LMFDB - Embedded newform 108.5.k.a.41.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 5]))

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

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

chi := DirichletCharacter("108.5");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	108.k (of order $18$, degree $6$, 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:	$72$
Relative dimension:	$12$ over $\Q(\zeta_{18})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			41.8
Character	$\chi$	$=$	108.41
Dual form			108.5.k.a.29.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.39288 - 7.85510i) q^{3} +(6.32074 - 7.53276i) q^{5} +(66.1128 + 24.0631i) q^{7} +(-42.4052 - 69.0131i) q^{9} +O(q^{10})$	\(q+(4.39288 - 7.85510i) q^{3} +(6.32074 - 7.53276i) q^{5} +(66.1128 + 24.0631i) q^{7} +(-42.4052 - 69.0131i) q^{9} +(90.1574 + 107.445i) q^{11} +(37.0033 - 209.856i) q^{13} +(-31.4043 - 82.7405i) q^{15} +(-45.6573 - 26.3602i) q^{17} +(-163.587 - 283.340i) q^{19} +(479.444 - 413.617i) q^{21} +(-193.229 - 530.892i) q^{23} +(91.7393 + 520.280i) q^{25} +(-728.385 + 29.9305i) q^{27} +(109.946 - 19.3864i) q^{29} +(868.674 - 316.171i) q^{31} +(1240.05 - 236.200i) q^{33} +(599.143 - 345.916i) q^{35} +(154.394 - 267.419i) q^{37} +(-1485.89 - 1212.54i) q^{39} +(-478.016 - 84.2871i) q^{41} +(1943.12 - 1630.47i) q^{43} +(-787.891 - 116.785i) q^{45} +(-1391.66 + 3823.56i) q^{47} +(1952.60 + 1638.43i) q^{49} +(-407.629 + 242.845i) q^{51} +4272.88i q^{53} +1379.22 q^{55} +(-2944.28 + 40.3082i) q^{57} +(-3174.41 + 3783.11i) q^{59} +(-1472.97 - 536.118i) q^{61} +(-1142.86 - 5583.05i) q^{63} +(-1346.91 - 1605.18i) q^{65} +(-527.119 + 2989.44i) q^{67} +(-5019.04 - 814.314i) q^{69} +(-5046.35 - 2913.51i) q^{71} +(3497.66 + 6058.12i) q^{73} +(4489.85 + 1564.91i) q^{75} +(3375.09 + 9272.99i) q^{77} +(54.1811 + 307.276i) q^{79} +(-2964.60 + 5853.02i) q^{81} +(11818.7 - 2083.95i) q^{83} +(-487.153 + 177.309i) q^{85} +(330.697 - 948.798i) q^{87} +(-8394.50 + 4846.57i) q^{89} +(7496.17 - 12983.8i) q^{91} +(1332.42 - 8212.42i) q^{93} +(-3168.32 - 558.661i) q^{95} +(3647.23 - 3060.39i) q^{97} +(3592.00 - 10778.3i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q + 9 q^{5} - 102 q^{9}+O(q^{10}) $ 72 * q + 9 * q^5 - 102 * q^9	$ 72 q + 9 q^{5} - 102 q^{9} + 18 q^{11} - 225 q^{15} - 282 q^{21} - 1278 q^{23} + 441 q^{25} + 54 q^{27} + 1854 q^{29} - 1665 q^{31} - 45 q^{33} - 2673 q^{35} + 6951 q^{39} - 5472 q^{41} + 1260 q^{43} + 5553 q^{45} + 5103 q^{47} - 5904 q^{49} + 1899 q^{51} + 1107 q^{57} - 10944 q^{59} + 8352 q^{61} - 11985 q^{63} + 8757 q^{65} + 378 q^{67} + 5607 q^{69} - 19764 q^{71} + 6111 q^{73} - 3453 q^{75} - 5679 q^{77} - 5652 q^{79} - 20466 q^{81} - 20061 q^{83} + 26100 q^{85} + 40545 q^{87} + 15633 q^{89} - 6039 q^{91} + 40179 q^{93} + 48024 q^{95} - 37530 q^{97} + 12177 q^{99}+O(q^{100}) $ 72 * q + 9 * q^5 - 102 * q^9 + 18 * q^11 - 225 * q^15 - 282 * q^21 - 1278 * q^23 + 441 * q^25 + 54 * q^27 + 1854 * q^29 - 1665 * q^31 - 45 * q^33 - 2673 * q^35 + 6951 * q^39 - 5472 * q^41 + 1260 * q^43 + 5553 * q^45 + 5103 * q^47 - 5904 * q^49 + 1899 * q^51 + 1107 * q^57 - 10944 * q^59 + 8352 * q^61 - 11985 * q^63 + 8757 * q^65 + 378 * q^67 + 5607 * q^69 - 19764 * q^71 + 6111 * q^73 - 3453 * q^75 - 5679 * q^77 - 5652 * q^79 - 20466 * q^81 - 20061 * q^83 + 26100 * q^85 + 40545 * q^87 + 15633 * q^89 - 6039 * q^91 + 40179 * q^93 + 48024 * q^95 - 37530 * q^97 + 12177 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{17}{18}\right)$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	4.39288	−	7.85510i	0.488098	−	0.872789i
$3$	4.39288	−	7.85510i	0.488098	−	0.872789i
$4$		0			0
$5$	6.32074	−	7.53276i	0.252829	−	0.301310i	−0.624669	−	0.780889i	$-0.714766\pi$
$5$	6.32074	−	7.53276i	0.252829	−	0.301310i	0.877499	+	0.479579i	$0.159211\pi$
$6$		0			0
$7$	66.1128	+	24.0631i	1.34924	+	0.491084i	0.912711	−	0.408606i	$-0.133985\pi$
$7$	66.1128	+	24.0631i	1.34924	+	0.491084i	0.436531	+	0.899689i	$0.356207\pi$
$8$		0			0
$9$	−42.4052	−	69.0131i	−0.523521	−	0.852013i
$10$		0			0
$11$	90.1574	+	107.445i	0.745103	+	0.887979i	0.996809	−	0.0798249i	$-0.0254361\pi$
$11$	90.1574	+	107.445i	0.745103	+	0.887979i	−0.251706	+	0.967804i	$0.580992\pi$
$12$		0			0
$13$	37.0033	−	209.856i	0.218954	−	1.24175i	−0.654957	−	0.755666i	$-0.727313\pi$
$13$	37.0033	−	209.856i	0.218954	−	1.24175i	0.873912	−	0.486085i	$-0.161575\pi$
$14$		0			0
$15$	−31.4043	−	82.7405i	−0.139575	−	0.367736i
$16$		0			0
$17$	−45.6573	−	26.3602i	−0.157984	−	0.0912119i	0.418924	−	0.908021i	$-0.362407\pi$
$17$	−45.6573	−	26.3602i	−0.157984	−	0.0912119i	−0.576908	+	0.816809i	$0.695741\pi$
$18$		0			0
$19$	−163.587	−	283.340i	−0.453148	−	0.784876i	0.545431	−	0.838156i	$-0.316366\pi$
$19$	−163.587	−	283.340i	−0.453148	−	0.784876i	−0.998580	+	0.0532795i	$0.983033\pi$
$20$		0			0
$21$	479.444	−	413.617i	1.08717	−	0.937906i
$22$		0			0
$23$	−193.229	−	530.892i	−0.365272	−	1.00358i	−0.977136	−	0.212614i	$-0.931802\pi$
$23$	−193.229	−	530.892i	−0.365272	−	1.00358i	0.611864	−	0.790963i	$-0.290420\pi$
$24$		0			0
$25$	91.7393	+	520.280i	0.146783	+	0.832447i
$26$		0			0
$27$	−728.385	+	29.9305i	−0.999157	+	0.0410569i
$28$		0			0
$29$	109.946	−	19.3864i	0.130732	−	0.0230516i	−0.107899	−	0.994162i	$-0.534412\pi$
$29$	109.946	−	19.3864i	0.130732	−	0.0230516i	0.238631	+	0.971110i	$0.423301\pi$
$30$		0			0
$31$	868.674	−	316.171i	0.903927	−	0.329002i	0.152102	−	0.988365i	$-0.451396\pi$
$31$	868.674	−	316.171i	0.903927	−	0.329002i	0.751825	+	0.659362i	$0.229174\pi$
$32$		0			0
$33$	1240.05	−	236.200i	1.13870	−	0.216897i
$34$		0			0
$35$	599.143	−	345.916i	0.489097	−	0.282380i
$36$		0			0
$37$	154.394	−	267.419i	0.112779	−	0.195339i	−0.804111	−	0.594480i	$-0.797358\pi$
$37$	154.394	−	267.419i	0.112779	−	0.195339i	0.916890	+	0.399141i	$0.130691\pi$
$38$		0			0
$39$	−1485.89	−	1212.54i	−0.976915	−	0.797197i
$40$		0			0
$41$	−478.016	−	84.2871i	−0.284364	−	0.0501411i	0.0296468	−	0.999560i	$-0.490562\pi$
$41$	−478.016	−	84.2871i	−0.284364	−	0.0501411i	−0.314011	+	0.949419i	$0.601673\pi$
$42$		0			0
$43$	1943.12	−	1630.47i	1.05090	−	0.881813i	0.0577150	−	0.998333i	$-0.481619\pi$
$43$	1943.12	−	1630.47i	1.05090	−	0.881813i	0.993188	+	0.116521i	$0.0371741\pi$
$44$		0			0
$45$	−787.891	−	116.785i	−0.389082	−	0.0576718i
$46$		0			0
$47$	−1391.66	+	3823.56i	−0.629997	+	1.73090i	0.0510841	+	0.998694i	$0.483732\pi$
$47$	−1391.66	+	3823.56i	−0.629997	+	1.73090i	−0.681081	+	0.732208i	$0.738490\pi$
$48$		0			0
$49$	1952.60	+	1638.43i	0.813245	+	0.682394i
$50$		0			0
$51$	−407.629	+	242.845i	−0.156720	+	0.0933660i
$52$		0			0
$53$			4272.88i			1.52114i	0.649256	+	0.760570i	$0.275080\pi$
$53$			4272.88i			1.52114i	−0.649256	+	0.760570i	$0.724920\pi$
$54$		0			0
$55$	1379.22			0.455941
$56$		0			0
$57$	−2944.28	+	40.3082i	−0.906212	+	0.0124064i
$58$		0			0
$59$	−3174.41	+	3783.11i	−0.911924	+	1.08679i	0.0839884	+	0.996467i	$0.473234\pi$
$59$	−3174.41	+	3783.11i	−0.911924	+	1.08679i	−0.995913	+	0.0903222i	$0.971210\pi$
$60$		0			0
