LMFDB - Embedded newform 144.3.j.a.53.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,3,Mod(53,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1, 2]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("144.53");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	144.j (of order $4$, 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:	$3.92371580679$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			53.1
Character	$\chi$	$=$	144.53
Dual form			144.3.j.a.125.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.99710 - 0.107698i) q^{2} +(3.97680 + 0.430168i) q^{4} +(-1.92848 + 1.92848i) q^{5} +2.43162i q^{7} +(-7.89574 - 1.28738i) q^{8} +O(q^{10})$	\(q+(-1.99710 - 0.107698i) q^{2} +(3.97680 + 0.430168i) q^{4} +(-1.92848 + 1.92848i) q^{5} +2.43162i q^{7} +(-7.89574 - 1.28738i) q^{8} +(4.05906 - 3.64367i) q^{10} +(-2.79147 + 2.79147i) q^{11} +(-1.54064 + 1.54064i) q^{13} +(0.261881 - 4.85619i) q^{14} +(15.6299 + 3.42138i) q^{16} +20.3746i q^{17} +(-25.0179 + 25.0179i) q^{19} +(-8.49876 + 6.83962i) q^{20} +(5.87548 - 5.27421i) q^{22} -3.55894 q^{23} +17.5619i q^{25} +(3.24273 - 2.91088i) q^{26} +(-1.04601 + 9.67008i) q^{28} +(23.4539 + 23.4539i) q^{29} -13.3241 q^{31} +(-30.8460 - 8.51615i) q^{32} +(2.19430 - 40.6900i) q^{34} +(-4.68934 - 4.68934i) q^{35} +(-25.4406 - 25.4406i) q^{37} +(52.6575 - 47.2687i) q^{38} +(17.7095 - 12.7441i) q^{40} +64.0726 q^{41} +(-24.6791 - 24.6791i) q^{43} +(-12.3019 + 9.90033i) q^{44} +(7.10756 + 0.383292i) q^{46} -79.5718i q^{47} +43.0872 q^{49} +(1.89139 - 35.0729i) q^{50} +(-6.78954 + 5.46408i) q^{52} +(-39.8061 + 39.8061i) q^{53} -10.7666i q^{55} +(3.13043 - 19.1994i) q^{56} +(-44.3138 - 49.3657i) q^{58} +(19.2371 - 19.2371i) q^{59} +(63.5441 - 63.5441i) q^{61} +(26.6094 + 1.43498i) q^{62} +(60.6853 + 20.3297i) q^{64} -5.94218i q^{65} +(-65.1837 + 65.1837i) q^{67} +(-8.76448 + 81.0256i) q^{68} +(8.86003 + 9.87010i) q^{70} -84.6981 q^{71} +39.7473i q^{73} +(48.0675 + 53.5473i) q^{74} +(-110.253 + 88.7292i) q^{76} +(-6.78781 - 6.78781i) q^{77} +109.386 q^{79} +(-36.7401 + 23.5439i) q^{80} +(-127.959 - 6.90050i) q^{82} +(14.7433 + 14.7433i) q^{83} +(-39.2920 - 39.2920i) q^{85} +(46.6287 + 51.9444i) q^{86} +(25.6344 - 18.4470i) q^{88} +32.3085 q^{89} +(-3.74625 - 3.74625i) q^{91} +(-14.1532 - 1.53094i) q^{92} +(-8.56974 + 158.913i) q^{94} -96.4929i q^{95} +123.467 q^{97} +(-86.0494 - 4.64042i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q + 40 q^{10} + 48 q^{16} + 64 q^{19} - 88 q^{22} - 120 q^{28} - 248 q^{34} - 184 q^{40} + 128 q^{43} + 24 q^{46} - 224 q^{49} + 632 q^{52} + 456 q^{58} + 64 q^{61} - 48 q^{64} - 64 q^{67} - 312 q^{70} - 576 q^{76} - 512 q^{79} - 720 q^{82} + 320 q^{85} - 400 q^{88} - 192 q^{91} + 696 q^{94}+O(q^{100}) $ 32 * q + 40 * q^10 + 48 * q^16 + 64 * q^19 - 88 * q^22 - 120 * q^28 - 248 * q^34 - 184 * q^40 + 128 * q^43 + 24 * q^46 - 224 * q^49 + 632 * q^52 + 456 * q^58 + 64 * q^61 - 48 * q^64 - 64 * q^67 - 312 * q^70 - 576 * q^76 - 512 * q^79 - 720 * q^82 + 320 * q^85 - 400 * q^88 - 192 * q^91 + 696 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−1.99710	−	0.107698i	−0.998549	−	0.0538491i
$2$	−1.99710	−	0.107698i	−0.998549	−	0.0538491i
$3$		0			0
$3$		0			0
$4$	3.97680	+	0.430168i	0.994201	+	0.107542i
$5$	−1.92848	+	1.92848i	−0.385696	+	0.385696i	−0.873149	−	0.487453i	$-0.837926\pi$
$5$	−1.92848	+	1.92848i	−0.385696	+	0.385696i	0.487453	+	0.873149i	$0.337926\pi$
$6$		0			0
$7$			2.43162i			0.347375i	0.984801	+	0.173687i	$0.0555682\pi$
$7$			2.43162i			0.347375i	−0.984801	+	0.173687i	$0.944432\pi$
$8$	−7.89574	−	1.28738i	−0.986967	−	0.160923i
$9$		0			0
$10$	4.05906	−	3.64367i	0.405906	−	0.364367i
$11$	−2.79147	+	2.79147i	−0.253770	+	0.253770i	−0.822514	−	0.568744i	$-0.807429\pi$
$11$	−2.79147	+	2.79147i	−0.253770	+	0.253770i	0.568744	+	0.822514i	$0.307429\pi$
$12$		0			0
$13$	−1.54064	+	1.54064i	−0.118511	+	0.118511i	−0.763875	−	0.645364i	$-0.776706\pi$
$13$	−1.54064	+	1.54064i	−0.118511	+	0.118511i	0.645364	+	0.763875i	$0.276706\pi$
$14$	0.261881	−	4.85619i	0.0187058	−	0.346871i
$15$		0			0
$16$	15.6299	+	3.42138i	0.976869	+	0.213837i
$17$			20.3746i			1.19850i	0.800561	+	0.599252i	$0.204535\pi$
$17$			20.3746i			1.19850i	−0.800561	+	0.599252i	$0.795465\pi$
$18$		0			0
$19$	−25.0179	+	25.0179i	−1.31673	+	1.31673i	−0.400380	+	0.916349i	$0.631122\pi$
$19$	−25.0179	+	25.0179i	−1.31673	+	1.31673i	−0.916349	+	0.400380i	$0.868878\pi$
$20$	−8.49876	+	6.83962i	−0.424938	+	0.341981i
$21$		0			0
$22$	5.87548	−	5.27421i	0.267067	−	0.239737i
$23$	−3.55894			−0.154737			−0.0773683	−	0.997003i	$-0.524652\pi$
$23$	−3.55894			−0.154737			−0.0773683	+	0.997003i	$0.524652\pi$
$24$		0			0
$25$			17.5619i			0.702477i
$26$	3.24273	−	2.91088i	0.124720	−	0.111957i
$27$		0			0
$28$	−1.04601	+	9.67008i	−0.0373573	+	0.345360i
$29$	23.4539	+	23.4539i	0.808756	+	0.808756i	0.984446	−	0.175690i	$-0.0562156\pi$
$29$	23.4539	+	23.4539i	0.808756	+	0.808756i	−0.175690	+	0.984446i	$0.556216\pi$
$30$		0			0
$31$	−13.3241			−0.429808			−0.214904	−	0.976635i	$-0.568944\pi$
$31$	−13.3241			−0.429808			−0.214904	+	0.976635i	$0.568944\pi$
$32$	−30.8460	−	8.51615i	−0.963937	−	0.266130i
$33$		0			0
$34$	2.19430	−	40.6900i	0.0645384	−	1.19676i
$35$	−4.68934	−	4.68934i	−0.133981	−	0.133981i
$36$		0			0
$37$	−25.4406	−	25.4406i	−0.687584	−	0.687584i	0.274113	−	0.961697i	$-0.411616\pi$
$37$	−25.4406	−	25.4406i	−0.687584	−	0.687584i	−0.961697	+	0.274113i	$0.911616\pi$
$38$	52.6575	−	47.2687i	1.38572	−	1.24391i
$39$		0			0
$40$	17.7095	−	12.7441i	0.442737	−	0.318602i
$41$	64.0726			1.56275			0.781373	−	0.624064i	$-0.214520\pi$
$41$	64.0726			1.56275			0.781373	+	0.624064i	$0.214520\pi$
$42$		0			0
$43$	−24.6791	−	24.6791i	−0.573932	−	0.573932i	0.359293	−	0.933225i	$-0.383018\pi$
$43$	−24.6791	−	24.6791i	−0.573932	−	0.573932i	−0.933225	+	0.359293i	$0.883018\pi$
$44$	−12.3019	+	9.90033i	−0.279589	+	0.225008i
$45$		0			0
$46$	7.10756	+	0.383292i	0.154512	+	0.00833243i
$47$		−	79.5718i		−	1.69302i	−0.532375	−	0.846508i	$-0.678701\pi$
$47$		−	79.5718i		−	1.69302i	0.532375	−	0.846508i	$-0.321299\pi$
$48$		0			0
$49$	43.0872			0.879331
$50$	1.89139	−	35.0729i	0.0378277	−	0.701457i
$51$		0			0
$52$	−6.78954	+	5.46408i	−0.130568	+	0.105078i
$53$	−39.8061	+	39.8061i	−0.751058	+	0.751058i	−0.974677	−	0.223619i	$-0.928213\pi$
$53$	−39.8061	+	39.8061i	−0.751058	+	0.751058i	0.223619	+	0.974677i	$0.428213\pi$
$54$		0			0
$55$		−	10.7666i		−	0.195757i
$56$	3.13043	−	19.1994i	0.0559005	−	0.342847i
$57$		0			0
$58$	−44.3138	−	49.3657i	−0.764031	−	0.851133i
$59$	19.2371	−	19.2371i	0.326053	−	0.326053i	−0.525030	−	0.851083i	$-0.675946\pi$
$59$	19.2371	−	19.2371i	0.326053	−	0.326053i	0.851083	+	0.525030i	$0.175946\pi$
$60$		0			0
$61$	63.5441	−	63.5441i	1.04171	−	1.04171i	0.0426158	−	0.999092i	$-0.486431\pi$
$61$	63.5441	−	63.5441i	1.04171	−	1.04171i	0.999092	−	0.0426158i	$-0.0135691\pi$
$62$	26.6094	+	1.43498i	0.429185	+	0.0231448i
$63$		0			0
$64$	60.6853	+	20.3297i	0.948208	+	0.317651i
$65$		−	5.94218i		−	0.0914182i
$66$		0			0
$67$	−65.1837	+	65.1837i	−0.972891	+	0.972891i	−0.999642	−	0.0267514i	$-0.991484\pi$
$67$	−65.1837	+	65.1837i	−0.972891	+	0.972891i	0.0267514	+	0.999642i	$0.491484\pi$
$68$	−8.76448	+	81.0256i	−0.128889	+	1.19155i
$69$		0			0
$70$	8.86003	+	9.87010i	0.126572	+	0.141001i
