LMFDB - Embedded newform 144.4.l.a.35.20

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)

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")

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);

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			35.20
Character	$\chi$	$=$	144.35
Dual form			144.4.l.a.107.20

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(2.19729 + 1.78098i) q^{2} +(1.65621 + 7.82668i) q^{4} +(-3.62348 - 3.62348i) q^{5} -33.7361 q^{7} +(-10.3000 + 20.1472i) q^{8} +(-1.50850 - 14.4152i) q^{10} +(-44.1694 + 44.1694i) q^{11} +(42.1143 + 42.1143i) q^{13} +(-74.1283 - 60.0835i) q^{14} +(-58.5139 + 25.9252i) q^{16} +17.6392i q^{17} +(99.2533 - 99.2533i) q^{19} +(22.3586 - 34.3611i) q^{20} +(-175.718 + 18.3883i) q^{22} +83.8215i q^{23} -98.7408i q^{25} +(17.5327 + 167.542i) q^{26} +(-55.8741 - 264.042i) q^{28} +(-89.7300 + 89.7300i) q^{29} +46.2516i q^{31} +(-174.745 - 47.2469i) q^{32} +(-31.4150 + 38.7584i) q^{34} +(122.242 + 122.242i) q^{35} +(-7.47930 + 7.47930i) q^{37} +(394.857 - 41.3205i) q^{38} +(110.325 - 35.6811i) q^{40} +299.247 q^{41} +(-56.3439 - 56.3439i) q^{43} +(-418.854 - 272.546i) q^{44} +(-149.285 + 184.181i) q^{46} -280.235 q^{47} +795.127 q^{49} +(175.856 - 216.963i) q^{50} +(-259.865 + 399.365i) q^{52} +(8.15996 + 8.15996i) q^{53} +320.094 q^{55} +(347.482 - 679.689i) q^{56} +(-356.971 + 37.3558i) q^{58} +(-193.284 + 193.284i) q^{59} +(127.057 + 127.057i) q^{61} +(-82.3732 + 101.628i) q^{62} +(-299.820 - 415.033i) q^{64} -305.201i q^{65} +(-110.221 + 110.221i) q^{67} +(-138.056 + 29.2141i) q^{68} +(50.8910 + 486.313i) q^{70} +1070.35i q^{71} +1015.84i q^{73} +(-29.7547 + 3.11373i) q^{74} +(941.209 + 612.440i) q^{76} +(1490.11 - 1490.11i) q^{77} +161.531i q^{79} +(305.964 + 118.084i) q^{80} +(657.533 + 532.953i) q^{82} +(-64.4215 - 64.4215i) q^{83} +(63.9151 - 63.9151i) q^{85} +(-23.4567 - 224.152i) q^{86} +(-434.945 - 1344.84i) q^{88} -177.502 q^{89} +(-1420.77 - 1420.77i) q^{91} +(-656.045 + 138.826i) q^{92} +(-615.758 - 499.093i) q^{94} -719.285 q^{95} +559.412 q^{97} +(1747.13 + 1416.11i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.19729	+	1.78098i	0.776861	+	0.629672i
$2$	2.19729	+	1.78098i	0.776861	+	0.629672i
$3$		0			0
$3$		0			0
$4$	1.65621	+	7.82668i	0.207026	+	0.978335i
$5$	−3.62348	−	3.62348i	−0.324094	−	0.324094i	0.526241	−	0.850335i	$-0.323601\pi$
$5$	−3.62348	−	3.62348i	−0.324094	−	0.324094i	−0.850335	+	0.526241i	$0.823601\pi$
$6$		0			0
$7$	−33.7361			−1.82158			−0.910790	−	0.412869i	$-0.864527\pi$
$7$	−33.7361			−1.82158			−0.910790	+	0.412869i	$0.864527\pi$
$8$	−10.3000	+	20.1472i	−0.455200	+	0.890389i
$9$		0			0
$10$	−1.50850	−	14.4152i	−0.0477030	−	0.455849i
$11$	−44.1694	+	44.1694i	−1.21069	+	1.21069i	−0.239889	+	0.970800i	$0.577111\pi$
$11$	−44.1694	+	44.1694i	−1.21069	+	1.21069i	−0.970800	+	0.239889i	$0.922889\pi$
$12$		0			0
$13$	42.1143	+	42.1143i	0.898493	+	0.898493i	0.995303	−	0.0968099i	$-0.0308639\pi$
$13$	42.1143	+	42.1143i	0.898493	+	0.898493i	−0.0968099	+	0.995303i	$0.530864\pi$
$14$	−74.1283	−	60.0835i	−1.41511	−	1.14700i
$15$		0			0
$16$	−58.5139	+	25.9252i	−0.914280	+	0.405082i
$17$			17.6392i			0.251654i	0.992052	+	0.125827i	$0.0401585\pi$
$17$			17.6392i			0.251654i	−0.992052	+	0.125827i	$0.959842\pi$
$18$		0			0
$19$	99.2533	−	99.2533i	1.19844	−	1.19844i	0.223801	−	0.974635i	$-0.428153\pi$
$19$	99.2533	−	99.2533i	1.19844	−	1.19844i	0.974635	−	0.223801i	$-0.0718465\pi$
$20$	22.3586	−	34.3611i	0.249977	−	0.384168i
$21$		0			0
$22$	−175.718	+	18.3883i	−1.70287	+	0.178200i
$23$			83.8215i			0.759913i	0.925004	+	0.379956i	$0.124061\pi$
$23$			83.8215i			0.759913i	−0.925004	+	0.379956i	$0.875939\pi$
$24$		0			0
$25$		−	98.7408i		−	0.789926i
$26$	17.5327	+	167.542i	0.132248	+	1.26376i
$27$		0			0
$28$	−55.8741	−	264.042i	−0.377115	−	1.78212i
$29$	−89.7300	+	89.7300i	−0.574567	+	0.574567i	−0.933401	−	0.358834i	$-0.883174\pi$
$29$	−89.7300	+	89.7300i	−0.574567	+	0.574567i	0.358834	+	0.933401i	$0.383174\pi$
$30$		0			0
$31$			46.2516i			0.267969i	0.990983	+	0.133984i	$0.0427772\pi$
$31$			46.2516i			0.267969i	−0.990983	+	0.133984i	$0.957223\pi$
$32$	−174.745	−	47.2469i	−0.965338	−	0.261005i
$33$		0			0
$34$	−31.4150	+	38.7584i	−0.158460	+	0.195500i
$35$	122.242	+	122.242i	0.590363	+	0.590363i
$36$		0			0
$37$	−7.47930	+	7.47930i	−0.0332321	+	0.0332321i	−0.723528	−	0.690295i	$-0.757481\pi$
$37$	−7.47930	+	7.47930i	−0.0332321	+	0.0332321i	0.690295	+	0.723528i	$0.257481\pi$
$38$	394.857	−	41.3205i	1.68564	−	0.176396i
$39$		0			0
$40$	110.325	−	35.6811i	0.436097	−	0.141042i
$41$	299.247			1.13986			0.569932	−	0.821692i	$-0.306969\pi$
$41$	299.247			1.13986			0.569932	+	0.821692i	$0.306969\pi$
$42$		0			0
$43$	−56.3439	−	56.3439i	−0.199823	−	0.199823i	0.600101	−	0.799924i	$-0.295127\pi$
$43$	−56.3439	−	56.3439i	−0.199823	−	0.199823i	−0.799924	+	0.600101i	$0.795127\pi$
$44$	−418.854	−	272.546i	−1.43510	−	0.933816i
$45$		0			0
$46$	−149.285	+	184.181i	−0.478496	+	0.590347i
$47$	−280.235			−0.869711			−0.434855	−	0.900500i	$-0.643201\pi$
$47$	−280.235			−0.869711			−0.434855	+	0.900500i	$0.643201\pi$
$48$		0			0
$49$	795.127			2.31816
$50$	175.856	−	216.963i	0.497395	−	0.613663i
$51$		0			0
$52$	−259.865	+	399.365i	−0.693016	+	1.06504i
$53$	8.15996	+	8.15996i	0.0211482	+	0.0211482i	0.717602	−	0.696454i	$-0.245240\pi$
$53$	8.15996	+	8.15996i	0.0211482	+	0.0211482i	−0.696454	+	0.717602i	$0.745240\pi$
$54$		0			0
$55$	320.094			0.784754
$56$	347.482	−	679.689i	0.829184	−	1.62192i
$57$		0			0
$58$	−356.971	+	37.3558i	−0.808148	+	0.0845699i
$59$	−193.284	+	193.284i	−0.426499	+	0.426499i	−0.887434	−	0.460935i	$-0.847514\pi$
$59$	−193.284	+	193.284i	−0.426499	+	0.426499i	0.460935	+	0.887434i	$0.347514\pi$
$60$		0			0
$61$	127.057	+	127.057i	0.266688	+	0.266688i	0.827764	−	0.561076i	$-0.189612\pi$
$61$	127.057	+	127.057i	0.266688	+	0.266688i	−0.561076	+	0.827764i	$0.689612\pi$
$62$	−82.3732	+	101.628i	−0.168732	+	0.208174i
$63$		0			0
$64$	−299.820	−	415.033i	−0.585586	−	0.810610i
$65$		−	305.201i		−	0.582392i
$66$		0			0
$67$	−110.221	+	110.221i	−0.200980	+	0.200980i	−0.800420	−	0.599440i	$-0.795390\pi$
$67$	−110.221	+	110.221i	−0.200980	+	0.200980i	0.599440	+	0.800420i	$0.295390\pi$
$68$	−138.056	+	29.2141i	−0.246202	+	0.0520990i
$69$		0			0
$70$	50.8910	+	486.313i	0.0868949	+	0.830365i
$71$			1070.35i			1.78911i	0.446955	+	0.894556i	$0.352508\pi$
$71$			1070.35i			1.78911i	−0.446955	+	0.894556i	$0.647492\pi$
$72$		0			0
$73$			1015.84i			1.62870i	0.580373	+	0.814351i	$0.302907\pi$
$73$			1015.84i			1.62870i	−0.580373	+	0.814351i	$0.697093\pi$
$74$	−29.7547	+	3.11373i	−0.0467421	+	0.00489140i
$75$		0			0
$76$	941.209	+	612.440i	1.42058	+	0.924365i
$77$	1490.11	−	1490.11i	2.20537	−	2.20537i
$78$		0			0
$79$			161.531i			0.230047i	0.993363	+	0.115023i	$0.0366943\pi$
$79$			161.531i			0.230047i	−0.993363	+	0.115023i	$0.963306\pi$
$80$	305.964	+	118.084i	0.427597	+	0.165028i
$81$		0			0
$82$	657.533	+	532.953i	0.885517	+	0.717741i
$83$	−64.4215	−	64.4215i	−0.0851950	−	0.0851950i	0.663225	−	0.748420i	$-0.269187\pi$
