LMFDB - Embedded newform 69.8.c.b.68.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,8,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, 8, names="a")

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

chi := DirichletCharacter("69.68");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 8 $
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:	$21.5545667584$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			68.2
Character	$\chi$	$=$	69.68
Dual form			69.8.c.b.68.48

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-20.7157i q^{2} +(11.8197 + 45.2471i) q^{3} -301.139 q^{4} +470.966 q^{5} +(937.323 - 244.852i) q^{6} +481.517i q^{7} +3586.70i q^{8} +(-1907.59 + 1069.61i) q^{9} +O(q^{10})$	\(q-20.7157i q^{2} +(11.8197 + 45.2471i) q^{3} -301.139 q^{4} +470.966 q^{5} +(937.323 - 244.852i) q^{6} +481.517i q^{7} +3586.70i q^{8} +(-1907.59 + 1069.61i) q^{9} -9756.38i q^{10} -6886.26 q^{11} +(-3559.36 - 13625.7i) q^{12} -10688.0 q^{13} +9974.96 q^{14} +(5566.66 + 21309.8i) q^{15} +35755.0 q^{16} -21621.3 q^{17} +(22157.7 + 39517.0i) q^{18} -7536.84i q^{19} -141826. q^{20} +(-21787.2 + 5691.37i) q^{21} +142654. i q^{22} +(-55075.6 - 19274.4i) q^{23} +(-162287. + 42393.5i) q^{24} +143684. q^{25} +221409. i q^{26} +(-70943.7 - 73670.5i) q^{27} -145004. i q^{28} -52802.1i q^{29} +(441447. - 115317. i) q^{30} +21912.7 q^{31} -281592. i q^{32} +(-81393.3 - 311583. i) q^{33} +447900. i q^{34} +226778. i q^{35} +(574451. - 322101. i) q^{36} +292394. i q^{37} -156131. q^{38} +(-126328. - 483600. i) q^{39} +1.68921e6i q^{40} +129303. i q^{41} +(117901. + 451337. i) q^{42} -901214. i q^{43} +2.07372e6 q^{44} +(-898411. + 503750. i) q^{45} +(-399283. + 1.14093e6i) q^{46} +1.13897e6i q^{47} +(422612. + 1.61781e6i) q^{48} +591684. q^{49} -2.97651e6i q^{50} +(-255556. - 978300. i) q^{51} +3.21857e6 q^{52} -79786.1 q^{53} +(-1.52613e6 + 1.46965e6i) q^{54} -3.24320e6 q^{55} -1.72706e6 q^{56} +(341020. - 89082.9i) q^{57} -1.09383e6 q^{58} +480616. i q^{59} +(-1.67634e6 - 6.41722e6i) q^{60} -234873. i q^{61} -453937. i q^{62} +(-515035. - 918538. i) q^{63} -1.25673e6 q^{64} -5.03368e6 q^{65} +(-6.45465e6 + 1.68612e6i) q^{66} -812954. i q^{67} +6.51102e6 q^{68} +(221136. - 2.71983e6i) q^{69} +4.69787e6 q^{70} +3.86363e6i q^{71} +(-3.83636e6 - 6.84195e6i) q^{72} +2.91162e6 q^{73} +6.05714e6 q^{74} +(1.69830e6 + 6.50128e6i) q^{75} +2.26964e6i q^{76} -3.31586e6i q^{77} +(-1.00181e7 + 2.61698e6i) q^{78} +7.24966e6i q^{79} +1.68394e7 q^{80} +(2.49484e6 - 4.08076e6i) q^{81} +2.67860e6 q^{82} -4.03139e6 q^{83} +(6.56099e6 - 1.71390e6i) q^{84} -1.01829e7 q^{85} -1.86692e7 q^{86} +(2.38914e6 - 624103. i) q^{87} -2.46989e7i q^{88} -1.11969e6 q^{89} +(1.04355e7 + 1.86112e7i) q^{90} -5.14646e6i q^{91} +(1.65854e7 + 5.80429e6i) q^{92} +(259001. + 991486. i) q^{93} +2.35946e7 q^{94} -3.54960e6i q^{95} +(1.27412e7 - 3.32832e6i) q^{96} +336972. i q^{97} -1.22571e7i q^{98} +(1.31362e7 - 7.36561e6i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 24 q^{3} - 2616 q^{4} + 4416 q^{6} + 648 q^{9}+O(q^{10}) $ 48 * q + 24 * q^3 - 2616 * q^4 + 4416 * q^6 + 648 * q^9	$ 48 q + 24 q^{3} - 2616 q^{4} + 4416 q^{6} + 648 q^{9} + 16668 q^{12} - 4992 q^{13} + 76296 q^{16} + 100548 q^{18} - 620268 q^{24} + 1257120 q^{25} - 105408 q^{27} - 738480 q^{31} - 1227096 q^{36} + 3832248 q^{39} - 489768 q^{46} + 733404 q^{48} - 5569584 q^{49} - 5646984 q^{52} + 1260252 q^{54} - 305760 q^{55} + 16920336 q^{58} + 6032664 q^{64} + 4496304 q^{69} + 23272152 q^{70} - 26866080 q^{72} - 2542752 q^{73} + 487176 q^{75} - 47724612 q^{78} + 25279128 q^{81} - 68451480 q^{82} - 18046992 q^{85} + 42338184 q^{87} - 59017272 q^{93} + 106197744 q^{94} + 67778460 q^{96}+O(q^{100}) $ 48 * q + 24 * q^3 - 2616 * q^4 + 4416 * q^6 + 648 * q^9 + 16668 * q^12 - 4992 * q^13 + 76296 * q^16 + 100548 * q^18 - 620268 * q^24 + 1257120 * q^25 - 105408 * q^27 - 738480 * q^31 - 1227096 * q^36 + 3832248 * q^39 - 489768 * q^46 + 733404 * q^48 - 5569584 * q^49 - 5646984 * q^52 + 1260252 * q^54 - 305760 * q^55 + 16920336 * q^58 + 6032664 * q^64 + 4496304 * q^69 + 23272152 * q^70 - 26866080 * q^72 - 2542752 * q^73 + 487176 * q^75 - 47724612 * q^78 + 25279128 * q^81 - 68451480 * q^82 - 18046992 * q^85 + 42338184 * q^87 - 59017272 * q^93 + 106197744 * q^94 + 67778460 * 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 $

$2$		−	20.7157i		−	1.83102i	−0.402290	−	0.915512i	$-0.631786\pi$
$2$		−	20.7157i		−	1.83102i	0.402290	−	0.915512i	$-0.368214\pi$
$3$	11.8197	+	45.2471i	0.252744	+	0.967533i
$3$	11.8197	+	45.2471i	0.252744	+	0.967533i
$4$	−301.139			−2.35265
$5$	470.966			1.68498			0.842490	−	0.538713i	$-0.181089\pi$
$5$	470.966			1.68498			0.842490	+	0.538713i	$0.181089\pi$
$6$	937.323	−	244.852i	1.77158	−	0.462780i
$7$			481.517i			0.530602i	0.964166	+	0.265301i	$0.0854713\pi$
$7$			481.517i			0.530602i	−0.964166	+	0.265301i	$0.914529\pi$
$8$			3586.70i			2.47674i
$9$	−1907.59	+	1069.61i	−0.872241	+	0.489076i
$10$		−	9756.38i		−	3.08524i
$11$	−6886.26			−1.55994			−0.779972	−	0.625814i	$-0.784767\pi$
$11$	−6886.26			−1.55994			−0.779972	+	0.625814i	$0.784767\pi$
$12$	−3559.36	−	13625.7i	−0.594618	−	2.27627i
$13$	−10688.0			−1.34926			−0.674628	−	0.738158i	$-0.735696\pi$
$13$	−10688.0			−1.34926			−0.674628	+	0.738158i	$0.735696\pi$
$14$	9974.96			0.971545
$15$	5566.66	+	21309.8i	0.425868	+	1.63027i
$16$	35755.0			2.18231
$17$	−21621.3			−1.06736			−0.533680	−	0.845687i	$-0.679191\pi$
$17$	−21621.3			−1.06736			−0.533680	+	0.845687i	$0.679191\pi$
$18$	22157.7	+	39517.0i	0.895510	+	1.59709i
$19$		−	7536.84i		−	0.252088i	−0.992025	−	0.126044i	$-0.959772\pi$
$19$		−	7536.84i		−	0.252088i	0.992025	−	0.126044i	$-0.0402280\pi$
$20$	−141826.			−3.96417
$21$	−21787.2	+	5691.37i	−0.513375	+	0.134106i
$22$			142654.i			2.85630i
$23$	−55075.6	−	19274.4i	−0.943869	−	0.330319i
$23$	−55075.6	−	19274.4i	−0.943869	−	0.330319i
$24$	−162287.	+	42393.5i	−2.39632	+	0.625980i
$25$	143684.			1.83916
$26$			221409.i			2.47052i
$27$	−70943.7	−	73670.5i	−0.693651	−	0.720311i
$28$		−	145004.i		−	1.24832i
$29$		−	52802.1i		−	0.402030i	−0.979588	−	0.201015i	$-0.935576\pi$
$29$		−	52802.1i		−	0.402030i	0.979588	−	0.201015i	$-0.0644240\pi$
$30$	441447.	−	115317.i	2.98507	−	0.779775i
$31$	21912.7			0.132108			0.0660542	−	0.997816i	$-0.478959\pi$
$31$	21912.7			0.132108			0.0660542	+	0.997816i	$0.478959\pi$
$32$		−	281592.i		−	1.51913i
$33$	−81393.3	−	311583.i	−0.394266	−	1.50930i
$34$			447900.i			1.95436i
$35$			226778.i			0.894053i
$36$	574451.	−	322101.i	2.05208	−	1.15062i
$37$			292394.i			0.948991i	0.880258	+	0.474496i	$0.157370\pi$
$37$			292394.i			0.948991i	−0.880258	+	0.474496i	$0.842630\pi$
$38$	−156131.			−0.461579
$39$	−126328.	−	483600.i	−0.341016	−	1.30545i
$40$			1.68921e6i			4.17325i
$41$			129303.i			0.292999i	0.989211	+	0.146499i	$0.0468007\pi$
$41$			129303.i			0.292999i	−0.989211	+	0.146499i	$0.953199\pi$
$42$	117901.	+	451337.i	0.245552	+	0.940002i
$43$		−	901214.i		−	1.72857i	−0.502998	−	0.864287i	$-0.667770\pi$
$43$		−	901214.i		−	1.72857i	0.502998	−	0.864287i	$-0.332230\pi$
$44$	2.07372e6			3.67000
$45$	−898411.	+	503750.i	−1.46971	+	0.824083i
$46$	−399283.	+	1.14093e6i	−0.604823	+	1.72825i
$47$			1.13897e6i			1.60019i	0.599875	+	0.800093i	$0.295217\pi$
$47$			1.13897e6i			1.60019i	−0.599875	+	0.800093i	$0.704783\pi$
$48$	422612.	+	1.61781e6i	0.551566	+	2.11146i
$49$	591684.			0.718462
$50$		−	2.97651e6i		−	3.36754i
$51$	−255556.	−	978300.i	−0.269768	−	1.03271i
$52$	3.21857e6			3.17433
$53$	−79786.1			−0.0736142			−0.0368071	−	0.999322i	$-0.511719\pi$
$53$	−79786.1			−0.0736142			−0.0368071	+	0.999322i	$0.511719\pi$
$54$	−1.52613e6	+	1.46965e6i	−1.31891	+	1.27009i
$55$	−3.24320e6			−2.62847
$56$	−1.72706e6			−1.31416
$57$	341020.	−	89082.9i	0.243903	−	0.0637136i
$58$	−1.09383e6			−0.736127
$59$			480616.i			0.304661i	0.988330	+	0.152330i	$0.0486778\pi$
$59$			480616.i			0.304661i	−0.988330	+	0.152330i	$0.951322\pi$
