0.984086\pi$
$313$		−	173231.i		−	0.0999460i	0.998751	−	0.0499730i	$-0.0159135\pi$
$314$	−754181.			−0.431669
$315$	−259356.	+	627456.i	−0.147272	+	0.356293i
$316$		−	98028.7i		−	0.0552250i
$317$		−	2.83762e6i		−	1.58601i	−0.609214	−	0.793006i	$-0.708515\pi$
$317$		−	2.83762e6i		−	1.58601i	0.609214	−	0.793006i	$-0.291485\pi$
$318$	3.03823e6	+	603171.i	1.68482	+	0.334482i
$319$			1.25416e6i			0.690042i
$320$	1.44171e6			0.787049
$321$	105777.	−	532809.i	0.0572966	−	0.288609i
$322$	−548962.	+	753936.i	−0.295055	+	0.405224i
$323$		−	2.11102e6i		−	1.12586i
$324$	110452.	+	110125.i	0.0584538	+	0.0582808i
$325$	24315.8			0.0127697
$326$			693844.i			0.361591i
$327$	−3.47765e6	−	690408.i	−1.79852	−	0.357056i
$328$	−940603.			−0.482750
$329$	1.39209e6			0.709050
$330$	−2.45255e6	−	486897.i	−1.23974	−	0.246123i
$331$	−22793.4			−0.0114351			−0.00571755	−	0.999984i	$-0.501820\pi$
$331$	−22793.4			−0.0114351			−0.00571755	+	0.999984i	$0.501820\pi$
$332$	173713.			0.0864942
$333$	−856711.	+	2.07263e6i	−0.423373	+	1.02426i
$334$	−993550.			−0.487331
$335$		−	1.75560e6i		−	0.854700i
$336$	−967329.	−	192041.i	−0.467440	−	0.0927995i
$337$		−	399386.i		−	0.191566i	−0.995402	−	0.0957829i	$-0.969465\pi$
$337$		−	399386.i		−	0.191566i	0.995402	−	0.0957829i	$-0.0305355\pi$
$338$			2.01023e6i			0.957092i
$339$	89494.5	−	450792.i	0.0422958	−	0.213048i
$340$	−201588.			−0.0945729
$341$	−2.81292e6			−1.31000
$342$	1.38611e6	+	572940.i	0.640812	+	0.264877i
$343$			1.96825e6i			0.903328i
$344$	−1.03340e6			−0.470838
$345$	−1.10475e6	−	1.19671e6i	−0.499707	−	0.541303i
$346$	−1.86147e6			−0.835921
$347$			1.58642e6i			0.707287i	0.935380	+	0.353644i	$0.115057\pi$
$347$			1.58642e6i			0.707287i	−0.935380	+	0.353644i	$0.884943\pi$
$348$	−70464.0	−	13989.0i	−0.0311903	−	0.00619211i
$349$	−932063.			−0.409620			−0.204810	−	0.978802i	$-0.565658\pi$
$349$	−932063.			−0.409620			−0.204810	+	0.978802i	$0.565658\pi$
$350$	−525316.			−0.229219
$351$	35750.1	−	53632.8i	0.0154885	−	0.0232361i
$352$			685608.i			0.294930i
$353$			1.30567e6i			0.557696i	0.960335	+	0.278848i	$0.0899526\pi$
$353$			1.30567e6i			0.557696i	−0.960335	+	0.278848i	$0.910047\pi$
$354$	−494539.	+	2.49104e6i	−0.209745	+	1.05651i
$355$		−	1.76609e6i		−	0.743776i
$356$	−285056.			−0.119208
$357$	1.92240e6	+	381649.i	0.798314	+	0.158487i
$358$	1.00501e6			0.414441
$359$	2.20403e6			0.902569			0.451285	−	0.892380i	$-0.350966\pi$
$359$	2.20403e6			0.902569			0.451285	+	0.892380i	$0.350966\pi$
$360$	717534.	−	1.73592e6i	0.291801	−	0.705949i
$361$	1.17848e6			0.475942
$362$	3.61527e6			1.45000
$363$	1.07964e6	−	5.43825e6i	0.430044	−	2.16617i
$364$		−	3049.33i		−	0.00120629i
$365$	−431886.			−0.169683
$366$	−499908.	+	2.51808e6i	−0.195068	+	0.982578i
$367$			3.53355e6i			1.36945i	0.728802	+	0.684725i	$0.240078\pi$
$367$			3.53355e6i			1.36945i	−0.728802	+	0.684725i	$0.759922\pi$
$368$	1.39253e6	−	1.91248e6i	0.536026	−	0.736170i
$369$	−465170.	+	1.12538e6i	−0.177847	+	0.430261i
$370$	2.05942e6			0.782062
$371$		−	2.48806e6i		−	0.938483i
$372$	31375.6	−	158042.i	0.0117553	−	0.0592128i
$373$		−	4.42433e6i		−	1.64655i	−0.567642	−	0.823275i	$-0.692144\pi$
$373$		−	4.42433e6i		−	1.64655i	0.567642	−	0.823275i	$-0.307856\pi$
$374$			7.21796e6i			2.66830i
$375$	569291.	−	2.86757e6i	0.209053	−	1.05302i
$376$	−3.85134e6			−1.40489
$377$			29687.6i			0.0107578i
$378$	−772342.	+	1.15868e6i	−0.278023	+	0.417094i
$379$		−	4.48833e6i		−	1.60504i	−0.596623	−	0.802522i	$-0.703491\pi$
$379$		−	4.48833e6i		−	1.60504i	0.596623	−	0.802522i	$-0.296509\pi$
$380$			123914.i			0.0440211i
$381$	662635.	−	3.33776e6i	0.233863	−	1.17799i
$382$		−	1.51997e6i		−	0.532938i
$383$	−781954.			−0.272386			−0.136193	−	0.990682i	$-0.543487\pi$
$383$	−781954.			−0.272386			−0.136193	+	0.990682i	$0.543487\pi$
$384$	2.43363e6	+	483141.i	0.842221	+	0.167204i
$385$			2.00844e6i			0.690567i
$386$		−	3.04073e6i		−	1.03875i
$387$	−511061.	+	1.23640e6i	−0.173458	+	0.419645i
$388$			182859.i			0.0616648i
$389$	−3.16228e6			−1.05956			−0.529780	−	0.848135i	$-0.677726\pi$
$389$	−3.16228e6			−1.05956			−0.529780	+	0.848135i	$0.677726\pi$
$390$	−58055.1	−	11525.5i	−0.0193277	−	0.00383706i
$391$	−2.76742e6	+	3.80073e6i	−0.915447	+	1.25726i
$392$		−	2.29070e6i		−	0.752926i
$393$	5.61071e6	+	1.11388e6i	1.83247	+	0.363795i
$394$	604660.			0.196232
$395$			1.52837e6i			0.492875i
$396$	426402.	+	176251.i	0.136641	+	0.0564799i
$397$	2.47813e6			0.789129			0.394564	−	0.918868i	$-0.370896\pi$
$397$	2.47813e6			0.789129			0.394564	+	0.918868i	$0.370896\pi$
$398$	328729.			0.104023
$399$	234596.	−	1.18168e6i	0.0737713	−	0.371593i
$400$	1.33255e6			0.416422
$401$	993255.			0.308461			0.154230	−	0.988035i	$-0.450710\pi$
$401$	993255.			0.308461			0.154230	+	0.988035i	$0.450710\pi$
$402$	701150.	−	3.53176e6i	0.216394	−	1.09000i
$403$	−66585.7			−0.0204230
$404$		−	48564.6i		−	0.0148036i
$405$	−1.72207e6	−	1.71698e6i	−0.521692	−	0.520148i
$406$		−	641369.i		−	0.193105i
$407$			6.63430e6i			1.98522i
$408$	−5.31851e6	−	1.05587e6i	−1.58176	−	0.314021i
