LMFDB - Embedded newform 109.8.b.a.108.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [109,8,Mod(108,109)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(109, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("109.108");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 109 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	109.b (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:	$34.0499677778$
Analytic rank:	$0$
Dimension:	$62$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			108.5
Character	$\chi$	$=$	109.108
Dual form			109.8.b.a.108.58

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-19.1107i q^{2} +70.4891 q^{3} -237.218 q^{4} -288.848 q^{5} -1347.09i q^{6} -109.637 q^{7} +2087.22i q^{8} +2781.72 q^{9} +O(q^{10})$	\(q-19.1107i q^{2} +70.4891 q^{3} -237.218 q^{4} -288.848 q^{5} -1347.09i q^{6} -109.637 q^{7} +2087.22i q^{8} +2781.72 q^{9} +5520.08i q^{10} +6231.73i q^{11} -16721.3 q^{12} +5635.67i q^{13} +2095.23i q^{14} -20360.7 q^{15} +9524.34 q^{16} +8505.86i q^{17} -53160.5i q^{18} +15195.3i q^{19} +68519.9 q^{20} -7728.20 q^{21} +119093. q^{22} -11730.4i q^{23} +147126. i q^{24} +5308.23 q^{25} +107701. q^{26} +41921.1 q^{27} +26007.8 q^{28} +11831.9 q^{29} +389106. i q^{30} -85185.7 q^{31} +85147.9i q^{32} +439270. i q^{33} +162553. q^{34} +31668.4 q^{35} -659872. q^{36} +90991.7i q^{37} +290393. q^{38} +397253. i q^{39} -602890. i q^{40} +42050.4i q^{41} +147691. i q^{42} +825275. q^{43} -1.47828e6i q^{44} -803494. q^{45} -224176. q^{46} +470759. i q^{47} +671362. q^{48} -811523. q^{49} -101444. i q^{50} +599571. i q^{51} -1.33688e6i q^{52} -1.14520e6i q^{53} -801140. i q^{54} -1.80002e6i q^{55} -228836. i q^{56} +1.07111e6i q^{57} -226116. i q^{58} +1.60549e6i q^{59} +4.82990e6 q^{60} +1.34053e6 q^{61} +1.62796e6i q^{62} -304979. q^{63} +2.84635e6 q^{64} -1.62785e6i q^{65} +8.39473e6 q^{66} +1.98728e6i q^{67} -2.01774e6i q^{68} -826866. i q^{69} -605204. i q^{70} +1.45817e6 q^{71} +5.80606e6i q^{72} -2.69579e6 q^{73} +1.73891e6 q^{74} +374172. q^{75} -3.60460e6i q^{76} -683227. i q^{77} +7.59178e6 q^{78} -2.99739e6i q^{79} -2.75109e6 q^{80} -3.12863e6 q^{81} +803611. q^{82} -9.05906e6 q^{83} +1.83327e6 q^{84} -2.45690e6i q^{85} -1.57716e7i q^{86} +834022. q^{87} -1.30070e7 q^{88} -5.17372e6 q^{89} +1.53553e7i q^{90} -617877. i q^{91} +2.78266e6i q^{92} -6.00467e6 q^{93} +8.99653e6 q^{94} -4.38915e6i q^{95} +6.00200e6i q^{96} -1.46853e7 q^{97} +1.55087e7i q^{98} +1.73349e7i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 62 q - 56 q^{3} - 3584 q^{4} + 194 q^{5} + 370 q^{7} + 40462 q^{9}+O(q^{10}) $ 62 * q - 56 * q^3 - 3584 * q^4 + 194 * q^5 + 370 * q^7 + 40462 * q^9	$ 62 q - 56 q^{3} - 3584 q^{4} + 194 q^{5} + 370 q^{7} + 40462 q^{9} + 6248 q^{12} - 21376 q^{15} + 227856 q^{16} - 216666 q^{20} + 42914 q^{21} - 101446 q^{22} + 788644 q^{25} - 282610 q^{26} - 235946 q^{27} - 69142 q^{28} - 213678 q^{29} + 300350 q^{31} - 401602 q^{34} - 377134 q^{35} - 2176902 q^{36} - 2421542 q^{38} + 2022550 q^{43} + 2223056 q^{45} + 2665174 q^{46} + 1060208 q^{48} + 7702792 q^{49} + 9894904 q^{60} + 278126 q^{61} - 6010300 q^{63} - 9362182 q^{64} - 23291020 q^{66} + 15373744 q^{71} - 1378934 q^{73} - 19097780 q^{74} - 41528062 q^{75} + 5705654 q^{78} + 42766270 q^{80} + 29291446 q^{81} + 14605464 q^{82} + 1191422 q^{83} - 6959552 q^{84} - 22121732 q^{87} + 6337872 q^{88} + 29157388 q^{89} + 14091936 q^{93} - 17482078 q^{94} - 22763314 q^{97}+O(q^{100}) $ 62 * q - 56 * q^3 - 3584 * q^4 + 194 * q^5 + 370 * q^7 + 40462 * q^9 + 6248 * q^12 - 21376 * q^15 + 227856 * q^16 - 216666 * q^20 + 42914 * q^21 - 101446 * q^22 + 788644 * q^25 - 282610 * q^26 - 235946 * q^27 - 69142 * q^28 - 213678 * q^29 + 300350 * q^31 - 401602 * q^34 - 377134 * q^35 - 2176902 * q^36 - 2421542 * q^38 + 2022550 * q^43 + 2223056 * q^45 + 2665174 * q^46 + 1060208 * q^48 + 7702792 * q^49 + 9894904 * q^60 + 278126 * q^61 - 6010300 * q^63 - 9362182 * q^64 - 23291020 * q^66 + 15373744 * q^71 - 1378934 * q^73 - 19097780 * q^74 - 41528062 * q^75 + 5705654 * q^78 + 42766270 * q^80 + 29291446 * q^81 + 14605464 * q^82 + 1191422 * q^83 - 6959552 * q^84 - 22121732 * q^87 + 6337872 * q^88 + 29157388 * q^89 + 14091936 * q^93 - 17482078 * q^94 - 22763314 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/109\mathbb{Z}\right)^\times$.

$n$	$6$
$\chi(n)$	$-1$

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 $

$2$		−	19.1107i		−	1.68916i	−0.535429	−	0.844580i	$-0.679850\pi$
$2$		−	19.1107i		−	1.68916i	0.535429	−	0.844580i	$-0.320150\pi$
$3$	70.4891			1.50729			0.753647	−	0.657280i	$-0.228293\pi$
$3$	70.4891			1.50729			0.753647	+	0.657280i	$0.228293\pi$
$4$	−237.218			−1.85326
$5$	−288.848			−1.03341			−0.516707	−	0.856162i	$-0.672842\pi$
$5$	−288.848			−1.03341			−0.516707	+	0.856162i	$0.672842\pi$
$6$		−	1347.09i		−	2.54606i
$7$	−109.637			−0.120813			−0.0604064	−	0.998174i	$-0.519240\pi$
$7$	−109.637			−0.120813			−0.0604064	+	0.998174i	$0.519240\pi$
$8$			2087.22i			1.44130i
$9$	2781.72			1.27193
$10$			5520.08i			1.74560i
$11$			6231.73i			1.41167i	0.708374	+	0.705837i	$0.249429\pi$
$11$			6231.73i			1.41167i	−0.708374	+	0.705837i	$0.750571\pi$
$12$	−16721.3			−2.79341
$13$			5635.67i			0.711449i	0.934591	+	0.355725i	$0.115766\pi$
$13$			5635.67i			0.711449i	−0.934591	+	0.355725i	$0.884234\pi$
$14$			2095.23i			0.204072i
$15$	−20360.7			−1.55766
$16$	9524.34			0.581319
$17$			8505.86i			0.419901i	0.977712	+	0.209951i	$0.0673303\pi$
$17$			8505.86i			0.419901i	−0.977712	+	0.209951i	$0.932670\pi$
$18$		−	53160.5i		−	2.14850i
$19$			15195.3i			0.508245i	0.967172	+	0.254122i	$0.0817866\pi$
$19$			15195.3i			0.508245i	−0.967172	+	0.254122i	$0.918213\pi$
$20$	68519.9			1.91519
$21$	−7728.20			−0.182100
$22$	119093.			2.38454
$23$		−	11730.4i		−	0.201032i	−0.994935	−	0.100516i	$-0.967951\pi$
$23$		−	11730.4i		−	0.201032i	0.994935	−	0.100516i	$-0.0320494\pi$
$24$			147126.i			2.17246i
$25$	5308.23			0.0679453
$26$	107701.			1.20175
$27$	41921.1			0.409883
$28$	26007.8			0.223898
$29$	11831.9			0.0900870			0.0450435	−	0.998985i	$-0.485657\pi$
$29$	11831.9			0.0900870			0.0450435	+	0.998985i	$0.485657\pi$
$30$			389106.i			2.63113i
$31$	−85185.7			−0.513571			−0.256786	−	0.966468i	$-0.582663\pi$
$31$	−85185.7			−0.513571			−0.256786	+	0.966468i	$0.582663\pi$
$32$			85147.9i			0.459356i
$33$			439270.i			2.12781i
$34$	162553.			0.709280
$35$	31668.4			0.124850
$36$	−659872.			−2.35723
$37$			90991.7i			0.295322i	0.989038	+	0.147661i	$0.0471744\pi$
$37$			90991.7i			0.295322i	−0.989038	+	0.147661i	$0.952826\pi$
$38$	290393.			0.858507
$39$			397253.i			1.07236i
$40$		−	602890.i		−	1.48946i
$41$			42050.4i			0.0952854i	0.998864	+	0.0476427i	$0.0151709\pi$
$41$			42050.4i			0.0952854i	−0.998864	+	0.0476427i	$0.984829\pi$
$42$			147691.i			0.307597i
$43$	825275.			1.58292			0.791460	−	0.611221i	$-0.209321\pi$
$43$	825275.			1.58292			0.791460	+	0.611221i	$0.209321\pi$
$44$		−	1.47828e6i		−	2.61620i
$45$	−803494.			−1.31443
$46$	−224176.			−0.339575
