LMFDB - Embedded newform 108.5.k.a.41.2

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			41.2
Character	$\chi$	$=$	108.41
Dual form			108.5.k.a.29.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-8.23149 - 3.63904i) q^{3} +(18.1168 - 21.5908i) q^{5} +(22.9581 + 8.35608i) q^{7} +(54.5147 + 59.9095i) q^{9} +O(q^{10})$	\(q+(-8.23149 - 3.63904i) q^{3} +(18.1168 - 21.5908i) q^{5} +(22.9581 + 8.35608i) q^{7} +(54.5147 + 59.9095i) q^{9} +(-73.4456 - 87.5290i) q^{11} +(-0.427402 + 2.42392i) q^{13} +(-227.698 + 111.797i) q^{15} +(-208.105 - 120.150i) q^{17} +(-246.932 - 427.698i) q^{19} +(-158.572 - 152.329i) q^{21} +(-179.414 - 492.937i) q^{23} +(-29.4131 - 166.810i) q^{25} +(-230.724 - 691.525i) q^{27} +(-95.9782 + 16.9236i) q^{29} +(-1499.65 + 545.827i) q^{31} +(286.044 + 987.765i) q^{33} +(596.344 - 344.299i) q^{35} +(414.251 - 717.504i) q^{37} +(12.3389 - 18.3971i) q^{39} +(2401.19 + 423.394i) q^{41} +(-1294.95 + 1086.59i) q^{43} +(2281.13 - 91.6475i) q^{45} +(-379.164 + 1041.74i) q^{47} +(-1382.02 - 1159.65i) q^{49} +(1275.79 + 1746.31i) q^{51} -3880.06i q^{53} -3220.42 q^{55} +(476.203 + 4419.19i) q^{57} +(2429.52 - 2895.39i) q^{59} +(1823.41 + 663.667i) q^{61} +(750.949 + 1830.94i) q^{63} +(44.5912 + 53.1417i) q^{65} +(-145.480 + 825.059i) q^{67} +(-316.971 + 4710.50i) q^{69} +(1966.62 + 1135.43i) q^{71} +(2940.78 + 5093.57i) q^{73} +(-364.915 + 1480.13i) q^{75} +(-954.774 - 2623.22i) q^{77} +(-988.613 - 5606.70i) q^{79} +(-617.285 + 6531.90i) q^{81} +(7283.42 - 1284.26i) q^{83} +(-6364.34 + 2316.43i) q^{85} +(851.629 + 209.963i) q^{87} +(10189.8 - 5883.10i) q^{89} +(-30.0668 + 52.0773i) q^{91} +(14330.6 + 964.312i) q^{93} +(-13708.0 - 2417.09i) q^{95} +(2197.40 - 1843.84i) q^{97} +(1239.95 - 9171.70i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−8.23149	−	3.63904i	−0.914610	−	0.404338i
$3$	−8.23149	−	3.63904i	−0.914610	−	0.404338i
$4$		0			0
$5$	18.1168	−	21.5908i	0.724674	−	0.863632i	−0.270402	−	0.962747i	$-0.587157\pi$
$5$	18.1168	−	21.5908i	0.724674	−	0.863632i	0.995076	+	0.0991149i	$0.0316011\pi$
$6$		0			0
$7$	22.9581	+	8.35608i	0.468534	+	0.170532i	0.565488	−	0.824757i	$-0.308688\pi$
$7$	22.9581	+	8.35608i	0.468534	+	0.170532i	−0.0969544	+	0.995289i	$0.530910\pi$
$8$		0			0
$9$	54.5147	+	59.9095i	0.673022	+	0.739623i
$10$		0			0
$11$	−73.4456	−	87.5290i	−0.606988	−	0.723380i	0.371787	−	0.928318i	$-0.378745\pi$
$11$	−73.4456	−	87.5290i	−0.606988	−	0.723380i	−0.978775	+	0.204938i	$0.934301\pi$
$12$		0			0
$13$	−0.427402	+	2.42392i	−0.00252901	+	0.0143427i	−0.986046	−	0.166472i	$-0.946762\pi$
$13$	−0.427402	+	2.42392i	−0.00252901	+	0.0143427i	0.983517	+	0.180815i	$0.0578735\pi$
$14$		0			0
$15$	−227.698	+	111.797i	−1.01199	+	0.496873i
$16$		0			0
$17$	−208.105	−	120.150i	−0.720088	−	0.415743i	0.0946973	−	0.995506i	$-0.469812\pi$
$17$	−208.105	−	120.150i	−0.720088	−	0.415743i	−0.814785	+	0.579763i	$0.803145\pi$
$18$		0			0
$19$	−246.932	−	427.698i	−0.684021	−	1.18476i	−0.973743	−	0.227649i	$-0.926896\pi$
$19$	−246.932	−	427.698i	−0.684021	−	1.18476i	0.289722	−	0.957111i	$-0.406437\pi$
$20$		0			0
$21$	−158.572	−	152.329i	−0.359573	−	0.345416i
$22$		0			0
$23$	−179.414	−	492.937i	−0.339158	−	0.931828i	−0.985634	−	0.168894i	$-0.945980\pi$
$23$	−179.414	−	492.937i	−0.339158	−	0.931828i	0.646477	−	0.762934i	$-0.276242\pi$
$24$		0			0
$25$	−29.4131	−	166.810i	−0.0470609	−	0.266896i
$26$		0			0
$27$	−230.724	−	691.525i	−0.316494	−	0.948594i
$28$		0			0
$29$	−95.9782	+	16.9236i	−0.114124	+	0.0201231i	−0.230418	−	0.973092i	$-0.574009\pi$
$29$	−95.9782	+	16.9236i	−0.114124	+	0.0201231i	0.116294	+	0.993215i	$0.462898\pi$
$30$		0			0
$31$	−1499.65	+	545.827i	−1.56051	+	0.567978i	−0.970852	−	0.239679i	$-0.922958\pi$
$31$	−1499.65	+	545.827i	−1.56051	+	0.567978i	−0.589654	+	0.807656i	$0.700736\pi$
$32$		0			0
$33$	286.044	+	987.765i	0.262667	+	0.907039i
$34$		0			0
$35$	596.344	−	344.299i	0.486811	−	0.281061i
$36$		0			0
$37$	414.251	−	717.504i	0.302594	−	0.524108i	−0.674129	−	0.738614i	$-0.735481\pi$
$37$	414.251	−	717.504i	0.302594	−	0.524108i	0.976723	+	0.214506i	$0.0688141\pi$
$38$		0			0
$39$	12.3389	−	18.3971i	0.00811236	−	0.0120954i
$40$		0			0
$41$	2401.19	+	423.394i	1.42843	+	0.251870i	0.833772	−	0.552110i	$-0.186177\pi$
$41$	2401.19	+	423.394i	1.42843	+	0.251870i	0.594656	+	0.803980i	$0.297288\pi$
$42$		0			0
$43$	−1294.95	+	1086.59i	−0.700351	+	0.587664i	−0.921873	−	0.387491i	$-0.873342\pi$
$43$	−1294.95	+	1086.59i	−0.700351	+	0.587664i	0.221523	+	0.975155i	$0.428897\pi$
$44$		0			0
$45$	2281.13	−	91.6475i	1.12648	−	0.0452580i
$46$		0			0
$47$	−379.164	+	1041.74i	−0.171645	+	0.471591i	−0.995450	−	0.0952820i	$-0.969625\pi$
$47$	−379.164	+	1041.74i	−0.171645	+	0.471591i	0.823805	+	0.566873i	$0.191847\pi$
$48$		0			0
$49$	−1382.02	−	1159.65i	−0.575602	−	0.482987i
$50$		0			0
$51$	1275.79	+	1746.31i	0.490498	+	0.671401i
$52$		0			0
$53$		−	3880.06i		−	1.38130i	−0.723191	−	0.690648i	$-0.757325\pi$
$53$		−	3880.06i		−	1.38130i	0.723191	−	0.690648i	$-0.242675\pi$
$54$		0			0
$55$	−3220.42			−1.06460
$56$		0			0
$57$	476.203	+	4419.19i	0.146569	+	1.36017i
$58$		0			0
$59$	2429.52	−	2895.39i	0.697939	−	0.831771i	−0.294353	−	0.955697i	$-0.595104\pi$
$59$	2429.52	−	2895.39i	0.697939	−	0.831771i	0.992291	+	0.123926i	$0.0395485\pi$
$60$		0			0
$61$	1823.41	+	663.667i	0.490032	+	0.178357i	0.575205	−	0.818009i	$-0.304922\pi$
$61$	1823.41	+	663.667i	0.490032	+	0.178357i	−0.0851732	+	0.996366i	$0.527144\pi$
$62$		0			0
$63$	750.949	+	1830.94i	0.189204	+	0.461310i
$64$		0			0
$65$	44.5912	+	53.1417i	0.0105541	+	0.0125779i
$66$		0			0
$67$	−145.480	+	825.059i	−0.0324082	+	0.183796i	−0.996715	−	0.0809926i	$-0.974191\pi$
$67$	−145.480	+	825.059i	−0.0324082	+	0.183796i	0.964307	+	0.264788i	$0.0853021\pi$
$68$		0			0
$69$	−316.971	+	4710.50i	−0.0665766	+	0.989393i
$70$		0			0
$71$	1966.62	+	1135.43i	0.390126	+	0.225239i	0.682215	−	0.731152i	$-0.261017\pi$
$71$	1966.62	+	1135.43i	0.390126	+	0.225239i	−0.292089	+	0.956391i	$0.594350\pi$
$72$		0			0
$73$	2940.78	+	5093.57i	0.551844	+	0.955822i	0.998142	+	0.0609373i	$0.0194090\pi$
$73$	2940.78	+	5093.57i	0.551844	+	0.955822i	−0.446298	+	0.894885i	$0.647258\pi$
$74$		0			0
$75$	−364.915	+	1480.13i	−0.0648737	+	0.263134i
$76$		0			0
$77$	−954.774	−	2623.22i	−0.161035	−	0.442439i
$78$		0			0
$79$	−988.613	−	5606.70i	−0.158406	−	0.898366i	−0.955605	−	0.294650i	$-0.904797\pi$
$79$	−988.613	−	5606.70i	−0.158406	−	0.898366i	0.797199	−	0.603717i	$-0.206314\pi$
$80$		0			0
$81$	−617.285	+	6531.90i	−0.0940840	+	0.995564i
$82$		0			0
$83$	7283.42	−	1284.26i	1.05725	−	0.186422i	0.382116	−	0.924114i	$-0.375196\pi$