$60$	−1.67634e6	−	6.41722e6i	−1.00192	−	3.83546i
$61$		−	234873.i		−	0.132488i	−0.997803	−	0.0662442i	$-0.978898\pi$
$61$		−	234873.i		−	0.132488i	0.997803	−	0.0662442i	$-0.0211016\pi$
$62$		−	453937.i		−	0.241894i
$63$	−515035.	−	918538.i	−0.259505	−	0.462813i
$64$	−1.25673e6			−0.599257
$65$	−5.03368e6			−2.27347
$66$	−6.45465e6	+	1.68612e6i	−2.76356	+	0.721911i
$67$		−	812954.i		−	0.330221i	−0.986275	−	0.165110i	$-0.947202\pi$
$67$		−	812954.i		−	0.330221i	0.986275	−	0.165110i	$-0.0527980\pi$
$68$	6.51102e6			2.51112
$69$	221136.	−	2.71983e6i	0.0810379	−	0.996711i
$70$	4.69787e6			1.63703
$71$			3.86363e6i			1.28113i	0.767906	+	0.640563i	$0.221299\pi$
$71$			3.86363e6i			1.28113i	−0.767906	+	0.640563i	$0.778701\pi$
$72$	−3.83636e6	−	6.84195e6i	−1.21131	−	2.16031i
$73$	2.91162e6			0.876002			0.438001	−	0.898975i	$-0.355687\pi$
$73$	2.91162e6			0.876002			0.438001	+	0.898975i	$0.355687\pi$
$74$	6.05714e6			1.73763
$75$	1.69830e6	+	6.50128e6i	0.464835	+	1.77944i
$76$			2.26964e6i			0.593075i
$77$		−	3.31586e6i		−	0.827710i
$78$	−1.00181e7	+	2.61698e6i	−2.39031	+	0.624409i
$79$			7.24966e6i			1.65433i	0.561958	+	0.827166i	$0.310048\pi$
$79$			7.24966e6i			1.65433i	−0.561958	+	0.827166i	$0.689952\pi$
$80$	1.68394e7			3.67715
$81$	2.49484e6	−	4.08076e6i	0.521609	−	0.853184i
$82$	2.67860e6			0.536488
$83$	−4.03139e6			−0.773893			−0.386947	−	0.922102i	$-0.626470\pi$
$83$	−4.03139e6			−0.773893			−0.386947	+	0.922102i	$0.626470\pi$
$84$	6.56099e6	−	1.71390e6i	1.20779	−	0.315505i
$85$	−1.01829e7			−1.79848
$86$	−1.86692e7			−3.16506
$87$	2.38914e6	−	624103.i	0.388977	−	0.101611i
$88$		−	2.46989e7i		−	3.86357i
$89$	−1.11969e6			−0.168357			−0.0841786	−	0.996451i	$-0.526827\pi$
$89$	−1.11969e6			−0.168357			−0.0841786	+	0.996451i	$0.526827\pi$
$90$	1.04355e7	+	1.86112e7i	1.50892	+	2.69107i
$91$		−	5.14646e6i		−	0.715918i
$92$	1.65854e7	+	5.80429e6i	2.22059	+	0.777126i
$93$	259001.	+	991486.i	0.0333896	+	0.127819i
$94$	2.35946e7			2.92998
$95$		−	3.54960e6i		−	0.424763i
$96$	1.27412e7	−	3.32832e6i	1.46981	−	0.383951i
$97$			336972.i			0.0374880i	0.999824	+	0.0187440i	$0.00596675\pi$
$97$			336972.i			0.0374880i	−0.999824	+	0.0187440i	$0.994033\pi$
$98$		−	1.22571e7i		−	1.31552i
$99$	1.31362e7	−	7.36561e6i	1.36065	−	0.762932i
$100$	−4.32689e7			−4.32689
$101$		−	8.61978e6i		−	0.832476i	−0.909256	−	0.416238i	$-0.863348\pi$
$101$		−	8.61978e6i		−	0.832476i	0.909256	−	0.416238i	$-0.136652\pi$
$102$	−2.02662e7	+	5.29402e6i	−1.89091	+	0.493953i
$103$			2.76798e6i			0.249593i	0.992182	+	0.124796i	$0.0398278\pi$
$103$			2.76798e6i			0.249593i	−0.992182	+	0.124796i	$0.960172\pi$
$104$		−	3.83346e7i		−	3.34175i
$105$	−1.02611e7	+	2.68044e6i	−0.865026	+	0.225966i
$106$			1.65282e6i			0.134789i
$107$	−4.70378e6			−0.371196			−0.185598	−	0.982626i	$-0.559422\pi$
$107$	−4.70378e6			−0.371196			−0.185598	+	0.982626i	$0.559422\pi$
$108$	2.13639e7	+	2.21851e7i	1.63192	+	1.69464i
$109$		−	1.39439e7i		−	1.03132i	−0.856794	−	0.515658i	$-0.827547\pi$
$109$		−	1.39439e7i		−	1.03132i	0.856794	−	0.515658i	$-0.172453\pi$
$110$			6.71850e7i			4.81280i
$111$	−1.32300e7	+	3.45600e6i	−0.918181	+	0.239852i
$112$			1.72167e7i			1.15794i
$113$	−671251.			−0.0437633			−0.0218817	−	0.999761i	$-0.506966\pi$
$113$	−671251.			−0.0437633			−0.0218817	+	0.999761i	$0.506966\pi$
$114$	−1.84541e6	−	7.06446e6i	−0.116661	−	0.446593i
$115$	−2.59387e7	−	9.07760e6i	−1.59040	−	0.556581i
$116$			1.59008e7i			0.945836i
$117$	2.03883e7	−	1.14320e7i	1.17688	−	0.659889i
$118$	9.95629e6			0.557841
$119$		−	1.04110e7i		−	0.566343i
$120$	−7.64319e7	+	1.99659e7i	−4.03776	+	1.05476i
$121$	2.79334e7			1.43343
$122$	−4.86554e6			−0.242590
$123$	−5.85059e6	+	1.52832e6i	−0.283486	+	0.0740537i
$124$	−6.59878e6			−0.310805
$125$	3.08761e7			1.41396
$126$	−1.90281e7	+	1.06693e7i	−0.847422	+	0.475159i
$127$	2.12578e7			0.920886			0.460443	−	0.887689i	$-0.347691\pi$
$127$	2.12578e7			0.920886			0.460443	+	0.887689i	$0.347691\pi$
$128$		−	1.00098e7i		−	0.421880i
$129$	4.07773e7	−	1.06520e7i	1.67245	−	0.436887i
$130$			1.04276e8i			4.16278i
$131$			2.05564e7i			0.798908i	0.916753	+	0.399454i	$0.130800\pi$
$131$			2.05564e7i			0.798908i	−0.916753	+	0.399454i	$0.869200\pi$
$132$	2.45107e7	+	9.38299e7i	0.927571	+	3.55085i
$133$	3.62912e6			0.133758
$134$	−1.68409e7			−0.604642
$135$	−3.34121e7	−	3.46963e7i	−1.16879	−	1.21371i
$136$		−	7.75491e7i		−	2.64357i
$137$	1.66865e7			0.554425			0.277212	−	0.960809i	$-0.410589\pi$
$137$	1.66865e7			0.554425			0.277212	+	0.960809i	$0.410589\pi$
$138$	−5.63430e7	−	4.58099e6i	−1.82500	−	0.148382i
$139$	−3.01575e7			−0.952452			−0.476226	−	0.879323i	$-0.657996\pi$
$139$	−3.01575e7			−0.952452			−0.476226	+	0.879323i	$0.657996\pi$
$140$		−	6.82919e7i		−	2.10339i
$141$	−5.15351e7	+	1.34623e7i	−1.54823	+	0.404437i
$142$	8.00378e7			2.34577
$143$	7.36003e7			2.10477
$144$	−6.82060e7	+	3.82439e7i	−1.90350	+	1.06732i
$145$		−	2.48680e7i		−	0.677412i
$146$		−	6.03162e7i		−	1.60398i
$147$	6.99350e6	+	2.67720e7i	0.181587	+	0.695135i
$148$		−	8.80513e7i		−	2.23264i
$149$	−5.15309e7			−1.27619			−0.638096	−	0.769957i	$-0.720278\pi$
$149$	−5.15309e7			−1.27619			−0.638096	+	0.769957i	$0.720278\pi$
$150$	1.34678e8	−	3.51813e7i	3.25821	−	0.851124i
$151$	−1.70010e7			−0.401843			−0.200921	−	0.979607i	$-0.564394\pi$
$151$	−1.70010e7			−0.401843			−0.200921	+	0.979607i	$0.564394\pi$
$152$	2.70324e7			0.624355
$153$	4.12446e7	−	2.31264e7i	0.930995	−	0.522020i
$154$	−6.86902e7			−1.51556
$155$	1.03201e7			0.222600
$156$	3.80424e7	+	1.45631e8i	0.802292	+	3.07127i
$157$			4.40744e7i			0.908946i	0.890761	+	0.454473i	$0.150172\pi$
$157$			4.40744e7i			0.908946i	−0.890761	+	0.454473i	$0.849828\pi$
$158$	1.50182e8			3.02912
$159$	−943044.	−	3.61008e6i	−0.0186055	−	0.0712242i
$160$		−	1.32620e8i		−	2.55971i
$161$	9.28097e6	−	2.65199e7i	0.175268	−	0.500819i
$162$	−8.45356e7	−	5.16823e7i	−1.56220	−	0.955079i
$163$	−2.34480e7			−0.424082			−0.212041	−	0.977261i	$-0.568011\pi$
$163$	−2.34480e7			−0.424082			−0.212041	+	0.977261i	$0.568011\pi$
$164$		−	3.89383e7i		−	0.689324i
$165$	−3.83335e7	−	1.46745e8i	−0.664331	−	2.54314i
$166$			8.35129e7i			1.41702i
$167$			1.01651e8i			1.68889i	0.535640	+	0.844446i	$0.320070\pi$
$167$			1.01651e8i			1.68889i	−0.535640	+	0.844446i	$0.679930\pi$
$168$	−2.04132e7	−	7.81442e7i	−0.332146	−	1.27149i
$169$	5.14847e7			0.820493
$170$			2.10946e8i			3.29306i
$171$	8.06148e6	+	1.43772e7i	0.123290	+	0.219881i
$172$			2.71391e8i			4.06673i
$173$		−	7.68918e7i		−	1.12906i	−0.825411	−	0.564532i	$-0.809057\pi$
$173$		−	7.68918e7i		−	1.12906i	0.825411	−	0.564532i	$-0.190943\pi$
$174$	−1.29287e7	−	4.94927e7i	−0.186052	−	0.712227i
$175$			6.91863e7i			0.975859i
$176$	−2.46218e8			−3.40429
$177$	−2.17465e7	+	5.68072e6i	−0.294769	+	0.0770011i
$178$			2.31951e7i			0.308266i
$179$			1.14478e7i			0.149188i	0.997214	+	0.0745942i	$0.0237662\pi$
$179$			1.14478e7i			0.149188i	−0.997214	+	0.0745942i	$0.976234\pi$
$180$	2.70547e8	−	1.51699e8i	3.45771	−	1.93878i
$181$			5.66486e7i			0.710091i	0.934849	+	0.355046i	$0.115535\pi$
$181$			5.66486e7i			0.710091i	−0.934849	+	0.355046i	$0.884465\pi$
$182$	−1.06612e8			−1.31086
$183$	1.06273e7	−	2.77611e6i	0.128187	−	0.0334856i
$184$	6.91315e7	−	1.97539e8i	0.818114	−	2.33771i
$185$			1.37708e8i			1.59903i
$186$	2.05393e7	−	5.36538e6i	0.234040	−	0.0611371i
$187$	1.48890e8			1.66502
$188$		−	3.42989e8i		−	3.76468i
$189$	3.54736e7	−	3.41606e7i	0.382199	−	0.368052i
$190$	−7.35323e7			−0.777751
$191$	−1.60623e8			−1.66798			−0.833991	−	0.551778i	$-0.813950\pi$
$191$	−1.60623e8			−1.66798			−0.833991	+	0.551778i	$0.813950\pi$
$192$	−1.48541e7	−	5.68634e7i	−0.151458	−	0.579801i
$193$	1.13967e8			1.14111			0.570554	−	0.821260i	$-0.306729\pi$
$193$	1.13967e8			1.14111			0.570554	+	0.821260i	$0.306729\pi$
$194$	6.98059e6			0.0686414
$195$	−5.94964e7	−	2.27759e8i	−0.574605	−	2.19966i
$196$	−1.78179e8			−1.69029
$197$		−	1.57294e8i		−	1.46582i	−0.680324	−	0.732912i	$-0.738161\pi$
