LMFDB - Embedded newform 39.4.b.a.25.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [39,4,Mod(25,39)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(39, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("39.25");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 39 = 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	39.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$2.30107449022$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.5054412.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 29x^{2} + 48 $ x^4 + 29*x^2 + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			25.2
Root			$-1.32750i$ of defining polynomial
Character	$\chi$	$=$	39.25
Dual form			39.4.b.a.25.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.32750i q^{2} -3.00000 q^{3} +6.23774 q^{4} -15.4241i q^{5} +3.98251i q^{6} -7.96501i q^{7} -18.9006i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-1.32750i q^{2} -3.00000 q^{3} +6.23774 q^{4} -15.4241i q^{5} +3.98251i q^{6} -7.96501i q^{7} -18.9006i q^{8} +9.00000 q^{9} -20.4755 q^{10} +12.7691i q^{11} -18.7132 q^{12} +(7.47548 + 46.2722i) q^{13} -10.5736 q^{14} +46.2722i q^{15} +24.8113 q^{16} +54.0000 q^{17} -11.9475i q^{18} +84.5794i q^{19} -96.2113i q^{20} +23.8950i q^{21} +16.9510 q^{22} -122.853 q^{23} +56.7019i q^{24} -112.902 q^{25} +(61.4264 - 9.92371i) q^{26} -27.0000 q^{27} -49.6837i q^{28} +140.853 q^{29} +61.4264 q^{30} +116.439i q^{31} -184.142i q^{32} -38.3072i q^{33} -71.6851i q^{34} -122.853 q^{35} +56.1397 q^{36} -433.898i q^{37} +112.279 q^{38} +(-22.4264 - 138.817i) q^{39} -291.525 q^{40} +205.823i q^{41} +31.7207 q^{42} +418.853 q^{43} +79.6501i q^{44} -138.817i q^{45} +163.087i q^{46} +485.861i q^{47} -74.4339 q^{48} +279.559 q^{49} +149.877i q^{50} -162.000 q^{51} +(46.6301 + 288.634i) q^{52} -674.559 q^{53} +35.8425i q^{54} +196.951 q^{55} -150.544 q^{56} -253.738i q^{57} -186.982i q^{58} +186.226i q^{59} +288.634i q^{60} -671.902 q^{61} +154.574 q^{62} -71.6851i q^{63} -45.9584 q^{64} +(713.706 - 115.302i) q^{65} -50.8529 q^{66} -14.0364i q^{67} +336.838 q^{68} +368.559 q^{69} +163.087i q^{70} +346.789i q^{71} -170.106i q^{72} -832.900i q^{73} -576.000 q^{74} +338.706 q^{75} +527.584i q^{76} +101.706 q^{77} +(-184.279 + 29.7711i) q^{78} -335.608 q^{79} -382.691i q^{80} +81.0000 q^{81} +273.230 q^{82} -568.797i q^{83} +149.051i q^{84} -832.900i q^{85} -556.028i q^{86} -422.559 q^{87} +241.343 q^{88} -236.671i q^{89} -184.279 q^{90} +(368.559 - 59.5423i) q^{91} -766.324 q^{92} -349.318i q^{93} +644.981 q^{94} +1304.56 q^{95} +552.426i q^{96} +1278.94i q^{97} -371.115i q^{98} +114.922i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} - 26 q^{4} + 36 q^{9}+O(q^{10}) $ 4 * q - 12 * q^3 - 26 * q^4 + 36 * q^9	$ 4 q - 12 q^{3} - 26 q^{4} + 36 q^{9} + 20 q^{10} + 78 q^{12} - 72 q^{13} - 348 q^{14} + 354 q^{16} + 216 q^{17} - 136 q^{22} + 120 q^{23} - 44 q^{25} - 60 q^{26} - 108 q^{27} - 48 q^{29} - 60 q^{30} + 120 q^{35} - 234 q^{36} - 468 q^{38} + 216 q^{39} - 1268 q^{40} + 1044 q^{42} + 1064 q^{43} - 1062 q^{48} - 716 q^{49} - 648 q^{51} + 1766 q^{52} - 864 q^{53} + 584 q^{55} + 3372 q^{56} - 2280 q^{61} + 924 q^{62} - 1050 q^{64} + 1632 q^{65} + 408 q^{66} - 1404 q^{68} - 360 q^{69} - 2304 q^{74} + 132 q^{75} - 816 q^{77} + 180 q^{78} + 288 q^{79} + 324 q^{81} - 28 q^{82} + 144 q^{87} + 2392 q^{88} + 180 q^{90} - 360 q^{91} - 8568 q^{92} + 6656 q^{94} + 3384 q^{95}+O(q^{100}) $ 4 * q - 12 * q^3 - 26 * q^4 + 36 * q^9 + 20 * q^10 + 78 * q^12 - 72 * q^13 - 348 * q^14 + 354 * q^16 + 216 * q^17 - 136 * q^22 + 120 * q^23 - 44 * q^25 - 60 * q^26 - 108 * q^27 - 48 * q^29 - 60 * q^30 + 120 * q^35 - 234 * q^36 - 468 * q^38 + 216 * q^39 - 1268 * q^40 + 1044 * q^42 + 1064 * q^43 - 1062 * q^48 - 716 * q^49 - 648 * q^51 + 1766 * q^52 - 864 * q^53 + 584 * q^55 + 3372 * q^56 - 2280 * q^61 + 924 * q^62 - 1050 * q^64 + 1632 * q^65 + 408 * q^66 - 1404 * q^68 - 360 * q^69 - 2304 * q^74 + 132 * q^75 - 816 * q^77 + 180 * q^78 + 288 * q^79 + 324 * q^81 - 28 * q^82 + 144 * q^87 + 2392 * q^88 + 180 * q^90 - 360 * q^91 - 8568 * q^92 + 6656 * q^94 + 3384 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/39\mathbb{Z}\right)^\times$.

$n$	$14$	$28$
$\chi(n)$	$1$	$-1$

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 $

$2$		−	1.32750i		−	0.469343i	−0.972075	−	0.234671i	$-0.924599\pi$
$2$		−	1.32750i		−	0.469343i	0.972075	−	0.234671i	$-0.0754014\pi$
$3$	−3.00000			−0.577350
$3$	−3.00000			−0.577350
$4$	6.23774			0.779717
$5$		−	15.4241i		−	1.37957i	−0.724014	−	0.689785i	$-0.757705\pi$
$5$		−	15.4241i		−	1.37957i	0.724014	−	0.689785i	$-0.242295\pi$
$6$			3.98251i			0.270975i
$7$		−	7.96501i		−	0.430070i	−0.976606	−	0.215035i	$-0.931013\pi$
$7$		−	7.96501i		−	0.430070i	0.976606	−	0.215035i	$-0.0689866\pi$
$8$		−	18.9006i		−	0.835297i
$9$	9.00000			0.333333
$10$	−20.4755			−0.647491
$11$			12.7691i			0.350002i	0.984568	+	0.175001i	$0.0559928\pi$
$11$			12.7691i			0.350002i	−0.984568	+	0.175001i	$0.944007\pi$
$12$	−18.7132			−0.450170
$13$	7.47548	+	46.2722i	0.159487	+	0.987200i
$13$	7.47548	+	46.2722i	0.159487	+	0.987200i
$14$	−10.5736			−0.201850
$15$			46.2722i			0.796496i
$16$	24.8113			0.387677
$17$	54.0000			0.770407			0.385204	−	0.922832i	$-0.374131\pi$
$17$	54.0000			0.770407			0.385204	+	0.922832i	$0.374131\pi$
$18$		−	11.9475i		−	0.156448i
$19$			84.5794i			1.02126i	0.859802	+	0.510628i	$0.170587\pi$
$19$			84.5794i			1.02126i	−0.859802	+	0.510628i	$0.829413\pi$
$20$		−	96.2113i		−	1.07568i
$21$			23.8950i			0.248301i
$22$	16.9510			0.164271
$23$	−122.853			−1.11376			−0.556882	−	0.830591i	$-0.688003\pi$
$23$	−122.853			−1.11376			−0.556882	+	0.830591i	$0.688003\pi$
$24$			56.7019i			0.482259i
$25$	−112.902			−0.903215
$26$	61.4264	−	9.92371i	0.463335	−	0.0748538i
$27$	−27.0000			−0.192450
$28$		−	49.6837i		−	0.335333i
$29$	140.853			0.901921			0.450961	−	0.892544i	$-0.351081\pi$
$29$	140.853			0.901921			0.450961	+	0.892544i	$0.351081\pi$
$30$	61.4264			0.373829
$31$			116.439i			0.674617i	0.941394	+	0.337309i	$0.109517\pi$
$31$			116.439i			0.674617i	−0.941394	+	0.337309i	$0.890483\pi$
$32$		−	184.142i		−	1.01725i
$33$		−	38.3072i		−	0.202074i
$34$		−	71.6851i		−	0.361585i
$35$	−122.853			−0.593312
$36$	56.1397			0.259906
$37$		−	433.898i		−	1.92790i	−0.266081	−	0.963951i	$-0.585729\pi$
$37$		−	433.898i		−	1.92790i	0.266081	−	0.963951i	$-0.414271\pi$
$38$	112.279			0.479319
$39$	−22.4264	−	138.817i	−0.0920796	−	0.569960i
$40$	−291.525			−1.15235
$41$			205.823i			0.784003i	0.919965	+	0.392002i	$0.128217\pi$
$41$			205.823i			0.784003i	−0.919965	+	0.392002i	$0.871783\pi$
$42$	31.7207			0.116538
$43$	418.853			1.48545			0.742726	−	0.669595i	$-0.233532\pi$
$43$	418.853			1.48545			0.742726	+	0.669595i	$0.233532\pi$
$44$			79.6501i			0.272902i
$45$		−	138.817i		−	0.459857i
$46$			163.087i			0.522737i
$47$			485.861i			1.50787i	0.656947	+	0.753937i	$0.271848\pi$
