LMFDB - Embedded newform 90.18.c.b.19.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [90,18,Mod(19,90)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("90.19"); S:= CuspForms(chi, 18); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 18, names="a")

Level:	$ N $	$=$	$ 90 = 2 \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	90.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,-524288,1225560] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$164.899878610$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 556201x^{6} + 76870744104x^{4} + 1868329791349729x^{2} + 78074963590050625 $ x^8 + 556201x^6 + 76870744104x^4 + 1868329791349729*x^2 + 78074963590050625
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{40}\cdot 3^{6}\cdot 5^{11} $
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			19.8
Root			$-594.906i$ of defining polynomial
Character	$\chi$	$=$	90.19
Dual form			90.18.c.b.19.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+256.000i q^{2} -65536.0 q^{4} +(765001. + 421560. i) q^{5} +1.76306e7i q^{7} -1.67772e7i q^{8} +(-1.07919e8 + 1.95840e8i) q^{10} -4.98950e8 q^{11} -5.00327e9i q^{13} -4.51344e9 q^{14} +4.29497e9 q^{16} +1.61697e10i q^{17} -3.43950e10 q^{19} +(-5.01351e10 - 2.76274e10i) q^{20} -1.27731e11i q^{22} -4.72599e11i q^{23} +(4.07513e11 + 6.44988e11i) q^{25} +1.28084e12 q^{26} -1.15544e12i q^{28} +3.30807e12 q^{29} +3.23730e12 q^{31} +1.09951e12i q^{32} -4.13944e12 q^{34} +(-7.43238e12 + 1.34875e13i) q^{35} +5.76908e11i q^{37} -8.80513e12i q^{38} +(7.07261e12 - 1.28346e13i) q^{40} +5.68215e13 q^{41} +5.07446e12i q^{43} +3.26992e13 q^{44} +1.20985e14 q^{46} -3.32325e13i q^{47} -7.82089e13 q^{49} +(-1.65117e14 + 1.04323e14i) q^{50} +3.27894e14i q^{52} -6.11026e14i q^{53} +(-3.81697e14 - 2.10338e14i) q^{55} +2.95793e14 q^{56} +8.46865e14i q^{58} -8.63122e14 q^{59} -8.57703e14 q^{61} +8.28749e14i q^{62} -2.81475e14 q^{64} +(2.10918e15 - 3.82751e15i) q^{65} -4.06389e14i q^{67} -1.05970e15i q^{68} +(-3.45279e15 - 1.90269e15i) q^{70} +9.53517e15 q^{71} -6.07142e15i q^{73} -1.47688e14 q^{74} +2.25411e15 q^{76} -8.79681e15i q^{77} -1.38305e16 q^{79} +(3.28565e15 + 1.81059e15i) q^{80} +1.45463e16i q^{82} +1.96416e16i q^{83} +(-6.81651e15 + 1.23698e16i) q^{85} -1.29906e15 q^{86} +8.37099e15i q^{88} +2.45918e16 q^{89} +8.82108e16 q^{91} +3.09722e16i q^{92} +8.50752e15 q^{94} +(-2.63122e16 - 1.44996e16i) q^{95} -3.63292e16i q^{97} -2.00215e16i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 524288 q^{4} + 1225560 q^{5} - 140779520 q^{10} - 146232096 q^{11} - 14260494336 q^{14} + 34359738368 q^{16} - 54264178080 q^{19} - 80318300160 q^{20} + 1013225778600 q^{25} + 2693383569408 q^{26}+ \cdots + 13\!\cdots\!00 q^{95}+O(q^{100}) $ 8 * q - 524288 * q^4 + 1225560 * q^5 - 140779520 * q^10 - 146232096 * q^11 - 14260494336 * q^14 + 34359738368 * q^16 - 54264178080 * q^19 - 80318300160 * q^20 + 1013225778600 * q^25 + 2693383569408 * q^26 - 1660243083120 * q^29 - 6055476993664 * q^31 + 14158246445056 * q^34 + 8725233780960 * q^35 + 9226126622720 * q^40 + 219921829971984 * q^41 + 9583466643456 * q^44 - 136753191067648 * q^46 + 797208944041464 * q^49 - 535409908531200 * q^50 + 1370380127176480 * q^55 + 934575756804096 * q^56 + 155410557761760 * q^59 - 6350444068968944 * q^61 - 2251799813685248 * q^64 - 10508144084782080 * q^65 - 1051117677240320 * q^70 + 33149026543733184 * q^71 - 6017648345726976 * q^74 + 3556257174650880 * q^76 - 19354668791708160 * q^79 + 5263740119285760 * q^80 - 71530234876895360 * q^85 - 29616028444434432 * q^86 + 43569230299558320 * q^89 + 333420307889971776 * q^91 - 11440296922292224 * q^94 + 131478039235197600 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/90\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$1$	$-1$

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 $

$2$			256.000i			0.707107i
$2$			256.000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−65536.0			−0.500000
$5$	765001.	+	421560.i	0.875824	+	0.482630i
$5$	765001.	+	421560.i	0.875824	+	0.482630i
$6$		0			0
$7$			1.76306e7i			1.15594i	0.816059	+	0.577969i	$0.196155\pi$
$7$			1.76306e7i			1.15594i	−0.816059	+	0.577969i	$0.803845\pi$
$8$		−	1.67772e7i		−	0.353553i
$9$		0			0
$10$	−1.07919e8	+	1.95840e8i	−0.341271	+	0.619301i
$11$	−4.98950e8			−0.701810			−0.350905	−	0.936411i	$-0.614126\pi$
$11$	−4.98950e8			−0.701810			−0.350905	+	0.936411i	$0.614126\pi$
$12$		0			0
$13$		−	5.00327e9i		−	1.70112i	−0.525877	−	0.850561i	$-0.676263\pi$
$13$		−	5.00327e9i		−	1.70112i	0.525877	−	0.850561i	$-0.323737\pi$
$14$	−4.51344e9			−0.817372
$15$		0			0
$16$	4.29497e9			0.250000
$17$			1.61697e10i			0.562194i	0.959679	+	0.281097i	$0.0906983\pi$
$17$			1.61697e10i			0.562194i	−0.959679	+	0.281097i	$0.909302\pi$
$18$		0			0
$19$	−3.43950e10			−0.464612			−0.232306	−	0.972643i	$-0.574627\pi$
$19$	−3.43950e10			−0.464612			−0.232306	+	0.972643i	$0.574627\pi$
$20$	−5.01351e10	−	2.76274e10i	−0.437912	−	0.241315i
$21$		0			0
$22$		−	1.27731e11i		−	0.496254i
$23$		−	4.72599e11i		−	1.25836i	−0.777258	−	0.629182i	$-0.783390\pi$
$23$		−	4.72599e11i		−	1.25836i	0.777258	−	0.629182i	$-0.216610\pi$
$24$		0			0
$25$	4.07513e11	+	6.44988e11i	0.534136	+	0.845399i
$26$	1.28084e12			1.20287
$27$		0			0
$28$		−	1.15544e12i		−	0.577969i
$29$	3.30807e12			1.22798			0.613990	−	0.789314i	$-0.289563\pi$
$29$	3.30807e12			1.22798			0.613990	+	0.789314i	$0.289563\pi$
$30$		0			0
$31$	3.23730e12			0.681724			0.340862	−	0.940113i	$-0.389281\pi$
$31$	3.23730e12			0.681724			0.340862	+	0.940113i	$0.389281\pi$
$32$			1.09951e12i			0.176777i
$33$		0			0
$34$	−4.13944e12			−0.397531
$35$	−7.43238e12	+	1.34875e13i	−0.557891	+	1.01240i
$36$		0			0
$37$			5.76908e11i			0.0270017i	0.999909	+	0.0135009i	$0.00429759\pi$
$37$			5.76908e11i			0.0270017i	−0.999909	+	0.0135009i	$0.995702\pi$
$38$		−	8.80513e12i		−	0.328530i
$39$		0			0
$40$	7.07261e12	−	1.28346e13i	0.170636	−	0.309651i
$41$	5.68215e13			1.11135			0.555674	−	0.831400i	$-0.312460\pi$
$41$	5.68215e13			1.11135			0.555674	+	0.831400i	$0.312460\pi$
$42$		0			0
$43$			5.07446e12i			0.0662076i	0.999452	+	0.0331038i	$0.0105392\pi$
$43$			5.07446e12i			0.0662076i	−0.999452	+	0.0331038i	$0.989461\pi$
$44$	3.26992e13			0.350905
$45$		0			0
$46$	1.20985e14			0.889798
$47$		−	3.32325e13i		−	0.203578i	−0.994806	−	0.101789i	$-0.967543\pi$
$47$		−	3.32325e13i		−	0.203578i	0.994806	−	0.101789i	$-0.0324567\pi$
$48$		0			0
$49$	−7.82089e13			−0.336193
$50$	−1.65117e14	+	1.04323e14i	−0.597787	+	0.377691i
$51$		0			0
$52$			3.27894e14i			0.850561i
$53$		−	6.11026e14i		−	1.34808i	−0.738696	−	0.674039i	$-0.764558\pi$
