LMFDB - Embedded newform 72.4.i.a.25.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,4,Mod(25,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("72.25");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	72.i (of order $3$, degree $2$, 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:	$4.24813752041$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.5206055409.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} + x^{6} + 9x^{5} - 23x^{4} + 27x^{3} + 9x^{2} - 81x + 81 $ x^8 - 3x^7 + x^6 + 9x^5 - 23x^4 + 27x^3 + 9x^2 - 81x + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			25.4
Root			$1.35516 + 1.07868i$ of defining polynomial
Character	$\chi$	$=$	72.25
Dual form			72.4.i.a.49.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(3.96518 + 3.35817i) q^{3} +(4.00813 + 6.94228i) q^{5} +(-0.468615 + 0.811666i) q^{7} +(4.44536 + 26.6315i) q^{9} +(-2.66835 + 4.62171i) q^{11} +(1.55794 + 2.69842i) q^{13} +(-7.42041 + 40.9874i) q^{15} +132.704 q^{17} -86.6565 q^{19} +(-4.58386 + 1.64471i) q^{21} +(-20.6212 - 35.7170i) q^{23} +(30.3698 - 52.6021i) q^{25} +(-71.8066 + 120.527i) q^{27} +(101.397 - 175.625i) q^{29} +(-159.367 - 276.031i) q^{31} +(-26.1010 + 9.36517i) q^{33} -7.51308 q^{35} -363.510 q^{37} +(-2.88427 + 15.9316i) q^{39} +(-4.91729 - 8.51700i) q^{41} +(181.230 - 313.899i) q^{43} +(-167.066 + 137.604i) q^{45} +(37.5880 - 65.1043i) q^{47} +(171.061 + 296.286i) q^{49} +(526.197 + 445.644i) q^{51} +403.981 q^{53} -42.7803 q^{55} +(-343.609 - 291.007i) q^{57} +(215.351 + 372.999i) q^{59} +(-160.783 + 278.484i) q^{61} +(-23.6991 - 8.87180i) q^{63} +(-12.4888 + 21.6313i) q^{65} +(363.324 + 629.296i) q^{67} +(38.1769 - 210.874i) q^{69} -829.702 q^{71} -160.106 q^{73} +(297.069 - 106.590i) q^{75} +(-2.50086 - 4.33161i) q^{77} +(462.314 - 800.752i) q^{79} +(-689.478 + 236.773i) q^{81} +(-255.843 + 443.134i) q^{83} +(531.896 + 921.271i) q^{85} +(991.838 - 355.876i) q^{87} -320.459 q^{89} -2.92029 q^{91} +(295.042 - 1629.69i) q^{93} +(-347.330 - 601.594i) q^{95} +(300.821 - 521.038i) q^{97} +(-134.945 - 50.5170i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 3 q^{3} - 5 q^{5} + 3 q^{7} - 15 q^{9} + 16 q^{11} + 29 q^{13} + 141 q^{15} - 34 q^{17} - 218 q^{19} + 27 q^{21} + 37 q^{23} + 97 q^{25} - 216 q^{27} - 3 q^{29} + 331 q^{31} + 468 q^{33} - 342 q^{35}+ \cdots - 5133 q^{99}+O(q^{100}) $ 8 * q - 3 * q^3 - 5 * q^5 + 3 * q^7 - 15 * q^9 + 16 * q^11 + 29 * q^13 + 141 * q^15 - 34 * q^17 - 218 * q^19 + 27 * q^21 + 37 * q^23 + 97 * q^25 - 216 * q^27 - 3 * q^29 + 331 * q^31 + 468 * q^33 - 342 * q^35 - 732 * q^37 - 375 * q^39 - 378 * q^41 + 506 * q^43 - 993 * q^45 + 171 * q^47 + 829 * q^49 + 2193 * q^51 + 820 * q^53 - 2326 * q^55 + 507 * q^57 + 616 * q^59 + 1331 * q^61 - 1899 * q^63 - 815 * q^65 + 1162 * q^67 + 1113 * q^69 - 688 * q^71 - 2614 * q^73 - 1755 * q^75 - 741 * q^77 + 1853 * q^79 - 2547 * q^81 + 1421 * q^83 + 2074 * q^85 + 5877 * q^87 + 1632 * q^89 - 3990 * q^91 + 537 * q^93 + 1292 * q^95 + 2506 * q^97 - 5133 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/72\mathbb{Z}\right)^\times$.

$n$	$37$	$55$	$65$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$	3.96518	+	3.35817i	0.763100	+	0.646281i
$3$	3.96518	+	3.35817i	0.763100	+	0.646281i
$4$		0			0
$5$	4.00813	+	6.94228i	0.358498	+	0.620937i	0.987710	−	0.156297i	$-0.0499557\pi$
$5$	4.00813	+	6.94228i	0.358498	+	0.620937i	−0.629212	+	0.777234i	$0.716622\pi$
$6$		0			0
$7$	−0.468615	+	0.811666i	−0.0253028	+	0.0438258i	−0.878400	−	0.477927i	$-0.841388\pi$
$7$	−0.468615	+	0.811666i	−0.0253028	+	0.0438258i	0.853097	+	0.521753i	$0.174722\pi$
$8$		0			0
$9$	4.44536	+	26.6315i	0.164643	+	0.986353i
$10$		0			0
$11$	−2.66835	+	4.62171i	−0.0731397	+	0.126682i	−0.900276	−	0.435320i	$-0.856635\pi$
$11$	−2.66835	+	4.62171i	−0.0731397	+	0.126682i	0.827136	+	0.562002i	$0.189969\pi$
$12$		0			0
$13$	1.55794	+	2.69842i	0.0332380	+	0.0575699i	0.882166	−	0.470939i	$-0.156085\pi$
$13$	1.55794	+	2.69842i	0.0332380	+	0.0575699i	−0.848928	+	0.528509i	$0.822751\pi$
$14$		0			0
$15$	−7.42041	+	40.9874i	−0.127730	+	0.705527i
$16$		0			0
$17$	132.704			1.89327			0.946634	−	0.322311i	$-0.104460\pi$
$17$	132.704			1.89327			0.946634	+	0.322311i	$0.104460\pi$
$18$		0			0
$19$	−86.6565			−1.04634			−0.523168	−	0.852230i	$-0.675250\pi$
$19$	−86.6565			−1.04634			−0.523168	+	0.852230i	$0.675250\pi$
$20$		0			0
$21$	−4.58386	+	1.64471i	−0.0476324	+	0.0170907i
$22$		0			0
$23$	−20.6212	−	35.7170i	−0.186949	−	0.323805i	0.757283	−	0.653087i	$-0.226526\pi$
$23$	−20.6212	−	35.7170i	−0.186949	−	0.323805i	−0.944231	+	0.329282i	$0.893193\pi$
$24$		0			0
$25$	30.3698	−	52.6021i	0.242959	−	0.420817i
$26$		0			0
$27$	−71.8066	+	120.527i	−0.511822	+	0.859092i
$28$		0			0
$29$	101.397	−	175.625i	0.649275	−	1.12458i	−0.334021	−	0.942566i	$-0.608406\pi$
$29$	101.397	−	175.625i	0.649275	−	1.12458i	0.983296	−	0.182012i	$-0.0582610\pi$
$30$		0			0
$31$	−159.367	−	276.031i	−0.923325	−	1.59925i	−0.794233	−	0.607614i	$-0.792127\pi$
$31$	−159.367	−	276.031i	−0.923325	−	1.59925i	−0.129093	−	0.991633i	$-0.541206\pi$
$32$		0			0
$33$	−26.1010	+	9.36517i	−0.137685	+	0.0494020i
$34$		0			0
$35$	−7.51308			−0.0362841
$36$		0			0
$37$	−363.510			−1.61515			−0.807577	−	0.589762i	$-0.799222\pi$
$37$	−363.510			−1.61515			−0.807577	+	0.589762i	$0.799222\pi$
$38$		0			0
$39$	−2.88427	+	15.9316i	−0.0118424	+	0.0654126i
$40$		0			0
$41$	−4.91729	−	8.51700i	−0.0187305	−	0.0324423i	0.856508	−	0.516133i	$-0.172629\pi$
$41$	−4.91729	−	8.51700i	−0.0187305	−	0.0324423i	−0.875239	+	0.483691i	$0.839296\pi$
$42$		0			0
$43$	181.230	−	313.899i	0.642728	−	1.11324i	−0.342094	−	0.939666i	$-0.611136\pi$
$43$	181.230	−	313.899i	0.642728	−	1.11324i	0.984821	−	0.173571i	$-0.0555307\pi$
$44$		0			0
$45$	−167.066	+	137.604i	−0.553439	+	0.455838i
$46$		0			0
$47$	37.5880	−	65.1043i	0.116655	−	0.202052i	−0.801785	−	0.597612i	$-0.796116\pi$
$47$	37.5880	−	65.1043i	0.116655	−	0.202052i	0.918440	+	0.395560i	$0.129450\pi$
$48$		0			0
$49$	171.061	+	296.286i	0.498720	+	0.863808i
$50$		0			0
$51$	526.197	+	445.644i	1.44475	+	1.22358i
$52$		0			0
$53$	403.981			1.04700			0.523501	−	0.852025i	$-0.324626\pi$
$53$	403.981			1.04700			0.523501	+	0.852025i	$0.324626\pi$
$54$		0			0
$55$	−42.7803			−0.104882
$56$		0			0
$57$	−343.609	−	291.007i	−0.798458	−	0.676226i
$58$		0			0
$59$	215.351	+	372.999i	0.475192	+	0.823057i	0.999596	−	0.0284128i	$-0.00904529\pi$
$59$	215.351	+	372.999i	0.475192	+	0.823057i	−0.524404	+	0.851469i	$0.675712\pi$
$60$		0			0
$61$	−160.783	+	278.484i	−0.337477	+	0.584527i	−0.983957	−	0.178403i	$-0.942907\pi$
$61$	−160.783	+	278.484i	−0.337477	+	0.584527i	0.646480	+	0.762931i	$0.276240\pi$
$62$		0			0