$71$	−84.6981			−1.19293			−0.596466	−	0.802639i	$-0.703429\pi$
$71$	−84.6981			−1.19293			−0.596466	+	0.802639i	$0.703429\pi$
$72$		0			0
$73$			39.7473i			0.544483i	0.962229	+	0.272242i	$0.0877650\pi$
$73$			39.7473i			0.544483i	−0.962229	+	0.272242i	$0.912235\pi$
$74$	48.0675	+	53.5473i	0.649561	+	0.723612i
$75$		0			0
$76$	−110.253	+	88.7292i	−1.45070	+	1.16749i
$77$	−6.78781	−	6.78781i	−0.0881533	−	0.0881533i
$78$		0			0
$79$	109.386			1.38463			0.692317	−	0.721593i	$-0.256590\pi$
$79$	109.386			1.38463			0.692317	+	0.721593i	$0.256590\pi$
$80$	−36.7401	+	23.5439i	−0.459251	+	0.294299i
$81$		0			0
$82$	−127.959	−	6.90050i	−1.56048	−	0.0841525i
$83$	14.7433	+	14.7433i	0.177630	+	0.177630i	0.790322	−	0.612692i	$-0.209913\pi$
$83$	14.7433	+	14.7433i	0.177630	+	0.177630i	−0.612692	+	0.790322i	$0.709913\pi$
$84$		0			0
$85$	−39.2920	−	39.2920i	−0.462258	−	0.462258i
$86$	46.6287	+	51.9444i	0.542194	+	0.604005i
$87$		0			0
$88$	25.6344	−	18.4470i	0.291300	−	0.209625i
$89$	32.3085			0.363017			0.181508	−	0.983389i	$-0.441902\pi$
$89$	32.3085			0.363017			0.181508	+	0.983389i	$0.441902\pi$
$90$		0			0
$91$	−3.74625	−	3.74625i	−0.0411676	−	0.0411676i
$92$	−14.1532	−	1.53094i	−0.153839	−	0.0166407i
$93$		0			0
$94$	−8.56974	+	158.913i	−0.0911674	+	1.69056i
$95$		−	96.4929i		−	1.01572i
$96$		0			0
$97$	123.467			1.27286			0.636428	−	0.771336i	$-0.280411\pi$
$97$	123.467			1.27286			0.636428	+	0.771336i	$0.280411\pi$
$98$	−86.0494	−	4.64042i	−0.878055	−	0.0473512i
$99$		0			0
$100$	−7.55457	+	69.8403i	−0.0755457	+	0.698403i
$101$	−83.5696	+	83.5696i	−0.827422	+	0.827422i	−0.987160	−	0.159737i	$-0.948935\pi$
$101$	−83.5696	+	83.5696i	−0.827422	+	0.827422i	0.159737	+	0.987160i	$0.448935\pi$
$102$		0			0
$103$		−	38.0120i		−	0.369049i	−0.982828	−	0.184524i	$-0.940926\pi$
$103$		−	38.0120i		−	0.369049i	0.982828	−	0.184524i	$-0.0590744\pi$
$104$	14.1479	−	10.1811i	0.136037	−	0.0978950i
$105$		0			0
$106$	83.7837	−	75.2096i	0.790412	−	0.709524i
$107$	−69.3864	+	69.3864i	−0.648471	+	0.648471i	−0.952623	−	0.304153i	$-0.901627\pi$
$107$	−69.3864	+	69.3864i	−0.648471	+	0.648471i	0.304153	+	0.952623i	$0.401627\pi$
$108$		0			0
$109$	46.2338	−	46.2338i	0.424163	−	0.424163i	−0.462471	−	0.886634i	$-0.653037\pi$
$109$	46.2338	−	46.2338i	0.424163	−	0.424163i	0.886634	+	0.462471i	$0.153037\pi$
$110$	−1.15954	+	21.5020i	−0.0105413	+	0.195472i
$111$		0			0
$112$	−8.31951	+	38.0060i	−0.0742814	+	0.339340i
$113$		−	97.1603i		−	0.859826i	−0.902870	−	0.429913i	$-0.858544\pi$
$113$		−	97.1603i		−	0.859826i	0.902870	−	0.429913i	$-0.141456\pi$
$114$		0			0
$115$	6.86335	−	6.86335i	0.0596813	−	0.0596813i
$116$	83.1825	+	103.361i	0.717090	+	0.891040i
$117$		0			0
$118$	−40.4902	+	36.3466i	−0.343138	+	0.308022i
$119$	−49.5432			−0.416330
$120$		0			0
$121$			105.415i			0.871201i
$122$	−133.747	+	120.060i	−1.09629	+	0.984101i
$123$		0			0
$124$	−52.9871	−	5.73158i	−0.427316	−	0.0462224i
$125$	−82.0799	−	82.0799i	−0.656639	−	0.656639i
$126$		0			0
$127$	187.433			1.47585			0.737926	−	0.674881i	$-0.235805\pi$
$127$	187.433			1.47585			0.737926	+	0.674881i	$0.235805\pi$
$128$	−119.005	−	47.1360i	−0.929727	−	0.368250i
$129$		0			0
$130$	−0.639962	+	11.8671i	−0.00492279	+	0.0912855i
$131$	79.2297	+	79.2297i	0.604807	+	0.604807i	0.941584	−	0.336778i	$-0.109337\pi$
$131$	79.2297	+	79.2297i	0.604807	+	0.604807i	−0.336778	+	0.941584i	$0.609337\pi$
$132$		0			0
$133$	−60.8340	−	60.8340i	−0.457398	−	0.457398i
$134$	137.198	−	123.158i	1.02387	−	0.919090i
$135$		0			0
$136$	26.2298	−	160.872i	0.192866	−	1.18288i
$137$	3.46186			0.0252691			0.0126345	−	0.999920i	$-0.495978\pi$
$137$	3.46186			0.0252691			0.0126345	+	0.999920i	$0.495978\pi$
$138$		0			0
$139$	−14.1054	−	14.1054i	−0.101478	−	0.101478i	0.654545	−	0.756023i	$-0.272860\pi$
$139$	−14.1054	−	14.1054i	−0.101478	−	0.101478i	−0.756023	+	0.654545i	$0.772860\pi$
$140$	−16.6314	−	20.6658i	−0.118795	−	0.147613i
$141$		0			0
$142$	169.150	+	9.12183i	1.19120	+	0.0642383i
$143$		−	8.60130i		−	0.0601489i
$144$		0			0
$145$	−90.4609			−0.623868
$146$	4.28071	−	79.3793i	0.0293200	−	0.543693i
$147$		0			0
$148$	−90.2285	−	112.116i	−0.609652	−	0.757541i
$149$	85.7760	−	85.7760i	0.575678	−	0.575678i	−0.358032	−	0.933709i	$-0.616552\pi$
$149$	85.7760	−	85.7760i	0.575678	−	0.575678i	0.933709	+	0.358032i	$0.116552\pi$
$150$		0			0
$151$		−	221.163i		−	1.46465i	−0.680953	−	0.732327i	$-0.738434\pi$
$151$		−	221.163i		−	1.46465i	0.680953	−	0.732327i	$-0.261566\pi$
$152$	229.742	−	165.327i	1.51146	−	1.08768i
$153$		0			0
$154$	12.8249	+	14.2870i	0.0832784	+	0.0927724i
$155$	25.6952	−	25.6952i	0.165775	−	0.165775i
$156$		0			0
$157$	−114.407	+	114.407i	−0.728709	+	0.728709i	−0.970363	−	0.241654i	$-0.922310\pi$
$157$	−114.407	+	114.407i	−0.728709	+	0.728709i	0.241654	+	0.970363i	$0.422310\pi$
$158$	−218.455	−	11.7807i	−1.38263	−	0.0745613i
$159$		0			0
$160$	75.9092	−	43.0627i	0.474432	−	0.269142i
$161$		−	8.65400i		−	0.0537516i
$162$		0			0
$163$	−124.266	+	124.266i	−0.762366	+	0.762366i	−0.976750	−	0.214383i	$-0.931226\pi$
$163$	−124.266	+	124.266i	−0.762366	+	0.762366i	0.214383	+	0.976750i	$0.431226\pi$
$164$	254.804	+	27.5620i	1.55368	+	0.168061i
$165$		0			0
$166$	−27.8559	−	31.0316i	−0.167807	−	0.186937i
$167$	230.474			1.38008			0.690041	−	0.723770i	$-0.257592\pi$
$167$	230.474			1.38008			0.690041	+	0.723770i	$0.257592\pi$
$168$		0			0
$169$			164.253i			0.971910i
$170$	74.2382	+	82.7016i	0.436696	+	0.486480i
$171$		0			0
$172$	−87.5277	−	108.760i	−0.508882	−	0.632325i
$173$	−43.4882	−	43.4882i	−0.251377	−	0.251377i	0.570158	−	0.821535i	$-0.306882\pi$
$173$	−43.4882	−	43.4882i	−0.251377	−	0.251377i	−0.821535	+	0.570158i	$0.806882\pi$
$174$		0			0
$175$	−42.7039			−0.244023
$176$	−53.1812	+	34.0798i	−0.302166	+	0.193635i
$177$		0			0
$178$	−64.5233	−	3.47957i	−0.362490	−	0.0195481i
$179$	145.904	+	145.904i	0.815104	+	0.815104i	0.985394	−	0.170290i	$-0.0544704\pi$
$179$	145.904	+	145.904i	0.815104	+	0.815104i	−0.170290	+	0.985394i	$0.554470\pi$
$180$		0			0
$181$	−184.775	−	184.775i	−1.02085	−	1.02085i	−0.999778	−	0.0210758i	$-0.993291\pi$
$181$	−184.775	−	184.775i	−1.02085	−	1.02085i	−0.0210758	−	0.999778i	$-0.506709\pi$
$182$	7.07816	+	7.88509i	0.0388910	+	0.0433247i
$183$		0			0
$184$	28.1005	+	4.58172i	0.152720	+	0.0249006i
$185$	98.1235			0.530397
$186$		0			0
$187$	−56.8750	−	56.8750i	−0.304145	−	0.304145i
$188$	34.2292	−	316.441i	0.182070	−	1.68320i
$189$		0			0
$190$	−10.3921	+	192.706i	−0.0546954	+	1.01424i
$191$			352.205i			1.84401i	0.387182	+	0.922003i	$0.373449\pi$
$191$			352.205i			1.84401i	−0.387182	+	0.922003i	$0.626551\pi$
$192$		0			0
$193$	−145.660			−0.754715			−0.377357	−	0.926068i	$-0.623167\pi$
$193$	−145.660			−0.754715			−0.377357	+	0.926068i	$0.623167\pi$
$194$	−246.576	−	13.2972i	−1.27101	−	0.0685422i
$195$		0			0
$196$	171.349	+	18.5347i	0.874231	+	0.0945650i
$197$	−18.7746	+	18.7746i	−0.0953027	+	0.0953027i	−0.753151	−	0.657848i	$-0.771467\pi$
$197$	−18.7746	+	18.7746i	−0.0953027	+	0.0953027i	0.657848	+	0.753151i	$0.271467\pi$
$198$		0			0
$199$			260.124i			1.30716i	0.756858	+	0.653579i	$0.226733\pi$
$199$			260.124i			1.30716i	−0.756858	+	0.653579i	$0.773267\pi$
$200$	22.6089	−	138.664i	0.113044	−	0.693321i
$201$		0			0
$202$	175.897	−	157.896i	0.870778	−	0.781666i