$61$	−1472.97	−	536.118i	−0.395854	−	0.144079i	0.136420	−	0.990651i	$-0.456440\pi$
$61$	−1472.97	−	536.118i	−0.395854	−	0.144079i	−0.532274	+	0.846572i	$0.678662\pi$
$62$		0			0
$63$	−1142.86	−	5583.05i	−0.287946	−	1.40666i
$64$		0			0
$65$	−1346.91	−	1605.18i	−0.318794	−	0.379924i
$66$		0			0
$67$	−527.119	+	2989.44i	−0.117425	+	0.665948i	0.868097	+	0.496395i	$0.165343\pi$
$67$	−527.119	+	2989.44i	−0.117425	+	0.665948i	−0.985521	+	0.169553i	$0.945768\pi$
$68$		0			0
$69$	−5019.04	−	814.314i	−1.05420	−	0.171039i
$70$		0			0
$71$	−5046.35	−	2913.51i	−1.00106	−	0.577963i	−0.0924992	−	0.995713i	$-0.529486\pi$
$71$	−5046.35	−	2913.51i	−1.00106	−	0.577963i	−0.908562	+	0.417750i	$0.862819\pi$
$72$		0			0
$73$	3497.66	+	6058.12i	0.656344	+	1.13682i	0.981555	+	0.191180i	$0.0612314\pi$
$73$	3497.66	+	6058.12i	0.656344	+	1.13682i	−0.325211	+	0.945642i	$0.605435\pi$
$74$		0			0
$75$	4489.85	+	1564.91i	0.798195	+	0.278206i
$76$		0			0
$77$	3375.09	+	9272.99i	0.569252	+	1.56401i
$78$		0			0
$79$	54.1811	+	307.276i	0.00868148	+	0.0492351i	0.988841	−	0.148976i	$-0.0475978\pi$
$79$	54.1811	+	307.276i	0.00868148	+	0.0492351i	−0.980159	+	0.198211i	$0.936487\pi$
$80$		0			0
$81$	−2964.60	+	5853.02i	−0.451852	+	0.892093i
$82$		0			0
$83$	11818.7	−	2083.95i	1.71559	−	0.302504i	0.772491	−	0.635025i	$-0.219010\pi$
$83$	11818.7	−	2083.95i	1.71559	−	0.302504i	0.943096	+	0.332521i	$0.107899\pi$
$84$		0			0
$85$	−487.153	+	177.309i	−0.0674260	+	0.0245411i
$86$		0			0
$87$	330.697	−	948.798i	0.0436910	−	0.125353i
$88$		0			0
$89$	−8394.50	+	4846.57i	−1.05978	+	0.611863i	−0.925371	−	0.379063i	$-0.876246\pi$
$89$	−8394.50	+	4846.57i	−1.05978	+	0.611863i	−0.134407	+	0.990926i	$0.542913\pi$
$90$		0			0
$91$	7496.17	−	12983.8i	0.905226	−	1.56790i
$92$		0			0
$93$	1332.42	−	8212.42i	0.154055	−	0.949523i
$94$		0			0
$95$	−3168.32	−	558.661i	−0.351061	−	0.0619015i
$96$		0			0
$97$	3647.23	−	3060.39i	0.387632	−	0.325262i	−0.428058	−	0.903751i	$-0.640802\pi$
$97$	3647.23	−	3060.39i	0.387632	−	0.325262i	0.815690	+	0.578490i	$0.196358\pi$
$98$		0			0
$99$	3592.00	−	10778.3i	0.366493	−	1.09971i
$100$		0			0
$101$	−3495.98	+	9605.13i	−0.342710	+	0.941587i	0.641895	+	0.766792i	$0.278148\pi$
$101$	−3495.98	+	9605.13i	−0.342710	+	0.941587i	−0.984605	+	0.174795i	$0.944074\pi$
$102$		0			0
$103$	1825.20	+	1531.53i	0.172043	+	0.144361i	0.724743	−	0.689019i	$-0.241958\pi$
$103$	1825.20	+	1531.53i	0.172043	+	0.144361i	−0.552701	+	0.833380i	$0.686403\pi$
$104$		0			0
$105$	−85.2347	−	6225.90i	−0.00773104	−	0.564707i
$106$		0			0
$107$		−	15855.4i		−	1.38487i	−0.721479	−	0.692436i	$-0.756538\pi$
$107$		−	15855.4i		−	1.38487i	0.721479	−	0.692436i	$-0.243462\pi$
$108$		0			0
$109$	18526.7			1.55935			0.779676	−	0.626183i	$-0.215384\pi$
$109$	18526.7			1.55935			0.779676	+	0.626183i	$0.215384\pi$
$110$		0			0
$111$	−1422.37	−	2387.52i	−0.115442	−	0.193777i
$112$		0			0
$113$	−4943.81	+	5891.80i	−0.387173	+	0.461415i	−0.924064	−	0.382237i	$-0.875154\pi$
$113$	−4943.81	+	5891.80i	−0.387173	+	0.461415i	0.536892	+	0.843651i	$0.319598\pi$
$114$		0			0
$115$	−5220.43	−	1900.08i	−0.394740	−	0.143673i
$116$		0			0
$117$	−16051.9	+	6345.26i	−1.17261	+	0.463530i
$118$		0			0
$119$	−2384.22	−	2841.41i	−0.168365	−	0.200650i
$120$		0			0
$121$	−873.777	+	4955.44i	−0.0596801	+	0.338463i
$122$		0			0
$123$	−2761.95	+	3384.60i	−0.182560	+	0.223716i
$124$		0			0
$125$	9821.44	+	5670.41i	0.628572	+	0.362906i
$126$		0			0
$127$	−1907.85	−	3304.49i	−0.118287	−	0.204879i	0.800802	−	0.598929i	$-0.204407\pi$
$127$	−1907.85	−	3304.49i	−0.118287	−	0.204879i	−0.919089	+	0.394050i	$0.871074\pi$
$128$		0			0
$129$	−4271.62	−	22425.9i	−0.256692	−	1.34763i
$130$		0			0
$131$	−4284.95	−	11772.8i	−0.249691	−	0.686021i	−0.999698	−	0.0245903i	$-0.992172\pi$
$131$	−4284.95	−	11772.8i	−0.249691	−	0.686021i	0.750006	−	0.661430i	$-0.230050\pi$
$132$		0			0
$133$	−3997.13	−	22668.8i	−0.225967	−	1.28152i
$134$		0			0
$135$	−4378.47	+	5675.93i	−0.240245	+	0.311437i
$136$		0			0
$137$	9048.16	−	1595.44i	0.482080	−	0.0850038i	0.0726736	−	0.997356i	$-0.476847\pi$
$137$	9048.16	−	1595.44i	0.482080	−	0.0850038i	0.409407	+	0.912352i	$0.365736\pi$
$138$		0			0
$139$	38.1456	−	13.8839i	0.00197431	−	0.000718590i	−0.341033	−	0.940051i	$-0.610777\pi$
$139$	38.1456	−	13.8839i	0.00197431	−	0.000718590i	0.343007	+	0.939333i	$0.388555\pi$
$140$		0			0
$141$	23921.1	+	27728.1i	1.20321	+	1.39470i
$142$		0			0
$143$	25884.2	−	14944.2i	1.26579	−	0.730805i
$144$		0			0
$145$	548.906	−	950.732i	0.0261073	−	0.0452191i
$146$		0			0
$147$	21447.6	−	8140.46i	0.992529	−	0.376716i
$148$		0			0
$149$	−6325.77	−	1115.40i	−0.284932	−	0.0502412i	0.0293557	−	0.999569i	$-0.490654\pi$
$149$	−6325.77	−	1115.40i	−0.284932	−	0.0502412i	−0.314287	+	0.949328i	$0.601766\pi$
$150$		0			0
$151$	−33859.8	+	28411.8i	−1.48501	+	1.24608i	−0.584394	+	0.811470i	$0.698668\pi$
$151$	−33859.8	+	28411.8i	−1.48501	+	1.24608i	−0.900621	+	0.434605i	$0.856888\pi$
$152$		0			0
$153$	116.903	+	4268.76i	0.00499395	+	0.182355i
$154$		0			0
$155$	3109.01	−	8541.94i	0.129407	−	0.355544i
$156$		0			0
$157$	9697.94	+	8137.54i	0.393442	+	0.330137i	0.817952	−	0.575286i	$-0.195109\pi$
$157$	9697.94	+	8137.54i	0.393442	+	0.330137i	−0.424511	+	0.905423i	$0.639554\pi$
$158$		0			0
$159$	33563.9	+	18770.3i	1.32763	+	0.742465i
$160$		0			0
$161$		−	39748.5i		−	1.53345i
$162$		0			0
$163$	−34222.6			−1.28806			−0.644032	−	0.764999i	$-0.722740\pi$
$163$	−34222.6			−1.28806			−0.644032	+	0.764999i	$0.722740\pi$
$164$		0			0
$165$	6058.76	−	10833.9i	0.222544	−	0.397940i
$166$		0			0
$167$	−26227.0	+	31256.2i	−0.940407	+	1.12073i	0.0521112	+	0.998641i	$0.483405\pi$
$167$	−26227.0	+	31256.2i	−0.940407	+	1.12073i	−0.992519	+	0.122093i	$0.961039\pi$
$168$		0			0
$169$	−15831.7	−	5762.26i	−0.554311	−	0.201753i
$170$		0			0
$171$	−12617.3	+	23304.7i	−0.431492	+	0.796987i
$172$		0			0
$173$	26785.1	+	31921.3i	0.894956	+	1.06657i	0.997417	+	0.0718297i	$0.0228838\pi$
$173$	26785.1	+	31921.3i	0.894956	+	1.06657i	−0.102461	+	0.994737i	$0.532672\pi$
$174$		0			0
$175$	−6454.40	+	36604.7i	−0.210756	+	1.19526i
$176$		0			0
$177$	15771.9	+	41554.1i	0.503429	+	1.32638i
$178$		0			0
$179$	−31745.4	−	18328.2i	−0.990773	−	0.572023i	−0.0852679	−	0.996358i	$-0.527175\pi$
$179$	−31745.4	−	18328.2i	−0.990773	−	0.572023i	−0.905505	+	0.424335i	$0.860508\pi$
$180$		0			0
$181$	−1763.07	−	3053.72i	−0.0538160	−	0.0932121i	0.837862	−	0.545881i	$-0.183805\pi$
$181$	−1763.07	−	3053.72i	−0.0538160	−	0.0932121i	−0.891678	+	0.452669i	$0.850472\pi$
$182$		0			0
$183$	−10681.8	+	9215.23i	−0.318966	+	0.275172i
$184$		0			0
$185$	−1038.52	−	2853.30i	−0.0303438	−	0.0833689i
$186$		0			0
$187$	−1284.05	−	7282.23i	−0.0367198	−	0.208248i
$188$		0			0
$189$	−48875.8	−	15548.4i	−1.36827	−	0.435274i
$190$		0			0
$191$	43588.6	−	7685.85i	1.19483	−	0.210681i	0.459368	−	0.888246i	$-0.348076\pi$
$191$	43588.6	−	7685.85i	1.19483	−	0.210681i	0.735462	+	0.677565i	$0.236965\pi$
$192$		0			0
$193$	−47996.6	+	17469.3i	−1.28853	+	0.468988i	−0.893245	−	0.449570i	$-0.851577\pi$
