LMFDB - Embedded newform 144.4.l.a.35.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [144,4,Mod(35,144)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("144.35"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(144, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3, 2])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	144.l (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$8.49627504083$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			35.1
Character	$\chi$	$=$	144.35
Dual form			144.4.l.a.107.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-2.81788 - 0.244009i) q^{2} +(7.88092 + 1.37518i) q^{4} +(15.2065 + 15.2065i) q^{5} +24.4971 q^{7} +(-21.8719 - 5.79811i) q^{8} +(-39.1395 - 46.5605i) q^{10} +(-20.1941 + 20.1941i) q^{11} +(-26.7706 - 26.7706i) q^{13} +(-69.0301 - 5.97753i) q^{14} +(60.2178 + 21.6753i) q^{16} +85.9084i q^{17} +(53.2541 - 53.2541i) q^{19} +(98.9292 + 140.752i) q^{20} +(61.8322 - 51.9771i) q^{22} -119.986i q^{23} +337.472i q^{25} +(68.9040 + 81.9686i) q^{26} +(193.060 + 33.6879i) q^{28} +(78.5071 - 78.5071i) q^{29} +200.984i q^{31} +(-164.398 - 75.7722i) q^{32} +(20.9624 - 242.080i) q^{34} +(372.515 + 372.515i) q^{35} +(-76.9819 + 76.9819i) q^{37} +(-163.058 + 137.069i) q^{38} +(-244.426 - 420.763i) q^{40} -279.001 q^{41} +(15.7637 + 15.7637i) q^{43} +(-186.919 + 131.378i) q^{44} +(-29.2778 + 338.108i) q^{46} +470.913 q^{47} +257.110 q^{49} +(82.3464 - 950.957i) q^{50} +(-174.162 - 247.791i) q^{52} +(-112.686 - 112.686i) q^{53} -614.162 q^{55} +(-535.800 - 142.037i) q^{56} +(-240.380 + 202.067i) q^{58} +(241.182 - 241.182i) q^{59} +(6.00098 + 6.00098i) q^{61} +(49.0420 - 566.349i) q^{62} +(444.764 + 253.632i) q^{64} -814.171i q^{65} +(273.801 - 273.801i) q^{67} +(-118.139 + 677.037i) q^{68} +(-958.805 - 1140.60i) q^{70} +448.812i q^{71} +54.7653i q^{73} +(235.710 - 198.142i) q^{74} +(492.925 - 346.457i) q^{76} +(-494.698 + 494.698i) q^{77} -29.1159i q^{79} +(586.094 + 1245.30i) q^{80} +(786.193 + 68.0789i) q^{82} +(-893.890 - 893.890i) q^{83} +(-1306.36 + 1306.36i) q^{85} +(-40.5738 - 48.2668i) q^{86} +(558.773 - 324.597i) q^{88} -281.264 q^{89} +(-655.802 - 655.802i) q^{91} +(165.003 - 945.604i) q^{92} +(-1326.98 - 114.907i) q^{94} +1619.61 q^{95} +188.720 q^{97} +(-724.506 - 62.7373i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 120 q^{10} - 144 q^{16} - 48 q^{19} + 72 q^{22} + 72 q^{28} - 984 q^{34} - 1272 q^{40} + 864 q^{43} - 1416 q^{46} + 2352 q^{49} - 648 q^{52} - 576 q^{55} + 1128 q^{58} + 1824 q^{61} + 3024 q^{64}+ \cdots - 11304 q^{94}+O(q^{100}) $ 48 * q - 120 * q^10 - 144 * q^16 - 48 * q^19 + 72 * q^22 + 72 * q^28 - 984 * q^34 - 1272 * q^40 + 864 * q^43 - 1416 * q^46 + 2352 * q^49 - 648 * q^52 - 576 * q^55 + 1128 * q^58 + 1824 * q^61 + 3024 * q^64 + 816 * q^67 + 2664 * q^70 + 1920 * q^76 + 1200 * q^82 - 480 * q^85 - 4560 * q^88 - 3600 * q^91 - 11304 * q^94

Character values

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

$n$	$37$	$65$	$127$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$	$-1$

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 $

$2$	−2.81788	−	0.244009i	−0.996272	−	0.0862703i
$2$	−2.81788	−	0.244009i	−0.996272	−	0.0862703i
$3$		0			0
$3$		0			0
$4$	7.88092	+	1.37518i	0.985115	+	0.171897i
$5$	15.2065	+	15.2065i	1.36011	+	1.36011i	0.873774	+	0.486333i	$0.161666\pi$
$5$	15.2065	+	15.2065i	1.36011	+	1.36011i	0.486333	+	0.873774i	$0.338334\pi$
$6$		0			0
$7$	24.4971			1.32272			0.661361	−	0.750068i	$-0.269979\pi$
$7$	24.4971			1.32272			0.661361	+	0.750068i	$0.269979\pi$
$8$	−21.8719	−	5.79811i	−0.966613	−	0.256243i
$9$		0			0
$10$	−39.1395	−	46.5605i	−1.23770	−	1.47237i
$11$	−20.1941	+	20.1941i	−0.553524	+	0.553524i	−0.927456	−	0.373932i	$-0.878009\pi$
$11$	−20.1941	+	20.1941i	−0.553524	+	0.553524i	0.373932	+	0.927456i	$0.378009\pi$
$12$		0			0
$13$	−26.7706	−	26.7706i	−0.571140	−	0.571140i	0.361307	−	0.932447i	$-0.382331\pi$
$13$	−26.7706	−	26.7706i	−0.571140	−	0.571140i	−0.932447	+	0.361307i	$0.882331\pi$
$14$	−69.0301	−	5.97753i	−1.31779	−	0.114112i
$15$		0			0
$16$	60.2178	+	21.6753i	0.940903	+	0.338677i
$17$			85.9084i			1.22564i	0.790223	+	0.612819i	$0.209965\pi$
$17$			85.9084i			1.22564i	−0.790223	+	0.612819i	$0.790035\pi$
$18$		0			0
$19$	53.2541	−	53.2541i	0.643017	−	0.643017i	−0.308279	−	0.951296i	$-0.599753\pi$
$19$	53.2541	−	53.2541i	0.643017	−	0.643017i	0.951296	+	0.308279i	$0.0997530\pi$
$20$	98.9292	+	140.752i	1.10606	+	1.57366i
$21$		0			0
$22$	61.8322	−	51.9771i	0.599212	−	0.503707i
$23$		−	119.986i		−	1.08778i	−0.839157	−	0.543889i	$-0.816951\pi$
$23$		−	119.986i		−	1.08778i	0.839157	−	0.543889i	$-0.183049\pi$
$24$		0			0
$25$			337.472i			2.69978i
$26$	68.9040	+	81.9686i	0.519738	+	0.618283i
$27$		0			0
$28$	193.060	+	33.6879i	1.30303	+	0.227372i
$29$	78.5071	−	78.5071i	0.502703	−	0.502703i	−0.409574	−	0.912277i	$-0.634323\pi$
$29$	78.5071	−	78.5071i	0.502703	−	0.502703i	0.912277	+	0.409574i	$0.134323\pi$
$30$		0			0
$31$			200.984i			1.16445i	0.813029	+	0.582223i	$0.197817\pi$
$31$			200.984i			1.16445i	−0.813029	+	0.582223i	$0.802183\pi$
$32$	−164.398	−	75.7722i	−0.908177	−	0.418586i
$33$		0			0
$34$	20.9624	−	242.080i	0.105736	−	1.22107i
$35$	372.515	+	372.515i	1.79904	+	1.79904i
$36$		0			0
$37$	−76.9819	+	76.9819i	−0.342047	+	0.342047i	−0.857137	−	0.515089i	$-0.827759\pi$
$37$	−76.9819	+	76.9819i	−0.342047	+	0.342047i	0.515089	+	0.857137i	$0.327759\pi$
$38$	−163.058	+	137.069i	−0.696093	+	0.585146i
$39$		0			0
$40$	−244.426	−	420.763i	−0.966179	−	1.66321i
$41$	−279.001			−1.06275			−0.531374	−	0.847137i	$-0.678324\pi$
$41$	−279.001			−1.06275			−0.531374	+	0.847137i	$0.678324\pi$
$42$		0			0
$43$	15.7637	+	15.7637i	0.0559057	+	0.0559057i	0.734507	−	0.678601i	$-0.237414\pi$
$43$	15.7637	+	15.7637i	0.0559057	+	0.0559057i	−0.678601	+	0.734507i	$0.737414\pi$
$44$	−186.919	+	131.378i	−0.640433	+	0.450135i
$45$		0			0
$46$	−29.2778	+	338.108i	−0.0938430	+	1.08372i
$47$	470.913			1.46148			0.730742	−	0.682654i	$-0.239174\pi$
$47$	470.913			1.46148			0.730742	+	0.682654i	$0.239174\pi$
$48$		0			0
$49$	257.110			0.749592
$50$	82.3464	−	950.957i	0.232911	−	2.68971i
$51$		0			0
$52$	−174.162	−	247.791i	−0.464461	−	0.660816i
$53$	−112.686	−	112.686i	−0.292050	−	0.292050i	0.545840	−	0.837889i	$-0.316211\pi$
$53$	−112.686	−	112.686i	−0.292050	−	0.292050i	−0.837889	+	0.545840i	$0.816211\pi$
$54$		0			0
$55$	−614.162			−1.50570
$56$	−535.800	−	142.037i	−1.27856	−	0.338938i
$57$		0			0
$58$	−240.380	+	202.067i	−0.544197	+	0.457461i
$59$	241.182	−	241.182i	0.532190	−	0.532190i	−0.389033	−	0.921224i	$-0.627191\pi$
$59$	241.182	−	241.182i	0.532190	−	0.532190i	0.921224	+	0.389033i	$0.127191\pi$
$60$		0			0
$61$	6.00098	+	6.00098i	0.0125958	+	0.0125958i	0.713377	−	0.700781i	$-0.247165\pi$
$61$	6.00098	+	6.00098i	0.0125958	+	0.0125958i	−0.700781	+	0.713377i	$0.747165\pi$
$62$	49.0420	−	566.349i	0.100457	−	1.16010i
$63$		0			0
$64$	444.764	+	253.632i	0.868679	+	0.495375i
$65$		−	814.171i		−	1.55362i
$66$		0			0
$67$	273.801	−	273.801i	0.499255	−	0.499255i	−0.411951	−	0.911206i	$-0.635153\pi$
$67$	273.801	−	273.801i	0.499255	−	0.499255i	0.911206	+	0.411951i	$0.135153\pi$
$68$	−118.139	+	677.037i	−0.210684	+	1.20739i
$69$		0			0
