LMFDB - Embedded newform 69.8.c.b.68.3

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			68.3
Character	$\chi$	$=$	69.68
Dual form			69.8.c.b.68.45

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-20.0107i q^{2} +(-46.5037 - 4.94059i) q^{3} -272.428 q^{4} -368.180 q^{5} +(-98.8646 + 930.570i) q^{6} -90.4608i q^{7} +2890.10i q^{8} +(2138.18 + 459.511i) q^{9} +O(q^{10})$	\(q-20.0107i q^{2} +(-46.5037 - 4.94059i) q^{3} -272.428 q^{4} -368.180 q^{5} +(-98.8646 + 930.570i) q^{6} -90.4608i q^{7} +2890.10i q^{8} +(2138.18 + 459.511i) q^{9} +7367.54i q^{10} -7616.73 q^{11} +(12668.9 + 1345.95i) q^{12} +684.265 q^{13} -1810.18 q^{14} +(17121.7 + 1819.03i) q^{15} +22962.1 q^{16} -17158.8 q^{17} +(9195.13 - 42786.5i) q^{18} -40907.8i q^{19} +100302. q^{20} +(-446.929 + 4206.76i) q^{21} +152416. i q^{22} +(-29933.2 - 50088.2i) q^{23} +(14278.8 - 134400. i) q^{24} +57431.5 q^{25} -13692.6i q^{26} +(-97163.0 - 31932.8i) q^{27} +24644.0i q^{28} +161238. i q^{29} +(36400.0 - 342617. i) q^{30} +40761.8 q^{31} -89555.3i q^{32} +(354206. + 37631.1i) q^{33} +343359. i q^{34} +33305.8i q^{35} +(-582500. - 125184. i) q^{36} -372565. i q^{37} -818593. q^{38} +(-31820.8 - 3380.67i) q^{39} -1.06408e6i q^{40} -412245. i q^{41} +(84180.1 + 8943.37i) q^{42} +434516. i q^{43} +2.07501e6 q^{44} +(-787235. - 169183. i) q^{45} +(-1.00230e6 + 598984. i) q^{46} -950678. i q^{47} +(-1.06782e6 - 113446. i) q^{48} +815360. q^{49} -1.14924e6i q^{50} +(797946. + 84774.5i) q^{51} -186413. q^{52} +39686.6 q^{53} +(-638998. + 1.94430e6i) q^{54} +2.80433e6 q^{55} +261441. q^{56} +(-202108. + 1.90236e6i) q^{57} +3.22648e6 q^{58} +1.08872e6i q^{59} +(-4.66443e6 - 495553. i) q^{60} +2.86600e6i q^{61} -815671. i q^{62} +(41567.7 - 193422. i) q^{63} +1.14709e6 q^{64} -251933. q^{65} +(753024. - 7.08790e6i) q^{66} +1.08818e6i q^{67} +4.67453e6 q^{68} +(1.14454e6 + 2.47717e6i) q^{69} +666473. q^{70} -2.31687e6i q^{71} +(-1.32803e6 + 6.17956e6i) q^{72} +175387. q^{73} -7.45528e6 q^{74} +(-2.67077e6 - 283745. i) q^{75} +1.11444e7i q^{76} +689015. i q^{77} +(-67649.6 + 636757. i) q^{78} -1.62299e6i q^{79} -8.45419e6 q^{80} +(4.36067e6 + 1.96504e6i) q^{81} -8.24930e6 q^{82} +5.37877e6 q^{83} +(121756. - 1.14604e6i) q^{84} +6.31752e6 q^{85} +8.69496e6 q^{86} +(796611. - 7.49816e6i) q^{87} -2.20131e7i q^{88} -1.03324e7 q^{89} +(-3.38546e6 + 1.57531e7i) q^{90} -61899.1i q^{91} +(8.15463e6 + 1.36454e7i) q^{92} +(-1.89557e6 - 201387. i) q^{93} -1.90237e7 q^{94} +1.50614e7i q^{95} +(-442456. + 4.16465e6i) q^{96} -1.54727e7i q^{97} -1.63159e7i q^{98} +(-1.62859e7 - 3.49997e6i) 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 $

$2$		−	20.0107i		−	1.76871i	−0.466813	−	0.884356i	$-0.654598\pi$
$2$		−	20.0107i		−	1.76871i	0.466813	−	0.884356i	$-0.345402\pi$
$3$	−46.5037	−	4.94059i	−0.994404	−	0.105646i
$3$	−46.5037	−	4.94059i	−0.994404	−	0.105646i
$4$	−272.428			−2.12834
$5$	−368.180			−1.31724			−0.658620	−	0.752475i	$-0.728860\pi$
$5$	−368.180			−1.31724			−0.658620	+	0.752475i	$0.728860\pi$
$6$	−98.8646	+	930.570i	−0.186858	+	1.75881i
$7$		−	90.4608i		−	0.0996821i	−0.998757	−	0.0498411i	$-0.984129\pi$
$7$		−	90.4608i		−	0.0996821i	0.998757	−	0.0498411i	$-0.0158715\pi$
$8$			2890.10i			1.99571i
$9$	2138.18	+	459.511i	0.977678	+	0.210110i
$10$			7367.54i			2.32982i
$11$	−7616.73			−1.72542			−0.862708	−	0.505702i	$-0.831233\pi$
$11$	−7616.73			−1.72542			−0.862708	+	0.505702i	$0.831233\pi$
$12$	12668.9	+	1345.95i	2.11643	+	0.224851i
$13$	684.265			0.0863818			0.0431909	−	0.999067i	$-0.486248\pi$
$13$	684.265			0.0863818			0.0431909	+	0.999067i	$0.486248\pi$
$14$	−1810.18			−0.176309
$15$	17121.7	+	1819.03i	1.30987	+	0.139162i
$16$	22962.1			1.40150
$17$	−17158.8			−0.847062			−0.423531	−	0.905882i	$-0.639210\pi$
$17$	−17158.8			−0.847062			−0.423531	+	0.905882i	$0.639210\pi$
$18$	9195.13	−	42786.5i	0.371624	−	1.72923i
$19$		−	40907.8i		−	1.36826i	−0.729361	−	0.684129i	$-0.760182\pi$
$19$		−	40907.8i		−	1.36826i	0.729361	−	0.684129i	$-0.239818\pi$
$20$	100302.			2.80354
$21$	−446.929	+	4206.76i	−0.0105310	+	0.0991243i
$22$			152416.i			3.05176i
$23$	−29933.2	−	50088.2i	−0.512986	−	0.858397i
$23$	−29933.2	−	50088.2i	−0.512986	−	0.858397i
$24$	14278.8	−	134400.i	0.210840	−	1.98454i
$25$	57431.5			0.735123
$26$		−	13692.6i		−	0.152785i
$27$	−97163.0	−	31932.8i	−0.950009	−	0.312222i
$28$			24644.0i			0.212158i
$29$			161238.i			1.22765i	0.789442	+	0.613825i	$0.210370\pi$
$29$			161238.i			1.22765i	−0.789442	+	0.613825i	$0.789630\pi$
$30$	36400.0	−	342617.i	0.246137	−	2.31678i
$31$	40761.8			0.245746			0.122873	−	0.992422i	$-0.460789\pi$
$31$	40761.8			0.245746			0.122873	+	0.992422i	$0.460789\pi$
$32$		−	89555.3i		−	0.483133i
$33$	354206.	+	37631.1i	1.71576	+	0.182284i
$34$			343359.i			1.49821i
$35$			33305.8i			0.131305i
$36$	−582500.	−	125184.i	−2.08083	−	0.447186i
$37$		−	372565.i		−	1.20919i	−0.796532	−	0.604597i	$-0.793334\pi$
$37$		−	372565.i		−	1.20919i	0.796532	−	0.604597i	$-0.206666\pi$
$38$	−818593.			−2.42006
$39$	−31820.8	−	3380.67i	−0.0858984	−	0.00912592i
$40$		−	1.06408e6i		−	2.62883i
$41$		−	412245.i		−	0.934139i	−0.884221	−	0.467069i	$-0.845310\pi$
$41$		−	412245.i		−	0.934139i	0.884221	−	0.467069i	$-0.154690\pi$
$42$	84180.1	+	8943.37i	0.175322	+	0.0186264i
$43$			434516.i			0.833424i	0.909039	+	0.416712i	$0.136818\pi$
$43$			434516.i			0.833424i	−0.909039	+	0.416712i	$0.863182\pi$
$44$	2.07501e6			3.67228
$45$	−787235.	−	169183.i	−1.28784	−	0.276766i
$46$	−1.00230e6	+	598984.i	−1.51826	+	0.907324i
$47$		−	950678.i		−	1.33564i	−0.744321	−	0.667822i	$-0.767227\pi$
$47$		−	950678.i		−	1.33564i	0.744321	−	0.667822i	$-0.232773\pi$
$48$	−1.06782e6	−	113446.i	−1.39365	−	0.148063i
$49$	815360.			0.990063
$50$		−	1.14924e6i		−	1.30022i
$51$	797946.	+	84774.5i	0.842322	+	0.0894889i
$52$	−186413.			−0.183850
$53$	39686.6			0.0366167			0.0183083	−	0.999832i	$-0.494172\pi$
$53$	39686.6			0.0366167			0.0183083	+	0.999832i	$0.494172\pi$
$54$	−638998.	+	1.94430e6i	−0.552231	+	1.68029i
$55$	2.80433e6			2.27279
$56$	261441.			0.198937
$57$	−202108.	+	1.90236e6i	−0.144551	+	1.36060i
$58$	3.22648e6			2.17136
$59$			1.08872e6i			0.690136i	0.938578	+	0.345068i	$0.112144\pi$
$59$			1.08872e6i			0.690136i	−0.938578	+	0.345068i	$0.887856\pi$
$60$	−4.66443e6	−	495553.i	−2.78785	−	0.296183i
$61$			2.86600e6i			1.61667i	0.588722	+	0.808336i	$0.299631\pi$
$61$			2.86600e6i			1.61667i	−0.588722	+	0.808336i	$0.700369\pi$
$62$		−	815671.i		−	0.434654i
$63$	41567.7	−	193422.i	0.0209442	−	0.0974570i
$64$	1.14709e6			0.546974
$65$	−251933.			−0.113786
$66$	753024.	−	7.08790e6i	0.322408	−	3.03469i
$67$			1.08818e6i			0.442017i	0.975272	+	0.221008i	$0.0709348\pi$
$67$			1.08818e6i			0.442017i	−0.975272	+	0.221008i	$0.929065\pi$
$68$	4.67453e6			1.80284
$69$	1.14454e6	+	2.47717e6i	0.419429	+	0.907788i
$70$	666473.			0.232241
$71$		−	2.31687e6i		−	0.768242i	−0.923283	−	0.384121i	$-0.874505\pi$
$71$		−	2.31687e6i		−	0.768242i	0.923283	−	0.384121i	$-0.125495\pi$
$72$	−1.32803e6	+	6.17956e6i	−0.419319	+	1.95116i
$73$	175387.			0.0527675			0.0263837	−	0.999652i	$-0.491601\pi$
$73$	175387.			0.0527675			0.0263837	+	0.999652i	$0.491601\pi$
$74$	−7.45528e6			−2.13871
$75$	−2.67077e6	−	283745.i	−0.731009	−	0.0776630i
$76$			1.11444e7i			2.91212i
$77$			689015.i			0.171993i
$78$	−67649.6	+	636757.i	−0.0161411	+	0.151930i
$79$		−	1.62299e6i		−	0.370356i	−0.982705	−	0.185178i	$-0.940714\pi$
$79$		−	1.62299e6i		−	0.370356i	0.982705	−	0.185178i	$-0.0592862\pi$
$80$	−8.45419e6			−1.84611
$81$	4.36067e6	+	1.96504e6i	0.911707	+	0.410840i
$82$	−8.24930e6			−1.65222