$197$		−	1.57294e8i		−	1.46582i	0.680324	−	0.732912i	$-0.261839\pi$
$198$	−1.52584e8	−	2.72125e8i	−1.39695	−	2.49138i
$199$		−	1.77663e8i		−	1.59812i	−0.601249	−	0.799062i	$-0.705330\pi$
$199$		−	1.77663e8i		−	1.59812i	0.601249	−	0.799062i	$-0.294670\pi$
$200$			5.15351e8i			4.55510i
$201$	3.67838e7	−	9.60884e6i	0.319499	−	0.0834612i
$202$	−1.78565e8			−1.52428
$203$	2.54252e7			0.213318
$204$	7.69581e7	+	2.94605e8i	0.634671	+	2.42960i
$205$			6.08974e7i			0.493697i
$206$	5.73405e7			0.457011
$207$	1.25678e8	−	2.21416e7i	0.984833	−	0.173506i
$208$	−3.82149e8			−2.94450
$209$			5.19007e7i			0.393243i
$210$	5.55272e7	+	2.12565e8i	0.413750	+	1.58388i
$211$	−1.35113e8			−0.990168			−0.495084	−	0.868845i	$-0.664863\pi$
$211$	−1.35113e8			−0.990168			−0.495084	+	0.868845i	$0.664863\pi$
$212$	2.40267e7			0.173188
$213$	−1.74818e8	+	4.56668e7i	−1.23953	+	0.323797i
$214$			9.74420e7i			0.679669i
$215$		−	4.24441e8i		−	2.91261i
$216$	2.64234e8	−	2.54454e8i	1.78402	−	1.71799i
$217$			1.05514e7i			0.0700970i
$218$	−2.88858e8			−1.88837
$219$	3.44144e7	+	1.31742e8i	0.221404	+	0.847561i
$220$	9.76653e8			6.18388
$221$	2.31088e8			1.44014
$222$	7.15933e7	+	2.74068e8i	0.439174	+	1.68121i
$223$	−1.29884e8			−0.784311			−0.392156	−	0.919899i	$-0.628271\pi$
$223$	−1.29884e8			−0.784311			−0.392156	+	0.919899i	$0.628271\pi$
$224$	1.35592e8			0.806055
$225$	−2.74090e8	+	1.53686e8i	−1.60419	+	0.899487i
$226$			1.39054e7i			0.0801317i
$227$	−3.25671e8			−1.84794			−0.923971	−	0.382462i	$-0.875076\pi$
$227$	−3.25671e8			−1.84794			−0.923971	+	0.382462i	$0.875076\pi$
$228$	−1.02694e8	+	2.68264e7i	−0.573819	+	0.149896i
$229$		−	2.58566e8i		−	1.42281i	−0.702782	−	0.711406i	$-0.748059\pi$
$229$		−	2.58566e8i		−	1.42281i	0.702782	−	0.711406i	$-0.251941\pi$
$230$	−1.88049e8	+	5.37338e8i	−1.01911	+	2.91206i
$231$	1.50033e8	−	3.91923e7i	0.800837	−	0.209198i
$232$	1.89385e8			0.995722
$233$			2.08070e8i			1.07761i	0.842430	+	0.538806i	$0.181125\pi$
$233$			2.08070e8i			1.07761i	−0.842430	+	0.538806i	$0.818875\pi$
$234$	−2.36821e8	−	4.22358e8i	−1.20827	−	2.15489i
$235$			5.36417e8i			2.69628i
$236$		−	1.44732e8i		−	0.716760i
$237$	−3.28026e8	+	8.56884e7i	−1.60062	+	0.418122i
$238$	−2.15672e8			−1.03699
$239$			5.80955e7i			0.275264i	0.990483	+	0.137632i	$0.0439491\pi$
$239$			5.80955e7i			0.275264i	−0.990483	+	0.137632i	$0.956051\pi$
$240$	1.99036e8	+	7.61933e8i	0.929378	+	3.55777i
$241$			1.78799e8i			0.822819i	0.911451	+	0.411410i	$0.134963\pi$
$241$			1.78799e8i			0.822819i	−0.911451	+	0.411410i	$0.865037\pi$
$242$		−	5.78660e8i		−	2.62464i
$243$	2.14130e8	+	6.46511e7i	0.957318	+	0.289037i
$244$			7.07294e7i			0.311699i
$245$	2.78663e8			1.21059
$246$	3.16602e7	+	1.21199e8i	0.135594	+	0.519070i
$247$			8.05537e7i			0.340131i
$248$			7.85943e7i			0.327198i
$249$	−4.76496e7	−	1.82408e8i	−0.195597	−	0.748767i
$250$		−	6.39619e8i		−	2.58899i
$251$	3.11626e8			1.24387			0.621936	−	0.783068i	$-0.286347\pi$
$251$	3.11626e8			1.24387			0.621936	+	0.783068i	$0.286347\pi$
$252$	1.55097e8	+	2.76608e8i	0.610524	+	1.08884i
$253$	3.79265e8	+	1.32729e8i	1.47238	+	0.515280i
$254$		−	4.40370e8i		−	1.68616i
$255$	−1.20358e8	−	4.60746e8i	−0.454554	−	1.74009i
$256$	−3.68221e8			−1.37173
$257$			2.92478e8i			1.07480i	0.843328	+	0.537400i	$0.180593\pi$
$257$			2.92478e8i			1.07480i	−0.843328	+	0.537400i	$0.819407\pi$
$258$	−2.20664e8	−	8.44728e8i	−0.799950	−	3.06230i
$259$	−1.40793e8			−0.503537
$260$	1.51584e9			5.34868
$261$	5.64777e7	+	1.00725e8i	0.196623	+	0.350667i
$262$	4.25839e8			1.46282
$263$	−4.88330e8			−1.65527			−0.827634	−	0.561269i	$-0.810313\pi$
$263$	−4.88330e8			−1.65527			−0.827634	+	0.561269i	$0.810313\pi$
$264$	1.11755e9	−	2.91933e8i	3.73813	−	0.976493i
$265$	−3.75765e7			−0.124038
$266$		−	7.51797e7i		−	0.244915i
$267$	−1.32343e7	−	5.06625e7i	−0.0425512	−	0.162891i
$268$			2.44812e8i			0.776894i
$269$			4.55111e8i			1.42556i	0.701389	+	0.712779i	$0.252564\pi$
$269$			4.55111e8i			1.42556i	−0.701389	+	0.712779i	$0.747436\pi$
$270$	−7.18757e8	+	6.92154e8i	−2.22233	+	2.14008i
$271$	2.68720e7			0.0820178			0.0410089	−	0.999159i	$-0.486943\pi$
$271$	2.68720e7			0.0820178			0.0410089	+	0.999159i	$0.486943\pi$
$272$	−7.73070e8			−2.32931
$273$	2.32862e8	−	6.08293e7i	0.692675	−	0.180944i
$274$		−	3.45672e8i		−	1.01517i
$275$	−9.89446e8			−2.86898
$276$	−6.65928e7	+	8.19046e8i	−0.190654	+	2.34491i
$277$	5.68022e7			0.160578			0.0802891	−	0.996772i	$-0.474416\pi$
$277$	5.68022e7			0.160578			0.0802891	+	0.996772i	$0.474416\pi$
$278$			6.24732e8i			1.74396i
$279$	−4.18005e7	+	2.34380e7i	−0.115230	+	0.0646110i
$280$	−8.13385e8			−2.21433
$281$	−2.21096e8			−0.594440			−0.297220	−	0.954809i	$-0.596059\pi$
$281$	−2.21096e8			−0.594440			−0.297220	+	0.954809i	$0.596059\pi$
$282$	2.78880e8	+	1.06759e9i	0.740535	+	2.83485i
$283$		−	3.66953e7i		−	0.0962406i	−0.998842	−	0.0481203i	$-0.984677\pi$
$283$		−	3.66953e7i		−	0.0962406i	0.998842	−	0.0481203i	$-0.0153231\pi$
$284$		−	1.16349e9i		−	3.01404i
$285$	1.60609e8	−	4.19550e7i	0.410972	−	0.107356i
$286$		−	1.52468e9i		−	3.85388i
$287$	−6.22618e7			−0.155466
$288$	3.01194e8	+	5.37163e8i	0.742972	+	1.32505i
$289$	5.71423e7			0.139256
$290$	−5.15158e8			−1.24036
$291$	−1.52470e7	+	3.98289e6i	−0.0362709	+	0.00947486i
$292$	−8.76803e8			−2.06093
$293$	−2.60342e8			−0.604655			−0.302327	−	0.953204i	$-0.597764\pi$
$293$	−2.60342e8			−0.604655			−0.302327	+	0.953204i	$0.597764\pi$
$294$	5.54599e8	−	1.44875e8i	1.27281	−	0.332490i
$295$			2.26354e8i			0.513347i
$296$	−1.04873e9			−2.35040
$297$	4.88537e8	+	5.07314e8i	1.08206	+	1.12365i
$298$			1.06750e9i			2.33674i
$299$	5.88648e8	+	2.06005e8i	1.27352	+	0.445686i
$300$	−5.11423e8	−	1.95779e9i	−1.09359	−	4.18641i
$301$	4.33950e8			0.917185
$302$			3.52188e8i			0.735784i
$303$	3.90020e8	−	1.01883e8i	0.805448	−	0.210403i
$304$		−	2.69480e8i		−	0.550135i
$305$		−	1.10617e8i		−	0.223240i
$306$	−4.79078e8	−	8.54410e8i	−0.955831	−	1.70467i
$307$	−2.05947e8			−0.406229			−0.203114	−	0.979155i	$-0.565106\pi$
$307$	−2.05947e8			−0.406229			−0.203114	+	0.979155i	$0.565106\pi$
$308$			9.98534e8i			1.94731i
$309$	−1.25243e8	+	3.27165e7i	−0.241489	+	0.0630830i
$310$		−	2.13789e8i		−	0.407586i
$311$			2.95936e8i			0.557875i	0.960309	+	0.278937i	$0.0899822\pi$
$311$			2.95936e8i			0.557875i	−0.960309	+	0.278937i	$0.910018\pi$
$312$	1.73453e9	−	4.53102e8i	3.23326	−	0.844607i
$313$			1.03514e8i			0.190807i	0.995439	+	0.0954036i	$0.0304142\pi$
$313$			1.03514e8i			0.190807i	−0.995439	+	0.0954036i	$0.969586\pi$
$314$	9.13032e8			1.66430
$315$	−2.42564e8	−	4.32600e8i	−0.437260	−	0.779830i
$316$		−	2.18316e9i		−	3.89206i
$317$		−	5.12694e8i		−	0.903964i	−0.892027	−	0.451982i	$-0.850717\pi$
$317$		−	5.12694e8i		−	0.903964i	0.892027	−	0.451982i	$-0.149283\pi$
$318$	−7.47853e7	+	1.95358e7i	−0.130413	+	0.0340672i
$319$			3.63609e8i			0.627145i
$320$	−5.91878e8			−1.00973
$321$	−5.55971e7	−	2.12832e8i	−0.0938176	−	0.359145i
$322$	−5.49377e8	−	1.92262e8i	−0.917012	−	0.320920i
$323$			1.62956e8i			0.269068i
$324$	−7.51294e8	+	1.22888e9i	−1.22716	+	2.00724i
$325$	−1.53569e9			−2.48149
$326$			4.85741e8i			0.776504i
$327$	6.30921e8	−	1.64812e8i	0.997833	−	0.260659i
$328$	−4.63772e8			−0.725681
$329$	−5.48435e8			−0.849062
$330$	−3.03992e9	+	7.94104e8i	−4.65654	+	1.21641i
$331$	4.84414e8			0.734207			0.367104	−	0.930180i	$-0.380349\pi$
$331$	4.84414e8			0.734207			0.367104	+	0.930180i	$0.380349\pi$
$332$	1.21401e9			1.82070
$333$	−3.12747e8	−	5.57768e8i	−0.464129	−	0.827749i
$334$	2.10576e9			3.09240
$335$		−	3.82874e8i		−	0.556415i
$336$	−7.79003e8	+	2.03495e8i	−1.12035	+	0.292662i
$337$		−	2.75666e8i		−	0.392355i	−0.980568	−	0.196177i	$-0.937147\pi$
$337$		−	2.75666e8i		−	0.392355i	0.980568	−	0.196177i	$-0.0628528\pi$
$338$		−	1.06654e9i		−	1.50234i
$339$	−7.93395e6	−	3.03721e7i	−0.0110609	−	0.0423425i
$340$	3.06647e9			4.23119
$341$	−1.50897e8			−0.206082
$342$	2.97834e8	−	1.66999e8i	0.402608	−	0.225747i
$343$			6.81456e8i			0.911819i