$47$			470759.i			0.661388i	0.943738	+	0.330694i	$0.107283\pi$
$47$			470759.i			0.661388i	−0.943738	+	0.330694i	$0.892717\pi$
$48$	671362.			0.876219
$49$	−811523.			−0.985404
$50$		−	101444.i		−	0.114771i
$51$			599571.i			0.632914i
$52$		−	1.33688e6i		−	1.31850i
$53$		−	1.14520e6i		−	1.05661i	−0.849055	−	0.528305i	$-0.822828\pi$
$53$		−	1.14520e6i		−	1.05661i	0.849055	−	0.528305i	$-0.177172\pi$
$54$		−	801140.i		−	0.692358i
$55$		−	1.80002e6i		−	1.45884i
$56$		−	228836.i		−	0.174127i
$57$			1.07111e6i			0.766074i
$58$		−	226116.i		−	0.152171i
$59$			1.60549e6i			1.01771i	0.860852	+	0.508856i	$0.169931\pi$
$59$			1.60549e6i			1.01771i	−0.860852	+	0.508856i	$0.830069\pi$
$60$	4.82990e6			2.88675
$61$	1.34053e6			0.756173			0.378087	−	0.925770i	$-0.376582\pi$
$61$	1.34053e6			0.756173			0.378087	+	0.925770i	$0.376582\pi$
$62$			1.62796e6i			0.867505i
$63$	−304979.			−0.153666
$64$	2.84635e6			1.35724
$65$		−	1.62785e6i		−	0.735222i
$66$	8.39473e6			3.59421
$67$			1.98728e6i			0.807229i	0.914929	+	0.403614i	$0.132246\pi$
$67$			1.98728e6i			0.807229i	−0.914929	+	0.403614i	$0.867754\pi$
$68$		−	2.01774e6i		−	0.778187i
$69$		−	826866.i		−	0.303014i
$70$		−	605204.i		−	0.210891i
$71$	1.45817e6			0.483510			0.241755	−	0.970337i	$-0.422277\pi$
$71$	1.45817e6			0.483510			0.241755	+	0.970337i	$0.422277\pi$
$72$			5.80606e6i			1.83323i
$73$	−2.69579e6			−0.811064			−0.405532	−	0.914081i	$-0.632914\pi$
$73$	−2.69579e6			−0.811064			−0.405532	+	0.914081i	$0.632914\pi$
$74$	1.73891e6			0.498846
$75$	374172.			0.102414
$76$		−	3.60460e6i		−	0.941911i
$77$		−	683227.i		−	0.170548i
$78$	7.59178e6			1.81139
$79$		−	2.99739e6i		−	0.683989i	−0.939702	−	0.341994i	$-0.888898\pi$
$79$		−	2.99739e6i		−	0.683989i	0.939702	−	0.341994i	$-0.111102\pi$
$80$	−2.75109e6			−0.600744
$81$	−3.12863e6			−0.654119
$82$	803611.			0.160952
$83$	−9.05906e6			−1.73904			−0.869520	−	0.493898i	$-0.835572\pi$
$83$	−9.05906e6			−1.73904			−0.869520	+	0.493898i	$0.835572\pi$
$84$	1.83327e6			0.337480
$85$		−	2.45690e6i		−	0.433932i
$86$		−	1.57716e7i		−	2.67381i
$87$	834022.			0.135788
$88$	−1.30070e7			−2.03464
$89$	−5.17372e6			−0.777925			−0.388962	−	0.921254i	$-0.627166\pi$
$89$	−5.17372e6			−0.777925			−0.388962	+	0.921254i	$0.627166\pi$
$90$			1.53553e7i			2.22029i
$91$		−	617877.i		−	0.0859522i
$92$			2.78266e6i			0.372565i
$93$	−6.00467e6			−0.774103
$94$	8.99653e6			1.11719
$95$		−	4.38915e6i		−	0.525228i
$96$			6.00200e6i			0.692384i
$97$	−1.46853e7			−1.63374			−0.816868	−	0.576825i	$-0.804292\pi$
$97$	−1.46853e7			−1.63374			−0.816868	+	0.576825i	$0.804292\pi$
$98$			1.55087e7i			1.66451i
$99$			1.73349e7i			1.79556i
$100$	−1.25920e6			−0.125920
$101$			1.70701e7i			1.64859i	0.566164	+	0.824293i	$0.308427\pi$
$101$			1.70701e7i			1.64859i	−0.566164	+	0.824293i	$0.691573\pi$
$102$	1.14582e7			1.06909
$103$			8.55768e6i			0.771660i	0.922570	+	0.385830i	$0.126085\pi$
$103$			8.55768e6i			0.771660i	−0.922570	+	0.385830i	$0.873915\pi$
$104$	−1.17629e7			−1.02541
$105$	2.23228e6			0.188185
$106$	−2.18855e7			−1.78478
$107$		−	8.17253e6i		−	0.644931i	−0.946581	−	0.322466i	$-0.895488\pi$
$107$		−	8.17253e6i		−	0.644931i	0.946581	−	0.322466i	$-0.104512\pi$
$108$	−9.94443e6			−0.759620
$109$	1.34581e7	−	1.29795e6i	0.995381	−	0.0959985i
$109$	1.34581e7	−	1.29795e6i	0.995381	−	0.0959985i
$110$	−3.43997e7			−2.46422
$111$			6.41393e6i			0.445137i
$112$	−1.04422e6			−0.0702309
$113$	−8.65189e6			−0.564074			−0.282037	−	0.959403i	$-0.591010\pi$
$113$	−8.65189e6			−0.564074			−0.282037	+	0.959403i	$0.591010\pi$
$114$	2.04696e7			1.29402
$115$			3.38830e6i			0.207749i
$116$	−2.80674e6			−0.166955
$117$			1.56768e7i			0.904915i
$118$	3.06819e7			1.71908
$119$		−	932555.i		−	0.0507295i
$120$		−	4.24972e7i		−	2.24505i
$121$	−1.93473e7			−0.992824
$122$		−	2.56184e7i		−	1.27730i
$123$			2.96409e6i			0.143623i
$124$	2.02076e7			0.951783
$125$	2.10330e7			0.963199
$126$			5.82835e6i			0.259566i
$127$			1.62596e7i			0.704362i	0.935932	+	0.352181i	$0.114560\pi$
$127$			1.62596e7i			0.704362i	−0.935932	+	0.352181i	$0.885440\pi$
$128$		−	4.34967e7i		−	1.83325i
$129$	5.81729e7			2.38593
$130$	−3.11093e7			−1.24191
$131$	−1.89113e7			−0.734974			−0.367487	−	0.930029i	$-0.619782\pi$
$131$	−1.89113e7			−0.734974			−0.367487	+	0.930029i	$0.619782\pi$
$132$		−	1.04202e8i		−	3.94339i
$133$		−	1.66597e6i		−	0.0614025i
$134$	3.79782e7			1.36354
$135$	−1.21088e7			−0.423579
$136$	−1.77536e7			−0.605202
$137$	−72067.0			−0.00239450			−0.00119725	−	0.999999i	$-0.500381\pi$
$137$	−72067.0			−0.00239450			−0.00119725	+	0.999999i	$0.500381\pi$
$138$	−1.58020e7			−0.511840
$139$			5.07596e7i			1.60312i	0.597913	+	0.801561i	$0.295997\pi$
$139$			5.07596e7i			1.60312i	−0.597913	+	0.801561i	$0.704003\pi$
$140$	−7.51230e6			−0.231379
$141$			3.31834e7i			0.996906i
$142$		−	2.78667e7i		−	0.816725i
$143$	−3.51200e7			−1.00433
$144$	2.64940e7			0.739399
$145$	−3.41763e6			−0.0930972
$146$			5.15183e7i			1.37002i
$147$	−5.72035e7			−1.48529
$148$		−	2.15848e7i		−	0.547309i
$149$		−	1.62383e7i		−	0.402151i	−0.979576	−	0.201075i	$-0.935556\pi$
$149$		−	1.62383e7i		−	0.402151i	0.979576	−	0.201075i	$-0.0644436\pi$
$150$		−	7.15068e6i		−	0.172993i
$151$			7.16484e7i			1.69351i	0.531985	+	0.846754i	$0.321446\pi$
$151$			7.16484e7i			1.69351i	−0.531985	+	0.846754i	$0.678554\pi$
$152$	−3.17160e7			−0.732532
$153$			2.36609e7i			0.534086i
$154$	−1.30569e7			−0.288084
$155$	2.46057e7			0.530732
$156$		−	9.42355e7i		−	1.98737i
$157$	8.88389e7			1.83212			0.916061	−	0.401039i	$-0.131351\pi$
$157$	8.88389e7			1.83212			0.916061	+	0.401039i	$0.131351\pi$
$158$	−5.72822e7			−1.15537
$159$		−	8.07239e7i		−	1.59262i
$160$		−	2.45948e7i		−	0.474705i
$161$			1.28608e6i			0.0242873i
$162$			5.97903e7i			1.10491i
$163$		−	3.77303e7i		−	0.682391i	−0.939992	−	0.341196i	$-0.889168\pi$
$163$		−	3.77303e7i		−	0.682391i	0.939992	−	0.341196i	$-0.110832\pi$
$164$		−	9.97509e6i		−	0.176589i
$165$		−	1.26882e8i		−	2.19891i
$166$			1.73125e8i			2.93752i
$167$		−	8.54862e7i		−	1.42033i	−0.704037	−	0.710164i	$-0.748621\pi$
$167$		−	8.54862e7i		−	1.42033i	0.704037	−	0.710164i	$-0.251379\pi$
$168$		−	1.61305e7i		−	0.262461i
$169$	3.09877e7			0.493840
$170$	−4.69530e7			−0.732980
$171$			4.22692e7i			0.646453i
$172$	−1.95770e8			−2.93357
$173$	−7.69749e7			−1.13028			−0.565142	−	0.824994i	$-0.691179\pi$
$173$	−7.69749e7			−1.13028			−0.565142	+	0.824994i	$0.691179\pi$
$174$		−	1.59387e7i		−	0.229367i
$175$	−581977.			−0.00820867
$176$			5.93531e7i			0.820634i
$177$			1.13169e8i			1.53399i
$178$			9.88732e7i			1.31404i
$179$		−	1.16022e8i		−	1.51201i	−0.654565	−	0.756006i	$-0.727148\pi$
$179$		−	1.16022e8i		−	1.51201i	0.654565	−	0.756006i	$-0.272852\pi$
$180$	1.90603e8			2.43599
$181$			5.42072e7i			0.679488i	0.940518	+	0.339744i	$0.110340\pi$
$181$			5.42072e7i			0.679488i	−0.940518	+	0.339744i	$0.889660\pi$
$182$	−1.18080e7			−0.145187
$183$	9.44926e7			1.13977
$184$	2.44839e7			0.289747
$185$		−	2.62828e7i		−	0.305190i
$186$			1.14753e8i			1.30758i