$83$	7283.42	−	1284.26i	1.05725	−	0.186422i	0.675137	+	0.737692i	$0.264084\pi$
$84$		0			0
$85$	−6364.34	+	2316.43i	−0.880877	+	0.320613i
$86$		0			0
$87$	851.629	+	209.963i	0.112515	+	0.0277398i
$88$		0			0
$89$	10189.8	−	5883.10i	1.28643	−	0.742722i	0.308416	−	0.951252i	$-0.400201\pi$
$89$	10189.8	−	5883.10i	1.28643	−	0.742722i	0.978016	+	0.208530i	$0.0668678\pi$
$90$		0			0
$91$	−30.0668	+	52.0773i	−0.00363082	+	0.00628877i
$92$		0			0
$93$	14330.6	+	964.312i	1.65691	+	0.111494i
$94$		0			0
$95$	−13708.0	−	2417.09i	−1.51889	−	0.267821i
$96$		0			0
$97$	2197.40	−	1843.84i	0.233542	−	0.195965i	−0.518505	−	0.855075i	$-0.673511\pi$
$97$	2197.40	−	1843.84i	0.233542	−	0.195965i	0.752047	+	0.659110i	$0.229067\pi$
$98$		0			0
$99$	1239.95	−	9171.70i	0.126513	−	0.935793i
$100$		0			0
$101$	573.994	−	1577.03i	0.0562684	−	0.154596i	−0.908374	−	0.418159i	$-0.862675\pi$
$101$	573.994	−	1577.03i	0.0562684	−	0.154596i	0.964642	+	0.263563i	$0.0848977\pi$
$102$		0			0
$103$	−11083.9	−	9300.48i	−1.04476	−	0.876659i	−0.0522289	−	0.998635i	$-0.516633\pi$
$103$	−11083.9	−	9300.48i	−1.04476	−	0.876659i	−0.992533	+	0.121976i	$0.961077\pi$
$104$		0			0
$105$	−6161.71	+	663.974i	−0.558886	+	0.0602244i
$106$		0			0
$107$			6102.13i			0.532983i	0.963837	+	0.266492i	$0.0858645\pi$
$107$			6102.13i			0.532983i	−0.963837	+	0.266492i	$0.914136\pi$
$108$		0			0
$109$	−19211.9			−1.61703			−0.808513	−	0.588479i	$-0.799727\pi$
$109$	−19211.9			−1.61703			−0.808513	+	0.588479i	$0.799727\pi$
$110$		0			0
$111$	−6020.93	+	4398.65i	−0.488672	+	0.357004i
$112$		0			0
$113$	176.048	−	209.806i	0.0137871	−	0.0164309i	−0.759107	−	0.650966i	$-0.774364\pi$
$113$	176.048	−	209.806i	0.0137871	−	0.0164309i	0.772894	+	0.634535i	$0.218808\pi$
$114$		0			0
$115$	−13893.3	−	5056.76i	−1.05054	−	0.382364i
$116$		0			0
$117$	−168.515	+	106.534i	−0.0123103	+	0.00778244i
$118$		0			0
$119$	−3773.73	−	4497.36i	−0.266488	−	0.317588i
$120$		0			0
$121$	275.306	−	1561.34i	0.0188037	−	0.106641i
$122$		0			0
$123$	−18224.6	−	12223.2i	−1.20461	−	0.807931i
$124$		0			0
$125$	11121.0	+	6420.74i	0.711747	+	0.410927i
$126$		0			0
$127$	14641.1	+	25359.1i	0.907749	+	1.57227i	0.817183	+	0.576378i	$0.195534\pi$
$127$	14641.1	+	25359.1i	0.907749	+	1.57227i	0.0905660	+	0.995890i	$0.471132\pi$
$128$		0			0
$129$	14613.5	−	4231.88i	0.878162	−	0.254305i
$130$		0			0
$131$	−8692.53	−	23882.5i	−0.506528	−	1.39167i	−0.884796	−	0.465979i	$-0.845702\pi$
$131$	−8692.53	−	23882.5i	−0.506528	−	1.39167i	0.378268	−	0.925696i	$-0.376520\pi$
$132$		0			0
$133$	−2095.21	−	11882.5i	−0.118447	−	0.671747i
$134$		0			0
$135$	−19110.6	−	7546.73i	−1.04859	−	0.414087i
$136$		0			0
$137$	−12879.6	+	2271.01i	−0.686214	+	0.120998i	−0.505875	−	0.862607i	$-0.668830\pi$
$137$	−12879.6	+	2271.01i	−0.686214	+	0.120998i	−0.180339	+	0.983605i	$0.557719\pi$
$138$		0			0
$139$	8423.60	−	3065.94i	0.435982	−	0.158684i	−0.114699	−	0.993400i	$-0.536590\pi$
$139$	8423.60	−	3065.94i	0.435982	−	0.158684i	0.550681	+	0.834716i	$0.314368\pi$
$140$		0			0
$141$	6912.04	−	7195.31i	0.347670	−	0.361919i
$142$		0			0
$143$	243.554	−	140.616i	0.0119103	−	0.00687642i
$144$		0			0
$145$	−1373.43	+	2378.85i	−0.0653236	+	0.113144i
$146$		0			0
$147$	7156.06	+	14574.9i	0.331161	+	0.674483i
$148$		0			0
$149$	2532.71	+	446.586i	0.114081	+	0.0201156i	0.230397	−	0.973097i	$-0.425997\pi$
$149$	2532.71	+	446.586i	0.114081	+	0.0201156i	−0.116316	+	0.993212i	$0.537109\pi$
$150$		0			0
$151$	27267.3	−	22880.0i	1.19588	−	1.00347i	0.196146	−	0.980575i	$-0.437157\pi$
$151$	27267.3	−	22880.0i	1.19588	−	1.00347i	0.999738	−	0.0228908i	$-0.00728701\pi$
$152$		0			0
$153$	−4146.71	−	19017.4i	−0.177142	−	0.812397i
$154$		0			0
$155$	−15384.0	+	42267.2i	−0.640334	+	1.75930i
$156$		0			0
$157$	6637.42	+	5569.46i	0.269277	+	0.225951i	0.767420	−	0.641144i	$-0.221540\pi$
$157$	6637.42	+	5569.46i	0.269277	+	0.225951i	−0.498143	+	0.867095i	$0.665984\pi$
$158$		0			0
$159$	−14119.7	+	31938.7i	−0.558511	+	1.26335i
$160$		0			0
$161$		−	12816.1i		−	0.494430i
$162$		0			0
$163$	44505.1			1.67508			0.837538	−	0.546380i	$-0.183994\pi$
$163$	44505.1			1.67508			0.837538	+	0.546380i	$0.183994\pi$
$164$		0			0
$165$	26508.9	+	11719.3i	0.973696	+	0.430459i
$166$		0			0
$167$	−20696.9	+	24665.6i	−0.742116	+	0.884420i	−0.996578	−	0.0826633i	$-0.973657\pi$
$167$	−20696.9	+	24665.6i	−0.742116	+	0.884420i	0.254461	+	0.967083i	$0.418102\pi$
$168$		0			0
$169$	26832.9	+	9766.37i	0.939493	+	0.341948i
$170$		0			0
$171$	12161.7	−	38109.4i	0.415914	−	1.30329i
$172$		0			0
$173$	19826.3	+	23628.1i	0.662445	+	0.789472i	0.987734	−	0.156143i	$-0.0499060\pi$
$173$	19826.3	+	23628.1i	0.662445	+	0.789472i	−0.325289	+	0.945615i	$0.605462\pi$
$174$		0			0
$175$	718.607	−	4075.42i	0.0234647	−	0.133075i
$176$		0			0
$177$	−30535.1	+	14992.3i	−0.974658	+	0.478543i
$178$		0			0
$179$	26159.6	+	15103.2i	0.816441	+	0.471372i	0.849187	−	0.528091i	$-0.177092\pi$
$179$	26159.6	+	15103.2i	0.816441	+	0.471372i	−0.0327469	+	0.999464i	$0.510426\pi$
$180$		0			0
$181$	26379.0	+	45689.8i	0.805196	+	1.39464i	0.916159	+	0.400815i	$0.131273\pi$
$181$	26379.0	+	45689.8i	0.805196	+	1.39464i	−0.110963	+	0.993824i	$0.535394\pi$
$182$		0			0
$183$	−12594.3	−	12098.4i	−0.376072	−	0.361266i
$184$		0			0
$185$	−7986.57	−	21942.9i	−0.233355	−	0.641137i
$186$		0			0
$187$	4767.83	+	27039.7i	0.136344	+	0.773248i
$188$		0			0
$189$	481.439	−	17804.1i	0.0134777	−	0.498421i
$190$		0			0
$191$	−43117.1	+	7602.71i	−1.18191	+	0.208402i	−0.729862	−	0.683594i	$-0.760416\pi$
$191$	−43117.1	+	7602.71i	−1.18191	+	0.208402i	−0.452043	+	0.891996i	$0.649305\pi$
$192$		0			0
$193$	−65952.9	+	24004.9i	−1.77060	+	0.644444i	−0.770621	+	0.637293i	$0.780054\pi$
$193$	−65952.9	+	24004.9i	−1.77060	+	0.644444i	−0.999974	+	0.00715076i	$0.997724\pi$
$194$		0			0
$195$	−173.667	−	599.705i	−0.00456718	−	0.0157713i
$196$		0			0
$197$	24057.2	−	13889.5i	0.619888	−	0.357893i	−0.156937	−	0.987609i	$-0.550162\pi$
$197$	24057.2	−	13889.5i	0.619888	−	0.357893i	0.776825	+	0.629716i	$0.216829\pi$
$198$		0			0
$199$	15512.8	−	26868.9i	0.391727	−	0.678491i	−0.600951	−	0.799286i	$-0.705211\pi$
$199$	15512.8	−	26868.9i	0.391727	−	0.678491i	0.992677	+	0.120795i	$0.0385445\pi$
$200$		0			0
$201$	4199.94	−	6262.05i	0.103956	−	0.154998i
$202$		0			0
$203$	−2344.90	−	413.469i	−0.0569025	−	0.0100335i
$204$		0			0
$205$	52643.4	−	44173.0i	1.25267	−	1.05111i
$206$		0			0
$207$	19750.9	−	37620.9i	0.460941	−	0.877989i
$208$		0			0
$209$	−19300.0	+	53026.2i	−0.441839	+	1.21394i
$210$		0			0
$211$	−19010.7	−	15951.9i	−0.427005	−	0.358300i	0.403815	−	0.914841i	$-0.367684\pi$
$211$	−19010.7	−	15951.9i	−0.427005	−	0.358300i	−0.830820	+	0.556541i	$0.812128\pi$
