LMFDB - Embedded newform 144.4.l.a.107.9

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			107.9
Character	$\chi$	$=$	144.107
Dual form			144.4.l.a.35.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.64440 + 2.30129i) q^{2} +(-2.59187 - 7.56850i) q^{4} +(-7.28730 + 7.28730i) q^{5} +29.9574 q^{7} +(21.6794 + 6.48105i) q^{8} +(-4.78691 - 28.7535i) q^{10} +(-0.408325 - 0.408325i) q^{11} +(-26.6262 + 26.6262i) q^{13} +(-49.2621 + 68.9406i) q^{14} +(-50.5645 + 39.2331i) q^{16} +83.0075i q^{17} +(-51.6479 - 51.6479i) q^{19} +(74.0416 + 36.2662i) q^{20} +(1.61113 - 0.268222i) q^{22} +173.664i q^{23} +18.7906i q^{25} +(-17.4903 - 105.059i) q^{26} +(-77.6455 - 226.732i) q^{28} +(-167.311 - 167.311i) q^{29} +191.279i q^{31} +(-7.13826 - 180.879i) q^{32} +(-191.024 - 136.498i) q^{34} +(-218.308 + 218.308i) q^{35} +(185.381 + 185.381i) q^{37} +(203.787 - 33.9267i) q^{38} +(-205.214 + 110.755i) q^{40} -62.7651 q^{41} +(-193.397 + 193.397i) q^{43} +(-2.03209 + 4.14873i) q^{44} +(-399.650 - 285.573i) q^{46} +93.1177 q^{47} +554.445 q^{49} +(-43.2425 - 30.8993i) q^{50} +(270.532 + 132.509i) q^{52} +(249.010 - 249.010i) q^{53} +5.95117 q^{55} +(649.458 + 194.155i) q^{56} +(660.156 - 109.904i) q^{58} +(-24.6042 - 24.6042i) q^{59} +(451.553 - 451.553i) q^{61} +(-440.189 - 314.541i) q^{62} +(427.992 + 281.010i) q^{64} -388.066i q^{65} +(453.974 + 453.974i) q^{67} +(628.242 - 215.144i) q^{68} +(-143.403 - 861.378i) q^{70} +348.649i q^{71} +923.229i q^{73} +(-731.458 + 121.774i) q^{74} +(-257.033 + 524.762i) q^{76} +(-12.2323 - 12.2323i) q^{77} -989.875i q^{79} +(82.5751 - 654.382i) q^{80} +(103.211 - 144.441i) q^{82} +(-325.077 + 325.077i) q^{83} +(-604.900 - 604.900i) q^{85} +(-127.039 - 763.084i) q^{86} +(-6.20586 - 11.4986i) q^{88} -1005.33 q^{89} +(-797.651 + 797.651i) q^{91} +(1314.37 - 450.113i) q^{92} +(-153.123 + 214.291i) q^{94} +752.747 q^{95} +997.579 q^{97} +(-911.731 + 1275.94i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$.