$409$	69730.4			0.0206117			0.0103059	−	0.999947i	$-0.496719\pi$
$409$	69730.4			0.0206117			0.0103059	+	0.999947i	$0.496719\pi$
$410$	1.11821e6			0.328521
$411$	385298.	−	1.94079e6i	0.112510	−	0.566726i
$412$		−	8577.71i		−	0.00248959i
$413$	2.03996e6			0.588499
$414$	−1.74449e6	−	2.84864e6i	−0.500228	−	0.816840i
$415$	−2.70837e6			−0.771949
$416$			16229.3i			0.00459797i
$417$	446278.	−	2.24794e6i	0.125680	−	0.633061i
$418$	4.43681e6			1.24202
$419$	4.63224e6			1.28901			0.644504	−	0.764601i	$-0.277064\pi$
$419$	4.63224e6			1.28901			0.644504	+	0.764601i	$0.277064\pi$
$420$	−112843.	−	22402.3i	−0.0312140	−	0.00619683i
$421$			355118.i			0.0976490i	0.998807	+	0.0488245i	$0.0155475\pi$
$421$			355118.i			0.0976490i	−0.998807	+	0.0488245i	$0.984453\pi$
$422$			3.91720e6i			1.07077i
$423$	−1.90466e6	+	4.60791e6i	−0.517566	+	1.25214i
$424$			6.88347e6i			1.85948i
$425$	−2.64822e6			−0.711183
$426$	705339.	−	3.55286e6i	0.188310	−	0.948537i
$427$	2.06210e6			0.547319
$428$	92044.5			0.0242878
$429$	37128.8	−	187021.i	0.00974019	−	0.0490623i
$430$	1.22852e6			0.320415
$431$	−1.55734e6			−0.403823			−0.201912	−	0.979404i	$-0.564715\pi$
$431$	−1.55734e6			−0.403823			−0.201912	+	0.979404i	$0.564715\pi$
$432$	1.95917e6	−	2.93918e6i	0.505084	−	0.757734i
$433$		−	5.82917e6i		−	1.49412i	−0.664754	−	0.747062i	$-0.731464\pi$
$433$		−	5.82917e6i		−	1.49412i	0.664754	−	0.747062i	$-0.268536\pi$
$434$	1.43851e6			0.366597
$435$	1.09861e6	+	218104.i	0.278369	+	0.0552637i
$436$		−	600775.i		−	0.151355i
$437$	2.33627e6	+	1.70110e6i	0.585221	+	0.426116i
$438$	−868830.	−	172486.i	−0.216396	−	0.0429605i
$439$	2.18077e6			0.540067			0.270034	−	0.962851i	$-0.412965\pi$
$439$	2.18077e6			0.540067			0.270034	+	0.962851i	$0.412965\pi$
$440$		−	5.55653e6i		−	1.36827i
$441$	−2.74069e6	−	1.13285e6i	−0.671063	−	0.277381i
$442$			170859.i			0.0415989i
$443$			3.48490e6i			0.843687i	0.906669	+	0.421844i	$0.138617\pi$
$443$			3.48490e6i			0.843687i	−0.906669	+	0.421844i	$0.861383\pi$
$444$	−372744.	−	73999.7i	−0.0897332	−	0.0178145i
$445$	4.44434e6			1.06391
$446$			4.70738e6i			1.12058i
$447$	588710.	−	2.96539e6i	0.139358	−	0.701961i
$448$		−	2.37510e6i		−	0.559098i
$449$		−	3.61570e6i		−	0.846401i	−0.906036	−	0.423201i	$-0.860907\pi$
$449$		−	3.61570e6i		−	0.846401i	0.906036	−	0.423201i	$-0.139093\pi$
$450$	718739.	−	1.73883e6i	0.167317	−	0.404787i
$451$			3.60224e6i			0.833933i
$452$	77875.8			0.0179290
$453$	−67827.8	+	341655.i	−0.0155297	+	0.0782244i
$454$		−	4.24260e6i		−	0.966034i
$455$			47542.4i			0.0107660i
$456$	−649031.	+	3.26923e6i	−0.146168	+	0.736264i
$457$		−	4.19748e6i		−	0.940153i	−0.882626	−	0.470076i	$-0.844226\pi$
$457$		−	4.19748e6i		−	0.940153i	0.882626	−	0.470076i	$-0.155774\pi$
$458$	−2.03560e6			−0.453449
$459$	−3.89352e6	+	5.84112e6i	−0.862603	+	1.29409i
$460$	162444.	−	223098.i	0.0357939	−	0.0491588i
$461$		−	8.43215e6i		−	1.84793i	−0.382474	−	0.923966i	$-0.624928\pi$
$461$		−	8.43215e6i		−	1.84793i	0.382474	−	0.923966i	$-0.375072\pi$
$462$	−802126.	+	4.04039e6i	−0.174839	+	0.880680i
$463$	4.96492e6			1.07636			0.538182	−	0.842828i	$-0.319111\pi$
$463$	4.96492e6			1.07636			0.538182	+	0.842828i	$0.319111\pi$
$464$			1.62694e6i			0.350813i
$465$	−489180.	+	2.46405e6i	−0.104915	+	0.528466i
$466$	−7.60380e6			−1.62206
$467$	−4.73084e6			−1.00380			−0.501898	−	0.864927i	$-0.667365\pi$
$467$	−4.73084e6			−1.00380			−0.501898	+	0.864927i	$0.667365\pi$
$468$	10093.5	+	4172.11i	0.00213024	+	0.000880523i
$469$	−2.89222e6			−0.607155
$470$	4.57855e6			0.956057
$471$	−2.12822e6	−	422510.i	−0.442043	−	0.0877575i
$472$	−5.64374e6			−1.16604
$473$			3.95762e6i			0.813356i
$474$	−610400.	+	3.07464e6i	−0.124787	+	0.628563i
$475$			1.62783e6i			0.331036i
$476$			332101.i			0.0671820i
$477$	8.23567e6	+	3.40418e6i	1.65731	+	0.685040i
$478$	1.42296e6			0.284855
$479$	7.96225e6			1.58561			0.792807	−	0.609473i	$-0.208619\pi$
$479$	7.96225e6			1.58561			0.792807	+	0.609473i	$0.208619\pi$
$480$	600575.	+	119230.i	0.118977	+	0.0236202i
$481$			157043.i			0.0309497i
$482$	295355.			0.0579064
$483$	−1.97149e6	+	1.81999e6i	−0.384527	+	0.354978i
$484$	939475.			0.182294
$485$		−	2.85098e6i		−	0.550350i
$486$	−2.77859e6	−	4.14182e6i	−0.533622	−	0.795427i
$487$	−7.38263e6			−1.41055			−0.705276	−	0.708933i	$-0.749177\pi$
$487$	−7.38263e6			−1.41055			−0.705276	+	0.708933i	$0.749177\pi$
$488$	−5.70501e6			−1.08444
$489$	−388707.	+	1.95796e6i	−0.0735107	+	0.370281i
$490$			2.72323e6i			0.512382i
$491$			8.19011e6i			1.53315i	0.642152	+	0.766577i	$0.278042\pi$
$491$			8.19011e6i			1.53315i	−0.642152	+	0.766577i	$0.721958\pi$
$492$	−202389.	−	40179.8i	−0.0376942	−	0.00748332i
$493$		−	3.23326e6i		−	0.599133i
$494$	105025.			0.0193632
$495$	−6.64807e6	−	2.74795e6i	−1.21950	−	0.504075i
$496$	−3.64902e6			−0.665997
$497$	−2.90950e6			−0.528357
$498$	−5.44846e6	−	1.08167e6i	−0.984466	−	0.195443i
$499$	4.71417e6			0.847527			0.423764	−	0.905773i	$-0.360709\pi$
$499$	4.71417e6			0.847527			0.423764	+	0.905773i	$0.360709\pi$
$500$	495382.			0.0886167
$501$	−2.80370e6	−	556610.i	−0.499042	−	0.0990733i