$193$	−47996.6	+	17469.3i	−1.28853	+	0.468988i	−0.395288	+	0.918557i	$0.629355\pi$
$194$		0			0
$195$	−18525.6	+	3528.71i	−0.487197	+	0.0927998i
$196$		0			0
$197$	−17477.7	+	10090.8i	−0.450353	+	0.260011i	−0.707979	−	0.706233i	$-0.750393\pi$
$197$	−17477.7	+	10090.8i	−0.450353	+	0.260011i	0.257626	+	0.966245i	$0.417060\pi$
$198$		0			0
$199$	14185.5	−	24570.0i	0.358211	−	0.620439i	−0.629451	−	0.777040i	$-0.716720\pi$
$199$	14185.5	−	24570.0i	0.358211	−	0.620439i	0.987662	+	0.156601i	$0.0500536\pi$
$200$		0			0
$201$	21166.8	+	17272.8i	0.523917	+	0.427535i
$202$		0			0
$203$	7735.33	+	1363.95i	0.187710	+	0.0330983i
$204$		0			0
$205$	−3656.33	+	3068.02i	−0.0870036	+	0.0730047i
$206$		0			0
$207$	−28444.6	+	35847.9i	−0.663833	+	0.836610i
$208$		0			0
$209$	15695.1	−	43121.9i	0.359311	−	0.987200i
$210$		0			0
$211$	2986.01	+	2505.56i	0.0670697	+	0.0562782i	0.675706	−	0.737171i	$-0.263839\pi$
$211$	2986.01	+	2505.56i	0.0670697	+	0.0562782i	−0.608637	+	0.793449i	$0.708283\pi$
$212$		0			0
$213$	−45053.9	+	26840.9i	−0.993056	+	0.591612i
$214$		0			0
$215$		−	24942.8i		−	0.539596i
$216$		0			0
$217$	65038.5			1.38118
$218$		0			0
$219$	62951.9	−	861.834i	1.31257	−	0.0179695i
$220$		0			0
$221$	−7221.32	+	8606.03i	−0.147854	+	0.176205i
$222$		0			0
$223$	−87864.5	−	31980.1i	−1.76687	−	0.643087i	−0.999999	−	0.00128111i	$-0.999592\pi$
$223$	−87864.5	−	31980.1i	−1.76687	−	0.643087i	−0.766867	−	0.641806i	$-0.778186\pi$
$224$		0			0
$225$	32015.9	−	28393.8i	0.632412	−	0.560864i
$226$		0			0
$227$	28076.5	+	33460.3i	0.544868	+	0.649349i	0.966272	−	0.257525i	$-0.0829069\pi$
$227$	28076.5	+	33460.3i	0.544868	+	0.649349i	−0.421404	+	0.906873i	$0.638462\pi$
$228$		0			0
$229$	−11334.4	+	64280.4i	−0.216136	+	1.22577i	0.662789	+	0.748806i	$0.269373\pi$
$229$	−11334.4	+	64280.4i	−0.216136	+	1.22577i	−0.878925	+	0.476961i	$0.841738\pi$
$230$		0			0
$231$	87666.6	+	14223.5i	1.64290	+	0.266552i
$232$		0			0
$233$	−65540.1	−	37839.6i	−1.20724	−	0.697003i	−0.245088	−	0.969501i	$-0.578817\pi$
$233$	−65540.1	−	37839.6i	−1.20724	−	0.697003i	−0.962156	+	0.272498i	$0.912150\pi$
$234$		0			0
$235$	20005.6	+	34650.8i	0.362257	+	0.627448i
$236$		0			0
$237$	2651.70	+	924.231i	0.0472093	+	0.0164545i
$238$		0			0
$239$	−10375.8	−	28507.3i	−0.181646	−	0.499069i	0.815132	−	0.579275i	$-0.196664\pi$
$239$	−10375.8	−	28507.3i	−0.181646	−	0.499069i	−0.996778	+	0.0802063i	$0.974442\pi$
$240$		0			0
$241$	−15861.1	−	89953.0i	−0.273087	−	1.54875i	−0.744975	−	0.667093i	$-0.767538\pi$
$241$	−15861.1	−	89953.0i	−0.273087	−	1.54875i	0.471888	−	0.881659i	$-0.343573\pi$
$242$		0			0
$243$	32952.9	+	48998.9i	0.558060	+	0.829801i
$244$		0			0
$245$	24683.8	−	4352.41i	0.411225	−	0.0725100i
$246$		0			0
$247$	−65513.9	+	23845.1i	−1.07384	+	0.390846i
$248$		0			0
$249$	35548.4	−	101991.i	0.573352	−	1.64500i
$250$		0			0
$251$	67801.8	−	39145.4i	1.07620	−	0.621346i	0.146333	−	0.989235i	$-0.453253\pi$
$251$	67801.8	−	39145.4i	1.07620	−	0.621346i	0.929869	+	0.367890i	$0.119920\pi$
$252$		0			0
$253$	39620.9	−	68625.4i	0.618990	−	1.07212i
$254$		0			0
$255$	−747.224	+	4605.53i	−0.0114913	+	0.0708271i
$256$		0			0
$257$	5822.68	+	1026.70i	0.0881570	+	0.0155445i	0.217553	−	0.976049i	$-0.430193\pi$
$257$	5822.68	+	1026.70i	0.0881570	+	0.0155445i	−0.129396	+	0.991593i	$0.541304\pi$
$258$		0			0
$259$	16642.4	−	13964.6i	0.248094	−	0.208175i
$260$		0			0
$261$	−6000.19	−	6765.62i	−0.0880814	−	0.0993176i
$262$		0			0
$263$	42306.2	−	116235.i	0.611635	−	1.68045i	−0.114946	−	0.993372i	$-0.536670\pi$
$263$	42306.2	−	116235.i	0.611635	−	1.68045i	0.726581	−	0.687081i	$-0.241108\pi$
$264$		0			0
$265$	32186.6	+	27007.8i	0.458335	+	0.384589i
$266$		0			0
$267$	1194.21	+	87230.0i	0.0167517	+	1.22361i
$268$		0			0
$269$		−	31192.9i		−	0.431073i	−0.976496	−	0.215536i	$-0.930850\pi$
$269$		−	31192.9i		−	0.431073i	0.976496	−	0.215536i	$-0.0691500\pi$
$270$		0			0
$271$	49482.4			0.673770			0.336885	−	0.941546i	$-0.390627\pi$
$271$	49482.4			0.673770			0.336885	+	0.941546i	$0.390627\pi$
$272$		0			0
$273$	−69058.9	−	115919.i	−0.926604	−	1.55536i
$274$		0			0
$275$	−47630.7	+	56764.0i	−0.629827	+	0.750599i
$276$		0			0
$277$	−53816.5	−	19587.6i	−0.701385	−	0.255283i	−0.0333826	−	0.999443i	$-0.510628\pi$
$277$	−53816.5	−	19587.6i	−0.701385	−	0.255283i	−0.668002	+	0.744160i	$0.732850\pi$
$278$		0			0
$279$	−58656.2	−	46542.5i	−0.753539	−	0.597918i
$280$		0			0
$281$	68511.2	+	81648.5i	0.867659	+	1.03404i	0.999088	+	0.0427098i	$0.0135991\pi$
$281$	68511.2	+	81648.5i	0.867659	+	1.03404i	−0.131429	+	0.991326i	$0.541956\pi$
$282$		0			0
$283$	−8701.95	+	49351.2i	−0.108654	+	0.616205i	0.881044	+	0.473034i	$0.156841\pi$
$283$	−8701.95	+	49351.2i	−0.108654	+	0.616205i	−0.989698	+	0.143171i	$0.954270\pi$
$284$		0			0
$285$	−18306.4	+	22433.4i	−0.225379	+	0.276188i
$286$		0			0
$287$	−29574.8	−	17075.0i	−0.359052	−	0.207299i
$288$		0			0
$289$	−40370.8	−	69924.2i	−0.483361	−	0.837205i
$290$		0			0
$291$	−8017.80	−	42093.3i	−0.0946824	−	0.497080i
$292$		0			0
$293$	32359.9	+	88908.2i	0.376940	+	1.03563i	0.972618	+	0.232411i	$0.0746614\pi$
$293$	32359.9	+	88908.2i	0.376940	+	1.03563i	−0.595677	+	0.803224i	$0.703116\pi$
$294$		0			0
$295$	8432.68	+	47824.1i	0.0968995	+	0.549544i
$296$		0			0
$297$	−68885.2	−	75563.2i	−0.780932	−	0.856638i
$298$		0			0
$299$	−118561.	+	20905.5i	−1.32617	+	0.233840i
$300$		0			0
$301$	167699.	−	61037.6i	1.85097	−	0.673697i
$302$		0			0
$303$	60091.8	+	69655.5i	0.654531	+	0.758700i
$304$		0			0
$305$	−13348.7	+	7706.88i	−0.143496	+	0.0828474i
$306$		0			0
$307$	−39692.4	+	68749.2i	−0.421143	+	0.729442i	−0.996052	−	0.0887760i	$-0.971704\pi$
$307$	−39692.4	+	68749.2i	−0.421143	+	0.729442i	0.574908	+	0.818218i	$0.305038\pi$
$308$		0			0
$309$	20048.2	−	7609.33i	0.209970	−	0.0796947i
$310$		0			0
$311$	7118.06	+	1255.11i	0.0735937	+	0.0129766i	0.210324	−	0.977632i	$-0.432548\pi$
$311$	7118.06	+	1255.11i	0.0735937	+	0.0129766i	−0.136730	+	0.990608i	$0.543659\pi$
$312$		0			0
$313$	63912.8	−	53629.2i	0.652378	−	0.547410i	−0.255414	−	0.966832i	$-0.582212\pi$
$313$	63912.8	−	53629.2i	0.652378	−	0.547410i	0.907791	+	0.419422i	$0.137767\pi$
$314$		0			0
$315$	−49279.5	−	26680.1i	−0.496644	−	0.268885i
$316$		0			0
$317$	−34347.7	+	94369.5i	−0.341805	+	0.939103i	0.643065	+	0.765811i	$0.277662\pi$
$317$	−34347.7	+	94369.5i	−0.341805	+	0.939103i	−0.984871	+	0.173291i	$0.944560\pi$
$318$		0			0
$319$	11995.4	+	10065.4i	0.117878	+	0.0989117i
$320$		0			0
$321$	−124546.	−	69650.9i	−1.20870	−	0.675953i
$322$		0			0
$323$			17248.7i			0.165330i
$324$		0			0
$325$	112578.			1.06583
$326$		0			0
$327$	81385.5	−	145529.i	0.761117	−	1.36099i
$328$		0			0
$329$	−184014.	+	219299.i	−1.70004	+	2.02602i
$330$		0			0
$331$	52213.1	+	19004.0i	0.476567	+	0.173456i	0.569125	−	0.822251i	$-0.307282\pi$
$331$	52213.1	+	19004.0i	0.476567	+	0.173456i	−0.0925579	+	0.995707i	$0.529504\pi$
$332$		0			0
$333$	−25002.5	+	684.714i	−0.225473	+	0.00617477i
$334$		0			0
