LMFDB - Embedded newform 144.3.o.c.31.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,3,Mod(31,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 2]))

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

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

chi := DirichletCharacter("144.31");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	144.o (of order $6$, 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:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.856615824.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 $ x^8 + 11x^6 + 36x^4 + 32*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			31.2
Root			$2.06288i$ of defining polynomial
Character	$\chi$	$=$	144.31
Dual form			144.3.o.c.79.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.456412 + 2.96508i) q^{3} +(4.61660 - 7.99619i) q^{5} +(5.33093 - 3.07781i) q^{7} +(-8.58338 + 2.70659i) q^{9} +O(q^{10})$	\(q+(0.456412 + 2.96508i) q^{3} +(4.61660 - 7.99619i) q^{5} +(5.33093 - 3.07781i) q^{7} +(-8.58338 + 2.70659i) q^{9} +(3.70016 - 2.13629i) q^{11} +(0.869235 - 1.50556i) q^{13} +(25.8164 + 10.0390i) q^{15} +12.3476 q^{17} +33.9338i q^{19} +(11.5590 + 14.4019i) q^{21} +(3.35035 + 1.93433i) q^{23} +(-30.1260 - 52.1798i) q^{25} +(-11.9428 - 24.2151i) q^{27} +(17.8409 + 30.9014i) q^{29} +(-38.8262 - 22.4163i) q^{31} +(8.02306 + 9.99624i) q^{33} -56.8361i q^{35} -32.7130 q^{37} +(4.86083 + 1.89019i) q^{39} +(21.8565 - 37.8565i) q^{41} +(-33.9339 + 19.5918i) q^{43} +(-17.9836 + 81.1296i) q^{45} +(-39.8784 + 23.0238i) q^{47} +(-5.55415 + 9.62007i) q^{49} +(5.63558 + 36.6116i) q^{51} +46.3143 q^{53} -39.4496i q^{55} +(-100.617 + 15.4878i) q^{57} +(-23.2710 - 13.4355i) q^{59} +(23.4545 + 40.6243i) q^{61} +(-37.4270 + 40.8467i) q^{63} +(-8.02582 - 13.9011i) q^{65} +(56.9984 + 32.9080i) q^{67} +(-4.20629 + 10.8169i) q^{69} -96.7955i q^{71} -14.0622 q^{73} +(140.967 - 113.142i) q^{75} +(13.1502 - 22.7768i) q^{77} +(-34.3954 + 19.8582i) q^{79} +(66.3487 - 46.4634i) q^{81} +(-81.7202 + 47.1812i) q^{83} +(57.0039 - 98.7336i) q^{85} +(-83.4822 + 67.0034i) q^{87} -81.8478 q^{89} -10.7014i q^{91} +(48.7454 - 125.354i) q^{93} +(271.341 + 156.659i) q^{95} +(-7.99028 - 13.8396i) q^{97} +(-25.9778 + 28.3514i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 3 q^{3} + 3 q^{5} - 3 q^{7} - 3 q^{9}+O(q^{10}) $ 8 * q + 3 * q^3 + 3 * q^5 - 3 * q^7 - 3 * q^9	$ 8 q + 3 q^{3} + 3 q^{5} - 3 q^{7} - 3 q^{9} - 18 q^{11} + 5 q^{13} + 21 q^{15} + 6 q^{17} - 33 q^{21} + 81 q^{23} - 23 q^{25} - 108 q^{27} + 69 q^{29} - 45 q^{31} + 72 q^{33} - 20 q^{37} + 141 q^{39} + 54 q^{41} - 117 q^{45} - 207 q^{47} + 41 q^{49} + 141 q^{51} - 252 q^{53} - 273 q^{57} + 306 q^{59} + 7 q^{61} - 441 q^{63} + 93 q^{65} - 12 q^{67} + 189 q^{69} + 74 q^{73} + 387 q^{75} + 207 q^{77} - 33 q^{79} + 117 q^{81} - 549 q^{83} - 30 q^{85} + 87 q^{87} - 168 q^{89} - 27 q^{93} + 684 q^{95} - 10 q^{97} - 585 q^{99}+O(q^{100}) $ 8 * q + 3 * q^3 + 3 * q^5 - 3 * q^7 - 3 * q^9 - 18 * q^11 + 5 * q^13 + 21 * q^15 + 6 * q^17 - 33 * q^21 + 81 * q^23 - 23 * q^25 - 108 * q^27 + 69 * q^29 - 45 * q^31 + 72 * q^33 - 20 * q^37 + 141 * q^39 + 54 * q^41 - 117 * q^45 - 207 * q^47 + 41 * q^49 + 141 * q^51 - 252 * q^53 - 273 * q^57 + 306 * q^59 + 7 * q^61 - 441 * q^63 + 93 * q^65 - 12 * q^67 + 189 * q^69 + 74 * q^73 + 387 * q^75 + 207 * q^77 - 33 * q^79 + 117 * q^81 - 549 * q^83 - 30 * q^85 + 87 * q^87 - 168 * q^89 - 27 * q^93 + 684 * q^95 - 10 * q^97 - 585 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$37$	$65$	$127$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	0.456412	+	2.96508i	0.152137	+	0.988359i
$3$	0.456412	+	2.96508i	0.152137	+	0.988359i
$4$		0			0
$5$	4.61660	−	7.99619i	0.923321	−	1.59924i	0.129080	−	0.991634i	$-0.458797\pi$
$5$	4.61660	−	7.99619i	0.923321	−	1.59924i	0.794240	−	0.607604i	$-0.207869\pi$
$6$		0			0
$7$	5.33093	−	3.07781i	0.761561	−	0.439687i	−0.0682950	−	0.997665i	$-0.521756\pi$
$7$	5.33093	−	3.07781i	0.761561	−	0.439687i	0.829856	+	0.557978i	$0.188423\pi$
$8$		0			0
$9$	−8.58338	+	2.70659i	−0.953709	+	0.300732i
$10$		0			0
$11$	3.70016	−	2.13629i	0.336378	−	0.194208i	−0.322291	−	0.946641i	$-0.604453\pi$
$11$	3.70016	−	2.13629i	0.336378	−	0.194208i	0.658669	+	0.752433i	$0.271120\pi$
$12$		0			0
$13$	0.869235	−	1.50556i	0.0668642	−	0.115812i	−0.830655	−	0.556787i	$-0.812034\pi$
$13$	0.869235	−	1.50556i	0.0668642	−	0.115812i	0.897519	+	0.440975i	$0.145367\pi$
$14$		0			0
$15$	25.8164	+	10.0390i	1.72109	+	0.669269i
$16$		0			0
$17$	12.3476			0.726329			0.363164	−	0.931725i	$-0.381696\pi$
$17$	12.3476			0.726329			0.363164	+	0.931725i	$0.381696\pi$
$18$		0			0
$19$			33.9338i			1.78599i	0.450065	+	0.892996i	$0.351401\pi$
$19$			33.9338i			1.78599i	−0.450065	+	0.892996i	$0.648599\pi$
$20$		0			0
$21$	11.5590	+	14.4019i	0.550431	+	0.685803i
$22$		0			0
$23$	3.35035	+	1.93433i	0.145668	+	0.0841012i	0.571062	−	0.820907i	$-0.306531\pi$
$23$	3.35035	+	1.93433i	0.145668	+	0.0841012i	−0.425395	+	0.905008i	$0.639865\pi$
$24$		0			0
$25$	−30.1260	−	52.1798i	−1.20504	−	2.08719i
$26$		0			0
$27$	−11.9428	−	24.2151i	−0.442326	−	0.896854i
$28$		0			0
$29$	17.8409	+	30.9014i	0.615204	+	1.06556i	0.990349	+	0.138598i	$0.0442596\pi$
$29$	17.8409	+	30.9014i	0.615204	+	1.06556i	−0.375145	+	0.926966i	$0.622407\pi$
$30$		0			0
$31$	−38.8262	−	22.4163i	−1.25246	−	0.723107i	−0.280861	−	0.959748i	$-0.590620\pi$
$31$	−38.8262	−	22.4163i	−1.25246	−	0.723107i	−0.971597	+	0.236641i	$0.923953\pi$
$32$		0			0
$33$	8.02306	+	9.99624i	0.243123	+	0.302916i
$34$		0			0
$35$		−	56.8361i		−	1.62389i
$36$		0			0
$37$	−32.7130			−0.884134			−0.442067	−	0.896982i	$-0.645755\pi$
$37$	−32.7130			−0.884134			−0.442067	+	0.896982i	$0.645755\pi$
$38$		0			0
$39$	4.86083	+	1.89019i	0.124637	+	0.0484665i
$40$		0			0
$41$	21.8565	−	37.8565i	0.533085	−	0.923330i	−0.466168	−	0.884696i	$-0.654366\pi$
$41$	21.8565	−	37.8565i	0.533085	−	0.923330i	0.999253	−	0.0386343i	$-0.0123007\pi$
$42$		0			0
$43$	−33.9339	+	19.5918i	−0.789161	+	0.455622i	−0.839667	−	0.543102i	$-0.817250\pi$
$43$	−33.9339	+	19.5918i	−0.789161	+	0.455622i	0.0505063	+	0.998724i	$0.483916\pi$
$44$		0			0
$45$	−17.9836	+	81.1296i	−0.399636	+	1.80288i
$46$		0			0
$47$	−39.8784	+	23.0238i	−0.848477	+	0.489868i	−0.860137	−	0.510064i	$-0.829622\pi$
$47$	−39.8784	+	23.0238i	−0.848477	+	0.489868i	0.0116600	+	0.999932i	$0.496288\pi$
$48$		0			0
$49$	−5.55415	+	9.62007i	−0.113350	+	0.196328i
$50$		0			0
$51$	5.63558	+	36.6116i	0.110502	+	0.717874i
$52$		0			0
$53$	46.3143			0.873854			0.436927	−	0.899497i	$-0.356067\pi$
$53$	46.3143			0.873854			0.436927	+	0.899497i	$0.356067\pi$
$54$		0			0
$55$		−	39.4496i		−	0.717265i
$56$		0			0
$57$	−100.617	+	15.4878i	−1.76520	+	0.271716i
$58$		0			0
$59$	−23.2710	−	13.4355i	−0.394423	−	0.227720i	0.289652	−	0.957132i	$-0.406461\pi$
$59$	−23.2710	−	13.4355i	−0.394423	−	0.227720i	−0.684075	+	0.729412i	$0.739794\pi$