$83$	5.37877e6			1.03255			0.516273	−	0.856424i	$-0.327319\pi$
$83$	5.37877e6			1.03255			0.516273	+	0.856424i	$0.327319\pi$
$84$	121756.	−	1.14604e6i	0.0224137	−	0.210970i
$85$	6.31752e6			1.11578
$86$	8.69496e6			1.47409
$87$	796611.	−	7.49816e6i	0.129697	−	1.22078i
$88$		−	2.20131e7i		−	3.44343i
$89$	−1.03324e7			−1.55360			−0.776798	−	0.629750i	$-0.783157\pi$
$89$	−1.03324e7			−1.55360			−0.776798	+	0.629750i	$0.783157\pi$
$90$	−3.38546e6	+	1.57531e7i	−0.489519	+	2.27781i
$91$		−	61899.1i		−	0.00861072i
$92$	8.15463e6	+	1.36454e7i	1.09181	+	1.82696i
$93$	−1.89557e6	−	201387.i	−0.244371	−	0.0259622i
$94$	−1.90237e7			−2.36237
$95$			1.50614e7i			1.80233i
$96$	−442456.	+	4.16465e6i	−0.0510412	+	0.480429i
$97$		−	1.54727e7i		−	1.72134i	−0.509165	−	0.860669i	$-0.670046\pi$
$97$		−	1.54727e7i		−	1.72134i	0.509165	−	0.860669i	$-0.329954\pi$
$98$		−	1.63159e7i		−	1.75114i
$99$	−1.62859e7	−	3.49997e6i	−1.68690	−	0.362527i
$100$	−1.56459e7			−1.56459
$101$			1.57569e7i			1.52176i	0.648892	+	0.760880i	$0.275233\pi$
$101$			1.57569e7i			1.52176i	−0.648892	+	0.760880i	$0.724767\pi$
$102$	1.69640e6	−	1.59675e7i	0.158280	−	1.48982i
$103$			1.52334e7i			1.37362i	0.726838	+	0.686809i	$0.240989\pi$
$103$			1.52334e7i			1.37362i	−0.726838	+	0.686809i	$0.759011\pi$
$104$			1.97759e6i			0.172393i
$105$	164550.	−	1.54884e6i	0.0138719	−	0.130571i
$106$		−	794157.i		−	0.0647643i
$107$	−2.67982e6			−0.211477			−0.105738	−	0.994394i	$-0.533721\pi$
$107$	−2.67982e6			−0.211477			−0.105738	+	0.994394i	$0.533721\pi$
$108$	2.64699e7	+	8.69938e6i	2.02194	+	0.664516i
$109$		−	9.07406e6i		−	0.671134i	−0.942016	−	0.335567i	$-0.891072\pi$
$109$		−	9.07406e6i		−	0.671134i	0.942016	−	0.335567i	$-0.108928\pi$
$110$		−	5.61165e7i		−	4.01991i
$111$	−1.84069e6	+	1.73256e7i	−0.127747	+	1.20243i
$112$		−	2.07717e6i		−	0.139704i
$113$	3.81729e6			0.248875			0.124437	−	0.992227i	$-0.460287\pi$
$113$	3.81729e6			0.248875			0.124437	+	0.992227i	$0.460287\pi$
$114$	3.80676e7	+	4.04433e6i	2.40651	+	0.255670i
$115$	1.10208e7	+	1.84415e7i	0.675726	+	1.13072i
$116$		−	4.39257e7i		−	2.61286i
$117$	1.46308e6	+	314427.i	0.0844536	+	0.0181497i
$118$	2.17861e7			1.22065
$119$			1.55220e6i			0.0844369i
$120$	−5.25716e6	+	4.94835e7i	−0.277726	+	2.61412i
$121$	3.85273e7			1.97706
$122$	5.73506e7			2.85943
$123$	−2.03673e6	+	1.91709e7i	−0.0986883	+	0.928911i
$124$	−1.11046e7			−0.523032
$125$	7.61894e6			0.348907
$126$	−3.87050e6	−	831799.i	−0.172373	−	0.0370443i
$127$	−3.89547e7			−1.68751			−0.843755	−	0.536728i	$-0.819660\pi$
$127$	−3.89547e7			−1.68751			−0.843755	+	0.536728i	$0.819660\pi$
$128$		−	3.44171e7i		−	1.45057i
$129$	2.14676e6	−	2.02066e7i	0.0880481	−	0.828760i
$130$			5.04135e6i			0.201254i
$131$		−	1.93004e6i		−	0.0750094i	−0.999296	−	0.0375047i	$-0.988059\pi$
$131$		−	1.93004e6i		−	0.0750094i	0.999296	−	0.0375047i	$-0.0119409\pi$
$132$	−9.64954e7	−	1.02518e7i	−3.65172	−	0.387962i
$133$	−3.70055e6			−0.136391
$134$	2.17752e7			0.781800
$135$	3.57735e7	+	1.17570e7i	1.25139	+	0.411272i
$136$		−	4.95906e7i		−	1.69049i
$137$	−4.38878e7			−1.45821			−0.729107	−	0.684399i	$-0.760064\pi$
$137$	−4.38878e7			−1.45821			−0.729107	+	0.684399i	$0.760064\pi$
$138$	4.95699e7	−	2.29030e7i	1.60562	−	0.741849i
$139$	920797.			0.0290812			0.0145406	−	0.999894i	$-0.495371\pi$
$139$	920797.			0.0290812			0.0145406	+	0.999894i	$0.495371\pi$
$140$		−	9.07344e6i		−	0.279463i
$141$	−4.69691e6	+	4.42100e7i	−0.141106	+	1.32817i
$142$	−4.63622e7			−1.35880
$143$	−5.21186e6			−0.149045
$144$	4.90972e7	+	1.05513e7i	1.37021	+	0.294469i
$145$		−	5.93646e7i		−	1.61711i
$146$		−	3.50961e6i		−	0.0933305i
$147$	−3.79172e7	−	4.02836e6i	−0.984523	−	0.104597i
$148$			1.01497e8i			2.57358i
$149$	−1.06742e7			−0.264353			−0.132177	−	0.991226i	$-0.542197\pi$
$149$	−1.06742e7			−0.264353			−0.132177	+	0.991226i	$0.542197\pi$
$150$	−5.67794e6	+	5.34440e7i	−0.137364	+	1.29294i
$151$	−4.84219e7			−1.14452			−0.572259	−	0.820073i	$-0.693933\pi$
$151$	−4.84219e7			−1.14452			−0.572259	+	0.820073i	$0.693933\pi$
$152$	1.18227e8			2.73065
$153$	−3.66886e7	−	7.88465e6i	−0.828154	−	0.177976i
$154$	1.37877e7			0.304206
$155$	−1.50077e7			−0.323707
$156$	8.66887e6	+	920988.i	0.182821	+	0.0194231i
$157$			3.53697e7i			0.729428i	0.931120	+	0.364714i	$0.118833\pi$
$157$			3.53697e7i			0.729428i	−0.931120	+	0.364714i	$0.881167\pi$
$158$	−3.24771e7			−0.655054
$159$	−1.84557e6	−	196075.i	−0.0364117	−	0.00386841i
$160$			3.29725e7i			0.636402i
$161$	−4.53102e6	+	2.70778e6i	−0.0855668	+	0.0511355i
$162$	3.93217e7	−	8.72600e7i	0.726658	−	1.61255i
$163$	4.89792e6			0.0885839			0.0442919	−	0.999019i	$-0.485897\pi$
$163$	4.89792e6			0.0885839			0.0442919	+	0.999019i	$0.485897\pi$
$164$			1.12307e8i			1.98817i
$165$	−1.30411e8	−	1.38550e7i	−2.26007	−	0.240112i
$166$		−	1.07633e8i		−	1.82628i
$167$		−	2.08843e7i		−	0.346986i	−0.984835	−	0.173493i	$-0.944495\pi$
$167$		−	2.08843e7i		−	0.346986i	0.984835	−	0.173493i	$-0.0555055\pi$
$168$	−1.21579e7	−	1.29167e6i	−0.197823	−	0.0210169i
$169$	−6.22803e7			−0.992538
$170$		−	1.26418e8i		−	1.97350i
$171$	1.87976e7	−	8.74682e7i	0.287485	−	1.33772i
$172$		−	1.18374e8i		−	1.77381i
$173$			7.30309e7i			1.07237i	0.844100	+	0.536186i	$0.180135\pi$
$173$			7.30309e7i			1.07237i	−0.844100	+	0.536186i	$0.819865\pi$
$174$	−1.50043e8	−	1.59407e7i	−2.15921	−	0.229396i
$175$		−	5.19530e6i		−	0.0732786i
$176$	−1.74896e8			−2.41817
$177$	5.37892e6	−	5.06295e7i	0.0729103	−	0.686274i
$178$			2.06759e8i			2.74786i
$179$		−	1.56526e7i		−	0.203986i	−0.994785	−	0.101993i	$-0.967478\pi$
$179$		−	1.56526e7i		−	0.203986i	0.994785	−	0.101993i	$-0.0325220\pi$
$180$	2.14465e8	+	4.60901e7i	2.74096	+	0.589052i
$181$			1.29131e8i			1.61866i	0.587352	+	0.809331i	$0.300170\pi$
$181$			1.29131e8i			1.61866i	−0.587352	+	0.809331i	$0.699830\pi$
$182$	−1.23864e6			−0.0152299
$183$	1.41597e7	−	1.33279e8i	0.170795	−	1.60762i
$184$	1.44760e8	−	8.65099e7i	1.71311	−	1.02377i
$185$			1.37171e8i			1.59280i
$186$	−4.02989e6	+	3.79317e7i	−0.0459196	+	0.432222i
$187$	1.30694e8			1.46153
$188$			2.58991e8i			2.84271i
$189$	−2.88867e6	+	8.78944e6i	−0.0311230	+	0.0946989i
$190$	3.01389e8			3.18779
$191$	7.49922e6			0.0778752			0.0389376	−	0.999242i	$-0.487603\pi$
$191$	7.49922e6			0.0778752			0.0389376	+	0.999242i	$0.487603\pi$
$192$	−5.33438e7	−	5.66729e6i	−0.543913	−	0.0577858i
$193$	1.60086e8			1.60289			0.801443	−	0.598071i	$-0.204066\pi$
$193$	1.60086e8			1.60289			0.801443	+	0.598071i	$0.204066\pi$
$194$	−3.09620e8			−3.04455
$195$	1.17158e7	+	1.24470e6i	0.113149	+	0.0120210i
$196$	−2.22127e8			−2.10719
$197$		−	1.09022e8i		−	1.01597i	−0.861365	−	0.507987i	$-0.830390\pi$
$197$		−	1.09022e8i		−	1.01597i	0.861365	−	0.507987i	$-0.169610\pi$
$198$	−7.00368e7	+	3.25893e8i	−0.641207	+	2.98364i
$199$		−	4.68410e7i		−	0.421348i	−0.977556	−	0.210674i	$-0.932434\pi$
$199$		−	4.68410e7i		−	0.421348i	0.977556	−	0.210674i	$-0.0675658\pi$
$200$			1.65983e8i			1.46709i
$201$	5.37625e6	−	5.06043e7i	0.0466974	−	0.439543i
$202$	3.15307e8			2.69156
$203$	1.45857e7			0.122375
$204$	−2.17383e8	−	2.30949e7i	−1.79275	−	0.190463i
$205$			1.51780e8i			1.23049i
$206$	3.04830e8			2.42953
$207$	−4.09865e7	−	1.20852e8i	−0.321177	−	0.947019i
$208$	1.57122e7			0.121064
$209$			3.11583e8i			2.36082i
$210$	−3.09934e7	−	3.29277e6i	−0.230942	−	0.0245354i
$211$	−5.90831e7			−0.432987			−0.216494	−	0.976284i	$-0.569462\pi$
$211$	−5.90831e7			−0.432987			−0.216494	+	0.976284i	$0.569462\pi$
$212$	−1.08117e7			−0.0779327
$213$	−1.14467e7	+	1.07743e8i	−0.0811619	+	0.763942i
$214$			5.36251e7i			0.374041i
$215$		−	1.59980e8i		−	1.09782i
