LMFDB - Embedded newform 69.8.c.b.68.12

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			68.12
Character	$\chi$	$=$	69.68
Dual form			69.8.c.b.68.38

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-13.3830i q^{2} +(20.4540 - 42.0551i) q^{3} -51.1052 q^{4} +506.030 q^{5} +(-562.825 - 273.736i) q^{6} +1171.85i q^{7} -1029.08i q^{8} +(-1350.27 - 1720.39i) q^{9} +O(q^{10})$	\(q-13.3830i q^{2} +(20.4540 - 42.0551i) q^{3} -51.1052 q^{4} +506.030 q^{5} +(-562.825 - 273.736i) q^{6} +1171.85i q^{7} -1029.08i q^{8} +(-1350.27 - 1720.39i) q^{9} -6772.20i q^{10} +775.848 q^{11} +(-1045.30 + 2149.23i) q^{12} +13619.1 q^{13} +15682.9 q^{14} +(10350.3 - 21281.1i) q^{15} -20313.7 q^{16} +8018.06 q^{17} +(-23024.0 + 18070.7i) q^{18} -33310.2i q^{19} -25860.7 q^{20} +(49282.4 + 23969.0i) q^{21} -10383.2i q^{22} +(-55825.9 - 16979.2i) q^{23} +(-43278.3 - 21048.9i) q^{24} +177941. q^{25} -182265. i q^{26} +(-99969.6 + 21596.9i) q^{27} -59887.7i q^{28} +112479. i q^{29} +(-284806. - 138519. i) q^{30} -237088. q^{31} +140136. i q^{32} +(15869.2 - 32628.4i) q^{33} -107306. i q^{34} +592992. i q^{35} +(69005.7 + 87920.8i) q^{36} +294893. i q^{37} -445791. q^{38} +(278565. - 572753. i) q^{39} -520747. i q^{40} -124615. i q^{41} +(320778. - 659547. i) q^{42} +115874. i q^{43} -39649.8 q^{44} +(-683276. - 870568. i) q^{45} +(-227233. + 747119. i) q^{46} +339232. i q^{47} +(-415497. + 854296. i) q^{48} -549694. q^{49} -2.38139e6i q^{50} +(164001. - 337201. i) q^{51} -696007. q^{52} -466953. q^{53} +(289032. + 1.33790e6i) q^{54} +392602. q^{55} +1.20594e6 q^{56} +(-1.40086e6 - 681326. i) q^{57} +1.50530e6 q^{58} -1.61607e6i q^{59} +(-528955. + 1.08758e6i) q^{60} +2.63326e6i q^{61} +3.17295e6i q^{62} +(2.01604e6 - 1.58232e6i) q^{63} -724713. q^{64} +6.89167e6 q^{65} +(-436666. - 212378. i) q^{66} +1.91699e6i q^{67} -409764. q^{68} +(-1.85592e6 + 2.00047e6i) q^{69} +7.93602e6 q^{70} -1.12014e6i q^{71} +(-1.77043e6 + 1.38954e6i) q^{72} -3.61533e6 q^{73} +3.94655e6 q^{74} +(3.63960e6 - 7.48333e6i) q^{75} +1.70232e6i q^{76} +909179. i q^{77} +(-7.66517e6 - 3.72804e6i) q^{78} -3.05043e6i q^{79} -1.02793e7 q^{80} +(-1.13652e6 + 4.64598e6i) q^{81} -1.66773e6 q^{82} +2.18281e6 q^{83} +(-2.51859e6 - 1.22494e6i) q^{84} +4.05738e6 q^{85} +1.55074e6 q^{86} +(4.73031e6 + 2.30064e6i) q^{87} -798413. i q^{88} +1.19178e7 q^{89} +(-1.16508e7 + 9.14430e6i) q^{90} +1.59596e7i q^{91} +(2.85299e6 + 867725. i) q^{92} +(-4.84939e6 + 9.97077e6i) q^{93} +4.53994e6 q^{94} -1.68559e7i q^{95} +(5.89344e6 + 2.86634e6i) q^{96} -7.15167e6i q^{97} +7.35657e6i q^{98} +(-1.04760e6 - 1.33476e6i) 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 $

$2$		−	13.3830i		−	1.18290i	−0.806341	−	0.591451i	$-0.798555\pi$
$2$		−	13.3830i		−	1.18290i	0.806341	−	0.591451i	$-0.201445\pi$
$3$	20.4540	−	42.0551i	0.437375	−	0.899279i
$3$	20.4540	−	42.0551i	0.437375	−	0.899279i
$4$	−51.1052			−0.399259
$5$	506.030			1.81043			0.905213	−	0.424958i	$-0.139711\pi$
$5$	506.030			1.81043			0.905213	+	0.424958i	$0.139711\pi$
$6$	−562.825	−	273.736i	−1.06376	−	0.517372i
$7$			1171.85i			1.29131i	0.763630	+	0.645654i	$0.223415\pi$
$7$			1171.85i			1.29131i	−0.763630	+	0.645654i	$0.776585\pi$
$8$		−	1029.08i		−	0.710618i
$9$	−1350.27	−	1720.39i	−0.617407	−	0.786644i
$10$		−	6772.20i		−	2.14156i
$11$	775.848			0.175753			0.0878764	−	0.996131i	$-0.471992\pi$
$11$	775.848			0.175753			0.0878764	+	0.996131i	$0.471992\pi$
$12$	−1045.30	+	2149.23i	−0.174626	+	0.359046i
$13$	13619.1			1.71928			0.859640	−	0.510900i	$-0.170688\pi$
$13$	13619.1			1.71928			0.859640	+	0.510900i	$0.170688\pi$
$14$	15682.9			1.52749
$15$	10350.3	−	21281.1i	0.791835	−	1.62808i
$16$	−20313.7			−1.23985
$17$	8018.06			0.395820			0.197910	−	0.980220i	$-0.436585\pi$
$17$	8018.06			0.395820			0.197910	+	0.980220i	$0.436585\pi$
$18$	−23024.0	+	18070.7i	−0.930523	+	0.730332i
$19$		−	33310.2i		−	1.11414i	−0.830466	−	0.557070i	$-0.811926\pi$
$19$		−	33310.2i		−	1.11414i	0.830466	−	0.557070i	$-0.188074\pi$
$20$	−25860.7			−0.722829
$21$	49282.4	+	23969.0i	1.16125	+	0.564785i
$22$		−	10383.2i		−	0.207898i
$23$	−55825.9	−	16979.2i	−0.956728	−	0.290984i
$23$	−55825.9	−	16979.2i	−0.956728	−	0.290984i
$24$	−43278.3	−	21048.9i	−0.639044	−	0.310806i
$25$	177941.			2.27764
$26$		−	182265.i		−	2.03374i
$27$	−99969.6	+	21596.9i	−0.977451	+	0.211163i
$28$		−	59887.7i		−	0.515566i
$29$			112479.i			0.856401i	0.903684	+	0.428201i	$0.140852\pi$
$29$			112479.i			0.856401i	−0.903684	+	0.428201i	$0.859148\pi$
$30$	−284806.	−	138519.i	−1.92586	−	0.936663i
$31$	−237088.			−1.42937			−0.714683	−	0.699448i	$-0.753429\pi$
$31$	−237088.			−1.42937			−0.714683	+	0.699448i	$0.753429\pi$
$32$			140136.i			0.756005i
$33$	15869.2	−	32628.4i	0.0768698	−	0.158051i
$34$		−	107306.i		−	0.468217i
$35$			592992.i			2.33782i
$36$	69005.7	+	87920.8i	0.246505	+	0.314075i
$37$			294893.i			0.957101i	0.878060	+	0.478550i	$0.158838\pi$
$37$			294893.i			0.957101i	−0.878060	+	0.478550i	$0.841162\pi$
$38$	−445791.			−1.31792
$39$	278565.	−	572753.i	0.751970	−	1.54611i
$40$		−	520747.i		−	1.28652i
$41$		−	124615.i		−	0.282376i	−0.989983	−	0.141188i	$-0.954908\pi$
$41$		−	124615.i		−	0.282376i	0.989983	−	0.141188i	$-0.0450923\pi$
$42$	320778.	−	659547.i	0.668086	−	1.37364i
$43$			115874.i			0.222252i	0.993806	+	0.111126i	$0.0354458\pi$
$43$			115874.i			0.222252i	−0.993806	+	0.111126i	$0.964554\pi$
$44$	−39649.8			−0.0701709
$45$	−683276.	−	870568.i	−1.11777	−	1.42416i
$46$	−227233.	+	747119.i	−0.344206	+	1.13172i
$47$			339232.i			0.476600i	0.971192	+	0.238300i	$0.0765901\pi$
$47$			339232.i			0.476600i	−0.971192	+	0.238300i	$0.923410\pi$
$48$	−415497.	+	854296.i	−0.542279	+	1.11497i
$49$	−549694.			−0.667475
$50$		−	2.38139e6i		−	2.69423i
$51$	164001.	−	337201.i	0.173122	−	0.355953i
$52$	−696007.			−0.686439
$53$	−466953.			−0.430832			−0.215416	−	0.976522i	$-0.569111\pi$
$53$	−466953.			−0.430832			−0.215416	+	0.976522i	$0.569111\pi$
$54$	289032.	+	1.33790e6i	0.249786	+	1.15623i
$55$	392602.			0.318187
$56$	1.20594e6			0.917626
$57$	−1.40086e6	−	681326.i	−1.00192	−	0.487296i
$58$	1.50530e6			1.01304
$59$		−	1.61607e6i		−	1.02442i	−0.858860	−	0.512210i	$-0.828827\pi$
$59$		−	1.61607e6i		−	1.02442i	0.858860	−	0.512210i	$-0.171173\pi$
$60$	−528955.	+	1.08758e6i	−0.316147	+	0.650025i
$61$			2.63326e6i			1.48539i	0.669631	+	0.742694i	$0.266452\pi$
$61$			2.63326e6i			1.48539i	−0.669631	+	0.742694i	$0.733548\pi$
$62$			3.17295e6i			1.69080i
$63$	2.01604e6	−	1.58232e6i	1.01580	−	0.797262i
$64$	−724713.			−0.345570
$65$	6.89167e6			3.11263
$66$	−436666.	−	212378.i	−0.186959	−	0.0909295i
$67$			1.91699e6i			0.778677i	0.921095	+	0.389339i	$0.127296\pi$
$67$			1.91699e6i			0.778677i	−0.921095	+	0.389339i	$0.872704\pi$
$68$	−409764.			−0.158035
$69$	−1.85592e6	+	2.00047e6i	−0.680125	+	0.733096i
$70$	7.93602e6			2.76541
$71$		−	1.12014e6i		−	0.371424i	−0.982604	−	0.185712i	$-0.940541\pi$
$71$		−	1.12014e6i		−	0.371424i	0.982604	−	0.185712i	$-0.0594591\pi$
$72$	−1.77043e6	+	1.38954e6i	−0.559003	+	0.438741i
$73$	−3.61533e6			−1.08772			−0.543861	−	0.839176i	$-0.683038\pi$
$73$	−3.61533e6			−1.08772			−0.543861	+	0.839176i	$0.683038\pi$
$74$	3.94655e6			1.13216
$75$	3.63960e6	−	7.48333e6i	0.996184	−	2.04824i
$76$			1.70232e6i			0.444830i
$77$			909179.i			0.226951i
$78$	−7.66517e6	−	3.72804e6i	−1.82890	−	0.889507i
$79$		−	3.05043e6i		−	0.696091i	−0.937478	−	0.348045i	$-0.886846\pi$
$79$		−	3.05043e6i		−	0.696091i	0.937478	−	0.348045i	$-0.113154\pi$
$80$	−1.02793e7			−2.24466
$81$	−1.13652e6	+	4.64598e6i	−0.237617	+	0.971359i
$82$	−1.66773e6			−0.334024