$63$	−23.6991	−	8.87180i	−0.0473937	−	0.0177419i
$64$		0			0
$65$	−12.4888	+	21.6313i	−0.0238315	+	0.0412774i
$66$		0			0
$67$	363.324	+	629.296i	0.662494	+	1.14747i	0.979958	+	0.199203i	$0.0638353\pi$
$67$	363.324	+	629.296i	0.662494	+	1.14747i	−0.317464	+	0.948270i	$0.602831\pi$
$68$		0			0
$69$	38.1769	−	210.874i	0.0666081	−	0.367917i
$70$		0			0
$71$	−829.702			−1.38687			−0.693433	−	0.720521i	$-0.743903\pi$
$71$	−829.702			−1.38687			−0.693433	+	0.720521i	$0.743903\pi$
$72$		0			0
$73$	−160.106			−0.256699			−0.128349	−	0.991729i	$-0.540968\pi$
$73$	−160.106			−0.256699			−0.128349	+	0.991729i	$0.540968\pi$
$74$		0			0
$75$	297.069	−	106.590i	0.457367	−	0.164106i
$76$		0			0
$77$	−2.50086	−	4.33161i	−0.00370129	−	0.00641081i
$78$		0			0
$79$	462.314	−	800.752i	0.658410	−	1.14040i	−0.322617	−	0.946530i	$-0.604563\pi$
$79$	462.314	−	800.752i	0.658410	−	1.14040i	0.981027	−	0.193870i	$-0.0621041\pi$
$80$		0			0
$81$	−689.478	+	236.773i	−0.945785	+	0.324792i
$82$		0			0
$83$	−255.843	+	443.134i	−0.338343	+	0.586027i	−0.984121	−	0.177498i	$-0.943200\pi$
$83$	−255.843	+	443.134i	−0.338343	+	0.586027i	0.645778	+	0.763525i	$0.276533\pi$
$84$		0			0
$85$	531.896	+	921.271i	0.678732	+	1.17560i
$86$		0			0
$87$	991.838	−	355.876i	1.22225	−	0.438551i
$88$		0			0
$89$	−320.459			−0.381669			−0.190835	−	0.981622i	$-0.561119\pi$
$89$	−320.459			−0.381669			−0.190835	+	0.981622i	$0.561119\pi$
$90$		0			0
$91$	−2.92029			−0.00336406
$92$		0			0
$93$	295.042	−	1629.69i	0.328972	−	1.81711i
$94$		0			0
$95$	−347.330	−	601.594i	−0.375109	−	0.649708i
$96$		0			0
$97$	300.821	−	521.038i	0.314884	−	0.545396i	−0.664529	−	0.747263i	$-0.731368\pi$
$97$	300.821	−	521.038i	0.314884	−	0.545396i	0.979413	+	0.201867i	$0.0647009\pi$
$98$		0			0
$99$	−134.945	−	50.5170i	−0.136995	−	0.0512843i
$100$		0			0
$101$	−438.162	+	758.920i	−0.431671	+	0.747676i	−0.997017	−	0.0771773i	$-0.975409\pi$
$101$	−438.162	+	758.920i	−0.431671	+	0.747676i	0.565346	+	0.824854i	$0.308743\pi$
$102$		0			0
$103$	111.719	+	193.502i	0.106873	+	0.185110i	0.914502	−	0.404581i	$-0.132583\pi$
$103$	111.719	+	193.502i	0.106873	+	0.185110i	−0.807629	+	0.589691i	$0.799249\pi$
$104$		0			0
$105$	−29.7907	−	25.2302i	−0.0276884	−	0.0234497i
$106$		0			0
$107$	−1265.34			−1.14322			−0.571612	−	0.820524i	$-0.693682\pi$
$107$	−1265.34			−1.14322			−0.571612	+	0.820524i	$0.693682\pi$
$108$		0			0
$109$	−213.009			−0.187180			−0.0935898	−	0.995611i	$-0.529834\pi$
$109$	−213.009			−0.187180			−0.0935898	+	0.995611i	$0.529834\pi$
$110$		0			0
$111$	−1441.39	−	1220.73i	−1.23252	−	1.04384i
$112$		0			0
$113$	−806.988	−	1397.74i	−0.671814	−	1.16362i	−0.977389	−	0.211449i	$-0.932182\pi$
$113$	−806.988	−	1397.74i	−0.671814	−	1.16362i	0.305575	−	0.952168i	$-0.401151\pi$
$114$		0			0
$115$	165.305	−	286.317i	0.134041	−	0.232167i
$116$		0			0
$117$	−64.9376	+	53.4857i	−0.0513118	+	0.0422629i
$118$		0			0
$119$	−62.1873	+	107.712i	−0.0479051	+	0.0829740i
$120$		0			0
$121$	651.260	+	1128.02i	0.489301	+	0.847494i
$122$		0			0
$123$	9.10359	−	50.2846i	0.00667352	−	0.0368619i
$124$		0			0
$125$	1488.94			1.06540
$126$		0			0
$127$	−1234.19			−0.862333			−0.431167	−	0.902272i	$-0.641898\pi$
$127$	−1234.19			−0.862333			−0.431167	+	0.902272i	$0.641898\pi$
$128$		0			0
$129$	1772.74	−	636.067i	1.20993	−	0.434129i
$130$		0			0
$131$	723.720	+	1253.52i	0.482685	+	0.836034i	0.999802	−	0.0198800i	$-0.00632843\pi$
$131$	723.720	+	1253.52i	0.482685	+	0.836034i	−0.517118	+	0.855914i	$0.672995\pi$
$132$		0			0
$133$	40.6086	−	70.3361i	0.0264753	−	0.0458565i
$134$		0			0
$135$	−1124.54	−	15.4133i	−0.716928	−	0.00982643i
$136$		0			0
$137$	−276.891	+	479.589i	−0.172674	+	0.299081i	−0.939354	−	0.342949i	$-0.888574\pi$
$137$	−276.891	+	479.589i	−0.172674	+	0.299081i	0.766680	+	0.642030i	$0.221907\pi$
$138$		0			0
$139$	794.528	+	1376.16i	0.484827	+	0.839745i	0.999848	−	0.0174326i	$-0.00554925\pi$
$139$	794.528	+	1376.16i	0.484827	+	0.839745i	−0.515021	+	0.857178i	$0.672216\pi$
$140$		0			0
$141$	367.675	−	131.924i	0.219601	−	0.0787941i
$142$		0			0
$143$	−16.6285			−0.00972406
$144$		0			0
$145$	1625.65			0.931055
$146$		0			0
$147$	−316.692	+	1749.28i	−0.177689	+	0.981484i
$148$		0			0
$149$	606.406	+	1050.33i	0.333414	+	0.577490i	0.983179	−	0.182645i	$-0.0584660\pi$
$149$	606.406	+	1050.33i	0.333414	+	0.577490i	−0.649765	+	0.760135i	$0.725133\pi$
$150$		0			0
$151$	−265.929	+	460.603i	−0.143318	+	0.248234i	−0.928744	−	0.370721i	$-0.879110\pi$
$151$	−265.929	+	460.603i	−0.143318	+	0.248234i	0.785426	+	0.618955i	$0.212444\pi$
$152$		0			0
$153$	589.919	+	3534.12i	0.311713	+	1.86743i
$154$		0			0
$155$	1277.52	−	2212.74i	0.662020	−	1.14665i
$156$		0			0
$157$	−1081.47	−	1873.16i	−0.549749	−	0.952194i	−0.998291	−	0.0584320i	$-0.981390\pi$
$157$	−1081.47	−	1873.16i	−0.549749	−	0.952194i	0.448542	−	0.893762i	$-0.351943\pi$
$158$		0			0
$159$	1601.86	+	1356.64i	0.798967	+	0.676657i
$160$		0			0
$161$	38.6537			0.0189213
$162$		0			0
$163$	−1602.11			−0.769859			−0.384930	−	0.922946i	$-0.625774\pi$
$163$	−1602.11			−0.769859			−0.384930	+	0.922946i	$0.625774\pi$
$164$		0			0
$165$	−169.632	−	143.664i	−0.0800352	−	0.0677830i
$166$		0			0
$167$	866.800	+	1501.34i	0.401647	+	0.695673i	0.993925	−	0.110061i	$-0.0351046\pi$
$167$	866.800	+	1501.34i	0.401647	+	0.695673i	−0.592278	+	0.805734i	$0.701771\pi$
$168$		0			0
$169$	1093.65	−	1894.25i	0.497790	−	0.862198i
$170$		0			0
$171$	−385.219	−	2307.80i	−0.172272	−	1.03206i
$172$		0			0
$173$	−937.323	+	1623.49i	−0.411927	+	0.713478i	−0.995100	−	0.0988692i	$-0.968477\pi$
$173$	−937.323	+	1623.49i	−0.411927	+	0.713478i	0.583173	+	0.812348i	$0.301811\pi$
$174$		0			0
$175$	28.4635	+	49.3003i	0.0122951	+	0.0212957i
$176$		0			0
$177$	−398.688	+	2202.20i	−0.169307	+	0.935182i
$178$		0			0
$179$	2058.01			0.859344			0.429672	−	0.902985i	$-0.358629\pi$
$179$	2058.01			0.859344			0.429672	+	0.902985i	$0.358629\pi$
$180$		0			0
$181$	−3938.45			−1.61736			−0.808681	−	0.588248i	$-0.799818\pi$
$181$	−3938.45			−1.61736			−0.808681	+	0.588248i	$0.799818\pi$
$182$		0			0
$183$	−1572.73	+	564.303i	−0.635297	+	0.227948i
$184$		0			0
$185$	−1457.00	−	2523.59i	−0.579030	−	1.00291i
$186$		0			0
$187$	−354.101	+	613.322i	−0.138473	+	0.239842i
$188$		0			0
$189$	−64.1781	−	114.764i	−0.0246998	−	0.0441685i
$190$		0			0
$191$	2232.54	−	3866.88i	0.845765	−	1.46491i	−0.0391899	−	0.999232i	$-0.512478\pi$
$191$	2232.54	−	3866.88i	0.845765	−	1.46491i	0.884955	−	0.465676i	$-0.154189\pi$
$192$		0			0
$193$	−2075.02	−	3594.03i	−0.773901	−	1.34044i	−0.935410	−	0.353564i	$-0.884970\pi$
$193$	−2075.02	−	3594.03i	−0.773901	−	1.34044i	0.161509	−	0.986871i	$-0.448364\pi$
$194$		0			0
$195$	−122.162	+	43.8323i	−0.0448626	+	0.0160969i