$203$	−57.0310	+	57.0310i	−0.280941	+	0.280941i
$204$		0			0
$205$	−123.563	+	123.563i	−0.602745	+	0.602745i
$206$	−4.09382	+	75.9137i	−0.0198729	+	0.368513i
$207$		0			0
$208$	−29.3511	+	18.8089i	−0.141111	+	0.0904275i
$209$		−	139.673i		−	0.668293i
$210$		0			0
$211$	127.020	−	127.020i	0.601991	−	0.601991i	−0.338850	−	0.940840i	$-0.610038\pi$
$211$	127.020	−	127.020i	0.601991	−	0.601991i	0.940840	+	0.338850i	$0.110038\pi$
$212$	−175.424	+	141.178i	−0.827472	+	0.665932i
$213$		0			0
$214$	146.044	−	131.099i	0.682449	−	0.612610i
$215$	95.1863			0.442727
$216$		0			0
$217$		−	32.3991i		−	0.149304i
$218$	−97.3127	+	87.3541i	−0.446388	+	0.400707i
$219$		0			0
$220$	4.63145	−	42.8167i	0.0210520	−	0.194621i
$221$	−31.3898	−	31.3898i	−0.142035	−	0.142035i
$222$		0			0
$223$	−70.2119			−0.314852			−0.157426	−	0.987531i	$-0.550320\pi$
$223$	−70.2119			−0.314852			−0.157426	+	0.987531i	$0.550320\pi$
$224$	20.7081	−	75.0058i	0.0924467	−	0.334847i
$225$		0			0
$226$	−10.4640	+	194.039i	−0.0463009	+	0.858579i
$227$	57.6438	+	57.6438i	0.253938	+	0.253938i	0.822583	−	0.568645i	$-0.192532\pi$
$227$	57.6438	+	57.6438i	0.253938	+	0.253938i	−0.568645	+	0.822583i	$0.692532\pi$
$228$		0			0
$229$	136.406	+	136.406i	0.595660	+	0.595660i	0.939155	−	0.343495i	$-0.111611\pi$
$229$	136.406	+	136.406i	0.595660	+	0.595660i	−0.343495	+	0.939155i	$0.611611\pi$
$230$	−14.4460	+	12.9676i	−0.0628085	+	0.0563810i
$231$		0			0
$232$	−154.992	−	215.380i	−0.668068	−	0.928362i
$233$	−171.152			−0.734556			−0.367278	−	0.930111i	$-0.619710\pi$
$233$	−171.152			−0.734556			−0.367278	+	0.930111i	$0.619710\pi$
$234$		0			0
$235$	153.453	+	153.453i	0.652990	+	0.652990i
$236$	84.7774	−	68.2270i	0.359226	−	0.289098i
$237$		0			0
$238$	98.9427	+	5.33572i	0.415726	+	0.0224190i
$239$		−	79.7138i		−	0.333531i	−0.985997	−	0.166765i	$-0.946668\pi$
$239$		−	79.7138i		−	0.333531i	0.985997	−	0.166765i	$-0.0533322\pi$
$240$		0			0
$241$	−68.5687			−0.284517			−0.142259	−	0.989830i	$-0.545436\pi$
$241$	−68.5687			−0.284517			−0.142259	+	0.989830i	$0.545436\pi$
$242$	11.3530	−	210.525i	0.0469134	−	0.869937i
$243$		0			0
$244$	280.037	−	225.368i	1.14769	−	0.923639i
$245$	−83.0929	+	83.0929i	−0.339155	+	0.339155i
$246$		0			0
$247$		−	77.0869i		−	0.312093i
$248$	105.203	+	17.1531i	0.424207	+	0.0691659i
$249$		0			0
$250$	155.082	+	172.761i	0.620327	+	0.691046i
$251$	−11.6232	+	11.6232i	−0.0463074	+	0.0463074i	−0.729881	−	0.683574i	$-0.760425\pi$
$251$	−11.6232	+	11.6232i	−0.0463074	+	0.0463074i	0.683574	+	0.729881i	$0.260425\pi$
$252$		0			0
$253$	9.93469	−	9.93469i	0.0392675	−	0.0392675i
$254$	−374.323	−	20.1862i	−1.47371	−	0.0794733i
$255$		0			0
$256$	232.588	+	106.952i	0.908548	+	0.417781i
$257$			44.8750i			0.174611i	0.996182	+	0.0873055i	$0.0278256\pi$
$257$			44.8750i			0.174611i	−0.996182	+	0.0873055i	$0.972174\pi$
$258$		0			0
$259$	61.8619	−	61.8619i	0.238849	−	0.238849i
$260$	2.55614	−	23.6309i	0.00983129	−	0.0908880i
$261$		0			0
$262$	−149.697	−	166.762i	−0.571361	−	0.636498i
$263$	−384.441			−1.46175			−0.730876	−	0.682510i	$-0.760888\pi$
$263$	−384.441			−1.46175			−0.730876	+	0.682510i	$0.760888\pi$
$264$		0			0
$265$		−	153.531i		−	0.579360i
$266$	114.940	+	128.043i	0.432104	+	0.481365i
$267$		0			0
$268$	−287.263	+	231.183i	−1.07188	+	0.862622i
$269$	−0.844060	−	0.844060i	−0.00313777	−	0.00313777i	0.705536	−	0.708674i	$-0.250706\pi$
$269$	−0.844060	−	0.844060i	−0.00313777	−	0.00313777i	−0.708674	+	0.705536i	$0.750706\pi$
$270$		0			0
$271$	−37.3816			−0.137939			−0.0689697	−	0.997619i	$-0.521971\pi$
$271$	−37.3816			−0.137939			−0.0689697	+	0.997619i	$0.521971\pi$
$272$	−69.7092	+	318.453i	−0.256284	+	1.17078i
$273$		0			0
$274$	−6.91368	−	0.372837i	−0.0252324	−	0.00136072i
$275$	−49.0236	−	49.0236i	−0.178268	−	0.178268i
$276$		0			0
$277$	342.799	+	342.799i	1.23754	+	1.23754i	0.961002	+	0.276541i	$0.0891882\pi$
$277$	342.799	+	342.799i	1.23754	+	1.23754i	0.276541	+	0.961002i	$0.410812\pi$
$278$	26.6507	+	29.6890i	0.0958659	+	0.106795i
$279$		0			0
$280$	30.9888	+	43.0627i	0.110674	+	0.153796i
$281$	−303.846			−1.08130			−0.540651	−	0.841247i	$-0.681822\pi$
$281$	−303.846			−1.08130			−0.540651	+	0.841247i	$0.681822\pi$
$282$		0			0
$283$	−10.3395	−	10.3395i	−0.0365354	−	0.0365354i	0.688603	−	0.725138i	$-0.258224\pi$
$283$	−10.3395	−	10.3395i	−0.0365354	−	0.0365354i	−0.725138	+	0.688603i	$0.758224\pi$
$284$	−336.828	−	36.4344i	−1.18601	−	0.128290i
$285$		0			0
$286$	−0.926344	+	17.1776i	−0.00323897	+	0.0600617i
$287$			155.800i			0.542858i
$288$		0			0
$289$	−126.123			−0.436411
$290$	180.659	+	9.74248i	0.622963	+	0.0335947i
$291$		0			0
$292$	−17.0980	+	158.067i	−0.0585548	+	0.541326i
$293$	230.634	−	230.634i	0.787148	−	0.787148i	−0.193877	−	0.981026i	$-0.562106\pi$
$293$	230.634	−	230.634i	0.787148	−	0.787148i	0.981026	+	0.193877i	$0.0621064\pi$
$294$		0			0
$295$			74.1969i			0.251515i
$296$	168.121	+	233.624i	0.567975	+	0.789271i
$297$		0			0
$298$	−180.541	+	162.065i	−0.605842	+	0.543843i
$299$	5.48304	−	5.48304i	0.0183379	−	0.0183379i
$300$		0			0
$301$	60.0102	−	60.0102i	0.199369	−	0.199369i
$302$	−23.8188	+	441.684i	−0.0788703	+	1.46253i
$303$		0			0
$304$	−476.623	+	305.431i	−1.56784	+	1.00471i
$305$			245.087i			0.803565i
$306$		0			0
$307$	255.002	−	255.002i	0.830625	−	0.830625i	−0.156978	−	0.987602i	$-0.550175\pi$
$307$	255.002	−	255.002i	0.830625	−	0.830625i	0.987602	+	0.156978i	$0.0501750\pi$
$308$	−24.0739	−	29.9137i	−0.0781619	−	0.0971223i
$309$		0			0
$310$	−54.0831	+	48.5485i	−0.174462	+	0.156608i
$311$	333.987			1.07391			0.536957	−	0.843610i	$-0.319574\pi$
$311$	333.987			1.07391			0.536957	+	0.843610i	$0.319574\pi$
$312$		0			0
$313$		−	159.757i		−	0.510405i	−0.966888	−	0.255203i	$-0.917858\pi$
$313$		−	159.757i		−	0.510405i	0.966888	−	0.255203i	$-0.0821421\pi$
$314$	240.804	−	216.161i	0.766892	−	0.688411i
$315$		0			0
$316$	435.007	+	47.0544i	1.37660	+	0.148906i
$317$	265.214	+	265.214i	0.836637	+	0.836637i	0.988415	−	0.151778i	$-0.0484998\pi$
$317$	265.214	+	265.214i	0.836637	+	0.836637i	−0.151778	+	0.988415i	$0.548500\pi$
$318$		0			0
$319$	−130.942			−0.410476
$320$	−156.236	+	77.8251i	−0.488237	+	0.243203i
$321$		0			0
$322$	−0.932020	+	17.2829i	−0.00289447	+	0.0536736i
$323$	−509.728	−	509.728i	−1.57810	−	1.57810i
$324$		0			0
$325$	−27.0566	−	27.0566i	−0.0832509	−	0.0832509i
$326$	261.554	−	234.788i	0.802313	−	0.720207i
$327$		0			0
$328$	−505.900	−	82.4859i	−1.54238	−	0.251481i
$329$	193.488			0.588111
$330$		0			0
$331$	−83.6950	−	83.6950i	−0.252855	−	0.252855i	0.569285	−	0.822140i	$-0.307220\pi$
$331$	−83.6950	−	83.6950i	−0.252855	−	0.252855i	−0.822140	+	0.569285i	$0.807220\pi$
$332$	52.2890	+	64.9731i	0.157497	+	0.195702i
$333$		0			0
$334$	−460.279	−	24.8216i	−1.37808	−	0.0743162i
$335$		−	251.411i		−	0.750481i
$336$		0			0
$337$	346.530			1.02828			0.514140	−	0.857706i	$-0.328111\pi$
$337$	346.530			1.02828			0.514140	+	0.857706i	$0.328111\pi$
$338$	17.6897	−	328.029i	0.0523365	−	0.970500i
$339$		0			0
$340$	−139.354	−	173.159i	−0.409865	−	0.509290i
$341$	37.1937	−	37.1937i	0.109073	−	0.109073i
$342$		0			0
$343$			223.921i			0.652832i
$344$	163.088	+	226.631i	0.474093	+	0.658811i
$345$		0			0
$346$	82.1665	+	91.5337i	0.237475	+	0.264548i
$347$	394.994	−	394.994i	1.13831	−	1.13831i	0.149559	−	0.988753i	$-0.452214\pi$
