LMFDB - Embedded newform 810.4.e.bd.271.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [810,4,Mod(271,810)] 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("810.271"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(810, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	810.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,4,0,-8,10,0,-1,-32,0,40,33] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$47.7915471046$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{401})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 101x^{2} + 100x + 10000 $ x^4 - x^3 + 101x^2 + 100x + 10000
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 270)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			271.2
Root			$-4.75625 - 8.23806i$ of defining polynomial
Character	$\chi$	$=$	810.271
Dual form			810.4.e.bd.541.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(2.50000 - 4.33013i) q^{5} +(14.7687 + 25.5802i) q^{7} -8.00000 q^{8} +10.0000 q^{10} +(23.2687 + 40.3026i) q^{11} +(-46.0375 + 79.7392i) q^{13} +(-29.5375 + 51.1604i) q^{14} +(-8.00000 - 13.8564i) q^{16} +4.53748 q^{17} +87.5375 q^{19} +(10.0000 + 17.3205i) q^{20} +(-46.5375 + 80.6053i) q^{22} +(80.3437 - 139.159i) q^{23} +(-12.5000 - 21.6506i) q^{25} -184.150 q^{26} -118.150 q^{28} +(-120.881 - 209.372i) q^{29} +(1.34369 - 2.32734i) q^{31} +(16.0000 - 27.7128i) q^{32} +(4.53748 + 7.85914i) q^{34} +147.687 q^{35} -20.6124 q^{37} +(87.5375 + 151.619i) q^{38} +(-20.0000 + 34.6410i) q^{40} +(-250.612 + 434.073i) q^{41} +(-147.344 - 255.207i) q^{43} -186.150 q^{44} +321.375 q^{46} +(239.494 + 414.815i) q^{47} +(-264.731 + 458.528i) q^{49} +(25.0000 - 43.3013i) q^{50} +(-184.150 - 318.957i) q^{52} +243.075 q^{53} +232.687 q^{55} +(-118.150 - 204.642i) q^{56} +(241.762 - 418.745i) q^{58} +(-191.850 + 332.294i) q^{59} +(-66.3062 - 114.846i) q^{61} +5.37477 q^{62} +64.0000 q^{64} +(230.187 + 398.696i) q^{65} +(-291.306 + 504.557i) q^{67} +(-9.07495 + 15.7183i) q^{68} +(147.687 + 255.802i) q^{70} +566.775 q^{71} -839.388 q^{73} +(-20.6124 - 35.7018i) q^{74} +(-175.075 + 303.239i) q^{76} +(-687.300 + 1190.44i) q^{77} +(-225.888 - 391.249i) q^{79} -80.0000 q^{80} -1002.45 q^{82} +(-150.525 - 260.716i) q^{83} +(11.3437 - 19.6478i) q^{85} +(294.687 - 510.414i) q^{86} +(-186.150 - 322.421i) q^{88} -739.349 q^{89} -2719.66 q^{91} +(321.375 + 556.637i) q^{92} +(-478.987 + 829.630i) q^{94} +(218.844 - 379.048i) q^{95} +(573.069 + 992.584i) q^{97} -1058.93 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 8 q^{4} + 10 q^{5} - q^{7} - 32 q^{8} + 40 q^{10} + 33 q^{11} - 64 q^{13} + 2 q^{14} - 32 q^{16} - 102 q^{17} + 230 q^{19} + 40 q^{20} - 66 q^{22} + 21 q^{23} - 50 q^{25} - 256 q^{26} + 8 q^{28}+ \cdots - 4476 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 - 8 * q^4 + 10 * q^5 - q^7 - 32 * q^8 + 40 * q^10 + 33 * q^11 - 64 * q^13 + 2 * q^14 - 32 * q^16 - 102 * q^17 + 230 * q^19 + 40 * q^20 - 66 * q^22 + 21 * q^23 - 50 * q^25 - 256 * q^26 + 8 * q^28 - 63 * q^29 - 295 * q^31 + 64 * q^32 - 102 * q^34 - 10 * q^35 + 278 * q^37 + 230 * q^38 - 80 * q^40 - 642 * q^41 - 289 * q^43 - 264 * q^44 + 84 * q^46 + 177 * q^47 - 1119 * q^49 + 100 * q^50 - 256 * q^52 + 732 * q^53 + 330 * q^55 + 8 * q^56 + 126 * q^58 - 1248 * q^59 - 85 * q^61 - 1180 * q^62 + 256 * q^64 + 320 * q^65 - 985 * q^67 + 204 * q^68 - 10 * q^70 + 2988 * q^71 - 3718 * q^73 + 278 * q^74 - 460 * q^76 - 1788 * q^77 - 1264 * q^79 - 320 * q^80 - 2568 * q^82 + 1080 * q^83 - 255 * q^85 + 578 * q^86 - 264 * q^88 + 1368 * q^89 - 7154 * q^91 + 84 * q^92 - 354 * q^94 + 575 * q^95 + 1271 * q^97 - 4476 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

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 $

$2$	1.00000	+	1.73205i	0.353553	+	0.612372i
$2$	1.00000	+	1.73205i	0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−2.00000	+	3.46410i	−0.250000	+	0.433013i
$5$	2.50000	−	4.33013i	0.223607	−	0.387298i
$5$	2.50000	−	4.33013i	0.223607	−	0.387298i
$6$		0			0
$7$	14.7687	+	25.5802i	0.797437	+	1.38120i	0.921280	+	0.388899i	$0.127145\pi$
$7$	14.7687	+	25.5802i	0.797437	+	1.38120i	−0.123843	+	0.992302i	$0.539522\pi$
$8$	−8.00000			−0.353553
$9$		0			0
$10$	10.0000			0.316228
$11$	23.2687	+	40.3026i	0.637799	+	1.10470i	0.985915	+	0.167249i	$0.0534883\pi$
$11$	23.2687	+	40.3026i	0.637799	+	1.10470i	−0.348116	+	0.937452i	$0.613178\pi$
$12$		0			0
$13$	−46.0375	+	79.7392i	−0.982192	+	1.70121i	−0.328388	+	0.944543i	$0.606505\pi$
$13$	−46.0375	+	79.7392i	−0.982192	+	1.70121i	−0.653804	+	0.756664i	$0.726828\pi$
$14$	−29.5375	+	51.1604i	−0.563873	+	0.976657i
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.125000	−	0.216506i
$17$	4.53748			0.0647353			0.0323676	−	0.999476i	$-0.489695\pi$
$17$	4.53748			0.0647353			0.0323676	+	0.999476i	$0.489695\pi$
$18$		0			0
$19$	87.5375			1.05697			0.528486	−	0.848942i	$-0.322760\pi$
$19$	87.5375			1.05697			0.528486	+	0.848942i	$0.322760\pi$
$20$	10.0000	+	17.3205i	0.111803	+	0.193649i
$21$		0			0
$22$	−46.5375	+	80.6053i	−0.450992	+	0.781141i
$23$	80.3437	−	139.159i	0.728383	−	1.26160i	−0.229183	−	0.973383i	$-0.573605\pi$
$23$	80.3437	−	139.159i	0.728383	−	1.26160i	0.957566	−	0.288214i	$-0.0930613\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.100000	−	0.173205i
$26$	−184.150			−1.38903
$27$		0			0
$28$	−118.150			−0.797437
$29$	−120.881	−	209.372i	−0.774037	−	1.34067i	−0.935334	−	0.353765i	$-0.884901\pi$
$29$	−120.881	−	209.372i	−0.774037	−	1.34067i	0.161297	−	0.986906i	$-0.448432\pi$
$30$		0			0
$31$	1.34369	−	2.32734i	0.00778497	−	0.0134840i	−0.862107	−	0.506727i	$-0.830855\pi$
$31$	1.34369	−	2.32734i	0.00778497	−	0.0134840i	0.869892	+	0.493243i	$0.164189\pi$
$32$	16.0000	−	27.7128i	0.0883883	−	0.153093i
$33$		0			0
$34$	4.53748	+	7.85914i	0.0228874	+	0.0396421i
$35$	147.687			0.713249
$36$		0			0
$37$	−20.6124			−0.0915855			−0.0457927	−	0.998951i	$-0.514581\pi$
$37$	−20.6124			−0.0915855			−0.0457927	+	0.998951i	$0.514581\pi$
$38$	87.5375	+	151.619i	0.373696	+	0.647261i
$39$		0			0
$40$	−20.0000	+	34.6410i	−0.0790569	+	0.136931i
$41$	−250.612	+	434.073i	−0.954612	+	1.65344i	−0.219358	+	0.975644i	$0.570396\pi$
$41$	−250.612	+	434.073i	−0.954612	+	1.65344i	−0.735254	+	0.677792i	$0.762937\pi$
$42$		0			0
$43$	−147.344	−	255.207i	−0.522551	−	0.905085i	−0.999656	−	0.0262387i	$-0.991647\pi$
$43$	−147.344	−	255.207i	−0.522551	−	0.905085i	0.477104	−	0.878847i	$-0.341686\pi$
$44$	−186.150			−0.637799
$45$		0			0
$46$	321.375			1.03009
$47$	239.494	+	414.815i	0.743271	+	1.28738i	0.950998	+	0.309196i	$0.100060\pi$
$47$	239.494	+	414.815i	0.743271	+	1.28738i	−0.207727	+	0.978187i	$0.566607\pi$
$48$		0			0
$49$	−264.731	+	458.528i	−0.771811	+	1.33682i
$50$	25.0000	−	43.3013i	0.0707107	−	0.122474i