$70$	−958.805	−	1140.60i	−1.63713	−	1.94754i
$71$			448.812i			0.750201i	0.926984	+	0.375100i	$0.122392\pi$
$71$			448.812i			0.750201i	−0.926984	+	0.375100i	$0.877608\pi$
$72$		0			0
$73$			54.7653i			0.0878053i	0.999036	+	0.0439027i	$0.0139791\pi$
$73$			54.7653i			0.0878053i	−0.999036	+	0.0439027i	$0.986021\pi$
$74$	235.710	−	198.142i	0.370281	−	0.311264i
$75$		0			0
$76$	492.925	−	346.457i	0.743978	−	0.522913i
$77$	−494.698	+	494.698i	−0.732157	+	0.732157i
$78$		0			0
$79$		−	29.1159i		−	0.0414658i	−0.999785	−	0.0207329i	$-0.993400\pi$
$79$		−	29.1159i		−	0.0414658i	0.999785	−	0.0207329i	$-0.00659996\pi$
$80$	586.094	+	1245.30i	0.819091	+	1.74036i
$81$		0			0
$82$	786.193	+	68.0789i	1.05879	+	0.0916836i
$83$	−893.890	−	893.890i	−1.18213	−	1.18213i	−0.979190	−	0.202945i	$-0.934949\pi$
$83$	−893.890	−	893.890i	−1.18213	−	1.18213i	−0.202945	−	0.979190i	$-0.565051\pi$
$84$		0			0
$85$	−1306.36	+	1306.36i	−1.66700	+	1.66700i
$86$	−40.5738	−	48.2668i	−0.0508743	−	0.0605203i
$87$		0			0
$88$	558.773	−	324.597i	0.676879	−	0.393206i
$89$	−281.264			−0.334988			−0.167494	−	0.985873i	$-0.553567\pi$
$89$	−281.264			−0.334988			−0.167494	+	0.985873i	$0.553567\pi$
$90$		0			0
$91$	−655.802	−	655.802i	−0.755459	−	0.755459i
$92$	165.003	−	945.604i	0.186986	−	1.07159i
$93$		0			0
$94$	−1326.98	−	114.907i	−1.45603	−	0.126083i
$95$	1619.61			1.74914
$96$		0			0
$97$	188.720			0.197542			0.0987709	−	0.995110i	$-0.468509\pi$
$97$	188.720			0.197542			0.0987709	+	0.995110i	$0.468509\pi$
$98$	−724.506	−	62.7373i	−0.746798	−	0.0646675i
$99$		0			0
$100$	−464.085	+	2659.59i	−0.464085	+	2.65959i
$101$	−218.904	−	218.904i	−0.215661	−	0.215661i	0.591006	−	0.806667i	$-0.298731\pi$
$101$	−218.904	−	218.904i	−0.215661	−	0.215661i	−0.806667	+	0.591006i	$0.798731\pi$
$102$		0			0
$103$	1897.36			1.81507			0.907536	−	0.419973i	$-0.137961\pi$
$103$	1897.36			1.81507			0.907536	+	0.419973i	$0.137961\pi$
$104$	430.306	+	740.743i	0.405721	+	0.698421i
$105$		0			0
$106$	290.040	+	345.033i	0.265766	+	0.316156i
$107$	400.476	−	400.476i	0.361827	−	0.361827i	−0.502658	−	0.864485i	$-0.667645\pi$
$107$	400.476	−	400.476i	0.361827	−	0.361827i	0.864485	+	0.502658i	$0.167645\pi$
$108$		0			0
$109$	−206.240	−	206.240i	−0.181231	−	0.181231i	0.610661	−	0.791892i	$-0.290904\pi$
$109$	−206.240	−	206.240i	−0.181231	−	0.181231i	−0.791892	+	0.610661i	$0.790904\pi$
$110$	1730.64	+	149.861i	1.50009	+	0.129897i
$111$		0			0
$112$	1475.16	+	530.984i	1.24455	+	0.447976i
$113$		−	385.068i		−	0.320568i	−0.987071	−	0.160284i	$-0.948759\pi$
$113$		−	385.068i		−	0.320568i	0.987071	−	0.160284i	$-0.0512410\pi$
$114$		0			0
$115$	1824.57	−	1824.57i	1.47949	−	1.47949i
$116$	726.669	−	510.747i	0.581634	−	0.408807i
$117$		0			0
$118$	−738.473	+	620.772i	−0.576119	+	0.484294i
$119$			2104.51i			1.62118i
$120$		0			0
$121$			515.394i			0.387223i
$122$	−15.4458	−	18.3743i	−0.0114622	−	0.0136355i
$123$		0			0
$124$	−276.389	+	1583.94i	−0.200165	+	1.14711i
$125$	−3230.95	+	3230.95i	−2.31188	+	2.31188i
$126$		0			0
$127$		−	1420.04i		−	0.992189i	−0.868268	−	0.496095i	$-0.834767\pi$
$127$		−	1420.04i		−	0.992189i	0.868268	−	0.496095i	$-0.165233\pi$
$128$	−1191.40	−	823.231i	−0.822705	−	0.568469i
$129$		0			0
$130$	−198.665	+	2294.24i	−0.134031	+	1.54783i
$131$	−202.131	−	202.131i	−0.134811	−	0.134811i	0.636481	−	0.771292i	$-0.280389\pi$
$131$	−202.131	−	202.131i	−0.134811	−	0.134811i	−0.771292	+	0.636481i	$0.780389\pi$
$132$		0			0
$133$	1304.57	−	1304.57i	0.850532	−	0.850532i
$134$	−838.348	+	704.728i	−0.540464	+	0.454323i
$135$		0			0
$136$	498.106	−	1878.98i	0.314061	−	1.18472i
$137$	985.515			0.614586			0.307293	−	0.951615i	$-0.400577\pi$
$137$	985.515			0.614586			0.307293	+	0.951615i	$0.400577\pi$
$138$		0			0
$139$	−1199.58	−	1199.58i	−0.731992	−	0.731992i	0.239022	−	0.971014i	$-0.423173\pi$
$139$	−1199.58	−	1199.58i	−0.731992	−	0.731992i	−0.971014	+	0.239022i	$0.923173\pi$
$140$	2423.48	+	3448.03i	1.46301	+	2.08151i
$141$		0			0
$142$	109.514	−	1264.70i	0.0647200	−	0.747404i
$143$	1081.22			0.632279
$144$		0			0
$145$	2387.63			1.36746
$146$	13.3632	−	154.322i	0.00757499	−	0.0874780i
$147$		0			0
$148$	−712.552	+	500.824i	−0.395753	+	0.278159i
$149$	−1257.05	−	1257.05i	−0.691150	−	0.691150i	0.271335	−	0.962485i	$-0.412535\pi$
$149$	−1257.05	−	1257.05i	−0.691150	−	0.691150i	−0.962485	+	0.271335i	$0.912535\pi$
$150$		0			0
$151$	−1832.83			−0.987770			−0.493885	−	0.869527i	$-0.664424\pi$
$151$	−1832.83			−0.987770			−0.493885	+	0.869527i	$0.664424\pi$
$152$	−1473.54	+	855.997i	−0.786317	+	0.456780i
$153$		0			0
$154$	1514.71	−	1273.29i	0.792591	−	0.666264i
$155$	−3056.25	+	3056.25i	−1.58377	+	1.58377i
$156$		0			0
$157$	−2101.87	−	2101.87i	−1.06845	−	1.06845i	−0.997478	−	0.0709767i	$-0.977388\pi$
$157$	−2101.87	−	2101.87i	−1.06845	−	1.06845i	−0.0709767	−	0.997478i	$-0.522612\pi$
$158$	−7.10455	+	82.0452i	−0.00357727	+	0.0413112i
$159$		0			0
$160$	−1347.68	−	3652.13i	−0.665895	−	1.80454i
$161$		−	2939.33i		−	1.43883i
$162$		0			0
$163$	−154.434	+	154.434i	−0.0742099	+	0.0742099i	−0.743238	−	0.669028i	$-0.766711\pi$
$163$	−154.434	+	154.434i	−0.0742099	+	0.0742099i	0.669028	+	0.743238i	$0.266711\pi$
$164$	−2198.79	−	383.676i	−1.04693	−	0.182684i
$165$		0			0
$166$	2300.76	+	2736.99i	1.07574	+	1.27971i
$167$		−	3482.63i		−	1.61374i	−0.590730	−	0.806869i	$-0.701160\pi$
$167$		−	3482.63i		−	1.61374i	0.590730	−	0.806869i	$-0.298840\pi$
$168$		0			0
$169$		−	763.674i		−	0.347598i
$170$	3999.94	−	3362.41i	1.80460	−	1.51697i
$171$		0			0
$172$	102.555	+	145.911i	0.0454635	+	0.0646836i
$173$	2563.59	−	2563.59i	1.12663	−	1.12663i	0.135903	−	0.990722i	$-0.456606\pi$
$173$	2563.59	−	2563.59i	1.12663	−	1.12663i	0.990722	−	0.135903i	$-0.0433936\pi$
$174$		0			0
$175$			8267.11i			3.57106i
$176$	−1653.76	+	778.331i	−0.708278	+	0.333346i
$177$		0			0
$178$	792.568	+	68.6310i	0.333739	+	0.0288995i
$179$	−580.106	−	580.106i	−0.242230	−	0.242230i	0.575542	−	0.817772i	$-0.304791\pi$
$179$	−580.106	−	580.106i	−0.242230	−	0.242230i	−0.817772	+	0.575542i	$0.804791\pi$
$180$		0			0
$181$	18.6302	−	18.6302i	0.00765067	−	0.00765067i	−0.703271	−	0.710922i	$-0.748278\pi$
$181$	18.6302	−	18.6302i	0.00765067	−	0.00765067i	0.710922	+	0.703271i	$0.248278\pi$
$182$	1687.95	+	2008.00i	0.687469	+	0.817816i
$183$		0			0
$184$	−695.695	+	2624.34i	−0.278735	+	1.05146i
$185$	−2341.24			−0.930441
$186$		0			0
$187$	−1734.85	−	1734.85i	−0.678420	−	0.678420i
$188$	3711.23	+	647.589i	1.43973	+	0.251225i
$189$		0			0
$190$	−4563.87	−	395.200i	−1.74262	−	0.150899i
$191$	−1727.83			−0.654561			−0.327281	−	0.944927i	$-0.606132\pi$
$191$	−1727.83			−0.654561			−0.327281	+	0.944927i	$0.606132\pi$
$192$		0			0
$193$	4097.40			1.52817			0.764087	−	0.645113i	$-0.223190\pi$
$193$	4097.40			1.52817			0.764087	+	0.645113i	$0.223190\pi$
$194$	−531.789	−	46.0493i	−0.196805	−	0.0170420i
$195$		0			0
$196$	2026.26	+	353.572i	0.738435	+	0.128853i
$197$	3889.19	+	3889.19i	1.40656	+	1.40656i	0.776729	+	0.629835i	$0.216877\pi$
$197$	3889.19	+	3889.19i	1.40656	+	1.40656i	0.629835	+	0.776729i	$0.283123\pi$
$198$		0			0
$199$	−3332.92			−1.18726			−0.593629	−	0.804739i	$-0.702305\pi$
$199$	−3332.92			−1.18726			−0.593629	+	0.804739i	$0.702305\pi$
$200$	1956.70	−	7381.18i	0.691798	−	2.60964i
$201$		0			0