$47$			485.861i			1.50787i	−0.656947	+	0.753937i	$0.728152\pi$
$48$	−74.4339			−0.223825
$49$	279.559			0.815040
$50$			149.877i			0.423918i
$51$	−162.000			−0.444795
$52$	46.6301	+	288.634i	0.124354	+	0.769737i
$53$	−674.559			−1.74826			−0.874130	−	0.485693i	$-0.838567\pi$
$53$	−674.559			−1.74826			−0.874130	+	0.485693i	$0.838567\pi$
$54$			35.8425i			0.0903251i
$55$	196.951			0.482852
$56$	−150.544			−0.359236
$57$		−	253.738i		−	0.589622i
$58$		−	186.982i		−	0.423310i
$59$			186.226i			0.410925i	0.978665	+	0.205462i	$0.0658698\pi$
$59$			186.226i			0.410925i	−0.978665	+	0.205462i	$0.934130\pi$
$60$			288.634i			0.621041i
$61$	−671.902			−1.41030			−0.705149	−	0.709059i	$-0.749120\pi$
$61$	−671.902			−1.41030			−0.705149	+	0.709059i	$0.749120\pi$
$62$	154.574			0.316627
$63$		−	71.6851i		−	0.143357i
$64$	−45.9584			−0.0897626
$65$	713.706	−	115.302i	1.36191	−	0.220023i
$66$	−50.8529			−0.0948418
$67$		−	14.0364i		−	0.0255944i	−0.999918	−	0.0127972i	$-0.995926\pi$
$67$		−	14.0364i		−	0.0255944i	0.999918	−	0.0127972i	$-0.00407358\pi$
$68$	336.838			0.600700
$69$	368.559			0.643033
$70$			163.087i			0.278467i
$71$			346.789i			0.579665i	0.957077	+	0.289833i	$0.0935996\pi$
$71$			346.789i			0.579665i	−0.957077	+	0.289833i	$0.906400\pi$
$72$		−	170.106i		−	0.278432i
$73$		−	832.900i		−	1.33539i	−0.744435	−	0.667695i	$-0.767281\pi$
$73$		−	832.900i		−	1.33539i	0.744435	−	0.667695i	$-0.232719\pi$
$74$	−576.000			−0.904846
$75$	338.706			0.521472
$76$			527.584i			0.796290i
$77$	101.706			0.150525
$78$	−184.279	+	29.7711i	−0.267507	+	0.0432169i
$79$	−335.608			−0.477960			−0.238980	−	0.971025i	$-0.576813\pi$
$79$	−335.608			−0.477960			−0.238980	+	0.971025i	$0.576813\pi$
$80$		−	382.691i		−	0.534827i
$81$	81.0000			0.111111
$82$	273.230			0.367966
$83$		−	568.797i		−	0.752212i	−0.926577	−	0.376106i	$-0.877263\pi$
$83$		−	568.797i		−	0.752212i	0.926577	−	0.376106i	$-0.122737\pi$
$84$			149.051i			0.193605i
$85$		−	832.900i		−	1.06283i
$86$		−	556.028i		−	0.697186i
$87$	−422.559			−0.520725
$88$	241.343			0.292355
$89$		−	236.671i		−	0.281877i	−0.990018	−	0.140939i	$-0.954988\pi$
$89$		−	236.671i		−	0.281877i	0.990018	−	0.140939i	$-0.0450120\pi$
$90$	−184.279			−0.215830
$91$	368.559	−	59.5423i	0.424565	−	0.0685904i
$92$	−766.324			−0.868422
$93$		−	349.318i		−	0.389491i
$94$	644.981			0.707710
$95$	1304.56			1.40889
$96$			552.426i			0.587310i
$97$			1278.94i			1.33873i	0.742934	+	0.669365i	$0.233434\pi$
$97$			1278.94i			1.33873i	−0.742934	+	0.669365i	$0.766566\pi$
$98$		−	371.115i		−	0.382533i
$99$			114.922i			0.116667i
$100$	−704.253			−0.704253
$101$	−632.264			−0.622898			−0.311449	−	0.950263i	$-0.600814\pi$
$101$	−632.264			−0.622898			−0.311449	+	0.950263i	$0.600814\pi$
$102$			215.055i			0.208761i
$103$	−1506.26			−1.44094			−0.720469	−	0.693487i	$-0.756073\pi$
$103$	−1506.26			−1.44094			−0.720469	+	0.693487i	$0.756073\pi$
$104$	874.574	−	141.291i	0.824606	−	0.133219i
$105$	368.559			0.342549
$106$			895.478i			0.820533i
$107$	−1268.56			−1.14613			−0.573066	−	0.819509i	$-0.694246\pi$
$107$	−1268.56			−1.14613			−0.573066	+	0.819509i	$0.694246\pi$
$108$	−168.419			−0.150057
$109$		−	347.425i		−	0.305296i	−0.988281	−	0.152648i	$-0.951220\pi$
$109$		−	347.425i		−	0.305296i	0.988281	−	0.152648i	$-0.0487801\pi$
$110$		−	261.453i		−	0.226623i
$111$			1301.69i			1.11307i
$112$		−	197.622i		−	0.166728i
$113$	659.706			0.549203			0.274601	−	0.961558i	$-0.411454\pi$
$113$	659.706			0.549203			0.274601	+	0.961558i	$0.411454\pi$
$114$	−336.838			−0.276735
$115$			1894.89i			1.53652i
$116$	878.603			0.703244
$117$	67.2793	+	416.450i	0.0531622	+	0.329067i
$118$	247.215			0.192865
$119$		−	430.111i		−	0.331329i
$120$	874.574			0.665311
$121$	1167.95			0.877499
$122$			891.951i			0.661913i
$123$		−	617.469i		−	0.452645i
$124$			726.319i			0.526011i
$125$		−	186.602i		−	0.133521i
$126$	−95.1621			−0.0672834
$127$	−275.019			−0.192157			−0.0960787	−	0.995374i	$-0.530630\pi$
$127$	−275.019			−0.192157			−0.0960787	+	0.995374i	$0.530630\pi$
$128$		−	1412.13i		−	0.975121i
$129$	−1256.56			−0.857626
$130$	−153.064	−	947.446i	−0.103266	−	0.639204i
$131$	−1183.97			−0.789648			−0.394824	−	0.918757i	$-0.629195\pi$
$131$	−1183.97			−0.789648			−0.394824	+	0.918757i	$0.629195\pi$
$132$		−	238.950i		−	0.157560i
$133$	673.676			0.439211
$134$	−18.6334			−0.0120125
$135$			416.450i			0.265499i
$136$		−	1020.63i		−	0.643519i
$137$			2557.36i			1.59482i	0.603438	+	0.797410i	$0.293797\pi$
$137$			2557.36i			1.59482i	−0.603438	+	0.797410i	$0.706203\pi$
$138$		−	489.262i		−	0.301803i
$139$	545.736			0.333012			0.166506	−	0.986040i	$-0.446751\pi$
$139$	545.736			0.333012			0.166506	+	0.986040i	$0.446751\pi$
$140$	−766.324			−0.462616
$141$		−	1457.58i		−	0.870571i
$142$	460.362			0.272062
$143$	−590.853	+	95.4549i	−0.345522	+	0.0558205i
$144$	223.302			0.129226
$145$		−	2172.52i		−	1.24426i
$146$	−1105.68			−0.626756
$147$	−838.676			−0.470563
$148$		−	2706.54i		−	1.50322i
$149$		−	1376.78i		−	0.756981i	−0.925605	−	0.378491i	$-0.876443\pi$
$149$		−	1376.78i		−	0.756981i	0.925605	−	0.378491i	$-0.123557\pi$
$150$		−	449.632i		−	0.244749i
$151$			2733.47i			1.47316i	0.676352	+	0.736579i	$0.263560\pi$
$151$			2733.47i			1.47316i	−0.676352	+	0.736579i	$0.736440\pi$
$152$	1598.60			0.853052
$153$	486.000			0.256802
$154$		−	135.015i		−	0.0706479i
$155$	1795.97			0.930683
$156$	−139.890	−	865.902i	−0.0717961	−	0.444408i
$157$	1029.97			0.523570			0.261785	−	0.965126i	$-0.415689\pi$
$157$	1029.97			0.523570			0.261785	+	0.965126i	$0.415689\pi$
$158$			445.520i			0.224327i
$159$	2023.68			1.00936
$160$	−2840.22			−1.40337
$161$			978.524i			0.478997i
$162$		−	107.528i		−	0.0521492i
$163$		−	2882.91i		−	1.38532i	−0.721264	−	0.692660i	$-0.756439\pi$
$163$		−	2882.91i		−	1.38532i	0.721264	−	0.692660i	$-0.243561\pi$
$164$			1283.87i			0.611301i
$165$	−590.853			−0.278775
$166$	−755.079			−0.353045
$167$			1153.90i			0.534679i	0.963602	+	0.267340i	$0.0861446\pi$
$167$			1153.90i			0.534679i	−0.963602	+	0.267340i	$0.913855\pi$
$168$	451.631			0.207405
$169$	−2085.23	+	691.814i	−0.949128	+	0.314890i
$170$	−1105.68			−0.498832
$171$			761.215i			0.340418i
$172$	2612.69			1.15823
$173$	1688.85			0.742203			0.371101	−	0.928592i	$-0.378980\pi$
$173$	1688.85			0.742203			0.371101	+	0.928592i	$0.378980\pi$
$174$			560.947i			0.244398i
$175$			899.265i			0.388446i
$176$			316.817i			0.135687i
$177$		−	558.678i		−	0.237247i
$178$	−314.181			−0.132297
$179$	942.793			0.393674			0.196837	−	0.980436i	$-0.436933\pi$
$179$	942.793			0.393674			0.196837	+	0.980436i	$0.436933\pi$
$180$		−	865.902i		−	0.358558i
$181$	−482.030			−0.197950			−0.0989751	−	0.995090i	$-0.531556\pi$
$181$	−482.030			−0.197950			−0.0989751	+	0.995090i	$0.531556\pi$
$182$	−79.0425	−	489.262i	−0.0321924	−	0.199267i
$183$	2015.71			0.814236
$184$			2322.00i			0.930325i
$185$	−6692.47			−2.65968