$51$		0			0
$52$	−184.150	−	318.957i	−0.491096	−	0.850603i
$53$	243.075			0.629979			0.314990	−	0.949095i	$-0.397999\pi$
$53$	243.075			0.629979			0.314990	+	0.949095i	$0.397999\pi$
$54$		0			0
$55$	232.687			0.570465
$56$	−118.150	−	204.642i	−0.281937	−	0.488328i
$57$		0			0
$58$	241.762	−	418.745i	0.547327	−	0.947998i
$59$	−191.850	+	332.294i	−0.423335	+	0.733237i	−0.996263	−	0.0863679i	$-0.972474\pi$
$59$	−191.850	+	332.294i	−0.423335	+	0.733237i	0.572928	+	0.819605i	$0.305807\pi$
$60$		0			0
$61$	−66.3062	−	114.846i	−0.139174	−	0.241057i	0.788010	−	0.615663i	$-0.211112\pi$
$61$	−66.3062	−	114.846i	−0.139174	−	0.241057i	−0.927184	+	0.374605i	$0.877778\pi$
$62$	5.37477			0.0110096
$63$		0			0
$64$	64.0000			0.125000
$65$	230.187	+	398.696i	0.439250	+	0.760803i
$66$		0			0
$67$	−291.306	+	504.557i	−0.531175	+	0.920022i	0.468163	+	0.883642i	$0.344916\pi$
$67$	−291.306	+	504.557i	−0.531175	+	0.920022i	−0.999338	+	0.0363799i	$0.988417\pi$
$68$	−9.07495	+	15.7183i	−0.0161838	+	0.0280312i
$69$		0			0
$70$	147.687	+	255.802i	0.252172	+	0.436774i
$71$	566.775			0.947378			0.473689	−	0.880692i	$-0.342922\pi$
$71$	566.775			0.947378			0.473689	+	0.880692i	$0.342922\pi$
$72$		0			0
$73$	−839.388			−1.34579			−0.672896	−	0.739737i	$-0.734950\pi$
$73$	−839.388			−1.34579			−0.672896	+	0.739737i	$0.734950\pi$
$74$	−20.6124	−	35.7018i	−0.0323804	−	0.0560844i
$75$		0			0
$76$	−175.075	+	303.239i	−0.264243	+	0.457682i
$77$	−687.300	+	1190.44i	−1.01721	+	1.76186i
$78$		0			0
$79$	−225.888	−	391.249i	−0.321700	−	0.557202i	0.659139	−	0.752022i	$-0.270921\pi$
$79$	−225.888	−	391.249i	−0.321700	−	0.557202i	−0.980839	+	0.194820i	$0.937588\pi$
$80$	−80.0000			−0.111803
$81$		0			0
$82$	−1002.45			−1.35003
$83$	−150.525	−	260.716i	−0.199063	−	0.344787i	0.749162	−	0.662387i	$-0.230457\pi$
$83$	−150.525	−	260.716i	−0.199063	−	0.344787i	−0.948225	+	0.317600i	$0.897123\pi$
$84$		0			0
$85$	11.3437	−	19.6478i	0.0144752	−	0.0250719i
$86$	294.687	−	510.414i	0.369500	−	0.639992i
$87$		0			0
$88$	−186.150	−	322.421i	−0.225496	−	0.390571i
$89$	−739.349			−0.880571			−0.440286	−	0.897858i	$-0.645123\pi$
$89$	−739.349			−0.880571			−0.440286	+	0.897858i	$0.645123\pi$
$90$		0			0
$91$	−2719.66			−3.13295
$92$	321.375	+	556.637i	0.364192	+	0.630799i
$93$		0			0
$94$	−478.987	+	829.630i	−0.525572	+	0.910317i
$95$	218.844	−	379.048i	0.236346	−	0.409364i
$96$		0			0
$97$	573.069	+	992.584i	0.599859	+	1.03899i	0.992841	+	0.119440i	$0.0381099\pi$
$97$	573.069	+	992.584i	0.599859	+	1.03899i	−0.392983	+	0.919546i	$0.628557\pi$
$98$	−1058.93			−1.09151
$99$		0			0
$100$	100.000			0.100000
$101$	−255.731	−	442.940i	−0.251943	−	0.436378i	0.712118	−	0.702060i	$-0.247736\pi$
$101$	−255.731	−	442.940i	−0.251943	−	0.436378i	−0.964061	+	0.265682i	$0.914403\pi$
$102$		0			0
$103$	−210.756	+	365.040i	−0.201616	+	0.349208i	−0.949049	−	0.315128i	$-0.897952\pi$
$103$	−210.756	+	365.040i	−0.201616	+	0.349208i	0.747434	+	0.664337i	$0.231286\pi$
$104$	368.300	−	637.914i	0.347257	−	0.601467i
$105$		0			0
$106$	243.075	+	421.018i	0.222731	+	0.385782i
$107$	1639.50			1.48127			0.740637	−	0.671905i	$-0.234524\pi$
$107$	1639.50			1.48127			0.740637	+	0.671905i	$0.234524\pi$
$108$		0			0
$109$	−690.475			−0.606748			−0.303374	−	0.952872i	$-0.598113\pi$
$109$	−690.475			−0.606748			−0.303374	+	0.952872i	$0.598113\pi$
$110$	232.687	+	403.026i	0.201690	+	0.349337i
$111$		0			0
$112$	236.300	−	409.283i	0.199359	−	0.345300i
$113$	485.119	−	840.250i	0.403860	−	0.699505i	−0.590328	−	0.807163i	$-0.701002\pi$
$113$	485.119	−	840.250i	0.403860	−	0.699505i	0.994188	+	0.107658i	$0.0343351\pi$
$114$		0			0
$115$	−401.718	−	695.797i	−0.325743	−	0.564203i
$116$	967.049			0.774037
$117$		0			0
$118$	−767.400			−0.598686
$119$	67.0128	+	116.070i	0.0516223	+	0.0894124i
$120$		0			0
$121$	−417.368	+	722.903i	−0.313575	+	0.543128i
$122$	132.612	−	229.691i	0.0984112	−	0.170453i
$123$		0			0
$124$	5.37477	+	9.30937i	0.00389249	+	0.00674198i
$125$	−125.000			−0.0894427
$126$		0			0
$127$	805.024			0.562475			0.281237	−	0.959638i	$-0.409255\pi$
$127$	805.024			0.562475			0.281237	+	0.959638i	$0.409255\pi$
$128$	64.0000	+	110.851i	0.0441942	+	0.0765466i
$129$		0			0
$130$	−460.375	+	797.392i	−0.310596	+	0.537969i
$131$	107.344	−	185.925i	0.0715928	−	0.124002i	−0.828007	−	0.560718i	$-0.810525\pi$
$131$	107.344	−	185.925i	0.0715928	−	0.124002i	0.899600	+	0.436716i	$0.143858\pi$
$132$		0			0
$133$	1292.82	+	2239.23i	0.842869	+	1.45989i
$134$	−1165.22			−0.751195
$135$		0			0
$136$	−36.2998			−0.0228874
$137$	183.899	+	318.523i	0.114683	+	0.198637i	0.917653	−	0.397383i	$-0.130081\pi$
$137$	183.899	+	318.523i	0.114683	+	0.198637i	−0.802970	+	0.596020i	$0.796748\pi$
$138$		0			0
$139$	713.655	−	1236.09i	0.435478	−	0.754270i	−0.561856	−	0.827235i	$-0.689913\pi$
$139$	713.655	−	1236.09i	0.435478	−	0.754270i	0.997335	+	0.0729645i	$0.0232460\pi$
$140$	−295.375	+	511.604i	−0.178312	+	0.308846i
$141$		0			0
$142$	566.775	+	981.683i	0.334949	+	0.580148i
$143$	−4284.94			−2.50576
$144$		0			0
$145$	−1208.81			−0.692320
$146$	−839.388	−	1453.86i	−0.475810	−	0.824126i
$147$		0			0
$148$	41.2249	−	71.4036i	0.0228964	−	0.0396577i
$149$	−878.868	+	1522.24i	−0.483219	+	0.836961i	−0.999814	−	0.0192693i	$-0.993866\pi$
$149$	−878.868	+	1522.24i	−0.483219	+	0.836961i	0.516595	+	0.856230i	$0.327199\pi$
$150$		0			0
$151$	268.888	+	465.727i	0.144912	+	0.250995i	0.929340	−	0.369225i	$-0.120377\pi$
$151$	268.888	+	465.727i	0.144912	+	0.250995i	−0.784428	+	0.620220i	$0.787043\pi$
$152$	−700.300			−0.373696
$153$		0			0
$154$	−2749.20			−1.43855
$155$	−6.71846	−	11.6367i	−0.00348155	−	0.00603021i
$156$		0			0
$157$	−335.894	+	581.785i	−0.170747	+	0.295742i	−0.938681	−	0.344786i	$-0.887951\pi$
$157$	−335.894	+	581.785i	−0.170747	+	0.295742i	0.767934	+	0.640529i	$0.221285\pi$
$158$	451.775	−	782.497i	0.227477	−	0.394001i
$159$		0			0
$160$	−80.0000	−	138.564i	−0.0395285	−	0.0684653i
$161$	4746.30			2.32336
$162$		0			0
$163$	−2501.42			−1.20200			−0.601002	−	0.799248i	$-0.705232\pi$
$163$	−2501.42			−1.20200			−0.601002	+	0.799248i	$0.705232\pi$
$164$	−1002.45	−	1736.29i	−0.477306	−	0.826718i
$165$		0			0
$166$	301.049	−	521.433i	0.140759	−	0.243801i
$167$	375.824	−	650.947i	0.174145	−	0.301628i	−0.765720	−	0.643174i	$-0.777617\pi$
$167$	375.824	−	650.947i	0.174145	−	0.301628i	0.939865	+	0.341546i	$0.110951\pi$
$168$		0			0
$169$	−3140.40	−	5439.33i	−1.42940	−	2.47580i
$170$	45.3748			0.0204711
$171$		0			0
$172$	1178.75			0.522551
$173$	1187.14	+	2056.18i	0.521713	+	0.903634i	0.999681	+	0.0252563i	$0.00804018\pi$
$173$	1187.14	+	2056.18i	0.521713	+	0.903634i	−0.477968	+	0.878377i	$0.658626\pi$
$174$		0			0
$175$	369.218	−	639.505i	0.159487	−	0.276240i