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

Coefficient data

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

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

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

$2$	−1.64440	+	2.30129i	−0.581385	+	0.813629i
$2$	−1.64440	+	2.30129i	−0.581385	+	0.813629i
$3$		0			0
$3$		0			0
$4$	−2.59187	−	7.56850i	−0.323983	−	0.946063i
$5$	−7.28730	+	7.28730i	−0.651796	+	0.651796i	−0.953425	−	0.301629i	$-0.902469\pi$
$5$	−7.28730	+	7.28730i	−0.651796	+	0.651796i	0.301629	+	0.953425i	$0.402469\pi$
$6$		0			0
$7$	29.9574			1.61755			0.808773	−	0.588121i	$-0.200132\pi$
$7$	29.9574			1.61755			0.808773	+	0.588121i	$0.200132\pi$
$8$	21.6794	+	6.48105i	0.958103	+	0.286425i
$9$		0			0
$10$	−4.78691	−	28.7535i	−0.151376	−	0.909264i
$11$	−0.408325	−	0.408325i	−0.0111922	−	0.0111922i	0.701489	−	0.712681i	$-0.252519\pi$
$11$	−0.408325	−	0.408325i	−0.0111922	−	0.0111922i	−0.712681	+	0.701489i	$0.752519\pi$
$12$		0			0
$13$	−26.6262	+	26.6262i	−0.568060	+	0.568060i	−0.931584	−	0.363525i	$-0.881573\pi$
$13$	−26.6262	+	26.6262i	−0.568060	+	0.568060i	0.363525	+	0.931584i	$0.381573\pi$
$14$	−49.2621	+	68.9406i	−0.940417	+	1.31608i
$15$		0			0
$16$	−50.5645	+	39.2331i	−0.790070	+	0.613017i
$17$			83.0075i			1.18425i	0.805846	+	0.592126i	$0.201711\pi$
$17$			83.0075i			1.18425i	−0.805846	+	0.592126i	$0.798289\pi$
$18$		0			0
$19$	−51.6479	−	51.6479i	−0.623623	−	0.623623i	0.322833	−	0.946456i	$-0.395365\pi$
$19$	−51.6479	−	51.6479i	−0.623623	−	0.623623i	−0.946456	+	0.322833i	$0.895365\pi$
$20$	74.0416	+	36.2662i	0.827811	+	0.405469i
$21$		0			0
$22$	1.61113	−	0.268222i	0.0156133	−	0.00259933i
$23$			173.664i			1.57441i	0.616693	+	0.787204i	$0.288472\pi$
$23$			173.664i			1.57441i	−0.616693	+	0.787204i	$0.711528\pi$
$24$		0			0
$25$			18.7906i			0.150325i
$26$	−17.4903	−	105.059i	−0.131928	−	0.792451i
$27$		0			0
$28$	−77.6455	−	226.732i	−0.524058	−	1.53030i
$29$	−167.311	−	167.311i	−1.07134	−	1.07134i	−0.997252	−	0.0740856i	$-0.976396\pi$
$29$	−167.311	−	167.311i	−1.07134	−	1.07134i	−0.0740856	−	0.997252i	$-0.523604\pi$
$30$		0			0
$31$			191.279i			1.10822i	0.832444	+	0.554109i	$0.186941\pi$
$31$			191.279i			1.10822i	−0.832444	+	0.554109i	$0.813059\pi$
$32$	−7.13826	−	180.879i	−0.0394337	−	0.999222i
$33$		0			0
$34$	−191.024	−	136.498i	−0.963541	−	0.688506i
$35$	−218.308	+	218.308i	−1.05431	+	1.05431i
$36$		0			0
$37$	185.381	+	185.381i	0.823689	+	0.823689i	0.986635	−	0.162946i	$-0.0520997\pi$
$37$	185.381	+	185.381i	0.823689	+	0.823689i	−0.162946	+	0.986635i	$0.552100\pi$
$38$	203.787	−	33.9267i	0.869963	−	0.144833i
$39$		0			0
$40$	−205.214	+	110.755i	−0.811178	+	0.437797i
$41$	−62.7651			−0.239080			−0.119540	−	0.992829i	$-0.538142\pi$
$41$	−62.7651			−0.239080			−0.119540	+	0.992829i	$0.538142\pi$
$42$		0			0
$43$	−193.397	+	193.397i	−0.685877	+	0.685877i	−0.961318	−	0.275441i	$-0.911176\pi$
$43$	−193.397	+	193.397i	−0.685877	+	0.685877i	0.275441	+	0.961318i	$0.411176\pi$
$44$	−2.03209	+	4.14873i	−0.00696246	+	0.0142147i
$45$		0			0
$46$	−399.650	−	285.573i	−1.28098	−	0.915336i
$47$	93.1177			0.288992			0.144496	−	0.989505i	$-0.453844\pi$
$47$	93.1177			0.288992			0.144496	+	0.989505i	$0.453844\pi$
$48$		0			0
$49$	554.445			1.61646
$50$	−43.2425	−	30.8993i	−0.122308	−	0.0873964i
$51$		0			0
$52$	270.532	+	132.509i	0.721462	+	0.353378i
$53$	249.010	−	249.010i	0.645360	−	0.645360i	−0.306508	−	0.951868i	$-0.599161\pi$
$53$	249.010	−	249.010i	0.645360	−	0.645360i	0.951868	+	0.306508i	$0.0991606\pi$
$54$		0			0
$55$	5.95117			0.0145901
$56$	649.458	+	194.155i	1.54978	+	0.463305i
$57$		0			0
$58$	660.156	−	109.904i	1.49453	−	0.248812i
$59$	−24.6042	−	24.6042i	−0.0542913	−	0.0542913i	0.679440	−	0.733731i	$-0.262223\pi$
$59$	−24.6042	−	24.6042i	−0.0542913	−	0.0542913i	−0.733731	+	0.679440i	$0.762223\pi$
$60$		0			0
$61$	451.553	−	451.553i	0.947795	−	0.947795i	−0.0509083	−	0.998703i	$-0.516212\pi$
$61$	451.553	−	451.553i	0.947795	−	0.947795i	0.998703	+	0.0509083i	$0.0162116\pi$
$62$	−440.189	−	314.541i	−0.901678	−	0.644301i
$63$		0			0
$64$	427.992	+	281.010i	0.835922	+	0.548848i
$65$		−	388.066i		−	0.740518i
$66$		0			0
$67$	453.974	+	453.974i	0.827788	+	0.827788i	0.987210	−	0.159423i	$-0.0509632\pi$
$67$	453.974	+	453.974i	0.827788	+	0.827788i	−0.159423	+	0.987210i	$0.550963\pi$
$68$	628.242	−	215.144i	1.12038	−	0.383678i
$69$		0			0
$70$	−143.403	−	861.378i	−0.244857	−	1.47078i
$71$			348.649i			0.582774i	0.956605	+	0.291387i	$0.0941168\pi$
$71$			348.649i			0.582774i	−0.956605	+	0.291387i	$0.905883\pi$
$72$		0			0
$73$			923.229i			1.48022i	0.672488	+	0.740108i	$0.265226\pi$
$73$			923.229i			1.48022i	−0.672488	+	0.740108i	$0.734774\pi$
$74$	−731.458	+	121.774i	−1.14906	+	0.191297i
$75$		0			0
$76$	−257.033	+	524.762i	−0.387943	+	0.792030i
$77$	−12.2323	−	12.2323i	−0.0181040	−	0.0181040i
$78$		0			0
$79$		−	989.875i		−	1.40974i	−0.709335	−	0.704871i	$-0.751005\pi$
$79$		−	989.875i		−	1.40974i	0.709335	−	0.704871i	$-0.248995\pi$
$80$	82.5751	−	654.382i	0.115402	−	0.914526i
$81$		0			0
$82$	103.211	−	144.441i	0.138997	−	0.194522i
$83$	−325.077	+	325.077i	−0.429901	+	0.429901i	−0.888595	−	0.458693i	$-0.848318\pi$
$83$	−325.077	+	325.077i	−0.429901	+	0.429901i	0.458693	+	0.888595i	$0.348318\pi$
$84$		0			0
$85$	−604.900	−	604.900i	−0.771890	−	0.771890i
$86$	−127.039	−	763.084i	−0.159291	−	0.956808i
$87$		0			0
$88$	−6.20586	−	11.4986i	−0.00751758	−	0.0139290i
$89$	−1005.33			−1.19735			−0.598676	−	0.800991i	$-0.704306\pi$
$89$	−1005.33			−1.19735			−0.598676	+	0.800991i	$0.704306\pi$
$90$		0			0
$91$	−797.651	+	797.651i	−0.918863	+	0.918863i
$92$	1314.37	−	450.113i	1.48949	−	0.510082i
$93$		0			0
$94$	−153.123	+	214.291i	−0.168015	+	0.235132i
$95$	752.747			0.812950
$96$		0			0
$97$	997.579			1.04421			0.522107	−	0.852880i	$-0.325146\pi$
$97$	997.579			1.04421			0.522107	+	0.852880i	$0.325146\pi$
$98$	−911.731	+	1275.94i	−0.939783	+	1.31520i
$99$		0			0
$100$	142.216	−	48.7026i	0.142216	−	0.0487026i
$101$	1176.16	−	1176.16i	1.15874	−	1.15874i	0.173993	−	0.984747i	$-0.444333\pi$
$101$	1176.16	−	1176.16i	1.15874	−	1.15874i	0.984747	−	0.173993i	$-0.0556671\pi$
$102$		0			0
$103$	−979.265			−0.936794			−0.468397	−	0.883518i	$-0.655168\pi$
$103$	−979.265			−0.936794			−0.468397	+	0.883518i	$0.655168\pi$
$104$	−749.805	+	404.674i	−0.706966	+	0.381553i
$105$		0			0
$106$	163.571	+	982.515i	0.149881	+	0.900286i
$107$	484.233	+	484.233i	0.437501	+	0.437501i	0.891170	−	0.453669i	$-0.149885\pi$
$107$	484.233	+	484.233i	0.437501	+	0.437501i	−0.453669	+	0.891170i	$0.649885\pi$
$108$		0			0
$109$	−719.387	+	719.387i	−0.632154	+	0.632154i	−0.948608	−	0.316454i	$-0.897508\pi$
$109$	−719.387	+	719.387i	−0.632154	+	0.632154i	0.316454	+	0.948608i	$0.397508\pi$
$110$	−9.78614	+	13.6954i	−0.00848247	+	0.0118709i
$111$		0			0
$112$	−1514.78	+	1175.32i	−1.27797	+	0.991584i
$113$		−	1396.36i		−	1.16246i	−0.813738	−	0.581232i	$-0.802571\pi$
$113$		−	1396.36i		−	1.16246i	0.813738	−	0.581232i	$-0.197429\pi$
$114$		0			0
$115$	−1265.54	−	1265.54i	−1.02619	−	1.02619i
$116$	−832.644	+	1699.94i	−0.666457	+	1.36065i
$117$		0			0
$118$	97.0805	−	16.1621i	0.0757371	−	0.0126088i
$119$			2486.69i			1.91558i
$120$		0			0
$121$		−	1330.67i		−	0.999749i
$122$	296.619	+	1781.69i	0.220120	+	1.32219i
$123$		0			0
$124$	1447.70	−	495.770i	1.04844	−	0.359044i
$125$	−1047.84	−	1047.84i	−0.749777	−	0.749777i
$126$		0			0
$127$		−	2325.65i		−	1.62494i	−0.583001	−	0.812471i	$-0.698122\pi$
$127$		−	2325.65i		−	1.62494i	0.583001	−	0.812471i	$-0.301878\pi$
$128$	−1350.48	+	522.839i	−0.932551	+	0.361038i
$129$		0			0
$130$	893.052	+	638.137i	0.602506	+	0.430526i
$131$	407.380	−	407.380i	0.271702	−	0.271702i	−0.558083	−	0.829785i	$-0.688463\pi$
$131$	407.380	−	407.380i	0.271702	−	0.271702i	0.829785	+	0.558083i	$0.188463\pi$
$132$		0			0
$133$	−1547.24	−	1547.24i	−1.00874	−	1.00874i
$134$	−1791.24	+	298.209i	−1.15478	+	0.192249i
$135$		0			0
$136$	−537.976	+	1799.55i	−0.339199	+	1.13463i
$137$	1751.03			1.09197			0.545987	−	0.837794i	$-0.316155\pi$
$137$	1751.03			1.09197			0.545987	+	0.837794i	$0.316155\pi$
$138$		0			0
$139$	−2031.19	+	2031.19i	−1.23945	+	1.23945i	−0.279220	+	0.960227i	$0.590076\pi$
$139$	−2031.19	+	2031.19i	−1.23945	+	1.23945i	−0.960227	+	0.279220i	$0.909924\pi$
$140$	2218.09	+	1086.44i	1.33902	+	0.655865i
$141$		0			0
$142$	−802.341	−	573.319i	−0.474162	−	0.338816i
$143$	21.7443			0.0127157
$144$		0			0
$145$	2438.48			1.39659
$146$	−2124.62	−	1518.16i	−1.20435	−	0.860575i
$147$		0			0
$148$	922.575	−	1883.54i	0.512400	−	1.04612i
$149$	14.2055	−	14.2055i	0.00781049	−	0.00781049i	−0.703191	−	0.711001i	$-0.748242\pi$
$149$	14.2055	−	14.2055i	0.00781049	−	0.00781049i	0.711001	+	0.703191i	$0.248242\pi$
$150$		0			0