$186$	−463.721			−0.182805
$187$			689.530i			0.269644i
$188$			3030.67i			1.17572i
$189$			215.055i			0.0827670i
$190$		−	1731.80i		−	0.661254i
$191$	4223.32			1.59994			0.799971	−	0.600039i	$-0.204848\pi$
$191$	4223.32			1.59994			0.799971	+	0.600039i	$0.204848\pi$
$192$	137.875			0.0518244
$193$		−	229.092i		−	0.0854424i	−0.999087	−	0.0427212i	$-0.986397\pi$
$193$		−	229.092i		−	0.0854424i	0.999087	−	0.0427212i	$-0.0136027\pi$
$194$	1697.80			0.628323
$195$	−2141.12	+	345.907i	−0.786300	+	0.127030i
$196$	1743.81			0.635501
$197$			228.335i			0.0825798i	0.999147	+	0.0412899i	$0.0131467\pi$
$197$			228.335i			0.0825798i	−0.999147	+	0.0412899i	$0.986853\pi$
$198$	152.559			0.0547569
$199$	−2939.02			−1.04694			−0.523471	−	0.852043i	$-0.675363\pi$
$199$	−2939.02			−1.04694			−0.523471	+	0.852043i	$0.675363\pi$
$200$			2133.92i			0.754453i
$201$			42.1093i			0.0147769i
$202$			839.332i			0.292352i
$203$		−	1121.89i		−	0.387889i
$204$	−1010.51			−0.346814
$205$	3174.63			1.08159
$206$			1999.57i			0.676294i
$207$	−1105.68			−0.371255
$208$	185.476	+	1148.07i	0.0618292	+	0.382714i
$209$	−1080.00			−0.357441
$210$		−	489.262i		−	0.160773i
$211$	−1607.02			−0.524321			−0.262161	−	0.965024i	$-0.584435\pi$
$211$	−1607.02			−0.524321			−0.262161	+	0.965024i	$0.584435\pi$
$212$	−4207.72			−1.36315
$213$		−	1040.37i		−	0.334670i
$214$			1684.01i			0.537929i
$215$		−	6460.42i		−	2.04929i
$216$			510.317i			0.160753i
$217$	927.441			0.290133
$218$	−461.207			−0.143288
$219$			2498.70i			0.770988i
$220$	1228.53			0.376488
$221$	403.676	+	2498.70i	0.122870	+	0.760546i
$222$	1728.00			0.522413
$223$		−	130.867i		−	0.0392981i	−0.999807	−	0.0196490i	$-0.993745\pi$
$223$		−	130.867i		−	0.0392981i	0.999807	−	0.0196490i	$-0.00625488\pi$
$224$	−1466.69			−0.437489
$225$	−1016.12			−0.301072
$226$		−	875.761i		−	0.257764i
$227$		−	4325.19i		−	1.26464i	−0.774708	−	0.632319i	$-0.782103\pi$
$227$		−	4325.19i		−	1.26464i	0.774708	−	0.632319i	$-0.217897\pi$
$228$		−	1582.75i		−	0.459738i
$229$		−	2621.57i		−	0.756499i	−0.925704	−	0.378250i	$-0.876526\pi$
$229$		−	2621.57i		−	0.756499i	0.925704	−	0.378250i	$-0.123474\pi$
$230$	2515.47			0.721153
$231$	−305.117			−0.0869058
$232$		−	2662.21i		−	0.753373i
$233$	4643.12			1.30550			0.652748	−	0.757575i	$-0.273616\pi$
$233$	4643.12			1.30550			0.652748	+	0.757575i	$0.273616\pi$
$234$	552.838	−	89.3134i	0.154445	−	0.0249513i
$235$	7493.95			2.08022
$236$			1161.63i			0.320405i
$237$	1006.82			0.275950
$238$	−570.972			−0.155507
$239$		−	6696.69i		−	1.81244i	−0.422809	−	0.906219i	$-0.638956\pi$
$239$		−	6696.69i		−	1.81244i	0.422809	−	0.906219i	$-0.361044\pi$
$240$			1148.07i			0.308783i
$241$			2301.47i			0.615148i	0.951524	+	0.307574i	$0.0995171\pi$
$241$			2301.47i			0.615148i	−0.951524	+	0.307574i	$0.900483\pi$
$242$		−	1550.46i		−	0.411848i
$243$	−243.000			−0.0641500
$244$	−4191.15			−1.09963
$245$		−	4311.93i		−	1.12440i
$246$	−819.691			−0.212445
$247$	−3913.68	+	632.272i	−1.00818	+	0.162876i
$248$	2200.78			0.563506
$249$			1706.39i			0.434290i
$250$	−247.714			−0.0626673
$251$	−828.000			−0.208219			−0.104109	−	0.994566i	$-0.533199\pi$
$251$	−828.000			−0.208219			−0.104109	+	0.994566i	$0.533199\pi$
$252$		−	447.153i		−	0.111778i
$253$		−	1568.72i		−	0.389820i
$254$			365.088i			0.0901877i
$255$			2498.70i			0.613626i
$256$	−2242.27			−0.547429
$257$	−884.763			−0.214747			−0.107374	−	0.994219i	$-0.534244\pi$
$257$	−884.763			−0.214747			−0.107374	+	0.994219i	$0.534244\pi$
$258$			1668.08i			0.402521i
$259$	−3456.00			−0.829133
$260$	4451.91	−	719.226i	1.06191	−	0.171556i
$261$	1267.68			0.300640
$262$			1571.72i			0.370616i
$263$	−8343.94			−1.95631			−0.978155	−	0.207878i	$-0.933344\pi$
$263$	−8343.94			−1.95631			−0.978155	+	0.207878i	$0.933344\pi$
$264$	−724.030			−0.168792
$265$			10404.4i			2.41185i
$266$		−	894.306i		−	0.206141i
$267$			710.013i			0.162742i
$268$		−	87.5556i		−	0.0199564i
$269$	2762.56			0.626157			0.313078	−	0.949727i	$-0.398640\pi$
$269$	2762.56			0.626157			0.313078	+	0.949727i	$0.398640\pi$
$270$	552.838			0.124610
$271$		−	3116.54i		−	0.698585i	−0.937014	−	0.349293i	$-0.886422\pi$
$271$		−	3116.54i		−	0.698585i	0.937014	−	0.349293i	$-0.113578\pi$
$272$	1339.81			0.298669
$273$	−1105.68	+	178.627i	−0.245123	+	0.0396007i
$274$	3394.91			0.748517
$275$		−	1441.65i		−	0.316127i
$276$	2298.97			0.501384
$277$	502.060			0.108902			0.0544510	−	0.998516i	$-0.482659\pi$
$277$	502.060			0.108902			0.0544510	+	0.998516i	$0.482659\pi$
$278$		−	724.465i		−	0.156297i
$279$			1047.96i			0.224872i
$280$			2322.00i			0.495592i
$281$			6607.56i			1.40275i	0.712791	+	0.701377i	$0.247431\pi$
$281$			6607.56i			1.40275i	−0.712791	+	0.701377i	$0.752569\pi$
$282$	−1934.94			−0.408596
$283$	4368.98			0.917699			0.458850	−	0.888514i	$-0.348262\pi$
$283$	4368.98			0.917699			0.458850	+	0.888514i	$0.348262\pi$
$284$			2163.18i			0.451975i
$285$	−3913.68			−0.813425
$286$	126.717	+	784.358i	0.0261990	+	0.162168i
$287$	1639.38			0.337176
$288$		−	1657.28i		−	0.339084i
$289$	−1997.00			−0.406473
$290$	−2884.03			−0.583986
$291$		−	3836.82i		−	0.772916i
$292$		−	5195.41i		−	1.04123i
$293$			5348.12i			1.06635i	0.846005	+	0.533175i	$0.179001\pi$
$293$			5348.12i			1.06635i	−0.846005	+	0.533175i	$0.820999\pi$
$294$			1113.34i			0.220856i
$295$	2872.36			0.566900
$296$	−8200.94			−1.61037
$297$		−	344.765i		−	0.0673579i
$298$	−1827.68			−0.355284
$299$	−918.384	−	5684.67i	−0.177630	−	1.09951i
$300$	2112.76			0.406600
$301$		−	3336.17i		−	0.638849i
$302$	3628.69			0.691416
$303$	1896.79			0.359630
$304$			2098.53i			0.395917i
$305$			10363.5i			1.94561i
$306$		−	645.166i		−	0.120528i
$307$			4502.46i			0.837032i	0.908210	+	0.418516i	$0.137450\pi$
$307$			4502.46i			0.837032i	−0.908210	+	0.418516i	$0.862550\pi$
$308$	634.414			0.117367
$309$	4518.79			0.831926
$310$		−	2384.15i		−	0.436809i
$311$	7447.20			1.35785			0.678926	−	0.734207i	$-0.262446\pi$
$311$	7447.20			1.35785			0.678926	+	0.734207i	$0.262446\pi$
$312$	−2623.72	+	423.874i	−0.476086	+	0.0769138i
$313$	−6508.93			−1.17542			−0.587710	−	0.809072i	$-0.699970\pi$
$313$	−6508.93			−1.17542			−0.587710	+	0.809072i	$0.699970\pi$
$314$		−	1367.29i		−	0.245734i
$315$	−1105.68			−0.197771
$316$	−2093.43			−0.372673
$317$			2465.57i			0.436846i	0.975854	+	0.218423i	$0.0700913\pi$
$317$			2465.57i			0.436846i	−0.975854	+	0.218423i	$0.929909\pi$
$318$		−	2686.43i		−	0.473735i
$319$			1798.56i			0.315674i
$320$			708.866i			0.123834i
$321$	3805.68			0.661720
$322$	1298.99			0.224814
$323$			4567.29i			0.786782i
$324$	505.257			0.0866353
$325$	−843.996	−	5224.22i	−0.144051	−	0.891654i
$326$	−3827.07			−0.650190
$327$			1042.27i			0.176263i
$328$	3890.18			0.654876
$329$	3869.89			0.648491
$330$			784.358i			0.130841i
$331$		−	4114.84i		−	0.683300i	−0.939827	−	0.341650i	$-0.889014\pi$
$331$		−	4114.84i		−	0.683300i	0.939827	−	0.341650i	$-0.110986\pi$