$196$		0			0
$197$	536.823			0.194148			0.0970738	−	0.995277i	$-0.469052\pi$
$197$	536.823			0.194148			0.0970738	+	0.995277i	$0.469052\pi$
$198$		0			0
$199$	1953.59			0.695911			0.347956	−	0.937511i	$-0.386876\pi$
$199$	1953.59			0.695911			0.347956	+	0.937511i	$0.386876\pi$
$200$		0			0
$201$	−672.637	+	3715.38i	−0.236040	+	1.30379i
$202$		0			0
$203$	95.0325	+	164.601i	0.0328570	+	0.0569100i
$204$		0			0
$205$	39.4183	−	68.2745i	0.0134297	−	0.0232610i
$206$		0			0
$207$	859.530	−	707.950i	0.288606	−	0.237710i
$208$		0			0
$209$	231.230	−	400.501i	0.0765286	−	0.132551i
$210$		0			0
$211$	−1613.64	−	2794.91i	−0.526482	−	0.911894i	−0.999524	−	0.0308540i	$-0.990177\pi$
$211$	−1613.64	−	2794.91i	−0.526482	−	0.911894i	0.473042	−	0.881040i	$-0.343156\pi$
$212$		0			0
$213$	−3289.92	−	2786.28i	−1.05832	−	0.896305i
$214$		0			0
$215$	2905.57			0.921666
$216$		0			0
$217$	298.726			0.0934510
$218$		0			0
$219$	−634.850	−	537.664i	−0.195887	−	0.165899i
$220$		0			0
$221$	206.745	+	358.093i	0.0629284	+	0.108995i
$222$		0			0
$223$	−2327.99	+	4032.19i	−0.699075	+	1.21083i	0.269713	+	0.962941i	$0.413071\pi$
$223$	−2327.99	+	4032.19i	−0.699075	+	1.21083i	−0.968788	+	0.247892i	$0.920262\pi$
$224$		0			0
$225$	1535.88	+	574.960i	0.455075	+	0.170359i
$226$		0			0
$227$	2040.40	−	3534.08i	0.596592	−	1.03333i	−0.396728	−	0.917936i	$-0.629854\pi$
$227$	2040.40	−	3534.08i	0.596592	−	1.03333i	0.993320	−	0.115391i	$-0.0368122\pi$
$228$		0			0
$229$	−649.220	−	1124.48i	−0.187344	−	0.324489i	0.757020	−	0.653391i	$-0.226654\pi$
$229$	−649.220	−	1124.48i	−0.187344	−	0.324489i	−0.944364	+	0.328903i	$0.893321\pi$
$230$		0			0
$231$	4.62994	−	25.5739i	0.00131873	−	0.00728416i
$232$		0			0
$233$	−3073.26			−0.864103			−0.432052	−	0.901849i	$-0.642210\pi$
$233$	−3073.26			−0.864103			−0.432052	+	0.901849i	$0.642210\pi$
$234$		0			0
$235$	602.630			0.167282
$236$		0			0
$237$	4522.22	−	1622.60i	1.23945	−	0.444721i
$238$		0			0
$239$	1787.76	+	3096.49i	0.483852	+	0.838056i	0.999828	−	0.0185469i	$-0.00590399\pi$
$239$	1787.76	+	3096.49i	0.483852	+	0.838056i	−0.515976	+	0.856603i	$0.672571\pi$
$240$		0			0
$241$	1358.79	−	2353.50i	0.363185	−	0.629055i	−0.625298	−	0.780386i	$-0.715023\pi$
$241$	1358.79	−	2353.50i	0.363185	−	0.629055i	0.988483	+	0.151331i	$0.0483560\pi$
$242$		0			0
$243$	−3529.03	−	1376.53i	−0.931636	−	0.363394i
$244$		0			0
$245$	−1371.27	+	2375.10i	−0.357580	+	0.619346i
$246$		0			0
$247$	−135.005	−	233.836i	−0.0347781	−	0.0602374i
$248$		0			0
$249$	−2502.59	+	897.941i	−0.636928	+	0.228533i
$250$		0			0
$251$	−4583.36			−1.15259			−0.576293	−	0.817243i	$-0.695501\pi$
$251$	−4583.36			−1.15259			−0.576293	+	0.817243i	$0.695501\pi$
$252$		0			0
$253$	220.098			0.0546935
$254$		0			0
$255$	−984.722	+	5439.21i	−0.241826	+	1.33575i
$256$		0			0
$257$	−2987.71	−	5174.87i	−0.725169	−	1.25603i	−0.958904	−	0.283730i	$-0.908428\pi$
$257$	−2987.71	−	5174.87i	−0.725169	−	1.25603i	0.233735	−	0.972300i	$-0.424905\pi$
$258$		0			0
$259$	170.346	−	295.049i	0.0408680	−	0.0707855i
$260$		0			0
$261$	5127.91	+	1919.65i	1.21613	+	0.455261i
$262$		0			0
$263$	−3087.59	+	5347.86i	−0.723912	+	1.25385i	0.235509	+	0.971872i	$0.424324\pi$
$263$	−3087.59	+	5347.86i	−0.723912	+	1.25385i	−0.959420	+	0.281980i	$0.909009\pi$
$264$		0			0
$265$	1619.21	+	2804.55i	0.375348	+	0.650121i
$266$		0			0
$267$	−1270.68	−	1076.16i	−0.291252	−	0.246665i
$268$		0			0
$269$	6580.02			1.49142			0.745708	−	0.666273i	$-0.232111\pi$
$269$	6580.02			1.49142			0.745708	+	0.666273i	$0.232111\pi$
$270$		0			0
$271$	5817.31			1.30397			0.651986	−	0.758231i	$-0.273936\pi$
$271$	5817.31			1.30397			0.651986	+	0.758231i	$0.273936\pi$
$272$		0			0
$273$	−11.5795	−	9.80684i	−0.00256712	−	0.00217413i
$274$		0			0
$275$	162.074	+	280.721i	0.0355398	+	0.0615568i
$276$		0			0
$277$	−4325.53	+	7492.03i	−0.938252	+	1.62510i	−0.169521	+	0.985527i	$0.554222\pi$
$277$	−4325.53	+	7492.03i	−0.938252	+	1.62510i	−0.768730	+	0.639573i	$0.779111\pi$
$278$		0			0
$279$	6642.69	−	5471.23i	1.42540	−	1.17403i
$280$		0			0
$281$	−3648.14	+	6318.76i	−0.774483	+	1.34144i	0.160602	+	0.987019i	$0.448656\pi$
$281$	−3648.14	+	6318.76i	−0.774483	+	1.34144i	−0.935085	+	0.354424i	$0.884677\pi$
$282$		0			0
$283$	405.958	+	703.140i	0.0852710	+	0.147694i	0.905507	−	0.424332i	$-0.139491\pi$
$283$	405.958	+	703.140i	0.0852710	+	0.147694i	−0.820236	+	0.572026i	$0.806158\pi$
$284$		0			0
$285$	643.027	−	3551.82i	0.133648	−	0.738217i
$286$		0			0
$287$	9.21728			0.00189574
$288$		0			0
$289$	12697.5			2.58446
$290$		0			0
$291$	2942.55	−	1055.80i	0.592767	−	0.212688i
$292$		0			0
$293$	2400.96	+	4158.58i	0.478722	+	0.829170i	0.999702	−	0.0243984i	$-0.00776702\pi$
$293$	2400.96	+	4158.58i	0.478722	+	0.829170i	−0.520981	+	0.853568i	$0.674434\pi$
$294$		0			0
$295$	−1726.31	+	2990.06i	−0.340711	+	0.590128i
$296$		0			0
$297$	−365.437	−	653.478i	−0.0713967	−	0.127672i
$298$		0			0
$299$	64.2531	−	111.290i	0.0124276	−	0.0215252i
$300$		0			0
$301$	169.854	+	294.196i	0.0325257	+	0.0563361i
$302$		0			0
$303$	−4285.98	+	1537.83i	−0.812617	+	0.291571i
$304$		0			0
$305$	−2577.75			−0.483939
$306$		0			0
$307$	−755.147			−0.140386			−0.0701930	−	0.997533i	$-0.522362\pi$
$307$	−755.147			−0.140386			−0.0701930	+	0.997533i	$0.522362\pi$
$308$		0			0
$309$	−206.829	+	1142.44i	−0.0380780	+	0.210328i
$310$		0			0
$311$	−2633.03	−	4560.55i	−0.480082	−	0.831527i	0.519657	−	0.854375i	$-0.326060\pi$
$311$	−2633.03	−	4560.55i	−0.480082	−	0.831527i	−0.999739	+	0.0228482i	$0.992727\pi$
$312$		0			0
$313$	−697.053	+	1207.33i	−0.125878	+	0.218027i	−0.922076	−	0.387009i	$-0.873508\pi$
$313$	−697.053	+	1207.33i	−0.125878	+	0.218027i	0.796198	+	0.605036i	$0.206841\pi$
$314$		0			0
$315$	−33.3983	−	200.085i	−0.00597391	−	0.0357889i
$316$		0			0
$317$	2558.55	−	4431.55i	0.453321	−	0.785175i	−0.545269	−	0.838261i	$-0.683572\pi$
$317$	2558.55	−	4431.55i	0.453321	−	0.785175i	0.998590	+	0.0530862i	$0.0169058\pi$
$318$		0			0
$319$	541.126	+	937.257i	0.0949756	+	0.164503i
$320$		0			0
$321$	−5017.30	−	4249.23i	−0.872394	−	0.738843i
$322$		0			0
$323$	−11499.7			−1.98099
$324$		0			0
$325$	189.257			0.0323018
$326$		0			0
$327$	−844.620	−	715.321i	−0.142837	−	0.120970i
$328$		0			0
$329$	35.2286	+	61.0177i	0.00590339	+	0.0102250i
$330$		0			0
$331$	121.400	−	210.271i	0.0201593	−	0.0349170i	−0.855770	−	0.517357i	$-0.826916\pi$
$331$	121.400	−	210.271i	0.0201593	−	0.0349170i	0.875929	+	0.482440i	$0.160249\pi$
$332$		0			0
$333$	−1615.93	−	9680.84i	−0.265924	−	1.59311i
$334$		0			0
$335$	−2912.50	+	5044.59i	−0.475005	+	0.822733i
$336$		0			0
$337$	25.6472	+	44.4223i	0.00414568	+	0.00718052i	0.868091	−	0.496405i	$-0.165347\pi$