$216$	9.22890e7	−	2.80811e8i	0.623106	−	1.89594i
$217$		−	3.68734e6i		−	0.0244965i
$218$	−1.81578e8			−1.18704
$219$	−8.15612e6	−	866513.i	−0.0524722	−	0.00557469i
$220$	−7.63976e8			−4.83727
$221$	−1.17411e7			−0.0731708
$222$	3.46698e8	+	3.68335e7i	2.12675	+	0.225947i
$223$	2.78192e8			1.67988			0.839938	−	0.542683i	$-0.182592\pi$
$223$	2.78192e8			1.67988			0.839938	+	0.542683i	$0.182592\pi$
$224$	−8.10124e6			−0.0481597
$225$	1.22799e8	+	2.63904e7i	0.718713	+	0.154457i
$226$		−	7.63866e7i		−	0.440187i
$227$	1.06556e8			0.604624			0.302312	−	0.953209i	$-0.402242\pi$
$227$	1.06556e8			0.604624			0.302312	+	0.953209i	$0.402242\pi$
$228$	5.50599e7	−	5.18256e8i	0.307655	−	2.89582i
$229$			6.86772e7i			0.377910i	0.981986	+	0.188955i	$0.0605100\pi$
$229$			6.86772e7i			0.377910i	−0.981986	+	0.188955i	$0.939490\pi$
$230$	3.69027e8	−	2.20534e8i	1.99991	−	1.19516i
$231$	3.40414e6	−	3.20417e7i	0.0181704	−	0.171031i
$232$	−4.65994e8			−2.45004
$233$		−	2.45593e8i		−	1.27195i	−0.771709	−	0.635975i	$-0.780598\pi$
$233$		−	2.45593e8i		−	1.27195i	0.771709	−	0.635975i	$-0.219402\pi$
$234$	6.29190e6	−	2.92773e7i	0.0321016	−	0.149374i
$235$			3.50021e8i			1.75937i
$236$		−	2.96598e8i		−	1.46884i
$237$	−8.01851e6	+	7.54748e7i	−0.0391268	+	0.368284i
$238$	3.10605e7			0.149345
$239$		−	2.53785e7i		−	0.120247i	−0.998191	−	0.0601233i	$-0.980851\pi$
$239$		−	2.53785e7i		−	0.120247i	0.998191	−	0.0601233i	$-0.0191494\pi$
$240$	3.93151e8	+	4.17687e7i	1.83578	+	0.195035i
$241$		−	3.10711e8i		−	1.42987i	−0.699190	−	0.714936i	$-0.746456\pi$
$241$		−	3.10711e8i		−	1.42987i	0.699190	−	0.714936i	$-0.253544\pi$
$242$		−	7.70958e8i		−	3.49685i
$243$	−1.93079e8	−	1.12926e8i	−0.863202	−	0.504859i
$244$		−	7.80778e8i		−	3.44083i
$245$	−3.00199e8			−1.30415
$246$	3.83623e8	+	4.07564e7i	1.64298	+	0.174551i
$247$		−	2.79917e7i		−	0.118193i
$248$			1.17806e8i			0.490439i
$249$	−2.50132e8	−	2.65743e7i	−1.02677	−	0.109085i
$250$		−	1.52460e8i		−	0.617116i
$251$	−7.81013e7			−0.311745			−0.155873	−	0.987777i	$-0.549819\pi$
$251$	−7.81013e7			−0.311745			−0.155873	+	0.987777i	$0.549819\pi$
$252$	−1.13242e7	+	5.26934e7i	−0.0445765	+	0.207422i
$253$	2.27993e8	+	3.81508e8i	0.885114	+	1.48109i
$254$			7.79510e8i			2.98472i
$255$	−2.93788e8	−	3.12123e7i	−1.10954	−	0.117878i
$256$	−5.41883e8			−2.01867
$257$			1.75961e8i			0.646623i	0.946293	+	0.323312i	$0.104796\pi$
$257$			1.75961e8i			0.646623i	−0.946293	+	0.323312i	$0.895204\pi$
$258$	−4.04347e8	−	4.29582e7i	−1.46584	−	0.155732i
$259$	−3.37025e7			−0.120535
$260$	6.86334e7			0.242175
$261$	−7.40906e7	+	3.44756e8i	−0.257942	+	1.20025i
$262$	−3.86214e7			−0.132670
$263$	5.49511e8			1.86265			0.931324	−	0.364191i	$-0.118654\pi$
$263$	5.49511e8			1.86265			0.931324	+	0.364191i	$0.118654\pi$
$264$	−1.08758e8	+	1.02369e9i	−0.363786	+	3.42416i
$265$	−1.46118e7			−0.0482329
$266$			7.40505e7i			0.241236i
$267$	4.80496e8	+	5.10483e7i	1.54490	+	0.164132i
$268$		−	2.96450e8i		−	0.940763i
$269$			3.56591e8i			1.11696i	0.829519	+	0.558479i	$0.188615\pi$
$269$			3.56591e8i			1.11696i	−0.829519	+	0.558479i	$0.811385\pi$
$270$	2.35266e8	−	7.15852e8i	0.727422	−	2.21335i
$271$	3.36726e8			1.02774			0.513870	−	0.857868i	$-0.328211\pi$
$271$	3.36726e8			1.02774			0.513870	+	0.857868i	$0.328211\pi$
$272$	−3.94002e8			−1.18715
$273$	−305818.	+	2.87854e6i	−0.000909691	+	0.00856254i
$274$			8.78224e8i			2.57916i
$275$	−4.37440e8			−1.26839
$276$	−3.11804e8	−	6.74851e8i	−0.892688	−	1.93208i
$277$	−3.65354e7			−0.103284			−0.0516422	−	0.998666i	$-0.516446\pi$
$277$	−3.65354e7			−0.103284			−0.0516422	+	0.998666i	$0.516446\pi$
$278$		−	1.84258e7i		−	0.0514362i
$279$	8.71560e7	+	1.87305e7i	0.240261	+	0.0516338i
$280$	−9.62572e7			−0.262048
$281$	3.55061e8			0.954621			0.477311	−	0.878735i	$-0.341612\pi$
$281$	3.55061e8			0.954621			0.477311	+	0.878735i	$0.341612\pi$
$282$	8.84673e8	+	9.39884e7i	2.34915	+	0.249576i
$283$		−	4.40887e8i		−	1.15631i	−0.815927	−	0.578155i	$-0.803773\pi$
$283$		−	4.40887e8i		−	1.15631i	0.815927	−	0.578155i	$-0.196227\pi$
$284$			6.31180e8i			1.63508i
$285$	7.44123e7	−	7.00411e8i	0.190409	−	1.79224i
$286$			1.04293e8i			0.263617i
$287$	−3.72920e7			−0.0931169
$288$	4.11516e7	−	1.91486e8i	0.101511	−	0.472348i
$289$	−1.15915e8			−0.282486
$290$	−1.18793e9			−2.86020
$291$	−7.64444e7	+	7.19539e8i	−0.181853	+	1.71171i
$292$	−4.77802e7			−0.112307
$293$	−2.44920e8			−0.568836			−0.284418	−	0.958700i	$-0.591800\pi$
$293$	−2.44920e8			−0.568836			−0.284418	+	0.958700i	$0.591800\pi$
$294$	−8.06102e7	+	7.58750e8i	−0.185001	+	1.74134i
$295$		−	4.00845e8i		−	0.909075i
$296$	1.07675e9			2.41320
$297$	7.40064e8	+	2.43223e8i	1.63916	+	0.538714i
$298$			2.13599e8i			0.467565i
$299$	−2.04822e7	−	3.42736e7i	−0.0443127	−	0.0741499i
$300$	7.27593e8	+	7.73001e7i	1.55584	+	0.165293i
$301$	3.93066e7			0.0830774
$302$			9.68955e8i			2.02432i
$303$	7.78484e7	−	7.32754e8i	0.160768	−	1.51324i
$304$		−	9.39329e8i		−	1.91761i
$305$		−	1.05520e9i		−	2.12954i
$306$	−1.57777e8	+	7.34164e8i	−0.314789	+	1.46476i
$307$	4.77523e8			0.941911			0.470955	−	0.882157i	$-0.343909\pi$
$307$	4.77523e8			0.941911			0.470955	+	0.882157i	$0.343909\pi$
$308$		−	1.87707e8i		−	0.366060i
$309$	7.52619e7	−	7.08408e8i	0.145118	−	1.36593i
$310$			3.00314e8i			0.572544i
$311$		−	4.45406e8i		−	0.839642i	−0.907607	−	0.419821i	$-0.862093\pi$
$311$		−	4.45406e8i		−	0.839642i	0.907607	−	0.419821i	$-0.137907\pi$
$312$	9.77047e6	−	9.19653e7i	0.0182127	−	0.171428i
$313$			6.05543e8i			1.11619i	0.829776	+	0.558097i	$0.188468\pi$
$313$			6.05543e8i			1.11619i	−0.829776	+	0.558097i	$0.811532\pi$
$314$	7.07772e8			1.29015
$315$	−1.53044e7	+	7.12139e7i	−0.0275886	+	0.128374i
$316$			4.42146e8i			0.788245i
$317$			2.14401e7i			0.0378024i	0.999821	+	0.0189012i	$0.00601679\pi$
$317$			2.14401e7i			0.0378024i	−0.999821	+	0.0189012i	$0.993983\pi$
$318$	−3.92360e6	+	3.69312e7i	−0.00684211	+	0.0644019i
$319$		−	1.22811e9i		−	2.11821i
$320$	−4.22335e8			−0.720497
$321$	1.24621e8	+	1.32399e7i	0.210293	+	0.0223417i
$322$	5.41845e7	+	9.06688e7i	0.0904440	+	0.151343i
$323$			7.01927e8i			1.15900i
$324$	−1.18797e9	−	5.35330e8i	−1.94043	−	0.874408i
$325$	3.92983e7			0.0635013
$326$		−	9.80107e7i		−	0.156679i
$327$	−4.48312e7	+	4.21977e8i	−0.0709028	+	0.667378i
$328$	1.19143e9			1.86427
$329$	−8.59991e7			−0.133140
$330$	−2.77248e8	+	2.60962e9i	−0.424688	+	3.99741i
$331$	−3.30715e8			−0.501252			−0.250626	−	0.968084i	$-0.580636\pi$
$331$	−3.30715e8			−0.501252			−0.250626	+	0.968084i	$0.580636\pi$
$332$	−1.46533e9			−2.19761
$333$	1.71198e8	−	7.96611e8i	0.254064	−	1.18220i
$334$	−4.17910e8			−0.613719
$335$		−	4.00646e8i		−	0.582242i
$336$	−1.02625e7	+	9.65961e7i	−0.0147592	+	0.138922i
$337$			1.07841e9i			1.53490i	0.641111	+	0.767448i	$0.278474\pi$
$337$			1.07841e9i			1.53490i	−0.641111	+	0.767448i	$0.721526\pi$
$338$			1.24627e9i			1.75551i
$339$	−1.77518e8	−	1.88596e7i	−0.247482	−	0.0262927i
$340$	−1.72107e9			−2.37477
$341$	−3.10471e8			−0.424015
$342$	−1.75030e9	−	3.76152e8i	−2.36603	−	0.508478i
$343$		−	1.48256e8i		−	0.198374i
$344$	−1.25579e9			−1.66327
$345$	−4.21396e8	−	9.12046e8i	−0.552489	−	1.19578i
$346$	1.46140e9			1.89672
$347$		−	7.91053e8i		−	1.01637i	−0.861247	−	0.508186i	$-0.830316\pi$
$347$		−	7.91053e8i		−	1.01637i	0.861247	−	0.508186i	$-0.169684\pi$
$348$	−2.17019e8	+	2.04271e9i	−0.276039	+	2.59824i
$349$	−1.07694e9			−1.35614			−0.678069	−	0.734998i	$-0.737183\pi$
$349$	−1.07694e9			−1.35614			−0.678069	+	0.734998i	$0.737183\pi$
$350$	−1.03961e8			−0.129609
$351$	−6.64852e7	−	2.18505e7i	−0.0820635	−	0.0269703i
