LMFDB - Embedded newform 69.7.d.a.22.22

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,7,Mod(22,69)]

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([0, 1]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("69.22");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	69.d (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:	$15.8737317698$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			22.22
Character	$\chi$	$=$	69.22
Dual form			69.7.d.a.22.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+13.3624 q^{2} -15.5885 q^{3} +114.554 q^{4} +40.7784i q^{5} -208.299 q^{6} +469.379i q^{7} +675.521 q^{8} +243.000 q^{9} +O(q^{10})$	\(q+13.3624 q^{2} -15.5885 q^{3} +114.554 q^{4} +40.7784i q^{5} -208.299 q^{6} +469.379i q^{7} +675.521 q^{8} +243.000 q^{9} +544.897i q^{10} +1567.76i q^{11} -1785.72 q^{12} +3791.51 q^{13} +6272.03i q^{14} -635.672i q^{15} +1695.14 q^{16} -9021.27i q^{17} +3247.06 q^{18} +7102.08i q^{19} +4671.32i q^{20} -7316.90i q^{21} +20949.1i q^{22} +(-6001.12 + 10584.1i) q^{23} -10530.3 q^{24} +13962.1 q^{25} +50663.7 q^{26} -3788.00 q^{27} +53769.2i q^{28} -3651.94 q^{29} -8494.11i q^{30} -27537.6 q^{31} -20582.2 q^{32} -24439.0i q^{33} -120546. i q^{34} -19140.5 q^{35} +27836.6 q^{36} -75148.6i q^{37} +94900.8i q^{38} -59103.9 q^{39} +27546.7i q^{40} -1160.82 q^{41} -97771.3i q^{42} -77082.8i q^{43} +179593. i q^{44} +9909.15i q^{45} +(-80189.3 + 141429. i) q^{46} +128089. q^{47} -26424.6 q^{48} -102668. q^{49} +186568. q^{50} +140628. i q^{51} +434333. q^{52} -149342. i q^{53} -50616.7 q^{54} -63930.8 q^{55} +317076. i q^{56} -110710. i q^{57} -48798.7 q^{58} -55950.5 q^{59} -72818.7i q^{60} +167059. i q^{61} -367968. q^{62} +114059. i q^{63} -383517. q^{64} +154612. i q^{65} -326564. i q^{66} -199627. i q^{67} -1.03342e6i q^{68} +(93548.1 - 164989. i) q^{69} -255763. q^{70} +120126. q^{71} +164152. q^{72} +659602. q^{73} -1.00417e6i q^{74} -217648. q^{75} +813570. i q^{76} -735875. q^{77} -789770. q^{78} -458548. i q^{79} +69125.1i q^{80} +59049.0 q^{81} -15511.3 q^{82} -293064. i q^{83} -838179. i q^{84} +367873. q^{85} -1.03001e6i q^{86} +56928.1 q^{87} +1.05906e6i q^{88} +43046.0i q^{89} +132410. i q^{90} +1.77966e6i q^{91} +(-687451. + 1.21245e6i) q^{92} +429269. q^{93} +1.71158e6 q^{94} -289611. q^{95} +320845. q^{96} +856674. i q^{97} -1.37189e6 q^{98} +380966. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 20 q^{2} + 816 q^{4} - 324 q^{6} - 940 q^{8} + 5832 q^{9}+O(q^{10}) $ 24 * q - 20 * q^2 + 816 * q^4 - 324 * q^6 - 940 * q^8 + 5832 * q^9	$ 24 q - 20 q^{2} + 816 q^{4} - 324 q^{6} - 940 q^{8} + 5832 q^{9} + 384 q^{13} + 29544 q^{16} - 4860 q^{18} + 29336 q^{23} - 39204 q^{24} - 61272 q^{25} + 10088 q^{26} + 64672 q^{29} + 9696 q^{31} - 319620 q^{32} - 225744 q^{35} + 198288 q^{36} - 11664 q^{39} + 135280 q^{41} + 233232 q^{46} - 74336 q^{47} + 552096 q^{48} - 722136 q^{49} + 619324 q^{50} + 1059720 q^{52} - 78732 q^{54} - 1019328 q^{55} - 694344 q^{58} + 1057648 q^{59} - 488776 q^{62} - 273888 q^{64} - 23328 q^{69} + 2785512 q^{70} - 255392 q^{71} - 228420 q^{72} - 322560 q^{73} - 365472 q^{75} - 1002960 q^{77} - 171072 q^{78} + 1417176 q^{81} - 5732712 q^{82} - 2704704 q^{85} + 611712 q^{87} - 1611444 q^{92} + 2484432 q^{93} - 147720 q^{94} - 1672656 q^{95} - 1818612 q^{96} + 9104212 q^{98}+O(q^{100}) $ 24 * q - 20 * q^2 + 816 * q^4 - 324 * q^6 - 940 * q^8 + 5832 * q^9 + 384 * q^13 + 29544 * q^16 - 4860 * q^18 + 29336 * q^23 - 39204 * q^24 - 61272 * q^25 + 10088 * q^26 + 64672 * q^29 + 9696 * q^31 - 319620 * q^32 - 225744 * q^35 + 198288 * q^36 - 11664 * q^39 + 135280 * q^41 + 233232 * q^46 - 74336 * q^47 + 552096 * q^48 - 722136 * q^49 + 619324 * q^50 + 1059720 * q^52 - 78732 * q^54 - 1019328 * q^55 - 694344 * q^58 + 1057648 * q^59 - 488776 * q^62 - 273888 * q^64 - 23328 * q^69 + 2785512 * q^70 - 255392 * q^71 - 228420 * q^72 - 322560 * q^73 - 365472 * q^75 - 1002960 * q^77 - 171072 * q^78 + 1417176 * q^81 - 5732712 * q^82 - 2704704 * q^85 + 611712 * q^87 - 1611444 * q^92 + 2484432 * q^93 - 147720 * q^94 - 1672656 * q^95 - 1818612 * q^96 + 9104212 * q^98

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 $

$2$	13.3624			1.67030			0.835150	−	0.550022i	$-0.185381\pi$
$2$	13.3624			1.67030			0.835150	+	0.550022i	$0.185381\pi$
$3$	−15.5885			−0.577350
$3$	−15.5885			−0.577350
$4$	114.554			1.78990
$5$			40.7784i			0.326227i	0.986607	+	0.163114i	$0.0521537\pi$
$5$			40.7784i			0.326227i	−0.986607	+	0.163114i	$0.947846\pi$
$6$	−208.299			−0.964348
$7$			469.379i			1.36845i	0.729270	+	0.684226i	$0.239860\pi$
$7$			469.379i			1.36845i	−0.729270	+	0.684226i	$0.760140\pi$
$8$	675.521			1.31938
$9$	243.000			0.333333
$10$			544.897i			0.544897i
$11$			1567.76i			1.17788i	0.808176	+	0.588941i	$0.200455\pi$
$11$			1567.76i			1.17788i	−0.808176	+	0.588941i	$0.799545\pi$
$12$	−1785.72			−1.03340
$13$	3791.51			1.72577			0.862884	−	0.505401i	$-0.168656\pi$
$13$	3791.51			1.72577			0.862884	+	0.505401i	$0.168656\pi$
$14$			6272.03i			2.28573i
$15$		−	635.672i		−	0.188347i
$16$	1695.14			0.413852
$17$		−	9021.27i		−	1.83620i	−0.396343	−	0.918102i	$-0.629721\pi$
$17$		−	9021.27i		−	1.83620i	0.396343	−	0.918102i	$-0.370279\pi$
$18$	3247.06			0.556767
$19$			7102.08i			1.03544i	0.855550	+	0.517720i	$0.173219\pi$
$19$			7102.08i			1.03544i	−0.855550	+	0.517720i	$0.826781\pi$
$20$			4671.32i			0.583915i
$21$		−	7316.90i		−	0.790076i
$22$			20949.1i			1.96742i
$23$	−6001.12	+	10584.1i	−0.493229	+	0.869900i
$23$	−6001.12	+	10584.1i	−0.493229	+	0.869900i
$24$	−10530.3			−0.761743
$25$	13962.1			0.893576
$26$	50663.7			2.88255
$27$	−3788.00			−0.192450
$28$			53769.2i			2.44940i
$29$	−3651.94			−0.149737			−0.0748685	−	0.997193i	$-0.523854\pi$
$29$	−3651.94			−0.149737			−0.0748685	+	0.997193i	$0.523854\pi$
$30$		−	8494.11i		−	0.314597i
$31$	−27537.6			−0.924359			−0.462180	−	0.886786i	$-0.652932\pi$
$31$	−27537.6			−0.924359			−0.462180	+	0.886786i	$0.652932\pi$
$32$	−20582.2			−0.628119
$33$		−	24439.0i		−	0.680051i
$34$		−	120546.i		−	3.06701i
$35$	−19140.5			−0.446426
$36$	27836.6			0.596635
$37$		−	75148.6i		−	1.48360i	−0.670623	−	0.741799i	$-0.733973\pi$
$37$		−	75148.6i		−	1.48360i	0.670623	−	0.741799i	$-0.266027\pi$
$38$			94900.8i			1.72949i
$39$	−59103.9			−0.996373
$40$			27546.7i			0.430417i
$41$	−1160.82			−0.0168427			−0.00842136	−	0.999965i	$-0.502681\pi$
$41$	−1160.82			−0.0168427			−0.00842136	+	0.999965i	$0.502681\pi$
$42$		−	97771.3i		−	1.31966i
$43$		−	77082.8i		−	0.969510i	−0.874650	−	0.484755i	$-0.838909\pi$
$43$		−	77082.8i		−	0.969510i	0.874650	−	0.484755i	$-0.161091\pi$
$44$			179593.i			2.10830i
$45$			9909.15i			0.108742i
$46$	−80189.3	+	141429.i	−0.823840	+	1.45299i
$47$	128089.			1.23372			0.616862	−	0.787071i	$-0.288404\pi$
$47$	128089.			1.23372			0.616862	+	0.787071i	$0.288404\pi$
$48$	−26424.6			−0.238938
$49$	−102668.			−0.872662
$50$	186568.			1.49254
$51$			140628.i			1.06013i
$52$	434333.			3.08896
$53$		−	149342.i		−	1.00312i	−0.865123	−	0.501560i	$-0.832760\pi$