$187$	−5.30063e7			−0.592764
$188$		−	1.11672e8i		−	1.22573i
$189$	−4.59610e6			−0.0495191
$190$	−8.38795e7			−0.887194
$191$	−1.07781e8			−1.11924			−0.559622	−	0.828748i	$-0.689054\pi$
$191$	−1.07781e8			−1.11924			−0.559622	+	0.828748i	$0.689054\pi$
$192$	2.00637e8			2.04577
$193$	−1.26358e7			−0.126518			−0.0632589	−	0.997997i	$-0.520149\pi$
$193$	−1.26358e7			−0.126518			−0.0632589	+	0.997997i	$0.520149\pi$
$194$			2.80646e8i			2.75964i
$195$		−	1.14746e8i		−	1.10819i
$196$	1.92507e8			1.82621
$197$	1.79992e8			1.67734			0.838670	−	0.544640i	$-0.183334\pi$
$197$	1.79992e8			1.67734			0.838670	+	0.544640i	$0.183334\pi$
$198$	3.31282e8			3.03298
$199$			1.05569e8i			0.949622i	0.880088	+	0.474811i	$0.157484\pi$
$199$			1.05569e8i			0.949622i	−0.880088	+	0.474811i	$0.842516\pi$
$200$			1.10794e7i			0.0979294i
$201$			1.40081e8i			1.21673i
$202$	3.26221e8			2.78472
$203$	−1.29721e6			−0.0108837
$204$		−	1.42229e8i		−	1.17296i
$205$		−	1.21462e7i		−	0.0984693i
$206$	1.63543e8			1.30346
$207$		−	3.26307e7i		−	0.255699i
$208$			5.36760e7i			0.413579i
$209$	−9.46934e7			−0.717476
$210$		−	4.26603e7i		−	0.317875i
$211$	−1.23048e8			−0.901750			−0.450875	−	0.892587i	$-0.648888\pi$
$211$	−1.23048e8			−0.901750			−0.450875	+	0.892587i	$0.648888\pi$
$212$			2.71661e8i			1.95818i
$213$	1.02785e8			0.728791
$214$	−1.56183e8			−1.08939
$215$	−2.38379e8			−1.63581
$216$			8.74986e7i			0.590763i
$217$	9.33949e6			0.0620461
$218$	−2.48046e7	−	2.57192e8i	−0.162157	−	1.68136i
$219$	−1.90024e8			−1.22251
$220$			4.26998e8i			2.70362i
$221$	−4.79362e7			−0.298738
$222$	1.22574e8			0.751907
$223$	−9.22012e7			−0.556762			−0.278381	−	0.960471i	$-0.589798\pi$
$223$	−9.22012e7			−0.556762			−0.278381	+	0.960471i	$0.589798\pi$
$224$		−	9.33534e6i		−	0.0554961i
$225$	1.47660e7			0.0864219
$226$			1.65343e8i			0.952812i
$227$	1.89125e7			0.107315			0.0536574	−	0.998559i	$-0.482912\pi$
$227$	1.89125e7			0.107315			0.0536574	+	0.998559i	$0.482912\pi$
$228$		−	2.54085e8i		−	1.41974i
$229$		−	3.01779e8i		−	1.66060i	−0.557316	−	0.830301i	$-0.688169\pi$
$229$		−	3.01779e8i		−	1.66060i	0.557316	−	0.830301i	$-0.311831\pi$
$230$	6.47527e7			0.350922
$231$		−	4.81601e7i		−	0.257067i
$232$			2.46958e7i			0.129842i
$233$	2.89056e8			1.49705			0.748526	−	0.663105i	$-0.230762\pi$
$233$	2.89056e8			1.49705			0.748526	+	0.663105i	$0.230762\pi$
$234$	2.99595e8			1.52855
$235$		−	1.35978e8i		−	0.683488i
$236$		−	3.80850e8i		−	1.88609i
$237$		−	2.11284e8i		−	1.03097i
$238$	−1.78218e7			−0.0856902
$239$	−8.49929e6			−0.0402708			−0.0201354	−	0.999797i	$-0.506410\pi$
$239$	−8.49929e6			−0.0402708			−0.0201354	+	0.999797i	$0.506410\pi$
$240$	−1.93922e8			−0.905497
$241$			1.54669e8i			0.711777i	0.934528	+	0.355888i	$0.115822\pi$
$241$			1.54669e8i			0.711777i	−0.934528	+	0.355888i	$0.884178\pi$
$242$			3.69741e8i			1.67704i
$243$	−3.12216e8			−1.39583
$244$	−3.17997e8			−1.40139
$245$	2.34407e8			1.01833
$246$	5.66458e7			0.242602
$247$	−8.56359e7			−0.361590
$248$		−	1.77801e8i		−	0.740209i
$249$	−6.38565e8			−2.62124
$250$		−	4.01954e8i		−	1.62700i
$251$		−	1.56014e8i		−	0.622739i	−0.950289	−	0.311370i	$-0.899212\pi$
$251$		−	1.56014e8i		−	0.622739i	0.950289	−	0.311370i	$-0.100788\pi$
$252$	7.23463e7			0.284783
$253$	7.31007e7			0.283792
$254$	3.10731e8			1.18978
$255$		−	1.73185e8i		−	0.654063i
$256$	−4.66918e8			−1.73941
$257$		−	3.42159e8i		−	1.25737i	−0.777661	−	0.628683i	$-0.783594\pi$
$257$		−	3.42159e8i		−	1.25737i	0.777661	−	0.628683i	$-0.216406\pi$
$258$		−	1.11172e9i		−	4.03021i
$259$		−	9.97604e6i		−	0.0356787i
$260$			3.86155e8i			1.36256i
$261$	3.29131e7			0.114585
$262$			3.61408e8i			1.24149i
$263$	−3.92394e7			−0.133008			−0.0665040	−	0.997786i	$-0.521185\pi$
$263$	−3.92394e7			−0.133008			−0.0665040	+	0.997786i	$0.521185\pi$
$264$	−9.16853e8			−3.06680
$265$			3.30788e8i			1.09192i
$266$	−3.18378e7			−0.103719
$267$	−3.64691e8			−1.17256
$268$		−	4.71417e8i		−	1.49601i
$269$		−	5.32048e8i		−	1.66655i	−0.552861	−	0.833274i	$-0.686464\pi$
$269$		−	5.32048e8i		−	1.66655i	0.552861	−	0.833274i	$-0.313536\pi$
$270$			2.31408e8i			0.715492i
$271$			2.71211e8i			0.827779i	0.910327	+	0.413889i	$0.135830\pi$
$271$			2.71211e8i			0.827779i	−0.910327	+	0.413889i	$0.864170\pi$
$272$			8.10127e7i			0.244097i
$273$		−	4.35536e7i		−	0.129555i
$274$			1.37725e6i			0.00404469i
$275$			3.30795e7i			0.0959167i
$276$			1.96147e8i			0.561565i
$277$			6.05389e8i			1.71142i	0.517459	+	0.855708i	$0.326878\pi$
$277$			6.05389e8i			1.71142i	−0.517459	+	0.855708i	$0.673122\pi$
$278$	9.70050e8			2.70793
$279$	−2.36963e8			−0.653228
$280$			6.60989e7i			0.179946i
$281$	1.11966e8			0.301034			0.150517	−	0.988607i	$-0.451906\pi$
$281$	1.11966e8			0.301034			0.150517	+	0.988607i	$0.451906\pi$
$282$	6.34157e8			1.68393
$283$		−	1.19684e8i		−	0.313895i	−0.987607	−	0.156948i	$-0.949835\pi$
$283$		−	1.19684e8i		−	0.313895i	0.987607	−	0.156948i	$-0.0501654\pi$
$284$	−3.45905e8			−0.896070
$285$		−	3.09387e8i		−	0.791672i
$286$			6.71167e8i			1.69648i
$287$		−	4.61027e6i		−	0.0115117i
$288$			2.36857e8i			0.584270i
$289$	3.37989e8			0.823683
$290$			6.53131e7i			0.157256i
$291$	−1.03515e9			−2.46252
$292$	6.39488e8			1.50312
$293$	4.34023e8			1.00804			0.504018	−	0.863693i	$-0.331854\pi$
$293$	4.34023e8			1.00804			0.504018	+	0.863693i	$0.331854\pi$
$294$			1.09320e9i			2.50890i
$295$		−	4.63742e8i		−	1.05172i
$296$	−1.89920e8			−0.425647
$297$			2.61241e8i			0.578621i
$298$	−3.10325e8			−0.679297
$299$	6.61086e7			0.143024
$300$	−8.87603e7			−0.189799
$301$	−9.04805e7			−0.191237
$302$	1.36925e9			2.86061
$303$			1.20326e9i			2.48490i
$304$			1.44726e8i			0.295453i
$305$	−3.87209e8			−0.781440
$306$	4.52176e8			0.902157
$307$	−1.65036e8			−0.325533			−0.162767	−	0.986665i	$-0.552042\pi$
$307$	−1.65036e8			−0.325533			−0.162767	+	0.986665i	$0.552042\pi$
$308$			1.62074e8i			0.316071i
$309$			6.03223e8i			1.16312i
$310$		−	4.70232e8i		−	0.896492i
$311$	2.39286e7			0.0451082			0.0225541	−	0.999746i	$-0.492820\pi$
$311$	2.39286e7			0.0451082			0.0225541	+	0.999746i	$0.492820\pi$
$312$	−8.29156e8			−1.54559
$313$		−	7.51264e8i		−	1.38480i	−0.721514	−	0.692400i	$-0.756553\pi$
$313$		−	7.51264e8i		−	1.38480i	0.721514	−	0.692400i	$-0.243447\pi$
$314$		−	1.69777e9i		−	3.09475i
$315$	8.80925e7			0.158801
$316$			7.11034e8i			1.26761i
$317$		−	2.16105e8i		−	0.381029i	−0.981684	−	0.190514i	$-0.938984\pi$
$317$		−	2.16105e8i		−	0.381029i	0.981684	−	0.190514i	$-0.0610156\pi$
$318$	−1.54269e9			−2.69019
$319$			7.37334e7i			0.127174i
$320$	−8.22162e8			−1.40260
$321$		−	5.76075e8i		−	0.972101i
$322$	2.45779e7			0.0410251
$323$	−1.29249e8			−0.213413
$324$	7.42167e8			1.21226
$325$			2.99154e7i			0.0483396i
$326$	−7.21051e8			−1.15267
$327$	9.48647e8	−	9.14912e7i	1.50033	−	0.144698i
$328$	−8.77684e7			−0.137335
$329$		−	5.16126e7i		−	0.0799042i
$330$	−2.42480e9			−3.71431
$331$	1.07419e8			0.162811			0.0814056	−	0.996681i	$-0.474059\pi$
$331$	1.07419e8			0.162811			0.0814056	+	0.996681i	$0.474059\pi$