$60$		0			0
$61$	23.4545	+	40.6243i	0.384500	+	0.665973i	0.991700	−	0.128576i	$-0.0410407\pi$
$61$	23.4545	+	40.6243i	0.384500	+	0.665973i	−0.607200	+	0.794549i	$0.707707\pi$
$62$		0			0
$63$	−37.4270	+	40.8467i	−0.594079	+	0.648360i
$64$		0			0
$65$	−8.02582	−	13.9011i	−0.123474	−	0.213864i
$66$		0			0
$67$	56.9984	+	32.9080i	0.850722	+	0.491164i	0.860894	−	0.508784i	$-0.169905\pi$
$67$	56.9984	+	32.9080i	0.850722	+	0.491164i	−0.0101725	+	0.999948i	$0.503238\pi$
$68$		0			0
$69$	−4.20629	+	10.8169i	−0.0609607	+	0.156767i
$70$		0			0
$71$		−	96.7955i		−	1.36332i	−0.731671	−	0.681658i	$-0.761259\pi$
$71$		−	96.7955i		−	1.36332i	0.731671	−	0.681658i	$-0.238741\pi$
$72$		0			0
$73$	−14.0622			−0.192633			−0.0963163	−	0.995351i	$-0.530706\pi$
$73$	−14.0622			−0.192633			−0.0963163	+	0.995351i	$0.530706\pi$
$74$		0			0
$75$	140.967	−	113.142i	1.87957	−	1.50855i
$76$		0			0
$77$	13.1502	−	22.7768i	0.170782	−	0.295803i
$78$		0			0
$79$	−34.3954	+	19.8582i	−0.435385	+	0.251369i	−0.701638	−	0.712534i	$-0.747548\pi$
$79$	−34.3954	+	19.8582i	−0.435385	+	0.251369i	0.266253	+	0.963903i	$0.414214\pi$
$80$		0			0
$81$	66.3487	−	46.4634i	0.819120	−	0.573622i
$82$		0			0
$83$	−81.7202	+	47.1812i	−0.984581	+	0.568448i	−0.903650	−	0.428272i	$-0.859123\pi$
$83$	−81.7202	+	47.1812i	−0.984581	+	0.568448i	−0.0809306	+	0.996720i	$0.525789\pi$
$84$		0			0
$85$	57.0039	−	98.7336i	0.670634	−	1.16157i
$86$		0			0
$87$	−83.4822	+	67.0034i	−0.959565	+	0.770154i
$88$		0			0
$89$	−81.8478			−0.919639			−0.459819	−	0.888012i	$-0.652086\pi$
$89$	−81.8478			−0.919639			−0.459819	+	0.888012i	$0.652086\pi$
$90$		0			0
$91$		−	10.7014i		−	0.117597i
$92$		0			0
$93$	48.7454	−	125.354i	0.524144	−	1.34789i
$94$		0			0
$95$	271.341	+	156.659i	2.85623	+	1.64904i
$96$		0			0
$97$	−7.99028	−	13.8396i	−0.0823741	−	0.142676i	0.821895	−	0.569639i	$-0.192917\pi$
$97$	−7.99028	−	13.8396i	−0.0823741	−	0.142676i	−0.904269	+	0.426963i	$0.859584\pi$
$98$		0			0
$99$	−25.9778	+	28.3514i	−0.262402	+	0.286378i
$100$		0			0
$101$	50.6344	+	87.7014i	0.501331	+	0.868330i	0.999999	+	0.00153723i	$0.000489314\pi$
$101$	50.6344	+	87.7014i	0.501331	+	0.868330i	−0.498668	+	0.866793i	$0.666177\pi$
$102$		0			0
$103$	88.4092	+	51.0431i	0.858341	+	0.495564i	0.863456	−	0.504423i	$-0.168295\pi$
$103$	88.4092	+	51.0431i	0.858341	+	0.495564i	−0.00511517	+	0.999987i	$0.501628\pi$
$104$		0			0
$105$	168.524	−	25.9407i	1.60499	−	0.247054i
$106$		0			0
$107$			73.1463i			0.683610i	0.939771	+	0.341805i	$0.111038\pi$
$107$			73.1463i			0.683610i	−0.939771	+	0.341805i	$0.888962\pi$
$108$		0			0
$109$	−33.9344			−0.311325			−0.155663	−	0.987810i	$-0.549751\pi$
$109$	−33.9344			−0.311325			−0.155663	+	0.987810i	$0.549751\pi$
$110$		0			0
$111$	−14.9306	−	96.9965i	−0.134510	−	0.873842i
$112$		0			0
$113$	−13.9292	+	24.1262i	−0.123268	+	0.213506i	−0.921054	−	0.389434i	$-0.872671\pi$
$113$	−13.9292	+	24.1262i	−0.123268	+	0.213506i	0.797787	+	0.602940i	$0.206004\pi$
$114$		0			0
$115$	30.9345	−	17.8600i	0.268996	−	0.155305i
$116$		0			0
$117$	−3.38604	+	15.2754i	−0.0289405	+	0.130559i
$118$		0			0
$119$	65.8241	−	38.0035i	0.553143	−	0.319358i
$120$		0			0
$121$	−51.3725	+	88.9798i	−0.424566	+	0.735371i
$122$		0			0
$123$	122.223	+	47.5280i	0.993684	+	0.386407i
$124$		0			0
$125$	−325.490			−2.60392
$126$		0			0
$127$		−	117.905i		−	0.928387i	−0.885734	−	0.464193i	$-0.846344\pi$
$127$		−	117.905i		−	0.928387i	0.885734	−	0.464193i	$-0.153656\pi$
$128$		0			0
$129$	−73.5789	−	91.6748i	−0.570379	−	0.710657i
$130$		0			0
$131$	−73.5214	−	42.4476i	−0.561232	−	0.324027i	0.192408	−	0.981315i	$-0.438370\pi$
$131$	−73.5214	−	42.4476i	−0.561232	−	0.324027i	−0.753640	+	0.657288i	$0.771704\pi$
$132$		0			0
$133$	104.442	+	180.899i	0.785278	+	1.36014i
$134$		0			0
$135$	−248.763	−	16.2943i	−1.84269	−	0.120699i
$136$		0			0
$137$	3.97975	+	6.89314i	0.0290493	+	0.0503149i	0.880185	−	0.474632i	$-0.157419\pi$
$137$	3.97975	+	6.89314i	0.0290493	+	0.0503149i	−0.851135	+	0.524946i	$0.824085\pi$
$138$		0			0
$139$	−17.9239	−	10.3484i	−0.128949	−	0.0744488i	0.434138	−	0.900846i	$-0.357053\pi$
$139$	−17.9239	−	10.3484i	−0.128949	−	0.0744488i	−0.563087	+	0.826398i	$0.690386\pi$
$140$		0			0
$141$	−86.4683	−	107.734i	−0.613251	−	0.764073i
$142$		0			0
$143$		−	7.42775i		−	0.0519423i
$144$		0			0
$145$	329.458			2.27212
$146$		0			0
$147$	−31.0592	−	12.0778i	−0.211287	−	0.0821617i
$148$		0			0
$149$	65.6122	−	113.644i	0.440350	−	0.762709i	−0.557365	−	0.830267i	$-0.688188\pi$
$149$	65.6122	−	113.644i	0.440350	−	0.762709i	0.997715	+	0.0675588i	$0.0215210\pi$
$150$		0			0
$151$	204.949	−	118.328i	1.35728	−	0.783627i	0.368025	−	0.929816i	$-0.380034\pi$
$151$	204.949	−	118.328i	1.35728	−	0.783627i	0.989257	+	0.146189i	$0.0467009\pi$
$152$		0			0
$153$	−105.984	+	33.4199i	−0.692706	+	0.218431i
$154$		0			0
$155$	−358.490	+	206.974i	−2.31284	+	1.33532i
$156$		0			0
$157$	−74.8892	+	129.712i	−0.477001	+	0.826190i	−0.999653	−	0.0263562i	$-0.991610\pi$
$157$	−74.8892	+	129.712i	−0.477001	+	0.826190i	0.522651	+	0.852546i	$0.324943\pi$
$158$		0			0
$159$	21.1384	+	137.325i	0.132946	+	0.863682i
$160$		0			0
$161$	23.8140			0.147913
$162$		0			0
$163$			152.365i			0.934756i	0.884057	+	0.467378i	$0.154801\pi$
$163$			152.365i			0.934756i	−0.884057	+	0.467378i	$0.845199\pi$
$164$		0			0
$165$	116.971	−	18.0053i	0.708916	−	0.109123i
$166$		0			0
$167$	21.5631	+	12.4494i	0.129120	+	0.0745476i	0.563169	−	0.826342i	$-0.309582\pi$
$167$	21.5631	+	12.4494i	0.129120	+	0.0745476i	−0.434049	+	0.900889i	$0.642915\pi$
$168$		0			0
$169$	82.9889	+	143.741i	0.491058	+	0.850538i
$170$		0			0
$171$	−91.8451	−	291.267i	−0.537106	−	1.70332i
$172$		0			0
$173$	−54.0452	−	93.6091i	−0.312400	−	0.541093i	0.666481	−	0.745522i	$-0.267800\pi$
$173$	−54.0452	−	93.6091i	−0.312400	−	0.541093i	−0.978881	+	0.204429i	$0.934466\pi$
$174$		0			0
$175$	−321.199	−	185.445i	−1.83543	−	1.05968i
$176$		0			0
$177$	29.2162	−	75.1323i	0.165063	−	0.424477i
$178$		0			0
$179$		−	313.318i		−	1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$179$		−	313.318i		−	1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$180$		0			0
$181$	20.5886			0.113749			0.0568746	−	0.998381i	$-0.481886\pi$
$181$	20.5886			0.113749			0.0568746	+	0.998381i	$0.481886\pi$
$182$		0			0
$183$	−109.749	+	88.0858i	−0.599724	+	0.481343i
$184$		0			0
$185$	−151.023	+	261.579i	−0.816339	+	1.41394i
$186$		0			0
$187$	45.6881	−	26.3780i	0.244321	−	0.141059i
$188$		0			0
$189$	−138.196	−	92.3310i	−0.731194	−	0.488524i
$190$		0			0
$191$	23.6619	−	13.6612i	0.123884	−	0.0715247i	−0.436777	−	0.899570i	$-0.643880\pi$
$191$	23.6619	−	13.6612i	0.123884	−	0.0715247i	0.560662	+	0.828045i	$0.310547\pi$
$192$		0			0
$193$	−65.7227	+	113.835i	−0.340532	+	0.589819i	−0.984532	−	0.175207i	$-0.943941\pi$