$53$		−	149342.i		−	1.00312i	0.865123	−	0.501560i	$-0.167240\pi$
$54$	−50616.7			−0.321449
$55$	−63930.8			−0.384257
$56$			317076.i			1.80550i
$57$		−	110710.i		−	0.597811i
$58$	−48798.7			−0.250106
$59$	−55950.5			−0.272425			−0.136213	−	0.990680i	$-0.543493\pi$
$59$	−55950.5			−0.272425			−0.136213	+	0.990680i	$0.543493\pi$
$60$		−	72818.7i		−	0.337124i
$61$			167059.i			0.736003i	0.929825	+	0.368002i	$0.119958\pi$
$61$			167059.i			0.736003i	−0.929825	+	0.368002i	$0.880042\pi$
$62$	−367968.			−1.54396
$63$			114059.i			0.456151i
$64$	−383517.			−1.46300
$65$			154612.i			0.562993i
$66$		−	326564.i		−	1.13589i
$67$		−	199627.i		−	0.663735i	−0.943326	−	0.331868i	$-0.892321\pi$
$67$		−	199627.i		−	0.663735i	0.943326	−	0.331868i	$-0.107679\pi$
$68$		−	1.03342e6i		−	3.28663i
$69$	93548.1	−	164989.i	0.284766	−	0.502237i
$70$	−255763.			−0.745666
$71$	120126.			0.335631			0.167816	−	0.985818i	$-0.446329\pi$
$71$	120126.			0.335631			0.167816	+	0.985818i	$0.446329\pi$
$72$	164152.			0.439792
$73$	659602.			1.69556			0.847781	−	0.530347i	$-0.177938\pi$
$73$	659602.			1.69556			0.847781	+	0.530347i	$0.177938\pi$
$74$		−	1.00417e6i		−	2.47805i
$75$	−217648.			−0.515906
$76$			813570.i			1.85334i
$77$	−735875.			−1.61188
$78$	−789770.			−1.66424
$79$		−	458548.i		−	0.930045i	−0.885299	−	0.465022i	$-0.846046\pi$
$79$		−	458548.i		−	0.930045i	0.885299	−	0.465022i	$-0.153954\pi$
$80$			69125.1i			0.135010i
$81$	59049.0			0.111111
$82$	−15511.3			−0.0281324
$83$		−	293064.i		−	0.512541i	−0.966605	−	0.256270i	$-0.917506\pi$
$83$		−	293064.i		−	0.512541i	0.966605	−	0.256270i	$-0.0824938\pi$
$84$		−	838179.i		−	1.41416i
$85$	367873.			0.599020
$86$		−	1.03001e6i		−	1.61937i
$87$	56928.1			0.0864507
$88$			1.05906e6i			1.55407i
$89$			43046.0i			0.0610608i	0.999534	+	0.0305304i	$0.00971964\pi$
$89$			43046.0i			0.0610608i	−0.999534	+	0.0305304i	$0.990280\pi$
$90$			132410.i			0.181632i
$91$			1.77966e6i			2.36163i
$92$	−687451.	+	1.21245e6i	−0.882832	+	1.55704i
$93$	429269.			0.533679
$94$	1.71158e6			2.06069
$95$	−289611.			−0.337788
$96$	320845.			0.362645
$97$			856674.i			0.938643i	0.883027	+	0.469321i	$0.155501\pi$
$97$			856674.i			0.938643i	−0.883027	+	0.469321i	$0.844499\pi$
$98$	−1.37189e6			−1.45761
$99$			380966.i			0.392628i
$100$	1.59942e6			1.59942
$101$	−292195.			−0.283602			−0.141801	−	0.989895i	$-0.545289\pi$
$101$	−292195.			−0.283602			−0.141801	+	0.989895i	$0.545289\pi$
$102$			1.87912e6i			1.77074i
$103$			712314.i			0.651869i	0.945392	+	0.325934i	$0.105679\pi$
$103$			712314.i			0.651869i	−0.945392	+	0.325934i	$0.894321\pi$
$104$	2.56125e6			2.27694
$105$	298371.			0.257744
$106$		−	1.99556e6i		−	1.67551i
$107$		−	562846.i		−	0.459450i	−0.973256	−	0.229725i	$-0.926217\pi$
$107$		−	562846.i		−	0.459450i	0.973256	−	0.229725i	$-0.0737827\pi$
$108$	−433929.			−0.344467
$109$			2.02076e6i			1.56040i	0.625529	+	0.780201i	$0.284883\pi$
$109$			2.02076e6i			1.56040i	−0.625529	+	0.780201i	$0.715117\pi$
$110$	−854269.			−0.641825
$111$			1.17145e6i			0.856555i
$112$			795663.i			0.566337i
$113$		−	1.65496e6i		−	1.14697i	−0.819217	−	0.573484i	$-0.805591\pi$
$113$		−	1.65496e6i		−	1.14697i	0.819217	−	0.573484i	$-0.194409\pi$
$114$		−	1.47936e6i		−	0.998524i
$115$	−431601.	−	244716.i	−0.283785	−	0.160905i
$116$	−418343.			−0.268015
$117$	921338.			0.575256
$118$	−747633.			−0.455032
$119$	4.23440e6			2.51276
$120$		−	429410.i		−	0.248501i
$121$	−686316.			−0.387407
$122$			2.23231e6i			1.22935i
$123$	18095.4			0.00972415
$124$	−3.15454e6			−1.65451
$125$			1.20652e6i			0.617736i
$126$			1.52410e6i			0.761909i
$127$	3.69029e6			1.80156			0.900781	−	0.434274i	$-0.142995\pi$
$127$	3.69029e6			1.80156			0.900781	+	0.434274i	$0.142995\pi$
$128$	−3.80744e6			−1.81553
$129$			1.20160e6i			0.559747i
$130$			2.06599e6i			0.940367i
$131$	−337756.			−0.150241			−0.0751207	−	0.997174i	$-0.523934\pi$
$131$	−337756.			−0.150241			−0.0751207	+	0.997174i	$0.523934\pi$
$132$		−	2.79958e6i		−	1.21723i
$133$	−3.33357e6			−1.41695
$134$		−	2.66750e6i		−	1.10864i
$135$		−	154468.i		−	0.0627824i
$136$		−	6.09406e6i		−	2.42265i
$137$			737585.i			0.286847i	0.989661	+	0.143424i	$0.0458111\pi$
$137$			737585.i			0.286847i	−0.989661	+	0.143424i	$0.954189\pi$
$138$	1.25003e6	−	2.20465e6i	0.475645	−	0.838886i
$139$	495563.			0.184525			0.0922624	−	0.995735i	$-0.470590\pi$
$139$	495563.			0.184525			0.0922624	+	0.995735i	$0.470590\pi$
$140$	−2.19262e6			−0.799060
$141$	−1.99671e6			−0.712291
$142$	1.60517e6			0.560605
$143$			5.94419e6i			2.03275i
$144$	411919.			0.137951
$145$		−	148920.i		−	0.0488483i
$146$	8.81387e6			2.83210
$147$	1.60043e6			0.503831
$148$		−	8.60857e6i		−	2.65550i
$149$		−	3.68998e6i		−	1.11549i	−0.830013	−	0.557744i	$-0.811667\pi$
$149$		−	3.68998e6i		−	1.11549i	0.830013	−	0.557744i	$-0.188333\pi$
$150$	−2.90830e6			−0.861719
$151$	2.42810e6			0.705239			0.352619	−	0.935767i	$-0.385291\pi$
$151$	2.42810e6			0.705239			0.352619	+	0.935767i	$0.385291\pi$
$152$			4.79760e6i			1.36613i
$153$		−	2.19217e6i		−	0.612068i
$154$	−9.83305e6			−2.69232
$155$		−	1.12294e6i		−	0.301551i
$156$	−6.77057e6			−1.78341
$157$			3.65883e6i			0.945461i	0.881207	+	0.472730i	$0.156732\pi$
$157$			3.65883e6i			0.945461i	−0.881207	+	0.472730i	$0.843268\pi$
$158$		−	6.12731e6i		−	1.55345i
$159$			2.32800e6i			0.579152i
$160$		−	839309.i		−	0.204909i
$161$	−4.96794e6	−	2.81680e6i	−1.19042	−	0.674960i
$162$	789037.			0.185589
$163$	−2.54229e6			−0.587032			−0.293516	−	0.955954i	$-0.594825\pi$
$163$	−2.54229e6			−0.587032			−0.293516	+	0.955954i	$0.594825\pi$
$164$	−132976.			−0.0301469
$165$	996582.			0.221851
$166$		−	3.91604e6i		−	0.856097i
$167$	−5.08973e6			−1.09281			−0.546406	−	0.837520i	$-0.684005\pi$
$167$	−5.08973e6			−1.09281			−0.546406	+	0.837520i	$0.684005\pi$
$168$		−	4.94272e6i		−	1.04241i
$169$	9.54877e6			1.97828
$170$	4.91567e6			1.00054
$171$			1.72580e6i			0.345146i
$172$		−	8.83014e6i		−	1.73533i
$173$	214344.			0.0413975			0.0206987	−	0.999786i	$-0.493411\pi$
$173$	214344.			0.0413975			0.0206987	+	0.999786i	$0.493411\pi$
$174$	760696.			0.144399
$175$			6.55353e6i			1.22282i
$176$			2.65758e6i			0.487470i
$177$	872182.			0.157285
$178$			575198.i			0.101990i
$179$	−9.94625e6			−1.73420			−0.867102	−	0.498131i	$-0.834020\pi$
$179$	−9.94625e6			−1.73420			−0.867102	+	0.498131i	$0.834020\pi$
$180$			1.13513e6i			0.194638i
$181$		−	504979.i		−	0.0851604i	−0.999093	−	0.0425802i	$-0.986442\pi$
$181$		−	504979.i		−	0.0851604i	0.999093	−	0.0425802i	$-0.0135578\pi$
$182$			2.37805e7i			3.94464i
$183$		−	2.60419e6i		−	0.424932i
$184$	−4.05388e6	+	7.14976e6i	−0.650755	+	1.14773i
$185$	3.06444e6			0.483990
$186$	5.73606e6			0.891405
$187$	1.41432e7			2.16283
$188$	1.46731e7			2.20825
$189$		−	1.77801e6i		−	0.263359i
$190$	−3.86990e6			−0.564208
$191$		−	7.85205e6i		−	1.12689i	−0.826152	−	0.563447i	$-0.809475\pi$
$191$		−	7.85205e6i		−	1.12689i	0.826152	−	0.563447i	$-0.190525\pi$
$192$	5.97843e6			0.844664
$193$	−1.28145e7			−1.78250			−0.891248	−	0.453516i	$-0.850169\pi$