$332$	2.14897e9			3.22290
$333$			2.53113e8i			0.375630i
$334$	−1.63370e9			−2.39916
$335$		−	5.74021e8i		−	0.834202i
$336$	−7.36060e7			−0.105859
$337$			5.32067e8i			0.757289i	0.925542	+	0.378644i	$0.123610\pi$
$337$			5.32067e8i			0.757289i	−0.925542	+	0.378644i	$0.876390\pi$
$338$		−	5.92196e8i		−	0.834175i
$339$	−6.09864e8			−0.850225
$340$			5.82820e8i			0.804190i
$341$		−	5.30855e8i		−	0.724996i
$342$	8.07792e8			1.09196
$343$	1.79263e8			0.239862
$344$			1.72253e9i			2.28146i
$345$			2.38839e8i			0.313139i
$346$			1.47104e9i			1.90923i
$347$	1.44182e9			1.85249			0.926247	−	0.376918i	$-0.123016\pi$
$347$	1.44182e9			1.85249			0.926247	+	0.376918i	$0.123016\pi$
$348$	−1.97845e8			−0.251650
$349$	9.53241e8			1.20037			0.600183	−	0.799863i	$-0.295094\pi$
$349$	9.53241e8			1.20037			0.600183	+	0.799863i	$0.295094\pi$
$350$			1.11220e7i			0.0138658i
$351$			2.36254e8i			0.291611i
$352$	−5.30619e8			−0.648461
$353$	−2.62707e7			−0.0317877			−0.0158939	−	0.999874i	$-0.505059\pi$
$353$	−2.62707e7			−0.0317877			−0.0158939	+	0.999874i	$0.505059\pi$
$354$	2.16274e9			2.59115
$355$	−4.21191e8			−0.499666
$356$	1.22730e9			1.44170
$357$		−	6.57350e7i		−	0.0764642i
$358$	−2.21726e9			−2.55403
$359$			1.96193e8i			0.223796i	0.993720	+	0.111898i	$0.0356930\pi$
$359$			1.96193e8i			0.223796i	−0.993720	+	0.111898i	$0.964307\pi$
$360$		−	1.67707e9i		−	1.89449i
$361$	6.62973e8			0.741687
$362$	1.03594e9			1.14776
$363$	−1.36378e9			−1.49648
$364$			1.46571e8i			0.159292i
$365$	7.78673e8			0.838166
$366$		−	1.80582e9i		−	1.92526i
$367$			4.78670e7i			0.0505481i	0.999681	+	0.0252741i	$0.00804584\pi$
$367$			4.78670e7i			0.0505481i	−0.999681	+	0.0252741i	$0.991954\pi$
$368$		−	1.11724e8i		−	0.116864i
$369$			1.16972e8i			0.121197i
$370$	−5.02281e8			−0.515515
$371$			1.25556e8i			0.127652i
$372$	1.42441e9			1.43462
$373$	−1.03742e9			−1.03508			−0.517541	−	0.855658i	$-0.673153\pi$
$373$	−1.03742e9			−1.03508			−0.517541	+	0.855658i	$0.673153\pi$
$374$			1.01298e9i			1.00127i
$375$	1.48260e9			1.45182
$376$	−9.82579e8			−0.953257
$377$			6.66808e7i			0.0640923i
$378$			8.78345e7i			0.0836457i
$379$			9.82124e7i			0.0926678i	0.998926	+	0.0463339i	$0.0147538\pi$
$379$			9.82124e7i			0.0926678i	−0.998926	+	0.0463339i	$0.985246\pi$
$380$			1.04118e9i			0.973385i
$381$			1.14612e9i			1.06168i
$382$			2.05976e9i			1.89058i
$383$			2.75613e8i			0.250671i	0.992114	+	0.125336i	$0.0400007\pi$
$383$			2.75613e8i			0.250671i	−0.992114	+	0.125336i	$0.959999\pi$
$384$		−	3.06604e9i		−	2.76324i
$385$			1.97349e8i			0.176247i
$386$			2.41478e8i			0.213709i
$387$	2.29568e9			2.01337
$388$	3.48361e9			3.02774
$389$		−	2.37877e8i		−	0.204894i	−0.994738	−	0.102447i	$-0.967333\pi$
$389$		−	2.37877e8i		−	0.204894i	0.994738	−	0.102447i	$-0.0326672\pi$
$390$	−2.19287e9			−1.87192
$391$	9.97771e7			0.0844136
$392$		−	1.69383e9i		−	1.42026i
$393$	−1.33304e9			−1.10782
$394$		−	3.43977e9i		−	2.83330i
$395$			8.65791e8i			0.706844i
$396$		−	4.11215e9i		−	3.32763i
$397$		−	1.60162e9i		−	1.28467i	−0.766424	−	0.642335i	$-0.777966\pi$
$397$		−	1.60162e9i		−	1.28467i	0.766424	−	0.642335i	$-0.222034\pi$
$398$	2.01749e9			1.60406
$399$		−	1.17433e8i		−	0.0925516i
$400$	5.05573e7			0.0394979
$401$	−2.20747e9			−1.70958			−0.854790	−	0.518974i	$-0.826314\pi$
$401$	−2.20747e9			−1.70958			−0.854790	+	0.518974i	$0.826314\pi$
$402$	2.67705e9			2.05525
$403$		−	4.80079e8i		−	0.365380i
$404$		−	4.04933e9i		−	3.05526i
$405$	9.03700e8			0.675976
$406$			2.47906e7i			0.0183843i
$407$	−5.67036e8			−0.416898
$408$	−1.25144e9			−0.912217
$409$	6.35277e8			0.459126			0.229563	−	0.973294i	$-0.426270\pi$
$409$	6.35277e8			0.459126			0.229563	+	0.973294i	$0.426270\pi$
$410$	−2.32121e8			−0.166330
$411$	−5.07994e6			−0.00360921
$412$		−	2.03003e9i		−	1.43009i
$413$		−	1.76020e8i		−	0.122953i
$414$	−6.23594e8			−0.431917
$415$	2.61669e9			1.79715
$416$	−4.79865e8			−0.326808
$417$			3.57800e9i			2.41637i
$418$			1.80965e9i			1.21193i
$419$			1.65017e8i			0.109592i	0.998498	+	0.0547960i	$0.0174508\pi$
$419$			1.65017e8i			0.109592i	−0.998498	+	0.0547960i	$0.982549\pi$
$420$	−5.29535e8			−0.348757
$421$	−1.43180e9			−0.935183			−0.467591	−	0.883945i	$-0.654878\pi$
$421$	−1.43180e9			−0.935183			−0.467591	+	0.883945i	$0.654878\pi$
$422$			2.35153e9i			1.52320i
$423$			1.30952e9i			0.841242i
$424$	2.39028e9			1.52289
$425$			4.51510e7i			0.0285303i
$426$		−	1.96430e9i		−	1.23104i
$427$	−1.46971e8			−0.0913555
$428$			1.93867e9i			1.19523i
$429$	−2.47558e9			−1.51383
$430$			4.55558e9i			2.76315i
$431$	−3.89030e8			−0.234052			−0.117026	−	0.993129i	$-0.537336\pi$
$431$	−3.89030e8			−0.234052			−0.117026	+	0.993129i	$0.537336\pi$
$432$	3.99271e8			0.238273
$433$	1.66592e9			0.986157			0.493079	−	0.869985i	$-0.335871\pi$
$433$	1.66592e9			0.986157			0.493079	+	0.869985i	$0.335871\pi$
$434$		−	1.78484e8i		−	0.104806i
$435$	−2.40906e8			−0.140325
$436$	−3.19249e9	+	3.07896e8i	−1.84470	+	0.177910i
$437$	1.78247e8			0.102174
$438$			3.63148e9i			2.06502i
$439$	−8.16250e8			−0.460466			−0.230233	−	0.973136i	$-0.573949\pi$
$439$	−8.16250e8			−0.460466			−0.230233	+	0.973136i	$0.573949\pi$
$440$	3.75705e9			2.10263
$441$	−2.25743e9			−1.25337
$442$			9.16093e8i			0.504617i
$443$	−6.67662e8			−0.364875			−0.182437	−	0.983217i	$-0.558399\pi$
$443$	−6.67662e8			−0.364875			−0.182437	+	0.983217i	$0.558399\pi$
$444$		−	1.52150e9i		−	0.824955i
$445$	1.49442e9			0.803919
$446$			1.76203e9i			0.940460i
$447$		−	1.14462e9i		−	0.606159i
$448$	−3.12065e8			−0.163973
$449$			3.01199e9i			1.57033i	0.619287	+	0.785165i	$0.287422\pi$
$449$			3.01199e9i			1.57033i	−0.619287	+	0.785165i	$0.712578\pi$
$450$		−	2.82188e8i		−	0.145980i
$451$	−2.62047e8			−0.134512
$452$	2.05238e9			1.04538
$453$			5.05043e9i			2.55261i
$454$		−	3.61431e8i		−	0.181272i
$455$			1.78473e8i			0.0888243i
$456$	−2.23564e9			−1.10414
$457$	2.94053e9			1.44118			0.720591	−	0.693361i	$-0.243871\pi$
$457$	2.94053e9			1.44118			0.720591	+	0.693361i	$0.243871\pi$
$458$	−5.76720e9			−2.80502
$459$			3.56575e8i			0.172110i
$460$		−	8.03765e8i		−	0.385014i
$461$	−1.21988e9			−0.579916			−0.289958	−	0.957039i	$-0.593641\pi$
$461$	−1.21988e9			−0.579916			−0.289958	+	0.957039i	$0.593641\pi$
$462$	−9.20372e8			−0.434227
$463$	1.25892e9			0.589476			0.294738	−	0.955578i	$-0.404768\pi$
$463$	1.25892e9			0.589476			0.294738	+	0.955578i	$0.404768\pi$
$464$	1.12691e8			0.0523693
$465$	1.73444e9			0.799969
$466$		−	5.52406e9i		−	2.52876i
$467$	−2.64923e9			−1.20368			−0.601840	−	0.798617i	$-0.705565\pi$
$467$	−2.64923e9			−1.20368			−0.601840	+	0.798617i	$0.705565\pi$
$468$		−	3.71882e9i		−	1.67705i
$469$		−	2.17879e8i		−	0.0975236i
$470$	−2.59863e9			−1.15452
$471$	6.26218e9			2.76155
$472$	−3.35100e9			−1.46682
$473$			5.14289e9i			2.23457i
$474$	−4.03777e9			−1.74148
$475$			8.06604e7i			0.0345329i
$476$			2.21219e8i			0.0940150i
$477$		−	3.18561e9i		−	1.34394i
$478$			1.62427e8i			0.0680238i
$479$	2.57179e9			1.06921			0.534603	−	0.845103i	$-0.320461\pi$