$347$	394.994	−	394.994i	1.13831	−	1.13831i	0.988753	−	0.149559i	$-0.0477855\pi$
$348$		0			0
$349$	100.763	−	100.763i	0.288719	−	0.288719i	−0.547854	−	0.836574i	$-0.684555\pi$
$349$	100.763	−	100.763i	0.288719	−	0.288719i	0.836574	+	0.547854i	$0.184555\pi$
$350$	85.2840	+	4.59914i	0.243668	+	0.0131404i
$351$		0			0
$352$	109.878	−	62.3331i	0.312154	−	0.177083i
$353$		−	211.250i		−	0.598442i	−0.954184	−	0.299221i	$-0.903273\pi$
$353$		−	211.250i		−	0.598442i	0.954184	−	0.299221i	$-0.0967268\pi$
$354$		0			0
$355$	163.339	−	163.339i	0.460109	−	0.460109i
$356$	128.485	+	13.8981i	0.360912	+	0.0390396i
$357$		0			0
$358$	−275.670	−	307.097i	−0.770029	−	0.857814i
$359$	−180.058			−0.501555			−0.250777	−	0.968045i	$-0.580686\pi$
$359$	−180.058			−0.501555			−0.250777	+	0.968045i	$0.580686\pi$
$360$		0			0
$361$		−	890.786i		−	2.46755i
$362$	349.113	+	388.913i	0.964400	+	1.07434i
$363$		0			0
$364$	−13.2866	−	16.5096i	−0.0365016	−	0.0453561i
$365$	−76.6519	−	76.6519i	−0.210005	−	0.210005i
$366$		0			0
$367$	−611.746			−1.66688			−0.833442	−	0.552608i	$-0.813633\pi$
$367$	−611.746			−1.66688			−0.833442	+	0.552608i	$0.813633\pi$
$368$	−55.6259	−	12.1765i	−0.151157	−	0.0330883i
$369$		0			0
$370$	−195.962	−	10.5677i	−0.529628	−	0.0285614i
$371$	−96.7933	−	96.7933i	−0.260898	−	0.260898i
$372$		0			0
$373$	250.082	+	250.082i	0.670462	+	0.670462i	0.957823	−	0.287360i	$-0.0927777\pi$
$373$	250.082	+	250.082i	0.670462	+	0.670462i	−0.287360	+	0.957823i	$0.592778\pi$
$374$	107.460	+	119.710i	0.287325	+	0.320081i
$375$		0			0
$376$	−102.439	+	628.278i	−0.272445	+	1.67095i
$377$	−72.2680			−0.191692
$378$		0			0
$379$	196.949	+	196.949i	0.519655	+	0.519655i	0.917467	−	0.397812i	$-0.130230\pi$
$379$	196.949	+	196.949i	0.519655	+	0.519655i	−0.397812	+	0.917467i	$0.630230\pi$
$380$	41.5082	−	383.733i	0.109232	−	1.00982i
$381$		0			0
$382$	37.9319	−	703.389i	0.0992981	−	1.84133i
$383$		−	27.7580i		−	0.0724752i	−0.999343	−	0.0362376i	$-0.988463\pi$
$383$		−	27.7580i		−	0.0724752i	0.999343	−	0.0362376i	$-0.0115373\pi$
$384$		0			0
$385$	26.1803			0.0680008
$386$	290.897	+	15.6873i	0.753620	+	0.0406407i
$387$		0			0
$388$	491.004	+	53.1115i	1.26547	+	0.136885i
$389$	250.685	−	250.685i	0.644434	−	0.644434i	−0.307208	−	0.951642i	$-0.599395\pi$
$389$	250.685	−	250.685i	0.644434	−	0.644434i	0.951642	+	0.307208i	$0.0993947\pi$
$390$		0			0
$391$		−	72.5119i		−	0.185452i
$392$	−340.205	−	55.4697i	−0.867871	−	0.141504i
$393$		0			0
$394$	39.5168	−	35.4728i	0.100296	−	0.0900324i
$395$	−210.949	+	210.949i	−0.534048	+	0.534048i
$396$		0			0
$397$	211.973	−	211.973i	0.533936	−	0.533936i	−0.387805	−	0.921741i	$-0.626767\pi$
$397$	211.973	−	211.973i	0.533936	−	0.533936i	0.921741	+	0.387805i	$0.126767\pi$
$398$	28.0149	−	519.494i	0.0703893	−	1.30526i
$399$		0			0
$400$	−60.0861	+	274.491i	−0.150215	+	0.686228i
$401$			603.861i			1.50589i	0.658085	+	0.752944i	$0.271367\pi$
$401$			603.861i			1.50589i	−0.658085	+	0.752944i	$0.728633\pi$
$402$		0			0
$403$	20.5275	−	20.5275i	0.0509368	−	0.0509368i
$404$	−368.289	+	296.391i	−0.911606	+	0.733641i
$405$		0			0
$406$	120.039	−	107.754i	0.295662	−	0.265405i
$407$	142.034			0.348977
$408$		0			0
$409$		−	229.079i		−	0.560094i	−0.959986	−	0.280047i	$-0.909650\pi$
$409$		−	229.079i		−	0.560094i	0.959986	−	0.280047i	$-0.0903501\pi$
$410$	260.075	−	233.460i	0.634328	−	0.569414i
$411$		0			0
$412$	16.3515	−	151.166i	0.0396882	−	0.366908i
$413$	46.7774	+	46.7774i	0.113262	+	0.113262i
$414$		0			0
$415$	−56.8642			−0.137022
$416$	60.6428	−	34.4022i	0.145776	−	0.0826976i
$417$		0			0
$418$	−15.0426	+	278.941i	−0.0359870	+	0.667324i
$419$	−581.441	−	581.441i	−1.38769	−	1.38769i	−0.830156	−	0.557531i	$-0.811749\pi$
$419$	−581.441	−	581.441i	−1.38769	−	1.38769i	−0.557531	−	0.830156i	$-0.688251\pi$
$420$		0			0
$421$	212.037	+	212.037i	0.503650	+	0.503650i	0.912570	−	0.408920i	$-0.134095\pi$
$421$	212.037	+	212.037i	0.503650	+	0.503650i	−0.408920	+	0.912570i	$0.634095\pi$
$422$	−267.351	+	239.992i	−0.633534	+	0.568700i
$423$		0			0
$424$	365.544	−	263.053i	0.862132	−	0.620407i
$425$	−357.816			−0.841921
$426$		0			0
$427$	154.515	+	154.515i	0.361863	+	0.361863i
$428$	−305.784	+	246.088i	−0.714448	+	0.574972i
$429$		0			0
$430$	−190.096	−	10.2514i	−0.442085	−	0.0238405i
$431$		−	71.0623i		−	0.164878i	−0.996596	−	0.0824388i	$-0.973729\pi$
$431$		−	71.0623i		−	0.164878i	0.996596	−	0.0824388i	$-0.0262709\pi$
$432$		0			0
$433$	−71.8456			−0.165925			−0.0829626	−	0.996553i	$-0.526438\pi$
$433$	−71.8456			−0.165925			−0.0829626	+	0.996553i	$0.526438\pi$
$434$	−3.48932	+	64.7041i	−0.00803991	+	0.149088i
$435$		0			0
$436$	203.751	−	163.974i	0.467318	−	0.376088i
$437$	89.0371	−	89.0371i	0.203746	−	0.203746i
$438$		0			0
$439$			75.0323i			0.170916i	0.996342	+	0.0854582i	$0.0272354\pi$
$439$			75.0323i			0.170916i	−0.996342	+	0.0854582i	$0.972765\pi$
$440$	−13.8607	+	85.0103i	−0.0315017	+	0.193205i
$441$		0			0
$442$	59.3079	+	66.0692i	0.134181	+	0.149478i
$443$	407.393	−	407.393i	0.919623	−	0.919623i	−0.0773783	−	0.997002i	$-0.524655\pi$
$443$	407.393	−	407.393i	0.919623	−	0.919623i	0.997002	+	0.0773783i	$0.0246549\pi$
$444$		0			0
$445$	−62.3064	+	62.3064i	−0.140014	+	0.140014i
$446$	140.220	+	7.56170i	0.314395	+	0.0169545i
$447$		0			0
$448$	−49.4340	+	147.564i	−0.110344	+	0.329383i
$449$			537.780i			1.19773i	0.800851	+	0.598864i	$0.204381\pi$
$449$			537.780i			1.19773i	−0.800851	+	0.598864i	$0.795619\pi$
$450$		0			0
$451$	−178.857	+	178.857i	−0.396578	+	0.396578i
$452$	41.7953	−	386.387i	0.0924674	−	0.854840i
$453$		0			0
$454$	−108.912	−	121.329i	−0.239895	−	0.267244i
$455$	14.4491			0.0317564
$456$		0			0
$457$			670.548i			1.46728i	0.679537	+	0.733641i	$0.262180\pi$
$457$			670.548i			1.46728i	−0.679537	+	0.733641i	$0.737820\pi$
$458$	−257.726	−	287.107i	−0.562720	−	0.626871i
$459$		0			0
$460$	30.2466	−	24.3418i	0.0657535	−	0.0529170i
$461$	624.560	+	624.560i	1.35479	+	1.35479i	0.880210	+	0.474584i	$0.157401\pi$
$461$	624.560	+	624.560i	1.35479	+	1.35479i	0.474584	+	0.880210i	$0.342599\pi$
$462$		0			0
$463$	−556.663			−1.20230			−0.601148	−	0.799138i	$-0.705290\pi$
$463$	−556.663			−1.20230			−0.601148	+	0.799138i	$0.705290\pi$
$464$	286.338	+	446.827i	0.617107	+	0.962990i
$465$		0			0
$466$	341.807	+	18.4327i	0.733491	+	0.0395552i
$467$	135.903	+	135.903i	0.291013	+	0.291013i	0.837480	−	0.546468i	$-0.184028\pi$
$467$	135.903	+	135.903i	0.291013	+	0.291013i	−0.546468	+	0.837480i	$0.684028\pi$
$468$		0			0
$469$	−158.502	−	158.502i	−0.337957	−	0.337957i
$470$	−289.934	−	322.987i	−0.616880	−	0.687206i
$471$		0			0
$472$	−176.657	+	127.126i	−0.374273	+	0.269334i
$473$	137.782			0.291294
$474$		0			0
$475$	−439.362	−	439.362i	−0.924972	−	0.924972i
$476$	−197.024	−	21.3119i	−0.413915	−	0.0447729i
$477$		0			0
$478$	−8.58504	+	159.196i	−0.0179603	+	0.333047i
$479$			754.590i			1.57534i	0.616095	+	0.787672i	$0.288714\pi$
$479$			754.590i			1.57534i	−0.616095	+	0.787672i	$0.711286\pi$
$480$		0			0
$481$	78.3895			0.162972
$482$	136.938	+	7.38472i	0.284104	+	0.0153210i
$483$		0			0
$484$	−45.3463	+	419.216i	−0.0936907	+	0.866149i
$485$	−238.104	+	238.104i	−0.490936	+	0.490936i
$486$		0			0
$487$			752.536i			1.54525i	0.634863	+	0.772624i	$0.281056\pi$
$487$			752.536i			1.54525i	−0.634863	+	0.772624i	$0.718944\pi$