$193$	−1.28145e7			−1.78250			−0.891248	+	0.453516i	$0.850169\pi$
$194$			1.14472e7i			1.56782i
$195$		−	2.41016e6i		−	0.325044i
$196$	−1.17610e7			−1.56198
$197$	1.12045e7			1.46552			0.732762	−	0.680485i	$-0.238231\pi$
$197$	1.12045e7			1.46552			0.732762	+	0.680485i	$0.238231\pi$
$198$			5.09062e6i			0.655806i
$199$		−	8.48481e6i		−	1.07667i	−0.842731	−	0.538335i	$-0.819053\pi$
$199$		−	8.48481e6i		−	1.07667i	0.842731	−	0.538335i	$-0.180947\pi$
$200$	9.43171e6			1.17896
$201$			3.11188e6i			0.383208i
$202$	−3.90443e6			−0.473700
$203$		−	1.71414e6i		−	0.204908i
$204$			1.61095e7i			1.89754i
$205$		−	47336.3i		−	0.00549455i
$206$			9.51823e6i			1.08882i
$207$	−1.45827e6	+	2.57193e6i	−0.164410	+	0.289967i
$208$	6.42715e6			0.714214
$209$	−1.11344e7			−1.21963
$210$	3.98696e6			0.430510
$211$	−2.24487e6			−0.238971			−0.119485	−	0.992836i	$-0.538124\pi$
$211$	−2.24487e6			−0.238971			−0.119485	+	0.992836i	$0.538124\pi$
$212$		−	1.71077e7i		−	1.79549i
$213$	−1.87258e6			−0.193777
$214$		−	7.52098e6i		−	0.767420i
$215$	3.14331e6			0.316280
$216$	−2.55887e6			−0.253914
$217$		−	1.29256e7i		−	1.26494i
$218$			2.70023e7i			2.60634i
$219$	−1.02822e7			−0.978933
$220$	−7.32352e6			−0.687784
$221$		−	3.42043e7i		−	3.16886i
$222$			1.56534e7i			1.43070i
$223$	−7.16592e6			−0.646185			−0.323093	−	0.946367i	$-0.604723\pi$
$223$	−7.16592e6			−0.646185			−0.323093	+	0.946367i	$0.604723\pi$
$224$		−	9.66086e6i		−	0.859551i
$225$	3.39280e6			0.297859
$226$		−	2.21142e7i		−	1.91578i
$227$			6.95534e6i			0.594622i	0.954781	+	0.297311i	$0.0960898\pi$
$227$			6.95534e6i			0.594622i	−0.954781	+	0.297311i	$0.903910\pi$
$228$		−	1.26823e7i		−	1.07002i
$229$			2.26736e6i			0.188805i	0.995534	+	0.0944026i	$0.0300941\pi$
$229$			2.26736e6i			0.188805i	−0.995534	+	0.0944026i	$0.969906\pi$
$230$	−5.76723e6	−	3.26999e6i	−0.474006	−	0.268759i
$231$	1.14712e7			0.930617
$232$	−2.46696e6			−0.197560
$233$	−8.37836e6			−0.662356			−0.331178	−	0.943568i	$-0.607446\pi$
$233$	−8.37836e6			−0.662356			−0.331178	+	0.943568i	$0.607446\pi$
$234$	1.23113e7			0.960851
$235$			5.22326e6i			0.402474i
$236$	−6.40934e6			−0.487615
$237$			7.14806e6i			0.536961i
$238$	5.65817e7			4.19706
$239$	−1.84843e7			−1.35397			−0.676986	−	0.735996i	$-0.736714\pi$
$239$	−1.84843e7			−1.35397			−0.676986	+	0.735996i	$0.736714\pi$
$240$		−	1.07755e6i		−	0.0779480i
$241$			1.13802e7i			0.813018i	0.913647	+	0.406509i	$0.133254\pi$
$241$			1.13802e7i			0.813018i	−0.913647	+	0.406509i	$0.866746\pi$
$242$	−9.17083e6			−0.647087
$243$	−920483.			−0.0641500
$244$			1.91372e7i			1.31738i
$245$		−	4.18663e6i		−	0.284686i
$246$	241797.			0.0162423
$247$			2.69276e7i			1.78693i
$248$	−1.86022e7			−1.21958
$249$			4.56842e6i			0.295916i
$250$			1.61219e7i			1.03180i
$251$		−	1.24948e7i		−	0.790149i	−0.918649	−	0.395074i	$-0.870719\pi$
$251$		−	1.24948e7i		−	0.790149i	0.918649	−	0.395074i	$-0.129281\pi$
$252$			1.30659e7i			0.816466i
$253$	−1.65933e7	−	9.40832e6i	−1.02464	−	0.580966i
$254$	4.93111e7			3.00915
$255$	−5.73457e6			−0.345844
$256$	−2.63315e7			−1.56948
$257$	−2.74092e6			−0.161472			−0.0807360	−	0.996736i	$-0.525727\pi$
$257$	−2.74092e6			−0.161472			−0.0807360	+	0.996736i	$0.525727\pi$
$258$			1.60563e7i			0.934945i
$259$	3.52732e7			2.03023
$260$			1.77114e7i			1.00770i
$261$	−887421.			−0.0499124
$262$	−4.51323e6			−0.250948
$263$		−	3.24876e7i		−	1.78587i	−0.450181	−	0.892937i	$-0.648641\pi$
$263$		−	3.24876e7i		−	1.78587i	0.450181	−	0.892937i	$-0.351359\pi$
$264$		−	1.65091e7i		−	0.897244i
$265$	6.08991e6			0.327245
$266$	−4.45445e7			−2.36673
$267$		−	671020.i		−	0.0352535i
$268$		−	2.28680e7i		−	1.18802i
$269$	3.23907e7			1.66404			0.832020	−	0.554746i	$-0.187185\pi$
$269$	3.23907e7			1.66404			0.832020	+	0.554746i	$0.187185\pi$
$270$		−	2.06407e6i		−	0.104866i
$271$	4.38123e6			0.220135			0.110067	−	0.993924i	$-0.464893\pi$
$271$	4.38123e6			0.220135			0.110067	+	0.993924i	$0.464893\pi$
$272$		−	1.52923e7i		−	0.759918i
$273$		−	2.77421e7i		−	1.36349i
$274$			9.85592e6i			0.479121i
$275$			2.18893e7i			1.05253i
$276$	1.07163e7	−	1.89002e7i	0.509703	−	0.898956i
$277$	−1.89203e7			−0.890203			−0.445102	−	0.895480i	$-0.646832\pi$
$277$	−1.89203e7			−0.890203			−0.445102	+	0.895480i	$0.646832\pi$
$278$	6.62192e6			0.308212
$279$	−6.69163e6			−0.308120
$280$	−1.29298e7			−0.589005
$281$		−	1.82191e7i		−	0.821124i	−0.911833	−	0.410562i	$-0.865333\pi$
$281$		−	1.82191e7i		−	0.821124i	0.911833	−	0.410562i	$-0.134667\pi$
$282$	−2.66808e7			−1.18974
$283$			1.04736e7i			0.462100i	0.972942	+	0.231050i	$0.0742162\pi$
$283$			1.04736e7i			0.462100i	−0.972942	+	0.231050i	$0.925784\pi$
$284$	1.37609e7			0.600748
$285$	4.51459e6			0.195022
$286$			7.94287e7i			3.39531i
$287$		−	544863.i		−	0.0230485i
$288$	−5.00148e6			−0.209373
$289$	−5.72458e7			−2.37165
$290$		−	1.98993e6i		−	0.0815913i
$291$		−	1.33542e7i		−	0.541926i
$292$	7.55600e7			3.03489
$293$			6.92202e6i			0.275188i	0.990489	+	0.137594i	$0.0439370\pi$
$293$			6.92202e6i			0.275188i	−0.990489	+	0.137594i	$0.956063\pi$
$294$	2.13856e7			0.841550
$295$		−	2.28157e6i		−	0.0888726i
$296$		−	5.07645e7i		−	1.95742i
$297$		−	5.93867e6i		−	0.226684i
$298$		−	4.93070e7i		−	1.86320i
$299$	−2.27533e7	+	4.01296e7i	−0.851199	+	1.50125i
$300$	−2.49324e7			−0.923423
$301$	3.61811e7			1.32673
$302$	3.24453e7			1.17796
$303$	4.55487e6			0.163738
$304$			1.20390e7i			0.428519i
$305$	−6.81239e6			−0.240104
$306$		−	2.92927e7i		−	1.02234i
$307$	−4.92649e6			−0.170264			−0.0851319	−	0.996370i	$-0.527131\pi$
$307$	−4.92649e6			−0.170264			−0.0851319	+	0.996370i	$0.527131\pi$
$308$	−8.42973e7			−2.88510
$309$		−	1.11039e7i		−	0.376356i
$310$		−	1.50052e7i		−	0.503681i
$311$	−2.07147e7			−0.688647			−0.344323	−	0.938851i	$-0.611892\pi$
$311$	−2.07147e7			−0.688647			−0.344323	+	0.938851i	$0.611892\pi$
$312$	−3.99259e7			−1.31459
$313$			2.78344e6i			0.0907712i	0.998970	+	0.0453856i	$0.0144517\pi$
$313$			2.78344e6i			0.0907712i	−0.998970	+	0.0453856i	$0.985548\pi$
$314$			4.88908e7i			1.57920i
$315$	−4.65115e6			−0.148809
$316$		−	5.25285e7i		−	1.66469i
$317$	1.45950e7			0.458171			0.229085	−	0.973406i	$-0.426426\pi$
$317$	1.45950e7			0.458171			0.229085	+	0.973406i	$0.426426\pi$
$318$			3.11077e7i			0.967358i
$319$		−	5.72537e6i		−	0.176373i
$320$		−	1.56392e7i		−	0.477270i
$321$			8.77390e6i			0.265264i
$322$	−6.63836e7	−	3.76392e7i	−1.98835	−	1.12739i
$323$	6.40698e7			1.90128
$324$	6.76429e6			0.198878
$325$	5.29376e7			1.54211
$326$	−3.39711e7			−0.980521
$327$		−	3.15006e7i		−	0.900898i
$328$	−784157.			−0.0222219
$329$			6.01223e7i			1.68829i
$330$	1.33167e7			0.370558
$331$	2.33621e6			0.0644211			0.0322105	−	0.999481i	$-0.489745\pi$
$331$	2.33621e6			0.0644211			0.0322105	+	0.999481i	$0.489745\pi$
$332$		−	3.35716e7i		−	0.917399i
$333$		−	1.82611e7i		−	0.494532i
$334$	−6.80111e7			−1.82533
$335$	8.14047e6			0.216528
$336$		−	1.24032e7i		−	0.326975i
$337$			1.06893e7i			0.279293i	0.990201	+	0.139647i	$0.0445967\pi$
$337$			1.06893e7i			0.279293i	−0.990201	+	0.139647i	$0.955403\pi$