$479$	2.57179e9			1.06921			0.534603	+	0.845103i	$0.320461\pi$
$480$		−	1.73367e9i		−	0.715519i
$481$	−5.12799e8			−0.210107
$482$	2.95583e9			1.20230
$483$			9.06549e7i			0.0366080i
$484$	4.58953e9			1.83996
$485$	4.24182e9			1.68833
$486$			5.96666e9i			2.35779i
$487$			2.30168e8i			0.0903012i	0.998980	+	0.0451506i	$0.0143768\pi$
$487$			2.30168e8i			0.0903012i	−0.998980	+	0.0451506i	$0.985623\pi$
$488$			2.79798e9i			1.08987i
$489$		−	2.65957e9i		−	1.02856i
$490$		−	4.47967e9i		−	1.72012i
$491$			1.79156e9i			0.683040i	0.939874	+	0.341520i	$0.110942\pi$
$491$			1.79156e9i			0.683040i	−0.939874	+	0.341520i	$0.889058\pi$
$492$		−	7.03135e8i		−	0.266171i
$493$			1.00641e8i			0.0378276i
$494$			1.63656e9i			0.610784i
$495$		−	5.00716e9i		−	1.85555i
$496$	−8.11337e8			−0.298549
$497$	−1.59870e8			−0.0584142
$498$			1.22034e10i			4.42770i
$499$	−2.47034e9			−0.890032			−0.445016	−	0.895523i	$-0.646802\pi$
$499$	−2.47034e9			−0.890032			−0.445016	+	0.895523i	$0.646802\pi$
$500$	−4.98939e9			−1.78506
$501$		−	6.02585e9i		−	2.14085i
$502$	−2.98154e9			−1.05191
$503$		−	4.16423e8i		−	0.145897i	−0.997336	−	0.0729485i	$-0.976759\pi$
$503$		−	4.16423e8i		−	0.145897i	0.997336	−	0.0729485i	$-0.0232409\pi$
$504$		−	6.36558e8i		−	0.221478i
$505$		−	4.93067e9i		−	1.70367i
$506$		−	1.39700e9i		−	0.479370i
$507$	2.18430e9			0.744362
$508$		−	3.85705e9i		−	1.30537i
$509$	−3.46676e9			−1.16523			−0.582615	−	0.812748i	$-0.697970\pi$
$509$	−3.46676e9			−1.16523			−0.582615	+	0.812748i	$0.697970\pi$
$510$	−3.30968e9			−1.10482
$511$	2.95557e8			0.0979870
$512$			3.35554e9i			1.10489i
$513$			6.37006e8i			0.208321i
$514$	−6.53889e9			−2.12389
$515$		−	2.47187e9i		−	0.797444i
$516$	−1.37996e10			−4.42175
$517$	−2.93365e9			−0.933665
$518$	−1.90649e8			−0.0602670
$519$	−5.42589e9			−1.70367
$520$	3.39769e9			1.05967
$521$		−	4.01889e9i		−	1.24501i	−0.782614	−	0.622507i	$-0.786114\pi$
$521$		−	4.01889e9i		−	1.24501i	0.782614	−	0.622507i	$-0.213886\pi$
$522$		−	6.28991e8i		−	0.193552i
$523$	1.48260e9			0.453178			0.226589	−	0.973990i	$-0.427243\pi$
$523$	1.48260e9			0.453178			0.226589	+	0.973990i	$0.427243\pi$
$524$	4.48610e9			1.36210
$525$	−4.10231e7			−0.0123729
$526$			7.49892e8i			0.224672i
$527$		−	7.24578e8i		−	0.215649i
$528$			4.18375e9i			1.23694i
$529$	3.26722e9			0.959586
$530$	6.32158e9			1.84442
$531$			4.46601e9i			1.29446i
$532$			3.95197e8i			0.113795i
$533$	−2.36982e8			−0.0677907
$534$			6.96948e9i			1.98064i
$535$			2.36062e9i			0.666481i
$536$	−4.14789e9			−1.16346
$537$		−	8.17830e9i		−	2.27905i
$538$	−1.01678e10			−2.81507
$539$		−	5.05719e9i		−	1.39107i
$540$	2.87243e9			0.785003
$541$	3.91359e9			1.06264			0.531319	−	0.847172i	$-0.321697\pi$
$541$	3.91359e9			1.06264			0.531319	+	0.847172i	$0.321697\pi$
$542$	5.18302e9			1.39825
$543$			3.82102e9i			1.02419i
$544$	−7.24256e8			−0.192884
$545$	−3.88733e9	+	3.74910e8i	−1.02864	+	0.0992062i
$546$	−8.32338e8			−0.218839
$547$		−	4.47999e9i		−	1.17037i	−0.810902	−	0.585183i	$-0.801023\pi$
$547$		−	4.47999e9i		−	1.17037i	0.810902	−	0.585183i	$-0.198977\pi$
$548$	1.70956e7			0.00443763
$549$	3.72897e9			0.961801
$550$	6.32171e8			0.162019
$551$			1.79790e8i			0.0457863i
$552$	1.72585e9			0.436734
$553$			3.28625e8i			0.0826346i
$554$	1.15694e10			2.89086
$555$		−	1.85265e9i		−	0.460011i
$556$		−	1.20411e10i		−	2.97100i
$557$	−2.81429e9			−0.690042			−0.345021	−	0.938595i	$-0.612128\pi$
$557$	−2.81429e9			−0.690042			−0.345021	+	0.938595i	$0.612128\pi$
$558$			4.52851e9i			1.10341i
$559$			4.65098e9i			1.12617i
$560$	3.01620e8			0.0725776
$561$	−3.73636e9			−0.893469
$562$		−	2.13975e9i		−	0.508494i
$563$			6.29803e8i			0.148739i	0.997231	+	0.0743696i	$0.0236944\pi$
$563$			6.29803e8i			0.148739i	−0.997231	+	0.0743696i	$0.976306\pi$
$564$		−	7.87169e9i		−	1.84753i
$565$	2.49908e9			0.582923
$566$	−2.28725e9			−0.530220
$567$	3.43013e8			0.0790261
$568$			3.04353e9i			0.696881i
$569$		−	4.69761e9i		−	1.06902i	−0.845164	−	0.534508i	$-0.820497\pi$
$569$		−	4.69761e9i		−	1.06902i	0.845164	−	0.534508i	$-0.179503\pi$
$570$	−5.91259e9			−1.33726
$571$	−4.80983e9			−1.08119			−0.540597	−	0.841282i	$-0.681802\pi$
$571$	−4.80983e9			−1.08119			−0.540597	+	0.841282i	$0.681802\pi$
$572$	8.33108e9			1.86130
$573$	−7.59737e9			−1.68703
$574$	−8.81053e7			−0.0194451
$575$		−	6.22676e7i		−	0.0136592i
$576$	7.91774e9			1.72632
$577$		−	4.95896e8i		−	0.107467i	−0.998555	−	0.0537335i	$-0.982888\pi$
$577$		−	4.95896e8i		−	0.107467i	0.998555	−	0.0537335i	$-0.0171121\pi$
$578$		−	6.45920e9i		−	1.39133i
$579$	−8.90685e8			−0.190699
$580$	8.10721e8			0.172534
$581$	9.93206e8			0.210098
$582$			1.97825e10i			4.15959i
$583$	7.13656e9			1.49159
$584$		−	5.62670e9i		−	1.16898i
$585$		−	4.52823e9i		−	0.935153i
$586$		−	8.29448e9i		−	1.70274i
$587$		−	4.62007e9i		−	0.942791i	−0.881922	−	0.471396i	$-0.843750\pi$
$587$		−	4.62007e9i		−	0.942791i	0.881922	−	0.471396i	$-0.156250\pi$
$588$	1.35697e10			2.75264
$589$		−	1.29443e9i		−	0.261020i
$590$	−8.86241e9			−1.77652
$591$	1.26875e10			2.52824
$592$			8.66635e8i			0.171676i
$593$	6.28448e9			1.23759			0.618797	−	0.785551i	$-0.287620\pi$
$593$	6.28448e9			1.23759			0.618797	+	0.785551i	$0.287620\pi$
$594$	4.99249e9			0.977384
$595$			2.69367e8i			0.0524246i
$596$			3.85201e9i			0.745291i
$597$			7.44147e9i			1.43136i
$598$		−	1.26338e9i		−	0.241591i
$599$			2.66782e9i			0.507181i	0.967312	+	0.253591i	$0.0816116\pi$
$599$			2.66782e9i			0.507181i	−0.967312	+	0.253591i	$0.918388\pi$
$600$			7.80980e8i			0.147608i
$601$			6.38397e9i			1.19958i	0.800157	+	0.599791i	$0.204750\pi$
$601$			6.38397e9i			1.19958i	−0.800157	+	0.599791i	$0.795250\pi$
$602$			1.72914e9i			0.323030i
$603$			5.52804e9i			1.02674i
$604$		−	1.69963e10i		−	3.13851i
$605$	5.58844e9			1.02600
$606$	2.29950e10			4.19740
$607$		−	5.66958e9i		−	1.02894i	−0.857508	−	0.514471i	$-0.827988\pi$
$607$		−	5.66958e9i		−	1.02894i	0.857508	−	0.514471i	$-0.172012\pi$
$608$	−1.29385e9			−0.233465
$609$	−9.14395e7			−0.0164049
$610$			7.39982e9i			1.31998i
$611$	−2.65304e9			−0.470544
$612$		−	5.61278e9i		−	0.989802i
$613$			5.95866e9i			1.04481i	0.852698	+	0.522404i	$0.174965\pi$
$613$			5.95866e9i			1.04481i	−0.852698	+	0.522404i	$0.825035\pi$
$614$			3.15396e9i			0.549878i
$615$		−	8.56173e8i		−	0.148422i
$616$	1.42605e9			0.245811
$617$			8.65037e9i			1.48264i	0.671150	+	0.741322i	$0.265801\pi$
$617$			8.65037e9i			1.48264i	−0.671150	+	0.741322i	$0.734199\pi$
$618$	1.15280e10			1.96469
$619$	3.20204e9			0.542637			0.271318	−	0.962490i	$-0.412540\pi$
$619$	3.20204e9			0.542637			0.271318	+	0.962490i	$0.412540\pi$
$620$	−5.83691e9			−0.983586
$621$		−	4.91751e8i		−	0.0823996i
$622$		−	4.57291e8i		−	0.0761950i
$623$	5.67230e8			0.0939833
$624$			3.78357e9i			0.623385i
$625$	−6.49004e9			−1.06333
$626$	−1.43572e10			−2.33915
$627$	−6.67485e9			−1.08145
$628$	−2.10742e10			−3.39540
$629$	−7.73963e8			−0.124006
$630$		−	1.68351e9i		−	0.268240i
$631$			9.11387e9i			1.44411i	0.691836	+	0.722055i	$0.256802\pi$
