LMFDB - Embedded newform 108.5.d.b.55.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,5,Mod(55,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("108.55");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	108.d (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:	$11.1639560131$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 38x^{14} + 1016x^{12} + 13512x^{10} + 130640x^{8} + 569472x^{6} + 1783808x^{4} + 352256x^{2} + 65536 $ x^16 + 38x^14 + 1016x^12 + 13512x^10 + 130640x^8 + 569472x^6 + 1783808x^4 + 352256*x^2 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{24} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			55.1
Root			$-0.222504 - 0.385387i$ of defining polynomial
Character	$\chi$	$=$	108.55
Dual form			108.5.d.b.55.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.98139 - 0.385387i) q^{2} +(15.7030 + 3.06876i) q^{4} +22.8093 q^{5} +82.1204i q^{7} +(-61.3369 - 18.2696i) q^{8} +O(q^{10})$	\(q+(-3.98139 - 0.385387i) q^{2} +(15.7030 + 3.06876i) q^{4} +22.8093 q^{5} +82.1204i q^{7} +(-61.3369 - 18.2696i) q^{8} +(-90.8127 - 8.79041i) q^{10} -107.306i q^{11} +57.1351 q^{13} +(31.6482 - 326.954i) q^{14} +(237.165 + 96.3771i) q^{16} -291.903 q^{17} +304.924i q^{19} +(358.173 + 69.9961i) q^{20} +(-41.3543 + 427.227i) q^{22} +982.813i q^{23} -104.737 q^{25} +(-227.477 - 22.0191i) q^{26} +(-252.008 + 1289.53i) q^{28} +1049.98 q^{29} +783.041i q^{31} +(-907.106 - 475.115i) q^{32} +(1162.18 + 112.496i) q^{34} +1873.11i q^{35} +1404.40 q^{37} +(117.514 - 1214.02i) q^{38} +(-1399.05 - 416.717i) q^{40} +363.412 q^{41} -64.7059i q^{43} +(329.295 - 1685.02i) q^{44} +(378.764 - 3912.96i) q^{46} +3495.66i q^{47} -4342.77 q^{49} +(416.998 + 40.3642i) q^{50} +(897.189 + 175.334i) q^{52} -3268.87 q^{53} -2447.57i q^{55} +(1500.31 - 5037.02i) q^{56} +(-4180.37 - 404.648i) q^{58} -867.469i q^{59} +6034.94 q^{61} +(301.774 - 3117.59i) q^{62} +(3428.44 + 2241.21i) q^{64} +1303.21 q^{65} -3421.54i q^{67} +(-4583.74 - 895.780i) q^{68} +(721.872 - 7457.58i) q^{70} +3188.67i q^{71} -4267.98 q^{73} +(-5591.47 - 541.239i) q^{74} +(-935.737 + 4788.20i) q^{76} +8812.00 q^{77} -7530.43i q^{79} +(5409.57 + 2198.29i) q^{80} +(-1446.89 - 140.055i) q^{82} -586.258i q^{83} -6658.11 q^{85} +(-24.9368 + 257.619i) q^{86} +(-1960.44 + 6581.81i) q^{88} -5789.27 q^{89} +4691.96i q^{91} +(-3016.01 + 15433.1i) q^{92} +(1347.18 - 13917.6i) q^{94} +6955.09i q^{95} +10837.8 q^{97} +(17290.3 + 1673.65i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 28 q^{4}+O(q^{10}) $ 16 * q + 28 * q^4	$ 16 q + 28 q^{4} + 176 q^{10} + 176 q^{13} + 88 q^{16} + 384 q^{22} + 2736 q^{25} + 1812 q^{28} + 1520 q^{34} + 80 q^{37} - 688 q^{40} - 1824 q^{46} - 7904 q^{49} - 5236 q^{52} - 11584 q^{58} - 1648 q^{61} + 5056 q^{64} + 26688 q^{70} + 80 q^{73} - 8388 q^{76} - 38464 q^{82} - 16832 q^{85} - 29520 q^{88} - 4512 q^{94} + 14864 q^{97}+O(q^{100}) $ 16 * q + 28 * q^4 + 176 * q^10 + 176 * q^13 + 88 * q^16 + 384 * q^22 + 2736 * q^25 + 1812 * q^28 + 1520 * q^34 + 80 * q^37 - 688 * q^40 - 1824 * q^46 - 7904 * q^49 - 5236 * q^52 - 11584 * q^58 - 1648 * q^61 + 5056 * q^64 + 26688 * q^70 + 80 * q^73 - 8388 * q^76 - 38464 * q^82 - 16832 * q^85 - 29520 * q^88 - 4512 * q^94 + 14864 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$1$	$-1$

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 $

$2$	−3.98139	−	0.385387i	−0.995348	−	0.0963469i
$2$	−3.98139	−	0.385387i	−0.995348	−	0.0963469i
$3$		0			0
$3$		0			0
$4$	15.7030	+	3.06876i	0.981435	+	0.191797i
$5$	22.8093			0.912371			0.456186	−	0.889885i	$-0.349215\pi$
$5$	22.8093			0.912371			0.456186	+	0.889885i	$0.349215\pi$
$6$		0			0
$7$			82.1204i			1.67593i	0.545726	+	0.837964i	$0.316254\pi$
$7$			82.1204i			1.67593i	−0.545726	+	0.837964i	$0.683746\pi$
$8$	−61.3369	−	18.2696i	−0.958390	−	0.285463i
$9$		0			0
$10$	−90.8127	−	8.79041i	−0.908127	−	0.0879041i
$11$		−	107.306i		−	0.886825i	−0.896318	−	0.443413i	$-0.853768\pi$
$11$		−	107.306i		−	0.886825i	0.896318	−	0.443413i	$-0.146232\pi$
$12$		0			0
$13$	57.1351			0.338077			0.169039	−	0.985609i	$-0.445934\pi$
$13$	57.1351			0.338077			0.169039	+	0.985609i	$0.445934\pi$
$14$	31.6482	−	326.954i	0.161470	−	1.66813i
$15$		0			0
$16$	237.165	+	96.3771i	0.926428	+	0.376473i
$17$	−291.903			−1.01005			−0.505023	−	0.863106i	$-0.668516\pi$
$17$	−291.903			−1.01005			−0.505023	+	0.863106i	$0.668516\pi$
$18$		0			0
$19$			304.924i			0.844664i	0.906441	+	0.422332i	$0.138788\pi$
$19$			304.924i			0.844664i	−0.906441	+	0.422332i	$0.861212\pi$
$20$	358.173	+	69.9961i	0.895433	+	0.174990i
$21$		0			0
$22$	−41.3543	+	427.227i	−0.0854428	+	0.882699i
$23$			982.813i			1.85787i	0.370243	+	0.928935i	$0.379274\pi$
$23$			982.813i			1.85787i	−0.370243	+	0.928935i	$0.620726\pi$
$24$		0			0
$25$	−104.737			−0.167579
$26$	−227.477	−	22.0191i	−0.336505	−	0.0325727i
$27$		0			0
$28$	−252.008	+	1289.53i	−0.321438	+	1.64481i
$29$	1049.98			1.24849			0.624244	−	0.781230i	$-0.285407\pi$
$29$	1049.98			1.24849			0.624244	+	0.781230i	$0.285407\pi$
$30$		0			0
$31$			783.041i			0.814819i	0.913246	+	0.407409i	$0.133568\pi$
$31$			783.041i			0.814819i	−0.913246	+	0.407409i	$0.866432\pi$
$32$	−907.106	−	475.115i	−0.885846	−	0.463980i
$33$		0			0
$34$	1162.18	+	112.496i	1.00535	+	0.0973148i
$35$			1873.11i			1.52907i
$36$		0			0
$37$	1404.40			1.02586			0.512930	−	0.858431i	$-0.328560\pi$
$37$	1404.40			1.02586			0.512930	+	0.858431i	$0.328560\pi$
$38$	117.514	−	1214.02i	0.0813808	−	0.840735i
$39$		0			0
$40$	−1399.05	−	416.717i	−0.874407	−	0.260448i
$41$	363.412			0.216188			0.108094	−	0.994141i	$-0.465525\pi$
$41$	363.412			0.216188			0.108094	+	0.994141i	$0.465525\pi$
$42$		0			0
$43$		−	64.7059i		−	0.0349951i	−0.999847	−	0.0174975i	$-0.994430\pi$
$43$		−	64.7059i		−	0.0349951i	0.999847	−	0.0174975i	$-0.00556992\pi$
$44$	329.295	−	1685.02i	0.170091	−	0.870361i
$45$		0			0
$46$	378.764	−	3912.96i	0.179000	−	1.84923i
$47$			3495.66i			1.58246i	0.611518	+	0.791230i	$0.290559\pi$
$47$			3495.66i			1.58246i	−0.611518	+	0.791230i	$0.709441\pi$
$48$		0			0
$49$	−4342.77			−1.80873
$50$	416.998	+	40.3642i	0.166799	+	0.0161457i
$51$		0			0
$52$	897.189	+	175.334i	0.331801	+	0.0648423i
$53$	−3268.87			−1.16371			−0.581856	−	0.813292i	$-0.697673\pi$
$53$	−3268.87			−1.16371			−0.581856	+	0.813292i	$0.697673\pi$
$54$		0			0
$55$		−	2447.57i		−	0.809114i
$56$	1500.31	−	5037.02i	0.478415	−	1.60619i
$57$		0			0
$58$	−4180.37	−	404.648i	−1.24268	−	0.120288i
$59$		−	867.469i		−	0.249201i	−0.992207	−	0.124601i	$-0.960235\pi$
$59$		−	867.469i		−	0.249201i	0.992207	−	0.124601i	$-0.0397650\pi$
$60$		0			0