$193$	−65.7227	+	113.835i	−0.340532	+	0.589819i	0.644000	+	0.765026i	$0.277274\pi$
$194$		0			0
$195$	37.5549	−	30.1418i	0.192589	−	0.154574i
$196$		0			0
$197$	−126.466			−0.641961			−0.320981	−	0.947086i	$-0.604012\pi$
$197$	−126.466			−0.641961			−0.320981	+	0.947086i	$0.604012\pi$
$198$		0			0
$199$		−	76.0070i		−	0.381945i	−0.981595	−	0.190972i	$-0.938836\pi$
$199$		−	76.0070i		−	0.381945i	0.981595	−	0.190972i	$-0.0611641\pi$
$200$		0			0
$201$	−71.5601	+	184.024i	−0.356021	+	0.915543i
$202$		0			0
$203$	190.217	+	109.822i	0.937030	+	0.540995i
$204$		0			0
$205$	−201.805	−	349.537i	−0.984417	−	1.70506i
$206$		0			0
$207$	−33.9928	−	7.53502i	−0.164216	−	0.0364011i
$208$		0			0
$209$	72.4925	+	125.561i	0.346854	+	0.600769i
$210$		0			0
$211$	91.8563	+	53.0332i	0.435338	+	0.251342i	0.701618	−	0.712553i	$-0.252461\pi$
$211$	91.8563	+	53.0332i	0.435338	+	0.251342i	−0.266280	+	0.963896i	$0.585795\pi$
$212$		0			0
$213$	287.006	−	44.1786i	1.34745	−	0.207411i
$214$		0			0
$215$			361.789i			1.68274i
$216$		0			0
$217$	−275.973			−1.27176
$218$		0			0
$219$	−6.41814	−	41.6954i	−0.0293066	−	0.190390i
$220$		0			0
$221$	10.7330	−	18.5900i	0.0485654	−	0.0841177i
$222$		0			0
$223$	−55.1700	+	31.8524i	−0.247399	+	0.142836i	−0.618573	−	0.785728i	$-0.712289\pi$
$223$	−55.1700	+	31.8524i	−0.247399	+	0.142836i	0.371174	+	0.928563i	$0.378955\pi$
$224$		0			0
$225$	399.813	+	366.340i	1.77695	+	1.62818i
$226$		0			0
$227$	304.643	−	175.886i	1.34204	−	0.774827i	0.354933	−	0.934892i	$-0.384504\pi$
$227$	304.643	−	175.886i	1.34204	−	0.774827i	0.987106	+	0.160065i	$0.0511704\pi$
$228$		0			0
$229$	102.232	−	177.072i	0.446429	−	0.773238i	−0.551721	−	0.834029i	$-0.686029\pi$
$229$	102.232	−	177.072i	0.446429	−	0.773238i	0.998151	+	0.0607903i	$0.0193621\pi$
$230$		0			0
$231$	73.5369	+	28.5958i	0.318342	+	0.123791i
$232$		0			0
$233$	236.626			1.01556			0.507782	−	0.861486i	$-0.330466\pi$
$233$	236.626			1.01556			0.507782	+	0.861486i	$0.330466\pi$
$234$		0			0
$235$			425.167i			1.80922i
$236$		0			0
$237$	−74.5795	−	92.9215i	−0.314682	−	0.392074i
$238$		0			0
$239$	−16.1578	−	9.32873i	−0.0676060	−	0.0390324i	0.465816	−	0.884882i	$-0.345761\pi$
$239$	−16.1578	−	9.32873i	−0.0676060	−	0.0390324i	−0.533422	+	0.845849i	$0.679094\pi$
$240$		0			0
$241$	−37.2290	−	64.4826i	−0.154477	−	0.267562i	0.778391	−	0.627779i	$-0.216036\pi$
$241$	−37.2290	−	64.4826i	−0.154477	−	0.267562i	−0.932869	+	0.360217i	$0.882703\pi$
$242$		0			0
$243$	168.050	+	175.523i	0.691564	+	0.722316i
$244$		0			0
$245$	51.2826	+	88.8241i	0.209317	+	0.362547i
$246$		0			0
$247$	51.0894	+	29.4965i	0.206840	+	0.119419i
$248$		0			0
$249$	−177.194	−	220.773i	−0.711622	−	0.886637i
$250$		0			0
$251$		−	206.637i		−	0.823257i	−0.911352	−	0.411628i	$-0.864960\pi$
$251$		−	206.637i		−	0.823257i	0.911352	−	0.411628i	$-0.135040\pi$
$252$		0			0
$253$	16.5291			0.0653325
$254$		0			0
$255$	318.770	+	123.958i	1.25008	+	0.486109i
$256$		0			0
$257$	148.678	−	257.517i	0.578513	−	1.00201i	−0.417138	−	0.908843i	$-0.636967\pi$
$257$	148.678	−	257.517i	0.578513	−	1.00201i	0.995650	−	0.0931698i	$-0.0296999\pi$
$258$		0			0
$259$	−174.390	+	100.684i	−0.673322	+	0.388743i
$260$		0			0
$261$	−236.773	−	216.950i	−0.907175	−	0.831226i
$262$		0			0
$263$	−123.730	+	71.4357i	−0.470457	+	0.271619i	−0.716431	−	0.697658i	$-0.754226\pi$
$263$	−123.730	+	71.4357i	−0.470457	+	0.271619i	0.245974	+	0.969276i	$0.420892\pi$
$264$		0			0
$265$	213.815	−	370.338i	0.806847	−	1.39750i
$266$		0			0
$267$	−37.3563	−	242.685i	−0.139911	−	0.908934i
$268$		0			0
$269$	370.517			1.37738			0.688692	−	0.725054i	$-0.258185\pi$
$269$	370.517			1.37738			0.688692	+	0.725054i	$0.258185\pi$
$270$		0			0
$271$		−	368.022i		−	1.35801i	−0.734132	−	0.679007i	$-0.762411\pi$
$271$		−	368.022i		−	1.35801i	0.734132	−	0.679007i	$-0.237589\pi$
$272$		0			0
$273$	31.7304	−	4.88423i	0.116229	−	0.0178909i
$274$		0			0
$275$	−222.942	−	128.716i	−0.810700	−	0.468058i
$276$		0			0
$277$	−15.3234	−	26.5409i	−0.0553191	−	0.0958154i	0.837040	−	0.547142i	$-0.184284\pi$
$277$	−15.3234	−	26.5409i	−0.0553191	−	0.0958154i	−0.892359	+	0.451327i	$0.850951\pi$
$278$		0			0
$279$	393.932	+	87.3210i	1.41194	+	0.312979i
$280$		0			0
$281$	−231.041	−	400.176i	−0.822212	−	1.42411i	−0.904032	−	0.427465i	$-0.859407\pi$
$281$	−231.041	−	400.176i	−0.822212	−	1.42411i	0.0818203	−	0.996647i	$-0.473927\pi$
$282$		0			0
$283$	422.693	+	244.042i	1.49361	+	0.862339i	0.999973	−	0.00732653i	$-0.00233213\pi$
$283$	422.693	+	244.042i	1.49361	+	0.862339i	0.493642	+	0.869665i	$0.335665\pi$
$284$		0			0
$285$	−340.663	+	876.050i	−1.19531	+	3.07386i
$286$		0			0
$287$		−	269.081i		−	0.937563i
$288$		0			0
$289$	−136.537			−0.472447
$290$		0			0
$291$	37.3886	−	30.0084i	0.128483	−	0.103122i
$292$		0			0
$293$	215.030	−	372.442i	0.733890	−	1.27113i	−0.221319	−	0.975201i	$-0.571036\pi$
$293$	215.030	−	372.442i	0.733890	−	1.27113i	0.955209	−	0.295933i	$-0.0956305\pi$
$294$		0			0
$295$	−214.866	+	124.053i	−0.728358	+	0.420518i
$296$		0			0
$297$	−95.9207	−	64.0863i	−0.322965	−	0.215779i
$298$		0			0
$299$	5.82449	−	3.36277i	0.0194799	−	0.0112467i
$300$		0			0
$301$	−120.599	+	208.884i	−0.400663	+	0.693968i
$302$		0			0
$303$	−236.931	+	190.163i	−0.781951	+	0.627600i
$304$		0			0
$305$	433.120			1.42007
$306$		0			0
$307$		−	276.184i		−	0.899621i	−0.893124	−	0.449810i	$-0.851492\pi$
$307$		−	276.184i		−	0.899621i	0.893124	−	0.449810i	$-0.148508\pi$
$308$		0			0
$309$	−110.996	+	285.437i	−0.359209	+	0.923743i
$310$		0			0
$311$	−172.998	−	99.8806i	−0.556264	−	0.321159i	0.195380	−	0.980728i	$-0.437406\pi$
$311$	−172.998	−	99.8806i	−0.556264	−	0.321159i	−0.751645	+	0.659568i	$0.770739\pi$
$312$		0			0
$313$	59.3385	+	102.777i	0.189580	+	0.328362i	0.945110	−	0.326752i	$-0.105954\pi$
$313$	59.3385	+	102.777i	0.189580	+	0.328362i	−0.755530	+	0.655114i	$0.772621\pi$
$314$		0			0
$315$	153.832	+	487.846i	0.488356	+	1.54872i
$316$		0			0
$317$	193.261	+	334.738i	0.609657	+	1.05596i	0.991297	+	0.131645i	$0.0420260\pi$
$317$	193.261	+	334.738i	0.609657	+	1.05596i	−0.381640	+	0.924311i	$0.624641\pi$
$318$		0			0
$319$	132.028	+	76.2267i	0.413882	+	0.238955i
$320$		0			0
$321$	−216.884	+	33.3848i	−0.675652	+	0.104003i
$322$		0			0
$323$			419.001i			1.29722i
$324$		0			0
$325$	−104.746			−0.322297
$326$		0			0
$327$	−15.4881	−	100.618i	−0.0473641	−	0.307701i
$328$		0			0
$329$	−141.726	+	245.476i	−0.430778	+	0.746129i
$330$		0			0
$331$	−282.733	+	163.236i	−0.854179	+	0.493161i	−0.862059	−	0.506808i	$-0.830825\pi$
$331$	−282.733	+	163.236i	−0.854179	+	0.493161i	0.00787942	+	0.999969i	$0.497492\pi$
$332$		0			0
$333$	280.788	−	88.5407i	0.843206	−	0.265888i
$334$		0			0
$335$	526.278	−	303.847i	1.57098	−	0.907005i
$336$		0			0
$337$	57.4906	−	99.5766i	0.170595	−	0.295479i	−0.768033	−	0.640410i	$-0.778764\pi$
$337$	57.4906	−	99.5766i	0.170595	−	0.295479i	0.938628	+	0.344931i	$0.112098\pi$