$332$		−	3548.01i		−	0.586513i
$333$		−	3905.08i		−	0.642634i
$334$	1531.80			0.250948
$335$	−216.499			−0.0353092
$336$			592.867i			0.0962605i
$337$	−4798.05			−0.775568			−0.387784	−	0.921750i	$-0.626759\pi$
$337$	−4798.05			−0.775568			−0.387784	+	0.921750i	$0.626759\pi$
$338$	918.384	+	2768.15i	0.147791	+	0.445466i
$339$	−1979.12			−0.317082
$340$		−	5195.41i		−	0.828708i
$341$	−1486.82			−0.236117
$342$	1010.51			0.159773
$343$		−	4958.69i		−	0.780594i
$344$		−	7916.58i		−	1.24079i
$345$		−	5684.67i		−	0.887109i
$346$		−	2241.96i		−	0.348348i
$347$	−3314.76			−0.512812			−0.256406	−	0.966569i	$-0.582538\pi$
$347$	−3314.76			−0.512812			−0.256406	+	0.966569i	$0.582538\pi$
$348$	−2635.81			−0.406018
$349$			371.740i			0.0570166i	0.999594	+	0.0285083i	$0.00907570\pi$
$349$			371.740i			0.0570166i	−0.999594	+	0.0285083i	$0.990924\pi$
$350$	1193.78			0.182314
$351$	−201.838	−	1249.35i	−0.0306932	−	0.189987i
$352$	2351.32			0.356039
$353$			7539.10i			1.13673i	0.822776	+	0.568365i	$0.192424\pi$
$353$			7539.10i			1.13673i	−0.822776	+	0.568365i	$0.807576\pi$
$354$	−741.646			−0.111350
$355$	5348.89			0.799689
$356$		−	1476.29i		−	0.219785i
$357$			1290.33i			0.191293i
$358$		−	1251.56i		−	0.184768i
$359$		−	12741.5i		−	1.87317i	−0.350437	−	0.936586i	$-0.613967\pi$
$359$		−	12741.5i		−	1.87317i	0.350437	−	0.936586i	$-0.386033\pi$
$360$	−2623.72			−0.384117
$361$	−294.676			−0.0429619
$362$			639.896i			0.0929065i
$363$	−3503.85			−0.506624
$364$	2298.97	−	371.409i	0.331041	−	0.0534811i
$365$	−12846.7			−1.84227
$366$		−	2675.85i		−	0.382156i
$367$	7187.26			1.02227			0.511134	−	0.859501i	$-0.329226\pi$
$367$	7187.26			1.02227			0.511134	+	0.859501i	$0.329226\pi$
$368$	−3048.14			−0.431781
$369$			1852.41i			0.261334i
$370$			8884.26i			1.24830i
$371$			5372.87i			0.751874i
$372$		−	2178.96i		−	0.303693i
$373$	−2087.99			−0.289845			−0.144922	−	0.989443i	$-0.546293\pi$
$373$	−2087.99			−0.289845			−0.144922	+	0.989443i	$0.546293\pi$
$374$	915.352			0.126555
$375$			559.805i			0.0770886i
$376$	9183.07			1.25952
$377$	1052.94	+	6517.57i	0.143844	+	0.890377i
$378$	285.486			0.0388461
$379$			3982.08i			0.539699i	0.962902	+	0.269850i	$0.0869740\pi$
$379$			3982.08i			0.539699i	−0.962902	+	0.269850i	$0.913026\pi$
$380$	8137.50			1.09854
$381$	825.057			0.110942
$382$		−	5606.47i		−	0.750921i
$383$			8638.43i			1.15249i	0.817278	+	0.576244i	$0.195482\pi$
$383$			8638.43i			1.15249i	−0.817278	+	0.576244i	$0.804518\pi$
$384$			4236.38i			0.562987i
$385$		−	1568.72i		−	0.207660i
$386$	−304.120			−0.0401018
$387$	3769.68			0.495151
$388$			7977.70i			1.04383i
$389$	1275.74			0.166279			0.0831393	−	0.996538i	$-0.473505\pi$
$389$	1275.74			0.166279			0.0831393	+	0.996538i	$0.473505\pi$
$390$	459.192	+	2842.34i	0.0596207	+	0.369044i
$391$	−6634.06			−0.858053
$392$		−	5283.83i		−	0.680801i
$393$	3551.91			0.455904
$394$	303.115			0.0387582
$395$			5176.44i			0.659379i
$396$			716.851i			0.0909675i
$397$			4622.65i			0.584394i	0.956358	+	0.292197i	$0.0943862\pi$
$397$			4622.65i			0.584394i	−0.956358	+	0.292197i	$0.905614\pi$
$398$			3901.55i			0.491375i
$399$	−2021.03			−0.253579
$400$	−2801.24			−0.350155
$401$			138.075i			0.0171949i	0.999963	+	0.00859743i	$0.00273668\pi$
$401$			138.075i			0.0171949i	−0.999963	+	0.00859743i	$0.997263\pi$
$402$	55.9001			0.00693544
$403$	−5387.91	+	870.441i	−0.665982	+	0.107592i
$404$	−3943.90			−0.485684
$405$		−	1249.35i		−	0.153286i
$406$	−1489.32			−0.182053
$407$	5540.47			0.674769
$408$			3061.90i			0.371536i
$409$		−	1204.64i		−	0.145637i	−0.997345	−	0.0728186i	$-0.976801\pi$
$409$		−	1204.64i		−	0.145637i	0.997345	−	0.0728186i	$-0.0231994\pi$
$410$		−	4214.32i		−	0.507635i
$411$		−	7672.09i		−	0.920770i
$412$	−9395.68			−1.12352
$413$	1483.29			0.176726
$414$			1467.79i			0.174246i
$415$	−8773.16			−1.03773
$416$	8520.66	−	1376.55i	1.00423	−	0.162238i
$417$	−1637.21			−0.192265
$418$			1433.70i			0.167762i
$419$	5199.85			0.606275			0.303138	−	0.952947i	$-0.401966\pi$
$419$	5199.85			0.606275			0.303138	+	0.952947i	$0.401966\pi$
$420$	2298.97			0.267091
$421$		−	14136.5i		−	1.63651i	−0.574854	−	0.818256i	$-0.694941\pi$
$421$		−	14136.5i		−	1.63651i	0.574854	−	0.818256i	$-0.305059\pi$
$422$			2133.32i			0.246086i
$423$			4372.75i			0.502625i
$424$			12749.6i			1.46032i
$425$	−6096.70			−0.695844
$426$	−1381.09			−0.157075
$427$			5351.71i			0.606527i
$428$	−7912.94			−0.893660
$429$	1772.56	−	286.365i	0.199487	−	0.0322280i
$430$	−8576.21			−0.961818
$431$		−	2279.83i		−	0.254793i	−0.991852	−	0.127396i	$-0.959338\pi$
$431$		−	2279.83i		−	0.254793i	0.991852	−	0.127396i	$-0.0406620\pi$
$432$	−669.905			−0.0746084
$433$	13298.7			1.47597			0.737984	−	0.674819i	$-0.235778\pi$
$433$	13298.7			1.47597			0.737984	+	0.674819i	$0.235778\pi$
$434$		−	1231.18i		−	0.136172i
$435$			6517.57i			0.718376i
$436$		−	2167.14i		−	0.238045i
$437$		−	10390.8i		−	1.13744i
$438$	3317.03			0.361858
$439$	10452.3			1.13635			0.568177	−	0.822907i	$-0.307649\pi$
$439$	10452.3			1.13635			0.568177	+	0.822907i	$0.307649\pi$
$440$		−	3722.50i		−	0.403325i
$441$	2516.03			0.271680
$442$	3317.03	−	535.880i	0.356957	−	0.0576679i
$443$	5363.50			0.575232			0.287616	−	0.957746i	$-0.407137\pi$
$443$	5363.50			0.575232			0.287616	+	0.957746i	$0.407137\pi$
$444$			8119.62i			0.867883i
$445$	−3650.43			−0.388870
$446$	−173.726			−0.0184443
$447$			4130.34i			0.437043i
$448$			366.059i			0.0386042i
$449$		−	9681.73i		−	1.01762i	−0.860880	−	0.508808i	$-0.830086\pi$
$449$		−	9681.73i		−	1.01762i	0.860880	−	0.508808i	$-0.169914\pi$
$450$			1348.90i			0.141306i
$451$	−2628.17			−0.274402
$452$	4115.07			0.428223
$453$		−	8200.42i		−	0.850528i
$454$	−5741.69			−0.593548
$455$	−918.384	−	5684.67i	−0.0946253	−	0.585718i
$456$	−4795.81			−0.492510
$457$		−	3537.94i		−	0.362139i	−0.983470	−	0.181070i	$-0.942044\pi$
$457$		−	3537.94i		−	0.362139i	0.983470	−	0.181070i	$-0.0579560\pi$
$458$	−3480.14			−0.355057
$459$	−1458.00			−0.148265
$460$			11819.8i			1.19805i
$461$		−	15074.9i		−	1.52302i	−0.648156	−	0.761508i	$-0.724459\pi$
$461$		−	15074.9i		−	1.52302i	0.648156	−	0.761508i	$-0.275541\pi$
$462$			405.044i			0.0407886i
$463$			11070.0i			1.11116i	0.831463	+	0.555580i	$0.187504\pi$
$463$			11070.0i			1.11116i	−0.831463	+	0.555580i	$0.812496\pi$
$464$	3494.74			0.349654
$465$	−5387.91			−0.537330
$466$		−	6163.75i		−	0.612725i
$467$	−13252.8			−1.31320			−0.656600	−	0.754239i	$-0.728006\pi$
$467$	−13252.8			−1.31320			−0.656600	+	0.754239i	$0.728006\pi$
$468$	419.671	+	2597.71i	0.0414515	+	0.256579i
$469$	−111.800			−0.0110074
$470$		−	9948.23i		−	0.976335i
$471$	−3089.91			−0.302284
$472$	3519.79			0.343244
$473$			5348.36i			0.519911i
$474$		−	1336.56i		−	0.129515i
$475$		−	9549.18i		−	0.922413i
$476$		−	2682.92i		−	0.258343i
$477$	−6071.03			−0.582753
$478$	−8889.86			−0.850654
$479$		−	12241.4i		−	1.16769i	−0.811866	−	0.583843i	$-0.801548\pi$