$488$	−583.533	+	419.922i	−1.19577	+	0.860496i
$489$		0			0
$490$	174.894	−	156.996i	0.356926	−	0.320399i
$491$	−577.864	+	577.864i	−1.17691	+	1.17691i	−0.196387	+	0.980527i	$0.562921\pi$
$491$	−577.864	+	577.864i	−1.17691	+	1.17691i	−0.980527	+	0.196387i	$0.937079\pi$
$492$		0			0
$493$	−477.863	+	477.863i	−0.969297	+	0.969297i
$494$	−8.30212	+	153.950i	−0.0168059	+	0.311640i
$495$		0			0
$496$	−208.254	−	45.5867i	−0.419866	−	0.0919087i
$497$		−	205.954i		−	0.414394i
$498$		0			0
$499$	46.1997	−	46.1997i	0.0925846	−	0.0925846i	−0.659298	−	0.751882i	$-0.729146\pi$
$499$	46.1997	−	46.1997i	0.0925846	−	0.0925846i	0.751882	+	0.659298i	$0.229146\pi$
$500$	−291.107	−	361.724i	−0.582215	−	0.723447i
$501$		0			0
$502$	24.4644	−	21.9608i	0.0487339	−	0.0437466i
$503$	442.969			0.880655			0.440327	−	0.897837i	$-0.354862\pi$
$503$	442.969			0.880655			0.440327	+	0.897837i	$0.354862\pi$
$504$		0			0
$505$		−	322.325i		−	0.638267i
$506$	−20.9105	+	18.7706i	−0.0413251	+	0.0370960i
$507$		0			0
$508$	745.385	+	80.6278i	1.46729	+	0.158716i
$509$	−166.024	−	166.024i	−0.326178	−	0.326178i	0.524953	−	0.851131i	$-0.324083\pi$
$509$	−166.024	−	166.024i	−0.326178	−	0.326178i	−0.851131	+	0.524953i	$0.824083\pi$
$510$		0			0
$511$	−96.6504			−0.189140
$512$	−452.983	−	238.643i	−0.884733	−	0.466099i
$513$		0			0
$514$	4.83296	−	89.6198i	0.00940264	−	0.174358i
$515$	73.3054	+	73.3054i	0.142341	+	0.142341i
$516$		0			0
$517$	222.122	+	222.122i	0.429637	+	0.429637i
$518$	−130.207	+	116.882i	−0.251364	+	0.225641i
$519$		0			0
$520$	−7.64986	+	46.9179i	−0.0147113	+	0.0902267i
$521$	7.42117			0.0142441			0.00712204	−	0.999975i	$-0.497733\pi$
$521$	7.42117			0.0142441			0.00712204	+	0.999975i	$0.497733\pi$
$522$		0			0
$523$	653.700	+	653.700i	1.24990	+	1.24990i	0.955761	+	0.294143i	$0.0950343\pi$
$523$	653.700	+	653.700i	1.24990	+	1.24990i	0.294143	+	0.955761i	$0.404966\pi$
$524$	280.999	+	349.163i	0.536257	+	0.666341i
$525$		0			0
$526$	767.766	+	41.4036i	1.45963	+	0.0787140i
$527$		−	271.472i		−	0.515127i
$528$		0			0
$529$	−516.334			−0.976057
$530$	−16.5350	+	306.616i	−0.0311980	+	0.578520i
$531$		0			0
$532$	−215.756	−	268.093i	−0.405556	−	0.503935i
$533$	−98.7126	+	98.7126i	−0.185202	+	0.185202i
$534$		0			0
$535$		−	267.621i		−	0.500226i
$536$	598.589	−	430.757i	1.11677	−	0.803651i
$537$		0			0
$538$	1.59477	+	1.77657i	0.00296425	+	0.00330218i
$539$	−120.277	+	120.277i	−0.223148	+	0.223148i
$540$		0			0
$541$	−192.630	+	192.630i	−0.356063	+	0.356063i	−0.862360	−	0.506296i	$-0.831014\pi$
$541$	−192.630	+	192.630i	−0.356063	+	0.356063i	0.506296	+	0.862360i	$0.331014\pi$
$542$	74.6547	+	4.02593i	0.137739	+	0.00742792i
$543$		0			0
$544$	173.513	−	628.474i	0.318958	−	1.15528i
$545$			178.322i			0.327196i
$546$		0			0
$547$	−131.598	+	131.598i	−0.240582	+	0.240582i	−0.817091	−	0.576509i	$-0.804415\pi$
$547$	−131.598	+	131.598i	−0.240582	+	0.240582i	0.576509	+	0.817091i	$0.304415\pi$
$548$	13.7671	+	1.48918i	0.0251225	+	0.00271749i
$549$		0			0
$550$	92.6252	+	103.185i	0.168409	+	0.187609i
$551$	−1173.53			−2.12982
$552$		0			0
$553$			265.986i			0.480987i
$554$	−647.685	−	721.523i	−1.16911	−	1.30239i
$555$		0			0
$556$	−50.0266	−	62.1620i	−0.0899760	−	0.111802i
$557$	−109.222	−	109.222i	−0.196090	−	0.196090i	0.602231	−	0.798322i	$-0.294278\pi$
$557$	−109.222	−	109.222i	−0.196090	−	0.196090i	−0.798322	+	0.602231i	$0.794278\pi$
$558$		0			0
$559$	76.0430			0.136034
$560$	−57.2499	−	89.3380i	−0.102232	−	0.159532i
$561$		0			0
$562$	606.810	+	32.7236i	1.07973	+	0.0582271i
$563$	640.646	+	640.646i	1.13791	+	1.13791i	0.988824	+	0.149090i	$0.0476346\pi$
$563$	640.646	+	640.646i	1.13791	+	1.13791i	0.149090	+	0.988824i	$0.452365\pi$
$564$		0			0
$565$	187.372	+	187.372i	0.331632	+	0.331632i
$566$	19.5355	+	21.7626i	0.0345150	+	0.0384498i
$567$		0			0
$568$	668.754	+	109.039i	1.17738	+	0.191970i
$569$	273.069			0.479911			0.239955	−	0.970784i	$-0.422867\pi$
$569$	273.069			0.479911			0.239955	+	0.970784i	$0.422867\pi$
$570$		0			0
$571$	−663.916	−	663.916i	−1.16273	−	1.16273i	−0.983877	−	0.178849i	$-0.942763\pi$
$571$	−663.916	−	663.916i	−1.16273	−	1.16273i	−0.178849	−	0.983877i	$-0.557237\pi$
$572$	3.70000	−	34.2057i	0.00646853	−	0.0598001i
$573$		0			0
$574$	16.7794	−	311.149i	0.0292324	−	0.542071i
$575$		−	62.5018i		−	0.108699i
$576$		0			0
$577$	−403.711			−0.699672			−0.349836	−	0.936811i	$-0.613763\pi$
$577$	−403.711			−0.699672			−0.349836	+	0.936811i	$0.613763\pi$
$578$	251.880	+	13.5832i	0.435778	+	0.0235003i
$579$		0			0
$580$	−359.745	−	38.9134i	−0.620250	−	0.0670920i
$581$	−35.8501	+	35.8501i	−0.0617041	+	0.0617041i
$582$		0			0
$583$		−	222.235i		−	0.381192i
$584$	51.1699	−	313.834i	0.0876198	−	0.537387i
$585$		0			0
$586$	−485.439	+	435.761i	−0.828394	+	0.743619i
$587$	703.189	−	703.189i	1.19794	−	1.19794i	0.223154	−	0.974783i	$-0.428365\pi$
$587$	703.189	−	703.189i	1.19794	−	1.19794i	0.974783	−	0.223154i	$-0.0716353\pi$
$588$		0			0
$589$	333.339	−	333.339i	0.565941	−	0.565941i
$590$	7.99087	−	148.178i	0.0135438	−	0.251150i
$591$		0			0
$592$	−310.592	−	484.677i	−0.524649	−	0.818710i
$593$		−	668.046i		−	1.12655i	−0.826268	−	0.563277i	$-0.809540\pi$
$593$		−	668.046i		−	1.12655i	0.826268	−	0.563277i	$-0.190460\pi$
$594$		0			0
$595$	95.5432	−	95.5432i	0.160577	−	0.160577i
$596$	378.012	−	304.216i	0.634248	−	0.510429i
$597$		0			0
$598$	−11.5407	+	10.3597i	−0.0192988	+	0.0173238i
$599$	677.684			1.13136			0.565679	−	0.824625i	$-0.308614\pi$
$599$	677.684			1.13136			0.565679	+	0.824625i	$0.308614\pi$
$600$		0			0
$601$		−	170.941i		−	0.284427i	−0.989836	−	0.142214i	$-0.954578\pi$
$601$		−	170.941i		−	0.284427i	0.989836	−	0.142214i	$-0.0454220\pi$
$602$	−126.309	+	113.383i	−0.209816	+	0.188344i
$603$		0			0
$604$	95.1371	−	879.520i	0.157512	−	1.45616i
$605$	−203.292	−	203.292i	−0.336019	−	0.336019i
$606$		0			0
$607$	1112.69			1.83310			0.916548	−	0.399925i	$-0.130964\pi$
$607$	1112.69			1.83310			0.916548	+	0.399925i	$0.130964\pi$
$608$	984.756	−	558.645i	1.61967	−	0.918823i
$609$		0			0
$610$	26.3955	−	489.464i	0.0432713	−	0.802399i
$611$	122.591	+	122.591i	0.200640	+	0.200640i
$612$		0			0
$613$	−455.272	−	455.272i	−0.742696	−	0.742696i	0.230400	−	0.973096i	$-0.425996\pi$
$613$	−455.272	−	455.272i	−0.742696	−	0.742696i	−0.973096	+	0.230400i	$0.925996\pi$
$614$	−536.727	+	481.800i	−0.874148	+	0.784691i
$615$		0			0
$616$	44.8562	+	62.3332i	0.0728186	+	0.101190i
$617$	−229.336			−0.371695			−0.185847	−	0.982579i	$-0.559503\pi$
$617$	−229.336			−0.371695			−0.185847	+	0.982579i	$0.559503\pi$
$618$		0			0
$619$	163.423	+	163.423i	0.264011	+	0.264011i	0.826681	−	0.562670i	$-0.190226\pi$
$619$	163.423	+	163.423i	0.264011	+	0.264011i	−0.562670	+	0.826681i	$0.690226\pi$
$620$	113.238	−	91.1315i	0.182642	−	0.146986i
$621$		0			0
$622$	−667.005	−	35.9698i	−1.07235	−	0.0578293i
$623$			78.5621i			0.126103i
$624$		0			0
$625$	−122.469			−0.195950
$626$	−17.2055	+	319.050i	−0.0274849	+	0.509664i
$627$		0			0
$628$	−504.190	+	405.761i	−0.802850	+	0.646116i
$629$	518.341	−	518.341i	0.824072	−	0.824072i
$630$		0			0
$631$		−	555.849i		−	0.880902i	−0.897777	−	0.440451i	$-0.854819\pi$
$631$		−	555.849i		−	0.880902i	0.897777	−	0.440451i	$-0.145181\pi$
$632$	−863.684	−	140.822i	−1.36659	−	0.222819i
$633$		0			0
$634$	−501.095	−	558.221i	−0.790371	−	0.880475i