$83$	−64.4215	−	64.4215i	−0.0851950	−	0.0851950i	−0.748420	+	0.663225i	$0.769187\pi$
$84$		0			0
$85$	63.9151	−	63.9151i	0.0815596	−	0.0815596i
$86$	−23.4567	−	224.152i	−0.0294117	−	0.281057i
$87$		0			0
$88$	−434.945	−	1344.84i	−0.526879	−	1.62909i
$89$	−177.502			−0.211406			−0.105703	−	0.994398i	$-0.533709\pi$
$89$	−177.502			−0.211406			−0.105703	+	0.994398i	$0.533709\pi$
$90$		0			0
$91$	−1420.77	−	1420.77i	−1.63668	−	1.63668i
$92$	−656.045	+	138.826i	−0.743450	+	0.157322i
$93$		0			0
$94$	−615.758	−	499.093i	−0.675644	−	0.547633i
$95$	−719.285			−0.776811
$96$		0			0
$97$	559.412			0.585564			0.292782	−	0.956179i	$-0.405419\pi$
$97$	559.412			0.585564			0.292782	+	0.956179i	$0.405419\pi$
$98$	1747.13	+	1416.11i	1.80088	+	1.45968i
$99$		0			0
$100$	772.813	−	163.535i	0.772813	−	0.163535i
$101$	−850.608	−	850.608i	−0.838006	−	0.838006i	0.150590	−	0.988596i	$-0.451883\pi$
$101$	−850.608	−	850.608i	−0.838006	−	0.838006i	−0.988596	+	0.150590i	$0.951883\pi$
$102$		0			0
$103$	−757.707			−0.724845			−0.362423	−	0.932014i	$-0.618050\pi$
$103$	−757.707			−0.724845			−0.362423	+	0.932014i	$0.618050\pi$
$104$	−1282.26	+	414.708i	−1.20900	+	0.391014i
$105$		0			0
$106$	3.39710	+	32.4626i	0.00311279	+	0.0297457i
$107$	364.469	−	364.469i	0.329295	−	0.329295i	−0.523024	−	0.852318i	$-0.675196\pi$
$107$	364.469	−	364.469i	0.329295	−	0.329295i	0.852318	+	0.523024i	$0.175196\pi$
$108$		0			0
$109$	−206.900	−	206.900i	−0.181812	−	0.181812i	0.610333	−	0.792145i	$-0.291036\pi$
$109$	−206.900	−	206.900i	−0.181812	−	0.181812i	−0.792145	+	0.610333i	$0.791036\pi$
$110$	703.341	+	570.082i	0.609645	+	0.494138i
$111$		0			0
$112$	1974.03	−	874.618i	1.66544	−	0.737889i
$113$			1379.24i			1.14821i	0.818781	+	0.574105i	$0.194650\pi$
$113$			1379.24i			1.14821i	−0.818781	+	0.574105i	$0.805350\pi$
$114$		0			0
$115$	303.726	−	303.726i	0.246283	−	0.246283i
$116$	−850.900	−	553.677i	−0.681070	−	0.443169i
$117$		0			0
$118$	−768.938	+	80.4667i	−0.599885	+	0.0627759i
$119$		−	595.077i		−	0.458409i
$120$		0			0
$121$		−	2570.88i		−	1.93154i
$122$	52.8955	+	505.468i	0.0392535	+	0.375106i
$123$		0			0
$124$	−361.996	+	76.6022i	−0.262163	+	0.0554765i
$125$	−810.720	+	810.720i	−0.580104	+	0.580104i
$126$		0			0
$127$			1260.84i			0.880958i	0.897763	+	0.440479i	$0.145191\pi$
$127$			1260.84i			0.880958i	−0.897763	+	0.440479i	$0.854809\pi$
$128$	80.3725	−	1445.92i	0.0555000	−	0.998459i
$129$		0			0
$130$	543.557	−	670.616i	0.366716	−	0.452438i
$131$	1015.28	+	1015.28i	0.677138	+	0.677138i	0.959352	−	0.282214i	$-0.0910687\pi$
$131$	1015.28	+	1015.28i	0.677138	+	0.677138i	−0.282214	+	0.959352i	$0.591069\pi$
$132$		0			0
$133$	−3348.43	+	3348.43i	−2.18305	+	2.18305i
$134$	−438.491	+	45.8866i	−0.282686	+	0.0295821i
$135$		0			0
$136$	−355.380	−	181.683i	−0.224070	−	0.114553i
$137$	1875.46			1.16957			0.584785	−	0.811189i	$-0.301179\pi$
$137$	1875.46			1.16957			0.584785	+	0.811189i	$0.301179\pi$
$138$		0			0
$139$	−1048.87	−	1048.87i	−0.640031	−	0.640031i	0.310532	−	0.950563i	$-0.399493\pi$
$139$	−1048.87	−	1048.87i	−0.640031	−	0.640031i	−0.950563	+	0.310532i	$0.899493\pi$
$140$	−754.293	+	1159.21i	−0.455353	+	0.699794i
$141$		0			0
$142$	−1906.27	+	2351.87i	−1.12655	+	1.38989i
$143$	−3720.33			−2.17559
$144$		0			0
$145$	650.270			0.372427
$146$	−1809.19	+	2232.10i	−1.02555	+	1.26527i
$147$		0			0
$148$	−70.9254	−	46.1508i	−0.0393921	−	0.0256323i
$149$	−1943.41	−	1943.41i	−1.06852	−	1.06852i	−0.997473	−	0.0710521i	$-0.977364\pi$
$149$	−1943.41	−	1943.41i	−1.06852	−	1.06852i	−0.0710521	−	0.997473i	$-0.522636\pi$
$150$		0			0
$151$	−1790.47			−0.964945			−0.482472	−	0.875911i	$-0.660261\pi$
$151$	−1790.47			−0.964945			−0.482472	+	0.875911i	$0.660261\pi$
$152$	977.368	+	3021.99i	0.521546	+	1.61260i
$153$		0			0
$154$	5928.05	−	620.351i	3.10192	−	0.324606i
$155$	167.592	−	167.592i	0.0868470	−	0.0868470i
$156$		0			0
$157$	−872.163	−	872.163i	−0.443351	−	0.443351i	0.449785	−	0.893137i	$-0.351500\pi$
$157$	−872.163	−	872.163i	−0.443351	−	0.443351i	−0.893137	+	0.449785i	$0.851500\pi$
$158$	−287.685	+	354.932i	−0.144854	+	0.178714i
$159$		0			0
$160$	461.986	+	804.382i	0.228270	+	0.397450i
$161$		−	2827.82i		−	1.38424i
$162$		0			0
$163$	541.663	−	541.663i	0.260284	−	0.260284i	−0.564885	−	0.825169i	$-0.691080\pi$
$163$	541.663	−	541.663i	0.260284	−	0.260284i	0.825169	+	0.564885i	$0.191080\pi$
$164$	495.615	+	2342.11i	0.235982	+	1.11517i
$165$		0			0
$166$	−26.8195	−	256.287i	−0.0125398	−	0.119830i
$167$		−	271.228i		−	0.125678i	−0.998024	−	0.0628390i	$-0.979985\pi$
$167$		−	271.228i		−	0.125678i	0.998024	−	0.0628390i	$-0.0200155\pi$
$168$		0			0
$169$			1350.23i			0.614579i
$170$	254.272	−	26.6087i	0.114716	−	0.0120047i
$171$		0			0
$172$	347.669	−	534.303i	0.154125	−	0.236862i
$173$	1396.64	−	1396.64i	0.613783	−	0.613783i	−0.330147	−	0.943930i	$-0.607098\pi$
$173$	1396.64	−	1396.64i	0.613783	−	0.613783i	0.943930	+	0.330147i	$0.107098\pi$
$174$		0			0
$175$			3331.13i			1.43891i
$176$	1439.42	−	3729.63i	0.616481	−	1.59734i
$177$		0			0
$178$	−390.023	−	316.127i	−0.164233	−	0.133116i
$179$	675.124	+	675.124i	0.281906	+	0.281906i	0.833869	−	0.551963i	$-0.186121\pi$
$179$	675.124	+	675.124i	0.281906	+	0.281906i	−0.551963	+	0.833869i	$0.686121\pi$
$180$		0			0
$181$	1306.81	−	1306.81i	0.536655	−	0.536655i	−0.385890	−	0.922545i	$-0.626106\pi$
$181$	1306.81	−	1306.81i	0.536655	−	0.536655i	0.922545	+	0.385890i	$0.126106\pi$
$182$	−591.487	−	5652.23i	−0.240901	−	2.30204i
$183$		0			0
$184$	−1688.77	−	863.362i	−0.676618	−	0.345912i
$185$	54.2021			0.0215407
$186$		0			0
$187$	−779.111	−	779.111i	−0.304675	−	0.304675i
$188$	−464.127	−	2193.31i	−0.180053	−	0.850869i
$189$		0			0
$190$	−1580.48	−	1281.03i	−0.603474	−	0.489136i
$191$	1942.70			0.735961			0.367980	−	0.929834i	$-0.380049\pi$
$191$	1942.70			0.735961			0.367980	+	0.929834i	$0.380049\pi$
$192$		0			0
$193$	4145.80			1.54623			0.773113	−	0.634269i	$-0.218699\pi$
$193$	4145.80			1.54623			0.773113	+	0.634269i	$0.218699\pi$
$194$	1229.19	+	996.302i	0.454901	+	0.368713i
$195$		0			0
$196$	1316.90	+	6223.21i	0.479919	+	2.26793i
$197$	1926.84	+	1926.84i	0.696860	+	0.696860i	0.963732	−	0.266872i	$-0.0859902\pi$
$197$	1926.84	+	1926.84i	0.696860	+	0.696860i	−0.266872	+	0.963732i	$0.585990\pi$
$198$		0			0
$199$	3804.49			1.35524			0.677622	−	0.735411i	$-0.263011\pi$
$199$	3804.49			1.35524			0.677622	+	0.735411i	$0.263011\pi$
$200$	1989.35	+	1017.03i	0.703342	+	0.359575i
$201$		0			0
$202$	−354.119	−	3383.95i	−0.123345	−	1.17868i
$203$	3027.15	−	3027.15i	1.04662	−	1.04662i
$204$		0			0
$205$	−1084.31	−	1084.31i	−0.369423	−	0.369423i
$206$	−1664.91	−	1349.46i	−0.563104	−	0.456415i
$207$		0			0
$208$	−3556.10	−	1372.45i	−1.18544	−	0.457511i
$209$			8767.93i			2.90187i
$210$		0			0
$211$	2274.25	−	2274.25i	0.742019	−	0.742019i	−0.230947	−	0.972966i	$-0.574182\pi$
$211$	2274.25	−	2274.25i	0.742019	−	0.742019i	0.972966	+	0.230947i	$0.0741825\pi$
$212$	−50.3508	+	77.3800i	−0.0163118	+	0.0250683i
$213$		0			0
$214$	1449.96	−	151.733i	0.463164	−	0.0484685i
$215$			408.322i			0.129523i
$216$		0			0
$217$		−	1560.35i		−	0.488126i
$218$	−86.1353	−	823.107i	−0.0267606	−	0.255724i