$53$		−	6.11026e14i		−	1.34808i	0.738696	−	0.674039i	$-0.235442\pi$
$54$		0			0
$55$	−3.81697e14	−	2.10338e14i	−0.614662	−	0.338715i
$56$	2.95793e14			0.408686
$57$		0			0
$58$			8.46865e14i			0.868313i
$59$	−8.63122e14			−0.765297			−0.382648	−	0.923894i	$-0.624988\pi$
$59$	−8.63122e14			−0.765297			−0.382648	+	0.923894i	$0.624988\pi$
$60$		0			0
$61$	−8.57703e14			−0.572840			−0.286420	−	0.958104i	$-0.592465\pi$
$61$	−8.57703e14			−0.572840			−0.286420	+	0.958104i	$0.592465\pi$
$62$			8.28749e14i			0.482051i
$63$		0			0
$64$	−2.81475e14			−0.125000
$65$	2.10918e15	−	3.82751e15i	0.821013	−	1.48988i
$66$		0			0
$67$		−	4.06389e14i		−	0.122266i	−0.998130	−	0.0611331i	$-0.980529\pi$
$67$		−	4.06389e14i		−	0.122266i	0.998130	−	0.0611331i	$-0.0194714\pi$
$68$		−	1.05970e15i		−	0.281097i
$69$		0			0
$70$	−3.45279e15	−	1.90269e15i	−0.715874	−	0.394488i
$71$	9.53517e15			1.75240			0.876198	−	0.481951i	$-0.160072\pi$
$71$	9.53517e15			1.75240			0.876198	+	0.481951i	$0.160072\pi$
$72$		0			0
$73$		−	6.07142e15i		−	0.881142i	−0.897718	−	0.440571i	$-0.854776\pi$
$73$		−	6.07142e15i		−	0.881142i	0.897718	−	0.440571i	$-0.145224\pi$
$74$	−1.47688e14			−0.0190931
$75$		0			0
$76$	2.25411e15			0.232306
$77$		−	8.79681e15i		−	0.811249i
$78$		0			0
$79$	−1.38305e16			−1.02567			−0.512836	−	0.858486i	$-0.671405\pi$
$79$	−1.38305e16			−1.02567			−0.512836	+	0.858486i	$0.671405\pi$
$80$	3.28565e15	+	1.81059e15i	0.218956	+	0.120658i
$81$		0			0
$82$			1.45463e16i			0.785842i
$83$			1.96416e16i			0.957222i	0.878027	+	0.478611i	$0.158860\pi$
$83$			1.96416e16i			0.957222i	−0.878027	+	0.478611i	$0.841140\pi$
$84$		0			0
$85$	−6.81651e15	+	1.23698e16i	−0.271332	+	0.492383i
$86$	−1.29906e15			−0.0468158
$87$		0			0
$88$			8.37099e15i			0.248127i
$89$	2.45918e16			0.662178			0.331089	−	0.943600i	$-0.392584\pi$
$89$	2.45918e16			0.662178			0.331089	+	0.943600i	$0.392584\pi$
$90$		0			0
$91$	8.82108e16			1.96639
$92$			3.09722e16i			0.629182i
$93$		0			0
$94$	8.50752e15			0.143951
$95$	−2.63122e16	−	1.44996e16i	−0.406918	−	0.224236i
$96$		0			0
$97$		−	3.63292e16i		−	0.470647i	−0.971917	−	0.235324i	$-0.924385\pi$
$97$		−	3.63292e16i		−	0.470647i	0.971917	−	0.235324i	$-0.0756150\pi$
$98$		−	2.00215e16i		−	0.237725i
$99$		0			0
$100$	−2.67068e16	−	4.22699e16i	−0.267068	−	0.422699i
$101$	1.67053e17			1.53505			0.767527	−	0.641017i	$-0.221487\pi$
$101$	1.67053e17			1.53505			0.767527	+	0.641017i	$0.221487\pi$
$102$		0			0
$103$			1.13488e16i			0.0882739i	0.999025	+	0.0441370i	$0.0140538\pi$
$103$			1.13488e16i			0.0882739i	−0.999025	+	0.0441370i	$0.985946\pi$
$104$	−8.39409e16			−0.601437
$105$		0			0
$106$	1.56423e17			0.953235
$107$		−	1.72393e17i		−	0.969970i	−0.874522	−	0.484985i	$-0.838825\pi$
$107$		−	1.72393e17i		−	0.969970i	0.874522	−	0.484985i	$-0.161175\pi$
$108$		0			0
$109$	3.17806e17			1.52770			0.763848	−	0.645396i	$-0.223307\pi$
$109$	3.17806e17			1.52770			0.763848	+	0.645396i	$0.223307\pi$
$110$	5.38464e16	−	9.77145e16i	0.239507	−	0.434632i
$111$		0			0
$112$			7.57230e16i			0.288985i
$113$			3.07271e17i			1.08731i	0.839307	+	0.543657i	$0.182961\pi$
$113$			3.07271e17i			1.08731i	−0.839307	+	0.543657i	$0.817039\pi$
$114$		0			0
$115$	1.99229e17	−	3.61539e17i	0.607325	−	1.10211i
$116$	−2.16798e17			−0.613990
$117$		0			0
$118$		−	2.20959e17i		−	0.541147i
$119$	−2.85082e17			−0.649862
$120$		0			0
$121$	−2.56496e17			−0.507463
$122$		−	2.19572e17i		−	0.405059i
$123$		0			0
$124$	−2.12160e17			−0.340862
$125$	3.98468e16	+	6.65208e17i	0.0597941	+	0.998211i
$126$		0			0
$127$		−	8.22015e17i		−	1.07782i	−0.842362	−	0.538912i	$-0.818835\pi$
$127$		−	8.22015e17i		−	1.07782i	0.842362	−	0.538912i	$-0.181165\pi$
$128$		−	7.20576e16i		−	0.0883883i
$129$		0			0
$130$	9.79841e17	+	5.39950e17i	1.05351	+	0.580544i
$131$	−2.64662e17			−0.266616			−0.133308	−	0.991075i	$-0.542560\pi$
$131$	−2.64662e17			−0.266616			−0.133308	+	0.991075i	$0.542560\pi$
$132$		0			0
$133$		−	6.06406e17i		−	0.537062i
$134$	1.04036e17			0.0864553
$135$		0			0
$136$	2.71283e17			0.198766
$137$			1.71830e17i			0.118297i	0.998249	+	0.0591486i	$0.0188386\pi$
$137$			1.71830e17i			0.118297i	−0.998249	+	0.0591486i	$0.981161\pi$
$138$		0			0
$139$	−2.86799e18			−1.74563			−0.872814	−	0.488053i	$-0.837707\pi$
$139$	−2.86799e18			−1.74563			−0.872814	+	0.488053i	$0.837707\pi$
$140$	4.87088e17	−	8.83914e17i	0.278945	−	0.506199i
$141$		0			0
$142$			2.44100e18i			1.23913i
$143$			2.49638e18i			1.19386i
$144$		0			0
$145$	2.53067e18	+	1.39455e18i	1.07549	+	0.592661i
$146$	1.55428e18			0.623061
$147$		0			0
$148$		−	3.78082e16i		−	0.0135009i
$149$	2.17323e18			0.732863			0.366431	−	0.930445i	$-0.380579\pi$
$149$	2.17323e18			0.732863			0.366431	+	0.930445i	$0.380579\pi$
$150$		0			0
$151$	1.81870e18			0.547591			0.273796	−	0.961788i	$-0.411721\pi$
$151$	1.81870e18			0.547591			0.273796	+	0.961788i	$0.411721\pi$
$152$			5.77053e17i			0.164265i
$153$		0			0
$154$	2.25198e18			0.573639
$155$	2.47654e18	+	1.36472e18i	0.597070	+	0.329021i
$156$		0			0
$157$			6.73380e18i			1.45584i	0.685663	+	0.727919i	$0.259512\pi$
$157$			6.73380e18i			1.45584i	−0.685663	+	0.727919i	$0.740488\pi$
$158$		−	3.54062e18i		−	0.725260i
$159$		0			0
$160$	−4.63510e17	+	8.41127e17i	−0.0853178	+	0.154825i
$161$	8.33222e18			1.45459
$162$		0			0
$163$		−	1.37505e18i		−	0.216134i	−0.994144	−	0.108067i	$-0.965534\pi$
$163$		−	1.37505e18i		−	0.216134i	0.994144	−	0.108067i	$-0.0344661\pi$
$164$	−3.72386e18			−0.555674
$165$		0			0
$166$	−5.02825e18			−0.676858
$167$			5.07885e18i			0.649644i	0.945775	+	0.324822i	$0.105304\pi$
$167$			5.07885e18i			0.649644i	−0.945775	+	0.324822i	$0.894696\pi$
$168$		0			0
$169$	−1.63823e19			−1.89381
$170$	−3.16668e18	−	1.74503e18i	−0.348168	−	0.191861i
$171$		0			0
$172$		−	3.32560e17i		−	0.0331038i
$173$			1.55546e19i			1.47390i	0.675948	+	0.736949i	$0.263734\pi$
$173$			1.55546e19i			1.47390i	−0.675948	+	0.736949i	$0.736266\pi$
$174$		0			0
$175$	−1.13715e19	+	7.18472e18i	−0.977229	+	0.617428i
$176$	−2.14297e18			−0.175452
$177$		0			0
$178$			6.29549e18i			0.468230i
$179$	−6.70360e18			−0.475398			−0.237699	−	0.971339i	$-0.576393\pi$
$179$	−6.70360e18			−0.475398			−0.237699	+	0.971339i	$0.576393\pi$
$180$		0			0
$181$	1.39268e19			0.898637			0.449318	−	0.893372i	$-0.351667\pi$
$181$	1.39268e19			0.898637			0.449318	+	0.893372i	$0.351667\pi$
$182$			2.25820e19i			1.39045i
$183$		0			0
$184$	−7.92890e18			−0.444899
$185$	−2.43201e17	+	4.41335e17i	−0.0130319	+	0.0236488i
$186$		0			0
$187$		−	8.06788e18i		−	0.394553i
$188$			2.17792e18i			0.101789i
$189$		0			0