$502$			3.90663e6i			0.691899i
$503$	−4.20494e6			−0.741037			−0.370518	−	0.928825i	$-0.620820\pi$
$503$	−4.20494e6			−0.741037			−0.370518	+	0.928825i	$0.620820\pi$
$504$	−2.85980e6	−	1.18208e6i	−0.501486	−	0.207287i
$505$			757175.i			0.132120i
$506$	−7.98815e6	−	5.81640e6i	−1.38698	−	1.00990i
$507$	−1.12618e6	+	5.67266e6i	−0.194575	+	0.980093i
$508$	576608.			0.0991338
$509$		−	1.13142e7i		−	1.93567i	−0.251595	−	0.967833i	$-0.580955\pi$
$509$		−	1.13142e7i		−	1.93567i	0.251595	−	0.967833i	$-0.419045\pi$
$510$	6.32275e6	+	1.25524e6i	1.07642	+	0.213698i
$511$			711501.i			0.120538i
$512$			6.49034e6i			1.09419i
$513$	3.59047e6	+	2.39331e6i	0.602363	+	0.401518i
$514$	3.94806e6			0.659137
$515$			133736.i			0.0222193i
$516$	−222356.	−	44143.7i	−0.0367641	−	0.00729868i
$517$			1.47495e7i			2.42690i
$518$		−	3.39275e6i		−	0.555555i
$519$	−5.25287e6	−	1.04284e6i	−0.856010	−	0.169941i
$520$		−	131531.i		−	0.0213314i
$521$	6.03040e6			0.973311			0.486656	−	0.873594i	$-0.338217\pi$
$521$	6.03040e6			0.973311			0.486656	+	0.873594i	$0.338217\pi$
$522$	2.12298e6	+	877523.i	0.341012	+	0.140956i
$523$		−	7.18045e6i		−	1.14788i	−0.818897	−	0.573941i	$-0.805414\pi$
$523$		−	7.18045e6i		−	1.14788i	0.818897	−	0.573941i	$-0.194586\pi$
$524$			969268.i			0.154211i
$525$	−1.48239e6	−	294294.i	−0.234727	−	0.0465998i
$526$			5.05148e6i			0.796075i
$527$	7.25181e6			1.13742
$528$	2.03473e6	−	1.02491e7i	0.317630	−	1.59993i
$529$	−1.97624e6	−	6.12544e6i	−0.307044	−	0.951695i
$530$		−	8.18320e6i		−	1.26542i
$531$	−2.79108e6	+	6.75241e6i	−0.429571	+	1.03926i
$532$	204139.			0.0312714
$533$			85270.0i			0.0130010i
$534$	8.94071e6	+	1.77497e6i	1.35681	+	0.269363i
$535$	−1.43507e6			−0.216765
$536$	8.00162e6			1.20300
$537$	2.83604e6	+	563030.i	0.424401	+	0.0842551i
$538$	9.31234e6			1.38708
$539$	−8.77271e6			−1.30065
$540$	228545.	−	342866.i	0.0337277	−	0.0505988i
$541$	−5.52819e6			−0.812063			−0.406032	−	0.913859i	$-0.633088\pi$
$541$	−5.52819e6			−0.812063			−0.406032	+	0.913859i	$0.633088\pi$
$542$			1.37740e6i			0.201401i
$543$	1.02019e7	+	2.02536e6i	1.48485	+	0.294783i
$544$		−	1.76752e6i		−	0.256075i
$545$			9.36673e6i			1.35082i
$546$	−18987.4	+	95641.5i	−0.00272574	+	0.0137298i
$547$	−9.72255e6			−1.38935			−0.694675	−	0.719324i	$-0.744452\pi$
$547$	−9.72255e6			−1.38935			−0.694675	+	0.719324i	$0.744452\pi$
$548$	335277.			0.0476927
$549$	−2.82138e6	+	6.82571e6i	−0.399512	+	0.966534i
$550$		−	5.56586e6i		−	0.784559i
$551$	−1.98745e6			−0.278880
$552$	5.45431e6	−	5.03518e6i	0.761890	−	0.703343i
$553$	2.51788e6			0.350125
$554$			8.76465e6i			1.21328i
$555$	5.81148e6	+	1.15374e6i	0.800856	+	0.158992i
$556$	388339.			0.0532751
$557$	−7.29824e6			−0.996736			−0.498368	−	0.866966i	$-0.666067\pi$
$557$	−7.29824e6			−0.996736			−0.498368	+	0.866966i	$0.666067\pi$
$558$	−1.96818e6	+	4.76158e6i	−0.267595	+	0.647389i
$559$			93682.2i			0.0126802i
$560$			2.60541e6i			0.351080i
$561$	−4.04367e6	+	2.03684e7i	−0.542461	+	2.73243i
$562$		−	1.17676e7i		−	1.57162i
$563$	1.13785e7			1.51291			0.756457	−	0.654043i	$-0.226928\pi$
$563$	1.13785e7			1.51291			0.756457	+	0.654043i	$0.226928\pi$
$564$	−828692.	−	164518.i	−0.109697	−	0.0217779i
$565$	−1.21417e6			−0.160014
$566$	−4.53070e6			−0.594463
$567$	−2.82859e6	+	2.83699e6i	−0.369498	+	0.370595i
$568$	8.04942e6			1.04687
$569$	−1.31430e7			−1.70182			−0.850909	−	0.525313i	$-0.823948\pi$
$569$	−1.31430e7			−1.70182			−0.850909	+	0.525313i	$0.823948\pi$
$570$	771581.	−	3.88653e6i	0.0994705	−	0.501043i
$571$			7.25737e6i			0.931513i	0.884913	+	0.465757i	$0.154218\pi$
$571$			7.25737e6i			0.931513i	−0.884913	+	0.465757i	$0.845782\pi$
$572$	32308.5			0.00412883
$573$	851523.	−	4.28920e6i	0.108345	−	0.545745i
$574$		−	1.84216e6i		−	0.233372i
$575$	2.13399e6	−	2.93080e6i	0.269168	−	0.369671i
$576$	7.86177e6	+	3.24962e6i	0.987334	+	0.408110i
$577$	−6.61354e6			−0.826979			−0.413490	−	0.910509i	$-0.635690\pi$
$577$	−6.61354e6			−0.826979			−0.413490	+	0.910509i	$0.635690\pi$
$578$		−	1.09149e7i		−	1.35893i
$579$	1.70349e6	−	8.58064e6i	0.211175	−	1.06371i
$580$			189788.i			0.0234261i
$581$			4.46185e6i			0.548371i
$582$	1.13862e6	−	5.73533e6i	0.139338	−	0.701861i
$583$	2.63617e7			3.21219
$584$		−	1.96844e6i		−	0.238830i
$585$	−157369.	−	65047.7i	−0.0190121	−	0.00785855i
$586$			4.13206e6i			0.497076i
$587$		−	1.11235e7i		−	1.33244i	−0.745756	−	0.666219i	$-0.767912\pi$
$587$		−	1.11235e7i		−	1.33244i	0.745756	−	0.666219i	$-0.232088\pi$
$588$	97851.7	−	492888.i	0.0116715	−	0.0587903i
$589$		−	4.45761e6i		−	0.529437i
$590$	6.70939e6			0.793511
$591$	1.70629e6	+	338744.i	0.200948	+	0.0398936i
$592$			8.60626e6i			1.00928i
$593$		−	1.11287e7i		−	1.29959i	−0.760110	−	0.649795i	$-0.774855\pi$
$593$		−	1.11287e7i		−	1.29959i	0.760110	−	0.649795i	$-0.225145\pi$
$594$	−1.22765e7	−	8.18317e6i	−1.42761	−	0.951602i
$595$		−	5.17781e6i		−	0.599590i
$596$	512280.			0.0590734
$597$	927640.	+	184162.i	0.106523	+	0.0211477i
$598$	−189091.	−	137682.i	−0.0216231	−	0.0157443i
$599$		−	9.52232e6i		−	1.08437i	−0.840261	−	0.542183i	$-0.817598\pi$