$219$		0			0
$220$	530.142	+	2505.27i	0.162465	+	0.767753i
$221$	−742.861	+	742.861i	−0.226110	+	0.226110i
$222$		0			0
$223$			2370.49i			0.711836i	0.934517	+	0.355918i	$0.115832\pi$
$223$			2370.49i			0.711836i	−0.934517	+	0.355918i	$0.884168\pi$
$224$	5895.21	+	1593.93i	1.75844	+	0.475441i
$225$		0			0
$226$	−2456.40	+	3030.59i	−0.722996	+	0.892000i
$227$	−3494.40	−	3494.40i	−1.02172	−	1.02172i	−0.999759	−	0.0219659i	$-0.993007\pi$
$227$	−3494.40	−	3494.40i	−1.02172	−	1.02172i	−0.0219659	−	0.999759i	$-0.506993\pi$
$228$		0			0
$229$	−2251.16	+	2251.16i	−0.649611	+	0.649611i	−0.952899	−	0.303288i	$-0.901916\pi$
$229$	−2251.16	+	2251.16i	−0.649611	+	0.649611i	0.303288	+	0.952899i	$0.401916\pi$
$230$	1208.30	−	126.445i	0.346405	−	0.0362501i
$231$		0			0
$232$	−883.590	−	2732.03i	−0.250045	−	0.773132i
$233$	−3104.86			−0.872989			−0.436494	−	0.899707i	$-0.643780\pi$
$233$	−3104.86			−0.872989			−0.436494	+	0.899707i	$0.643780\pi$
$234$		0			0
$235$	1015.42	+	1015.42i	0.281868	+	0.281868i
$236$	−1832.89	−	1192.65i	−0.505556	−	0.328963i
$237$		0			0
$238$	1059.82	−	1307.56i	0.288647	−	0.356120i
$239$	5602.27			1.51624			0.758119	−	0.652117i	$-0.226119\pi$
$239$	5602.27			1.51624			0.758119	+	0.652117i	$0.226119\pi$
$240$		0			0
$241$	−278.260			−0.0743748			−0.0371874	−	0.999308i	$-0.511840\pi$
$241$	−278.260			−0.0743748			−0.0371874	+	0.999308i	$0.511840\pi$
$242$	4578.68	−	5648.97i	1.21624	−	1.50054i
$243$		0			0
$244$	−784.002	+	1204.87i	−0.205699	+	0.316122i
$245$	−2881.13	−	2881.13i	−0.751300	−	0.751300i
$246$		0			0
$247$	8359.97			2.15357
$248$	−931.840	−	476.391i	−0.238596	−	0.121979i
$249$		0			0
$250$	−3225.27	+	337.513i	−0.815936	+	0.0853849i
$251$	4432.83	−	4432.83i	1.11473	−	1.11473i	0.122231	−	0.992502i	$-0.460995\pi$
$251$	4432.83	−	4432.83i	1.11473	−	1.11473i	0.992502	−	0.122231i	$-0.0390048\pi$
$252$		0			0
$253$	−3702.35	−	3702.35i	−0.920019	−	0.920019i
$254$	−2245.54	+	2770.44i	−0.554715	+	0.684382i
$255$		0			0
$256$	2751.76	−	3033.98i	0.671817	−	0.740717i
$257$			3828.87i			0.929332i	0.885486	+	0.464666i	$0.153826\pi$
$257$			3828.87i			0.929332i	−0.885486	+	0.464666i	$0.846174\pi$
$258$		0			0
$259$	252.323	−	252.323i	0.0605350	−	0.0605350i
$260$	2388.71	−	505.476i	0.569775	−	0.120570i
$261$		0			0
$262$	422.673	+	4039.05i	0.0996672	+	0.952417i
$263$			2299.83i			0.539216i	0.962970	+	0.269608i	$0.0868941\pi$
$263$			2299.83i			0.539216i	−0.962970	+	0.269608i	$0.913106\pi$
$264$		0			0
$265$		−	59.1349i		−	0.0137080i
$266$	−13321.0	+	1393.99i	−3.07053	+	0.321320i
$267$		0			0
$268$	−1045.22	−	680.118i	−0.238234	−	0.155018i
$269$	−3350.78	+	3350.78i	−0.759481	+	0.759481i	−0.976228	−	0.216747i	$-0.930455\pi$
$269$	−3350.78	+	3350.78i	−0.759481	+	0.759481i	0.216747	+	0.976228i	$0.430455\pi$
$270$		0			0
$271$			3778.66i			0.847000i	0.905896	+	0.423500i	$0.139199\pi$
$271$			3778.66i			0.847000i	−0.905896	+	0.423500i	$0.860801\pi$
$272$	−457.299	−	1032.14i	−0.101941	−	0.230083i
$273$		0			0
$274$	4120.93	+	3340.15i	0.908593	+	0.736445i
$275$	4361.32	+	4361.32i	0.956356	+	0.956356i
$276$		0			0
$277$	−161.818	+	161.818i	−0.0351000	+	0.0351000i	−0.724439	−	0.689339i	$-0.757901\pi$
$277$	−161.818	+	161.818i	−0.0351000	+	0.0351000i	0.689339	+	0.724439i	$0.257901\pi$
$278$	−436.660	−	4172.71i	−0.0942054	−	0.900224i
$279$		0			0
$280$	−3721.93	+	1203.74i	−0.794386	+	0.256920i
$281$	−4259.63			−0.904299			−0.452149	−	0.891942i	$-0.649343\pi$
$281$	−4259.63			−0.904299			−0.452149	+	0.891942i	$0.649343\pi$
$282$		0			0
$283$	−769.856	−	769.856i	−0.161707	−	0.161707i	0.621615	−	0.783323i	$-0.286477\pi$
$283$	−769.856	−	769.856i	−0.161707	−	0.161707i	−0.783323	+	0.621615i	$0.786477\pi$
$284$	−8377.28	+	1772.72i	−1.75035	+	0.370393i
$285$		0			0
$286$	−8174.66	−	6625.84i	−1.69013	−	1.36991i
$287$	−10095.4			−2.07636
$288$		0			0
$289$	4601.86			0.936670
$290$	1428.83	+	1158.12i	0.289324	+	0.234507i
$291$		0			0
$292$	−7950.66	+	1682.44i	−1.59342	+	0.337184i
$293$	4375.88	+	4375.88i	0.872497	+	0.872497i	0.992744	−	0.120247i	$-0.0383685\pi$
$293$	4375.88	+	4375.88i	0.872497	+	0.872497i	−0.120247	+	0.992744i	$0.538369\pi$
$294$		0			0
$295$	1400.72			0.276451
$296$	−73.6502	−	227.724i	−0.0144623	−	0.0447168i
$297$		0			0
$298$	−809.066	−	7731.41i	−0.157275	−	1.50292i
$299$	−3530.09	+	3530.09i	−0.682776	+	0.682776i
$300$		0			0
$301$	1900.83	+	1900.83i	0.363993	+	0.363993i
$302$	−3934.20	−	3188.80i	−0.749628	−	0.607599i
$303$		0			0
$304$	−3234.54	+	8380.87i	−0.610242	+	1.58117i
$305$		−	920.776i		−	0.172864i
$306$		0			0
$307$	−1892.89	+	1892.89i	−0.351899	+	0.351899i	−0.860816	−	0.508917i	$-0.830046\pi$
$307$	−1892.89	+	1892.89i	−0.351899	+	0.351899i	0.508917	+	0.860816i	$0.330046\pi$
$308$	14130.5	+	9194.66i	2.61416	+	1.70102i
$309$		0			0
$310$	666.726	−	69.7706i	0.122153	−	0.0127829i
$311$		−	4969.36i		−	0.906067i	−0.891494	−	0.453033i	$-0.850342\pi$
$311$		−	4969.36i		−	0.906067i	0.891494	−	0.453033i	$-0.149658\pi$
$312$		0			0
$313$		−	431.319i		−	0.0778901i	−0.999241	−	0.0389450i	$-0.987600\pi$
$313$		−	431.319i		−	0.0778901i	0.999241	−	0.0389450i	$-0.0123997\pi$
$314$	−363.093	−	3469.70i	−0.0652564	−	0.623588i
$315$		0			0
$316$	−1264.26	+	267.530i	−0.225063	+	0.0476257i
$317$	−4151.14	+	4151.14i	−0.735493	+	0.735493i	−0.971702	−	0.236209i	$-0.924095\pi$
$317$	−4151.14	+	4151.14i	−0.735493	+	0.735493i	0.236209	+	0.971702i	$0.424095\pi$
$318$		0			0
$319$		−	7926.65i		−	1.39125i
$320$	−417.470	+	2590.25i	−0.0729291	+	0.452499i
$321$		0			0
$322$	5036.29	−	6213.54i	0.871619	−	1.07536i
$323$	1750.74	+	1750.74i	0.301591	+	0.301591i
$324$		0			0
$325$	4158.40	−	4158.40i	0.709743	−	0.709743i
$326$	2154.88	−	225.501i	0.366098	−	0.0383109i
$327$		0			0
$328$	−3082.24	+	6028.98i	−0.518867	+	1.01492i
$329$	9454.03			1.58425
$330$		0			0
$331$	3350.31	+	3350.31i	0.556343	+	0.556343i	0.928264	−	0.371921i	$-0.121301\pi$
$331$	3350.31	+	3350.31i	0.556343	+	0.556343i	−0.371921	+	0.928264i	$0.621301\pi$
$332$	397.511	−	610.902i	0.0657117	−	0.100987i
$333$		0			0
$334$	483.051	−	595.967i	0.0791359	−	0.0976343i
$335$	798.770			0.130273
$336$		0			0
$337$	−9901.48			−1.60050			−0.800249	−	0.599668i	$-0.795299\pi$
$337$	−9901.48			−1.60050			−0.800249	+	0.599668i	$0.795299\pi$
$338$	−2404.74	+	2966.86i	−0.386983	+	0.477443i
$339$		0			0
$340$	606.100	+	394.386i	0.0966776	+	0.0629077i
$341$	−2042.90	−	2042.90i	−0.324427	−	0.324427i
$342$		0			0
$343$	−15253.0			−2.40113
$344$	1715.52	−	554.830i	0.268879	−	0.0869606i
$345$		0			0
$346$	5556.21	−	581.439i	0.863306	−	0.0903420i
$347$	−1244.90	+	1244.90i	−0.192593	+	0.192593i	−0.796816	−	0.604222i	$-0.793484\pi$
$347$	−1244.90	+	1244.90i	−0.192593	+	0.192593i	0.604222	+	0.796816i	$0.293484\pi$
$348$		0			0
$349$	84.2939	+	84.2939i	0.0129288	+	0.0129288i	0.713542	−	0.700613i	$-0.247090\pi$
$349$	84.2939	+	84.2939i	0.0129288	+	0.0129288i	−0.700613	+	0.713542i	$0.747090\pi$
$350$	−5932.69	+	7319.48i	−0.906044	+	1.11784i
$351$		0			0
$352$	9805.24	−	5631.51i	1.48472	−	0.852729i
$353$		−	366.806i		−	0.0553063i	−0.999618	−	0.0276531i	$-0.991197\pi$