$190$	3.71189e18	−	6.73593e18i	0.158559	−	0.287735i
$191$	1.88045e18			0.0768205			0.0384103	−	0.999262i	$-0.487771\pi$
$191$	1.88045e18			0.0768205			0.0384103	+	0.999262i	$0.487771\pi$
$192$		0			0
$193$		−	2.08963e18i		−	0.0781325i	−0.999237	−	0.0390663i	$-0.987562\pi$
$193$		−	2.08963e18i		−	0.0781325i	0.999237	−	0.0390663i	$-0.0124383\pi$
$194$	9.30026e18			0.332798
$195$		0			0
$196$	5.12550e18			0.168097
$197$		−	3.11134e18i		−	0.0977202i	−0.998806	−	0.0488601i	$-0.984441\pi$
$197$		−	3.11134e18i		−	0.0977202i	0.998806	−	0.0488601i	$-0.0155588\pi$
$198$		0			0
$199$	2.90090e19			0.836144			0.418072	−	0.908414i	$-0.362706\pi$
$199$	2.90090e19			0.836144			0.418072	+	0.908414i	$0.362706\pi$
$200$	1.08211e19	−	6.83694e18i	0.298894	−	0.188846i
$201$		0			0
$202$			4.27656e19i			1.08545i
$203$			5.83233e19i			1.41947i
$204$		0			0
$205$	4.34685e19	+	2.39537e19i	0.973345	+	0.536370i
$206$	−2.90529e18			−0.0624191
$207$		0			0
$208$		−	2.14889e19i		−	0.425280i
$209$	1.71614e19			0.326069
$210$		0			0
$211$	3.27307e19			0.573528			0.286764	−	0.958001i	$-0.407420\pi$
$211$	3.27307e19			0.573528			0.286764	+	0.958001i	$0.407420\pi$
$212$			4.00442e19i			0.674039i
$213$		0			0
$214$	4.41327e19			0.685873
$215$	−2.13919e18	+	3.88197e18i	−0.0319538	+	0.0579862i
$216$		0			0
$217$			5.70756e19i			0.788031i
$218$			8.13584e19i			1.08024i
$219$		0			0
$220$	2.50149e19	+	1.37847e19i	0.307331	+	0.169357i
$221$	8.09014e19			0.956361
$222$		0			0
$223$			4.94295e19i			0.541246i	0.962685	+	0.270623i	$0.0872297\pi$
$223$			4.94295e19i			0.541246i	−0.962685	+	0.270623i	$0.912770\pi$
$224$	−1.93851e19			−0.204343
$225$		0			0
$226$	−7.86615e19			−0.768848
$227$			1.02192e20i			0.962052i	0.876706	+	0.481026i	$0.159736\pi$
$227$			1.02192e20i			0.962052i	−0.876706	+	0.481026i	$0.840264\pi$
$228$		0			0
$229$	1.74856e20			1.52785			0.763923	−	0.645308i	$-0.223271\pi$
$229$	1.74856e20			1.52785			0.763923	+	0.645308i	$0.223271\pi$
$230$	9.25539e19	+	5.10026e19i	0.779306	+	0.429443i
$231$		0			0
$232$		−	5.55002e19i		−	0.434157i
$233$			1.22044e20i			0.920430i	0.887807	+	0.460215i	$0.152228\pi$
$233$			1.22044e20i			0.920430i	−0.887807	+	0.460215i	$0.847772\pi$
$234$		0			0
$235$	1.40095e19	−	2.54229e19i	0.0982530	−	0.178299i
$236$	5.65656e19			0.382648
$237$		0			0
$238$		−	7.29811e19i		−	0.459522i
$239$	2.03446e20			1.23614			0.618069	−	0.786124i	$-0.287915\pi$
$239$	2.03446e20			1.23614			0.618069	+	0.786124i	$0.287915\pi$
$240$		0			0
$241$	−2.15084e20			−1.21749			−0.608743	−	0.793368i	$-0.708326\pi$
$241$	−2.15084e20			−1.21749			−0.608743	+	0.793368i	$0.708326\pi$
$242$		−	6.56629e19i		−	0.358831i
$243$		0			0
$244$	5.62104e19			0.286420
$245$	−5.98298e19	−	3.29697e19i	−0.294446	−	0.162257i
$246$		0			0
$247$			1.72088e20i			0.790361i
$248$		−	5.43129e19i		−	0.241026i
$249$		0			0
$250$	−1.70293e20	+	1.02008e19i	−0.705842	+	0.0422808i
$251$	1.40847e19			0.0564315			0.0282158	−	0.999602i	$-0.491017\pi$
$251$	1.40847e19			0.0564315			0.0282158	+	0.999602i	$0.491017\pi$
$252$		0			0
$253$			2.35803e20i			0.883132i
$254$	2.10436e20			0.762137
$255$		0			0
$256$	1.84467e19			0.0625000
$257$			1.66925e20i			0.547129i	0.961854	+	0.273564i	$0.0882026\pi$
$257$			1.66925e20i			0.547129i	−0.961854	+	0.273564i	$0.911797\pi$
$258$		0			0
$259$	−1.01712e19			−0.0312123
$260$	−1.38227e20	+	2.50839e20i	−0.410506	+	0.744942i
$261$		0			0
$262$		−	6.77536e19i		−	0.188526i
$263$		−	2.85711e20i		−	0.769667i	−0.922986	−	0.384834i	$-0.874259\pi$
$263$		−	2.85711e20i		−	0.769667i	0.922986	−	0.384834i	$-0.125741\pi$
$264$		0			0
$265$	2.57584e20	−	4.67435e20i	0.650623	−	1.18068i
$266$	1.55240e20			0.379760
$267$		0			0
$268$			2.66331e19i			0.0611331i
$269$	4.15461e20			0.923923			0.461962	−	0.886900i	$-0.347146\pi$
$269$	4.15461e20			0.923923			0.461962	+	0.886900i	$0.347146\pi$
$270$		0			0
$271$	2.65437e20			0.554272			0.277136	−	0.960831i	$-0.410615\pi$
$271$	2.65437e20			0.554272			0.277136	+	0.960831i	$0.410615\pi$
$272$			6.94484e19i			0.140549i
$273$		0			0
$274$	−4.39885e19			−0.0836487
$275$	−2.03329e20	−	3.21817e20i	−0.374862	−	0.593309i
$276$		0			0
$277$			1.04764e21i			1.81608i	0.418886	+	0.908039i	$0.362421\pi$
$277$			1.04764e21i			1.81608i	−0.418886	+	0.908039i	$0.637579\pi$
$278$		−	7.34205e20i		−	1.23435i
$279$		0			0
$280$	2.26282e20	+	1.24695e20i	0.357937	+	0.197244i
$281$	4.31210e20			0.661736			0.330868	−	0.943677i	$-0.392658\pi$
$281$	4.31210e20			0.661736			0.330868	+	0.943677i	$0.392658\pi$
$282$		0			0
$283$		−	5.09227e20i		−	0.735745i	−0.929876	−	0.367872i	$-0.880086\pi$
$283$		−	5.09227e20i		−	0.735745i	0.929876	−	0.367872i	$-0.119914\pi$
$284$	−6.24897e20			−0.876198
$285$		0			0
$286$	−6.39074e20			−0.844189
$287$			1.00180e21i			1.28465i
$288$		0			0
$289$	5.65781e20			0.683938
$290$	−3.57005e20	+	6.47853e20i	−0.419074	+	0.760490i
$291$		0			0
$292$			3.97896e20i			0.440571i
$293$			6.39057e20i			0.687329i	0.939093	+	0.343664i	$0.111668\pi$
$293$			6.39057e20i			0.687329i	−0.939093	+	0.343664i	$0.888332\pi$
$294$		0			0
$295$	−6.60289e20	−	3.63858e20i	−0.670266	−	0.369356i
$296$	9.67890e18			0.00954655
$297$		0			0
$298$			5.56347e20i			0.518212i
$299$	−2.36454e21			−2.14063
$300$		0			0
$301$	−8.94659e19			−0.0765319
$302$			4.65587e20i			0.387205i
$303$		0			0
$304$	−1.47726e20			−0.116153
$305$	−6.56144e20	−	3.61574e20i	−0.501707	−	0.276470i
$306$		0			0
$307$		−	2.65100e21i		−	1.91749i	−0.284265	−	0.958746i	$-0.591749\pi$
$307$		−	2.65100e21i		−	1.91749i	0.284265	−	0.958746i	$-0.408251\pi$
$308$			5.76508e20i			0.405624i
$309$		0			0
$310$	−3.49367e20	+	6.33993e20i	−0.232653	+	0.422192i
$311$	2.17669e21			1.41037			0.705185	−	0.709023i	$-0.250864\pi$
$311$	2.17669e21			1.41037			0.705185	+	0.709023i	$0.250864\pi$
$312$		0			0
$313$			6.47134e19i			0.0397070i	0.999803	+	0.0198535i	$0.00631998\pi$
$313$			6.47134e19i			0.0397070i	−0.999803	+	0.0198535i	$0.993680\pi$
$314$	−1.72385e21			−1.02943
$315$		0			0
$316$	9.06398e20			0.512836
$317$		−	3.20908e21i		−	1.76757i	−0.467893	−	0.883785i	$-0.654987\pi$
$317$		−	3.20908e21i		−	1.76757i	0.467893	−	0.883785i	$-0.345013\pi$
$318$		0			0
$319$	−1.65056e21			−0.861808
$320$	−2.15329e20	−	1.18659e20i	−0.109478	−	0.0603288i
$321$		0			0
$322$			2.13305e21i			1.02855i
$323$		−	5.56158e20i		−	0.261202i
$324$		0			0
$325$	3.22705e21	−	2.03890e21i	1.43813	−	0.908630i
$326$	3.52012e20			0.152830
$327$		0			0
$328$		−	9.53307e20i		−	0.392921i
$329$	5.85910e20			0.235324
$330$		0			0
$331$	−4.59189e21			−1.75167			−0.875837	−	0.482608i	$-0.839690\pi$
$331$	−4.59189e21			−1.75167			−0.875837	+	0.482608i	$0.839690\pi$