$202$	563.430	+	670.259i	0.196251	+	0.233462i
$203$	1923.20	−	1923.20i	0.664936	−	0.664936i
$204$		0			0
$205$	−4242.62	−	4242.62i	−1.44545	−	1.44545i
$206$	−5346.54	−	462.974i	−1.80831	−	0.156587i
$207$		0			0
$208$	−1031.80	−	2192.32i	−0.343955	−	0.730819i
$209$			2150.84i			0.711850i
$210$		0			0
$211$	3673.14	−	3673.14i	1.19843	−	1.19843i	0.223796	−	0.974636i	$-0.428155\pi$
$211$	3673.14	−	3673.14i	1.19843	−	1.19843i	0.974636	−	0.223796i	$-0.0718448\pi$
$212$	−733.107	−	1043.03i	−0.237500	−	0.337905i
$213$		0			0
$214$	−1226.21	+	1030.77i	−0.391693	+	0.329263i
$215$			479.421i			0.152075i
$216$		0			0
$217$			4923.54i			1.54024i
$218$	530.834	+	631.483i	0.164920	+	0.196190i
$219$		0			0
$220$	−4840.16	−	844.582i	−1.48329	−	0.258826i
$221$	2299.82	−	2299.82i	0.700011	−	0.700011i
$222$		0			0
$223$		−	867.243i		−	0.260426i	−0.991486	−	0.130213i	$-0.958434\pi$
$223$		−	867.243i		−	0.260426i	0.991486	−	0.130213i	$-0.0415660\pi$
$224$	−4027.27	−	1856.20i	−1.20127	−	0.553673i
$225$		0			0
$226$	−93.9602	+	1085.08i	−0.0276555	+	0.319373i
$227$	−331.276	−	331.276i	−0.0968614	−	0.0968614i	0.657016	−	0.753877i	$-0.271819\pi$
$227$	−331.276	−	331.276i	−0.0968614	−	0.0968614i	−0.753877	+	0.657016i	$0.771819\pi$
$228$		0			0
$229$	−4331.27	+	4331.27i	−1.24986	+	1.24986i	−0.294080	+	0.955781i	$0.595013\pi$
$229$	−4331.27	+	4331.27i	−1.24986	+	1.24986i	−0.955781	+	0.294080i	$0.904987\pi$
$230$	−5586.63	+	4696.21i	−1.60162	+	1.34634i
$231$		0			0
$232$	−2172.29	+	1261.91i	−0.614733	+	0.357105i
$233$	3952.98			1.11145			0.555726	−	0.831365i	$-0.312440\pi$
$233$	3952.98			1.11145			0.555726	+	0.831365i	$0.312440\pi$
$234$		0			0
$235$	7160.91	+	7160.91i	1.98777	+	1.98777i
$236$	2232.41	−	1569.07i	0.615751	−	0.432787i
$237$		0			0
$238$	513.520	−	5930.26i	0.139860	−	1.61513i
$239$	2530.55			0.684887			0.342443	−	0.939538i	$-0.388745\pi$
$239$	2530.55			0.684887			0.342443	+	0.939538i	$0.388745\pi$
$240$		0			0
$241$	−315.543			−0.0843398			−0.0421699	−	0.999110i	$-0.513427\pi$
$241$	−315.543			−0.0843398			−0.0421699	+	0.999110i	$0.513427\pi$
$242$	125.761	−	1452.32i	0.0334059	−	0.385780i
$243$		0			0
$244$	39.0408	+	55.5456i	0.0102432	+	0.0145735i
$245$	3909.73	+	3909.73i	1.01953	+	1.01953i
$246$		0			0
$247$	−2851.28			−0.734505
$248$	1165.33	−	4395.91i	0.298380	−	1.12557i
$249$		0			0
$250$	9892.82	−	8316.06i	2.50271	−	2.10381i
$251$	−871.154	+	871.154i	−0.219071	+	0.219071i	−0.808107	−	0.589036i	$-0.799508\pi$
$251$	−871.154	+	871.154i	−0.219071	+	0.219071i	0.589036	+	0.808107i	$0.299508\pi$
$252$		0			0
$253$	2423.02	+	2423.02i	0.602111	+	0.602111i
$254$	−346.503	+	4001.50i	−0.0855965	+	0.988490i
$255$		0			0
$256$	3156.36	+	2610.48i	0.770596	+	0.637325i
$257$		−	1822.51i		−	0.442353i	−0.975234	−	0.221177i	$-0.929010\pi$
$257$		−	1822.51i		−	0.442353i	0.975234	−	0.221177i	$-0.0709898\pi$
$258$		0			0
$259$	−1885.84	+	1885.84i	−0.452433	+	0.452433i
$260$	1119.63	−	6416.41i	0.267063	−	1.53050i
$261$		0			0
$262$	520.259	+	618.902i	0.122678	+	0.145939i
$263$		−	1120.24i		−	0.262651i	−0.991339	−	0.131325i	$-0.958077\pi$
$263$		−	1120.24i		−	0.262651i	0.991339	−	0.131325i	$-0.0419233\pi$
$264$		0			0
$265$		−	3427.11i		−	0.794437i
$266$	−3994.46	+	3357.80i	−0.920737	+	0.773986i
$267$		0			0
$268$	2534.33	−	1781.28i	0.577644	−	0.406003i
$269$	−1477.67	+	1477.67i	−0.334926	+	0.334926i	−0.854454	−	0.519527i	$-0.826108\pi$
$269$	−1477.67	+	1477.67i	−0.334926	+	0.334926i	0.519527	+	0.854454i	$0.326108\pi$
$270$		0			0
$271$			5865.36i			1.31474i	0.753566	+	0.657372i	$0.228332\pi$
$271$			5865.36i			1.31474i	−0.753566	+	0.657372i	$0.771668\pi$
$272$	−1862.09	+	5173.21i	−0.415096	+	1.15321i
$273$		0			0
$274$	−2777.07	−	240.475i	−0.612294	−	0.0530205i
$275$	−6814.96	−	6814.96i	−1.49439	−	1.49439i
$276$		0			0
$277$	197.049	−	197.049i	0.0427420	−	0.0427420i	−0.685413	−	0.728155i	$-0.740378\pi$
$277$	197.049	−	197.049i	0.0427420	−	0.0427420i	0.728155	+	0.685413i	$0.240378\pi$
$278$	3087.56	+	3672.98i	0.666114	+	0.792412i
$279$		0			0
$280$	−5987.74	−	10307.5i	−1.27799	−	2.19997i
$281$	4436.82			0.941917			0.470958	−	0.882155i	$-0.343908\pi$
$281$	4436.82			0.941917			0.470958	+	0.882155i	$0.343908\pi$
$282$		0			0
$283$	−6047.06	−	6047.06i	−1.27018	−	1.27018i	−0.945994	−	0.324186i	$-0.894910\pi$
$283$	−6047.06	−	6047.06i	−1.27018	−	1.27018i	−0.324186	−	0.945994i	$-0.605090\pi$
$284$	−617.197	+	3537.05i	−0.128957	+	0.739034i
$285$		0			0
$286$	−3046.74	−	263.827i	−0.629921	−	0.0545469i
$287$	−6834.73			−1.40572
$288$		0			0
$289$	−2467.25			−0.502189
$290$	−6728.05	−	582.603i	−1.36236	−	0.117971i
$291$		0			0
$292$	−75.3120	+	431.601i	−0.0150935	+	0.0864984i
$293$	365.261	+	365.261i	0.0728287	+	0.0728287i	0.742583	−	0.669754i	$-0.233601\pi$
$293$	365.261	+	365.261i	0.0728287	+	0.0728287i	−0.669754	+	0.742583i	$0.733601\pi$
$294$		0			0
$295$	7335.05			1.44767
$296$	2130.09	−	1237.39i	0.418274	−	0.242980i
$297$		0			0
$298$	3235.48	+	3848.94i	0.628948	+	0.748199i
$299$	−3212.11	+	3212.11i	−0.621274	+	0.621274i
$300$		0			0
$301$	386.166	+	386.166i	0.0739477	+	0.0739477i
$302$	5164.69	+	447.226i	0.984087	+	0.0852152i
$303$		0			0
$304$	4361.14	−	2052.54i	0.822792	−	0.387241i
$305$			182.507i			0.0342634i
$306$		0			0
$307$	−4339.31	+	4339.31i	−0.806701	+	0.806701i	−0.984133	−	0.177432i	$-0.943221\pi$
$307$	−4339.31	+	4339.31i	−0.806701	+	0.806701i	0.177432	+	0.984133i	$0.443221\pi$
$308$	−4578.98	+	3218.38i	−0.847115	+	0.595403i
$309$		0			0
$310$	9357.92	−	7866.41i	1.71450	−	1.44123i
$311$			8077.43i			1.47276i	0.676567	+	0.736382i	$0.263467\pi$
$311$			8077.43i			1.47276i	−0.676567	+	0.736382i	$0.736533\pi$
$312$		0			0
$313$		−	6259.69i		−	1.13041i	−0.824950	−	0.565206i	$-0.808797\pi$
$313$		−	6259.69i		−	1.13041i	0.824950	−	0.565206i	$-0.191203\pi$
$314$	5409.94	+	6435.69i	0.972295	+	1.15665i
$315$		0			0
$316$	40.0396	−	229.460i	0.00712786	−	0.0408486i
$317$	−4411.87	+	4411.87i	−0.781689	+	0.781689i	−0.980116	−	0.198427i	$-0.936417\pi$
$317$	−4411.87	+	4411.87i	−0.781689	+	0.781689i	0.198427	+	0.980116i	$0.436417\pi$
$318$		0			0
$319$			3170.76i			0.556516i
$320$	2906.44	+	10620.1i	0.507734	+	1.85526i
$321$		0			0
$322$	−717.223	+	8282.67i	−0.124128	+	1.43346i
$323$	4574.97	+	4574.97i	0.788106	+	0.788106i
$324$		0			0
$325$	9034.32	−	9034.32i	1.54195	−	1.54195i
$326$	472.860	−	397.494i	0.0803353	−	0.0675311i
$327$		0			0
$328$	6102.30	+	1617.68i	1.02727	+	0.272321i
$329$	11536.0			1.93314
$330$		0			0
$331$	5055.03	+	5055.03i	0.839424	+	0.839424i	0.988783	−	0.149359i	$-0.0477210\pi$
$331$	5055.03	+	5055.03i	0.839424	+	0.839424i	−0.149359	+	0.988783i	$0.547721\pi$
$332$	−5815.42	−	8273.94i	−0.961333	−	1.36774i
$333$		0			0
$334$	−849.795	+	9813.65i	−0.139218	+	1.60772i
$335$	8327.07			1.35808
$336$		0			0
$337$	9698.71			1.56772			0.783861	−	0.620936i	$-0.213247\pi$
$337$	9698.71			1.56772			0.783861	+	0.620936i	$0.213247\pi$
$338$	−186.343	+	2151.94i	−0.0299874	+	0.346302i
$339$		0			0
$340$	−12091.8	+	8498.85i	−1.92874	+	1.35563i
$341$	−4058.70	−	4058.70i	−0.644548	−	0.644548i
$342$		0			0
$343$	−2104.06			−0.331220
$344$	−253.384	−	436.183i	−0.0397137	−	0.0683646i
$345$		0			0
$346$	−7849.44	+	6598.36i	−1.21962	+	1.02523i
$347$	1095.25	−	1095.25i	0.169441	−	0.169441i	−0.617293	−	0.786734i	$-0.711771\pi$