$353$		−	366.806i		−	0.0553063i	0.999618	−	0.0276531i	$-0.00880339\pi$
$354$		0			0
$355$	3878.39	−	3878.39i	0.579840	−	0.579840i
$356$	−293.980	−	1389.25i	−0.0437665	−	0.206826i
$357$		0			0
$358$	281.063	+	2685.83i	0.0414934	+	0.396510i
$359$		−	6841.31i		−	1.00577i	−0.864354	−	0.502884i	$-0.832272\pi$
$359$		−	6841.31i		−	1.00577i	0.864354	−	0.502884i	$-0.167728\pi$
$360$		0			0
$361$		−	12843.5i		−	1.87250i
$362$	5198.85	−	544.042i	0.754822	−	0.0789896i
$363$		0			0
$364$	8766.85	−	13473.1i	1.26238	−	1.94005i
$365$	3680.88	−	3680.88i	0.527852	−	0.527852i
$366$		0			0
$367$			1916.32i			0.272564i	0.990670	+	0.136282i	$0.0435153\pi$
$367$			1916.32i			0.272564i	−0.990670	+	0.136282i	$0.956485\pi$
$368$	−2173.09	−	4904.73i	−0.307827	−	0.694774i
$369$		0			0
$370$	119.098	+	96.5330i	0.0167341	+	0.0135636i
$371$	−275.286	−	275.286i	−0.0385232	−	0.0385232i
$372$		0			0
$373$	−7158.65	+	7158.65i	−0.993728	+	0.993728i	−0.999980	−	0.00625218i	$-0.998010\pi$
$373$	−7158.65	+	7158.65i	−0.993728	+	0.993728i	0.00625218	+	0.999980i	$0.498010\pi$
$374$	−324.354	−	3099.52i	−0.0448448	−	0.428536i
$375$		0			0
$376$	2886.42	−	5645.94i	0.395892	−	0.774381i
$377$	−7557.84			−1.03249
$378$		0			0
$379$	5108.00	+	5108.00i	0.692297	+	0.692297i	0.962737	−	0.270440i	$-0.0871692\pi$
$379$	5108.00	+	5108.00i	0.692297	+	0.692297i	−0.270440	+	0.962737i	$0.587169\pi$
$380$	−1191.29	−	5629.61i	−0.160820	−	0.759982i
$381$		0			0
$382$	4268.68	+	3459.91i	0.571739	+	0.463414i
$383$	10334.0			1.37870			0.689349	−	0.724429i	$-0.257897\pi$
$383$	10334.0			1.37870			0.689349	+	0.724429i	$0.257897\pi$
$384$		0			0
$385$	−10798.7			−1.42949
$386$	9109.55	+	7383.60i	1.20120	+	0.973615i
$387$		0			0
$388$	926.503	+	4378.34i	0.121227	+	0.572878i
$389$	−4119.25	−	4119.25i	−0.536900	−	0.536900i	0.385717	−	0.922617i	$-0.373954\pi$
$389$	−4119.25	−	4119.25i	−0.536900	−	0.536900i	−0.922617	+	0.385717i	$0.873954\pi$
$390$		0			0
$391$	−1478.54			−0.191235
$392$	−8189.81	+	16019.6i	−1.05522	+	2.06406i
$393$		0			0
$394$	802.167	+	7665.49i	0.102570	+	0.980156i
$395$	585.306	−	585.306i	0.0745568	−	0.0745568i
$396$		0			0
$397$	6024.12	+	6024.12i	0.761566	+	0.761566i	0.976605	−	0.215039i	$-0.0689880\pi$
$397$	6024.12	+	6024.12i	0.761566	+	0.761566i	−0.215039	+	0.976605i	$0.568988\pi$
$398$	8359.60	+	6775.73i	1.05284	+	0.853359i
$399$		0			0
$400$	2559.88	+	5777.71i	0.319985	+	0.722214i
$401$		−	11848.6i		−	1.47554i	−0.675050	−	0.737772i	$-0.735878\pi$
$401$		−	11848.6i		−	1.47554i	0.675050	−	0.737772i	$-0.264122\pi$
$402$		0			0
$403$	−1947.85	+	1947.85i	−0.240768	+	0.240768i
$404$	5248.65	−	8066.22i	0.646362	−	0.993341i
$405$		0			0
$406$	12042.8	−	1260.24i	1.47211	−	0.154051i
$407$		−	660.713i		−	0.0804676i
$408$		0			0
$409$			12692.3i			1.53446i	0.641374	+	0.767228i	$0.278365\pi$
$409$			12692.3i			1.53446i	−0.641374	+	0.767228i	$0.721635\pi$
$410$	−451.414	−	4313.70i	−0.0543750	−	0.519606i
$411$		0			0
$412$	−1254.92	−	5930.33i	−0.150062	−	0.709142i
$413$	6520.66	−	6520.66i	0.776903	−	0.776903i
$414$		0			0
$415$			466.860i			0.0552223i
$416$	−5369.49	−	9349.03i	−0.632838	−	1.10186i
$417$		0			0
$418$	−15615.5	+	19265.7i	−1.82722	+	2.25435i
$419$	−9974.97	−	9974.97i	−1.16303	−	1.16303i	−0.983809	−	0.179220i	$-0.942643\pi$
$419$	−9974.97	−	9974.97i	−1.16303	−	1.16303i	−0.179220	−	0.983809i	$-0.557357\pi$
$420$		0			0
$421$	6979.64	−	6979.64i	0.807997	−	0.807997i	−0.176333	−	0.984331i	$-0.556424\pi$
$421$	6979.64	−	6979.64i	0.807997	−	0.807997i	0.984331	+	0.176333i	$0.0564236\pi$
$422$	9047.61	−	946.801i	1.04367	−	0.109217i
$423$		0			0
$424$	−248.448	+	80.3528i	−0.0284568	+	0.00920348i
$425$	1741.70			0.198788
$426$		0			0
$427$	−4286.41	−	4286.41i	−0.485794	−	0.485794i
$428$	3456.22	+	2248.95i	0.390333	+	0.253988i
$429$		0			0
$430$	−727.214	+	897.204i	−0.0815567	+	0.100621i
$431$	6433.80			0.719038			0.359519	−	0.933138i	$-0.382941\pi$
$431$	6433.80			0.719038			0.359519	+	0.933138i	$0.382941\pi$
$432$		0			0
$433$	2276.81			0.252694			0.126347	−	0.991986i	$-0.459675\pi$
$433$	2276.81			0.252694			0.126347	+	0.991986i	$0.459675\pi$
$434$	2778.95	−	3428.55i	0.307360	−	0.379206i
$435$		0			0
$436$	1276.67	−	1962.01i	0.140233	−	0.215512i
$437$	8319.57	+	8319.57i	0.910707	+	0.910707i
$438$		0			0
$439$	8874.90			0.964866			0.482433	−	0.875933i	$-0.339753\pi$
$439$	8874.90			0.964866			0.482433	+	0.875933i	$0.339753\pi$
$440$	−3296.97	+	6449.00i	−0.357220	+	0.698736i
$441$		0			0
$442$	−2955.31	+	309.263i	−0.318031	+	0.0332808i
$443$	−801.170	+	801.170i	−0.0859249	+	0.0859249i	−0.748763	−	0.662838i	$-0.769352\pi$
$443$	−801.170	+	801.170i	−0.0859249	+	0.0859249i	0.662838	+	0.748763i	$0.269352\pi$
$444$		0			0
$445$	643.173	+	643.173i	0.0685154	+	0.0685154i
$446$	−4221.79	+	5208.66i	−0.448224	+	0.552998i
$447$		0			0
$448$	10114.8	+	14001.6i	1.06669	+	1.47659i
$449$			453.819i			0.0476995i	0.999716	+	0.0238497i	$0.00759232\pi$
$449$			453.819i			0.0476995i	−0.999716	+	0.0238497i	$0.992408\pi$
$450$		0			0
$451$	−13217.6	+	13217.6i	−1.38002	+	1.38002i
$452$	−10794.9	+	2284.31i	−1.12334	+	0.237710i
$453$		0			0
$454$	−1454.76	−	13901.7i	−0.150387	−	1.43709i
$455$			10296.3i			1.06087i
$456$		0			0
$457$			17430.4i			1.78416i	0.451876	+	0.892081i	$0.350755\pi$
$457$			17430.4i			1.78416i	−0.451876	+	0.892081i	$0.649245\pi$
$458$	−8955.74	+	937.188i	−0.913700	+	0.0956156i
$459$		0			0
$460$	2880.20	+	1874.13i	0.291934	+	0.189960i
$461$	−2812.24	+	2812.24i	−0.284119	+	0.284119i	−0.834749	−	0.550630i	$-0.814387\pi$
$461$	−2812.24	+	2812.24i	−0.284119	+	0.284119i	0.550630	+	0.834749i	$0.314387\pi$
$462$		0			0
$463$		−	8580.37i		−	0.861260i	−0.902528	−	0.430630i	$-0.858291\pi$
$463$		−	8580.37i		−	0.861260i	0.902528	−	0.430630i	$-0.141709\pi$
$464$	2924.19	−	7576.73i	0.292569	−	0.758062i
$465$		0			0
$466$	−6822.30	−	5529.70i	−0.678191	−	0.549697i
$467$	−4028.61	−	4028.61i	−0.399190	−	0.399190i	0.478757	−	0.877947i	$-0.341087\pi$
$467$	−4028.61	−	4028.61i	−0.399190	−	0.399190i	−0.877947	+	0.478757i	$0.841087\pi$
$468$		0			0
$469$	3718.45	−	3718.45i	0.366102	−	0.366102i
$470$	422.734	+	4039.64i	0.0414878	+	0.396457i
$471$		0			0
$472$	−1903.31	−	5884.96i	−0.185608	−	0.573893i
$473$	4977.36			0.483846
$474$		0			0
$475$	−9800.35	−	9800.35i	−0.946676	−	0.946676i
$476$	4657.48	−	985.572i	0.448477	−	0.0949025i
$477$		0			0
$478$	12309.8	+	9977.54i	1.17791	+	0.954732i
$479$	−7073.65			−0.674746			−0.337373	−	0.941371i	$-0.609538\pi$
$479$	−7073.65			−0.674746			−0.337373	+	0.941371i	$0.609538\pi$
$480$		0			0
$481$	−629.971			−0.0597177
$482$	−611.420	−	495.577i	−0.0577789	−	0.0468317i
$483$		0			0
$484$	20121.4	−	4257.91i	1.88969	−	0.399879i
$485$	−2027.02	−	2027.02i	−0.189778	−	0.189778i
$486$		0			0
$487$	−16066.6			−1.49497			−0.747483	−	0.664281i	$-0.768738\pi$
$487$	−16066.6			−1.49497			−0.747483	+	0.664281i	$0.768738\pi$
$488$	−3868.53	+	1251.16i	−0.358853	+	0.116060i
$489$		0			0
$490$	−1199.45	−	11461.9i	−0.110583	−	1.05673i
$491$	−4947.42	+	4947.42i	−0.454733	+	0.454733i	−0.896922	−	0.442189i	$-0.854202\pi$
$491$	−4947.42	+	4947.42i	−0.454733	+	0.454733i	0.442189	+	0.896922i	$0.354202\pi$