$332$		−	1.28723e21i		−	0.478611i
$333$		0			0
$334$	−1.30019e21			−0.459368
$335$	1.71318e20	−	3.10888e20i	0.0590094	−	0.107084i
$336$		0			0
$337$		−	3.38339e21i		−	1.10789i	−0.832552	−	0.553946i	$-0.813121\pi$
$337$		−	3.38339e21i		−	1.10789i	0.832552	−	0.553946i	$-0.186879\pi$
$338$		−	4.19386e21i		−	1.33913i
$339$		0			0
$340$	4.46727e20	−	8.10670e20i	0.135666	−	0.246192i
$341$	−1.61525e21			−0.478440
$342$		0			0
$343$			2.72255e21i			0.767319i
$344$	8.51353e19			0.0234079
$345$		0			0
$346$	−3.98198e21			−1.04220
$347$			7.94764e20i			0.202973i	0.994837	+	0.101486i	$0.0323598\pi$
$347$			7.94764e20i			0.202973i	−0.994837	+	0.101486i	$0.967640\pi$
$348$		0			0
$349$	5.52679e20			0.134418			0.0672089	−	0.997739i	$-0.478591\pi$
$349$	5.52679e20			0.134418			0.0672089	+	0.997739i	$0.478591\pi$
$350$	−1.83929e21	−	2.91112e21i	−0.436588	−	0.691005i
$351$		0			0
$352$		−	5.48601e20i		−	0.124064i
$353$		−	4.24483e21i		−	0.937076i	−0.883443	−	0.468538i	$-0.844781\pi$
$353$		−	4.24483e21i		−	0.937076i	0.883443	−	0.468538i	$-0.155219\pi$
$354$		0			0
$355$	7.29441e21	+	4.01965e21i	1.53479	+	0.845760i
$356$	−1.61165e21			−0.331089
$357$		0			0
$358$		−	1.71612e21i		−	0.336157i
$359$	−6.15257e21			−1.17694			−0.588469	−	0.808520i	$-0.700269\pi$
$359$	−6.15257e21			−1.17694			−0.588469	+	0.808520i	$0.700269\pi$
$360$		0			0
$361$	−4.29737e21			−0.784136
$362$			3.56527e21i			0.635432i
$363$		0			0
$364$	−5.78098e21			−0.983196
$365$	2.55947e21	−	4.64464e21i	0.425266	−	0.771725i
$366$		0			0
$367$			1.53580e21i			0.243598i	0.992555	+	0.121799i	$0.0388664\pi$
$367$			1.53580e21i			0.243598i	−0.992555	+	0.121799i	$0.961134\pi$
$368$		−	2.02980e21i		−	0.314591i
$369$		0			0
$370$	−1.12982e20	−	6.22595e19i	−0.0167222	−	0.00921491i
$371$	1.07728e22			1.55829
$372$		0			0
$373$		−	1.00414e22i		−	1.38762i	−0.720160	−	0.693808i	$-0.755931\pi$
$373$		−	1.00414e22i		−	1.38762i	0.720160	−	0.693808i	$-0.244069\pi$
$374$	2.06538e21			0.278991
$375$		0			0
$376$	−5.57549e20			−0.0719757
$377$		−	1.65511e22i		−	2.08894i
$378$		0			0
$379$	1.19831e22			1.44589			0.722944	−	0.690907i	$-0.242788\pi$
$379$	1.19831e22			1.44589			0.722944	+	0.690907i	$0.242788\pi$
$380$	1.72440e21	+	9.50244e20i	0.203459	+	0.112118i
$381$		0			0
$382$			4.81394e20i			0.0543203i
$383$			1.12305e22i			1.23940i	0.784839	+	0.619699i	$0.212745\pi$
$383$			1.12305e22i			1.23940i	−0.784839	+	0.619699i	$0.787255\pi$
$384$		0			0
$385$	3.70839e21	−	6.72957e21i	0.391533	−	0.710511i
$386$	5.34945e20			0.0552480
$387$		0			0
$388$			2.38087e21i			0.235324i
$389$	−2.54158e21			−0.245772			−0.122886	−	0.992421i	$-0.539215\pi$
$389$	−2.54158e21			−0.245772			−0.122886	+	0.992421i	$0.539215\pi$
$390$		0			0
$391$	7.64179e21			0.707445
$392$			1.31213e21i			0.118862i
$393$		0			0
$394$	7.96503e20			0.0690986
$395$	−1.05804e22	−	5.83041e21i	−0.898309	−	0.495021i
$396$		0			0
$397$		−	3.06633e21i		−	0.249402i	−0.992194	−	0.124701i	$-0.960203\pi$
$397$		−	3.06633e21i		−	0.249402i	0.992194	−	0.124701i	$-0.0397972\pi$
$398$			7.42630e21i			0.591243i
$399$		0			0
$400$	1.75026e21	+	2.77020e21i	0.133534	+	0.211350i
$401$	1.31178e22			0.979793			0.489897	−	0.871781i	$-0.337035\pi$
$401$	1.31178e22			0.979793			0.489897	+	0.871781i	$0.337035\pi$
$402$		0			0
$403$		−	1.61971e22i		−	1.15969i
$404$	−1.09480e22			−0.767527
$405$		0			0
$406$	−1.49308e22			−1.00372
$407$		−	2.87848e20i		−	0.0189501i
$408$		0			0
$409$	−2.29036e21			−0.144629			−0.0723146	−	0.997382i	$-0.523039\pi$
$409$	−2.29036e21			−0.144629			−0.0723146	+	0.997382i	$0.523039\pi$
$410$	−6.13215e21	+	1.11279e22i	−0.379271	+	0.688259i
$411$		0			0
$412$		−	7.43753e20i		−	0.0441370i
$413$		−	1.52174e22i		−	0.884636i
$414$		0			0
$415$	−8.28011e21	+	1.50258e22i	−0.461984	+	0.838358i
$416$	5.50115e21			0.300719
$417$		0			0
$418$			4.39332e21i			0.230566i
$419$	−9.99153e21			−0.513822			−0.256911	−	0.966435i	$-0.582705\pi$
$419$	−9.99153e21			−0.513822			−0.256911	+	0.966435i	$0.582705\pi$
$420$		0			0
$421$	2.42321e22			1.19672			0.598361	−	0.801227i	$-0.295819\pi$
$421$	2.42321e22			1.19672			0.598361	+	0.801227i	$0.295819\pi$
$422$			8.37906e21i			0.405546i
$423$		0			0
$424$	−1.02513e22			−0.476617
$425$	−1.04293e22	+	6.58937e21i	−0.475278	+	0.300288i
$426$		0			0
$427$		−	1.51219e22i		−	0.662168i
$428$			1.12980e22i			0.484985i
$429$		0			0
$430$	−9.93783e20	−	5.47633e20i	−0.0410024	−	0.0225947i
$431$	3.74076e21			0.151322			0.0756611	−	0.997134i	$-0.475893\pi$
$431$	3.74076e21			0.151322			0.0756611	+	0.997134i	$0.475893\pi$
$432$		0			0
$433$			3.00276e22i			1.16781i	0.811820	+	0.583907i	$0.198477\pi$
$433$			3.00276e22i			1.16781i	−0.811820	+	0.583907i	$0.801523\pi$
$434$	−1.46114e22			−0.557222
$435$		0			0
$436$	−2.08277e22			−0.763848
$437$			1.62551e22i			0.584650i
$438$		0			0
$439$	4.88091e22			1.68870			0.844349	−	0.535793i	$-0.179987\pi$
$439$	4.88091e22			1.68870			0.844349	+	0.535793i	$0.179987\pi$
$440$	−3.52888e21	+	6.40382e21i	−0.119754	+	0.217316i
$441$		0			0
$442$			2.07108e22i			0.676249i
$443$			8.85676e20i			0.0283689i	0.999899	+	0.0141845i	$0.00451521\pi$
$443$			8.85676e20i			0.0283689i	−0.999899	+	0.0141845i	$0.995485\pi$
$444$		0			0
$445$	1.88127e22	+	1.03669e22i	0.579951	+	0.319587i
$446$	−1.26540e22			−0.382719
$447$		0			0
$448$		−	4.96258e21i		−	0.144492i
$449$	1.07587e18			3.07374e−5			1.53687e−5	−	1.00000i	$-0.499995\pi$
$449$	1.07587e18			3.07374e−5			1.53687e−5		1.00000i	$0.499995\pi$
$450$		0			0
$451$	−2.83511e22			−0.779955
$452$		−	2.01373e22i		−	0.543657i
$453$		0			0
$454$	−2.61612e22			−0.680274
$455$	6.74814e22	+	3.71862e22i	1.72221	+	0.949040i
$456$		0			0
$457$			4.30315e21i			0.105803i	0.998600	+	0.0529016i	$0.0168470\pi$
$457$			4.30315e21i			0.105803i	−0.998600	+	0.0529016i	$0.983153\pi$
$458$			4.47632e22i			1.08035i
$459$		0			0
$460$	−1.30567e22	+	2.36938e22i	−0.303662	+	0.551053i
$461$	3.48054e22			0.794674			0.397337	−	0.917673i	$-0.369934\pi$
$461$	3.48054e22			0.794674			0.397337	+	0.917673i	$0.369934\pi$
$462$		0			0
$463$		−	2.06561e21i		−	0.0454579i	−0.999742	−	0.0227290i	$-0.992765\pi$
$463$		−	2.06561e21i		−	0.0454579i	0.999742	−	0.0227290i	$-0.00723548\pi$
$464$	1.42080e22			0.306995
$465$		0			0
$466$	−3.12432e22			−0.650843
$467$			2.27698e22i			0.465763i	0.972505	+	0.232882i	$0.0748154\pi$
$467$			2.27698e22i			0.465763i	−0.972505	+	0.232882i	$0.925185\pi$
$468$		0			0
$469$	7.16490e21			0.141332
$470$	6.50826e21	+	3.58643e21i	0.126076	+	0.0694753i
$471$		0			0
$472$			1.44808e22i			0.270573i
$473$		−	2.53190e21i		−	0.0464651i
$474$		0			0