$338$	1.27595e8			3.30432
$339$			2.57982e7i			0.662203i
$340$	4.21413e7			1.07219
$341$		−	4.31724e7i		−	1.08879i
$342$			2.30609e7i			0.576498i
$343$			7.03188e6i			0.174257i
$344$		−	5.20711e7i		−	1.27915i
$345$	6.72800e6	+	3.81474e6i	0.163843	+	0.0928983i
$346$	2.86416e6			0.0691462
$347$	−2.24727e7			−0.537857			−0.268929	−	0.963160i	$-0.586670\pi$
$347$	−2.24727e7			−0.537857			−0.268929	+	0.963160i	$0.586670\pi$
$348$	6.52133e6			0.154739
$349$	4.55946e7			1.07260			0.536299	−	0.844028i	$-0.319822\pi$
$349$	4.55946e7			1.07260			0.536299	+	0.844028i	$0.319822\pi$
$350$			8.75709e7i			2.04247i
$351$	−1.43622e7			−0.332124
$352$		−	3.22680e7i		−	0.739851i
$353$	−6.29900e7			−1.43201			−0.716007	−	0.698093i	$-0.754032\pi$
$353$	−6.29900e7			−1.43201			−0.716007	+	0.698093i	$0.754032\pi$
$354$	1.16544e7			0.262713
$355$			4.89855e6i			0.109492i
$356$			4.93108e6i			0.109293i
$357$	−6.60077e7			−1.45074
$358$	−1.32906e8			−2.89664
$359$			6.83335e7i			1.47690i	0.674309	+	0.738449i	$0.264441\pi$
$359$			6.83335e7i			1.47690i	−0.674309	+	0.738449i	$0.735559\pi$
$360$			6.69384e6i			0.143472i
$361$	−3.39362e6			−0.0721342
$362$		−	6.74774e6i		−	0.142244i
$363$	1.06986e7			0.223670
$364$			2.03867e8i			4.22709i
$365$			2.68975e7i			0.553138i
$366$		−	3.47982e7i		−	0.709764i
$367$		−	7.33714e7i		−	1.48432i	−0.670221	−	0.742162i	$-0.733801\pi$
$367$		−	7.33714e7i		−	1.48432i	0.670221	−	0.742162i	$-0.266199\pi$
$368$	−1.01727e7	+	1.79415e7i	−0.204124	+	0.360010i
$369$	−282079.			−0.00561424
$370$	4.09483e7			0.808408
$371$	7.00978e7			1.37272
$372$	4.91744e7			0.955234
$373$			8.30679e7i			1.60069i	0.599541	+	0.800344i	$0.295350\pi$
$373$			8.30679e7i			1.60069i	−0.599541	+	0.800344i	$0.704650\pi$
$374$	1.88987e8			3.61258
$375$		−	1.88077e7i		−	0.356650i
$376$	8.65268e7			1.62775
$377$	−1.38464e7			−0.258412
$378$		−	2.37584e7i		−	0.439888i
$379$			6.89177e7i			1.26594i	0.774176	+	0.632970i	$0.218164\pi$
$379$			6.89177e7i			1.26594i	−0.774176	+	0.632970i	$0.781836\pi$
$380$	−3.31761e7			−0.604609
$381$	−5.75259e7			−1.04013
$382$		−	1.04922e8i		−	1.88225i
$383$			7.07466e7i			1.25924i	0.776902	+	0.629621i	$0.216790\pi$
$383$			7.07466e7i			1.25924i	−0.776902	+	0.629621i	$0.783210\pi$
$384$	5.93522e7			1.04820
$385$		−	3.00078e7i		−	0.525838i
$386$	−1.71232e8			−2.97730
$387$		−	1.87311e7i		−	0.323170i
$388$			9.81353e7i			1.68008i
$389$		−	1.19445e7i		−	0.202917i	−0.994840	−	0.101458i	$-0.967649\pi$
$389$		−	1.19445e7i		−	0.202917i	0.994840	−	0.101458i	$-0.0323509\pi$
$390$		−	3.22055e7i		−	0.542921i
$391$	9.54818e7	+	5.41377e7i	1.59731	+	0.905669i
$392$	−6.93542e7			−1.15137
$393$	5.26510e6			0.0867419
$394$	1.49719e8			2.44787
$395$	1.86989e7			0.303406
$396$			4.36411e7i			0.702766i
$397$	−1.35684e7			−0.216849			−0.108425	−	0.994105i	$-0.534581\pi$
$397$	−1.35684e7			−0.216849			−0.108425	+	0.994105i	$0.534581\pi$
$398$		−	1.13377e8i		−	1.79836i
$399$	5.19652e7			0.818076
$400$	2.36677e7			0.369809
$401$			6.30703e7i			0.978119i	0.872250	+	0.489059i	$0.162660\pi$
$401$			6.30703e7i			0.978119i	−0.872250	+	0.489059i	$0.837340\pi$
$402$			4.15822e7i			0.640072i
$403$	−1.04409e8			−1.59523
$404$	−3.34721e7			−0.507620
$405$			2.40792e6i			0.0362475i
$406$		−	2.29051e7i		−	0.342258i
$407$	1.17815e8			1.74750
$408$			9.49970e7i			1.39872i
$409$	6.92270e6			0.101183			0.0505913	−	0.998719i	$-0.483889\pi$
$409$	6.92270e6			0.101183			0.0505913	+	0.998719i	$0.483889\pi$
$410$		−	632526.i		−	0.00917756i
$411$		−	1.14978e7i		−	0.165611i
$412$			8.15984e7i			1.16678i
$413$		−	2.62620e7i		−	0.372801i
$414$	−1.94860e7	+	3.43672e7i	−0.274613	+	0.484331i
$415$	1.19507e7			0.167205
$416$	−7.80377e7			−1.08399
$417$	−7.72507e6			−0.106535
$418$	−1.48782e8			−2.03714
$419$		−	3.86181e7i		−	0.524987i	−0.964934	−	0.262494i	$-0.915455\pi$
$419$		−	3.86181e7i		−	0.524987i	0.964934	−	0.262494i	$-0.0845449\pi$
$420$	3.41796e7			0.461338
$421$			5.59677e7i			0.750052i	0.927014	+	0.375026i	$0.122366\pi$
$421$			5.59677e7i			0.750052i	−0.927014	+	0.375026i	$0.877634\pi$
$422$	−2.99969e7			−0.399153
$423$	3.11256e7			0.411241
$424$		−	1.00883e8i		−	1.32349i
$425$		−	1.25956e8i		−	1.64079i
$426$	−2.50222e7			−0.323666
$427$	−7.84139e7			−1.00719
$428$		−	6.44762e7i		−	0.822372i
$429$		−	9.26608e7i		−	1.17361i
$430$	4.20022e7			0.528283
$431$		−	9.77946e7i		−	1.22147i	−0.791835	−	0.610735i	$-0.790874\pi$
$431$		−	9.77946e7i		−	1.22147i	0.791835	−	0.610735i	$-0.209126\pi$
$432$	−6.42118e6			−0.0796459
$433$		−	1.55568e8i		−	1.91627i	−0.286315	−	0.958136i	$-0.592430\pi$
$433$		−	1.55568e8i		−	1.91627i	0.286315	−	0.958136i	$-0.407570\pi$
$434$		−	1.72717e8i		−	2.11283i
$435$			2.32143e6i			0.0282026i
$436$			2.31486e8i			2.79297i
$437$	−7.51689e7	−	4.26204e7i	−0.900728	−	0.510708i
$438$	−1.37395e8			−1.63511
$439$	1.32635e7			0.156770			0.0783851	−	0.996923i	$-0.475024\pi$
$439$	1.32635e7			0.156770			0.0783851	+	0.996923i	$0.475024\pi$
$440$	−4.31866e7			−0.506980
$441$	−2.49483e7			−0.290887
$442$		−	4.57052e8i		−	5.29296i
$443$	−1.32324e6			−0.0152205			−0.00761023	−	0.999971i	$-0.502422\pi$
$443$	−1.32324e6			−0.0152205			−0.00761023	+	0.999971i	$0.502422\pi$
$444$			1.34194e8i			1.53315i
$445$	−1.75535e6			−0.0199197
$446$	−9.57539e7			−1.07932
$447$			5.75211e7i			0.644027i
$448$		−	1.80015e8i		−	2.00205i
$449$	−8.63709e7			−0.954176			−0.477088	−	0.878855i	$-0.658308\pi$
$449$	−8.63709e7			−0.954176			−0.477088	+	0.878855i	$0.658308\pi$
$450$	4.53359e7			0.497513
$451$		−	1.81989e6i		−	0.0198388i
$452$		−	1.89582e8i		−	2.05296i
$453$	−3.78504e7			−0.407170
$454$			9.29401e7i			0.993197i
$455$	−7.25716e7			−0.770428
$456$		−	7.47872e7i		−	0.788738i
$457$			5.19478e7i			0.544275i	0.962258	+	0.272138i	$0.0877306\pi$
$457$			5.19478e7i			0.544275i	−0.962258	+	0.272138i	$0.912269\pi$
$458$			3.02974e7i			0.315362i
$459$			3.41725e7i			0.353378i
$460$	−4.94416e7	−	2.80331e7i	−0.507948	−	0.288004i
$461$	−4.67094e7			−0.476762			−0.238381	−	0.971172i	$-0.576617\pi$
$461$	−4.67094e7			−0.476762			−0.238381	+	0.971172i	$0.576617\pi$
$462$	1.53282e8			1.55441
$463$	5.63404e7			0.567646			0.283823	−	0.958877i	$-0.408397\pi$
$463$	5.63404e7			0.567646			0.283823	+	0.958877i	$0.408397\pi$
$464$	−6.19054e6			−0.0619690
$465$			1.75049e7i			0.174101i
$466$	−1.11955e8			−1.10633
$467$			1.36113e8i			1.33643i	0.743966	+	0.668217i	$0.232942\pi$
$467$			1.36113e8i			1.33643i	−0.743966	+	0.668217i	$0.767058\pi$
$468$	1.05543e8			1.02965
$469$	9.37008e7			0.908290
$470$			6.97953e7i			0.672253i
$471$		−	5.70356e7i		−	0.545862i
$472$	−3.77957e7			−0.359432
$473$	1.20848e8			1.14197
$474$			9.55153e7i			0.896887i
$475$			9.91601e7i			0.925243i
$476$	4.85067e8			4.49760
$477$		−	3.62900e7i		−	0.334373i
$478$	−2.46995e8			−2.26154
$479$			5.92042e7i			0.538699i	0.963043	+	0.269349i	$0.0868086\pi$
$479$			5.92042e7i			0.538699i	−0.963043	+	0.269349i	$0.913191\pi$
$480$			1.30835e7i			0.118305i
$481$		−	2.84927e8i		−	2.56035i
$482$			1.52067e8i			1.35798i