$61$	6034.94			1.62186			0.810929	−	0.585144i	$-0.198962\pi$
$61$	6034.94			1.62186			0.810929	+	0.585144i	$0.198962\pi$
$62$	301.774	−	3117.59i	0.0785052	−	0.811028i
$63$		0			0
$64$	3428.44	+	2241.21i	0.837022	+	0.547170i
$65$	1303.21			0.308452
$66$		0			0
$67$		−	3421.54i		−	0.762205i	−0.924533	−	0.381103i	$-0.875544\pi$
$67$		−	3421.54i		−	0.762205i	0.924533	−	0.381103i	$-0.124456\pi$
$68$	−4583.74	−	895.780i	−0.991294	−	0.193724i
$69$		0			0
$70$	721.872	−	7457.58i	0.147321	−	1.52195i
$71$			3188.67i			0.632547i	0.948668	+	0.316273i	$0.102432\pi$
$71$			3188.67i			0.632547i	−0.948668	+	0.316273i	$0.897568\pi$
$72$		0			0
$73$	−4267.98			−0.800896			−0.400448	−	0.916319i	$-0.631146\pi$
$73$	−4267.98			−0.800896			−0.400448	+	0.916319i	$0.631146\pi$
$74$	−5591.47	−	541.239i	−1.02109	−	0.0988383i
$75$		0			0
$76$	−935.737	+	4788.20i	−0.162004	+	0.828983i
$77$	8812.00			1.48625
$78$		0			0
$79$		−	7530.43i		−	1.20661i	−0.797512	−	0.603303i	$-0.793851\pi$
$79$		−	7530.43i		−	1.20661i	0.797512	−	0.603303i	$-0.206149\pi$
$80$	5409.57	+	2198.29i	0.845246	+	0.343483i
$81$		0			0
$82$	−1446.89	−	140.055i	−0.215182	−	0.0208290i
$83$		−	586.258i		−	0.0851006i	−0.999094	−	0.0425503i	$-0.986452\pi$
$83$		−	586.258i		−	0.0851006i	0.999094	−	0.0425503i	$-0.0135483\pi$
$84$		0			0
$85$	−6658.11			−0.921537
$86$	−24.9368	+	257.619i	−0.00337167	+	0.0348323i
$87$		0			0
$88$	−1960.44	+	6581.81i	−0.253156	+	0.849924i
$89$	−5789.27			−0.730876			−0.365438	−	0.930836i	$-0.619081\pi$
$89$	−5789.27			−0.730876			−0.365438	+	0.930836i	$0.619081\pi$
$90$		0			0
$91$			4691.96i			0.566593i
$92$	−3016.01	+	15433.1i	−0.356334	+	1.82338i
$93$		0			0
$94$	1347.18	−	13917.6i	0.152465	−	1.57510i
$95$			6955.09i			0.770647i
$96$		0			0
$97$	10837.8			1.15185			0.575927	−	0.817501i	$-0.304641\pi$
$97$	10837.8			1.15185			0.575927	+	0.817501i	$0.304641\pi$
$98$	17290.3	+	1673.65i	1.80032	+	0.174266i
$99$		0			0
$100$	−1644.68	−	321.411i	−0.164468	−	0.0321411i
$101$	−11502.9			−1.12763			−0.563813	−	0.825903i	$-0.690666\pi$
$101$	−11502.9			−1.12763			−0.563813	+	0.825903i	$0.690666\pi$
$102$		0			0
$103$			13839.0i			1.30446i	0.758020	+	0.652231i	$0.226167\pi$
$103$			13839.0i			1.30446i	−0.758020	+	0.652231i	$0.773833\pi$
$104$	−3504.49	−	1043.84i	−0.324010	−	0.0965086i
$105$		0			0
$106$	13014.6	+	1259.78i	1.15830	+	0.112120i
$107$		−	14731.9i		−	1.28674i	−0.765555	−	0.643371i	$-0.777535\pi$
$107$		−	14731.9i		−	1.28674i	0.765555	−	0.643371i	$-0.222465\pi$
$108$		0			0
$109$	3545.18			0.298391			0.149195	−	0.988808i	$-0.452332\pi$
$109$	3545.18			0.298391			0.149195	+	0.988808i	$0.452332\pi$
$110$	−943.262	+	9744.73i	−0.0779556	+	0.805350i
$111$		0			0
$112$	−7914.53	+	19476.1i	−0.630941	+	1.55263i
$113$	12472.0			0.976742			0.488371	−	0.872636i	$-0.337591\pi$
$113$	12472.0			0.976742			0.488371	+	0.872636i	$0.337591\pi$
$114$		0			0
$115$			22417.3i			1.69507i
$116$	16487.8	+	3222.13i	1.22531	+	0.239456i
$117$		0			0
$118$	−334.312	+	3453.73i	−0.0240098	+	0.248042i
$119$		−	23971.2i		−	1.69276i
$120$		0			0
$121$	3126.46			0.213541
$122$	−24027.4	−	2325.79i	−1.61431	−	0.156261i
$123$		0			0
$124$	−2402.96	+	12296.1i	−0.156280	+	0.799691i
$125$	−16644.8			−1.06527
$126$		0			0
$127$		−	1002.07i		−	0.0621284i	−0.999517	−	0.0310642i	$-0.990110\pi$
$127$		−	1002.07i		−	0.0621284i	0.999517	−	0.0310642i	$-0.00988964\pi$
$128$	−12786.2	−	10244.4i	−0.780410	−	0.625269i
$129$		0			0
$130$	−5188.59	−	502.241i	−0.307017	−	0.0297184i
$131$		−	33549.8i		−	1.95500i	−0.210933	−	0.977501i	$-0.567650\pi$
$131$		−	33549.8i		−	1.95500i	0.210933	−	0.977501i	$-0.432350\pi$
$132$		0			0
$133$	−25040.5			−1.41560
$134$	−1318.62	+	13622.5i	−0.0734361	+	0.758659i
$135$		0			0
$136$	17904.5	+	5332.97i	0.968018	+	0.288331i
$137$	19700.7			1.04964			0.524821	−	0.851213i	$-0.324132\pi$
$137$	19700.7			1.04964			0.524821	+	0.851213i	$0.324132\pi$
$138$		0			0
$139$		−	12634.4i		−	0.653922i	−0.945038	−	0.326961i	$-0.893975\pi$
$139$		−	12634.4i		−	0.653922i	0.945038	−	0.326961i	$-0.106025\pi$
$140$	−5748.11	+	29413.3i	−0.293271	+	1.50068i
$141$		0			0
$142$	1228.87	−	12695.3i	0.0609439	−	0.629604i
$143$		−	6130.93i		−	0.299816i
$144$		0			0
$145$	23949.2			1.13908
$146$	16992.5	+	1644.82i	0.797171	+	0.0771639i
$147$		0			0
$148$	22053.2	+	4309.76i	1.00681	+	0.196757i
$149$	24054.7			1.08350			0.541749	−	0.840541i	$-0.317762\pi$
$149$	24054.7			1.08350			0.541749	+	0.840541i	$0.317762\pi$
$150$		0			0
$151$		−	29000.6i		−	1.27190i	−0.771730	−	0.635951i	$-0.780608\pi$
$151$		−	29000.6i		−	1.27190i	0.771730	−	0.635951i	$-0.219392\pi$
$152$	5570.85	−	18703.1i	0.241121	−	0.809518i
$153$		0			0
$154$	−35084.0	−	3396.04i	−1.47934	−	0.143196i
$155$			17860.6i			0.743417i
$156$		0			0
$157$	11143.2			0.452074			0.226037	−	0.974119i	$-0.427423\pi$
$157$	11143.2			0.452074			0.226037	+	0.974119i	$0.427423\pi$
$158$	−2902.13	+	29981.6i	−0.116253	+	1.20099i
$159$		0			0
$160$	−20690.4	−	10837.0i	−0.808220	−	0.423322i
$161$	−80709.0			−3.11365
$162$		0			0
$163$			3212.14i			0.120898i	0.998171	+	0.0604490i	$0.0192533\pi$
$163$			3212.14i			0.120898i	−0.998171	+	0.0604490i	$0.980747\pi$
$164$	5706.65	+	1115.22i	0.212175	+	0.0414643i
$165$		0			0
$166$	−225.937	+	2334.12i	−0.00819918	+	0.0847047i
$167$		−	22589.6i		−	0.809983i	−0.914320	−	0.404992i	$-0.867274\pi$
$167$		−	22589.6i		−	0.809983i	0.914320	−	0.404992i	$-0.132726\pi$
$168$		0			0
$169$	−25296.6			−0.885704
$170$	26508.5	+	2565.95i	0.917250	+	0.0887872i
$171$		0			0
$172$	198.567	−	1016.07i	0.00671196	−	0.0343454i
$173$	20741.2			0.693013			0.346506	−	0.938048i	$-0.387368\pi$
$173$	20741.2			0.693013			0.346506	+	0.938048i	$0.387368\pi$
$174$		0			0
$175$		−	8601.02i		−	0.280850i
$176$	10341.8	−	25449.2i	0.333866	−	0.821579i
$177$		0			0
$178$	23049.3	+	2231.11i	0.727476	+	0.0704176i
$179$			37551.2i			1.17197i	0.810321	+	0.585986i	$0.199293\pi$
$179$			37551.2i			1.17197i	−0.810321	+	0.585986i	$0.800707\pi$
$180$		0			0
$181$	−38019.3			−1.16050			−0.580252	−	0.814437i	$-0.697046\pi$
$181$	−38019.3			−1.16050			−0.580252	+	0.814437i	$0.697046\pi$
$182$	1808.22	−	18680.5i	0.0545895	−	0.563957i
$183$		0			0
$184$	17955.6	−	60282.7i	0.530353	−	1.78056i
$185$	32033.4			0.935964
$186$		0			0
$187$			31322.9i			0.895734i
$188$	−10727.3	+	54892.1i	−0.303512	+	1.55308i
$189$		0			0
$190$	2680.41	−	27690.9i	0.0742495	−	0.767062i
$191$			7313.44i			0.200473i	0.994964	+	0.100236i	$0.0319599\pi$
$191$			7313.44i			0.200473i	−0.994964	+	0.100236i	$0.968040\pi$
$192$		0			0
$193$	60987.7			1.63730			0.818649	−	0.574294i	$-0.194724\pi$
$193$	60987.7			1.63730			0.818649	+	0.574294i	$0.194724\pi$