$631$			9.11387e9i			1.44411i	−0.691836	+	0.722055i	$0.743198\pi$
$632$	6.25622e9			0.985831
$633$	−8.67355e9			−1.35920
$634$	−4.12991e9			−0.643618
$635$		−	4.69654e9i		−	0.727898i
$636$			1.91491e10i			2.95154i
$637$		−	4.57347e9i		−	0.701065i
$638$	1.40909e9			0.214816
$639$	4.05623e9			0.614992
$640$			1.25639e10i			1.89450i
$641$			2.65939e9i			0.398822i	0.979916	+	0.199411i	$0.0639029\pi$
$641$			2.65939e9i			0.398822i	−0.979916	+	0.199411i	$0.936097\pi$
$642$	−1.10092e10			−1.64203
$643$			5.34520e9i			0.792913i	0.918054	+	0.396457i	$0.129760\pi$
$643$			5.34520e9i			0.792913i	−0.918054	+	0.396457i	$0.870240\pi$
$644$		−	3.05082e8i		−	0.0450107i
$645$	−1.68031e10			−2.46565
$646$			2.47004e9i			0.360488i
$647$	1.07598e10			1.56185			0.780926	−	0.624624i	$-0.214748\pi$
$647$	1.07598e10			1.56185			0.780926	+	0.624624i	$0.214748\pi$
$648$		−	6.53015e9i		−	0.942780i
$649$	−1.00050e10			−1.43668
$650$	5.71704e8			0.0816534
$651$	6.58333e8			0.0935216
$652$			8.95029e9i			1.26465i
$653$	1.38828e10			1.95111			0.975554	−	0.219761i	$-0.0705279\pi$
$653$	1.38828e10			1.95111			0.975554	+	0.219761i	$0.0705279\pi$
$654$	−1.74846e9	−	1.81293e10i	−0.244418	−	2.53430i
$655$	5.46250e9			0.759533
$656$			4.00502e8i			0.0553912i
$657$	−7.49891e9			−1.03162
$658$	−9.86350e8			−0.134971
$659$	7.85491e9			1.06916			0.534579	−	0.845118i	$-0.320470\pi$
$659$	7.85491e9			1.06916			0.534579	+	0.845118i	$0.320470\pi$
$660$			3.00987e10i			4.07515i
$661$	−9.70948e9			−1.30765			−0.653824	−	0.756647i	$-0.726836\pi$
$661$	−9.70948e9			−1.30765			−0.653824	+	0.756647i	$0.726836\pi$
$662$		−	2.05286e9i		−	0.275014i
$663$	−3.37898e9			−0.450286
$664$		−	1.89083e10i		−	2.50647i
$665$			4.81212e8i			0.0634543i
$666$	4.83716e9			0.634499
$667$		−	1.38793e8i		−	0.0181104i
$668$			2.02788e10i			2.63224i
$669$	−6.49918e9			−0.839203
$670$	−1.09699e10			−1.40910
$671$			8.35381e9i			1.06747i
$672$		−	6.58040e8i		−	0.0836489i
$673$		−	1.48079e10i		−	1.87258i	−0.351230	−	0.936289i	$-0.614236\pi$
$673$		−	1.48079e10i		−	1.87258i	0.351230	−	0.936289i	$-0.385764\pi$
$674$	1.01682e10			1.27918
$675$	2.22527e8			0.0278496
$676$	−7.35084e9			−0.915216
$677$			5.04997e8i			0.0625501i	0.999511	+	0.0312751i	$0.00995678\pi$
$677$			5.04997e8i			0.0625501i	−0.999511	+	0.0312751i	$0.990043\pi$
$678$			1.16549e10i			1.43617i
$679$	1.61005e9			0.197376
$680$	5.12810e9			0.625425
$681$	1.33313e9			0.161755
$682$	−1.01450e10			−1.22463
$683$	1.29185e10			1.55146			0.775728	−	0.631068i	$-0.217383\pi$
$683$	1.29185e10			1.55146			0.775728	+	0.631068i	$0.217383\pi$
$684$		−	1.00270e10i		−	1.19805i
$685$	2.08164e7			0.00247451
$686$		−	3.42584e9i		−	0.405166i
$687$		−	2.12722e10i		−	2.50301i
$688$	7.86020e9			0.920182
$689$	6.45395e9			0.751724
$690$	4.56436e9			0.528942
$691$			1.23764e10i			1.42699i	0.700660	+	0.713495i	$0.252889\pi$
$691$			1.23764e10i			1.42699i	−0.700660	+	0.713495i	$0.747111\pi$
$692$	1.82598e10			2.09471
$693$		−	1.90055e9i		−	0.216926i
$694$		−	2.75541e10i		−	3.12916i
$695$		−	1.46618e10i		−	1.65669i
$696$			1.74079e9i			0.195710i
$697$	−3.57675e8			−0.0400104
$698$		−	1.82171e10i		−	2.02761i
$699$	2.03753e10			2.25650
$700$	1.38055e8			0.0152128
$701$		−	2.50576e9i		−	0.274743i	−0.990520	−	0.137371i	$-0.956135\pi$
$701$		−	2.50576e9i		−	0.274743i	0.990520	−	0.137371i	$-0.0438654\pi$
$702$	4.51496e9			0.492577
$703$	−1.38265e9			−0.150096
$704$			1.77377e10i			1.91599i
$705$		−	9.58497e9i		−	1.03022i
$706$			5.02050e8i			0.0536946i
$707$		−	1.87151e9i		−	0.199170i
$708$		−	2.68458e10i		−	2.84288i
$709$			5.25830e9i			0.554094i	0.960856	+	0.277047i	$0.0893557\pi$
$709$			5.25830e9i			0.554094i	−0.960856	+	0.277047i	$0.910644\pi$
$710$			8.04924e9i			0.844016i
$711$		−	8.33790e9i		−	0.869988i
$712$		−	1.07987e10i		−	1.12122i
$713$			9.99262e8i			0.103244i
$714$	−1.25624e9			−0.129160
$715$	1.01443e10			1.03789
$716$			2.75225e10i			2.80215i
$717$	−5.99108e8			−0.0606999
$718$	3.74937e9			0.378027
$719$			9.50535e9i			0.953711i	0.878982	+	0.476856i	$0.158224\pi$
$719$			9.50535e9i			0.953711i	−0.878982	+	0.476856i	$0.841776\pi$
$720$	−7.65274e9			−0.764106
$721$		−	9.38237e8i		−	0.0932264i
$722$		−	1.26699e10i		−	1.25283i
$723$			1.09025e10i			1.07286i
$724$		−	1.28589e10i		−	1.25927i
$725$	6.28065e7			0.00612099
$726$			2.60627e10i			2.52779i
$727$	3.00737e9			0.290280			0.145140	−	0.989411i	$-0.453637\pi$
$727$	3.00737e9			0.290280			0.145140	+	0.989411i	$0.453637\pi$
$728$	1.28965e9			0.123883
$729$	−1.51655e10			−1.44981
$730$		−	1.48810e10i		−	1.41580i
$731$			7.01967e9i			0.664670i
$732$	−2.24153e10			−2.11230
$733$			6.74937e9i			0.632993i	0.948594	+	0.316497i	$0.102507\pi$
$733$			6.74937e9i			0.632993i	−0.948594	+	0.316497i	$0.897493\pi$
$734$	9.14770e8			0.0853839
$735$	1.65231e10			1.53492
$736$	9.98819e8			0.0923452
$737$	−1.23842e10			−1.13954
$738$	2.23542e9			0.204721
$739$			1.47005e10i			1.33991i	0.742401	+	0.669956i	$0.233687\pi$
$739$			1.47005e10i			1.33991i	−0.742401	+	0.669956i	$0.766313\pi$
$740$			6.23474e9i			0.565597i
$741$	−6.03640e9			−0.545023
$742$	2.39945e9			0.215625
$743$	−5.14438e9			−0.460121			−0.230061	−	0.973176i	$-0.573892\pi$
$743$	−5.14438e9			−0.460121			−0.230061	+	0.973176i	$0.573892\pi$
$744$		−	1.25331e10i		−	1.11571i
$745$			4.69040e9i			0.415588i
$746$			1.98258e10i			1.74842i
$747$	−2.51997e10			−2.21194
$748$	1.25740e10			1.09855
$749$			8.96010e8i			0.0779160i
$750$		−	2.83334e10i		−	2.45236i
$751$	5.88362e9			0.506879			0.253440	−	0.967351i	$-0.418438\pi$
$751$	5.88362e9			0.506879			0.253440	+	0.967351i	$0.418438\pi$
$752$			4.48367e9i			0.384478i
$753$		−	1.09973e10i		−	0.938650i
$754$	1.27431e9			0.108262
$755$		−	2.06955e10i		−	1.75010i
$756$	1.09028e9			0.0917719
$757$		−	4.61667e9i		−	0.386806i	−0.981119	−	0.193403i	$-0.938048\pi$
$757$		−	4.61667e9i		−	0.386806i	0.981119	−	0.193403i	$-0.0619524\pi$
$758$	1.87690e9			0.156531
$759$	5.15281e9			0.427757
$760$	9.16112e9			0.757009
$761$			1.71644e9i			0.141183i	0.997505	+	0.0705916i	$0.0224887\pi$
$761$			1.71644e9i			0.141183i	−0.997505	+	0.0705916i	$0.977511\pi$
$762$	2.19032e10			1.79335
$763$	−1.47550e9	+	1.42303e8i	−0.120255	+	0.0115979i
$764$	2.55675e10			2.07425
$765$		−	6.83441e9i		−	0.551932i
$766$	5.26715e9			0.423424
$767$	−9.04799e9			−0.724050
$768$	−3.29126e10			−2.62179
$769$			1.95456e10i			1.54991i	0.632016	+	0.774956i	$0.282228\pi$
$769$			1.95456e10i			1.54991i	−0.632016	+	0.774956i	$0.717772\pi$
$770$	3.77147e9			0.297710
$771$		−	2.41185e10i		−	1.89522i
$772$	2.99743e9			0.234471
$773$			1.69093e10i			1.31673i	0.752698	+	0.658366i	$0.228752\pi$
$773$			1.69093e10i			1.31673i	−0.752698	+	0.658366i	$0.771248\pi$
$774$		−	4.38720e10i		−	3.40090i
$775$	−4.52185e8			−0.0348948
$776$		−	3.06515e10i		−	2.35470i
$777$		−	7.03202e8i		−	0.0537783i
$778$	−4.54599e9			−0.346099
$779$	−6.38970e8			−0.0484283
$780$			2.72197e10i			2.05378i
$781$			9.08695e9i			0.682558i
$782$		−	1.90681e9i		−	0.142588i
$783$	4.96007e8			0.0369251
$784$	−7.72921e9			−0.572834