$151$	2215.34			1.19392			0.596959	−	0.802272i	$-0.296376\pi$
$151$	2215.34			1.19392			0.596959	+	0.802272i	$0.296376\pi$
$152$	−784.963	−	1454.43i	−0.418874	−	0.776116i
$153$		0			0
$154$	48.2651	−	8.03524i	0.0252553	−	0.00420453i
$155$	−1393.91	−	1393.91i	−0.722332	−	0.722332i
$156$		0			0
$157$	−360.380	+	360.380i	−0.183194	+	0.183194i	−0.792746	−	0.609552i	$-0.791349\pi$
$157$	−360.380	+	360.380i	−0.183194	+	0.183194i	0.609552	+	0.792746i	$0.291349\pi$
$158$	2277.99	+	1627.76i	1.14701	+	0.819603i
$159$		0			0
$160$	1370.13	+	1266.10i	0.676992	+	0.625586i
$161$			5202.51i			2.54668i
$162$		0			0
$163$	−205.489	−	205.489i	−0.0987432	−	0.0987432i	0.656009	−	0.754753i	$-0.272243\pi$
$163$	−205.489	−	205.489i	−0.0987432	−	0.0987432i	−0.754753	+	0.656009i	$0.772243\pi$
$164$	162.679	+	475.038i	0.0774578	+	0.226184i
$165$		0			0
$166$	−213.538	−	1282.65i	−0.0998420	−	0.599718i
$167$		−	1245.76i		−	0.577246i	−0.957443	−	0.288623i	$-0.906803\pi$
$167$		−	1245.76i		−	0.577246i	0.957443	−	0.288623i	$-0.0931975\pi$
$168$		0			0
$169$			779.093i			0.354617i
$170$	2386.75	−	397.350i	1.07680	−	0.179267i
$171$		0			0
$172$	1964.98	+	962.465i	0.871095	+	0.426670i
$173$	1094.91	+	1094.91i	0.481181	+	0.481181i	0.905509	−	0.424328i	$-0.139490\pi$
$173$	1094.91	+	1094.91i	0.481181	+	0.481181i	−0.424328	+	0.905509i	$0.639490\pi$
$174$		0			0
$175$			562.916i			0.243157i
$176$	36.6666	+	4.62688i	0.0157037	+	0.00198162i
$177$		0			0
$178$	1653.16	−	2313.55i	0.696123	−	0.974201i
$179$	340.980	−	340.980i	0.142380	−	0.142380i	−0.632324	−	0.774704i	$-0.717899\pi$
$179$	340.980	−	340.980i	0.142380	−	0.142380i	0.774704	+	0.632324i	$0.217899\pi$
$180$		0			0
$181$	−2257.36	−	2257.36i	−0.927009	−	0.927009i	0.0705027	−	0.997512i	$-0.477540\pi$
$181$	−2257.36	−	2257.36i	−0.927009	−	0.927009i	−0.997512	+	0.0705027i	$0.977540\pi$
$182$	−523.964	−	3147.29i	−0.213400	−	1.28183i
$183$		0			0
$184$	−1125.52	+	3764.92i	−0.450949	+	1.50844i
$185$	−2701.86			−1.07375
$186$		0			0
$187$	33.8940	−	33.8940i	0.0132544	−	0.0132544i
$188$	−241.349	−	704.762i	−0.0936285	−	0.273404i
$189$		0			0
$190$	−1237.82	+	1732.29i	−0.472637	+	0.661440i
$191$	4673.62			1.77053			0.885265	−	0.465087i	$-0.153977\pi$
$191$	4673.62			1.77053			0.885265	+	0.465087i	$0.153977\pi$
$192$		0			0
$193$	−1023.43			−0.381701			−0.190850	−	0.981619i	$-0.561125\pi$
$193$	−1023.43			−0.381701			−0.190850	+	0.981619i	$0.561125\pi$
$194$	−1640.42	+	2295.72i	−0.607090	+	0.849603i
$195$		0			0
$196$	−1437.05	−	4196.31i	−0.523705	−	1.52927i
$197$	3664.48	−	3664.48i	1.32530	−	1.32530i	0.415877	−	0.909421i	$-0.363475\pi$
$197$	3664.48	−	3664.48i	1.32530	−	1.32530i	0.909421	−	0.415877i	$-0.136525\pi$
$198$		0			0
$199$	2442.52			0.870079			0.435040	−	0.900411i	$-0.356734\pi$
$199$	2442.52			0.870079			0.435040	+	0.900411i	$0.356734\pi$
$200$	−121.783	+	407.368i	−0.0430566	+	0.144026i
$201$		0			0
$202$	772.605	+	4640.79i	0.269110	+	1.61646i
$203$	−5012.19	−	5012.19i	−1.73294	−	1.73294i
$204$		0			0
$205$	457.388	−	457.388i	0.155831	−	0.155831i
$206$	1610.31	−	2253.57i	0.544638	−	0.762203i
$207$		0			0
$208$	301.711	−	2390.97i	0.100576	−	0.797037i
$209$			42.1783i			0.0139595i
$210$		0			0
$211$	2619.72	+	2619.72i	0.854736	+	0.854736i	0.990712	−	0.135976i	$-0.0434172\pi$
$211$	2619.72	+	2619.72i	0.854736	+	0.854736i	−0.135976	+	0.990712i	$0.543417\pi$
$212$	−2530.03	−	1239.23i	−0.819637	−	0.401465i
$213$		0			0
$214$	−1910.63	+	318.085i	−0.610319	+	0.101607i
$215$		−	2818.68i		−	0.894103i
$216$		0			0
$217$			5730.23i			1.79259i
$218$	−472.555	−	2838.48i	−0.146814	−	0.881864i
$219$		0			0
$220$	−15.4246	−	45.0415i	−0.00472695	−	0.0138032i
$221$	−2210.17	−	2210.17i	−0.672725	−	0.672725i
$222$		0			0
$223$		−	3002.85i		−	0.901730i	−0.892592	−	0.450865i	$-0.851116\pi$
$223$		−	3002.85i		−	0.901730i	0.892592	−	0.450865i	$-0.148884\pi$
$224$	−213.844	−	5418.65i	−0.0637858	−	1.61629i
$225$		0			0
$226$	3213.42	+	2296.18i	0.945813	+	0.675838i
$227$	−2191.51	+	2191.51i	−0.640774	+	0.640774i	−0.950746	−	0.309971i	$-0.899680\pi$
$227$	−2191.51	+	2191.51i	−0.640774	+	0.640774i	0.309971	+	0.950746i	$0.399680\pi$
$228$		0			0
$229$	1807.23	+	1807.23i	0.521508	+	0.521508i	0.918027	−	0.396519i	$-0.129782\pi$
$229$	1807.23	+	1807.23i	0.521508	+	0.521508i	−0.396519	+	0.918027i	$0.629782\pi$
$230$	4993.43	−	831.313i	1.43155	−	0.238327i
$231$		0			0
$232$	−2542.84	−	4711.54i	−0.719594	−	1.33331i
$233$	−1109.72			−0.312018			−0.156009	−	0.987756i	$-0.549863\pi$
$233$	−1109.72			−0.312018			−0.156009	+	0.987756i	$0.549863\pi$
$234$		0			0
$235$	−678.576	+	678.576i	−0.188364	+	0.188364i
$236$	−122.446	+	249.987i	−0.0337735	+	0.0689525i
$237$		0			0
$238$	−5722.59	−	4089.12i	−1.55857	−	1.11369i
$239$	−471.822			−0.127697			−0.0638486	−	0.997960i	$-0.520337\pi$
$239$	−471.822			−0.127697			−0.0638486	+	0.997960i	$0.520337\pi$
$240$		0			0
$241$	1506.72			0.402723			0.201362	−	0.979517i	$-0.435463\pi$
$241$	1506.72			0.402723			0.201362	+	0.979517i	$0.435463\pi$
$242$	3062.25	+	2188.15i	0.813425	+	0.581239i
$243$		0			0
$244$	−4587.95	−	2247.22i	−1.20374	−	0.589604i
$245$	−4040.40	+	4040.40i	−1.05360	+	1.05360i
$246$		0			0
$247$	2750.37			0.708510
$248$	−1239.69	+	4146.82i	−0.317421	+	1.06179i
$249$		0			0
$250$	4134.47	−	688.313i	1.04595	−	0.174131i
$251$	3490.17	+	3490.17i	0.877679	+	0.877679i	0.993294	−	0.115616i	$-0.0368840\pi$
$251$	3490.17	+	3490.17i	0.877679	+	0.877679i	−0.115616	+	0.993294i	$0.536884\pi$
$252$		0			0
$253$	70.9112	−	70.9112i	0.0176211	−	0.0176211i
$254$	5351.99	+	3824.30i	1.32210	+	0.944717i
$255$		0			0
$256$	1017.53	−	3967.60i	0.248420	−	0.968652i
$257$			4623.43i			1.12219i	0.827753	+	0.561093i	$0.189619\pi$
$257$			4623.43i			1.12219i	−0.827753	+	0.561093i	$0.810381\pi$
$258$		0			0
$259$	5553.53	+	5553.53i	1.33235	+	1.33235i
$260$	−2937.08	+	1005.81i	−0.700576	+	0.239915i
$261$		0			0
$262$	267.602	+	1607.40i	0.0631011	+	0.379028i
$263$			1480.27i			0.347063i	0.984828	+	0.173532i	$0.0555179\pi$
$263$			1480.27i			0.347063i	−0.984828	+	0.173532i	$0.944482\pi$
$264$		0			0
$265$			3629.21i			0.841286i
$266$	6104.92	−	1016.36i	1.40721	−	0.234274i
$267$		0			0
$268$	2259.27	−	4612.55i	0.514950	−	1.05133i
$269$	4722.72	+	4722.72i	1.07044	+	1.07044i	0.997323	+	0.0731197i	$0.0232955\pi$
$269$	4722.72	+	4722.72i	1.07044	+	1.07044i	0.0731197	+	0.997323i	$0.476704\pi$
$270$		0			0
$271$		−	76.7667i		−	0.0172076i	−0.999963	−	0.00860378i	$-0.997261\pi$
$271$		−	76.7667i		−	0.0172076i	0.999963	−	0.00860378i	$-0.00273870\pi$
$272$	−3256.64	−	4197.23i	−0.725966	−	0.935641i
$273$		0			0
$274$	−2879.40	+	4029.62i	−0.634857	+	0.888461i
$275$	7.67266	−	7.67266i	0.00168247	−	0.00168247i
$276$		0			0
$277$	−3935.81	−	3935.81i	−0.853718	−	0.853718i	0.136871	−	0.990589i	$-0.456295\pi$
$277$	−3935.81	−	3935.81i	−0.853718	−	0.853718i	−0.990589	+	0.136871i	$0.956295\pi$
$278$	−1334.26	−	8014.45i	−0.287854	−	1.72905i
$279$		0			0
$280$	−6147.66	+	3317.93i	−1.31212	+	0.708157i
$281$	−2951.92			−0.626678			−0.313339	−	0.949641i	$-0.601448\pi$
$281$	−2951.92			−0.626678			−0.313339	+	0.949641i	$0.601448\pi$
$282$		0			0
$283$	2479.89	−	2479.89i	0.520898	−	0.520898i	−0.396944	−	0.917843i	$-0.629929\pi$
$283$	2479.89	−	2479.89i	0.520898	−	0.520898i	0.917843	+	0.396944i	$0.129929\pi$
$284$	2638.75	−	903.651i	0.551341	−	0.188809i
$285$		0			0
$286$	−35.7564	+	50.0399i	−0.00739272	+	0.0103459i
$287$	−1880.28			−0.386723
$288$		0			0
$289$	−1977.25			−0.402452
$290$	−4009.85	+	5611.66i	−0.811954	+	1.13630i
$291$		0			0
$292$	6987.46	−	2392.89i	1.40038	−	0.479565i
$293$	−3083.66	+	3083.66i	−0.614844	+	0.614844i	−0.944204	−	0.329360i	$-0.893167\pi$
$293$	−3083.66	+	3083.66i	−0.614844	+	0.614844i	0.329360	+	0.944204i	$0.393167\pi$
$294$		0			0
$295$	358.596			0.0707737
$296$	2817.49	+	5220.42i	0.553254	+	1.02510i
$297$		0			0
$298$	9.33140	+	56.0507i	0.00181394	+	0.0108957i
$299$	−4624.00	−	4624.00i	−0.894357	−	0.894357i
$300$		0			0
$301$	−5793.66	+	5793.66i	−1.10944	+	1.10944i
$302$	−3642.91	+	5098.13i	−0.694126	+	0.971406i
$303$		0			0
$304$	4637.86	+	585.242i	0.874998	+	0.110414i
$305$			6581.21i			1.23554i
$306$		0			0
$307$	−3541.41	−	3541.41i	−0.658367	−	0.658367i	0.296627	−	0.954994i	$-0.404138\pi$
$307$	−3541.41	−	3541.41i	−0.658367	−	0.658367i	−0.954994	+	0.296627i	$0.904138\pi$
$308$	−60.8759	+	124.285i	−0.0112621	+	0.0229929i
$309$		0			0
$310$	5499.94	−	915.638i	1.00766	−	0.167757i
$311$		−	3223.24i		−	0.587694i	−0.955852	−	0.293847i	$-0.905064\pi$
$311$		−	3223.24i		−	0.587694i	0.955852	−	0.293847i	$-0.0949357\pi$
$312$		0			0
$313$		−	8312.33i		−	1.50109i	−0.660820	−	0.750545i	$-0.729791\pi$
$313$		−	8312.33i		−	1.50109i	0.660820	−	0.750545i	$-0.270209\pi$
$314$	−236.728	−	1421.95i	−0.0425456	−	0.255558i
$315$		0			0
$316$	−7491.87	+	2565.62i	−1.33371	+	0.456733i