$83$	2.18281e6			0.419028			0.209514	−	0.977806i	$-0.432812\pi$
$83$	2.18281e6			0.419028			0.209514	+	0.977806i	$0.432812\pi$
$84$	−2.51859e6	−	1.22494e6i	−0.463638	−	0.225496i
$85$	4.05738e6			0.716604
$86$	1.55074e6			0.262903
$87$	4.73031e6	+	2.30064e6i	0.770144	+	0.374568i
$88$		−	798413.i		−	0.124893i
$89$	1.19178e7			1.79197			0.895985	−	0.444084i	$-0.146471\pi$
$89$	1.19178e7			1.79197			0.895985	+	0.444084i	$0.146471\pi$
$90$	−1.16508e7	+	9.14430e6i	−1.68464	+	1.32221i
$91$			1.59596e7i			2.22012i
$92$	2.85299e6	+	867725.i	0.381982	+	0.116178i
$93$	−4.84939e6	+	9.97077e6i	−0.625169	+	1.28540i
$94$	4.53994e6			0.563771
$95$		−	1.68559e7i		−	2.01707i
$96$	5.89344e6	+	2.86634e6i	0.679860	+	0.330658i
$97$		−	7.15167e6i		−	0.795621i	−0.917468	−	0.397811i	$-0.869770\pi$
$97$		−	7.15167e6i		−	0.795621i	0.917468	−	0.397811i	$-0.130230\pi$
$98$			7.35657e6i			0.789558i
$99$	−1.04760e6	−	1.33476e6i	−0.108511	−	0.138255i
$100$	−9.09370e6			−0.909370
$101$			1.24031e7i			1.19786i	0.800802	+	0.598930i	$0.204407\pi$
$101$			1.24031e7i			1.19786i	−0.800802	+	0.598930i	$0.795593\pi$
$102$	−4.51276e6	−	2.19483e6i	−0.421058	−	0.204786i
$103$			7.29638e6i			0.657926i	0.944343	+	0.328963i	$0.106699\pi$
$103$			7.29638e6i			0.657926i	−0.944343	+	0.328963i	$0.893301\pi$
$104$		−	1.40152e7i		−	1.22175i
$105$	2.49383e7	+	1.21290e7i	2.10235	+	1.02250i
$106$			6.24924e6i			0.509632i
$107$	−5.55089e6			−0.438046			−0.219023	−	0.975720i	$-0.570287\pi$
$107$	−5.55089e6			−0.438046			−0.219023	+	0.975720i	$0.570287\pi$
$108$	5.10896e6	−	1.10371e6i	0.390256	−	0.0843089i
$109$		−	2.10470e7i		−	1.55667i	−0.627848	−	0.778336i	$-0.716064\pi$
$109$		−	2.10470e7i		−	1.55667i	0.627848	−	0.778336i	$-0.283936\pi$
$110$		−	5.25420e6i		−	0.376385i
$111$	1.24017e7	+	6.03173e6i	0.860701	+	0.418612i
$112$		−	2.38047e7i		−	1.60103i
$113$	−1.95105e7			−1.27202			−0.636011	−	0.771680i	$-0.719417\pi$
$113$	−1.95105e7			−1.27202			−0.636011	+	0.771680i	$0.719417\pi$
$114$	−9.11820e6	+	1.87478e7i	−0.576424	+	1.18518i
$115$	−2.82496e7	−	8.59197e6i	−1.73209	−	0.526806i
$116$		−	5.74824e6i		−	0.341926i
$117$	−1.83895e7	−	2.34302e7i	−1.06150	−	1.35246i
$118$	−2.16279e7			−1.21179
$119$			9.39598e6i			0.511126i
$120$	−2.19001e7	−	1.06514e7i	−1.15694	−	0.562692i
$121$	−1.88852e7			−0.969111
$122$	3.52410e7			1.75707
$123$	−5.24072e6	−	2.54888e6i	−0.253935	−	0.123504i
$124$	1.21164e7			0.570688
$125$	5.05098e7			2.31308
$126$	−2.11762e7	−	2.69807e7i	−0.943084	−	1.20159i
$127$	1.29898e7			0.562717			0.281359	−	0.959603i	$-0.409215\pi$
$127$	1.29898e7			0.562717			0.281359	+	0.959603i	$0.409215\pi$
$128$			2.76363e7i			1.16478i
$129$	4.87309e6	+	2.37008e6i	0.199867	+	0.0972075i
$130$		−	9.22313e7i		−	3.68194i
$131$		−	2.52265e7i		−	0.980409i	−0.871608	−	0.490204i	$-0.836922\pi$
$131$		−	2.52265e7i		−	0.980409i	0.871608	−	0.490204i	$-0.163078\pi$
$132$	−810997.	+	1.66748e6i	−0.0306910	+	0.0631032i
$133$	3.90346e7			1.43870
$134$	2.56551e7			0.921100
$135$	−5.05876e7	+	1.09287e7i	−1.76960	+	0.382296i
$136$		−	8.25127e6i		−	0.281277i
$137$	−6.68929e6			−0.222258			−0.111129	−	0.993806i	$-0.535447\pi$
$137$	−6.68929e6			−0.222258			−0.111129	+	0.993806i	$0.535447\pi$
$138$	2.67724e7	+	2.48379e7i	0.867182	+	0.804521i
$139$	1.98885e7			0.628130			0.314065	−	0.949401i	$-0.398309\pi$
$139$	1.98885e7			0.628130			0.314065	+	0.949401i	$0.398309\pi$
$140$		−	3.03049e7i		−	0.933395i
$141$	1.42664e7	+	6.93864e6i	0.428596	+	0.208453i
$142$	−1.49909e7			−0.439359
$143$	1.05664e7			0.302168
$144$	2.74290e7	+	3.49475e7i	0.765493	+	0.975322i
$145$			5.69176e7i			1.55045i
$146$			4.83840e7i			1.28667i
$147$	−1.12434e7	+	2.31175e7i	−0.291936	+	0.600246i
$148$		−	1.50705e7i		−	0.382131i
$149$	2.96819e7			0.735088			0.367544	−	0.930006i	$-0.380199\pi$
$149$	2.96819e7			0.735088			0.367544	+	0.930006i	$0.380199\pi$
$150$	−1.00150e8	−	4.87089e7i	−2.42287	−	1.17839i
$151$	−1.15541e7			−0.273096			−0.136548	−	0.990633i	$-0.543601\pi$
$151$	−1.15541e7			−0.273096			−0.136548	+	0.990633i	$0.543601\pi$
$152$	−3.42790e7			−0.791727
$153$	−1.08265e7	−	1.37942e7i	−0.244382	−	0.311370i
$154$	1.21676e7			0.268461
$155$	−1.19974e8			−2.58776
$156$	−1.42361e7	+	2.92707e7i	−0.300231	+	0.617300i
$157$			7.55373e7i			1.55780i	0.627145	+	0.778902i	$0.284223\pi$
$157$			7.55373e7i			1.55780i	−0.627145	+	0.778902i	$0.715777\pi$
$158$	−4.08239e7			−0.823408
$159$	−9.55105e6	+	1.96378e7i	−0.188435	+	0.387438i
$160$			7.09130e7i			1.36869i
$161$	1.98971e7	−	6.54197e7i	0.375750	−	1.23543i
$162$	6.21772e7	+	1.52100e7i	1.14902	+	0.281078i
$163$	−1.16224e7			−0.210203			−0.105102	−	0.994461i	$-0.533517\pi$
$163$	−1.16224e7			−0.210203			−0.105102	+	0.994461i	$0.533517\pi$
$164$			6.36849e6i			0.112741i
$165$	8.03028e6	−	1.65109e7i	0.139167	−	0.286139i
$166$		−	2.92126e7i		−	0.495669i
$167$		−	3.56868e6i		−	0.0592925i	−0.999560	−	0.0296462i	$-0.990562\pi$
$167$		−	3.56868e6i		−	0.0592925i	0.999560	−	0.0296462i	$-0.00943807\pi$
$168$	2.46662e7	−	5.07158e7i	0.401346	−	0.825202i
$169$	1.22731e8			1.95593
$170$		−	5.42999e7i		−	0.847673i
$171$	−5.73065e7	+	4.49777e7i	−0.876431	+	0.687877i
$172$		−	5.92176e6i		−	0.0887362i
$173$			5.68840e7i			0.835273i	0.908614	+	0.417636i	$0.137141\pi$
$173$			5.68840e7i			0.835273i	−0.908614	+	0.417636i	$0.862859\pi$
$174$	3.07895e7	−	6.33058e7i	0.443078	−	0.911006i
$175$			2.08520e8i			2.94114i
$176$	−1.57604e7			−0.217907
$177$	−6.79641e7	−	3.30551e7i	−0.921240	−	0.448056i
$178$		−	1.59496e8i		−	2.11973i
$179$			1.02688e7i			0.133824i	0.997759	+	0.0669119i	$0.0213146\pi$
$179$			1.02688e7i			0.133824i	−0.997759	+	0.0669119i	$0.978685\pi$
$180$	3.49189e7	+	4.44905e7i	0.446280	+	0.568609i
$181$			7.02801e7i			0.880963i	0.897762	+	0.440481i	$0.145192\pi$
$181$			7.02801e7i			0.880963i	−0.897762	+	0.440481i	$0.854808\pi$
$182$	2.13587e8			2.62619
$183$	1.10742e8	+	5.38607e7i	1.33578	+	0.649671i
$184$	−1.74730e7	+	5.74496e7i	−0.206779	+	0.679868i
$185$			1.49224e8i			1.73276i
$186$	1.33439e8	+	6.48995e7i	1.52050	+	0.739514i
$187$	6.22080e6			0.0695665
$188$		−	1.73365e7i		−	0.190287i
$189$	−2.53084e7	−	1.17150e8i	−0.272677	−	1.26219i
$190$	−2.25583e8			−2.38599
$191$	7.46193e7			0.774880			0.387440	−	0.921895i	$-0.373359\pi$
$191$	7.46193e7			0.774880			0.387440	+	0.921895i	$0.373359\pi$
$192$	−1.48233e7	+	3.04779e7i	−0.151144	+	0.310764i
$193$	1.85452e7			0.185687			0.0928435	−	0.995681i	$-0.470404\pi$
$193$	1.85452e7			0.185687			0.0928435	+	0.995681i	$0.470404\pi$
$194$	−9.57109e7			−0.941143
$195$	1.40962e8	−	2.89830e8i	1.36139	−	2.79913i
$196$	2.80922e7			0.266495
$197$		−	4.07771e6i		−	0.0380000i	−0.999819	−	0.0190000i	$-0.993952\pi$
$197$		−	4.07771e6i		−	0.0380000i	0.999819	−	0.0190000i	$-0.00604826\pi$
$198$	−1.78631e7	+	1.40201e7i	−0.163542	+	0.128358i
$199$			1.28729e8i			1.15795i	0.815346	+	0.578974i	$0.196547\pi$
$199$			1.28729e8i			1.15795i	−0.815346	+	0.578974i	$0.803453\pi$
$200$		−	1.83116e8i		−	1.61854i
$201$	8.06192e7	+	3.92100e7i	0.700249	+	0.340574i
$202$	1.65991e8			1.41695
$203$	−1.31808e8			−1.10588
$204$	−8.38132e6	+	1.72327e7i	−0.0691205	+	0.142118i
$205$		−	6.30591e7i		−	0.511222i
$206$	9.76476e7			0.778263
$207$	4.61692e7	+	1.18969e8i	0.361789	+	0.932260i
$208$	−2.76655e8			−2.13165
$209$		−	2.58436e7i		−	0.195813i
$210$	1.62323e8	−	3.33750e8i	1.20952	−	2.48688i
$211$	2.40827e7			0.176489			0.0882443	−	0.996099i	$-0.471874\pi$
$211$	2.40827e7			0.176489			0.0882443	+	0.996099i	$0.471874\pi$
$212$	2.38637e7			0.172013
$213$	−4.71078e7	−	2.29114e7i	−0.334014	−	0.162451i
$214$			7.42876e7i			0.518165i
$215$			5.86356e7i			0.402371i