$212$		0			0
$213$	−12056.4	−	16502.9i	−0.265740	−	0.363749i
$214$		0			0
$215$			47644.6i			1.03071i
$216$		0			0
$217$	−38990.1			−0.828008
$218$		0			0
$219$	−5671.23	−	52629.3i	−0.118247	−	1.09734i
$220$		0			0
$221$	380.178	−	453.078i	0.00778399	−	0.00927659i
$222$		0			0
$223$	74226.5	+	27016.2i	1.49262	+	0.543269i	0.954138	−	0.299368i	$-0.0967760\pi$
$223$	74226.5	+	27016.2i	1.49262	+	0.543269i	0.538482	+	0.842637i	$0.318998\pi$
$224$		0			0
$225$	8390.04	−	10855.7i	0.165729	−	0.214434i
$226$		0			0
$227$	48220.6	+	57467.1i	0.935795	+	1.11524i	0.993146	+	0.116883i	$0.0372903\pi$
$227$	48220.6	+	57467.1i	0.935795	+	1.11524i	−0.0573506	+	0.998354i	$0.518265\pi$
$228$		0			0
$229$	−2505.13	+	14207.3i	−0.0477705	+	0.270920i	−0.999332	−	0.0365377i	$-0.988367\pi$
$229$	−2505.13	+	14207.3i	−0.0477705	+	0.270920i	0.951562	+	0.307458i	$0.0994782\pi$
$230$		0			0
$231$	−1686.80	+	25067.5i	−0.0316111	+	0.469771i
$232$		0			0
$233$	−21793.0	−	12582.2i	−0.401425	−	0.231763i	0.285674	−	0.958327i	$-0.407783\pi$
$233$	−21793.0	−	12582.2i	−0.401425	−	0.231763i	−0.687099	+	0.726564i	$0.741116\pi$
$234$		0			0
$235$	15622.9	+	27059.6i	0.282895	+	0.489988i
$236$		0			0
$237$	−12265.3	+	49749.1i	−0.218364	+	0.885704i
$238$		0			0
$239$	−12305.3	−	33808.6i	−0.215425	−	0.591876i	0.784163	−	0.620554i	$-0.213092\pi$
$239$	−12305.3	−	33808.6i	−0.215425	−	0.591876i	−0.999589	+	0.0286783i	$0.990870\pi$
$240$		0			0
$241$	−11393.0	−	64612.9i	−0.196157	−	1.11246i	−0.910762	−	0.412933i	$-0.864504\pi$
$241$	−11393.0	−	64612.9i	−0.196157	−	1.11246i	0.714605	−	0.699529i	$-0.246607\pi$
$242$		0			0
$243$	28851.0	−	51520.9i	0.488595	−	0.872511i
$244$		0			0
$245$	−50075.7	+	8829.70i	−0.834247	+	0.147100i
$246$		0			0
$247$	1142.24	−	415.743i	0.0187226	−	0.00681445i
$248$		0			0
$249$	−64626.8	−	15933.3i	−1.04235	−	0.256984i
$250$		0			0
$251$	35938.0	−	20748.8i	0.570435	−	0.329341i	−0.186888	−	0.982381i	$-0.559840\pi$
$251$	35938.0	−	20748.8i	0.570435	−	0.329341i	0.757323	+	0.653040i	$0.226507\pi$
$252$		0			0
$253$	−29969.1	+	51908.0i	−0.468201	+	0.810948i
$254$		0			0
$255$	60817.6	+	4092.44i	0.935295	+	0.0629364i
$256$		0			0
$257$	−67142.0	−	11838.9i	−1.01655	−	0.179245i	−0.359541	−	0.933129i	$-0.617067\pi$
$257$	−67142.0	−	11838.9i	−1.01655	−	0.179245i	−0.657007	+	0.753884i	$0.728178\pi$
$258$		0			0
$259$	15506.0	−	13011.0i	0.231153	−	0.193960i
$260$		0			0
$261$	−6246.11	−	4827.42i	−0.0916914	−	0.0708654i
$262$		0			0
$263$	−10711.9	+	29430.8i	−0.154866	+	0.425492i	−0.992726	−	0.120394i	$-0.961584\pi$
$263$	−10711.9	+	29430.8i	−0.154866	+	0.425492i	0.837860	+	0.545885i	$0.183807\pi$
$264$		0			0
$265$	−83773.7	−	70294.5i	−1.19293	−	1.00099i
$266$		0			0
$267$	−105286.	+	11345.4i	−1.47689	+	0.159147i
$268$		0			0
$269$		−	136677.i		−	1.88882i	−0.328766	−	0.944411i	$-0.606633\pi$
$269$		−	136677.i		−	1.88882i	0.328766	−	0.944411i	$-0.393367\pi$
$270$		0			0
$271$	122527.			1.66837			0.834187	−	0.551481i	$-0.185937\pi$
$271$	122527.			1.66837			0.834187	+	0.551481i	$0.185937\pi$
$272$		0			0
$273$	437.006	−	319.259i	0.00586357	−	0.00428369i
$274$		0			0
$275$	−12440.4	+	14825.9i	−0.164502	+	0.196046i
$276$		0			0
$277$	−39521.0	−	14384.5i	−0.515073	−	0.187471i	0.0713884	−	0.997449i	$-0.477257\pi$
$277$	−39521.0	−	14384.5i	−0.515073	−	0.187471i	−0.586461	+	0.809978i	$0.699479\pi$
$278$		0			0
$279$	−114453.	−	60087.4i	−1.47034	−	0.771925i
$280$		0			0
$281$	−36512.2	−	43513.5i	−0.462408	−	0.551076i	0.483571	−	0.875305i	$-0.339340\pi$
$281$	−36512.2	−	43513.5i	−0.462408	−	0.551076i	−0.945979	+	0.324229i	$0.894895\pi$
$282$		0			0
$283$	1441.88	−	8177.30i	0.0180035	−	0.102103i	−0.974482	−	0.224466i	$-0.927936\pi$
$283$	1441.88	−	8177.30i	0.0180035	−	0.102103i	0.992485	+	0.122363i	$0.0390473\pi$
$284$		0			0
$285$	104041.	+	69780.1i	1.28090	+	0.859096i
$286$		0			0
$287$	51588.9	+	29784.9i	0.626314	+	0.361603i
$288$		0			0
$289$	−12888.6	−	22323.7i	−0.154316	−	0.267283i
$290$		0			0
$291$	−24797.7	+	7181.09i	−0.292836	+	0.0848017i
$292$		0			0
$293$	−51954.9	−	142745.i	−0.605189	−	1.66274i	−0.740593	−	0.671954i	$-0.765455\pi$
$293$	−51954.9	−	142745.i	−0.605189	−	1.66274i	0.135404	−	0.990791i	$-0.456767\pi$
$294$		0			0
$295$	−18498.6	−	104911.i	−0.212567	−	1.20553i
$296$		0			0
$297$	−43582.8	+	70984.5i	−0.494086	+	0.804731i
$298$		0			0
$299$	1271.52	−	224.203i	0.0142227	−	0.00250784i
$300$		0			0
$301$	−38809.2	+	14125.4i	−0.428353	+	0.155908i
$302$		0			0
$303$	−10463.7	+	10892.5i	−0.113973	+	0.118644i
$304$		0			0
$305$	47363.5	−	27345.3i	0.509148	−	0.293957i
$306$		0			0
$307$	−90294.2	+	156394.i	−0.958039	+	1.65937i	−0.230783	+	0.973005i	$0.574129\pi$
$307$	−90294.2	+	156394.i	−0.958039	+	1.65937i	−0.727255	+	0.686367i	$0.759204\pi$
$308$		0			0
$309$	57392.0	+	116891.i	0.601083	+	1.22424i
$310$		0			0
$311$	71897.1	+	12677.4i	0.743345	+	0.131072i	0.532479	−	0.846443i	$-0.321260\pi$
$311$	71897.1	+	12677.4i	0.743345	+	0.131072i	0.210866	+	0.977515i	$0.432372\pi$
$312$		0			0
$313$	−74410.2	+	62437.5i	−0.759528	+	0.637319i	−0.938004	−	0.346625i	$-0.887328\pi$
$313$	−74410.2	+	62437.5i	−0.759528	+	0.637319i	0.178476	+	0.983944i	$0.442883\pi$
$314$		0			0
$315$	53136.3	+	16957.2i	0.535513	+	0.170897i
$316$		0			0
$317$	27314.7	−	75046.5i	0.271818	−	0.746813i	−0.726408	−	0.687264i	$-0.758812\pi$
$317$	27314.7	−	75046.5i	0.271818	−	0.746813i	0.998225	−	0.0595488i	$-0.0189662\pi$
$318$		0			0
$319$	8530.48	+	7157.92i	0.0838286	+	0.0703405i
$320$		0			0
$321$	22205.9	−	50229.6i	0.215505	−	0.487472i
$322$		0			0
$323$			118675.i			1.13751i
$324$		0			0
$325$	416.905			0.00394703
$326$		0			0
$327$	158142.	+	69912.8i	1.47895	+	0.653825i
$328$		0			0
$329$	−17409.8	+	20748.2i	−0.160843	+	0.191685i
$330$		0			0
$331$	−51094.6	−	18596.9i	−0.466357	−	0.169740i	0.0981442	−	0.995172i	$-0.468709\pi$
$331$	−51094.6	−	18596.9i	−0.466357	−	0.169740i	−0.564502	+	0.825432i	$0.690932\pi$
$332$		0			0
$333$	65568.0	−	14297.0i	0.591294	−	0.128931i
$334$		0			0
$335$	15178.1	+	18088.5i	0.135247	+	0.161181i
$336$		0			0
$337$	3454.56	−	19591.8i	0.0304181	−	0.172510i	−0.965814	−	0.259236i	$-0.916529\pi$
$337$	3454.56	−	19591.8i	0.0304181	−	0.172510i	0.996232	+	0.0867260i	$0.0276405\pi$
$338$		0			0
$339$	−2212.63	+	1086.37i	−0.0192535	+	0.00945317i
$340$		0			0
$341$	157918.	+	91174.0i	1.35807	+	0.784083i
$342$		0			0
$343$	−51368.5	−	88972.9i	−0.436625	−	0.756257i
$344$		0			0
$345$	95961.0	+	92183.1i	0.806226	+	0.774485i
$346$		0			0
$347$	−19743.2	−	54243.9i	−0.163967	−	0.450497i	0.830313	−	0.557297i	$-0.188162\pi$
$347$	−19743.2	−	54243.9i	−0.163967	−	0.450497i	−0.994280	+	0.106801i	$0.965939\pi$