$317$	4342.41	+	4342.41i	0.769382	+	0.769382i	0.977998	−	0.208616i	$-0.0668958\pi$
$317$	4342.41	+	4342.41i	0.769382	+	0.769382i	−0.208616	+	0.977998i	$0.566896\pi$
$318$		0			0
$319$			136.634i			0.0239813i
$320$	−5166.71	+	1071.10i	−0.902587	+	0.187113i
$321$		0			0
$322$	−11972.5	−	8555.03i	−2.07205	−	1.48060i
$323$	4287.16	−	4287.16i	0.738527	−	0.738527i
$324$		0			0
$325$	−500.321	−	500.321i	−0.0853933	−	0.0853933i
$326$	810.797	−	134.983i	0.137748	−	0.0229325i
$327$		0			0
$328$	−1360.71	−	406.784i	−0.229063	−	0.0684783i
$329$	2789.56			0.467458
$330$		0			0
$331$	−4075.14	+	4075.14i	−0.676706	+	0.676706i	−0.959253	−	0.282547i	$-0.908821\pi$
$331$	−4075.14	+	4075.14i	−0.676706	+	0.676706i	0.282547	+	0.959253i	$0.408821\pi$
$332$	3302.90	+	1617.79i	0.545995	+	0.267433i
$333$		0			0
$334$	2866.86	+	2048.54i	0.469664	+	0.335602i
$335$	−6616.49			−1.07910
$336$		0			0
$337$	936.542			0.151385			0.0756925	−	0.997131i	$-0.475883\pi$
$337$	936.542			0.151385			0.0756925	+	0.997131i	$0.475883\pi$
$338$	−1792.92	−	1281.14i	−0.288526	−	0.206169i
$339$		0			0
$340$	−3010.37	+	6146.01i	−0.480177	+	0.980336i
$341$	78.1041	−	78.1041i	0.0124034	−	0.0124034i
$342$		0			0
$343$	6334.32			0.997147
$344$	−5446.13	+	2939.31i	−0.853593	+	0.460689i
$345$		0			0
$346$	−4320.17	+	719.228i	−0.671254	+	0.111751i
$347$	−793.398	−	793.398i	−0.122743	−	0.122743i	0.643067	−	0.765810i	$-0.277662\pi$
$347$	−793.398	−	793.398i	−0.122743	−	0.122743i	−0.765810	+	0.643067i	$0.777662\pi$
$348$		0			0
$349$	396.244	−	396.244i	0.0607749	−	0.0607749i	−0.676066	−	0.736841i	$-0.736317\pi$
$349$	396.244	−	396.244i	0.0607749	−	0.0607749i	0.736841	+	0.676066i	$0.236317\pi$
$350$	−1295.43	−	925.662i	−0.197839	−	0.141368i
$351$		0			0
$352$	−70.9425	+	76.7720i	−0.0107422	+	0.0116249i
$353$		−	5035.58i		−	0.759254i	−0.925140	−	0.379627i	$-0.876052\pi$
$353$		−	5035.58i		−	0.759254i	0.925140	−	0.379627i	$-0.123948\pi$
$354$		0			0
$355$	−2540.71	−	2540.71i	−0.379850	−	0.379850i
$356$	2605.67	+	7608.82i	0.387922	+	1.13277i
$357$		0			0
$358$	223.984	+	1345.40i	0.0330669	+	0.198622i
$359$			3252.36i			0.478142i	0.971002	+	0.239071i	$0.0768428\pi$
$359$			3252.36i			0.478142i	−0.971002	+	0.239071i	$0.923157\pi$
$360$		0			0
$361$		−	1523.99i		−	0.222188i
$362$	8906.87	−	1482.83i	1.29319	−	0.215292i
$363$		0			0
$364$	8104.42	+	3969.62i	1.16700	+	0.571606i
$365$	−6727.85	−	6727.85i	−0.964799	−	0.964799i
$366$		0			0
$367$			5829.07i			0.829087i	0.910030	+	0.414543i	$0.136059\pi$
$367$			5829.07i			0.829087i	−0.910030	+	0.414543i	$0.863941\pi$
$368$	−6813.36	−	8781.21i	−0.965138	−	1.24389i
$369$		0			0
$370$	4442.95	−	6217.75i	0.624264	−	0.873637i
$371$	7459.67	−	7459.67i	1.04390	−	1.04390i
$372$		0			0
$373$	3187.28	+	3187.28i	0.442442	+	0.442442i	0.892832	−	0.450390i	$-0.148715\pi$
$373$	3187.28	+	3187.28i	0.442442	+	0.442442i	−0.450390	+	0.892832i	$0.648715\pi$
$374$	22.2645	+	133.736i	0.00307826	+	0.0184901i
$375$		0			0
$376$	2018.74	+	603.500i	0.276884	+	0.0827743i
$377$	8909.68			1.21717
$378$		0			0
$379$	159.188	−	159.188i	0.0215751	−	0.0215751i	−0.696237	−	0.717812i	$-0.745144\pi$
$379$	159.188	−	159.188i	0.0215751	−	0.0215751i	0.717812	+	0.696237i	$0.245144\pi$
$380$	−1951.02	−	5697.17i	−0.263382	−	0.769102i
$381$		0			0
$382$	−7685.32	+	10755.3i	−1.02936	+	1.44055i
$383$	32.8699			0.00438530			0.00219265	−	0.999998i	$-0.499302\pi$
$383$	32.8699			0.00438530			0.00219265	+	0.999998i	$0.499302\pi$
$384$		0			0
$385$	178.282			0.0236002
$386$	1682.94	−	2355.21i	0.221915	−	0.310563i
$387$		0			0
$388$	−2585.59	−	7550.18i	−0.338308	−	0.987892i
$389$	−350.837	+	350.837i	−0.0457278	+	0.0457278i	−0.729601	−	0.683873i	$-0.760294\pi$
$389$	−350.837	+	350.837i	−0.0457278	+	0.0457278i	0.683873	+	0.729601i	$0.260294\pi$
$390$		0			0
$391$	−14415.4			−1.86449
$392$	12020.0	+	3593.38i	1.54873	+	0.462993i
$393$		0			0
$394$	2407.14	+	14458.9i	0.307792	+	1.84881i
$395$	7213.52	+	7213.52i	0.918864	+	0.918864i
$396$		0			0
$397$	1334.06	−	1334.06i	0.168652	−	0.168652i	−0.617735	−	0.786387i	$-0.711949\pi$
$397$	1334.06	−	1334.06i	0.168652	−	0.168652i	0.786387	+	0.617735i	$0.211949\pi$
$398$	−4016.49	+	5620.95i	−0.505851	+	0.707921i
$399$		0			0
$400$	−737.212	−	950.135i	−0.0921515	−	0.118767i
$401$		−	15289.0i		−	1.90398i	−0.306137	−	0.951988i	$-0.599037\pi$
$401$		−	15289.0i		−	1.90398i	0.306137	−	0.951988i	$-0.400963\pi$
$402$		0			0
$403$	−5093.04	−	5093.04i	−0.629534	−	0.629534i
$404$	−11950.3	−	5853.34i	−1.47165	−	0.720829i
$405$		0			0
$406$	19776.6	−	3292.43i	2.41747	−	0.402464i
$407$		−	151.392i		−	0.0184378i
$408$		0			0
$409$		−	8690.12i		−	1.05061i	−0.850915	−	0.525304i	$-0.823952\pi$
$409$		−	8690.12i		−	1.05061i	0.850915	−	0.525304i	$-0.176048\pi$
$410$	300.451	+	1804.71i	0.0361908	+	0.217387i
$411$		0			0
$412$	2538.12	+	7411.57i	0.303506	+	0.886266i
$413$	−737.076	−	737.076i	−0.0878188	−	0.0878188i
$414$		0			0
$415$		−	4737.86i		−	0.560416i
$416$	5006.17	+	4626.04i	0.590018	+	0.545217i
$417$		0			0
$418$	−97.0644	−	69.3581i	−0.0113578	−	0.00811583i
$419$	−5393.05	+	5393.05i	−0.628802	+	0.628802i	−0.947766	−	0.318965i	$-0.896665\pi$
$419$	−5393.05	+	5393.05i	−0.628802	+	0.628802i	0.318965	+	0.947766i	$0.396665\pi$
$420$		0			0
$421$	1229.98	+	1229.98i	0.142389	+	0.142389i	0.774708	−	0.632319i	$-0.217897\pi$
$421$	1229.98	+	1229.98i	0.142389	+	0.142389i	−0.632319	+	0.774708i	$0.717897\pi$
$422$	−10336.6	+	1720.86i	−1.19237	+	0.198507i
$423$		0			0
$424$	7012.22	−	3784.53i	0.803168	−	0.433474i
$425$	−1559.76			−0.178022
$426$		0			0
$427$	13527.4	−	13527.4i	1.53310	−	1.53310i
$428$	2409.85	−	4919.98i	0.272160	−	0.555646i
$429$		0			0
$430$	6486.59	+	4635.05i	0.727468	+	0.519818i
$431$	5167.35			0.577500			0.288750	−	0.957405i	$-0.406760\pi$
$431$	5167.35			0.577500			0.288750	+	0.957405i	$0.406760\pi$
$432$		0			0
$433$	−8025.47			−0.890715			−0.445357	−	0.895353i	$-0.646923\pi$
$433$	−8025.47			−0.890715			−0.445357	+	0.895353i	$0.646923\pi$
$434$	−13186.9	−	9422.81i	−1.45851	−	1.04219i
$435$		0			0
$436$	7309.24	+	3580.13i	0.802865	+	0.393250i
$437$	8969.36	−	8969.36i	0.981837	−	0.981837i
$438$		0			0
$439$	−2849.28			−0.309770			−0.154885	−	0.987933i	$-0.549501\pi$
$439$	−2849.28			−0.309770			−0.154885	+	0.987933i	$0.549501\pi$
$440$	129.018	+	38.5698i	0.0139788	+	0.00417896i
$441$		0			0
$442$	8720.67	−	1451.83i	0.938461	−	0.156236i
$443$	11748.5	+	11748.5i	1.26001	+	1.26001i	0.951085	+	0.308928i	$0.0999703\pi$
$443$	11748.5	+	11748.5i	1.26001	+	1.26001i	0.308928	+	0.951085i	$0.400030\pi$
$444$		0			0
$445$	7326.11	−	7326.11i	0.780429	−	0.780429i
$446$	6910.43	+	4937.90i	0.733673	+	0.524252i
$447$		0			0
$448$	12821.5	+	8418.33i	1.35214	+	0.887788i
$449$		−	14375.4i		−	1.51096i	−0.655174	−	0.755478i	$-0.727405\pi$
$449$		−	14375.4i		−	1.51096i	0.655174	−	0.755478i	$-0.272595\pi$
$450$		0			0
$451$	25.6286	+	25.6286i	0.00267584	+	0.00267584i
$452$	−10568.3	+	3619.17i	−1.09976	+	0.376619i
$453$		0			0
$454$	−1439.57	−	8647.04i	−0.148816	−	0.893889i
$455$		−	11625.4i		−	1.19782i
$456$		0			0
$457$			125.930i			0.0128900i	0.999979	+	0.00644502i	$0.00205153\pi$
$457$			125.930i			0.0128900i	−0.999979	+	0.00644502i	$0.997948\pi$
$458$	−7130.79	+	1187.14i	−0.727511	+	0.121117i
$459$		0			0
$460$	−6298.12	+	12858.3i	−0.638373	+	1.30331i
$461$	179.001	+	179.001i	0.0180844	+	0.0180844i	0.716091	−	0.698007i	$-0.245929\pi$
$461$	179.001	+	179.001i	0.0180844	+	0.0180844i	−0.698007	+	0.716091i	$0.745929\pi$
$462$		0			0
$463$			1590.83i			0.159681i	0.996808	+	0.0798403i	$0.0254410\pi$
$463$			1590.83i			0.159681i	−0.996808	+	0.0798403i	$0.974559\pi$
$464$	15024.1	+	1895.86i	1.50318	+	0.189683i
$465$		0			0
$466$	1824.83	−	2553.79i	0.181403	−	0.253867i
$467$	−11457.8	+	11457.8i	−1.13534	+	1.13534i	−0.146061	+	0.989276i	$0.546660\pi$
$467$	−11457.8	+	11457.8i	−1.13534	+	1.13534i	−0.989276	+	0.146061i	$0.953340\pi$
$468$		0			0
$469$	13599.9	+	13599.9i	1.33899	+	1.33899i
$470$	−445.746	−	2677.46i	−0.0437463	−	0.262770i
$471$		0			0
$472$	−373.943	−	692.864i	−0.0364663	−	0.0675671i
$473$	157.937			0.0153530
$474$		0			0
$475$	970.493	−	970.493i	0.0937459	−	0.0937459i
$476$	18820.5	−	6445.16i	1.81226	−	0.620616i
$477$		0			0
$478$	775.866	−	1085.80i	0.0742412	−	0.103898i
$479$	7077.38			0.675102			0.337551	−	0.941307i	$-0.390401\pi$
$479$	7077.38			0.675102			0.337551	+	0.941307i	$0.390401\pi$
$480$		0			0
$481$	−9871.99			−0.935808
$482$	−2477.66	+	3467.40i	−0.234137	+	0.327667i
$483$		0			0
$484$	−10071.2	+	3448.91i	−0.945826	+	0.323902i
$485$	−7269.65	+	7269.65i	−0.680614	+	0.680614i
$486$		0			0
$487$	−16187.6			−1.50622			−0.753112	−	0.657892i	$-0.771448\pi$
$487$	−16187.6			−1.50622			−0.753112	+	0.657892i	$0.771448\pi$
$488$	12715.9	−	6862.86i	1.17956	−	0.636613i
$489$		0			0