$216$	2.22250e7	+	1.02877e8i	0.150056	+	0.694594i
$217$		−	2.77832e8i		−	1.84575i
$218$	−2.81672e8			−1.84139
$219$	−7.39478e7	+	1.52043e8i	−0.475742	+	0.978165i
$220$	−2.00640e7			−0.127039
$221$	1.09199e8			0.680526
$222$	8.07227e7	−	1.65973e8i	0.495177	−	1.01813i
$223$	2.76494e8			1.66963			0.834813	−	0.550534i	$-0.185576\pi$
$223$	2.76494e8			1.66963			0.834813	+	0.550534i	$0.185576\pi$
$224$	−1.64219e8			−0.976235
$225$	−2.40268e8	−	3.06128e8i	−1.40623	−	1.79169i
$226$			2.61110e8i			1.50468i
$227$	−7.04638e7			−0.399830			−0.199915	−	0.979813i	$-0.564067\pi$
$227$	−7.04638e7			−0.399830			−0.199915	+	0.979813i	$0.564067\pi$
$228$	7.15914e7	+	3.48193e7i	0.400027	+	0.194557i
$229$		−	1.25974e8i		−	0.693198i	−0.938013	−	0.346599i	$-0.887336\pi$
$229$		−	1.25974e8i		−	0.693198i	0.938013	−	0.346599i	$-0.112664\pi$
$230$	−1.14987e8	+	3.78064e8i	−0.623160	+	2.04889i
$231$	3.82356e7	+	1.85963e7i	0.204092	+	0.0992625i
$232$	1.15750e8			0.608574
$233$		−	2.69733e8i		−	1.39698i	−0.715622	−	0.698488i	$-0.753857\pi$
$233$		−	2.69733e8i		−	1.39698i	0.715622	−	0.698488i	$-0.246143\pi$
$234$	−3.13566e8	+	2.46106e8i	−1.59983	+	1.25565i
$235$			1.71661e8i			0.862849i
$236$			8.25896e7i			0.409009i
$237$	−1.28286e8	−	6.23934e7i	−0.625980	−	0.304452i
$238$	1.25747e8			0.604612
$239$		−	9.55359e6i		−	0.0452662i	−0.999744	−	0.0226331i	$-0.992795\pi$
$239$		−	9.55359e6i		−	0.0452662i	0.999744	−	0.0226331i	$-0.00720495\pi$
$240$	−2.10254e8	+	4.32299e8i	−0.981757	+	2.01858i
$241$			1.17790e8i			0.542064i	0.962570	+	0.271032i	$0.0873649\pi$
$241$			1.17790e8i			0.542064i	−0.962570	+	0.271032i	$0.912635\pi$
$242$			2.52741e8i			1.14636i
$243$	1.72141e8	+	1.42825e8i	0.769595	+	0.638532i
$244$		−	1.34573e8i		−	0.593054i
$245$	−2.78161e8			−1.20841
$246$	−3.41117e7	+	7.01366e7i	−0.146094	+	0.300381i
$247$		−	4.53655e8i		−	1.91552i
$248$			2.43984e8i			1.01573i
$249$	4.46472e7	−	9.17984e7i	0.183272	−	0.376823i
$250$		−	6.75974e8i		−	2.73615i
$251$	−3.01962e8			−1.20530			−0.602649	−	0.798006i	$-0.705888\pi$
$251$	−3.01962e8			−1.20530			−0.602649	+	0.798006i	$0.705888\pi$
$252$	−1.03030e8	+	8.08645e7i	−0.405567	+	0.318314i
$253$	−4.33124e7	−	1.31733e7i	−0.168148	−	0.0511413i
$254$		−	1.73843e8i		−	0.665640i
$255$	8.29895e7	−	1.70634e8i	0.313424	−	0.644427i
$256$	2.77093e8			1.03225
$257$			1.37003e8i			0.503459i	0.967798	+	0.251729i	$0.0809993\pi$
$257$			1.37003e8i			0.503459i	−0.967798	+	0.251729i	$0.919001\pi$
$258$	3.17189e7	−	6.52167e7i	0.114987	−	0.236423i
$259$	−3.45570e8			−1.23591
$260$	−3.52200e8			−1.24275
$261$	1.93507e8	−	1.51877e8i	0.673683	−	0.528748i
$262$	−3.37606e8			−1.15973
$263$	3.96731e8			1.34478			0.672390	−	0.740197i	$-0.265268\pi$
$263$	3.96731e8			1.34478			0.672390	+	0.740197i	$0.265268\pi$
$264$	−3.35774e7	−	1.63307e7i	−0.112314	−	0.0546251i
$265$	−2.36292e8			−0.779989
$266$		−	5.22401e8i		−	1.70184i
$267$	2.43766e8	−	5.01204e8i	0.783762	−	1.61148i
$268$		−	9.79680e7i		−	0.310894i
$269$		−	1.02190e8i		−	0.320092i	−0.987110	−	0.160046i	$-0.948836\pi$
$269$		−	1.02190e8i		−	0.320092i	0.987110	−	0.160046i	$-0.0511642\pi$
$270$	1.46259e8	+	6.77015e8i	0.452218	+	2.09327i
$271$	−4.65690e8			−1.42136			−0.710680	−	0.703515i	$-0.751613\pi$
$271$	−4.65690e8			−1.42136			−0.710680	+	0.703515i	$0.751613\pi$
$272$	−1.62877e8			−0.490758
$273$	6.71182e8	+	3.26437e8i	1.99651	+	0.971024i
$274$			8.95229e7i			0.262910i
$275$	1.38055e8			0.400302
$276$	9.48473e7	−	1.02235e8i	0.271546	−	0.292695i
$277$	1.17525e8			0.332240			0.166120	−	0.986106i	$-0.446876\pi$
$277$	1.17525e8			0.332240			0.166120	+	0.986106i	$0.446876\pi$
$278$		−	2.66168e8i		−	0.743017i
$279$	3.20133e8	+	4.07884e8i	0.882501	+	1.12440i
$280$	6.10239e8			1.66130
$281$	−4.65958e8			−1.25278			−0.626389	−	0.779510i	$-0.715468\pi$
$281$	−4.65958e8			−1.25278			−0.626389	+	0.779510i	$0.715468\pi$
$282$	9.28599e7	−	1.90928e8i	0.246579	−	0.506988i
$283$			3.19092e8i			0.836881i	0.908244	+	0.418440i	$0.137423\pi$
$283$			3.19092e8i			0.836881i	−0.908244	+	0.418440i	$0.862577\pi$
$284$			5.72452e7i			0.148294i
$285$	−7.08879e8	−	3.44771e8i	−1.81391	−	0.882214i
$286$		−	1.41410e8i		−	0.357436i
$287$	1.46031e8			0.364635
$288$	2.41089e8	−	1.89221e8i	0.594707	−	0.466763i
$289$	−3.46049e8			−0.843326
$290$	7.61729e8			1.83403
$291$	−3.00764e8	−	1.46280e8i	−0.715486	−	0.347985i
$292$	1.84762e8			0.434283
$293$	6.51600e8			1.51337			0.756684	−	0.653781i	$-0.226818\pi$
$293$	6.51600e8			1.51337			0.756684	+	0.653781i	$0.226818\pi$
$294$	3.09381e8	+	1.50471e8i	0.710033	+	0.345332i
$295$		−	8.17780e8i		−	1.85464i
$296$	3.03469e8			0.680133
$297$	−7.75612e7	+	1.67559e7i	−0.171790	+	0.0371125i
$298$		−	3.97233e8i		−	0.869538i
$299$	−7.60299e8	−	2.31241e8i	−1.64488	−	0.500284i
$300$	−1.86002e8	+	3.82437e8i	−0.397735	+	0.817778i
$301$	−1.35787e8			−0.286996
$302$			1.54628e8i			0.323046i
$303$	5.21615e8	+	2.53693e8i	1.07721	+	0.523913i
$304$			6.76654e8i			1.38137i
$305$			1.33251e9i			2.68918i
$306$	−1.84608e8	+	1.44892e8i	−0.368320	+	0.289080i
$307$	−5.03373e8			−0.992899			−0.496450	−	0.868066i	$-0.665363\pi$
$307$	−5.03373e8			−0.992899			−0.496450	+	0.868066i	$0.665363\pi$
$308$		−	4.64637e7i		−	0.0906122i
$309$	3.06850e8	+	1.49240e8i	0.591660	+	0.287760i
$310$			1.60561e9i			3.06107i
$311$			5.73753e8i			1.08159i	0.841154	+	0.540796i	$0.181877\pi$
$311$			5.73753e8i			1.08159i	−0.841154	+	0.540796i	$0.818123\pi$
$312$	−5.89412e8	−	2.86667e8i	−1.09870	−	0.534363i
$313$			5.98966e7i			0.110407i	0.998475	+	0.0552036i	$0.0175808\pi$
$313$			5.98966e7i			0.110407i	−0.998475	+	0.0552036i	$0.982419\pi$
$314$	1.01092e9			1.84273
$315$	1.02018e9	−	8.00698e8i	1.83903	−	1.44338i
$316$			1.55893e8i			0.277921i
$317$		−	9.21070e8i		−	1.62400i	−0.583659	−	0.811999i	$-0.698380\pi$
$317$		−	9.21070e8i		−	1.62400i	0.583659	−	0.811999i	$-0.301620\pi$
$318$	2.62813e8	+	1.27822e8i	0.458301	+	0.222900i
$319$			8.72664e7i			0.150515i
$320$	−3.66726e8			−0.625629
$321$	−1.13538e8	+	2.33443e8i	−0.191590	+	0.393925i
$322$	−8.75513e8	−	2.66283e8i	−1.46139	−	0.444476i
$323$		−	2.67083e8i		−	0.440999i
$324$	5.80819e7	−	2.37434e8i	0.0948709	−	0.387824i
$325$	2.42340e9			3.91591
$326$			1.55543e8i			0.248650i
$327$	−8.85134e8	−	4.30495e8i	−1.39988	−	0.680849i
$328$	−1.28240e8			−0.200662
$329$	−3.97529e8			−0.615437
$330$	−2.20966e8	−	1.07469e8i	−0.338475	−	0.164621i
$331$	−1.56329e7			−0.0236942			−0.0118471	−	0.999930i	$-0.503771\pi$
$331$	−1.56329e7			−0.0236942			−0.0118471	+	0.999930i	$0.503771\pi$
$332$	−1.11553e8			−0.167301
$333$	5.07330e8	−	3.98184e8i	0.752898	−	0.590921i
$334$	−4.77597e7			−0.0701372
$335$			9.70053e8i			1.40974i
$336$	−1.00111e9	−	4.86901e8i	−1.43977	−	0.700249i
$337$			1.40491e9i			1.99961i	0.0198378	+	0.999803i	$0.493685\pi$
$337$			1.40491e9i			1.99961i	−0.0198378	+	0.999803i	$0.506315\pi$
$338$		−	1.64252e9i		−	2.31367i
$339$	−3.99068e8	+	8.20518e8i	−0.556350	+	1.14390i
$340$	−2.07353e8			−0.286111
$341$	−1.83944e8			−0.251215
$342$	6.01937e8	+	7.66934e8i	0.813692	+	1.03673i
$343$			3.20910e8i			0.429392i
$344$	1.19244e8			0.157936
$345$	−9.39153e8	+	1.01230e9i	−1.23132	+	1.32722i
$346$	7.61279e8			0.988047
$347$		−	1.09466e9i		−	1.40645i	−0.710967	−	0.703225i	$-0.751743\pi$
$347$		−	1.09466e9i		−	1.40645i	0.710967	−	0.703225i	$-0.248257\pi$
$348$	−2.41743e8	−	1.17574e8i	−0.307487	−	0.149550i
$349$	−5.39607e8			−0.679499			−0.339749	−	0.940516i	$-0.610342\pi$
$349$	−5.39607e8			−0.679499			−0.339749	+	0.940516i	$0.610342\pi$
$350$	2.79063e9			3.47908
$351$	−1.36150e9	+	2.94130e8i	−1.68051	+	0.363049i
$352$			1.08724e8i			0.132870i