$347$	1095.25	−	1095.25i	0.169441	−	0.169441i	0.786734	+	0.617293i	$0.211771\pi$
$348$		0			0
$349$	2265.51	+	2265.51i	0.347478	+	0.347478i	0.859169	−	0.511691i	$-0.170981\pi$
$349$	2265.51	+	2265.51i	0.347478	+	0.347478i	−0.511691	+	0.859169i	$0.670981\pi$
$350$	2017.25	−	23295.7i	0.308076	−	3.55774i
$351$		0			0
$352$	4850.02	−	1789.71i	0.734395	−	0.271000i
$353$			1670.69i			0.251904i	0.992036	+	0.125952i	$0.0401985\pi$
$353$			1670.69i			0.251904i	−0.992036	+	0.125952i	$0.959801\pi$
$354$		0			0
$355$	−6824.85	+	6824.85i	−1.02035	+	1.02035i
$356$	−2216.62	−	386.788i	−0.330001	−	0.0575835i
$357$		0			0
$358$	1493.12	+	1776.22i	0.220430	+	0.262224i
$359$		−	753.959i		−	0.110842i	−0.998463	−	0.0554212i	$-0.982350\pi$
$359$		−	753.959i		−	0.110842i	0.998463	−	0.0554212i	$-0.0176502\pi$
$360$		0			0
$361$			1187.01i			0.173058i
$362$	−57.0436	+	47.9517i	−0.00828217	+	0.00696212i
$363$		0			0
$364$	−4266.48	−	6070.17i	−0.614353	−	0.874075i
$365$	−832.785	+	832.785i	−0.119425	+	0.119425i
$366$		0			0
$367$		−	5281.79i		−	0.751246i	−0.926773	−	0.375623i	$-0.877429\pi$
$367$		−	5281.79i		−	0.751246i	0.926773	−	0.375623i	$-0.122571\pi$
$368$	2600.75	−	7225.32i	0.368406	−	1.02349i
$369$		0			0
$370$	6597.35	+	571.285i	0.926972	+	0.0802694i
$371$	−2760.49	−	2760.49i	−0.386300	−	0.386300i
$372$		0			0
$373$	−1795.08	+	1795.08i	−0.249184	+	0.249184i	−0.820636	−	0.571452i	$-0.806380\pi$
$373$	−1795.08	+	1795.08i	−0.249184	+	0.249184i	0.571452	+	0.820636i	$0.306380\pi$
$374$	4465.27	+	5311.91i	0.617363	+	0.734418i
$375$		0			0
$376$	−10299.8	−	2730.40i	−1.41269	−	0.374494i
$377$	−4203.36			−0.574228
$378$		0			0
$379$	−9113.85	−	9113.85i	−1.23522	−	1.23522i	−0.961934	−	0.273283i	$-0.911890\pi$
$379$	−9113.85	−	9113.85i	−1.23522	−	1.23522i	−0.273283	−	0.961934i	$-0.588110\pi$
$380$	12764.0	+	2227.25i	1.72311	+	0.300673i
$381$		0			0
$382$	4868.81	+	421.606i	0.652121	+	0.0564692i
$383$	−2975.11			−0.396922			−0.198461	−	0.980109i	$-0.563594\pi$
$383$	−2975.11			−0.396922			−0.198461	+	0.980109i	$0.563594\pi$
$384$		0			0
$385$	−15045.2			−1.99162
$386$	−11546.0	−	999.804i	−1.52248	−	0.131836i
$387$		0			0
$388$	1487.28	+	259.523i	0.194601	+	0.0339569i
$389$	−4056.95	−	4056.95i	−0.528780	−	0.528780i	0.391429	−	0.920209i	$-0.371981\pi$
$389$	−4056.95	−	4056.95i	−0.528780	−	0.528780i	−0.920209	+	0.391429i	$0.871981\pi$
$390$		0			0
$391$	10307.8			1.33322
$392$	−5623.50	−	1490.75i	−0.724565	−	0.192077i
$393$		0			0
$394$	−10010.3	−	11908.3i	−1.27998	−	1.52266i
$395$	442.750	−	442.750i	0.0563979	−	0.0563979i
$396$		0			0
$397$	2409.95	+	2409.95i	0.304665	+	0.304665i	0.842836	−	0.538171i	$-0.180885\pi$
$397$	2409.95	+	2409.95i	0.304665	+	0.304665i	−0.538171	+	0.842836i	$0.680885\pi$
$398$	9391.78	+	813.263i	1.18283	+	0.102425i
$399$		0			0
$400$	−7314.83	+	20321.8i	−0.914353	+	2.54023i
$401$		−	7830.00i		−	0.975091i	−0.873098	−	0.487545i	$-0.837892\pi$
$401$		−	7830.00i		−	0.975091i	0.873098	−	0.487545i	$-0.162108\pi$
$402$		0			0
$403$	5380.46	−	5380.46i	0.665061	−	0.665061i
$404$	−1424.13	−	2026.19i	−0.175379	−	0.249522i
$405$		0			0
$406$	−5888.63	+	4950.07i	−0.719822	+	0.605093i
$407$		−	3109.17i		−	0.378662i
$408$		0			0
$409$		−	4518.97i		−	0.546329i	−0.961967	−	0.273164i	$-0.911930\pi$
$409$		−	4518.97i		−	0.546329i	0.961967	−	0.273164i	$-0.0880703\pi$
$410$	10920.0	+	12990.4i	1.31536	+	1.56476i
$411$		0			0
$412$	14952.9	+	2609.21i	1.78806	+	0.312006i
$413$	5908.27	−	5908.27i	0.703940	−	0.703940i
$414$		0			0
$415$		−	27185.8i		−	3.21566i
$416$	2372.55	+	6429.48i	0.279625	+	0.757768i
$417$		0			0
$418$	524.825	−	6060.81i	0.0614115	−	0.709196i
$419$	−427.092	−	427.092i	−0.0497967	−	0.0497967i	0.681770	−	0.731567i	$-0.261211\pi$
$419$	−427.092	−	427.092i	−0.0497967	−	0.0497967i	−0.731567	+	0.681770i	$0.761211\pi$
$420$		0			0
$421$	−3815.11	+	3815.11i	−0.441656	+	0.441656i	−0.892568	−	0.450913i	$-0.851099\pi$
$421$	−3815.11	+	3815.11i	−0.441656	+	0.441656i	0.450913	+	0.892568i	$0.351099\pi$
$422$	−11246.7	+	9454.18i	−1.29735	+	1.09057i
$423$		0			0
$424$	1811.30	+	3118.03i	0.207463	+	0.357134i
$425$	−28991.7			−3.30895
$426$		0			0
$427$	147.007	+	147.007i	0.0166608	+	0.0166608i
$428$	3706.84	−	2605.39i	0.418638	−	0.294244i
$429$		0			0
$430$	116.983	−	1350.95i	0.0131196	−	0.151508i
$431$	−12742.0			−1.42404			−0.712022	−	0.702157i	$-0.752220\pi$
$431$	−12742.0			−1.42404			−0.712022	+	0.702157i	$0.752220\pi$
$432$		0			0
$433$	−14363.7			−1.59417			−0.797085	−	0.603867i	$-0.793626\pi$
$433$	−14363.7			−1.59417			−0.797085	+	0.603867i	$0.793626\pi$
$434$	1201.39	−	13873.9i	0.132877	−	1.53449i
$435$		0			0
$436$	−1341.74	−	1908.97i	−0.147380	−	0.209686i
$437$	−6389.77	−	6389.77i	−0.699460	−	0.699460i
$438$		0			0
$439$	−3156.53			−0.343173			−0.171586	−	0.985169i	$-0.554889\pi$
$439$	−3156.53			−0.343173			−0.171586	+	0.985169i	$0.554889\pi$
$440$	13432.9	+	3560.98i	1.45543	+	0.385825i
$441$		0			0
$442$	−7041.79	+	5919.44i	−0.757791	+	0.637011i
$443$	8113.23	−	8113.23i	0.870139	−	0.870139i	−0.122349	−	0.992487i	$-0.539043\pi$
$443$	8113.23	−	8113.23i	0.870139	−	0.870139i	0.992487	+	0.122349i	$0.0390426\pi$
$444$		0			0
$445$	−4277.03	−	4277.03i	−0.455619	−	0.455619i
$446$	−211.615	+	2443.79i	−0.0224670	+	0.259455i
$447$		0			0
$448$	10895.4	+	6213.25i	1.14902	+	0.655243i
$449$			18120.9i			1.90463i	0.305112	+	0.952316i	$0.401306\pi$
$449$			18120.9i			1.90463i	−0.305112	+	0.952316i	$0.598694\pi$
$450$		0			0
$451$	5634.19	−	5634.19i	0.588256	−	0.588256i
$452$	529.537	−	3034.69i	0.0551047	−	0.315796i
$453$		0			0
$454$	852.662	+	1014.33i	0.0881441	+	0.104857i
$455$		−	19944.9i		−	2.05501i
$456$		0			0
$457$			11253.4i			1.15189i	0.817490	+	0.575943i	$0.195365\pi$
$457$			11253.4i			1.15189i	−0.817490	+	0.575943i	$0.804635\pi$
$458$	13261.9	−	11148.1i	1.35303	−	1.13738i
$459$		0			0
$460$	16888.4	−	11870.2i	1.71179	−	1.20315i
$461$	1886.34	−	1886.34i	0.190576	−	0.190576i	−0.605369	−	0.795945i	$-0.706975\pi$
$461$	1886.34	−	1886.34i	0.190576	−	0.190576i	0.795945	+	0.605369i	$0.206975\pi$
$462$		0			0
$463$		−	4690.87i		−	0.470849i	−0.971893	−	0.235424i	$-0.924352\pi$
$463$		−	4690.87i		−	0.470849i	0.971893	−	0.235424i	$-0.0756480\pi$
$464$	6429.19	−	3025.85i	0.643249	−	0.302741i
$465$		0			0
$466$	−11139.0	−	964.564i	−1.10731	−	0.0958853i
$467$	7780.38	+	7780.38i	0.770948	+	0.770948i	0.978272	−	0.207324i	$-0.0664754\pi$
$467$	7780.38	+	7780.38i	0.770948	+	0.770948i	−0.207324	+	0.978272i	$0.566475\pi$
$468$		0			0
$469$	6707.34	−	6707.34i	0.660375	−	0.660375i
$470$	−18431.3	−	21925.9i	−1.80888	−	2.15185i
$471$		0			0
$472$	−6673.52	+	3876.72i	−0.650792	+	0.378052i
$473$	−636.669			−0.0618902
$474$		0			0
$475$	17971.8	+	17971.8i	1.73600	+	1.73600i
$476$	−2894.08	+	16585.5i	−0.278676	+	1.59705i
$477$		0			0
$478$	−7130.80	−	617.479i	−0.682333	−	0.0590854i
$479$	1769.08			0.168750			0.0843749	−	0.996434i	$-0.473111\pi$
$479$	1769.08			0.168750			0.0843749	+	0.996434i	$0.473111\pi$
$480$		0			0
$481$	4121.70			0.390714
$482$	889.162	+	76.9953i	0.0840253	+	0.00727602i
$483$		0			0
$484$	−708.759	+	4061.78i	−0.0665627	+	0.381460i
$485$	2869.75	+	2869.75i	0.268678	+	0.268678i
$486$		0			0
$487$	−18479.1			−1.71944			−0.859721	−	0.510765i	$-0.829362\pi$