$599$		−	9.52232e6i		−	1.08437i	0.840261	−	0.542183i	$-0.182402\pi$
$600$	4.10117e6	+	814194.i	0.465082	+	0.0923314i
$601$	6.82958e6			0.771272			0.385636	−	0.922651i	$-0.373982\pi$
$601$	6.82958e6			0.771272			0.385636	+	0.922651i	$0.373982\pi$
$602$		−	2.02390e6i		−	0.227614i
$603$	3.95715e6	−	9.57347e6i	0.443189	−	1.07220i
$604$	−59022.0			−0.00658296
$605$	−1.46474e7			−1.62695
$606$	−302400.	+	1.52322e6i	−0.0334503	+	0.168492i
$607$	−1.59142e7			−1.75312			−0.876562	−	0.481290i	$-0.840169\pi$
$607$	−1.59142e7			−1.75312			−0.876562	+	0.481290i	$0.840169\pi$
$608$	−1.08648e6			−0.119196
$609$	359310.	−	1.80988e6i	0.0392578	−	0.197745i
$610$	6.78223e6			0.737985
$611$			349142.i			0.0378354i
$612$	−1.09928e6	−	454382.i	−0.118639	−	0.0490391i
$613$		−	508854.i		−	0.0546943i	−0.999626	−	0.0273471i	$-0.991294\pi$
$613$		−	508854.i		−	0.0546943i	0.999626	−	0.0273471i	$-0.00870595\pi$
$614$			1.19209e7i			1.27611i
$615$	3.15547e6	+	626446.i	0.336416	+	0.0667876i
$616$	−9.15397e6			−0.971981
$617$	−7.01562e6			−0.741913			−0.370957	−	0.928650i	$-0.620970\pi$
$617$	−7.01562e6			−0.741913			−0.370957	+	0.928650i	$0.620970\pi$
$618$	−53411.3	+	269038.i	−0.00562551	+	0.0283362i
$619$			7.74825e6i			0.812788i	0.913698	+	0.406394i	$0.133214\pi$
$619$			7.74825e6i			0.812788i	−0.913698	+	0.406394i	$0.866786\pi$
$620$	−425672.			−0.0444729
$621$	−3.32690e6	−	9.01589e6i	−0.346187	−	0.938165i
$622$	7.37241e6			0.764071
$623$		−	7.32171e6i		−	0.755775i
$624$	48164.8	−	242610.i	0.00495185	−	0.0249430i
$625$	−3.25789e6			−0.333608
$626$	−938629.			−0.0957322
$627$	1.25202e7	+	2.48560e6i	1.27187	+	0.252501i
$628$		−	367657.i		−	0.0372001i
$629$		−	1.71035e7i		−	1.72368i
$630$	3.39978e6	+	1.40528e6i	0.341271	+	0.141063i
$631$			6.56263e6i			0.656152i	0.944651	+	0.328076i	$0.106400\pi$
$631$			6.56263e6i			0.656152i	−0.944651	+	0.328076i	$0.893600\pi$
$632$	−6.96596e6			−0.693727
$633$	−2.19451e6	+	1.10539e7i	−0.217685	+	1.09650i
$634$	−1.53752e7			−1.51914
$635$	−8.98995e6			−0.884755
$636$	−294041.	+	1.48111e6i	−0.0288247	+	0.145193i
$637$	−207662.			−0.0202772
$638$	6.79547e6			0.660949
$639$	3.98079e6	−	9.63067e6i	0.385671	−	0.933048i
$640$		−	6.55476e6i		−	0.632567i
$641$	1.76495e7			1.69663			0.848316	−	0.529490i	$-0.177617\pi$
$641$	1.76495e7			1.69663			0.848316	+	0.529490i	$0.177617\pi$
$642$	−2.88695e6	−	573138.i	−0.276441	−	0.0548809i
$643$		−	1.16866e7i		−	1.11470i	−0.830277	−	0.557351i	$-0.811818\pi$
$643$		−	1.16866e7i		−	1.11470i	0.830277	−	0.557351i	$-0.188182\pi$
$644$	−367538.	−	267615.i	−0.0349211	−	0.0254270i
$645$	3.46677e6	+	688248.i	0.328115	+	0.0651397i
$646$	−1.14382e7			−1.07839
$647$		−	3.55429e6i		−	0.333805i	−0.985973	−	0.166902i	$-0.946624\pi$
$647$		−	3.55429e6i		−	0.333805i	0.985973	−	0.166902i	$-0.0533764\pi$
$648$	7.82557e6	−	7.84881e6i	0.732113	−	0.734287i
$649$			2.16139e7i			2.01429i
$650$		−	131752.i		−	0.0122313i
$651$	4.05933e6	+	805888.i	0.375407	+	0.0745285i
$652$	−338243.			−0.0311609
$653$			6.10634e6i			0.560400i	0.959942	+	0.280200i	$0.0904007\pi$
$653$			6.10634e6i			0.560400i	−0.959942	+	0.280200i	$0.909599\pi$
$654$	−3.74087e6	+	1.88431e7i	−0.342002	+	1.72270i
$655$		−	1.51119e7i		−	1.37631i
$656$			4.67295e6i			0.423967i
$657$	−2.35512e6	−	973478.i	−0.212863	−	0.0879858i
$658$		−	7.54283e6i		−	0.679156i
$659$	−1.03682e7			−0.930015			−0.465008	−	0.885307i	$-0.653948\pi$
$659$	−1.03682e7			−0.930015			−0.465008	+	0.885307i	$0.653948\pi$
$660$	237358.	−	1.19560e6i	0.0212102	−	0.106838i
$661$			1.12713e7i			1.00339i	0.865043	+	0.501697i	$0.167291\pi$
$661$			1.12713e7i			1.00339i	−0.865043	+	0.501697i	$0.832709\pi$
$662$			123503.i			0.0109530i
$663$	−95719.2	+	482147.i	−0.00845698	+	0.0425986i
$664$		−	1.23441e7i		−	1.08653i
$665$	−3.18275e6			−0.279093
$666$	1.12302e7	+	4.64196e6i	0.981077	+	0.405524i
$667$	3.57827e6	+	2.60544e6i	0.311428	+	0.226760i
$668$		−	484348.i		−	0.0419968i
$669$	−2.63719e6	+	1.32838e7i	−0.227811	+	1.14751i
$670$	−9.51248e6			−0.818666
$671$			2.18485e7i			1.87334i
$672$	196423.	−	989403.i	0.0167791	−	0.0845181i
$673$	1.22876e7			1.04576			0.522878	−	0.852407i	$-0.324858\pi$
$673$	1.22876e7			1.04576			0.522878	+	0.852407i	$0.324858\pi$
$674$	−2.16402e6			−0.183489
$675$	3.00234e6	−	4.50416e6i	0.253630	−	0.380500i
$676$	−979970.			−0.0824795
$677$	1.93499e7			1.62259			0.811293	−	0.584639i	$-0.198764\pi$
$677$	1.93499e7			1.62259			0.811293	+	0.584639i	$0.198764\pi$
$678$	−2.44255e6	−	484913.i	−0.204066	−	0.0405126i
$679$	−4.69677e6			−0.390953
$680$			1.43249e7i			1.18801i
$681$	2.37680e6	−	1.19722e7i	0.196393	−	0.989249i
$682$			1.52414e7i			1.25477i
$683$			1.67467e7i			1.37365i	0.726822	+	0.686826i	$0.240996\pi$
$683$			1.67467e7i			1.37365i	−0.726822	+	0.686826i	$0.759004\pi$
$684$	−279304.	+	675715.i	−0.0228263	+	0.0552234i
$685$	−5.22733e6			−0.425651
$686$	1.06647e7			0.865243
$687$	−5.74425e6	−	1.14039e6i	−0.464346	−	0.0921852i
$688$			5.13396e6i			0.413506i
$689$	624017.			0.0500782
$690$	−6.48419e6	+	5.98592e6i	−0.518481	+	0.478639i