$635$	−361.462	+	361.462i	−0.569231	+	0.569231i
$636$		0			0
$637$	−66.3818	+	66.3818i	−0.104210	+	0.104210i
$638$	261.504	+	14.1022i	0.409881	+	0.0221038i
$639$		0			0
$640$	320.400	−	138.598i	0.500625	−	0.216559i
$641$		−	982.994i		−	1.53353i	−0.641927	−	0.766766i	$-0.721865\pi$
$641$		−	982.994i		−	1.53353i	0.641927	−	0.766766i	$-0.278135\pi$
$642$		0			0
$643$	−163.762	+	163.762i	−0.254684	+	0.254684i	−0.822888	−	0.568204i	$-0.807639\pi$
$643$	−163.762	+	163.762i	−0.254684	+	0.254684i	0.568204	+	0.822888i	$0.307639\pi$
$644$	3.72267	−	34.4152i	0.00578055	−	0.0534398i
$645$		0			0
$646$	963.080	+	1072.87i	1.49084	+	1.66079i
$647$	655.404			1.01299			0.506495	−	0.862243i	$-0.330941\pi$
$647$	655.404			1.01299			0.506495	+	0.862243i	$0.330941\pi$
$648$		0			0
$649$			107.400i			0.165485i
$650$	51.1206	+	56.9485i	0.0786471	+	0.0876131i
$651$		0			0
$652$	−547.635	+	440.725i	−0.839931	+	0.675959i
$653$	−469.752	−	469.752i	−0.719376	−	0.719376i	0.249102	−	0.968477i	$-0.419865\pi$
$653$	−469.752	−	469.752i	−0.719376	−	0.719376i	−0.968477	+	0.249102i	$0.919865\pi$
$654$		0			0
$655$	−305.586			−0.466543
$656$	1001.45	+	219.217i	1.52660	+	0.334172i
$657$		0			0
$658$	−386.415	−	20.8384i	−0.587258	−	0.0316692i
$659$	576.834	+	576.834i	0.875317	+	0.875317i	0.993046	−	0.117728i	$-0.0375612\pi$
$659$	576.834	+	576.834i	0.875317	+	0.875317i	−0.117728	+	0.993046i	$0.537561\pi$
$660$		0			0
$661$	−679.417	−	679.417i	−1.02786	−	1.02786i	−0.999601	−	0.0282616i	$-0.991003\pi$
$661$	−679.417	−	679.417i	−1.02786	−	1.02786i	−0.0282616	−	0.999601i	$-0.508997\pi$
$662$	158.133	+	176.161i	0.238872	+	0.266104i
$663$		0			0
$664$	−97.4287	−	135.389i	−0.146730	−	0.203899i
$665$	234.634			0.352834
$666$		0			0
$667$	−83.4711	−	83.4711i	−0.125144	−	0.125144i
$668$	916.549	+	99.1424i	1.37208	+	0.148417i
$669$		0			0
$670$	−27.0765	+	502.093i	−0.0404127	+	0.749392i
$671$			354.764i			0.528709i
$672$		0			0
$673$	929.909			1.38174			0.690868	−	0.722981i	$-0.257228\pi$
$673$	929.909			1.38174			0.690868	+	0.722981i	$0.257228\pi$
$674$	−692.055	−	37.3207i	−1.02679	−	0.0553720i
$675$		0			0
$676$	−70.6563	+	653.201i	−0.104521	+	0.966274i
$677$	−597.623	+	597.623i	−0.882751	+	0.882751i	−0.993813	−	0.111062i	$-0.964575\pi$
$677$	−597.623	+	597.623i	−0.882751	+	0.882751i	0.111062	+	0.993813i	$0.464575\pi$
$678$		0			0
$679$			300.225i			0.442158i
$680$	259.655	+	360.823i	0.381846	+	0.530622i
$681$		0			0
$682$	−78.2852	+	70.2738i	−0.114788	+	0.103041i
$683$	−286.314	+	286.314i	−0.419200	+	0.419200i	−0.884928	−	0.465728i	$-0.845793\pi$
$683$	−286.314	+	286.314i	−0.419200	+	0.419200i	0.465728	+	0.884928i	$0.345793\pi$
$684$		0			0
$685$	−6.67614	+	6.67614i	−0.00974619	+	0.00974619i
$686$	24.1159	−	447.193i	0.0351544	−	0.651885i
$687$		0			0
$688$	−301.295	−	470.168i	−0.437929	−	0.683384i
$689$		−	122.653i		−	0.178017i
$690$		0			0
$691$	483.066	−	483.066i	0.699083	−	0.699083i	−0.265130	−	0.964213i	$-0.585415\pi$
$691$	483.066	−	483.066i	0.699083	−	0.699083i	0.964213	+	0.265130i	$0.0854149\pi$
$692$	−154.237	−	191.651i	−0.222885	−	0.276952i
$693$		0			0
$694$	−831.383	+	746.302i	−1.19796	+	1.07536i
$695$	54.4039			0.0782790
$696$		0			0
$697$			1305.45i			1.87296i
$698$	−212.086	+	190.382i	−0.303848	+	0.272753i
$699$		0			0
$700$	−169.825	−	18.3699i	−0.242607	−	0.0262427i
$701$	−349.679	−	349.679i	−0.498829	−	0.498829i	0.412244	−	0.911073i	$-0.364745\pi$
$701$	−349.679	−	349.679i	−0.498829	−	0.498829i	−0.911073	+	0.412244i	$0.864745\pi$
$702$		0			0
$703$	1272.94			1.81072
$704$	−226.151	+	112.652i	−0.321237	+	0.160017i
$705$		0			0
$706$	−22.7512	+	421.887i	−0.0322256	+	0.597574i
$707$	−203.210	−	203.210i	−0.287425	−	0.287425i
$708$		0			0
$709$	−374.404	−	374.404i	−0.528074	−	0.528074i	0.391924	−	0.919998i	$-0.371810\pi$
$709$	−374.404	−	374.404i	−0.528074	−	0.528074i	−0.919998	+	0.391924i	$0.871810\pi$
$710$	−343.795	+	308.612i	−0.484218	+	0.434665i
$711$		0			0
$712$	−255.099	−	41.5934i	−0.358286	−	0.0584177i
$713$	47.4195			0.0665071
$714$		0			0
$715$	16.5874	+	16.5874i	0.0231992	+	0.0231992i
$716$	517.467	+	642.993i	0.722719	+	0.898035i
$717$		0			0
$718$	359.594	+	19.3920i	0.500827	+	0.0270083i
$719$		−	819.536i		−	1.13983i	−0.821704	−	0.569914i	$-0.806976\pi$
$719$		−	819.536i		−	1.13983i	0.821704	−	0.569914i	$-0.193024\pi$
$720$		0			0
$721$	92.4308			0.128198
$722$	−95.9361	+	1778.99i	−0.132875	+	2.46397i
$723$		0			0
$724$	−655.328	−	814.296i	−0.905149	−	1.12472i
$725$	−411.896	+	411.896i	−0.568132	+	0.568132i
$726$		0			0
$727$		−	1089.99i		−	1.49930i	−0.661836	−	0.749649i	$-0.730222\pi$
$727$		−	1089.99i		−	1.49930i	0.661836	−	0.749649i	$-0.269778\pi$
$728$	24.7565	+	34.4022i	0.0340062	+	0.0472558i
$729$		0			0
$730$	144.826	+	161.337i	0.198392	+	0.221009i
$731$	502.825	−	502.825i	0.687860	−	0.687860i
$732$		0			0
$733$	165.917	−	165.917i	0.226354	−	0.226354i	−0.584814	−	0.811168i	$-0.698832\pi$
$733$	165.917	−	165.917i	0.226354	−	0.226354i	0.811168	+	0.584814i	$0.198832\pi$
$734$	1221.72	+	65.8840i	1.66446	+	0.0897602i
$735$		0			0
$736$	109.779	+	30.3085i	0.149156	+	0.0411800i
$737$		−	363.917i		−	0.493781i
$738$		0			0
$739$	−316.451	+	316.451i	−0.428215	+	0.428215i	−0.888020	−	0.459805i	$-0.847919\pi$
$739$	−316.451	+	316.451i	−0.428215	+	0.428215i	0.459805	+	0.888020i	$0.347919\pi$
$740$	390.218	+	42.2096i	0.527321	+	0.0570400i
$741$		0			0
$742$	182.881	+	203.730i	0.246471	+	0.274569i
$743$	−1074.58			−1.44627			−0.723134	−	0.690708i	$-0.757299\pi$
$743$	−1074.58			−1.44627			−0.723134	+	0.690708i	$0.757299\pi$
$744$		0			0
$745$			330.835i			0.444073i
$746$	−472.506	−	526.372i	−0.633386	−	0.705593i
$747$		0			0
$748$	−201.715	−	250.647i	−0.269672	−	0.335089i
$749$	−168.721	−	168.721i	−0.225262	−	0.225262i
$750$		0			0
$751$	425.307			0.566320			0.283160	−	0.959073i	$-0.408617\pi$
$751$	425.307			0.566320			0.283160	+	0.959073i	$0.408617\pi$
$752$	272.246	−	1243.70i	0.362029	−	1.65386i
$753$		0			0
$754$	144.326	+	7.78313i	0.191414	+	0.0103225i
$755$	426.508	+	426.508i	0.564912	+	0.564912i
$756$		0			0
$757$	93.5146	+	93.5146i	0.123533	+	0.123533i	0.766170	−	0.642637i	$-0.222160\pi$
$757$	93.5146	+	93.5146i	0.123533	+	0.123533i	−0.642637	+	0.766170i	$0.722160\pi$
$758$	−372.116	−	414.538i	−0.490919	−	0.546885i
$759$		0			0
$760$	−124.223	+	761.883i	−0.163452	+	1.00248i
$761$	−430.074			−0.565144			−0.282572	−	0.959246i	$-0.591188\pi$
$761$	−430.074			−0.565144			−0.282572	+	0.959246i	$0.591188\pi$
$762$		0			0
$763$	112.423	+	112.423i	0.147343	+	0.147343i
$764$	−151.507	+	1400.65i	−0.198308	+	1.83331i
$765$		0			0
$766$	−2.98949	+	55.4355i	−0.00390273	+	0.0723701i
$767$			59.2749i			0.0772814i
$768$		0			0
$769$	−307.931			−0.400430			−0.200215	−	0.979752i	$-0.564164\pi$
$769$	−307.931			−0.400430			−0.200215	+	0.979752i	$0.564164\pi$
$770$	−52.2847	−	2.81957i	−0.0679022	−	0.00366178i
$771$		0			0
$772$	−579.261	−	62.6582i	−0.750338	−	0.0811635i
$773$	531.375	−	531.375i	0.687419	−	0.687419i	−0.274241	−	0.961661i	$-0.588427\pi$
$773$	531.375	−	531.375i	0.687419	−	0.687419i	0.961661	+	0.274241i	$0.0884268\pi$
$774$		0			0
$775$		−	233.996i		−	0.301930i
$776$	−974.863	−	158.949i	−1.25627	−	0.204831i
$777$		0			0
$778$	−527.641	+	473.644i	−0.678202	+	0.608797i
$779$	−1602.96	+	1602.96i	−2.05771	+	2.05771i
$780$		0			0