$483$	7.74425e7	+	4.39095e7i	0.687287	+	0.389688i
$484$	−7.86201e7			−0.693422
$485$	−3.49338e7			−0.306211
$486$	−1.22999e7			−0.107150
$487$	−1.88221e7			−0.162960			−0.0814798	−	0.996675i	$-0.525965\pi$
$487$	−1.88221e7			−0.162960			−0.0814798	+	0.996675i	$0.525965\pi$
$488$			1.12852e8i			0.971066i
$489$	3.96304e7			0.338923
$490$		−	5.59434e7i		−	0.475511i
$491$	5.97827e7			0.505046			0.252523	−	0.967591i	$-0.418740\pi$
$491$	5.97827e7			0.505046			0.252523	+	0.967591i	$0.418740\pi$
$492$	2.07289e6			0.0174053
$493$			3.29451e7i			0.274948i
$494$			3.59818e8i			2.98471i
$495$	−1.55352e7			−0.128086
$496$	−4.66801e7			−0.382548
$497$			5.63847e7i			0.459296i
$498$			6.10451e7i			0.494268i
$499$	−7.60944e7			−0.612423			−0.306211	−	0.951964i	$-0.599061\pi$
$499$	−7.60944e7			−0.612423			−0.306211	+	0.951964i	$0.599061\pi$
$500$			1.38211e8i			1.10569i
$501$	7.93411e7			0.630936
$502$		−	1.66961e8i		−	1.31979i
$503$			1.17172e8i			0.920703i	0.887737	+	0.460352i	$0.152277\pi$
$503$			1.17172e8i			0.920703i	−0.887737	+	0.460352i	$0.847723\pi$
$504$			7.70493e7i			0.601835i
$505$		−	1.19152e7i		−	0.0925186i
$506$	−2.21726e8	−	1.25718e8i	−1.71146	−	0.970387i
$507$	−1.48851e8			−1.14216
$508$	4.22737e8			3.22462
$509$	1.36371e8			1.03412			0.517059	−	0.855950i	$-0.327027\pi$
$509$	1.36371e8			1.03412			0.517059	+	0.855950i	$0.327027\pi$
$510$	−7.66277e7			−0.577664
$511$			3.09604e8i			2.32030i
$512$	−1.08176e8			−0.805976
$513$		−	2.69026e7i		−	0.199270i
$514$	−3.66253e7			−0.269707
$515$	−2.90470e7			−0.212657
$516$			1.37648e8i			1.00189i
$517$			2.00813e8i			1.45318i
$518$	4.71335e8			3.39110
$519$	−3.34130e6			−0.0239008
$520$			1.04444e8i			0.742799i
$521$		−	1.08762e8i		−	0.769068i	−0.923111	−	0.384534i	$-0.874362\pi$
$521$		−	1.08762e8i		−	0.769068i	0.923111	−	0.384534i	$-0.125638\pi$
$522$	−1.18581e7			−0.0833686
$523$			1.00555e7i			0.0702908i	0.999382	+	0.0351454i	$0.0111894\pi$
$523$			1.00555e7i			0.0702908i	−0.999382	+	0.0351454i	$0.988811\pi$
$524$	−3.86913e7			−0.268918
$525$		−	1.02159e8i		−	0.705993i
$526$		−	4.34113e8i		−	2.98295i
$527$			2.48424e8i			1.69731i
$528$		−	4.14275e7i		−	0.281441i
$529$	−7.60091e7	−	1.27032e8i	−0.513451	−	0.858119i
$530$	8.13758e7			0.546598
$531$	−1.35960e7			−0.0908085
$532$	−3.81873e8			−2.53620
$533$	−4.40126e6			−0.0290666
$534$		−	8.96644e6i		−	0.0588839i
$535$	2.29520e7			0.149885
$536$		−	1.34852e8i		−	0.875717i
$537$	1.55047e8			1.00124
$538$	4.32818e8			2.77945
$539$		−	1.60959e8i		−	1.02789i
$540$		−	1.76949e7i		−	0.112375i
$541$	−1.74284e8			−1.10069			−0.550347	−	0.834936i	$-0.685505\pi$
$541$	−1.74284e8			−1.10069			−0.550347	+	0.834936i	$0.685505\pi$
$542$	5.85438e7			0.367691
$543$			7.87185e6i			0.0491674i
$544$			1.85678e8i			1.15336i
$545$	−8.24035e7			−0.509045
$546$		−	3.70701e8i		−	2.27744i
$547$	−1.67420e8			−1.02293			−0.511465	−	0.859304i	$-0.670897\pi$
$547$	−1.67420e8			−1.02293			−0.511465	+	0.859304i	$0.670897\pi$
$548$			8.44933e7i			0.513429i
$549$			4.05953e7i			0.245334i
$550$			2.92493e8i			1.75804i
$551$		−	2.59363e7i		−	0.155044i
$552$	6.31937e7	−	1.11454e8i	0.375714	−	0.662640i
$553$	2.15233e8			1.27272
$554$	−2.52821e8			−1.48691
$555$	−4.77699e7			−0.279432
$556$	5.67687e7			0.330282
$557$		−	1.38029e8i		−	0.798736i	−0.916791	−	0.399368i	$-0.869230\pi$
$557$		−	1.38029e8i		−	0.798736i	0.916791	−	0.399368i	$-0.130770\pi$
$558$	−8.94163e7			−0.514653
$559$		−	2.92261e8i		−	1.67315i
$560$	−3.24459e7			−0.184755
$561$	−2.20471e8			−1.24871
$562$		−	2.43451e8i		−	1.37152i
$563$		−	3.21701e8i		−	1.80271i	−0.433077	−	0.901357i	$-0.642572\pi$
$563$		−	3.21701e8i		−	1.80271i	0.433077	−	0.901357i	$-0.357428\pi$
$564$	−2.28731e8			−1.27493
$565$	6.74865e7			0.374172
$566$			1.39952e8i			0.771847i
$567$			2.77164e7i			0.152050i
$568$	8.11478e7			0.442824
$569$			7.25247e7i			0.393685i	0.980435	+	0.196843i	$0.0630688\pi$
$569$			7.25247e7i			0.393685i	−0.980435	+	0.196843i	$0.936931\pi$
$570$	6.03258e7			0.325746
$571$			3.26457e8i			1.75355i	0.480903	+	0.876774i	$0.340309\pi$
$571$			3.26457e8i			1.75355i	−0.480903	+	0.876774i	$0.659691\pi$
$572$			6.80930e8i			3.63843i
$573$			1.22401e8i			0.650612i
$574$		−	7.28069e6i		−	0.0384979i
$575$	−8.37883e7	+	1.47776e8i	−0.440737	+	0.777321i
$576$	−9.31946e7			−0.487667
$577$	2.03247e8			1.05803			0.529014	−	0.848613i	$-0.322562\pi$
$577$	2.03247e8			1.05803			0.529014	+	0.848613i	$0.322562\pi$
$578$	−7.64942e8			−3.96136
$579$	1.99758e8			1.02912
$580$		−	1.70594e7i		−	0.0874337i
$581$	1.37558e8			0.701388
$582$		−	1.78445e8i		−	0.905179i
$583$	2.34132e8			1.18156
$584$	4.45575e8			2.23709
$585$			3.75707e7i			0.187664i
$586$			9.24948e7i			0.459647i
$587$	3.41225e8			1.68704			0.843522	−	0.537094i	$-0.180478\pi$
$587$	3.41225e8			1.68704			0.843522	+	0.537094i	$0.180478\pi$
$588$	1.83336e8			0.901810
$589$		−	1.95574e8i		−	0.957118i
$590$		−	3.04873e7i		−	0.148444i
$591$	−1.74661e8			−0.846121
$592$		−	1.27387e8i		−	0.613990i
$593$	−4.85029e7			−0.232597			−0.116298	−	0.993214i	$-0.537103\pi$
$593$	−4.85029e7			−0.232597			−0.116298	+	0.993214i	$0.537103\pi$
$594$		−	7.93550e7i		−	0.378630i
$595$			1.72672e8i			0.819730i
$596$		−	4.22701e8i		−	1.99662i
$597$			1.32265e8i			0.621616i
$598$	−3.04039e8	+	5.36229e8i	−1.42176	+	2.50753i
$599$	−2.09120e8			−0.973005			−0.486502	−	0.873679i	$-0.661728\pi$
$599$	−2.09120e8			−0.973005			−0.486502	+	0.873679i	$0.661728\pi$
$600$	−1.47026e8			−0.680675
$601$	−4.08234e8			−1.88056			−0.940278	−	0.340409i	$-0.889435\pi$
$601$	−4.08234e8			−1.88056			−0.940278	+	0.340409i	$0.889435\pi$
$602$	4.83466e8			2.21603
$603$		−	4.85094e7i		−	0.221245i
$604$	2.78149e8			1.26231
$605$		−	2.79869e7i		−	0.126383i
$606$	6.08641e7			0.273491
$607$	3.04590e8			1.36191			0.680957	−	0.732323i	$-0.261564\pi$
$607$	3.04590e8			1.36191			0.680957	+	0.732323i	$0.261564\pi$
$608$		−	1.46176e8i		−	0.650379i
$609$			2.67208e7i			0.118304i
$610$	−9.10299e7			−0.401046
$611$	4.85651e8			2.12912
$612$		−	2.51121e8i		−	1.09554i
$613$		−	5.70085e7i		−	0.247491i	−0.992314	−	0.123745i	$-0.960509\pi$
$613$		−	5.70085e7i		−	0.247491i	0.992314	−	0.123745i	$-0.0394906\pi$
$614$	−6.58298e7			−0.284392
$615$			737899.i			0.00317228i
$616$	−4.97099e8			−2.12667
$617$			3.77637e7i			0.160775i	0.996764	+	0.0803877i	$0.0256158\pi$
$617$			3.77637e7i			0.160775i	−0.996764	+	0.0803877i	$0.974384\pi$
$618$		−	1.48375e8i		−	0.628628i
$619$		−	3.40594e7i		−	0.143604i	−0.997419	−	0.0718018i	$-0.977125\pi$
$619$		−	3.40594e7i		−	0.143604i	0.997419	−	0.0718018i	$-0.0228749\pi$
$620$		−	1.28637e8i		−	0.539747i
$621$	2.27322e7	−	4.00924e7i	0.0949219	−	0.167412i
$622$	−2.76798e8			−1.15025
$623$	−2.02049e7			−0.0835588
$624$	−1.00189e8			−0.412351
$625$	1.68958e8			0.692054
$626$			3.71934e7i			0.151615i
$627$	1.73568e8			0.704151
$628$			4.19133e8i			1.69228i
$629$	−6.77937e8			−2.72419
$630$	−6.21505e7			−0.248555
$631$		−	2.37604e8i		−	0.945725i	−0.881136	−	0.472863i	$-0.843221\pi$
$631$		−	2.37604e8i		−	0.945725i	0.881136	−	0.472863i	$-0.156779\pi$