$353$		−	3.39022e8i		−	0.410219i	−0.978739	−	0.205110i	$-0.934245\pi$
$353$		−	3.39022e8i		−	0.410219i	0.978739	−	0.205110i	$-0.0657551\pi$
$354$	−4.42377e8	+	9.09564e8i	−0.530006	+	1.08974i
$355$		−	5.66826e8i		−	0.672436i
$356$	−6.09061e8			−0.715460
$357$	3.95149e8	+	1.92185e8i	0.459645	+	0.223553i
$358$	1.37427e8			0.158301
$359$	−1.67241e9			−1.90771			−0.953857	−	0.300261i	$-0.902926\pi$
$359$	−1.67241e9			−1.90771			−0.953857	+	0.300261i	$0.902926\pi$
$360$	−8.95889e8	+	7.03149e8i	−1.01203	+	0.794307i
$361$	−2.15697e8			−0.241306
$362$	9.40560e8			1.04209
$363$	−3.86278e8	+	7.94221e8i	−0.423865	+	0.871502i
$364$		−	8.15617e8i		−	0.886403i
$365$	−1.82946e9			−1.96924
$366$	7.20819e8	−	1.48206e9i	0.768497	−	1.58010i
$367$		−	1.09907e9i		−	1.16063i	−0.814393	−	0.580314i	$-0.802930\pi$
$367$		−	1.09907e9i		−	1.16063i	0.814393	−	0.580314i	$-0.197070\pi$
$368$	1.13403e9	+	3.44911e8i	1.18620	+	0.360777i
$369$	−2.14387e8	+	1.68264e8i	−0.222130	+	0.174341i
$370$	1.99707e9			2.04969
$371$		−	5.47200e8i		−	0.556336i
$372$	2.47829e8	−	5.09558e8i	0.249604	−	0.513208i
$373$			4.10955e8i			0.410028i	0.978759	+	0.205014i	$0.0657240\pi$
$373$			4.10955e8i			0.410028i	−0.978759	+	0.205014i	$0.934276\pi$
$374$		−	8.32530e7i		−	0.0822905i
$375$	1.03313e9	−	2.12420e9i	1.01168	−	2.08011i
$376$	3.49098e8			0.338680
$377$			1.53186e9i			1.47239i
$378$	−1.56782e9	+	3.38702e8i	−1.49305	+	0.322550i
$379$			3.99705e8i			0.377140i	0.982060	+	0.188570i	$0.0603852\pi$
$379$			3.99705e8i			0.377140i	−0.982060	+	0.188570i	$0.939615\pi$
$380$			8.61426e8i			0.805332i
$381$	2.65694e8	−	5.46289e8i	0.246118	−	0.506040i
$382$		−	9.98632e8i		−	0.916608i
$383$	1.46497e9			1.33240			0.666198	−	0.745775i	$-0.267921\pi$
$383$	1.46497e9			1.33240			0.666198	+	0.745775i	$0.267921\pi$
$384$	1.16225e9	+	5.65272e8i	1.04746	+	0.509446i
$385$			4.60071e8i			0.410878i
$386$		−	2.48191e8i		−	0.219650i
$387$	1.99348e8	−	1.56461e8i	0.174833	−	0.137220i
$388$			3.65487e8i			0.317659i
$389$	−5.53507e8			−0.476759			−0.238380	−	0.971172i	$-0.576616\pi$
$389$	−5.53507e8			−0.476759			−0.238380	+	0.971172i	$0.576616\pi$
$390$	−3.87880e9	−	1.88650e9i	−3.31109	−	1.61039i
$391$	−4.47616e8	−	1.36140e8i	−0.378692	−	0.115178i
$392$			5.65682e8i			0.474320i
$393$	−1.06090e9	−	5.15982e8i	−0.881661	−	0.428806i
$394$	−5.45720e7			−0.0449504
$395$		−	1.54361e9i		−	1.26022i
$396$	5.35379e7	+	6.82132e7i	0.0433240	+	0.0551995i
$397$	1.03478e9			0.830009			0.415004	−	0.909819i	$-0.363780\pi$
$397$	1.03478e9			0.830009			0.415004	+	0.909819i	$0.363780\pi$
$398$	1.72278e9			1.36974
$399$	7.98413e8	−	1.64161e9i	0.629249	−	1.29379i
$400$	−3.61464e9			−2.82394
$401$	9.00473e8			0.697373			0.348687	−	0.937239i	$-0.386628\pi$
$401$	9.00473e8			0.697373			0.348687	+	0.937239i	$0.386628\pi$
$402$	5.24749e8	−	1.07893e9i	0.402866	−	0.828326i
$403$	−3.22893e9			−2.45748
$404$		−	6.33863e8i		−	0.478256i
$405$	−5.75111e8	+	2.35100e9i	−0.430189	+	1.75857i
$406$			1.76399e9i			1.30815i
$407$			2.28792e8i			0.168213i
$408$	−3.47008e8	−	1.68771e8i	−0.252947	−	0.123023i
$409$	1.27749e9			0.923261			0.461631	−	0.887072i	$-0.347265\pi$
$409$	1.27749e9			0.923261			0.461631	+	0.887072i	$0.347265\pi$
$410$	−8.43921e8			−0.604725
$411$	−1.36823e8	+	2.81319e8i	−0.0972101	+	0.199872i
$412$		−	3.72883e8i		−	0.262683i
$413$	1.89380e9			1.32284
$414$	1.59216e9	−	6.17883e8i	1.10277	−	0.427962i
$415$	1.10457e9			0.758619
$416$			1.90853e9i			1.29979i
$417$	4.06799e8	−	8.36413e8i	0.274728	−	0.564865i
$418$	−3.45866e8			−0.231628
$419$	1.23307e7			0.00818915			0.00409458	−	0.999992i	$-0.498697\pi$
$419$	1.23307e7			0.00818915			0.00409458	+	0.999992i	$0.498697\pi$
$420$	−1.27448e9	−	6.19857e8i	−0.839383	−	0.408243i
$421$		−	1.97588e9i		−	1.29055i	−0.763951	−	0.645274i	$-0.776743\pi$
$421$		−	1.97588e9i		−	1.29055i	0.763951	−	0.645274i	$-0.223257\pi$
$422$		−	3.22299e8i		−	0.208769i
$423$	5.83611e8	−	4.58054e8i	0.374914	−	0.294256i
$424$			4.80534e8i			0.306157i
$425$	1.42674e9			0.901538
$426$	−3.06624e8	+	6.30445e8i	−0.192164	+	0.395106i
$427$	−3.08579e9			−1.91809
$428$	2.83679e8			0.174894
$429$	2.16124e8	−	4.44369e8i	0.132161	−	0.271734i
$430$	7.84722e8			0.475966
$431$	−1.64215e9			−0.987967			−0.493984	−	0.869471i	$-0.664460\pi$
$431$	−1.64215e9			−0.987967			−0.493984	+	0.869471i	$0.664460\pi$
$432$	2.03076e9	−	4.38714e8i	1.21189	−	0.261811i
$433$		−	2.46058e9i		−	1.45657i	−0.685276	−	0.728284i	$-0.740318\pi$
$433$		−	2.46058e9i		−	1.45657i	0.685276	−	0.728284i	$-0.259682\pi$
$434$	−3.71823e9			−2.18334
$435$	2.39368e9	+	1.16419e9i	1.39429	+	0.678128i
$436$			1.07561e9i			0.621516i
$437$	−5.65580e8	+	1.85957e9i	−0.324197	+	1.06593i
$438$	2.03480e9	+	9.89645e8i	1.15707	+	0.562756i
$439$	−1.01768e9			−0.574099			−0.287049	−	0.957916i	$-0.592674\pi$
$439$	−1.01768e9			−0.574099			−0.287049	+	0.957916i	$0.592674\pi$
$440$		−	4.04021e8i		−	0.226110i
$441$	7.42235e8	+	9.45688e8i	0.412103	+	0.525065i
$442$		−	1.46141e9i		−	0.804997i
$443$		−	3.26319e9i		−	1.78332i	−0.452708	−	0.891659i	$-0.649542\pi$
$443$		−	3.26319e9i		−	1.78332i	0.452708	−	0.891659i	$-0.350458\pi$
$444$	−6.33793e8	−	3.08252e8i	−0.343643	−	0.167135i
$445$	6.03075e9			3.24423
$446$		−	3.70033e9i		−	1.97501i
$447$	6.07112e8	−	1.24827e9i	0.321509	−	0.661049i
$448$		−	8.49257e8i		−	0.446237i
$449$		−	1.26943e9i		−	0.661828i	−0.943661	−	0.330914i	$-0.892643\pi$
$449$		−	1.26943e9i		−	0.661828i	0.943661	−	0.330914i	$-0.107357\pi$
$450$	−4.09691e9	+	3.21551e9i	−2.11940	+	1.66344i
$451$		−	9.66826e7i		−	0.0496284i
$452$	9.97089e8			0.507866
$453$	−2.36327e8	+	4.85908e8i	−0.119445	+	0.245590i
$454$			9.43018e8i			0.472960i
$455$			8.07602e9i			4.01936i
$456$	−7.01142e8	+	1.44161e9i	−0.346281	+	0.711984i
$457$			2.71396e8i			0.133014i	0.997786	+	0.0665070i	$0.0211855\pi$
$457$			2.71396e8i			0.133014i	−0.997786	+	0.0665070i	$0.978815\pi$
$458$	−1.68591e9			−0.819986
$459$	−8.01563e8	+	1.73165e8i	−0.386895	+	0.0835827i
$460$	1.44370e9	+	4.39094e8i	0.691551	+	0.210332i
$461$			2.11774e8i			0.100675i	0.998732	+	0.0503374i	$0.0160297\pi$
$461$			2.11774e8i			0.100675i	−0.998732	+	0.0503374i	$0.983970\pi$
$462$	2.48875e8	−	5.11708e8i	0.117418	−	0.241421i
$463$	1.53947e9			0.720837			0.360419	−	0.932791i	$-0.382634\pi$
$463$	1.53947e9			0.720837			0.360419	+	0.932791i	$0.382634\pi$
$464$		−	2.28486e9i		−	1.06181i
$465$	−2.45394e9	+	5.04550e9i	−1.13182	+	2.32712i
$466$	−3.60985e9			−1.65249
$467$	−9.24419e8			−0.420010			−0.210005	−	0.977700i	$-0.567348\pi$
$467$	−9.24419e8			−0.420010			−0.210005	+	0.977700i	$0.567348\pi$
$468$	9.39796e8	+	1.19740e9i	0.423812	+	0.539983i
$469$	−2.24643e9			−1.00551
$470$	2.29734e9			1.02067
$471$	3.17673e9	+	1.54504e9i	1.40090	+	0.681344i
$472$	−1.66307e9			−0.727972
$473$			8.99005e7i			0.0390614i
$474$	−8.35012e8	+	1.71686e9i	−0.360138	+	0.740473i
$475$		−	5.92725e9i		−	2.53761i
$476$		−	4.80183e8i		−	0.204072i
$477$	6.30512e8	+	8.03341e8i	0.265998	+	0.338911i
$478$	−1.27856e8			−0.0535455
$479$	−1.68698e9			−0.701352			−0.350676	−	0.936497i	$-0.614048\pi$
$479$	−1.68698e9			−0.701352			−0.350676	+	0.936497i	$0.614048\pi$
$480$	2.98226e9	+	1.45045e9i	1.23084	+	0.598631i
$481$			4.01617e9i			1.64553i
$482$	1.57639e9			0.641209
$483$	−2.34426e9	−	2.17487e9i	−0.946653	−	0.878250i
$484$	9.65133e8			0.386926
$485$		−	3.61896e9i		−	1.44041i
$486$	1.91143e9	−	2.30377e9i	0.755321	−	0.910356i
$487$	8.14433e8			0.319524			0.159762	−	0.987156i	$-0.448927\pi$
$487$	8.14433e8			0.319524			0.159762	+	0.987156i	$0.448927\pi$
$488$	2.70985e9			1.05554
$489$	−2.37725e8	+	4.88782e8i	−0.0919377	+	0.189032i