$785$	−2.56609e10			−1.89334
$786$			2.54753e10i			1.87129i
$787$			1.86979e10i			1.36735i	0.729785	+	0.683677i	$0.239620\pi$
$787$			1.86979e10i			1.36735i	−0.729785	+	0.683677i	$0.760380\pi$
$788$	−4.26973e10			−3.10855
$789$	−2.76595e9			−0.200482
$790$	1.65458e10			1.19397
$791$	9.48565e8			0.0681474
$792$	−3.61818e10			−2.58793
$793$			7.55477e9i			0.537979i
$794$	−3.06080e10			−2.17001
$795$			2.33170e10i			1.64584i
$796$		−	2.50428e10i		−	1.75990i
$797$	−7.07123e8			−0.0494755			−0.0247378	−	0.999694i	$-0.507875\pi$
$797$	−7.07123e8			−0.0494755			−0.0247378	+	0.999694i	$0.507875\pi$
$798$	−2.24422e9			−0.156335
$799$	−4.00421e9			−0.277718
$800$			4.51984e8i			0.0312111i
$801$	−1.43918e10			−0.989468
$802$			4.21862e10i			2.88776i
$803$		−	1.67994e10i		−	1.14496i
$804$		−	3.32298e10i		−	2.25492i
$805$		−	3.71483e8i		−	0.0250988i
$806$	−9.17462e9			−0.617185
$807$		−	3.75036e10i		−	2.51198i
$808$	−3.56291e10			−2.37610
$809$	−1.09048e10			−0.724097			−0.362049	−	0.932159i	$-0.617923\pi$
$809$	−1.09048e10			−0.724097			−0.362049	+	0.932159i	$0.617923\pi$
$810$		−	1.72703e10i		−	1.14183i
$811$	−1.63700e10			−1.07765			−0.538823	−	0.842419i	$-0.681131\pi$
$811$	−1.63700e10			−1.07765			−0.538823	+	0.842419i	$0.681131\pi$
$812$	3.07722e8			0.0201703
$813$			1.91174e10i			1.24771i
$814$			1.08364e10i			0.704208i
$815$			1.08983e10i			0.705193i
$816$			5.71051e9i			0.367925i
$817$			1.25403e10i			0.804511i
$818$		−	1.21406e10i		−	0.775537i
$819$		−	1.71876e9i		−	0.109325i
$820$			2.88129e9i			0.182489i
$821$		−	1.67452e10i		−	1.05606i	−0.849226	−	0.528030i	$-0.822931\pi$
$821$		−	1.67452e10i		−	1.05606i	0.849226	−	0.528030i	$-0.177069\pi$
$822$			9.70810e7i			0.00609653i
$823$	−2.19243e10			−1.37096			−0.685482	−	0.728090i	$-0.740408\pi$
$823$	−2.19243e10			−1.37096			−0.685482	+	0.728090i	$0.740408\pi$
$824$	−1.78618e10			−1.11219
$825$			2.33174e9i			0.144575i
$826$	−3.36387e9			−0.207687
$827$	6.87816e9			0.422866			0.211433	−	0.977392i	$-0.432187\pi$
$827$	6.87816e9			0.422866			0.211433	+	0.977392i	$0.432187\pi$
$828$			7.74056e9i			0.473878i
$829$	2.35835e10			1.43770			0.718848	−	0.695167i	$-0.244670\pi$
$829$	2.35835e10			1.43770			0.718848	+	0.695167i	$0.244670\pi$
$830$		−	5.00067e10i		−	3.03567i
$831$			4.26734e10i			2.57961i
$832$			1.60411e10i			0.965610i
$833$		−	6.90270e9i		−	0.413772i
$834$	6.83780e10			4.08164
$835$			2.46925e10i			1.46779i
$836$	2.24629e10			1.32967
$837$	−3.57108e9			−0.210504
$838$	3.15358e9			0.185118
$839$			9.12850e9i			0.533621i	0.963749	+	0.266810i	$0.0859697\pi$
$839$			9.12850e9i			0.533621i	−0.963749	+	0.266810i	$0.914030\pi$
$840$			4.65926e9i			0.271231i
$841$	−1.71099e10			−0.991884
$842$			2.73627e10i			1.57967i
$843$	7.89240e9			0.453746
$844$	2.91892e10			1.67118
$845$	−8.95075e9			−0.510342
$846$	2.50258e10			1.42099
$847$	2.12118e9			0.119946
$848$		−	1.09072e10i		−	0.614228i
$849$		−	8.43645e9i		−	0.473132i
$850$	8.62866e8			0.0481923
$851$	1.06737e9			0.0593692
$852$	−2.43825e10			−1.35064
$853$			2.27462e10i			1.25484i	0.778683	+	0.627418i	$0.215888\pi$
$853$			2.27462e10i			1.25484i	−0.778683	+	0.627418i	$0.784112\pi$
$854$			2.80872e9i			0.154314i
$855$		−	1.22094e10i		−	0.668054i
$856$	1.70579e10			0.929538
$857$	−2.62495e10			−1.42458			−0.712292	−	0.701883i	$-0.752343\pi$
$857$	−2.62495e10			−1.42458			−0.712292	+	0.701883i	$0.752343\pi$
$858$			4.73099e10i			2.55710i
$859$			1.44031e10i			0.775320i	0.921802	+	0.387660i	$0.126717\pi$
$859$			1.44031e10i			0.775320i	−0.921802	+	0.387660i	$0.873283\pi$
$860$	5.65477e10			3.03159
$861$		−	3.24974e8i		−	0.0173515i
$862$			7.43461e9i			0.395351i
$863$	1.99980e10			1.05913			0.529564	−	0.848270i	$-0.322356\pi$
$863$	1.99980e10			1.05913			0.529564	+	0.848270i	$0.322356\pi$
$864$			3.56949e9i			0.188282i
$865$	2.22340e10			1.16805
$866$		−	3.18368e10i		−	1.66578i
$867$	2.38246e10			1.24153
$868$	−2.21549e9			−0.114988
$869$	1.86790e10			0.965569
$870$			4.60387e9i			0.237031i
$871$	−1.11996e10			−0.574302
$872$	2.70910e9	+	2.80899e10i	0.138362	+	1.43464i
$873$	−4.08504e10			−2.07800
$874$		−	3.40643e9i		−	0.172587i
$875$	−2.30599e9			−0.116367
$876$	4.50769e10			2.26564
$877$	1.01704e10			0.509145			0.254572	−	0.967054i	$-0.418065\pi$
$877$	1.01704e10			0.509145			0.254572	+	0.967054i	$0.418065\pi$
$878$			1.55991e10i			0.777801i
$879$	3.05939e10			1.51941
$880$		−	1.71440e10i		−	0.848054i
$881$	−3.45637e10			−1.70296			−0.851481	−	0.524386i	$-0.824295\pi$
$881$	−3.45637e10			−1.70296			−0.851481	+	0.524386i	$0.824295\pi$
$882$			4.31409e10i			2.11714i
$883$		−	1.61470e10i		−	0.789275i	−0.918837	−	0.394638i	$-0.870870\pi$
$883$		−	1.61470e10i		−	0.789275i	0.918837	−	0.394638i	$-0.129130\pi$
$884$	1.13713e10			0.553640
$885$		−	3.26887e10i		−	1.58525i
$886$			1.27595e10i			0.616332i
$887$	−2.15373e10			−1.03624			−0.518118	−	0.855309i	$-0.673367\pi$
$887$	−2.15373e10			−1.03624			−0.518118	+	0.855309i	$0.673367\pi$
$888$	−1.33873e10			−0.641574
$889$		−	1.78265e9i		−	0.0850960i
$890$		−	2.85593e10i		−	1.35795i
$891$		−	1.94968e10i		−	0.923404i
$892$	2.18717e10			1.03183
$893$	−7.15335e9			−0.336147
$894$	−2.18745e10			−1.02390
$895$			3.35128e10i			1.56253i
$896$			4.76884e9i			0.221480i
$897$	4.65994e9			0.215579
$898$	5.75611e10			2.65254
$899$	−1.00791e9			−0.0462661
$900$	−3.50275e9			−0.160162
$901$	9.74089e9			0.443672
$902$			5.00789e9i			0.227212i
$903$	−6.37789e9			−0.288251
$904$		−	1.80584e10i		−	0.812999i
$905$		−	1.56576e10i		−	0.702193i
$906$	9.65171e10			4.31177
$907$	−8.78413e9			−0.390907			−0.195453	−	0.980713i	$-0.562618\pi$
$907$	−8.78413e9			−0.390907			−0.195453	+	0.980713i	$0.562618\pi$
$908$	−4.48639e9			−0.198882
$909$			4.74842e10i			2.09689i
$910$	3.41073e9			0.150038
$911$		−	4.05945e10i		−	1.77891i	−0.457026	−	0.889454i	$-0.651085\pi$
$911$		−	4.05945e10i		−	1.77891i	0.457026	−	0.889454i	$-0.348915\pi$
$912$			1.02016e10i			0.445334i
$913$		−	5.64536e10i		−	2.45496i
$914$		−	5.61955e10i		−	2.43439i
$915$	−2.72940e10			−1.17786
$916$			7.15874e10i			3.07753i
$917$	2.07338e9			0.0887944
$918$	6.81439e9			0.290722
$919$			4.45729e10i			1.89438i	0.320677	+	0.947189i	$0.396089\pi$
$919$			4.45729e10i			1.89438i	−0.320677	+	0.947189i	$0.603911\pi$
$920$	−7.07214e9			−0.299429
$921$	−1.16333e10			−0.490674
$922$			2.33128e10i			0.979572i
$923$			8.21779e9i			0.343993i
$924$			1.14244e10i			0.476412i
$925$			4.83005e8i			0.0200657i
$926$		−	2.40589e10i		−	0.995719i
$927$			2.38050e10i			0.981499i
$928$			1.00746e9i			0.0413820i
$929$		−	1.34669e10i		−	0.551079i	−0.961290	−	0.275540i	$-0.911143\pi$
$929$		−	1.34669e10i		−	0.551079i	0.961290	−	0.275540i	$-0.0888565\pi$
$930$		−	3.31462e10i		−	1.35128i
$931$		−	1.23314e10i		−	0.500827i
$932$	−6.85693e10			−2.77443
$933$	1.68670e9			0.0679913
$934$			5.06286e10i			2.03321i
$935$	1.53108e10			0.612570
$936$	−3.27210e10			−1.30425
$937$		−	1.03936e10i		−	0.412739i	−0.978474	−	0.206370i	$-0.933835\pi$
$937$		−	1.03936e10i		−	0.412739i	0.978474	−	0.206370i	$-0.0661649\pi$