$348$		0			0
$349$	4802.56	+	27236.6i	0.0394295	+	0.223616i	0.998155	−	0.0607175i	$-0.0193389\pi$
$349$	4802.56	+	27236.6i	0.0394295	+	0.223616i	−0.958725	+	0.284333i	$0.908228\pi$
$350$		0			0
$351$	1774.81	−	263.698i	0.0144058	−	0.00214039i
$352$		0			0
$353$	163139.	−	28765.8i	1.30921	−	0.230848i	0.524869	−	0.851183i	$-0.324114\pi$
$353$	163139.	−	28765.8i	1.30921	−	0.230848i	0.784337	+	0.620334i	$0.213003\pi$
$354$		0			0
$355$	60143.9	−	21890.6i	0.477238	−	0.173700i
$356$		0			0
$357$	14697.3	+	50752.7i	0.115319	+	0.398220i
$358$		0			0
$359$	20895.8	−	12064.2i	0.162133	−	0.0936074i	−0.416738	−	0.909027i	$-0.636827\pi$
$359$	20895.8	−	12064.2i	0.162133	−	0.0936074i	0.578871	+	0.815419i	$0.303493\pi$
$360$		0			0
$361$	−56789.9	+	98363.1i	−0.435770	+	0.754775i
$362$		0			0
$363$	−7947.94	+	11850.3i	−0.0603172	+	0.0899321i
$364$		0			0
$365$	163252.	+	28785.7i	1.22539	+	0.216069i
$366$		0			0
$367$	7464.93	−	6263.82i	0.0554235	−	0.0465058i	−0.614655	−	0.788796i	$-0.710705\pi$
$367$	7464.93	−	6263.82i	0.0554235	−	0.0465058i	0.670079	+	0.742290i	$0.266260\pi$
$368$		0			0
$369$	105535.	+	166935.i	0.775074	+	1.22601i
$370$		0			0
$371$	32422.1	−	89079.0i	0.235556	−	0.647184i
$372$		0			0
$373$	19173.2	+	16088.2i	0.137809	+	0.115635i	0.709086	−	0.705122i	$-0.249108\pi$
$373$	19173.2	+	16088.2i	0.137809	+	0.115635i	−0.571278	+	0.820757i	$0.693552\pi$
$374$		0			0
$375$	−68177.4	−	93322.1i	−0.484817	−	0.663624i
$376$		0			0
$377$		−	239.877i		−	0.00168774i
$378$		0			0
$379$	−57436.8			−0.399863			−0.199932	−	0.979810i	$-0.564072\pi$
$379$	−57436.8			−0.399863			−0.199932	+	0.979810i	$0.564072\pi$
$380$		0			0
$381$	−28235.1	−	262023.i	−0.194509	−	1.80505i
$382$		0			0
$383$	−82991.0	+	98904.8i	−0.565762	+	0.674249i	−0.970755	−	0.240071i	$-0.922829\pi$
$383$	−82991.0	+	98904.8i	−0.565762	+	0.674249i	0.404994	+	0.914319i	$0.367274\pi$
$384$		0			0
$385$	−73935.0	−	26910.1i	−0.498802	−	0.181549i
$386$		0			0
$387$	−135691.	−	18344.4i	−0.906001	−	0.122485i
$388$		0			0
$389$	24505.0	+	29203.9i	0.161941	+	0.192993i	0.840913	−	0.541171i	$-0.182019\pi$
$389$	24505.0	+	29203.9i	0.161941	+	0.192993i	−0.678972	+	0.734164i	$0.737574\pi$
$390$		0			0
$391$	−21889.1	+	124139.i	−0.143177	+	0.812000i
$392$		0			0
$393$	−15357.1	+	228221.i	−0.0994315	+	1.47765i
$394$		0			0
$395$	−138964.	−	80230.8i	−0.890651	−	0.514218i
$396$		0			0
$397$	5111.47	+	8853.33i	0.0324313	+	0.0561727i	0.881786	−	0.471651i	$-0.156342\pi$
$397$	5111.47	+	8853.33i	0.0324313	+	0.0561727i	−0.849354	+	0.527823i	$0.823008\pi$
$398$		0			0
$399$	−25994.3	+	105435.i	−0.163280	+	0.662279i
$400$		0			0
$401$	−92985.8	−	255477.i	−0.578267	−	1.58877i	−0.791101	−	0.611686i	$-0.790492\pi$
$401$	−92985.8	−	255477.i	−0.578267	−	1.58877i	0.212834	−	0.977088i	$-0.431731\pi$
$402$		0			0
$403$	−682.087	−	3868.31i	−0.00419981	−	0.0238183i
$404$		0			0
$405$	129846.	+	131665.i	0.791621	+	0.802713i
$406$		0			0
$407$	−93227.3	+	16438.5i	−0.562800	+	0.0992368i
$408$		0			0
$409$	−236318.	+	86012.7i	−1.41270	+	0.514181i	−0.931921	−	0.362661i	$-0.881868\pi$
$409$	−236318.	+	86012.7i	−1.41270	+	0.514181i	−0.480779	+	0.876842i	$0.659646\pi$
$410$		0			0
$411$	114282.	+	28175.4i	0.676542	+	0.166796i
$412$		0			0
$413$	79971.5	−	46171.6i	0.468851	−	0.270692i
$414$		0			0
$415$	104224.	−	180522.i	0.605163	−	1.04817i
$416$		0			0
$417$	−80495.9	−	5416.60i	−0.462915	−	0.0311497i
$418$		0			0
$419$	226077.	+	39863.4i	1.28774	+	0.227063i	0.775263	−	0.631638i	$-0.217617\pi$
$419$	226077.	+	39863.4i	1.28774	+	0.227063i	0.512476	+	0.858702i	$0.328728\pi$
$420$		0			0
$421$	34300.2	−	28781.3i	0.193523	−	0.162385i	−0.540878	−	0.841101i	$-0.681908\pi$
$421$	34300.2	−	28781.3i	0.193523	−	0.162385i	0.734401	+	0.678716i	$0.237463\pi$
$422$		0			0
$423$	−83080.4	+	34074.9i	−0.464320	+	0.190438i
$424$		0			0
$425$	−13921.1	+	38248.0i	−0.0770720	+	0.211754i
$426$		0			0
$427$	36316.4	+	30473.1i	0.199181	+	0.167133i
$428$		0			0
$429$	−2516.52	+	271.175i	−0.0136737	+	0.00147345i
$430$		0			0
$431$		−	128951.i		−	0.694176i	−0.937833	−	0.347088i	$-0.887170\pi$
$431$		−	128951.i		−	0.694176i	0.937833	−	0.347088i	$-0.112830\pi$
$432$		0			0
$433$	68399.9			0.364821			0.182410	−	0.983222i	$-0.441610\pi$
$433$	68399.9			0.364821			0.182410	+	0.983222i	$0.441610\pi$
$434$		0			0
$435$	19962.1	−	14583.5i	0.105494	−	0.0770696i
$436$		0			0
$437$	−166525.	+	198457.i	−0.872001	+	1.03921i
$438$		0			0
$439$	6048.33	+	2201.41i	0.0313839	+	0.0114228i	0.357664	−	0.933850i	$-0.383573\pi$
$439$	6048.33	+	2201.41i	0.0313839	+	0.0114228i	−0.326281	+	0.945273i	$0.605795\pi$
$440$		0			0
$441$	−5866.32	−	146014.i	−0.0301640	−	0.750789i
$442$		0			0
$443$	−181179.	−	215921.i	−0.923212	−	1.10024i	−0.994702	−	0.102800i	$-0.967220\pi$
$443$	−181179.	−	215921.i	−0.923212	−	1.10024i	0.0714898	−	0.997441i	$-0.477225\pi$
$444$		0			0
$445$	57586.6	−	326590.i	0.290805	−	1.64924i
$446$		0			0
$447$	−19222.8	−	12892.7i	−0.0962061	−	0.0645252i
$448$		0			0
$449$	−250048.	−	144365.i	−1.24031	−	0.716094i	−0.271154	−	0.962536i	$-0.587405\pi$
$449$	−250048.	−	144365.i	−1.24031	−	0.716094i	−0.969158	+	0.246442i	$0.920739\pi$
$450$		0			0
$451$	−139297.	−	241270.i	−0.684841	−	1.18618i
$452$		0			0
$453$	−307712.	+	89109.6i	−1.49951	+	0.434238i
$454$		0			0
$455$	579.675	+	1592.64i	0.00280002	+	0.00769300i
$456$		0			0
$457$	56283.2	+	319198.i	0.269492	+	1.52837i	0.755930	+	0.654652i	$0.227185\pi$
$457$	56283.2	+	319198.i	0.269492	+	1.52837i	−0.486438	+	0.873715i	$0.661704\pi$
$458$		0			0
$459$	−35071.6	+	171632.i	−0.166468	+	0.814651i
$460$		0			0
$461$	254561.	−	44886.0i	1.19782	−	0.211207i	0.461062	−	0.887368i	$-0.347469\pi$
$461$	254561.	−	44886.0i	1.19782	−	0.211207i	0.736754	+	0.676161i	$0.236358\pi$
$462$		0			0
$463$	249634.	−	90859.5i	1.16451	−	0.423846i	0.313802	−	0.949488i	$-0.398397\pi$
$463$	249634.	−	90859.5i	1.16451	−	0.423846i	0.850706	+	0.525642i	$0.176175\pi$
$464$		0			0
$465$	280446.	−	291939.i	1.29701	−	1.35016i
$466$		0			0
$467$	−234369.	+	135313.i	−1.07465	+	0.620448i	−0.929448	−	0.368954i	$-0.879716\pi$
$467$	−234369.	+	135313.i	−1.07465	+	0.620448i	−0.145200	+	0.989402i	$0.546383\pi$
$468$		0			0
$469$	−10234.2	+	17726.2i	−0.0465274	+	0.0805878i
$470$		0			0
$471$	−34368.3	−	69998.7i	−0.154923	−	0.315536i
$472$		0			0
$473$	190216.	+	33540.3i	0.850209	+	0.149915i
$474$		0			0
$475$	−64081.2	+	53770.5i	−0.284017	+	0.238318i
$476$		0			0
$477$	232452.	−	211521.i	1.02164	−	0.929643i
$478$		0			0
$479$	−43361.6	+	119135.i	−0.188988	+	0.519240i	−0.997611	−	0.0690874i	$-0.977991\pi$
$479$	−43361.6	+	119135.i	−0.188988	+	0.519240i	0.808623	+	0.588328i	$0.200213\pi$