$335$	19187.0	+	22866.1i	0.170969	+	0.203752i
$336$		0			0
$337$	22332.3	−	126653.i	0.196641	−	1.11520i	−0.713422	−	0.700734i	$-0.752856\pi$
$337$	22332.3	−	126653.i	0.196641	−	1.11520i	0.910063	−	0.414470i	$-0.136033\pi$
$338$		0			0
$339$	24563.1	+	64716.1i	0.213739	+	0.563136i
$340$		0			0
$341$	112289.	+	64829.8i	0.965665	+	0.557527i
$342$		0			0
$343$	5204.24	+	9014.01i	0.0442353	+	0.0766178i
$344$		0			0
$345$	−37858.1	+	32660.2i	−0.318068	+	0.274398i
$346$		0			0
$347$	−73160.9	−	201008.i	−0.607603	−	1.66938i	−0.735447	−	0.677582i	$-0.763028\pi$
$347$	−73160.9	−	201008.i	−0.607603	−	1.66938i	0.127844	−	0.991794i	$-0.459194\pi$
$348$		0			0
$349$	35289.9	+	200139.i	0.289734	+	1.64316i	0.687869	+	0.725835i	$0.258546\pi$
$349$	35289.9	+	200139.i	0.289734	+	1.64316i	−0.398135	+	0.917327i	$0.630343\pi$
$350$		0			0
$351$	−20671.5	+	153963.i	−0.167787	+	1.24969i
$352$		0			0
$353$	72699.0	−	12818.8i	0.583417	−	0.102872i	0.125853	−	0.992049i	$-0.459833\pi$
$353$	72699.0	−	12818.8i	0.583417	−	0.102872i	0.457564	+	0.889177i	$0.348722\pi$
$354$		0			0
$355$	−53843.4	+	19597.4i	−0.427244	+	0.155504i
$356$		0			0
$357$	−32793.1	+	6246.34i	−0.257304	+	0.0490105i
$358$		0			0
$359$	−40092.8	+	23147.6i	−0.311084	+	0.179605i	−0.647412	−	0.762141i	$-0.724148\pi$
$359$	−40092.8	+	23147.6i	−0.311084	+	0.179605i	0.336327	+	0.941745i	$0.390815\pi$
$360$		0			0
$361$	11639.4	−	20160.0i	0.0893130	−	0.154695i
$362$		0			0
$363$	35087.0	+	28632.3i	0.266277	+	0.217291i
$364$		0			0
$365$	67742.1	+	11944.8i	0.508479	+	0.0896586i
$366$		0			0
$367$	−71661.3	+	60131.0i	−0.532050	+	0.446443i	−0.868809	−	0.495148i	$-0.835114\pi$
$367$	−71661.3	+	60131.0i	−0.532050	+	0.446443i	0.336758	+	0.941591i	$0.390669\pi$
$368$		0			0
$369$	14453.4	+	36563.6i	0.106150	+	0.268532i
$370$		0			0
$371$	−102819.	+	282492.i	−0.747007	+	2.05238i
$372$		0			0
$373$	62546.6	+	52482.8i	0.449558	+	0.377224i	0.839272	−	0.543712i	$-0.182982\pi$
$373$	62546.6	+	52482.8i	0.449558	+	0.377224i	−0.389714	+	0.920936i	$0.627426\pi$
$374$		0			0
$375$	87686.1	−	52238.9i	0.623546	−	0.371477i
$376$		0			0
$377$		−	23790.2i		−	0.167384i
$378$		0			0
$379$	−38147.7			−0.265577			−0.132788	−	0.991144i	$-0.542393\pi$
$379$	−38147.7			−0.265577			−0.132788	+	0.991144i	$0.542393\pi$
$380$		0			0
$381$	−34338.0	+	470.100i	−0.236551	+	0.00323847i
$382$		0			0
$383$	14867.8	−	17718.8i	0.101356	−	0.120792i	−0.712982	−	0.701183i	$-0.752656\pi$
$383$	14867.8	−	17718.8i	0.101356	−	0.120792i	0.814338	+	0.580391i	$0.197100\pi$
$384$		0			0
$385$	91184.3	+	33188.4i	0.615175	+	0.223905i
$386$		0			0
$387$	−194922.	−	64960.3i	−1.30149	−	0.433736i
$388$		0			0
$389$	−95185.9	−	113438.i	−0.629033	−	0.749652i	0.353563	−	0.935411i	$-0.384970\pi$
$389$	−95185.9	−	113438.i	−0.629033	−	0.749652i	−0.982595	+	0.185759i	$0.940526\pi$
$390$		0			0
$391$	−5172.14	+	29332.6i	−0.0338311	+	0.191866i
$392$		0			0
$393$	−111300.	−	18057.8i	−0.720625	−	0.116918i
$394$		0			0
$395$	2657.10	+	1534.08i	0.0170300	+	0.00983227i
$396$		0			0
$397$	39001.3	+	67552.2i	0.247456	+	0.428606i	0.962819	−	0.270146i	$-0.0870721\pi$
$397$	39001.3	+	67552.2i	0.247456	+	0.428606i	−0.715363	+	0.698753i	$0.753739\pi$
$398$		0			0
$399$	−195625.	−	68183.7i	−1.22879	−	0.428287i
$400$		0			0
$401$	−28243.3	−	77597.8i	−0.175641	−	0.482570i	0.820366	−	0.571838i	$-0.193770\pi$
$401$	−28243.3	−	77597.8i	−0.175641	−	0.482570i	−0.996008	+	0.0892678i	$0.971547\pi$
$402$		0			0
$403$	−34206.7	−	193996.i	−0.210620	−	1.19449i
$404$		0			0
$405$	25350.9	+	59327.0i	0.154555	+	0.361695i
$406$		0			0
$407$	42652.7	−	7520.83i	0.257489	−	0.0454022i
$408$		0			0
$409$	−61270.7	+	22300.7i	−0.366274	+	0.133313i	−0.518599	−	0.855018i	$-0.673546\pi$
$409$	−61270.7	+	22300.7i	−0.366274	+	0.133313i	0.152325	+	0.988330i	$0.451324\pi$
$410$		0			0
$411$	27215.2	−	78082.8i	0.161112	−	0.462244i
$412$		0			0
$413$	−300903.	+	173726.i	−1.76411	+	1.01851i
$414$		0			0
$415$	59004.8	−	102199.i	0.342603	−	0.593406i
$416$		0			0
$417$	58.5101	−	360.628i	0.000336479	−	0.00207390i
$418$		0			0
$419$	269322.	+	47488.7i	1.53406	+	0.270497i	0.875943	−	0.482415i	$-0.160240\pi$
$419$	269322.	+	47488.7i	1.53406	+	0.270497i	0.658121	+	0.752912i	$0.271352\pi$
$420$		0			0
$421$	−18999.3	+	15942.3i	−0.107195	+	0.0899471i	−0.694810	−	0.719194i	$-0.744511\pi$
$421$	−18999.3	+	15942.3i	−0.107195	+	0.0899471i	0.587615	+	0.809141i	$0.300067\pi$
$422$		0			0
$423$	322890.	−	66095.9i	1.80457	−	0.369397i
$424$		0			0
$425$	9526.13	−	26172.8i	0.0527398	−	0.144901i
$426$		0			0
$427$	−84481.7	−	70888.5i	−0.463347	−	0.388795i
$428$		0			0
$429$	−3682.31	−	268971.i	−0.0200081	−	1.46147i
$430$		0			0
$431$			124866.i			0.672186i	0.941829	+	0.336093i	$0.109106\pi$
$431$			124866.i			0.672186i	−0.941829	+	0.336093i	$0.890894\pi$
$432$		0			0
$433$	125519.			0.669475			0.334738	−	0.942311i	$-0.391352\pi$
$433$	125519.			0.669475			0.334738	+	0.942311i	$0.391352\pi$
$434$		0			0
$435$	−5056.82	−	8488.16i	−0.0267238	−	0.0448575i
$436$		0			0
$437$	−118813.	+	141596.i	−0.622161	+	0.741463i
$438$		0			0
$439$	156512.	+	56965.8i	0.812118	+	0.295587i	0.714498	−	0.699637i	$-0.246655\pi$
$439$	156512.	+	56965.8i	0.812118	+	0.295587i	0.0976196	+	0.995224i	$0.468877\pi$
$440$		0			0
$441$	30272.5	−	204233.i	0.155658	−	1.05014i
$442$		0			0
$443$	−94109.7	−	112156.i	−0.479542	−	0.571496i	0.470983	−	0.882142i	$-0.343899\pi$
$443$	−94109.7	−	112156.i	−0.479542	−	0.571496i	−0.950526	+	0.310646i	$0.899455\pi$
$444$		0			0
$445$	−16551.4	+	93867.6i	−0.0835824	+	0.474019i
$446$		0			0
$447$	−36550.0	+	44789.7i	−0.182925	+	0.224163i
$448$		0			0
$449$	146812.	+	84762.1i	0.728231	+	0.420445i	0.817775	−	0.575538i	$-0.195207\pi$
$449$	146812.	+	84762.1i	0.728231	+	0.420445i	−0.0895434	+	0.995983i	$0.528541\pi$
$450$		0			0
$451$	−34040.4	−	58959.7i	−0.167356	−	0.289869i
$452$		0			0
$453$	74435.0	+	390782.i	0.362728	+	1.90431i
$454$		0			0
$455$	−50422.2	−	138534.i	−0.243556	−	0.669164i
$456$		0			0
$457$	−58409.4	−	331256.i	−0.279673	−	1.58610i	−0.723717	−	0.690097i	$-0.757568\pi$
$457$	−58409.4	−	331256.i	−0.279673	−	1.58610i	0.444044	−	0.896005i	$-0.353543\pi$
$458$		0			0
$459$	34045.1	+	17833.9i	0.161595	+	0.0846486i
$460$		0			0
$461$	97008.3	−	17105.2i	0.456465	−	0.0804870i	0.0593118	−	0.998240i	$-0.481109\pi$
$461$	97008.3	−	17105.2i	0.456465	−	0.0804870i	0.397153	+	0.917752i	$0.369998\pi$
$462$		0			0
$463$	312412.	−	113709.i	1.45736	−	0.530434i	0.512720	−	0.858556i	$-0.328638\pi$
$463$	312412.	−	113709.i	1.45736	−	0.530434i	0.944635	+	0.328122i	$0.106416\pi$
$464$		0			0
$465$	−53440.3	−	61945.4i	−0.247151	−	0.286486i
$466$		0			0
$467$	311347.	−	179756.i	1.42761	−	0.824233i	0.430681	−	0.902504i	$-0.358273\pi$
$467$	311347.	−	179756.i	1.42761	−	0.824233i	0.996932	+	0.0782717i	$0.0249402\pi$
$468$		0			0
$469$	−106785.	+	184956.i	−0.485470	+	0.840859i
$470$		0			0
$471$	106523.	−	40431.0i	0.480178	−	0.182252i
$472$		0			0
$473$	350373.	+	61780.3i	1.56606	+	0.276139i
$474$		0			0
$475$	132409.	−	111104.i	0.586854	−	0.492429i