$492$		0			0
$493$	−1582.76	−	1582.76i	−0.144592	−	0.144592i
$494$	18369.3	+	14889.0i	1.67303	+	1.35604i
$495$		0			0
$496$	−1199.08	−	2706.36i	−0.108549	−	0.244998i
$497$		−	36109.4i		−	3.25901i
$498$		0			0
$499$	−10229.8	+	10229.8i	−0.917733	+	0.917733i	−0.996864	−	0.0791313i	$-0.974785\pi$
$499$	−10229.8	+	10229.8i	−0.917733	+	0.917733i	0.0791313	+	0.996864i	$0.474785\pi$
$500$	−7687.97	−	5002.53i	−0.687633	−	0.447440i
$501$		0			0
$502$	17635.0	−	1845.45i	1.56791	−	0.164076i
$503$			6091.42i			0.539966i	0.962865	+	0.269983i	$0.0870181\pi$
$503$			6091.42i			0.539966i	−0.962865	+	0.269983i	$0.912982\pi$
$504$		0			0
$505$			6164.32i			0.543185i
$506$	−1541.34	−	14729.0i	−0.135417	−	1.29404i
$507$		0			0
$508$	−9868.21	+	2088.22i	−0.861872	+	0.182381i
$509$	10933.5	−	10933.5i	0.952097	−	0.952097i	−0.0468068	−	0.998904i	$-0.514904\pi$
$509$	10933.5	−	10933.5i	0.952097	−	0.952097i	0.998904	+	0.0468068i	$0.0149045\pi$
$510$		0			0
$511$		−	34270.6i		−	2.96681i
$512$	11449.9	−	1765.70i	0.988317	−	0.152409i
$513$		0			0
$514$	−6819.15	+	8413.15i	−0.585174	+	0.721962i
$515$	2745.53	+	2745.53i	0.234918	+	0.234918i
$516$		0			0
$517$	12377.8	−	12377.8i	1.05295	−	1.05295i
$518$	1003.81	−	105.045i	0.0851445	−	0.00891008i
$519$		0			0
$520$	6148.94	+	3143.57i	0.518556	+	0.265105i
$521$	−8966.41			−0.753984			−0.376992	−	0.926217i	$-0.623042\pi$
$521$	−8966.41			−0.753984			−0.376992	+	0.926217i	$0.623042\pi$
$522$		0			0
$523$	14085.5	+	14085.5i	1.17766	+	1.17766i	0.980340	+	0.197318i	$0.0632230\pi$
$523$	14085.5	+	14085.5i	1.17766	+	1.17766i	0.197318	+	0.980340i	$0.436777\pi$
$524$	−6264.73	+	9627.75i	−0.522283	+	0.802653i
$525$		0			0
$526$	−4095.96	+	5053.41i	−0.339529	+	0.418896i
$527$	−815.838			−0.0674354
$528$		0			0
$529$	5140.95			0.422532
$530$	105.318	−	129.937i	0.00863156	−	0.0106492i
$531$		0			0
$532$	−31752.8	−	20661.4i	−2.58770	−	1.68380i
$533$	12602.6	+	12602.6i	1.02416	+	1.02416i
$534$		0			0
$535$	−2641.29			−0.213445
$536$	−1085.37	−	3355.93i	−0.0874645	−	0.270437i
$537$		0			0
$538$	−13330.3	+	1394.97i	−1.06824	+	0.111787i
$539$	−35120.3	+	35120.3i	−2.80657	+	2.80657i
$540$		0			0
$541$	3398.59	+	3398.59i	0.270087	+	0.270087i	0.829135	−	0.559048i	$-0.188833\pi$
$541$	3398.59	+	3398.59i	0.270087	+	0.270087i	−0.559048	+	0.829135i	$0.688833\pi$
$542$	−6729.72	+	8302.82i	−0.533332	+	0.658001i
$543$		0			0
$544$	833.395	−	3082.35i	0.0656829	−	0.242931i
$545$			1499.40i			0.117848i
$546$		0			0
$547$	15912.7	−	15912.7i	1.24383	−	1.24383i	0.285438	−	0.958397i	$-0.407861\pi$
$547$	15912.7	−	15912.7i	1.24383	−	1.24383i	0.958397	−	0.285438i	$-0.0921389\pi$
$548$	3106.15	+	14678.6i	0.242131	+	1.14423i
$549$		0			0
$550$	1815.68	+	17350.6i	0.140765	+	1.34515i
$551$			17812.0i			1.37716i
$552$		0			0
$553$		−	5449.45i		−	0.419049i
$554$	−643.757	+	67.3669i	−0.0493693	+	0.00516633i
$555$		0			0
$556$	6472.04	−	9946.35i	0.493662	−	0.758668i
$557$	−12309.2	+	12309.2i	−0.936371	+	0.936371i	−0.998093	−	0.0617225i	$-0.980341\pi$
$557$	−12309.2	+	12309.2i	−0.936371	+	0.936371i	0.0617225	+	0.998093i	$0.480341\pi$
$558$		0			0
$559$		−	4745.77i		−	0.359078i
$560$	−10322.0	−	3983.72i	−0.778903	−	0.300612i
$561$		0			0
$562$	−9359.65	−	7586.32i	−0.702515	−	0.569412i
$563$	13010.7	+	13010.7i	0.973955	+	0.973955i	0.999669	−	0.0257144i	$-0.00818604\pi$
$563$	13010.7	+	13010.7i	0.973955	+	0.973955i	−0.0257144	+	0.999669i	$0.508186\pi$
$564$		0			0
$565$	4997.64	−	4997.64i	0.372128	−	0.372128i
$566$	−320.501	−	3062.70i	−0.0238015	−	0.227447i
$567$		0			0
$568$	−21564.5	−	11024.6i	−1.59301	−	0.814404i
$569$	20663.0			1.52238			0.761192	−	0.648527i	$-0.224615\pi$
$569$	20663.0			1.52238			0.761192	+	0.648527i	$0.224615\pi$
$570$		0			0
$571$	6847.53	+	6847.53i	0.501857	+	0.501857i	0.912015	−	0.410158i	$-0.134526\pi$
$571$	6847.53	+	6847.53i	0.501857	+	0.501857i	−0.410158	+	0.912015i	$0.634526\pi$
$572$	−6161.64	−	29117.8i	−0.450404	−	2.12846i
$573$		0			0
$574$	−22182.6	−	17979.8i	−1.61304	−	1.30742i
$575$	8276.60			0.600275
$576$		0			0
$577$	−12587.4			−0.908178			−0.454089	−	0.890956i	$-0.650035\pi$
$577$	−12587.4			−0.908178			−0.454089	+	0.890956i	$0.650035\pi$
$578$	10111.6	+	8195.83i	0.727663	+	0.589795i
$579$		0			0
$580$	1076.98	+	5089.46i	0.0771022	+	0.364359i
$581$	2173.33	+	2173.33i	0.155189	+	0.155189i
$582$		0			0
$583$	−720.841			−0.0512079
$584$	−20466.4	−	10463.2i	−1.45018	−	0.741385i
$585$		0			0
$586$	1821.74	+	17408.5i	0.128422	+	1.22720i
$587$	10285.9	−	10285.9i	0.723241	−	0.723241i	−0.246023	−	0.969264i	$-0.579124\pi$
$587$	10285.9	−	10285.9i	0.723241	−	0.723241i	0.969264	+	0.246023i	$0.0791238\pi$
$588$		0			0
$589$	4590.62	+	4590.62i	0.321143	+	0.321143i
$590$	3077.80	+	2494.66i	0.214764	+	0.174074i
$591$		0			0
$592$	243.741	−	631.546i	0.0169218	−	0.0438452i
$593$			19189.3i			1.32886i	0.747353	+	0.664428i	$0.231325\pi$
$593$			19189.3i			1.32886i	−0.747353	+	0.664428i	$0.768675\pi$
$594$		0			0
$595$	−2156.25	+	2156.25i	−0.148567	+	0.148567i
$596$	11991.8	−	18429.1i	0.824163	−	1.26659i
$597$		0			0
$598$	−14043.7	+	1469.62i	−0.960348	+	0.100497i
$599$		−	25709.1i		−	1.75366i	−0.480796	−	0.876832i	$-0.659652\pi$
$599$		−	25709.1i		−	1.75366i	0.480796	−	0.876832i	$-0.340348\pi$
$600$		0			0
$601$		−	6062.56i		−	0.411476i	−0.978607	−	0.205738i	$-0.934041\pi$
$601$		−	6062.56i		−	0.411476i	0.978607	−	0.205738i	$-0.0659595\pi$
$602$	791.339	+	7562.02i	0.0535757	+	0.511968i
$603$		0			0
$604$	−2965.40	−	14013.5i	−0.199769	−	0.944040i
$605$	−9315.52	+	9315.52i	−0.625999	+	0.625999i
$606$		0			0
$607$			21062.9i			1.40843i	0.709989	+	0.704213i	$0.248700\pi$
$607$			21062.9i			1.40843i	−0.709989	+	0.704213i	$0.751300\pi$
$608$	−22033.4	+	12654.6i	−1.46969	+	0.844098i
$609$		0			0
$610$	1639.89	−	2023.22i	0.108848	−	0.134291i
$611$	−11801.9	−	11801.9i	−0.781429	−	0.781429i
$612$		0			0
$613$	5955.84	−	5955.84i	0.392421	−	0.392421i	−0.483128	−	0.875550i	$-0.660500\pi$
$613$	5955.84	−	5955.84i	0.392421	−	0.392421i	0.875550	+	0.483128i	$0.160500\pi$
$614$	−7530.43	+	788.034i	−0.494957	+	0.0517956i
$615$		0			0
$616$	14673.4	+	45369.6i	0.959752	+	2.96752i
$617$	−28289.5			−1.84586			−0.922929	−	0.384970i	$-0.874212\pi$
$617$	−28289.5			−1.84586			−0.922929	+	0.384970i	$0.874212\pi$
$618$		0			0
$619$	9943.57	+	9943.57i	0.645664	+	0.645664i	0.951942	−	0.306278i	$-0.0990838\pi$
$619$	9943.57	+	9943.57i	0.645664	+	0.645664i	−0.306278	+	0.951942i	$0.599084\pi$
$620$	1589.25	+	1034.12i	0.102945	+	0.0669859i
$621$		0			0
$622$	8850.35	−	10919.2i	0.570525	−	0.703888i
$623$	5988.22			0.385093
$624$		0			0
$625$	−6467.35			−0.413910
$626$	768.171	−	947.735i	0.0490452	−	0.0605098i
$627$		0			0
$628$	5381.66	−	8270.62i	0.341961	−	0.525532i
$629$	−131.928	−	131.928i	−0.00836301	−	0.00836301i
$630$		0			0
$631$	9728.67			0.613775			0.306888	−	0.951746i	$-0.400712\pi$
$631$	9728.67			0.613775			0.306888	+	0.951746i	$0.400712\pi$
$632$	−3254.41	−	1663.77i	−0.204831	−	0.104717i
$633$		0			0
$634$	−16514.4	+	1728.17i	−1.03450	+	0.108256i
$635$	4568.64	−	4568.64i	0.285513	−	0.285513i
$636$		0			0
$637$	33486.2	+	33486.2i	2.08285	+	2.08285i