$337$	25.6472	+	44.4223i	0.00414568	+	0.00718052i	−0.863945	+	0.503586i	$0.832014\pi$
$338$		0			0
$339$	1494.01	−	8252.31i	0.239361	−	1.32214i
$340$		0			0
$341$	1700.98			0.270127
$342$		0			0
$343$	−642.117			−0.101082
$344$		0			0
$345$	1616.96	−	580.175i	0.252332	−	0.0905379i
$346$		0			0
$347$	430.722	+	746.032i	0.0666350	+	0.115415i	0.897418	−	0.441181i	$-0.145440\pi$
$347$	430.722	+	746.032i	0.0666350	+	0.115415i	−0.830783	+	0.556596i	$0.812107\pi$
$348$		0			0
$349$	−1562.68	+	2706.64i	−0.239680	+	0.415138i	−0.960623	−	0.277857i	$-0.910376\pi$
$349$	−1562.68	+	2706.64i	−0.239680	+	0.415138i	0.720942	+	0.692995i	$0.243709\pi$
$350$		0			0
$351$	−437.104	−	5.99107i	−0.0664697	−	0.000911053i
$352$		0			0
$353$	4162.36	−	7209.43i	0.627593	−	1.08702i	−0.360440	−	0.932782i	$-0.617374\pi$
$353$	4162.36	−	7209.43i	0.627593	−	1.08702i	0.988033	−	0.154241i	$-0.0492931\pi$
$354$		0			0
$355$	−3325.55	−	5760.02i	−0.497189	−	0.861156i
$356$		0			0
$357$	−608.298	+	218.261i	−0.0901808	+	0.0323573i
$358$		0			0
$359$	−5819.46			−0.855542			−0.427771	−	0.903887i	$-0.640701\pi$
$359$	−5819.46			−0.855542			−0.427771	+	0.903887i	$0.640701\pi$
$360$		0			0
$361$	650.350			0.0948170
$362$		0			0
$363$	−1205.70	+	6659.83i	−0.174334	+	0.962949i
$364$		0			0
$365$	−641.725	−	1111.50i	−0.0920259	−	0.159393i
$366$		0			0
$367$	3129.64	−	5420.69i	0.445138	−	0.771002i	−0.552924	−	0.833232i	$-0.686488\pi$
$367$	3129.64	−	5420.69i	0.445138	−	0.771002i	0.998062	+	0.0622299i	$0.0198212\pi$
$368$		0			0
$369$	204.962	−	168.816i	0.0289157	−	0.0238163i
$370$		0			0
$371$	−189.312	+	327.898i	−0.0264921	+	0.0458857i
$372$		0			0
$373$	−240.335	−	416.273i	−0.0333622	−	0.0577850i	0.848862	−	0.528614i	$-0.177288\pi$
$373$	−240.335	−	416.273i	−0.0333622	−	0.0577850i	−0.882224	+	0.470829i	$0.843955\pi$
$374$		0			0
$375$	5903.91	+	5000.11i	0.813004	+	0.688545i
$376$		0			0
$377$	631.881			0.0863224
$378$		0			0
$379$	7321.57			0.992305			0.496152	−	0.868235i	$-0.334746\pi$
$379$	7321.57			0.992305			0.496152	+	0.868235i	$0.334746\pi$
$380$		0			0
$381$	−4893.77	−	4144.61i	−0.658046	−	0.557309i
$382$		0			0
$383$	−1323.45	−	2292.28i	−0.176567	−	0.305823i	0.764135	−	0.645056i	$-0.223166\pi$
$383$	−1323.45	−	2292.28i	−0.176567	−	0.305823i	−0.940702	+	0.339233i	$0.889833\pi$
$384$		0			0
$385$	20.0475	−	34.7233i	0.00265381	−	0.00459653i
$386$		0			0
$387$	9165.25	+	3431.03i	1.20387	+	0.450670i
$388$		0			0
$389$	−2430.72	+	4210.12i	−0.316818	+	0.548745i	−0.979822	−	0.199871i	$-0.935948\pi$
$389$	−2430.72	+	4210.12i	−0.316818	+	0.548745i	0.663004	+	0.748616i	$0.269281\pi$
$390$		0			0
$391$	−2736.53	−	4739.80i	−0.353944	−	0.613049i
$392$		0			0
$393$	−1339.85	+	7400.81i	−0.171976	+	0.949927i
$394$		0			0
$395$	7412.06			0.944155
$396$		0			0
$397$	−7253.21			−0.916948			−0.458474	−	0.888708i	$-0.651604\pi$
$397$	−7253.21			−0.916948			−0.458474	+	0.888708i	$0.651604\pi$
$398$		0			0
$399$	397.221	−	142.525i	0.0498394	−	0.0178826i
$400$		0			0
$401$	1150.69	+	1993.05i	0.143298	+	0.248200i	0.928737	−	0.370740i	$-0.120896\pi$
$401$	1150.69	+	1993.05i	0.143298	+	0.248200i	−0.785438	+	0.618940i	$0.787562\pi$
$402$		0			0
$403$	496.566	−	860.078i	0.0613789	−	0.106311i
$404$		0			0
$405$	−4407.26	−	3837.53i	−0.540737	−	0.470835i
$406$		0			0
$407$	969.971	−	1680.04i	0.118132	−	0.204611i
$408$		0			0
$409$	902.216	+	1562.68i	0.109075	+	0.188924i	0.915396	−	0.402555i	$-0.131878\pi$
$409$	902.216	+	1562.68i	0.109075	+	0.188924i	−0.806321	+	0.591479i	$0.798544\pi$
$410$		0			0
$411$	−2708.47	+	971.811i	−0.325058	+	0.116632i
$412$		0			0
$413$	−403.667			−0.0480948
$414$		0			0
$415$	−4101.81			−0.485181
$416$		0			0
$417$	−1470.94	+	8124.90i	−0.172739	+	0.954143i
$418$		0			0
$419$	3738.18	+	6474.71i	0.435852	+	0.754917i	0.997365	−	0.0725506i	$-0.0231139\pi$
$419$	3738.18	+	6474.71i	0.435852	+	0.754917i	−0.561513	+	0.827468i	$0.689781\pi$
$420$		0			0
$421$	−6443.01	+	11159.6i	−0.745875	+	1.29189i	0.203910	+	0.978990i	$0.434635\pi$
$421$	−6443.01	+	11159.6i	−0.745875	+	1.29189i	−0.949785	+	0.312903i	$0.898698\pi$
$422$		0			0
$423$	1900.92	+	711.614i	0.218501	+	0.0817963i
$424$		0			0
$425$	4030.21	−	6980.53i	0.459986	−	0.796718i
$426$		0			0
$427$	−150.690	−	261.003i	−0.0170783	−	0.0295804i
$428$		0			0
$429$	−65.9349	−	55.8412i	−0.00742043	−	0.00628447i
$430$		0			0
$431$	13065.8			1.46022			0.730110	−	0.683329i	$-0.239469\pi$
$431$	13065.8			1.46022			0.730110	+	0.683329i	$0.239469\pi$
$432$		0			0
$433$	−6190.38			−0.687046			−0.343523	−	0.939144i	$-0.611620\pi$
$433$	−6190.38			−0.687046			−0.343523	+	0.939144i	$0.611620\pi$
$434$		0			0
$435$	6446.01	+	5459.22i	0.710488	+	0.601723i
$436$		0			0
$437$	1786.96	+	3095.11i	0.195611	+	0.338808i
$438$		0			0
$439$	7271.78	−	12595.1i	0.790576	−	1.36932i	−0.135034	−	0.990841i	$-0.543114\pi$
$439$	7271.78	−	12595.1i	0.790576	−	1.36932i	0.925610	−	0.378478i	$-0.123552\pi$
$440$		0			0
$441$	−7130.13	+	5872.71i	−0.769909	+	0.634133i
$442$		0			0
$443$	426.857	−	739.339i	0.0457802	−	0.0792935i	−0.842227	−	0.539123i	$-0.818756\pi$
$443$	426.857	−	739.339i	0.0457802	−	0.0792935i	0.888007	+	0.459829i	$0.152089\pi$
$444$		0			0
$445$	−1284.44	−	2224.71i	−0.136828	−	0.236992i
$446$		0			0
$447$	−1122.66	+	6201.15i	−0.118792	+	0.656162i
$448$		0			0
$449$	6187.67			0.650366			0.325183	−	0.945651i	$-0.394574\pi$
$449$	6187.67			0.650366			0.325183	+	0.945651i	$0.394574\pi$
$450$		0			0
$451$	52.4842			0.00547979
$452$		0			0
$453$	−2601.24	+	933.339i	−0.269795	+	0.0968037i
$454$		0			0
$455$	−11.7049	−	20.2735i	−0.00120601	−	0.00208887i
$456$		0			0
$457$	−2389.06	+	4137.98i	−0.244542	+	0.423559i	−0.962003	−	0.273040i	$-0.911971\pi$
$457$	−2389.06	+	4137.98i	−0.244542	+	0.423559i	0.717461	+	0.696599i	$0.245304\pi$
$458$		0			0
$459$	−9529.06	+	15994.5i	−0.969016	+	1.62649i
$460$		0			0
$461$	4544.60	−	7871.47i	0.459139	−	0.795252i	−0.539777	−	0.841808i	$-0.681491\pi$
$461$	4544.60	−	7871.47i	0.459139	−	0.795252i	0.998916	+	0.0465564i	$0.0148247\pi$
$462$		0			0
$463$	3674.11	+	6363.74i	0.368791	+	0.638765i	0.989377	−	0.145374i	$-0.0464385\pi$
$463$	3674.11	+	6363.74i	0.368791	+	0.638765i	−0.620586	+	0.784139i	$0.713105\pi$
$464$		0			0
$465$	12496.4	−	4483.76i	1.24625	−	0.447160i
$466$		0			0
$467$	948.436			0.0939794			0.0469897	−	0.998895i	$-0.485037\pi$
$467$	948.436			0.0939794			0.0469897	+	0.998895i	$0.485037\pi$
$468$		0			0
$469$	−681.037			−0.0670519
$470$		0			0
$471$	2002.17	−	11059.2i	0.195871	−	1.08191i
$472$		0			0
$473$	967.168	+	1675.18i	0.0940178	+	0.162844i
$474$		0			0
$475$	−2631.74	+	4558.31i	−0.254216	+	0.440315i
$476$		0			0
$477$	1795.84	+	10758.6i	0.172381	+	1.03271i
$478$		0			0