$476$		0			0
$477$	294885.	−	181192.i	1.29603	−	0.796348i
$478$		0			0
$479$	100143.	−	275140.i	0.436464	−	1.19917i	−0.505313	−	0.862936i	$-0.668623\pi$
$479$	100143.	−	275140.i	0.436464	−	1.19917i	0.941777	−	0.336238i	$-0.109155\pi$
$480$		0			0
$481$	−50406.3	−	42295.9i	−0.217869	−	0.182814i
$482$		0			0
$483$	−312228.	−	174610.i	−1.33838	−	0.748472i
$484$		0			0
$485$		−	46817.6i		−	0.199033i
$486$		0			0
$487$	−468342.			−1.97472			−0.987359	−	0.158502i	$-0.949334\pi$
$487$	−468342.			−1.97472			−0.987359	+	0.158502i	$0.949334\pi$
$488$		0			0
$489$	−150336.	+	268822.i	−0.628701	+	1.12421i
$490$		0			0
$491$	122177.	−	145605.i	0.506788	−	0.603967i	−0.450616	−	0.892718i	$-0.648796\pi$
$491$	122177.	−	145605.i	0.506788	−	0.603967i	0.957404	+	0.288751i	$0.0932400\pi$
$492$		0			0
$493$	−5530.86	−	2013.07i	−0.0227561	−	0.00828256i
$494$		0			0
$495$	−58486.1	−	95184.3i	−0.238695	−	0.388468i
$496$		0			0
$497$	−263520.	−	314051.i	−1.06685	−	1.27142i
$498$		0			0
$499$	71474.9	−	405354.i	0.287047	−	1.62792i	−0.410836	−	0.911709i	$-0.634763\pi$
$499$	71474.9	−	405354.i	0.287047	−	1.62792i	0.697882	−	0.716213i	$-0.254126\pi$
$500$		0			0
$501$	130308.	+	343320.i	0.519153	+	1.36781i
$502$		0			0
$503$	−254905.	−	147169.i	−1.00749	−	0.581676i	−0.0970362	−	0.995281i	$-0.530936\pi$
$503$	−254905.	−	147169.i	−1.00749	−	0.581676i	−0.910457	+	0.413605i	$0.864270\pi$
$504$		0			0
$505$	50256.0	+	87045.9i	0.197063	+	0.341323i
$506$		0			0
$507$	−114810.	+	99046.5i	−0.446646	+	0.385321i
$508$		0			0
$509$	−91361.9	−	251015.i	−0.352638	−	0.968866i	−0.981519	−	0.191364i	$-0.938709\pi$
$509$	−91361.9	−	251015.i	−0.352638	−	0.968866i	0.628881	−	0.777502i	$-0.283513\pi$
$510$		0			0
$511$	85462.9	+	484684.i	0.327292	+	1.85617i
$512$		0			0
$513$	127635.	+	201485.i	0.484991	+	0.765609i
$514$		0			0
$515$	23073.2	−	4068.44i	0.0869950	−	0.0153396i
$516$		0			0
$517$	−536293.	+	195195.i	−2.00642	+	0.730276i
$518$		0			0
$519$	368409.	−	70173.4i	1.36771	−	0.260518i
$520$		0			0
$521$	−32279.1	+	18636.3i	−0.118918	+	0.0686571i	−0.558279	−	0.829653i	$-0.688538\pi$
$521$	−32279.1	+	18636.3i	−0.118918	+	0.0686571i	0.439361	+	0.898310i	$0.355205\pi$
$522$		0			0
$523$	−237983.	+	412199.i	−0.870048	+	1.50697i	−0.00810215	+	0.999967i	$0.502579\pi$
$523$	−237983.	+	412199.i	−0.870048	+	1.50697i	−0.861946	+	0.507000i	$0.830754\pi$
$524$		0			0
$525$	259180.	+	211500.i	0.940336	+	0.767347i
$526$		0			0
$527$	−47995.6	−	8462.92i	−0.172815	−	0.0304719i
$528$		0			0
$529$	−30138.4	+	25289.1i	−0.107698	+	0.0903695i
$530$		0			0
$531$	395695.	+	58652.1i	1.40337	+	0.208015i
$532$		0			0
$533$	−35376.3	+	97195.6i	−0.124525	+	0.342131i
$534$		0			0
$535$	−119435.	−	100218.i	−0.417276	−	0.350136i
$536$		0			0
$537$	−283423.	+	168849.i	−0.982850	+	0.585532i
$538$		0			0
$539$			357515.i			1.23060i
$540$		0			0
$541$	−312630.			−1.06816			−0.534081	−	0.845434i	$-0.679342\pi$
$541$	−312630.			−1.06816			−0.534081	+	0.845434i	$0.679342\pi$
$542$		0			0
$543$	−31732.2	+	434.425i	−0.107622	+	0.00147338i
$544$		0			0
$545$	117102.	−	139557.i	0.394250	−	0.469849i
$546$		0			0
$547$	92448.8	+	33648.6i	0.308977	+	0.112459i	0.491855	−	0.870677i	$-0.336319\pi$
$547$	92448.8	+	33648.6i	0.308977	+	0.112459i	−0.182878	+	0.983136i	$0.558541\pi$
$548$		0			0
$549$	25462.5	+	124388.i	0.0844804	+	0.412701i
$550$		0			0
$551$	−23478.6	−	27980.7i	−0.0773338	−	0.0921629i
$552$		0			0
$553$	−3811.96	+	21618.7i	−0.0124652	+	0.0706934i
$554$		0			0
$555$	−26975.0	−	4376.56i	−0.0875742	−	0.0142085i
$556$		0			0
$557$	13215.8	+	7630.14i	0.0425973	+	0.0245936i	0.521147	−	0.853467i	$-0.325504\pi$
$557$	13215.8	+	7630.14i	0.0425973	+	0.0245936i	−0.478550	+	0.878060i	$0.658837\pi$
$558$		0			0
$559$	−270262.	−	468108.i	−0.864892	−	1.49804i
$560$		0			0
$561$	−62843.4	−	21903.6i	−0.199680	−	0.0695970i
$562$		0			0
$563$	−207977.	−	571412.i	−0.656143	−	1.80274i	−0.593696	−	0.804690i	$-0.702332\pi$
$563$	−207977.	−	571412.i	−0.656143	−	1.80274i	−0.0624470	−	0.998048i	$-0.519890\pi$
$564$		0			0
$565$	13133.0	+	74481.1i	0.0411403	+	0.233318i
$566$		0			0
$567$	−336840.	+	315622.i	−1.04775	+	0.981751i
$568$		0			0
$569$	−232026.	+	40912.4i	−0.716658	+	0.126366i	−0.520074	−	0.854121i	$-0.674096\pi$
$569$	−232026.	+	40912.4i	−0.716658	+	0.126366i	−0.196584	+	0.980487i	$0.562985\pi$
$570$		0			0
$571$	462951.	−	168501.i	1.41992	−	0.516808i	0.485894	−	0.874018i	$-0.338494\pi$
$571$	462951.	−	168501.i	1.41992	−	0.516808i	0.934024	+	0.357210i	$0.116272\pi$
$572$		0			0
$573$	131107.	−	376156.i	0.399314	−	1.14567i
$574$		0			0
$575$	258486.	−	149237.i	0.781809	−	0.451378i
$576$		0			0
$577$	92113.0	−	159544.i	0.276675	−	0.479214i	−0.693882	−	0.720089i	$-0.744101\pi$
$577$	92113.0	−	159544.i	0.276675	−	0.479214i	0.970556	+	0.240875i	$0.0774343\pi$
$578$		0			0
$579$	−73620.0	+	453759.i	−0.219603	+	1.35353i
$580$		0			0
$581$	831513.	+	146618.i	2.46330	+	0.434346i
$582$		0			0
$583$	−459102.	+	385232.i	−1.35074	+	1.13341i
$584$		0			0
$585$	−53662.6	+	161022.i	−0.156805	+	0.470515i
$586$		0			0
$587$	−172208.	+	473138.i	−0.499779	+	1.37313i	0.391710	+	0.920089i	$0.371884\pi$
$587$	−172208.	+	473138.i	−0.499779	+	1.37313i	−0.891489	+	0.453042i	$0.850339\pi$
$588$		0			0
$589$	−231687.	−	194409.i	−0.667839	−	0.560384i
$590$		0			0
$591$	2486.40	+	181617.i	0.00711862	+	0.519974i
$592$		0			0
$593$			427298.i			1.21513i	0.794271	+	0.607563i	$0.207853\pi$
$593$			427298.i			1.21513i	−0.794271	+	0.607563i	$0.792147\pi$
$594$		0			0
$595$	−36473.7			−0.103026
$596$		0			0
$597$	−130685.	−	219362.i	−0.366670	−	0.615478i
$598$		0			0
$599$	−142612.	+	169958.i	−0.397467	+	0.473683i	−0.927246	−	0.374453i	$-0.877831\pi$
$599$	−142612.	+	169958.i	−0.397467	+	0.473683i	0.529779	+	0.848136i	$0.322275\pi$
$600$		0			0
$601$	196054.	+	71357.7i	0.542783	+	0.197557i	0.598837	−	0.800871i	$-0.295630\pi$
$601$	196054.	+	71357.7i	0.542783	+	0.197557i	−0.0560543	+	0.998428i	$0.517852\pi$
$602$		0			0
$603$	228663.	−	90389.6i	0.628870	−	0.248590i
$604$		0			0
$605$	31805.2	+	37904.0i	0.0868935	+	0.103556i
$606$		0			0
$607$	37410.2	−	212164.i	0.101534	−	0.575830i	−0.891014	−	0.453976i	$-0.850005\pi$
$607$	37410.2	−	212164.i	0.101534	−	0.575830i	0.992548	−	0.121853i	$-0.0388838\pi$
$608$		0			0
$609$	44694.4	−	54770.1i	0.120509	−	0.147676i
$610$		0			0
$611$	750901.	+	433533.i	2.01141	+	1.16129i
$612$		0			0
$613$	275443.	+	477081.i	0.733011	+	1.26961i	0.955590	+	0.294698i	$0.0952192\pi$
$613$	275443.	+	477081.i	0.733011	+	1.26961i	−0.222579	+	0.974915i	$0.571447\pi$
$614$		0			0
$615$	8037.81	+	42198.3i	0.0212514	+	0.111569i
$616$		0			0
$617$	−236748.	−	650459.i	−0.621892	−	1.70863i	−0.702308	−	0.711874i	$-0.747847\pi$
$617$	−236748.	−	650459.i	−0.621892	−	1.70863i	0.0804154	−	0.996761i	$-0.474375\pi$
$618$		0			0
$619$	116854.	+	662713.i	0.304974	+	1.72959i	0.623626	+	0.781723i	$0.285659\pi$
$619$	116854.	+	662713.i	0.304974	+	1.72959i	−0.318652	+	0.947872i	$0.603230\pi$
$620$		0			0
$621$	156635.	+	380911.i	0.406168	+	0.987734i