$338$		0			0
$339$	−77.8934	−	30.2898i	−0.229774	−	0.0893506i
$340$		0			0
$341$	−191.551			−0.561733
$342$		0			0
$343$			370.004i			1.07873i
$344$		0			0
$345$	67.0753	+	83.5717i	0.194421	+	0.242237i
$346$		0			0
$347$	−502.945	−	290.375i	−1.44941	−	0.836817i	−0.450963	−	0.892543i	$-0.648919\pi$
$347$	−502.945	−	290.375i	−1.44941	−	0.836817i	−0.998446	+	0.0557260i	$0.982253\pi$
$348$		0			0
$349$	−175.463	−	303.912i	−0.502761	−	0.870807i	−0.999995	−	0.00319067i	$-0.998984\pi$
$349$	−175.463	−	303.912i	−0.502761	−	0.870807i	0.497234	−	0.867616i	$-0.334349\pi$
$350$		0			0
$351$	−46.8383	−	3.06797i	−0.133442	−	0.00874066i
$352$		0			0
$353$	−67.7870	−	117.411i	−0.192031	−	0.332608i	0.753892	−	0.656998i	$-0.228174\pi$
$353$	−67.7870	−	117.411i	−0.192031	−	0.332608i	−0.945923	+	0.324390i	$0.894841\pi$
$354$		0			0
$355$	−773.995	−	446.866i	−2.18027	−	1.25878i
$356$		0			0
$357$	142.726	+	177.828i	0.399794	+	0.498118i
$358$		0			0
$359$			108.852i			0.303210i	0.988441	+	0.151605i	$0.0484441\pi$
$359$			108.852i			0.303210i	−0.988441	+	0.151605i	$0.951556\pi$
$360$		0			0
$361$	−790.506			−2.18977
$362$		0			0
$363$	−287.279	−	111.712i	−0.791403	−	0.307747i
$364$		0			0
$365$	−64.9195	+	112.444i	−0.177862	+	0.308065i
$366$		0			0
$367$	−14.6619	+	8.46503i	−0.0399506	+	0.0230655i	−0.519842	−	0.854262i	$-0.674009\pi$
$367$	−14.6619	+	8.46503i	−0.0399506	+	0.0230655i	0.479892	+	0.877328i	$0.340676\pi$
$368$		0			0
$369$	−85.1402	+	384.094i	−0.230732	+	1.04090i
$370$		0			0
$371$	246.898	−	142.547i	0.665493	−	0.384223i
$372$		0			0
$373$	−18.5300	+	32.0949i	−0.0496783	+	0.0860454i	−0.889795	−	0.456360i	$-0.849153\pi$
$373$	−18.5300	+	32.0949i	−0.0496783	+	0.0860454i	0.840117	+	0.542405i	$0.182486\pi$
$374$		0			0
$375$	−148.557	−	965.102i	−0.396153	−	2.57361i
$376$		0			0
$377$	62.0318			0.164540
$378$		0			0
$379$		−	531.193i		−	1.40156i	−0.713375	−	0.700782i	$-0.752834\pi$
$379$		−	531.193i		−	1.40156i	0.713375	−	0.700782i	$-0.247166\pi$
$380$		0			0
$381$	349.598	−	53.8133i	0.917580	−	0.141242i
$382$		0			0
$383$	−324.004	−	187.064i	−0.845963	−	0.488417i	0.0133235	−	0.999911i	$-0.495759\pi$
$383$	−324.004	−	187.064i	−0.845963	−	0.488417i	−0.859287	+	0.511494i	$0.829092\pi$
$384$		0			0
$385$	−121.418	−	210.303i	−0.315373	−	0.546241i
$386$		0			0
$387$	238.241	−	260.009i	0.615609	−	0.671857i
$388$		0			0
$389$	271.593	+	470.412i	0.698182	+	1.20929i	0.969096	+	0.246683i	$0.0793406\pi$
$389$	271.593	+	470.412i	0.698182	+	1.20929i	−0.270915	+	0.962603i	$0.587326\pi$
$390$		0			0
$391$	41.3688	+	23.8843i	0.105802	+	0.0610851i
$392$		0			0
$393$	92.3044	−	237.370i	0.234871	−	0.603995i
$394$		0			0
$395$			366.709i			0.928378i
$396$		0			0
$397$	606.097			1.52669			0.763346	−	0.645990i	$-0.223555\pi$
$397$	606.097			1.52669			0.763346	+	0.645990i	$0.223555\pi$
$398$		0			0
$399$	−488.711	+	392.243i	−1.22484	+	0.983065i
$400$		0			0
$401$	−293.529	+	508.408i	−0.731994	+	1.26785i	0.224036	+	0.974581i	$0.428077\pi$
$401$	−293.529	+	508.408i	−0.731994	+	1.26785i	−0.956030	+	0.293269i	$0.905257\pi$
$402$		0			0
$403$	−67.4982	+	38.9701i	−0.167489	+	0.0967000i
$404$		0			0
$405$	−65.2245	−	745.040i	−0.161048	−	1.83961i
$406$		0			0
$407$	−121.043	+	69.8844i	−0.297404	+	0.171706i
$408$		0			0
$409$	129.882	−	224.961i	0.317559	−	0.550028i	−0.662419	−	0.749133i	$-0.730470\pi$
$409$	129.882	−	224.961i	0.317559	−	0.550028i	0.979978	+	0.199105i	$0.0638036\pi$
$410$		0			0
$411$	−18.6223	+	14.9464i	−0.0453097	+	0.0363659i
$412$		0			0
$413$	−165.408			−0.400503
$414$		0			0
$415$			871.267i			2.09944i
$416$		0			0
$417$	22.5031	−	57.8689i	0.0539642	−	0.138774i
$418$		0			0
$419$	297.997	+	172.049i	0.711210	+	0.410617i	0.811509	−	0.584340i	$-0.198647\pi$
$419$	297.997	+	172.049i	0.711210	+	0.410617i	−0.100299	+	0.994957i	$0.531980\pi$
$420$		0			0
$421$	153.263	+	265.460i	0.364046	+	0.630546i	0.988623	−	0.150417i	$-0.0480617\pi$
$421$	153.263	+	265.460i	0.364046	+	0.630546i	−0.624576	+	0.780964i	$0.714728\pi$
$422$		0			0
$423$	279.975	−	305.557i	0.661880	−	0.722356i
$424$		0			0
$425$	−371.984	−	644.295i	−0.875256	−	1.51599i
$426$		0			0
$427$	250.068	+	144.377i	0.585640	+	0.338119i
$428$		0			0
$429$	22.0239	−	3.39011i	0.0513376	−	0.00790235i
$430$		0			0
$431$		−	208.029i		−	0.482667i	−0.970442	−	0.241333i	$-0.922415\pi$
$431$		−	208.029i		−	0.482667i	0.970442	−	0.241333i	$-0.0775847\pi$
$432$		0			0
$433$	353.874			0.817260			0.408630	−	0.912700i	$-0.366007\pi$
$433$	353.874			0.817260			0.408630	+	0.912700i	$0.366007\pi$
$434$		0			0
$435$	150.368	+	976.867i	0.345674	+	2.24567i
$436$		0			0
$437$	−65.6392	+	113.690i	−0.150204	+	0.260161i
$438$		0			0
$439$	200.356	−	115.675i	0.456391	−	0.263497i	−0.254135	−	0.967169i	$-0.581791\pi$
$439$	200.356	−	115.675i	0.456391	−	0.263497i	0.710525	+	0.703672i	$0.248457\pi$
$440$		0			0
$441$	21.6358	−	97.6055i	0.0490607	−	0.221328i
$442$		0			0
$443$	−221.434	+	127.845i	−0.499851	+	0.288589i	−0.728652	−	0.684884i	$-0.759853\pi$
$443$	−221.434	+	127.845i	−0.499851	+	0.288589i	0.228801	+	0.973473i	$0.426520\pi$
$444$		0			0
$445$	−377.859	+	654.471i	−0.849121	+	1.47072i
$446$		0			0
$447$	366.908	+	142.677i	0.820824	+	0.319188i
$448$		0			0
$449$	323.060			0.719509			0.359755	−	0.933047i	$-0.382860\pi$
$449$	323.060			0.719509			0.359755	+	0.933047i	$0.382860\pi$
$450$		0			0
$451$		−	186.767i		−	0.414118i
$452$		0			0
$453$	444.392	+	553.685i	0.980998	+	1.22226i
$454$		0			0
$455$	−85.5702	−	49.4039i	−0.188066	−	0.108580i
$456$		0			0
$457$	277.264	+	480.235i	0.606704	+	1.05084i	0.991780	+	0.127957i	$0.0408418\pi$
$457$	277.264	+	480.235i	0.606704	+	1.05084i	−0.385076	+	0.922885i	$0.625825\pi$
$458$		0			0
$459$	−147.465	−	298.998i	−0.321274	−	0.651411i
$460$		0			0
$461$	−368.753	−	638.699i	−0.799898	−	1.38546i	−0.919682	−	0.392664i	$-0.871553\pi$
$461$	−368.753	−	638.699i	−0.799898	−	1.38546i	0.119784	−	0.992800i	$-0.461780\pi$
$462$		0			0
$463$	116.461	+	67.2386i	0.251535	+	0.145224i	0.620467	−	0.784233i	$-0.286943\pi$
$463$	116.461	+	67.2386i	0.251535	+	0.145224i	−0.368932	+	0.929456i	$0.620276\pi$
$464$		0			0
$465$	−777.315	−	968.486i	−1.67164	−	2.08277i
$466$		0			0
$467$			595.000i			1.27409i	0.770827	+	0.637045i	$0.219844\pi$
$467$			595.000i			1.27409i	−0.770827	+	0.637045i	$0.780156\pi$
$468$		0			0
$469$	405.139			0.863835
$470$		0			0
$471$	−418.786	−	162.850i	−0.889143	−	0.345754i
$472$		0			0
$473$	−83.7073	+	144.985i	−0.176971	+	0.306523i
$474$		0			0
$475$	1770.66	−	1022.29i	3.72771	−	2.15219i
$476$		0			0
$477$	−397.533	+	125.354i	−0.833402	+	0.262796i
$478$		0			0
$479$	388.924	−	224.545i	0.811950	−	0.468779i	−0.0356829	−	0.999363i	$-0.511361\pi$
$479$	388.924	−	224.545i	0.811950	−	0.468779i	0.847632	+	0.530584i	$0.178027\pi$
$480$		0			0
$481$	−28.4352	+	49.2513i	−0.0591169	+	0.102394i
$482$		0			0