$638$	14117.2	−	17417.2i	0.876028	−	1.08080i
$639$		0			0
$640$	−5530.50	+	4948.04i	−0.341581	+	0.305607i
$641$		−	445.753i		−	0.0274667i	−0.999906	−	0.0137334i	$-0.995628\pi$
$641$		−	445.753i		−	0.0274667i	0.999906	−	0.0137334i	$-0.00437160\pi$
$642$		0			0
$643$	879.050	−	879.050i	0.0539134	−	0.0539134i	−0.679636	−	0.733549i	$-0.737862\pi$
$643$	879.050	−	879.050i	0.0539134	−	0.0539134i	0.733549	+	0.679636i	$0.237862\pi$
$644$	22132.4	−	4683.45i	1.35425	−	0.286574i
$645$		0			0
$646$	728.858	+	6964.95i	0.0443909	+	0.424198i
$647$			4067.41i			0.247150i	0.992335	+	0.123575i	$0.0394360\pi$
$647$			4067.41i			0.247150i	−0.992335	+	0.123575i	$0.960564\pi$
$648$		0			0
$649$		−	17074.5i		−	1.03272i
$650$	16543.3	−	1731.20i	0.998278	−	0.104466i
$651$		0			0
$652$	5136.53	+	3342.32i	0.308531	+	0.200760i
$653$	595.450	−	595.450i	0.0356842	−	0.0356842i	−0.689040	−	0.724724i	$-0.741967\pi$
$653$	595.450	−	595.450i	0.0356842	−	0.0356842i	0.724724	+	0.689040i	$0.241967\pi$
$654$		0			0
$655$		−	7357.66i		−	0.438912i
$656$	−17510.1	+	7758.04i	−1.04216	+	0.461739i
$657$		0			0
$658$	20773.3	+	16837.5i	1.23074	+	0.997557i
$659$	−9075.08	−	9075.08i	−0.536441	−	0.536441i	0.386041	−	0.922482i	$-0.373842\pi$
$659$	−9075.08	−	9075.08i	−0.536441	−	0.536441i	−0.922482	+	0.386041i	$0.873842\pi$
$660$		0			0
$661$	1351.64	−	1351.64i	0.0795353	−	0.0795353i	−0.666220	−	0.745755i	$-0.732089\pi$
$661$	1351.64	−	1351.64i	0.0795353	−	0.0795353i	0.745755	+	0.666220i	$0.232089\pi$
$662$	1394.78	+	13328.4i	0.0818875	+	0.782515i
$663$		0			0
$664$	1961.46	−	634.372i	0.114637	−	0.0370759i
$665$	24265.9			1.41502
$666$		0			0
$667$	−7521.31	−	7521.31i	−0.436621	−	0.436621i
$668$	2122.81	−	449.210i	0.122955	−	0.0260186i
$669$		0			0
$670$	1755.13	+	1422.59i	0.101204	+	0.0820293i
$671$	−11224.1			−0.645753
$672$		0			0
$673$	−10501.3			−0.601482			−0.300741	−	0.953706i	$-0.597234\pi$
$673$	−10501.3			−0.601482			−0.300741	+	0.953706i	$0.597234\pi$
$674$	−21756.5	−	17634.3i	−1.24336	−	1.00779i
$675$		0			0
$676$	−10567.8	+	2236.26i	−0.601265	+	0.127234i
$677$	−10590.4	−	10590.4i	−0.601215	−	0.601215i	0.339420	−	0.940635i	$-0.389769\pi$
$677$	−10590.4	−	10590.4i	−0.601215	−	0.601215i	−0.940635	+	0.339420i	$0.889769\pi$
$678$		0			0
$679$	−18872.4			−1.06665
$680$	629.385	+	1946.04i	0.0354939	+	0.109746i
$681$		0			0
$682$	−850.488	−	8127.24i	−0.0477520	−	0.456317i
$683$	−2888.12	+	2888.12i	−0.161802	+	0.161802i	−0.783365	−	0.621563i	$-0.786498\pi$
$683$	−2888.12	+	2888.12i	−0.161802	+	0.161802i	0.621563	+	0.783365i	$0.286498\pi$
$684$		0			0
$685$	−6795.68	−	6795.68i	−0.379050	−	0.379050i
$686$	−33515.4	−	27165.4i	−1.86534	−	1.51192i
$687$		0			0
$688$	4757.64	+	1836.18i	0.263638	+	0.101749i
$689$			687.302i			0.0380031i
$690$		0			0
$691$	12657.6	−	12657.6i	0.696844	−	0.696844i	−0.266885	−	0.963728i	$-0.585994\pi$
$691$	12657.6	−	12657.6i	0.696844	−	0.696844i	0.963728	+	0.266885i	$0.0859943\pi$
$692$	13244.2	+	8617.92i	0.727555	+	0.473416i
$693$		0			0
$694$	−4952.57	+	518.269i	−0.270889	+	0.0283476i
$695$			7601.14i			0.414860i
$696$		0			0
$697$			5278.46i			0.286852i
$698$	35.0927	+	335.344i	0.00190298	+	0.0181848i
$699$		0			0
$700$	−26071.7	+	5517.05i	−1.40774	+	0.297893i
$701$	18873.8	−	18873.8i	1.01691	−	1.01691i	0.0170555	−	0.999855i	$-0.494571\pi$
$701$	18873.8	−	18873.8i	1.01691	−	1.01691i	0.999855	−	0.0170555i	$-0.00542920\pi$
$702$		0			0
$703$			1484.69i			0.0796531i
$704$	31574.6	+	5088.87i	1.69036	+	0.272435i
$705$		0			0
$706$	653.275	−	805.981i	0.0348248	−	0.0429653i
$707$	28696.2	+	28696.2i	1.52650	+	1.52650i
$708$		0			0
$709$	−21572.6	+	21572.6i	−1.14270	+	1.14270i	−0.154747	+	0.987954i	$0.549456\pi$
$709$	−21572.6	+	21572.6i	−1.14270	+	1.14270i	−0.987954	+	0.154747i	$0.950544\pi$
$710$	15429.3	−	1614.62i	0.815565	−	0.0853461i
$711$		0			0
$712$	1828.27	−	3576.16i	0.0962320	−	0.188234i
$713$	−3876.88			−0.203633
$714$		0			0
$715$	13480.5	+	13480.5i	0.705096	+	0.705096i
$716$	−4165.84	+	6402.13i	−0.217437	+	0.334160i
$717$		0			0
$718$	12184.2	−	15032.4i	0.633304	−	0.781341i
$719$	29101.6			1.50947			0.754733	−	0.656032i	$-0.227766\pi$
$719$	29101.6			1.50947			0.754733	+	0.656032i	$0.227766\pi$
$720$		0			0
$721$	25562.1			1.32036
$722$	22874.0	−	28220.9i	1.17906	−	1.45467i
$723$		0			0
$724$	12392.3	+	8063.64i	0.636130	+	0.413927i
$725$	8860.02	+	8860.02i	0.453866	+	0.453866i
$726$		0			0
$727$	23683.7			1.20822			0.604112	−	0.796899i	$-0.293528\pi$
$727$	23683.7			1.20822			0.604112	+	0.796899i	$0.293528\pi$
$728$	43258.6	−	13990.7i	2.20230	−	0.712264i
$729$		0			0
$730$	14643.6	−	1532.40i	0.742441	−	0.0776940i
$731$	993.859	−	993.859i	0.0502862	−	0.0502862i
$732$		0			0
$733$	−1650.17	−	1650.17i	−0.0831521	−	0.0831521i	0.664307	−	0.747460i	$-0.268727\pi$
$733$	−1650.17	−	1650.17i	−0.0831521	−	0.0831521i	−0.747460	+	0.664307i	$0.768727\pi$
$734$	−3412.92	+	4210.71i	−0.171626	+	0.211744i
$735$		0			0
$736$	3960.31	−	14647.4i	0.198341	−	0.733573i
$737$		−	9736.83i		−	0.486650i
$738$		0			0
$739$	−9966.72	+	9966.72i	−0.496119	+	0.496119i	−0.910227	−	0.414109i	$-0.864093\pi$
$739$	−9966.72	+	9966.72i	−0.496119	+	0.496119i	0.414109	+	0.910227i	$0.364093\pi$
$740$	89.7701	+	424.223i	0.00445948	+	0.0210740i
$741$		0			0
$742$	−114.605	−	1095.16i	−0.00567019	−	0.0541842i
$743$		−	13180.6i		−	0.650806i	−0.945576	−	0.325403i	$-0.894500\pi$
$743$		−	13180.6i		−	0.650806i	0.945576	−	0.325403i	$-0.105500\pi$
$744$		0			0
$745$			14083.8i			0.692605i
$746$	−28479.1	+	2980.24i	−1.39771	+	0.146266i
$747$		0			0
$748$	4807.49	−	7388.23i	0.234999	−	0.361150i
$749$	−12295.8	+	12295.8i	−0.599837	+	0.599837i
$750$		0			0
$751$		−	18926.8i		−	0.919641i	−0.888012	−	0.459821i	$-0.847914\pi$
$751$		−	18926.8i		−	0.919641i	0.888012	−	0.459821i	$-0.152086\pi$
$752$	16397.6	−	7265.15i	0.795159	−	0.352304i
$753$		0			0
$754$	−16606.8	−	13460.4i	−0.802101	−	0.650130i
$755$	6487.74	+	6487.74i	0.312733	+	0.312733i
$756$		0			0
$757$	23156.4	−	23156.4i	1.11180	−	1.11180i	0.118896	−	0.992907i	$-0.462065\pi$
$757$	23156.4	−	23156.4i	1.11180	−	1.11180i	0.992907	−	0.118896i	$-0.0379354\pi$
$758$	2126.53	+	20321.0i	0.101898	+	0.973738i
$759$		0			0
$760$	7408.63	−	14491.6i	0.353605	−	0.691664i
$761$	−6699.86			−0.319146			−0.159573	−	0.987186i	$-0.551012\pi$
$761$	−6699.86			−0.319146			−0.159573	+	0.987186i	$0.551012\pi$
$762$		0			0
$763$	6980.02	+	6980.02i	0.331184	+	0.331184i
$764$	3217.51	+	15204.9i	0.152363	+	0.720017i
$765$		0			0
$766$	22706.8	+	18404.6i	1.07106	+	0.868128i
$767$	−16280.1			−0.766413
$768$		0			0
$769$	−13009.6			−0.610064			−0.305032	−	0.952342i	$-0.598667\pi$
$769$	−13009.6			−0.610064			−0.305032	+	0.952342i	$0.598667\pi$
$770$	−23728.0	−	19232.4i	−1.11052	−	0.900112i
$771$		0			0
$772$	6866.32	+	32447.9i	0.320109	+	1.51273i
$773$	−991.579	−	991.579i	−0.0461379	−	0.0461379i	0.683661	−	0.729799i	$-0.260387\pi$
$773$	−991.579	−	991.579i	−0.0461379	−	0.0461379i	−0.729799	+	0.683661i	$0.760387\pi$
$774$		0			0
$775$	4566.92			0.211675
$776$	−5761.94	+	11270.6i	−0.266549	+	0.521379i
$777$		0			0
$778$	−1714.90	−	16387.5i	−0.0790258	−	0.755168i