$176$	372.300	−	644.842i	0.159450	−	0.276175i
$177$		0			0
$178$	−739.349	−	1280.59i	−0.311329	−	0.539238i
$179$	2424.52			1.01239			0.506194	−	0.862420i	$-0.331052\pi$
$179$	2424.52			1.01239			0.506194	+	0.862420i	$0.331052\pi$
$180$		0			0
$181$	1179.46			0.484357			0.242179	−	0.970232i	$-0.422138\pi$
$181$	1179.46			0.484357			0.242179	+	0.970232i	$0.422138\pi$
$182$	−2719.66	−	4710.59i	−1.10766	−	1.91853i
$183$		0			0
$184$	−642.750	+	1113.27i	−0.257522	+	0.446042i
$185$	−51.5311	+	89.2544i	−0.0204791	+	0.0354709i
$186$		0			0
$187$	105.581	+	182.872i	0.0412881	+	0.0715131i
$188$	−1915.95			−0.743271
$189$		0			0
$190$	875.375			0.334244
$191$	−172.238	−	298.324i	−0.0652496	−	0.113016i	0.831555	−	0.555442i	$-0.187451\pi$
$191$	−172.238	−	298.324i	−0.0652496	−	0.113016i	−0.896805	+	0.442427i	$0.854118\pi$
$192$		0			0
$193$	−970.919	+	1681.68i	−0.362115	+	0.627202i	−0.988309	−	0.152466i	$-0.951279\pi$
$193$	−970.919	+	1681.68i	−0.362115	+	0.627202i	0.626193	+	0.779668i	$0.284612\pi$
$194$	−1146.14	+	1985.17i	−0.424164	+	0.734674i
$195$		0			0
$196$	−1058.93	−	1834.11i	−0.385906	−	0.668408i
$197$	1780.95			0.644099			0.322049	−	0.946723i	$-0.395628\pi$
$197$	1780.95			0.644099			0.322049	+	0.946723i	$0.395628\pi$
$198$		0			0
$199$	−808.375			−0.287961			−0.143980	−	0.989581i	$-0.545990\pi$
$199$	−808.375			−0.287961			−0.143980	+	0.989581i	$0.545990\pi$
$200$	100.000	+	173.205i	0.0353553	+	0.0612372i
$201$		0			0
$202$	511.463	−	885.879i	0.178150	−	0.308566i
$203$	3570.52	−	6184.33i	1.23449	−	2.13820i
$204$		0			0
$205$	1253.06	+	2170.37i	0.426915	+	0.739439i
$206$	−843.024			−0.285127
$207$		0			0
$208$	1473.20			0.491096
$209$	2036.89	+	3527.99i	0.674136	+	1.16764i
$210$		0			0
$211$	−611.518	+	1059.18i	−0.199520	+	0.345578i	−0.948373	−	0.317158i	$-0.897272\pi$
$211$	−611.518	+	1059.18i	−0.199520	+	0.345578i	0.748853	+	0.662736i	$0.230605\pi$
$212$	−486.150	+	842.036i	−0.157495	+	0.272789i
$213$		0			0
$214$	1639.50	+	2839.70i	0.523710	+	0.907092i
$215$	−1473.44			−0.467384
$216$		0			0
$217$	79.3785			0.0248321
$218$	−690.475	−	1195.94i	−0.214518	−	0.371556i
$219$		0			0
$220$	−465.375	+	806.053i	−0.142616	+	0.247018i
$221$	−208.894	+	361.815i	−0.0635825	+	0.110128i
$222$		0			0
$223$	−530.501	−	918.855i	−0.159305	−	0.275924i	0.775313	−	0.631577i	$-0.217592\pi$
$223$	−530.501	−	918.855i	−0.159305	−	0.275924i	−0.934618	+	0.355653i	$0.884259\pi$
$224$	945.199			0.281937
$225$		0			0
$226$	1940.48			0.571144
$227$	−1786.54	−	3094.37i	−0.522364	−	0.904760i	−0.999661	−	0.0260187i	$-0.991717\pi$
$227$	−1786.54	−	3094.37i	−0.522364	−	0.904760i	0.477298	−	0.878742i	$-0.341616\pi$
$228$		0			0
$229$	−500.912	+	867.605i	−0.144547	+	0.250362i	−0.929204	−	0.369568i	$-0.879506\pi$
$229$	−500.912	+	867.605i	−0.144547	+	0.250362i	0.784657	+	0.619930i	$0.212839\pi$
$230$	803.437	−	1391.59i	0.230335	−	0.398952i
$231$		0			0
$232$	967.049	+	1674.98i	0.273663	+	0.473999i
$233$	6082.42			1.71018			0.855091	−	0.518477i	$-0.173501\pi$
$233$	6082.42			1.71018			0.855091	+	0.518477i	$0.173501\pi$
$234$		0			0
$235$	2394.94			0.664802
$236$	−767.400	−	1329.18i	−0.211667	−	0.366619i
$237$		0			0
$238$	−134.026	+	232.139i	−0.0365025	+	0.0632241i
$239$	106.199	−	183.942i	0.0287425	−	0.0497835i	−0.851296	−	0.524685i	$-0.824183\pi$
$239$	106.199	−	183.942i	0.0287425	−	0.0497835i	0.880039	+	0.474902i	$0.157516\pi$
$240$		0			0
$241$	1472.22	+	2549.97i	0.393503	+	0.681568i	0.992909	−	0.118878i	$-0.0379297\pi$
$241$	1472.22	+	2549.97i	0.393503	+	0.681568i	−0.599406	+	0.800445i	$0.704596\pi$
$242$	−1669.47			−0.443462
$243$		0			0
$244$	530.450			0.139174
$245$	1323.66	+	2292.64i	0.345164	+	0.597842i
$246$		0			0
$247$	−4030.00	+	6980.17i	−1.03815	+	1.79813i
$248$	−10.7495	+	18.6187i	−0.00275240	+	0.00476730i
$249$		0			0
$250$	−125.000	−	216.506i	−0.0316228	−	0.0547723i
$251$	1130.36			0.284254			0.142127	−	0.989848i	$-0.454606\pi$
$251$	1130.36			0.284254			0.142127	+	0.989848i	$0.454606\pi$
$252$		0			0
$253$	7477.99			1.85825
$254$	805.024	+	1394.34i	0.198865	+	0.344444i
$255$		0			0
$256$	−128.000	+	221.703i	−0.0312500	+	0.0541266i
$257$	−609.168	+	1055.11i	−0.147856	+	0.256093i	−0.930435	−	0.366458i	$-0.880570\pi$
$257$	−609.168	+	1055.11i	−0.147856	+	0.256093i	0.782579	+	0.622551i	$0.213904\pi$
$258$		0			0
$259$	−304.420	−	527.270i	−0.0730336	−	0.126498i
$260$	−1841.50			−0.439250
$261$		0			0
$262$	429.375			0.101248
$263$	3468.75	+	6008.05i	0.813279	+	1.40864i	0.910557	+	0.413383i	$0.135653\pi$
$263$	3468.75	+	6008.05i	0.813279	+	1.40864i	−0.0972789	+	0.995257i	$0.531014\pi$
$264$		0			0
$265$	607.687	−	1052.55i	0.140868	−	0.243990i
$266$	−2585.64	+	4478.45i	−0.595998	+	1.03230i
$267$		0			0
$268$	−1165.22	−	2018.23i	−0.265587	−	0.460011i
$269$	515.214			0.116778			0.0583888	−	0.998294i	$-0.481404\pi$
$269$	515.214			0.116778			0.0583888	+	0.998294i	$0.481404\pi$
$270$		0			0
$271$	−652.411			−0.146240			−0.0731202	−	0.997323i	$-0.523296\pi$
$271$	−652.411			−0.146240			−0.0731202	+	0.997323i	$0.523296\pi$
$272$	−36.2998	−	62.8731i	−0.00809191	−	0.0140156i
$273$		0			0
$274$	−367.799	+	637.046i	−0.0810932	+	0.140458i
$275$	581.718	−	1007.57i	0.127560	−	0.220940i
$276$		0			0
$277$	−1469.42	−	2545.12i	−0.318733	−	0.552062i	0.661491	−	0.749953i	$-0.269924\pi$
$277$	−1469.42	−	2545.12i	−0.318733	−	0.552062i	−0.980224	+	0.197891i	$0.936591\pi$
$278$	2854.62			0.615859
$279$		0			0
$280$	−1181.50			−0.252172
$281$	−2675.89	−	4634.77i	−0.568078	−	0.983941i	−0.996756	−	0.0804830i	$-0.974354\pi$
$281$	−2675.89	−	4634.77i	−0.568078	−	0.983941i	0.428678	−	0.903458i	$-0.358980\pi$
$282$		0			0
$283$	−739.974	+	1281.67i	−0.155431	+	0.269214i	−0.933216	−	0.359316i	$-0.883010\pi$
$283$	−739.974	+	1281.67i	−0.155431	+	0.269214i	0.777785	+	0.628530i	$0.216343\pi$
$284$	−1133.55	+	1963.37i	−0.236844	+	0.410227i
$285$		0			0
$286$	−4284.94	−	7421.73i	−0.885922	−	1.53446i
$287$	−14804.9			−3.04497
$288$		0			0
$289$	−4892.41			−0.995809
$290$	−1208.81	−	2093.72i	−0.244772	−	0.423957i
$291$		0			0
$292$	1678.78	−	2907.72i	0.336448	−	0.582745i
$293$	2589.45	−	4485.06i	0.516305	−	0.894266i	−0.483516	−	0.875335i	$-0.660641\pi$
$293$	2589.45	−	4485.06i	0.516305	−	0.894266i	0.999821	−	0.0189305i	$-0.00602612\pi$
$294$		0			0
$295$	959.250	+	1661.47i	0.189321	+	0.327914i
$296$	164.899			0.0323804
$297$		0			0
$298$	−3515.47			−0.683376
$299$	7397.64	+	12813.1i	1.43082	+	2.47826i
$300$		0			0
$301$	4352.16	−	7538.16i	0.833403	−	1.44350i
$302$	−537.775	+	931.454i	−0.102468	+	0.177481i
$303$		0			0
$304$	−700.300	−	1212.95i	−0.132122	−	0.228841i
$305$	−663.062			−0.124481
$306$		0			0
$307$	10306.0			1.91594			0.957970	−	0.286868i	$-0.0926141\pi$