$352$			6.82118e8i			0.833605i
$353$			1.41121e9i			1.70758i	0.520618	+	0.853789i	$0.325701\pi$
$353$			1.41121e9i			1.70758i	−0.520618	+	0.853789i	$0.674299\pi$
$354$	−1.01313e9	−	1.07636e8i	−1.21382	−	0.128957i
$355$			8.53026e8i			1.01196i
$356$	2.81484e9			3.30658
$357$	7.66876e6	−	7.21828e7i	0.00892045	−	0.0839644i
$358$	−3.13219e8			−0.360793
$359$	8.56798e8			0.977345			0.488672	−	0.872467i	$-0.337481\pi$
$359$	8.56798e8			0.977345			0.488672	+	0.872467i	$0.337481\pi$
$360$	4.88955e8	−	2.27519e9i	0.552344	−	2.57015i
$361$	−7.79573e8			−0.872131
$362$	2.58401e9			2.86295
$363$	−1.79166e9	−	1.90348e8i	−1.96600	−	0.208869i
$364$			1.68630e7i			0.0183266i
$365$	−6.45738e7			−0.0695075
$366$	−2.66701e9	−	2.83346e8i	−2.84342	−	0.302088i
$367$		−	7.47471e8i		−	0.789338i	−0.918823	−	0.394669i	$-0.870859\pi$
$367$		−	7.47471e8i		−	0.789338i	0.918823	−	0.394669i	$-0.129141\pi$
$368$	−6.87330e8	−	1.15013e9i	−0.718948	−	1.20304i
$369$	1.89431e8	−	8.81454e8i	0.196272	−	0.913287i
$370$	2.74488e9			2.81720
$371$		−	3.59008e6i		−	0.00365002i
$372$	5.16406e8	+	5.48634e7i	0.520105	+	0.0552564i
$373$		−	1.57437e8i		−	0.157082i	−0.996911	−	0.0785408i	$-0.974974\pi$
$373$		−	1.57437e8i		−	0.157082i	0.996911	−	0.0785408i	$-0.0250261\pi$
$374$		−	2.61527e9i		−	2.58503i
$375$	−3.54308e8	−	3.76420e7i	−0.346954	−	0.0368607i
$376$	2.74755e9			2.66556
$377$			1.10330e8i			0.106047i
$378$	1.75883e8	+	5.78042e7i	0.167495	+	0.0550476i
$379$		−	1.20207e9i		−	1.13420i	−0.823648	−	0.567102i	$-0.808065\pi$
$379$		−	1.20207e9i		−	1.13420i	0.823648	−	0.567102i	$-0.191935\pi$
$380$		−	4.10315e9i		−	3.83597i
$381$	1.81153e9	+	1.92459e8i	1.67807	+	0.178279i
$382$		−	1.50064e8i		−	0.137739i
$383$	5.11397e8			0.465117			0.232559	−	0.972582i	$-0.425290\pi$
$383$	5.11397e8			0.465117			0.232559	+	0.972582i	$0.425290\pi$
$384$	−1.70041e8	+	1.60052e9i	−0.153248	+	1.44245i
$385$		−	2.53681e8i		−	0.226556i
$386$		−	3.20343e9i		−	2.83504i
$387$	−1.99665e8	+	9.29073e8i	−0.175111	+	0.814820i
$388$			4.21520e9i			3.66360i
$389$	1.77899e9			1.53232			0.766159	−	0.642652i	$-0.222166\pi$
$389$	1.77899e9			1.53232			0.766159	+	0.642652i	$0.222166\pi$
$390$	2.49072e7	−	2.34441e8i	0.0212617	−	0.200128i
$391$	5.13617e8	+	8.59453e8i	0.434531	+	0.727115i
$392$			2.35647e9i			1.97588i
$393$	−9.53551e6	+	8.97538e7i	−0.00792447	+	0.0745897i
$394$	−2.18161e9			−1.79697
$395$			5.97551e8i			0.487849i
$396$	4.43674e9	+	9.53488e8i	3.59030	+	0.771582i
$397$	−1.51126e9			−1.21219			−0.606097	−	0.795391i	$-0.707266\pi$
$397$	−1.51126e9			−1.21219			−0.606097	+	0.795391i	$0.707266\pi$
$398$	−9.37321e8			−0.745242
$399$	1.72089e8	+	1.82829e7i	0.135628	+	0.0144092i
$400$	1.31875e9			1.03027
$401$	−1.56483e9			−1.21188			−0.605942	−	0.795509i	$-0.707204\pi$
$401$	−1.56483e9			−1.21188			−0.605942	+	0.795509i	$0.707204\pi$
$402$	−1.01263e9	−	1.07582e8i	−0.777425	−	0.0825943i
$403$	2.78918e7			0.0212280
$404$		−	4.29262e9i		−	3.23883i
$405$	−1.60551e9	−	7.23487e8i	−1.20094	−	0.541175i
$406$		−	2.91870e8i		−	0.216446i
$407$			2.83772e9i			2.08636i
$408$	−2.45007e8	+	2.30614e9i	−0.178594	+	1.68103i
$409$	−5.13245e8			−0.370931			−0.185466	−	0.982651i	$-0.559379\pi$
$409$	−5.13245e8			−0.370931			−0.185466	+	0.982651i	$0.559379\pi$
$410$	3.03723e9			2.17637
$411$	2.04094e9	+	2.16831e8i	1.45005	+	0.154055i
$412$		−	4.15000e9i		−	2.92353i
$413$	9.84866e7			0.0687942
$414$	−2.41834e9	+	8.20168e8i	−1.67500	+	0.568070i
$415$	−1.98035e9			−1.36011
$416$		−	6.12796e7i		−	0.0417339i
$417$	−4.28204e7	−	4.54928e6i	−0.0289184	−	0.00307232i
$418$	6.23500e9			4.17560
$419$	−1.19495e9			−0.793601			−0.396800	−	0.917905i	$-0.629879\pi$
$419$	−1.19495e9			−0.793601			−0.396800	+	0.917905i	$0.629879\pi$
$420$	−4.48281e7	+	4.21948e8i	−0.0295242	+	0.277899i
$421$		−	2.22222e9i		−	1.45145i	−0.687987	−	0.725723i	$-0.741506\pi$
$421$		−	2.22222e9i		−	1.45145i	0.687987	−	0.725723i	$-0.258494\pi$
$422$			1.18229e9i			0.765830i
$423$	4.36847e8	−	2.03272e9i	0.280633	−	1.30583i
$424$			1.14698e8i			0.0730763i
$425$	−9.85454e8			−0.622695
$426$	2.15601e9	+	2.29057e8i	1.35119	+	0.143552i
$427$	2.59261e8			0.161153
$428$	7.30058e8			0.450095
$429$	2.42370e8	+	2.57496e7i	0.148211	+	0.0157460i
$430$	−3.20131e9			−1.94173
$431$	−1.93633e9			−1.16495			−0.582476	−	0.812848i	$-0.697916\pi$
$431$	−1.93633e9			−1.16495			−0.582476	+	0.812848i	$0.697916\pi$
$432$	−2.23107e9	−	7.33245e8i	−1.33143	−	0.437579i
$433$			1.19810e9i			0.709230i	0.935012	+	0.354615i	$0.115388\pi$
$433$			1.19810e9i			0.709230i	−0.935012	+	0.354615i	$0.884612\pi$
$434$	−7.37862e7			−0.0433273
$435$	−2.93296e8	+	2.76067e9i	−0.170842	+	1.60806i
$436$			2.47203e9i			1.42840i
$437$	−2.04900e9	+	1.22450e9i	−1.17451	+	0.701897i
$438$	−1.73395e7	+	1.63210e8i	−0.00986002	+	0.0928082i
$439$	−3.21256e8			−0.181228			−0.0906140	−	0.995886i	$-0.528883\pi$
$439$	−3.21256e8			−0.181228			−0.0906140	+	0.995886i	$0.528883\pi$
$440$			8.10478e9i			4.53583i
$441$	1.74339e9	+	3.74667e8i	0.967963	+	0.208022i
$442$			2.34948e8i			0.129418i
$443$		−	1.41043e9i		−	0.770792i	−0.922751	−	0.385396i	$-0.874065\pi$
$443$		−	1.41043e9i		−	0.770792i	0.922751	−	0.385396i	$-0.125935\pi$
$444$	5.01455e8	−	4.71998e9i	0.271889	−	2.55917i
$445$	3.80420e9			2.04646
$446$		−	5.56681e9i		−	2.97122i
$447$	4.96391e8	+	5.27370e7i	0.262874	+	0.0279280i
$448$		−	1.03766e8i		−	0.0545235i
$449$			1.71634e9i			0.894830i	0.894327	+	0.447415i	$0.147655\pi$
$449$			1.71634e9i			0.894830i	−0.894327	+	0.447415i	$0.852345\pi$
$450$	5.28090e8	−	2.45729e9i	0.273190	−	1.27120i
$451$			3.13995e9i			1.61178i
$452$	−1.03994e9			−0.529690
$453$	2.25179e9	+	2.39233e8i	1.13811	+	0.120914i
$454$		−	2.13225e9i		−	1.06941i
$455$			2.27900e7i			0.0113424i
$456$	−5.49801e9	−	5.84113e8i	−2.71537	−	0.288483i
$457$			3.45672e8i			0.169417i	0.996406	+	0.0847087i	$0.0269960\pi$
$457$			3.45672e8i			0.169417i	−0.996406	+	0.0847087i	$0.973004\pi$
$458$	1.37428e9			0.668414
$459$	1.66720e9	+	5.47928e8i	0.804716	+	0.264472i
$460$	−3.00237e9	−	5.02397e9i	−1.43818	−	2.40655i
$461$			1.14537e9i			0.544494i	0.962227	+	0.272247i	$0.0877668\pi$
$461$			1.14537e9i			0.544494i	−0.962227	+	0.272247i	$0.912233\pi$
$462$	−6.41177e8	−	6.81192e7i	−0.302504	−	0.0321383i
$463$	−1.23913e9			−0.580206			−0.290103	−	0.956995i	$-0.593690\pi$
$463$	−1.23913e9			−0.580206			−0.290103	+	0.956995i	$0.593690\pi$
$464$			3.70237e9i			1.72055i
$465$	6.97911e8	+	7.41467e7i	0.321895	+	0.0341984i
$466$	−4.91449e9			−2.24971
$467$	2.41421e9			1.09690			0.548449	−	0.836184i	$-0.315219\pi$
$467$	2.41421e9			1.09690			0.548449	+	0.836184i	$0.315219\pi$
$468$	−3.98584e8	−	8.56587e7i	−0.179746	−	0.0386288i
$469$	9.84376e7			0.0440612
$470$	7.00415e9			3.11181
$471$	1.74747e8	−	1.64482e9i	0.0770614	−	0.725346i
$472$	−3.14651e9			−1.37731
$473$		−	3.30959e9i		−	1.43800i
$474$	1.51030e9	+	1.60456e8i	0.651388	+	0.0692040i
$475$		−	2.34939e9i		−	1.00584i
$476$		−	4.22861e8i		−	0.179711i
$477$	8.48572e7	+	1.82364e7i	0.0357993	+	0.00769353i
$478$	−5.07840e8			−0.212681
$479$	−1.87939e9			−0.781346			−0.390673	−	0.920530i	$-0.627758\pi$
$479$	−1.87939e9			−0.781346			−0.390673	+	0.920530i	$0.627758\pi$
$480$	1.62903e8	−	1.53334e9i	0.0672335	−	0.632841i
$481$		−	2.54933e8i		−	0.104452i
$482$	−6.21754e9			−2.52903
$483$	2.24087e8	−	1.03536e8i	0.0904902	−	0.0418095i
$484$	−1.04959e10			−4.20786
$485$			5.69675e9i			2.26742i
$486$	−2.25972e9	+	3.86364e9i	−0.892951	+	1.52675i
$487$	2.97796e9			1.16833			0.584167	−	0.811633i	$-0.301421\pi$
$487$	2.97796e9			1.16833			0.584167	+	0.811633i	$0.301421\pi$
$488$	−8.28302e9			−3.22641