$194$	−43149.5	−	4176.75i	−1.14650	−	0.110978i
$195$		0			0
$196$	−68194.3	−	13326.9i	−1.77515	−	0.346910i
$197$	13891.0			0.357934			0.178967	−	0.983855i	$-0.442725\pi$
$197$	13891.0			0.357934			0.178967	+	0.983855i	$0.442725\pi$
$198$		0			0
$199$		−	43065.2i		−	1.08748i	−0.839255	−	0.543738i	$-0.817008\pi$
$199$		−	43065.2i		−	1.08748i	0.839255	−	0.543738i	$-0.182992\pi$
$200$	6424.23	+	1913.50i	0.160606	+	0.0478375i
$201$		0			0
$202$	45797.6	+	4433.08i	1.12238	+	0.108643i
$203$			86224.6i			2.09237i
$204$		0			0
$205$	8289.17			0.197244
$206$	5333.39	−	55098.6i	0.125681	−	1.29839i
$207$		0			0
$208$	13550.5	+	5506.51i	0.313204	+	0.127277i
$209$	32720.1			0.749070
$210$		0			0
$211$		−	23061.3i		−	0.517987i	−0.965879	−	0.258993i	$-0.916609\pi$
$211$		−	23061.3i		−	0.517987i	0.965879	−	0.258993i	$-0.0833908\pi$
$212$	−51330.9	−	10031.4i	−1.14211	−	0.223197i
$213$		0			0
$214$	−5677.49	+	58653.5i	−0.123973	+	1.28076i
$215$		−	1475.89i		−	0.0319285i
$216$		0			0
$217$	−64303.6			−1.36558
$218$	−14114.8	−	1366.27i	−0.297003	−	0.0287490i
$219$		0			0
$220$	7510.99	−	38434.1i	0.155186	−	0.794092i
$221$	−16677.9			−0.341474
$222$		0			0
$223$		−	59739.3i		−	1.20130i	−0.799514	−	0.600648i	$-0.794909\pi$
$223$		−	59739.3i		−	1.20130i	0.799514	−	0.600648i	$-0.205091\pi$
$224$	39016.7	−	74491.9i	0.777597	−	1.48461i
$225$		0			0
$226$	−49656.0	−	4806.56i	−0.972198	−	0.0941060i
$227$		−	33998.0i		−	0.659784i	−0.944019	−	0.329892i	$-0.892988\pi$
$227$		−	33998.0i		−	0.659784i	0.944019	−	0.329892i	$-0.107012\pi$
$228$		0			0
$229$	14052.1			0.267959			0.133980	−	0.990984i	$-0.457224\pi$
$229$	14052.1			0.267959			0.133980	+	0.990984i	$0.457224\pi$
$230$	8639.33	−	89251.9i	0.163314	−	1.68718i
$231$		0			0
$232$	−64402.4	−	19182.7i	−1.19654	−	0.356397i
$233$	−9622.25			−0.177241			−0.0886206	−	0.996065i	$-0.528246\pi$
$233$	−9622.25			−0.177241			−0.0886206	+	0.996065i	$0.528246\pi$
$234$		0			0
$235$			79733.4i			1.44379i
$236$	2662.05	−	13621.8i	0.0477961	−	0.244575i
$237$		0			0
$238$	−9238.21	+	95438.9i	−0.163092	+	1.68489i
$239$			28426.1i			0.497647i	0.968549	+	0.248824i	$0.0800439\pi$
$239$			28426.1i			0.497647i	−0.968549	+	0.248824i	$0.919956\pi$
$240$		0			0
$241$	−9489.78			−0.163389			−0.0816943	−	0.996657i	$-0.526033\pi$
$241$	−9489.78			−0.163389			−0.0816943	+	0.996657i	$0.526033\pi$
$242$	−12447.6	−	1204.90i	−0.212548	−	0.0205740i
$243$		0			0
$244$	94766.3	+	18519.7i	1.59175	+	0.311068i
$245$	−99055.4			−1.65024
$246$		0			0
$247$			17421.8i			0.285562i
$248$	14305.9	−	48029.3i	0.232601	−	0.780914i
$249$		0			0
$250$	66269.3	+	6414.68i	1.06031	+	0.102635i
$251$			29529.8i			0.468720i	0.972150	+	0.234360i	$0.0752994\pi$
$251$			29529.8i			0.468720i	−0.972150	+	0.234360i	$0.924701\pi$
$252$		0			0
$253$	105462.			1.64761
$254$	−386.185	+	3989.63i	−0.00598588	+	0.0618394i
$255$		0			0
$256$	46958.9	+	45714.6i	0.716536	+	0.697550i
$257$	14174.7			0.214609			0.107304	−	0.994226i	$-0.465778\pi$
$257$	14174.7			0.214609			0.107304	+	0.994226i	$0.465778\pi$
$258$		0			0
$259$			115330.i			1.71927i
$260$	20464.2	+	3999.23i	0.302726	+	0.0591603i
$261$		0			0
$262$	−12929.7	+	133575.i	−0.188358	+	1.94591i
$263$			106157.i			1.53475i	0.641197	+	0.767377i	$0.278438\pi$
$263$			106157.i			1.53475i	−0.641197	+	0.767377i	$0.721562\pi$
$264$		0			0
$265$	−74560.5			−1.06174
$266$	99695.9	+	9650.29i	1.40901	+	0.136388i
$267$		0			0
$268$	10499.9	−	53728.3i	0.146189	−	0.748055i
$269$	57391.1			0.793122			0.396561	−	0.918008i	$-0.370204\pi$
$269$	57391.1			0.793122			0.396561	+	0.918008i	$0.370204\pi$
$270$		0			0
$271$		−	22027.2i		−	0.299930i	−0.988691	−	0.149965i	$-0.952084\pi$
$271$		−	22027.2i		−	0.299930i	0.988691	−	0.149965i	$-0.0479161\pi$
$272$	−69229.4	−	28132.8i	−0.935735	−	0.380255i
$273$		0			0
$274$	−78436.3	−	7592.41i	−1.04476	−	0.101130i
$275$			11238.9i			0.148613i
$276$		0			0
$277$	30261.0			0.394388			0.197194	−	0.980364i	$-0.436817\pi$
$277$	30261.0			0.394388			0.197194	+	0.980364i	$0.436817\pi$
$278$	−4869.15	+	50302.6i	−0.0630034	+	0.650880i
$279$		0			0
$280$	34221.0	−	114891.i	0.436492	−	1.46544i
$281$	30448.5			0.385615			0.192807	−	0.981237i	$-0.438241\pi$
$281$	30448.5			0.385615			0.192807	+	0.981237i	$0.438241\pi$
$282$		0			0
$283$			102769.i			1.28318i	0.767046	+	0.641592i	$0.221726\pi$
$283$			102769.i			1.28318i	−0.767046	+	0.641592i	$0.778274\pi$
$284$	−9785.25	+	50071.5i	−0.121321	+	0.620803i
$285$		0			0
$286$	−2362.78	+	24409.6i	−0.0288863	+	0.298421i
$287$			29843.6i			0.362316i
$288$		0			0
$289$	1686.57			0.0201934
$290$	−95351.3	−	9229.73i	−1.13378	−	0.109747i
$291$		0			0
$292$	−67019.8	−	13097.4i	−0.786027	−	0.153610i
$293$	40053.4			0.466557			0.233278	−	0.972410i	$-0.425055\pi$
$293$	40053.4			0.466557			0.233278	+	0.972410i	$0.425055\pi$
$294$		0			0
$295$		−	19786.4i		−	0.227364i
$296$	−86141.7	−	25657.9i	−0.983173	−	0.292845i
$297$		0			0
$298$	−95771.3	−	9270.39i	−1.07846	−	0.104392i
$299$			56153.1i			0.628104i
$300$		0			0
$301$	5313.68			0.0586492
$302$	−11176.5	+	115463.i	−0.122544	+	1.26598i
$303$		0			0
$304$	−29387.7	+	72317.4i	−0.317993	+	0.782520i
$305$	137653.			1.47974
$306$		0			0
$307$			17196.5i			0.182458i	0.995830	+	0.0912291i	$0.0290796\pi$
$307$			17196.5i			0.182458i	−0.995830	+	0.0912291i	$0.970920\pi$
$308$	138374.	+	27041.9i	1.45866	+	0.285060i
$309$		0			0
$310$	6883.25	−	71110.0i	0.0716259	−	0.739958i
$311$		−	135448.i		−	1.40039i	−0.713949	−	0.700197i	$-0.753095\pi$
$311$		−	135448.i		−	1.40039i	0.713949	−	0.700197i	$-0.246905\pi$
$312$		0			0
$313$	−29263.8			−0.298705			−0.149352	−	0.988784i	$-0.547719\pi$
$313$	−29263.8			−0.298705			−0.149352	+	0.988784i	$0.547719\pi$
$314$	−44365.3	−	4294.44i	−0.449971	−	0.0435559i
$315$		0			0
$316$	23109.0	−	118250.i	0.231424	−	1.18420i
$317$	−58389.3			−0.581051			−0.290526	−	0.956867i	$-0.593830\pi$
$317$	−58389.3			−0.581051			−0.290526	+	0.956867i	$0.593830\pi$
$318$		0			0
$319$		−	112669.i		−	1.10719i
$320$	78200.3	+	51120.3i	0.763674	+	0.499222i
$321$		0			0
$322$	321334.	+	31104.2i	3.09917	+	0.299991i
$323$		−	89008.3i		−	0.853150i
$324$		0			0
$325$	−5984.14			−0.0566546
$326$	1237.92	−	12788.8i	0.0116481	−	0.120336i
$327$		0			0
$328$	−22290.6	−	6639.41i	−0.207193	−	0.0617137i
$329$	−287065.			−2.65209
$330$		0			0
$331$		−	92155.6i		−	0.841135i	−0.907261	−	0.420568i	$-0.861831\pi$
$331$		−	92155.6i		−	0.841135i	0.907261	−	0.420568i	$-0.138169\pi$
$332$	1799.08	−	9205.98i	0.0163221	−	0.0835207i
$333$		0			0
$334$	−8705.76	+	89938.1i	−0.0780393	+	0.806215i
$335$		−	78042.8i		−	0.695414i
$336$		0			0