$490$			3.72264e9i			1.42944i
$491$		−	1.50895e9i		−	0.575293i	−0.957737	−	0.287647i	$-0.907127\pi$
$491$		−	1.50895e9i		−	0.575293i	0.957737	−	0.287647i	$-0.0928729\pi$
$492$	2.67828e8	+	1.30261e8i	0.101386	+	0.0493102i
$493$			9.01861e8i			0.338981i
$494$	−6.07127e9			−2.26587
$495$	−5.30118e8	−	6.75429e8i	−0.196451	−	0.250300i
$496$	4.81614e9			1.77220
$497$	1.31264e9			0.479623
$498$	−1.22854e9	−	5.97514e8i	−0.445745	−	0.216793i
$499$	1.65870e9			0.597608			0.298804	−	0.954314i	$-0.403412\pi$
$499$	1.65870e9			0.597608			0.298804	+	0.954314i	$0.403412\pi$
$500$	−2.58131e9			−0.923518
$501$	−1.50081e8	−	7.29937e7i	−0.0533205	−	0.0259330i
$502$			4.04117e9i			1.42575i
$503$	1.82705e9			0.640123			0.320062	−	0.947397i	$-0.396296\pi$
$503$	1.82705e9			0.640123			0.320062	+	0.947397i	$0.396296\pi$
$504$	−1.62834e9	−	2.07468e9i	−0.566549	−	0.721845i
$505$			6.27634e9i			2.16864i
$506$	−1.76298e8	+	5.79651e8i	−0.0604952	+	0.198902i
$507$	2.51035e9	−	5.16149e9i	0.855472	−	1.75892i
$508$	−6.63847e8			−0.224670
$509$			4.07196e9i			1.36864i	0.729179	+	0.684322i	$0.239902\pi$
$509$			4.07196e9i			1.36864i	−0.729179	+	0.684322i	$0.760098\pi$
$510$	−2.28359e9	−	1.11065e9i	−0.762295	−	0.370750i
$511$		−	4.23663e9i		−	1.40458i
$512$		−	1.70903e8i		−	0.0562736i
$513$	7.19397e8	+	3.33001e9i	0.235265	+	1.08902i
$514$	1.83351e9			0.595543
$515$			3.69219e9i			1.19113i
$516$	−2.49040e8	−	1.21124e8i	−0.0797987	−	0.0388110i
$517$			2.63192e8i			0.0837637i
$518$			4.62478e9i			1.46196i
$519$	2.39226e9	+	1.16350e9i	0.751144	+	0.365327i
$520$		−	7.09211e9i		−	2.21189i
$521$	1.58594e9			0.491310			0.245655	−	0.969357i	$-0.420997\pi$
$521$	1.58594e9			0.491310			0.245655	+	0.969357i	$0.420997\pi$
$522$	−2.03257e9	−	2.58971e9i	−0.625458	−	0.796901i
$523$			4.41727e8i			0.135020i	0.997719	+	0.0675100i	$0.0215055\pi$
$523$			4.41727e8i			0.135020i	−0.997719	+	0.0675100i	$0.978495\pi$
$524$			1.28920e9i			0.391437i
$525$	8.76936e9	+	4.26507e9i	2.64491	+	1.28638i
$526$		−	5.30946e9i		−	1.59074i
$527$	−1.90099e9			−0.565773
$528$	−3.22362e8	+	6.62804e8i	−0.0953071	+	0.195960i
$529$	2.82824e9	+	1.89576e9i	0.830656	+	0.556786i
$530$			3.16230e9i			0.922651i
$531$	−2.78027e9	+	2.18213e9i	−0.805854	+	0.632484i
$532$	−1.99487e9			−0.574413
$533$		−	1.69715e9i		−	0.485484i
$534$	−6.70763e9	−	3.26233e9i	−1.90623	−	0.927115i
$535$	−2.80891e9			−0.793049
$536$	1.97274e9			0.553342
$537$	4.31855e8	+	2.10038e8i	0.120345	+	0.0585311i
$538$	−1.36761e9			−0.378637
$539$	−4.26479e8			−0.117311
$540$	2.58529e9	−	5.58512e8i	0.706530	−	0.152635i
$541$	−6.56335e9			−1.78211			−0.891056	−	0.453893i	$-0.850035\pi$
$541$	−6.56335e9			−1.78211			−0.891056	+	0.453893i	$0.850035\pi$
$542$			6.23234e9i			1.68133i
$543$	2.95564e9	+	1.43751e9i	0.792232	+	0.385311i
$544$			1.12362e9i			0.299242i
$545$		−	1.06504e10i		−	2.81824i
$546$	4.36871e9	−	8.98244e9i	1.14863	−	2.36168i
$547$	3.43377e9			0.897049			0.448525	−	0.893771i	$-0.351950\pi$
$547$	3.43377e9			0.897049			0.448525	+	0.893771i	$0.351950\pi$
$548$	3.41857e8			0.0887387
$549$	4.53024e9	−	3.55561e9i	1.16847	−	0.917088i
$550$		−	1.84759e9i		−	0.473519i
$551$	3.74669e9			0.954150
$552$	2.05866e9	+	1.90990e9i	0.520952	+	0.483309i
$553$	3.57465e9			0.898867
$554$		−	1.57284e9i		−	0.393007i
$555$	6.27565e9	+	3.05223e9i	1.55824	+	0.757866i
$556$	−1.01640e9			−0.250787
$557$	1.05557e8			0.0258817			0.0129409	−	0.999916i	$-0.495881\pi$
$557$	1.05557e8			0.0258817			0.0129409	+	0.999916i	$0.495881\pi$
$558$	5.45872e9	−	4.28434e9i	1.33006	−	1.04391i
$559$			1.57810e9i			0.382114i
$560$		−	1.20459e10i		−	2.89855i
$561$	1.27240e8	−	2.61616e8i	0.0304266	−	0.0625598i
$562$			6.23592e9i			1.48192i
$563$	−7.23388e9			−1.70841			−0.854204	−	0.519937i	$-0.825955\pi$
$563$	−7.23388e9			−1.70841			−0.854204	+	0.519937i	$0.825955\pi$
$564$	−7.29088e8	−	3.54600e8i	−0.171121	−	0.0832266i
$565$	−9.87291e9			−2.30290
$566$	4.27042e9			0.989949
$567$	−5.44440e9	−	1.33183e9i	−1.25432	−	0.306837i
$568$	−1.15272e9			−0.263941
$569$	6.07419e8			0.138228			0.0691139	−	0.997609i	$-0.477983\pi$
$569$	6.07419e8			0.138228			0.0691139	+	0.997609i	$0.477983\pi$
$570$	−4.61408e9	+	9.48694e9i	−1.04357	+	2.14568i
$571$		−	6.16510e9i		−	1.38584i	−0.721014	−	0.692921i	$-0.756323\pi$
$571$		−	6.16510e9i		−	1.38584i	0.721014	−	0.692921i	$-0.243677\pi$
$572$	−5.39995e8			−0.120643
$573$	1.52626e9	−	3.13813e9i	0.338913	−	0.696834i
$574$		−	1.95433e9i		−	0.431327i
$575$	−9.93371e9	−	3.02129e9i	−2.17909	−	0.662759i
$576$	9.78558e8	+	1.24679e9i	0.213357	+	0.271841i
$577$	−4.11066e9			−0.890834			−0.445417	−	0.895323i	$-0.646945\pi$
$577$	−4.11066e9			−0.890834			−0.445417	+	0.895323i	$0.646945\pi$
$578$			4.63118e9i			0.997573i
$579$	3.79324e8	−	7.79922e8i	0.0812148	−	0.166985i
$580$		−	2.90878e9i		−	0.619032i
$581$			2.55793e9i			0.541094i
$582$	−1.95767e9	+	4.02514e9i	−0.411632	+	0.846350i
$583$	−3.62284e8			−0.0757198
$584$			3.72048e9i			0.772954i
$585$	−9.30561e9	−	1.18564e10i	−1.92176	−	2.44853i
$586$		−	8.72038e9i		−	1.79017i
$587$		−	2.83278e9i		−	0.578068i	−0.957319	−	0.289034i	$-0.906666\pi$
$587$		−	2.83278e9i		−	0.578068i	0.957319	−	0.289034i	$-0.0933341\pi$
$588$	5.74598e8	−	1.18142e9i	0.116558	−	0.239654i
$589$			7.89745e9i			1.59251i
$590$	−1.09444e10			−2.19386
$591$	−1.71488e8	−	8.34053e7i	−0.0341727	−	0.0166203i
$592$		−	5.99037e9i		−	1.18666i
$593$		−	3.18583e9i		−	0.627380i	−0.949525	−	0.313690i	$-0.898435\pi$
$593$		−	3.18583e9i		−	0.627380i	0.949525	−	0.313690i	$-0.101565\pi$
$594$	2.24245e8	+	1.03800e9i	0.0439005	+	0.203211i
$595$			4.75464e9i			0.925356i
$596$	−1.51690e9			−0.293491
$597$	5.41370e9	+	2.63301e9i	1.04132	+	0.506457i
$598$	−3.09471e9	+	1.01751e10i	−0.591787	+	1.94574i
$599$			3.44478e9i			0.654889i	0.944870	+	0.327444i	$0.106187\pi$
$599$			3.44478e9i			0.654889i	−0.944870	+	0.327444i	$0.893813\pi$
$600$	−7.70098e9	−	3.74546e9i	−1.45552	−	0.707906i
$601$	7.77569e8			0.146109			0.0730547	−	0.997328i	$-0.476725\pi$
$601$	7.77569e8			0.146109			0.0730547	+	0.997328i	$0.476725\pi$
$602$			1.81724e9i			0.339488i
$603$	3.29797e9	−	2.58845e9i	0.612542	−	0.480761i
$604$	5.90472e8			0.109036
$605$	−9.55649e9			−1.75450
$606$	3.39518e9	−	6.98078e9i	0.619739	−	1.27424i
$607$	2.43856e9			0.442560			0.221280	−	0.975210i	$-0.428977\pi$
$607$	2.43856e9			0.442560			0.221280	+	0.975210i	$0.428977\pi$
$608$	4.66796e9			0.842295
$609$	−2.69601e9	+	5.54322e9i	−0.483683	+	0.994493i
$610$	1.78330e10			3.18104
$611$			4.62003e9i			0.819409i
$612$	5.53292e8	+	7.04955e8i	0.0975718	+	0.124317i
$613$		−	1.75921e9i		−	0.308464i	−0.988035	−	0.154232i	$-0.950710\pi$
$613$		−	1.75921e9i		−	0.308464i	0.988035	−	0.154232i	$-0.0492904\pi$
$614$			6.73664e9i			1.17450i
$615$	−2.65196e9	−	1.28981e9i	−0.459731	−	0.223595i
$616$	9.35622e8			0.161275
$617$	−8.58878e9			−1.47209			−0.736043	−	0.676935i	$-0.763308\pi$
$617$	−8.58878e9			−1.47209			−0.736043	+	0.676935i	$0.763308\pi$
$618$	1.99728e9	−	4.10658e9i	0.340392	−	0.699876i
$619$			4.89173e9i			0.828982i	0.910053	+	0.414491i	$0.136040\pi$
$619$			4.89173e9i			0.828982i	−0.910053	+	0.414491i	$0.863960\pi$
$620$	6.13127e9			1.03319
$621$	5.94759e9	+	4.91737e8i	0.996600	+	0.0823972i
$622$	7.67854e9			1.27942
$623$			1.39659e10i			2.31398i
$624$	−5.65869e9	+	1.16348e10i	−0.932331	+	1.91695i
$625$	1.16578e10			1.91002
$626$	8.01598e8			0.130601
$627$	−1.08686e9	−	5.28605e8i	−0.176091	−	0.0856437i
$628$		−	3.86035e9i		−	0.621968i
$629$			2.36447e9i			0.378840i
$630$	−1.07158e10	−	1.36531e10i	−1.70738	−	2.17539i
$631$			2.95537e9i			0.468283i	0.972203	+	0.234142i	$0.0752279\pi$