$480$		0			0
$481$	1562.12	+	1310.77i	0.00675187	+	0.00566549i
$482$		0			0
$483$	−46638.4	+	105496.i	−0.199917	+	0.452210i
$484$		0			0
$485$		−	80848.2i		−	0.343706i
$486$		0			0
$487$	−99411.6			−0.419159			−0.209580	−	0.977792i	$-0.567210\pi$
$487$	−99411.6			−0.419159			−0.209580	+	0.977792i	$0.567210\pi$
$488$		0			0
$489$	−366343.	−	161956.i	−1.53204	−	0.677297i
$490$		0			0
$491$	59612.1	−	71043.0i	0.247270	−	0.294685i	−0.628106	−	0.778128i	$-0.716169\pi$
$491$	59612.1	−	71043.0i	0.247270	−	0.294685i	0.875376	+	0.483443i	$0.160614\pi$
$492$		0			0
$493$	22006.9	+	8009.87i	0.0905453	+	0.0329558i
$494$		0			0
$495$	−175561.	−	192934.i	−0.716501	−	0.787405i
$496$		0			0
$497$	35662.3	+	42500.7i	0.144376	+	0.172061i
$498$		0			0
$499$	−58901.1	+	334045.i	−0.236550	+	1.34154i	0.602776	+	0.797910i	$0.294061\pi$
$499$	−58901.1	+	334045.i	−0.236550	+	1.34154i	−0.839326	+	0.543629i	$0.817050\pi$
$500$		0			0
$501$	260125.	−	127718.i	1.03635	−	0.508833i
$502$		0			0
$503$	−22048.1	−	12729.5i	−0.0871435	−	0.0503123i	0.455795	−	0.890085i	$-0.349355\pi$
$503$	−22048.1	−	12729.5i	−0.0871435	−	0.0503123i	−0.542939	+	0.839772i	$0.682688\pi$
$504$		0			0
$505$	−23650.5	−	40963.9i	−0.0927380	−	0.160627i
$506$		0			0
$507$	−185334.	−	178038.i	−0.721007	−	0.692621i
$508$		0			0
$509$	149571.	+	410944.i	0.577315	+	1.58616i	0.792688	+	0.609628i	$0.208681\pi$
$509$	149571.	+	410944.i	0.577315	+	1.58616i	−0.215373	+	0.976532i	$0.569097\pi$
$510$		0			0
$511$	24952.5	+	141512.i	0.0955590	+	0.541942i
$512$		0			0
$513$	−238791.	+	269440.i	−0.907367	+	1.02383i
$514$		0			0
$515$	−401610.	+	70814.6i	−1.51422	+	0.266998i
$516$		0			0
$517$	119031.	−	43323.7i	0.445326	−	0.162085i
$518$		0			0
$519$	−77216.5	−	266643.i	−0.286666	−	0.989910i
$520$		0			0
$521$	−389228.	+	224721.i	−1.43393	+	0.827882i	−0.997418	−	0.0718152i	$-0.977121\pi$
$521$	−389228.	+	224721.i	−1.43393	+	0.827882i	−0.436515	+	0.899697i	$0.643787\pi$
$522$		0			0
$523$	179231.	−	310437.i	0.655254	−	1.13493i	−0.326576	−	0.945171i	$-0.605895\pi$
$523$	179231.	−	310437.i	0.655254	−	1.13493i	0.981830	−	0.189762i	$-0.0607717\pi$
$524$		0			0
$525$	−20745.8	+	30931.7i	−0.0752683	+	0.112224i
$526$		0			0
$527$	377665.	+	66592.6i	1.35983	+	0.239775i
$528$		0			0
$529$	3573.37	−	2998.41i	0.0127693	−	0.0107147i
$530$		0			0
$531$	305906.	−	12290.2i	1.08492	−	0.0435883i
$532$		0			0
$533$	−2052.55	+	5639.32i	−0.00722501	+	0.0198506i
$534$		0			0
$535$	131750.	+	110551.i	0.460302	+	0.386239i
$536$		0			0
$537$	−160371.	−	219518.i	−0.556131	−	0.761240i
$538$		0			0
$539$			206138.i			0.709547i
$540$		0			0
$541$	116585.			0.398333			0.199167	−	0.979966i	$-0.436177\pi$
$541$	116585.			0.398333			0.199167	+	0.979966i	$0.436177\pi$
$542$		0			0
$543$	−50871.4	−	472089.i	−0.172534	−	1.60112i
$544$		0			0
$545$	−348059.	+	414800.i	−1.17182	+	1.39652i
$546$		0			0
$547$	−51478.9	−	18736.8i	−0.172050	−	0.0626210i	0.254559	−	0.967057i	$-0.418070\pi$
$547$	−51478.9	−	18736.8i	−0.172050	−	0.0626210i	−0.426609	+	0.904436i	$0.640292\pi$
$548$		0			0
$549$	59642.8	+	145419.i	0.197885	+	0.482477i
$550$		0			0
$551$	30938.2	+	36870.8i	0.101904	+	0.121445i
$552$		0			0
$553$	24153.4	−	136980.i	0.0789818	−	0.447928i
$554$		0			0
$555$	−14109.9	+	209686.i	−0.0458075	+	0.680744i
$556$		0			0
$557$	−30508.0	−	17613.8i	−0.0983339	−	0.0567731i	0.450027	−	0.893015i	$-0.351414\pi$
$557$	−30508.0	−	17613.8i	−0.0983339	−	0.0567731i	−0.548360	+	0.836242i	$0.684748\pi$
$558$		0			0
$559$	−2080.34	−	3603.26i	−0.00665750	−	0.0115311i
$560$		0			0
$561$	59152.3	−	239927.i	0.187952	−	0.762349i
$562$		0			0
$563$	−61925.1	−	170138.i	−0.195366	−	0.536765i	0.802868	−	0.596156i	$-0.203306\pi$
$563$	−61925.1	−	170138.i	−0.195366	−	0.536765i	−0.998235	+	0.0593917i	$0.981084\pi$
$564$		0			0
$565$	−1340.44	−	7602.04i	−0.00419906	−	0.0238140i
$566$		0			0
$567$	−68752.8	+	144802.i	−0.213857	+	0.450411i
$568$		0			0
$569$	368377.	−	64954.7i	1.13780	−	0.200626i	0.427159	−	0.904176i	$-0.359514\pi$
$569$	368377.	−	64954.7i	1.13780	−	0.200626i	0.710645	+	0.703551i	$0.248403\pi$
$570$		0			0
$571$	459788.	−	167349.i	1.41022	−	0.513277i	0.479023	−	0.877803i	$-0.340991\pi$
$571$	459788.	−	167349.i	1.41022	−	0.513277i	0.931193	+	0.364526i	$0.118769\pi$
$572$		0			0
$573$	382584.	+	94323.3i	1.16525	+	0.287283i
$574$		0			0
$575$	−76949.6	+	44426.9i	−0.232740	+	0.134372i
$576$		0			0
$577$	113291.	−	196225.i	0.340285	−	0.589390i	−0.644201	−	0.764856i	$-0.722810\pi$
$577$	113291.	−	196225.i	0.340285	−	0.589390i	0.984485	+	0.175466i	$0.0561433\pi$
$578$		0			0
$579$	630245.	+	42409.4i	1.87998	+	0.126504i
$580$		0			0
$581$	177945.	+	31376.5i	0.527149	+	0.0929507i
$582$		0			0
$583$	−339618.	+	284973.i	−0.999203	+	0.838431i
$584$		0			0
$585$	−752.814	+	5568.44i	−0.00219976	+	0.0162713i
$586$		0			0
$587$	136938.	−	376234.i	0.397418	−	1.09190i	−0.566120	−	0.824323i	$-0.691556\pi$
$587$	136938.	−	376234.i	0.397418	−	1.09190i	0.963538	−	0.267573i	$-0.0862217\pi$
$588$		0			0
$589$	603759.	+	506614.i	1.74034	+	1.46032i
$590$		0			0
$591$	−248571.	+	26785.5i	−0.711665	+	0.0766877i
$592$		0			0
$593$			332164.i			0.944590i	0.881441	+	0.472295i	$0.156574\pi$
$593$			332164.i			0.944590i	−0.881441	+	0.472295i	$0.843426\pi$
$594$		0			0
$595$	−165470.			−0.467396
$596$		0			0
$597$	−225470.	+	164719.i	−0.632617	+	0.462164i
$598$		0			0
$599$	−403032.	+	480315.i	−1.12327	+	1.33867i	−0.189056	+	0.981966i	$0.560543\pi$
$599$	−403032.	+	480315.i	−1.12327	+	1.33867i	−0.934219	+	0.356700i	$0.883902\pi$
$600$		0			0
$601$	565600.	+	205862.i	1.56589	+	0.569936i	0.972076	−	0.234668i	$-0.0754003\pi$
$601$	565600.	+	205862.i	1.56589	+	0.569936i	0.593811	+	0.804604i	$0.297623\pi$
$602$		0			0
$603$	−57359.7	+	36262.2i	−0.157751	+	0.0997287i
$604$		0			0
$605$	−28722.8	−	34230.6i	−0.0784724	−	0.0935197i
$606$		0			0
$607$	−72553.0	+	411469.i	−0.196915	+	1.11676i	0.712751	+	0.701417i	$0.247449\pi$
$607$	−72553.0	+	411469.i	−0.196915	+	1.11676i	−0.909665	+	0.415342i	$0.863662\pi$
$608$		0			0
$609$	17797.4	+	11936.6i	0.0479867	+	0.0321846i
$610$		0			0
$611$	−2363.05	−	1364.31i	−0.00632980	−	0.00365451i
$612$		0			0
$613$	165625.	+	286871.i	0.440764	+	0.763425i	0.997746	−	0.0670991i	$-0.0213744\pi$
$613$	165625.	+	286871.i	0.440764	+	0.763425i	−0.556983	+	0.830524i	$0.688041\pi$
$614$		0			0
$615$	−594081.	+	172038.i	−1.57071	+	0.454857i
$616$		0			0
$617$	−182752.	−	502108.i	−0.480057	−	1.31895i	−0.909444	−	0.415826i	$-0.863493\pi$
$617$	−182752.	−	502108.i	−0.480057	−	1.31895i	0.429387	−	0.903120i	$-0.358730\pi$
$618$		0			0
$619$	18382.4	+	104252.i	0.0479756	+	0.272083i	0.999354	−	0.0359418i	$-0.0114431\pi$