$483$	10.8690	+	70.6103i	0.0225031	+	0.146191i
$484$		0			0
$485$	−147.552			−0.304231
$486$		0			0
$487$		−	120.044i		−	0.246497i	−0.992376	−	0.123249i	$-0.960669\pi$
$487$		−	120.044i		−	0.246497i	0.992376	−	0.123249i	$-0.0393312\pi$
$488$		0			0
$489$	−451.775	+	69.5413i	−0.923875	+	0.142211i
$490$		0			0
$491$	372.302	+	214.949i	0.758252	+	0.437777i	0.828668	−	0.559741i	$-0.189099\pi$
$491$	372.302	+	214.949i	0.758252	+	0.437777i	−0.0704158	+	0.997518i	$0.522433\pi$
$492$		0			0
$493$	220.292	+	381.557i	0.446840	+	0.773950i
$494$		0			0
$495$	106.774	+	338.611i	0.215705	+	0.684062i
$496$		0			0
$497$	−297.918	−	516.010i	−0.599433	−	1.03825i
$498$		0			0
$499$	−639.117	−	368.994i	−1.28080	−	0.739467i	−0.303802	−	0.952735i	$-0.598256\pi$
$499$	−639.117	−	368.994i	−1.28080	−	0.739467i	−0.976994	+	0.213268i	$0.931589\pi$
$500$		0			0
$501$	−27.0719	+	69.6182i	−0.0540358	+	0.138959i
$502$		0			0
$503$			951.782i			1.89221i	0.323859	+	0.946105i	$0.395020\pi$
$503$			951.782i			1.89221i	−0.323859	+	0.946105i	$0.604980\pi$
$504$		0			0
$505$	935.036			1.85156
$506$		0			0
$507$	−388.326	+	311.673i	−0.765929	+	0.614741i
$508$		0			0
$509$	37.8898	−	65.6271i	0.0744398	−	0.128933i	−0.826403	−	0.563079i	$-0.809616\pi$
$509$	37.8898	−	65.6271i	0.0744398	−	0.128933i	0.900843	+	0.434146i	$0.142950\pi$
$510$		0			0
$511$	−74.9644	+	43.2807i	−0.146701	+	0.0846981i
$512$		0			0
$513$	821.710	−	405.265i	1.60177	−	0.789991i
$514$		0			0
$515$	816.300	−	471.291i	1.58505	−	0.915128i
$516$		0			0
$517$	−98.3710	+	170.384i	−0.190273	+	0.329562i
$518$		0			0
$519$	252.891	−	202.973i	0.487267	−	0.391084i
$520$		0			0
$521$	24.6152			0.0472461			0.0236230	−	0.999721i	$-0.492480\pi$
$521$	24.6152			0.0472461			0.0236230	+	0.999721i	$0.492480\pi$
$522$		0			0
$523$			165.798i			0.317013i	0.987358	+	0.158506i	$0.0506678\pi$
$523$			165.798i			0.317013i	−0.987358	+	0.158506i	$0.949332\pi$
$524$		0			0
$525$	403.259	−	1037.02i	0.768111	−	1.97528i
$526$		0			0
$527$	−479.410	−	276.787i	−0.909696	−	0.525213i
$528$		0			0
$529$	−257.017	−	445.166i	−0.485854	−	0.841524i
$530$		0			0
$531$	236.108	+	52.3369i	0.444648	+	0.0985630i
$532$		0			0
$533$	−37.9968	−	65.8124i	−0.0712886	−	0.123476i
$534$		0			0
$535$	584.892	+	337.687i	1.09326	+	0.631191i
$536$		0			0
$537$	929.013	−	143.002i	1.73001	−	0.266298i
$538$		0			0
$539$			47.4611i			0.0880539i
$540$		0			0
$541$	−184.323			−0.340708			−0.170354	−	0.985383i	$-0.554491\pi$
$541$	−184.323			−0.340708			−0.170354	+	0.985383i	$0.554491\pi$
$542$		0			0
$543$	9.39688	+	61.0469i	0.0173055	+	0.112425i
$544$		0			0
$545$	−156.662	+	271.346i	−0.287453	+	0.497883i
$546$		0			0
$547$	−803.354	+	463.817i	−1.46865	+	0.847928i	−0.999383	−	0.0351259i	$-0.988817\pi$
$547$	−803.354	+	463.817i	−1.46865	+	0.847928i	−0.469272	+	0.883054i	$0.655483\pi$
$548$		0			0
$549$	−311.272	−	285.212i	−0.566980	−	0.519512i
$550$		0			0
$551$	−1048.60	+	605.411i	−1.90309	+	1.09875i
$552$		0			0
$553$	−122.240	+	211.725i	−0.221048	+	0.382866i
$554$		0			0
$555$	−844.531	−	328.407i	−1.52168	−	0.591724i
$556$		0			0
$557$	−492.087			−0.883459			−0.441730	−	0.897148i	$-0.645635\pi$
$557$	−492.087			−0.883459			−0.441730	+	0.897148i	$0.645635\pi$
$558$		0			0
$559$			68.1193i			0.121859i
$560$		0			0
$561$	99.0654	+	123.429i	0.176587	+	0.220017i
$562$		0			0
$563$	626.453	+	361.683i	1.11271	+	0.642421i	0.939529	−	0.342470i	$-0.111264\pi$
$563$	626.453	+	361.683i	1.11271	+	0.642421i	0.173177	+	0.984891i	$0.444597\pi$
$564$		0			0
$565$	128.612	+	222.762i	0.227631	+	0.394269i
$566$		0			0
$567$	210.695	−	451.902i	0.371595	−	0.797005i
$568$		0			0
$569$	435.816	+	754.855i	0.765933	+	1.32664i	0.939752	+	0.341857i	$0.111056\pi$
$569$	435.816	+	754.855i	0.765933	+	1.32664i	−0.173819	+	0.984778i	$0.555611\pi$
$570$		0			0
$571$	210.649	+	121.618i	0.368912	+	0.212991i	0.672983	−	0.739658i	$-0.265013\pi$
$571$	210.649	+	121.618i	0.368912	+	0.212991i	−0.304071	+	0.952649i	$0.598346\pi$
$572$		0			0
$573$	51.3061	+	63.9243i	0.0895395	+	0.111561i
$574$		0			0
$575$		−	233.094i		−	0.405382i
$576$		0			0
$577$	−201.625			−0.349436			−0.174718	−	0.984619i	$-0.555901\pi$
$577$	−201.625			−0.349436			−0.174718	+	0.984619i	$0.555901\pi$
$578$		0			0
$579$	−367.526	−	142.917i	−0.634761	−	0.246835i
$580$		0			0
$581$	−290.430	+	503.039i	−0.499879	+	0.865815i
$582$		0			0
$583$	171.370	−	98.9406i	0.293945	−	0.169710i
$584$		0			0
$585$	106.513	+	97.5960i	0.182074	+	0.166831i
$586$		0			0
$587$	688.983	−	397.784i	1.17374	−	0.677656i	0.219178	−	0.975685i	$-0.429662\pi$
$587$	688.983	−	397.784i	1.17374	−	0.677656i	0.954557	+	0.298029i	$0.0963291\pi$
$588$		0			0
$589$	760.672	−	1317.52i	1.29146	−	2.23688i
$590$		0			0
$591$	−57.7207	−	374.983i	−0.0976662	−	0.634489i
$592$		0			0
$593$	−1078.05			−1.81796			−0.908980	−	0.416839i	$-0.863138\pi$
$593$	−1078.05			−1.81796			−0.908980	+	0.416839i	$0.863138\pi$
$594$		0			0
$595$		−	701.789i		−	1.17948i
$596$		0			0
$597$	225.367	−	34.6905i	0.377499	−	0.0581080i
$598$		0			0
$599$	−209.699	−	121.070i	−0.350082	−	0.202120i	0.314639	−	0.949211i	$-0.398117\pi$
$599$	−209.699	−	121.070i	−0.350082	−	0.202120i	−0.664722	+	0.747091i	$0.731450\pi$
$600$		0			0
$601$	135.406	+	234.529i	0.225300	+	0.390232i	0.956410	−	0.292029i	$-0.0943303\pi$
$601$	135.406	+	234.529i	0.225300	+	0.390232i	−0.731109	+	0.682261i	$0.760997\pi$
$602$		0			0
$603$	−578.307	−	128.191i	−0.959050	−	0.212588i
$604$		0			0
$605$	474.333	+	821.569i	0.784022	+	1.35797i
$606$		0			0
$607$	−335.657	−	193.792i	−0.552977	−	0.319261i	0.197345	−	0.980334i	$-0.436768\pi$
$607$	−335.657	−	193.792i	−0.552977	−	0.319261i	−0.750322	+	0.661073i	$0.770101\pi$
$608$		0			0
$609$	−238.813	+	614.133i	−0.392140	+	1.00843i
$610$		0			0
$611$			80.0524i			0.131019i
$612$		0			0
$613$	−1120.09			−1.82722			−0.913610	−	0.406591i	$-0.866718\pi$
$613$	−1120.09			−1.82722			−0.913610	+	0.406591i	$0.866718\pi$
$614$		0			0
$615$	944.299	−	757.902i	1.53545	−	1.23236i
$616$		0			0
$617$	−266.289	+	461.226i	−0.431587	+	0.747530i	−0.997010	−	0.0772706i	$-0.975379\pi$
$617$	−266.289	+	461.226i	−0.431587	+	0.747530i	0.565423	+	0.824801i	$0.308713\pi$
$618$		0			0
$619$	−761.814	+	439.833i	−1.23072	+	0.710555i	−0.967179	−	0.254095i	$-0.918223\pi$
$619$	−761.814	+	439.833i	−1.23072	+	0.710555i	−0.263537	+	0.964649i	$0.584889\pi$
$620$		0			0
$621$	6.82723	−	104.230i	0.0109939	−	0.167843i
$622$		0			0
$623$	−436.325	+	251.912i	−0.700361	+	0.404354i
$624$		0			0
$625$	−749.506	+	1298.18i	−1.19921	+	2.07709i
$626$		0			0
$627$	−339.211	+	272.253i	−0.541006	+	0.434216i
$628$		0			0
$629$	−403.926			−0.642172
$630$		0			0
$631$		−	310.499i		−	0.492075i	−0.969260	−	0.246037i	$-0.920871\pi$
$631$		−	310.499i		−	0.492075i	0.969260	−	0.246037i	$-0.0791286\pi$
$632$		0			0
$633$	−115.323	+	296.566i	−0.182186	+	0.468509i
$634$		0			0