$779$	29701.2	−	29701.2i	1.36605	−	1.36605i
$780$		0			0
$781$	−47276.7	−	47276.7i	−2.16606	−	2.16606i
$782$	−3248.79	−	2633.25i	−0.148563	−	0.120416i
$783$		0			0
$784$	−46526.0	+	20613.9i	−2.11944	+	0.939043i
$785$			6320.53i			0.287375i
$786$		0			0
$787$	−19796.5	+	19796.5i	−0.896659	+	0.896659i	−0.995139	−	0.0984801i	$-0.968602\pi$
$787$	−19796.5	+	19796.5i	−0.896659	+	0.896659i	0.0984801	+	0.995139i	$0.468602\pi$
$788$	−11889.5	+	18272.0i	−0.537494	+	0.826031i
$789$		0			0
$790$	2328.51	−	243.670i	0.104867	−	0.0109739i
$791$		−	46530.2i		−	2.09156i
$792$		0			0
$793$			10701.8i			0.479235i
$794$	2507.92	+	23965.6i	0.112094	+	1.07117i
$795$		0			0
$796$	6301.04	+	29776.6i	0.280571	+	1.32588i
$797$	−13170.1	+	13170.1i	−0.585331	+	0.585331i	−0.936363	−	0.351032i	$-0.885831\pi$
$797$	−13170.1	+	13170.1i	−0.585331	+	0.585331i	0.351032	+	0.936363i	$0.385831\pi$
$798$		0			0
$799$		−	4943.10i		−	0.218866i
$800$	−4665.19	+	17254.4i	−0.206174	+	0.762546i
$801$		0			0
$802$	21102.2	−	26035.0i	0.929109	−	1.14629i
$803$	−44869.1	−	44869.1i	−1.97185	−	1.97185i
$804$		0			0
$805$	−10246.5	+	10246.5i	−0.448624	+	0.448624i
$806$	−7749.10	+	810.917i	−0.338648	+	0.0354384i
$807$		0			0
$808$	25898.6	−	8376.11i	1.12761	−	0.364691i
$809$	−19658.9			−0.854352			−0.427176	−	0.904169i	$-0.640491\pi$
$809$	−19658.9			−0.854352			−0.427176	+	0.904169i	$0.640491\pi$
$810$		0			0
$811$	−26123.4	−	26123.4i	−1.13109	−	1.13109i	−0.989996	−	0.141099i	$-0.954936\pi$
$811$	−26123.4	−	26123.4i	−1.13109	−	1.13109i	−0.141099	−	0.989996i	$-0.545064\pi$
$812$	28706.1	+	18678.9i	1.24062	+	0.807268i
$813$		0			0
$814$	1176.72	−	1451.78i	0.0506682	−	0.0625121i
$815$	−3925.41			−0.168713
$816$		0			0
$817$	−11184.6			−0.478949
$818$	−22604.7	+	27888.7i	−0.966205	+	1.19206i
$819$		0			0
$820$	6690.73	−	10282.4i	0.284940	−	0.437900i
$821$	8922.97	+	8922.97i	0.379310	+	0.379310i	0.870853	−	0.491543i	$-0.163567\pi$
$821$	8922.97	+	8922.97i	0.379310	+	0.379310i	−0.491543	+	0.870853i	$0.663567\pi$
$822$		0			0
$823$	19943.0			0.844677			0.422339	−	0.906438i	$-0.361209\pi$
$823$	19943.0			0.844677			0.422339	+	0.906438i	$0.361209\pi$
$824$	7804.38	−	15265.7i	0.329950	−	0.645394i
$825$		0			0
$826$	25941.0	−	2714.64i	1.09274	−	0.114351i
$827$	18332.6	−	18332.6i	0.770843	−	0.770843i	−0.207411	−	0.978254i	$-0.566504\pi$
$827$	18332.6	−	18332.6i	0.770843	−	0.770843i	0.978254	+	0.207411i	$0.0665037\pi$
$828$		0			0
$829$	−26778.5	−	26778.5i	−1.12190	−	1.12190i	−0.991455	−	0.130446i	$-0.958359\pi$
$829$	−26778.5	−	26778.5i	−1.12190	−	1.12190i	−0.130446	−	0.991455i	$-0.541641\pi$
$830$	−831.469	+	1025.83i	−0.0347720	+	0.0429001i
$831$		0			0
$832$	4852.10	−	30105.5i	0.202183	−	1.25447i
$833$			14025.4i			0.583374i
$834$		0			0
$835$	−982.788	+	982.788i	−0.0407315	+	0.0407315i
$836$	−68623.8	+	14521.5i	−2.83900	+	0.600762i
$837$		0			0
$838$	−4152.71	−	39683.2i	−0.171185	−	1.63584i
$839$			28064.1i			1.15480i	0.816461	+	0.577401i	$0.195933\pi$
$839$			28064.1i			1.15480i	−0.816461	+	0.577401i	$0.804067\pi$
$840$		0			0
$841$			8286.04i			0.339745i
$842$	27766.9	−	2905.72i	1.13647	−	0.118928i
$843$		0			0
$844$	21566.5	+	14033.2i	0.879561	+	0.572326i
$845$	4892.53	−	4892.53i	0.199181	−	0.199181i
$846$		0			0
$847$			86731.5i			3.51845i
$848$	−689.020	−	265.923i	−0.0279022	−	0.0107687i
$849$		0			0
$850$	3827.04	+	3101.94i	0.154431	+	0.125171i
$851$	−626.926	−	626.926i	−0.0252535	−	0.0252535i
$852$		0			0
$853$	−25918.5	+	25918.5i	−1.04037	+	1.04037i	−0.0412172	+	0.999150i	$0.513124\pi$
$853$	−25918.5	+	25918.5i	−1.04037	+	1.04037i	−0.999150	+	0.0412172i	$0.986876\pi$
$854$	−1784.49	−	17052.5i	−0.0715035	−	0.683285i
$855$		0			0
$856$	3589.00	+	11097.1i	0.143305	+	0.443095i
$857$	−23507.5			−0.936992			−0.468496	−	0.883466i	$-0.655204\pi$
$857$	−23507.5			−0.936992			−0.468496	+	0.883466i	$0.655204\pi$
$858$		0			0
$859$	−11981.1	−	11981.1i	−0.475890	−	0.475890i	0.427925	−	0.903814i	$-0.359245\pi$
$859$	−11981.1	−	11981.1i	−0.475890	−	0.475890i	−0.903814	+	0.427925i	$0.859245\pi$
$860$	−3195.81	+	676.267i	−0.126716	+	0.0268145i
$861$		0			0
$862$	14137.0	+	11458.5i	0.558593	+	0.452758i
$863$	−8191.53			−0.323109			−0.161554	−	0.986864i	$-0.551651\pi$
$863$	−8191.53			−0.323109			−0.161554	+	0.986864i	$0.551651\pi$
$864$		0			0
$865$	−10121.4			−0.397846
$866$	5002.81	+	4054.95i	0.196308	+	0.159114i
$867$		0			0
$868$	12212.4	−	2584.26i	0.477551	−	0.101055i
$869$	−7134.75	−	7134.75i	−0.278515	−	0.278515i
$870$		0			0
$871$	−9283.80			−0.361159
$872$	6299.54	−	2037.39i	0.244644	−	0.0791224i
$873$		0			0
$874$	3463.54	+	33097.5i	0.134046	+	1.28094i
$875$	27350.6	−	27350.6i	1.05671	−	1.05671i
$876$		0			0
$877$	−7896.06	−	7896.06i	−0.304026	−	0.304026i	0.538561	−	0.842587i	$-0.318968\pi$
$877$	−7896.06	−	7896.06i	−0.304026	−	0.304026i	−0.842587	+	0.538561i	$0.818968\pi$
$878$	19500.8	+	15806.0i	0.749566	+	0.607549i
$879$		0			0
$880$	−18730.0	+	8298.51i	−0.717485	+	0.317890i
$881$			43097.6i			1.64812i	0.566501	+	0.824061i	$0.308297\pi$
$881$			43097.6i			1.64812i	−0.566501	+	0.824061i	$0.691703\pi$
$882$		0			0
$883$	−21106.8	+	21106.8i	−0.804418	+	0.804418i	−0.983783	−	0.179364i	$-0.942596\pi$
$883$	−21106.8	+	21106.8i	−0.804418	+	0.804418i	0.179364	+	0.983783i	$0.442596\pi$
$884$	−7044.47	−	4583.80i	−0.268022	−	0.174400i
$885$		0			0
$886$	−3187.27	+	333.537i	−0.120856	+	0.0126472i
$887$			5713.71i			0.216288i	0.994135	+	0.108144i	$0.0344908\pi$
$887$			5713.71i			0.216288i	−0.994135	+	0.108144i	$0.965509\pi$
$888$		0			0
$889$		−	42536.0i		−	1.60474i
$890$	267.761	+	2558.72i	0.0100847	+	0.0963691i
$891$		0			0
$892$	−18553.0	+	3926.02i	−0.696415	+	0.147369i
$893$	−27814.2	+	27814.2i	−1.04229	+	1.04229i
$894$		0			0
$895$		−	4892.59i		−	0.182728i
$896$	−2711.46	+	48779.9i	−0.101098	+	1.81877i
$897$		0			0
$898$	−808.244	+	997.175i	−0.0300350	+	0.0370558i
$899$	−4150.15	−	4150.15i	−0.153966	−	0.153966i
$900$		0			0
$901$	−143.935	+	143.935i	−0.00532204	+	0.00532204i
$902$	−52583.1	+	5502.64i	−1.94105	+	0.203124i
$903$		0			0
$904$	−27787.8	−	14206.2i	−1.02235	−	0.522666i
$905$	−9470.40			−0.347853
$906$		0			0
$907$	−2440.45	−	2440.45i	−0.0893426	−	0.0893426i	0.661023	−	0.750366i	$-0.270123\pi$
$907$	−2440.45	−	2440.45i	−0.0893426	−	0.0893426i	−0.750366	+	0.661023i	$0.770123\pi$
$908$	21562.1	−	33137.0i	0.788066	−	1.21111i
$909$		0			0
$910$	−18337.5	+	22624.0i	−0.668003	+	0.824152i
$911$	41215.3			1.49893			0.749463	−	0.662046i	$-0.230312\pi$
$911$	41215.3			1.49893			0.749463	+	0.662046i	$0.230312\pi$
$912$		0			0
$913$	5690.92			0.206289
$914$	−31043.3	+	38299.8i	−1.12344	+	1.38605i
$915$		0			0
$916$	−21347.5	−	13890.7i	−0.770024	−	0.501051i
$917$	−34251.5	−	34251.5i	−1.23346	−	1.23346i
$918$		0			0
$919$	4600.68			0.165139			0.0825694	−	0.996585i	$-0.473687\pi$
$919$	4600.68			0.165139			0.0825694	+	0.996585i	$0.473687\pi$
$920$	2990.85	+	9247.60i	0.107180	+	0.331396i
$921$		0			0
$922$	−11187.9	+	1170.77i	−0.399623	+	0.0418192i
$923$	−45077.0	+	45077.0i	−1.60751	+	1.60751i
$924$		0			0
$925$	738.512	+	738.512i	0.0262509	+	0.0262509i