$691$	5.86496e6			0.467272			0.233636	−	0.972324i	$-0.424938\pi$
$691$	5.86496e6			0.467272			0.233636	+	0.972324i	$0.424938\pi$
$692$		−	907451.i		−	0.0720374i
$693$	−4.52704e6	+	1.09522e7i	−0.358081	+	0.866300i
$694$	8.59582e6			0.677467
$695$	−6.05463e6			−0.475473
$696$	−994066.	+	5.00720e6i	−0.0777843	+	0.391807i
$697$		−	9.28669e6i		−	0.724068i
$698$			5.05025e6i			0.392351i
$699$	−2.14572e7	−	4.25983e6i	−1.66104	−	0.329761i
$700$		−	256087.i		−	0.0197534i
$701$	5.62134e6			0.432061			0.216030	−	0.976387i	$-0.430689\pi$
$701$	5.62134e6			0.432061			0.216030	+	0.976387i	$0.430689\pi$
$702$	−290602.	−	193707.i	−0.0222564	−	0.0148355i
$703$	−1.05133e7			−0.802328
$704$	2.51649e7			1.91365
$705$	1.29202e7	+	2.56501e6i	0.979032	+	0.194364i
$706$	7.07460e6			0.534183
$707$	1.24739e6			0.0938542
$708$	−1.21436e6	−	241084.i	−0.0910468	−	0.0180753i
$709$		−	2.00990e7i		−	1.50162i	−0.660520	−	0.750809i	$-0.729664\pi$
$709$		−	2.00990e7i		−	1.50162i	0.660520	−	0.750809i	$-0.270336\pi$
$710$	−9.56931e6			−0.712418
$711$	−3.44497e6	+	8.33437e6i	−0.255571	+	0.618300i
$712$			2.02562e7i			1.49747i
$713$	−5.84367e6	+	8.02561e6i	−0.430489	+	0.591227i
$714$	2.06791e6	−	1.04163e7i	0.151805	−	0.764656i
$715$	−503725.			−0.0368492
$716$			489934.i			0.0357154i
$717$	4.01546e6	+	797178.i	0.291701	+	0.0579105i
$718$		−	1.19422e7i		−	0.864516i
$719$		−	2.86471e6i		−	0.206661i	−0.994647	−	0.103330i	$-0.967050\pi$
$719$		−	2.86471e6i		−	0.206661i	0.994647	−	0.103330i	$-0.0329499\pi$
$720$	−8.62411e6	−	3.56473e6i	−0.619988	−	0.256269i
$721$	220320.			0.0157839
$722$		−	6.38542e6i		−	0.455876i
$723$	833462.	+	165465.i	0.0592980	+	0.0117723i
$724$			1.76242e6i			0.124957i
$725$			2.49321e6i			0.176163i
$726$	−2.94664e7	−	5.84988e6i	−2.07484	−	0.411913i
$727$			1.23443e7i			0.866224i	0.901340	+	0.433112i	$0.142585\pi$
$727$			1.23443e7i			0.866224i	−0.901340	+	0.433112i	$0.857415\pi$
$728$	−216687.			−0.0151532
$729$	−5.52056e6	−	1.32444e7i	−0.384737	−	0.923026i
$730$			2.34011e6i			0.162529i
$731$		−	1.02029e7i		−	0.706202i
$732$	−1.22754e6	−	243701.i	−0.0846759	−	0.0168104i
$733$			2.73775e7i			1.88206i	0.338322	+	0.941030i	$0.390141\pi$
$733$			2.73775e7i			1.88206i	−0.338322	+	0.941030i	$0.609859\pi$
$734$	1.91460e7			1.31171
$735$	−1.52561e6	+	7.68467e6i	−0.104166	+	0.524695i
$736$	1.95612e6	+	1.42431e6i	0.133107	+	0.0969192i
$737$		−	3.06439e7i		−	2.07814i
$738$	6.09770e6	+	2.52045e6i	0.412121	+	0.170348i
$739$	1.27646e7			0.859794			0.429897	−	0.902878i	$-0.358550\pi$
$739$	1.27646e7			0.859794			0.429897	+	0.902878i	$0.358550\pi$
$740$			1.00395e6i			0.0673959i
$741$	296371.	+	58837.6i	0.0198285	+	0.00393649i
$742$	−1.34812e7			−0.898916
$743$	1.30516e7			0.867346			0.433673	−	0.901070i	$-0.357217\pi$
$743$	1.30516e7			0.867346			0.433673	+	0.901070i	$0.357217\pi$
$744$	−1.12305e7	−	2.22957e6i	−0.743821	−	0.147669i
$745$	−7.98700e6			−0.527222
$746$	−2.39726e7			−1.57713
$747$	−1.47690e7	−	6.10471e6i	−0.968391	−	0.400280i
$748$	−3.51870e6			−0.229947
$749$			2.36418e6i			0.153984i
$750$	−1.55375e7	−	3.08462e6i	−1.00862	−	0.200239i
$751$		−	1.49278e7i		−	0.965817i	−0.875671	−	0.482909i	$-0.839580\pi$
$751$		−	1.49278e7i		−	0.965817i	0.875671	−	0.482909i	$-0.160420\pi$
$752$			1.91336e7i			1.23382i
$753$	−2.18859e6	+	1.10241e7i	−0.140662	+	0.708527i
$754$	160858.			0.0103042
$755$	920217.			0.0587520
$756$	−564846.	−	376510.i	−0.0359440	−	0.0239592i
$757$		−	8.72424e6i		−	0.553335i	−0.960966	−	0.276667i	$-0.910770\pi$
$757$		−	8.72424e6i		−	0.553335i	0.960966	−	0.276667i	$-0.0892300\pi$
$758$	−2.43194e7			−1.53737
$759$	−1.92833e7	−	2.08884e7i	−1.21500	−	1.31614i
$760$	8.80538e6			0.552986
$761$		−	849979.i		−	0.0532043i	−0.999646	−	0.0266022i	$-0.991531\pi$
$761$		−	849979.i		−	0.0532043i	0.999646	−	0.0266022i	$-0.00846873\pi$
$762$	−1.80852e7	−	3.59040e6i	−1.12833	−	0.224004i
$763$	1.54310e7			0.959583
$764$	740973.			0.0459271
$765$	1.71389e7	+	7.08430e6i	1.05884	+	0.437667i
$766$			4.23690e6i			0.260902i
$767$			511631.i			0.0314028i
$768$	−782679.	+	3.94243e6i	−0.0478829	+	0.241191i
$769$			1.44206e7i			0.879365i	0.898153	+	0.439682i	$0.144909\pi$
$769$			1.44206e7i			0.879365i	−0.898153	+	0.439682i	$0.855091\pi$
$770$	1.08824e7			0.661452
$771$	1.11410e7	+	2.21179e6i	0.674977	+	0.134001i
$772$	1.48233e6			0.0895163
$773$	−1.08132e7			−0.650885			−0.325443	−	0.945562i	$-0.605513\pi$
$773$	−1.08132e7			−0.650885			−0.325443	+	0.945562i	$0.605513\pi$
$774$	6.69927e6	+	2.76911e6i	0.401953	+	0.166145i
$775$	−5.59196e6			−0.334434
$776$	1.29941e7			0.774624
$777$	1.90070e6	−	9.57398e6i	0.112943	−	0.568906i
$778$			1.71343e7i			1.01489i
$779$	−5.70844e6			−0.337034
$780$	5618.60	−	28301.4i	0.000330667	−	0.00166560i
$781$		−	3.08269e7i		−	1.80844i
$782$	2.05937e7	+	1.49949e7i	1.20425	+	0.876851i
$783$	5.49922e6	+	3.66562e6i	0.320551	+	0.213670i
$784$	−1.13803e7			−0.661245
$785$			5.73217e6i			0.332005i
$786$	6.03539e6	−	3.04008e7i	0.348457	−	1.75521i
$787$		−	1.90163e7i		−	1.09443i	−0.836992	−	0.547215i	$-0.815688\pi$