$344$	3.23238e9			4.28122
$345$	1.04148e8	−	1.28095e9i	0.136547	−	1.67944i
$346$	−1.59287e9			−2.06734
$347$		−	2.23848e8i		−	0.287607i	−0.989606	−	0.143804i	$-0.954067\pi$
$347$		−	2.23848e8i		−	0.287607i	0.989606	−	0.143804i	$-0.0459333\pi$
$348$	−7.19464e8	+	1.87942e8i	−0.915128	+	0.239054i
$349$	−2.23857e8			−0.281892			−0.140946	−	0.990017i	$-0.545014\pi$
$349$	−2.23857e8			−0.281892			−0.140946	+	0.990017i	$0.545014\pi$
$350$	1.43324e9			1.78682
$351$	7.58246e8	+	7.87390e8i	0.935913	+	0.971885i
$352$			1.93912e9i			2.36976i
$353$		−	1.19215e9i		−	1.44252i	−0.692667	−	0.721258i	$-0.743564\pi$
$353$		−	1.19215e9i		−	1.44252i	0.692667	−	0.721258i	$-0.256436\pi$
$354$	1.17680e8	+	4.50493e8i	0.140991	+	0.539730i
$355$			1.81964e9i			2.15867i
$356$	3.37182e8			0.396086
$357$	4.71069e8	−	1.23055e8i	0.547956	−	0.143140i
$358$	2.37148e8			0.273168
$359$	4.01614e8			0.458119			0.229059	−	0.973412i	$-0.426435\pi$
$359$	4.01614e8			0.458119			0.229059	+	0.973412i	$0.426435\pi$
$360$	−1.80680e9	−	3.22233e9i	−2.04104	−	3.64008i
$361$	8.37068e8			0.936452
$362$	1.17351e9			1.30019
$363$	3.30164e8	+	1.26391e9i	0.362290	+	1.38689i
$364$			1.54980e9i			1.68430i
$365$	1.37127e9			1.47604
$366$	−5.75091e7	−	2.20152e8i	−0.0613130	−	0.234713i
$367$		−	1.56099e9i		−	1.64842i	−0.566284	−	0.824210i	$-0.691620\pi$
$367$		−	1.56099e9i		−	1.64842i	0.566284	−	0.824210i	$-0.308380\pi$
$368$	−1.96923e9	−	6.89158e8i	−2.05982	−	0.720860i
$369$	−1.38304e8	−	2.46658e8i	−0.143299	−	0.255566i
$370$	2.85271e9			2.92786
$371$		−	3.84184e7i		−	0.0390598i
$372$	−7.79953e7	−	2.98575e8i	−0.0785540	−	0.300714i
$373$			7.23735e7i			0.0722103i	0.999348	+	0.0361051i	$0.0114951\pi$
$373$			7.23735e7i			0.0722103i	−0.999348	+	0.0361051i	$0.988505\pi$
$374$		−	3.08436e9i		−	3.04870i
$375$	3.64945e8	+	1.39705e9i	0.357369	+	1.36805i
$376$	−4.08515e9			−3.96324
$377$			5.64349e8i			0.542442i
$378$	−7.07661e8	−	7.34860e8i	−0.673913	−	0.699815i
$379$			7.93487e8i			0.748691i	0.927289	+	0.374345i	$0.122133\pi$
$379$			7.93487e8i			0.748691i	−0.927289	+	0.374345i	$0.877867\pi$
$380$			1.06892e9i			0.999318i
$381$	2.51260e8	+	9.61854e8i	0.232748	+	0.890987i
$382$			3.32742e9i			3.05412i
$383$	9.27019e7			0.0843127			0.0421563	−	0.999111i	$-0.486577\pi$
$383$	9.27019e7			0.0843127			0.0421563	+	0.999111i	$0.486577\pi$
$384$	4.52912e8	−	1.18312e8i	0.408183	−	0.106627i
$385$		−	1.56166e9i		−	1.39467i
$386$		−	2.36089e9i		−	2.08940i
$387$	9.63946e8	+	1.71915e9i	0.845404	+	1.50773i
$388$		−	1.01475e8i		−	0.0881961i
$389$	3.52678e8			0.303777			0.151889	−	0.988398i	$-0.451464\pi$
$389$	3.52678e8			0.303777			0.151889	+	0.988398i	$0.451464\pi$
$390$	−4.71819e9	+	1.23251e9i	−4.02763	+	1.05212i
$391$	1.19081e9	+	4.16738e8i	1.00745	+	0.352570i
$392$			2.12219e9i			1.77944i
$393$	−9.30115e8	+	2.42969e8i	−0.772970	+	0.201919i
$394$	−3.25846e9			−2.68396
$395$			3.41434e9i			2.78751i
$396$	−3.95582e9	+	2.21807e9i	−3.20113	+	1.79491i
$397$	−1.34328e8			−0.107746			−0.0538729	−	0.998548i	$-0.517157\pi$
$397$	−1.34328e8			−0.107746			−0.0538729	+	0.998548i	$0.517157\pi$
$398$	−3.68040e9			−2.92620
$399$	4.28950e7	+	1.64207e8i	0.0338066	+	0.129416i
$400$	5.13742e9			4.01361
$401$	−2.14164e9			−1.65860			−0.829300	−	0.558804i	$-0.811260\pi$
$401$	−2.14164e9			−1.65860			−0.829300	+	0.558804i	$0.811260\pi$
$402$	−1.99054e8	−	7.62001e8i	−0.152820	−	0.585011i
$403$	−2.34203e8			−0.178248
$404$			2.59576e9i			1.95852i
$405$	1.17499e9	−	1.92190e9i	0.878901	−	1.43760i
$406$		−	5.26699e8i		−	0.390590i
$407$		−	2.01350e9i		−	1.48037i
$408$	3.50887e9	−	9.16603e8i	2.55774	−	0.668145i
$409$	1.25547e9			0.907350			0.453675	−	0.891167i	$-0.350113\pi$
$409$	1.25547e9			0.907350			0.453675	+	0.891167i	$0.350113\pi$
$410$	1.26153e9			0.903971
$411$	1.97228e8	+	7.55014e8i	0.140127	+	0.536424i
$412$		−	8.33546e8i		−	0.587205i
$413$	−2.31425e8			−0.161654
$414$	−4.58679e8	−	2.60350e9i	−0.317693	−	1.80325i
$415$	−1.89865e9			−1.30399
$416$			3.00966e9i			2.04970i
$417$	−3.56451e8	−	1.36454e9i	−0.240726	−	0.921529i
$418$	1.07516e9			0.720038
$419$	1.98214e9			1.31639			0.658195	−	0.752848i	$-0.271320\pi$
$419$	1.98214e9			1.31639			0.658195	+	0.752848i	$0.271320\pi$
$420$	3.09001e9	−	8.07186e8i	2.03510	−	0.531620i
$421$		−	1.05203e9i		−	0.687130i	−0.939129	−	0.343565i	$-0.888365\pi$
$421$		−	1.05203e9i		−	0.687130i	0.939129	−	0.343565i	$-0.111635\pi$
$422$			2.79896e9i			1.81302i
$423$	−1.21826e9	−	2.17269e9i	−0.782613	−	1.39575i
$424$		−	2.86168e8i		−	0.182323i
$425$	−3.10664e9			−1.96304
$426$	9.46019e8	+	3.62147e9i	0.592879	+	2.26961i
$427$	1.13095e8			0.0702986
$428$	1.41649e9			0.873295
$429$	8.69931e8	+	3.33020e9i	0.531966	+	2.03643i
$430$	−8.79258e9			−5.33306
$431$	1.59104e8			0.0957217			0.0478609	−	0.998854i	$-0.484760\pi$
$431$	1.59104e8			0.0957217			0.0478609	+	0.998854i	$0.484760\pi$
$432$	−2.53660e9	−	2.63409e9i	−1.51376	−	1.57195i
$433$		−	1.86159e9i		−	1.10199i	−0.834510	−	0.550993i	$-0.814249\pi$
$433$		−	1.86159e9i		−	1.10199i	0.834510	−	0.550993i	$-0.185751\pi$
$434$	2.18578e8			0.128349
$435$	1.12520e9	−	2.93931e8i	0.655419	−	0.171212i
$436$			4.19906e9i			2.42633i
$437$	−1.45268e8	+	4.15096e8i	−0.0832695	+	0.237938i
$438$	2.72913e9	−	7.12917e8i	1.55190	−	0.405396i
$439$	2.44240e8			0.137781			0.0688907	−	0.997624i	$-0.478054\pi$
$439$	2.44240e8			0.137781			0.0688907	+	0.997624i	$0.478054\pi$
$440$		−	1.16324e10i		−	6.51004i
$441$	−1.12869e9	+	6.32871e8i	−0.626672	+	0.351382i
$442$		−	4.78715e9i		−	2.63693i
$443$			1.71997e9i			0.939955i	0.882678	+	0.469977i	$0.155738\pi$
$443$			1.71997e9i			0.939955i	−0.882678	+	0.469977i	$0.844262\pi$
$444$	3.98406e9	−	1.04074e9i	2.16016	−	0.564287i
$445$	−5.27335e8			−0.283678
$446$			2.69063e9i			1.43609i
$447$	−6.09078e8	−	2.33162e9i	−0.322550	−	1.23476i
$448$		−	6.05138e8i		−	0.317967i
$449$		−	2.34383e9i		−	1.22198i	−0.791639	−	0.610989i	$-0.790772\pi$
$449$		−	2.34383e9i		−	1.22198i	0.791639	−	0.610989i	$-0.209228\pi$
$450$	3.18370e9	+	5.67797e9i	1.64698	+	2.93731i
$451$		−	8.90416e8i		−	0.457062i
$452$	2.02140e8			0.102960
$453$	−2.00946e8	−	7.69247e8i	−0.101563	−	0.388796i
$454$			6.74649e9i			3.38363i
$455$		−	2.42381e9i		−	1.20631i
$456$	3.19513e8	+	1.22314e9i	0.157802	+	0.604084i
$457$		−	2.92025e9i		−	1.43124i	−0.698487	−	0.715622i	$-0.746143\pi$
$457$		−	2.92025e9i		−	1.43124i	0.698487	−	0.715622i	$-0.253857\pi$
$458$	−5.35637e9			−2.60520
$459$	1.53390e9	+	1.59285e9i	0.740375	+	0.768831i
$460$	7.81117e9	+	2.73362e9i	3.74166	+	1.30944i
$461$		−	2.26835e9i		−	1.07834i	−0.842197	−	0.539171i	$-0.818738\pi$
$461$		−	2.26835e9i		−	1.07834i	0.842197	−	0.539171i	$-0.181262\pi$
$462$	−8.11894e8	−	3.10803e9i	−0.383047	−	1.46635i
$463$	2.29406e9			1.07417			0.537083	−	0.843529i	$-0.319526\pi$
$463$	2.29406e9			1.07417			0.537083	+	0.843529i	$0.319526\pi$
$464$		−	1.88794e9i		−	0.877356i
$465$	1.21981e8	+	4.66956e8i	0.0562607	+	0.215373i
$466$	4.31030e9			1.97314
$467$	−2.13733e9			−0.971099			−0.485549	−	0.874209i	$-0.661380\pi$
$467$	−2.13733e9			−0.971099			−0.485549	+	0.874209i	$0.661380\pi$
$468$	−6.13973e9	+	3.44262e9i	−2.76878	+	1.55249i
$469$	3.91452e8			0.175216
$470$	1.11122e10			4.93696
$471$	−1.99424e9	+	5.20945e8i	−0.879435	+	0.229730i
$472$	−1.72382e9			−0.754564
$473$			6.20599e9i			2.69648i
$474$	1.77509e9	+	6.79527e9i	0.765592	+	2.93078i
$475$		−	1.08292e9i		−	0.463629i
$476$			3.13517e9i			1.33241i
$477$	1.52199e8	−	8.53399e7i	0.0642093	−	0.0360029i
$478$	1.20349e9			0.504015
$479$	−5.77172e8			−0.239956			−0.119978	−	0.992777i	$-0.538282\pi$
$479$	−5.77172e8			−0.239956			−0.119978	+	0.992777i	$0.538282\pi$
$480$	6.00068e9	−	1.56753e9i	2.47660	−	0.646950i
$481$		−	3.12510e9i		−	1.28043i
$482$	3.70393e9			1.50660
$483$	1.30964e9	+	1.06481e8i	0.528857	+	0.0429989i
$484$	−8.41185e9			−3.37235
$485$			1.58702e8i			0.0631665i
$486$	1.33929e9	−	4.43585e9i	0.529234	−	1.75287i
$487$	−5.06475e9			−1.98704			−0.993521	−	0.113649i	$-0.963746\pi$