$337$	−44787.4			−0.394363			−0.197182	−	0.980367i	$-0.563179\pi$
$337$	−44787.4			−0.394363			−0.197182	+	0.980367i	$0.563179\pi$
$338$	100716.	+	9748.98i	0.881583	+	0.0853348i
$339$		0			0
$340$	−104552.	−	20432.1i	−0.904428	−	0.176748i
$341$	84024.8			0.722602
$342$		0			0
$343$		−	159459.i		−	1.35538i
$344$	−1182.15	+	3968.86i	−0.00998980	+	0.0335389i
$345$		0			0
$346$	−82578.8	−	7993.39i	−0.689789	−	0.0667696i
$347$		−	105966.i		−	0.880049i	−0.897986	−	0.440024i	$-0.854970\pi$
$347$		−	105966.i		−	0.880049i	0.897986	−	0.440024i	$-0.145030\pi$
$348$		0			0
$349$	66799.1			0.548428			0.274214	−	0.961669i	$-0.411582\pi$
$349$	66799.1			0.548428			0.274214	+	0.961669i	$0.411582\pi$
$350$	−3314.73	+	34244.0i	−0.0270590	+	0.279543i
$351$		0			0
$352$	−50982.7	+	97337.8i	−0.411469	+	0.785590i
$353$	−37451.3			−0.300551			−0.150275	−	0.988644i	$-0.548016\pi$
$353$	−37451.3			−0.300551			−0.150275	+	0.988644i	$0.548016\pi$
$354$		0			0
$355$			72731.2i			0.577118i
$356$	−90908.6	−	17765.8i	−0.717307	−	0.140180i
$357$		0			0
$358$	14471.7	−	149506.i	0.112916	−	1.16652i
$359$		−	117783.i		−	0.913893i	−0.889494	−	0.456947i	$-0.848943\pi$
$359$		−	117783.i		−	0.913893i	0.889494	−	0.456947i	$-0.151057\pi$
$360$		0			0
$361$	37342.4			0.286542
$362$	151370.	+	14652.2i	1.15511	+	0.111811i
$363$		0			0
$364$	−14398.5	+	73677.6i	−0.108671	+	0.556074i
$365$	−97349.5			−0.730715
$366$		0			0
$367$		−	119975.i		−	0.890753i	−0.895343	−	0.445376i	$-0.853070\pi$
$367$		−	119975.i		−	0.890753i	0.895343	−	0.445376i	$-0.146930\pi$
$368$	−94720.6	+	233089.i	−0.699438	+	1.72118i
$369$		0			0
$370$	−127537.	−	12345.3i	−0.931610	−	0.0901772i
$371$		−	268441.i		−	1.95030i
$372$		0			0
$373$	−261060.			−1.87638			−0.938192	−	0.346114i	$-0.887501\pi$
$373$	−261060.			−1.87638			−0.938192	+	0.346114i	$0.887501\pi$
$374$	12071.5	−	124709.i	0.0863012	−	0.891567i
$375$		0			0
$376$	63864.4	−	214413.i	0.451734	−	1.51661i
$377$	59990.6			0.422085
$378$		0			0
$379$			138313.i			0.962907i	0.876472	+	0.481454i	$0.159891\pi$
$379$			138313.i			0.962907i	−0.876472	+	0.481454i	$0.840109\pi$
$380$	−21343.5	+	109216.i	−0.147808	+	0.756340i
$381$		0			0
$382$	2818.51	−	29117.7i	0.0193149	−	0.199540i
$383$			182153.i			1.24176i	0.783905	+	0.620880i	$0.213225\pi$
$383$			182153.i			1.24176i	−0.783905	+	0.620880i	$0.786775\pi$
$384$		0			0
$385$	200995.			1.35602
$386$	−242816.	−	23503.9i	−1.62968	−	0.157748i
$387$		0			0
$388$	170185.	+	33258.6i	1.13047	+	0.220922i
$389$	143024.			0.945171			0.472586	−	0.881285i	$-0.343321\pi$
$389$	143024.			0.945171			0.472586	+	0.881285i	$0.343321\pi$
$390$		0			0
$391$		−	286886.i		−	1.87653i
$392$	266372.	+	79340.8i	1.73347	+	0.516326i
$393$		0			0
$394$	−55305.7	−	5353.43i	−0.356268	−	0.0344858i
$395$		−	171764.i		−	1.10087i
$396$		0			0
$397$	−55426.8			−0.351673			−0.175837	−	0.984419i	$-0.556263\pi$
$397$	−55426.8			−0.351673			−0.175837	+	0.984419i	$0.556263\pi$
$398$	−16596.8	+	171459.i	−0.104775	+	1.08242i
$399$		0			0
$400$	−24839.9	−	10094.2i	−0.155250	−	0.0630889i
$401$	70455.5			0.438154			0.219077	−	0.975708i	$-0.429695\pi$
$401$	70455.5			0.438154			0.219077	+	0.975708i	$0.429695\pi$
$402$		0			0
$403$			44739.1i			0.275472i
$404$	−180630.	−	35299.6i	−1.10669	−	0.216275i
$405$		0			0
$406$	33229.9	−	343294.i	0.201594	−	2.08264i
$407$		−	150700.i		−	0.909758i
$408$		0			0
$409$	157877.			0.943785			0.471892	−	0.881656i	$-0.343571\pi$
$409$	157877.			0.943785			0.471892	+	0.881656i	$0.343571\pi$
$410$	−33002.4	−	3194.54i	−0.196326	−	0.0190038i
$411$		0			0
$412$	−42468.6	+	217314.i	−0.250192	+	1.28024i
$413$	71237.0			0.417643
$414$		0			0
$415$		−	13372.1i		−	0.0776434i
$416$	−51827.6	−	27145.8i	−0.299484	−	0.156861i
$417$		0			0
$418$	−130272.	−	12609.9i	−0.745585	−	0.0721705i
$419$		−	22310.6i		−	0.127082i	−0.997979	−	0.0635409i	$-0.979761\pi$
$419$		−	22310.6i		−	0.127082i	0.997979	−	0.0635409i	$-0.0202393\pi$
$420$		0			0
$421$	−106342.			−0.599985			−0.299992	−	0.953942i	$-0.596984\pi$
$421$	−106342.			−0.599985			−0.299992	+	0.953942i	$0.596984\pi$
$422$	−8887.53	+	91816.0i	−0.0499064	+	0.515577i
$423$		0			0
$424$	200502.	+	59721.0i	1.11529	+	0.332197i
$425$	30573.0			0.169262
$426$		0			0
$427$			495592.i			2.71812i
$428$	45208.6	−	231334.i	0.246793	−	1.26285i
$429$		0			0
$430$	−568.791	+	5876.12i	−0.00307621	+	0.0317800i
$431$			270251.i			1.45483i	0.686197	+	0.727416i	$0.259279\pi$
$431$			270251.i			1.45483i	−0.686197	+	0.727416i	$0.740721\pi$
$432$		0			0
$433$	−115964.			−0.618513			−0.309256	−	0.950979i	$-0.600080\pi$
$433$	−115964.			−0.618513			−0.309256	+	0.950979i	$0.600080\pi$
$434$	256018.	+	24781.8i	1.35922	+	0.131569i
$435$		0			0
$436$	55669.8	+	10879.3i	0.292851	+	0.0572306i
$437$	−299683.			−1.56928
$438$		0			0
$439$			285409.i			1.48095i	0.672087	+	0.740473i	$0.265398\pi$
$439$			285409.i			1.48095i	−0.672087	+	0.740473i	$0.734602\pi$
$440$	−44716.2	+	150126.i	−0.230972	+	0.775446i
$441$		0			0
$442$	66401.3	+	6427.46i	0.339885	+	0.0328999i
$443$			269606.i			1.37380i	0.726753	+	0.686899i	$0.241028\pi$
$443$			269606.i			1.37380i	−0.726753	+	0.686899i	$0.758972\pi$
$444$		0			0
$445$	−132049.			−0.666830
$446$	−23022.8	+	237845.i	−0.115741	+	1.19571i
$447$		0			0
$448$	−184049.	+	281545.i	−0.917017	+	1.40279i
$449$	−12218.4			−0.0606070			−0.0303035	−	0.999541i	$-0.509647\pi$
$449$	−12218.4			−0.0606070			−0.0303035	+	0.999541i	$0.509647\pi$
$450$		0			0
$451$		−	38996.3i		−	0.191721i
$452$	195848.	+	38273.6i	0.958608	+	0.187336i
$453$		0			0
$454$	−13102.4	+	135359.i	−0.0635681	+	0.656715i
$455$			107020.i			0.516943i
$456$		0			0
$457$	296484.			1.41961			0.709805	−	0.704398i	$-0.248783\pi$
$457$	296484.			1.41961			0.709805	+	0.704398i	$0.248783\pi$
$458$	−55946.8	−	5415.49i	−0.266713	−	0.0258171i
$459$		0			0
$460$	−68793.1	+	352017.i	−0.325109	+	1.66360i
$461$	−38233.3			−0.179904			−0.0899519	−	0.995946i	$-0.528671\pi$
$461$	−38233.3			−0.179904			−0.0899519	+	0.995946i	$0.528671\pi$
$462$		0			0
$463$		−	210426.i		−	0.981607i	−0.871270	−	0.490804i	$-0.836703\pi$
$463$		−	210426.i		−	0.981607i	0.871270	−	0.490804i	$-0.163297\pi$
$464$	249018.	+	101194.i	1.15663	+	0.470022i
$465$		0			0
$466$	38309.9	+	3708.29i	0.176417	+	0.0170766i
$467$			58599.8i			0.268697i	0.990934	+	0.134348i	$0.0428941\pi$
$467$			58599.8i			0.268697i	−0.990934	+	0.134348i	$0.957106\pi$
$468$		0			0
$469$	280978.			1.27740
$470$	30728.2	−	317450.i	0.139105	−	1.43707i
$471$		0			0
$472$	−15848.4	+	53207.9i	−0.0711378	+	0.238832i
$473$	−6943.32			−0.0310345
$474$		0			0
$475$		−	31936.7i		−	0.141548i
$476$	73561.9	−	376419.i	0.324668	−	1.66134i