$479$		−	12241.4i		−	1.16769i	0.811866	−	0.583843i	$-0.198452\pi$
$480$	8520.66			0.810236
$481$	20077.4	−	3243.59i	1.90322	−	0.307474i
$482$	3055.20			0.288715
$483$		−	2935.57i		−	0.276549i
$484$	7285.37			0.684201
$485$	19726.5			1.84687
$486$			322.583i			0.0301084i
$487$		−	13413.3i		−	1.24808i	−0.781392	−	0.624041i	$-0.785490\pi$
$487$		−	13413.3i		−	1.24808i	0.781392	−	0.624041i	$-0.214510\pi$
$488$			12699.4i			1.17802i
$489$			8648.74i			0.799815i
$490$	−5724.10			−0.527731
$491$	737.885			0.0678214			0.0339107	−	0.999425i	$-0.489204\pi$
$491$	737.885			0.0678214			0.0339107	+	0.999425i	$0.489204\pi$
$492$		−	3851.61i		−	0.352935i
$493$	7606.06			0.694847
$494$	839.342	+	5195.41i	0.0764449	+	0.473183i
$495$	1772.56			0.160951
$496$			2889.01i			0.261533i
$497$	2762.17			0.249297
$498$	2265.24			0.203831
$499$			1865.65i			0.167370i	0.996492	+	0.0836852i	$0.0266690\pi$
$499$			1865.65i			0.167370i	−0.996492	+	0.0836852i	$0.973331\pi$
$500$		−	1163.97i		−	0.104109i
$501$		−	3461.70i		−	0.308697i
$502$			1099.17i			0.0977259i
$503$	−16632.0			−1.47432			−0.737161	−	0.675717i	$-0.763834\pi$
$503$	−16632.0			−1.47432			−0.737161	+	0.675717i	$0.763834\pi$
$504$	−1354.89			−0.119745
$505$			9752.09i			0.859331i
$506$	−2082.47			−0.182959
$507$	6255.70	−	2075.44i	0.547979	−	0.181802i
$508$	−1715.50			−0.149829
$509$			4128.91i			0.359549i	0.983708	+	0.179775i	$0.0575368\pi$
$509$			4128.91i			0.359549i	−0.983708	+	0.179775i	$0.942463\pi$
$510$	3317.03			0.288001
$511$	−6634.06			−0.574312
$512$		−	8320.40i		−	0.718190i
$513$		−	2283.64i		−	0.196541i
$514$			1174.52i			0.100790i
$515$			23232.7i			1.98788i
$516$	−7838.08			−0.668706
$517$	−6203.99			−0.527758
$518$			4587.85i			0.389147i
$519$	−5066.56			−0.428511
$520$	−2179.29	−	13489.5i	−0.183785	−	1.13760i
$521$	−988.234			−0.0831005			−0.0415502	−	0.999136i	$-0.513230\pi$
$521$	−988.234			−0.0831005			−0.0415502	+	0.999136i	$0.513230\pi$
$522$		−	1682.84i		−	0.141103i
$523$	−9441.62			−0.789394			−0.394697	−	0.918811i	$-0.629150\pi$
$523$	−9441.62			−0.789394			−0.394697	+	0.918811i	$0.629150\pi$
$524$	−7385.30			−0.615703
$525$		−	2697.79i		−	0.224269i
$526$			11076.6i			0.918180i
$527$			6287.73i			0.519730i
$528$		−	950.452i		−	0.0783392i
$529$	2925.83			0.240472
$530$	13811.9			1.13198
$531$			1676.03i			0.136975i
$532$	4202.21			0.342461
$533$	−9523.88	+	1538.62i	−0.773968	+	0.125038i
$534$	942.544			0.0763817
$535$			19566.3i			1.58117i
$536$	−265.297			−0.0213789
$537$	−2828.38			−0.227288
$538$		−	3667.30i		−	0.293882i
$539$			3569.70i			0.285265i
$540$			2597.71i			0.207014i
$541$		−	14001.5i		−	1.11270i	−0.830948	−	0.556351i	$-0.812201\pi$
$541$		−	14001.5i		−	1.11270i	0.830948	−	0.556351i	$-0.187799\pi$
$542$	−4137.22			−0.327876
$543$	1446.09			0.114287
$544$		−	9943.67i		−	0.783697i
$545$	−5358.70			−0.421177
$546$	237.127	+	1467.79i	0.0185863	+	0.115047i
$547$	−4244.85			−0.331804			−0.165902	−	0.986142i	$-0.553053\pi$
$547$	−4244.85			−0.331804			−0.165902	+	0.986142i	$0.553053\pi$
$548$			15952.2i			1.24351i
$549$	−6047.12			−0.470100
$550$	−1913.80			−0.148372
$551$			11913.3i			0.921092i
$552$		−	6965.99i		−	0.537123i
$553$			2673.12i			0.205556i
$554$		−	666.485i		−	0.0511124i
$555$	20077.4			1.53556
$556$	3404.16			0.259655
$557$			11732.2i			0.892473i	0.894915	+	0.446236i	$0.147236\pi$
$557$			11732.2i			0.892473i	−0.894915	+	0.446236i	$0.852764\pi$
$558$	1391.16			0.105542
$559$	3131.13	+	19381.2i	0.236910	+	1.46644i
$560$	−3048.14			−0.230013
$561$		−	2068.59i		−	0.155679i
$562$	8771.54			0.658372
$563$	9941.80			0.744222			0.372111	−	0.928188i	$-0.378634\pi$
$563$	9941.80			0.744222			0.372111	+	0.928188i	$0.378634\pi$
$564$		−	9092.02i		−	0.678800i
$565$		−	10175.3i		−	0.757664i
$566$		−	5799.83i		−	0.430716i
$567$		−	645.166i		−	0.0477856i
$568$	6554.52			0.484193
$569$	−3690.77			−0.271925			−0.135962	−	0.990714i	$-0.543413\pi$
$569$	−3690.77			−0.271925			−0.135962	+	0.990714i	$0.543413\pi$
$570$			5195.41i			0.381775i
$571$	5685.09			0.416661			0.208331	−	0.978058i	$-0.433197\pi$
$571$	5685.09			0.416661			0.208331	+	0.978058i	$0.433197\pi$
$572$	−3685.59	+	595.423i	−0.269409	+	0.0435242i
$573$	−12670.0			−0.923727
$574$		−	2176.28i		−	0.158251i
$575$	13870.3			1.00597
$576$	−413.626			−0.0299209
$577$			7746.50i			0.558910i	0.960159	+	0.279455i	$0.0901537\pi$
$577$			7746.50i			0.558910i	−0.960159	+	0.279455i	$0.909846\pi$
$578$			2651.02i			0.190775i
$579$			687.275i			0.0493302i
$580$		−	13551.6i		−	0.970175i
$581$	−4530.47			−0.323504
$582$	−5093.39			−0.362762
$583$		−	8613.48i		−	0.611894i
$584$	−15742.3			−1.11545
$585$	6423.35	−	1037.72i	0.453971	−	0.0733410i
$586$	7099.64			0.500484
$587$		−	2766.54i		−	0.194527i	−0.995259	−	0.0972635i	$-0.968991\pi$
$587$		−	2766.54i		−	0.194527i	0.995259	−	0.0972635i	$-0.0310089\pi$
$588$	−5231.44			−0.366906
$589$	−9848.38			−0.688957
$590$		−	3813.07i		−	0.266070i
$591$		−	685.006i		−	0.0476774i
$592$		−	10765.6i		−	0.747402i
$593$			1440.79i			0.0997743i	0.998755	+	0.0498871i	$0.0158862\pi$
$593$			1440.79i			0.0997743i	−0.998755	+	0.0498871i	$0.984114\pi$
$594$	−457.676			−0.0316139
$595$	−6634.06			−0.457092
$596$		−	8587.99i		−	0.590231i
$597$	8817.06			0.604452
$598$	−7546.41	+	1219.16i	−0.516047	+	0.0833696i
$599$	23837.5			1.62600			0.813001	−	0.582263i	$-0.197832\pi$
$599$	23837.5			1.62600			0.813001	+	0.582263i	$0.197832\pi$
$600$		−	6401.75i		−	0.435584i
$601$	−6694.23			−0.454348			−0.227174	−	0.973854i	$-0.572949\pi$
$601$	−6694.23			−0.454348			−0.227174	+	0.973854i	$0.572949\pi$
$602$	−4428.77			−0.299839
$603$		−	126.328i		−	0.00853145i
$604$			17050.7i			1.14865i
$605$		−	18014.6i		−	1.21057i
$606$		−	2518.00i		−	0.168790i
$607$	3330.50			0.222703			0.111352	−	0.993781i	$-0.464482\pi$
$607$	3330.50			0.222703			0.111352	+	0.993781i	$0.464482\pi$
$608$	15574.6			1.03887
$609$			3365.68i			0.223948i
$610$	13757.5			0.913156
$611$	−22481.8	+	3632.04i	−1.48857	+	0.240486i
$612$	3031.54			0.200233
$613$		−	13490.3i		−	0.888857i	−0.895814	−	0.444428i	$-0.853407\pi$
$613$		−	13490.3i		−	0.888857i	0.895814	−	0.444428i	$-0.146593\pi$
$614$	5977.02			0.392855
$615$	−9523.88			−0.624455
$616$		−	1922.30i		−	0.125733i
$617$			7470.76i			0.487458i	0.969843	+	0.243729i	$0.0783707\pi$
$617$			7470.76i			0.487458i	−0.969843	+	0.243729i	$0.921629\pi$
$618$		−	5998.71i		−	0.390458i
$619$		−	24806.9i		−	1.61078i	−0.592746	−	0.805389i	$-0.701956\pi$
$619$		−	24806.9i		−	1.61078i	0.592746	−	0.805389i	$-0.298044\pi$
$620$	11202.8			0.725669
$621$	3317.03			0.214344
$622$		−	9886.17i		−	0.637298i
$623$	−1885.09			−0.121227
$624$	−556.429	−	3444.22i	−0.0356971	−	0.220960i
$625$	−16990.9			−1.08742
$626$			8640.61i			0.551675i
$627$	3240.00			0.206369
$628$	6424.68			0.408237
$629$		−	23430.5i		−	1.48527i
$630$			1467.79i			0.0928222i
$631$		−	314.333i		−	0.0198311i	−0.999951	−	0.00991554i	$-0.996844\pi$