$632$		−	3.09759e8i		−	1.22708i
$633$	3.49941e7			0.137970
$634$	1.95025e8			0.765283
$635$			1.50484e8i			0.587718i
$636$			2.66682e8i			1.03663i
$637$	−3.89266e8			−1.50601
$638$		−	7.65047e7i		−	0.294595i
$639$	2.91907e7			0.111877
$640$		−	1.55261e8i		−	0.592275i
$641$		−	4.50113e8i		−	1.70902i	−0.519433	−	0.854511i	$-0.673857\pi$
$641$		−	4.50113e8i		−	1.70902i	0.519433	−	0.854511i	$-0.326143\pi$
$642$			1.17240e8i			0.443070i
$643$		−	1.93945e8i		−	0.729533i	−0.931099	−	0.364767i	$-0.881149\pi$
$643$		−	1.93945e8i		−	0.729533i	0.931099	−	0.364767i	$-0.118851\pi$
$644$	−5.69097e8	−	3.22675e8i	−2.13073	−	1.20811i
$645$	−4.89994e7			−0.182605
$646$	8.56126e8			3.17571
$647$	−2.17106e8			−0.801601			−0.400800	−	0.916165i	$-0.631268\pi$
$647$	−2.17106e8			−0.801601			−0.400800	+	0.916165i	$0.631268\pi$
$648$	3.98888e7			0.146597
$649$		−	8.77170e7i		−	0.320885i
$650$	7.07373e8			2.57578
$651$			2.01490e8i			0.730314i
$652$	−2.91229e8			−1.05073
$653$	−4.36163e8			−1.56642			−0.783212	−	0.621755i	$-0.786420\pi$
$653$	−4.36163e8			−1.56642			−0.783212	+	0.621755i	$0.786420\pi$
$654$		−	4.20924e8i		−	1.50477i
$655$		−	1.37732e7i		−	0.0490128i
$656$	−1.96775e6			−0.00697040
$657$	1.60283e8			0.565187
$658$			8.03378e8i			2.81996i
$659$		−	1.46383e8i		−	0.511486i	−0.966745	−	0.255743i	$-0.917680\pi$
$659$		−	1.46383e8i		−	0.511486i	0.966745	−	0.255743i	$-0.0823200\pi$
$660$	1.14162e8			0.397092
$661$			3.35799e8i			1.16272i	0.813647	+	0.581359i	$0.197479\pi$
$661$			3.35799e8i			1.16272i	−0.813647	+	0.581359i	$0.802521\pi$
$662$	3.12174e7			0.107603
$663$			5.33192e8i			1.82954i
$664$		−	1.97971e8i		−	0.676235i
$665$		−	1.35937e8i		−	0.462247i
$666$		−	2.44012e8i		−	0.826018i
$667$	2.19157e7	−	3.86523e7i	0.0738546	−	0.130256i
$668$	−5.83049e8			−1.95603
$669$	1.11706e8			0.373075
$670$	1.08776e8			0.361668
$671$	−2.61908e8			−0.866925
$672$			1.50598e8i			0.496262i
$673$	−1.42716e8			−0.468196			−0.234098	−	0.972213i	$-0.575214\pi$
$673$	−1.42716e8			−0.468196			−0.234098	+	0.972213i	$0.575214\pi$
$674$			1.42835e8i			0.466504i
$675$	−5.28885e7			−0.171969
$676$	1.09385e9			3.54093
$677$			4.50766e8i			1.45273i	0.687308	+	0.726366i	$0.258792\pi$
$677$			4.50766e8i			1.45273i	−0.687308	+	0.726366i	$0.741208\pi$
$678$			3.44726e8i			1.10608i
$679$	−4.02105e8			−1.28449
$680$	2.48506e8			0.790333
$681$		−	1.08423e8i		−	0.343305i
$682$		−	5.76887e8i		−	1.81860i
$683$	2.27098e8			0.712773			0.356387	−	0.934339i	$-0.384009\pi$
$683$	2.27098e8			0.712773			0.356387	+	0.934339i	$0.384009\pi$
$684$			1.97698e8i			0.617779i
$685$	−3.00775e7			−0.0935773
$686$			9.39628e7i			0.291061i
$687$		−	3.53447e7i		−	0.109007i
$688$		−	1.30666e8i		−	0.401234i
$689$		−	5.66231e8i		−	1.73115i
$690$	8.99022e7	+	5.09741e7i	0.273667	+	0.155168i
$691$	5.21427e8			1.58037			0.790186	−	0.612867i	$-0.209984\pi$
$691$	5.21427e8			1.58037			0.790186	+	0.612867i	$0.209984\pi$
$692$	2.45540e7			0.0740975
$693$	−1.78818e8			−0.537292
$694$	−3.00289e8			−0.898383
$695$			2.02083e7i			0.0601970i
$696$	3.84561e7			0.114061
$697$			1.04721e7i			0.0309267i
$698$	6.09254e8			1.79156
$699$	1.30606e8			0.382411
$700$			7.50732e8i			2.18872i
$701$		−	5.08788e7i		−	0.147701i	−0.997269	−	0.0738503i	$-0.976471\pi$
$701$		−	5.08788e7i		−	0.147701i	0.997269	−	0.0738503i	$-0.0235287\pi$
$702$	−1.91914e8			−0.554748
$703$	5.33711e8			1.53617
$704$		−	6.01263e8i		−	1.72324i
$705$		−	8.14226e7i		−	0.232369i
$706$	−8.41697e8			−2.39189
$707$		−	1.37150e8i		−	0.388096i
$708$	9.99118e7			0.281525
$709$		−	4.19730e8i		−	1.17769i	−0.808245	−	0.588846i	$-0.799583\pi$
$709$		−	4.19730e8i		−	1.17769i	0.808245	−	0.588846i	$-0.200417\pi$
$710$			6.54564e7i			0.182885i
$711$		−	1.11427e8i		−	0.310015i
$712$			2.90785e7i			0.0805622i
$713$	1.65256e8	−	2.91460e8i	0.455921	−	0.804100i
$714$	−8.82022e8			−2.42317
$715$	−2.42394e8			−0.663139
$716$	−1.13938e9			−3.10406
$717$	2.88142e8			0.781716
$718$			9.13100e8i			2.46686i
$719$	−2.31060e8			−0.621639			−0.310820	−	0.950469i	$-0.600604\pi$
$719$	−2.31060e8			−0.621639			−0.310820	+	0.950469i	$0.600604\pi$
$720$			1.67974e7i			0.0450033i
$721$	−3.34346e8			−0.892051
$722$	−4.53469e7			−0.120486
$723$		−	1.77400e8i		−	0.469396i
$724$		−	5.78473e7i		−	0.152429i
$725$	−5.09888e7			−0.133801
$726$	1.42959e8			0.373596
$727$		−	6.06964e8i		−	1.57965i	−0.613335	−	0.789823i	$-0.710172\pi$
$727$		−	6.06964e8i		−	1.57965i	0.613335	−	0.789823i	$-0.289828\pi$
$728$			1.20220e9i			3.11588i
$729$	1.43489e7			0.0370370
$730$			3.59415e8i			0.923907i
$731$	−6.95385e8			−1.78022
$732$		−	2.98320e8i		−	0.760587i
$733$			9.92159e7i			0.251924i	0.992035	+	0.125962i	$0.0402017\pi$
$733$			9.92159e7i			0.251924i	−0.992035	+	0.125962i	$0.959798\pi$
$734$		−	9.80418e8i		−	2.47927i
$735$			6.52630e7i			0.164363i
$736$	1.23516e8	−	2.17844e8i	0.309807	−	0.546401i
$737$	3.12968e8			0.781802
$738$	−3.76925e6			−0.00937747
$739$	−9.18743e7			−0.227646			−0.113823	−	0.993501i	$-0.536310\pi$
$739$	−9.18743e7			−0.227646			−0.113823	+	0.993501i	$0.536310\pi$
$740$	3.51043e8			0.866295
$741$		−	4.19760e8i		−	1.03168i
$742$	9.36675e8			2.29286
$743$		−	6.62329e8i		−	1.61476i	−0.590034	−	0.807379i	$-0.700886\pi$
$743$		−	6.62329e8i		−	1.61476i	0.590034	−	0.807379i	$-0.299114\pi$
$744$	2.89980e8			0.704124
$745$	1.50471e8			0.363902
$746$			1.10999e9i			2.67363i
$747$		−	7.12146e7i		−	0.170847i
$748$	1.62016e9			3.87126
$749$	2.64188e8			0.628736
$750$		−	2.51316e8i		−	0.595713i
$751$		−	3.19510e8i		−	0.754335i	−0.926145	−	0.377167i	$-0.876898\pi$
$751$		−	3.19510e8i		−	0.754335i	0.926145	−	0.377167i	$-0.123102\pi$
$752$	2.17129e8			0.510580
$753$			1.94775e8i			0.456193i
$754$	−1.85021e8			−0.431625
$755$			9.90141e7i			0.230068i
$756$		−	2.03677e8i		−	0.471387i
$757$			5.19511e8i			1.19759i	0.800903	+	0.598794i	$0.204353\pi$
$757$			5.19511e8i			1.19759i	−0.800903	+	0.598794i	$0.795647\pi$
$758$			9.20906e8i			2.11450i
$759$	2.58664e8	+	1.46661e8i	0.591576	+	0.335421i
$760$	−1.95638e8			−0.445670
$761$	3.04486e8			0.690896			0.345448	−	0.938438i	$-0.387727\pi$
$761$	3.04486e8			0.690896			0.345448	+	0.938438i	$0.387727\pi$
$762$	−7.68684e8			−1.73733
$763$	−9.48505e8			−2.13533
$764$		−	8.99482e8i		−	2.01703i
$765$	8.93931e7			0.199673
$766$			9.45345e8i			2.10331i
$767$	−2.12137e8			−0.470143
$768$	4.10468e8			0.906141
$769$			1.99259e8i			0.438167i	0.975706	+	0.219083i	$0.0703067\pi$
$769$			1.99259e8i			0.438167i	−0.975706	+	0.219083i	$0.929693\pi$
$770$		−	4.00976e8i		−	0.878307i
$771$	4.27268e7			0.0932260
$772$	−1.46795e9			−3.19050
$773$		−	3.83867e7i		−	0.0831079i	−0.999136	−	0.0415539i	$-0.986769\pi$
$773$		−	3.83867e7i		−	0.0831079i	0.999136	−	0.0415539i	$-0.0132308\pi$
$774$		−	2.50293e8i		−	0.539791i
$775$	−3.84483e8			−0.825985
$776$			5.78701e8i			1.23842i
$777$	−5.49855e8			−1.17215
$778$		−	1.59607e8i		−	0.338932i
$779$		−	8.24421e6i		−	0.0174396i
$780$		−	2.76093e8i		−	0.581797i
$781$			1.88329e8i			0.395334i
$782$	1.27587e9	+	7.23410e8i	2.66799	+	1.51274i
$783$	1.38335e7			0.0288169