$631$			2.95537e9i			0.468283i	−0.972203	+	0.234142i	$0.924772\pi$
$632$	−3.13915e9			−0.494655
$633$	4.92587e8	−	1.01280e9i	0.0771916	−	0.158713i
$634$	−1.23267e10			−1.92103
$635$	6.57324e9			1.01876
$636$	4.88108e8	−	1.00359e9i	0.0752343	−	0.154688i
$637$	−7.48634e9			−1.14758
$638$	1.16789e9			0.178045
$639$	−1.92709e9	+	1.51250e9i	−0.292178	+	0.229320i
$640$			1.39848e10i			2.10875i
$641$	−2.91645e9			−0.437373			−0.218686	−	0.975795i	$-0.570177\pi$
$641$	−2.91645e9			−0.437373			−0.218686	+	0.975795i	$0.570177\pi$
$642$	3.12418e9	+	1.51948e9i	0.465975	+	0.226632i
$643$			4.60777e9i			0.683521i	0.939787	+	0.341761i	$0.111023\pi$
$643$			4.60777e9i			0.683521i	−0.939787	+	0.341761i	$0.888977\pi$
$644$	−1.01684e9	+	3.34329e9i	−0.150022	+	0.493257i
$645$	2.46593e9	+	1.19933e9i	0.361844	+	0.175987i
$646$	−3.57438e9			−0.521659
$647$			8.01581e9i			1.16354i	0.813352	+	0.581771i	$0.197640\pi$
$647$			8.01581e9i			1.16354i	−0.813352	+	0.581771i	$0.802360\pi$
$648$	4.78111e9	+	1.16957e9i	0.690265	+	0.168855i
$649$		−	1.25382e9i		−	0.180045i
$650$		−	3.24324e10i		−	4.63214i
$651$	−1.16843e10	−	5.68277e9i	−1.65985	−	0.807285i
$652$	5.93966e8			0.0839257
$653$		−	1.09676e10i		−	1.54139i	−0.637202	−	0.770697i	$-0.719908\pi$
$653$		−	1.09676e10i		−	1.54139i	0.637202	−	0.770697i	$-0.280092\pi$
$654$	−5.76132e9	+	1.18458e10i	−0.805378	+	1.65593i
$655$		−	1.27653e10i		−	1.77496i
$656$			2.53140e9i			0.350105i
$657$	4.88166e9	+	6.21977e9i	0.671567	+	0.855649i
$658$			5.32014e9i			0.728002i
$659$	4.51703e9			0.614829			0.307414	−	0.951576i	$-0.400536\pi$
$659$	4.51703e9			0.614829			0.307414	+	0.951576i	$0.400536\pi$
$660$	−4.10389e8	+	8.43794e8i	−0.0555637	+	0.114244i
$661$		−	3.78426e9i		−	0.509654i	−0.966987	−	0.254827i	$-0.917981\pi$
$661$		−	3.78426e9i		−	0.509654i	0.966987	−	0.254827i	$-0.0820185\pi$
$662$			2.09215e8i			0.0280279i
$663$	2.23355e9	−	4.59237e9i	0.297645	−	0.611983i
$664$		−	2.24630e9i		−	0.297769i
$665$	1.97527e10			2.60465
$666$	−5.32891e9	−	6.78961e9i	−0.699002	−	0.890605i
$667$	1.90980e9	−	6.27923e9i	0.249199	−	0.819343i
$668$			1.82378e8i			0.0236731i
$669$	5.65541e9	−	1.16280e10i	0.730252	−	1.50146i
$670$	1.29822e10			1.66758
$671$			2.04301e9i			0.261061i
$672$	−3.35893e9	+	6.90624e9i	−0.426981	+	0.877908i
$673$	−1.00486e9			−0.127073			−0.0635364	−	0.997980i	$-0.520238\pi$
$673$	−1.00486e9			−0.127073			−0.0635364	+	0.997980i	$0.520238\pi$
$674$	1.88020e10			2.36534
$675$	−1.77887e10	+	3.84297e9i	−2.22628	+	0.480955i
$676$	−6.27221e9			−0.780921
$677$	4.00345e9			0.495878			0.247939	−	0.968776i	$-0.420247\pi$
$677$	4.00345e9			0.495878			0.247939	+	0.968776i	$0.420247\pi$
$678$	1.09810e10	+	5.34074e9i	1.35313	+	0.658108i
$679$	8.38070e9			1.02739
$680$		−	4.17539e9i		−	0.509232i
$681$	−1.44127e9	+	2.96336e9i	−0.174876	+	0.359559i
$682$			2.46173e9i			0.297163i
$683$		−	1.47567e10i		−	1.77221i	−0.463483	−	0.886106i	$-0.653400\pi$
$683$		−	1.47567e10i		−	1.77221i	0.463483	−	0.886106i	$-0.346600\pi$
$684$	2.92866e9	−	2.29859e9i	0.349923	−	0.274641i
$685$	−3.38498e9			−0.402382
$686$	4.29475e9			0.507929
$687$	−5.29786e9	−	2.57667e9i	−0.623379	−	0.303187i
$688$		−	2.35383e9i		−	0.275560i
$689$	−6.35948e9			−0.740720
$690$	1.35476e10	+	1.25687e10i	1.56997	+	1.45653i
$691$	−7.43761e9			−0.857552			−0.428776	−	0.903411i	$-0.641055\pi$
$691$	−7.43761e9			−0.857552			−0.428776	+	0.903411i	$0.641055\pi$
$692$		−	2.90706e9i		−	0.333490i
$693$	1.56414e9	−	1.22764e9i	0.178530	−	0.140121i
$694$	−1.46498e10			−1.66369
$695$	1.00642e10			1.13718
$696$	2.36755e9	−	4.86789e9i	0.266175	−	0.547278i
$697$		−	9.99174e8i		−	0.111770i
$698$			7.22157e9i			0.803781i
$699$	−1.13437e10	−	5.51712e9i	−1.25627	−	0.611002i
$700$		−	1.06565e10i		−	1.17428i
$701$	9.72898e9			1.06673			0.533365	−	0.845885i	$-0.320927\pi$
$701$	9.72898e9			1.06673			0.533365	+	0.845885i	$0.320927\pi$
$702$	3.93635e9	+	1.82209e10i	0.429452	+	1.98788i
$703$	9.82293e9			1.06634
$704$	−5.62267e8			−0.0607349
$705$	7.21924e9	+	3.51116e9i	0.775942	+	0.377388i
$706$	−4.53713e9			−0.485249
$707$	−1.45346e10			−1.54680
$708$	3.47332e9	+	1.68929e9i	0.367814	+	0.178890i
$709$			7.02092e8i			0.0739830i	0.999316	+	0.0369915i	$0.0117774\pi$
$709$			7.02092e8i			0.0739830i	−0.999316	+	0.0369915i	$0.988223\pi$
$710$	−7.58585e9			−0.795426
$711$	−5.24792e9	+	4.11890e9i	−0.547575	+	0.429771i
$712$		−	1.22644e10i		−	1.27341i
$713$	1.32357e10	+	4.02556e9i	1.36751	+	0.415923i
$714$	2.57202e9	−	5.28829e9i	0.264442	−	0.543715i
$715$	5.34689e9			0.547054
$716$		−	5.24788e8i		−	0.0534304i
$717$	−4.01778e8	−	1.95409e8i	−0.0407070	−	0.0197983i
$718$			2.23820e10i			2.25664i
$719$		−	1.26941e9i		−	0.127365i	−0.997970	−	0.0636827i	$-0.979715\pi$
$719$		−	1.26941e9i		−	0.127365i	0.997970	−	0.0636827i	$-0.0202846\pi$
$720$	1.38799e10	+	1.76845e10i	1.38587	+	1.76575i
$721$	−8.55028e9			−0.849585
$722$			2.88667e9i			0.285442i
$723$	4.95369e9	+	2.40928e9i	0.487467	+	0.237085i
$724$		−	3.59168e9i		−	0.351732i
$725$			2.00146e10i			1.95058i
$726$	1.06291e10	+	5.16957e9i	1.03090	+	0.501391i
$727$		−	2.56479e9i		−	0.247561i	−0.992310	−	0.123780i	$-0.960498\pi$
$727$		−	2.56479e9i		−	0.247561i	0.992310	−	0.123780i	$-0.0395018\pi$
$728$	1.64238e10			1.57766
$729$	9.52750e9	−	4.31807e9i	0.910820	−	0.412803i
$730$			2.44837e10i			2.32942i
$731$			9.29084e8i			0.0879720i
$732$	−5.65950e9	−	2.75256e9i	−0.533322	−	0.259387i
$733$			2.08134e10i			1.95200i	0.217768	+	0.976001i	$0.430122\pi$
$733$			2.08134e10i			1.95200i	−0.217768	+	0.976001i	$0.569878\pi$
$734$	−1.47088e10			−1.37291
$735$	−5.68951e9	+	1.16981e10i	−0.528530	+	1.08670i
$736$	2.37940e9	−	7.82322e9i	0.219986	−	0.723291i
$737$			1.48729e9i			0.136855i
$738$	2.25188e9	+	2.86915e9i	0.206229	+	0.262758i
$739$	−1.90616e10			−1.73742			−0.868710	−	0.495321i	$-0.835050\pi$
$739$	−1.90616e10			−1.73742			−0.868710	+	0.495321i	$0.835050\pi$
$740$		−	7.62614e9i		−	0.691821i
$741$	−1.90785e10	−	9.27905e9i	−1.72259	−	0.837799i
$742$	−7.32318e9			−0.658091
$743$	−5.91672e9			−0.529201			−0.264600	−	0.964358i	$-0.585240\pi$
$743$	−5.91672e9			−0.529201			−0.264600	+	0.964358i	$0.585240\pi$
$744$	1.02608e10	+	4.99044e9i	0.913429	+	0.444256i
$745$	1.50199e10			1.33082
$746$	5.49982e9			0.485023
$747$	−2.94738e9	−	3.75529e9i	−0.258711	−	0.329626i
$748$	−3.17915e8			−0.0277751
$749$		−	6.50482e9i		−	0.565651i
$750$	−2.84282e10	−	1.38264e10i	−2.46056	−	1.19672i
$751$		−	5.69173e9i		−	0.490349i	−0.969479	−	0.245174i	$-0.921155\pi$
$751$		−	5.69173e9i		−	0.490349i	0.969479	−	0.245174i	$-0.0788452\pi$
$752$		−	6.89106e9i		−	0.590913i
$753$	−6.17633e9	+	1.26991e10i	−0.527167	+	1.08390i
$754$	2.05009e10			1.74170
$755$	−5.84670e9			−0.494420
$756$	1.29339e9	+	5.98695e9i	0.108869	+	0.503941i
$757$		−	1.39349e10i		−	1.16753i	−0.811922	−	0.583766i	$-0.801578\pi$
$757$		−	1.39349e10i		−	1.16753i	0.811922	−	0.583766i	$-0.198422\pi$
$758$	5.34926e9			0.446119
$759$	−1.43992e9	+	1.55206e9i	−0.119534	+	0.128844i
$760$	−1.73462e10			−1.43336
$761$		−	5.03175e9i		−	0.413878i	−0.978354	−	0.206939i	$-0.933650\pi$
$761$		−	5.03175e9i		−	0.413878i	0.978354	−	0.206939i	$-0.0663502\pi$
$762$	−7.31099e9	−	3.55578e9i	−0.598596	−	0.291134i
$763$	2.46640e10			2.01014
$764$	−3.81343e9			−0.309378
$765$	−5.47855e9	−	6.98027e9i	−0.442436	−	0.563712i
$766$		−	1.96057e10i		−	1.57610i
$767$		−	2.20094e10i		−	1.76127i
$768$	5.66766e9	−	1.16532e10i	0.451481	−	0.928284i
$769$		−	8.13763e9i		−	0.645291i	−0.946520	−	0.322646i	$-0.895428\pi$
$769$		−	8.13763e9i		−	0.645291i	0.946520	−	0.322646i	$-0.104572\pi$
$770$	6.15714e9			0.486029