$487$	−18479.1			−1.71944			−0.859721	+	0.510765i	$0.829362\pi$
$488$	−96.4587	−	166.047i	−0.00894771	−	0.0154029i
$489$		0			0
$490$	−10063.2	−	11971.2i	−0.927769	−	1.10368i
$491$	3394.93	−	3394.93i	0.312039	−	0.312039i	−0.533660	−	0.845699i	$-0.679184\pi$
$491$	3394.93	−	3394.93i	0.312039	−	0.312039i	0.845699	+	0.533660i	$0.179184\pi$
$492$		0			0
$493$	6744.42	+	6744.42i	0.616132	+	0.616132i
$494$	8034.58	+	695.739i	0.731767	+	0.0633660i
$495$		0			0
$496$	−4356.40	+	12102.8i	−0.394371	+	1.09563i
$497$			10994.6i			0.992307i
$498$		0			0
$499$	7006.50	−	7006.50i	0.628565	−	0.628565i	−0.319142	−	0.947707i	$-0.603395\pi$
$499$	7006.50	−	7006.50i	0.628565	−	0.628565i	0.947707	+	0.319142i	$0.103395\pi$
$500$	−29906.0	+	21019.7i	−2.67487	+	1.88006i
$501$		0			0
$502$	2667.38	−	2242.24i	0.237153	−	0.199355i
$503$			15005.5i			1.33014i	0.746781	+	0.665070i	$0.231598\pi$
$503$			15005.5i			1.33014i	−0.746781	+	0.665070i	$0.768402\pi$
$504$		0			0
$505$		−	6657.49i		−	0.586643i
$506$	−6236.55	−	7419.03i	−0.547922	−	0.651811i
$507$		0			0
$508$	1952.81	−	11191.2i	0.170555	−	0.977421i
$509$	15473.1	−	15473.1i	1.34742	−	1.34742i	0.458958	−	0.888458i	$-0.348223\pi$
$509$	15473.1	−	15473.1i	1.34742	−	1.34742i	0.888458	−	0.458958i	$-0.151777\pi$
$510$		0			0
$511$			1341.59i			0.116142i
$512$	−8257.27	−	8126.21i	−0.712740	−	0.701428i
$513$		0			0
$514$	−444.708	+	5135.61i	−0.0381620	+	0.440704i
$515$	28852.1	+	28852.1i	2.46869	+	2.46869i
$516$		0			0
$517$	−9509.68	+	9509.68i	−0.808965	+	0.808965i
$518$	5774.23	−	4853.91i	0.489778	−	0.411715i
$519$		0			0
$520$	−4720.65	+	17807.5i	−0.398104	+	1.50175i
$521$	−8687.96			−0.730569			−0.365284	−	0.930896i	$-0.619028\pi$
$521$	−8687.96			−0.730569			−0.365284	+	0.930896i	$0.619028\pi$
$522$		0			0
$523$	−6047.61	−	6047.61i	−0.505628	−	0.505628i	0.407553	−	0.913182i	$-0.366382\pi$
$523$	−6047.61	−	6047.61i	−0.505628	−	0.505628i	−0.913182	+	0.407553i	$0.866382\pi$
$524$	−1315.01	−	1870.94i	−0.109631	−	0.155978i
$525$		0			0
$526$	−273.350	+	3156.71i	−0.0226590	+	0.261672i
$527$	−17266.2			−1.42719
$528$		0			0
$529$	−2229.75			−0.183262
$530$	−836.247	+	9657.20i	−0.0685363	+	0.791475i
$531$		0			0
$532$	12075.3	−	8487.21i	0.984076	−	0.691668i
$533$	7469.02	+	7469.02i	0.606978	+	0.606978i
$534$		0			0
$535$	12179.6			0.984246
$536$	−7576.08	+	4401.03i	−0.610516	+	0.354656i
$537$		0			0
$538$	4524.47	−	3803.34i	0.362572	−	0.304783i
$539$	−5192.12	+	5192.12i	−0.414917	+	0.414917i
$540$		0			0
$541$	16062.4	+	16062.4i	1.27648	+	1.27648i	0.942625	+	0.333854i	$0.108349\pi$
$541$	16062.4	+	16062.4i	1.27648	+	1.27648i	0.333854	+	0.942625i	$0.391651\pi$
$542$	1431.20	−	16527.9i	0.113423	−	1.30984i
$543$		0			0
$544$	6509.47	−	14123.1i	0.513036	−	1.11310i
$545$		−	6272.34i		−	0.492987i
$546$		0			0
$547$	−10435.4	+	10435.4i	−0.815697	+	0.815697i	−0.985481	−	0.169784i	$-0.945693\pi$
$547$	−10435.4	+	10435.4i	−0.815697	+	0.815697i	0.169784	+	0.985481i	$0.445693\pi$
$548$	7766.77	+	1355.26i	0.605438	+	0.105646i
$549$		0			0
$550$	17540.8	+	20866.7i	1.35990	+	1.61774i
$551$		−	8361.64i		−	0.646493i
$552$		0			0
$553$		−	713.257i		−	0.0548477i
$554$	−603.343	+	507.180i	−0.0462700	+	0.0388953i
$555$		0			0
$556$	−7804.15	−	11103.4i	−0.595269	−	0.846924i
$557$	−6335.69	+	6335.69i	−0.481960	+	0.481960i	−0.905757	−	0.423797i	$-0.860697\pi$
$557$	−6335.69	+	6335.69i	−0.481960	+	0.481960i	0.423797	+	0.905757i	$0.360697\pi$
$558$		0			0
$559$		−	844.008i		−	0.0638600i
$560$	14357.6	+	30506.4i	1.08343	+	2.30202i
$561$		0			0
$562$	−12502.4	−	1082.63i	−0.938405	−	0.0812594i
$563$	9187.42	+	9187.42i	0.687750	+	0.687750i	0.961734	−	0.273984i	$-0.0883415\pi$
$563$	9187.42	+	9187.42i	0.687750	+	0.687750i	−0.273984	+	0.961734i	$0.588342\pi$
$564$		0			0
$565$	5855.52	−	5855.52i	0.436006	−	0.436006i
$566$	15564.4	+	18515.5i	1.15586	+	1.37502i
$567$		0			0
$568$	2602.26	−	9816.40i	0.192233	−	0.725153i
$569$	−11816.3			−0.870588			−0.435294	−	0.900288i	$-0.643356\pi$
$569$	−11816.3			−0.870588			−0.435294	+	0.900288i	$0.643356\pi$
$570$		0			0
$571$	2344.11	+	2344.11i	0.171800	+	0.171800i	0.787770	−	0.615970i	$-0.211236\pi$
$571$	2344.11	+	2344.11i	0.171800	+	0.171800i	−0.615970	+	0.787770i	$0.711236\pi$
$572$	8520.98	+	1486.87i	0.622867	+	0.108687i
$573$		0			0
$574$	19259.5	+	1667.74i	1.40048	+	0.121272i
$575$	40492.1			2.93676
$576$		0			0
$577$	16615.2			1.19878			0.599392	−	0.800456i	$-0.295409\pi$
$577$	16615.2			1.19878			0.599392	+	0.800456i	$0.295409\pi$
$578$	6952.43	+	602.033i	0.500317	+	0.0433240i
$579$		0			0
$580$	18816.7	+	3283.41i	1.34711	+	0.235063i
$581$	−21897.8	−	21897.8i	−1.56364	−	1.56364i
$582$		0			0
$583$	4551.20			0.323313
$584$	317.535	−	1197.82i	0.0224995	−	0.0848737i
$585$		0			0
$586$	−940.136	−	1118.39i	−0.0662742	−	0.0788401i
$587$	1186.35	−	1186.35i	0.0834173	−	0.0834173i	−0.664167	−	0.747584i	$-0.731214\pi$
$587$	1186.35	−	1186.35i	0.0834173	−	0.0834173i	0.747584	+	0.664167i	$0.231214\pi$
$588$		0			0
$589$	10703.2	+	10703.2i	0.748758	+	0.748758i
$590$	−20669.3	−	1789.82i	−1.44227	−	0.124891i
$591$		0			0
$592$	−6304.29	+	2967.07i	−0.437677	+	0.205990i
$593$			1478.23i			0.102367i	0.998689	+	0.0511833i	$0.0162993\pi$
$593$			1478.23i			0.102367i	−0.998689	+	0.0511833i	$0.983701\pi$
$594$		0			0
$595$	−32002.1	+	32002.1i	−2.20497	+	2.20497i
$596$	−8178.03	−	11635.4i	−0.562055	−	0.799669i
$597$		0			0
$598$	9835.12	−	8267.55i	0.672555	−	0.565360i
$599$			10548.3i			0.719522i	0.933044	+	0.359761i	$0.117142\pi$
$599$			10548.3i			0.719522i	−0.933044	+	0.359761i	$0.882858\pi$
$600$		0			0
$601$			21596.7i			1.46580i	0.680335	+	0.732901i	$0.261834\pi$
$601$			21596.7i			1.46580i	−0.680335	+	0.732901i	$0.738166\pi$
$602$	−993.943	−	1182.40i	−0.0672925	−	0.0800515i
$603$		0			0
$604$	−14444.3	−	2520.46i	−0.973067	−	0.169795i
$605$	−7837.32	+	7837.32i	−0.526665	+	0.526665i
$606$		0			0
$607$		−	12331.9i		−	0.824604i	−0.911047	−	0.412302i	$-0.864725\pi$
$607$		−	12331.9i		−	0.824604i	0.911047	−	0.412302i	$-0.135275\pi$
$608$	−12790.0	+	4719.66i	−0.853131	+	0.314815i
$609$		0			0
$610$	44.5334	−	514.284i	0.00295591	−	0.0341356i
$611$	−12606.6	−	12606.6i	−0.834711	−	0.834711i
$612$		0			0
$613$	6110.69	−	6110.69i	0.402624	−	0.402624i	−0.476533	−	0.879157i	$-0.658107\pi$
$613$	6110.69	−	6110.69i	0.402624	−	0.402624i	0.879157	+	0.476533i	$0.158107\pi$
$614$	13286.5	−	11168.8i	0.873288	−	0.734099i
$615$		0			0
$616$	13688.3	−	7951.70i	0.895323	−	0.520103i
$617$	9428.72			0.615212			0.307606	−	0.951514i	$-0.400472\pi$
$617$	9428.72			0.615212			0.307606	+	0.951514i	$0.400472\pi$
$618$		0			0
$619$	−209.732	−	209.732i	−0.0136185	−	0.0136185i	0.700265	−	0.713883i	$-0.253065\pi$
$619$	−209.732	−	209.732i	−0.0136185	−	0.0136185i	−0.713883	+	0.700265i	$0.753065\pi$
$620$	−28289.0	+	19883.2i	−1.83244	+	1.28795i
$621$		0			0
$622$	1970.97	−	22761.3i	0.127056	−	1.46727i
$623$	−6890.16			−0.443096
$624$		0			0
$625$	−56078.5			−3.58903
$626$	−1527.42	+	17639.1i	−0.0975209	+	1.12620i
$627$		0			0
$628$	−13674.2	−	19455.1i	−0.868886	−	1.23622i
$629$	−6613.39	−	6613.39i	−0.419226	−	0.419226i
$630$		0			0
$631$	17914.3			1.13020			0.565101	−	0.825022i	$-0.308837\pi$
$631$	17914.3			1.13020			0.565101	+	0.825022i	$0.308837\pi$