$787$		−	1.90163e7i		−	1.09443i	0.836992	−	0.547215i	$-0.184312\pi$
$788$			294767.i			0.0169107i
$789$	−2.82996e6	+	1.42548e7i	−0.161840	+	0.815206i
$790$	8.28127e6			0.472095
$791$			2.00025e6i			0.113669i
$792$	1.25245e7	−	3.03003e7i	0.709491	−	1.71646i
$793$			517185.i			0.0292054i
$794$		−	1.34274e7i		−	0.755858i
$795$	4.58442e6	−	2.30922e7i	0.257257	−	1.29583i
$796$			160253.i			0.00896443i
$797$	−2.11079e7			−1.17706			−0.588532	−	0.808474i	$-0.700294\pi$
$797$	−2.11079e7			−1.17706			−0.588532	+	0.808474i	$0.700294\pi$
$798$	−6.40277e6	−	1.27112e6i	−0.355927	−	0.0706611i
$799$		−	3.80248e7i		−	2.10717i
$800$			1.36296e6i			0.0752935i
$801$	2.42354e7	+	1.00176e7i	1.33465	+	0.551673i
$802$		−	5.38181e6i		−	0.295456i
$803$	−7.53854e6			−0.412571
$804$	1.72170e6	+	341805.i	0.0939331	+	0.0186483i
$805$	5.73031e6	+	4.17240e6i	0.311665	+	0.226932i
$806$			360785.i			0.0195619i
$807$	2.62785e7	+	5.21699e6i	1.42042	+	0.281991i
$808$	−3.45102e6			−0.185960
$809$			3.04678e7i			1.63671i	0.574717	+	0.818353i	$0.305112\pi$
$809$			3.04678e7i			1.63671i	−0.574717	+	0.818353i	$0.694888\pi$
$810$	−9.30319e6	+	9.33081e6i	−0.498218	+	0.499697i
$811$	2.63906e7			1.40895			0.704477	−	0.709727i	$-0.251182\pi$
$811$	2.63906e7			1.40895			0.704477	+	0.709727i	$0.251182\pi$
$812$	312662.			0.0166412
$813$	−771652.	+	3.88688e6i	−0.0409444	+	0.206241i
$814$	3.59470e7			1.90152
$815$	5.27358e6			0.278107
$816$	−5.24559e6	+	2.64226e7i	−0.275784	+	1.38915i
$817$	−6.27160e6			−0.328718
$818$		−	377824.i		−	0.0197427i
$819$	−107161.	+	259254.i	−0.00558249	+	0.0135056i
$820$			545117.i			0.0283110i
$821$		−	3.38268e6i		−	0.175147i	−0.996158	−	0.0875735i	$-0.972089\pi$
$821$		−	3.38268e6i		−	0.175147i	0.996158	−	0.0875735i	$-0.0279113\pi$
$822$	−1.05159e7	−	2.08768e6i	−0.542832	−	0.107767i
$823$	−1.39567e7			−0.718263			−0.359131	−	0.933287i	$-0.616927\pi$
$823$	−1.39567e7			−0.718263			−0.359131	+	0.933287i	$0.616927\pi$
$824$	−609536.			−0.0312739
$825$	3.11813e6	−	1.57063e7i	0.159499	−	0.803413i
$826$		−	1.10532e7i		−	0.563688i
$827$	1.63880e7			0.833224			0.416612	−	0.909084i	$-0.363217\pi$
$827$	1.63880e7			0.833224			0.416612	+	0.909084i	$0.363217\pi$
$828$	1.38869e6	−	850426.i	0.0703930	−	0.0431083i
$829$	−1.58295e7			−0.799982			−0.399991	−	0.916519i	$-0.630987\pi$
$829$	−1.58295e7			−0.799982			−0.399991	+	0.916519i	$0.630987\pi$
$830$			1.46749e7i			0.739403i
$831$	−4.91016e6	+	2.47329e7i	−0.246657	+	1.24243i
$832$	595687.			0.0298339
$833$	2.26163e7			1.12930
$834$	−1.21802e7	−	2.41809e6i	−0.606370	−	0.120381i
$835$			7.55151e6i			0.374815i
$836$			2.16291e6i			0.107034i
$837$	−8.22154e6	+	1.23341e7i	−0.405639	+	0.608545i
$838$		−	2.50991e7i		−	1.23466i
$839$	−1.29784e7			−0.636527			−0.318263	−	0.948002i	$-0.603100\pi$
$839$	−1.29784e7			−0.636527			−0.318263	+	0.948002i	$0.603100\pi$
$840$	−1.59192e6	+	8.01864e6i	−0.0778435	+	0.392105i
$841$	1.74671e7			0.851592
$842$	1.92416e6			0.0935320
$843$	6.59248e6	−	3.32070e7i	0.319507	−	1.60939i
$844$	−1.90960e6			−0.0922757
$845$	1.52788e7			0.736118
$846$	2.49673e7	+	1.03201e7i	1.19935	+	0.495745i
$847$			2.41306e7i			1.15574i
$848$	3.41973e7			1.63306
$849$	−1.27852e7	−	2.53821e6i	−0.608748	−	0.120853i
$850$			1.43490e7i			0.681199i
$851$	1.89285e7	+	1.37824e7i	0.895967	+	0.652379i
$852$	1.73199e6	+	343847.i	0.0817422	+	0.0162280i
$853$	962677.			0.0453010			0.0226505	−	0.999743i	$-0.492790\pi$
$853$	962677.			0.0453010			0.0226505	+	0.999743i	$0.492790\pi$
$854$		−	1.11732e7i		−	0.524244i
$855$	4.35465e6	−	1.05351e7i	0.203722	−	0.492861i
$856$		−	6.54072e6i		−	0.305099i
$857$		−	1.42123e7i		−	0.661017i	−0.943803	−	0.330508i	$-0.892780\pi$
$857$		−	1.42123e7i		−	0.661017i	0.943803	−	0.330508i	$-0.107220\pi$
$858$	−1.01335e6	−	201177.i	−0.0469938	−	0.00932953i
$859$	1.32244e7			0.611495			0.305747	−	0.952113i	$-0.401094\pi$
$859$	1.32244e7			0.611495			0.305747	+	0.952113i	$0.401094\pi$
$860$			598896.i			0.0276125i
$861$	1.03202e6	−	5.19840e6i	0.0474440	−	0.238980i
$862$			8.43824e6i			0.386798i
$863$		−	1.09746e7i		−	0.501606i	−0.968038	−	0.250803i	$-0.919305\pi$
$863$		−	1.09746e7i		−	0.501606i	0.968038	−	0.250803i	$-0.0806946\pi$
$864$	3.00625e6	+	2.00388e6i	0.137006	+	0.0913245i
$865$			1.41481e7i			0.642923i
$866$	−3.15845e7			−1.43113
$867$	6.11476e6	−	3.08006e7i	0.276269	−	1.39159i
$868$			701263.i			0.0315923i
$869$			2.66776e7i			1.19839i
$870$	1.18176e6	−	5.95266e6i	0.0529338	−	0.266632i
$871$		−	725383.i		−	0.0323983i
$872$	−4.26913e7			−1.90129
$873$	6.42614e6	−	1.55466e7i	0.285374	−	0.690401i
$874$	9.21719e6	−	1.26588e7i	0.408150	−	0.560548i
$875$			1.27240e7i			0.561827i
$876$	84085.7	−	423548.i	0.00370222	−	0.0186484i
$877$	9.50830e6			0.417449			0.208725	−	0.977974i	$-0.433069\pi$
$877$	9.50830e6			0.417449			0.208725	+	0.977974i	$0.433069\pi$
$878$		−	1.18162e7i		−	0.517297i
$879$	−2.31488e6	+	1.16603e7i	−0.101055	+	0.509022i
$880$	−2.76050e7			−1.20166
$881$	1.11727e7			0.484974			0.242487	−	0.970155i	$-0.422037\pi$
$881$	1.11727e7			0.484974			0.242487	+	0.970155i	$0.422037\pi$