$477$		0			0
$478$	10955.1	−	113175.i	0.0479467	−	0.495332i
$479$			123721.i			0.539227i	0.962969	+	0.269614i	$0.0868960\pi$
$479$			123721.i			0.539227i	−0.962969	+	0.269614i	$0.913104\pi$
$480$		0			0
$481$	80240.6			0.346820
$482$	37782.5	+	3657.24i	0.162629	+	0.0157420i
$483$		0			0
$484$	49094.6	+	9594.33i	0.209577	+	0.0409566i
$485$	247202.			1.05092
$486$		0			0
$487$			39541.5i			0.166723i	0.996519	+	0.0833614i	$0.0265656\pi$
$487$			39541.5i			0.166723i	−0.996519	+	0.0833614i	$0.973434\pi$
$488$	−370164.	−	110256.i	−1.55437	−	0.462981i
$489$		0			0
$490$	394378.	+	38174.7i	1.64256	+	0.158995i
$491$		−	97311.7i		−	0.403647i	−0.979422	−	0.201824i	$-0.935313\pi$
$491$		−	97311.7i		−	0.403647i	0.979422	−	0.201824i	$-0.0646868\pi$
$492$		0			0
$493$	−306492.			−1.26103
$494$	6714.16	−	69363.2i	0.0275130	−	0.284233i
$495$		0			0
$496$	−75467.2	+	185710.i	−0.306757	+	0.754870i
$497$	−261855.			−1.06010
$498$		0			0
$499$			217620.i			0.873974i	0.899468	+	0.436987i	$0.143954\pi$
$499$			217620.i			0.873974i	−0.899468	+	0.436987i	$0.856046\pi$
$500$	−261372.	−	51078.7i	−1.04549	−	0.204315i
$501$		0			0
$502$	11380.4	−	117570.i	0.0451597	−	0.466539i
$503$		−	190305.i		−	0.752169i	−0.926585	−	0.376084i	$-0.877270\pi$
$503$		−	190305.i		−	0.752169i	0.926585	−	0.376084i	$-0.122730\pi$
$504$		0			0
$505$	−262373.			−1.02881
$506$	−419884.	−	40643.6i	−1.63994	−	0.158742i
$507$		0			0
$508$	3075.11	−	15735.4i	0.0119161	−	0.0609750i
$509$	100460.			0.387755			0.193878	−	0.981026i	$-0.437894\pi$
$509$	100460.			0.387755			0.193878	+	0.981026i	$0.437894\pi$
$510$		0			0
$511$		−	350488.i		−	1.34224i
$512$	−169344.	−	200105.i	−0.645996	−	0.763341i
$513$		0			0
$514$	−56435.1	−	5462.75i	−0.213611	−	0.0206769i
$515$			315658.i			1.19015i
$516$		0			0
$517$	375104.			1.40337
$518$	44446.7	−	459174.i	0.165646	−	1.71127i
$519$		0			0
$520$	−79934.9	−	23809.2i	−0.295617	−	0.0880517i
$521$	417246.			1.53715			0.768575	−	0.639759i	$-0.220966\pi$
$521$	417246.			1.53715			0.768575	+	0.639759i	$0.220966\pi$
$522$		0			0
$523$			246542.i			0.901339i	0.892691	+	0.450669i	$0.148815\pi$
$523$			246542.i			0.901339i	−0.892691	+	0.450669i	$0.851185\pi$
$524$	102956.	−	526831.i	0.374964	−	1.91871i
$525$		0			0
$526$	40911.7	−	422654.i	0.147869	−	1.52761i
$527$		−	228572.i		−	0.823004i
$528$		0			0
$529$	−686081.			−2.45168
$530$	296855.	+	28734.7i	1.05680	+	0.102295i
$531$		0			0
$532$	−393209.	−	76843.1i	−1.38931	−	0.271507i
$533$	20763.6			0.0730883
$534$		0			0
$535$		−	336024.i		−	1.17399i
$536$	−62510.3	+	209867.i	−0.217581	+	0.730490i
$537$		0			0
$538$	−228496.	−	22117.8i	−0.789432	−	0.0764148i
$539$			466004.i			1.60403i
$540$		0			0
$541$	65384.7			0.223399			0.111700	−	0.993742i	$-0.464371\pi$
$541$	65384.7			0.223399			0.111700	+	0.993742i	$0.464371\pi$
$542$	−8489.00	+	87698.8i	−0.0288973	+	0.298535i
$543$		0			0
$544$	264787.	+	138688.i	0.894745	+	0.468641i
$545$	80863.1			0.272243
$546$		0			0
$547$			238533.i			0.797213i	0.917122	+	0.398607i	$0.130506\pi$
$547$			238533.i			0.797213i	−0.917122	+	0.398607i	$0.869494\pi$
$548$	309360.	+	60456.7i	1.03016	+	0.201318i
$549$		0			0
$550$	4331.32	−	44746.3i	0.0143184	−	0.147922i
$551$			320163.i			1.05455i
$552$		0			0
$553$	618402.			2.02218
$554$	−120481.	−	11662.2i	−0.392554	−	0.0379981i
$555$		0			0
$556$	38772.0	−	198398.i	0.125421	−	0.641782i
$557$	−412118.			−1.32835			−0.664173	−	0.747579i	$-0.731216\pi$
$557$	−412118.			−1.32835			−0.664173	+	0.747579i	$0.731216\pi$
$558$		0			0
$559$		−	3696.98i		−	0.0118310i
$560$	−180525.	+	444237.i	−0.575653	+	1.41657i
$561$		0			0
$562$	−121227.	−	11734.5i	−0.383821	−	0.0371528i
$563$			451490.i			1.42440i	0.701978	+	0.712199i	$0.252301\pi$
$563$			451490.i			1.42440i	−0.701978	+	0.712199i	$0.747699\pi$
$564$		0			0
$565$	284478.			0.891151
$566$	39605.8	−	409163.i	0.123631	−	1.27721i
$567$		0			0
$568$	58255.8	−	195583.i	0.180569	−	0.606226i
$569$	−125491.			−0.387603			−0.193801	−	0.981041i	$-0.562082\pi$
$569$	−125491.			−0.387603			−0.193801	+	0.981041i	$0.562082\pi$
$570$		0			0
$571$		−	295508.i		−	0.906352i	−0.891421	−	0.453176i	$-0.850291\pi$
$571$		−	295508.i		−	0.906352i	0.891421	−	0.453176i	$-0.149709\pi$
$572$	18814.3	−	96273.7i	0.0575038	−	0.294249i
$573$		0			0
$574$	11501.3	−	118819.i	0.0349080	−	0.360630i
$575$		−	102937.i		−	0.311339i
$576$		0			0
$577$	−169123.			−0.507986			−0.253993	−	0.967206i	$-0.581744\pi$
$577$	−169123.			−0.507986			−0.253993	+	0.967206i	$0.581744\pi$
$578$	−6714.89	−	649.983i	−0.0200994	−	0.00194557i
$579$		0			0
$580$	376074.	+	73494.4i	1.11794	+	0.218473i
$581$	48143.8			0.142622
$582$		0			0
$583$			350769.i			1.03201i
$584$	261785.	+	77974.4i	0.767571	+	0.228626i
$585$		0			0
$586$	−159468.	−	15436.1i	−0.464386	−	0.0449513i
$587$		−	287012.i		−	0.832959i	−0.909145	−	0.416480i	$-0.863264\pi$
$587$		−	287012.i		−	0.832959i	0.909145	−	0.416480i	$-0.136736\pi$
$588$		0			0
$589$	−238768.			−0.688248
$590$	−7625.41	+	78777.2i	−0.0219058	+	0.226306i
$591$		0			0
$592$	333075.	+	135352.i	0.950384	+	0.386208i
$593$	−275747.			−0.784155			−0.392078	−	0.919932i	$-0.628244\pi$
$593$	−275747.			−0.784155			−0.392078	+	0.919932i	$0.628244\pi$
$594$		0			0
$595$		−	546767.i		−	1.54443i
$596$	377730.	+	73818.1i	1.06338	+	0.207812i
$597$		0			0
$598$	21640.7	−	223567.i	0.0605158	−	0.625182i
$599$			498296.i			1.38878i	0.719598	+	0.694391i	$0.244326\pi$
$599$			498296.i			1.38878i	−0.719598	+	0.694391i	$0.755674\pi$
$600$		0			0
$601$	−382620.			−1.05930			−0.529650	−	0.848216i	$-0.677677\pi$
$601$	−382620.			−1.05930			−0.529650	+	0.848216i	$0.677677\pi$
$602$	−21155.8	−	2047.82i	−0.0583764	−	0.00565067i
$603$		0			0
$604$	88995.9	−	455395.i	0.243947	−	1.24829i
$605$	71312.2			0.194829
$606$		0			0
$607$			11197.4i			0.0303907i	0.999885	+	0.0151954i	$0.00483702\pi$
$607$			11197.4i			0.0303907i	−0.999885	+	0.0151954i	$0.995163\pi$
$608$	144874.	−	276598.i	0.391907	−	0.748242i
$609$		0			0
$610$	−548049.	−	53049.6i	−1.47285	−	0.142568i
$611$			199725.i			0.534994i
$612$		0			0
$613$	289927.			0.771558			0.385779	−	0.922591i	$-0.373933\pi$
$613$	289927.			0.771558			0.385779	+	0.922591i	$0.373933\pi$
$614$	6627.32	−	68466.0i	0.0175793	−	0.181609i
$615$		0			0
$616$	−540501.	−	160992.i	−1.42441	−	0.424271i
$617$	477462.			1.25420			0.627102	−	0.778937i	$-0.284241\pi$
$617$	477462.			1.25420			0.627102	+	0.778937i	$0.284241\pi$
$618$		0			0
$619$		−	254686.i		−	0.664698i	−0.943156	−	0.332349i	$-0.892159\pi$
$619$		−	254686.i		−	0.664698i	0.943156	−	0.332349i	$-0.107841\pi$
$620$	−54809.8	+	280464.i	−0.142585	+	0.729615i
$621$		0			0