$489$	−2.27771e8	−	2.41986e7i	−0.0880882	−	0.00935856i
$490$			6.00719e9i			2.30667i
$491$			9.35734e8i			0.356752i	0.983962	+	0.178376i	$0.0570844\pi$
$491$			9.35734e8i			0.356752i	−0.983962	+	0.178376i	$0.942916\pi$
$492$	5.54862e8	−	5.22268e9i	0.210042	−	1.97704i
$493$		−	2.76665e9i		−	1.03990i
$494$	−5.60134e8			−0.209049
$495$	5.99616e9	+	1.28862e9i	2.22205	+	0.477536i
$496$	9.35977e8			0.344413
$497$	−2.09586e8			−0.0765799
$498$	−5.31770e8	+	5.00532e9i	−0.192939	+	1.81606i
$499$	−3.83678e9			−1.38234			−0.691170	−	0.722693i	$-0.742904\pi$
$499$	−3.83678e9			−1.38234			−0.691170	+	0.722693i	$0.742904\pi$
$500$	−2.07561e9			−0.742593
$501$	−1.03181e8	+	9.71197e8i	−0.0366578	+	0.345045i
$502$			1.56286e9i			0.551388i
$503$	−3.94391e8			−0.138178			−0.0690891	−	0.997610i	$-0.522009\pi$
$503$	−3.94391e8			−0.138178			−0.0690891	+	0.997610i	$0.522009\pi$
$504$	5.59007e8	+	1.20135e8i	0.194496	+	0.0417986i
$505$		−	5.80138e9i		−	2.00453i
$506$	7.63424e9	−	4.56229e9i	2.61963	−	1.56551i
$507$	2.89626e9	+	3.07701e8i	0.986984	+	0.104858i
$508$	1.06123e10			3.59160
$509$			1.61647e9i			0.543318i	0.962394	+	0.271659i	$0.0875723\pi$
$509$			1.61647e9i			0.543318i	−0.962394	+	0.271659i	$0.912428\pi$
$510$	−6.24579e8	+	5.87890e9i	−0.208493	+	1.96246i
$511$		−	1.58656e7i		−	0.00525997i
$512$			6.43806e9i			2.11987i
$513$	−1.30630e9	+	3.97472e9i	−0.427201	+	1.29986i
$514$	3.52111e9			1.14369
$515$		−	5.60862e9i		−	1.80939i
$516$	−5.84838e8	+	5.50483e9i	−0.187396	+	1.76388i
$517$			7.24105e9i			2.30454i
$518$			6.74410e8i			0.213192i
$519$	3.60816e8	−	3.39621e9i	0.113292	−	1.06637i
$520$		−	7.28110e8i		−	0.227083i
$521$	−3.32798e8			−0.103098			−0.0515488	−	0.998670i	$-0.516416\pi$
$521$	−3.32798e8			−0.103098			−0.0515488	+	0.998670i	$0.516416\pi$
$522$	6.89881e9	+	1.48260e9i	2.12289	+	0.456225i
$523$			2.69369e9i			0.823363i	0.911328	+	0.411682i	$0.135058\pi$
$523$			2.69369e9i			0.823363i	−0.911328	+	0.411682i	$0.864942\pi$
$524$			5.25795e8i			0.159646i
$525$	−2.56678e7	+	2.41600e8i	−0.00774161	+	0.0728685i
$526$		−	1.09961e10i		−	3.29449i
$527$	−6.99422e8			−0.208162
$528$	8.13331e9	+	8.64090e8i	2.40463	+	0.255470i
$529$	−1.61283e9	+	2.99860e9i	−0.473691	+	0.880691i
$530$			2.92393e8i			0.0853102i
$531$	−5.00279e8	+	2.32788e9i	−0.145005	+	0.674730i
$532$	1.00813e9			0.290286
$533$		−	2.82084e8i		−	0.0806926i
$534$	1.02151e9	−	9.61506e9i	0.290301	−	2.73248i
$535$	9.86656e8			0.278566
$536$	−3.14495e9			−0.882138
$537$	−7.73330e7	+	7.27903e8i	−0.0215504	+	0.202845i
$538$	7.13562e9			1.97558
$539$	−6.21037e9			−1.70827
$540$	−9.74569e9	−	3.20294e9i	−2.66339	−	0.875327i
$541$	−3.53599e9			−0.960110			−0.480055	−	0.877238i	$-0.659383\pi$
$541$	−3.53599e9			−0.960110			−0.480055	+	0.877238i	$0.659383\pi$
$542$		−	6.73811e9i		−	1.81778i
$543$	6.37984e8	−	6.00508e9i	0.171006	−	1.60960i
$544$			1.53666e9i			0.409243i
$545$			3.34089e9i			0.884045i
$546$	5.76015e7	+	6.11963e6i	0.0151447	+	0.00160898i
$547$	−4.47881e8			−0.117006			−0.0585028	−	0.998287i	$-0.518633\pi$
$547$	−4.47881e8			−0.117006			−0.0585028	+	0.998287i	$0.518633\pi$
$548$	1.19562e10			3.10358
$549$	−1.31696e9	+	6.12803e9i	−0.339679	+	1.58058i
$550$			8.75347e9i			2.24342i
$551$	6.59589e9			1.67974
$552$	−7.15928e9	+	3.30783e9i	−1.81168	+	0.837059i
$553$	−1.46817e8			−0.0369179
$554$			7.31099e8i			0.182680i
$555$	6.77705e8	−	6.37895e9i	0.168273	−	1.58388i
$556$	−2.50851e8			−0.0618947
$557$	−3.84440e9			−0.942616			−0.471308	−	0.881969i	$-0.656218\pi$
$557$	−3.84440e9			−0.942616			−0.471308	+	0.881969i	$0.656218\pi$
$558$	3.74810e8	−	1.74405e9i	0.0913253	−	0.424952i
$559$			2.97324e8i			0.0719927i
$560$			7.64773e8i			0.184024i
$561$	−6.07774e9	−	6.45704e8i	−1.45336	−	0.154406i
$562$		−	7.10502e9i		−	1.68845i
$563$	5.24212e9			1.23802			0.619010	−	0.785383i	$-0.287534\pi$
$563$	5.24212e9			1.23802			0.619010	+	0.785383i	$0.287534\pi$
$564$	1.27957e9	−	1.20440e10i	0.300322	−	2.82680i
$565$	−1.40545e9			−0.327828
$566$	−8.82244e9			−2.04518
$567$	1.77759e8	−	3.94469e8i	0.0409534	−	0.0908809i
$568$	6.69599e9			1.53319
$569$	7.03623e9			1.60120			0.800602	−	0.599196i	$-0.204513\pi$
$569$	7.03623e9			1.60120			0.800602	+	0.599196i	$0.204513\pi$
$570$	−1.40157e10	−	1.48904e9i	−3.16996	−	0.336779i
$571$		−	5.29543e9i		−	1.19035i	−0.803596	−	0.595175i	$-0.797083\pi$
$571$		−	5.29543e9i		−	1.19035i	0.803596	−	0.595175i	$-0.202917\pi$
$572$	1.41985e9			0.317218
$573$	−3.48741e8	−	3.70505e7i	−0.0774394	−	0.00822722i
$574$			7.46238e8i			0.164697i
$575$	−1.71911e9	−	2.87664e9i	−0.377108	−	0.631027i
$576$	2.45268e9	+	5.27099e8i	0.534765	+	0.114925i
$577$	6.06963e9			1.31537			0.657684	−	0.753294i	$-0.271536\pi$
$577$	6.06963e9			1.31537			0.657684	+	0.753294i	$0.271536\pi$
$578$			2.31954e9i			0.499637i
$579$	−7.44458e9	−	7.90919e8i	−1.59392	−	0.169339i
$580$			1.61726e10i			3.44176i
$581$		−	4.86568e8i		−	0.102926i
$582$	1.43985e10	+	1.52971e9i	3.02751	+	0.321646i
$583$	−3.02282e8			−0.0631790
$584$			5.06885e8i			0.105309i
$585$	−5.38678e8	−	1.15766e8i	−0.111246	−	0.0239075i
$586$			4.90101e9i			1.00611i
$587$			6.31855e9i			1.28939i	0.764440	+	0.644695i	$0.223015\pi$
$587$			6.31855e9i			1.28939i	−0.764440	+	0.644695i	$0.776985\pi$
$588$	1.03297e10	+	1.09744e9i	2.09540	+	0.222617i
$589$		−	1.66747e9i		−	0.336244i
$590$	−8.02119e9			−1.60789
$591$	−5.38633e8	+	5.06993e9i	−0.107334	+	1.01029i
$592$		−	8.55488e9i		−	1.69468i
$593$			3.19869e9i			0.629914i	0.949106	+	0.314957i	$0.101990\pi$
$593$			3.19869e9i			0.629914i	−0.949106	+	0.314957i	$0.898010\pi$
$594$	4.86707e9	−	1.48092e10i	0.952829	−	2.89920i
$595$		−	5.71488e8i		−	0.111224i
$596$	2.90796e9			0.562634
$597$	−2.31422e8	+	2.17828e9i	−0.0445138	+	0.418990i
$598$	−6.85839e8	+	4.09863e8i	−0.131150	+	0.0783764i
$599$			2.93410e9i			0.557804i	0.960320	+	0.278902i	$0.0899704\pi$
$599$			2.93410e9i			0.557804i	−0.960320	+	0.278902i	$0.910030\pi$
$600$	8.20052e8	−	7.71880e9i	0.154993	−	1.45888i
$601$	2.88156e9			0.541461			0.270731	−	0.962655i	$-0.412735\pi$
$601$	2.88156e9			0.541461			0.270731	+	0.962655i	$0.412735\pi$
$602$		−	7.86553e8i		−	0.146940i
$603$	−5.00030e8	+	2.32672e9i	−0.0928722	+	0.432150i
$604$	1.31915e10			2.43592
$605$	−1.41850e10			−2.60427
$606$	−1.46629e10	−	1.55780e9i	−2.67649	−	0.284353i
$607$	1.63381e9			0.296511			0.148256	−	0.988949i	$-0.452634\pi$
$607$	1.63381e9			0.296511			0.148256	+	0.988949i	$0.452634\pi$
$608$	−3.66351e9			−0.661051
$609$	−6.78289e8	−	7.20620e7i	−0.121690	−	0.0129284i
$610$	−2.11154e10			−3.76655
$611$		−	6.50516e8i		−	0.115375i
$612$	9.99499e9	+	2.14800e9i	1.76259	+	0.378794i
$613$		−	2.10829e9i		−	0.369674i	−0.982769	−	0.184837i	$-0.940824\pi$
$613$		−	2.10829e9i		−	0.369674i	0.982769	−	0.184837i	$-0.0591757\pi$
$614$		−	9.55556e9i		−	1.66597i
$615$	7.49883e8	−	7.05834e9i	0.129996	−	1.22360i
$616$	−1.99132e9			−0.343249
$617$	−5.07357e9			−0.869593			−0.434796	−	0.900529i	$-0.643180\pi$
$617$	−5.07357e9			−0.869593			−0.434796	+	0.900529i	$0.643180\pi$
$618$	−1.41757e10	−	1.50604e9i	−2.41594	−	0.256671i
$619$		−	9.85062e9i		−	1.66935i	−0.550747	−	0.834673i	$-0.685657\pi$
$619$		−	9.85062e9i		−	1.66935i	0.550747	−	0.834673i	$-0.314343\pi$
$620$	4.08850e9			0.688959
$621$	1.30894e9	+	5.82257e9i	0.219331	+	0.975651i
$622$	−8.91287e9			−1.48509
$623$			9.34681e8i			0.154866i
$624$	−7.30674e8	−	7.76274e7i	−0.120386	−	0.0127900i
$625$	−7.29197e9			−1.19472
$626$	1.21173e10			1.97423
$627$	1.53940e9	−	1.44898e10i	0.249411	−	2.34760i
$628$		−	9.63569e9i		−	1.55247i
$629$			6.39276e9i			1.02426i
$630$	1.42504e9	+	3.06252e8i	0.227057	+	0.0487963i