$487$	−5.06475e9			−1.98704			−0.993521	+	0.113649i	$0.963746\pi$
$488$	8.42417e8			0.328139
$489$	−2.77147e8	−	1.06095e9i	−0.107184	−	0.410313i
$490$		−	5.77269e9i		−	2.21663i
$491$			2.07659e9i			0.791710i	0.918313	+	0.395855i	$0.129552\pi$
$491$			2.07659e9i			0.791710i	−0.918313	+	0.395855i	$0.870448\pi$
$492$	1.76184e9	−	4.60237e8i	0.666944	−	0.174222i
$493$			1.14165e9i			0.429111i
$494$	1.66873e9			0.622788
$495$	6.18669e9	−	3.46895e9i	2.29266	−	1.28552i
$496$	7.83490e8			0.288302
$497$	−1.86041e9			−0.679768
$498$	−3.77871e9	+	9.87094e8i	−1.37101	+	0.358142i
$499$	−1.22325e8			−0.0440719			−0.0220359	−	0.999757i	$-0.507015\pi$
$499$	−1.22325e8			−0.0440719			−0.0220359	+	0.999757i	$0.507015\pi$
$500$	−9.29800e9			−3.32655
$501$	−4.59939e9	+	1.20147e9i	−1.63406	+	0.426857i
$502$		−	6.45554e9i		−	2.27756i
$503$	−2.73270e9			−0.957424			−0.478712	−	0.877972i	$-0.658896\pi$
$503$	−2.73270e9			−0.957424			−0.478712	+	0.877972i	$0.658896\pi$
$504$	3.29452e9	−	1.84728e9i	1.14627	−	0.642725i
$505$		−	4.05963e9i		−	1.40270i
$506$	2.74957e9	−	7.85673e9i	0.943490	−	2.69597i
$507$	6.08532e8	+	2.32953e9i	0.207374	+	0.793854i
$508$	−6.40156e9			−2.16652
$509$			4.35891e9i			1.46509i	0.680716	+	0.732547i	$0.261669\pi$
$509$			4.35891e9i			1.46509i	−0.680716	+	0.732547i	$0.738331\pi$
$510$	−9.54467e9	+	2.49331e9i	−3.18614	+	0.832300i
$511$			1.40200e9i			0.464808i
$512$			6.34669e9i			2.08979i
$513$	−5.55243e8	+	5.34692e8i	−0.181582	+	0.174861i
$514$	6.05888e9			1.96798
$515$			1.30362e9i			0.420559i
$516$	−1.22796e10	+	3.20775e9i	−3.93470	+	1.02784i
$517$		−	7.84326e9i		−	2.49620i
$518$			2.91662e9i			0.921988i
$519$	3.47913e9	−	9.08835e8i	1.09241	−	0.285364i
$520$		−	1.80543e10i		−	5.63078i
$521$	2.91772e9			0.903882			0.451941	−	0.892048i	$-0.350732\pi$
$521$	2.91772e9			0.903882			0.451941	+	0.892048i	$0.350732\pi$
$522$	2.08658e9	−	1.16997e9i	0.642080	−	0.360022i
$523$		−	7.63721e8i		−	0.233442i	−0.993165	−	0.116721i	$-0.962762\pi$
$523$		−	7.63721e8i		−	0.233442i	0.993165	−	0.116721i	$-0.0372383\pi$
$524$		−	6.19033e9i		−	1.87955i
$525$	−3.13048e9	+	8.17759e8i	−0.944176	+	0.246642i
$526$			1.01161e10i			3.03084i
$527$	−4.73782e8			−0.141007
$528$	−2.91022e9	−	1.11407e10i	−0.860413	−	3.29376i
$529$	2.66182e9	+	2.12310e9i	0.781778	+	0.623557i
$530$			7.78423e8i			0.227117i
$531$	−5.14072e8	−	9.16820e8i	−0.149002	−	0.265738i
$532$	−1.09287e9			−0.314686
$533$		−	1.38199e9i		−	0.395331i
$534$	−1.04951e9	+	2.74158e8i	−0.298258	+	0.0779123i
$535$	−2.21532e9			−0.625458
$536$	2.91582e9			0.817869
$537$	−5.17978e8	+	1.35309e8i	−0.144345	+	0.0377065i
$538$	9.42794e9			2.61023
$539$	−4.07449e9			−1.12076
$540$	1.00617e10	+	1.04484e10i	2.74975	+	2.85543i
$541$	−4.54330e9			−1.23362			−0.616809	−	0.787113i	$-0.711575\pi$
$541$	−4.54330e9			−1.23362			−0.616809	+	0.787113i	$0.711575\pi$
$542$		−	5.56672e8i		−	0.150177i
$543$	−2.56318e9	+	6.69567e8i	−0.687037	+	0.179471i
$544$			6.08839e9i			1.62146i
$545$		−	6.56711e9i		−	1.73775i
$546$	−1.26012e9	−	4.82389e9i	−0.331313	−	1.26830i
$547$	−1.54026e9			−0.402383			−0.201191	−	0.979552i	$-0.564481\pi$
$547$	−1.54026e9			−0.402383			−0.201191	+	0.979552i	$0.564481\pi$
$548$	−5.02495e9			−1.30437
$549$	2.51222e8	+	4.48041e8i	0.0647969	+	0.115562i
$550$			2.04970e10i			5.25317i
$551$	−3.97961e8			−0.101347
$552$	9.75519e9	+	7.93149e8i	2.46859	+	0.200709i
$553$	−3.49084e9			−0.877792
$554$		−	1.17670e9i		−	0.294022i
$555$	−6.23086e9	+	1.62766e9i	−1.54712	+	0.404145i
$556$	9.08159e9			2.24079
$557$	−9.43747e8			−0.231399			−0.115700	−	0.993284i	$-0.536911\pi$
$557$	−9.43747e8			−0.231399			−0.115700	+	0.993284i	$0.536911\pi$
$558$	4.85535e8	+	8.65926e8i	0.118304	+	0.210990i
$559$			9.63217e9i			2.33229i
$560$			8.10846e9i			1.95110i
$561$	1.75983e9	+	6.73683e9i	0.420824	+	1.61096i
$562$			4.58014e9i			1.08843i
$563$	5.37774e9			1.27005			0.635024	−	0.772492i	$-0.280990\pi$
$563$	5.37774e9			1.27005			0.635024	+	0.772492i	$0.280990\pi$
$564$	1.55193e10	−	4.05402e9i	3.64245	−	0.951500i
$565$	−3.16136e8			−0.0737403
$566$	−7.60168e8			−0.176219
$567$	1.96495e9	+	1.20131e9i	0.452701	+	0.276767i
$568$	−1.38577e10			−3.17301
$569$	1.47361e9			0.335342			0.167671	−	0.985843i	$-0.446375\pi$
$569$	1.47361e9			0.335342			0.167671	+	0.985843i	$0.446375\pi$
$570$	−8.69127e8	−	3.32712e9i	−0.196572	−	0.752500i
$571$			1.48308e9i			0.333378i	0.986009	+	0.166689i	$0.0533076\pi$
$571$			1.48308e9i			0.333378i	−0.986009	+	0.166689i	$0.946692\pi$
$572$	−2.21639e10			−4.95178
$573$	−1.89851e9	−	7.26772e9i	−0.421572	−	1.61383i
$574$			1.28979e9i			0.284662i
$575$	−7.91348e9	−	2.76943e9i	−1.73592	−	0.607509i
$576$	2.39733e9	−	1.34421e9i	0.522696	−	0.293082i
$577$	3.96923e9			0.860183			0.430091	−	0.902785i	$-0.358481\pi$
$577$	3.96923e9			0.860183			0.430091	+	0.902785i	$0.358481\pi$
$578$		−	1.18374e9i		−	0.254982i
$579$	1.34705e9	+	5.15665e9i	0.288408	+	1.10406i
$580$			7.48874e9i			1.59371i
$581$		−	1.94118e9i		−	0.410629i
$582$	8.25082e7	+	3.15851e8i	0.0173487	+	0.0664129i
$583$	5.49428e8			0.114834
$584$			1.04431e10i			2.16962i
$585$	9.60221e9	−	5.38407e9i	1.98301	−	1.11190i
$586$			5.39316e9i			1.10714i
$587$			1.64685e9i			0.336064i	0.985782	+	0.168032i	$0.0537412\pi$
$587$			1.64685e9i			0.336064i	−0.985782	+	0.168032i	$0.946259\pi$
$588$	−2.10602e9	−	8.06209e9i	−0.427210	−	1.63541i
$589$		−	1.65153e8i		−	0.0333029i
$590$	4.68908e9			0.939951
$591$	7.11711e9	−	1.85917e9i	1.41823	−	0.370478i
$592$			1.04545e10i			2.07100i
$593$			6.54796e9i			1.28948i	0.764402	+	0.644740i	$0.223034\pi$
$593$			6.54796e9i			1.28948i	−0.764402	+	0.644740i	$0.776966\pi$
$594$	1.05094e10	−	1.01204e10i	2.05742	−	1.98127i
$595$		−	4.90324e9i		−	0.954276i
$596$	1.55180e10			3.00243
$597$	8.03871e9	−	2.09991e9i	1.54624	−	0.403916i
$598$	4.26753e9	−	1.21942e10i	0.816061	−	2.33185i
$599$			2.22978e9i			0.423904i	0.977280	+	0.211952i	$0.0679821\pi$
$599$			2.22978e9i			0.423904i	−0.977280	+	0.211952i	$0.932018\pi$
$600$	−2.33181e10	+	6.09127e9i	−4.40721	+	1.15127i
$601$	−3.73936e9			−0.702647			−0.351323	−	0.936254i	$-0.614268\pi$
$601$	−3.73936e9			−0.702647			−0.351323	+	0.936254i	$0.614268\pi$
$602$		−	8.98957e9i		−	1.67939i
$603$	8.69543e8	+	1.55078e9i	0.161503	+	0.288032i
$604$	5.11968e9			0.945396
$605$	1.31557e10			2.41529
$606$	−2.11057e9	−	8.07952e9i	−0.385253	−	1.47479i
$607$	6.99600e9			1.26967			0.634833	−	0.772650i	$-0.281069\pi$
$607$	6.99600e9			1.26967			0.634833	+	0.772650i	$0.281069\pi$
$608$	−2.12232e9			−0.382955
$609$	3.00517e8	+	1.15041e9i	0.0539148	+	0.206392i
$610$	−2.29151e9			−0.408758
$611$		−	1.21733e10i		−	2.15906i
$612$	−1.24204e10	+	6.96425e9i	−2.19031	+	1.22813i
$613$		−	2.33246e9i		−	0.408981i	−0.978869	−	0.204491i	$-0.934446\pi$
$613$		−	2.33246e9i		−	0.408981i	0.978869	−	0.204491i	$-0.0655538\pi$
$614$			4.26632e9i			0.743814i
$615$	−2.75543e9	+	7.19787e8i	−0.477668	+	0.124779i
$616$	1.18930e10			2.05002
$617$	−5.53297e9			−0.948331			−0.474165	−	0.880436i	$-0.657250\pi$
$617$	−5.53297e9			−0.948331			−0.474165	+	0.880436i	$0.657250\pi$
$618$	6.77745e8	+	2.59449e9i	0.115507	+	0.442173i
$619$		−	4.06234e9i		−	0.688429i	−0.938891	−	0.344215i	$-0.888145\pi$
$619$		−	4.06234e9i		−	0.688429i	0.938891	−	0.344215i	$-0.111855\pi$
$620$	−3.10780e9			−0.523700
$621$	2.48731e9	+	5.42485e9i	0.416783	+	0.909006i
$622$	6.13052e9			1.02148
$623$		−	5.39149e8i		−	0.0893306i
$624$	−4.51688e9	−	1.72911e10i	−0.744204	−	2.84890i
$625$	3.31627e9			0.543337
$626$	2.14437e9			0.349373
$627$	−2.34835e9	+	6.13448e8i	−0.380476	+	0.0993897i
$628$		−	1.32725e10i		−	2.13843i
$629$		−	6.32194e9i		−	1.01291i
$630$	−8.96161e9	+	5.02488e9i	−1.42789	+	0.800634i
$631$			9.78171e9i			1.54993i	0.632005	+	0.774965i	$0.282232\pi$
$631$			9.78171e9i			1.54993i	−0.632005	+	0.774965i	$0.717768\pi$
$632$	−2.60023e10			−4.09734
$633$	−1.59699e9	−	6.11347e9i	−0.250259	−	0.958020i
$634$	−1.06208e10			−1.65518
$635$	1.00117e10			1.55167
$636$	2.83988e8	+	1.08714e9i	0.0437723	+	0.167566i