$781$	236.432	−	236.432i	0.302730	−	0.302730i
$782$	−7.80940	+	144.813i	−0.00998644	+	0.185183i
$783$		0			0
$784$	673.449	+	147.418i	0.858992	+	0.188033i
$785$		−	441.265i		−	0.562121i
$786$		0			0
$787$	−114.817	+	114.817i	−0.145892	+	0.145892i	−0.776280	−	0.630388i	$-0.782896\pi$
$787$	−114.817	+	114.817i	−0.145892	+	0.145892i	0.630388	+	0.776280i	$0.282896\pi$
$788$	−82.7392	+	66.5867i	−0.104999	+	0.0845009i
$789$		0			0
$790$	444.005	−	398.567i	0.562031	−	0.504515i
$791$	236.257			0.298682
$792$		0			0
$793$			195.797i			0.246907i
$794$	−446.159	+	400.501i	−0.561913	+	0.504409i
$795$		0			0
$796$	−111.897	+	1034.46i	−0.140574	+	1.29958i
$797$	−699.220	−	699.220i	−0.877315	−	0.877315i	0.115941	−	0.993256i	$-0.463012\pi$
$797$	−699.220	−	699.220i	−0.877315	−	0.877315i	−0.993256	+	0.115941i	$0.963012\pi$
$798$		0			0
$799$	1621.24			2.02909
$800$	149.560	−	541.715i	0.186950	−	0.677143i
$801$		0			0
$802$	65.0348	−	1205.97i	0.0810907	−	1.50370i
$803$	−110.953	−	110.953i	−0.138174	−	0.138174i
$804$		0			0
$805$	16.6891	+	16.6891i	0.0207318	+	0.0207318i
$806$	−43.2063	+	38.7847i	−0.0536058	+	0.0481200i
$807$		0			0
$808$	767.430	−	552.258i	0.949789	−	0.683487i
$809$	1160.71			1.43475			0.717376	−	0.696686i	$-0.245343\pi$
$809$	1160.71			1.43475			0.717376	+	0.696686i	$0.245343\pi$
$810$		0			0
$811$	297.004	+	297.004i	0.366220	+	0.366220i	0.866097	−	0.499877i	$-0.166621\pi$
$811$	297.004	+	297.004i	0.366220	+	0.366220i	−0.499877	+	0.866097i	$0.666621\pi$
$812$	−251.334	+	202.268i	−0.309525	+	0.249099i
$813$		0			0
$814$	−283.655	−	15.2968i	−0.348470	−	0.0187921i
$815$		−	479.288i		−	0.588084i
$816$		0			0
$817$	1234.84			1.51143
$818$	−24.6714	+	457.492i	−0.0301606	+	0.559282i
$819$		0			0
$820$	−544.538	+	438.232i	−0.664070	+	0.534429i
$821$	−606.627	+	606.627i	−0.738888	+	0.738888i	−0.972363	−	0.233475i	$-0.924990\pi$
$821$	−606.627	+	606.627i	−0.738888	+	0.738888i	0.233475	+	0.972363i	$0.424990\pi$
$822$		0			0
$823$		−	320.341i		−	0.389236i	−0.980879	−	0.194618i	$-0.937653\pi$
$823$		−	320.341i		−	0.389236i	0.980879	−	0.194618i	$-0.0623467\pi$
$824$	−48.9360	+	300.133i	−0.0593883	+	0.364239i
$825$		0			0
$826$	−88.3812	−	98.4569i	−0.106999	−	0.119197i
$827$	686.333	−	686.333i	0.829907	−	0.829907i	−0.157597	−	0.987504i	$-0.550375\pi$
$827$	686.333	−	686.333i	0.829907	−	0.829907i	0.987504	+	0.157597i	$0.0503746\pi$
$828$		0			0
$829$	317.721	−	317.721i	0.383258	−	0.383258i	−0.489017	−	0.872275i	$-0.662644\pi$
$829$	317.721	−	317.721i	0.383258	−	0.383258i	0.872275	+	0.489017i	$0.162644\pi$
$830$	113.563	+	6.12418i	0.136823	+	0.00737853i
$831$		0			0
$832$	−124.815	+	62.1734i	−0.150018	+	0.0747277i
$833$			877.883i			1.05388i
$834$		0			0
$835$	−444.465	+	444.465i	−0.532293	+	0.532293i
$836$	60.0830	−	555.453i	0.0718696	−	0.664418i
$837$		0			0
$838$	1098.57	+	1223.81i	1.31095	+	1.46040i
$839$	−999.424			−1.19121			−0.595605	−	0.803278i	$-0.703087\pi$
$839$	−999.424			−1.19121			−0.595605	+	0.803278i	$0.703087\pi$
$840$		0			0
$841$			259.172i			0.308171i
$842$	−400.622	−	446.294i	−0.475798	−	0.530040i
$843$		0			0
$844$	559.773	−	450.494i	0.663239	−	0.533760i
$845$	−316.759	−	316.759i	−0.374862	−	0.374862i
$846$		0			0
$847$	−256.330			−0.302633
$848$	−758.357	+	485.973i	−0.894289	+	0.573082i
$849$		0			0
$850$	714.594	+	38.5362i	0.840699	+	0.0453367i
$851$	90.5416	+	90.5416i	0.106394	+	0.106394i
$852$		0			0
$853$	−437.602	−	437.602i	−0.513015	−	0.513015i	0.402434	−	0.915449i	$-0.368164\pi$
$853$	−437.602	−	437.602i	−0.513015	−	0.513015i	−0.915449	+	0.402434i	$0.868164\pi$
$854$	−291.941	−	325.223i	−0.341852	−	0.380824i
$855$		0			0
$856$	637.183	−	458.530i	0.744373	−	0.535666i
$857$	−734.356			−0.856891			−0.428445	−	0.903568i	$-0.640939\pi$
$857$	−734.356			−0.856891			−0.428445	+	0.903568i	$0.640939\pi$
$858$		0			0
$859$	−473.257	−	473.257i	−0.550939	−	0.550939i	0.375773	−	0.926712i	$-0.377377\pi$
$859$	−473.257	−	473.257i	−0.550939	−	0.550939i	−0.926712	+	0.375773i	$0.877377\pi$
$860$	378.537	+	40.9461i	0.440159	+	0.0476117i
$861$		0			0
$862$	−7.65328	+	141.918i	−0.00887851	+	0.164638i
$863$			415.374i			0.481315i	0.970610	+	0.240657i	$0.0773630\pi$
$863$			415.374i			0.481315i	−0.970610	+	0.240657i	$0.922637\pi$
$864$		0			0
$865$	167.732			0.193910
$866$	143.483	+	7.73765i	0.165684	+	0.00893493i
$867$		0			0
$868$	13.9370	−	128.845i	0.0160565	−	0.148439i
$869$	−305.348	+	305.348i	−0.351379	+	0.351379i
$870$		0			0
$871$		−	200.849i		−	0.230596i
$872$	−424.570	+	305.529i	−0.486892	+	0.350377i
$873$		0			0
$874$	−187.405	+	168.227i	−0.214422	+	0.192479i
$875$	199.587	−	199.587i	0.228100	−	0.228100i
$876$		0			0
$877$	−639.234	+	639.234i	−0.728887	+	0.728887i	−0.970398	−	0.241511i	$-0.922357\pi$
$877$	−639.234	+	639.234i	−0.728887	+	0.728887i	0.241511	+	0.970398i	$0.422357\pi$
$878$	8.08084	−	149.847i	0.00920369	−	0.170668i
$879$		0			0
$880$	36.8367	−	168.281i	0.0418599	−	0.191229i
$881$			434.503i			0.493193i	0.969118	+	0.246596i	$0.0793122\pi$
$881$			434.503i			0.493193i	−0.969118	+	0.246596i	$0.920688\pi$
$882$		0			0
$883$	−597.662	+	597.662i	−0.676854	+	0.676854i	−0.959287	−	0.282433i	$-0.908858\pi$
$883$	−597.662	+	597.662i	−0.676854	+	0.676854i	0.282433	+	0.959287i	$0.408858\pi$
$884$	−111.328	−	138.334i	−0.125937	−	0.156486i
$885$		0			0
$886$	−857.480	+	769.729i	−0.967810	+	0.868768i
$887$	1100.10			1.24025			0.620125	−	0.784503i	$-0.287082\pi$
$887$	1100.10			1.24025			0.620125	+	0.784503i	$0.287082\pi$
$888$		0			0
$889$			455.767i			0.512674i
$890$	131.142	−	117.722i	0.147351	−	0.132272i
$891$		0			0
$892$	−279.219	−	30.2029i	−0.313026	−	0.0338598i
$893$	1990.72	+	1990.72i	2.22924	+	2.22924i
$894$		0			0
$895$	−562.745			−0.628765
$896$	114.617	−	289.375i	0.127921	−	0.322963i
$897$		0			0
$898$	57.9179	−	1074.00i	0.0644966	−	1.19599i
$899$	−312.501	−	312.501i	−0.347610	−	0.347610i
$900$		0			0
$901$	−811.031	−	811.031i	−0.900146	−	0.900146i
$902$	376.457	−	337.932i	0.417358	−	0.374648i
$903$		0			0
$904$	−125.082	+	767.152i	−0.138366	+	0.848620i
$905$	712.669			0.787479
$906$		0			0
$907$	−344.291	−	344.291i	−0.379593	−	0.379593i	0.491362	−	0.870955i	$-0.336499\pi$
$907$	−344.291	−	344.291i	−0.379593	−	0.379593i	−0.870955	+	0.491362i	$0.836499\pi$
$908$	204.442	+	254.035i	0.225156	+	0.279774i
$909$		0			0
$910$	−28.8564	−	1.55615i	−0.0317103	−	0.00171005i
$911$		−	803.034i		−	0.881486i	−0.897633	−	0.440743i	$-0.854715\pi$
$911$		−	803.034i		−	0.881486i	0.897633	−	0.440743i	$-0.145285\pi$
$912$		0			0
$913$	−82.3109			−0.0901543
$914$	72.2168	−	1339.15i	0.0790118	−	1.46515i
$915$		0			0
$916$	483.782	+	601.137i	0.528147	+	0.656264i
$917$	−192.657	+	192.657i	−0.210094	+	0.210094i
$918$		0			0
$919$			1119.49i			1.21816i	0.793107	+	0.609082i	$0.208462\pi$
$919$			1119.49i			1.21816i	−0.793107	+	0.609082i	$0.791538\pi$
$920$	−63.0270	+	45.3555i	−0.0685076	+	0.0492994i
$921$		0			0
$922$	−1180.04	−	1314.57i	−1.27987	−	1.42578i
$923$	130.489	−	130.489i	0.141375	−	0.141375i
$924$		0			0
$925$	446.786	−	446.786i	0.483012	−	0.483012i
$926$	1111.71	+	59.9516i	1.20055	+	0.0647426i
$927$		0			0
$928$	−523.722	−	923.196i	−0.564356	−	0.994824i
$929$		−	1069.69i		−	1.15144i	−0.817647	−	0.575721i	$-0.804722\pi$
$929$		−	1069.69i		−	1.15144i	0.817647	−	0.575721i	$-0.195278\pi$
$930$		0			0