$622$		0			0
$623$	−671608.	+	118423.i	−1.73037	+	0.305111i
$624$		0			0
$625$	−205486.	+	74790.6i	−0.526043	+	0.191464i
$626$		0			0
$627$	−269780.	−	312716.i	−0.686238	−	0.795453i
$628$		0			0
$629$	−14098.5	+	8139.74i	−0.0356345	+	0.0205736i
$630$		0			0
$631$	−196798.	+	340864.i	−0.494268	+	0.856097i	−0.999978	−	0.00660642i	$-0.997897\pi$
$631$	−196798.	+	340864.i	−0.494268	+	0.856097i	0.505710	+	0.862703i	$0.331230\pi$
$632$		0			0
$633$	32798.6	−	12448.8i	0.0818556	−	0.0310684i
$634$		0			0
$635$	−36950.9	−	6515.45i	−0.0916385	−	0.0161583i
$636$		0			0
$637$	416086.	−	349138.i	1.02543	−	0.860435i
$638$		0			0
$639$	12921.0	+	471812.i	0.0316441	+	1.15549i
$640$		0			0
$641$	240188.	−	659912.i	0.584569	−	1.60609i	−0.195713	−	0.980661i	$-0.562702\pi$
$641$	240188.	−	659912.i	0.584569	−	1.60609i	0.780282	−	0.625428i	$-0.215076\pi$
$642$		0			0
$643$	−208490.	−	174944.i	−0.504270	−	0.423133i	0.354837	−	0.934928i	$-0.384536\pi$
$643$	−208490.	−	174944.i	−0.504270	−	0.423133i	−0.859107	+	0.511795i	$0.828981\pi$
$644$		0			0
$645$	−195928.	−	109571.i	−0.470954	−	0.263376i
$646$		0			0
$647$		−	337710.i		−	0.806743i	−0.915036	−	0.403372i	$-0.867838\pi$
$647$		−	337710.i		−	0.806743i	0.915036	−	0.403372i	$-0.132162\pi$
$648$		0			0
$649$	−692675.			−1.64452
$650$		0			0
$651$	285707.	−	510884.i	0.674153	−	1.20548i
$652$		0			0
$653$	−88035.2	+	104916.i	−0.206457	+	0.246046i	−0.859330	−	0.511421i	$-0.829119\pi$
$653$	−88035.2	+	104916.i	−0.206457	+	0.246046i	0.652873	+	0.757468i	$0.273564\pi$
$654$		0			0
$655$	−115766.	−	42135.3i	−0.269834	−	0.0982117i
$656$		0			0
$657$	269771.	−	498280.i	0.624977	−	1.15436i
$658$		0			0
$659$	82186.9	+	97946.5i	0.189248	+	0.225537i	0.852323	−	0.523016i	$-0.175193\pi$
$659$	82186.9	+	97946.5i	0.189248	+	0.225537i	−0.663075	+	0.748553i	$0.730749\pi$
$660$		0			0
$661$	59270.9	−	336142.i	0.135656	−	0.769343i	−0.838745	−	0.544525i	$-0.816710\pi$
$661$	59270.9	−	336142.i	0.135656	−	0.769343i	0.974401	−	0.224818i	$-0.0721788\pi$
$662$		0			0
$663$	35878.8	+	94529.4i	0.0816227	+	0.215050i
$664$		0			0
$665$	−196024.	−	113174.i	−0.443267	−	0.255920i
$666$		0			0
$667$	−31536.8	−	54623.4i	−0.0708870	−	0.122780i
$668$		0			0
$669$	−637185.	+	549700.i	−1.42368	+	1.22821i
$670$		0			0
$671$	−75195.9	−	206599.i	−0.167013	−	0.458863i
$672$		0			0
$673$	34575.4	+	196087.i	0.0763375	+	0.432931i	0.998892	+	0.0470642i	$0.0149865\pi$
$673$	34575.4	+	196087.i	0.0763375	+	0.432931i	−0.922554	+	0.385867i	$0.873902\pi$
$674$		0			0
$675$	−82393.8	−	376218.i	−0.180837	−	0.825719i
$676$		0			0
$677$	−261985.	+	46195.1i	−0.571609	+	0.100790i	−0.451979	−	0.892028i	$-0.649282\pi$
$677$	−261985.	+	46195.1i	−0.571609	+	0.100790i	−0.119630	+	0.992819i	$0.538171\pi$
$678$		0			0
$679$	314771.	−	114567.i	0.682740	−	0.248497i
$680$		0			0
$681$	386171.	−	73556.7i	0.832693	−	0.158609i
$682$		0			0
$683$	−228548.	+	131952.i	−0.489933	+	0.282863i	−0.724547	−	0.689226i	$-0.757951\pi$
$683$	−228548.	+	131952.i	−0.489933	+	0.282863i	0.234614	+	0.972089i	$0.424617\pi$
$684$		0			0
$685$	45173.0	−	78242.0i	0.0962716	−	0.166747i
$686$		0			0
$687$	455139.	+	371409.i	0.964340	+	0.786935i
$688$		0			0
$689$	896689.	+	158110.i	1.88888	+	0.333060i
$690$		0			0
$691$	508520.	−	426699.i	1.06501	−	0.893646i	0.0704147	−	0.997518i	$-0.477568\pi$
$691$	508520.	−	426699.i	1.06501	−	0.893646i	0.994591	+	0.103872i	$0.0331233\pi$
$692$		0			0
$693$	496836.	−	626148.i	1.03454	−	1.30380i
$694$		0			0
$695$	136.525	−	375.098i	0.000282645	−	0.000776560i
$696$		0			0
$697$	19603.1	+	16448.9i	0.0403514	+	0.0338588i
$698$		0			0
$699$	−585144.	+	348599.i	−1.19759	+	0.713464i
$700$		0			0
$701$		−	214146.i		−	0.435788i	−0.975972	−	0.217894i	$-0.930081\pi$
$701$		−	214146.i		−	0.435788i	0.975972	−	0.217894i	$-0.0699186\pi$
$702$		0			0
$703$	−101027.			−0.204422
$704$		0			0
$705$	360068.	−	4929.45i	0.724446	−	0.00991792i
$706$		0			0
$707$	−462259.	+	550898.i	−0.924797	+	1.10213i
$708$		0			0
$709$	−283230.	−	103087.i	−0.563438	−	0.205075i	0.0445691	−	0.999006i	$-0.485809\pi$
$709$	−283230.	−	103087.i	−0.563438	−	0.205075i	−0.608007	+	0.793932i	$0.708031\pi$
$710$		0			0
$711$	18908.5	−	16769.3i	0.0374040	−	0.0331723i
$712$		0			0
$713$	−335706.	−	400079.i	−0.660358	−	0.786984i
$714$		0			0
$715$	51035.7	−	289438.i	0.0998302	−	0.566165i
$716$		0			0
$717$	−269508.	−	43726.2i	−0.524243	−	0.0850558i
$718$		0			0
$719$	249436.	+	144012.i	0.482505	+	0.278574i	0.721460	−	0.692456i	$-0.243472\pi$
$719$	249436.	+	144012.i	0.482505	+	0.278574i	−0.238955	+	0.971031i	$0.576805\pi$
$720$		0			0
$721$	83816.0	+	145174.i	0.161234	+	0.279265i
$722$		0			0
$723$	−776266.	−	270562.i	−1.48503	−	0.517596i
$724$		0			0
$725$	20172.7	+	55424.1i	0.0383786	+	0.105444i
$726$		0			0
$727$	−64865.9	−	367873.i	−0.122729	−	0.696032i	−0.982631	−	0.185572i	$-0.940586\pi$
$727$	−64865.9	−	367873.i	−0.122729	−	0.696032i	0.859901	−	0.510460i	$-0.170525\pi$
$728$		0			0
$729$	529649.	−	43601.9i	0.996629	−	0.0820446i
$730$		0			0
$731$	−131697.	+	23221.8i	−0.246457	+	0.0434571i
$732$		0			0
$733$	88629.6	−	32258.5i	0.164957	−	0.0600394i	−0.258222	−	0.966086i	$-0.583136\pi$
$733$	88629.6	−	32258.5i	0.164957	−	0.0600394i	0.423179	+	0.906046i	$0.360914\pi$
$734$		0			0
$735$	74244.2	−	213013.i	0.137432	−	0.394304i
$736$		0			0
$737$	−368725.	+	212884.i	−0.678841	+	0.391929i
$738$		0			0
$739$	419686.	−	726918.i	0.768486	−	1.33106i	−0.169898	−	0.985462i	$-0.554344\pi$
$739$	419686.	−	726918.i	0.768486	−	1.33106i	0.938384	−	0.345595i	$-0.112323\pi$
$740$		0			0
$741$	−100489.	+	619367.i	−0.183013	+	1.12801i
$742$		0			0
$743$	565336.	+	99684.0i	1.02407	+	0.180571i	0.660366	−	0.750944i	$-0.270401\pi$
$743$	565336.	+	99684.0i	1.02407	+	0.180571i	0.363703	+	0.931515i	$0.381512\pi$
$744$		0			0
$745$	−48385.6	+	40600.3i	−0.0871773	+	0.0731505i
$746$		0			0
$747$	−644993.	−	727273.i	−1.15588	−	1.30334i
$748$		0			0
$749$	381530.	−	1.04825e6i	0.680088	−	1.86853i
$750$		0			0
$751$	719252.	+	603524.i	1.27527	+	1.07008i	0.993878	+	0.110482i	$0.0352394\pi$
$751$	719252.	+	603524.i	1.27527	+	1.07008i	0.281388	+	0.959594i	$0.409205\pi$
$752$		0			0
$753$	−9645.55	−	704551.i	−0.0170113	−	1.24257i
$754$		0			0
$755$			434641.i			0.762495i
$756$		0			0
$757$	980425.			1.71089			0.855446	−	0.517891i	$-0.173283\pi$
$757$	980425.			1.71089			0.855446	+	0.517891i	$0.173283\pi$
$758$		0			0
$759$	−365010.	−	612690.i	−0.633608	−	1.06355i
$760$		0			0
$761$	−326111.	+	388644.i	−0.563113	+	0.671092i	−0.970202	−	0.242296i	$-0.922099\pi$
$761$	−326111.	+	388644.i	−0.563113	+	0.671092i	0.407089	+	0.913388i	$0.366544\pi$
$762$		0			0
$763$	1.22485e6	+	445809.i	2.10394	+	0.765773i
$764$		0			0
$765$	32894.4	+	26101.1i	0.0562082	+	0.0446001i
$766$		0			0
$767$	676445.	+	806156.i	1.14985	+	1.37034i
$768$		0			0
$769$	−63373.5	+	359409.i	−0.107166	+	0.607766i	0.883168	+	0.469057i	$0.155406\pi$
$769$	−63373.5	+	359409.i	−0.107166	+	0.607766i	−0.990333	+	0.138709i	$0.955705\pi$
$770$		0			0
$771$	33643.2	−	41227.6i	0.0565963	−	0.0693553i