$619$	18382.4	+	104252.i	0.0479756	+	0.272083i	−0.951378	+	0.308025i	$0.900332\pi$
$620$		0			0
$621$	−299483.	+	237802.i	−0.776585	+	0.616641i
$622$		0			0
$623$	283099.	−	49918.0i	0.729395	−	0.128612i
$624$		0			0
$625$	439587.	−	159997.i	1.12534	−	0.409591i
$626$		0			0
$627$	351832.	−	366251.i	0.894953	−	0.931631i
$628$		0			0
$629$	−172416.	+	99544.2i	−0.435788	+	0.251602i
$630$		0			0
$631$	288702.	−	500047.i	0.725089	−	1.25589i	−0.233848	−	0.972273i	$-0.575132\pi$
$631$	288702.	−	500047.i	0.725089	−	1.25589i	0.958937	−	0.283618i	$-0.0915348\pi$
$632$		0			0
$633$	98436.7	+	200488.i	0.245669	+	0.500359i
$634$		0			0
$635$	812774.	+	143314.i	2.01568	+	0.355419i
$636$		0			0
$637$	3401.58	−	2854.27i	0.00838305	−	0.00703422i
$638$		0			0
$639$	39187.0	+	179717.i	0.0959710	+	0.440137i
$640$		0			0
$641$	181206.	−	497859.i	0.441018	−	1.21169i	−0.497806	−	0.867288i	$-0.665861\pi$
$641$	181206.	−	497859.i	0.441018	−	1.21169i	0.938824	−	0.344397i	$-0.111917\pi$
$642$		0			0
$643$	68330.7	+	57336.3i	0.165270	+	0.138678i	0.721672	−	0.692235i	$-0.243374\pi$
$643$	68330.7	+	57336.3i	0.165270	+	0.138678i	−0.556402	+	0.830913i	$0.687818\pi$
$644$		0			0
$645$	173381.	−	392186.i	0.416755	−	0.942697i
$646$		0			0
$647$			690108.i			1.64857i	0.566172	+	0.824287i	$0.308424\pi$
$647$			690108.i			1.64857i	−0.566172	+	0.824287i	$0.691576\pi$
$648$		0			0
$649$	−431869.			−1.02533
$650$		0			0
$651$	320946.	+	141886.i	0.757304	+	0.334795i
$652$		0			0
$653$	−360793.	+	429977.i	−0.846121	+	1.00837i	0.153674	+	0.988122i	$0.450889\pi$
$653$	−360793.	+	429977.i	−0.846121	+	1.00837i	−0.999795	+	0.0202458i	$0.993555\pi$
$654$		0			0
$655$	−673125.	−	244997.i	−1.56896	−	0.571056i
$656$		0			0
$657$	−144838.	+	453855.i	−0.335545	+	1.05145i
$658$		0			0
$659$	−470549.	−	560778.i	−1.08351	−	1.29128i	−0.954034	−	0.299698i	$-0.903114\pi$
$659$	−470549.	−	560778.i	−1.08351	−	1.29128i	−0.129479	−	0.991582i	$-0.541330\pi$
$660$		0			0
$661$	118352.	−	671209.i	0.270878	−	1.53623i	−0.480880	−	0.876786i	$-0.659683\pi$
$661$	118352.	−	671209.i	0.270878	−	1.53623i	0.751758	−	0.659439i	$-0.229206\pi$
$662$		0			0
$663$	−4778.20	+	2346.02i	−0.0108702	+	0.00533710i
$664$		0			0
$665$	−294512.	−	170037.i	−0.665978	−	0.384503i
$666$		0			0
$667$	25562.1	+	44274.9i	0.0574573	+	0.0995190i
$668$		0			0
$669$	−512681.	−	492497.i	−1.14550	−	1.10040i
$670$		0			0
$671$	−75831.2	−	208345.i	−0.168424	−	0.462740i
$672$		0			0
$673$	−50789.8	−	288043.i	−0.112136	−	0.635957i	−0.988128	−	0.153631i	$-0.950903\pi$
$673$	−50789.8	−	288043.i	−0.112136	−	0.635957i	0.875992	−	0.482326i	$-0.160208\pi$
$674$		0			0
$675$	−108567.	+	58827.0i	−0.238281	+	0.129113i
$676$		0			0
$677$	−547224.	+	96490.4i	−1.19396	+	0.210527i	−0.735084	−	0.677976i	$-0.762857\pi$
$677$	−547224.	+	96490.4i	−1.19396	+	0.210527i	−0.458871	+	0.888503i	$0.651746\pi$
$678$		0			0
$679$	65855.5	−	23969.4i	0.142841	−	0.0519898i
$680$		0			0
$681$	−187802.	−	648516.i	−0.404954	−	1.39838i
$682$		0			0
$683$	−8476.38	+	4893.84i	−0.0181706	+	0.0104908i	−0.509058	−	0.860732i	$-0.670006\pi$
$683$	−8476.38	+	4893.84i	−0.0181706	+	0.0104908i	0.490887	+	0.871223i	$0.336673\pi$
$684$		0			0
$685$	−184304.	+	319224.i	−0.392783	+	0.680321i
$686$		0			0
$687$	72321.9	−	107831.i	0.153235	−	0.228470i
$688$		0			0
$689$	9404.96	+	1658.35i	0.0198115	+	0.00349331i
$690$		0			0
$691$	−604854.	+	507533.i	−1.26676	+	1.06294i	−0.271833	+	0.962344i	$0.587630\pi$
$691$	−604854.	+	507533.i	−1.26676	+	1.06294i	−0.994928	+	0.100594i	$0.967926\pi$
$692$		0			0
$693$	105106.	−	200204.i	0.218858	−	0.416876i
$694$		0			0
$695$	86412.9	−	237418.i	0.178900	−	0.491522i
$696$		0			0
$697$	−448829.	−	376612.i	−0.923880	−	0.775227i
$698$		0			0
$699$	133601.	+	182876.i	0.273437	+	0.374284i
$700$		0			0
$701$		−	765720.i		−	1.55824i	−0.626876	−	0.779119i	$-0.715667\pi$
$701$		−	765720.i		−	1.55824i	0.626876	−	0.779119i	$-0.284333\pi$
$702$		0			0
$703$	−409167.			−0.827922
$704$		0			0
$705$	−30128.4	−	279593.i	−0.0606174	−	0.562533i
$706$		0			0
$707$	26355.6	−	31409.4i	0.0527272	−	0.0628379i
$708$		0			0
$709$	−132595.	−	48260.7i	−0.263776	−	0.0960066i	0.206747	−	0.978394i	$-0.433712\pi$
$709$	−132595.	−	48260.7i	−0.263776	−	0.0960066i	−0.470523	+	0.882388i	$0.655935\pi$
$710$		0			0
$711$	282001.	−	364875.i	0.557842	−	0.721781i
$712$		0			0
$713$	538116.	+	641302.i	1.05851	+	1.26149i
$714$		0			0
$715$	1376.42	−	7806.05i	0.00269239	−	0.0152693i
$716$		0			0
$717$	−21739.8	+	323074.i	−0.0422880	+	0.628440i
$718$		0			0
$719$	76955.1	+	44430.1i	0.148861	+	0.0859447i	0.572580	−	0.819849i	$-0.305942\pi$
$719$	76955.1	+	44430.1i	0.148861	+	0.0859447i	−0.423720	+	0.905793i	$0.639276\pi$
$720$		0			0
$721$	−176750.	−	306140.i	−0.340007	−	0.588910i
$722$		0			0
$723$	−141348.	+	573319.i	−0.270403	+	1.09678i
$724$		0			0
$725$	5646.03	+	15512.3i	0.0107416	+	0.0295122i
$726$		0			0
$727$	−2153.14	−	12211.1i	−0.00407384	−	0.0231039i	0.982703	−	0.185188i	$-0.0592895\pi$
$727$	−2153.14	−	12211.1i	−0.00407384	−	0.0231039i	−0.986777	+	0.162084i	$0.948178\pi$
$728$		0			0
$729$	−424974.	+	319103.i	−0.799663	+	0.600449i
$730$		0			0
$731$	400039.	−	70537.7i	0.748631	−	0.132004i
$732$		0			0
$733$	−405994.	+	147770.i	−0.755634	+	0.275028i	−0.690975	−	0.722879i	$-0.742818\pi$
$733$	−405994.	+	147770.i	−0.755634	+	0.275028i	−0.0646595	+	0.997907i	$0.520596\pi$
$734$		0			0
$735$	444329.	+	109546.i	0.822489	+	0.202779i
$736$		0			0
$737$	82901.5	−	47863.2i	0.152626	−	0.0881184i
$738$		0			0
$739$	95170.2	−	164840.i	0.174266	−	0.301837i	−0.765641	−	0.643268i	$-0.777578\pi$
$739$	95170.2	−	164840.i	0.174266	−	0.301837i	0.939907	+	0.341431i	$0.110911\pi$
$740$		0			0
$741$	−10915.3	−	734.493i	−0.0198792	−	0.00133768i
$742$		0			0
$743$	920487.	+	162307.i	1.66740	+	0.294008i	0.926132	−	0.377200i	$-0.123113\pi$
$743$	920487.	+	162307.i	1.66740	+	0.294008i	0.741269	+	0.671208i	$0.234224\pi$
$744$		0			0
$745$	55526.9	−	46592.6i	0.100044	−	0.0839469i
$746$		0			0
$747$	473993.	+	366334.i	0.849436	+	0.656502i
$748$		0			0
$749$	−50989.9	+	140093.i	−0.0908908	+	0.249721i
$750$		0			0
$751$	193215.	+	162126.i	0.342578	+	0.287457i	0.797802	−	0.602920i	$-0.205996\pi$
$751$	193215.	+	162126.i	0.342578	+	0.287457i	−0.455223	+	0.890377i	$0.650441\pi$
$752$		0			0
$753$	−371329.	+	40013.7i	−0.654890	+	0.0705697i
$754$		0			0
$755$		−	1.00324e6i		−	1.75999i
$756$		0			0
$757$	−449009.			−0.783544			−0.391772	−	0.920062i	$-0.628138\pi$
$757$	−449009.			−0.783544			−0.391772	+	0.920062i	$0.628138\pi$
$758$		0			0
$759$	435586.	−	318221.i	0.756119	−	0.552390i
$760$		0			0