$631$		−	314.333i		−	0.0198311i	0.999951	−	0.00991554i	$-0.00315627\pi$
$632$			6343.19i			0.399238i
$633$	4821.06			0.302717
$634$	3273.05			0.205031
$635$			4241.91i			0.265095i
$636$	12623.2			0.787014
$637$	2089.83	+	12935.8i	0.129988	+	0.804607i
$638$	2387.59			0.148159
$639$			3121.10i			0.193222i
$640$	−21780.7			−1.34525
$641$	−5550.41			−0.342009			−0.171005	−	0.985270i	$-0.554701\pi$
$641$	−5550.41			−0.342009			−0.171005	+	0.985270i	$0.554701\pi$
$642$		−	5052.04i		−	0.310573i
$643$			5479.48i			0.336065i	0.985781	+	0.168032i	$0.0537413\pi$
$643$			5479.48i			0.336065i	−0.985781	+	0.168032i	$0.946259\pi$
$644$			6103.78i			0.373482i
$645$			19381.2i			1.18316i
$646$	6063.08			0.369271
$647$	−4724.83			−0.287098			−0.143549	−	0.989643i	$-0.545851\pi$
$647$	−4724.83			−0.287098			−0.143549	+	0.989643i	$0.545851\pi$
$648$		−	1530.95i		−	0.0928108i
$649$	−2377.93			−0.143824
$650$	−6935.16	+	1120.41i	−0.418491	+	0.0676091i
$651$	−2782.32			−0.167508
$652$		−	17982.9i		−	1.08016i
$653$	3463.91			0.207585			0.103793	−	0.994599i	$-0.466902\pi$
$653$	3463.91			0.207585			0.103793	+	0.994599i	$0.466902\pi$
$654$	1383.62			0.0827276
$655$			18261.6i			1.08938i
$656$			5106.73i			0.303940i
$657$		−	7496.10i		−	0.445130i
$658$		−	5137.28i		−	0.304365i
$659$	−2606.35			−0.154065			−0.0770327	−	0.997029i	$-0.524545\pi$
$659$	−2606.35			−0.154065			−0.0770327	+	0.997029i	$0.524545\pi$
$660$	−3685.59			−0.217366
$661$			22436.7i			1.32025i	0.751154	+	0.660127i	$0.229498\pi$
$661$			22436.7i			1.32025i	−0.751154	+	0.660127i	$0.770502\pi$
$662$	−5462.46			−0.320702
$663$	−1211.03	−	7496.10i	−0.0709388	−	0.439102i
$664$	−10750.6			−0.628321
$665$		−	10390.8i		−	0.605923i
$666$	−5184.00			−0.301615
$667$	−17304.2			−1.00453
$668$			7197.73i			0.416899i
$669$			392.600i			0.0226888i
$670$			287.403i			0.0165721i
$671$		−	8579.56i		−	0.493607i
$672$	4400.08			0.252584
$673$	633.970			0.0363117			0.0181558	−	0.999835i	$-0.494221\pi$
$673$	633.970			0.0363117			0.0181558	+	0.999835i	$0.494221\pi$
$674$			6369.42i			0.364007i
$675$	3048.35			0.173824
$676$	−13007.1	+	4315.35i	−0.740052	+	0.245525i
$677$	24457.4			1.38844			0.694221	−	0.719762i	$-0.255749\pi$
$677$	24457.4			1.38844			0.694221	+	0.719762i	$0.255749\pi$
$678$			2627.28i			0.148820i
$679$	10186.8			0.575747
$680$	−15742.3			−0.887780
$681$			12975.6i			0.730139i
$682$			1973.76i			0.110820i
$683$			12367.6i			0.692875i	0.938073	+	0.346437i	$0.112609\pi$
$683$			12367.6i			0.692875i	−0.938073	+	0.346437i	$0.887391\pi$
$684$			4748.26i			0.265430i
$685$	39445.0			2.20017
$686$	−6582.66			−0.366366
$687$			7864.71i			0.436765i
$688$	10392.3			0.575875
$689$	−5042.65	−	31213.3i	−0.278824	−	1.72588i
$690$	−7546.41			−0.416358
$691$			1050.99i			0.0578605i	0.999581	+	0.0289302i	$0.00921006\pi$
$691$			1050.99i			0.0578605i	−0.999581	+	0.0289302i	$0.990790\pi$
$692$	10534.6			0.578709
$693$	915.352			0.0501751
$694$			4400.35i			0.240685i
$695$		−	8417.46i		−	0.459414i
$696$			7986.62i			0.434960i
$697$			11114.4i			0.604002i
$698$	493.486			0.0267603
$699$	−13929.4			−0.753729
$700$			5609.38i			0.302878i
$701$	24294.1			1.30895			0.654476	−	0.756083i	$-0.272889\pi$
$701$	24294.1			1.30895			0.654476	+	0.756083i	$0.272889\pi$
$702$	−1658.51	+	267.940i	−0.0891689	+	0.0144056i
$703$	36698.8			1.96888
$704$		−	586.846i		−	0.0314170i
$705$	−22481.8			−1.20101
$706$	10008.2			0.533516
$707$			5035.99i			0.267890i
$708$		−	3484.89i		−	0.184986i
$709$			27465.9i			1.45487i	0.686176	+	0.727436i	$0.259288\pi$
$709$			27465.9i			1.45487i	−0.686176	+	0.727436i	$0.740712\pi$
$710$		−	7100.66i		−	0.375328i
$711$	−3020.47			−0.159320
$712$	−4473.23			−0.235451
$713$		−	14304.9i		−	0.751365i
$714$	1712.92			0.0897820
$715$	1472.30	+	9113.36i	0.0770084	+	0.476672i
$716$	5880.90			0.306955
$717$			20090.1i			1.04641i
$718$	−16914.3			−0.879160
$719$	−36433.5			−1.88976			−0.944882	−	0.327411i	$-0.893824\pi$
$719$	−36433.5			−1.88976			−0.944882	+	0.327411i	$0.893824\pi$
$720$		−	3444.22i		−	0.178276i
$721$			11997.4i			0.619704i
$722$			391.183i			0.0201639i
$723$		−	6904.40i		−	0.355156i
$724$	−3006.78			−0.154345
$725$	−15902.6			−0.814629
$726$			4651.37i			0.237780i
$727$	−551.608			−0.0281403			−0.0140701	−	0.999901i	$-0.504479\pi$
$727$	−551.608			−0.0281403			−0.0140701	+	0.999901i	$0.504479\pi$
$728$	−1125.39	−	6965.99i	−0.0572934	−	0.354638i
$729$	729.000			0.0370370
$730$			17054.0i			0.864654i
$731$	22618.1			1.14440
$732$	12573.4			0.634874
$733$		−	20317.2i		−	1.02379i	−0.859049	−	0.511893i	$-0.828945\pi$
$733$		−	20317.2i		−	1.02379i	0.859049	−	0.511893i	$-0.171055\pi$
$734$		−	9541.10i		−	0.479794i
$735$			12935.8i			0.649175i
$736$			22622.4i			1.13298i
$737$	179.232			0.00895807
$738$	2459.07			0.122655
$739$		−	29931.6i		−	1.48992i	−0.667107	−	0.744962i	$-0.732468\pi$
$739$		−	29931.6i		−	1.48992i	0.667107	−	0.744962i	$-0.267532\pi$
$740$	−41745.9			−2.07380
$741$	11741.0	−	1896.81i	0.582075	−	0.0940367i
$742$	7132.49			0.352887
$743$		−	24376.4i		−	1.20361i	−0.798643	−	0.601805i	$-0.794448\pi$
$743$		−	24376.4i		−	1.20361i	0.798643	−	0.601805i	$-0.205552\pi$
$744$	−6602.33			−0.325340
$745$	−21235.5			−1.04431
$746$			2771.81i			0.136037i
$747$		−	5119.17i		−	0.250737i
$748$			4301.11i			0.210246i
$749$			10104.1i			0.492917i
$750$	743.142			0.0361810
$751$	−22692.2			−1.10260			−0.551298	−	0.834308i	$-0.685867\pi$
$751$	−22692.2			−1.10260			−0.551298	+	0.834308i	$0.685867\pi$
$752$			12054.8i			0.584567i
$753$	2484.00			0.120215
$754$	8652.09	−	1397.78i	0.417892	−	0.0675123i
$755$	42161.3			2.03232
$756$			1341.46i			0.0645349i
$757$	−33063.5			−1.58747			−0.793734	−	0.608265i	$-0.791866\pi$
$757$	−33063.5			−1.58747			−0.793734	+	0.608265i	$0.791866\pi$
$758$	5286.22			0.253304
$759$			4706.15i			0.225062i
$760$		−	24657.0i		−	1.17685i
$761$		−	216.324i		−	0.0103045i	−0.999987	−	0.00515226i	$-0.998360\pi$
$761$		−	216.324i		−	0.0103045i	0.999987	−	0.00515226i	$-0.00164002\pi$
$762$		−	1095.27i		−	0.0520699i
$763$	−2767.24			−0.131299
$764$	26344.0			1.24750
$765$		−	7496.10i		−	0.354277i
$766$	11467.5			0.540912
$767$	−8617.09	+	1392.13i	−0.405665	+	0.0655369i
$768$	6726.80			0.316058
$769$		−	19214.4i		−	0.901025i	−0.892770	−	0.450512i	$-0.851241\pi$
$769$		−	19214.4i		−	0.901025i	0.892770	−	0.450512i	$-0.148759\pi$
$770$	−2082.47			−0.0974638
$771$	2654.29			0.123984
$772$		−	1429.01i		−	0.0666209i
$773$		−	33175.6i		−	1.54365i	−0.635834	−	0.771826i	$-0.719344\pi$
$773$		−	33175.6i		−	1.54365i	0.635834	−	0.771826i	$-0.280656\pi$
$774$		−	5004.25i		−	0.232395i
$775$		−	13146.2i		−	0.609325i
$776$	24172.8			1.11824
$777$	10368.0			0.478700
$778$		−	1693.54i		−	0.0780416i
$779$	−17408.4			−0.800667
$780$	−13355.7	+	2157.68i	−0.613092	+	0.0990477i
$781$	−4428.17			−0.202884
$782$			8806.72i			0.402721i
$783$	−3803.03			−0.173575
$784$	6936.21			0.315972