$475$	−1.40164e22	−	2.21844e22i	−0.248166	−	0.392782i
$476$	1.86831e22			0.324931
$477$		0			0
$478$			5.20822e22i			0.874081i
$479$	2.90207e22			0.478471			0.239236	−	0.970962i	$-0.423103\pi$
$479$	2.90207e22			0.478471			0.239236	+	0.970962i	$0.423103\pi$
$480$		0			0
$481$	2.88642e21			0.0459332
$482$		−	5.50616e22i		−	0.860892i
$483$		0			0
$484$	1.68097e22			0.253732
$485$	1.53149e22	−	2.77918e22i	0.227149	−	0.412204i
$486$		0			0
$487$		−	2.68073e22i		−	0.383934i	−0.981401	−	0.191967i	$-0.938513\pi$
$487$		−	2.68073e22i		−	0.383934i	0.981401	−	0.191967i	$-0.0614866\pi$
$488$			1.43899e22i			0.202530i
$489$		0			0
$490$	8.44026e21	−	1.53164e22i	0.114733	−	0.208205i
$491$	9.88548e22			1.32070			0.660351	−	0.750957i	$-0.270407\pi$
$491$	9.88548e22			1.32070			0.660351	+	0.750957i	$0.270407\pi$
$492$		0			0
$493$			5.34905e22i			0.690363i
$494$	−4.40544e22			−0.558869
$495$		0			0
$496$	1.39041e22			0.170431
$497$			1.68111e23i			2.02566i
$498$		0			0
$499$	−1.20742e23			−1.40606			−0.703030	−	0.711160i	$-0.748170\pi$
$499$	−1.20742e23			−1.40606			−0.703030	+	0.711160i	$0.748170\pi$
$500$	−2.61140e21	−	4.35951e22i	−0.0298970	−	0.499105i
$501$		0			0
$502$			3.60569e21i			0.0399031i
$503$			7.15920e22i			0.778998i	0.921027	+	0.389499i	$0.127352\pi$
$503$			7.15920e22i			0.778998i	−0.921027	+	0.389499i	$0.872648\pi$
$504$		0			0
$505$	1.27796e23	+	7.04230e22i	1.34444	+	0.740863i
$506$	−6.03657e22			−0.624469
$507$		0			0
$508$			5.38716e22i			0.538912i
$509$	−9.08719e22			−0.893981			−0.446991	−	0.894539i	$-0.647504\pi$
$509$	−9.08719e22			−0.893981			−0.446991	+	0.894539i	$0.647504\pi$
$510$		0			0
$511$	1.07043e23			1.01855
$512$			4.72237e21i			0.0441942i
$513$		0			0
$514$	−4.27328e22			−0.386878
$515$	−4.78419e21	+	8.68182e21i	−0.0426037	+	0.0773124i
$516$		0			0
$517$			1.65814e22i			0.142873i
$518$		−	2.60384e21i		−	0.0220705i
$519$		0			0
$520$	−6.42149e22	−	3.53862e22i	−0.526753	−	0.290272i
$521$	−3.85520e22			−0.311119			−0.155559	−	0.987827i	$-0.549718\pi$
$521$	−3.85520e22			−0.311119			−0.155559	+	0.987827i	$0.549718\pi$
$522$		0			0
$523$		−	1.28297e23i		−	1.00219i	−0.865391	−	0.501097i	$-0.832930\pi$
$523$		−	1.28297e23i		−	1.00219i	0.865391	−	0.501097i	$-0.167070\pi$
$524$	1.73449e22			0.133308
$525$		0			0
$526$	7.31420e22			0.544237
$527$			5.23462e22i			0.383261i
$528$		0			0
$529$	−8.22998e22			−0.583479
$530$	1.19663e23	+	6.59416e22i	0.834866	+	0.460060i
$531$		0			0
$532$			3.97414e22i			0.268531i
$533$		−	2.84293e23i		−	1.89054i
$534$		0			0
$535$	7.26742e22	−	1.31881e23i	0.468137	−	0.849523i
$536$	−6.81808e21			−0.0432276
$537$		0			0
$538$			1.06358e23i			0.653312i
$539$	3.90223e22			0.235944
$540$		0			0
$541$	3.40003e23			1.99207			0.996037	−	0.0889412i	$-0.0283483\pi$
$541$	3.40003e23			1.99207			0.996037	+	0.0889412i	$0.0283483\pi$
$542$			6.79519e22i			0.391929i
$543$		0			0
$544$	−1.77788e22			−0.0993828
$545$	2.43122e23	+	1.33974e23i	1.33799	+	0.737313i
$546$		0			0
$547$		−	2.66900e23i		−	1.42383i	−0.702268	−	0.711913i	$-0.747829\pi$
$547$		−	2.66900e23i		−	1.42383i	0.702268	−	0.711913i	$-0.252171\pi$
$548$		−	1.12611e22i		−	0.0591486i
$549$		0			0
$550$	8.23851e22	−	5.20522e22i	0.419533	−	0.265067i
$551$	−1.13781e23			−0.570534
$552$		0			0
$553$		−	2.43841e23i		−	1.18561i
$554$	−2.68196e23			−1.28416
$555$		0			0
$556$	1.87957e23			0.872814
$557$		−	2.58262e22i		−	0.118111i	−0.998255	−	0.0590555i	$-0.981191\pi$
$557$		−	2.58262e22i		−	0.118111i	0.998255	−	0.0590555i	$-0.0188089\pi$
$558$		0			0
$559$	2.53889e22			0.112627
$560$	−3.19218e22	+	5.79282e22i	−0.139473	+	0.253100i
$561$		0			0
$562$			1.10390e23i			0.467918i
$563$		−	1.41028e22i		−	0.0588822i	−0.999567	−	0.0294411i	$-0.990627\pi$
$563$		−	1.41028e22i		−	0.0588822i	0.999567	−	0.0294411i	$-0.00937275\pi$
$564$		0			0
$565$	−1.29533e23	+	2.35063e23i	−0.524771	+	0.952297i
$566$	1.30362e23			0.520250
$567$		0			0
$568$		−	1.59974e23i		−	0.619566i
$569$	−2.13811e23			−0.815785			−0.407892	−	0.913030i	$-0.633736\pi$
$569$	−2.13811e23			−0.815785			−0.407892	+	0.913030i	$0.633736\pi$
$570$		0			0
$571$	1.13718e23			0.421136			0.210568	−	0.977579i	$-0.432469\pi$
$571$	1.13718e23			0.421136			0.210568	+	0.977579i	$0.432469\pi$
$572$		−	1.63603e23i		−	0.596932i
$573$		0			0
$574$	−2.56461e23			−0.908385
$575$	3.04821e23	−	1.92590e23i	1.06382	−	0.672137i
$576$		0			0
$577$			2.79237e23i			0.946191i	0.881011	+	0.473096i	$0.156864\pi$
$577$			2.79237e23i			0.946191i	−0.881011	+	0.473096i	$0.843136\pi$
$578$			1.44840e23i			0.483617i
$579$		0			0
$580$	−1.65850e23	−	9.13932e22i	−0.537747	−	0.296330i
$581$	−3.46294e23			−1.10649
$582$		0			0
$583$			3.04872e23i			0.946094i
$584$	−1.01861e23			−0.311531
$585$		0			0
$586$	−1.63599e23			−0.486015
$587$		−	6.02342e23i		−	1.76368i	−0.471549	−	0.881840i	$-0.656305\pi$
$587$		−	6.02342e23i		−	1.76368i	0.471549	−	0.881840i	$-0.343695\pi$
$588$		0			0
$589$	−1.11347e23			−0.316737
$590$	9.31476e22	−	1.69034e23i	0.261174	−	0.473949i
$591$		0			0
$592$			2.47780e21i			0.00675043i
$593$		−	3.93286e23i		−	1.05619i	−0.849184	−	0.528097i	$-0.822906\pi$
$593$		−	3.93286e23i		−	1.05619i	0.849184	−	0.528097i	$-0.177094\pi$
$594$		0			0
$595$	−2.18088e23	−	1.20179e23i	−0.569165	−	0.313643i
$596$	−1.42425e23			−0.366431
$597$		0			0
$598$		−	6.05322e23i		−	1.51365i
$599$	−3.72267e23			−0.917754			−0.458877	−	0.888500i	$-0.651748\pi$
$599$	−3.72267e23			−0.917754			−0.458877	+	0.888500i	$0.651748\pi$
$600$		0			0
$601$	−2.05419e23			−0.492275			−0.246138	−	0.969235i	$-0.579162\pi$
$601$	−2.05419e23			−0.492275			−0.246138	+	0.969235i	$0.579162\pi$
$602$		−	2.29033e22i		−	0.0541162i
$603$		0			0
$604$	−1.19190e23			−0.273796
$605$	−1.96219e23	−	1.08128e23i	−0.444449	−	0.244917i
$606$		0			0
$607$		−	6.45041e22i		−	0.142064i	−0.997474	−	0.0710319i	$-0.977371\pi$
$607$		−	6.45041e22i		−	0.142064i	0.997474	−	0.0710319i	$-0.0226292\pi$
$608$		−	3.78177e22i		−	0.0821325i
$609$		0			0
$610$	9.25628e22	−	1.67973e23i	0.195494	−	0.354761i
$611$	−1.66271e23			−0.346311
$612$		0			0
$613$		−	8.33557e22i		−	0.168858i	−0.996430	−	0.0844289i	$-0.973093\pi$
$613$		−	8.33557e22i		−	0.168858i	0.996430	−	0.0844289i	$-0.0269066\pi$
$614$	6.78656e23			1.35587
$615$		0			0
$616$	−1.47586e23			−0.286820
$617$		−	1.53109e23i		−	0.293479i	−0.989175	−	0.146739i	$-0.953122\pi$
$617$		−	1.53109e23i		−	0.293479i	0.989175	−	0.146739i	$-0.0468778\pi$
$618$		0			0
$619$	−9.76593e23			−1.82114			−0.910569	−	0.413356i	$-0.864356\pi$
$619$	−9.76593e23			−1.82114			−0.910569	+	0.413356i	$0.864356\pi$
$620$	−1.62302e23	−	8.94381e22i	−0.298535	−	0.164510i
$621$		0			0