$632$	−168.817	+	636.822i	−0.0106253	+	0.0400813i
$633$		0			0
$634$	13508.7	−	11355.6i	0.846211	−	0.711338i
$635$	21593.7	−	21593.7i	1.34948	−	1.34948i
$636$		0			0
$637$	−6882.98	−	6882.98i	−0.428122	−	0.428122i
$638$	773.696	−	8934.84i	0.0480108	−	0.554441i
$639$		0			0
$640$	−5598.60	−	30635.4i	−0.345788	−	1.89214i
$641$			3322.93i			0.204755i	0.994746	+	0.102377i	$0.0326450\pi$
$641$			3322.93i			0.204755i	−0.994746	+	0.102377i	$0.967355\pi$
$642$		0			0
$643$	11199.4	−	11199.4i	0.686878	−	0.686878i	−0.274663	−	0.961541i	$-0.588566\pi$
$643$	11199.4	−	11199.4i	0.686878	−	0.686878i	0.961541	+	0.274663i	$0.0885662\pi$
$644$	4042.10	−	23164.6i	0.247331	−	1.41741i
$645$		0			0
$646$	−11775.4	−	14008.1i	−0.717178	−	0.853158i
$647$			21868.3i			1.32879i	0.747380	+	0.664397i	$0.231312\pi$
$647$			21868.3i			1.32879i	−0.747380	+	0.664397i	$0.768688\pi$
$648$		0			0
$649$			9740.93i			0.589160i
$650$	−27662.1	+	23253.2i	−1.66923	+	1.40318i
$651$		0			0
$652$	−1429.46	+	1004.71i	−0.0858617	+	0.0603488i
$653$	−2170.27	+	2170.27i	−0.130060	+	0.130060i	−0.769140	−	0.639080i	$-0.779315\pi$
$653$	−2170.27	+	2170.27i	−0.130060	+	0.130060i	0.639080	+	0.769140i	$0.279315\pi$
$654$		0			0
$655$		−	6147.38i		−	0.366715i
$656$	−16800.8	−	6047.45i	−0.999942	−	0.359929i
$657$		0			0
$658$	−32507.2	−	2814.90i	−1.92593	−	0.166772i
$659$	−9827.52	−	9827.52i	−0.580919	−	0.580919i	0.354237	−	0.935156i	$-0.384741\pi$
$659$	−9827.52	−	9827.52i	−0.580919	−	0.580919i	−0.935156	+	0.354237i	$0.884741\pi$
$660$		0			0
$661$	6022.78	−	6022.78i	0.354401	−	0.354401i	−0.507343	−	0.861744i	$-0.669372\pi$
$661$	6022.78	−	6022.78i	0.354401	−	0.354401i	0.861744	+	0.507343i	$0.169372\pi$
$662$	−13011.0	−	15477.9i	−0.763877	−	0.908712i
$663$		0			0
$664$	14368.2	+	24734.0i	0.839753	+	1.44558i
$665$	39675.8			2.31363
$666$		0			0
$667$	−9419.79	−	9419.79i	−0.546830	−	0.546830i
$668$	4789.24	−	27446.4i	0.277397	−	1.58972i
$669$		0			0
$670$	−23464.7	−	2031.88i	−1.35302	−	0.117162i
$671$	−242.369			−0.0139442
$672$		0			0
$673$	−15216.4			−0.871543			−0.435771	−	0.900057i	$-0.643524\pi$
$673$	−15216.4			−0.871543			−0.435771	+	0.900057i	$0.643524\pi$
$674$	−27329.8	−	2366.57i	−1.56188	−	0.135248i
$675$		0			0
$676$	1050.19	−	6018.45i	0.0597512	−	0.342424i
$677$	5754.92	+	5754.92i	0.326705	+	0.326705i	0.851332	−	0.524627i	$-0.175795\pi$
$677$	5754.92	+	5754.92i	0.326705	+	0.326705i	−0.524627	+	0.851332i	$0.675795\pi$
$678$		0			0
$679$	4623.09			0.261293
$680$	36147.1	−	20998.2i	2.03850	−	1.18419i
$681$		0			0
$682$	10446.6	+	12427.3i	0.586539	+	0.697750i
$683$	−10651.0	+	10651.0i	−0.596705	+	0.596705i	−0.939434	−	0.342729i	$-0.888649\pi$
$683$	−10651.0	+	10651.0i	−0.596705	+	0.596705i	0.342729	+	0.939434i	$0.388649\pi$
$684$		0			0
$685$	14986.2	+	14986.2i	0.835902	+	0.835902i
$686$	5928.98	+	513.409i	0.329985	+	0.0285744i
$687$		0			0
$688$	607.572	+	1290.94i	0.0336678	+	0.0715358i
$689$			6033.34i			0.333602i
$690$		0			0
$691$	−16585.3	+	16585.3i	−0.913073	+	0.913073i	−0.996513	−	0.0834401i	$-0.973409\pi$
$691$	−16585.3	+	16585.3i	−0.913073	+	0.913073i	0.0834401	+	0.996513i	$0.473409\pi$
$692$	23728.8	−	16678.1i	1.30352	−	0.916191i
$693$		0			0
$694$	−3353.53	+	2819.03i	−0.183427	+	0.154191i
$695$		−	36482.7i		−	1.99117i
$696$		0			0
$697$		−	23968.6i		−	1.30254i
$698$	−5831.12	−	6936.73i	−0.316205	−	0.376159i
$699$		0			0
$700$	−11368.8	+	65152.4i	−0.613855	+	3.51790i
$701$	6023.16	−	6023.16i	0.324524	−	0.324524i	−0.525975	−	0.850500i	$-0.676300\pi$
$701$	6023.16	−	6023.16i	0.324524	−	0.324524i	0.850500	+	0.525975i	$0.176300\pi$
$702$		0			0
$703$			8199.20i			0.439884i
$704$	−14103.5	+	3859.75i	−0.755036	+	0.206633i
$705$		0			0
$706$	407.665	−	4707.82i	0.0217318	−	0.250965i
$707$	−5362.51	−	5362.51i	−0.285259	−	0.285259i
$708$		0			0
$709$	−21886.4	+	21886.4i	−1.15932	+	1.15932i	−0.174701	+	0.984621i	$0.555896\pi$
$709$	−21886.4	+	21886.4i	−1.15932	+	1.15932i	−0.984621	+	0.174701i	$0.944104\pi$
$710$	20896.9	−	17566.3i	1.10457	−	0.928522i
$711$		0			0
$712$	6151.79	+	1630.80i	0.323803	+	0.0858381i
$713$	24115.4			1.26666
$714$		0			0
$715$	16441.5	+	16441.5i	0.859966	+	0.859966i
$716$	−3774.02	−	5369.52i	−0.196986	−	0.280263i
$717$		0			0
$718$	−183.973	+	2124.57i	−0.00956241	+	0.110429i
$719$	−24967.9			−1.29506			−0.647528	−	0.762041i	$-0.724197\pi$
$719$	−24967.9			−1.29506			−0.647528	+	0.762041i	$0.724197\pi$
$720$		0			0
$721$	46479.9			2.40084
$722$	289.641	−	3344.85i	0.0149298	−	0.172413i
$723$		0			0
$724$	172.443	−	121.203i	0.00885192	−	0.00622166i
$725$	26494.0	+	26494.0i	1.35719	+	1.35719i
$726$		0			0
$727$	31570.1			1.61055			0.805276	−	0.592900i	$-0.202017\pi$
$727$	31570.1			1.61055			0.805276	+	0.592900i	$0.202017\pi$
$728$	10541.3	+	18146.1i	0.536655	+	0.923817i
$729$		0			0
$730$	2549.90	−	2143.48i	0.129282	−	0.108677i
$731$	−1354.24	+	1354.24i	−0.0685202	+	0.0685202i
$732$		0			0
$733$	−14686.6	−	14686.6i	−0.740055	−	0.740055i	0.232533	−	0.972588i	$-0.425299\pi$
$733$	−14686.6	−	14686.6i	−0.740055	−	0.740055i	−0.972588	+	0.232533i	$0.925299\pi$
$734$	−1288.81	+	14883.5i	−0.0648102	+	0.748445i
$735$		0			0
$736$	−9091.64	+	19725.5i	−0.455329	+	0.987896i
$737$			11058.3i			0.552699i
$738$		0			0
$739$	16900.6	−	16900.6i	0.841271	−	0.841271i	−0.147753	−	0.989024i	$-0.547204\pi$
$739$	16900.6	−	16900.6i	0.841271	−	0.841271i	0.989024	+	0.147753i	$0.0472041\pi$
$740$	−18451.2	−	3219.63i	−0.916592	−	0.159940i
$741$		0			0
$742$	7105.14	+	8452.31i	0.351534	+	0.418186i
$743$			13888.3i			0.685750i	0.939381	+	0.342875i	$0.111401\pi$
$743$			13888.3i			0.685750i	−0.939381	+	0.342875i	$0.888599\pi$
$744$		0			0
$745$		−	38230.5i		−	1.88008i
$746$	5496.34	−	4620.31i	0.269752	−	0.226758i
$747$		0			0
$748$	−11286.5	−	16057.9i	−0.551703	−	0.784940i
$749$	9810.52	−	9810.52i	0.478596	−	0.478596i
$750$		0			0
$751$		−	8334.75i		−	0.404979i	−0.979284	−	0.202490i	$-0.935097\pi$
$751$		−	8334.75i		−	0.404979i	0.979284	−	0.202490i	$-0.0649032\pi$
$752$	28357.3	+	10207.2i	1.37511	+	0.494971i
$753$		0			0
$754$	11844.6	+	1025.66i	0.572087	+	0.0495388i
$755$	−27870.8	−	27870.8i	−1.34347	−	1.34347i
$756$		0			0
$757$	−14800.7	+	14800.7i	−0.710620	+	0.710620i	−0.966665	−	0.256045i	$-0.917580\pi$
$757$	−14800.7	+	14800.7i	−0.710620	+	0.710620i	0.256045	+	0.966665i	$0.417580\pi$
$758$	23457.9	+	27905.6i	1.12405	+	1.33717i
$759$		0			0
$760$	−35424.0	−	9390.68i	−1.69074	−	0.448205i
$761$	−5724.34			−0.272677			−0.136339	−	0.990662i	$-0.543533\pi$
$761$	−5724.34			−0.272677			−0.136339	+	0.990662i	$0.543533\pi$
$762$		0			0
$763$	−5052.28	−	5052.28i	−0.239718	−	0.239718i
$764$	−13616.9	−	2376.07i	−0.644818	−	0.112517i
$765$		0			0
$766$	8383.51	+	725.955i	0.395442	+	0.0342426i
$767$	−12913.2			−0.607910
$768$		0			0
$769$	−16257.1			−0.762348			−0.381174	−	0.924503i	$-0.624480\pi$
$769$	−16257.1			−0.762348			−0.381174	+	0.924503i	$0.624480\pi$
$770$	42395.6	+	3671.17i	1.98420	+	0.171818i
$771$		0			0
$772$	32291.3	+	5634.66i	1.50543	+	0.262689i
$773$	−9339.94	−	9339.94i	−0.434585	−	0.434585i	0.455600	−	0.890185i	$-0.349425\pi$
$773$	−9339.94	−	9339.94i	−0.434585	−	0.434585i	−0.890185	+	0.455600i	$0.849425\pi$
$774$		0			0
$775$	−67826.6			−3.14374
$776$	−4127.66	−	1094.22i	−0.190946	−	0.0506186i