$926$	15281.5	−	18853.6i	0.542312	−	0.669080i
$927$		0			0
$928$	19919.3	−	11440.4i	0.704616	−	0.404687i
$929$			22590.7i			0.797821i	0.916990	+	0.398911i	$0.130612\pi$
$929$			22590.7i			0.797821i	−0.916990	+	0.398911i	$0.869388\pi$
$930$		0			0
$931$	78919.1	−	78919.1i	2.77816	−	2.77816i
$932$	−5142.30	−	24300.8i	−0.180731	−	0.854076i
$933$		0			0
$934$	−1677.16	−	16026.9i	−0.0587563	−	0.561474i
$935$			5646.19i			0.197487i
$936$		0			0
$937$		−	37208.7i		−	1.29728i	−0.761094	−	0.648642i	$-0.775337\pi$
$937$		−	37208.7i		−	1.29728i	0.761094	−	0.648642i	$-0.224663\pi$
$938$	14793.0	−	1548.04i	0.514935	−	0.0538862i
$939$		0			0
$940$	−6265.65	+	9629.15i	−0.217407	+	0.334115i
$941$	29322.0	−	29322.0i	1.01580	−	1.01580i	0.0159300	−	0.999873i	$-0.494929\pi$
$941$	29322.0	−	29322.0i	1.01580	−	1.01580i	0.999873	−	0.0159300i	$-0.00507091\pi$
$942$		0			0
$943$			25083.3i			0.866198i
$944$	6298.88	−	16320.8i	0.217173	−	0.562707i
$945$		0			0
$946$	10936.7	+	8864.59i	0.375881	+	0.304664i
$947$	24343.1	+	24343.1i	0.835315	+	0.835315i	0.988238	−	0.152923i	$-0.0488688\pi$
$947$	24343.1	+	24343.1i	0.835315	+	0.835315i	−0.152923	+	0.988238i	$0.548869\pi$
$948$		0			0
$949$	−42781.4	+	42781.4i	−1.46338	+	1.46338i
$950$	−4080.02	−	38988.5i	−0.139340	−	1.33153i
$951$		0			0
$952$	11989.1	+	6129.29i	0.408162	+	0.208668i
$953$	12372.3			0.420542			0.210271	−	0.977643i	$-0.432565\pi$
$953$	12372.3			0.420542			0.210271	+	0.977643i	$0.432565\pi$
$954$		0			0
$955$	−7039.32	−	7039.32i	−0.238520	−	0.238520i
$956$	9278.53	+	43847.2i	0.313901	+	1.48339i
$957$		0			0
$958$	−15542.9	−	12598.0i	−0.524184	−	0.424869i
$959$	−63270.7			−2.13046
$960$		0			0
$961$	27651.8			0.928193
$962$	−1384.23	−	1121.97i	−0.0463923	−	0.0376026i
$963$		0			0
$964$	−460.857	−	2177.86i	−0.0153975	−	0.0727635i
$965$	−15022.2	−	15022.2i	−0.501122	−	0.501122i
$966$		0			0
$967$	−18615.7			−0.619069			−0.309534	−	0.950888i	$-0.600173\pi$
$967$	−18615.7			−0.619069			−0.309534	+	0.950888i	$0.600173\pi$
$968$	51796.0	+	26480.0i	1.71982	+	0.879236i
$969$		0			0
$970$	−843.874	−	8064.03i	−0.0279331	−	0.266928i
$971$	23920.0	−	23920.0i	0.790555	−	0.790555i	−0.191029	−	0.981584i	$-0.561183\pi$
$971$	23920.0	−	23920.0i	0.790555	−	0.790555i	0.981584	+	0.191029i	$0.0611825\pi$
$972$		0			0
$973$	35384.9	+	35384.9i	1.16587	+	1.16587i
$974$	−35303.1	−	28614.4i	−1.16138	−	0.941339i
$975$		0			0
$976$	−10728.6	−	4140.62i	−0.351858	−	0.135797i
$977$		−	3892.45i		−	0.127462i	−0.997967	−	0.0637311i	$-0.979700\pi$
$977$		−	3892.45i		−	0.127462i	0.997967	−	0.0637311i	$-0.0203000\pi$
$978$		0			0
$979$	7840.14	−	7840.14i	0.255947	−	0.255947i
$980$	17777.9	−	27321.4i	0.579485	−	0.890562i
$981$		0			0
$982$	−19682.2	+	2059.67i	−0.639596	+	0.0669316i
$983$		−	1771.35i		−	0.0574742i	−0.999587	−	0.0287371i	$-0.990851\pi$
$983$		−	1771.35i		−	0.0574742i	0.999587	−	0.0287371i	$-0.00914857\pi$
$984$		0			0
$985$		−	13963.7i		−	0.451696i
$986$	−658.925	−	6296.66i	−0.0212824	−	0.203374i
$987$		0			0
$988$	13845.9	+	65430.9i	0.445846	+	2.10692i
$989$	4722.84	−	4722.84i	0.151848	−	0.151848i
$990$		0			0
$991$			2828.01i			0.0906504i	0.998972	+	0.0453252i	$0.0144324\pi$
$991$			2828.01i			0.0906504i	−0.998972	+	0.0453252i	$0.985568\pi$
$992$	2185.24	−	8082.22i	0.0699410	−	0.258680i
$993$		0			0
$994$	64310.2	−	79343.1i	2.05211	−	2.53180i
$995$	−13785.5	−	13785.5i	−0.439226	−	0.439226i
$996$		0			0
$997$	−14851.0	+	14851.0i	−0.471750	+	0.471750i	−0.902480	−	0.430731i	$-0.858256\pi$
$997$	−14851.0	+	14851.0i	−0.471750	+	0.471750i	0.430731	+	0.902480i	$0.358256\pi$
$998$	−40697.0	+	4258.80i	−1.29082	+	0.135080i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.4.l.a.35.20	yes	48
3.2	odd	2	inner	144.4.l.a.35.5	✓	48
4.3	odd	2		576.4.l.a.431.9		48
8.3	odd	2		1152.4.l.a.863.16		48
8.5	even	2		1152.4.l.b.863.16		48
12.11	even	2		576.4.l.a.431.16		48
16.3	odd	4		1152.4.l.b.287.9		48
16.5	even	4		576.4.l.a.143.16		48
16.11	odd	4	inner	144.4.l.a.107.5	yes	48
16.13	even	4		1152.4.l.a.287.9		48
24.5	odd	2		1152.4.l.b.863.9		48
24.11	even	2		1152.4.l.a.863.9		48
48.5	odd	4		576.4.l.a.143.9		48
48.11	even	4	inner	144.4.l.a.107.20	yes	48
48.29	odd	4		1152.4.l.a.287.16		48
48.35	even	4		1152.4.l.b.287.16		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.4.l.a.35.5	✓	48	3.2	odd	2	inner
144.4.l.a.35.20	yes	48	1.1	even	1	trivial
144.4.l.a.107.5	yes	48	16.11	odd	4	inner
144.4.l.a.107.20	yes	48	48.11	even	4	inner
576.4.l.a.143.9		48	48.5	odd	4
576.4.l.a.143.16		48	16.5	even	4
576.4.l.a.431.9		48	4.3	odd	2
576.4.l.a.431.16		48	12.11	even	2
1152.4.l.a.287.9		48	16.13	even	4
1152.4.l.a.287.16		48	48.29	odd	4
1152.4.l.a.863.9		48	24.11	even	2
1152.4.l.a.863.16		48	8.3	odd	2
1152.4.l.b.287.9		48	16.3	odd	4
1152.4.l.b.287.16		48	48.35	even	4
1152.4.l.b.863.9		48	24.5	odd	2
1152.4.l.b.863.16		48	8.5	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+(2.19729 + 1.78098i) q^{2} +(1.65621 + 7.82668i) q^{4} +(-3.62348 - 3.62348i) q^{5} -33.7361 q^{7} +(-10.3000 + 20.1472i) q^{8} +(-1.50850 - 14.4152i) q^{10} +(-44.1694 + 44.1694i) q^{11} +(42.1143 + 42.1143i) q^{13} +(-74.1283 - 60.0835i) q^{14} +(-58.5139 + 25.9252i) q^{16} +17.6392i q^{17} +(99.2533 - 99.2533i) q^{19} +(22.3586 - 34.3611i) q^{20} +(-175.718 + 18.3883i) q^{22} +83.8215i q^{23} -98.7408i q^{25} +(17.5327 + 167.542i) q^{26} +(-55.8741 - 264.042i) q^{28} +(-89.7300 + 89.7300i) q^{29} +46.2516i q^{31} +(-174.745 - 47.2469i) q^{32} +(-31.4150 + 38.7584i) q^{34} +(122.242 + 122.242i) q^{35} +(-7.47930 + 7.47930i) q^{37} +(394.857 - 41.3205i) q^{38} +(110.325 - 35.6811i) q^{40} +299.247 q^{41} +(-56.3439 - 56.3439i) q^{43} +(-418.854 - 272.546i) q^{44} +(-149.285 + 184.181i) q^{46} -280.235 q^{47} +795.127 q^{49} +(175.856 - 216.963i) q^{50} +(-259.865 + 399.365i) q^{52} +(8.15996 + 8.15996i) q^{53} +320.094 q^{55} +(347.482 - 679.689i) q^{56} +(-356.971 + 37.3558i) q^{58} +(-193.284 + 193.284i) q^{59} +(127.057 + 127.057i) q^{61} +(-82.3732 + 101.628i) q^{62} +(-299.820 - 415.033i) q^{64} -305.201i q^{65} +(-110.221 + 110.221i) q^{67} +(-138.056 + 29.2141i) q^{68} +(50.8910 + 486.313i) q^{70} +1070.35i q^{71} +1015.84i q^{73} +(-29.7547 + 3.11373i) q^{74} +(941.209 + 612.440i) q^{76} +(1490.11 - 1490.11i) q^{77} +161.531i q^{79} +(305.964 + 118.084i) q^{80} +(657.533 + 532.953i) q^{82} +(-64.4215 - 64.4215i) q^{83} +(63.9151 - 63.9151i) q^{85} +(-23.4567 - 224.152i) q^{86} +(-434.945 - 1344.84i) q^{88} -177.502 q^{89} +(-1420.77 - 1420.77i) q^{91} +(-656.045 + 138.826i) q^{92} +(-615.758 - 499.093i) q^{94} -719.285 q^{95} +559.412 q^{97} +(1747.13 + 1416.11i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 120 q^{10} - 144 q^{16} - 48 q^{19} + 72 q^{22} + 72 q^{28} - 984 q^{34} - 1272 q^{40} + 864 q^{43} - 1416 q^{46} + 2352 q^{49} - 648 q^{52} - 576 q^{55} + 1128 q^{58} + 1824 q^{61} + 3024 q^{64}+ \cdots - 11304 q^{94}+O(q^{100}) \) 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

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Representations

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Database

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