$307$	10306.0			1.91594			0.957970	+	0.286868i	$0.0926141\pi$
$308$	−2749.20	−	4761.75i	−0.508604	−	0.880929i
$309$		0			0
$310$	13.4369	−	23.2734i	0.00246182	−	0.00426400i
$311$	−1031.32	+	1786.30i	−0.188042	+	0.325698i	−0.944597	−	0.328232i	$-0.893547\pi$
$311$	−1031.32	+	1786.30i	−0.188042	+	0.325698i	0.756556	+	0.653929i	$0.226881\pi$
$312$		0			0
$313$	−4623.98	−	8008.97i	−0.835025	−	1.44631i	−0.894010	−	0.448046i	$-0.852120\pi$
$313$	−4623.98	−	8008.97i	−0.835025	−	1.44631i	0.0589856	−	0.998259i	$-0.481213\pi$
$314$	−1343.58			−0.241473
$315$		0			0
$316$	1807.10			0.321700
$317$	−305.174	−	528.576i	−0.0540702	−	0.0936524i	0.837723	−	0.546095i	$-0.183886\pi$
$317$	−305.174	−	528.576i	−0.0540702	−	0.0936524i	−0.891794	+	0.452442i	$0.850553\pi$
$318$		0			0
$319$	5625.50	−	9743.66i	0.987360	−	1.71016i
$320$	160.000	−	277.128i	0.0279508	−	0.0484123i
$321$		0			0
$322$	4746.30	+	8220.83i	0.821432	+	1.42276i
$323$	397.199			0.0684234
$324$		0			0
$325$	2301.87			0.392877
$326$	−2501.42	−	4332.59i	−0.424973	−	0.736074i
$327$		0			0
$328$	2004.90	−	3472.59i	0.337506	−	0.584578i
$329$	−7074.04	+	12252.6i	−1.18542	+	2.05321i
$330$		0			0
$331$	1044.63	+	1809.36i	0.173469	+	0.300457i	0.939630	−	0.342191i	$-0.111169\pi$
$331$	1044.63	+	1809.36i	0.173469	+	0.300457i	−0.766162	+	0.642648i	$0.777836\pi$
$332$	1204.20			0.199063
$333$		0			0
$334$	1503.30			0.246278
$335$	1456.53	+	2522.79i	0.237549	+	0.411446i
$336$		0			0
$337$	4713.18	−	8163.46i	0.761849	−	1.31956i	−0.180047	−	0.983658i	$-0.557625\pi$
$337$	4713.18	−	8163.46i	0.761849	−	1.31956i	0.941896	−	0.335904i	$-0.109042\pi$
$338$	6280.80	−	10878.7i	1.01074	−	1.75065i
$339$		0			0
$340$	45.3748	+	78.5914i	0.00723762	+	0.0125359i
$341$	125.064			0.0198610
$342$		0			0
$343$	−5507.63			−0.867009
$344$	1178.75	+	2041.65i	0.184750	+	0.319996i
$345$		0			0
$346$	−2374.27	+	4112.36i	−0.368907	+	0.638965i
$347$	−610.612	+	1057.61i	−0.0944651	+	0.163618i	−0.909385	−	0.415955i	$-0.863447\pi$
$347$	−610.612	+	1057.61i	−0.0944651	+	0.163618i	0.814920	+	0.579573i	$0.196781\pi$
$348$		0			0
$349$	2094.14	+	3627.16i	0.321195	+	0.556325i	0.980735	−	0.195344i	$-0.0625823\pi$
$349$	2094.14	+	3627.16i	0.321195	+	0.556325i	−0.659540	+	0.751669i	$0.729249\pi$
$350$	1476.87			0.225549
$351$		0			0
$352$	1489.20			0.225496
$353$	−2234.68	−	3870.58i	−0.336941	−	0.583599i	0.646915	−	0.762562i	$-0.276059\pi$
$353$	−2234.68	−	3870.58i	−0.336941	−	0.583599i	−0.983856	+	0.178964i	$0.942726\pi$
$354$		0			0
$355$	1416.94	−	2454.21i	0.211840	−	0.366918i
$356$	1478.70	−	2561.18i	0.220143	−	0.381299i
$357$		0			0
$358$	2424.52	+	4199.40i	0.357933	+	0.619958i
$359$	643.572			0.0946140			0.0473070	−	0.998880i	$-0.484936\pi$
$359$	643.572			0.0946140			0.0473070	+	0.998880i	$0.484936\pi$
$360$		0			0
$361$	803.810			0.117191
$362$	1179.46	+	2042.89i	0.171246	+	0.296607i
$363$		0			0
$364$	5439.32	−	9421.18i	0.783236	−	1.35661i
$365$	−2098.47	+	3634.65i	−0.300928	+	0.521223i
$366$		0			0
$367$	2967.37	+	5139.63i	0.422058	+	0.731026i	0.996141	−	0.0877712i	$-0.0279744\pi$
$367$	2967.37	+	5139.63i	0.422058	+	0.731026i	−0.574082	+	0.818798i	$0.694641\pi$
$368$	−2571.00			−0.364192
$369$		0			0
$370$	−206.124			−0.0289619
$371$	3589.91	+	6217.91i	0.502369	+	0.870128i
$372$		0			0
$373$	3093.91	−	5358.81i	0.429482	−	0.743884i	−0.567345	−	0.823480i	$-0.692030\pi$
$373$	3093.91	−	5358.81i	0.429482	−	0.743884i	0.996827	+	0.0795955i	$0.0253629\pi$
$374$	−211.163	+	365.745i	−0.0291951	+	0.0505674i
$375$		0			0
$376$	−1915.95	−	3318.52i	−0.262786	−	0.455159i
$377$	22260.3			3.04101
$378$		0			0
$379$	10428.5			1.41340			0.706699	−	0.707515i	$-0.250184\pi$
$379$	10428.5			1.41340			0.706699	+	0.707515i	$0.250184\pi$
$380$	875.375	+	1516.19i	0.118173	+	0.204682i
$381$		0			0
$382$	344.475	−	596.649i	0.0461385	−	0.0799142i
$383$	−5123.34	+	8873.89i	−0.683526	+	1.18390i	0.290372	+	0.956914i	$0.406221\pi$
$383$	−5123.34	+	8873.89i	−0.683526	+	1.18390i	−0.973898	+	0.226988i	$0.927112\pi$
$384$		0			0
$385$	3436.50	+	5952.19i	0.454910	+	0.787927i
$386$	−3883.67			−0.512108
$387$		0			0
$388$	−4584.55			−0.599859
$389$	−4254.24	−	7368.56i	−0.554495	−	0.960413i	−0.997943	−	0.0641131i	$-0.979578\pi$
$389$	−4254.24	−	7368.56i	−0.554495	−	0.960413i	0.443448	−	0.896300i	$-0.353755\pi$
$390$		0			0
$391$	364.558	−	631.432i	0.0471521	−	0.0816698i
$392$	2117.85	−	3668.22i	0.272876	−	0.472636i
$393$		0			0
$394$	1780.95	+	3084.70i	0.227723	+	0.394428i
$395$	−2258.88			−0.287738
$396$		0			0
$397$	3775.96			0.477356			0.238678	−	0.971099i	$-0.423286\pi$
$397$	3775.96			0.477356			0.238678	+	0.971099i	$0.423286\pi$
$398$	−808.375	−	1400.15i	−0.101809	−	0.176339i
$399$		0			0
$400$	−200.000	+	346.410i	−0.0250000	+	0.0433013i
$401$	3778.01	−	6543.70i	0.470486	−	0.814905i	−0.528945	−	0.848656i	$-0.677412\pi$
$401$	3778.01	−	6543.70i	0.470486	−	0.814905i	0.999430	+	0.0337512i	$0.0107454\pi$
$402$		0			0
$403$	123.720	+	214.290i	0.0152927	+	0.0264877i
$404$	2045.85			0.251943
$405$		0			0
$406$	14282.1			1.74583
$407$	−479.625	−	830.735i	−0.0584131	−	0.101174i
$408$		0			0
$409$	−4481.61	+	7762.37i	−0.541812	+	0.938446i	0.456988	+	0.889473i	$0.348928\pi$
$409$	−4481.61	+	7762.37i	−0.541812	+	0.938446i	−0.998800	+	0.0489735i	$0.984405\pi$
$410$	−2506.12	+	4340.73i	−0.301875	+	0.522862i
$411$		0			0
$412$	−843.024	−	1460.16i	−0.100808	−	0.174604i
$413$	−11333.5			−1.35033
$414$		0			0
$415$	−1505.25			−0.178047
$416$	1473.20	+	2551.66i	0.173629	+	0.300734i
$417$		0			0
$418$	−4073.77	+	7055.98i	−0.476686	+	0.825644i
$419$	−7076.90	+	12257.6i	−0.825130	+	1.42917i	0.0766901	+	0.997055i	$0.475565\pi$
$419$	−7076.90	+	12257.6i	−0.825130	+	1.42917i	−0.901820	+	0.432112i	$0.857769\pi$
$420$		0			0
$421$	7363.64	+	12754.2i	0.852451	+	1.47649i	0.878990	+	0.476841i	$0.158218\pi$
$421$	7363.64	+	12754.2i	0.852451	+	1.47649i	−0.0265387	+	0.999648i	$0.508449\pi$
$422$	−2446.07			−0.282163
$423$		0			0
$424$	−1944.60			−0.222731
$425$	−56.7185	−	98.2392i	−0.00647353	−	0.0112125i
$426$		0			0
$427$	1958.52	−	3392.25i	0.221966	−	0.384456i
$428$	−3279.00	+	5679.39i	−0.370319	+	0.641411i
$429$		0			0
$430$	−1473.44	−	2552.07i	−0.165245	−	0.286213i
$431$	−3048.98			−0.340752			−0.170376	−	0.985379i	$-0.554498\pi$
$431$	−3048.98			−0.340752			−0.170376	+	0.985379i	$0.554498\pi$
$432$		0			0
$433$	17218.4			1.91100			0.955500	−	0.294990i	$-0.0953164\pi$
$433$	17218.4			1.91100			0.955500	+	0.294990i	$0.0953164\pi$
$434$	79.3785	+	137.488i	0.00877947	+	0.0152065i
$435$		0			0
$436$	1380.95	−	2391.88i	0.151687	−	0.262730i
$437$	7033.08	−	12181.7i	0.769881	−	1.33347i
$438$		0			0
$439$	−833.953	−	1444.45i	−0.0906660	−	0.157038i	0.817126	−	0.576460i	$-0.195566\pi$
$439$	−833.953	−	1444.45i	−0.0906660	−	0.157038i	−0.907792	+	0.419422i	$0.862233\pi$