$637$	−6.32392e9			−0.969389
$638$	7.53241e9			1.14832
$639$	−4.13258e9	−	7.37023e9i	−0.626568	−	1.11745i
$640$		−	4.71426e9i		−	0.710859i
$641$	9.94854e8			0.149196			0.0745979	−	0.997214i	$-0.476233\pi$
$641$	9.94854e8			0.149196			0.0745979	+	0.997214i	$0.476233\pi$
$642$	−4.40896e9	+	1.15173e9i	−0.657603	+	0.171782i
$643$			1.18935e9i			0.176430i	0.996101	+	0.0882151i	$0.0281163\pi$
$643$			1.18935e9i			0.176430i	−0.996101	+	0.0882151i	$0.971884\pi$
$644$	−2.79486e9	+	7.98617e9i	−0.412345	+	1.17825i
$645$	1.92047e10	−	5.01675e9i	2.81805	−	0.736145i
$646$	3.37575e9			0.492671
$647$			5.14188e9i			0.746375i	0.927756	+	0.373188i	$0.121735\pi$
$647$			5.14188e9i			0.746375i	−0.927756	+	0.373188i	$0.878265\pi$
$648$	1.46364e10	+	8.94824e9i	2.11311	+	1.29189i
$649$		−	3.30965e9i		−	0.475254i
$650$			3.18129e10i			4.54367i
$651$	−4.77418e8	+	1.24713e8i	−0.0678211	+	0.0177166i
$652$	7.06112e9			0.997716
$653$			2.54695e9i			0.357951i	0.983854	+	0.178976i	$0.0572783\pi$
$653$			2.54695e9i			0.357951i	−0.983854	+	0.178976i	$0.942722\pi$
$654$	−3.41420e9	−	1.30700e10i	−0.477273	−	1.82706i
$655$			9.68135e9i			1.34614i
$656$			4.62324e9i			0.639415i
$657$	−5.55418e9	+	3.11430e9i	−0.764085	+	0.428431i
$658$			1.13612e10i			1.55465i
$659$	−1.20708e10			−1.64300			−0.821501	−	0.570207i	$-0.806863\pi$
$659$	−1.20708e10			−1.64300			−0.821501	+	0.570207i	$0.806863\pi$
$660$	1.15437e10	+	4.41907e10i	1.56294	+	5.98311i
$661$			1.12830e10i			1.51956i	0.650179	+	0.759781i	$0.274694\pi$
$661$			1.12830e10i			1.51956i	−0.650179	+	0.759781i	$0.725306\pi$
$662$		−	1.00350e10i		−	1.34435i
$663$	2.73139e9	+	1.04561e10i	0.363987	+	1.39338i
$664$		−	1.44594e10i		−	1.91673i
$665$	1.70919e9			0.225380
$666$	−1.15545e10	+	6.47877e9i	−1.51563	+	0.849831i
$667$	−1.01773e9	+	2.90811e9i	−0.132798	+	0.379464i
$668$		−	3.06110e10i		−	3.97337i
$669$	−1.53518e9	−	5.87687e9i	−0.198230	−	0.758847i
$670$	−7.93149e9			−1.01881
$671$			1.61739e9i			0.206675i
$672$	1.60265e9	+	6.13512e9i	0.203725	+	0.779885i
$673$	−9.08705e9			−1.14913			−0.574566	−	0.818458i	$-0.694829\pi$
$673$	−9.08705e9			−1.14913			−0.574566	+	0.818458i	$0.694829\pi$
$674$	−5.71061e9			−0.718411
$675$	−1.01935e10	−	1.05853e10i	−1.27573	−	1.32476i
$676$	−1.55041e10			−1.93033
$677$	1.06160e10			1.31492			0.657459	−	0.753490i	$-0.271631\pi$
$677$	1.06160e10			1.31492			0.657459	+	0.753490i	$0.271631\pi$
$678$	−6.29179e8	+	1.64357e8i	−0.0775301	+	0.0202528i
$679$	−1.62258e8			−0.0198912
$680$		−	3.65230e10i		−	4.45436i
$681$	−3.84932e9	−	1.47356e10i	−0.467056	−	1.78795i
$682$			3.12593e9i			0.377341i
$683$		−	1.39171e9i		−	0.167138i	−0.996502	−	0.0835690i	$-0.973368\pi$
$683$		−	1.39171e9i		−	0.167138i	0.996502	−	0.0835690i	$-0.0266319\pi$
$684$	−2.42763e9	−	4.32954e9i	−0.290059	−	0.517304i
$685$	7.85876e9			0.934194
$686$	1.41168e10			1.66956
$687$	1.16993e10	−	3.05616e9i	1.37662	−	0.359607i
$688$		−	3.22229e10i		−	3.77229i
$689$	8.52753e8			0.0993244
$690$	−2.65357e10	−	2.15749e9i	−3.07509	−	0.250021i
$691$	1.11472e10			1.28527			0.642634	−	0.766173i	$-0.277842\pi$
$691$	1.11472e10			1.28527			0.642634	+	0.766173i	$0.277842\pi$
$692$			2.31551e10i			2.65629i
$693$	3.54667e9	+	6.32530e9i	0.404813	+	0.721962i
$694$	−4.63716e9			−0.526616
$695$	−1.42031e10			−1.60486
$696$	2.23847e9	+	8.56912e9i	0.251663	+	0.963394i
$697$		−	2.79571e9i		−	0.312735i
$698$			4.63735e9i			0.516150i
$699$	−9.41453e9	+	2.45931e9i	−1.04263	+	0.272360i
$700$		−	2.08347e10i		−	2.29586i
$701$	1.42660e10			1.56419			0.782094	−	0.623160i	$-0.214152\pi$
$701$	1.42660e10			1.56419			0.782094	+	0.623160i	$0.214152\pi$
$702$	1.63113e10	−	1.57076e10i	1.77954	−	1.71368i
$703$	2.20373e9			0.239229
$704$	8.65419e9			0.934807
$705$	−2.42713e10	+	6.34027e9i	−2.60874	+	0.681468i
$706$	−2.46963e10			−2.64128
$707$	4.15058e9			0.441713
$708$	6.54872e9	−	1.71069e9i	0.693489	−	0.181157i
$709$			1.57940e10i			1.66429i	0.554557	+	0.832146i	$0.312888\pi$
$709$			1.57940e10i			1.66429i	−0.554557	+	0.832146i	$0.687112\pi$
$710$	3.76951e10			3.95258
$711$	−7.75430e9	−	1.38294e10i	−0.809094	−	1.44298i
$712$		−	4.01598e9i		−	0.416976i
$713$	−1.20686e9	−	4.22355e8i	−0.124693	−	0.0436380i
$714$	−2.54916e9	−	9.75851e9i	−0.262092	−	1.00332i
$715$	3.46633e10			3.54649
$716$		−	3.44737e9i		−	0.350988i
$717$	−2.62865e9	+	6.86668e8i	−0.266327	+	0.0695713i
$718$		−	8.31970e9i		−	0.838827i
$719$			1.28542e10i			1.28972i	0.764302	+	0.644858i	$0.223084\pi$
$719$			1.28542e10i			1.28972i	−0.764302	+	0.644858i	$0.776916\pi$
$720$	−3.21227e10	+	1.80116e10i	−3.20736	+	1.79841i
$721$	−1.33283e9			−0.132434
$722$		−	1.73404e10i		−	1.71467i
$723$	−8.09011e9	+	2.11334e9i	−0.796105	+	0.207963i
$724$		−	1.70591e10i		−	1.67060i
$725$		−	7.58682e9i		−	0.739396i
$726$	2.61827e10	−	6.83956e9i	2.53943	−	0.663361i
$727$			6.01050e8i			0.0580150i	0.999579	+	0.0290075i	$0.00923467\pi$
$727$			6.01050e8i			0.0580150i	−0.999579	+	0.0290075i	$0.990765\pi$
$728$	1.84588e10			1.77314
$729$	−3.94323e8	+	1.04529e10i	−0.0376970	+	0.999289i
$730$		−	2.84069e10i		−	2.70267i
$731$			1.94854e10i			1.84501i
$732$	−3.20029e9	+	8.35997e8i	−0.301579	+	0.0787800i
$733$			9.87321e9i			0.925965i	0.886368	+	0.462982i	$0.153221\pi$
$733$			9.87321e9i			0.925965i	−0.886368	+	0.462982i	$0.846779\pi$
$734$	−3.23369e10			−3.01830
$735$	3.29370e9	+	1.26087e10i	0.305970	+	1.17129i
$736$	−5.42753e9	+	1.55089e10i	−0.501799	+	1.43386i
$737$			5.59821e9i			0.515126i
$738$	−5.10968e9	+	2.86506e9i	−0.467947	+	0.262384i
$739$	−3.43002e9			−0.312637			−0.156319	−	0.987707i	$-0.549963\pi$
$739$	−3.43002e9			−0.312637			−0.156319	+	0.987707i	$0.549963\pi$
$740$		−	4.14692e10i		−	3.76196i
$741$	−3.64482e9	+	9.52118e8i	−0.329088	+	0.0859660i
$742$	−7.95863e8			−0.0715195
$743$	5.58623e9			0.499641			0.249820	−	0.968292i	$-0.419628\pi$
$743$	5.58623e9			0.499641			0.249820	+	0.968292i	$0.419628\pi$
$744$	−3.55616e9	+	9.28957e8i	−0.316574	+	0.0826971i
$745$	−2.42693e10			−2.15036
$746$	1.49927e9			0.132219
$747$	7.69024e9	−	4.31201e9i	0.675021	−	0.378493i
$748$	−4.48366e10			−3.91721
$749$		−	2.26495e9i		−	0.196957i
$750$	2.89409e10	−	7.56007e9i	2.50494	−	0.654352i
$751$		−	5.15163e9i		−	0.443818i	−0.975067	−	0.221909i	$-0.928771\pi$
$751$		−	5.15163e9i		−	0.443818i	0.975067	−	0.221909i	$-0.0712288\pi$
$752$			4.07240e10i			3.49211i
$753$	3.68331e9	+	1.41002e10i	0.314381	+	1.20349i
$754$	1.16909e10			0.993224
$755$	−8.00691e9			−0.677097
$756$	−1.06825e10	+	1.02871e10i	−0.899180	+	0.865899i
$757$			7.00046e9i			0.586531i	0.956031	+	0.293265i	$0.0947419\pi$
$757$			7.00046e9i			0.586531i	−0.956031	+	0.293265i	$0.905258\pi$
$758$	1.64376e10			1.37087
$759$	−1.52280e9	+	1.87294e10i	−0.126415	+	1.55481i
$760$	1.27313e10			1.05203
$761$		−	1.32458e10i		−	1.08951i	−0.838594	−	0.544757i	$-0.816622\pi$
$761$		−	1.32458e10i		−	1.08951i	0.838594	−	0.544757i	$-0.183378\pi$
$762$	1.99254e10	−	5.20502e9i	1.63142	−	0.426167i
$763$	6.71424e9			0.547219
$764$	4.83699e10			3.92418
$765$	1.94248e10	−	1.08917e10i	1.56871	−	0.879593i
$766$		−	1.92038e9i		−	0.154379i
$767$		−	5.13682e9i		−	0.411065i
$768$	−4.35224e9	−	1.66609e10i	−0.346696	−	1.32719i
$769$			1.78034e10i			1.41176i	0.708332	+	0.705879i	$0.249448\pi$
$769$			1.78034e10i			1.41176i	−0.708332	+	0.705879i	$0.750552\pi$
$770$	−3.23507e10			−2.55368
$771$	−1.32338e10	+	3.45699e9i	−1.03990	+	0.271649i
$772$	−3.43198e10			−2.68463
$773$	−1.44442e10			−1.12477			−0.562386	−	0.826875i	$-0.690117\pi$
$773$	−1.44442e10			−1.12477			−0.562386	+	0.826875i	$0.690117\pi$
$774$	3.56133e10	−	1.99688e10i	2.76070	−	1.54796i
$775$	3.14851e9			0.242968
$776$	−1.20861e9			−0.0928478
$777$	−1.66412e9	−	6.37046e9i	−0.127266	−	0.487188i
$778$		−	7.30597e9i		−	0.556224i
$779$	9.74538e8			0.0738615
$780$	1.79167e10	+	6.85873e10i	1.35185	+	5.17502i
$781$		−	2.66060e10i		−	1.99848i
$782$	8.63302e9	−	2.46684e10i	0.645563	−	1.84466i
$783$	−3.88996e9	+	3.74598e9i	−0.289587	+	0.278869i
$784$	2.11557e10			1.56791