$635$	−942.792	−	544.321i	−1.48471	−	0.857199i
$636$		0			0
$637$	9.65572	+	16.7242i	0.0151581	+	0.0262546i
$638$		0			0
$639$	261.986	+	830.832i	0.409994	+	1.30021i
$640$		0			0
$641$	−275.610	−	477.371i	−0.429969	−	0.744728i	0.566901	−	0.823786i	$-0.308142\pi$
$641$	−275.610	−	477.371i	−0.429969	−	0.744728i	−0.996870	+	0.0790578i	$0.974809\pi$
$642$		0			0
$643$	352.044	+	203.253i	0.547503	+	0.316101i	0.748114	−	0.663570i	$-0.230959\pi$
$643$	352.044	+	203.253i	0.547503	+	0.316101i	−0.200611	+	0.979671i	$0.564293\pi$
$644$		0			0
$645$	−1072.73	+	165.125i	−1.66315	+	0.256008i
$646$		0			0
$647$		−	652.891i		−	1.00910i	−0.863381	−	0.504552i	$-0.831658\pi$
$647$		−	652.891i		−	1.00910i	0.863381	−	0.504552i	$-0.168342\pi$
$648$		0			0
$649$	−114.808			−0.176900
$650$		0			0
$651$	−125.957	−	818.281i	−0.193483	−	1.25696i
$652$		0			0
$653$	513.767	−	889.870i	0.786779	−	1.36274i	−0.141151	−	0.989988i	$-0.545080\pi$
$653$	513.767	−	889.870i	0.786779	−	1.36274i	0.927930	−	0.372754i	$-0.121586\pi$
$654$		0			0
$655$	−678.838	+	391.927i	−1.03639	+	0.598362i
$656$		0			0
$657$	120.701	−	38.0606i	0.183715	−	0.0579309i
$658$		0			0
$659$	−20.5780	+	11.8807i	−0.0312261	+	0.0180284i	−0.515532	−	0.856870i	$-0.672406\pi$
$659$	−20.5780	+	11.8807i	−0.0312261	+	0.0180284i	0.484306	+	0.874899i	$0.339072\pi$
$660$		0			0
$661$	328.164	−	568.396i	0.496465	−	0.859903i	−0.503526	−	0.863980i	$-0.667964\pi$
$661$	328.164	−	568.396i	0.496465	−	0.859903i	0.999992	+	0.00407665i	$0.00129764\pi$
$662$		0			0
$663$	60.0195	+	23.3393i	0.0905271	+	0.0352026i
$664$		0			0
$665$	1928.67			2.90025
$666$		0			0
$667$			138.041i			0.206957i
$668$		0			0
$669$	−119.625	−	149.046i	−0.178812	−	0.222789i
$670$		0			0
$671$	173.571	+	100.211i	0.258675	+	0.149346i
$672$		0			0
$673$	405.169	+	701.773i	0.602034	+	1.04275i	0.992513	+	0.122141i	$0.0389760\pi$
$673$	405.169	+	701.773i	0.602034	+	1.04275i	−0.390479	+	0.920612i	$0.627691\pi$
$674$		0			0
$675$	−903.748	+	1352.68i	−1.33889	+	2.00397i
$676$		0			0
$677$	−199.580	−	345.683i	−0.294801	−	0.510610i	0.680138	−	0.733084i	$-0.261920\pi$
$677$	−199.580	−	345.683i	−0.294801	−	0.510610i	−0.974939	+	0.222474i	$0.928587\pi$
$678$		0			0
$679$	−85.1912	−	49.1852i	−0.125466	−	0.0724377i
$680$		0			0
$681$	660.557	+	823.014i	0.969981	+	1.20854i
$682$		0			0
$683$			203.612i			0.298114i	0.988829	+	0.149057i	$0.0476238\pi$
$683$			203.612i			0.298114i	−0.988829	+	0.149057i	$0.952376\pi$
$684$		0			0
$685$	73.4918			0.107287
$686$		0			0
$687$	571.691	+	222.309i	0.832156	+	0.323594i
$688$		0			0
$689$	40.2580	−	69.7288i	0.0584296	−	0.101203i
$690$		0			0
$691$	255.646	−	147.597i	0.369965	−	0.213599i	−0.303478	−	0.952838i	$-0.598148\pi$
$691$	255.646	−	147.597i	0.369965	−	0.213599i	0.673443	+	0.739239i	$0.264815\pi$
$692$		0			0
$693$	−51.2255	+	231.094i	−0.0739185	+	0.333469i
$694$		0			0
$695$	−165.495	+	95.5487i	−0.238123	+	0.137480i
$696$		0			0
$697$	269.875	−	467.437i	0.387195	−	0.670641i
$698$		0			0
$699$	107.999	+	701.615i	0.154505	+	1.00374i
$700$		0			0
$701$	283.069			0.403808			0.201904	−	0.979405i	$-0.435287\pi$
$701$	283.069			0.403808			0.201904	+	0.979405i	$0.435287\pi$
$702$		0			0
$703$		−	1110.08i		−	1.57906i
$704$		0			0
$705$	−1260.65	+	194.051i	−1.78816	+	0.275250i
$706$		0			0
$707$	539.857	+	311.686i	0.763588	+	0.440858i
$708$		0			0
$709$	−209.399	−	362.690i	−0.295344	−	0.511551i	0.679721	−	0.733471i	$-0.262101\pi$
$709$	−209.399	−	362.690i	−0.295344	−	0.511551i	−0.975065	+	0.221920i	$0.928768\pi$
$710$		0			0
$711$	241.481	−	263.545i	0.339635	−	0.370667i
$712$		0			0
$713$	−86.7210	−	150.205i	−0.121628	−	0.210666i
$714$		0			0
$715$	−59.3937	−	34.2910i	−0.0830681	−	0.0479594i
$716$		0			0
$717$	20.2858	−	52.1670i	0.0282926	−	0.0727573i
$718$		0			0
$719$		−	454.879i		−	0.632655i	−0.948650	−	0.316328i	$-0.897550\pi$
$719$		−	454.879i		−	0.632655i	0.948650	−	0.316328i	$-0.102450\pi$
$720$		0			0
$721$	628.404			0.871572
$722$		0			0
$723$	174.204	−	139.818i	0.240946	−	0.193385i
$724$		0			0
$725$	1074.95	−	1861.87i	1.48269	−	2.56810i
$726$		0			0
$727$	207.375	−	119.728i	0.285247	−	0.164688i	−0.350549	−	0.936544i	$-0.614005\pi$
$727$	207.375	−	119.728i	0.285247	−	0.164688i	0.635797	+	0.771857i	$0.280672\pi$
$728$		0			0
$729$	−443.739	+	578.392i	−0.608695	+	0.793404i
$730$		0			0
$731$	−419.002	+	241.911i	−0.573190	+	0.330931i
$732$		0			0
$733$	−210.973	+	365.416i	−0.287822	+	0.498522i	−0.973289	−	0.229581i	$-0.926264\pi$
$733$	−210.973	+	365.416i	−0.287822	+	0.498522i	0.685468	+	0.728103i	$0.259598\pi$
$734$		0			0
$735$	−239.964	+	192.597i	−0.326482	+	0.262037i
$736$		0			0
$737$	281.204			0.381552
$738$		0			0
$739$		−	150.203i		−	0.203251i	−0.994823	−	0.101626i	$-0.967596\pi$
$739$		−	150.203i		−	0.203251i	0.994823	−	0.101626i	$-0.0324044\pi$
$740$		0			0
$741$	−64.1416	+	164.947i	−0.0865608	+	0.222600i
$742$		0			0
$743$	−726.756	−	419.593i	−0.978137	−	0.564728i	−0.0764298	−	0.997075i	$-0.524352\pi$
$743$	−726.756	−	419.593i	−0.978137	−	0.564728i	−0.901707	+	0.432347i	$0.857685\pi$
$744$		0			0
$745$	−605.810	−	1049.29i	−0.813168	−	1.40845i
$746$		0			0
$747$	573.735	−	626.157i	0.768052	−	0.838229i
$748$		0			0
$749$	225.130	+	389.937i	0.300575	+	0.520611i
$750$		0			0
$751$	634.488	+	366.322i	0.844858	+	0.487779i	0.858912	−	0.512122i	$-0.171141\pi$
$751$	634.488	+	366.322i	0.844858	+	0.487779i	−0.0140549	+	0.999901i	$0.504474\pi$
$752$		0			0
$753$	612.696	−	94.3117i	0.813673	−	0.125248i
$754$		0			0
$755$		−	2185.09i		−	2.89415i
$756$		0			0
$757$	−1455.19			−1.92231			−0.961153	−	0.276015i	$-0.910986\pi$
$757$	−1455.19			−1.92231			−0.961153	+	0.276015i	$0.910986\pi$
$758$		0			0
$759$	7.54409	+	49.0102i	0.00993951	+	0.0645720i
$760$		0			0
$761$	365.230	−	632.596i	0.479934	−	0.831270i	−0.519801	−	0.854287i	$-0.673994\pi$
$761$	365.230	−	632.596i	0.479934	−	0.831270i	0.999735	+	0.0230176i	$0.00732737\pi$
$762$		0			0
$763$	−180.902	+	104.444i	−0.237093	+	0.136886i
$764$		0			0
$765$	−222.054	+	1001.75i	−0.290267	+	1.30948i
$766$		0			0
$767$	−40.4559	+	23.3572i	−0.0527456	+	0.0304527i
$768$		0			0
$769$	430.746	−	746.074i	0.560138	−	0.970187i	−0.437346	−	0.899293i	$-0.644082\pi$
$769$	430.746	−	746.074i	0.560138	−	0.970187i	0.997484	−	0.0708938i	$-0.0225851\pi$
$770$		0			0
$771$	831.417	+	323.307i	1.07836	+	0.419335i
$772$		0			0
$773$	−981.517			−1.26975			−0.634875	−	0.772615i	$-0.718948\pi$
$773$	−981.517			−1.26975			−0.634875	+	0.772615i	$0.718948\pi$
$774$		0			0
$775$			2701.26i			3.48550i
$776$		0			0
$777$	−378.131	−	471.128i	−0.486655	−	0.606342i
$778$		0			0
$779$	1284.62	+	741.675i	1.64906	+	0.952085i
$780$		0			0
$781$	−206.783	−	358.159i	−0.264767	−	0.458590i
$782$		0			0
$783$	535.208	−	801.068i	0.683535	−	1.02308i
$784$		0			0
$785$	691.467	+	1197.66i	0.880850	+	1.52568i
$786$		0			0
$787$	−614.292	−	354.662i	−0.780549	−	0.450650i	0.0560758	−	0.998427i	$-0.482141\pi$