$440$	−1861.50			−0.201690
$441$		0			0
$442$	−835.576			−0.0899192
$443$	−4611.60	−	7987.53i	−0.494591	−	0.856657i	0.505390	−	0.862891i	$-0.331349\pi$
$443$	−4611.60	−	7987.53i	−0.494591	−	0.856657i	−0.999981	+	0.00623459i	$0.998015\pi$
$444$		0			0
$445$	−1848.37	+	3201.48i	−0.196902	+	0.341044i
$446$	1061.00	−	1837.71i	0.112646	−	0.195108i
$447$		0			0
$448$	945.199	+	1637.13i	0.0996796	+	0.172650i
$449$	2636.18			0.277080			0.138540	−	0.990357i	$-0.455759\pi$
$449$	2636.18			0.277080			0.138540	+	0.990357i	$0.455759\pi$
$450$		0			0
$451$	−23325.7			−2.43540
$452$	1940.48	+	3361.00i	0.201930	+	0.349753i
$453$		0			0
$454$	3573.07	−	6188.74i	0.369367	−	0.639762i
$455$	−6799.15	+	11776.5i	−0.700548	+	1.21338i
$456$		0			0
$457$	1929.84	+	3342.58i	0.197536	+	0.342142i	0.947729	−	0.319077i	$-0.103373\pi$
$457$	1929.84	+	3342.58i	0.197536	+	0.342142i	−0.750193	+	0.661219i	$0.770039\pi$
$458$	−2003.65			−0.204420
$459$		0			0
$460$	3213.75			0.325743
$461$	−3307.54	−	5728.82i	−0.334159	−	0.578780i	0.649164	−	0.760649i	$-0.275119\pi$
$461$	−3307.54	−	5728.82i	−0.334159	−	0.578780i	−0.983323	+	0.181868i	$0.941786\pi$
$462$		0			0
$463$	8785.87	−	15217.6i	0.881887	−	1.52747i	0.0326470	−	0.999467i	$-0.489606\pi$
$463$	8785.87	−	15217.6i	0.881887	−	1.52747i	0.849240	−	0.528007i	$-0.177060\pi$
$464$	−1934.10	+	3349.96i	−0.193509	+	0.335168i
$465$		0			0
$466$	6082.42	+	10535.1i	0.604641	+	1.04727i
$467$	10076.8			0.998502			0.499251	−	0.866457i	$-0.333609\pi$
$467$	10076.8			0.998502			0.499251	+	0.866457i	$0.333609\pi$
$468$		0			0
$469$	−17208.9			−1.69431
$470$	2394.94	+	4148.15i	0.235043	+	0.407106i
$471$		0			0
$472$	1534.80	−	2658.35i	0.149671	−	0.259239i
$473$	6857.00	−	11876.7i	0.666565	−	1.15452i
$474$		0			0
$475$	−1094.22	−	1895.24i	−0.105697	−	0.183073i
$476$	−536.102			−0.0516223
$477$		0			0
$478$	424.797			0.0406480
$479$	8972.89	+	15541.5i	0.855912	+	1.48248i	0.875797	+	0.482680i	$0.160337\pi$
$479$	8972.89	+	15541.5i	0.855912	+	1.48248i	−0.0198851	+	0.999802i	$0.506330\pi$
$480$		0			0
$481$	948.944	−	1643.62i	0.0899545	−	0.155806i
$482$	−2944.45	+	5099.93i	−0.278249	+	0.481941i
$483$		0			0
$484$	−1669.47	−	2891.61i	−0.156788	−	0.271564i
$485$	5730.69			0.536530
$486$		0			0
$487$	17356.9			1.61502			0.807511	−	0.589853i	$-0.200814\pi$
$487$	17356.9			1.61502			0.807511	+	0.589853i	$0.200814\pi$
$488$	530.450	+	918.766i	0.0492056	+	0.0852266i
$489$		0			0
$490$	−2647.31	+	4585.28i	−0.244068	+	0.422738i
$491$	4196.59	−	7268.70i	0.385722	−	0.668089i	−0.606147	−	0.795352i	$-0.707286\pi$
$491$	4196.59	−	7268.70i	0.385722	−	0.668089i	0.991869	+	0.127263i	$0.0406192\pi$
$492$		0			0
$493$	−548.495	−	950.022i	−0.0501075	−	0.0867887i
$494$	−16120.0			−1.46817
$495$		0			0
$496$	−42.9981			−0.00389249
$497$	8370.55	+	14498.2i	0.755474	+	1.30852i
$498$		0			0
$499$	−8210.57	+	14221.1i	−0.736585	+	1.27580i	0.217440	+	0.976074i	$0.430229\pi$
$499$	−8210.57	+	14221.1i	−0.736585	+	1.27580i	−0.954025	+	0.299728i	$0.903104\pi$
$500$	250.000	−	433.013i	0.0223607	−	0.0387298i
$501$		0			0
$502$	1130.36	+	1957.84i	0.100499	+	0.174069i
$503$	16014.9			1.41962			0.709808	−	0.704395i	$-0.248782\pi$
$503$	16014.9			1.41962			0.709808	+	0.704395i	$0.248782\pi$
$504$		0			0
$505$	−2557.31			−0.225344
$506$	7477.99	+	12952.3i	0.656990	+	1.13794i
$507$		0			0
$508$	−1610.05	+	2788.68i	−0.140619	+	0.243559i
$509$	6847.96	−	11861.0i	0.596327	−	1.03287i	−0.397031	−	0.917805i	$-0.629959\pi$
$509$	6847.96	−	11861.0i	0.596327	−	1.03287i	0.993358	−	0.115064i	$-0.0367073\pi$
$510$		0			0
$511$	−12396.7	−	21471.7i	−1.07318	−	1.85881i
$512$	−512.000			−0.0441942
$513$		0			0
$514$	−2436.67			−0.209099
$515$	1053.78	+	1825.20i	0.0901652	+	0.156171i
$516$		0			0
$517$	−11145.4	+	19304.4i	−0.948115	+	1.64218i
$518$	608.839	−	1054.54i	0.0516426	−	0.0894476i
$519$		0			0
$520$	−1841.50	−	3189.57i	−0.155298	−	0.268984i
$521$	5887.12			0.495047			0.247523	−	0.968882i	$-0.420383\pi$
$521$	5887.12			0.495047			0.247523	+	0.968882i	$0.420383\pi$
$522$		0			0
$523$	8544.57			0.714394			0.357197	−	0.934029i	$-0.383733\pi$
$523$	8544.57			0.714394			0.357197	+	0.934029i	$0.383733\pi$
$524$	429.375	+	743.699i	0.0357964	+	0.0620012i
$525$		0			0
$526$	−6937.50	+	12016.1i	−0.575075	+	0.996059i
$527$	6.09697	−	10.5603i	0.000503962	−	0.000872888i
$528$		0			0
$529$	−6826.72	−	11824.2i	−0.561085	−	0.971827i
$530$	2430.75			0.199217
$531$		0			0
$532$	−10342.5			−0.842869
$533$	−23075.1	−	39967.3i	−1.87522	−	3.24798i
$534$		0			0
$535$	4098.75	−	7099.24i	0.331223	−	0.573695i
$536$	2330.45	−	4036.46i	0.187799	−	0.325277i
$537$		0			0
$538$	515.214	+	892.377i	0.0412871	+	0.0715113i
$539$	−24639.8			−1.96904
$540$		0			0
$541$	10236.5			0.813499			0.406749	−	0.913540i	$-0.366662\pi$
$541$	10236.5			0.813499			0.406749	+	0.913540i	$0.366662\pi$
$542$	−652.411	−	1130.01i	−0.0517038	−	0.0895536i
$543$		0			0
$544$	72.5996	−	125.746i	0.00572184	−	0.00991052i
$545$	−1726.19	+	2989.85i	−0.135673	+	0.234993i
$546$		0			0
$547$	2392.66	+	4144.20i	0.187025	+	0.323937i	0.944257	−	0.329209i	$-0.106782\pi$
$547$	2392.66	+	4144.20i	0.187025	+	0.323937i	−0.757232	+	0.653146i	$0.773449\pi$
$548$	−1471.20			−0.114683
$549$		0			0
$550$	2326.87			0.180397
$551$	−10581.6	−	18327.9i	−0.818136	−	1.41705i
$552$		0			0
$553$	6672.15	−	11556.5i	0.513072	−	0.888666i
$554$	2938.85	−	5090.23i	0.225379	−	0.390367i
$555$		0			0
$556$	2854.62	+	4944.35i	0.217739	+	0.377135i
$557$	−2031.75			−0.154557			−0.0772783	−	0.997010i	$-0.524623\pi$
$557$	−2031.75			−0.154557			−0.0772783	+	0.997010i	$0.524623\pi$
$558$		0			0
$559$	27133.3			2.05298
$560$	−1181.50	−	2046.42i	−0.0891562	−	0.154423i
$561$		0			0
$562$	5351.77	−	9269.54i	0.401692	−	0.695751i
$563$	6081.18	−	10532.9i	0.455224	−	0.788472i	−0.543477	−	0.839424i	$-0.682892\pi$
$563$	6081.18	−	10532.9i	0.455224	−	0.788472i	0.998701	+	0.0509524i	$0.0162257\pi$
$564$		0			0
$565$	−2425.59	−	4201.25i	−0.180612	−	0.312828i
$566$	−2959.90			−0.219812
$567$		0			0
$568$	−4534.20			−0.334949
$569$	72.3766	+	125.360i	0.00533249	+	0.00923614i	0.868679	−	0.495375i	$-0.164969\pi$
$569$	72.3766	+	125.360i	0.00533249	+	0.00923614i	−0.863347	+	0.504611i	$0.831636\pi$
$570$		0			0
$571$	3739.64	−	6477.24i	0.274079	−	0.474718i	−0.695823	−	0.718213i	$-0.744960\pi$
$571$	3739.64	−	6477.24i	0.274079	−	0.474718i	0.969902	+	0.243494i	$0.0782937\pi$
$572$	8569.87	−	14843.5i	0.626441	−	1.08503i
$573$		0			0
$574$	−14804.9	−	25642.9i	−1.07656	−	1.86466i
$575$	−4017.18			−0.291353
$576$		0			0
$577$	−6562.94			−0.473516			−0.236758	−	0.971569i	$-0.576085\pi$
$577$	−6562.94			−0.473516			−0.236758	+	0.971569i	$0.576085\pi$
$578$	−4892.41	−	8473.90i	−0.352072	−	0.609806i