$622$			5.57233e23i			0.997283i
$623$			4.33568e23i			0.765437i
$624$		0			0
$625$	−2.49942e23	+	5.25682e23i	−0.429398	+	0.903116i
$626$	−1.65666e22			−0.0280771
$627$		0			0
$628$		−	4.41306e23i		−	0.727919i
$629$	−9.32843e21			−0.0151802
$630$		0			0
$631$	−6.02040e23			−0.953621			−0.476811	−	0.879006i	$-0.658207\pi$
$631$	−6.02040e23			−0.953621			−0.476811	+	0.879006i	$0.658207\pi$
$632$			2.32038e23i			0.362630i
$633$		0			0
$634$	8.21524e23			1.24986
$635$	3.46529e23	−	6.28842e23i	0.520191	−	0.943985i
$636$		0			0
$637$			3.91300e23i			0.571906i
$638$		−	4.22544e23i		−	0.609391i
$639$		0			0
$640$	3.03766e22	−	5.51241e22i	0.0426589	−	0.0774126i
$641$	−8.13646e23			−1.12757			−0.563783	−	0.825923i	$-0.690655\pi$
$641$	−8.13646e23			−1.12757			−0.563783	+	0.825923i	$0.690655\pi$
$642$		0			0
$643$		−	1.02063e23i		−	0.137745i	−0.997625	−	0.0688726i	$-0.978060\pi$
$643$		−	1.02063e23i		−	0.137745i	0.997625	−	0.0688726i	$-0.0219402\pi$
$644$	−5.46060e23			−0.727295
$645$		0			0
$646$	1.42376e23			0.184698
$647$			9.71752e23i			1.24414i	0.782962	+	0.622070i	$0.213708\pi$
$647$			9.71752e23i			1.24414i	−0.782962	+	0.622070i	$0.786292\pi$
$648$		0			0
$649$	4.30655e23			0.537093
$650$	5.21958e23	+	8.26124e23i	0.642498	+	1.01691i
$651$		0			0
$652$			9.01151e22i			0.108067i
$653$		−	5.73338e23i		−	0.678655i	−0.940668	−	0.339327i	$-0.889801\pi$
$653$		−	5.73338e23i		−	0.678655i	0.940668	−	0.339327i	$-0.110199\pi$
$654$		0			0
$655$	−2.02467e23	−	1.11571e23i	−0.233509	−	0.128677i
$656$	2.44047e23			0.277837
$657$		0			0
$658$			1.49993e23i			0.166399i
$659$	6.64180e22			0.0727377			0.0363689	−	0.999338i	$-0.488421\pi$
$659$	6.64180e22			0.0727377			0.0363689	+	0.999338i	$0.488421\pi$
$660$		0			0
$661$	−2.25960e23			−0.241168			−0.120584	−	0.992703i	$-0.538477\pi$
$661$	−2.25960e23			−0.241168			−0.120584	+	0.992703i	$0.538477\pi$
$662$		−	1.17552e24i		−	1.23862i
$663$		0			0
$664$	3.29531e23			0.338429
$665$	2.55637e23	−	4.63901e23i	0.259203	−	0.470372i
$666$		0			0
$667$		−	1.56339e24i		−	1.54525i
$668$		−	3.32848e23i		−	0.324822i
$669$		0			0
$670$	7.95874e22	+	4.38573e22i	0.0757196	+	0.0417259i
$671$	4.27951e23			0.402025
$672$		0			0
$673$			2.25201e23i			0.206273i	0.994667	+	0.103136i	$0.0328878\pi$
$673$			2.25201e23i			0.206273i	−0.994667	+	0.103136i	$0.967112\pi$
$674$	8.66147e23			0.783398
$675$		0			0
$676$	1.07363e24			0.946907
$677$			6.00722e23i			0.523203i	0.965176	+	0.261601i	$0.0842505\pi$
$677$			6.00722e23i			0.523203i	−0.965176	+	0.261601i	$0.915749\pi$
$678$		0			0
$679$	6.40506e23			0.544039
$680$	2.07531e23	+	1.14362e23i	0.174084	+	0.0959303i
$681$		0			0
$682$		−	4.13504e23i		−	0.338308i
$683$			2.10638e24i			1.70200i	0.525163	+	0.851002i	$0.324005\pi$
$683$			2.10638e24i			1.70200i	−0.525163	+	0.851002i	$0.675995\pi$
$684$		0			0
$685$	−7.24367e22	+	1.31450e23i	−0.0570938	+	0.103607i
$686$	−6.96973e23			−0.542577
$687$		0			0
$688$			2.17946e22i			0.0165519i
$689$	−3.05713e24			−2.29324
$690$		0			0
$691$	1.47665e24			1.08072			0.540360	−	0.841434i	$-0.318288\pi$
$691$	1.47665e24			1.08072			0.540360	+	0.841434i	$0.318288\pi$
$692$		−	1.01939e24i		−	0.736949i
$693$		0			0
$694$	−2.03460e23			−0.143523
$695$	−2.19401e24	−	1.20903e24i	−1.52886	−	0.842493i
$696$		0			0
$697$			9.18787e23i			0.624793i
$698$			1.41486e23i			0.0950478i
$699$		0			0
$700$	7.45246e23	−	4.70858e23i	0.488614	−	0.308714i
$701$	1.95160e24			1.26412			0.632059	−	0.774920i	$-0.282210\pi$
$701$	1.95160e24			1.26412			0.632059	+	0.774920i	$0.282210\pi$
$702$		0			0
$703$		−	1.98428e22i		−	0.0125453i
$704$	1.40442e23			0.0877262
$705$		0			0
$706$	1.08668e24			0.662613
$707$			2.94526e24i			1.77443i
$708$		0			0
$709$	−1.53951e24			−0.905505			−0.452752	−	0.891636i	$-0.649558\pi$
$709$	−1.53951e24			−0.905505			−0.452752	+	0.891636i	$0.649558\pi$
$710$	−1.02903e24	+	1.86737e24i	−0.598042	+	1.08526i
$711$		0			0
$712$		−	4.12581e23i		−	0.234115i
$713$		−	1.52994e24i		−	0.857856i
$714$		0			0
$715$	−1.05238e24	+	1.90973e24i	−0.576195	+	1.04561i
$716$	4.39327e23			0.237699
$717$		0			0
$718$		−	1.57506e24i		−	0.832221i
$719$	2.84029e24			1.48309			0.741545	−	0.670903i	$-0.234093\pi$
$719$	2.84029e24			1.48309			0.741545	+	0.670903i	$0.234093\pi$
$720$		0			0
$721$	−2.00086e23			−0.102039
$722$		−	1.10013e24i		−	0.554468i
$723$		0			0
$724$	−9.12708e23			−0.449318
$725$	1.34808e24	+	2.13366e24i	0.655908	+	1.03813i
$726$		0			0
$727$			6.98622e23i			0.332047i	0.986122	+	0.166024i	$0.0530928\pi$
$727$			6.98622e23i			0.332047i	−0.986122	+	0.166024i	$0.946907\pi$
$728$		−	1.47993e24i		−	0.695224i
$729$		0			0
$730$	1.18903e24	+	6.55224e23i	0.545692	+	0.300708i
$731$	−8.20525e22			−0.0372215
$732$		0			0
$733$		−	2.64577e24i		−	1.17265i	−0.810075	−	0.586326i	$-0.800574\pi$
$733$		−	2.64577e24i		−	1.17265i	0.810075	−	0.586326i	$-0.199426\pi$
$734$	−3.93166e23			−0.172250
$735$		0			0
$736$	5.19628e23			0.222449
$737$			2.02768e23i			0.0858076i
$738$		0			0
$739$	8.69279e22			0.0359486			0.0179743	−	0.999838i	$-0.494278\pi$
$739$	8.69279e22			0.0359486			0.0179743	+	0.999838i	$0.494278\pi$
$740$	1.59384e22	−	2.89233e22i	0.00651593	−	0.0118244i
$741$		0			0
$742$			2.75783e24i			1.10188i
$743$		−	5.50364e23i		−	0.217393i	−0.994075	−	0.108696i	$-0.965332\pi$
$743$		−	5.50364e23i		−	0.217393i	0.994075	−	0.108696i	$-0.0346676\pi$
$744$		0			0
$745$	1.66252e24	+	9.16148e23i	0.641859	+	0.353702i
$746$	2.57060e24			0.981193
$747$		0			0
$748$			5.28736e23i			0.197277i
$749$	3.03941e24			1.12123
$750$		0			0
$751$	−2.95235e24			−1.06470			−0.532352	−	0.846523i	$-0.678692\pi$
$751$	−2.95235e24			−1.06470			−0.532352	+	0.846523i	$0.678692\pi$
$752$		−	1.42732e23i		−	0.0508945i
$753$		0			0
$754$	4.23709e24			1.47711
$755$	1.39131e24	+	7.66691e23i	0.479594	+	0.264284i
$756$		0			0
$757$		−	4.84963e24i		−	1.63453i	−0.576261	−	0.817266i	$-0.695489\pi$
$757$		−	4.84963e24i		−	1.63453i	0.576261	−	0.817266i	$-0.304511\pi$
$758$			3.06766e24i			1.02240i
$759$		0			0
$760$	−2.43263e23	+	4.41446e23i	−0.0792793	+	0.143867i
$761$	4.97641e24			1.60379			0.801893	−	0.597468i	$-0.203827\pi$
$761$	4.97641e24			1.60379			0.801893	+	0.597468i	$0.203827\pi$
$762$		0			0
$763$			5.60313e24i			1.76592i
$764$	−1.23237e23			−0.0384103
$765$		0			0
$766$	−2.87502e24			−0.876387
$767$			4.31843e24i			1.30186i
$768$		0			0
$769$	4.91902e23			0.145046			0.0725228	−	0.997367i	$-0.476895\pi$
$769$	4.91902e23			0.145046			0.0725228	+	0.997367i	$0.476895\pi$
$770$	1.72277e24	+	9.49347e23i	0.502407	+	0.276856i
$771$		0			0
$772$			1.36946e23i			0.0390663i
$773$		−	2.61117e24i		−	0.736732i	−0.929681	−	0.368366i	$-0.879917\pi$