$785$			2.07576e10i			1.53156i
$786$	5.03327e9	+	1.92680e10i	0.369719	+	1.41533i
$787$			4.67154e9i			0.341624i	0.985304	+	0.170812i	$0.0546390\pi$
$787$			4.67154e9i			0.341624i	−0.985304	+	0.170812i	$0.945361\pi$
$788$			4.73675e10i			3.44857i
$789$	−5.77189e9	−	2.20955e10i	−0.418359	−	1.60153i
$790$	7.07304e10			5.10401
$791$		−	3.23219e8i		−	0.0232209i
$792$	2.64182e10	+	4.71155e10i	1.88958	+	3.36997i
$793$			2.51032e9i			0.178761i
$794$			2.78270e9i			0.197285i
$795$	−4.44142e8	−	1.70023e9i	−0.0313499	−	0.120011i
$796$			5.35012e10i			3.75983i
$797$	−1.04683e10			−0.732440			−0.366220	−	0.930528i	$-0.619348\pi$
$797$	−1.04683e10			−0.732440			−0.366220	+	0.930528i	$0.619348\pi$
$798$	3.40166e9	−	8.88598e8i	0.236963	−	0.0619007i
$799$		−	2.46261e10i		−	1.70797i
$800$		−	4.04603e10i		−	2.79392i
$801$	2.13591e9	−	1.19763e9i	0.146848	−	0.0823395i
$802$			4.43655e10i			3.03694i
$803$	−2.00502e10			−1.36651
$804$	−1.10770e10	+	2.89360e9i	−0.751670	+	0.196355i
$805$	4.37102e9	−	1.24900e10i	0.295323	−	0.843869i
$806$			4.85167e9i			0.326377i
$807$	−2.05925e10	+	5.37926e9i	−1.37927	+	0.360301i
$808$	3.09166e10			2.06182
$809$		−	2.10467e10i		−	1.39754i	−0.715346	−	0.698771i	$-0.753731\pi$
$809$		−	2.10467e10i		−	1.39754i	0.715346	−	0.698771i	$-0.246269\pi$
$810$	−3.98134e10	−	2.43406e10i	−2.63228	−	1.60929i
$811$	−1.11582e9			−0.0734549			−0.0367274	−	0.999325i	$-0.511693\pi$
$811$	−1.11582e9			−0.0734549			−0.0367274	+	0.999325i	$0.511693\pi$
$812$	−7.65651e9			−0.501863
$813$	3.17618e8	+	1.21588e9i	0.0207295	+	0.0793549i
$814$	−4.17110e10			−2.71060
$815$	−1.10432e10			−0.714569
$816$	−9.13743e9	−	3.49792e10i	−0.588719	−	2.25369i
$817$	−6.79231e9			−0.435753
$818$		−	2.60079e10i		−	1.66138i
$819$	5.50470e9	+	9.81733e9i	0.350138	+	0.624453i
$820$		−	1.83386e10i		−	1.16150i
$821$			8.03471e9i			0.506721i	0.967372	+	0.253361i	$0.0815359\pi$
$821$			8.03471e9i			0.506721i	−0.967372	+	0.253361i	$0.918464\pi$
$822$	1.56406e10	−	4.08572e9i	0.982206	−	0.256577i
$823$	−2.06163e10			−1.28917			−0.644586	−	0.764532i	$-0.722970\pi$
$823$	−2.06163e10			−1.28917			−0.644586	+	0.764532i	$0.722970\pi$
$824$	−9.92789e9			−0.618175
$825$	−1.16949e10	−	4.47695e10i	−0.725117	−	2.77583i
$826$			4.79413e9i			0.295992i
$827$	−1.75200e10			−1.07712			−0.538561	−	0.842586i	$-0.681032\pi$
$827$	−1.75200e10			−1.07712			−0.538561	+	0.842586i	$0.681032\pi$
$828$	−3.78465e10	+	6.66772e9i	−2.31697	+	0.408198i
$829$	−9.79252e9			−0.596971			−0.298486	−	0.954414i	$-0.596482\pi$
$829$	−9.79252e9			−0.596971			−0.298486	+	0.954414i	$0.596482\pi$
$830$			3.93317e10i			2.38764i
$831$	6.71383e8	+	2.57013e9i	0.0405851	+	0.155365i
$832$	1.34319e10			0.808551
$833$	−1.27930e10			−0.766857
$834$	−2.82673e10	+	7.38412e9i	−1.68734	+	0.440776i
$835$			4.78740e10i			2.84575i
$836$		−	1.56293e10i		−	0.925163i
$837$	−1.55457e9	−	1.61432e9i	−0.0916371	−	0.0951592i
$838$		−	4.10613e10i		−	2.41034i
$839$	2.82237e10			1.64986			0.824929	−	0.565236i	$-0.191215\pi$
$839$	2.82237e10			1.64986			0.824929	+	0.565236i	$0.191215\pi$
$840$	−9.61393e9	−	3.68033e10i	−0.559659	−	2.14244i
$841$	1.44618e10			0.838372
$842$	−2.17934e10			−1.25815
$843$	−2.61327e9	−	1.00039e10i	−0.150241	−	0.575140i
$844$	4.06878e10			2.32952
$845$	2.42475e10			1.38251
$846$	−4.50088e10	+	2.52370e10i	−2.55565	+	1.43298i
$847$			1.34504e10i			0.760579i
$848$	−2.85275e9			−0.160649
$849$	1.66036e9	−	4.33726e8i	0.0931159	−	0.0243242i
$850$			6.43561e10i			3.59437i
$851$	5.63572e9	−	1.61038e10i	0.313470	−	0.895724i
$852$	5.26446e10	−	1.37521e10i	2.91618	−	0.761780i
$853$	4.16062e9			0.229529			0.114764	−	0.993393i	$-0.463389\pi$
$853$	4.16062e9			0.229529			0.114764	+	0.993393i	$0.463389\pi$
$854$		−	2.34284e9i		−	0.128718i
$855$	3.79668e9	+	6.77118e9i	0.207741	+	0.370496i
$856$		−	1.68710e10i		−	0.919355i
$857$		−	1.27766e8i		−	0.00693399i	−0.999994	−	0.00346700i	$-0.998896\pi$
$857$		−	1.27766e8i		−	0.00693399i	0.999994	−	0.00346700i	$-0.00110358\pi$
$858$	6.89873e10	−	1.80212e10i	3.72875	−	0.974043i
$859$	1.15734e10			0.622993			0.311497	−	0.950247i	$-0.399170\pi$
$859$	1.15734e10			0.622993			0.311497	+	0.950247i	$0.399170\pi$
$860$			1.27816e11i			6.85236i
$861$	−7.35913e8	−	2.81716e9i	−0.0392930	−	0.150418i
$862$		−	3.29595e9i		−	0.175269i
$863$		−	2.29987e10i		−	1.21805i	−0.793150	−	0.609026i	$-0.791560\pi$
$863$		−	2.29987e10i		−	1.21805i	0.793150	−	0.609026i	$-0.208440\pi$
$864$	−2.07450e10	+	1.99772e10i	−1.09425	+	1.05375i
$865$		−	3.62134e10i		−	1.90245i
$866$	−3.85640e10			−2.01776
$867$	6.75402e8	+	2.58552e9i	0.0351962	+	0.134735i
$868$		−	3.17743e9i		−	0.164914i
$869$		−	4.99230e10i		−	2.58067i
$870$	−6.08899e9	−	2.33094e10i	−0.313493	−	1.20009i
$871$			8.68885e9i			0.445552i
$872$	5.00126e10			2.55430
$873$	−3.60428e8	−	6.42804e8i	−0.0183345	−	0.0326986i
$874$	8.59900e9	+	3.00933e9i	0.435670	+	0.152468i
$875$			1.48674e10i			0.750250i
$876$	−1.03635e10	−	3.96728e10i	−0.520886	−	1.99401i
$877$	9.16942e9			0.459032			0.229516	−	0.973305i	$-0.426286\pi$
$877$	9.16942e9			0.459032			0.229516	+	0.973305i	$0.426286\pi$
$878$		−	5.05960e9i		−	0.252281i
$879$	−3.07715e9	−	1.17797e10i	−0.152823	−	0.585024i
$880$	−1.15961e11			−5.73615
$881$	−1.52730e10			−0.752505			−0.376252	−	0.926517i	$-0.622787\pi$
$881$	−1.52730e10			−0.752505			−0.376252	+	0.926517i	$0.622787\pi$
$882$	1.31103e10	+	2.33816e10i	0.643390	+	1.14745i
$883$	2.07164e10			1.01263			0.506316	−	0.862348i	$-0.331007\pi$
$883$	2.07164e10			1.01263			0.506316	+	0.862348i	$0.331007\pi$
$884$	−6.95898e10			−3.38815
$885$	−1.02418e10	+	2.67543e9i	−0.496680	+	0.129745i
$886$	3.56303e10			1.72108
$887$			3.47510e10i			1.67200i	0.548733	+	0.835998i	$0.315110\pi$
$887$			3.47510e10i			1.67200i	−0.548733	+	0.835998i	$0.684890\pi$
$888$	−1.23956e10	−	4.74519e10i	−0.594049	−	2.27409i
$889$			1.02360e10i			0.488624i
$890$			1.09241e10i			0.519422i
$891$	−1.71801e10	+	2.81011e10i	−0.813681	+	1.33092i
$892$	3.91132e10			1.84521
$893$	8.58426e9			0.403388
$894$	−4.83012e10	+	1.26175e10i	−2.26087	+	0.590596i
$895$			5.39151e9i			0.251380i
$896$	4.81987e9			0.223850
$897$	−2.36350e9	+	2.90695e10i	−0.109341	+	1.34482i
$898$	−4.85540e10			−2.23747
$899$		−	1.15704e9i		−	0.0531115i
$900$	8.25394e10	−	4.62808e10i	3.77409	−	2.11618i
$901$	1.72508e9			0.0785728
$902$	−1.84456e10			−0.836892
$903$	5.12914e9	+	1.96350e10i	0.231813	+	0.887407i
$904$		−	2.40757e9i		−	0.108390i
$905$			2.66796e10i			1.19649i
$906$	−1.59355e10	+	4.16274e9i	−0.711895	+	0.185965i
$907$		−	5.98983e9i		−	0.266556i	−0.991079	−	0.133278i	$-0.957450\pi$
$907$		−	5.98983e9i		−	0.266556i	0.991079	−	0.133278i	$-0.0425504\pi$
$908$	9.80723e10			4.34756
$909$	9.21980e9	+	1.64430e10i	0.407144	+	0.726120i
$910$	−5.02108e10			−2.20878
$911$	2.49486e10			1.09328			0.546641	−	0.837367i	$-0.315906\pi$
$911$	2.49486e10			1.09328			0.546641	+	0.837367i	$0.315906\pi$
$912$	1.21932e10	−	3.18516e9i	0.532274	−	0.139043i
$913$	2.77612e10			1.20723
$914$	−6.04950e10			−2.62064
$915$	5.00509e9	−	1.30746e9i	0.215992	−	0.0564226i
$916$			7.78644e10i			3.34738i
$917$	−9.89825e9			−0.423902
$918$	3.29970e10	−	3.17757e10i	1.40775	−	1.35564i
$919$		−	1.51975e10i		−	0.645903i	−0.946415	−	0.322952i	$-0.895325\pi$
$919$		−	1.51975e10i		−	0.645903i	0.946415	−	0.322952i	$-0.104675\pi$
$920$	3.25586e10	−	9.30344e10i	1.37850	−	3.93900i
$921$	−2.43422e9	−	9.31848e9i	−0.102672	−	0.393040i
$922$	−4.69903e10			−1.97447
$923$		−	4.12945e10i		−	1.72857i
$924$	−4.51807e10	+	1.18023e10i	−1.88409	+	0.492171i
$925$			4.20123e10i			1.74534i
$926$		−	4.75230e10i		−	1.96682i
$927$	−2.96065e9	−	5.28017e9i	−0.122070	−	0.217705i
$928$	−1.48687e10			−0.610737
$929$		−	3.11145e8i		−	0.0127323i	−0.999980	−	0.00636616i	$-0.997974\pi$
$929$		−	3.11145e8i		−	0.0127323i	0.999980	−	0.00636616i	$-0.00202642\pi$
$930$	9.67331e9	−	2.52691e9i	0.394353	−	0.103015i
$931$		−	4.45943e9i		−	0.181115i
$932$		−	6.26579e10i		−	2.53525i
$933$	−1.33902e10	+	3.49786e9i	−0.539762	+	0.140999i