$771$	5.76168e9	+	2.80226e9i	0.452750	+	0.220200i
$772$	−9.47757e8			−0.0741372
$773$	−1.05844e10			−0.824207			−0.412103	−	0.911137i	$-0.635206\pi$
$773$	−1.05844e10			−0.824207			−0.412103	+	0.911137i	$0.635206\pi$
$774$	−2.09392e9	−	2.66788e9i	−0.162318	−	0.206811i
$775$	−4.21877e10			−3.25559
$776$	−7.35968e9			−0.565383
$777$	−7.06829e9	+	1.45330e10i	−0.540556	+	1.11143i
$778$			7.40759e9i			0.563960i
$779$	−4.15096e9			−0.314607
$780$	−7.20389e9	+	1.48118e10i	−0.543546	+	1.11758i
$781$		−	8.69062e8i		−	0.0652788i
$782$	−1.82197e9	+	5.99045e9i	−0.136244	+	0.447956i
$783$	−2.42919e9	−	1.12445e10i	−0.180840	−	0.837090i
$784$	1.11663e10			0.827569
$785$			3.82241e10i			2.82029i
$786$	−6.90540e9	+	1.41981e10i	−0.507236	+	1.04292i
$787$		−	1.60652e10i		−	1.17483i	−0.809287	−	0.587414i	$-0.800146\pi$
$787$		−	1.60652e10i		−	1.17483i	0.809287	−	0.587414i	$-0.199854\pi$
$788$			2.08392e8i			0.0151719i
$789$	8.11473e9	−	1.66846e10i	0.588173	−	1.20933i
$790$	−2.06581e10			−1.49072
$791$		−	2.28635e10i		−	1.64257i
$792$	−1.37358e9	+	1.07807e9i	−0.0982464	+	0.0771099i
$793$			3.58627e10i			2.55380i
$794$		−	1.38485e10i		−	0.981820i
$795$	−4.83311e9	+	9.93729e9i	−0.341147	+	0.701428i
$796$		−	6.57870e9i		−	0.462322i
$797$	1.18997e10			0.832588			0.416294	−	0.909230i	$-0.363329\pi$
$797$	1.18997e10			0.832588			0.416294	+	0.909230i	$0.363329\pi$
$798$	−2.19696e10	−	1.06852e10i	−1.53043	−	0.744341i
$799$			2.71998e9i			0.188648i
$800$			2.49359e10i			1.72191i
$801$	−1.60922e10	−	2.05033e10i	−1.10637	−	1.40964i
$802$		−	1.20510e10i		−	0.824925i
$803$	−2.80494e9			−0.191170
$804$	−4.12006e9	−	2.00384e9i	−0.279581	−	0.135977i
$805$	1.00685e10	−	3.31043e10i	0.680268	−	2.23665i
$806$			4.32128e10i			2.90696i
$807$	−4.29760e9	−	2.09019e9i	−0.287852	−	0.140000i
$808$	1.27639e10			0.851221
$809$		−	1.18092e10i		−	0.784150i	−0.919933	−	0.392075i	$-0.871757\pi$
$809$		−	1.18092e10i		−	0.784150i	0.919933	−	0.392075i	$-0.128243\pi$
$810$	3.14635e10	+	7.69672e9i	2.08022	+	0.508872i
$811$	1.43950e10			0.947632			0.473816	−	0.880624i	$-0.342876\pi$
$811$	1.43950e10			0.947632			0.473816	+	0.880624i	$0.342876\pi$
$812$	6.73609e9			0.441532
$813$	−9.52521e9	+	1.95847e10i	−0.621667	+	1.27820i
$814$	3.06192e9			0.198980
$815$	−5.88129e9			−0.380558
$816$	−3.33148e9	+	6.84980e9i	−0.214645	+	0.441329i
$817$	3.85978e9			0.247620
$818$		−	1.70966e10i		−	1.09213i
$819$	2.74567e10	−	2.15497e10i	1.74644	−	1.37072i
$820$			3.22265e9i			0.204110i
$821$		−	1.01443e10i		−	0.639766i	−0.947457	−	0.319883i	$-0.896356\pi$
$821$		−	1.01443e10i		−	0.639766i	0.947457	−	0.319883i	$-0.103644\pi$
$822$	3.76490e9	+	1.83110e9i	0.236430	+	0.114990i
$823$	2.53071e10			1.58250			0.791249	−	0.611494i	$-0.209431\pi$
$823$	2.53071e10			1.58250			0.791249	+	0.611494i	$0.209431\pi$
$824$	7.50860e9			0.467534
$825$	2.82378e9	−	5.80593e9i	0.175082	−	0.359984i
$826$		−	2.53447e10i		−	1.56479i
$827$	−1.23299e10			−0.758034			−0.379017	−	0.925390i	$-0.623738\pi$
$827$	−1.23299e10			−0.758034			−0.379017	+	0.925390i	$0.623738\pi$
$828$	−2.35948e9	−	6.07992e9i	−0.144448	−	0.372213i
$829$	1.12449e10			0.685510			0.342755	−	0.939425i	$-0.388640\pi$
$829$	1.12449e10			0.685510			0.342755	+	0.939425i	$0.388640\pi$
$830$		−	1.47824e10i		−	0.897373i
$831$	2.40386e9	−	4.94253e9i	0.145313	−	0.298776i
$832$	−9.86995e9			−0.594132
$833$	−4.40748e9			−0.264200
$834$	−1.11937e10	−	5.44419e9i	−0.668180	−	0.324977i
$835$		−	1.80586e9i		−	0.107345i
$836$			1.32074e9i			0.0781801i
$837$	2.37016e10	−	5.12037e9i	1.39714	−	0.301830i
$838$		−	1.65022e8i		−	0.00968697i
$839$	1.20386e10			0.703735			0.351867	−	0.936050i	$-0.385547\pi$
$839$	1.20386e10			0.703735			0.351867	+	0.936050i	$0.385547\pi$
$840$	1.24818e10	−	2.56637e10i	0.726608	−	1.49397i
$841$	4.59842e9			0.266577
$842$	−2.64433e10			−1.52659
$843$	−9.53069e9	+	1.95959e10i	−0.547934	+	1.12660i
$844$	−1.23075e9			−0.0704647
$845$	6.21058e10			3.54106
$846$	−6.13014e9	−	7.81047e9i	−0.348076	−	0.443487i
$847$		−	2.21307e10i		−	1.25142i
$848$	9.48555e9			0.534167
$849$	1.34195e10	+	6.52671e9i	0.752590	+	0.366030i
$850$		−	1.90941e10i		−	1.06643i
$851$	5.00704e9	−	1.64626e10i	0.278501	−	0.915685i
$852$	2.40745e9	+	1.17089e9i	0.133358	+	0.0648602i
$853$	−3.06309e10			−1.68981			−0.844904	−	0.534917i	$-0.820343\pi$
$853$	−3.06309e10			−1.68981			−0.844904	+	0.534917i	$0.820343\pi$
$854$			4.12972e10i			2.26892i
$855$	−2.89988e10	+	2.27601e10i	−1.58671	+	1.24535i
$856$			5.71234e9i			0.311283i
$857$		−	2.80638e10i		−	1.52305i	−0.648138	−	0.761523i	$-0.724452\pi$
$857$		−	2.80638e10i		−	1.52305i	0.648138	−	0.761523i	$-0.275548\pi$
$858$	−5.94700e9	−	2.89239e9i	−0.321435	−	0.156333i
$859$	−4.58300e9			−0.246702			−0.123351	−	0.992363i	$-0.539364\pi$
$859$	−4.58300e9			−0.246702			−0.123351	+	0.992363i	$0.539364\pi$
$860$		−	2.99658e9i		−	0.160650i
$861$	2.98691e9	−	6.14135e9i	0.159482	−	0.327908i
$862$			2.19769e10i			1.16867i
$863$			5.33780e9i			0.282699i	0.989960	+	0.141350i	$0.0451442\pi$
$863$			5.33780e9i			0.282699i	−0.989960	+	0.141350i	$0.954856\pi$
$864$	−3.02650e9	−	1.40093e10i	−0.159641	−	0.738958i
$865$			2.87850e10i			1.51220i
$866$	−3.29300e10			−1.72298
$867$	−7.07809e9	+	1.45532e10i	−0.368849	+	0.758386i
$868$			1.41987e10i			0.736933i
$869$		−	2.36667e9i		−	0.122340i
$870$	1.55804e10	−	3.20346e10i	0.802160	−	1.64931i
$871$			2.61077e10i			1.33877i
$872$	−2.16591e10			−1.10620
$873$	−1.23037e10	+	9.65668e9i	−0.625871	+	0.491222i
$874$	2.48867e10	+	7.56917e9i	1.26089	+	0.383494i
$875$			5.91900e10i			2.98690i
$876$	3.77912e9	−	7.77019e9i	0.189944	−	0.390541i
$877$	−6.27107e9			−0.313937			−0.156969	−	0.987604i	$-0.550172\pi$
$877$	−6.27107e9			−0.313937			−0.156969	+	0.987604i	$0.550172\pi$
$878$			1.36197e10i			0.679103i
$879$	1.33278e10	−	2.74031e10i	0.661909	−	1.36094i
$880$	−7.97521e9			−0.394505
$881$	−1.02562e10			−0.505325			−0.252662	−	0.967555i	$-0.581306\pi$
$881$	−1.02562e10			−0.505325			−0.252662	+	0.967555i	$0.581306\pi$
$882$	1.26562e10	−	9.93334e9i	0.621101	−	0.487478i
$883$	−6.17756e9			−0.301964			−0.150982	−	0.988537i	$-0.548243\pi$
$883$	−6.17756e9			−0.301964			−0.150982	+	0.988537i	$0.548243\pi$
$884$	−5.58062e9			−0.271706
$885$	−3.43918e10	−	1.67269e10i	−1.66784	−	0.811172i
$886$	−4.36713e10			−2.10949
$887$		−	1.39319e10i		−	0.670314i	−0.942162	−	0.335157i	$-0.891211\pi$
$887$		−	1.39319e10i		−	0.670314i	0.942162	−	0.335157i	$-0.108789\pi$
$888$	6.20716e9	−	1.27625e10i	0.297473	−	0.611630i
$889$			1.52222e10i			0.726641i
$890$		−	8.07097e10i		−	3.83761i
$891$	−8.81764e8	+	3.60457e9i	−0.0417619	+	0.170719i
$892$	−1.41303e10			−0.666613
$893$	1.12999e10			0.530998
$894$	−1.67057e10	−	8.12500e9i	−0.781957	−	0.380314i
$895$			5.19631e9i			0.242278i
$896$	−3.23856e10			−1.50409
$897$	−2.52760e10	+	2.72447e10i	−1.16933	+	1.26040i
$898$	−1.69887e10			−0.782878
$899$		−	2.66674e10i		−	1.22411i
$900$	1.22789e10	+	1.56447e10i	0.561451	+	0.715351i
$901$	−3.74406e9			−0.170532
$902$	−1.29391e9			−0.0587056
$903$	−2.77739e9	+	5.71054e9i	−0.125525	+	0.258090i
$904$			2.00780e10i			0.903922i
$905$			3.55638e10i			1.59492i
$906$	6.50291e9	+	3.16276e9i	0.290509	+	0.141292i
$907$			2.99037e10i			1.33076i	0.746505	+	0.665379i	$0.231730\pi$
$907$			2.99037e10i			1.33076i	−0.746505	+	0.665379i	$0.768270\pi$
$908$	3.60106e9			0.159636
$909$	2.13382e10	−	1.67475e10i	0.942289	−	0.739567i
$910$	1.08081e11			4.75452
$911$	1.10234e9			0.0483059			0.0241529	−	0.999708i	$-0.492311\pi$
$911$	1.10234e9			0.0483059			0.0241529	+	0.999708i	$0.492311\pi$
$912$	2.84568e10	+	1.38403e10i	1.24223	+	0.604175i
$913$	1.69353e9			0.0736453