$784$	−1.74036e8			−0.361153
$785$	−1.49201e8			−0.308435
$786$	7.03544e7			0.144885
$787$		−	2.15211e7i		−	0.0441510i	−0.999756	−	0.0220755i	$-0.992973\pi$
$787$		−	2.15211e7i		−	0.0441510i	0.999756	−	0.0220755i	$-0.00702743\pi$
$788$	1.28352e9			2.62315
$789$			5.06432e8i			1.03108i
$790$	2.49862e8			0.506779
$791$	7.76803e8			1.56957
$792$			2.57351e8i			0.518024i
$793$			6.33406e8i			1.27017i
$794$	−1.81307e8			−0.362203
$795$	−9.49323e7			−0.188935
$796$		−	9.71968e8i		−	1.92714i
$797$			2.04730e8i			0.404396i	0.979345	+	0.202198i	$0.0648085\pi$
$797$			2.04730e8i			0.404396i	−0.979345	+	0.202198i	$0.935191\pi$
$798$	6.94380e8			1.36643
$799$		−	1.15553e9i		−	2.26537i
$800$	−2.87371e8			−0.561272
$801$			1.04602e7i			0.0203536i
$802$			8.42771e8i			1.63375i
$803$			1.03410e9i			1.99717i
$804$			3.56478e8i			0.685905i
$805$	1.14864e8	−	2.02585e8i	0.220190	−	0.388346i
$806$	−1.39516e9			−2.66451
$807$	−5.04921e8			−0.960734
$808$	−1.97384e8			−0.374178
$809$	2.79419e8			0.527729			0.263865	−	0.964560i	$-0.415003\pi$
$809$	2.79419e8			0.527729			0.263865	+	0.964560i	$0.415003\pi$
$810$			3.21756e7i			0.0605441i
$811$	1.40447e8			0.263299			0.131650	−	0.991296i	$-0.457973\pi$
$811$	1.40447e8			0.263299			0.131650	+	0.991296i	$0.457973\pi$
$812$		−	1.96362e8i		−	0.366766i
$813$	−6.82966e7			−0.127095
$814$	1.57429e9			2.91886
$815$		−	1.03670e8i		−	0.191506i
$816$			2.38384e8i			0.438739i
$817$	5.47448e8			1.00387
$818$	9.25039e7			0.169005
$819$			4.32457e8i			0.787211i
$820$		−	5.42255e6i		−	0.00983472i
$821$	7.91944e8			1.43108			0.715542	−	0.698570i	$-0.246180\pi$
$821$	7.91944e8			1.43108			0.715542	+	0.698570i	$0.246180\pi$
$822$		−	1.53639e8i		−	0.276621i
$823$	−4.21544e8			−0.756212			−0.378106	−	0.925762i	$-0.623424\pi$
$823$	−4.21544e8			−0.756212			−0.378106	+	0.925762i	$0.623424\pi$
$824$			4.81183e8i			0.860061i
$825$		−	3.41220e8i		−	0.607677i
$826$		−	3.50923e8i		−	0.622690i
$827$			6.25183e8i			1.10533i	0.833404	+	0.552664i	$0.186389\pi$
$827$			6.25183e8i			1.10533i	−0.833404	+	0.552664i	$0.813611\pi$
$828$	−1.67051e8	+	2.94624e8i	−0.294277	+	0.519012i
$829$	5.46404e8			0.959069			0.479535	−	0.877523i	$-0.340805\pi$
$829$	5.46404e8			0.959069			0.479535	+	0.877523i	$0.340805\pi$
$830$	1.59690e8			0.279282
$831$	2.94939e8			0.513959
$832$	−1.45411e9			−2.52480
$833$			9.26194e8i			1.60239i
$834$	−1.03225e8			−0.177946
$835$		−	2.07551e8i		−	0.356505i
$836$	−1.27548e9			−2.18301
$837$	1.04312e8			0.177893
$838$		−	5.16031e8i		−	0.876887i
$839$			5.10945e8i			0.865143i	0.901600	+	0.432571i	$0.142394\pi$
$839$			5.10945e8i			0.865143i	−0.901600	+	0.432571i	$0.857606\pi$
$840$	2.01556e8			0.340062
$841$	−5.81487e8			−0.977579
$842$			7.47863e8i			1.25281i
$843$			2.84008e8i			0.474076i
$844$	−2.57159e8			−0.427734
$845$			3.89383e8i			0.645368i
$846$	4.15913e8			0.686897
$847$		−	3.22142e8i		−	0.530149i
$848$		−	2.53155e8i		−	0.415144i
$849$		−	1.63267e8i		−	0.266794i
$850$		−	1.68308e9i		−	2.74061i
$851$	7.95378e8	+	4.50976e8i	1.29058	+	0.731753i
$852$	−2.14511e8			−0.346842
$853$	5.93821e8			0.956772			0.478386	−	0.878150i	$-0.341222\pi$
$853$	5.93821e8			0.956772			0.478386	+	0.878150i	$0.341222\pi$
$854$	−1.04780e9			−1.68230
$855$	−7.03755e7			−0.112596
$856$		−	3.80214e8i		−	0.606188i
$857$	2.88254e8			0.457965			0.228983	−	0.973431i	$-0.426460\pi$
$857$	2.88254e8			0.457965			0.228983	+	0.973431i	$0.426460\pi$
$858$		−	1.23817e9i		−	1.96028i
$859$	−6.27706e8			−0.990323			−0.495161	−	0.868801i	$-0.664891\pi$
$859$	−6.27706e8			−0.990323			−0.495161	+	0.868801i	$0.664891\pi$
$860$	3.60079e8			0.566112
$861$			8.49358e6i			0.0133070i
$862$		−	1.30677e9i		−	2.04022i
$863$	−7.29580e8			−1.13512			−0.567558	−	0.823333i	$-0.692112\pi$
$863$	−7.29580e8			−1.13512			−0.567558	+	0.823333i	$0.692112\pi$
$864$	7.79653e7			0.120882
$865$			8.74062e6i			0.0135050i
$866$		−	2.07876e9i		−	3.20075i
$867$	8.92374e8			1.36927
$868$		−	1.48067e9i		−	2.26412i
$869$	7.18894e8			1.09548
$870$			3.10199e7i			0.0471068i
$871$		−	7.56889e8i		−	1.14545i
$872$			1.36507e9i			2.05876i
$873$			2.08172e8i			0.312881i
$874$	−1.00444e9	−	5.69511e8i	−1.50449	−	0.853037i
$875$	−5.66313e8			−0.845342
$876$	−1.17786e9			−1.75220
$877$	2.96672e8			0.439822			0.219911	−	0.975520i	$-0.429423\pi$
$877$	2.96672e8			0.439822			0.219911	+	0.975520i	$0.429423\pi$
$878$	1.77232e8			0.261853
$879$		−	1.07904e8i		−	0.158880i
$880$	−1.08372e8			−0.159026
$881$			1.13251e8i			0.165621i	0.996565	+	0.0828103i	$0.0263896\pi$
$881$			1.13251e8i			0.165621i	−0.996565	+	0.0828103i	$0.973610\pi$
$882$	−3.33369e8			−0.485869
$883$	1.21809e8			0.176929			0.0884643	−	0.996079i	$-0.471804\pi$
$883$	1.21809e8			0.176929			0.0884643	+	0.996079i	$0.471804\pi$
$884$		−	3.91823e9i		−	5.67196i
$885$			3.55662e7i			0.0513106i
$886$	−1.76817e7			−0.0254228
$887$	4.35800e8			0.624477			0.312238	−	0.950004i	$-0.398921\pi$
$887$	4.35800e8			0.624477			0.312238	+	0.950004i	$0.398921\pi$
$888$			7.91340e8i			1.13012i
$889$			1.73214e9i			2.46535i
$890$	−2.34556e7			−0.0332719
$891$			9.25748e7i			0.130876i
$892$	−8.20883e8			−1.15661
$893$			9.09698e8i			1.27745i
$894$			7.68620e8i			1.07572i
$895$		−	4.05592e8i		−	0.565744i
$896$		−	1.78714e9i		−	2.48447i
$897$	3.54689e8	−	6.25559e8i	0.491440	−	0.866745i
$898$	−1.15412e9			−1.59376
$899$	1.00566e8			0.138411
$900$	3.88658e8			0.533138
$901$	−1.34725e9			−1.84193
$902$		−	2.43180e7i		−	0.0331367i
$903$	−5.64007e8			−0.765987
$904$		−	1.11796e9i		−	1.51328i
$905$	2.05922e7			0.0277816
$906$	−5.05772e8			−0.680096
$907$			2.67950e8i			0.359114i	0.983748	+	0.179557i	$0.0574664\pi$
$907$			2.67950e8i			0.359114i	−0.983748	+	0.179557i	$0.942534\pi$
$908$			7.96761e8i			1.06432i
$909$	−7.10034e7			−0.0945339
$910$	−9.69731e8			−1.28685
$911$		−	2.28173e8i		−	0.301792i	−0.988550	−	0.150896i	$-0.951784\pi$
$911$		−	2.28173e8i		−	0.301792i	0.988550	−	0.150896i	$-0.0482159\pi$
$912$		−	1.87670e8i		−	0.247406i
$913$	4.59455e8			0.603713
$914$			6.94147e8i			0.909103i
$915$	1.06195e8			0.138624
$916$			2.59735e8i			0.337943i
$917$		−	1.58536e8i		−	0.205598i
$918$			4.56627e8i			0.590247i
$919$			1.06953e9i			1.37799i	0.724765	+	0.688996i	$0.241948\pi$
$919$			1.06953e9i			1.37799i	−0.724765	+	0.688996i	$0.758052\pi$
$920$	−2.91556e8	−	1.65311e8i	−0.374419	−	0.212294i
$921$	7.67964e7			0.0983019
$922$	−6.24150e8			−0.796336
$923$	4.55460e8			0.579222
$924$	1.31406e9			1.66572
$925$		−	1.04923e9i		−	1.32571i
$926$	7.52844e8			0.948139
$927$			1.73092e8i			0.217290i
$928$	7.51649e7			0.0940527
$929$	−9.78729e8			−1.22072			−0.610359	−	0.792125i	$-0.708975\pi$
$929$	−9.78729e8			−1.22072			−0.610359	+	0.792125i	$0.708975\pi$
$930$			2.33907e8i			0.290800i
$931$		−	7.29154e8i		−	0.903588i
$932$	−9.59773e8			−1.18555
$933$	3.22910e8			0.397591
$934$			1.81879e9i			2.23225i
$935$			5.76737e8i			0.705575i
$936$	6.22383e8			0.758980
$937$			1.06425e9i			1.29367i	0.762629	+	0.646836i	$0.223908\pi$
$937$			1.06425e9i			1.29367i	−0.762629	+	0.646836i	$0.776092\pi$