$934$			4.42763e10i			1.77811i
$935$	7.01221e10			2.80553
$936$	4.10030e10	+	7.31267e10i	1.63437	+	2.91481i
$937$		−	1.73891e10i		−	0.690540i	−0.938503	−	0.345270i	$-0.887787\pi$
$937$		−	1.73891e10i		−	0.690540i	0.938503	−	0.345270i	$-0.112213\pi$
$938$		−	8.10918e9i		−	0.320824i
$939$	−4.68371e9	+	1.22350e9i	−0.184612	+	0.0482253i
$940$		−	1.61536e11i		−	6.34341i
$941$	3.94712e10			1.54425			0.772123	−	0.635473i	$-0.219195\pi$
$941$	3.94712e10			1.54425			0.772123	+	0.635473i	$0.219195\pi$
$942$	1.07917e10	+	4.13120e10i	0.420642	+	1.61027i
$943$	2.49225e9	−	7.12146e9i	0.0967832	−	0.276553i
$944$			1.71844e10i			0.664865i
$945$	1.67069e10	−	1.60885e10i	0.643997	−	0.620161i
$946$	1.28561e11			4.93732
$947$			8.05069e9i			0.308041i	0.988068	+	0.154020i	$0.0492221\pi$
$947$			8.05069e9i			0.308041i	−0.988068	+	0.154020i	$0.950778\pi$
$948$	9.87814e10	−	2.58042e10i	3.76570	−	0.983695i
$949$	−3.11194e10			−1.18195
$950$	−2.24335e10			−0.848915
$951$	2.31979e10	−	6.05987e9i	0.874615	−	0.228471i
$952$	3.73412e10			1.40268
$953$	2.99772e10			1.12193			0.560965	−	0.827840i	$-0.310430\pi$
$953$	2.99772e10			1.12193			0.560965	+	0.827840i	$0.310430\pi$
$954$	−1.76787e9	−	3.15291e9i	−0.0659223	−	0.117569i
$955$	−7.56480e10			−2.81052
$956$		−	1.74948e10i		−	0.647600i
$957$	−1.64523e10	+	4.29774e9i	−0.606783	+	0.158507i
$958$			1.19565e10i			0.439365i
$959$			8.03483e9i			0.294179i
$960$	−6.99580e9	−	2.67807e10i	−0.255204	−	0.976952i
$961$	−2.70324e10			−0.982547
$962$	−6.47386e10			−2.34450
$963$	8.97289e9	−	5.03121e9i	0.323773	−	0.181543i
$964$		−	5.38433e10i		−	1.93581i
$965$	5.36744e10			1.92274
$966$	2.20583e9	−	2.71301e10i	0.0787320	−	0.968350i
$967$	7.51374e9			0.267217			0.133608	−	0.991034i	$-0.457344\pi$
$967$	7.51374e9			0.267217			0.133608	+	0.991034i	$0.457344\pi$
$968$			1.00189e11i			3.55022i
$969$	−7.37330e9	+	1.92609e9i	−0.260333	+	0.0680054i
$970$	3.28762e9			0.115659
$971$	4.03849e10			1.41564			0.707819	−	0.706394i	$-0.249679\pi$
$971$	4.03849e10			1.41564			0.707819	+	0.706394i	$0.249679\pi$
$972$	−6.44830e10	−	1.94690e10i	−2.25223	−	0.680003i
$973$		−	1.45213e10i		−	0.505373i
$974$			1.04920e11i			3.63832i
$975$	−1.81514e10	−	6.94856e10i	−0.627182	−	2.40093i
$976$		−	8.39788e9i		−	0.289131i
$977$	−4.64513e10			−1.59355			−0.796777	−	0.604274i	$-0.793463\pi$
$977$	−4.64513e10			−1.59355			−0.796777	+	0.604274i	$0.793463\pi$
$978$	−2.19784e10	+	5.74130e9i	−0.751293	+	0.196256i
$979$	7.71046e9			0.262628
$980$	−8.39164e10			−2.84810
$981$	1.49145e10	+	2.65993e10i	0.504392	+	0.899557i
$982$	4.30180e10			1.44964
$983$	−2.37706e10			−0.798184			−0.399092	−	0.916911i	$-0.630675\pi$
$983$	−2.37706e10			−0.798184			−0.399092	+	0.916911i	$0.630675\pi$
$984$	−5.48162e9	−	2.09843e10i	−0.183411	−	0.702120i
$985$		−	7.40804e10i		−	2.46988i
$986$	2.36501e10			0.785712
$987$	−6.48231e9	−	2.48151e10i	−0.214595	−	0.821496i
$988$		−	2.42579e10i		−	0.800210i
$989$	−1.73704e10	+	4.96349e10i	−0.570982	+	1.63155i
$990$	−7.18617e10	−	1.28162e11i	−2.35383	−	4.19792i
$991$	−2.49700e10			−0.815005			−0.407502	−	0.913204i	$-0.633600\pi$
$991$	−2.49700e10			−0.815005			−0.407502	+	0.913204i	$0.633600\pi$
$992$		−	6.17045e9i		−	0.200690i
$993$	5.72561e9	+	2.19183e10i	0.185566	+	0.710370i
$994$			3.85396e10i			1.24467i
$995$		−	8.36731e10i		−	2.69280i
$996$	1.43492e10	+	5.49303e10i	0.460171	+	1.76159i
$997$	9.06575e9			0.289715			0.144857	−	0.989453i	$-0.453728\pi$
$997$	9.06575e9			0.289715			0.144857	+	0.989453i	$0.453728\pi$
$998$			2.53404e9i			0.0806967i
$999$	2.15408e10	−	2.07435e10i	0.683569	−	0.658269i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.c.b.68.2	yes	48
3.2	odd	2	inner	69.8.c.b.68.47	yes	48
23.22	odd	2	inner	69.8.c.b.68.1	✓	48
69.68	even	2	inner	69.8.c.b.68.48	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.c.b.68.1	✓	48	23.22	odd	2	inner
69.8.c.b.68.2	yes	48	1.1	even	1	trivial
69.8.c.b.68.47	yes	48	3.2	odd	2	inner
69.8.c.b.68.48	yes	48	69.68	even	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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	69.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-20.7157i q^{2} +(11.8197 + 45.2471i) q^{3} -301.139 q^{4} +470.966 q^{5} +(937.323 - 244.852i) q^{6} +481.517i q^{7} +3586.70i q^{8} +(-1907.59 + 1069.61i) q^{9} +O(q^{10})\)	\(q-20.7157i q^{2} +(11.8197 + 45.2471i) q^{3} -301.139 q^{4} +470.966 q^{5} +(937.323 - 244.852i) q^{6} +481.517i q^{7} +3586.70i q^{8} +(-1907.59 + 1069.61i) q^{9} -9756.38i q^{10} -6886.26 q^{11} +(-3559.36 - 13625.7i) q^{12} -10688.0 q^{13} +9974.96 q^{14} +(5566.66 + 21309.8i) q^{15} +35755.0 q^{16} -21621.3 q^{17} +(22157.7 + 39517.0i) q^{18} -7536.84i q^{19} -141826. q^{20} +(-21787.2 + 5691.37i) q^{21} +142654. i q^{22} +(-55075.6 - 19274.4i) q^{23} +(-162287. + 42393.5i) q^{24} +143684. q^{25} +221409. i q^{26} +(-70943.7 - 73670.5i) q^{27} -145004. i q^{28} -52802.1i q^{29} +(441447. - 115317. i) q^{30} +21912.7 q^{31} -281592. i q^{32} +(-81393.3 - 311583. i) q^{33} +447900. i q^{34} +226778. i q^{35} +(574451. - 322101. i) q^{36} +292394. i q^{37} -156131. q^{38} +(-126328. - 483600. i) q^{39} +1.68921e6i q^{40} +129303. i q^{41} +(117901. + 451337. i) q^{42} -901214. i q^{43} +2.07372e6 q^{44} +(-898411. + 503750. i) q^{45} +(-399283. + 1.14093e6i) q^{46} +1.13897e6i q^{47} +(422612. + 1.61781e6i) q^{48} +591684. q^{49} -2.97651e6i q^{50} +(-255556. - 978300. i) q^{51} +3.21857e6 q^{52} -79786.1 q^{53} +(-1.52613e6 + 1.46965e6i) q^{54} -3.24320e6 q^{55} -1.72706e6 q^{56} +(341020. - 89082.9i) q^{57} -1.09383e6 q^{58} +480616. i q^{59} +(-1.67634e6 - 6.41722e6i) q^{60} -234873. i q^{61} -453937. i q^{62} +(-515035. - 918538. i) q^{63} -1.25673e6 q^{64} -5.03368e6 q^{65} +(-6.45465e6 + 1.68612e6i) q^{66} -812954. i q^{67} +6.51102e6 q^{68} +(221136. - 2.71983e6i) q^{69} +4.69787e6 q^{70} +3.86363e6i q^{71} +(-3.83636e6 - 6.84195e6i) q^{72} +2.91162e6 q^{73} +6.05714e6 q^{74} +(1.69830e6 + 6.50128e6i) q^{75} +2.26964e6i q^{76} -3.31586e6i q^{77} +(-1.00181e7 + 2.61698e6i) q^{78} +7.24966e6i q^{79} +1.68394e7 q^{80} +(2.49484e6 - 4.08076e6i) q^{81} +2.67860e6 q^{82} -4.03139e6 q^{83} +(6.56099e6 - 1.71390e6i) q^{84} -1.01829e7 q^{85} -1.86692e7 q^{86} +(2.38914e6 - 624103. i) q^{87} -2.46989e7i q^{88} -1.11969e6 q^{89} +(1.04355e7 + 1.86112e7i) q^{90} -5.14646e6i q^{91} +(1.65854e7 + 5.80429e6i) q^{92} +(259001. + 991486. i) q^{93} +2.35946e7 q^{94} -3.54960e6i q^{95} +(1.27412e7 - 3.32832e6i) q^{96} +336972. i q^{97} -1.22571e7i q^{98} +(1.31362e7 - 7.36561e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 24 q^{3} - 2616 q^{4} + 4416 q^{6} + 648 q^{9}+O(q^{10}) \) 48 * q + 24 * q^3 - 2616 * q^4 + 4416 * q^6 + 648 * q^9	\( 48 q + 24 q^{3} - 2616 q^{4} + 4416 q^{6} + 648 q^{9} + 16668 q^{12} - 4992 q^{13} + 76296 q^{16} + 100548 q^{18} - 620268 q^{24} + 1257120 q^{25} - 105408 q^{27} - 738480 q^{31} - 1227096 q^{36} + 3832248 q^{39} - 489768 q^{46} + 733404 q^{48} - 5569584 q^{49} - 5646984 q^{52} + 1260252 q^{54} - 305760 q^{55} + 16920336 q^{58} + 6032664 q^{64} + 4496304 q^{69} + 23272152 q^{70} - 26866080 q^{72} - 2542752 q^{73} + 487176 q^{75} - 47724612 q^{78} + 25279128 q^{81} - 68451480 q^{82} - 18046992 q^{85} + 42338184 q^{87} - 59017272 q^{93} + 106197744 q^{94} + 67778460 q^{96}+O(q^{100}) \) 48 * q + 24 * q^3 - 2616 * q^4 + 4416 * q^6 + 648 * q^9 + 16668 * q^12 - 4992 * q^13 + 76296 * q^16 + 100548 * q^18 - 620268 * q^24 + 1257120 * q^25 - 105408 * q^27 - 738480 * q^31 - 1227096 * q^36 + 3832248 * q^39 - 489768 * q^46 + 733404 * q^48 - 5569584 * q^49 - 5646984 * q^52 + 1260252 * q^54 - 305760 * q^55 + 16920336 * q^58 + 6032664 * q^64 + 4496304 * q^69 + 23272152 * q^70 - 26866080 * q^72 - 2542752 * q^73 + 487176 * q^75 - 47724612 * q^78 + 25279128 * q^81 - 68451480 * q^82 - 18046992 * q^85 + 42338184 * q^87 - 59017272 * q^93 + 106197744 * q^94 + 67778460 * q^96

Introduction

L-functions

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