$479$	−1493.94	+	2587.58i	−0.142505	+	0.246826i	−0.928439	−	0.371484i	$-0.878849\pi$
$479$	−1493.94	+	2587.58i	−0.142505	+	0.246826i	0.785934	+	0.618310i	$0.212182\pi$
$480$		0			0
$481$	−566.326	−	980.905i	−0.0536845	−	0.0929843i
$482$		0			0
$483$	153.269	+	129.806i	0.0144389	+	0.0122285i
$484$		0			0
$485$	4822.92			0.451542
$486$		0			0
$487$	−19555.8			−1.81963			−0.909814	−	0.415017i	$-0.863776\pi$
$487$	−19555.8			−1.81963			−0.909814	+	0.415017i	$0.863776\pi$
$488$		0			0
$489$	−6352.66	−	5380.16i	−0.587479	−	0.497545i
$490$		0			0
$491$	−429.064	−	743.160i	−0.0394366	−	0.0683062i	0.845633	−	0.533764i	$-0.179223\pi$
$491$	−429.064	−	743.160i	−0.0394366	−	0.0683062i	−0.885070	+	0.465458i	$0.845890\pi$
$492$		0			0
$493$	13455.9	−	23306.2i	1.22925	−	2.12913i
$494$		0			0
$495$	−190.174	−	1139.30i	−0.0172680	−	0.103450i
$496$		0			0
$497$	388.811	−	673.440i	0.0350917	−	0.0607806i
$498$		0			0
$499$	−1844.70	−	3195.12i	−0.165491	−	0.286640i	0.771338	−	0.636425i	$-0.219588\pi$
$499$	−1844.70	−	3195.12i	−0.165491	−	0.286640i	−0.936830	+	0.349786i	$0.886254\pi$
$500$		0			0
$501$	−1604.74	+	8863.96i	−0.143103	+	0.790444i
$502$		0			0
$503$	9034.08			0.800814			0.400407	−	0.916337i	$-0.368869\pi$
$503$	9034.08			0.800814			0.400407	+	0.916337i	$0.368869\pi$
$504$		0			0
$505$	−7024.84			−0.619013
$506$		0			0
$507$	10697.7	−	3838.40i	0.937086	−	0.336231i
$508$		0			0
$509$	3145.21	+	5447.66i	0.273888	+	0.474387i	0.969854	−	0.243687i	$-0.0783570\pi$
$509$	3145.21	+	5447.66i	0.273888	+	0.474387i	−0.695966	+	0.718075i	$0.745024\pi$
$510$		0			0
$511$	75.0281	−	129.953i	0.00649520	−	0.0112500i
$512$		0			0
$513$	6222.51	−	10444.5i	0.535537	−	0.898898i
$514$		0			0
$515$	−895.565	+	1551.16i	−0.0766278	+	0.132723i
$516$		0			0
$517$	200.595	+	347.442i	0.0170642	+	0.0295560i
$518$		0			0
$519$	−9168.62	+	3289.75i	−0.775449	+	0.278235i
$520$		0			0
$521$	−13010.4			−1.09404			−0.547019	−	0.837120i	$-0.684238\pi$
$521$	−13010.4			−1.09404			−0.547019	+	0.837120i	$0.684238\pi$
$522$		0			0
$523$	1108.69			0.0926956			0.0463478	−	0.998925i	$-0.485242\pi$
$523$	1108.69			0.0926956			0.0463478	+	0.998925i	$0.485242\pi$
$524$		0			0
$525$	−52.6957	+	291.070i	−0.00438063	+	0.0241968i
$526$		0			0
$527$	−21148.7	−	36630.5i	−1.74810	−	3.02780i
$528$		0			0
$529$	5233.03	−	9063.88i	0.430100	−	0.744956i
$530$		0			0
$531$	−8976.22	+	7393.25i	−0.733588	+	0.604218i
$532$		0			0
$533$	15.3217	−	26.5379i	0.00124513	−	0.00215663i
$534$		0			0
$535$	−5071.64	−	8784.34i	−0.409843	−	0.709869i
$536$		0			0
$537$	8160.37	+	6911.14i	0.655765	+	0.555377i
$538$		0			0
$539$	−1825.80			−0.145905
$540$		0			0
$541$	−38.8964			−0.00309110			−0.00154555	−	0.999999i	$-0.500492\pi$
$541$	−38.8964			−0.00309110			−0.00154555	+	0.999999i	$0.500492\pi$
$542$		0			0
$543$	−15616.7	−	13226.0i	−1.23421	−	1.04527i
$544$		0			0
$545$	−853.768	−	1478.77i	−0.0671035	−	0.116227i
$546$		0			0
$547$	−5014.91	+	8686.08i	−0.391997	+	0.678958i	−0.992713	−	0.120504i	$-0.961549\pi$
$547$	−5014.91	+	8686.08i	−0.391997	+	0.678958i	0.600716	+	0.799462i	$0.294882\pi$
$548$		0			0
$549$	−8131.18	−	3043.93i	−0.632114	−	0.236633i
$550$		0			0
$551$	−8786.73	+	15219.1i	−0.679359	+	1.17669i
$552$		0			0
$553$	433.295	+	750.489i	0.0333193	+	0.0577107i
$554$		0			0
$555$	2697.40	−	14899.3i	0.206303	−	1.13953i
$556$		0			0
$557$	−7355.74			−0.559556			−0.279778	−	0.960065i	$-0.590261\pi$
$557$	−7355.74			−0.559556			−0.279778	+	0.960065i	$0.590261\pi$
$558$		0			0
$559$	1129.38			0.0854519
$560$		0			0
$561$	−3463.72	+	1242.80i	−0.260674	+	0.0935312i
$562$		0			0
$563$	10729.9	+	18584.8i	0.803220	+	1.39122i	0.917486	+	0.397767i	$0.130215\pi$
$563$	10729.9	+	18584.8i	0.803220	+	1.39122i	−0.114267	+	0.993450i	$0.536452\pi$
$564$		0			0
$565$	6469.02	−	11204.7i	0.481688	−	0.834308i
$566$		0			0
$567$	130.919	−	670.581i	0.00969678	−	0.0496680i
$568$		0			0
$569$	728.962	−	1262.60i	0.0537077	−	0.0930245i	−0.837922	−	0.545791i	$-0.816229\pi$
$569$	728.962	−	1262.60i	0.0537077	−	0.0930245i	0.891629	+	0.452766i	$0.149563\pi$
$570$		0			0
$571$	11200.0	+	19399.0i	0.820853	+	1.42176i	0.905048	+	0.425310i	$0.139835\pi$
$571$	11200.0	+	19399.0i	0.820853	+	1.42176i	−0.0841950	+	0.996449i	$0.526832\pi$
$572$		0			0
$573$	21838.1	−	7835.62i	1.59215	−	0.571270i
$574$		0			0
$575$	−2505.05			−0.181683
$576$		0			0
$577$	−12833.5			−0.925936			−0.462968	−	0.886375i	$-0.653215\pi$
$577$	−12833.5			−0.925936			−0.462968	+	0.886375i	$0.653215\pi$
$578$		0			0
$579$	3841.56	−	21219.3i	0.275734	−	1.52304i
$580$		0			0
$581$	−239.784	−	415.319i	−0.0171221	−	0.0296563i
$582$		0			0
$583$	−1077.96	+	1867.08i	−0.0765774	+	0.132636i
$584$		0			0
$585$	−631.591	−	236.438i	−0.0446377	−	0.0167102i
$586$		0			0
$587$	10377.4	−	17974.2i	0.729677	−	1.26384i	−0.227343	−	0.973815i	$-0.573004\pi$
$587$	10377.4	−	17974.2i	0.729677	−	1.26384i	0.957020	−	0.290023i	$-0.0936629\pi$
$588$		0			0
$589$	13810.2	+	23919.9i	0.966108	+	1.67335i
$590$		0			0
$591$	2128.60	+	1802.75i	0.148154	+	0.125474i
$592$		0			0
$593$	113.272			0.00784406			0.00392203	−	0.999992i	$-0.498752\pi$
$593$	113.272			0.00784406			0.00392203	+	0.999992i	$0.498752\pi$
$594$		0			0
$595$	−997.019			−0.0686955
$596$		0			0
$597$	7746.34	+	6560.49i	0.531050	+	0.449754i
$598$		0			0
$599$	5581.01	+	9666.59i	0.380691	+	0.659376i	0.991161	−	0.132663i	$-0.0423529\pi$
$599$	5581.01	+	9666.59i	0.380691	+	0.659376i	−0.610470	+	0.792039i	$0.709020\pi$
$600$		0			0
$601$	939.910	−	1627.97i	0.0637932	−	0.110493i	−0.832365	−	0.554228i	$-0.813014\pi$
$601$	939.910	−	1627.97i	0.0637932	−	0.110493i	0.896158	+	0.443735i	$0.146347\pi$
$602$		0			0
$603$	−15144.0	+	12473.3i	−1.02274	+	0.842376i
$604$		0			0
$605$	−5220.67	+	9042.46i	−0.350827	+	0.607650i
$606$		0			0
$607$	10662.2	+	18467.4i	0.712956	+	1.23488i	0.963743	+	0.266834i	$0.0859774\pi$
$607$	10662.2	+	18467.4i	0.712956	+	1.23488i	−0.250787	+	0.968042i	$0.580689\pi$
$608$		0			0
$609$	−175.938	+	971.810i	−0.0117067	+	0.0646629i
$610$		0			0
$611$	234.239			0.0155095
$612$		0			0
$613$	26096.2			1.71944			0.859720	−	0.510765i	$-0.170638\pi$
$613$	26096.2			1.71944			0.859720	+	0.510765i	$0.170638\pi$
$614$		0			0
$615$	385.578	−	138.347i	0.0252813	−	0.00907107i
$616$		0			0
$617$	9590.63	+	16611.5i	0.625777	+	1.08388i	0.988390	+	0.151937i	$0.0485512\pi$
$617$	9590.63	+	16611.5i	0.625777	+	1.08388i	−0.362614	+	0.931940i	$0.618116\pi$
$618$		0			0
$619$	1681.58	−	2912.58i	0.109190	−	0.189122i	−0.806253	−	0.591571i	$-0.798508\pi$
$619$	1681.58	−	2912.58i	0.109190	−	0.189122i	0.915442	+	0.402450i	$0.131841\pi$
$620$		0			0
$621$	5785.61	+	79.2993i	0.373862	+	0.00512427i
$622$		0			0
$623$	150.172	−	260.105i	0.00965731	−	0.0167270i