$490$	−2654.08	−	15942.2i	−0.244692	−	1.46979i
$491$	−3058.51	−	3058.51i	−0.281118	−	0.281118i	0.552437	−	0.833555i	$-0.313698\pi$
$491$	−3058.51	−	3058.51i	−0.281118	−	0.281118i	−0.833555	+	0.552437i	$0.813698\pi$
$492$		0			0
$493$	13888.0	−	13888.0i	1.26873	−	1.26873i
$494$	−4522.73	+	6329.40i	−0.411917	+	0.576464i
$495$		0			0
$496$	−7504.48	−	9671.93i	−0.679357	−	0.875570i
$497$			10444.6i			0.942665i
$498$		0			0
$499$	−726.658	−	726.658i	−0.0651898	−	0.0651898i	0.673760	−	0.738950i	$-0.264678\pi$
$499$	−726.658	−	726.658i	−0.0651898	−	0.0651898i	−0.738950	+	0.673760i	$0.764678\pi$
$500$	−5214.74	+	10646.5i	−0.466421	+	0.952251i
$501$		0			0
$502$	−13771.1	+	2292.64i	−1.22437	+	0.203835i
$503$			4589.71i			0.406849i	0.979091	+	0.203424i	$0.0652071\pi$
$503$			4589.71i			0.406849i	−0.979091	+	0.203424i	$0.934793\pi$
$504$		0			0
$505$			17142.1i			1.51052i
$506$	46.5805	+	279.794i	0.00409240	+	0.0245817i
$507$		0			0
$508$	−17601.7	+	6027.76i	−1.53730	+	0.526454i
$509$	−4040.42	−	4040.42i	−0.351844	−	0.351844i	0.508951	−	0.860795i	$-0.330033\pi$
$509$	−4040.42	−	4040.42i	−0.351844	−	0.351844i	−0.860795	+	0.508951i	$0.830033\pi$
$510$		0			0
$511$			27657.5i			2.39432i
$512$	7457.37	+	8865.97i	0.643696	+	0.765282i
$513$		0			0
$514$	−10639.9	−	7602.79i	−0.913042	−	0.652422i
$515$	7136.20	−	7136.20i	0.610599	−	0.610599i
$516$		0			0
$517$	−38.0223	−	38.0223i	−0.00323446	−	0.00323446i
$518$	−21912.5	+	3648.03i	−1.85865	+	0.309431i
$519$		0			0
$520$	2515.07	−	8413.03i	0.212102	−	0.709492i
$521$	243.434			0.0204703			0.0102351	−	0.999948i	$-0.496742\pi$
$521$	243.434			0.0204703			0.0102351	+	0.999948i	$0.496742\pi$
$522$		0			0
$523$	7802.07	−	7802.07i	0.652315	−	0.652315i	−0.301235	−	0.953550i	$-0.597399\pi$
$523$	7802.07	−	7802.07i	0.652315	−	0.652315i	0.953550	+	0.301235i	$0.0973989\pi$
$524$	−4139.13	−	2027.38i	−0.345074	−	0.169020i
$525$		0			0
$526$	−3406.54	−	2434.17i	−0.282381	−	0.201777i
$527$	−15877.6			−1.31241
$528$		0			0
$529$	−17992.0			−1.47876
$530$	−8351.87	−	5967.90i	−0.684494	−	0.489111i
$531$		0			0
$532$	−7700.03	+	15720.5i	−0.627516	+	1.28115i
$533$	1671.20	−	1671.20i	0.135812	−	0.135812i
$534$		0			0
$535$	−7057.50			−0.570322
$536$	6899.66	+	12784.1i	0.556007	+	1.03020i
$537$		0			0
$538$	−18634.4	+	3102.28i	−1.49328	+	0.248604i
$539$	−226.394	−	226.394i	−0.0180918	−	0.0180918i
$540$		0			0
$541$	−8566.46	+	8566.46i	−0.680778	+	0.680778i	−0.960176	−	0.279397i	$-0.909865\pi$
$541$	−8566.46	+	8566.46i	−0.680778	+	0.680778i	0.279397	+	0.960176i	$0.409865\pi$
$542$	176.662	+	126.236i	0.0140006	+	0.0100042i
$543$		0			0
$544$	15014.3	−	592.529i	1.18333	−	0.0466994i
$545$		−	10484.8i		−	0.824071i
$546$		0			0
$547$	386.930	+	386.930i	0.0302448	+	0.0302448i	0.722067	−	0.691823i	$-0.243192\pi$
$547$	386.930	+	386.930i	0.0302448	+	0.0302448i	−0.691823	+	0.722067i	$0.743192\pi$
$548$	−4538.43	−	13252.7i	−0.353781	−	1.03308i
$549$		0			0
$550$	5.04005	+	30.2740i	0.000390743	+	0.00234707i
$551$			17282.5i			1.33622i
$552$		0			0
$553$		−	29654.1i		−	2.28032i
$554$	15529.5	−	2585.37i	1.19095	−	0.198271i
$555$		0			0
$556$	20637.6	+	10108.5i	1.57416	+	0.771035i
$557$	−2421.34	−	2421.34i	−0.184193	−	0.184193i	0.608987	−	0.793180i	$-0.291576\pi$
$557$	−2421.34	−	2421.34i	−0.184193	−	0.184193i	−0.793180	+	0.608987i	$0.791576\pi$
$558$		0			0
$559$		−	10298.8i		−	0.779238i
$560$	2473.73	−	19603.6i	0.186668	−	1.47929i
$561$		0			0
$562$	4854.14	−	6793.21i	0.364341	−	0.509883i
$563$	−13851.2	+	13851.2i	−1.03687	+	1.03687i	−0.0375777	+	0.999294i	$0.511964\pi$
$563$	−13851.2	+	13851.2i	−1.03687	+	1.03687i	−0.999294	+	0.0375777i	$0.988036\pi$
$564$		0			0
$565$	10175.7	+	10175.7i	0.757689	+	0.757689i
$566$	1629.00	+	9784.89i	0.120975	+	0.726660i
$567$		0			0
$568$	−2259.61	+	7558.49i	−0.166921	+	0.558358i
$569$	1993.40			0.146868			0.0734338	−	0.997300i	$-0.476604\pi$
$569$	1993.40			0.146868			0.0734338	+	0.997300i	$0.476604\pi$
$570$		0			0
$571$	−14755.5	+	14755.5i	−1.08143	+	1.08143i	−0.0850541	+	0.996376i	$0.527106\pi$
$571$	−14755.5	+	14755.5i	−1.08143	+	1.08143i	−0.996376	+	0.0850541i	$0.972894\pi$
$572$	−56.3582	−	164.572i	−0.00411968	−	0.0120299i
$573$		0			0
$574$	3091.94	−	4327.07i	0.224835	−	0.314649i
$575$	−3263.24			−0.236672
$576$		0			0
$577$	24614.5			1.77593			0.887967	−	0.459907i	$-0.152117\pi$
$577$	24614.5			1.77593			0.887967	+	0.459907i	$0.152117\pi$
$578$	3251.39	−	4550.21i	0.233979	−	0.327446i
$579$		0			0
$580$	−6320.22	−	18455.7i	−0.452471	−	1.32126i
$581$	−9738.45	+	9738.45i	−0.695386	+	0.695386i
$582$		0			0
$583$	−203.354			−0.0144460
$584$	−5983.49	+	20015.0i	−0.423970	+	1.41820i
$585$		0			0
$586$	−2025.61	−	12167.2i	−0.142794	−	0.857716i
$587$	16905.8	+	16905.8i	1.18871	+	1.18871i	0.977423	+	0.211291i	$0.0677668\pi$
$587$	16905.8	+	16905.8i	1.18871	+	1.18871i	0.211291	+	0.977423i	$0.432233\pi$
$588$		0			0
$589$	9879.17	−	9879.17i	0.691111	−	0.691111i
$590$	−589.676	+	825.233i	−0.0411468	+	0.0575835i
$591$		0			0
$592$	−16646.8	−	2100.62i	−1.15571	−	0.145836i
$593$		−	6650.14i		−	0.460520i	−0.973129	−	0.230260i	$-0.926042\pi$
$593$		−	6650.14i		−	0.460520i	0.973129	−	0.230260i	$-0.0739577\pi$
$594$		0			0
$595$	−18121.2	−	18121.2i	−1.24857	−	1.24857i
$596$	−144.333	−	70.6958i	−0.00991968	−	0.00485874i
$597$		0			0
$598$	18244.9	−	3037.43i	1.24764	−	0.207709i
$599$		−	662.505i		−	0.0451907i	−0.999745	−	0.0225953i	$-0.992807\pi$
$599$		−	662.505i		−	0.0451907i	0.999745	−	0.0225953i	$-0.00719293\pi$
$600$		0			0
$601$			5113.18i			0.347040i	0.984830	+	0.173520i	$0.0555141\pi$
$601$			5113.18i			0.347040i	−0.984830	+	0.173520i	$0.944486\pi$
$602$	−3805.76	−	22860.0i	−0.257660	−	1.54768i
$603$		0			0
$604$	−5741.86	−	16766.8i	−0.386809	−	1.12952i
$605$	9696.96	+	9696.96i	0.651632	+	0.651632i
$606$		0			0
$607$			28192.4i			1.88516i	0.333975	+	0.942582i	$0.391610\pi$
$607$			28192.4i			1.88516i	−0.333975	+	0.942582i	$0.608390\pi$
$608$	−8973.32	+	9710.67i	−0.598546	+	0.647730i
$609$		0			0
$610$	−15145.3	−	10822.2i	−1.00527	−	0.718323i
$611$	−2479.37	+	2479.37i	−0.164165	+	0.164165i
$612$		0			0
$613$	14861.4	+	14861.4i	0.979193	+	0.979193i	0.999788	−	0.0205946i	$-0.00655594\pi$
$613$	14861.4	+	14861.4i	0.979193	+	0.979193i	−0.0205946	+	0.999788i	$0.506556\pi$
$614$	13973.3	−	2326.30i	0.918431	−	0.152902i
$615$		0			0
$616$	−185.911	−	344.468i	−0.0121600	−	0.0225309i
$617$	−1149.18			−0.0749826			−0.0374913	−	0.999297i	$-0.511937\pi$
$617$	−1149.18			−0.0749826			−0.0374913	+	0.999297i	$0.511937\pi$
$618$		0			0
$619$	−572.280	+	572.280i	−0.0371598	+	0.0371598i	−0.725443	−	0.688283i	$-0.758365\pi$
$619$	−572.280	+	572.280i	−0.0371598	+	0.0371598i	0.688283	+	0.725443i	$0.258365\pi$
$620$	−6936.98	+	14162.6i	−0.449348	+	0.917395i
$621$		0			0
$622$	7417.60	+	5300.30i	0.478165	+	0.341677i
$623$	−30116.9			−1.93677
$624$		0			0
$625$	12923.1			0.827078
$626$	19129.1	+	13668.8i	1.22133	+	0.872711i
$627$		0			0
$628$	3661.59	+	1793.48i	0.232665	+	0.113961i
$629$	−15388.0	+	15388.0i	−0.975455	+	0.975455i
$630$		0			0
$631$	374.155			0.0236052			0.0118026	−	0.999930i	$-0.496243\pi$
$631$	374.155			0.0236052			0.0118026	+	0.999930i	$0.496243\pi$
$632$	6415.43	−	21459.9i	0.403785	−	1.35068i
$633$		0			0
$634$	−17133.8	+	2852.46i	−1.07330	+	0.178684i
$635$	16947.7	+	16947.7i	1.05913	+	1.05913i
$636$		0			0
$637$	−14762.7	+	14762.7i	−0.918243	+	0.918243i
$638$	−314.435	−	224.682i	−0.0195119	−	0.0139424i
$639$		0			0
$640$	6031.25	−	13651.4i	0.372510	−	0.843156i
$641$		−	11493.2i		−	0.708196i	−0.935208	−	0.354098i	$-0.884788\pi$
$641$		−	11493.2i		−	0.708196i	0.935208	−	0.354098i	$-0.115212\pi$
$642$		0			0
$643$	15400.0	+	15400.0i	0.944507	+	0.944507i	0.998539	−	0.0540325i	$-0.0172074\pi$
$643$	15400.0	+	15400.0i	0.944507	+	0.944507i	−0.0540325	+	0.998539i	$0.517207\pi$
$644$	39375.2	−	13484.2i	2.40932	−	0.825081i
$645$		0			0
$646$	2816.17	+	16915.8i	0.171518	+	1.03025i
$647$			18258.5i			1.10945i	0.832034	+	0.554725i	$0.187176\pi$
$647$			18258.5i			1.10945i	−0.832034	+	0.554725i	$0.812824\pi$
$648$		0			0
$649$			20.0930i			0.00121528i
$650$	1974.11	−	328.653i	0.119125	−	0.0198321i
$651$		0			0
$652$	−1022.64	+	2087.84i	−0.0614261	+	0.125408i
$653$	−15515.3	−	15515.3i	−0.929800	−	0.929800i	0.0678930	−	0.997693i	$-0.478372\pi$
$653$	−15515.3	−	15515.3i	−0.929800	−	0.929800i	−0.997693	+	0.0678930i	$0.978372\pi$
$654$		0			0
$655$			5937.40i			0.354188i
$656$	3173.69	−	2462.47i	0.188890	−	0.146560i
$657$		0			0
$658$	−4587.17	+	6419.59i	−0.271773	+	0.380337i
$659$	10830.2	−	10830.2i	0.640189	−	0.640189i	−0.310413	−	0.950602i	$-0.600467\pi$
$659$	10830.2	−	10830.2i	0.640189	−	0.640189i	0.950602	+	0.310413i	$0.100467\pi$
$660$		0			0
$661$	4221.15	+	4221.15i	0.248387	+	0.248387i	0.820308	−	0.571922i	$-0.193802\pi$