$882$	−6.13819e6	+	1.48500e7i	−0.265686	+	0.642770i
$883$	1.23123e7			0.531420			0.265710	−	0.964053i	$-0.414394\pi$
$883$	1.23123e7			0.531420			0.265710	+	0.964053i	$0.414394\pi$
$884$	−83292.4			−0.00358488
$885$	1.89332e7	+	3.75876e6i	0.812580	+	0.161319i
$886$	1.88825e7			0.808117
$887$			3.96976e6i			0.169416i	0.996406	+	0.0847082i	$0.0269958\pi$
$887$			3.96976e6i			0.169416i	−0.996406	+	0.0847082i	$0.973004\pi$
$888$	−5.25846e6	+	2.64874e7i	−0.223782	+	1.12721i
$889$			1.48103e7i			0.628505i
$890$		−	2.40810e7i		−	1.01906i
$891$	−3.00587e7	−	2.99697e7i	−1.26846	−	1.26470i
$892$	−2.29481e6			−0.0965683
$893$	−2.33735e7			−0.980831
$894$	−1.60675e7	−	3.18984e6i	−0.672365	−	0.133483i
$895$		−	7.63861e6i		−	0.318755i
$896$	−1.07985e7			−0.449358
$897$	−456462.	−	494458.i	−0.0189419	−	0.0205186i
$898$	−1.95911e7			−0.810716
$899$		−	6.82734e6i		−	0.281743i
$900$	847668.	+	350379.i	0.0348834	+	0.0144189i
$901$	−6.79613e7			−2.78901
$902$	1.95182e7			0.798774
$903$	1.13384e6	−	5.71125e6i	0.0462734	−	0.233084i
$904$		−	5.53389e6i		−	0.225221i
$905$		−	2.74780e7i		−	1.11523i
$906$	1.85121e6	+	367515.i	0.0749264	+	0.0148749i
$907$			1.50726e7i			0.608372i	0.952613	+	0.304186i	$0.0983845\pi$
$907$			1.50726e7i			0.608372i	−0.952613	+	0.304186i	$0.901616\pi$
$908$	2.06823e6			0.0832501
$909$	−1.70668e6	+	4.12895e6i	−0.0685083	+	0.165741i
$910$	257602.			0.0103121
$911$	4.25237e6			0.169760			0.0848799	−	0.996391i	$-0.472949\pi$
$911$	4.25237e6			0.169760			0.0848799	+	0.996391i	$0.472949\pi$
$912$	1.62417e7	+	3.22441e6i	0.646612	+	0.128370i
$913$	−4.72744e7			−1.87694
$914$	−2.27435e7			−0.900515
$915$	1.91388e7	+	3.79956e6i	0.755720	+	0.150031i
$916$		−	992337.i		−	0.0390769i
$917$	−2.48958e7			−0.977694
$918$	3.16493e7	+	2.10965e7i	1.23953	+	0.826235i
$919$			3.72731e7i			1.45582i	0.685674	+	0.727909i	$0.259508\pi$
$919$			3.72731e7i			1.45582i	−0.685674	+	0.727909i	$0.740492\pi$
$920$	−1.58535e7	−	1.15434e7i	−0.617525	−	0.449637i
$921$	−6.67836e6	+	3.36395e7i	−0.259430	+	1.30678i
$922$	−4.56884e7			−1.77002
$923$		−	729716.i		−	0.0281936i
$924$	−1.96966e6	−	391030.i	−0.0758945	−	0.0150671i
$925$			1.31887e7i			0.506813i
$926$		−	2.69017e7i		−	1.03098i
$927$	−301442.	+	729275.i	−0.0115214	+	0.0278735i
$928$	−1.66406e6			−0.0634308
$929$		−	4.35184e7i		−	1.65437i	−0.561927	−	0.827187i	$-0.689940\pi$
$929$		−	4.35184e7i		−	1.65437i	0.561927	−	0.827187i	$-0.310060\pi$
$930$	1.33511e7	+	2.65055e6i	0.506185	+	0.100491i
$931$		−	1.39020e7i		−	0.525659i
$932$		−	3.70679e6i		−	0.139784i
$933$	2.08042e7	+	4.13020e6i	0.782432	+	0.155334i
$934$			2.56334e7i			0.961476i
$935$	5.48603e7			2.05225
$936$	296472.	−	717250.i	0.0110610	−	0.0267597i
$937$			3.26678e7i			1.21554i	0.794112	+	0.607772i	$0.207936\pi$
$937$			3.26678e7i			1.21554i	−0.794112	+	0.607772i	$0.792064\pi$
$938$			1.56711e7i			0.581557i
$939$	−2.64872e6	−	525842.i	−0.0980328	−	0.0194622i
$940$			2.23201e6i			0.0823903i
$941$	3.00113e7			1.10487			0.552435	−	0.833556i	$-0.313699\pi$
$941$	3.00113e7			1.10487			0.552435	+	0.833556i	$0.313699\pi$
$942$	−2.28931e6	+	1.15315e7i	−0.0840576	+	0.423406i
$943$	1.02776e7	+	7.48343e6i	0.376369	+	0.274045i
$944$			2.80383e7i			1.02405i
$945$	8.80657e6	+	5.87021e6i	0.320795	+	0.213833i
$946$	2.14438e7			0.779065
$947$			3.32657e7i			1.20537i	0.797978	+	0.602686i	$0.205903\pi$
$947$			3.32657e7i			1.20537i	−0.797978	+	0.602686i	$0.794097\pi$
$948$	−1.49886e6	−	297565.i	−0.0541678	−	0.0107538i
$949$	−178448.			−0.00643199
$950$	8.82017e6			0.317079
$951$	−4.33874e7	−	8.61357e6i	−1.55565	−	0.308839i
$952$	2.35992e7			0.843929
$953$	−4.29067e7			−1.53036			−0.765178	−	0.643818i	$-0.777349\pi$
$953$	−4.29067e7			−1.53036			−0.765178	+	0.643818i	$0.777349\pi$
$954$	1.84450e7	−	4.46238e7i	0.656158	−	1.58743i
$955$	−1.15526e7			−0.409893
$956$			693683.i			0.0245480i
$957$	1.91761e7	+	3.80698e6i	0.676833	+	0.134370i
$958$		−	4.31423e7i		−	1.51876i
$959$			8.61164e6i			0.302370i
$960$	4.37628e6	−	2.20438e7i	0.153260	−	0.771983i
$961$	−1.33163e7			−0.465130
$962$	850916.			0.0296448
$963$	−7.82560e6	−	3.23467e6i	−0.271927	−	0.112400i
$964$			143983.i			0.00499021i
$965$	−2.31112e7			−0.798921
$966$	9.86136e6	+	1.06822e7i	0.340012	+	0.368315i
$967$	−3.23298e7			−1.11183			−0.555913	−	0.831241i	$-0.687631\pi$
$967$	−3.23298e7			−1.11183			−0.555913	+	0.831241i	$0.687631\pi$
$968$		−	6.67595e7i		−	2.28994i
$969$	−3.22775e7	−	6.40797e6i	−1.10431	−	0.219235i
$970$	−1.54476e7			−0.527147
$971$	−5.17661e7			−1.76197			−0.880983	−	0.473148i	$-0.843118\pi$
$971$	−5.17661e7			−1.76197			−0.880983	+	0.473148i	$0.843118\pi$
$972$	2.01910e6	−	1.35454e6i	0.0685476	−	0.0459861i
$973$			9.97456e6i			0.337763i
$974$			4.00017e7i			1.35108i
$975$	73810.3	−	371790.i	0.00248660	−	0.0125252i
$976$			2.83427e7i			0.952394i
$977$	−3.81832e6			−0.127978			−0.0639890	−	0.997951i	$-0.520382\pi$
$977$	−3.81832e6			−0.127978			−0.0639890	+	0.997951i	$0.520382\pi$
$978$	1.06089e7	+	2.10616e6i	0.354669	+	0.0704115i
$979$	7.75755e7			2.58683
$980$	−1.32755e6			−0.0441556