$622$	−52199.8	+	539270.i	−0.134924	+	1.39388i
$623$		−	475417.i		−	1.22489i
$624$		0			0
$625$	−314195.			−0.804339
$626$	116511.	+	11277.9i	0.297315	+	0.0287793i
$627$		0			0
$628$	174981.	+	34195.7i	0.443681	+	0.0867065i
$629$	−409949.			−1.03617
$630$		0			0
$631$		−	522734.i		−	1.31287i	−0.754383	−	0.656435i	$-0.772064\pi$
$631$		−	522734.i		−	1.31287i	0.754383	−	0.656435i	$-0.227936\pi$
$632$	−137578.	+	461893.i	−0.344441	+	1.15640i
$633$		0			0
$634$	232471.	+	22502.5i	0.578348	+	0.0559825i
$635$		−	22856.5i		−	0.0566842i
$636$		0			0
$637$	−248124.			−0.611492
$638$	−43421.1	+	448578.i	−0.106674	+	1.10204i
$639$		0			0
$640$	−291645.	−	233667.i	−0.712023	−	0.570477i
$641$	399966.			0.973434			0.486717	−	0.873560i	$-0.338194\pi$
$641$	399966.			0.973434			0.486717	+	0.873560i	$0.338194\pi$
$642$		0			0
$643$		−	95576.4i		−	0.231169i	−0.993298	−	0.115584i	$-0.963126\pi$
$643$		−	95576.4i		−	0.231169i	0.993298	−	0.115584i	$-0.0368740\pi$
$644$	−1.26737e6	−	247676.i	−3.05585	−	0.597190i
$645$		0			0
$646$	−34302.7	+	354377.i	−0.0821983	+	0.849181i
$647$		−	310762.i		−	0.742367i	−0.928560	−	0.371184i	$-0.878952\pi$
$647$		−	310762.i		−	0.742367i	0.928560	−	0.371184i	$-0.121048\pi$
$648$		0			0
$649$	−93084.5			−0.220998
$650$	23825.2	+	2306.21i	0.0563910	+	0.00545849i
$651$		0			0
$652$	−9857.28	+	50440.1i	−0.0231879	+	0.118654i
$653$	648837.			1.52163			0.760815	−	0.648969i	$-0.224799\pi$
$653$	648837.			1.52163			0.760815	+	0.648969i	$0.224799\pi$
$654$		0			0
$655$		−	765246.i		−	1.78369i
$656$	86188.8	+	35024.6i	0.200283	+	0.0813890i
$657$		0			0
$658$	1.14292e6	+	110631.i	2.63975	+	0.255520i
$659$		−	149570.i		−	0.344408i	−0.985061	−	0.172204i	$-0.944911\pi$
$659$		−	149570.i		−	0.344408i	0.985061	−	0.172204i	$-0.0550889\pi$
$660$		0			0
$661$	170849.			0.391029			0.195515	−	0.980701i	$-0.437362\pi$
$661$	170849.			0.391029			0.195515	+	0.980701i	$0.437362\pi$
$662$	−35515.6	+	366907.i	−0.0810407	+	0.837222i
$663$		0			0
$664$	−10710.7	+	35959.3i	−0.0242931	+	0.0815596i
$665$	−571155.			−1.29155
$666$		0			0
$667$			1.03193e6i			2.31953i
$668$	69322.0	−	354724.i	0.155353	−	0.794945i
$669$		0			0
$670$	−30076.7	+	310719.i	−0.0670010	+	0.692179i
$671$		−	647584.i		−	1.43830i
$672$		0			0
$673$	782980.			1.72870			0.864351	−	0.502888i	$-0.167729\pi$
$673$	782980.			1.72870			0.864351	+	0.502888i	$0.167729\pi$
$674$	178316.	+	17260.5i	0.392528	+	0.0379956i
$675$		0			0
$676$	−397231.	−	77629.0i	−0.869260	−	0.169876i
$677$	458234.			0.999792			0.499896	−	0.866085i	$-0.333372\pi$
$677$	458234.			0.999792			0.499896	+	0.866085i	$0.333372\pi$
$678$		0			0
$679$			890005.i			1.93042i
$680$	408388.	+	121641.i	0.883192	+	0.263065i
$681$		0			0
$682$	−334536.	−	32382.1i	−0.719240	−	0.0696204i
$683$		−	444767.i		−	0.953434i	−0.879057	−	0.476717i	$-0.841827\pi$
$683$		−	444767.i		−	0.953434i	0.879057	−	0.476717i	$-0.158173\pi$
$684$		0			0
$685$	449360.			0.957663
$686$	−61453.4	+	634868.i	−0.130586	+	1.34907i
$687$		0			0
$688$	6236.16	−	15346.0i	0.0131747	−	0.0324204i
$689$	−186767.			−0.393425
$690$		0			0
$691$			373248.i			0.781702i	0.920454	+	0.390851i	$0.127819\pi$
$691$			373248.i			0.781702i	−0.920454	+	0.390851i	$0.872181\pi$
$692$	325698.	+	63649.6i	0.680147	+	0.132918i
$693$		0			0
$694$	−40837.9	+	421891.i	−0.0847899	+	0.875955i
$695$		−	288182.i		−	0.596620i
$696$		0			0
$697$	−106081.			−0.218360
$698$	−265953.	−	25743.5i	−0.545877	−	0.0528393i
$699$		0			0
$700$	26394.4	−	135061.i	0.0538662	−	0.275636i
$701$	−699890.			−1.42427			−0.712137	−	0.702041i	$-0.752273\pi$
$701$	−699890.			−1.42427			−0.712137	+	0.702041i	$0.752273\pi$
$702$		0			0
$703$			428235.i			0.866507i
$704$	240495.	−	367892.i	0.485244	−	0.742292i
$705$		0			0
$706$	149108.	+	14433.3i	0.299153	+	0.0289571i
$707$		−	944624.i		−	1.88982i
$708$		0			0
$709$	305897.			0.608531			0.304265	−	0.952587i	$-0.401589\pi$
$709$	305897.			0.608531			0.304265	+	0.952587i	$0.401589\pi$
$710$	28029.7	−	289571.i	0.0556035	−	0.574433i
$711$		0			0
$712$	355096.	+	105768.i	0.700464	+	0.208638i
$713$	−769583.			−1.51383
$714$		0			0
$715$		−	139842.i		−	0.273543i
$716$	−115235.	+	589664.i	−0.224781	+	1.15021i
$717$		0			0
$718$	−45392.3	+	468942.i	−0.0880507	+	0.909641i
$719$			265534.i			0.513644i	0.966459	+	0.256822i	$0.0826755\pi$
$719$			265534.i			0.513644i	−0.966459	+	0.256822i	$0.917325\pi$
$720$		0			0
$721$	−1.13647e6			−2.18618
$722$	−148675.	−	14391.3i	−0.285209	−	0.0276074i
$723$		0			0
$724$	−597015.	−	116672.i	−1.13896	−	0.222582i
$725$	−109971.			−0.209220
$726$		0			0
$727$			901047.i			1.70482i	0.522874	+	0.852410i	$0.324860\pi$
$727$			901047.i			1.70482i	−0.522874	+	0.852410i	$0.675140\pi$
$728$	85720.4	−	287790.i	0.161741	−	0.543017i
$729$		0			0
$730$	387586.	+	37517.3i	0.727315	+	0.0704021i
$731$			18887.9i			0.0353466i
$732$		0			0
$733$	−887881.			−1.65252			−0.826261	−	0.563288i	$-0.809536\pi$
$733$	−887881.			−1.65252			−0.826261	+	0.563288i	$0.809536\pi$
$734$	−46236.7	+	477666.i	−0.0858212	+	0.886609i
$735$		0			0
$736$	466950.	−	891516.i	0.862014	−	1.64579i
$737$	−367151.			−0.675943
$738$		0			0
$739$		−	401381.i		−	0.734966i	−0.930030	−	0.367483i	$-0.880220\pi$
$739$		−	401381.i		−	0.734966i	0.930030	−	0.367483i	$-0.119780\pi$
$740$	503019.	+	98302.6i	0.918588	+	0.179515i
$741$		0			0
$742$	−103454.	+	1.06877e6i	−0.187905	+	1.94122i
$743$		−	314050.i		−	0.568880i	−0.958694	−	0.284440i	$-0.908192\pi$
$743$		−	314050.i		−	0.568880i	0.958694	−	0.284440i	$-0.0918077\pi$
$744$		0			0
$745$	548671.			0.988552
$746$	1.03938e6	+	100609.i	1.86766	+	0.180784i
$747$		0			0
$748$	−96122.5	+	491863.i	−0.171799	+	0.879105i
$749$	1.20979e6			2.15648
$750$		0			0
$751$			331743.i			0.588195i	0.955775	+	0.294098i	$0.0950191\pi$
$751$			331743.i			0.588195i	−0.955775	+	0.294098i	$0.904981\pi$
$752$	−336901.	+	829049.i	−0.595754	+	1.46604i
$753$		0			0
$754$	−238846.	−	23119.6i	−0.420122	−	0.0406666i
$755$		−	661483.i		−	1.16045i
$756$		0			0
$757$	300610.			0.524581			0.262290	−	0.964989i	$-0.415522\pi$
$757$	300610.			0.524581			0.262290	+	0.964989i	$0.415522\pi$
$758$	53304.1	−	550678.i	0.0927731	−	0.958428i
$759$		0			0
$760$	127067.	−	426604.i	0.219991	−	0.738581i
$761$	78374.5			0.135334			0.0676668	−	0.997708i	$-0.478445\pi$
$761$	78374.5			0.135334			0.0676668	+	0.997708i	$0.478445\pi$
$762$		0			0
$763$			291132.i			0.500081i
$764$	−22443.2	+	114843.i	−0.0384501	+	0.196751i
$765$		0			0
$766$	70199.3	−	725221.i	0.119640	−	1.23598i
$767$		−	49562.9i		−	0.0842493i
$768$		0			0
$769$	987499.			1.66988			0.834938	−	0.550345i	$-0.185504\pi$