$772$		0			0
$773$	−629772.	−	363599.i	−1.05396	−	0.608504i	−0.130205	−	0.991487i	$-0.541563\pi$
$773$	−629772.	−	363599.i	−1.05396	−	0.608504i	−0.923755	+	0.382983i	$0.874897\pi$
$774$		0			0
$775$	244189.	+	422948.i	0.406558	+	0.704180i
$776$		0			0
$777$	−36585.4	−	192072.i	−0.0605991	−	0.318144i
$778$		0			0
$779$	54315.1	+	149229.i	0.0895046	+	0.245912i
$780$		0			0
$781$	−141922.	−	804882.i	−0.232675	−	1.31956i
$782$		0			0
$783$	−79502.7	+	17411.5i	−0.129676	+	0.0283997i
$784$		0			0
$785$	122596.	−	21617.0i	0.198947	−	0.0350798i
$786$		0			0
$787$	342474.	−	124650.i	0.552940	−	0.201254i	−0.0504125	−	0.998728i	$-0.516054\pi$
$787$	342474.	−	124650.i	0.552940	−	0.201254i	0.603352	+	0.797475i	$0.293831\pi$
$788$		0			0
$789$	−727193.	−	842927.i	−1.16814	−	1.35405i
$790$		0			0
$791$	−468624.	+	270560.i	−0.748983	+	0.432425i
$792$		0			0
$793$	−167012.	+	289274.i	−0.265584	+	0.460005i
$794$		0			0
$795$	353540.	−	134187.i	0.559377	−	0.212313i
$796$		0			0
$797$	−291577.	−	51412.9i	−0.459025	−	0.0809386i	−0.0606461	−	0.998159i	$-0.519316\pi$
$797$	−291577.	−	51412.9i	−0.459025	−	0.0809386i	−0.398379	+	0.917221i	$0.630427\pi$
$798$		0			0
$799$	164330.	−	137889.i	0.257408	−	0.215991i
$800$		0			0
$801$	690447.	+	373811.i	1.07613	+	0.582622i
$802$		0			0
$803$	−335578.	+	921992.i	−0.520429	+	1.42987i
$804$		0			0
$805$	−299416.	−	251240.i	−0.462043	−	0.387700i
$806$		0			0
$807$	−245023.	−	137027.i	−0.376236	−	0.210406i
$808$		0			0
$809$			494809.i			0.756033i	0.925799	+	0.378017i	$0.123394\pi$
$809$			494809.i			0.756033i	−0.925799	+	0.378017i	$0.876606\pi$
$810$		0			0
$811$	466081.			0.708631			0.354315	−	0.935126i	$-0.384714\pi$
$811$	466081.			0.708631			0.354315	+	0.935126i	$0.384714\pi$
$812$		0			0
$813$	217370.	−	388689.i	0.328866	−	0.588059i
$814$		0			0
$815$	−216312.	+	257790.i	−0.325660	+	0.388107i
$816$		0			0
$817$	−779847.	−	283841.i	−1.16833	−	0.425237i
$818$		0			0
$819$	−1.21393e6	+	33244.3i	−1.80977	+	0.0495621i
$820$		0			0
$821$	105926.	+	126237.i	0.157150	+	0.187284i	0.838874	−	0.544325i	$-0.183214\pi$
$821$	105926.	+	126237.i	0.157150	+	0.187284i	−0.681724	+	0.731609i	$0.738770\pi$
$822$		0			0
$823$	−21965.1	+	124571.i	−0.0324291	+	0.183914i	−0.996720	−	0.0809323i	$-0.974210\pi$
$823$	−21965.1	+	124571.i	−0.0324291	+	0.183914i	0.964291	+	0.264847i	$0.0853214\pi$
$824$		0			0
$825$	236651.	+	623502.i	0.347697	+	0.916072i
$826$		0			0
$827$	776147.	+	448109.i	1.13484	+	0.655198i	0.945147	−	0.326646i	$-0.105919\pi$
$827$	776147.	+	448109.i	1.13484	+	0.655198i	0.189689	+	0.981844i	$0.439252\pi$
$828$		0			0
$829$	−195237.	−	338161.i	−0.284089	−	0.492056i	0.688299	−	0.725427i	$-0.258358\pi$
$829$	−195237.	−	338161.i	−0.284089	−	0.492056i	−0.972388	+	0.233371i	$0.925024\pi$
$830$		0			0
$831$	−390272.	+	336688.i	−0.565153	+	0.487557i
$832$		0			0
$833$	−45961.1	−	126277.i	−0.0662370	−	0.181985i
$834$		0			0
$835$	69671.0	+	395124.i	0.0999261	+	0.566709i
$836$		0			0
$837$	−623266.	+	256294.i	−0.889657	+	0.365837i
$838$		0			0
$839$	−955631.	+	168504.i	−1.35758	+	0.239378i	−0.804601	−	0.593815i	$-0.797621\pi$
$839$	−955631.	+	168504.i	−1.35758	+	0.239378i	−0.552981	+	0.833194i	$0.686510\pi$
$840$		0			0
$841$	−652914.	+	237641.i	−0.923133	+	0.335993i
$842$		0			0
$843$	942318.	−	179490.i	1.32600	−	0.252572i
$844$		0			0
$845$	−143474.	+	82834.5i	−0.200936	+	0.116011i
$846$		0			0
$847$	−177011.	+	306592.i	−0.246737	+	0.427360i
$848$		0			0
$849$	349432.	+	285149.i	0.484783	+	0.395600i
$850$		0			0
$851$	−171804.	−	30293.7i	−0.237233	−	0.0418305i
$852$		0			0
$853$	−578870.	+	485730.i	−0.795579	+	0.667570i	−0.947119	−	0.320881i	$-0.896021\pi$
$853$	−578870.	+	485730.i	−0.795579	+	0.667570i	0.151541	+	0.988451i	$0.451577\pi$
$854$		0			0
$855$	95798.3	+	242346.i	0.131047	+	0.331515i
$856$		0			0
$857$	−248575.	+	682955.i	−0.338452	+	0.929888i	0.647383	+	0.762165i	$0.275864\pi$
$857$	−248575.	+	682955.i	−0.338452	+	0.929888i	−0.985834	+	0.167723i	$0.946359\pi$
$858$		0			0
$859$	−284896.	−	239056.i	−0.386101	−	0.323977i	0.428991	−	0.903309i	$-0.358869\pi$
$859$	−284896.	−	239056.i	−0.386101	−	0.323977i	−0.815092	+	0.579332i	$0.803314\pi$
$860$		0			0
$861$	−264044.	+	157304.i	−0.356181	+	0.212195i
$862$		0			0
$863$		−	175442.i		−	0.235565i	−0.993039	−	0.117783i	$-0.962421\pi$
$863$		−	175442.i		−	0.235565i	0.993039	−	0.117783i	$-0.0375786\pi$
$864$		0			0
$865$	409757.			0.547639
$866$		0			0
$867$	−726606.	+	9947.48i	−0.966631	+	0.0132335i
$868$		0			0
$869$	−28130.6	+	33524.8i	−0.0372511	+	0.0443942i
$870$		0			0
$871$	607846.	+	221238.i	0.801231	+	0.291624i
$872$		0			0
$873$	−365868.	−	121930.i	−0.480061	−	0.159986i
$874$		0			0
$875$	512876.	+	611221.i	0.669878	+	0.798330i
$876$		0			0
$877$	−191527.	+	1.08620e6i	−0.249018	+	1.41225i	0.561956	+	0.827167i	$0.310049\pi$
$877$	−191527.	+	1.08620e6i	−0.249018	+	1.41225i	−0.810974	+	0.585083i	$0.801062\pi$
$878$		0			0
$879$	840536.	+	136373.i	1.08787	+	0.176502i
$880$		0			0
$881$	−564401.	−	325857.i	−0.727171	−	0.419832i	0.0902156	−	0.995922i	$-0.471244\pi$
$881$	−564401.	−	325857.i	−0.727171	−	0.419832i	−0.817386	+	0.576090i	$0.804578\pi$
$882$		0			0
$883$	−437795.	−	758282.i	−0.561499	−	0.972545i	−0.997366	−	0.0725337i	$-0.976892\pi$
$883$	−437795.	−	758282.i	−0.561499	−	0.972545i	0.435867	−	0.900011i	$-0.356442\pi$
$884$		0			0
$885$	412707.	+	143846.i	0.526933	+	0.183659i
$886$		0			0
$887$	−500450.	−	1.37498e6i	−0.636083	−	1.74762i	−0.663694	−	0.748004i	$-0.731012\pi$
$887$	−500450.	−	1.37498e6i	−0.636083	−	1.74762i	0.0276111	−	0.999619i	$-0.491210\pi$
$888$		0			0
$889$	−46617.0	−	264378.i	−0.0589848	−	0.334520i
$890$		0			0
$891$	−896161.	+	209160.i	−1.12884	+	0.263465i
$892$		0			0
$893$	1.31103e6	−	231169.i	1.64403	−	0.289886i
$894$		0			0
$895$	−338716.	+	123283.i	−0.422853	+	0.153906i
$896$		0			0
$897$	−356610.	+	1.02314e6i	−0.443208	+	1.27160i
$898$		0			0
$899$	89377.7	−	51602.2i	0.110588	−	0.0638482i
$900$		0			0
$901$	112634.	−	195088.i	0.138746	−	0.240315i
$902$		0			0
$903$	257227.	−	1.58543e6i	0.315458	−	1.94433i
$904$		0			0
$905$	−34146.8	−	6021.01i	−0.0416921	−	0.00735144i
$906$		0			0
$907$	−369828.	+	310322.i	−0.449557	+	0.377223i	−0.839271	−	0.543713i	$-0.817018\pi$
$907$	−369828.	+	310322.i	−0.449557	+	0.377223i	0.389715	+	0.920936i	$0.372574\pi$
$908$		0			0
$909$	811127.	−	166039.i	0.981660	−	0.200947i
$910$		0			0
$911$	−106730.	+	293238.i	−0.128602	+	0.353332i	−0.987237	−	0.159255i	$-0.949091\pi$
$911$	−106730.	+	293238.i	−0.128602	+	0.353332i	0.858635	+	0.512587i	$0.171313\pi$
$912$		0			0
$913$	1.28945e6	+	1.08198e6i	1.54691	+	1.29801i
$914$		0			0
$915$	1899.00	+	138711.i	0.00226821	+	0.165679i
$916$		0			0
$917$		−	881443.i		−	1.04823i
$918$		0			0
$919$	−527904.			−0.625063			−0.312532	−	0.949907i	$-0.601177\pi$
$919$	−527904.			−0.625063			−0.312532	+	0.949907i	$0.601177\pi$
$920$		0			0
$921$	365668.	+	613794.i	0.431089	+	0.723609i
$922$		0			0
$923$	−798149.	+	951197.i	−0.936873	+	1.11652i