$777$		0			0
$778$	10442.1	+	12421.9i	0.481191	+	0.572427i
$779$	−14858.0	+	14858.0i	−0.683365	+	0.683365i
$780$		0			0
$781$	−9063.38	−	9063.38i	−0.415254	−	0.415254i
$782$	−29046.3	−	2515.21i	−1.32825	−	0.115018i
$783$		0			0
$784$	15482.6	+	5572.95i	0.705293	+	0.253870i
$785$		−	63923.9i		−	2.90642i
$786$		0			0
$787$	−21944.5	+	21944.5i	−0.993950	+	0.993950i	−0.999982	−	0.00603172i	$-0.998080\pi$
$787$	−21944.5	+	21944.5i	−0.993950	+	0.993950i	0.00603172	+	0.999982i	$0.498080\pi$
$788$	25302.0	+	35998.7i	1.14384	+	1.62741i
$789$		0			0
$790$	−1355.65	+	1139.58i	−0.0610531	+	0.0513221i
$791$		−	9433.07i		−	0.424022i
$792$		0			0
$793$		−	321.299i		−	0.0143880i
$794$	−6202.90	−	7379.00i	−0.277245	−	0.329812i
$795$		0			0
$796$	−26266.5	−	4583.36i	−1.16959	−	0.204087i
$797$	−21112.7	+	21112.7i	−0.938331	+	0.938331i	−0.998206	−	0.0598747i	$-0.980930\pi$
$797$	−21112.7	+	21112.7i	−0.938331	+	0.938331i	0.0598747	+	0.998206i	$0.480930\pi$
$798$		0			0
$799$			40455.4i			1.79125i
$800$	25571.0	−	55479.6i	1.13009	−	2.45188i
$801$		0			0
$802$	−1910.59	+	22064.0i	−0.0841214	+	0.971455i
$803$	−1105.94	−	1105.94i	−0.0486023	−	0.0486023i
$804$		0			0
$805$	44696.7	−	44696.7i	1.95696	−	1.95696i
$806$	−16474.4	+	13848.6i	−0.719957	+	0.605207i
$807$		0			0
$808$	3518.62	+	6057.07i	0.153199	+	0.263722i
$809$	−39210.9			−1.70406			−0.852028	−	0.523496i	$-0.824628\pi$
$809$	−39210.9			−1.70406			−0.852028	+	0.523496i	$0.824628\pi$
$810$		0			0
$811$	−7512.78	−	7512.78i	−0.325289	−	0.325289i	0.525503	−	0.850792i	$-0.323877\pi$
$811$	−7512.78	−	7512.78i	−0.325289	−	0.325289i	−0.850792	+	0.525503i	$0.823877\pi$
$812$	17801.3	−	12511.8i	0.769340	−	0.540738i
$813$		0			0
$814$	−758.665	+	8761.26i	−0.0326673	+	0.377251i
$815$	−4696.79			−0.201867
$816$		0			0
$817$	1678.97			0.0718966
$818$	−1102.67	+	12733.9i	−0.0471320	+	0.544292i
$819$		0			0
$820$	−27601.4	−	39270.1i	−1.17547	−	1.67240i
$821$	12626.1	+	12626.1i	0.536727	+	0.536727i	0.922566	−	0.385839i	$-0.126088\pi$
$821$	12626.1	+	12626.1i	0.536727	+	0.536727i	−0.385839	+	0.922566i	$0.626088\pi$
$822$		0			0
$823$	−15641.0			−0.662468			−0.331234	−	0.943549i	$-0.607465\pi$
$823$	−15641.0			−0.662468			−0.331234	+	0.943549i	$0.607465\pi$
$824$	−41499.0	−	11001.1i	−1.75447	−	0.465099i
$825$		0			0
$826$	−18090.5	+	15207.1i	−0.762044	+	0.640586i
$827$	26716.0	−	26716.0i	1.12334	−	1.12334i	0.132109	−	0.991235i	$-0.457825\pi$
$827$	26716.0	−	26716.0i	1.12334	−	1.12334i	0.991235	−	0.132109i	$-0.0421750\pi$
$828$		0			0
$829$	−3654.34	−	3654.34i	−0.153101	−	0.153101i	0.626401	−	0.779501i	$-0.284527\pi$
$829$	−3654.34	−	3654.34i	−0.153101	−	0.153101i	−0.779501	+	0.626401i	$0.784527\pi$
$830$	−6633.59	+	76606.4i	−0.277416	+	3.20367i
$831$		0			0
$832$	−5116.71	−	18696.4i	−0.213209	−	0.779066i
$833$			22087.9i			0.918729i
$834$		0			0
$835$	52958.5	−	52958.5i	2.19486	−	2.19486i
$836$	−2957.79	+	16950.6i	−0.122365	+	0.701254i
$837$		0			0
$838$	1099.28	+	1307.71i	0.0453150	+	0.0539070i
$839$			7316.14i			0.301050i	0.988606	+	0.150525i	$0.0480965\pi$
$839$			7316.14i			0.301050i	−0.988606	+	0.150525i	$0.951904\pi$
$840$		0			0
$841$			12062.3i			0.494579i
$842$	11681.4	−	9819.60i	0.478111	−	0.401907i
$843$		0			0
$844$	33998.9	−	23896.5i	1.38660	−	0.974586i
$845$	11612.8	−	11612.8i	0.472771	−	0.472771i
$846$		0			0
$847$			12625.7i			0.512189i
$848$	−4343.20	−	9228.22i	−0.175880	−	0.373701i
$849$		0			0
$850$	81695.2	+	7074.25i	3.29662	+	0.285464i
$851$	9236.79	+	9236.79i	0.372072	+	0.372072i
$852$		0			0
$853$	4362.84	−	4362.84i	0.175124	−	0.175124i	−0.614102	−	0.789226i	$-0.710482\pi$
$853$	4362.84	−	4362.84i	0.175124	−	0.175124i	0.789226	+	0.614102i	$0.210482\pi$
$854$	−378.377	−	450.119i	−0.0151613	−	0.0180360i
$855$		0			0
$856$	−11081.2	+	6437.19i	−0.442462	+	0.257031i
$857$	−10598.4			−0.422442			−0.211221	−	0.977438i	$-0.567744\pi$
$857$	−10598.4			−0.422442			−0.211221	+	0.977438i	$0.567744\pi$
$858$		0			0
$859$	25041.9	+	25041.9i	0.994668	+	0.994668i	0.999986	−	0.00531813i	$-0.00169282\pi$
$859$	25041.9	+	25041.9i	0.994668	+	0.994668i	−0.00531813	+	0.999986i	$0.501693\pi$
$860$	−659.289	+	3778.28i	−0.0261414	+	0.149812i
$861$		0			0
$862$	35905.6	+	3109.18i	1.41873	+	0.122853i
$863$	11567.3			0.456265			0.228133	−	0.973630i	$-0.426738\pi$
$863$	11567.3			0.456265			0.228133	+	0.973630i	$0.426738\pi$
$864$		0			0
$865$	77966.2			3.06466
$866$	40475.2	+	3504.88i	1.58823	+	0.137530i
$867$		0			0
$868$	−6770.74	+	38802.0i	−0.264763	+	1.51731i
$869$	587.971	+	587.971i	0.0229523	+	0.0229523i
$870$		0			0
$871$	−14659.6			−0.570289
$872$	3315.06	+	5706.66i	0.128741	+	0.221619i
$873$		0			0
$874$	16446.5	+	19564.8i	0.636510	+	0.757195i
$875$	−79149.1	+	79149.1i	−3.05797	+	3.05797i
$876$		0			0
$877$	−29456.6	−	29456.6i	−1.13418	−	1.13418i	−0.989474	−	0.144710i	$-0.953775\pi$
$877$	−29456.6	−	29456.6i	−1.13418	−	1.13418i	−0.144710	−	0.989474i	$-0.546225\pi$
$878$	8894.72	+	770.221i	0.341893	+	0.0296056i
$879$		0			0
$880$	−36983.5	−	13312.2i	−1.41672	−	0.509947i
$881$			42700.5i			1.63294i	0.577390	+	0.816468i	$0.304071\pi$
$881$			42700.5i			1.63294i	−0.577390	+	0.816468i	$0.695929\pi$
$882$		0			0
$883$	135.261	−	135.261i	0.00515504	−	0.00515504i	−0.704525	−	0.709680i	$-0.748840\pi$
$883$	135.261	−	135.261i	0.00515504	−	0.00515504i	0.709680	+	0.704525i	$0.248840\pi$
$884$	21287.3	−	14962.0i	0.809921	−	0.569261i
$885$		0			0
$886$	−24841.8	+	20882.4i	−0.941962	+	0.791827i
$887$			27302.4i			1.03351i	0.856132	+	0.516757i	$0.172861\pi$
$887$			27302.4i			1.03351i	−0.856132	+	0.516757i	$0.827139\pi$
$888$		0			0
$889$		−	34786.9i		−	1.31239i
$890$	11008.5	+	13095.8i	0.414614	+	0.493227i
$891$		0			0
$892$	1192.61	−	6834.67i	0.0447665	−	0.256549i
$893$	25078.0	−	25078.0i	0.939758	−	0.939758i
$894$		0			0
$895$		−	17642.7i		−	0.658917i
$896$	−29186.0	−	20166.8i	−1.08821	−	0.751926i
$897$		0			0
$898$	4421.68	−	51062.7i	0.164313	−	1.89753i
$899$	15778.7	+	15778.7i	0.585370	+	0.585370i
$900$		0			0
$901$	9680.68	−	9680.68i	0.357947	−	0.357947i
$902$	−17251.3	+	14501.7i	−0.636812	+	0.535314i
$903$		0			0
$904$	−2232.67	+	8422.19i	−0.0821431	+	0.309865i
$905$	566.598			0.0208114
$906$		0			0
$907$	−22150.8	−	22150.8i	−0.810921	−	0.810921i	0.173851	−	0.984772i	$-0.444379\pi$
$907$	−22150.8	−	22150.8i	−0.810921	−	0.810921i	−0.984772	+	0.173851i	$0.944379\pi$
$908$	−2155.19	−	3066.32i	−0.0787694	−	0.112070i
$909$		0			0
$910$	−4866.73	+	56202.3i	−0.177286	+	2.04735i
$911$	−40596.8			−1.47643			−0.738217	−	0.674564i	$-0.764332\pi$
$911$	−40596.8			−1.47643			−0.738217	+	0.674564i	$0.764332\pi$
$912$		0			0
$913$	36102.7			1.30868
$914$	2745.93	−	31710.8i	0.0993736	−	1.14759i
$915$		0			0
$916$	−40090.6	+	28178.1i	−1.44610	+	1.01641i
$917$	−4951.63	−	4951.63i	−0.178317	−	0.178317i
$918$		0			0
$919$	33460.1			1.20103			0.600515	−	0.799613i	$-0.294962\pi$
$919$	33460.1			1.20103			0.600515	+	0.799613i	$0.294962\pi$
$920$	−50485.9	+	29327.8i	−1.80921	+	1.05099i
$921$		0			0
$922$	−5775.75	+	4855.19i	−0.206306	+	0.173424i
$923$	12015.0	−	12015.0i	0.428470	−	0.428470i
$924$		0			0
$925$	−25979.3	−	25979.3i	−0.923452	−	0.923452i
$926$	−1144.61	+	13218.3i	−0.0406203	+	0.469093i
$927$		0			0
$928$	−18855.0	+	6957.71i	−0.666968	+	0.246119i