$624$		0			0
$625$	2171.62	+	3761.36i	0.138984	+	0.240727i
$626$		0			0
$627$	2261.82	−	811.553i	0.144064	−	0.0516911i
$628$		0			0
$629$	−48239.4			−3.05792
$630$		0			0
$631$	1485.04			0.0936900			0.0468450	−	0.998902i	$-0.485083\pi$
$631$	1485.04			0.0936900			0.0468450	+	0.998902i	$0.485083\pi$
$632$		0			0
$633$	2987.40	−	16501.2i	0.187581	−	1.03612i
$634$		0			0
$635$	−4946.78	−	8568.07i	−0.309145	−	0.535454i
$636$		0			0
$637$	−533.004	+	923.189i	−0.0331529	+	0.0574224i
$638$		0			0
$639$	−3688.32	−	22096.2i	−0.228338	−	1.36794i
$640$		0			0
$641$	−8208.10	+	14216.8i	−0.505773	+	0.876024i	0.494205	+	0.869345i	$0.335459\pi$
$641$	−8208.10	+	14216.8i	−0.505773	+	0.876024i	−0.999978	+	0.00667861i	$0.997874\pi$
$642$		0			0
$643$	10200.0	+	17666.9i	0.625580	+	1.08354i	0.988428	+	0.151688i	$0.0484710\pi$
$643$	10200.0	+	17666.9i	0.625580	+	1.08354i	−0.362848	+	0.931848i	$0.618196\pi$
$644$		0			0
$645$	11521.1	+	9757.40i	0.703323	+	0.595655i
$646$		0			0
$647$	25011.7			1.51980			0.759900	−	0.650040i	$-0.225248\pi$
$647$	25011.7			1.51980			0.759900	+	0.650040i	$0.225248\pi$
$648$		0			0
$649$	−2298.53			−0.139022
$650$		0			0
$651$	1184.51	+	1003.17i	0.0713125	+	0.0603956i
$652$		0			0
$653$	−3763.77	−	6519.03i	−0.225555	−	0.390673i	0.730931	−	0.682452i	$-0.239086\pi$
$653$	−3763.77	−	6519.03i	−0.225555	−	0.390673i	−0.956486	+	0.291779i	$0.905753\pi$
$654$		0			0
$655$	−5801.52	+	10048.5i	−0.346083	+	0.599433i
$656$		0			0
$657$	−711.729	−	4263.87i	−0.0422636	−	0.253195i
$658$		0			0
$659$	−14984.0	+	25953.0i	−0.885725	+	1.53412i	−0.0408452	+	0.999165i	$0.513005\pi$
$659$	−14984.0	+	25953.0i	−0.885725	+	1.53412i	−0.844880	+	0.534956i	$0.820328\pi$
$660$		0			0
$661$	−6153.81	−	10658.7i	−0.362111	−	0.627195i	0.626197	−	0.779665i	$-0.284611\pi$
$661$	−6153.81	−	10658.7i	−0.362111	−	0.627195i	−0.988308	+	0.152470i	$0.951277\pi$
$662$		0			0
$663$	−382.756	+	2114.19i	−0.0224208	+	0.123844i
$664$		0			0
$665$	651.057			0.0379653
$666$		0			0
$667$	−8363.73			−0.485525
$668$		0			0
$669$	−22771.7	+	8170.60i	−1.31600	+	0.472188i
$670$		0			0
$671$	−858.047	−	1486.18i	−0.0493659	−	0.0855043i
$672$		0			0
$673$	−1385.42	+	2399.61i	−0.0793521	+	0.137442i	−0.902971	−	0.429702i	$-0.858618\pi$
$673$	−1385.42	+	2399.61i	−0.0793521	+	0.137442i	0.823618	+	0.567144i	$0.191952\pi$
$674$		0			0
$675$	4159.23	+	7437.57i	0.237168	+	0.424107i
$676$		0			0
$677$	11232.5	−	19455.2i	0.637666	−	1.10447i	−0.348278	−	0.937391i	$-0.613233\pi$
$677$	11232.5	−	19455.2i	0.637666	−	1.10447i	0.985944	−	0.167078i	$-0.0534332\pi$
$678$		0			0
$679$	281.939	+	488.333i	0.0159349	+	0.0276001i
$680$		0			0
$681$	19958.6	−	7161.26i	1.12308	−	0.402966i
$682$		0			0
$683$	−7924.28			−0.443945			−0.221972	−	0.975053i	$-0.571249\pi$
$683$	−7924.28			−0.443945			−0.221972	+	0.975053i	$0.571249\pi$
$684$		0			0
$685$	−4439.25			−0.247613
$686$		0			0
$687$	1201.93	−	6638.97i	0.0667488	−	0.368694i
$688$		0			0
$689$	629.377	+	1090.11i	0.0348002	+	0.0602757i
$690$		0			0
$691$	11390.2	−	19728.4i	0.627067	−	1.08611i	−0.361070	−	0.932539i	$-0.617589\pi$
$691$	11390.2	−	19728.4i	0.627067	−	1.08611i	0.988137	−	0.153573i	$-0.0490780\pi$
$692$		0			0
$693$	104.240	−	85.8572i	0.00571394	−	0.00470627i
$694$		0			0
$695$	−6369.14	+	11031.7i	−0.347619	+	0.602094i
$696$		0			0
$697$	−652.547	−	1130.24i	−0.0354619	−	0.0614219i
$698$		0			0
$699$	−12186.0	−	10320.5i	−0.659397	−	0.558453i
$700$		0			0
$701$	−13999.0			−0.754256			−0.377128	−	0.926161i	$-0.623088\pi$
$701$	−13999.0			−0.754256			−0.377128	+	0.926161i	$0.623088\pi$
$702$		0			0
$703$	31500.5			1.68999
$704$		0			0
$705$	2389.54	+	2023.73i	0.127653	+	0.108111i
$706$		0			0
$707$	−410.659	−	711.283i	−0.0218450	−	0.0378367i
$708$		0			0
$709$	4513.32	−	7817.30i	0.239071	−	0.414083i	−0.721377	−	0.692543i	$-0.756490\pi$
$709$	4513.32	−	7817.30i	0.239071	−	0.414083i	0.960448	+	0.278459i	$0.0898238\pi$
$710$		0			0
$711$	23380.4	+	8752.51i	1.23324	+	0.461666i
$712$		0			0
$713$	−6572.67	+	11384.2i	−0.345229	+	0.597954i
$714$		0			0
$715$	−66.6490	−	115.439i	−0.00348606	−	0.00603803i
$716$		0			0
$717$	−3309.76	+	18281.8i	−0.172392	+	0.952225i
$718$		0			0
$719$	18206.9			0.944371			0.472186	−	0.881499i	$-0.343465\pi$
$719$	18206.9			0.944371			0.472186	+	0.881499i	$0.343465\pi$
$720$		0			0
$721$	−209.412			−0.0108168
$722$		0			0
$723$	13291.3	−	4768.99i	0.683692	−	0.245312i
$724$		0			0
$725$	−6158.83	−	10667.4i	−0.315494	−	0.546452i
$726$		0			0
$727$	−5167.05	+	8949.60i	−0.263598	+	0.456564i	−0.967195	−	0.254034i	$-0.918242\pi$
$727$	−5167.05	+	8949.60i	−0.263598	+	0.456564i	0.703598	+	0.710599i	$0.251576\pi$
$728$		0			0
$729$	−9370.62	−	17309.3i	−0.476077	−	0.879404i
$730$		0			0
$731$	24050.0	−	41655.8i	1.21686	−	2.10766i
$732$		0			0
$733$	−11176.3	−	19358.0i	−0.563176	−	0.975449i	−0.997217	−	0.0745557i	$-0.976246\pi$
$733$	−11176.3	−	19358.0i	−0.563176	−	0.975449i	0.434041	−	0.900893i	$-0.357087\pi$
$734$		0			0
$735$	−13413.3	+	4812.77i	−0.673141	+	0.241526i
$736$		0			0
$737$	−3877.90			−0.193818
$738$		0			0
$739$	−22423.2			−1.11617			−0.558086	−	0.829783i	$-0.688464\pi$
$739$	−22423.2			−1.11617			−0.558086	+	0.829783i	$0.688464\pi$
$740$		0			0
$741$	249.941	−	1380.57i	0.0123911	−	0.0684435i
$742$		0			0
$743$	−5827.69	−	10093.9i	−0.287749	−	0.498395i	0.685523	−	0.728051i	$-0.259573\pi$
$743$	−5827.69	−	10093.9i	−0.287749	−	0.498395i	−0.973272	+	0.229655i	$0.926240\pi$
$744$		0			0
$745$	−4861.10	+	8419.68i	−0.239056	+	0.414058i
$746$		0			0
$747$	−12938.7	−	4843.62i	−0.633736	−	0.237240i
$748$		0			0
$749$	592.957	−	1027.03i	0.0289268	−	0.0501027i
$750$		0			0
$751$	−1319.06	−	2284.68i	−0.0640923	−	0.111011i	0.832199	−	0.554478i	$-0.187082\pi$
$751$	−1319.06	−	2284.68i	−0.0640923	−	0.111011i	−0.896291	+	0.443466i	$0.853749\pi$
$752$		0			0
$753$	−18173.9	−	15391.7i	−0.879539	−	0.744895i
$754$		0			0
$755$	−4263.51			−0.205517
$756$		0			0
$757$	−19939.5			−0.957350			−0.478675	−	0.877992i	$-0.658883\pi$
$757$	−19939.5			−0.957350			−0.478675	+	0.877992i	$0.658883\pi$
$758$		0			0
$759$	872.730	+	739.128i	0.0417366	+	0.0353473i
$760$		0			0
$761$	13380.7	+	23176.0i	0.637385	+	1.10398i	0.986005	+	0.166719i	$0.0533172\pi$
$761$	13380.7	+	23176.0i	0.637385	+	1.10398i	−0.348620	+	0.937264i	$0.613349\pi$
$762$		0			0
$763$	99.8193	−	172.892i	0.00473618	−	0.00820330i
$764$		0			0
$765$	−22170.4	+	18260.6i	−1.04781	+	0.863024i
$766$		0			0
$767$	−671.007	+	1162.22i	−0.0315888	+	0.0547135i
$768$		0			0
$769$	−3788.40	−	6561.71i	−0.177651	−	0.307700i	0.763425	−	0.645897i	$-0.223516\pi$
$769$	−3788.40	−	6561.71i	−0.177651	−	0.307700i	−0.941075	+	0.338197i	$0.890183\pi$
$770$		0			0
$771$	5531.28	−	30552.6i	0.258371	−	1.42714i