$661$	4221.15	+	4221.15i	0.248387	+	0.248387i	−0.571922	+	0.820308i	$0.693802\pi$
$662$	−2676.89	−	16079.2i	−0.157161	−	0.944014i
$663$		0			0
$664$	−9154.31	+	4940.63i	−0.535024	+	0.288755i
$665$	22550.3			1.31498
$666$		0			0
$667$	29055.8	−	29055.8i	1.68672	−	1.68672i
$668$	−9428.57	+	3228.85i	−0.546111	+	0.187018i
$669$		0			0
$670$	10880.2	−	15226.5i	0.627371	−	0.877985i
$671$	−368.761			−0.0212159
$672$		0			0
$673$	610.423			0.0349630			0.0174815	−	0.999847i	$-0.494435\pi$
$673$	610.423			0.0349630			0.0174815	+	0.999847i	$0.494435\pi$
$674$	−1540.05	+	2155.26i	−0.0880129	+	0.123171i
$675$		0			0
$676$	5896.57	−	2019.30i	0.335490	−	0.114890i
$677$	6186.42	−	6186.42i	0.351202	−	0.351202i	−0.509355	−	0.860557i	$-0.670116\pi$
$677$	6186.42	−	6186.42i	0.351202	−	0.351202i	0.860557	+	0.509355i	$0.170116\pi$
$678$		0			0
$679$	29884.8			1.68906
$680$	−9193.49	−	17034.3i	−0.518462	−	0.960638i
$681$		0			0
$682$	51.3054	+	308.175i	0.00288062	+	0.0173030i
$683$	−23478.3	−	23478.3i	−1.31533	−	1.31533i	−0.917425	−	0.397908i	$-0.869736\pi$
$683$	−23478.3	−	23478.3i	−1.31533	−	1.31533i	−0.397908	−	0.917425i	$-0.630264\pi$
$684$		0			0
$685$	−12760.3	+	12760.3i	−0.711744	+	0.711744i
$686$	−10416.2	+	14577.1i	−0.579726	+	0.811307i
$687$		0			0
$688$	2191.45	−	17366.5i	0.121436	−	0.962345i
$689$			13260.3i			0.733206i
$690$		0			0
$691$	19296.9	+	19296.9i	1.06235	+	1.06235i	0.997922	+	0.0644325i	$0.0205237\pi$
$691$	19296.9	+	19296.9i	1.06235	+	1.06235i	0.0644325	+	0.997922i	$0.479476\pi$
$692$	5448.96	−	11124.7i	0.299333	−	0.611122i
$693$		0			0
$694$	3130.51	−	521.171i	0.171228	−	0.0285063i
$695$		−	29603.8i		−	1.61573i
$696$		0			0
$697$		−	5209.98i		−	0.283131i
$698$	260.286	+	1563.46i	0.0141146	+	0.0847818i
$699$		0			0
$700$	4260.43	−	1459.00i	0.230042	−	0.0787788i
$701$	5595.13	+	5595.13i	0.301462	+	0.301462i	0.841586	−	0.540123i	$-0.181622\pi$
$701$	5595.13	+	5595.13i	0.301462	+	0.301462i	−0.540123	+	0.841586i	$0.681622\pi$
$702$		0			0
$703$		−	19149.1i		−	1.02734i
$704$	−60.0163	−	289.503i	−0.00321300	−	0.0154987i
$705$		0			0
$706$	11588.3	+	8280.53i	0.617751	+	0.441419i
$707$	35234.8	−	35234.8i	1.87432	−	1.87432i
$708$		0			0
$709$	−7035.82	−	7035.82i	−0.372688	−	0.372688i	0.495767	−	0.868455i	$-0.334887\pi$
$709$	−7035.82	−	7035.82i	−0.372688	−	0.372688i	−0.868455	+	0.495767i	$0.834887\pi$
$710$	10024.9	−	1668.95i	0.529896	−	0.0882178i
$711$		0			0
$712$	−21794.9	−	6515.57i	−1.14719	−	0.342951i
$713$	−33218.2			−1.74479
$714$		0			0
$715$	−158.457	+	158.457i	−0.00828805	+	0.00828805i
$716$	−3464.48	−	1696.93i	−0.180829	−	0.0885717i
$717$		0			0
$718$	−7484.61	−	5348.19i	−0.389030	−	0.277984i
$719$	20454.1			1.06093			0.530465	−	0.847707i	$-0.322017\pi$
$719$	20454.1			1.06093			0.530465	+	0.847707i	$0.322017\pi$
$720$		0			0
$721$	−29336.2			−1.51531
$722$	3507.14	+	2506.05i	0.180778	+	0.129177i
$723$		0			0
$724$	−11234.1	+	22935.7i	−0.576673	+	1.17734i
$725$	3143.86	−	3143.86i	0.161048	−	0.161048i
$726$		0			0
$727$	−4127.12			−0.210545			−0.105273	−	0.994443i	$-0.533572\pi$
$727$	−4127.12			−0.210545			−0.105273	+	0.994443i	$0.533572\pi$
$728$	−22462.2	+	12123.0i	−1.14355	+	0.617180i
$729$		0			0
$730$	26546.0	−	4419.42i	1.34591	−	0.224069i
$731$	−16053.4	−	16053.4i	−0.812251	−	0.812251i
$732$		0			0
$733$	4302.46	−	4302.46i	0.216801	−	0.216801i	−0.590348	−	0.807149i	$-0.701009\pi$
$733$	4302.46	−	4302.46i	0.216801	−	0.216801i	0.807149	+	0.590348i	$0.201009\pi$
$734$	−13414.4	−	9585.35i	−0.674569	−	0.482019i
$735$		0			0
$736$	31412.0	−	1239.66i	1.57318	−	0.0620847i
$737$		−	370.738i		−	0.0185296i
$738$		0			0
$739$	−12198.1	−	12198.1i	−0.607190	−	0.607190i	0.335021	−	0.942211i	$-0.391257\pi$
$739$	−12198.1	−	12198.1i	−0.607190	−	0.607190i	−0.942211	+	0.335021i	$0.891257\pi$
$740$	7002.85	+	20449.0i	0.347878	+	1.01584i
$741$		0			0
$742$	4900.14	+	29433.6i	0.242439	+	1.45625i
$743$		−	15022.3i		−	0.741742i	−0.928684	−	0.370871i	$-0.879059\pi$
$743$		−	15022.3i		−	0.741742i	0.928684	−	0.370871i	$-0.120941\pi$
$744$		0			0
$745$			207.040i			0.0101817i
$746$	−12576.0	+	2093.67i	−0.617213	+	0.102755i
$747$		0			0
$748$	−344.376	−	168.678i	−0.0168337	−	0.00824531i
$749$	14506.3	+	14506.3i	0.707677	+	0.707677i
$750$		0			0
$751$		−	20203.5i		−	0.981673i	−0.871252	−	0.490836i	$-0.836691\pi$
$751$		−	20203.5i		−	0.981673i	0.871252	−	0.490836i	$-0.163309\pi$
$752$	−4708.45	+	3653.30i	−0.228324	+	0.177157i
$753$		0			0
$754$	−14651.1	+	20503.8i	−0.707642	+	0.990322i
$755$	−16143.8	+	16143.8i	−0.778191	+	0.778191i
$756$		0			0
$757$	−8332.94	−	8332.94i	−0.400087	−	0.400087i	0.478177	−	0.878264i	$-0.341298\pi$
$757$	−8332.94	−	8332.94i	−0.400087	−	0.400087i	−0.878264	+	0.478177i	$0.841298\pi$
$758$	104.568	+	628.108i	0.00501067	+	0.0300975i
$759$		0			0
$760$	16319.1	+	4878.59i	0.778890	+	0.232849i
$761$	13643.0			0.649878			0.324939	−	0.945735i	$-0.394656\pi$
$761$	13643.0			0.649878			0.324939	+	0.945735i	$0.394656\pi$
$762$		0			0
$763$	−21551.0	+	21551.0i	−1.02254	+	1.02254i
$764$	−12113.4	−	35372.3i	−0.573622	−	1.67503i
$765$		0			0
$766$	−54.0513	+	75.6431i	−0.00254955	+	0.00356801i
$767$	1310.23			0.0616814
$768$		0			0
$769$	−18855.2			−0.884180			−0.442090	−	0.896971i	$-0.645763\pi$
$769$	−18855.2			−0.884180			−0.442090	+	0.896971i	$0.645763\pi$
$770$	−293.167	+	410.277i	−0.0137208	+	0.0192018i
$771$		0			0
$772$	2652.60	+	7745.85i	0.123665	+	0.361113i
$773$	−9480.13	+	9480.13i	−0.441108	+	0.441108i	−0.892384	−	0.451276i	$-0.850969\pi$
$773$	−9480.13	+	9480.13i	−0.441108	+	0.441108i	0.451276	+	0.892384i	$0.350969\pi$
$774$		0			0
$775$	−3594.25			−0.166592
$776$	21626.9	+	6465.35i	1.00046	+	0.299089i
$777$		0			0
$778$	−230.459	−	1384.29i	−0.0106200	−	0.0637909i
$779$	3241.69	+	3241.69i	0.149096	+	0.149096i
$780$		0			0
$781$	142.362	−	142.362i	0.00652255	−	0.00652255i
$782$	23704.7	−	33174.0i	1.08399	−	1.51701i
$783$		0			0
$784$	−28035.2	+	21752.6i	−1.27711	+	0.990915i
$785$		−	5252.39i		−	0.238810i
$786$		0			0
$787$	491.764	+	491.764i	0.0222738	+	0.0222738i	0.718156	−	0.695882i	$-0.244986\pi$
$787$	491.764	+	491.764i	0.0222738	+	0.0222738i	−0.695882	+	0.718156i	$0.744986\pi$
$788$	−37232.5	−	18236.8i	−1.68319	−	0.824441i
$789$		0			0
$790$	−28462.3	+	4738.45i	−1.28183	+	0.213401i
$791$		−	41831.2i		−	1.88034i
$792$		0			0
$793$			24046.3i			1.07681i
$794$	876.326	+	5263.81i	0.0391683	+	0.235272i
$795$		0			0
$796$	−6330.69	−	18486.2i	−0.281891	−	0.823149i
$797$	22555.1	+	22555.1i	1.00244	+	1.00244i	0.999997	+	0.00244147i	$0.000777144\pi$
$797$	22555.1	+	22555.1i	1.00244	+	1.00244i	0.00244147	+	0.999997i	$0.499223\pi$
$798$		0			0
$799$			7729.47i			0.342239i
$800$	3398.81	−	134.132i	0.150208	−	0.00592785i
$801$		0			0
$802$	35184.3	+	25141.2i	1.54913	+	1.10694i
$803$	376.978	−	376.978i	0.0165669	−	0.0165669i
$804$		0			0
$805$	−37912.2	−	37912.2i	−1.65991	−	1.65991i
$806$	20095.6	−	3345.54i	0.878209	−	0.146205i
$807$		0			0
$808$	33121.3	−	17875.8i	1.44208	−	0.778301i
$809$	2649.70			0.115153			0.0575764	−	0.998341i	$-0.481663\pi$
$809$	2649.70			0.115153			0.0575764	+	0.998341i	$0.481663\pi$
$810$		0			0
$811$	14726.8	−	14726.8i	0.637644	−	0.637644i	−0.312330	−	0.949974i	$-0.601109\pi$
$811$	14726.8	−	14726.8i	0.637644	−	0.637644i	0.949974	+	0.312330i	$0.101109\pi$
$812$	−24943.8	+	50925.6i	−1.07803	+	2.20091i
$813$		0			0
$814$	348.396	+	248.949i	0.0150016	+	0.0107195i
$815$	2994.92			0.128721
$816$		0			0
$817$	19977.1			0.855458
$818$	19998.5	+	14290.1i	0.854805	+	0.610808i
$819$		0			0
$820$	−4647.23	−	2276.26i	−0.197913	−	0.0969394i
$821$	19231.4	−	19231.4i	0.817518	−	0.817518i	−0.168230	−	0.985748i	$-0.553805\pi$
$821$	19231.4	−	19231.4i	0.817518	−	0.817518i	0.985748	+	0.168230i	$0.0538051\pi$
$822$		0			0
$823$	285.189			0.0120790			0.00603952	−	0.999982i	$-0.498078\pi$
$823$	285.189			0.0120790			0.00603952	+	0.999982i	$0.498078\pi$
$824$	−21229.9	−	6346.66i	−0.897545	−	0.268321i
$825$		0			0
$826$	2908.28	−	484.174i	0.122508	−	0.0203954i
$827$	7276.50	+	7276.50i	0.305960	+	0.305960i	0.843340	−	0.537380i	$-0.180586\pi$
$827$	7276.50	+	7276.50i	0.305960	+	0.305960i	−0.537380	+	0.843340i	$0.680586\pi$
$828$		0			0
$829$	14705.2	−	14705.2i	0.616084	−	0.616084i	−0.328441	−	0.944525i	$-0.606523\pi$
$829$	14705.2	−	14705.2i	0.616084	−	0.616084i	0.944525	+	0.328441i	$0.106523\pi$
$830$	10903.2	+	7790.97i	0.455970	+	0.325817i
$831$		0			0
$832$	−18878.0	+	3913.56i	−0.786632	+	0.163075i
$833$			46023.1i			1.91429i
$834$		0			0
$835$	9078.26	+	9078.26i	0.376247	+	0.376247i
$836$	319.226	−	109.320i	0.0132065	−	0.00452264i
$837$		0			0
$838$	−3542.61	−	21279.3i	−0.146035	−	0.877187i
$839$			42053.0i			1.73043i	0.501400	+	0.865216i	$0.332819\pi$
$839$			42053.0i			1.73043i	−0.501400	+	0.865216i	$0.667181\pi$
$840$		0			0
$841$			31596.6i			1.29553i