$769$	987499.			1.66988			0.834938	+	0.550345i	$0.185504\pi$
$770$	−800242.	−	77461.1i	−1.34971	−	0.130648i
$771$		0			0
$772$	957687.	+	187156.i	1.60690	+	0.314029i
$773$	−622600.			−1.04196			−0.520979	−	0.853569i	$-0.674433\pi$
$773$	−622600.			−1.04196			−0.520979	+	0.853569i	$0.674433\pi$
$774$		0			0
$775$		−	82013.1i		−	0.136546i
$776$	−664757.	−	198003.i	−1.10393	−	0.328812i
$777$		0			0
$778$	−569436.	−	55119.8i	−0.940774	−	0.0910643i
$779$			110813.i			0.182606i
$780$		0			0
$781$	342163.			0.560958
$782$	−110562.	+	1.14221e6i	−0.180798	+	1.86780i
$783$		0			0
$784$	−1.02995e6	−	418543.i	−1.67566	−	0.680939i
$785$	254168.			0.412459
$786$		0			0
$787$		−	1.09638e6i		−	1.77016i	−0.465437	−	0.885081i	$-0.654103\pi$
$787$		−	1.09638e6i		−	1.77016i	0.465437	−	0.885081i	$-0.345897\pi$
$788$	218130.	+	42628.2i	0.351288	+	0.0686507i
$789$		0			0
$790$	−66195.5	+	683858.i	−0.106066	+	1.09575i
$791$			1.02421e6i			1.63695i
$792$		0			0
$793$	344807.			0.548314
$794$	220676.	+	21360.8i	0.350037	+	0.0338826i
$795$		0			0
$796$	132156.	−	676250.i	0.208575	−	1.06729i
$797$	666106.			1.04864			0.524321	−	0.851521i	$-0.324319\pi$
$797$	666106.			1.04864			0.524321	+	0.851521i	$0.324319\pi$
$798$		0			0
$799$		−	1.02039e6i		−	1.59836i
$800$	95007.3	+	49762.0i	0.148449	+	0.0777532i
$801$		0			0
$802$	−280511.	−	27152.7i	−0.436115	−	0.0422147i
$803$			457979.i			0.710255i
$804$		0			0
$805$	−1.84092e6			−2.84081
$806$	17241.9	−	178124.i	0.0265408	−	0.274190i
$807$		0			0
$808$	705553.	+	210154.i	1.08070	+	0.321895i
$809$	1.18960e6			1.81762			0.908812	−	0.417206i	$-0.136991\pi$
$809$	1.18960e6			1.81762			0.908812	+	0.417206i	$0.136991\pi$
$810$		0			0
$811$		−	189071.i		−	0.287464i	−0.989617	−	0.143732i	$-0.954090\pi$
$811$		−	189071.i		−	0.287464i	0.989617	−	0.143732i	$-0.0459104\pi$
$812$	−264602.	+	1.35398e6i	−0.401312	+	2.05353i
$813$		0			0
$814$	−58078.1	+	599997.i	−0.0876523	+	0.905525i
$815$			73266.6i			0.110304i
$816$		0			0
$817$	19730.4			0.0295591
$818$	−628571.	−	60843.9i	−0.939394	−	0.0909307i
$819$		0			0
$820$	130164.	+	25437.5i	0.193582	+	0.0378308i
$821$	−1.32025e6			−1.95870			−0.979352	−	0.202164i	$-0.935203\pi$
$821$	−1.32025e6			−1.95870			−0.979352	+	0.202164i	$0.935203\pi$
$822$		0			0
$823$			498753.i			0.736353i	0.929756	+	0.368176i	$0.120018\pi$
$823$			498753.i			0.736353i	−0.929756	+	0.368176i	$0.879982\pi$
$824$	252834.	−	848844.i	0.372376	−	1.25018i
$825$		0			0
$826$	−283622.	−	27453.8i	−0.415700	−	0.0402386i
$827$		−	160812.i		−	0.235130i	−0.993065	−	0.117565i	$-0.962491\pi$
$827$		−	160812.i		−	0.235130i	0.993065	−	0.117565i	$-0.0375088\pi$
$828$		0			0
$829$	70245.8			0.102214			0.0511071	−	0.998693i	$-0.483725\pi$
$829$	70245.8			0.102214			0.0511071	+	0.998693i	$0.483725\pi$
$830$	−5153.45	+	53239.7i	−0.00748069	+	0.0772821i
$831$		0			0
$832$	195884.	+	128052.i	0.282978	+	0.184986i
$833$	1.26767e6			1.82690
$834$		0			0
$835$		−	515253.i		−	0.739005i
$836$	513802.	+	100410.i	0.735163	+	0.143670i
$837$		0			0
$838$	−8598.23	+	88827.2i	−0.0122439	+	0.126491i
$839$			568844.i			0.808108i	0.914735	+	0.404054i	$0.132399\pi$
$839$			568844.i			0.808108i	−0.914735	+	0.404054i	$0.867601\pi$
$840$		0			0
$841$	395172.			0.558720
$842$	423389.	+	40982.8i	0.597193	+	0.0578066i
$843$		0			0
$844$	70769.5	−	362130.i	0.0993484	−	0.508370i
$845$	−576997.			−0.808091
$846$		0			0
$847$			256746.i			0.357879i
$848$	−775263.	−	315044.i	−1.07810	−	0.438106i
$849$		0			0
$850$	−121723.	−	11782.4i	−0.168475	−	0.0163079i
$851$			1.38026e6i			1.90591i
$852$		0			0
$853$	−959646.			−1.31890			−0.659452	−	0.751747i	$-0.729212\pi$
$853$	−959646.			−1.31890			−0.659452	+	0.751747i	$0.729212\pi$
$854$	190995.	−	1.97314e6i	0.261882	−	2.70547i
$855$		0			0
$856$	−269147.	+	903610.i	−0.367317	+	1.23320i
$857$	171138.			0.233016			0.116508	−	0.993190i	$-0.462830\pi$
$857$	171138.			0.233016			0.116508	+	0.993190i	$0.462830\pi$
$858$		0			0
$859$		−	75902.7i		−	0.102866i	−0.998676	−	0.0514329i	$-0.983621\pi$
$859$		−	75902.7i		−	0.102866i	0.998676	−	0.0514329i	$-0.0163788\pi$
$860$	4529.16	−	23175.9i	0.00612380	−	0.0313357i
$861$		0			0
$862$	104151.	−	1.07597e6i	0.140168	−	1.44806i
$863$		−	727039.i		−	0.976194i	−0.872790	−	0.488097i	$-0.837691\pi$
$863$		−	727039.i		−	0.976194i	0.872790	−	0.488097i	$-0.162309\pi$
$864$		0			0
$865$	473091.			0.632285
$866$	461700.	+	44691.2i	0.615635	+	0.0595918i
$867$		0			0
$868$	−1.00976e6	−	197332.i	−1.34022	−	0.261914i
$869$	−808059.			−1.07005
$870$		0			0
$871$		−	195490.i		−	0.257684i
$872$	−217451.	−	64769.2i	−0.285975	−	0.0851796i
$873$		0			0
$874$	1.19316e6	+	115494.i	1.56198	+	0.151195i
$875$		−	1.36688e6i		−	1.78531i
$876$		0			0
$877$	5985.62			0.00778233			0.00389117	−	0.999992i	$-0.498761\pi$
$877$	5985.62			0.00778233			0.00389117	+	0.999992i	$0.498761\pi$
$878$	109993.	−	1.13633e6i	0.142684	−	1.47406i
$879$		0			0
$880$	235890.	−	580479.i	0.304609	−	0.749585i
$881$	−1.04403e6			−1.34512			−0.672559	−	0.740043i	$-0.734805\pi$
$881$	−1.04403e6			−1.34512			−0.672559	+	0.740043i	$0.734805\pi$
$882$		0			0
$883$		−	1.43819e6i		−	1.84457i	−0.386512	−	0.922284i	$-0.626320\pi$
$883$		−	1.43819e6i		−	1.84457i	0.386512	−	0.922284i	$-0.373680\pi$
$884$	−261893.	−	51180.5i	−0.335134	−	0.0654937i
$885$		0			0
$886$	103903.	−	1.07341e6i	0.132361	−	1.36741i
$887$			29669.6i			0.0377106i	0.999822	+	0.0188553i	$0.00600219\pi$
$887$			29669.6i			0.0377106i	−0.999822	+	0.0188553i	$0.993998\pi$
$888$		0			0
$889$	82290.4			0.104123
$890$	525739.	+	50890.0i	0.663728	+	0.0642470i
$891$		0			0
$892$	183325.	−	938083.i	0.230405	−	1.17899i
$893$	−1.06591e6			−1.33665
$894$		0			0
$895$			856515.i			1.06927i
$896$	841275.	−	1.05001e6i	1.04790	−	1.30791i
$897$		0			0
$898$	48646.4	+	4708.83i	0.0603251	+	0.00583930i
$899$			822175.i			1.01729i
$900$		0			0
$901$	954194.			1.17540
$902$	−15028.7	+	155259.i	−0.0184717	+	0.190829i
$903$		0			0
$904$	−764996.	−	227859.i	−0.936100	−	0.278824i
$905$	−867193.			−1.05881
$906$		0			0
$907$		−	110783.i		−	0.134666i	−0.997731	−	0.0673331i	$-0.978551\pi$
$907$		−	110783.i		−	0.134666i	0.997731	−	0.0673331i	$-0.0214490\pi$
$908$	104332.	−	533869.i	0.126545	−	0.647535i
$909$		0			0
$910$	41244.2	−	426089.i	0.0498059	−	0.514538i
$911$			951442.i			1.14642i	0.819407	+	0.573212i	$0.194303\pi$
$911$			951442.i			1.14642i	−0.819407	+	0.573212i	$0.805697\pi$
$912$		0			0
$913$	−62908.9			−0.0754694
$914$	−1.18042e6	−	114261.i	−1.41301	−	0.136775i
$915$		0			0
$916$	220659.	+	43122.4i	0.262985	+	0.0513939i
$917$	2.75512e6			3.27644
$918$		0			0
$919$			930589.i			1.10186i	0.834551	+	0.550931i	$0.185727\pi$