$773$		−	2.61117e24i		−	0.736732i	0.929681	−	0.368366i	$-0.120083\pi$
$774$		0			0
$775$	1.31924e24	+	2.08802e24i	0.364133	+	0.576328i
$776$	−6.09502e23			−0.166399
$777$		0			0
$778$		−	6.50645e23i		−	0.173787i
$779$	−1.95438e24			−0.516345
$780$		0			0
$781$	−4.75757e24			−1.22985
$782$			1.95630e24i			0.500239i
$783$		0			0
$784$	−3.35904e23			−0.0840484
$785$	−2.83870e24	+	5.15136e24i	−0.702632	+	1.27506i
$786$		0			0
$787$			4.37257e24i			1.05914i	0.848268	+	0.529568i	$0.177646\pi$
$787$			4.37257e24i			1.05914i	−0.848268	+	0.529568i	$0.822354\pi$
$788$			2.03905e23i			0.0488601i
$789$		0			0
$790$	1.49258e24	−	2.70858e24i	0.350033	−	0.635200i
$791$	−5.41739e24			−1.25687
$792$		0			0
$793$			4.29132e24i			0.974471i
$794$	7.84981e23			0.176354
$795$		0			0
$796$	−1.90113e24			−0.418072
$797$		−	1.47316e24i		−	0.320519i	−0.987075	−	0.160260i	$-0.948767\pi$
$797$		−	1.47316e24i		−	0.320519i	0.987075	−	0.160260i	$-0.0512331\pi$
$798$		0			0
$799$	5.37360e23			0.114450
$800$	−7.09172e23	+	4.48066e23i	−0.149447	+	0.0944228i
$801$		0			0
$802$			3.35816e24i			0.692818i
$803$			3.02934e24i			0.618394i
$804$		0			0
$805$	6.37416e24	+	3.51253e24i	1.27397	+	0.702030i
$806$	4.14645e24			0.820028
$807$		0			0
$808$		−	2.80269e24i		−	0.542723i
$809$	−1.91465e24			−0.366882			−0.183441	−	0.983031i	$-0.558724\pi$
$809$	−1.91465e24			−0.366882			−0.183441	+	0.983031i	$0.558724\pi$
$810$		0			0
$811$	−4.84935e24			−0.909927			−0.454964	−	0.890510i	$-0.650348\pi$
$811$	−4.84935e24			−0.909927			−0.454964	+	0.890510i	$0.650348\pi$
$812$		−	3.82228e24i		−	0.709735i
$813$		0			0
$814$	7.36891e22			0.0133997
$815$	5.79665e23	−	1.05191e24i	0.104313	−	0.189295i
$816$		0			0
$817$		−	1.74536e23i		−	0.0307608i
$818$		−	5.86332e23i		−	0.102268i
$819$		0			0
$820$	−2.84875e24	−	1.56983e24i	−0.486673	−	0.268185i
$821$	−2.84679e24			−0.481326			−0.240663	−	0.970609i	$-0.577365\pi$
$821$	−2.84679e24			−0.481326			−0.240663	+	0.970609i	$0.577365\pi$
$822$		0			0
$823$		−	9.76200e24i		−	1.61674i	−0.588674	−	0.808370i	$-0.700350\pi$
$823$		−	9.76200e24i		−	1.61674i	0.588674	−	0.808370i	$-0.299650\pi$
$824$	1.90401e23			0.0312095
$825$		0			0
$826$	3.89565e24			0.625532
$827$		−	4.92807e24i		−	0.783213i	−0.920133	−	0.391607i	$-0.871919\pi$
$827$		−	4.92807e24i		−	0.783213i	0.920133	−	0.391607i	$-0.128081\pi$
$828$		0			0
$829$	−2.49105e24			−0.387854			−0.193927	−	0.981016i	$-0.562123\pi$
$829$	−2.49105e24			−0.387854			−0.193927	+	0.981016i	$0.562123\pi$
$830$	−3.84661e24	−	2.11971e24i	−0.592809	−	0.326672i
$831$		0			0
$832$			1.40829e24i			0.212640i
$833$		−	1.26461e24i		−	0.189006i
$834$		0			0
$835$	−2.14104e24	+	3.88533e24i	−0.313538	+	0.568974i
$836$	−1.12469e24			−0.163034
$837$		0			0
$838$		−	2.55783e24i		−	0.363327i
$839$	9.44495e23			0.132808			0.0664038	−	0.997793i	$-0.478847\pi$
$839$	9.44495e23			0.132808			0.0664038	+	0.997793i	$0.478847\pi$
$840$		0			0
$841$	3.68616e24			0.507935
$842$			6.20341e24i			0.846210i
$843$		0			0
$844$	−2.14504e24			−0.286764
$845$	−1.25325e25	−	6.90612e24i	−1.65865	−	0.914012i
$846$		0			0
$847$		−	4.52218e24i		−	0.586596i
$848$		−	2.62434e24i		−	0.337019i
$849$		0			0
$850$	−1.68688e24	−	2.66989e24i	−0.212336	−	0.336072i
$851$	2.72646e23			0.0339780
$852$		0			0
$853$		−	1.00068e25i		−	1.22244i	−0.791460	−	0.611221i	$-0.790679\pi$
$853$		−	1.00068e25i		−	1.22244i	0.791460	−	0.611221i	$-0.209321\pi$
$854$	3.87120e24			0.468223
$855$		0			0
$856$	−2.89228e24			−0.342936
$857$			1.02788e25i			1.20671i	0.797472	+	0.603356i	$0.206170\pi$
$857$			1.02788e25i			1.20671i	−0.797472	+	0.603356i	$0.793830\pi$
$858$		0			0
$859$	9.78146e24			1.12580			0.562900	−	0.826525i	$-0.309685\pi$
$859$	9.78146e24			1.12580			0.562900	+	0.826525i	$0.309685\pi$
$860$	1.40194e23	−	2.54409e23i	0.0159769	−	0.0289931i
$861$		0			0
$862$			9.57634e23i			0.107001i
$863$		−	1.30891e25i		−	1.44816i	−0.689715	−	0.724081i	$-0.742264\pi$
$863$		−	1.30891e25i		−	1.44816i	0.689715	−	0.724081i	$-0.257736\pi$
$864$		0			0
$865$	−6.55721e24	+	1.18993e25i	−0.711348	+	1.29088i
$866$	−7.68707e24			−0.825770
$867$		0			0
$868$		−	3.74051e24i		−	0.394015i
$869$	6.90075e24			0.719827
$870$		0			0
$871$	−2.03327e24			−0.207990
$872$		−	5.33190e24i		−	0.540122i
$873$		0			0
$874$	−4.16129e24			−0.413410
$875$	−1.17280e25	+	7.02524e23i	−1.15387	+	0.0691183i
$876$		0			0
$877$			9.63915e22i			0.00930127i	0.999989	+	0.00465063i	$0.00148035\pi$
$877$			9.63915e22i			0.00930127i	−0.999989	+	0.00465063i	$0.998520\pi$
$878$			1.24951e25i			1.19409i
$879$		0			0
$880$	−1.63938e24	−	9.03393e23i	−0.153665	−	0.0846787i
$881$	1.24804e25			1.15859			0.579297	−	0.815116i	$-0.303327\pi$
$881$	1.24804e25			1.15859			0.579297	+	0.815116i	$0.303327\pi$
$882$		0			0
$883$		−	1.82649e25i		−	1.66322i	−0.555359	−	0.831611i	$-0.687419\pi$
$883$		−	1.82649e25i		−	1.66322i	0.555359	−	0.831611i	$-0.312581\pi$
$884$	−5.30195e24			−0.478180
$885$		0			0
$886$	−2.26733e23			−0.0200599
$887$		−	9.15990e24i		−	0.802675i	−0.915930	−	0.401338i	$-0.868545\pi$
$887$		−	9.15990e24i		−	0.802675i	0.915930	−	0.401338i	$-0.131455\pi$
$888$		0			0
$889$	1.44926e25			1.24590
$890$	−2.65393e24	+	4.81606e24i	−0.225982	+	0.410088i
$891$		0			0
$892$		−	3.23941e24i		−	0.270623i
$893$			1.14303e24i			0.0945847i
$894$		0			0
$895$	−5.12826e24	−	2.82597e24i	−0.416365	−	0.229441i
$896$	1.27042e24			0.102171
$897$		0			0
$898$			2.75424e20i			2.17346e-5i
$899$	1.07092e25			0.837143
$900$		0			0
$901$	9.88011e24			0.757881
$902$		−	7.25788e24i		−	0.551511i
$903$		0			0
$904$	5.15516e24			0.384424
$905$	1.06540e25	+	5.87100e24i	0.787048	+	0.433709i
$906$		0			0
$907$			2.04267e25i			1.48093i	0.672092	+	0.740467i	$0.265396\pi$
$907$			2.04267e25i			1.48093i	−0.672092	+	0.740467i	$0.734604\pi$
$908$		−	6.69728e24i		−	0.481026i
$909$		0			0
$910$	−9.51966e24	+	1.72752e25i	−0.671073	+	1.21779i
$911$	−7.78026e24			−0.543360			−0.271680	−	0.962388i	$-0.587579\pi$
$911$	−7.78026e24			−0.543360			−0.271680	+	0.962388i	$0.587579\pi$
$912$		0			0
$913$		−	9.80017e24i		−	0.671788i
$914$	−1.10161e24			−0.0748141
$915$		0			0
$916$	−1.14594e25			−0.763923
$917$		−	4.66617e24i		−	0.308192i
$918$		0			0
$919$	1.92668e25			1.24919			0.624594	−	0.780949i	$-0.285264\pi$
$919$	1.92668e25			1.24919			0.624594	+	0.780949i	$0.285264\pi$
$920$	−6.06561e24	−	3.34251e24i	−0.389653	−	0.214722i
$921$		0			0
$922$			8.91018e24i			0.561919i
$923$		−	4.77070e25i		−	2.98104i
$924$		0			0
$925$	−3.72098e23	+	2.35098e23i	−0.0228272	+	0.0144226i
$926$	5.28796e23			0.0321436
$927$		0			0
$928$			3.63726e24i			0.217078i