$842$	−4853.13	+	807.957i	−0.198634	+	0.0330689i
$843$		0			0
$844$	13037.4	−	26617.4i	0.531714	−	1.08555i
$845$	−5677.48	−	5677.48i	−0.231138	−	0.231138i
$846$		0			0
$847$		−	39863.3i		−	1.61714i
$848$	−2821.62	+	22360.4i	−0.114263	+	0.905496i
$849$		0			0
$850$	2564.87	−	3589.45i	0.103499	−	0.144844i
$851$	−32194.0	+	32194.0i	−1.29682	+	1.29682i
$852$		0			0
$853$	−21450.1	−	21450.1i	−0.861004	−	0.861004i	0.130450	−	0.991455i	$-0.458358\pi$
$853$	−21450.1	−	21450.1i	−0.861004	−	0.861004i	−0.991455	+	0.130450i	$0.958358\pi$
$854$	8885.91	+	53374.8i	0.356054	+	2.13870i
$855$		0			0
$856$	7359.54	+	13636.2i	0.293860	+	0.544481i
$857$	−24889.9			−0.992093			−0.496046	−	0.868296i	$-0.665215\pi$
$857$	−24889.9			−0.992093			−0.496046	+	0.868296i	$0.665215\pi$
$858$		0			0
$859$	−21885.6	+	21885.6i	−0.869296	+	0.869296i	−0.992395	−	0.123098i	$-0.960717\pi$
$859$	−21885.6	+	21885.6i	−0.869296	+	0.869296i	0.123098	+	0.992395i	$0.460717\pi$
$860$	−21333.2	+	7305.63i	−0.845878	+	0.289675i
$861$		0			0
$862$	−8497.21	+	11891.6i	−0.335750	+	0.469870i
$863$	45740.0			1.80418			0.902089	−	0.431549i	$-0.142033\pi$
$863$	45740.0			1.80418			0.902089	+	0.431549i	$0.142033\pi$
$864$		0			0
$865$	−15957.8			−0.627263
$866$	13197.1	−	18468.9i	0.517848	−	0.724711i
$867$		0			0
$868$	43369.2	−	14852.0i	1.69591	−	0.580771i
$869$	−404.191	+	404.191i	−0.0157782	+	0.0157782i
$870$		0			0
$871$	−24175.2			−0.940466
$872$	−20258.3	+	10933.5i	−0.786733	+	0.424604i
$873$		0			0
$874$	5891.84	+	35390.4i	0.228026	+	1.36968i
$875$	−31390.7	−	31390.7i	−1.21280	−	1.21280i
$876$		0			0
$877$	10214.3	−	10214.3i	0.393286	−	0.393286i	−0.482571	−	0.875857i	$-0.660297\pi$
$877$	10214.3	−	10214.3i	0.393286	−	0.393286i	0.875857	+	0.482571i	$0.160297\pi$
$878$	4685.38	−	6557.03i	0.180095	−	0.252037i
$879$		0			0
$880$	−300.918	+	233.483i	−0.0115272	+	0.00894398i
$881$			29167.7i			1.11542i	0.830036	+	0.557709i	$0.188320\pi$
$881$			29167.7i			1.11542i	−0.830036	+	0.557709i	$0.811680\pi$
$882$		0			0
$883$	2295.54	+	2295.54i	0.0874871	+	0.0874871i	0.749496	−	0.662009i	$-0.230296\pi$
$883$	2295.54	+	2295.54i	0.0874871	+	0.0874871i	−0.662009	+	0.749496i	$0.730296\pi$
$884$	−10999.2	+	22456.2i	−0.418489	+	0.854392i
$885$		0			0
$886$	−46355.8	+	7717.38i	−1.75774	+	0.292630i
$887$		−	25089.5i		−	0.949742i	−0.880055	−	0.474871i	$-0.842495\pi$
$887$		−	25089.5i		−	0.949742i	0.880055	−	0.474871i	$-0.157505\pi$
$888$		0			0
$889$		−	69670.3i		−	2.62842i
$890$	4812.41	+	28906.6i	0.181250	+	1.08871i
$891$		0			0
$892$	−22727.1	+	7782.99i	−0.853093	+	0.292145i
$893$	−4809.33	−	4809.33i	−0.180222	−	0.180222i
$894$		0			0
$895$			4969.64i			0.185605i
$896$	−40456.8	+	15662.9i	−1.50844	+	0.583996i
$897$		0			0
$898$	33082.1	+	23639.0i	1.22936	+	0.878447i
$899$	32003.0	−	32003.0i	1.18728	−	1.18728i
$900$		0			0
$901$	20669.7	+	20669.7i	0.764269	+	0.764269i
$902$	−101.123	+	16.8350i	−0.00373283	+	0.000621447i
$903$		0			0
$904$	9049.86	−	30272.2i	0.332958	−	1.11376i
$905$	32900.2			1.20844
$906$		0			0
$907$	20254.3	−	20254.3i	0.741492	−	0.741492i	−0.231373	−	0.972865i	$-0.574322\pi$
$907$	20254.3	−	20254.3i	0.741492	−	0.741492i	0.972865	+	0.231373i	$0.0743219\pi$
$908$	22266.6	+	10906.4i	0.813813	+	0.398613i
$909$		0			0
$910$	26753.5	+	19116.9i	0.974582	+	0.696395i
$911$	−17166.4			−0.624310			−0.312155	−	0.950031i	$-0.601051\pi$
$911$	−17166.4			−0.624310			−0.312155	+	0.950031i	$0.601051\pi$
$912$		0			0
$913$	265.474			0.00962312
$914$	−289.801	−	207.080i	−0.0104877	−	0.00749408i
$915$		0			0
$916$	8993.94	−	18362.2i	0.324419	−	0.662339i
$917$	12204.0	−	12204.0i	0.439490	−	0.439490i
$918$		0			0
$919$	23560.5			0.845690			0.422845	−	0.906202i	$-0.361031\pi$
$919$	23560.5			0.845690			0.422845	+	0.906202i	$0.361031\pi$
$920$	−19234.1	−	35638.1i	−0.689271	−	1.27712i
$921$		0			0
$922$	−706.282	+	117.583i	−0.0252279	+	0.00419998i
$923$	−9283.18	−	9283.18i	−0.331051	−	0.331051i
$924$		0			0
$925$	−3483.42	+	3483.42i	−0.123821	+	0.123821i
$926$	−3660.96	−	2615.97i	−0.129921	−	0.0928359i
$927$		0			0
$928$	−29068.6	+	31457.2i	−1.02826	+	1.11275i
$929$			10039.3i			0.354553i	0.984161	+	0.177276i	$0.0567287\pi$
$929$			10039.3i			0.354553i	−0.984161	+	0.177276i	$0.943271\pi$
$930$		0			0
$931$	−28635.9	−	28635.9i	−1.00806	−	1.00806i
$932$	2876.25	+	8398.93i	0.101089	+	0.295189i
$933$		0			0
$934$	−7526.43	−	45208.8i	−0.263675	−	1.58381i
$935$			493.992i			0.0172784i
$936$		0			0
$937$			6962.35i			0.242743i	0.992607	+	0.121371i	$0.0387292\pi$
$937$			6962.35i			0.242743i	−0.992607	+	0.121371i	$0.961271\pi$
$938$	−53661.0	+	8933.55i	−1.86790	+	0.310971i
$939$		0			0
$940$	6894.59	+	3377.03i	0.239230	+	0.117177i
$941$	−7675.75	−	7675.75i	−0.265911	−	0.265911i	0.561539	−	0.827450i	$-0.310209\pi$
$941$	−7675.75	−	7675.75i	−0.265911	−	0.265911i	−0.827450	+	0.561539i	$0.810209\pi$
$942$		0			0
$943$		−	10900.0i		−	0.376409i
$944$	2209.39	+	278.799i	0.0761755	+	0.00961242i
$945$		0			0
$946$	−259.713	+	363.460i	−0.00892600	+	0.0124916i
$947$	7623.19	−	7623.19i	0.261584	−	0.261584i	−0.564113	−	0.825697i	$-0.690782\pi$
$947$	7623.19	−	7623.19i	0.261584	−	0.261584i	0.825697	+	0.564113i	$0.190782\pi$
$948$		0			0
$949$	−24582.1	−	24582.1i	−0.840851	−	0.840851i
$950$	637.502	+	3829.27i	0.0217719	+	0.130777i
$951$		0			0
$952$	−16116.3	+	53909.9i	−0.548670	+	1.83532i
$953$	−6874.60			−0.233673			−0.116836	−	0.993151i	$-0.537275\pi$
$953$	−6874.60			−0.233673			−0.116836	+	0.993151i	$0.537275\pi$
$954$		0			0
$955$	−34058.0	+	34058.0i	−1.15402	+	1.15402i
$956$	1222.90	+	3570.98i	0.0413717	+	0.120809i
$957$		0			0
$958$	−11638.1	+	16287.1i	−0.392494	+	0.549282i
$959$	52456.2			1.76632
$960$		0			0
$961$	−6796.76			−0.228148
$962$	16233.5	−	22718.3i	0.544065	−	0.761401i
$963$		0			0
$964$	−3905.21	−	11403.6i	−0.130476	−	0.381001i
$965$	7458.05	−	7458.05i	0.248791	−	0.248791i
$966$		0			0
$967$	41309.2			1.37375			0.686874	−	0.726777i	$-0.258982\pi$
$967$	41309.2			1.37375			0.686874	+	0.726777i	$0.258982\pi$
$968$	8624.11	−	28848.0i	0.286353	−	0.957863i
$969$		0			0
$970$	−4775.32	−	28683.8i	−0.158068	−	0.949466i
$971$	−29768.1	−	29768.1i	−0.983835	−	0.983835i	0.0160360	−	0.999871i	$-0.494895\pi$
$971$	−29768.1	−	29768.1i	−0.983835	−	0.983835i	−0.999871	+	0.0160360i	$0.994895\pi$
$972$		0			0
$973$	−60849.1	+	60849.1i	−2.00486	+	2.00486i
$974$	26619.0	−	37252.4i	0.875696	−	1.22551i
$975$		0			0
$976$	−5116.72	+	40548.4i	−0.167810	+	1.32984i
$977$			38173.7i			1.25004i	0.780610	+	0.625018i	$0.214908\pi$
$977$			38173.7i			1.25004i	−0.780610	+	0.625018i	$0.785092\pi$
$978$		0			0
$979$	410.500	+	410.500i	0.0134011	+	0.0134011i
$980$	41052.0	+	20107.6i	1.33812	+	0.655423i
$981$		0			0
$982$	12068.0	−	2009.09i	0.392163	−	0.0652878i
$983$			22215.4i			0.720815i	0.932795	+	0.360408i	$0.117362\pi$
$983$			22215.4i			0.720815i	−0.932795	+	0.360408i	$0.882638\pi$
$984$		0			0
$985$			53408.4i			1.72765i
$986$	9122.83	+	54797.9i	0.294655	+	1.76990i
$987$		0			0
$988$	−7128.60	−	20816.2i	−0.229546	−	0.670295i
$989$	−33585.9	−	33585.9i	−1.07985	−	1.07985i
$990$		0			0
$991$			11237.5i			0.360213i	0.983647	+	0.180107i	$0.0576442\pi$
$991$			11237.5i			0.360213i	−0.983647	+	0.180107i	$0.942356\pi$
$992$	34598.3	−	1365.40i	1.10736	−	0.0437011i
$993$		0			0
$994$	−24036.0	−	17175.1i	−0.766979	−	0.548051i
$995$	−17799.4	+	17799.4i	−0.567114	+	0.567114i
$996$		0			0
$997$	−10953.4	−	10953.4i	−0.347941	−	0.347941i	0.511401	−	0.859342i	$-0.329127\pi$
$997$	−10953.4	−	10953.4i	−0.347941	−	0.347941i	−0.859342	+	0.511401i	$0.829127\pi$
$998$	2867.17	−	477.331i	0.0909406	−	0.0151399i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.4.l.a.107.9	yes	48
3.2	odd	2	inner	144.4.l.a.107.16	yes	48
4.3	odd	2		576.4.l.a.143.6		48
8.3	odd	2		1152.4.l.a.287.19		48
8.5	even	2		1152.4.l.b.287.19		48
12.11	even	2		576.4.l.a.143.19		48
16.3	odd	4	inner	144.4.l.a.35.16	yes	48
16.5	even	4		1152.4.l.a.863.6		48
16.11	odd	4		1152.4.l.b.863.6		48
16.13	even	4		576.4.l.a.431.19		48
24.5	odd	2		1152.4.l.b.287.6		48
24.11	even	2		1152.4.l.a.287.6		48
48.5	odd	4		1152.4.l.a.863.19		48
48.11	even	4		1152.4.l.b.863.19		48
48.29	odd	4		576.4.l.a.431.6		48
48.35	even	4	inner	144.4.l.a.35.9	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.4.l.a.35.9	✓	48	48.35	even	4	inner
144.4.l.a.35.16	yes	48	16.3	odd	4	inner
144.4.l.a.107.9	yes	48	1.1	even	1	trivial
144.4.l.a.107.16	yes	48	3.2	odd	2	inner
576.4.l.a.143.6		48	4.3	odd	2
576.4.l.a.143.19		48	12.11	even	2
576.4.l.a.431.6		48	48.29	odd	4
576.4.l.a.431.19		48	16.13	even	4
1152.4.l.a.287.6		48	24.11	even	2
1152.4.l.a.287.19		48	8.3	odd	2
1152.4.l.a.863.6		48	16.5	even	4
1152.4.l.a.863.19		48	48.5	odd	4
1152.4.l.b.287.6		48	24.5	odd	2
1152.4.l.b.287.19		48	8.5	even	2
1152.4.l.b.863.6		48	16.11	odd	4
1152.4.l.b.863.19		48	48.11	even	4