$931$	−1077.95	+	1077.95i	−1.15784	+	1.15784i
$932$	−680.636	−	73.6239i	−0.730296	−	0.0789956i
$933$		0			0
$934$	−256.775	−	286.048i	−0.274920	−	0.306261i
$935$	219.365			0.234615
$936$		0			0
$937$		−	61.7823i		−	0.0659363i	−0.999456	−	0.0329682i	$-0.989504\pi$
$937$		−	61.7823i		−	0.0659363i	0.999456	−	0.0329682i	$-0.0104960\pi$
$938$	299.474	+	333.615i	0.319268	+	0.355666i
$939$		0			0
$940$	544.241	+	676.262i	0.578979	+	0.719427i
$941$	−455.261	−	455.261i	−0.483806	−	0.483806i	0.422539	−	0.906345i	$-0.361139\pi$
$941$	−455.261	−	455.261i	−0.483806	−	0.483806i	−0.906345	+	0.422539i	$0.861139\pi$
$942$		0			0
$943$	−228.031			−0.241814
$944$	366.492	−	234.857i	0.388233	−	0.248789i
$945$		0			0
$946$	−275.164	−	14.8389i	−0.290871	−	0.0156859i
$947$	1145.09	+	1145.09i	1.20918	+	1.20918i	0.971294	+	0.237883i	$0.0764534\pi$
$947$	1145.09	+	1145.09i	1.20918	+	1.20918i	0.237883	+	0.971294i	$0.423547\pi$
$948$		0			0
$949$	−61.2362	−	61.2362i	−0.0645271	−	0.0645271i
$950$	830.130	+	924.767i	0.873821	+	0.973438i
$951$		0			0
$952$	391.180	+	63.7811i	0.410904	+	0.0669969i
$953$	1187.58			1.24615			0.623076	−	0.782161i	$-0.285883\pi$
$953$	1187.58			1.24615			0.623076	+	0.782161i	$0.285883\pi$
$954$		0			0
$955$	−679.221	−	679.221i	−0.711227	−	0.711227i
$956$	34.2903	−	317.006i	0.0358685	−	0.331596i
$957$		0			0
$958$	81.2680	−	1506.99i	0.0848309	−	1.57306i
$959$			8.41794i			0.00877783i
$960$		0			0
$961$	−783.470			−0.815265
$962$	−156.552	−	8.44241i	−0.162736	−	0.00877590i
$963$		0			0
$964$	−272.684	−	29.4960i	−0.282867	−	0.0305975i
$965$	280.903	−	280.903i	0.291091	−	0.291091i
$966$		0			0
$967$			464.902i			0.480767i	0.970678	+	0.240384i	$0.0772732\pi$
$967$			464.902i			0.480767i	−0.970678	+	0.240384i	$0.922727\pi$
$968$	135.710	−	832.332i	0.140196	−	0.859847i
$969$		0			0
$970$	501.160	−	449.873i	0.516660	−	0.463787i
$971$	−102.522	+	102.522i	−0.105584	+	0.105584i	−0.757925	−	0.652341i	$-0.773787\pi$
$971$	−102.522	+	102.522i	−0.105584	+	0.105584i	0.652341	+	0.757925i	$0.273787\pi$
$972$		0			0
$973$	34.2990	−	34.2990i	0.0352507	−	0.0352507i
$974$	81.0468	−	1502.89i	0.0832103	−	1.54301i
$975$		0			0
$976$	1210.60	−	775.780i	1.24037	−	0.794857i
$977$			1273.15i			1.30312i	0.758597	+	0.651560i	$0.225885\pi$
$977$			1273.15i			1.30312i	−0.758597	+	0.651560i	$0.774115\pi$
$978$		0			0
$979$	−90.1883	+	90.1883i	−0.0921229	+	0.0921229i
$980$	−366.188	+	294.700i	−0.373661	+	0.300714i
$981$		0			0
$982$	1216.29	−	1091.82i	1.23858	−	1.11183i
$983$	−523.931			−0.532992			−0.266496	−	0.963836i	$-0.585866\pi$
$983$	−523.931			−0.532992			−0.266496	+	0.963836i	$0.585866\pi$
$984$		0			0
$985$		−	72.4130i		−	0.0735158i
$986$	1005.80	−	902.875i	1.02009	−	0.915694i
$987$		0			0
$988$	33.1603	−	306.559i	0.0335631	−	0.310283i
$989$	87.8314	+	87.8314i	0.0888083	+	0.0888083i
$990$		0			0
$991$	−1675.46			−1.69067			−0.845336	−	0.534235i	$-0.820600\pi$
$991$	−1675.46			−1.69067			−0.845336	+	0.534235i	$0.820600\pi$
$992$	410.994	+	113.470i	0.414308	+	0.114385i
$993$		0			0
$994$	−22.1809	+	411.310i	−0.0223147	+	0.413793i
$995$	−501.645	−	501.645i	−0.504166	−	0.504166i
$996$		0			0
$997$	911.770	+	911.770i	0.914513	+	0.914513i	0.996623	−	0.0821099i	$-0.0261659\pi$
$997$	911.770	+	911.770i	0.914513	+	0.914513i	−0.0821099	+	0.996623i	$0.526166\pi$
$998$	−97.2410	+	87.2898i	−0.0974359	+	0.0874647i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.3.j.a.53.1	✓	32
3.2	odd	2	inner	144.3.j.a.53.16	yes	32
4.3	odd	2		576.3.j.a.305.5		32
8.3	odd	2		1152.3.j.b.737.12		32
8.5	even	2		1152.3.j.a.737.12		32
12.11	even	2		576.3.j.a.305.12		32
16.3	odd	4		576.3.j.a.17.12		32
16.5	even	4		1152.3.j.a.161.5		32
16.11	odd	4		1152.3.j.b.161.5		32
16.13	even	4	inner	144.3.j.a.125.16	yes	32
24.5	odd	2		1152.3.j.a.737.5		32
24.11	even	2		1152.3.j.b.737.5		32
48.5	odd	4		1152.3.j.a.161.12		32
48.11	even	4		1152.3.j.b.161.12		32
48.29	odd	4	inner	144.3.j.a.125.1	yes	32
48.35	even	4		576.3.j.a.17.5		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.3.j.a.53.1	✓	32	1.1	even	1	trivial
144.3.j.a.53.16	yes	32	3.2	odd	2	inner
144.3.j.a.125.1	yes	32	48.29	odd	4	inner
144.3.j.a.125.16	yes	32	16.13	even	4	inner
576.3.j.a.17.5		32	48.35	even	4
576.3.j.a.17.12		32	16.3	odd	4
576.3.j.a.305.5		32	4.3	odd	2
576.3.j.a.305.12		32	12.11	even	2
1152.3.j.a.161.5		32	16.5	even	4
1152.3.j.a.161.12		32	48.5	odd	4
1152.3.j.a.737.5		32	24.5	odd	2
1152.3.j.a.737.12		32	8.5	even	2
1152.3.j.b.161.5		32	16.11	odd	4
1152.3.j.b.161.12		32	48.11	even	4
1152.3.j.b.737.5		32	24.11	even	2
1152.3.j.b.737.12		32	8.3	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:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	144.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.99710 - 0.107698i) q^{2} +(3.97680 + 0.430168i) q^{4} +(-1.92848 + 1.92848i) q^{5} +2.43162i q^{7} +(-7.89574 - 1.28738i) q^{8} +O(q^{10})\)	\(q+(-1.99710 - 0.107698i) q^{2} +(3.97680 + 0.430168i) q^{4} +(-1.92848 + 1.92848i) q^{5} +2.43162i q^{7} +(-7.89574 - 1.28738i) q^{8} +(4.05906 - 3.64367i) q^{10} +(-2.79147 + 2.79147i) q^{11} +(-1.54064 + 1.54064i) q^{13} +(0.261881 - 4.85619i) q^{14} +(15.6299 + 3.42138i) q^{16} +20.3746i q^{17} +(-25.0179 + 25.0179i) q^{19} +(-8.49876 + 6.83962i) q^{20} +(5.87548 - 5.27421i) q^{22} -3.55894 q^{23} +17.5619i q^{25} +(3.24273 - 2.91088i) q^{26} +(-1.04601 + 9.67008i) q^{28} +(23.4539 + 23.4539i) q^{29} -13.3241 q^{31} +(-30.8460 - 8.51615i) q^{32} +(2.19430 - 40.6900i) q^{34} +(-4.68934 - 4.68934i) q^{35} +(-25.4406 - 25.4406i) q^{37} +(52.6575 - 47.2687i) q^{38} +(17.7095 - 12.7441i) q^{40} +64.0726 q^{41} +(-24.6791 - 24.6791i) q^{43} +(-12.3019 + 9.90033i) q^{44} +(7.10756 + 0.383292i) q^{46} -79.5718i q^{47} +43.0872 q^{49} +(1.89139 - 35.0729i) q^{50} +(-6.78954 + 5.46408i) q^{52} +(-39.8061 + 39.8061i) q^{53} -10.7666i q^{55} +(3.13043 - 19.1994i) q^{56} +(-44.3138 - 49.3657i) q^{58} +(19.2371 - 19.2371i) q^{59} +(63.5441 - 63.5441i) q^{61} +(26.6094 + 1.43498i) q^{62} +(60.6853 + 20.3297i) q^{64} -5.94218i q^{65} +(-65.1837 + 65.1837i) q^{67} +(-8.76448 + 81.0256i) q^{68} +(8.86003 + 9.87010i) q^{70} -84.6981 q^{71} +39.7473i q^{73} +(48.0675 + 53.5473i) q^{74} +(-110.253 + 88.7292i) q^{76} +(-6.78781 - 6.78781i) q^{77} +109.386 q^{79} +(-36.7401 + 23.5439i) q^{80} +(-127.959 - 6.90050i) q^{82} +(14.7433 + 14.7433i) q^{83} +(-39.2920 - 39.2920i) q^{85} +(46.6287 + 51.9444i) q^{86} +(25.6344 - 18.4470i) q^{88} +32.3085 q^{89} +(-3.74625 - 3.74625i) q^{91} +(-14.1532 - 1.53094i) q^{92} +(-8.56974 + 158.913i) q^{94} -96.4929i q^{95} +123.467 q^{97} +(-86.0494 - 4.64042i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q + 40 q^{10} + 48 q^{16} + 64 q^{19} - 88 q^{22} - 120 q^{28} - 248 q^{34} - 184 q^{40} + 128 q^{43} + 24 q^{46} - 224 q^{49} + 632 q^{52} + 456 q^{58} + 64 q^{61} - 48 q^{64} - 64 q^{67} - 312 q^{70} - 576 q^{76} - 512 q^{79} - 720 q^{82} + 320 q^{85} - 400 q^{88} - 192 q^{91} + 696 q^{94}+O(q^{100}) \) 32 * q + 40 * q^10 + 48 * q^16 + 64 * q^19 - 88 * q^22 - 120 * q^28 - 248 * q^34 - 184 * q^40 + 128 * q^43 + 24 * q^46 - 224 * q^49 + 632 * q^52 + 456 * q^58 + 64 * q^61 - 48 * q^64 - 64 * q^67 - 312 * q^70 - 576 * q^76 - 512 * q^79 - 720 * q^82 + 320 * q^85 - 400 * q^88 - 192 * q^91 + 696 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more