$981$	−2.11127e7	+	5.10777e7i	−0.700442	+	1.69457i
$982$	4.43769e7			1.46852
$983$	1.29294e7			0.426769			0.213385	−	0.976968i	$-0.431551\pi$
$983$	1.29294e7			0.426769			0.213385	+	0.976968i	$0.431551\pi$
$984$	−2.85519e6	+	1.43819e7i	−0.0940043	+	0.473509i
$985$		−	4.59573e6i		−	0.150926i
$986$	−1.75190e7			−0.573874
$987$	4.22567e6	−	2.12851e7i	0.138071	−	0.695477i
$988$			51199.0i			0.00166866i
$989$	1.12916e7	+	8.22171e6i	0.367082	+	0.267283i
$990$	−1.48894e7	+	3.60216e7i	−0.482823	+	1.16809i
$991$	−6.78538e6			−0.219477			−0.109739	−	0.993960i	$-0.535001\pi$
$991$	−6.78538e6			−0.219477			−0.109739	+	0.993960i	$0.535001\pi$
$992$		−	3.73229e6i		−	0.120419i
$993$	−69189.3	+	348513.i	−0.00222672	+	0.0112162i
$994$			1.57647e7i			0.506081i
$995$		−	2.49851e6i		−	0.0800063i
$996$	527304.	−	2.65608e6i	0.0168427	−	0.0848385i
$997$	−2.80264e7			−0.892956			−0.446478	−	0.894795i	$-0.647322\pi$
$997$	−2.80264e7			−0.892956			−0.446478	+	0.894795i	$0.647322\pi$
$998$		−	2.55430e7i		−	0.811795i
$999$	2.90900e7	+	1.93906e7i	0.922212	+	0.614720i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.6.c.b.68.9	✓	32
3.2	odd	2	inner	69.6.c.b.68.24	yes	32
23.22	odd	2	inner	69.6.c.b.68.10	yes	32
69.68	even	2	inner	69.6.c.b.68.23	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.c.b.68.9	✓	32	1.1	even	1	trivial
69.6.c.b.68.10	yes	32	23.22	odd	2	inner
69.6.c.b.68.23	yes	32	69.68	even	2	inner
69.6.c.b.68.24	yes	32	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	69.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.41836i q^{2} +(3.03549 - 15.2901i) q^{3} +2.64141 q^{4} -41.1824 q^{5} +(-82.8470 - 16.4474i) q^{6} +67.8449i q^{7} -187.700i q^{8} +(-224.572 - 92.8256i) q^{9} +O(q^{10})\)	\(q-5.41836i q^{2} +(3.03549 - 15.2901i) q^{3} +2.64141 q^{4} -41.1824 q^{5} +(-82.8470 - 16.4474i) q^{6} +67.8449i q^{7} -187.700i q^{8} +(-224.572 - 92.8256i) q^{9} +223.141i q^{10} -718.835 q^{11} +(8.01796 - 40.3872i) q^{12} -17.0158 q^{13} +367.608 q^{14} +(-125.009 + 629.681i) q^{15} -932.498 q^{16} +1853.18 q^{17} +(-502.962 + 1216.81i) q^{18} -1139.13i q^{19} -108.779 q^{20} +(1037.35 + 205.943i) q^{21} +3894.90i q^{22} +(-1493.34 + 2050.92i) q^{23} +(-2869.94 - 569.760i) q^{24} -1429.01 q^{25} +92.1977i q^{26} +(-2100.99 + 3151.94i) q^{27} +179.206i q^{28} -1744.71i q^{29} +(3411.84 + 677.342i) q^{30} +3913.17 q^{31} -953.777i q^{32} +(-2182.02 + 10991.0i) q^{33} -10041.2i q^{34} -2794.01i q^{35} +(-593.185 - 245.190i) q^{36} -9229.25i q^{37} -6172.22 q^{38} +(-51.6513 + 260.173i) q^{39} +7729.91i q^{40} -5011.22i q^{41} +(1115.87 - 5620.75i) q^{42} -5505.60i q^{43} -1898.73 q^{44} +(9248.39 + 3822.78i) q^{45} +(11112.6 + 8091.43i) q^{46} -20518.7i q^{47} +(-2830.59 + 14257.9i) q^{48} +12204.1 q^{49} +7742.90i q^{50} +(5625.31 - 28335.2i) q^{51} -44.9456 q^{52} -36672.8 q^{53} +(17078.3 + 11383.9i) q^{54} +29603.3 q^{55} +12734.5 q^{56} +(-17417.4 - 3457.82i) q^{57} -9453.46 q^{58} -30068.0i q^{59} +(-330.199 + 1663.24i) q^{60} -30394.4i q^{61} -21202.9i q^{62} +(6297.75 - 15236.0i) q^{63} -35007.8 q^{64} +700.751 q^{65} +(59553.3 + 11822.9i) q^{66} +42629.9i q^{67} +4895.00 q^{68} +(26825.7 + 29058.7i) q^{69} -15139.0 q^{70} +42884.6i q^{71} +(-17423.3 + 42152.0i) q^{72} +10487.2 q^{73} -50007.4 q^{74} +(-4337.75 + 21849.7i) q^{75} -3008.91i q^{76} -48769.3i q^{77} +(1409.71 + 279.865i) q^{78} -37112.3i q^{79} +38402.5 q^{80} +(41815.8 + 41692.0i) q^{81} -27152.6 q^{82} +65765.4 q^{83} +(2740.07 + 543.978i) q^{84} -76318.4 q^{85} -29831.3 q^{86} +(-26676.7 - 5296.05i) q^{87} +134925. i q^{88} -107918. q^{89} +(20713.2 - 50111.1i) q^{90} -1154.44i q^{91} +(-3944.50 + 5417.32i) q^{92} +(11878.4 - 59832.6i) q^{93} -111177. q^{94} +46912.1i q^{95} +(-14583.3 - 2895.18i) q^{96} +69228.0i q^{97} -66126.0i q^{98} +(161430. + 66726.3i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 408 q^{4} - 528 q^{6} - 444 q^{9}+O(q^{10}) \) 32 * q - 408 * q^4 - 528 * q^6 - 444 * q^9	\( 32 q - 408 q^{4} - 528 q^{6} - 444 q^{9} - 2484 q^{12} + 520 q^{13} + 4936 q^{16} + 7188 q^{18} + 18660 q^{24} + 36032 q^{25} - 22032 q^{27} + 6544 q^{31} - 33912 q^{36} - 63912 q^{39} + 54328 q^{46} + 88284 q^{48} - 207664 q^{49} + 46296 q^{52} - 38628 q^{54} - 139296 q^{55} - 184144 q^{58} + 486584 q^{64} - 113580 q^{69} + 37176 q^{70} - 15504 q^{72} - 93896 q^{73} + 249840 q^{75} + 368028 q^{78} - 339372 q^{81} - 23512 q^{82} + 259584 q^{85} + 509928 q^{87} + 82740 q^{93} - 562000 q^{94} + 1404 q^{96}+O(q^{100}) \) 32 * q - 408 * q^4 - 528 * q^6 - 444 * q^9 - 2484 * q^12 + 520 * q^13 + 4936 * q^16 + 7188 * q^18 + 18660 * q^24 + 36032 * q^25 - 22032 * q^27 + 6544 * q^31 - 33912 * q^36 - 63912 * q^39 + 54328 * q^46 + 88284 * q^48 - 207664 * q^49 + 46296 * q^52 - 38628 * q^54 - 139296 * q^55 - 184144 * q^58 + 486584 * q^64 - 113580 * q^69 + 37176 * q^70 - 15504 * q^72 - 93896 * q^73 + 249840 * q^75 + 368028 * q^78 - 339372 * q^81 - 23512 * q^82 + 259584 * q^85 + 509928 * q^87 + 82740 * q^93 - 562000 * q^94 + 1404 * q^96

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