$761$	−380749.	+	453759.i	−0.657460	+	0.783530i	−0.987019	−	0.160605i	$-0.948656\pi$
$761$	−380749.	+	453759.i	−0.657460	+	0.783530i	0.329559	+	0.944135i	$0.393100\pi$
$762$		0			0
$763$	−441069.	−	160536.i	−0.757631	−	0.275755i
$764$		0			0
$765$	−485726.	−	255005.i	−0.829982	−	0.435738i
$766$		0			0
$767$	5979.82	+	7126.47i	0.0101648	+	0.0121139i
$768$		0			0
$769$	−35286.0	+	200117.i	−0.0596692	+	0.338401i	−0.999998	−	0.00186555i	$-0.999406\pi$
$769$	−35286.0	+	200117.i	−0.0596692	+	0.338401i	0.940329	+	0.340267i	$0.110517\pi$
$770$		0			0
$771$	509596.	+	341785.i	0.857269	+	0.574968i
$772$		0			0
$773$	−93598.2	−	54039.0i	−0.156642	−	0.0904374i	0.419630	−	0.907695i	$-0.362160\pi$
$773$	−93598.2	−	54039.0i	−0.156642	−	0.0904374i	−0.576272	+	0.817258i	$0.695493\pi$
$774$		0			0
$775$	135158.	+	234101.i	0.225030	+	0.389763i
$776$		0			0
$777$	−174985.	+	50673.4i	−0.289840	+	0.0839340i
$778$		0			0
$779$	−411844.	−	1.13153e6i	−0.678669	−	1.86463i
$780$		0			0
$781$	−45056.7	−	255529.i	−0.0738681	−	0.418927i
$782$		0			0
$783$	33847.6	+	62466.7i	0.0552083	+	0.101888i
$784$		0			0
$785$	240498.	−	42406.3i	0.390277	−	0.0688163i
$786$		0			0
$787$	−819692.	+	298343.i	−1.32343	+	0.481689i	−0.904556	−	0.426355i	$-0.859797\pi$
$787$	−819692.	+	298343.i	−1.32343	+	0.481689i	−0.418874	+	0.908044i	$0.637575\pi$
$788$		0			0
$789$	195275.	−	203278.i	0.313685	−	0.326540i
$790$		0			0
$791$	5794.89	−	3345.68i	0.00926173	−	0.00534726i
$792$		0			0
$793$	−2388.00	+	4136.14i	−0.00379742	+	0.00657732i
$794$		0			0
$795$	433778.	+	883484.i	0.686330	+	1.39786i
$796$		0			0
$797$	105519.	+	18605.8i	0.166116	+	0.0292908i	0.256088	−	0.966654i	$-0.417566\pi$
$797$	105519.	+	18605.8i	0.166116	+	0.0292908i	−0.0899713	+	0.995944i	$0.528678\pi$
$798$		0			0
$799$	204071.	−	171236.i	0.319660	−	0.268227i
$800$		0			0
$801$	907949.	+	289751.i	1.41513	+	0.451607i
$802$		0			0
$803$	229849.	−	631504.i	0.356460	−	0.979366i
$804$		0			0
$805$	−276710.	−	232188.i	−0.427006	−	0.358300i
$806$		0			0
$807$	−497374.	+	1.12506e6i	−0.763723	+	1.72754i
$808$		0			0
$809$			15837.0i			0.0241977i	0.999927	+	0.0120989i	$0.00385128\pi$
$809$			15837.0i			0.0241977i	−0.999927	+	0.0120989i	$0.996149\pi$
$810$		0			0
$811$	77890.8			0.118425			0.0592127	−	0.998245i	$-0.481141\pi$
$811$	77890.8			0.118425			0.0592127	+	0.998245i	$0.481141\pi$
$812$		0			0
$813$	−1.00858e6	−	445881.i	−1.52591	−	0.674587i
$814$		0			0
$815$	806291.	−	960901.i	1.21388	−	1.44665i
$816$		0			0
$817$	784496.	+	285533.i	1.17529	+	0.427772i
$818$		0			0
$819$	−4759.01	+	1037.69i	−0.00709493	+	0.00154704i
$820$		0			0
$821$	−281744.	−	335769.i	−0.417992	−	0.498143i	0.515426	−	0.856934i	$-0.327634\pi$
$821$	−281744.	−	335769.i	−0.417992	−	0.498143i	−0.933418	+	0.358791i	$0.883189\pi$
$822$		0			0
$823$	43692.6	−	247793.i	0.0645072	−	0.365838i	−0.935417	−	0.353546i	$-0.884976\pi$
$823$	43692.6	−	247793.i	0.0645072	−	0.365838i	0.999924	−	0.0122928i	$-0.00391301\pi$
$824$		0			0
$825$	156356.	−	76768.2i	0.229724	−	0.112791i
$826$		0			0
$827$	317420.	+	183262.i	0.464112	+	0.267955i	0.713772	−	0.700379i	$-0.246985\pi$
$827$	317420.	+	183262.i	0.464112	+	0.267955i	−0.249660	+	0.968334i	$0.580319\pi$
$828$		0			0
$829$	−260535.	−	451260.i	−0.379103	−	0.656625i	0.611829	−	0.790990i	$-0.290434\pi$
$829$	−260535.	−	451260.i	−0.379103	−	0.656625i	−0.990932	+	0.134365i	$0.957101\pi$
$830$		0			0
$831$	272971.	+	262224.i	0.395289	+	0.379726i
$832$		0			0
$833$	148274.	+	407379.i	0.213685	+	0.587096i
$834$		0			0
$835$	157588.	+	893725.i	0.226021	+	1.28183i
$836$		0			0
$837$	723458.	+	911108.i	1.03267	+	1.30053i
$838$		0			0
$839$	−488823.	+	86192.7i	−0.694429	+	0.122447i	−0.509712	−	0.860345i	$-0.670248\pi$
$839$	−488823.	+	86192.7i	−0.694429	+	0.122447i	−0.184717	+	0.982792i	$0.559137\pi$
$840$		0			0
$841$	−655701.	+	238656.i	−0.927073	+	0.337427i
$842$		0			0
$843$	142202.	+	491050.i	0.200102	+	0.690989i
$844$		0			0
$845$	696991.	−	402408.i	0.976143	−	0.563577i
$846$		0			0
$847$	19367.2	−	33544.9i	0.0269960	−	0.0467584i
$848$		0			0
$849$	−41626.4	+	62064.3i	−0.0577502	+	0.0861046i
$850$		0			0
$851$	−428007.	−	75469.1i	−0.591005	−	0.104210i
$852$		0			0
$853$	740679.	−	621504.i	1.01796	−	0.854172i	0.0285928	−	0.999591i	$-0.490897\pi$
$853$	740679.	−	621504.i	1.01796	−	0.854172i	0.989370	+	0.145419i	$0.0464529\pi$
$854$		0			0
$855$	−602480.	−	953004.i	−0.824158	−	1.30365i
$856$		0			0
$857$	−396643.	+	1.08977e6i	−0.540055	+	1.48379i	0.306701	+	0.951806i	$0.400775\pi$
$857$	−396643.	+	1.08977e6i	−0.540055	+	1.48379i	−0.846755	+	0.531982i	$0.821447\pi$
$858$		0			0
$859$	−138587.	−	116288.i	−0.187817	−	0.157597i	0.544031	−	0.839065i	$-0.316897\pi$
$859$	−138587.	−	116288.i	−0.187817	−	0.157597i	−0.731848	+	0.681468i	$0.761342\pi$
$860$		0			0
$861$	−316265.	−	432908.i	−0.426623	−	0.583968i
$862$		0			0
$863$			941281.i			1.26386i	0.775027	+	0.631928i	$0.217736\pi$
$863$			941281.i			1.26386i	−0.775027	+	0.631928i	$0.782264\pi$
$864$		0			0
$865$	869340.			1.16187
$866$		0			0
$867$	24855.5	+	230660.i	0.0330661	+	0.306855i
$868$		0			0
$869$	−418140.	+	498320.i	−0.553710	+	0.659886i
$870$		0			0
$871$	−1937.70	−	705.264i	−0.00255417	−	0.000929642i
$872$		0			0
$873$	230254.	+	31128.7i	0.302120	+	0.0408444i
$874$		0			0
$875$	201666.	+	240336.i	0.263401	+	0.313909i
$876$		0			0
$877$	−123251.	+	698994.i	−0.160248	+	0.908812i	0.793582	+	0.608464i	$0.208214\pi$
$877$	−123251.	+	698994.i	−0.160248	+	0.908812i	−0.953830	+	0.300348i	$0.902897\pi$
$878$		0			0
$879$	−91788.7	+	1.36407e6i	−0.118799	+	1.76546i
$880$		0			0
$881$	−479040.	−	276574.i	−0.617191	−	0.356336i	0.158583	−	0.987346i	$-0.449307\pi$
$881$	−479040.	−	276574.i	−0.617191	−	0.356336i	−0.775775	+	0.631010i	$0.782641\pi$
$882$		0			0
$883$	−567258.	−	982519.i	−0.727543	−	1.26014i	−0.957919	−	0.287040i	$-0.907329\pi$
$883$	−567258.	−	982519.i	−0.727543	−	1.26014i	0.230375	−	0.973102i	$-0.426005\pi$
$884$		0			0
$885$	−229504.	+	930889.i	−0.293024	+	1.18853i
$886$		0			0
$887$	−201922.	−	554777.i	−0.256648	−	0.705133i	−0.999369	−	0.0355323i	$-0.988687\pi$
$887$	−201922.	−	554777.i	−0.256648	−	0.705133i	0.742721	−	0.669601i	$-0.233535\pi$
$888$		0			0
$889$	124229.	+	704540.i	0.157189	+	0.891461i
$890$		0			0
$891$	617067.	−	425708.i	0.777279	−	0.536237i
$892$		0			0
$893$	539180.	−	95071.9i	0.676131	−	0.119220i
$894$		0			0
$895$	800020.	−	291183.i	0.998745	−	0.363514i
$896$		0			0
$897$	−11282.4	−	2781.59i	−0.0140222	−	0.00345707i
$898$		0			0
$899$	134696.	−	77766.8i	0.166662	−	0.0962221i
$900$		0			0
$901$	−466188.	+	807462.i	−0.574264	+	0.994655i
$902$		0			0
$903$	370861.	+	24955.4i	0.454816	+	0.0306047i