$785$		−	15886.3i		−	0.722302i
$786$		−	4715.17i		−	0.213975i
$787$		−	14887.9i		−	0.674330i	−0.941446	−	0.337165i	$-0.890532\pi$
$787$		−	14887.9i		−	0.674330i	0.941446	−	0.337165i	$-0.109468\pi$
$788$			1424.30i			0.0643889i
$789$	25031.8			1.12948
$790$	6871.73			0.309475
$791$		−	5254.56i		−	0.236196i
$792$	2172.09			0.0974518
$793$	−5022.79	−	31090.4i	−0.224924	−	1.39225i
$794$	6136.58			0.274281
$795$		−	31213.3i		−	1.39248i
$796$	−18332.8			−0.816319
$797$	−12954.5			−0.575751			−0.287876	−	0.957668i	$-0.592949\pi$
$797$	−12954.5			−0.575751			−0.287876	+	0.957668i	$0.592949\pi$
$798$			2682.92i			0.119015i
$799$			26236.5i			1.16168i
$800$			20790.0i			0.918796i
$801$		−	2130.04i		−	0.0939591i
$802$	183.295			0.00807028
$803$	10635.4			0.467389
$804$			262.667i			0.0115218i
$805$	15092.8			0.660810
$806$	1155.51	+	7152.46i	0.0504977	+	0.312574i
$807$	−8287.68			−0.361512
$808$			11950.2i			0.520305i
$809$	8275.59			0.359647			0.179823	−	0.983699i	$-0.442447\pi$
$809$	8275.59			0.359647			0.179823	+	0.983699i	$0.442447\pi$
$810$	−1658.51			−0.0719435
$811$		−	26327.1i		−	1.13991i	−0.821675	−	0.569956i	$-0.806960\pi$
$811$		−	26327.1i		−	1.13991i	0.821675	−	0.569956i	$-0.193040\pi$
$812$		−	6998.09i		−	0.302444i
$813$			9349.63i			0.403328i
$814$		−	7354.98i		−	0.316698i
$815$	−44466.3			−1.91115
$816$	−4019.43			−0.172437
$817$			35426.3i			1.51703i
$818$	−1599.16			−0.0683538
$819$	3317.03	−	535.880i	0.141522	−	0.0228635i
$820$	19802.5			0.843333
$821$			34439.0i			1.46398i	0.681314	+	0.731992i	$0.261409\pi$
$821$			34439.0i			1.46398i	−0.681314	+	0.731992i	$0.738591\pi$
$822$	−10184.7			−0.432157
$823$	13870.5			0.587479			0.293739	−	0.955886i	$-0.405100\pi$
$823$	13870.5			0.587479			0.293739	+	0.955886i	$0.405100\pi$
$824$			28469.3i			1.20361i
$825$			4324.96i			0.182516i
$826$		−	1969.07i		−	0.0829453i
$827$			2132.30i			0.0896583i	0.998995	+	0.0448292i	$0.0142743\pi$
$827$			2132.30i			0.0896583i	−0.998995	+	0.0448292i	$0.985726\pi$
$828$	−6896.92			−0.289474
$829$	6212.39			0.260272			0.130136	−	0.991496i	$-0.458459\pi$
$829$	6212.39			0.260272			0.130136	+	0.991496i	$0.458459\pi$
$830$			11646.4i			0.487051i
$831$	−1506.18			−0.0628746
$832$	−343.561	−	2126.60i	−0.0143159	−	0.0886136i
$833$	15096.2			0.627913
$834$			2173.40i			0.0902380i
$835$	17797.8			0.737628
$836$	−6736.76			−0.278703
$837$		−	3143.87i		−	0.129830i
$838$		−	6902.81i		−	0.284551i
$839$		−	4550.52i		−	0.187248i	−0.995608	−	0.0936242i	$-0.970155\pi$
$839$		−	4550.52i		−	0.187248i	0.995608	−	0.0936242i	$-0.0298452\pi$
$840$		−	6965.99i		−	0.286130i
$841$	−4549.47			−0.186538
$842$	−18766.2			−0.768085
$843$		−	19822.7i		−	0.809880i
$844$	−10024.2			−0.408822
$845$	10670.6	+	32162.8i	0.434413	+	1.30939i
$846$	5804.83			0.235903
$847$		−	9302.74i		−	0.377386i
$848$	−16736.7			−0.677759
$849$	−13106.9			−0.529834
$850$			8093.38i			0.326589i
$851$			53305.6i			2.14723i
$852$		−	6489.53i		−	0.260948i
$853$			12262.8i			0.492228i	0.969241	+	0.246114i	$0.0791538\pi$
$853$			12262.8i			0.492228i	−0.969241	+	0.246114i	$0.920846\pi$
$854$	7104.40			0.284669
$855$	11741.0			0.469631
$856$			23976.5i			0.957362i
$857$	34949.1			1.39304			0.696521	−	0.717536i	$-0.254730\pi$
$857$	34949.1			1.39304			0.696521	+	0.717536i	$0.254730\pi$
$858$	−380.150	−	2353.07i	−0.0151260	−	0.0936278i
$859$	−21762.1			−0.864394			−0.432197	−	0.901779i	$-0.642261\pi$
$859$	−21762.1			−0.864394			−0.432197	+	0.901779i	$0.642261\pi$
$860$		−	40298.4i		−	1.59786i
$861$	−4918.14			−0.194669
$862$	−3026.48			−0.119585
$863$			19811.5i			0.781450i	0.920507	+	0.390725i	$0.127776\pi$
$863$			19811.5i			0.781450i	−0.920507	+	0.390725i	$0.872224\pi$
$864$			4971.83i			0.195770i
$865$		−	26049.0i		−	1.02392i
$866$		−	17654.0i		−	0.692735i
$867$	5991.00			0.234677
$868$	5785.14			0.226222
$869$		−	4285.40i		−	0.167287i
$870$	8652.09			0.337165
$871$	649.496	−	104.929i	0.0252668	−	0.00408196i
$872$	−6566.54			−0.255013
$873$			11510.5i			0.446243i
$874$	−13793.8			−0.533848
$875$	−1486.28			−0.0574235
$876$			15586.2i			0.601153i
$877$			25716.8i			0.990186i	0.868840	+	0.495093i	$0.164866\pi$
$877$			25716.8i			0.990186i	−0.868840	+	0.495093i	$0.835134\pi$
$878$		−	13875.4i		−	0.533339i
$879$		−	16044.4i		−	0.615658i
$880$	4886.61			0.187190
$881$	−34709.6			−1.32735			−0.663676	−	0.748020i	$-0.731005\pi$
$881$	−34709.6			−1.32735			−0.663676	+	0.748020i	$0.731005\pi$
$882$		−	3340.03i		−	0.127511i
$883$	3848.68			0.146680			0.0733400	−	0.997307i	$-0.476634\pi$
$883$	3848.68			0.146680			0.0733400	+	0.997307i	$0.476634\pi$
$884$	2518.02	+	15586.2i	0.0958035	+	0.593011i
$885$	−8617.09			−0.327300
$886$		−	7120.06i		−	0.269981i
$887$	32804.8			1.24180			0.620900	−	0.783890i	$-0.286767\pi$
$887$	32804.8			1.24180			0.620900	+	0.783890i	$0.286767\pi$
$888$	24602.8			0.929748
$889$			2190.53i			0.0826412i
$890$			4845.95i			0.182513i
$891$			1034.29i			0.0388891i
$892$		−	816.311i		−	0.0306414i
$893$	−41093.8			−1.53992
$894$	5483.03			0.205123
$895$		−	14541.7i		−	0.543101i
$896$	−11247.6			−0.419371
$897$	2755.15	+	17054.0i	0.102555	+	0.634802i
$898$	−12852.5			−0.477610
$899$			16400.8i			0.608452i
$900$	−6338.27			−0.234751
$901$	−36426.2			−1.34687
$902$			3488.90i			0.128789i
$903$			10008.5i			0.368840i
$904$		−	12468.8i		−	0.458748i
$905$			7434.86i			0.273086i
$906$	−10886.1			−0.399189
$907$	22262.2			0.814999			0.407500	−	0.913205i	$-0.366401\pi$
$907$	22262.2			0.814999			0.407500	+	0.913205i	$0.366401\pi$
$908$		−	26979.4i		−	0.986060i
$909$	−5690.38			−0.207633
$910$	−7546.41	+	1219.16i	−0.274902	+	0.0444117i
$911$	−13515.3			−0.491528			−0.245764	−	0.969330i	$-0.579039\pi$
$911$	−13515.3			−0.491528			−0.245764	+	0.969330i	$0.579039\pi$
$912$		−	6295.58i		−	0.228583i
$913$	7263.01			0.263275
$914$	−4696.62			−0.169968
$915$		−	31090.4i		−	1.12330i
$916$		−	16352.7i		−	0.589855i
$917$			9430.33i			0.339604i
$918$			1935.50i			0.0695871i
$919$	26600.6			0.954811			0.477405	−	0.878683i	$-0.341577\pi$
$919$	26600.6			0.954811			0.477405	+	0.878683i	$0.341577\pi$
$920$	35814.6			1.28345
$921$		−	13507.4i		−	0.483260i
$922$	−20012.0			−0.714816
$923$	−16046.7	+	2592.41i	−0.572246	+	0.0924488i
$924$	−1903.24			−0.0677620
$925$			48987.9i			1.74131i
$926$	14695.5			0.521515
$927$	−13556.4			−0.480313
$928$		−	25936.9i		−	0.917480i
$929$			32887.7i			1.16148i	0.814090	+	0.580738i	$0.197236\pi$
$929$			32887.7i			1.16148i	−0.814090	+	0.580738i	$0.802764\pi$
$930$			7152.46i			0.252192i
$931$			23644.9i			0.832363i
$932$	28962.6			1.01792
$933$	−22341.6			−0.783956
$934$			17593.1i			0.616341i
$935$	10635.4			0.371993
$936$	7871.16	−	1271.62i	0.274869	−	0.0444062i
$937$	−9261.78			−0.322913			−0.161456	−	0.986880i	$-0.551619\pi$
$937$	−9261.78			−0.322913			−0.161456	+	0.986880i	$0.551619\pi$
$938$			148.415i			0.00516623i
$939$	19526.8			0.678629
$940$	46745.3			1.62198