$929$		−	22691.6i		−	0.801387i	−0.916212	−	0.400693i	$-0.868769\pi$
$929$		−	22691.6i		−	0.801387i	0.916212	−	0.400693i	$-0.131231\pi$
$930$		0			0
$931$	13692.2	−	13692.2i	0.482001	−	0.482001i
$932$	31153.1	+	5436.05i	1.09491	+	0.191056i
$933$		0			0
$934$	−20025.7	−	23822.7i	−0.701564	−	0.834584i
$935$		−	52761.7i		−	1.84545i
$936$		0			0
$937$		−	5447.73i		−	0.189936i	−0.995480	−	0.0949678i	$-0.969725\pi$
$937$		−	5447.73i		−	0.189936i	0.995480	−	0.0949678i	$-0.0302748\pi$
$938$	−20537.1	+	17263.8i	−0.714884	+	0.600942i
$939$		0			0
$940$	46587.1	+	66282.1i	1.61649	+	2.29988i
$941$	−20191.7	+	20191.7i	−0.699501	+	0.699501i	−0.964303	−	0.264802i	$-0.914693\pi$
$941$	−20191.7	+	20191.7i	−0.699501	+	0.699501i	0.264802	+	0.964303i	$0.414693\pi$
$942$		0			0
$943$			33476.4i			1.15603i
$944$	19751.2	−	9295.74i	0.680980	−	0.320499i
$945$		0			0
$946$	1794.06	+	155.353i	0.0616595	+	0.00533929i
$947$	2444.37	+	2444.37i	0.0838768	+	0.0838768i	0.747800	−	0.663924i	$-0.231110\pi$
$947$	2444.37	+	2444.37i	0.0838768	+	0.0838768i	−0.663924	+	0.747800i	$0.731110\pi$
$948$		0			0
$949$	1466.10	−	1466.10i	0.0501491	−	0.0501491i
$950$	−46257.1	−	55027.6i	−1.57977	−	1.87930i
$951$		0			0
$952$	12202.2	−	46029.7i	0.415415	−	1.56705i
$953$	9299.20			0.316087			0.158043	−	0.987432i	$-0.449481\pi$
$953$	9299.20			0.316087			0.158043	+	0.987432i	$0.449481\pi$
$954$		0			0
$955$	−26274.1	−	26274.1i	−0.890273	−	0.890273i
$956$	19943.1	+	3479.96i	0.674692	+	0.117730i
$957$		0			0
$958$	−4985.05	−	431.671i	−0.168121	−	0.0145581i
$959$	24142.3			0.812926
$960$		0			0
$961$	−10603.6			−0.355933
$962$	−11614.5	−	1005.73i	−0.389257	−	0.0337070i
$963$		0			0
$964$	−2486.77	−	433.927i	−0.0830843	−	0.0144978i
$965$	62307.0	+	62307.0i	2.07848	+	2.07848i
$966$		0			0
$967$	−39928.8			−1.32784			−0.663922	−	0.747802i	$-0.731109\pi$
$967$	−39928.8			−1.32784			−0.663922	+	0.747802i	$0.731109\pi$
$968$	2988.31	−	11272.7i	0.0992231	−	0.374295i
$969$		0			0
$970$	−7386.38	−	8786.88i	−0.244497	−	0.290855i
$971$	18987.0	−	18987.0i	0.627521	−	0.627521i	−0.319923	−	0.947444i	$-0.603657\pi$
$971$	18987.0	−	18987.0i	0.627521	−	0.627521i	0.947444	+	0.319923i	$0.103657\pi$
$972$		0			0
$973$	−29386.2	−	29386.2i	−0.968222	−	0.968222i
$974$	52071.9	+	4509.07i	1.71303	+	0.148337i
$975$		0			0
$976$	231.292	+	491.439i	0.00758554	+	0.0161174i
$977$			26451.4i			0.866176i	0.901352	+	0.433088i	$0.142576\pi$
$977$			26451.4i			0.866176i	−0.901352	+	0.433088i	$0.857424\pi$
$978$		0			0
$979$	5679.88	−	5679.88i	0.185424	−	0.185424i
$980$	25435.7	+	36188.9i	0.829096	+	1.17960i
$981$		0			0
$982$	−10394.9	+	8738.12i	−0.337795	+	0.283956i
$983$			3516.98i			0.114114i	0.998371	+	0.0570571i	$0.0181717\pi$
$983$			3516.98i			0.114114i	−0.998371	+	0.0570571i	$0.981828\pi$
$984$		0			0
$985$			118281.i			3.82615i
$986$	−17359.3	−	20650.7i	−0.560681	−	0.666989i
$987$		0			0
$988$	−22470.7	−	3921.02i	−0.723572	−	0.126259i
$989$	1891.43	−	1891.43i	0.0608130	−	0.0608130i
$990$		0			0
$991$			55027.6i			1.76388i	0.471359	+	0.881942i	$0.343764\pi$
$991$			55027.6i			1.76388i	−0.471359	+	0.881942i	$0.656236\pi$
$992$	15229.0	−	33041.3i	0.487421	−	1.05752i
$993$		0			0
$994$	2682.79	−	30981.6i	0.0856066	−	0.988607i
$995$	−50681.9	−	50681.9i	−1.61480	−	1.61480i
$996$		0			0
$997$	−28913.9	+	28913.9i	−0.918466	+	0.918466i	−0.996918	−	0.0784515i	$-0.975002\pi$
$997$	−28913.9	+	28913.9i	−0.918466	+	0.918466i	0.0784515	+	0.996918i	$0.475002\pi$
$998$	−21453.2	+	18033.8i	−0.680448	+	0.571995i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.4.l.a.35.1	✓	48
3.2	odd	2	inner	144.4.l.a.35.24	yes	48
4.3	odd	2		576.4.l.a.431.24		48
8.3	odd	2		1152.4.l.a.863.1		48
8.5	even	2		1152.4.l.b.863.1		48
12.11	even	2		576.4.l.a.431.1		48
16.3	odd	4		1152.4.l.b.287.24		48
16.5	even	4		576.4.l.a.143.1		48
16.11	odd	4	inner	144.4.l.a.107.24	yes	48
16.13	even	4		1152.4.l.a.287.24		48
24.5	odd	2		1152.4.l.b.863.24		48
24.11	even	2		1152.4.l.a.863.24		48
48.5	odd	4		576.4.l.a.143.24		48
48.11	even	4	inner	144.4.l.a.107.1	yes	48
48.29	odd	4		1152.4.l.a.287.1		48
48.35	even	4		1152.4.l.b.287.1		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.4.l.a.35.1	✓	48	1.1	even	1	trivial
144.4.l.a.35.24	yes	48	3.2	odd	2	inner
144.4.l.a.107.1	yes	48	48.11	even	4	inner
144.4.l.a.107.24	yes	48	16.11	odd	4	inner
576.4.l.a.143.1		48	16.5	even	4
576.4.l.a.143.24		48	48.5	odd	4
576.4.l.a.431.1		48	12.11	even	2
576.4.l.a.431.24		48	4.3	odd	2
1152.4.l.a.287.1		48	48.29	odd	4
1152.4.l.a.287.24		48	16.13	even	4
1152.4.l.a.863.1		48	8.3	odd	2
1152.4.l.a.863.24		48	24.11	even	2
1152.4.l.b.287.1		48	48.35	even	4
1152.4.l.b.287.24		48	16.3	odd	4
1152.4.l.b.863.1		48	8.5	even	2
1152.4.l.b.863.24		48	24.5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	144.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.81788 - 0.244009i) q^{2} +(7.88092 + 1.37518i) q^{4} +(15.2065 + 15.2065i) q^{5} +24.4971 q^{7} +(-21.8719 - 5.79811i) q^{8} +(-39.1395 - 46.5605i) q^{10} +(-20.1941 + 20.1941i) q^{11} +(-26.7706 - 26.7706i) q^{13} +(-69.0301 - 5.97753i) q^{14} +(60.2178 + 21.6753i) q^{16} +85.9084i q^{17} +(53.2541 - 53.2541i) q^{19} +(98.9292 + 140.752i) q^{20} +(61.8322 - 51.9771i) q^{22} -119.986i q^{23} +337.472i q^{25} +(68.9040 + 81.9686i) q^{26} +(193.060 + 33.6879i) q^{28} +(78.5071 - 78.5071i) q^{29} +200.984i q^{31} +(-164.398 - 75.7722i) q^{32} +(20.9624 - 242.080i) q^{34} +(372.515 + 372.515i) q^{35} +(-76.9819 + 76.9819i) q^{37} +(-163.058 + 137.069i) q^{38} +(-244.426 - 420.763i) q^{40} -279.001 q^{41} +(15.7637 + 15.7637i) q^{43} +(-186.919 + 131.378i) q^{44} +(-29.2778 + 338.108i) q^{46} +470.913 q^{47} +257.110 q^{49} +(82.3464 - 950.957i) q^{50} +(-174.162 - 247.791i) q^{52} +(-112.686 - 112.686i) q^{53} -614.162 q^{55} +(-535.800 - 142.037i) q^{56} +(-240.380 + 202.067i) q^{58} +(241.182 - 241.182i) q^{59} +(6.00098 + 6.00098i) q^{61} +(49.0420 - 566.349i) q^{62} +(444.764 + 253.632i) q^{64} -814.171i q^{65} +(273.801 - 273.801i) q^{67} +(-118.139 + 677.037i) q^{68} +(-958.805 - 1140.60i) q^{70} +448.812i q^{71} +54.7653i q^{73} +(235.710 - 198.142i) q^{74} +(492.925 - 346.457i) q^{76} +(-494.698 + 494.698i) q^{77} -29.1159i q^{79} +(586.094 + 1245.30i) q^{80} +(786.193 + 68.0789i) q^{82} +(-893.890 - 893.890i) q^{83} +(-1306.36 + 1306.36i) q^{85} +(-40.5738 - 48.2668i) q^{86} +(558.773 - 324.597i) q^{88} -281.264 q^{89} +(-655.802 - 655.802i) q^{91} +(165.003 - 945.604i) q^{92} +(-1326.98 - 114.907i) q^{94} +1619.61 q^{95} +188.720 q^{97} +(-724.506 - 62.7373i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 120 q^{10} - 144 q^{16} - 48 q^{19} + 72 q^{22} + 72 q^{28} - 984 q^{34} - 1272 q^{40} + 864 q^{43} - 1416 q^{46} + 2352 q^{49} - 648 q^{52} - 576 q^{55} + 1128 q^{58} + 1824 q^{61} + 3024 q^{64}+ \cdots - 11304 q^{94}+O(q^{100}) \) 48 * q - 120 * q^10 - 144 * q^16 - 48 * q^19 + 72 * q^22 + 72 * q^28 - 984 * q^34 - 1272 * q^40 + 864 * q^43 - 1416 * q^46 + 2352 * q^49 - 648 * q^52 - 576 * q^55 + 1128 * q^58 + 1824 * q^61 + 3024 * q^64 + 816 * q^67 + 2664 * q^70 + 1920 * q^76 + 1200 * q^82 - 480 * q^85 - 4560 * q^88 - 3600 * q^91 - 11304 * q^94

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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