$631$			1.06463e9i			0.168693i	0.996436	+	0.0843464i	$0.0268802\pi$
$631$			1.06463e9i			0.168693i	−0.996436	+	0.0843464i	$0.973120\pi$
$632$	4.69059e9			0.739125
$633$	2.74758e9	+	2.91905e8i	0.430564	+	0.0457435i
$634$	4.29031e8			0.0668615
$635$	1.43423e10			2.22286
$636$	5.02785e8	+	5.34163e7i	0.0774966	+	0.00823331i
$637$	5.57922e8			0.0855235
$638$	−2.45752e10			−3.74650
$639$	1.06463e9	−	4.95389e9i	0.161415	−	0.751093i
$640$			1.26717e10i			1.91075i
$641$	4.74227e9			0.711186			0.355593	−	0.934641i	$-0.384279\pi$
$641$	4.74227e9			0.711186			0.355593	+	0.934641i	$0.384279\pi$
$642$	2.64939e8	−	2.49376e9i	0.0395161	−	0.371948i
$643$			1.18238e10i			1.75395i	0.480533	+	0.876977i	$0.340443\pi$
$643$			1.18238e10i			1.75395i	−0.480533	+	0.876977i	$0.659557\pi$
$644$	1.23438e9	−	7.37674e8i	0.182115	−	0.108834i
$645$	−7.90395e8	+	7.43965e9i	−0.115981	+	1.09168i
$646$	1.40460e10			2.04994
$647$		−	5.74291e9i		−	0.833618i	−0.908994	−	0.416809i	$-0.863148\pi$
$647$		−	5.74291e9i		−	0.833618i	0.908994	−	0.416809i	$-0.136852\pi$
$648$	−5.67915e9	+	1.26028e10i	−0.819918	+	1.81951i
$649$		−	8.29249e9i		−	1.19077i
$650$		−	7.86387e8i		−	0.112315i
$651$	−1.82176e7	+	1.71475e8i	−0.00258797	+	0.0243594i
$652$	−1.33433e9			−0.188537
$653$			1.01305e10i			1.42376i	0.702302	+	0.711879i	$0.252156\pi$
$653$			1.01305e10i			1.42376i	−0.702302	+	0.711879i	$0.747844\pi$
$654$	8.44405e9	+	8.97103e8i	1.18040	+	0.125407i
$655$			7.10601e8i			0.0988055i
$656$		−	9.46601e9i		−	1.30919i
$657$	3.75008e8	+	8.05920e7i	0.0515896	+	0.0110870i
$658$			1.72090e9i			0.235486i
$659$	−2.77377e9			−0.377547			−0.188774	−	0.982021i	$-0.560451\pi$
$659$	−2.77377e9			−0.377547			−0.188774	+	0.982021i	$0.560451\pi$
$660$	3.55277e10	+	3.77449e9i	4.81020	+	0.511040i
$661$			2.39823e9i			0.322988i	0.986874	+	0.161494i	$0.0516312\pi$
$661$			2.39823e9i			0.322988i	−0.986874	+	0.161494i	$0.948369\pi$
$662$			6.61784e9i			0.886571i
$663$	5.46006e8	+	5.80082e7i	0.0727613	+	0.00773022i
$664$			1.55452e10i			2.06066i
$665$	1.36247e9			0.179660
$666$	−1.59407e10	−	3.42578e9i	−2.09097	−	0.449366i
$667$	8.07613e9	−	4.82637e9i	1.05381	−	0.629767i
$668$			5.68947e9i			0.738506i
$669$	−1.29369e10	−	1.37443e9i	−1.67047	−	0.177473i
$670$	−8.01720e9			−1.02982
$671$		−	2.18295e10i		−	2.78943i
$672$	3.76738e8	+	4.00249e7i	0.0478902	+	0.00508789i
$673$	−1.22304e9			−0.154663			−0.0773316	−	0.997005i	$-0.524640\pi$
$673$	−1.22304e9			−0.154663			−0.0773316	+	0.997005i	$0.524640\pi$
$674$	2.15797e10			2.71479
$675$	−5.58022e9	−	1.83395e9i	−0.698374	−	0.229522i
$676$	1.69669e10			2.11246
$677$	−3.78140e9			−0.468374			−0.234187	−	0.972192i	$-0.575243\pi$
$677$	−3.78140e9			−0.468374			−0.234187	+	0.972192i	$0.575243\pi$
$678$	−3.77395e8	+	3.55226e9i	−0.0465042	+	0.437724i
$679$	−1.39968e9			−0.171587
$680$			1.82583e10i			2.22678i
$681$	−4.95522e9	−	5.26447e8i	−0.601241	−	0.0638763i
$682$			6.21274e9i			0.749960i
$683$		−	1.15323e10i		−	1.38498i	−0.721426	−	0.692491i	$-0.756513\pi$
$683$		−	1.15323e10i		−	1.38498i	0.721426	−	0.692491i	$-0.243487\pi$
$684$	−5.12098e9	+	2.38288e10i	−0.611866	+	2.84712i
$685$	1.61586e10			1.92082
$686$	−2.96671e9			−0.350866
$687$	3.39306e8	−	3.19374e9i	0.0399248	−	0.375795i
$688$			9.97740e9i			1.16804i
$689$	2.71562e7			0.00316301
$690$	−1.82507e10	+	8.43242e9i	−2.11498	+	0.977193i
$691$	−9.25952e9			−1.06762			−0.533808	−	0.845606i	$-0.679240\pi$
$691$	−9.25952e9			−1.06762			−0.533808	+	0.845606i	$0.679240\pi$
$692$		−	1.98957e10i		−	2.28237i
$693$	−3.16610e8	+	1.47324e9i	−0.0361375	+	0.168154i
$694$	−1.58295e10			−1.79767
$695$	−3.39019e8			−0.0383069
$696$	2.16704e10	+	2.30228e9i	2.43632	+	0.258837i
$697$			7.07361e9i			0.791273i
$698$			2.15504e10i			2.39862i
$699$	−1.21337e9	+	1.14210e10i	−0.134377	+	1.26483i
$700$			1.41534e9i			0.155962i
$701$	−7.53253e9			−0.825900			−0.412950	−	0.910754i	$-0.635502\pi$
$701$	−7.53253e9			−0.825900			−0.412950	+	0.910754i	$0.635502\pi$
$702$	−4.37244e8	+	1.33042e9i	−0.0477028	+	0.145147i
$703$	−1.52408e10			−1.65449
$704$	−8.73706e9			−0.943758
$705$	1.72931e9	−	1.62772e10i	0.185870	−	1.74952i
$706$	2.82393e10			3.02022
$707$	1.42538e9			0.151692
$708$	−1.46537e9	+	1.37929e10i	−0.155178	+	1.46062i
$709$			7.28403e9i			0.767555i	0.923426	+	0.383778i	$0.125377\pi$
$709$			7.28403e9i			0.767555i	−0.923426	+	0.383778i	$0.874623\pi$
$710$	1.70696e10			1.78986
$711$	7.45780e8	−	3.47024e9i	0.0778157	−	0.362089i
$712$		−	2.98618e10i		−	3.10053i
$713$	−1.22013e9	−	2.04168e9i	−0.126064	−	0.210948i
$714$	−1.44443e9	−	1.53457e8i	−0.148509	−	0.0157777i
$715$	1.91890e9			0.196328
$716$			4.26420e9i			0.434152i
$717$	−1.25384e8	+	1.18019e9i	−0.0127036	+	0.119574i
$718$		−	1.71451e10i		−	1.72864i
$719$		−	1.64416e9i		−	0.164965i	−0.996593	−	0.0824825i	$-0.973715\pi$
$719$		−	1.64416e9i		−	0.164965i	0.996593	−	0.0824825i	$-0.0262849\pi$
$720$	−1.80766e10	−	3.88479e9i	−1.80490	−	0.387886i
$721$	1.37802e9			0.136925
$722$			1.55998e10i			1.54255i
$723$	−1.53510e9	+	1.44492e10i	−0.151061	+	1.42187i
$724$		−	3.51789e10i		−	3.44507i
$725$			9.26014e9i			0.902474i
$726$	−3.80899e9	+	3.58524e10i	−0.369429	+	3.47728i
$727$			3.37994e9i			0.326241i	0.986606	+	0.163120i	$0.0521559\pi$
$727$			3.37994e9i			0.326241i	−0.986606	+	0.163120i	$0.947844\pi$
$728$	1.78895e8			0.0171845
$729$	8.42094e9	+	6.20538e9i	0.805034	+	0.593228i
$730$			1.29217e9i			0.122939i
$731$		−	7.45576e9i		−	0.705961i
$732$	−3.85750e9	+	3.63090e10i	−0.363511	+	3.42157i
$733$		−	1.18880e10i		−	1.11492i	−0.830203	−	0.557461i	$-0.811776\pi$
$733$		−	1.18880e10i		−	1.11492i	0.830203	−	0.557461i	$-0.188224\pi$
$734$	−1.49574e10			−1.39611
$735$	1.39604e10	+	1.48316e9i	1.29685	+	0.137779i
$736$	−4.48567e9	+	2.68068e9i	−0.414720	+	0.247840i
$737$		−	8.28836e9i		−	0.762663i
$738$	−1.76385e10	−	3.79064e9i	−1.61534	−	0.347149i
$739$	−2.36961e9			−0.215984			−0.107992	−	0.994152i	$-0.534442\pi$
$739$	−2.36961e9			−0.215984			−0.107992	+	0.994152i	$0.534442\pi$
$740$		−	3.73692e10i		−	3.39002i
$741$	−1.38296e8	+	1.30172e9i	−0.0124866	+	0.117531i
$742$	−7.18400e7			−0.00645584
$743$	6.19669e9			0.554242			0.277121	−	0.960835i	$-0.410620\pi$
$743$	6.19669e9			0.554242			0.277121	+	0.960835i	$0.410620\pi$
$744$	5.82029e8	−	5.47839e9i	0.0518130	−	0.487694i
$745$	3.93004e9			0.348217
$746$	−3.15042e9			−0.277832
$747$	1.15008e10	+	2.47160e9i	1.00950	+	0.216948i
$748$	−3.56046e10			−3.11064
$749$			2.42419e8i			0.0210804i
$750$	−7.53243e8	+	7.08996e9i	−0.0651960	+	0.613662i
$751$			1.78920e10i			1.54141i	0.637189	+	0.770707i	$0.280097\pi$
$751$			1.78920e10i			1.54141i	−0.637189	+	0.770707i	$0.719903\pi$
$752$		−	2.18296e10i		−	1.87190i
$753$	3.63200e9	+	3.85866e8i	0.310001	+	0.0329348i
$754$	2.20777e9			0.187566
$755$	1.78280e10			1.50760
$756$	7.86953e8	−	2.39449e9i	0.0662403	−	0.201552i
$757$		−	1.50289e10i		−	1.25919i	−0.776923	−	0.629596i	$-0.783220\pi$
$757$		−	1.50289e10i		−	1.25919i	0.776923	−	0.629596i	$-0.216780\pi$
$758$	−2.40542e10			−2.00608
$759$	−8.71763e9	−	1.88679e10i	−0.723689	−	1.56631i
$760$	−4.35290e10			−3.59692
$761$		−	2.77569e9i		−	0.228310i	−0.993463	−	0.114155i	$-0.963584\pi$
$761$		−	2.77569e9i		−	0.228310i	0.993463	−	0.114155i	$-0.0364160\pi$
$762$	3.85124e9	−	3.62501e10i	0.315325	−	2.96802i
$763$	−8.20847e8			−0.0669000
$764$	−2.04299e9			−0.165745
$765$	1.35080e10	+	2.90297e9i	1.09088	+	0.234438i
$766$		−	1.02334e10i		−	0.822658i
$767$			7.44973e8i			0.0596152i
$768$	2.51995e10	+	2.67722e9i	2.00737	+	0.213265i
$769$			1.49613e10i			1.18639i	0.805059	+	0.593195i	$0.202134\pi$
$769$			1.49613e10i			1.18639i	−0.805059	+	0.593195i	$0.797866\pi$