$941$		−	12054.9i		−	0.417619i	−0.977956	−	0.208810i	$-0.933041\pi$
$941$		−	12054.9i		−	0.417619i	0.977956	−	0.208810i	$-0.0669589\pi$
$942$			4101.86i			0.141875i
$943$		−	25285.9i		−	0.873195i
$944$			4620.51i			0.159306i
$945$	3317.03			0.114183
$946$	7099.96			0.244016
$947$			20221.4i			0.693885i	0.937886	+	0.346942i	$0.112780\pi$
$947$			20221.4i			0.693885i	−0.937886	+	0.346942i	$0.887220\pi$
$948$	6280.30			0.215163
$949$	38540.1	−	6226.32i	1.31830	−	0.212977i
$950$	−12676.5			−0.432928
$951$		−	7396.72i		−	0.252213i
$952$	−8129.36			−0.276758
$953$	20331.5			0.691083			0.345542	−	0.938403i	$-0.387695\pi$
$953$	20331.5			0.691083			0.345542	+	0.938403i	$0.387695\pi$
$954$			8059.30i			0.273511i
$955$		−	65140.8i		−	2.20723i
$956$		−	41772.2i		−	1.41319i
$957$		−	5395.68i		−	0.182254i
$958$	−16250.4			−0.548045
$959$	20369.4			0.685885
$960$		−	2126.60i		−	0.0714955i
$961$	16232.9			0.544891
$962$	−4305.88	−	26652.8i	−0.144311	−	0.893265i
$963$	−11417.0			−0.382044
$964$			14356.0i			0.479641i
$965$	−3533.53			−0.117874
$966$	−3896.98			−0.129796
$967$			11082.2i			0.368541i	0.982876	+	0.184270i	$0.0589922\pi$
$967$			11082.2i			0.368541i	−0.982876	+	0.184270i	$0.941008\pi$
$968$		−	22075.0i		−	0.732973i
$969$		−	13701.9i		−	0.454249i
$970$		−	26186.9i		−	0.866816i
$971$	−36694.1			−1.21274			−0.606369	−	0.795184i	$-0.707374\pi$
$971$	−36694.1			−1.21274			−0.606369	+	0.795184i	$0.707374\pi$
$972$	−1515.77			−0.0500189
$973$		−	4346.79i		−	0.143219i
$974$	−17806.2			−0.585778
$975$	2531.99	+	15672.7i	0.0831677	+	0.514797i
$976$	−16670.8			−0.546740
$977$		−	17155.3i		−	0.561767i	−0.959742	−	0.280883i	$-0.909373\pi$
$977$		−	17155.3i		−	0.561767i	0.959742	−	0.280883i	$-0.0906274\pi$
$978$	11481.2			0.375387
$979$	3022.07			0.0986575
$980$		−	26896.7i		−	0.876718i
$981$		−	3126.82i		−	0.101765i
$982$		−	979.544i		−	0.0318315i
$983$		−	38419.7i		−	1.24659i	−0.781986	−	0.623296i	$-0.785793\pi$
$983$		−	38419.7i		−	1.24659i	0.781986	−	0.623296i	$-0.214207\pi$
$984$	−11670.5			−0.378093
$985$	3521.86			0.113925
$986$		−	10097.1i		−	0.326121i
$987$	−11609.7			−0.374407
$988$	−24412.5	+	3943.94i	−0.786098	+	0.126998i
$989$	−51457.3			−1.65445
$990$		−	2353.07i		−	0.0755410i
$991$	51728.9			1.65815			0.829073	−	0.559140i	$-0.188869\pi$
$991$	51728.9			1.65815			0.829073	+	0.559140i	$0.188869\pi$
$992$	21441.4			0.686255
$993$			12344.5i			0.394503i
$994$		−	3666.79i		−	0.117006i
$995$			45331.6i			1.44433i
$996$			10644.0i			0.338623i
$997$	−26846.8			−0.852805			−0.426403	−	0.904533i	$-0.640219\pi$
$997$	−26846.8			−0.852805			−0.426403	+	0.904533i	$0.640219\pi$
$998$	2476.65			0.0785541
$999$			11715.2i			0.371025i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.4.b.a.25.2	✓	4
3.2	odd	2		117.4.b.d.64.3		4
4.3	odd	2		624.4.c.e.337.1		4
13.5	odd	4		507.4.a.j.1.2		4
13.8	odd	4		507.4.a.j.1.3		4
13.12	even	2	inner	39.4.b.a.25.3	yes	4
39.5	even	4		1521.4.a.x.1.3		4
39.8	even	4		1521.4.a.x.1.2		4
39.38	odd	2		117.4.b.d.64.2		4
52.51	odd	2		624.4.c.e.337.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.b.a.25.2	✓	4	1.1	even	1	trivial
39.4.b.a.25.3	yes	4	13.12	even	2	inner
117.4.b.d.64.2		4	39.38	odd	2
117.4.b.d.64.3		4	3.2	odd	2
507.4.a.j.1.2		4	13.5	odd	4
507.4.a.j.1.3		4	13.8	odd	4
624.4.c.e.337.1		4	4.3	odd	2
624.4.c.e.337.4		4	52.51	odd	2
1521.4.a.x.1.2		4	39.8	even	4
1521.4.a.x.1.3		4	39.5	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	39.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.32750i q^{2} -3.00000 q^{3} +6.23774 q^{4} -15.4241i q^{5} +3.98251i q^{6} -7.96501i q^{7} -18.9006i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-1.32750i q^{2} -3.00000 q^{3} +6.23774 q^{4} -15.4241i q^{5} +3.98251i q^{6} -7.96501i q^{7} -18.9006i q^{8} +9.00000 q^{9} -20.4755 q^{10} +12.7691i q^{11} -18.7132 q^{12} +(7.47548 + 46.2722i) q^{13} -10.5736 q^{14} +46.2722i q^{15} +24.8113 q^{16} +54.0000 q^{17} -11.9475i q^{18} +84.5794i q^{19} -96.2113i q^{20} +23.8950i q^{21} +16.9510 q^{22} -122.853 q^{23} +56.7019i q^{24} -112.902 q^{25} +(61.4264 - 9.92371i) q^{26} -27.0000 q^{27} -49.6837i q^{28} +140.853 q^{29} +61.4264 q^{30} +116.439i q^{31} -184.142i q^{32} -38.3072i q^{33} -71.6851i q^{34} -122.853 q^{35} +56.1397 q^{36} -433.898i q^{37} +112.279 q^{38} +(-22.4264 - 138.817i) q^{39} -291.525 q^{40} +205.823i q^{41} +31.7207 q^{42} +418.853 q^{43} +79.6501i q^{44} -138.817i q^{45} +163.087i q^{46} +485.861i q^{47} -74.4339 q^{48} +279.559 q^{49} +149.877i q^{50} -162.000 q^{51} +(46.6301 + 288.634i) q^{52} -674.559 q^{53} +35.8425i q^{54} +196.951 q^{55} -150.544 q^{56} -253.738i q^{57} -186.982i q^{58} +186.226i q^{59} +288.634i q^{60} -671.902 q^{61} +154.574 q^{62} -71.6851i q^{63} -45.9584 q^{64} +(713.706 - 115.302i) q^{65} -50.8529 q^{66} -14.0364i q^{67} +336.838 q^{68} +368.559 q^{69} +163.087i q^{70} +346.789i q^{71} -170.106i q^{72} -832.900i q^{73} -576.000 q^{74} +338.706 q^{75} +527.584i q^{76} +101.706 q^{77} +(-184.279 + 29.7711i) q^{78} -335.608 q^{79} -382.691i q^{80} +81.0000 q^{81} +273.230 q^{82} -568.797i q^{83} +149.051i q^{84} -832.900i q^{85} -556.028i q^{86} -422.559 q^{87} +241.343 q^{88} -236.671i q^{89} -184.279 q^{90} +(368.559 - 59.5423i) q^{91} -766.324 q^{92} -349.318i q^{93} +644.981 q^{94} +1304.56 q^{95} +552.426i q^{96} +1278.94i q^{97} -371.115i q^{98} +114.922i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} - 26 q^{4} + 36 q^{9}+O(q^{10}) \) 4 * q - 12 * q^3 - 26 * q^4 + 36 * q^9	\( 4 q - 12 q^{3} - 26 q^{4} + 36 q^{9} + 20 q^{10} + 78 q^{12} - 72 q^{13} - 348 q^{14} + 354 q^{16} + 216 q^{17} - 136 q^{22} + 120 q^{23} - 44 q^{25} - 60 q^{26} - 108 q^{27} - 48 q^{29} - 60 q^{30} + 120 q^{35} - 234 q^{36} - 468 q^{38} + 216 q^{39} - 1268 q^{40} + 1044 q^{42} + 1064 q^{43} - 1062 q^{48} - 716 q^{49} - 648 q^{51} + 1766 q^{52} - 864 q^{53} + 584 q^{55} + 3372 q^{56} - 2280 q^{61} + 924 q^{62} - 1050 q^{64} + 1632 q^{65} + 408 q^{66} - 1404 q^{68} - 360 q^{69} - 2304 q^{74} + 132 q^{75} - 816 q^{77} + 180 q^{78} + 288 q^{79} + 324 q^{81} - 28 q^{82} + 144 q^{87} + 2392 q^{88} + 180 q^{90} - 360 q^{91} - 8568 q^{92} + 6656 q^{94} + 3384 q^{95}+O(q^{100}) \) 4 * q - 12 * q^3 - 26 * q^4 + 36 * q^9 + 20 * q^10 + 78 * q^12 - 72 * q^13 - 348 * q^14 + 354 * q^16 + 216 * q^17 - 136 * q^22 + 120 * q^23 - 44 * q^25 - 60 * q^26 - 108 * q^27 - 48 * q^29 - 60 * q^30 + 120 * q^35 - 234 * q^36 - 468 * q^38 + 216 * q^39 - 1268 * q^40 + 1044 * q^42 + 1064 * q^43 - 1062 * q^48 - 716 * q^49 - 648 * q^51 + 1766 * q^52 - 864 * q^53 + 584 * q^55 + 3372 * q^56 - 2280 * q^61 + 924 * q^62 - 1050 * q^64 + 1632 * q^65 + 408 * q^66 - 1404 * q^68 - 360 * q^69 - 2304 * q^74 + 132 * q^75 - 816 * q^77 + 180 * q^78 + 288 * q^79 + 324 * q^81 - 28 * q^82 + 144 * q^87 + 2392 * q^88 + 180 * q^90 - 360 * q^91 - 8568 * q^92 + 6656 * q^94 + 3384 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more