$579$		0			0
$580$	2417.62	−	4187.45i	0.173080	−	0.299783i
$581$	4446.12	−	7700.90i	0.317480	−	0.549892i
$582$		0			0
$583$	5656.05	+	9796.56i	0.401800	+	0.695938i
$584$	6715.10			0.475810
$585$		0			0
$586$	10357.8			0.730165
$587$	2113.95	+	3661.46i	0.148640	+	0.257453i	0.930725	−	0.365719i	$-0.119177\pi$
$587$	2113.95	+	3661.46i	0.148640	+	0.257453i	−0.782085	+	0.623172i	$0.785844\pi$
$588$		0			0
$589$	117.623	−	203.730i	0.00822850	−	0.0142522i
$590$	−1918.50	+	3322.94i	−0.133870	+	0.231870i
$591$		0			0
$592$	164.899	+	285.614i	0.0114482	+	0.0198288i
$593$	28746.9			1.99071			0.995356	−	0.0962641i	$-0.0306893\pi$
$593$	28746.9			1.99071			0.995356	+	0.0962641i	$0.0306893\pi$
$594$		0			0
$595$	670.128			0.0461724
$596$	−3515.47	−	6088.98i	−0.241610	−	0.418480i
$597$		0			0
$598$	−14795.3	+	25626.2i	−1.01175	+	1.75240i
$599$	−10857.4	+	18805.7i	−0.740607	+	1.28277i	0.211613	+	0.977354i	$0.432128\pi$
$599$	−10857.4	+	18805.7i	−0.740607	+	1.28277i	−0.952219	+	0.305415i	$0.901205\pi$
$600$		0			0
$601$	−3027.91	−	5244.49i	−0.205509	−	0.355952i	0.744786	−	0.667304i	$-0.232552\pi$
$601$	−3027.91	−	5244.49i	−0.205509	−	0.355952i	−0.950295	+	0.311351i	$0.899218\pi$
$602$	17408.6			1.17861
$603$		0			0
$604$	−2151.10			−0.144912
$605$	2086.84	+	3614.52i	0.140235	+	0.242894i
$606$		0			0
$607$	−9129.98	+	15813.6i	−0.610502	+	1.05742i	0.380654	+	0.924717i	$0.375699\pi$
$607$	−9129.98	+	15813.6i	−0.610502	+	1.05742i	−0.991156	+	0.132702i	$0.957635\pi$
$608$	1400.60	−	2425.91i	0.0934240	−	0.161815i
$609$		0			0
$610$	−663.062	−	1148.46i	−0.0440108	−	0.0762290i
$611$	−44102.7			−2.92014
$612$		0			0
$613$	−16207.4			−1.06788			−0.533941	−	0.845521i	$-0.679290\pi$
$613$	−16207.4			−1.06788			−0.533941	+	0.845521i	$0.679290\pi$
$614$	10306.0	+	17850.5i	0.677387	+	1.17327i
$615$		0			0
$616$	5498.40	−	9523.51i	0.359638	−	0.622911i
$617$	−7001.56	+	12127.1i	−0.456843	+	0.791276i	−0.998792	−	0.0491358i	$-0.984353\pi$
$617$	−7001.56	+	12127.1i	−0.456843	+	0.791276i	0.541949	+	0.840411i	$0.317687\pi$
$618$		0			0
$619$	134.842	+	233.553i	0.00875565	+	0.0151652i	0.870370	−	0.492398i	$-0.163880\pi$
$619$	134.842	+	233.553i	0.00875565	+	0.0151652i	−0.861614	+	0.507563i	$0.830546\pi$
$620$	53.7477			0.00348155
$621$		0			0
$622$	−4125.29			−0.265931
$623$	−10919.3	−	18912.7i	−0.702200	−	1.21625i
$624$		0			0
$625$	−312.500	+	541.266i	−0.0200000	+	0.0346410i
$626$	9247.96	−	16017.9i	0.590452	−	1.02269i
$627$		0			0
$628$	−1343.58	−	2327.14i	−0.0853734	−	0.147871i
$629$	−93.5284			−0.00592881
$630$		0			0
$631$	−21240.2			−1.34003			−0.670016	−	0.742347i	$-0.733713\pi$
$631$	−21240.2			−1.34003			−0.670016	+	0.742347i	$0.733713\pi$
$632$	1807.10	+	3129.99i	0.113738	+	0.197001i
$633$		0			0
$634$	610.347	−	1057.15i	0.0382334	−	0.0662222i
$635$	2012.56	−	3485.86i	0.125773	−	0.217846i
$636$		0			0
$637$	−24375.1	−	42218.9i	−1.51613	−	2.62602i
$638$	22502.0			1.39634
$639$		0			0
$640$	640.000			0.0395285
$641$	9605.43	+	16637.1i	0.591875	+	1.02516i	0.993980	+	0.109563i	$0.0349453\pi$
$641$	9605.43	+	16637.1i	0.591875	+	1.02516i	−0.402105	+	0.915593i	$0.631721\pi$
$642$		0			0
$643$	−3865.54	+	6695.31i	−0.237079	+	0.410633i	−0.959875	−	0.280429i	$-0.909523\pi$
$643$	−3865.54	+	6695.31i	−0.237079	+	0.410633i	0.722796	+	0.691062i	$0.242857\pi$
$644$	−9492.60	+	16441.7i	−0.580840	+	1.00604i
$645$		0			0
$646$	397.199	+	687.969i	0.0241913	+	0.0419006i
$647$	−23347.4			−1.41867			−0.709337	−	0.704870i	$-0.751005\pi$
$647$	−23347.4			−1.41867			−0.709337	+	0.704870i	$0.751005\pi$
$648$		0			0
$649$	−17856.4			−1.08001
$650$	2301.87	+	3986.96i	0.138903	+	0.240587i
$651$		0			0
$652$	5002.85	−	8665.19i	0.300501	−	0.520483i
$653$	3833.85	−	6640.43i	0.229755	−	0.397948i	−0.727980	−	0.685598i	$-0.759541\pi$
$653$	3833.85	−	6640.43i	0.229755	−	0.397948i	0.957736	+	0.287650i	$0.0928740\pi$
$654$		0			0
$655$	−536.718	−	929.624i	−0.0320173	−	0.0554556i
$656$	8019.60			0.477306
$657$		0			0
$658$	−28296.1			−1.67644
$659$	−7336.80	−	12707.7i	−0.433689	−	0.751171i	0.563499	−	0.826117i	$-0.309455\pi$
$659$	−7336.80	−	12707.7i	−0.433689	−	0.751171i	−0.997188	+	0.0749456i	$0.976122\pi$
$660$		0			0
$661$	−12723.5	+	22037.7i	−0.748691	+	1.29677i	0.199759	+	0.979845i	$0.435984\pi$
$661$	−12723.5	+	22037.7i	−0.748691	+	1.29677i	−0.948450	+	0.316926i	$0.897349\pi$
$662$	−2089.26	+	3618.71i	−0.122661	+	0.212455i
$663$		0			0
$664$	1204.20	+	2085.73i	0.0703794	+	0.121901i
$665$	12928.2			0.753885
$666$		0			0
$667$	−38848.2			−2.25518
$668$	1503.30	+	2603.79i	0.0870724	+	0.150814i
$669$		0			0
$670$	−2913.06	+	5045.57i	−0.167972	+	0.290936i
$671$	3085.72	−	5344.63i	0.177531	−	0.307492i
$672$		0			0
$673$	6119.73	+	10599.7i	0.350518	+	0.607114i	0.986340	−	0.164721i	$-0.0526724\pi$
$673$	6119.73	+	10599.7i	0.350518	+	0.607114i	−0.635823	+	0.771835i	$0.719339\pi$
$674$	18852.7			1.07742
$675$		0			0
$676$	25123.2			1.42940
$677$	−5228.77	−	9056.49i	−0.296836	−	0.514135i	0.678574	−	0.734532i	$-0.262598\pi$
$677$	−5228.77	−	9056.49i	−0.296836	−	0.514135i	−0.975410	+	0.220397i	$0.929265\pi$
$678$		0			0
$679$	−16927.0	+	29318.4i	−0.956699	+	1.65705i
$680$	−90.7495	+	157.183i	−0.00511777	+	0.00886424i
$681$		0			0
$682$	125.064	+	216.617i	0.00702192	+	0.0121623i
$683$	−304.720			−0.0170714			−0.00853572	−	0.999964i	$-0.502717\pi$
$683$	−304.720			−0.0170714			−0.00853572	+	0.999964i	$0.502717\pi$
$684$		0			0
$685$	1838.99			0.102576
$686$	−5507.63	−	9539.50i	−0.306534	−	0.530933i
$687$		0			0
$688$	−2357.50	+	4083.31i	−0.130638	+	0.226271i
$689$	−11190.6	+	19382.6i	−0.618761	+	1.07173i
$690$		0			0
$691$	7468.88	+	12936.5i	0.411186	+	0.712196i	0.995020	−	0.0996778i	$-0.0317812\pi$
$691$	7468.88	+	12936.5i	0.411186	+	0.712196i	−0.583833	+	0.811873i	$0.698448\pi$
$692$	−9497.10			−0.521713
$693$		0			0
$694$	−2442.45			−0.133594
$695$	−3568.28	−	6180.44i	−0.194752	−	0.337320i
$696$		0			0
$697$	−1137.15	+	1969.60i	−0.0617971	+	0.107036i
$698$	−4188.29	+	7254.33i	−0.227119	+	0.393381i
$699$		0			0
$700$	1476.87	+	2558.02i	0.0797437	+	0.138120i
$701$	−13752.9			−0.741000			−0.370500	−	0.928832i	$-0.620814\pi$
$701$	−13752.9			−0.741000			−0.370500	+	0.928832i	$0.620814\pi$
$702$		0			0
$703$	−1804.36			−0.0968033
$704$	1489.20	+	2579.37i	0.0797249	+	0.138088i
$705$		0			0
$706$	4469.36	−	7741.17i	0.238253	−	0.412667i
$707$	7553.66	−	13083.3i	0.401817	−	0.695967i
$708$		0			0
$709$	15612.7	+	27042.1i	0.827008	+	1.43242i	0.900375	+	0.435115i	$0.143292\pi$
$709$	15612.7	+	27042.1i	0.827008	+	1.43242i	−0.0733670	+	0.997305i	$0.523374\pi$
$710$	5667.75			0.299587
$711$		0			0
$712$	5914.79			0.311329
$713$	−215.914	−	373.974i	−0.0113409	−	0.0196430i
$714$		0			0