$938$	−4.16381e9			−0.164733
$939$		−	5.29559e10i		−	2.08730i
$940$			3.22564e10i			1.26668i
$941$		−	2.70393e10i		−	1.05787i	−0.848663	−	0.528934i	$-0.822592\pi$
$941$		−	2.70393e10i		−	1.05787i	0.848663	−	0.528934i	$-0.177408\pi$
$942$		−	1.19674e11i		−	4.66469i
$943$	4.93268e8			0.0191554
$944$			1.52912e10i			0.591615i
$945$	1.32757e9			0.0511738
$946$	9.82841e10			3.77454
$947$	1.20985e10			0.462921			0.231460	−	0.972844i	$-0.425650\pi$
$947$	1.20985e10			0.462921			0.231460	+	0.972844i	$0.425650\pi$
$948$			5.01202e10i			1.91066i
$949$		−	1.51926e10i		−	0.577031i
$950$	1.54147e9			0.0583315
$951$		−	1.52331e10i		−	0.574322i
$952$	1.94645e9			0.0731162
$953$	1.99675e10			0.747305			0.373652	−	0.927569i	$-0.378105\pi$
$953$	1.99675e10			0.747305			0.373652	+	0.927569i	$0.378105\pi$
$954$	−6.08792e10			−2.27012
$955$	3.11323e10			1.15664
$956$	2.01618e9			0.0746324
$957$			5.19740e9i			0.191688i
$958$		−	4.91486e10i		−	1.80606i
$959$	7.90119e6			0.000289286
$960$	−5.79535e10			−2.11412
$961$	−2.02560e10			−0.736244
$962$			9.79993e9i			0.354904i
$963$		−	2.27337e10i		−	0.820309i
$964$		−	3.66902e10i		−	1.31911i
$965$	3.64982e9			0.130745
$966$	1.73248e9			0.0618368
$967$		−	1.16405e10i		−	0.413980i	−0.978343	−	0.206990i	$-0.933633\pi$
$967$		−	1.16405e10i		−	0.413980i	0.978343	−	0.206990i	$-0.0663668\pi$
$968$		−	4.03822e10i		−	1.43095i
$969$	−9.11068e9			−0.321675
$970$		−	8.10640e10i		−	2.85185i
$971$			4.15885e10i			1.45783i	0.684605	+	0.728914i	$0.259975\pi$
$971$			4.15885e10i			1.45783i	−0.684605	+	0.728914i	$0.740025\pi$
$972$	7.40632e10			2.58684
$973$		−	5.56512e9i		−	0.193678i
$974$	4.39866e9			0.152533
$975$			2.10871e9i			0.0728620i
$976$	1.27676e10			0.439578
$977$	−9.03915e9			−0.310096			−0.155048	−	0.987907i	$-0.549553\pi$
$977$	−9.03915e9			−0.310096			−0.155048	+	0.987907i	$0.549553\pi$
$978$	−5.08262e10			−1.73741
$979$		−	3.22412e10i		−	1.09818i
$980$	−5.56054e10			−1.88723
$981$	3.74365e10	−	3.61052e9i	1.26606	−	0.122104i
$982$	3.42379e10			1.15376
$983$			1.66615e9i			0.0559469i	0.999609	+	0.0279734i	$0.00890538\pi$
$983$			1.66615e9i			0.0559469i	−0.999609	+	0.0279734i	$0.991095\pi$
$984$	−6.18672e9			−0.207003
$985$	−5.19903e10			−1.73339
$986$	1.92331e9			0.0638969
$987$		−	3.63812e9i		−	0.120439i
$988$	2.03144e10			0.670122
$989$		−	9.68080e9i		−	0.318218i
$990$	−9.56902e10			−3.13433
$991$			4.06377e10i			1.32639i	0.748446	+	0.663196i	$0.230800\pi$
$991$			4.06377e10i			1.32639i	−0.748446	+	0.663196i	$0.769200\pi$
$992$		−	7.25339e9i		−	0.235912i
$993$	7.57190e9			0.245404
$994$			3.05521e9i			0.0986710i
$995$		−	3.04934e10i		−	0.981353i
$996$	1.51479e11			4.85785
$997$	2.45217e10			0.783642			0.391821	−	0.920041i	$-0.371845\pi$
$997$	2.45217e10			0.783642			0.391821	+	0.920041i	$0.371845\pi$
$998$			4.72099e10i			1.50341i
$999$			3.81447e9i			0.121047i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	109.8.b.a.108.5	✓	62
109.108	even	2	inner	109.8.b.a.108.58	yes	62

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
109.8.b.a.108.5	✓	62	1.1	even	1	trivial
109.8.b.a.108.58	yes	62	109.108	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:	\( N \)	\(=\)	\( 109 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	109.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-19.1107i q^{2} +70.4891 q^{3} -237.218 q^{4} -288.848 q^{5} -1347.09i q^{6} -109.637 q^{7} +2087.22i q^{8} +2781.72 q^{9} +O(q^{10})\)	\(q-19.1107i q^{2} +70.4891 q^{3} -237.218 q^{4} -288.848 q^{5} -1347.09i q^{6} -109.637 q^{7} +2087.22i q^{8} +2781.72 q^{9} +5520.08i q^{10} +6231.73i q^{11} -16721.3 q^{12} +5635.67i q^{13} +2095.23i q^{14} -20360.7 q^{15} +9524.34 q^{16} +8505.86i q^{17} -53160.5i q^{18} +15195.3i q^{19} +68519.9 q^{20} -7728.20 q^{21} +119093. q^{22} -11730.4i q^{23} +147126. i q^{24} +5308.23 q^{25} +107701. q^{26} +41921.1 q^{27} +26007.8 q^{28} +11831.9 q^{29} +389106. i q^{30} -85185.7 q^{31} +85147.9i q^{32} +439270. i q^{33} +162553. q^{34} +31668.4 q^{35} -659872. q^{36} +90991.7i q^{37} +290393. q^{38} +397253. i q^{39} -602890. i q^{40} +42050.4i q^{41} +147691. i q^{42} +825275. q^{43} -1.47828e6i q^{44} -803494. q^{45} -224176. q^{46} +470759. i q^{47} +671362. q^{48} -811523. q^{49} -101444. i q^{50} +599571. i q^{51} -1.33688e6i q^{52} -1.14520e6i q^{53} -801140. i q^{54} -1.80002e6i q^{55} -228836. i q^{56} +1.07111e6i q^{57} -226116. i q^{58} +1.60549e6i q^{59} +4.82990e6 q^{60} +1.34053e6 q^{61} +1.62796e6i q^{62} -304979. q^{63} +2.84635e6 q^{64} -1.62785e6i q^{65} +8.39473e6 q^{66} +1.98728e6i q^{67} -2.01774e6i q^{68} -826866. i q^{69} -605204. i q^{70} +1.45817e6 q^{71} +5.80606e6i q^{72} -2.69579e6 q^{73} +1.73891e6 q^{74} +374172. q^{75} -3.60460e6i q^{76} -683227. i q^{77} +7.59178e6 q^{78} -2.99739e6i q^{79} -2.75109e6 q^{80} -3.12863e6 q^{81} +803611. q^{82} -9.05906e6 q^{83} +1.83327e6 q^{84} -2.45690e6i q^{85} -1.57716e7i q^{86} +834022. q^{87} -1.30070e7 q^{88} -5.17372e6 q^{89} +1.53553e7i q^{90} -617877. i q^{91} +2.78266e6i q^{92} -6.00467e6 q^{93} +8.99653e6 q^{94} -4.38915e6i q^{95} +6.00200e6i q^{96} -1.46853e7 q^{97} +1.55087e7i q^{98} +1.73349e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 62 q - 56 q^{3} - 3584 q^{4} + 194 q^{5} + 370 q^{7} + 40462 q^{9}+O(q^{10}) \) 62 * q - 56 * q^3 - 3584 * q^4 + 194 * q^5 + 370 * q^7 + 40462 * q^9	\( 62 q - 56 q^{3} - 3584 q^{4} + 194 q^{5} + 370 q^{7} + 40462 q^{9} + 6248 q^{12} - 21376 q^{15} + 227856 q^{16} - 216666 q^{20} + 42914 q^{21} - 101446 q^{22} + 788644 q^{25} - 282610 q^{26} - 235946 q^{27} - 69142 q^{28} - 213678 q^{29} + 300350 q^{31} - 401602 q^{34} - 377134 q^{35} - 2176902 q^{36} - 2421542 q^{38} + 2022550 q^{43} + 2223056 q^{45} + 2665174 q^{46} + 1060208 q^{48} + 7702792 q^{49} + 9894904 q^{60} + 278126 q^{61} - 6010300 q^{63} - 9362182 q^{64} - 23291020 q^{66} + 15373744 q^{71} - 1378934 q^{73} - 19097780 q^{74} - 41528062 q^{75} + 5705654 q^{78} + 42766270 q^{80} + 29291446 q^{81} + 14605464 q^{82} + 1191422 q^{83} - 6959552 q^{84} - 22121732 q^{87} + 6337872 q^{88} + 29157388 q^{89} + 14091936 q^{93} - 17482078 q^{94} - 22763314 q^{97}+O(q^{100}) \) 62 * q - 56 * q^3 - 3584 * q^4 + 194 * q^5 + 370 * q^7 + 40462 * q^9 + 6248 * q^12 - 21376 * q^15 + 227856 * q^16 - 216666 * q^20 + 42914 * q^21 - 101446 * q^22 + 788644 * q^25 - 282610 * q^26 - 235946 * q^27 - 69142 * q^28 - 213678 * q^29 + 300350 * q^31 - 401602 * q^34 - 377134 * q^35 - 2176902 * q^36 - 2421542 * q^38 + 2022550 * q^43 + 2223056 * q^45 + 2665174 * q^46 + 1060208 * q^48 + 7702792 * q^49 + 9894904 * q^60 + 278126 * q^61 - 6010300 * q^63 - 9362182 * q^64 - 23291020 * q^66 + 15373744 * q^71 - 1378934 * q^73 - 19097780 * q^74 - 41528062 * q^75 + 5705654 * q^78 + 42766270 * q^80 + 29291446 * q^81 + 14605464 * q^82 + 1191422 * q^83 - 6959552 * q^84 - 22121732 * q^87 + 6337872 * q^88 + 29157388 * q^89 + 14091936 * q^93 - 17482078 * q^94 - 22763314 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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