$924$		0			0
$925$	153297.	+	55795.4i	0.179163	+	0.0652101i
$926$		0			0
$927$	28297.3	−	190907.i	0.0329296	−	0.222159i
$928$		0			0
$929$	−332213.	−	395916.i	−0.384933	−	0.458745i	0.538432	−	0.842669i	$-0.319017\pi$
$929$	−332213.	−	395916.i	−0.384933	−	0.458745i	−0.923365	+	0.383924i	$0.874572\pi$
$930$		0			0
$931$	144813.	−	821276.i	0.167074	−	0.947522i
$932$		0			0
$933$	41127.8	−	50399.5i	0.0472468	−	0.0578980i
$934$		0			0
$935$	−62971.5	−	36356.6i	−0.0720312	−	0.0415872i
$936$		0			0
$937$	383203.	+	663727.i	0.436465	+	0.755980i	0.997414	−	0.0718706i	$-0.0228969\pi$
$937$	383203.	+	663727.i	0.436465	+	0.755980i	−0.560949	+	0.827851i	$0.689564\pi$
$938$		0			0
$939$	−140501.	−	737628.i	−0.159349	−	0.836578i
$940$		0			0
$941$	279218.	+	767146.i	0.315329	+	0.866361i	0.991557	+	0.129668i	$0.0413912\pi$
$941$	279218.	+	767146.i	0.315329	+	0.866361i	−0.676228	+	0.736692i	$0.736387\pi$
$942$		0			0
$943$	47619.2	+	270062.i	0.0535498	+	0.303696i
$944$		0			0
$945$	−426054.	+	269892.i	−0.477091	+	0.302223i
$946$		0			0
$947$	845700.	−	149120.i	0.943010	−	0.166278i	0.319053	−	0.947737i	$-0.396635\pi$
$947$	845700.	−	149120.i	0.943010	−	0.166278i	0.623957	+	0.781459i	$0.285524\pi$
$948$		0			0
$949$	1.40076e6	−	509834.i	1.55536	−	0.566104i
$950$		0			0
$951$	590396.	+	684359.i	0.652804	+	0.756698i
$952$		0			0
$953$	1.27474e6	−	735974.i	1.40358	−	0.810358i	0.408823	−	0.912614i	$-0.365939\pi$
$953$	1.27474e6	−	735974.i	1.40358	−	0.810358i	0.994758	+	0.102256i	$0.0326061\pi$
$954$		0			0
$955$	217616.	−	376923.i	0.238608	−	0.413281i
$956$		0			0
$957$	131759.	−	50009.3i	0.143865	−	0.0546043i
$958$		0			0
$959$	636591.	+	112248.i	0.692187	+	0.122051i
$960$		0			0
$961$	−52828.6	+	44328.5i	−0.0572035	+	0.0479994i
$962$		0			0
$963$	−1.09423e6	+	672351.i	−1.17993	+	0.725009i
$964$		0			0
$965$	−171781.	+	471966.i	−0.184468	+	0.506822i
$966$		0			0
$967$	−194589.	−	163280.i	−0.208097	−	0.174614i	0.532782	−	0.846252i	$-0.321147\pi$
$967$	−194589.	−	163280.i	−0.208097	−	0.174614i	−0.740879	+	0.671638i	$0.765591\pi$
$968$		0			0
$969$	135490.	+	75771.6i	0.144298	+	0.0806973i
$970$		0			0
$971$		−	114390.i		−	0.121325i	−0.998158	−	0.0606625i	$-0.980679\pi$
$971$		−	114390.i		−	0.121325i	0.998158	−	0.0606625i	$-0.0193213\pi$
$972$		0			0
$973$	2856.00			0.00301671
$974$		0			0
$975$	494544.	−	884314.i	0.520230	−	0.930245i
$976$		0			0
$977$	−887683.	+	1.05790e6i	−0.929969	+	1.10829i	0.0639248	+	0.997955i	$0.479638\pi$
$977$	−887683.	+	1.05790e6i	−0.929969	+	1.10829i	−0.993894	+	0.110339i	$0.964806\pi$
$978$		0			0
$979$	−1.27757e6	−	464997.i	−1.33296	−	0.485160i
$980$		0			0
$981$	−785627.	−	1.27858e6i	−0.816353	−	1.32859i
$982$		0			0
$983$	−867499.	−	1.03385e6i	−0.897764	−	1.06991i	−0.997193	−	0.0748689i	$-0.976146\pi$
$983$	−867499.	−	1.03385e6i	−0.897764	−	1.06991i	0.0994294	−	0.995045i	$-0.468298\pi$
$984$		0			0
$985$	−34460.8	+	195437.i	−0.0355183	+	0.201434i
$986$		0			0
$987$	914264.	+	2.40880e6i	0.938507	+	2.47267i
$988$		0			0
$989$	−1.24107e6	−	716533.i	−1.26883	−	0.732561i
$990$		0			0
$991$	−290755.	−	503602.i	−0.296060	−	0.512791i	0.679171	−	0.733980i	$-0.262339\pi$
$991$	−290755.	−	503602.i	−0.296060	−	0.512791i	−0.975231	+	0.221189i	$0.929006\pi$
$992$		0			0
$993$	378645.	−	326657.i	0.384002	−	0.331279i
$994$		0			0
$995$	−95417.2	−	262157.i	−0.0963786	−	0.264798i
$996$		0			0
$997$	−201359.	−	1.14196e6i	−0.202572	−	1.14884i	−0.901214	−	0.433374i	$-0.857323\pi$
$997$	−201359.	−	1.14196e6i	−0.202572	−	1.14884i	0.698642	−	0.715471i	$-0.253788\pi$
$998$		0			0
$999$	−104455.	+	199405.i	−0.104664	+	0.199805i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.5.k.a.41.8	yes	72
3.2	odd	2		324.5.k.a.233.5		72
27.2	odd	18	inner	108.5.k.a.29.8	✓	72
27.25	even	9		324.5.k.a.89.5		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.k.a.29.8	✓	72	27.2	odd	18	inner
108.5.k.a.41.8	yes	72	1.1	even	1	trivial
324.5.k.a.89.5		72	27.25	even	9
324.5.k.a.233.5		72	3.2	odd	2

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.k (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(4.39288 - 7.85510i) q^{3} +(6.32074 - 7.53276i) q^{5} +(66.1128 + 24.0631i) q^{7} +(-42.4052 - 69.0131i) q^{9} +O(q^{10})\)	\(q+(4.39288 - 7.85510i) q^{3} +(6.32074 - 7.53276i) q^{5} +(66.1128 + 24.0631i) q^{7} +(-42.4052 - 69.0131i) q^{9} +(90.1574 + 107.445i) q^{11} +(37.0033 - 209.856i) q^{13} +(-31.4043 - 82.7405i) q^{15} +(-45.6573 - 26.3602i) q^{17} +(-163.587 - 283.340i) q^{19} +(479.444 - 413.617i) q^{21} +(-193.229 - 530.892i) q^{23} +(91.7393 + 520.280i) q^{25} +(-728.385 + 29.9305i) q^{27} +(109.946 - 19.3864i) q^{29} +(868.674 - 316.171i) q^{31} +(1240.05 - 236.200i) q^{33} +(599.143 - 345.916i) q^{35} +(154.394 - 267.419i) q^{37} +(-1485.89 - 1212.54i) q^{39} +(-478.016 - 84.2871i) q^{41} +(1943.12 - 1630.47i) q^{43} +(-787.891 - 116.785i) q^{45} +(-1391.66 + 3823.56i) q^{47} +(1952.60 + 1638.43i) q^{49} +(-407.629 + 242.845i) q^{51} +4272.88i q^{53} +1379.22 q^{55} +(-2944.28 + 40.3082i) q^{57} +(-3174.41 + 3783.11i) q^{59} +(-1472.97 - 536.118i) q^{61} +(-1142.86 - 5583.05i) q^{63} +(-1346.91 - 1605.18i) q^{65} +(-527.119 + 2989.44i) q^{67} +(-5019.04 - 814.314i) q^{69} +(-5046.35 - 2913.51i) q^{71} +(3497.66 + 6058.12i) q^{73} +(4489.85 + 1564.91i) q^{75} +(3375.09 + 9272.99i) q^{77} +(54.1811 + 307.276i) q^{79} +(-2964.60 + 5853.02i) q^{81} +(11818.7 - 2083.95i) q^{83} +(-487.153 + 177.309i) q^{85} +(330.697 - 948.798i) q^{87} +(-8394.50 + 4846.57i) q^{89} +(7496.17 - 12983.8i) q^{91} +(1332.42 - 8212.42i) q^{93} +(-3168.32 - 558.661i) q^{95} +(3647.23 - 3060.39i) q^{97} +(3592.00 - 10778.3i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{5} - 102 q^{9}+O(q^{10}) \) 72 * q + 9 * q^5 - 102 * q^9	\( 72 q + 9 q^{5} - 102 q^{9} + 18 q^{11} - 225 q^{15} - 282 q^{21} - 1278 q^{23} + 441 q^{25} + 54 q^{27} + 1854 q^{29} - 1665 q^{31} - 45 q^{33} - 2673 q^{35} + 6951 q^{39} - 5472 q^{41} + 1260 q^{43} + 5553 q^{45} + 5103 q^{47} - 5904 q^{49} + 1899 q^{51} + 1107 q^{57} - 10944 q^{59} + 8352 q^{61} - 11985 q^{63} + 8757 q^{65} + 378 q^{67} + 5607 q^{69} - 19764 q^{71} + 6111 q^{73} - 3453 q^{75} - 5679 q^{77} - 5652 q^{79} - 20466 q^{81} - 20061 q^{83} + 26100 q^{85} + 40545 q^{87} + 15633 q^{89} - 6039 q^{91} + 40179 q^{93} + 48024 q^{95} - 37530 q^{97} + 12177 q^{99}+O(q^{100}) \) 72 * q + 9 * q^5 - 102 * q^9 + 18 * q^11 - 225 * q^15 - 282 * q^21 - 1278 * q^23 + 441 * q^25 + 54 * q^27 + 1854 * q^29 - 1665 * q^31 - 45 * q^33 - 2673 * q^35 + 6951 * q^39 - 5472 * q^41 + 1260 * q^43 + 5553 * q^45 + 5103 * q^47 - 5904 * q^49 + 1899 * q^51 + 1107 * q^57 - 10944 * q^59 + 8352 * q^61 - 11985 * q^63 + 8757 * q^65 + 378 * q^67 + 5607 * q^69 - 19764 * q^71 + 6111 * q^73 - 3453 * q^75 - 5679 * q^77 - 5652 * q^79 - 20466 * q^81 - 20061 * q^83 + 26100 * q^85 + 40545 * q^87 + 15633 * q^89 - 6039 * q^91 + 40179 * q^93 + 48024 * q^95 - 37530 * q^97 + 12177 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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