$929$	1.92204e25			1.13666			0.568328	−	0.822802i	$-0.307590\pi$
$929$	1.92204e25			1.13666			0.568328	+	0.822802i	$0.307590\pi$
$930$		0			0
$931$	2.69000e24			0.156199
$932$		−	7.99827e24i		−	0.460215i
$933$		0			0
$934$	−5.82906e24			−0.329344
$935$	3.40110e24	−	6.17193e24i	0.190423	−	0.345559i
$936$		0			0
$937$		−	2.49742e25i		−	1.37311i	−0.727079	−	0.686554i	$-0.759122\pi$
$937$		−	2.49742e25i		−	1.37311i	0.727079	−	0.686554i	$-0.240878\pi$
$938$			1.83421e24i			0.0999370i
$939$		0			0
$940$	−9.18126e23	+	1.66611e24i	−0.0491265	+	0.0891493i
$941$	−2.13047e25			−1.12970			−0.564851	−	0.825193i	$-0.691066\pi$
$941$	−2.13047e25			−1.12970			−0.564851	+	0.825193i	$0.691066\pi$
$942$		0			0
$943$		−	2.68538e25i		−	1.39848i
$944$	−3.70708e24			−0.191324
$945$		0			0
$946$	6.48167e23			0.0328558
$947$			1.70570e25i			0.856893i	0.903567	+	0.428447i	$0.140939\pi$
$947$			1.70570e25i			0.856893i	−0.903567	+	0.428447i	$0.859061\pi$
$948$		0			0
$949$	−3.03769e25			−1.49893
$950$	5.67920e24	−	3.58821e24i	0.277739	−	0.175480i
$951$		0			0
$952$			4.78289e24i			0.229761i
$953$		−	6.64867e24i		−	0.316552i	−0.987395	−	0.158276i	$-0.949406\pi$
$953$		−	6.64867e24i		−	0.316552i	0.987395	−	0.158276i	$-0.0505935\pi$
$954$		0			0
$955$	1.43854e24	+	7.92721e23i	0.0672813	+	0.0370759i
$956$	−1.33330e25			−0.618069
$957$		0			0
$958$			7.42929e24i			0.338330i
$959$	−3.02947e24			−0.136744
$960$		0			0
$961$	−1.20700e25			−0.535253
$962$			7.38924e23i			0.0324797i
$963$		0			0
$964$	1.40958e25			0.608743
$965$	8.80904e23	−	1.59857e24i	0.0377091	−	0.0684303i
$966$		0			0
$967$			1.05112e25i			0.442105i	0.975262	+	0.221052i	$0.0709492\pi$
$967$			1.05112e25i			0.442105i	−0.975262	+	0.221052i	$0.929051\pi$
$968$			4.30328e24i			0.179415i
$969$		0			0
$970$	7.11471e24	+	3.92062e24i	0.291472	+	0.160618i
$971$	2.13371e25			0.866507			0.433254	−	0.901272i	$-0.357365\pi$
$971$	2.13371e25			0.866507			0.433254	+	0.901272i	$0.357365\pi$
$972$		0			0
$973$		−	5.05645e25i		−	2.01784i
$974$	6.86266e24			0.271482
$975$		0			0
$976$	−3.68381e24			−0.143210
$977$			1.33440e25i			0.514260i	0.966377	+	0.257130i	$0.0827768\pi$
$977$			1.33440e25i			0.514260i	−0.966377	+	0.257130i	$0.917223\pi$
$978$		0			0
$979$	−1.22701e25			−0.464723
$980$	3.92101e24	+	2.16071e24i	0.147223	+	0.0811286i
$981$		0			0
$982$			2.53068e25i			0.933878i
$983$		−	1.31969e25i		−	0.482800i	−0.970426	−	0.241400i	$-0.922393\pi$
$983$		−	1.31969e25i		−	0.482800i	0.970426	−	0.241400i	$-0.0776066\pi$
$984$		0			0
$985$	1.31162e24	−	2.38018e24i	0.0471627	−	0.0855857i
$986$	−1.36936e25			−0.488161
$987$		0			0
$988$		−	1.12779e25i		−	0.395180i
$989$	2.39818e24			0.0833132
$990$		0			0
$991$	4.80007e25			1.63916			0.819580	−	0.572964i	$-0.194207\pi$
$991$	4.80007e25			1.63916			0.819580	+	0.572964i	$0.194207\pi$
$992$			3.55945e24i			0.120513i
$993$		0			0
$994$	−4.30364e25			−1.43236
$995$	2.21919e25	+	1.22290e25i	0.732316	+	0.403549i
$996$		0			0
$997$			1.30477e25i			0.423276i	0.977348	+	0.211638i	$0.0678799\pi$
$997$			1.30477e25i			0.423276i	−0.977348	+	0.211638i	$0.932120\pi$
$998$		−	3.09100e25i		−	0.994234i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	90.18.c.b.19.8		8
3.2	odd	2		10.18.b.a.9.1	✓	8
5.4	even	2	inner	90.18.c.b.19.4		8
12.11	even	2		80.18.c.a.49.8		8
15.2	even	4		50.18.a.k.1.1		4
15.8	even	4		50.18.a.j.1.4		4
15.14	odd	2		10.18.b.a.9.8	yes	8
60.59	even	2		80.18.c.a.49.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.18.b.a.9.1	✓	8	3.2	odd	2
10.18.b.a.9.8	yes	8	15.14	odd	2
50.18.a.j.1.4		4	15.8	even	4
50.18.a.k.1.1		4	15.2	even	4
80.18.c.a.49.1		8	60.59	even	2
80.18.c.a.49.8		8	12.11	even	2
90.18.c.b.19.4		8	5.4	even	2	inner
90.18.c.b.19.8		8	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	90.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+256.000i q^{2} -65536.0 q^{4} +(765001. + 421560. i) q^{5} +1.76306e7i q^{7} -1.67772e7i q^{8} +(-1.07919e8 + 1.95840e8i) q^{10} -4.98950e8 q^{11} -5.00327e9i q^{13} -4.51344e9 q^{14} +4.29497e9 q^{16} +1.61697e10i q^{17} -3.43950e10 q^{19} +(-5.01351e10 - 2.76274e10i) q^{20} -1.27731e11i q^{22} -4.72599e11i q^{23} +(4.07513e11 + 6.44988e11i) q^{25} +1.28084e12 q^{26} -1.15544e12i q^{28} +3.30807e12 q^{29} +3.23730e12 q^{31} +1.09951e12i q^{32} -4.13944e12 q^{34} +(-7.43238e12 + 1.34875e13i) q^{35} +5.76908e11i q^{37} -8.80513e12i q^{38} +(7.07261e12 - 1.28346e13i) q^{40} +5.68215e13 q^{41} +5.07446e12i q^{43} +3.26992e13 q^{44} +1.20985e14 q^{46} -3.32325e13i q^{47} -7.82089e13 q^{49} +(-1.65117e14 + 1.04323e14i) q^{50} +3.27894e14i q^{52} -6.11026e14i q^{53} +(-3.81697e14 - 2.10338e14i) q^{55} +2.95793e14 q^{56} +8.46865e14i q^{58} -8.63122e14 q^{59} -8.57703e14 q^{61} +8.28749e14i q^{62} -2.81475e14 q^{64} +(2.10918e15 - 3.82751e15i) q^{65} -4.06389e14i q^{67} -1.05970e15i q^{68} +(-3.45279e15 - 1.90269e15i) q^{70} +9.53517e15 q^{71} -6.07142e15i q^{73} -1.47688e14 q^{74} +2.25411e15 q^{76} -8.79681e15i q^{77} -1.38305e16 q^{79} +(3.28565e15 + 1.81059e15i) q^{80} +1.45463e16i q^{82} +1.96416e16i q^{83} +(-6.81651e15 + 1.23698e16i) q^{85} -1.29906e15 q^{86} +8.37099e15i q^{88} +2.45918e16 q^{89} +8.82108e16 q^{91} +3.09722e16i q^{92} +8.50752e15 q^{94} +(-2.63122e16 - 1.44996e16i) q^{95} -3.63292e16i q^{97} -2.00215e16i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 524288 q^{4} + 1225560 q^{5} - 140779520 q^{10} - 146232096 q^{11} - 14260494336 q^{14} + 34359738368 q^{16} - 54264178080 q^{19} - 80318300160 q^{20} + 1013225778600 q^{25} + 2693383569408 q^{26}+ \cdots + 13\!\cdots\!00 q^{95}+O(q^{100}) \) 8 * q - 524288 * q^4 + 1225560 * q^5 - 140779520 * q^10 - 146232096 * q^11 - 14260494336 * q^14 + 34359738368 * q^16 - 54264178080 * q^19 - 80318300160 * q^20 + 1013225778600 * q^25 + 2693383569408 * q^26 - 1660243083120 * q^29 - 6055476993664 * q^31 + 14158246445056 * q^34 + 8725233780960 * q^35 + 9226126622720 * q^40 + 219921829971984 * q^41 + 9583466643456 * q^44 - 136753191067648 * q^46 + 797208944041464 * q^49 - 535409908531200 * q^50 + 1370380127176480 * q^55 + 934575756804096 * q^56 + 155410557761760 * q^59 - 6350444068968944 * q^61 - 2251799813685248 * q^64 - 10508144084782080 * q^65 - 1051117677240320 * q^70 + 33149026543733184 * q^71 - 6017648345726976 * q^74 + 3556257174650880 * q^76 - 19354668791708160 * q^79 + 5263740119285760 * q^80 - 71530234876895360 * q^85 - 29616028444434432 * q^86 + 43569230299558320 * q^89 + 333420307889971776 * q^91 - 11440296922292224 * q^94 + 131478039235197600 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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