$772$		0			0
$773$	27041.3			1.25823			0.629113	−	0.777314i	$-0.283418\pi$
$773$	27041.3			1.25823			0.629113	+	0.777314i	$0.283418\pi$
$774$		0			0
$775$	−19359.7			−0.897319
$776$		0			0
$777$	1666.28	−	597.870i	0.0769337	−	0.0276042i
$778$		0			0
$779$	426.116	+	738.054i	0.0195984	+	0.0339455i
$780$		0			0
$781$	2213.93	−	3834.64i	0.101435	−	0.175691i
$782$		0			0
$783$	13886.6	+	24832.2i	0.633802	+	1.13337i
$784$		0			0
$785$	8669.33	−	15015.7i	0.394168	−	0.682719i
$786$		0			0
$787$	−16802.1	−	29102.2i	−0.761031	−	1.31814i	−0.942320	−	0.334714i	$-0.891360\pi$
$787$	−16802.1	−	29102.2i	−0.761031	−	1.31814i	0.181289	−	0.983430i	$-0.441973\pi$
$788$		0			0
$789$	−30201.9	+	10836.6i	−1.36276	+	0.488964i
$790$		0			0
$791$	1512.67			0.0679953
$792$		0			0
$793$	−1001.96			−0.0448682
$794$		0			0
$795$	−2997.71	+	16558.1i	−0.133733	+	0.738688i
$796$		0			0
$797$	−21603.1	−	37417.7i	−0.960127	−	1.66299i	−0.722172	−	0.691714i	$-0.756856\pi$
$797$	−21603.1	−	37417.7i	−0.960127	−	1.66299i	−0.237955	−	0.971276i	$-0.576477\pi$
$798$		0			0
$799$	4988.09	−	8639.63i	0.220859	−	0.382538i
$800$		0			0
$801$	−1424.55	−	8534.30i	−0.0628391	−	0.376460i
$802$		0			0
$803$	427.218	−	739.964i	0.0187749	−	0.0325190i
$804$		0			0
$805$	154.929	+	268.345i	0.00678326	+	0.0117490i
$806$		0			0
$807$	26091.0	+	22096.8i	1.13810	+	0.963873i
$808$		0			0
$809$	33276.5			1.44616			0.723078	−	0.690766i	$-0.242727\pi$
$809$	33276.5			1.44616			0.723078	+	0.690766i	$0.242727\pi$
$810$		0			0
$811$	21917.4			0.948980			0.474490	−	0.880261i	$-0.342633\pi$
$811$	21917.4			0.948980			0.474490	+	0.880261i	$0.342633\pi$
$812$		0			0
$813$	23066.7	+	19535.5i	0.995061	+	0.842732i
$814$		0			0
$815$	−6421.47	−	11122.3i	−0.275993	−	0.478034i
$816$		0			0
$817$	−15704.7	+	27201.4i	−0.672509	+	1.16482i
$818$		0			0
$819$	−12.9817	−	77.7718i	−0.000553869	−	0.00331815i
$820$		0			0
$821$	536.732	−	929.647i	0.0228162	−	0.0395188i	−0.854392	−	0.519629i	$-0.826070\pi$
$821$	536.732	−	929.647i	0.0228162	−	0.0395188i	0.877208	+	0.480110i	$0.159403\pi$
$822$		0			0
$823$	−6229.21	−	10789.3i	−0.263835	−	0.456976i	0.703422	−	0.710772i	$-0.251654\pi$
$823$	−6229.21	−	10789.3i	−0.263835	−	0.456976i	−0.967258	+	0.253796i	$0.918321\pi$
$824$		0			0
$825$	−300.055	+	1657.38i	−0.0126625	+	0.0699427i
$826$		0			0
$827$	−902.282			−0.0379388			−0.0189694	−	0.999820i	$-0.506039\pi$
$827$	−902.282			−0.0379388			−0.0189694	+	0.999820i	$0.506039\pi$
$828$		0			0
$829$	29605.8			1.24035			0.620176	−	0.784462i	$-0.287061\pi$
$829$	29605.8			1.24035			0.620176	+	0.784462i	$0.287061\pi$
$830$		0			0
$831$	−42311.0	+	15181.4i	−1.76625	+	0.633740i
$832$		0			0
$833$	22700.5	+	39318.5i	0.944210	+	1.63542i
$834$		0			0
$835$	−6948.49	+	12035.1i	−0.287979	+	0.498794i
$836$		0			0
$837$	44712.8	+	612.847i	1.84648	+	0.0253084i
$838$		0			0
$839$	−22726.8	+	39363.9i	−0.935178	+	1.61978i	−0.160863	+	0.986977i	$0.551428\pi$
$839$	−22726.8	+	39363.9i	−0.935178	+	1.61978i	−0.774316	+	0.632799i	$0.781906\pi$
$840$		0			0
$841$	−8368.28	−	14494.3i	−0.343117	−	0.594296i
$842$		0			0
$843$	−35685.0	+	12804.0i	−1.45796	+	0.523122i
$844$		0			0
$845$	17533.9			0.713827
$846$		0			0
$847$	−1220.76			−0.0495229
$848$		0			0
$849$	−751.567	+	4151.36i	−0.0303813	+	0.167814i
$850$		0			0
$851$	7496.02	+	12983.5i	0.301951	+	0.522995i
$852$		0			0
$853$	−2795.92	+	4842.68i	−0.112228	+	0.194385i	−0.916668	−	0.399649i	$-0.869132\pi$
$853$	−2795.92	+	4842.68i	−0.112228	+	0.194385i	0.804440	+	0.594034i	$0.202465\pi$
$854$		0			0
$855$	14477.4	−	11924.2i	0.579082	−	0.476960i
$856$		0			0
$857$	20490.8	−	35491.1i	0.816747	−	1.41465i	−0.0913204	−	0.995822i	$-0.529109\pi$
$857$	20490.8	−	35491.1i	0.816747	−	1.41465i	0.908067	−	0.418825i	$-0.137558\pi$
$858$		0			0
$859$	−21149.0	−	36631.2i	−0.840040	−	1.45499i	−0.889860	−	0.456234i	$-0.849198\pi$
$859$	−21149.0	−	36631.2i	−0.840040	−	1.45499i	0.0498192	−	0.998758i	$-0.484135\pi$
$860$		0			0
$861$	36.5482	+	30.9532i	0.00144664	+	0.00122518i
$862$		0			0
$863$	19236.0			0.758749			0.379374	−	0.925243i	$-0.376139\pi$
$863$	19236.0			0.758749			0.379374	+	0.925243i	$0.376139\pi$
$864$		0			0
$865$	−15027.6			−0.590700
$866$		0			0
$867$	50347.8	+	42640.3i	1.97220	+	1.67029i
$868$		0			0
$869$	2467.23	+	4273.36i	0.0963118	+	0.166817i
$870$		0			0
$871$	−1132.07	+	1960.80i	−0.0440399	+	0.0762794i
$872$		0			0
$873$	15213.3	+	5695.14i	0.589796	+	0.220792i
$874$		0			0
$875$	−697.738	+	1208.52i	−0.0269576	+	0.0466919i
$876$		0			0
$877$	12060.4	+	20889.2i	0.464368	+	0.804308i	0.999173	−	0.0406670i	$-0.0129483\pi$
$877$	12060.4	+	20889.2i	0.464368	+	0.804308i	−0.534805	+	0.844975i	$0.679615\pi$
$878$		0			0
$879$	−4444.99	+	24552.4i	−0.170564	+	0.942128i
$880$		0			0
$881$	34292.4			1.31140			0.655698	−	0.755023i	$-0.272375\pi$
$881$	34292.4			1.31140			0.655698	+	0.755023i	$0.272375\pi$
$882$		0			0
$883$	−15818.9			−0.602887			−0.301444	−	0.953484i	$-0.597469\pi$
$883$	−15818.9			−0.602887			−0.301444	+	0.953484i	$0.597469\pi$
$884$		0			0
$885$	−16886.3	+	6058.87i	−0.641384	+	0.230132i
$886$		0			0
$887$	10381.8	+	17981.8i	0.392995	+	0.680688i	0.992843	−	0.119426i	$-0.0381055\pi$
$887$	10381.8	+	17981.8i	0.392995	+	0.680688i	−0.599848	+	0.800114i	$0.704772\pi$
$888$		0			0
$889$	578.359	−	1001.75i	0.0218195	−	0.0377925i
$890$		0			0
$891$	745.466	−	3818.36i	0.0280292	−	0.143569i
$892$		0			0
$893$	−3257.24	+	5641.71i	−0.122060	+	0.211414i
$894$		0			0
$895$	8248.75	+	14287.3i	0.308073	+	0.533598i
$896$		0			0
$897$	628.505	−	225.511i	0.0233948	−	0.00839419i
$898$		0			0
$899$	−64637.3			−2.39797
$900$		0			0
$901$	53610.1			1.98225
$902$		0			0
$903$	−314.458	+	1736.94i	−0.0115886	+	0.0640108i
$904$		0			0
$905$	−15785.8	−	27341.8i	−0.579821	−	1.00428i
$906$		0			0
$907$	6337.77	−	10977.3i	0.232020	−	0.401871i	−0.726382	−	0.687291i	$-0.758800\pi$
$907$	6337.77	−	10977.3i	0.232020	−	0.401871i	0.958402	+	0.285420i	$0.0921331\pi$
$908$		0			0
$909$	−22159.0	−	8295.27i	−0.808545	−	0.302681i
$910$		0			0
$911$	9162.83	−	15870.5i	0.333236	−	0.577182i	−0.649908	−	0.760013i	$-0.725193\pi$
$911$	9162.83	−	15870.5i	0.333236	−	0.577182i	0.983144	+	0.182831i	$0.0585261\pi$
$912$		0			0
$913$	−1365.36	−	2364.87i	−0.0494926	−	0.0857237i
$914$		0			0
$915$	−10221.2	−	8656.52i	−0.369294	−	0.312760i
$916$		0			0
$917$	−1356.58			−0.0488532
$918$		0			0
$919$	2706.30			0.0971409			0.0485705	−	0.998820i	$-0.484533\pi$
$919$	2706.30			0.0971409			0.0485705	+	0.998820i	$0.484533\pi$
$920$		0			0
$921$	−2994.30	−	2535.91i	−0.107129	−	0.0907288i
$922$		0			0
$923$	−1292.62	−	2238.89i	−0.0460966	−	0.0798417i
$924$		0			0
$925$	−11039.7	+	19121.4i	−0.392416	+	0.679684i
$926$		0			0