$715$	−10712.3	+	18554.3i	−0.560306	+	0.970478i
$716$	−4849.05	+	8398.79i	−0.253097	+	0.438377i
$717$		0			0
$718$	643.572	+	1114.70i	0.0334511	+	0.0579390i
$719$	−13945.5			−0.723337			−0.361669	−	0.932307i	$-0.617793\pi$
$719$	−13945.5			−0.723337			−0.361669	+	0.932307i	$0.617793\pi$
$720$		0			0
$721$	−12450.4			−0.643103
$722$	803.810	+	1392.24i	0.0414331	+	0.0717642i
$723$		0			0
$724$	−2358.92	+	4085.77i	−0.121089	+	0.209733i
$725$	−3022.03	+	5234.31i	−0.154807	+	0.268134i
$726$		0			0
$727$	18792.6	+	32549.7i	0.958704	+	1.66052i	0.725655	+	0.688058i	$0.241537\pi$
$727$	18792.6	+	32549.7i	0.958704	+	1.66052i	0.233048	+	0.972465i	$0.425130\pi$
$728$	21757.3			1.10766
$729$		0			0
$730$	−8393.88			−0.425577
$731$	−668.569	−	1157.99i	−0.0338275	−	0.0585909i
$732$		0			0
$733$	−1424.35	+	2467.05i	−0.0717729	+	0.124314i	−0.899678	−	0.436553i	$-0.856199\pi$
$733$	−1424.35	+	2467.05i	−0.0717729	+	0.124314i	0.827905	+	0.560868i	$0.189532\pi$
$734$	−5934.74	+	10279.3i	−0.298440	+	0.516914i
$735$		0			0
$736$	−2571.00	−	4453.10i	−0.128761	−	0.223021i
$737$	−27113.3			−1.35513
$738$		0			0
$739$	1828.99			0.0910428			0.0455214	−	0.998963i	$-0.485505\pi$
$739$	1828.99			0.0910428			0.0455214	+	0.998963i	$0.485505\pi$
$740$	−206.124	−	357.018i	−0.0102396	−	0.0177355i
$741$		0			0
$742$	−7179.82	+	12435.8i	−0.355228	+	0.615274i
$743$	16399.4	−	28404.6i	0.809740	−	1.40251i	−0.103304	−	0.994650i	$-0.532941\pi$
$743$	16399.4	−	28404.6i	0.809740	−	1.40251i	0.913044	−	0.407861i	$-0.133725\pi$
$744$		0			0
$745$	4394.34	+	7611.22i	0.216102	+	0.374300i
$746$	12375.7			0.607379
$747$		0			0
$748$	−844.651			−0.0412881
$749$	24213.3	+	41938.7i	1.18122	+	2.04594i
$750$		0			0
$751$	8112.61	−	14051.5i	0.394186	−	0.682749i	−0.598811	−	0.800890i	$-0.704360\pi$
$751$	8112.61	−	14051.5i	0.394186	−	0.682749i	0.992997	+	0.118141i	$0.0376934\pi$
$752$	3831.90	−	6637.04i	0.185818	−	0.321846i
$753$		0			0
$754$	22260.3	+	38555.9i	1.07516	+	1.86223i
$755$	2688.88			0.129614
$756$		0			0
$757$	−14461.0			−0.694312			−0.347156	−	0.937807i	$-0.612853\pi$
$757$	−14461.0			−0.694312			−0.347156	+	0.937807i	$0.612853\pi$
$758$	10428.5	+	18062.7i	0.499712	+	0.865526i
$759$		0			0
$760$	−1750.75	+	3032.39i	−0.0835610	+	0.144732i
$761$	11030.3	−	19105.0i	0.525422	−	0.910058i	−0.474139	−	0.880450i	$-0.657241\pi$
$761$	11030.3	−	19105.0i	0.525422	−	0.910058i	0.999562	−	0.0296081i	$-0.00942593\pi$
$762$		0			0
$763$	−10197.4	−	17662.5i	−0.483843	−	0.838041i
$764$	1377.90			0.0652496
$765$		0			0
$766$	−20493.4			−0.966652
$767$	−17664.6	−	30596.0i	−0.831592	−	1.44036i
$768$		0			0
$769$	837.871	−	1451.24i	0.0392905	−	0.0680532i	−0.845711	−	0.533640i	$-0.820824\pi$
$769$	837.871	−	1451.24i	0.0392905	−	0.0680532i	0.885002	+	0.465587i	$0.154157\pi$
$770$	−6873.00	+	11904.4i	−0.321670	+	0.557148i
$771$		0			0
$772$	−3883.67	−	6726.72i	−0.181058	−	0.313601i
$773$	−10878.6			−0.506179			−0.253089	−	0.967443i	$-0.581447\pi$
$773$	−10878.6			−0.506179			−0.253089	+	0.967443i	$0.581447\pi$
$774$		0			0
$775$	−67.1846			−0.00311399
$776$	−4584.55	−	7940.67i	−0.212082	−	0.367337i
$777$		0			0
$778$	8508.48	−	14737.1i	0.392087	−	0.679115i
$779$	−21938.0	+	37997.7i	−1.00900	+	1.74764i
$780$		0			0
$781$	13188.1	+	22842.5i	0.604237	+	1.04657i
$782$	1458.23			0.0666831
$783$		0			0
$784$	8471.40			0.385906
$785$	1679.47	+	2908.93i	0.0763603	+	0.132260i
$786$		0			0
$787$	13646.6	−	23636.6i	0.618105	−	1.07059i	−0.371727	−	0.928342i	$-0.621234\pi$
$787$	13646.6	−	23636.6i	0.618105	−	1.07059i	0.989831	−	0.142246i	$-0.0454326\pi$
$788$	−3561.90	+	6169.39i	−0.161025	+	0.278903i
$789$		0			0
$790$	−2258.88	−	3912.49i	−0.101731	−	0.176203i
$791$	28658.4			1.28821
$792$		0			0
$793$	12210.3			0.546784
$794$	3775.96	+	6540.16i	0.168771	+	0.292319i
$795$		0			0
$796$	1616.75	−	2800.29i	0.0719902	−	0.124691i
$797$	19462.7	−	33710.4i	0.865000	−	1.49822i	−0.00204738	−	0.999998i	$-0.500652\pi$
$797$	19462.7	−	33710.4i	0.865000	−	1.49822i	0.867047	−	0.498226i	$-0.166015\pi$
$798$		0			0
$799$	1086.70	+	1882.21i	0.0481158	+	0.0833391i
$800$	−800.000			−0.0353553
$801$		0			0
$802$	15112.0			0.665367
$803$	−19531.5	−	33829.5i	−0.858345	−	1.48670i
$804$		0			0
$805$	11865.7	−	20552.1i	0.519519	−	0.899833i
$806$	−247.441	+	428.580i	−0.0108136	+	0.0187296i
$807$		0			0
$808$	2045.85	+	3543.52i	0.0890752	+	0.154283i
$809$	22806.9			0.991159			0.495579	−	0.868563i	$-0.334956\pi$
$809$	22806.9			0.991159			0.495579	+	0.868563i	$0.334956\pi$
$810$		0			0
$811$	−42523.0			−1.84117			−0.920583	−	0.390547i	$-0.872286\pi$
$811$	−42523.0			−1.84117			−0.920583	+	0.390547i	$0.872286\pi$
$812$	14282.1	+	24737.3i	0.617246	+	1.06910i
$813$		0			0
$814$	959.250	−	1661.47i	0.0413043	−	0.0715412i
$815$	−6253.56	+	10831.5i	−0.268776	+	0.465534i
$816$		0			0
$817$	−12898.1	−	22340.2i	−0.552322	−	0.956650i
$818$	−17926.4			−0.766238
$819$		0			0
$820$	−10024.5			−0.426915
$821$	16322.2	+	28270.9i	0.693848	+	1.20178i	0.970568	+	0.240829i	$0.0774193\pi$
$821$	16322.2	+	28270.9i	0.693848	+	1.20178i	−0.276720	+	0.960951i	$0.589247\pi$
$822$		0			0
$823$	−2235.91	+	3872.70i	−0.0947008	+	0.164027i	−0.909484	−	0.415740i	$-0.863523\pi$
$823$	−2235.91	+	3872.70i	−0.0947008	+	0.164027i	0.814783	+	0.579766i	$0.196856\pi$
$824$	1686.05	−	2920.32i	0.0712819	−	0.123464i
$825$		0			0
$826$	−11333.5	−	19630.3i	−0.477414	−	0.826906i
$827$	−29667.5			−1.24745			−0.623725	−	0.781644i	$-0.714381\pi$
$827$	−29667.5			−1.24745			−0.623725	+	0.781644i	$0.714381\pi$
$828$		0			0
$829$	5130.74			0.214955			0.107478	−	0.994207i	$-0.465723\pi$
$829$	5130.74			0.214955			0.107478	+	0.994207i	$0.465723\pi$
$830$	−1505.25	−	2607.16i	−0.0629492	−	0.109031i
$831$		0			0
$832$	−2946.40	+	5103.31i	−0.122774	+	0.212651i
$833$	−1201.21	+	2080.56i	−0.0499634	+	0.0865392i
$834$		0			0
$835$	−1879.12	−	3254.74i	−0.0778799	−	0.134892i
$836$	−16295.1			−0.674136
$837$		0			0
$838$	−28307.6			−1.16691
$839$	12102.9	+	20962.9i	0.498021	+	0.862598i	0.999997	−	0.00228364i	$-0.000726906\pi$
$839$	12102.9	+	20962.9i	0.498021	+	0.862598i	−0.501976	+	0.864881i	$0.667394\pi$
$840$		0			0
$841$	−17030.0	+	29496.8i	−0.698266	+	1.20943i
$842$	−14727.3	+	25508.4i	−0.602774	+	1.04404i
$843$		0			0
$844$	−2446.07	−	4236.72i	−0.0997598	−	0.172789i
$845$	−31404.0			−1.27850
$846$		0			0
$847$	−24656.0			−1.00023
$848$	−1944.60	−	3368.15i	−0.0787474	−	0.136395i
$849$		0			0
$850$	113.437	−	196.478i	0.00457748	−	0.00792842i
$851$	−1656.08	+	2868.41i	−0.0667093	+	0.115544i
$852$		0			0
$853$	−10921.6	−	18916.8i	−0.438392	−	0.759317i	0.559174	−	0.829051i	$-0.311119\pi$
$853$	−10921.6	−	18916.8i	−0.438392	−	0.759317i	−0.997566	+	0.0697332i	$0.977785\pi$
$854$	7834.07			0.313907
$855$		0			0
$856$	−13116.0			−0.523710