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)

\(f(q)\)	\(=\)	\(q+(-1.64440 + 2.30129i) q^{2} +(-2.59187 - 7.56850i) q^{4} +(-7.28730 + 7.28730i) q^{5} +29.9574 q^{7} +(21.6794 + 6.48105i) q^{8} +(-4.78691 - 28.7535i) q^{10} +(-0.408325 - 0.408325i) q^{11} +(-26.6262 + 26.6262i) q^{13} +(-49.2621 + 68.9406i) q^{14} +(-50.5645 + 39.2331i) q^{16} +83.0075i q^{17} +(-51.6479 - 51.6479i) q^{19} +(74.0416 + 36.2662i) q^{20} +(1.61113 - 0.268222i) q^{22} +173.664i q^{23} +18.7906i q^{25} +(-17.4903 - 105.059i) q^{26} +(-77.6455 - 226.732i) q^{28} +(-167.311 - 167.311i) q^{29} +191.279i q^{31} +(-7.13826 - 180.879i) q^{32} +(-191.024 - 136.498i) q^{34} +(-218.308 + 218.308i) q^{35} +(185.381 + 185.381i) q^{37} +(203.787 - 33.9267i) q^{38} +(-205.214 + 110.755i) q^{40} -62.7651 q^{41} +(-193.397 + 193.397i) q^{43} +(-2.03209 + 4.14873i) q^{44} +(-399.650 - 285.573i) q^{46} +93.1177 q^{47} +554.445 q^{49} +(-43.2425 - 30.8993i) q^{50} +(270.532 + 132.509i) q^{52} +(249.010 - 249.010i) q^{53} +5.95117 q^{55} +(649.458 + 194.155i) q^{56} +(660.156 - 109.904i) q^{58} +(-24.6042 - 24.6042i) q^{59} +(451.553 - 451.553i) q^{61} +(-440.189 - 314.541i) q^{62} +(427.992 + 281.010i) q^{64} -388.066i q^{65} +(453.974 + 453.974i) q^{67} +(628.242 - 215.144i) q^{68} +(-143.403 - 861.378i) q^{70} +348.649i q^{71} +923.229i q^{73} +(-731.458 + 121.774i) q^{74} +(-257.033 + 524.762i) q^{76} +(-12.2323 - 12.2323i) q^{77} -989.875i q^{79} +(82.5751 - 654.382i) q^{80} +(103.211 - 144.441i) q^{82} +(-325.077 + 325.077i) q^{83} +(-604.900 - 604.900i) q^{85} +(-127.039 - 763.084i) q^{86} +(-6.20586 - 11.4986i) q^{88} -1005.33 q^{89} +(-797.651 + 797.651i) q^{91} +(1314.37 - 450.113i) q^{92} +(-153.123 + 214.291i) q^{94} +752.747 q^{95} +997.579 q^{97} +(-911.731 + 1275.94i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 120 q^{10} - 144 q^{16} - 48 q^{19} + 72 q^{22} + 72 q^{28} - 984 q^{34} - 1272 q^{40} + 864 q^{43} - 1416 q^{46} + 2352 q^{49} - 648 q^{52} - 576 q^{55} + 1128 q^{58} + 1824 q^{61} + 3024 q^{64}+ \cdots - 11304 q^{94}+O(q^{100}) \) 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

Fields

Representations

Groups

Database

Properties

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