$787$	−614.292	−	354.662i	−0.780549	−	0.450650i	−0.836625	+	0.547776i	$0.815474\pi$
$788$		0			0
$789$	−268.284	−	334.266i	−0.340031	−	0.423658i
$790$		0			0
$791$			171.486i			0.216797i
$792$		0			0
$793$	81.5498			0.102837
$794$		0			0
$795$	1195.67	+	464.950i	1.50398	+	0.584843i
$796$		0			0
$797$	−591.856	+	1025.12i	−0.742605	+	1.28623i	0.208701	+	0.977980i	$0.433076\pi$
$797$	−591.856	+	1025.12i	−0.742605	+	1.28623i	−0.951306	+	0.308249i	$0.900257\pi$
$798$		0			0
$799$	−492.402	+	284.288i	−0.616273	+	0.355805i
$800$		0			0
$801$	702.531	−	221.529i	0.877067	−	0.276565i
$802$		0			0
$803$	−52.0323	+	30.0409i	−0.0647974	+	0.0374108i
$804$		0			0
$805$	109.940	−	190.421i	0.136571	−	0.236548i
$806$		0			0
$807$	169.108	+	1098.61i	0.209551	+	1.36135i
$808$		0			0
$809$	522.491			0.645847			0.322924	−	0.946425i	$-0.395334\pi$
$809$	522.491			0.645847			0.322924	+	0.946425i	$0.395334\pi$
$810$		0			0
$811$			115.368i			0.142254i	0.997467	+	0.0711271i	$0.0226596\pi$
$811$			115.368i			0.142254i	−0.997467	+	0.0711271i	$0.977340\pi$
$812$		0			0
$813$	1091.21	−	167.969i	1.34221	−	0.206604i
$814$		0			0
$815$	1218.34	+	703.410i	1.49490	+	0.863080i
$816$		0			0
$817$	−664.823	−	1151.51i	−0.813737	−	1.40943i
$818$		0			0
$819$	28.9642	+	91.8538i	0.0353654	+	0.112154i
$820$		0			0
$821$	417.973	+	723.951i	0.509102	+	0.881791i	0.999944	+	0.0105427i	$0.00335589\pi$
$821$	417.973	+	723.951i	0.509102	+	0.881791i	−0.490842	+	0.871249i	$0.663311\pi$
$822$		0			0
$823$	629.895	+	363.670i	0.765364	+	0.441883i	0.831218	−	0.555946i	$-0.187644\pi$
$823$	629.895	+	363.670i	0.765364	+	0.441883i	−0.0658543	+	0.997829i	$0.520977\pi$
$824$		0			0
$825$	279.899	−	719.789i	0.339272	−	0.872472i
$826$		0			0
$827$			1271.70i			1.53773i	0.639413	+	0.768864i	$0.279178\pi$
$827$			1271.70i			1.53773i	−0.639413	+	0.768864i	$0.720822\pi$
$828$		0			0
$829$	577.896			0.697100			0.348550	−	0.937290i	$-0.386674\pi$
$829$	577.896			0.697100			0.348550	+	0.937290i	$0.386674\pi$
$830$		0			0
$831$	71.7020	−	57.5486i	0.0862840	−	0.0692522i
$832$		0			0
$833$	−68.5803	+	118.785i	−0.0823293	+	0.142599i
$834$		0			0
$835$	199.096	−	114.948i	0.238439	−	0.137663i
$836$		0			0
$837$	−79.1186	+	1207.89i	−0.0945264	+	1.44312i
$838$		0			0
$839$	620.279	−	358.118i	0.739308	−	0.426839i	−0.0825099	−	0.996590i	$-0.526294\pi$
$839$	620.279	−	358.118i	0.739308	−	0.426839i	0.821818	+	0.569751i	$0.192960\pi$
$840$		0			0
$841$	−216.096	+	374.289i	−0.256951	+	0.445053i
$842$		0			0
$843$	1081.10	−	867.701i	1.28245	−	1.02930i
$844$		0			0
$845$	1532.51			1.81362
$846$		0			0
$847$			632.460i			0.746706i
$848$		0			0
$849$	−530.681	+	1364.70i	−0.625066	+	1.60742i
$850$		0			0
$851$	−109.600	−	63.2776i	−0.128790	−	0.0743567i
$852$		0			0
$853$	−522.638	−	905.235i	−0.612705	−	1.06124i	−0.990782	−	0.135463i	$-0.956748\pi$
$853$	−522.638	−	905.235i	−0.612705	−	1.06124i	0.378077	−	0.925774i	$-0.376585\pi$
$854$		0			0
$855$	−2753.04	−	610.253i	−3.21993	−	0.713746i
$856$		0			0
$857$	−411.368	−	712.510i	−0.480009	−	0.831400i	0.519728	−	0.854332i	$-0.326033\pi$
$857$	−411.368	−	712.510i	−0.480009	−	0.831400i	−0.999737	+	0.0229317i	$0.992700\pi$
$858$		0			0
$859$	−53.7624	−	31.0397i	−0.0625871	−	0.0361347i	0.468380	−	0.883527i	$-0.344838\pi$
$859$	−53.7624	−	31.0397i	−0.0625871	−	0.0361347i	−0.530967	+	0.847392i	$0.678171\pi$
$860$		0			0
$861$	797.845	−	122.811i	0.926649	−	0.142638i
$862$		0			0
$863$		−	1250.20i		−	1.44867i	−0.689448	−	0.724335i	$-0.742147\pi$
$863$		−	1250.20i		−	1.44867i	0.689448	−	0.724335i	$-0.257853\pi$
$864$		0			0
$865$	−998.022			−1.15378
$866$		0			0
$867$	−62.3171	−	404.843i	−0.0718767	−	0.466947i
$868$		0			0
$869$	−84.8456	+	146.957i	−0.0976360	+	0.169110i
$870$		0			0
$871$	99.0899	−	57.2096i	0.113766	−	0.0656827i
$872$		0			0
$873$	106.042	+	97.1639i	0.121468	+	0.111299i
$874$		0			0
$875$	−1735.16	+	1001.80i	−1.98304	+	1.14491i
$876$		0			0
$877$	−105.302	+	182.388i	−0.120071	+	0.207968i	−0.919795	−	0.392398i	$-0.871645\pi$
$877$	−105.302	+	182.388i	−0.120071	+	0.207968i	0.799725	+	0.600367i	$0.204979\pi$
$878$		0			0
$879$	1202.46	+	467.593i	1.36799	+	0.531960i
$880$		0			0
$881$	−366.361			−0.415846			−0.207923	−	0.978145i	$-0.566670\pi$
$881$	−366.361			−0.415846			−0.207923	+	0.978145i	$0.566670\pi$
$882$		0			0
$883$			266.329i			0.301619i	0.988563	+	0.150809i	$0.0481879\pi$
$883$			266.329i			0.301619i	−0.988563	+	0.150809i	$0.951812\pi$
$884$		0			0
$885$	−465.893	−	580.474i	−0.526433	−	0.655903i
$886$		0			0
$887$	175.292	+	101.205i	0.197623	+	0.114098i	0.595546	−	0.803321i	$-0.296936\pi$
$887$	175.292	+	101.205i	0.197623	+	0.114098i	−0.397923	+	0.917419i	$0.630269\pi$
$888$		0			0
$889$	−362.890	−	628.544i	−0.408200	−	0.707023i
$890$		0			0
$891$	146.242	−	313.662i	0.164132	−	0.352034i
$892$		0			0
$893$	−781.286	−	1353.23i	−0.874901	−	1.51537i
$894$		0			0
$895$	−2505.35	−	1446.47i	−2.79928	−	1.61616i
$896$		0			0
$897$	12.6292	+	15.7353i	0.0140794	+	0.0175421i
$898$		0			0
$899$		−	1599.71i		−	1.77943i
$900$		0			0
$901$	571.869			0.634705
$902$		0			0
$903$	−674.401	−	262.250i	−0.746845	−	0.290420i
$904$		0			0
$905$	95.0495	−	164.630i	0.105027	−	0.181912i
$906$		0			0
$907$	1048.89	−	605.576i	1.15644	−	0.667669i	0.205989	−	0.978554i	$-0.433959\pi$
$907$	1048.89	−	605.576i	1.15644	−	0.667669i	0.950448	+	0.310885i	$0.100625\pi$
$908$		0			0
$909$	−671.986	−	615.727i	−0.739258	−	0.677368i
$910$		0			0
$911$	640.792	−	369.962i	0.703394	−	0.406105i	−0.105216	−	0.994449i	$-0.533553\pi$
$911$	640.792	−	369.962i	0.703394	−	0.406105i	0.808610	+	0.588344i	$0.200220\pi$
$912$		0			0
$913$	−201.585	+	349.156i	−0.220794	+	0.382427i
$914$		0			0
$915$	197.681	+	1284.23i	0.216045	+	1.40353i
$916$		0			0
$917$	−522.583			−0.569883
$918$		0			0
$919$		−	1080.71i		−	1.17596i	−0.808874	−	0.587982i	$-0.799922\pi$
$919$		−	1080.71i		−	1.17596i	0.808874	−	0.587982i	$-0.200078\pi$
$920$		0			0
$921$	818.906	−	126.053i	0.889149	−	0.136866i
$922$		0			0
$923$	−145.731	−	84.1380i	−0.157889	−	0.0911571i
$924$		0			0
$925$	985.512	+	1706.96i	1.06542	+	1.84536i
$926$		0			0
$927$	−897.002	−	198.834i	−0.967640	−	0.214492i
$928$		0			0
$929$	594.530	+	1029.76i	0.639968	+	1.10846i	0.985439	+	0.170027i	$0.0543856\pi$
$929$	594.530	+	1029.76i	0.639968	+	1.10846i	−0.345472	+	0.938429i	$0.612281\pi$
$930$		0			0
$931$	−326.446	−	188.474i	−0.350640	−	0.202442i
$932$		0			0
$933$	217.195	−	558.540i	0.232792	−	0.598649i
$934$		0			0
$935$		−	487.107i		−	0.520970i
$936$		0			0
$937$	1165.30			1.24364			0.621822	−	0.783158i	$-0.286393\pi$
$937$	1165.30			1.24364			0.621822	+	0.783158i	$0.286393\pi$
$938$		0			0
$939$	−277.660	+	222.852i	−0.295698	+	0.237329i
$940$		0			0
$941$	431.742	−	747.800i	0.458812	−	0.794686i	−0.540086	−	0.841610i	$-0.681608\pi$