$914$	3.63210e9			0.157343
$915$	5.60388e10	+	2.72551e10i	2.41833	+	1.17618i
$916$			6.43793e9i			0.276766i
$917$	2.95617e10			1.26601
$918$	2.31747e9	+	1.07273e10i	0.0988702	+	0.457659i
$919$		−	1.62322e10i		−	0.689879i	−0.938625	−	0.344940i	$-0.887899\pi$
$919$		−	1.62322e10i		−	0.689879i	0.938625	−	0.344940i	$-0.112101\pi$
$920$	−8.84187e9	+	2.90712e10i	−0.374358	+	1.23085i
$921$	−1.02960e10	+	2.11694e10i	−0.434269	+	0.892894i
$922$	2.83418e9			0.119088
$923$		−	1.52554e10i		−	0.638582i
$924$	−1.95404e9	−	9.50369e8i	−0.0814857	−	0.0396315i
$925$			5.24735e10i			2.17994i
$926$		−	2.06027e10i		−	0.852681i
$927$	1.25526e10	−	9.85208e9i	0.517554	−	0.406208i
$928$	−1.57623e10			−0.647444
$929$			3.57724e9i			0.146384i	0.997318	+	0.0731920i	$0.0233186\pi$
$929$			3.57724e9i			0.146384i	−0.997318	+	0.0731920i	$0.976681\pi$
$930$	6.75241e10	+	3.28411e10i	2.75276	+	1.33884i
$931$			1.83104e10i			0.743660i
$932$			1.37848e10i			0.557755i
$933$	2.41293e10	+	1.17355e10i	0.972653	+	0.473061i
$934$			1.23715e10i			0.496831i
$935$	3.14791e9			0.125945
$936$	−2.41116e10	+	1.89243e10i	−0.961084	+	0.754318i
$937$		−	4.17989e10i		−	1.65988i	−0.557853	−	0.829940i	$-0.688375\pi$
$937$		−	4.17989e10i		−	1.65988i	0.557853	−	0.829940i	$-0.311625\pi$
$938$			3.00640e10i			1.18942i
$939$	2.51896e9	+	1.22513e9i	0.0992869	+	0.0482893i
$940$		−	8.77277e9i		−	0.344500i
$941$	−2.78000e10			−1.08763			−0.543814	−	0.839205i	$-0.683020\pi$
$941$	−2.78000e10			−1.08763			−0.543814	+	0.839205i	$0.683020\pi$
$942$	2.06773e10	−	4.25143e10i	0.805964	−	1.65713i
$943$	−2.11587e9	+	6.95677e9i	−0.0821671	+	0.270157i
$944$			3.28284e10i			1.27013i
$945$	−1.28068e10	−	5.92812e10i	−0.493661	−	2.28510i
$946$	1.20314e9			0.0462059
$947$		−	1.51390e10i		−	0.579258i	−0.957139	−	0.289629i	$-0.906468\pi$
$947$		−	1.51390e10i		−	0.579258i	0.957139	−	0.289629i	$-0.0935320\pi$
$948$	6.55608e9	+	3.18862e9i	0.249928	+	0.121555i
$949$	−4.92375e10			−1.87010
$950$	−7.93244e10			−3.00175
$951$	−3.87357e10	−	1.88396e10i	−1.46043	−	0.710295i
$952$	9.66926e9			0.363215
$953$	−2.86419e10			−1.07196			−0.535978	−	0.844232i	$-0.680057\pi$
$953$	−2.86419e10			−1.07196			−0.535978	+	0.844232i	$0.680057\pi$
$954$	1.07511e10	−	8.43815e9i	0.400899	−	0.314650i
$955$	3.77596e10			1.40286
$956$			4.88238e8i			0.0180729i
$957$	3.67000e9	+	1.78495e9i	0.135355	+	0.0658314i
$958$			2.25769e10i			0.829632i
$959$		−	7.83886e9i		−	0.287004i
$960$	−7.50102e9	+	1.54227e10i	−0.273634	+	0.562616i
$961$	2.86981e10			1.04309
$962$	5.37485e10			1.94650
$963$	7.49519e9	+	9.54969e9i	0.270452	+	0.344586i
$964$		−	6.01970e9i		−	0.216424i
$965$	9.38443e9			0.336173
$966$	−2.91063e10	+	3.13733e10i	−1.03888	+	1.11980i
$967$	8.78496e9			0.312426			0.156213	−	0.987723i	$-0.450071\pi$
$967$	8.78496e9			0.312426			0.156213	+	0.987723i	$0.450071\pi$
$968$			1.94345e10i			0.688668i
$969$	−1.12322e10	−	5.46291e9i	−0.396581	−	0.192882i
$970$	−4.84326e10			−1.70387
$971$	1.43156e10			0.501812			0.250906	−	0.968011i	$-0.419272\pi$
$971$	1.43156e10			0.501812			0.250906	+	0.968011i	$0.419272\pi$
$972$	−8.79729e9	−	7.29910e9i	−0.307268	−	0.254940i
$973$			2.33064e10i			0.811109i
$974$		−	1.08996e10i		−	0.377966i
$975$	4.95681e10	−	1.01916e11i	1.71272	−	3.52150i
$976$		−	5.34914e10i		−	1.84166i
$977$	−2.38448e10			−0.818018			−0.409009	−	0.912530i	$-0.634125\pi$
$977$	−2.38448e10			−0.818018			−0.409009	+	0.912530i	$0.634125\pi$
$978$	6.54138e9	+	3.18148e9i	0.223606	+	0.108753i
$979$	9.24639e9			0.314944
$980$	1.42155e10			0.482470
$981$	−3.62090e10	+	2.84191e10i	−1.22455	+	0.961100i
$982$	−2.01943e10			−0.680516
$983$	−2.48854e10			−0.835619			−0.417809	−	0.908535i	$-0.637202\pi$
$983$	−2.48854e10			−0.835619			−0.417809	+	0.908535i	$0.637202\pi$
$984$	−2.62302e9	+	5.39314e9i	−0.0877643	+	0.180451i
$985$		−	2.06344e9i		−	0.0687963i
$986$	1.20696e10			0.400982
$987$	−8.13106e9	+	1.67181e10i	−0.269176	+	0.553450i
$988$			2.31841e10i			0.764788i
$989$	1.96745e9	−	6.46877e9i	0.0646719	−	0.212635i
$990$	−9.03927e9	+	7.09458e9i	−0.296081	+	0.232383i
$991$	3.33304e10			1.08788			0.543942	−	0.839123i	$-0.316931\pi$
$991$	3.33304e10			1.08788			0.543942	+	0.839123i	$0.316931\pi$
$992$		−	3.32246e10i		−	1.08061i
$993$	−3.19755e8	+	6.57444e8i	−0.0103632	+	0.0213077i
$994$		−	1.75671e10i		−	0.567347i
$995$			6.51405e10i			2.09638i
$996$	−2.28170e9	+	4.69137e9i	−0.0731731	+	0.150450i
$997$	−4.87622e10			−1.55830			−0.779149	−	0.626839i	$-0.784349\pi$
$997$	−4.87622e10			−1.55830			−0.779149	+	0.626839i	$0.784349\pi$
$998$		−	2.21984e10i		−	0.706912i
$999$	−6.36877e9	−	2.94803e10i	−0.202105	−	0.935519i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.c.b.68.12	yes	48
3.2	odd	2	inner	69.8.c.b.68.37	yes	48
23.22	odd	2	inner	69.8.c.b.68.11	✓	48
69.68	even	2	inner	69.8.c.b.68.38	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.c.b.68.11	✓	48	23.22	odd	2	inner
69.8.c.b.68.12	yes	48	1.1	even	1	trivial
69.8.c.b.68.37	yes	48	3.2	odd	2	inner
69.8.c.b.68.38	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)

\(f(q)\)	\(=\)	\(q-13.3830i q^{2} +(20.4540 - 42.0551i) q^{3} -51.1052 q^{4} +506.030 q^{5} +(-562.825 - 273.736i) q^{6} +1171.85i q^{7} -1029.08i q^{8} +(-1350.27 - 1720.39i) q^{9} +O(q^{10})\)	\(q-13.3830i q^{2} +(20.4540 - 42.0551i) q^{3} -51.1052 q^{4} +506.030 q^{5} +(-562.825 - 273.736i) q^{6} +1171.85i q^{7} -1029.08i q^{8} +(-1350.27 - 1720.39i) q^{9} -6772.20i q^{10} +775.848 q^{11} +(-1045.30 + 2149.23i) q^{12} +13619.1 q^{13} +15682.9 q^{14} +(10350.3 - 21281.1i) q^{15} -20313.7 q^{16} +8018.06 q^{17} +(-23024.0 + 18070.7i) q^{18} -33310.2i q^{19} -25860.7 q^{20} +(49282.4 + 23969.0i) q^{21} -10383.2i q^{22} +(-55825.9 - 16979.2i) q^{23} +(-43278.3 - 21048.9i) q^{24} +177941. q^{25} -182265. i q^{26} +(-99969.6 + 21596.9i) q^{27} -59887.7i q^{28} +112479. i q^{29} +(-284806. - 138519. i) q^{30} -237088. q^{31} +140136. i q^{32} +(15869.2 - 32628.4i) q^{33} -107306. i q^{34} +592992. i q^{35} +(69005.7 + 87920.8i) q^{36} +294893. i q^{37} -445791. q^{38} +(278565. - 572753. i) q^{39} -520747. i q^{40} -124615. i q^{41} +(320778. - 659547. i) q^{42} +115874. i q^{43} -39649.8 q^{44} +(-683276. - 870568. i) q^{45} +(-227233. + 747119. i) q^{46} +339232. i q^{47} +(-415497. + 854296. i) q^{48} -549694. q^{49} -2.38139e6i q^{50} +(164001. - 337201. i) q^{51} -696007. q^{52} -466953. q^{53} +(289032. + 1.33790e6i) q^{54} +392602. q^{55} +1.20594e6 q^{56} +(-1.40086e6 - 681326. i) q^{57} +1.50530e6 q^{58} -1.61607e6i q^{59} +(-528955. + 1.08758e6i) q^{60} +2.63326e6i q^{61} +3.17295e6i q^{62} +(2.01604e6 - 1.58232e6i) q^{63} -724713. q^{64} +6.89167e6 q^{65} +(-436666. - 212378. i) q^{66} +1.91699e6i q^{67} -409764. q^{68} +(-1.85592e6 + 2.00047e6i) q^{69} +7.93602e6 q^{70} -1.12014e6i q^{71} +(-1.77043e6 + 1.38954e6i) q^{72} -3.61533e6 q^{73} +3.94655e6 q^{74} +(3.63960e6 - 7.48333e6i) q^{75} +1.70232e6i q^{76} +909179. i q^{77} +(-7.66517e6 - 3.72804e6i) q^{78} -3.05043e6i q^{79} -1.02793e7 q^{80} +(-1.13652e6 + 4.64598e6i) q^{81} -1.66773e6 q^{82} +2.18281e6 q^{83} +(-2.51859e6 - 1.22494e6i) q^{84} +4.05738e6 q^{85} +1.55074e6 q^{86} +(4.73031e6 + 2.30064e6i) q^{87} -798413. i q^{88} +1.19178e7 q^{89} +(-1.16508e7 + 9.14430e6i) q^{90} +1.59596e7i q^{91} +(2.85299e6 + 867725. i) q^{92} +(-4.84939e6 + 9.97077e6i) q^{93} +4.53994e6 q^{94} -1.68559e7i q^{95} +(5.89344e6 + 2.86634e6i) q^{96} -7.15167e6i q^{97} +7.35657e6i q^{98} +(-1.04760e6 - 1.33476e6i) 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

Modular forms

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