$927$	−4656.64	+	3835.43i	−0.164988	+	0.135892i
$928$		0			0
$929$	−12442.9	+	21551.7i	−0.439437	+	0.761127i	−0.997646	−	0.0685731i	$-0.978155\pi$
$929$	−12442.9	+	21551.7i	−0.439437	+	0.761127i	0.558209	+	0.829700i	$0.311489\pi$
$930$		0			0
$931$	−14823.5	−	25675.1i	−0.521828	−	0.903832i
$932$		0			0
$933$	4874.64	−	26925.6i	0.171049	−	0.944806i
$934$		0			0
$935$	−5677.13			−0.198569
$936$		0			0
$937$	−33383.1			−1.16390			−0.581952	−	0.813223i	$-0.697711\pi$
$937$	−33383.1			−1.16390			−0.581952	+	0.813223i	$0.697711\pi$
$938$		0			0
$939$	−6818.37	+	2446.47i	−0.236964	+	0.0850238i
$940$		0			0
$941$	14589.7	+	25270.1i	0.505431	+	0.875433i	0.999980	+	0.00628300i	$0.00199996\pi$
$941$	14589.7	+	25270.1i	0.505431	+	0.875433i	−0.494549	+	0.869150i	$0.664667\pi$
$942$		0			0
$943$	−202.801	+	351.262i	−0.00700330	+	0.0121301i
$944$		0			0
$945$	539.489	−	905.531i	0.0185710	−	0.0311713i
$946$		0			0
$947$	19442.0	−	33674.6i	0.667140	−	1.15552i	−0.311561	−	0.950226i	$-0.600852\pi$
$947$	19442.0	−	33674.6i	0.667140	−	1.15552i	0.978700	−	0.205294i	$-0.0658149\pi$
$948$		0			0
$949$	−249.435	−	432.034i	−0.00853214	−	0.0147781i
$950$		0			0
$951$	25027.0	−	8979.83i	0.853372	−	0.306194i
$952$		0			0
$953$	−5321.13			−0.180869			−0.0904345	−	0.995902i	$-0.528826\pi$
$953$	−5321.13			−0.180869			−0.0904345	+	0.995902i	$0.528826\pi$
$954$		0			0
$955$	35793.3			1.21282
$956$		0			0
$957$	−1001.81	+	5533.59i	−0.0338389	+	0.186913i
$958$		0			0
$959$	−259.511	−	449.485i	−0.00873830	−	0.0151352i
$960$		0			0
$961$	−35899.9	+	62180.5i	−1.20506	+	2.08722i
$962$		0			0
$963$	−5624.88	−	33697.9i	−0.188224	−	1.12762i
$964$		0			0
$965$	16633.9	−	28810.7i	0.554884	−	0.961087i
$966$		0			0
$967$	−24510.2	−	42453.0i	−0.815094	−	1.41178i	−0.909261	−	0.416227i	$-0.863352\pi$
$967$	−24510.2	−	42453.0i	−0.815094	−	1.41178i	0.0941669	−	0.995556i	$-0.469981\pi$
$968$		0			0
$969$	−45598.4	−	38618.0i	−1.51170	−	1.28028i
$970$		0			0
$971$	−11409.6			−0.377087			−0.188544	−	0.982065i	$-0.560377\pi$
$971$	−11409.6			−0.377087			−0.188544	+	0.982065i	$0.560377\pi$
$972$		0			0
$973$	−1489.31			−0.0490700
$974$		0			0
$975$	750.439	+	635.558i	0.0246495	+	0.0208760i
$976$		0			0
$977$	−9228.33	−	15983.9i	−0.302191	−	0.523410i	0.674441	−	0.738329i	$-0.264385\pi$
$977$	−9228.33	−	15983.9i	−0.302191	−	0.523410i	−0.976632	+	0.214919i	$0.931051\pi$
$978$		0			0
$979$	855.094	−	1481.07i	0.0279152	−	0.0483505i
$980$		0			0
$981$	−946.902	−	5672.76i	−0.0308178	−	0.184625i
$982$		0			0
$983$	−17803.6	+	30836.8i	−0.577668	+	1.00055i	0.418078	+	0.908411i	$0.362704\pi$
$983$	−17803.6	+	30836.8i	−0.577668	+	1.00055i	−0.995746	+	0.0921393i	$0.970629\pi$
$984$		0			0
$985$	2151.66	+	3726.78i	0.0696015	+	0.120553i
$986$		0			0
$987$	−65.2202	+	360.250i	−0.00210332	+	0.0116179i
$988$		0			0
$989$	−14948.7			−0.480629
$990$		0			0
$991$	31299.9			1.00330			0.501652	−	0.865070i	$-0.332726\pi$
$991$	31299.9			1.00330			0.501652	+	0.865070i	$0.332726\pi$
$992$		0			0
$993$	1187.50	−	426.080i	0.0379497	−	0.0136166i
$994$		0			0
$995$	7830.24	+	13562.4i	0.249483	+	0.432117i
$996$		0			0
$997$	14389.2	−	24922.9i	0.457083	−	0.791691i	−0.541722	−	0.840557i	$-0.682228\pi$
$997$	14389.2	−	24922.9i	0.457083	−	0.791691i	0.998805	+	0.0488666i	$0.0155609\pi$
$998$		0			0
$999$	26102.4	−	43812.9i	0.826672	−	1.38757i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.4.i.a.25.4	✓	8
3.2	odd	2		216.4.i.a.73.1		8
4.3	odd	2		144.4.i.e.97.1		8
9.2	odd	6		648.4.a.h.1.4		4
9.4	even	3	inner	72.4.i.a.49.4	yes	8
9.5	odd	6		216.4.i.a.145.1		8
9.7	even	3		648.4.a.i.1.1		4
12.11	even	2		432.4.i.e.289.1		8
36.7	odd	6		1296.4.a.ba.1.1		4
36.11	even	6		1296.4.a.y.1.4		4
36.23	even	6		432.4.i.e.145.1		8
36.31	odd	6		144.4.i.e.49.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.4.i.a.25.4	✓	8	1.1	even	1	trivial
72.4.i.a.49.4	yes	8	9.4	even	3	inner
144.4.i.e.49.1		8	36.31	odd	6
144.4.i.e.97.1		8	4.3	odd	2
216.4.i.a.73.1		8	3.2	odd	2
216.4.i.a.145.1		8	9.5	odd	6
432.4.i.e.145.1		8	36.23	even	6
432.4.i.e.289.1		8	12.11	even	2
648.4.a.h.1.4		4	9.2	odd	6
648.4.a.i.1.1		4	9.7	even	3
1296.4.a.y.1.4		4	36.11	even	6
1296.4.a.ba.1.1		4	36.7	odd	6

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:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	72.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(3.96518 + 3.35817i) q^{3} +(4.00813 + 6.94228i) q^{5} +(-0.468615 + 0.811666i) q^{7} +(4.44536 + 26.6315i) q^{9} +(-2.66835 + 4.62171i) q^{11} +(1.55794 + 2.69842i) q^{13} +(-7.42041 + 40.9874i) q^{15} +132.704 q^{17} -86.6565 q^{19} +(-4.58386 + 1.64471i) q^{21} +(-20.6212 - 35.7170i) q^{23} +(30.3698 - 52.6021i) q^{25} +(-71.8066 + 120.527i) q^{27} +(101.397 - 175.625i) q^{29} +(-159.367 - 276.031i) q^{31} +(-26.1010 + 9.36517i) q^{33} -7.51308 q^{35} -363.510 q^{37} +(-2.88427 + 15.9316i) q^{39} +(-4.91729 - 8.51700i) q^{41} +(181.230 - 313.899i) q^{43} +(-167.066 + 137.604i) q^{45} +(37.5880 - 65.1043i) q^{47} +(171.061 + 296.286i) q^{49} +(526.197 + 445.644i) q^{51} +403.981 q^{53} -42.7803 q^{55} +(-343.609 - 291.007i) q^{57} +(215.351 + 372.999i) q^{59} +(-160.783 + 278.484i) q^{61} +(-23.6991 - 8.87180i) q^{63} +(-12.4888 + 21.6313i) q^{65} +(363.324 + 629.296i) q^{67} +(38.1769 - 210.874i) q^{69} -829.702 q^{71} -160.106 q^{73} +(297.069 - 106.590i) q^{75} +(-2.50086 - 4.33161i) q^{77} +(462.314 - 800.752i) q^{79} +(-689.478 + 236.773i) q^{81} +(-255.843 + 443.134i) q^{83} +(531.896 + 921.271i) q^{85} +(991.838 - 355.876i) q^{87} -320.459 q^{89} -2.92029 q^{91} +(295.042 - 1629.69i) q^{93} +(-347.330 - 601.594i) q^{95} +(300.821 - 521.038i) q^{97} +(-134.945 - 50.5170i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 3 q^{3} - 5 q^{5} + 3 q^{7} - 15 q^{9} + 16 q^{11} + 29 q^{13} + 141 q^{15} - 34 q^{17} - 218 q^{19} + 27 q^{21} + 37 q^{23} + 97 q^{25} - 216 q^{27} - 3 q^{29} + 331 q^{31} + 468 q^{33} - 342 q^{35}+ \cdots - 5133 q^{99}+O(q^{100}) \) 8 * q - 3 * q^3 - 5 * q^5 + 3 * q^7 - 15 * q^9 + 16 * q^11 + 29 * q^13 + 141 * q^15 - 34 * q^17 - 218 * q^19 + 27 * q^21 + 37 * q^23 + 97 * q^25 - 216 * q^27 - 3 * q^29 + 331 * q^31 + 468 * q^33 - 342 * q^35 - 732 * q^37 - 375 * q^39 - 378 * q^41 + 506 * q^43 - 993 * q^45 + 171 * q^47 + 829 * q^49 + 2193 * q^51 + 820 * q^53 - 2326 * q^55 + 507 * q^57 + 616 * q^59 + 1331 * q^61 - 1899 * q^63 - 815 * q^65 + 1162 * q^67 + 1113 * q^69 - 688 * q^71 - 2614 * q^73 - 1755 * q^75 - 741 * q^77 + 1853 * q^79 - 2547 * q^81 + 1421 * q^83 + 2074 * q^85 + 5877 * q^87 + 1632 * q^89 - 3990 * q^91 + 537 * q^93 + 1292 * q^95 + 2506 * q^97 - 5133 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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