$941$	431.742	−	747.800i	0.458812	−	0.794686i	0.998898	+	0.0469234i	$0.0149417\pi$
$942$		0			0
$943$	146.454	−	84.5552i	0.155306	−	0.0896662i
$944$		0			0
$945$	−1376.29	+	678.783i	−1.45639	+	0.718289i
$946$		0			0
$947$	−994.656	+	574.265i	−1.05032	+	0.606404i	−0.922740	−	0.385423i	$-0.874055\pi$
$947$	−994.656	+	574.265i	−1.05032	+	0.606404i	−0.127583	+	0.991828i	$0.540722\pi$
$948$		0			0
$949$	−12.2233	+	21.1714i	−0.0128802	+	0.0223092i
$950$		0			0
$951$	−904.318	+	725.813i	−0.950913	+	0.763210i
$952$		0			0
$953$	−1447.80			−1.51920			−0.759600	−	0.650391i	$-0.774605\pi$
$953$	−1447.80			−1.51920			−0.759600	+	0.650391i	$0.774605\pi$
$954$		0			0
$955$		−	252.274i		−	0.264161i
$956$		0			0
$957$	−165.759	+	426.266i	−0.173207	+	0.445419i
$958$		0			0
$959$	42.4316	+	24.4979i	0.0442456	+	0.0255452i
$960$		0			0
$961$	524.483	+	908.431i	0.545768	+	0.945297i
$962$		0			0
$963$	−197.977	−	627.842i	−0.205584	−	0.651965i
$964$		0			0
$965$	606.831	+	1051.06i	0.628840	+	1.08918i
$966$		0			0
$967$	1318.80	+	761.408i	1.36380	+	0.787392i	0.990128	−	0.140168i	$-0.0447641\pi$
$967$	1318.80	+	761.408i	1.36380	+	0.787392i	0.373675	+	0.927560i	$0.378097\pi$
$968$		0			0
$969$	−1242.37	+	191.237i	−1.28212	+	0.197355i
$970$		0			0
$971$			266.782i			0.274750i	0.990519	+	0.137375i	$0.0438666\pi$
$971$			266.782i			0.274750i	−0.990519	+	0.137375i	$0.956133\pi$
$972$		0			0
$973$	−127.401			−0.130937
$974$		0			0
$975$	−47.8075	−	310.581i	−0.0490333	−	0.318545i
$976$		0			0
$977$	874.915	−	1515.40i	0.895512	−	1.55107i	0.0623423	−	0.998055i	$-0.480143\pi$
$977$	874.915	−	1515.40i	0.895512	−	1.55107i	0.833170	−	0.553017i	$-0.186524\pi$
$978$		0			0
$979$	−302.850	+	174.851i	−0.309347	+	0.178601i
$980$		0			0
$981$	291.272	−	91.8467i	0.296913	−	0.0936256i
$982$		0			0
$983$	−1127.06	+	650.711i	−1.14656	+	0.661964i	−0.948046	−	0.318135i	$-0.896944\pi$
$983$	−1127.06	+	650.711i	−1.14656	+	0.661964i	−0.198510	+	0.980099i	$0.563610\pi$
$984$		0			0
$985$	−583.845	+	1011.25i	−0.592736	+	1.02665i
$986$		0			0
$987$	−792.542	−	308.190i	−0.802981	−	0.312249i
$988$		0			0
$989$	−151.587			−0.153273
$990$		0			0
$991$			136.009i			0.137245i	0.997643	+	0.0686223i	$0.0218603\pi$
$991$			136.009i			0.137245i	−0.997643	+	0.0686223i	$0.978140\pi$
$992$		0			0
$993$	−613.051	−	763.824i	−0.617372	−	0.769208i
$994$		0			0
$995$	−607.766	−	350.894i	−0.610820	−	0.352657i
$996$		0			0
$997$	637.348	+	1103.92i	0.639266	+	1.10724i	0.985594	+	0.169128i	$0.0540950\pi$
$997$	637.348	+	1103.92i	0.639266	+	1.10724i	−0.346328	+	0.938113i	$0.612572\pi$
$998$		0			0
$999$	390.685	+	792.147i	0.391076	+	0.792940i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.3.o.c.31.2	yes	8
3.2	odd	2		432.3.o.a.415.1		8
4.3	odd	2		144.3.o.a.31.3	✓	8
8.3	odd	2		576.3.o.f.319.2		8
8.5	even	2		576.3.o.d.319.3		8
9.2	odd	6		432.3.o.b.127.1		8
9.4	even	3		1296.3.g.j.1135.2		8
9.5	odd	6		1296.3.g.k.1135.8		8
9.7	even	3		144.3.o.a.79.3	yes	8
12.11	even	2		432.3.o.b.415.1		8
24.5	odd	2		1728.3.o.e.1279.4		8
24.11	even	2		1728.3.o.f.1279.4		8
36.7	odd	6	inner	144.3.o.c.79.2	yes	8
36.11	even	6		432.3.o.a.127.1		8
36.23	even	6		1296.3.g.k.1135.7		8
36.31	odd	6		1296.3.g.j.1135.1		8
72.11	even	6		1728.3.o.e.127.4		8
72.29	odd	6		1728.3.o.f.127.4		8
72.43	odd	6		576.3.o.d.511.3		8
72.61	even	6		576.3.o.f.511.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.3.o.a.31.3	✓	8	4.3	odd	2
144.3.o.a.79.3	yes	8	9.7	even	3
144.3.o.c.31.2	yes	8	1.1	even	1	trivial
144.3.o.c.79.2	yes	8	36.7	odd	6	inner
432.3.o.a.127.1		8	36.11	even	6
432.3.o.a.415.1		8	3.2	odd	2
432.3.o.b.127.1		8	9.2	odd	6
432.3.o.b.415.1		8	12.11	even	2
576.3.o.d.319.3		8	8.5	even	2
576.3.o.d.511.3		8	72.43	odd	6
576.3.o.f.319.2		8	8.3	odd	2
576.3.o.f.511.2		8	72.61	even	6
1296.3.g.j.1135.1		8	36.31	odd	6
1296.3.g.j.1135.2		8	9.4	even	3
1296.3.g.k.1135.7		8	36.23	even	6
1296.3.g.k.1135.8		8	9.5	odd	6
1728.3.o.e.127.4		8	72.11	even	6
1728.3.o.e.1279.4		8	24.5	odd	2
1728.3.o.f.127.4		8	72.29	odd	6
1728.3.o.f.1279.4		8	24.11	even	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.o (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.456412 + 2.96508i) q^{3} +(4.61660 - 7.99619i) q^{5} +(5.33093 - 3.07781i) q^{7} +(-8.58338 + 2.70659i) q^{9} +O(q^{10})\)	\(q+(0.456412 + 2.96508i) q^{3} +(4.61660 - 7.99619i) q^{5} +(5.33093 - 3.07781i) q^{7} +(-8.58338 + 2.70659i) q^{9} +(3.70016 - 2.13629i) q^{11} +(0.869235 - 1.50556i) q^{13} +(25.8164 + 10.0390i) q^{15} +12.3476 q^{17} +33.9338i q^{19} +(11.5590 + 14.4019i) q^{21} +(3.35035 + 1.93433i) q^{23} +(-30.1260 - 52.1798i) q^{25} +(-11.9428 - 24.2151i) q^{27} +(17.8409 + 30.9014i) q^{29} +(-38.8262 - 22.4163i) q^{31} +(8.02306 + 9.99624i) q^{33} -56.8361i q^{35} -32.7130 q^{37} +(4.86083 + 1.89019i) q^{39} +(21.8565 - 37.8565i) q^{41} +(-33.9339 + 19.5918i) q^{43} +(-17.9836 + 81.1296i) q^{45} +(-39.8784 + 23.0238i) q^{47} +(-5.55415 + 9.62007i) q^{49} +(5.63558 + 36.6116i) q^{51} +46.3143 q^{53} -39.4496i q^{55} +(-100.617 + 15.4878i) q^{57} +(-23.2710 - 13.4355i) q^{59} +(23.4545 + 40.6243i) q^{61} +(-37.4270 + 40.8467i) q^{63} +(-8.02582 - 13.9011i) q^{65} +(56.9984 + 32.9080i) q^{67} +(-4.20629 + 10.8169i) q^{69} -96.7955i q^{71} -14.0622 q^{73} +(140.967 - 113.142i) q^{75} +(13.1502 - 22.7768i) q^{77} +(-34.3954 + 19.8582i) q^{79} +(66.3487 - 46.4634i) q^{81} +(-81.7202 + 47.1812i) q^{83} +(57.0039 - 98.7336i) q^{85} +(-83.4822 + 67.0034i) q^{87} -81.8478 q^{89} -10.7014i q^{91} +(48.7454 - 125.354i) q^{93} +(271.341 + 156.659i) q^{95} +(-7.99028 - 13.8396i) q^{97} +(-25.9778 + 28.3514i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{3} + 3 q^{5} - 3 q^{7} - 3 q^{9}+O(q^{10}) \) 8 * q + 3 * q^3 + 3 * q^5 - 3 * q^7 - 3 * q^9	\( 8 q + 3 q^{3} + 3 q^{5} - 3 q^{7} - 3 q^{9} - 18 q^{11} + 5 q^{13} + 21 q^{15} + 6 q^{17} - 33 q^{21} + 81 q^{23} - 23 q^{25} - 108 q^{27} + 69 q^{29} - 45 q^{31} + 72 q^{33} - 20 q^{37} + 141 q^{39} + 54 q^{41} - 117 q^{45} - 207 q^{47} + 41 q^{49} + 141 q^{51} - 252 q^{53} - 273 q^{57} + 306 q^{59} + 7 q^{61} - 441 q^{63} + 93 q^{65} - 12 q^{67} + 189 q^{69} + 74 q^{73} + 387 q^{75} + 207 q^{77} - 33 q^{79} + 117 q^{81} - 549 q^{83} - 30 q^{85} + 87 q^{87} - 168 q^{89} - 27 q^{93} + 684 q^{95} - 10 q^{97} - 585 q^{99}+O(q^{100}) \) 8 * q + 3 * q^3 + 3 * q^5 - 3 * q^7 - 3 * q^9 - 18 * q^11 + 5 * q^13 + 21 * q^15 + 6 * q^17 - 33 * q^21 + 81 * q^23 - 23 * q^25 - 108 * q^27 + 69 * q^29 - 45 * q^31 + 72 * q^33 - 20 * q^37 + 141 * q^39 + 54 * q^41 - 117 * q^45 - 207 * q^47 + 41 * q^49 + 141 * q^51 - 252 * q^53 - 273 * q^57 + 306 * q^59 + 7 * q^61 - 441 * q^63 + 93 * q^65 - 12 * q^67 + 189 * q^69 + 74 * q^73 + 387 * q^75 + 207 * q^77 - 33 * q^79 + 117 * q^81 - 549 * q^83 - 30 * q^85 + 87 * q^87 - 168 * q^89 - 27 * q^93 + 684 * q^95 - 10 * q^97 - 585 * q^99

Introduction

L-functions

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