$770$	−5.07634e9			−0.400713
$771$	8.69353e8	−	8.18285e9i	0.0683134	−	0.643005i
$772$	−4.36118e10			−3.41149
$773$	1.10066e10			0.857087			0.428544	−	0.903521i	$-0.359027\pi$
$773$	1.10066e10			0.857087			0.428544	+	0.903521i	$0.359027\pi$
$774$	1.85914e10	+	3.99543e9i	1.44118	+	0.309720i
$775$	2.34101e9			0.180654
$776$	4.47178e10			3.43529
$777$	1.56729e9	+	1.66510e8i	0.119860	+	0.0127341i
$778$		−	3.55987e10i		−	2.71023i
$779$	−1.68640e10			−1.27814
$780$	−3.19171e9	−	3.39089e8i	−0.240820	−	0.0255849i
$781$			1.76470e10i			1.32554i
$782$	1.71982e10	−	1.02778e10i	1.28606	−	0.768560i
$783$	5.14878e9	−	1.56664e10i	0.383300	−	1.16628i
$784$	1.87224e10			1.38757
$785$		−	1.30224e10i		−	0.960833i
$786$	1.79603e9	+	1.90812e8i	0.131928	+	0.0140161i
$787$			4.42761e9i			0.323785i	0.986808	+	0.161893i	$0.0517599\pi$
$787$			4.42761e9i			0.323785i	−0.986808	+	0.161893i	$0.948240\pi$
$788$			2.97006e10i			2.16234i
$789$	−2.55543e10	−	2.71491e9i	−1.85222	−	0.196782i
$790$	1.19574e10			0.862864
$791$		−	3.45315e8i		−	0.0248083i
$792$	1.01153e10	−	4.70680e10i	0.723500	−	3.36657i
$793$			1.96110e9i			0.139651i
$794$			3.02413e10i			2.14402i
$795$	6.79503e8	+	7.21910e7i	0.0479630	+	0.00509563i
$796$			1.27608e10i			0.896772i
$797$	1.73415e10			1.21334			0.606670	−	0.794954i	$-0.292505\pi$
$797$	1.73415e10			1.21334			0.606670	+	0.794954i	$0.292505\pi$
$798$	3.65853e8	−	3.44362e9i	0.0254857	−	0.239886i
$799$			1.63125e10i			1.13137i
$800$		−	5.14330e9i		−	0.355162i
$801$	−2.20926e10	−	4.74787e9i	−1.51892	−	0.326426i
$802$			3.13133e10i			2.14348i
$803$	−1.33587e9			−0.0910459
$804$	−1.46464e9	+	1.37860e10i	−0.0993881	+	0.935498i
$805$	1.66823e9	−	9.96950e8i	0.112712	−	0.0673578i
$806$		−	5.58135e8i		−	0.0375462i
$807$	1.76177e9	−	1.65828e10i	0.118002	−	1.11071i
$808$	−4.55390e10			−3.03700
$809$			9.43072e8i			0.0626217i	0.999510	+	0.0313109i	$0.00996818\pi$
$809$			9.43072e8i			0.0626217i	−0.999510	+	0.0313109i	$0.990032\pi$
$810$	−1.44775e10	+	3.21274e10i	−0.957183	+	2.12411i
$811$	1.97403e10			1.29951			0.649756	−	0.760143i	$-0.274871\pi$
$811$	1.97403e10			1.29951			0.649756	+	0.760143i	$0.274871\pi$
$812$	−3.97355e9			−0.260455
$813$	−1.56590e10	−	1.66362e9i	−1.02199	−	0.108577i
$814$	5.67848e10			3.69017
$815$	−1.80331e9			−0.116686
$816$	1.83225e10	+	1.94660e9i	1.18051	+	0.125418i
$817$	1.77751e10			1.14034
$818$			1.02704e10i			0.656070i
$819$	2.84433e7	−	1.32352e8i	0.00180920	−	0.00841851i
$820$		−	4.13491e10i		−	2.61889i
$821$		−	4.28589e9i		−	0.270296i	−0.990825	−	0.135148i	$-0.956849\pi$
$821$		−	4.28589e9i		−	0.270296i	0.990825	−	0.135148i	$-0.0431510\pi$
$822$	4.33894e9	−	4.08406e10i	0.272479	−	2.56473i
$823$	2.00460e10			1.25351			0.626755	−	0.779217i	$-0.284383\pi$
$823$	2.00460e10			1.25351			0.626755	+	0.779217i	$0.284383\pi$
$824$	−4.40260e10			−2.74135
$825$	2.03426e10	+	2.16121e9i	1.26129	+	0.134001i
$826$		−	1.97078e9i		−	0.121677i
$827$	1.18345e10			0.727579			0.363789	−	0.931481i	$-0.381483\pi$
$827$	1.18345e10			0.727579			0.363789	+	0.931481i	$0.381483\pi$
$828$	1.11659e10	+	3.29235e10i	0.683575	+	2.01558i
$829$	1.74418e9			0.106328			0.0531642	−	0.998586i	$-0.483069\pi$
$829$	1.74418e9			0.106328			0.0531642	+	0.998586i	$0.483069\pi$
$830$			3.96283e10i			2.40565i
$831$	1.69903e9	+	1.80506e8i	0.102706	+	0.0109116i
$832$	7.84912e8			0.0472486
$833$	−1.39906e10			−0.838645
$834$	−9.10342e7	+	8.56866e8i	−0.00543405	+	0.0511484i
$835$			7.68919e9i			0.457065i
$836$		−	8.48839e10i		−	5.02462i
$837$	−3.96053e9	−	1.30164e9i	−0.233461	−	0.0767275i
$838$			2.39119e10i			1.40365i
$839$	7.51622e9			0.439372			0.219686	−	0.975571i	$-0.429497\pi$
$839$	7.51622e9			0.439372			0.219686	+	0.975571i	$0.429497\pi$
$840$	4.47631e9	+	4.75567e8i	0.260581	+	0.0276844i
$841$	−8.74783e9			−0.507124
$842$	−4.44683e10			−2.56719
$843$	−1.65116e10	−	1.75421e9i	−0.949279	−	0.100852i
$844$	1.60959e10			0.921545
$845$	2.29304e10			1.30741
$846$	−4.06762e10	−	8.74161e9i	−2.30964	−	0.496358i
$847$		−	3.48521e9i		−	0.197078i
$848$	9.11289e8			0.0513181
$849$	−2.17824e9	+	2.05028e10i	−0.122160	+	1.14984i
$850$			1.97196e10i			1.10137i
$851$	−1.86611e10	+	1.11520e10i	−1.03797	+	0.620299i
$852$	3.11840e9	−	2.93522e10i	0.172740	−	1.62593i
$853$	2.51834e10			1.38929			0.694644	−	0.719353i	$-0.255562\pi$
$853$	2.51834e10			1.38929			0.694644	+	0.719353i	$0.255562\pi$
$854$		−	5.18798e9i		−	0.285034i
$855$	−6.92089e9	+	3.22040e10i	−0.378687	+	1.76209i
$856$		−	7.74495e9i		−	0.422047i
$857$		−	1.51077e10i		−	0.819907i	−0.912106	−	0.409953i	$-0.865545\pi$
$857$		−	1.51077e10i		−	0.819907i	0.912106	−	0.409953i	$-0.134455\pi$
$858$	5.15268e8	−	4.85000e9i	0.0278502	−	0.262142i
$859$	2.09998e10			1.13042			0.565208	−	0.824948i	$-0.308796\pi$
$859$	2.09998e10			1.13042			0.565208	+	0.824948i	$0.308796\pi$
$860$			4.35830e10i			2.33654i
$861$	1.73421e9	+	1.84244e8i	0.0925958	+	0.00983746i
$862$			3.87473e10i			2.06047i
$863$			1.27804e10i			0.676874i	0.940989	+	0.338437i	$0.109898\pi$
$863$			1.27804e10i			0.676874i	−0.940989	+	0.338437i	$0.890102\pi$
$864$	−2.85975e9	+	8.70146e9i	−0.150845	+	0.458981i
$865$		−	2.68885e10i		−	1.41257i
$866$	2.39749e10			1.25442
$867$	5.39047e9	+	5.72688e8i	0.280905	+	0.0298436i
$868$			1.00453e9i			0.0521369i
$869$			1.23618e10i			0.639019i
$870$	5.52430e10	+	5.86906e9i	2.84420	+	0.302170i
$871$			7.44603e8i			0.0381822i
$872$	2.62249e10			1.33939
$873$	7.10989e9	−	3.30835e10i	0.361671	−	1.68291i
$874$	2.45031e10	+	4.10018e10i	1.24145	+	2.07737i
$875$		−	6.89215e8i		−	0.0347798i
$876$	2.22195e9	+	2.36062e8i	0.111679	+	0.0118648i
$877$	3.76738e9			0.188600			0.0942998	−	0.995544i	$-0.469939\pi$
$877$	3.76738e9			0.188600			0.0942998	+	0.995544i	$0.469939\pi$
$878$			6.42855e9i			0.320540i
$879$	1.13897e10	+	1.21005e9i	0.565652	+	0.0600954i
$880$	6.43933e10			3.18531
$881$	2.60108e10			1.28156			0.640778	−	0.767726i	$-0.278612\pi$
$881$	2.60108e10			1.28156			0.640778	+	0.767726i	$0.278612\pi$
$882$	7.49734e9	−	3.48864e10i	0.367932	−	1.71205i
$883$	−2.82195e10			−1.37939			−0.689694	−	0.724101i	$-0.742255\pi$
$883$	−2.82195e10			−1.37939			−0.689694	+	0.724101i	$0.742255\pi$
$884$	3.19861e9			0.155732
$885$	−1.98041e9	+	1.86408e10i	−0.0960404	+	0.903988i
$886$	−2.82236e10			−1.36331
$887$		−	1.10103e10i		−	0.529747i	−0.964283	−	0.264873i	$-0.914670\pi$
$887$		−	1.10103e10i		−	0.529747i	0.964283	−	0.264873i	$-0.0853301\pi$
$888$	−5.00728e10	−	5.31977e9i	−2.39970	−	0.254946i
$889$			3.52387e9i			0.168215i
$890$		−	7.61246e10i		−	3.61960i
$891$	−3.32140e10	−	1.49671e10i	−1.57307	−	0.708870i
$892$	−7.57871e10			−3.57535
$893$	−3.88901e10			−1.82751
$894$	1.05530e9	−	9.93313e9i	0.0493965	−	0.464948i
$895$			5.76297e9i			0.268699i
$896$	−3.11340e9			−0.144596
$897$	7.83167e8	+	1.69504e9i	0.0362310	+	0.0784164i
$898$	3.43451e10			1.58270
$899$			6.57235e9i			0.301690i
$900$	−3.34538e10	−	7.18948e9i	−1.52967	−	0.328737i
$901$	−6.80974e8			−0.0310166
$902$	6.28326e10			2.85077
$903$	−1.82790e9	−	1.94198e8i	−0.0826125	−	0.00877682i
$904$			1.10323e10i			0.496682i
$905$		−	4.75435e10i		−	2.13217i
$906$	4.78721e9	−	4.50600e10i	0.213862	−	2.01299i
$907$		−	3.21381e10i		−	1.43019i	−0.699027	−	0.715095i	$-0.746383\pi$
$907$		−	3.21381e10i		−	1.43019i	0.699027	−	0.715095i	$-0.253617\pi$
$908$	−2.90287e10			−1.28685
$909$	−7.24047e9	+	3.36911e10i	−0.319737	+	1.48779i
$910$	4.56044e8			0.0200614
$911$	2.11414e10			0.926443			0.463222	−	0.886242i	$-0.346693\pi$
$911$	2.11414e10			0.926443			0.463222	+	0.886242i	$0.346693\pi$
$912$	−4.64084e9	+	4.36822e10i	−0.202588	+	1.90688i
$913$	−4.09686e10			−1.78157