$904$		0			0
$905$	1.46438e6	+	258210.i	1.78796	+	0.315266i
$906$		0			0
$907$	−452398.	+	379607.i	−0.549928	+	0.461444i	−0.874917	−	0.484274i	$-0.839084\pi$
$907$	−452398.	+	379607.i	−0.549928	+	0.461444i	0.324989	+	0.945718i	$0.394639\pi$
$908$		0			0
$909$	125770.	−	51584.0i	0.152213	−	0.0624291i
$910$		0			0
$911$	56684.2	−	155738.i	0.0683007	−	0.187655i	−0.900846	−	0.434138i	$-0.857053\pi$
$911$	56684.2	−	155738.i	0.0683007	−	0.187655i	0.969147	+	0.246484i	$0.0792752\pi$
$912$		0			0
$913$	−647345.	−	543187.i	−0.776594	−	0.651640i
$914$		0			0
$915$	−489383.	+	52735.0i	−0.584530	+	0.0629878i
$916$		0			0
$917$		−	620934.i		−	0.738426i
$918$		0			0
$919$	1.06675e6			1.26309			0.631543	−	0.775341i	$-0.282422\pi$
$919$	1.06675e6			1.26309			0.631543	+	0.775341i	$0.282422\pi$
$920$		0			0
$921$	1.31238e6	−	958772.i	1.54718	−	1.13031i
$922$		0			0
$923$	−3592.73	+	4281.65i	−0.00421717	+	0.00502583i
$924$		0			0
$925$	−131871.	−	47997.2i	−0.154123	−	0.0560960i
$926$		0			0
$927$	−47048.2	−	1.17104e6i	−0.0547500	−	1.36274i
$928$		0			0
$929$	389468.	+	464150.i	0.451274	+	0.537807i	0.942934	−	0.332980i	$-0.108054\pi$
$929$	389468.	+	464150.i	0.451274	+	0.537807i	−0.491660	+	0.870787i	$0.663610\pi$
$930$		0			0
$931$	−154717.	+	877443.i	−0.178500	+	1.01232i
$932$		0			0
$933$	−545686.	−	365990.i	−0.626873	−	0.420442i
$934$		0			0
$935$	670187.	+	386933.i	0.766607	+	0.442601i
$936$		0			0
$937$	650792.	+	1.12720e6i	0.741247	+	1.28388i	0.951928	+	0.306323i	$0.0990987\pi$
$937$	650792.	+	1.12720e6i	0.741247	+	1.28388i	−0.210681	+	0.977555i	$0.567568\pi$
$938$		0			0
$939$	839719.	−	243172.i	0.952364	−	0.275793i
$940$		0			0
$941$	−285013.	−	783067.i	−0.321874	−	0.884341i	−0.990098	−	0.140381i	$-0.955167\pi$
$941$	−285013.	−	783067.i	−0.321874	−	0.884341i	0.668224	−	0.743960i	$-0.267055\pi$
$942$		0			0
$943$	−222101.	−	1.25960e6i	−0.249762	−	1.41647i
$944$		0			0
$945$	−375683.	−	332949.i	−0.420685	−	0.372832i
$946$		0			0
$947$	−1.02383e6	+	180529.i	−1.14164	+	0.201302i	−0.712323	−	0.701852i	$-0.752357\pi$
$947$	−1.02383e6	+	180529.i	−1.14164	+	0.201302i	−0.429317	+	0.903154i	$0.641246\pi$
$948$		0			0
$949$	−13603.3	+	4951.20i	−0.0151047	+	0.00549766i
$950$		0			0
$951$	−497938.	+	518345.i	−0.550572	+	0.573136i
$952$		0			0
$953$	1.26089e6	−	727976.i	1.38833	−	0.801551i	0.395200	−	0.918595i	$-0.370675\pi$
$953$	1.26089e6	−	727976.i	1.38833	−	0.801551i	0.993127	+	0.117044i	$0.0373418\pi$
$954$		0			0
$955$	−616997.	+	1.06867e6i	−0.676513	+	1.17176i
$956$		0			0
$957$	−44170.5	−	89963.1i	−0.0482291	−	0.0982292i
$958$		0			0
$959$	−314667.	−	55484.3i	−0.342148	−	0.0603300i
$960$		0			0
$961$	1.24355e6	−	1.04347e6i	1.34654	−	1.12988i
$962$		0			0
$963$	−365575.	+	332656.i	−0.394207	+	0.358709i
$964$		0			0
$965$	−676573.	+	1.85887e6i	−0.726541	+	1.99616i
$966$		0			0
$967$	−215918.	−	181176.i	−0.230906	−	0.193753i	0.519992	−	0.854171i	$-0.325935\pi$
$967$	−215918.	−	181176.i	−0.230906	−	0.193753i	−0.750898	+	0.660418i	$0.770379\pi$
$968$		0			0
$969$	431863.	−	976872.i	0.459937	−	1.04038i
$970$		0			0
$971$			450143.i			0.477433i	0.971089	+	0.238716i	$0.0767267\pi$
$971$			450143.i			0.477433i	−0.971089	+	0.238716i	$0.923273\pi$
$972$		0			0
$973$	219010.			0.231333
$974$		0			0
$975$	−3431.75	−	1517.13i	−0.00360999	−	0.00159593i
$976$		0			0
$977$	−601049.	+	716302.i	−0.629681	+	0.750424i	−0.982703	−	0.185191i	$-0.940710\pi$
$977$	−601049.	+	716302.i	−0.629681	+	0.750424i	0.353022	+	0.935615i	$0.385154\pi$
$978$		0			0
$979$	−1.26334e6	−	459818.i	−1.31812	−	0.479756i
$980$		0			0
$981$	−1.04733e6	−	1.15097e6i	−1.08829	−	1.19599i
$982$		0			0
$983$	−32996.0	−	39323.2i	−0.0341472	−	0.0406950i	0.748701	−	0.662908i	$-0.230678\pi$
$983$	−32996.0	−	39323.2i	−0.0341472	−	0.0406950i	−0.782848	+	0.622213i	$0.786234\pi$
$984$		0			0
$985$	135957.	−	771048.i	0.140129	−	0.794711i
$986$		0			0
$987$	218812.	−	107433.i	0.224614	−	0.110282i
$988$		0			0
$989$	767953.	+	443378.i	0.785131	+	0.453295i
$990$		0			0
$991$	124272.	+	215246.i	0.126540	+	0.219173i	0.922334	−	0.386394i	$-0.126280\pi$
$991$	124272.	+	215246.i	0.126540	+	0.219173i	−0.795794	+	0.605567i	$0.792946\pi$
$992$		0			0
$993$	352909.	+	339015.i	0.357902	+	0.343812i
$994$		0			0
$995$	−299079.	−	821713.i	−0.302093	−	0.829992i
$996$		0			0
$997$	−209863.	−	1.19019e6i	−0.211128	−	1.19737i	−0.887501	−	0.460807i	$-0.847560\pi$
$997$	−209863.	−	1.19019e6i	−0.211128	−	1.19737i	0.676373	−	0.736560i	$-0.263551\pi$
$998$		0			0
$999$	−591750.	−	120919.i	−0.592935	−	0.121162i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.5.k.a.41.2	yes	72
3.2	odd	2		324.5.k.a.233.3		72
27.2	odd	18	inner	108.5.k.a.29.2	✓	72
27.25	even	9		324.5.k.a.89.3		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.k.a.29.2	✓	72	27.2	odd	18	inner
108.5.k.a.41.2	yes	72	1.1	even	1	trivial
324.5.k.a.89.3		72	27.25	even	9
324.5.k.a.233.3		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)

\(f(q)\)	\(=\)	\(q+(-8.23149 - 3.63904i) q^{3} +(18.1168 - 21.5908i) q^{5} +(22.9581 + 8.35608i) q^{7} +(54.5147 + 59.9095i) q^{9} +O(q^{10})\)	\(q+(-8.23149 - 3.63904i) q^{3} +(18.1168 - 21.5908i) q^{5} +(22.9581 + 8.35608i) q^{7} +(54.5147 + 59.9095i) q^{9} +(-73.4456 - 87.5290i) q^{11} +(-0.427402 + 2.42392i) q^{13} +(-227.698 + 111.797i) q^{15} +(-208.105 - 120.150i) q^{17} +(-246.932 - 427.698i) q^{19} +(-158.572 - 152.329i) q^{21} +(-179.414 - 492.937i) q^{23} +(-29.4131 - 166.810i) q^{25} +(-230.724 - 691.525i) q^{27} +(-95.9782 + 16.9236i) q^{29} +(-1499.65 + 545.827i) q^{31} +(286.044 + 987.765i) q^{33} +(596.344 - 344.299i) q^{35} +(414.251 - 717.504i) q^{37} +(12.3389 - 18.3971i) q^{39} +(2401.19 + 423.394i) q^{41} +(-1294.95 + 1086.59i) q^{43} +(2281.13 - 91.6475i) q^{45} +(-379.164 + 1041.74i) q^{47} +(-1382.02 - 1159.65i) q^{49} +(1275.79 + 1746.31i) q^{51} -3880.06i q^{53} -3220.42 q^{55} +(476.203 + 4419.19i) q^{57} +(2429.52 - 2895.39i) q^{59} +(1823.41 + 663.667i) q^{61} +(750.949 + 1830.94i) q^{63} +(44.5912 + 53.1417i) q^{65} +(-145.480 + 825.059i) q^{67} +(-316.971 + 4710.50i) q^{69} +(1966.62 + 1135.43i) q^{71} +(2940.78 + 5093.57i) q^{73} +(-364.915 + 1480.13i) q^{75} +(-954.774 - 2623.22i) q^{77} +(-988.613 - 5606.70i) q^{79} +(-617.285 + 6531.90i) q^{81} +(7283.42 - 1284.26i) q^{83} +(-6364.34 + 2316.43i) q^{85} +(851.629 + 209.963i) q^{87} +(10189.8 - 5883.10i) q^{89} +(-30.0668 + 52.0773i) q^{91} +(14330.6 + 964.312i) q^{93} +(-13708.0 - 2417.09i) q^{95} +(2197.40 - 1843.84i) q^{97} +(1239.95 - 9171.70i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{5} - 102 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more