$938$	1.25207e9			1.51712
$939$		−	4.33895e7i		−	0.0524068i
$940$			5.98345e8i			0.720390i
$941$		−	2.04937e8i		−	0.245953i	−0.992410	−	0.122976i	$-0.960756\pi$
$941$		−	2.04937e8i		−	0.245953i	0.992410	−	0.122976i	$-0.0392439\pi$
$942$		−	7.62132e8i		−	0.911754i
$943$	6.96620e6	−	1.22862e7i	0.00830732	−	0.0146515i
$944$	−9.48439e7			−0.112744
$945$	7.25042e7			0.0859148
$946$	1.61481e9			1.90743
$947$	−1.19798e8			−0.141059			−0.0705296	−	0.997510i	$-0.522469\pi$
$947$	−1.19798e8			−0.141059			−0.0705296	+	0.997510i	$0.522469\pi$
$948$			8.18838e8i			0.961110i
$949$	2.50089e9			2.92615
$950$			1.32502e9i			1.54543i
$951$	−2.27514e8			−0.264525
$952$	2.86042e9			3.31528
$953$			1.31168e8i			0.151548i	0.997125	+	0.0757741i	$0.0241428\pi$
$953$			1.31168e8i			0.151548i	−0.997125	+	0.0757741i	$0.975857\pi$
$954$		−	4.84922e8i		−	0.558504i
$955$	3.20194e8			0.367623
$956$	−2.11745e9			−2.42348
$957$			8.92496e7i			0.101829i
$958$			7.91110e8i			0.899789i
$959$	−3.46207e8			−0.392537
$960$			2.43791e8i			0.275552i
$961$	−1.29185e8			−0.145560
$962$		−	3.80731e9i		−	4.27655i
$963$		−	1.36772e8i		−	0.153150i
$964$			1.30365e9i			1.45522i
$965$		−	5.22553e8i		−	0.581498i
$966$	1.03482e9	+	5.86737e8i	1.14798	+	0.650897i
$967$	2.06255e8			0.228100			0.114050	−	0.993475i	$-0.463618\pi$
$967$	2.06255e8			0.228100			0.114050	+	0.993475i	$0.463618\pi$
$968$	−4.63621e8			−0.511137
$969$	−9.98749e8			−1.09770
$970$	−4.66799e8			−0.511464
$971$			3.50529e8i			0.382883i	0.981504	+	0.191442i	$0.0613162\pi$
$971$			3.50529e8i			0.382883i	−0.981504	+	0.191442i	$0.938684\pi$
$972$	−1.05445e8			−0.114822
$973$			2.32607e8i			0.252513i
$974$	−2.51508e8			−0.272192
$975$	−8.25215e8			−0.890335
$976$			2.83188e8i			0.304597i
$977$			6.80555e8i			0.729760i	0.931055	+	0.364880i	$0.118890\pi$
$977$			6.80555e8i			0.729760i	−0.931055	+	0.364880i	$0.881110\pi$
$978$	5.29557e8			0.566104
$979$	−6.74858e7			−0.0719224
$980$		−	4.79594e8i		−	0.509560i
$981$			4.91046e8i			0.520134i
$982$	7.98840e8			0.843578
$983$		−	9.28861e8i		−	0.977890i	−0.872315	−	0.488945i	$-0.837382\pi$
$983$		−	9.28861e8i		−	0.977890i	0.872315	−	0.488945i	$-0.162618\pi$
$984$	1.22238e7			0.0128298
$985$			4.56901e8i			0.478094i
$986$			4.40226e8i			0.459246i
$987$		−	9.37214e8i		−	0.974736i
$988$			3.08466e9i			3.19843i
$989$	8.15850e8	+	4.62583e8i	0.843376	+	0.478190i
$990$	−2.07587e8			−0.213942
$991$	6.12594e8			0.629436			0.314718	−	0.949185i	$-0.398090\pi$
$991$	6.12594e8			0.629436			0.314718	+	0.949185i	$0.398090\pi$
$992$	5.66784e8			0.580608
$993$	−3.64179e7			−0.0371935
$994$			7.53435e8i			0.767162i
$995$	3.45997e8			0.351239
$996$			5.23330e8i			0.529661i
$997$	−1.65349e8			−0.166846			−0.0834231	−	0.996514i	$-0.526585\pi$
$997$	−1.65349e8			−0.166846			−0.0834231	+	0.996514i	$0.526585\pi$
$998$	−1.01680e9			−1.02293
$999$			2.84663e8i			0.285518i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.7.d.a.22.22	yes	24
3.2	odd	2		207.7.d.e.91.3		24
23.22	odd	2	inner	69.7.d.a.22.21	✓	24
69.68	even	2		207.7.d.e.91.4		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.7.d.a.22.21	✓	24	23.22	odd	2	inner
69.7.d.a.22.22	yes	24	1.1	even	1	trivial
207.7.d.e.91.3		24	3.2	odd	2
207.7.d.e.91.4		24	69.68	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	69.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+13.3624 q^{2} -15.5885 q^{3} +114.554 q^{4} +40.7784i q^{5} -208.299 q^{6} +469.379i q^{7} +675.521 q^{8} +243.000 q^{9} +O(q^{10})\)	\(q+13.3624 q^{2} -15.5885 q^{3} +114.554 q^{4} +40.7784i q^{5} -208.299 q^{6} +469.379i q^{7} +675.521 q^{8} +243.000 q^{9} +544.897i q^{10} +1567.76i q^{11} -1785.72 q^{12} +3791.51 q^{13} +6272.03i q^{14} -635.672i q^{15} +1695.14 q^{16} -9021.27i q^{17} +3247.06 q^{18} +7102.08i q^{19} +4671.32i q^{20} -7316.90i q^{21} +20949.1i q^{22} +(-6001.12 + 10584.1i) q^{23} -10530.3 q^{24} +13962.1 q^{25} +50663.7 q^{26} -3788.00 q^{27} +53769.2i q^{28} -3651.94 q^{29} -8494.11i q^{30} -27537.6 q^{31} -20582.2 q^{32} -24439.0i q^{33} -120546. i q^{34} -19140.5 q^{35} +27836.6 q^{36} -75148.6i q^{37} +94900.8i q^{38} -59103.9 q^{39} +27546.7i q^{40} -1160.82 q^{41} -97771.3i q^{42} -77082.8i q^{43} +179593. i q^{44} +9909.15i q^{45} +(-80189.3 + 141429. i) q^{46} +128089. q^{47} -26424.6 q^{48} -102668. q^{49} +186568. q^{50} +140628. i q^{51} +434333. q^{52} -149342. i q^{53} -50616.7 q^{54} -63930.8 q^{55} +317076. i q^{56} -110710. i q^{57} -48798.7 q^{58} -55950.5 q^{59} -72818.7i q^{60} +167059. i q^{61} -367968. q^{62} +114059. i q^{63} -383517. q^{64} +154612. i q^{65} -326564. i q^{66} -199627. i q^{67} -1.03342e6i q^{68} +(93548.1 - 164989. i) q^{69} -255763. q^{70} +120126. q^{71} +164152. q^{72} +659602. q^{73} -1.00417e6i q^{74} -217648. q^{75} +813570. i q^{76} -735875. q^{77} -789770. q^{78} -458548. i q^{79} +69125.1i q^{80} +59049.0 q^{81} -15511.3 q^{82} -293064. i q^{83} -838179. i q^{84} +367873. q^{85} -1.03001e6i q^{86} +56928.1 q^{87} +1.05906e6i q^{88} +43046.0i q^{89} +132410. i q^{90} +1.77966e6i q^{91} +(-687451. + 1.21245e6i) q^{92} +429269. q^{93} +1.71158e6 q^{94} -289611. q^{95} +320845. q^{96} +856674. i q^{97} -1.37189e6 q^{98} +380966. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 20 q^{2} + 816 q^{4} - 324 q^{6} - 940 q^{8} + 5832 q^{9}+O(q^{10}) \) 24 * q - 20 * q^2 + 816 * q^4 - 324 * q^6 - 940 * q^8 + 5832 * q^9	\( 24 q - 20 q^{2} + 816 q^{4} - 324 q^{6} - 940 q^{8} + 5832 q^{9} + 384 q^{13} + 29544 q^{16} - 4860 q^{18} + 29336 q^{23} - 39204 q^{24} - 61272 q^{25} + 10088 q^{26} + 64672 q^{29} + 9696 q^{31} - 319620 q^{32} - 225744 q^{35} + 198288 q^{36} - 11664 q^{39} + 135280 q^{41} + 233232 q^{46} - 74336 q^{47} + 552096 q^{48} - 722136 q^{49} + 619324 q^{50} + 1059720 q^{52} - 78732 q^{54} - 1019328 q^{55} - 694344 q^{58} + 1057648 q^{59} - 488776 q^{62} - 273888 q^{64} - 23328 q^{69} + 2785512 q^{70} - 255392 q^{71} - 228420 q^{72} - 322560 q^{73} - 365472 q^{75} - 1002960 q^{77} - 171072 q^{78} + 1417176 q^{81} - 5732712 q^{82} - 2704704 q^{85} + 611712 q^{87} - 1611444 q^{92} + 2484432 q^{93} - 147720 q^{94} - 1672656 q^{95} - 1818612 q^{96} + 9104212 q^{98}+O(q^{100}) \) 24 * q - 20 * q^2 + 816 * q^4 - 324 * q^6 - 940 * q^8 + 5832 * q^9 + 384 * q^13 + 29544 * q^16 - 4860 * q^18 + 29336 * q^23 - 39204 * q^24 - 61272 * q^25 + 10088 * q^26 + 64672 * q^29 + 9696 * q^31 - 319620 * q^32 - 225744 * q^35 + 198288 * q^36 - 11664 * q^39 + 135280 * q^41 + 233232 * q^46 - 74336 * q^47 + 552096 * q^48 - 722136 * q^49 + 619324 * q^50 + 1059720 * q^52 - 78732 * q^54 - 1019328 * q^55 - 694344 * q^58 + 1057648 * q^59 - 488776 * q^62 - 273888 * q^64 - 23328 * q^69 + 2785512 * q^70 - 255392 * q^71 - 228420 * q^72 - 322560 * q^73 - 365472 * q^75 - 1002960 * q^77 - 171072 * q^78 + 1417176 * q^81 - 5732712 * q^82 - 2704704 * q^85 + 611712 * q^87 - 1611444 * q^92 + 2484432 * q^93 - 147720 * q^94 - 1672656 * q^95 - 1818612 * q^96 + 9104212 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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