$919$			930589.i			1.10186i	−0.834551	+	0.550931i	$0.814273\pi$
$920$	409555.	−	1.37501e6i	0.483879	−	1.62453i
$921$		0			0
$922$	152222.	+	14734.6i	0.179067	+	0.0173332i
$923$			182185.i			0.213850i
$924$		0			0
$925$	−147092.			−0.171912
$926$	−81095.6	+	837789.i	−0.0945748	+	0.977041i
$927$		0			0
$928$	−952441.	−	498861.i	−1.10597	−	0.579273i
$929$	−188066.			−0.217911			−0.108955	−	0.994047i	$-0.534751\pi$
$929$	−188066.			−0.217911			−0.108955	+	0.994047i	$0.534751\pi$
$930$		0			0
$931$		−	1.32421e6i		−	1.52777i
$932$	−151098.	−	29528.3i	−0.173951	−	0.0339944i
$933$		0			0
$934$	22583.6	−	233309.i	0.0258881	−	0.267447i
$935$			714454.i			0.817242i
$936$		0			0
$937$	−1.37071e6			−1.56123			−0.780613	−	0.625014i	$-0.785093\pi$
$937$	−1.37071e6			−1.56123			−0.780613	+	0.625014i	$0.785093\pi$
$938$	−1.11868e6	−	108286.i	−1.27146	−	0.123074i
$939$		0			0
$940$	−244682.	+	1.25205e6i	−0.276915	+	1.41699i
$941$	256232.			0.289370			0.144685	−	0.989478i	$-0.453783\pi$
$941$	256232.			0.289370			0.144685	+	0.989478i	$0.453783\pi$
$942$		0			0
$943$			357166.i			0.401649i
$944$	83604.2	−	205734.i	0.0938175	−	0.230867i
$945$		0			0
$946$	27644.1	+	2675.87i	0.0308901	+	0.00299008i
$947$		−	397.662i		−	0.000443419i	−1.00000		0.000221709i	$-0.999929\pi$
$947$		−	397.662i		−	0.000443419i	1.00000		0.000221709i	$-7.05723e-5\pi$
$948$		0			0
$949$	−243851.			−0.270765
$950$	−12308.0	+	127153.i	−0.0136377	+	0.140889i
$951$		0			0
$952$	−437946.	+	1.47032e6i	−0.483222	+	1.62233i
$953$	709501.			0.781209			0.390605	−	0.920559i	$-0.372266\pi$
$953$	709501.			0.781209			0.390605	+	0.920559i	$0.372266\pi$
$954$		0			0
$955$			166814.i			0.182905i
$956$	−87232.8	+	446374.i	−0.0954474	+	0.488408i
$957$		0			0
$958$	47680.5	−	492581.i	0.0519529	−	0.536719i
$959$			1.61783e6i			1.75912i
$960$		0			0
$961$	310368.			0.336071
$962$	−319469.	−	30923.7i	−0.345206	−	0.0334150i
$963$		0			0
$964$	−149018.	−	29121.8i	−0.160355	−	0.0313375i
$965$	1.39109e6			1.49382
$966$		0			0
$967$			506695.i			0.541868i	0.962598	+	0.270934i	$0.0873326\pi$
$967$			506695.i			0.541868i	−0.962598	+	0.270934i	$0.912667\pi$
$968$	−191767.	−	57119.2i	−0.204656	−	0.0609581i
$969$		0			0
$970$	−984209.	−	95268.7i	−1.04603	−	0.101253i
$971$		−	27400.1i		−	0.0290612i	−0.999894	−	0.0145306i	$-0.995375\pi$
$971$		−	27400.1i		−	0.0290612i	0.999894	−	0.0145306i	$-0.00462540\pi$
$972$		0			0
$973$	1.03755e6			1.09593
$974$	15238.8	−	157430.i	0.0160632	−	0.165947i
$975$		0			0
$976$	1.43128e6	+	581629.i	1.50253	+	0.610586i
$977$	−1.08148e6			−1.13300			−0.566500	−	0.824062i	$-0.691703\pi$
$977$	−1.08148e6			−1.13300			−0.566500	+	0.824062i	$0.691703\pi$
$978$		0			0
$979$			621222.i			0.648159i
$980$	−1.55546e6	−	303977.i	−1.61960	−	0.316511i
$981$		0			0
$982$	−37502.7	+	387436.i	−0.0388902	+	0.401770i
$983$			1.10339e6i			1.14188i	0.820992	+	0.570940i	$0.193421\pi$
$983$			1.10339e6i			1.14188i	−0.820992	+	0.570940i	$0.806579\pi$
$984$		0			0
$985$	316845.			0.326568
$986$	1.22026e6	+	118118.i	1.25516	+	0.121496i
$987$		0			0
$988$	−53463.4	+	273574.i	−0.0547700	+	0.280260i
$989$	63593.8			0.0650163
$990$		0			0
$991$			52573.8i			0.0535331i	0.999642	+	0.0267666i	$0.00852108\pi$
$991$			52573.8i			0.0535331i	−0.999642	+	0.0267666i	$0.991479\pi$
$992$	372035.	−	710301.i	0.378059	−	0.721804i
$993$		0			0
$994$	1.04255e6	+	100916.i	1.05517	+	0.102138i
$995$		−	982285.i		−	0.992182i
$996$		0			0
$997$	−87335.5			−0.0878619			−0.0439309	−	0.999035i	$-0.513988\pi$
$997$	−87335.5			−0.0878619			−0.0439309	+	0.999035i	$0.513988\pi$
$998$	83868.2	−	866432.i	0.0842047	−	0.869908i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.5.d.b.55.1	✓	16
3.2	odd	2	inner	108.5.d.b.55.16	yes	16
4.3	odd	2	inner	108.5.d.b.55.2	yes	16
12.11	even	2	inner	108.5.d.b.55.15	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.d.b.55.1	✓	16	1.1	even	1	trivial
108.5.d.b.55.2	yes	16	4.3	odd	2	inner
108.5.d.b.55.15	yes	16	12.11	even	2	inner
108.5.d.b.55.16	yes	16	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	108.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-3.98139 - 0.385387i) q^{2} +(15.7030 + 3.06876i) q^{4} +22.8093 q^{5} +82.1204i q^{7} +(-61.3369 - 18.2696i) q^{8} +O(q^{10})\)	\(q+(-3.98139 - 0.385387i) q^{2} +(15.7030 + 3.06876i) q^{4} +22.8093 q^{5} +82.1204i q^{7} +(-61.3369 - 18.2696i) q^{8} +(-90.8127 - 8.79041i) q^{10} -107.306i q^{11} +57.1351 q^{13} +(31.6482 - 326.954i) q^{14} +(237.165 + 96.3771i) q^{16} -291.903 q^{17} +304.924i q^{19} +(358.173 + 69.9961i) q^{20} +(-41.3543 + 427.227i) q^{22} +982.813i q^{23} -104.737 q^{25} +(-227.477 - 22.0191i) q^{26} +(-252.008 + 1289.53i) q^{28} +1049.98 q^{29} +783.041i q^{31} +(-907.106 - 475.115i) q^{32} +(1162.18 + 112.496i) q^{34} +1873.11i q^{35} +1404.40 q^{37} +(117.514 - 1214.02i) q^{38} +(-1399.05 - 416.717i) q^{40} +363.412 q^{41} -64.7059i q^{43} +(329.295 - 1685.02i) q^{44} +(378.764 - 3912.96i) q^{46} +3495.66i q^{47} -4342.77 q^{49} +(416.998 + 40.3642i) q^{50} +(897.189 + 175.334i) q^{52} -3268.87 q^{53} -2447.57i q^{55} +(1500.31 - 5037.02i) q^{56} +(-4180.37 - 404.648i) q^{58} -867.469i q^{59} +6034.94 q^{61} +(301.774 - 3117.59i) q^{62} +(3428.44 + 2241.21i) q^{64} +1303.21 q^{65} -3421.54i q^{67} +(-4583.74 - 895.780i) q^{68} +(721.872 - 7457.58i) q^{70} +3188.67i q^{71} -4267.98 q^{73} +(-5591.47 - 541.239i) q^{74} +(-935.737 + 4788.20i) q^{76} +8812.00 q^{77} -7530.43i q^{79} +(5409.57 + 2198.29i) q^{80} +(-1446.89 - 140.055i) q^{82} -586.258i q^{83} -6658.11 q^{85} +(-24.9368 + 257.619i) q^{86} +(-1960.44 + 6581.81i) q^{88} -5789.27 q^{89} +4691.96i q^{91} +(-3016.01 + 15433.1i) q^{92} +(1347.18 - 13917.6i) q^{94} +6955.09i q^{95} +10837.8 q^{97} +(17290.3 + 1673.65i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 28 q^{4}+O(q^{10}) \) 16 * q + 28 * q^4	\( 16 q + 28 q^{4} + 176 q^{10} + 176 q^{13} + 88 q^{16} + 384 q^{22} + 2736 q^{25} + 1812 q^{28} + 1520 q^{34} + 80 q^{37} - 688 q^{40} - 1824 q^{46} - 7904 q^{49} - 5236 q^{52} - 11584 q^{58} - 1648 q^{61} + 5056 q^{64} + 26688 q^{70} + 80 q^{73} - 8388 q^{76} - 38464 q^{82} - 16832 q^{85} - 29520 q^{88} - 4512 q^{94} + 14864 q^{97}+O(q^{100}) \) 16 * q + 28 * q^4 + 176 * q^10 + 176 * q^13 + 88 * q^16 + 384 * q^22 + 2736 * q^25 + 1812 * q^28 + 1520 * q^34 + 80 * q^37 - 688 * q^40 - 1824 * q^46 - 7904 * q^49 - 5236 * q^52 - 11584 * q^58 - 1648 * q^61 + 5056 * q^64 + 26688 * q^70 + 80 * q^73 - 8388 * q^76 - 38464 * q^82 - 16832 * q^85 - 29520 * q^88 - 4512 * q^94 + 14864 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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