$914$	6.91714e9			0.299650
$915$	−5.21333e9	+	4.90708e10i	−0.224979	+	2.11763i
$916$		−	1.87096e10i		−	0.804322i
$917$	−1.74593e8			−0.00747710
$918$	1.09644e10	−	3.33618e10i	0.467774	−	1.42331i
$919$			1.27035e10i			0.539908i	0.962873	+	0.269954i	$0.0870085\pi$
$919$			1.27035e10i			0.539908i	−0.962873	+	0.269954i	$0.912991\pi$
$920$	−5.32977e10	+	3.18512e10i	−2.25658	+	1.34855i
$921$	−2.22066e10	−	2.35924e9i	−0.936640	−	0.0995094i
$922$	2.29197e10			0.963053
$923$		−	1.58535e9i		−	0.0663621i
$924$	−9.27382e8	+	8.72905e9i	−0.0386729	+	0.364012i
$925$		−	2.13969e10i		−	0.888906i
$926$			2.47958e10i			1.02622i
$927$	−6.99990e9	+	3.25717e10i	−0.288611	+	1.34296i
$928$	1.44397e10			0.593118
$929$			2.43340e10i			0.995770i	0.867243	+	0.497885i	$0.165890\pi$
$929$			2.43340e10i			0.995770i	−0.867243	+	0.497885i	$0.834110\pi$
$930$	1.48373e9	−	1.39657e10i	0.0604872	−	0.569340i
$931$		−	3.33545e10i		−	1.35466i
$932$			6.69063e10i			2.70715i
$933$	−2.20057e9	+	2.07130e10i	−0.0887051	+	0.834943i
$934$		−	4.83100e10i		−	1.94010i
$935$	−4.81188e10			−1.92519
$936$	−9.08726e8	+	4.22845e9i	−0.0362216	+	0.168545i
$937$		−	3.42457e9i		−	0.135993i	−0.997686	−	0.0679966i	$-0.978339\pi$
$937$		−	3.42457e9i		−	0.135993i	0.997686	−	0.0679966i	$-0.0216607\pi$
$938$		−	1.96980e9i		−	0.0779315i
$939$	2.99174e9	−	2.81600e10i	0.117922	−	1.10995i
$940$		−	9.53553e10i		−	3.74453i
$941$	3.23545e10			1.26582			0.632909	−	0.774226i	$-0.281861\pi$
$941$	3.23545e10			1.26582			0.632909	+	0.774226i	$0.281861\pi$
$942$	−3.29140e10	−	3.49681e9i	−1.28293	−	0.136299i
$943$	−2.06486e10	+	1.23398e10i	−0.801862	+	0.479200i
$944$			2.49994e10i			0.967223i
$945$	1.06355e9	−	3.23610e9i	0.0409965	−	0.124741i
$946$	−6.62271e10			−2.54341
$947$			4.22400e10i			1.61621i	0.589036	+	0.808107i	$0.299508\pi$
$947$			4.22400e10i			1.61621i	−0.589036	+	0.808107i	$0.700492\pi$
$948$	2.18446e9	−	2.05614e10i	0.0832752	−	0.783834i
$949$	1.20011e8			0.00455815
$950$	−4.70130e10			−1.77904
$951$	1.05927e8	−	9.97042e8i	0.00399368	−	0.0375908i
$952$	−4.48600e9			−0.168512
$953$	2.53321e10			0.948081			0.474041	−	0.880503i	$-0.342795\pi$
$953$	2.53321e10			0.948081			0.474041	+	0.880503i	$0.342795\pi$
$954$	3.64924e8	−	1.69805e9i	0.0136076	−	0.0633186i
$955$	−2.76106e9			−0.102580
$956$			6.91380e9i			0.255926i
$957$	−6.06757e9	+	5.71114e10i	−0.223781	+	2.10635i
$958$			3.76079e10i			1.38198i
$959$			3.97012e9i			0.145358i
$960$	1.96401e10	+	2.08658e9i	0.716465	+	0.0761178i
$961$	−2.58511e10			−0.939609
$962$	−5.10138e9			−0.184746
$963$	−5.72994e9	−	1.23141e9i	−0.206756	−	0.0444334i
$964$			8.46463e10i			3.04326i
$965$	−5.89404e10			−2.11139
$966$	−2.07182e9	−	4.48414e9i	−0.0739490	−	0.160051i
$967$	−4.24595e10			−1.51002			−0.755009	−	0.655714i	$-0.772368\pi$
$967$	−4.24595e10			−1.51002			−0.755009	+	0.655714i	$0.772368\pi$
$968$			1.11348e11i			3.94564i
$969$	3.46793e9	−	3.26422e10i	0.122444	−	1.15251i
$970$	1.13996e11			4.01041
$971$	−2.51440e10			−0.881388			−0.440694	−	0.897657i	$-0.645268\pi$
$971$	−2.51440e10			−0.881388			−0.440694	+	0.897657i	$0.645268\pi$
$972$	5.26000e10	+	3.07641e10i	1.83719	+	1.07451i
$973$		−	8.32960e7i		−	0.00289887i
$974$		−	5.95910e10i		−	2.06645i
$975$	−1.82752e9	−	1.94157e8i	−0.0631459	−	0.00670867i
$976$			6.58094e10i			2.26576i
$977$	8.41925e9			0.288830			0.144415	−	0.989517i	$-0.453870\pi$
$977$	8.41925e9			0.288830			0.144415	+	0.989517i	$0.453870\pi$
$978$	−4.84230e8	+	4.55786e9i	−0.0165526	+	0.155803i
$979$	7.86994e10			2.68060
$980$	8.17826e10			2.77568
$981$	4.16963e9	−	1.94020e10i	0.141012	−	0.656152i
$982$	1.87247e10			0.630992
$983$	−3.24623e10			−1.09004			−0.545019	−	0.838423i	$-0.683478\pi$
$983$	−3.24623e10			−1.09004			−0.545019	+	0.838423i	$0.683478\pi$
$984$	−5.54058e10	−	5.88635e9i	−1.85384	−	0.196953i
$985$			4.01398e10i			1.33828i
$986$	−5.53625e10			−1.83928
$987$	3.99927e9	+	4.24886e8i	0.132395	+	0.0140657i
$988$			7.62573e9i			0.251554i
$989$	2.17641e10	−	1.30064e10i	0.715408	−	0.427535i
$990$	2.57861e10	−	1.19987e11i	0.844623	−	3.93017i
$991$	−3.93779e10			−1.28527			−0.642635	−	0.766172i	$-0.722159\pi$
$991$	−3.93779e10			−1.28527			−0.642635	+	0.766172i	$0.722159\pi$
$992$		−	3.65043e9i		−	0.118728i
$993$	1.53795e10	+	1.63393e9i	0.498447	+	0.0529554i
$994$			4.19396e9i			0.135448i
$995$			1.72459e10i			0.555016i
$996$	6.81430e10	+	7.23957e9i	2.18531	+	0.232170i
$997$	2.18138e10			0.697105			0.348553	−	0.937289i	$-0.386673\pi$
$997$	2.18138e10			0.697105			0.348553	+	0.937289i	$0.386673\pi$
$998$			7.67766e10i			2.44496i
$999$	−1.18970e10	+	3.61995e10i	−0.377537	+	1.14874i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.c.b.68.3	✓	48
3.2	odd	2	inner	69.8.c.b.68.46	yes	48
23.22	odd	2	inner	69.8.c.b.68.4	yes	48
69.68	even	2	inner	69.8.c.b.68.45	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.c.b.68.3	✓	48	1.1	even	1	trivial
69.8.c.b.68.4	yes	48	23.22	odd	2	inner
69.8.c.b.68.45	yes	48	69.68	even	2	inner
69.8.c.b.68.46	yes	48	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	69.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-20.0107i q^{2} +(-46.5037 - 4.94059i) q^{3} -272.428 q^{4} -368.180 q^{5} +(-98.8646 + 930.570i) q^{6} -90.4608i q^{7} +2890.10i q^{8} +(2138.18 + 459.511i) q^{9} +O(q^{10})\)	\(q-20.0107i q^{2} +(-46.5037 - 4.94059i) q^{3} -272.428 q^{4} -368.180 q^{5} +(-98.8646 + 930.570i) q^{6} -90.4608i q^{7} +2890.10i q^{8} +(2138.18 + 459.511i) q^{9} +7367.54i q^{10} -7616.73 q^{11} +(12668.9 + 1345.95i) q^{12} +684.265 q^{13} -1810.18 q^{14} +(17121.7 + 1819.03i) q^{15} +22962.1 q^{16} -17158.8 q^{17} +(9195.13 - 42786.5i) q^{18} -40907.8i q^{19} +100302. q^{20} +(-446.929 + 4206.76i) q^{21} +152416. i q^{22} +(-29933.2 - 50088.2i) q^{23} +(14278.8 - 134400. i) q^{24} +57431.5 q^{25} -13692.6i q^{26} +(-97163.0 - 31932.8i) q^{27} +24644.0i q^{28} +161238. i q^{29} +(36400.0 - 342617. i) q^{30} +40761.8 q^{31} -89555.3i q^{32} +(354206. + 37631.1i) q^{33} +343359. i q^{34} +33305.8i q^{35} +(-582500. - 125184. i) q^{36} -372565. i q^{37} -818593. q^{38} +(-31820.8 - 3380.67i) q^{39} -1.06408e6i q^{40} -412245. i q^{41} +(84180.1 + 8943.37i) q^{42} +434516. i q^{43} +2.07501e6 q^{44} +(-787235. - 169183. i) q^{45} +(-1.00230e6 + 598984. i) q^{46} -950678. i q^{47} +(-1.06782e6 - 113446. i) q^{48} +815360. q^{49} -1.14924e6i q^{50} +(797946. + 84774.5i) q^{51} -186413. q^{52} +39686.6 q^{53} +(-638998. + 1.94430e6i) q^{54} +2.80433e6 q^{55} +261441. q^{56} +(-202108. + 1.90236e6i) q^{57} +3.22648e6 q^{58} +1.08872e6i q^{59} +(-4.66443e6 - 495553. i) q^{60} +2.86600e6i q^{61} -815671. i q^{62} +(41567.7 - 193422. i) q^{63} +1.14709e6 q^{64} -251933. q^{65} +(753024. - 7.08790e6i) q^{66} +1.08818e6i q^{67} +4.67453e6 q^{68} +(1.14454e6 + 2.47717e6i) q^{69} +666473. q^{70} -2.31687e6i q^{71} +(-1.32803e6 + 6.17956e6i) q^{72} +175387. q^{73} -7.45528e6 q^{74} +(-2.67077e6 - 283745. i) q^{75} +1.11444e7i q^{76} +689015. i q^{77} +(-67649.6 + 636757. i) q^{78} -1.62299e6i q^{79} -8.45419e6 q^{80} +(4.36067e6 + 1.96504e6i) q^{81} -8.24930e6 q^{82} +5.37877e6 q^{83} +(121756. - 1.14604e6i) q^{84} +6.31752e6 q^{85} +8.69496e6 q^{86} +(796611. - 7.49816e6i) q^{87} -2.20131e7i q^{88} -1.03324e7 q^{89} +(-3.38546e6 + 1.57531e7i) q^{90} -61899.1i q^{91} +(8.15463e6 + 1.36454e7i) q^{92} +(-1.89557e6 - 201387. i) q^{93} -1.90237e7 q^{94} +1.50614e7i q^{95} +(-442456. + 4.16465e6i) q^{96} -1.54727e7i q^{97} -1.63159e7i q^{98} +(-1.62859e7 - 3.49997e6i) 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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