$857$	1385.25	+	2399.33i	0.0552151	+	0.0956353i	0.892312	−	0.451420i	$-0.149082\pi$
$857$	1385.25	+	2399.33i	0.0552151	+	0.0956353i	−0.837097	+	0.547055i	$0.815749\pi$
$858$		0			0
$859$	9219.78	−	15969.1i	0.366211	−	0.634295i	−0.622759	−	0.782414i	$-0.713988\pi$
$859$	9219.78	−	15969.1i	0.366211	−	0.634295i	0.988970	+	0.148118i	$0.0473217\pi$
$860$	2946.87	−	5104.14i	0.116846	−	0.202383i
$861$		0			0
$862$	−3048.98	−	5280.98i	−0.120474	−	0.208667i
$863$	−22073.1			−0.870659			−0.435329	−	0.900271i	$-0.643368\pi$
$863$	−22073.1			−0.870659			−0.435329	+	0.900271i	$0.643368\pi$
$864$		0			0
$865$	11871.4			0.466634
$866$	17218.4	+	29823.1i	0.675641	+	1.17024i
$867$		0			0
$868$	−158.757	+	274.975i	−0.00620802	+	0.0107526i
$869$	10512.2	−	18207.7i	0.410360	−	0.710765i
$870$		0			0
$871$	−26822.0	−	46457.1i	−1.04343	−	1.80728i
$872$	5523.80			0.214518
$873$		0			0
$874$	28132.3			1.08878
$875$	−1846.09	−	3197.53i	−0.0713249	−	0.123538i
$876$		0			0
$877$	4223.34	−	7315.03i	0.162613	−	0.281655i	−0.773192	−	0.634172i	$-0.781341\pi$
$877$	4223.34	−	7315.03i	0.162613	−	0.281655i	0.935805	+	0.352518i	$0.114674\pi$
$878$	1667.91	−	2888.90i	0.0641106	−	0.111043i
$879$		0			0
$880$	−1861.50	−	3224.21i	−0.0713081	−	0.123509i
$881$	16665.8			0.637325			0.318663	−	0.947868i	$-0.396766\pi$
$881$	16665.8			0.637325			0.318663	+	0.947868i	$0.396766\pi$
$882$		0			0
$883$	37742.6			1.43844			0.719219	−	0.694784i	$-0.244500\pi$
$883$	37742.6			1.43844			0.719219	+	0.694784i	$0.244500\pi$
$884$	−835.576	−	1447.26i	−0.0317912	−	0.0550640i
$885$		0			0
$886$	9223.20	−	15975.1i	0.349729	−	0.605748i
$887$	−9723.59	+	16841.7i	−0.368079	+	0.637531i	−0.989265	−	0.146132i	$-0.953318\pi$
$887$	−9723.59	+	16841.7i	−0.368079	+	0.637531i	0.621186	+	0.783663i	$0.286651\pi$
$888$		0			0
$889$	11889.2	+	20592.7i	0.448538	+	0.776891i
$890$	−7393.49			−0.278461
$891$		0			0
$892$	4244.01			0.159305
$893$	20964.7	+	36311.9i	0.785617	+	1.36073i
$894$		0			0
$895$	6061.31	−	10498.5i	0.226377	−	0.392096i
$896$	−1890.40	+	3274.27i	−0.0704841	+	0.122082i
$897$		0			0
$898$	2636.18	+	4565.99i	0.0979625	+	0.169676i
$899$	−649.708			−0.0241034
$900$		0			0
$901$	1102.95			0.0407819
$902$	−23325.7	−	40401.4i	−0.861045	−	1.49137i
$903$		0			0
$904$	−3880.95	+	6722.00i	−0.142786	+	0.247313i
$905$	2948.65	−	5107.21i	0.108306	−	0.187591i
$906$		0			0
$907$	−7714.45	−	13361.8i	−0.282419	−	0.489164i	0.689561	−	0.724228i	$-0.257804\pi$
$907$	−7714.45	−	13361.8i	−0.282419	−	0.489164i	−0.971980	+	0.235063i	$0.924470\pi$
$908$	14292.3			0.522364
$909$		0			0
$910$	−27196.6			−0.990724
$911$	344.095	+	595.990i	0.0125141	+	0.0216751i	0.872215	−	0.489123i	$-0.162683\pi$
$911$	344.095	+	595.990i	0.0125141	+	0.0216751i	−0.859701	+	0.510798i	$0.829350\pi$
$912$		0			0
$913$	7005.04	−	12133.1i	0.253924	−	0.439810i
$914$	−3859.67	+	6685.15i	−0.139679	+	0.241931i
$915$		0			0
$916$	−2003.65	−	3470.42i	−0.0722734	−	0.125181i
$917$	6341.32			0.228363
$918$		0			0
$919$	6809.05			0.244407			0.122203	−	0.992505i	$-0.461004\pi$
$919$	6809.05			0.244407			0.122203	+	0.992505i	$0.461004\pi$
$920$	3213.75	+	5566.37i	0.115168	+	0.199476i
$921$		0			0
$922$	6615.07	−	11457.6i	0.236286	−	0.409259i
$923$	−26092.9	+	45194.2i	−0.930507	+	1.61169i
$924$		0			0
$925$	257.655	+	446.272i	0.00915855	+	0.0158631i
$926$	35143.5			1.24718
$927$		0			0
$928$	−7736.39			−0.273663
$929$	23841.1	+	41294.0i	0.841982	+	1.45835i	0.888217	+	0.459424i	$0.151944\pi$
$929$	23841.1	+	41294.0i	0.841982	+	1.45835i	−0.0462352	+	0.998931i	$0.514722\pi$
$930$		0			0
$931$	−23173.9	+	40138.4i	−0.815783	+	1.41298i
$932$	−12164.8	+	21070.1i	−0.427546	+	0.740531i
$933$		0			0
$934$	10076.8	+	17453.6i	0.353024	+	0.611455i
$935$	1055.81			0.0369292
$936$		0			0
$937$	−30516.3			−1.06395			−0.531977	−	0.846759i	$-0.678551\pi$
$937$	−30516.3			−1.06395			−0.531977	+	0.846759i	$0.678551\pi$
$938$	−17208.9	−	29806.7i	−0.599030	−	1.03755i
$939$		0			0
$940$	−4789.87	+	8296.30i	−0.166200	+	0.287868i
$941$	15674.4	−	27148.9i	0.543009	−	0.940519i	−0.455720	−	0.890123i	$-0.650618\pi$
$941$	15674.4	−	27148.9i	0.543009	−	0.940519i	0.998729	−	0.0503962i	$-0.0160484\pi$
$942$		0			0
$943$	40270.3	+	69750.1i	1.39065	+	2.40867i
$944$	6139.20			0.211667
$945$		0			0
$946$	27428.0			0.942666
$947$	−17299.3	−	29963.3i	−0.593614	−	1.02817i	−0.993741	−	0.111709i	$-0.964367\pi$
$947$	−17299.3	−	29963.3i	−0.593614	−	1.02817i	0.400127	−	0.916460i	$-0.368966\pi$
$948$		0			0
$949$	38643.3	−	66932.1i	1.32183	−	2.28947i
$950$	2188.44	−	3790.48i	0.0747392	−	0.129452i
$951$		0			0
$952$	−536.102	−	928.557i	−0.0182512	−	0.0316121i
$953$	26766.7			0.909820			0.454910	−	0.890537i	$-0.349671\pi$
$953$	26766.7			0.909820			0.454910	+	0.890537i	$0.349671\pi$
$954$		0			0
$955$	−1722.38			−0.0583610
$956$	424.797	+	735.770i	0.0143713	+	0.0248917i
$957$		0			0
$958$	−17945.8	+	31083.0i	−0.605221	+	1.04827i
$959$	−5431.93	+	9408.37i	−0.182905	+	0.316801i
$960$		0			0
$961$	14891.9	+	25793.5i	0.499879	+	0.865815i
$962$	3795.78			0.127215
$963$		0			0
$964$	−11777.8			−0.393503
$965$	4854.59	+	8408.40i	0.161943	+	0.280493i
$966$		0			0
$967$	−4382.13	+	7590.08i	−0.145729	+	0.252410i	−0.929645	−	0.368457i	$-0.879886\pi$
$967$	−4382.13	+	7590.08i	−0.145729	+	0.252410i	0.783916	+	0.620867i	$0.213219\pi$
$968$	3338.95	−	5783.23i	0.110866	−	0.192025i
$969$		0			0
$970$	5730.69	+	9925.84i	0.189692	+	0.328556i
$971$	35417.3			1.17054			0.585270	−	0.810838i	$-0.300988\pi$
$971$	35417.3			1.17054			0.585270	+	0.810838i	$0.300988\pi$
$972$		0			0
$973$	42159.2			1.38907
$974$	17356.9	+	30063.0i	0.570996	+	0.988995i
$975$		0			0
$976$	−1060.90	+	1837.53i	−0.0347936	+	0.0602643i
$977$	−13314.5	+	23061.4i	−0.435997	+	0.755168i	−0.997376	−	0.0723900i	$-0.976937\pi$
$977$	−13314.5	+	23061.4i	−0.435997	+	0.755168i	0.561380	+	0.827558i	$0.310271\pi$
$978$		0			0
$979$	−17203.7	−	29797.7i	−0.561628	−	0.972767i
$980$	−10589.3			−0.345164
$981$		0			0
$982$	16786.3			0.545493
$983$	8404.59	+	14557.2i	0.272701	+	0.472332i	0.969552	−	0.244884i	$-0.0787498\pi$
$983$	8404.59	+	14557.2i	0.272701	+	0.472332i	−0.696852	+	0.717215i	$0.745416\pi$
$984$		0			0
$985$	4452.38	−	7711.74i	0.144025	−	0.249458i
$986$	1096.99	−	1900.04i	0.0354313	−	0.0613689i
$987$		0			0
$988$	−16120.0	−	27920.7i	−0.519075	−	0.899064i
$989$	−47352.5			−1.52247
$990$		0			0
$991$	53758.7			1.72321			0.861606	−	0.507578i	$-0.169459\pi$
$991$	53758.7			1.72321			0.861606	+	0.507578i	$0.169459\pi$
$992$	−42.9981	−	74.4749i	−0.00137620	−	0.00238365i
$993$		0			0
$994$	−16741.1	+	28996.4i	−0.534201	+	0.925263i
$995$	−2020.94	+	3500.37i	−0.0643900	+	0.111527i
$996$		0			0
$997$	−20787.6	−	36005.1i	−0.660330	−	1.14372i	−0.980529	−	0.196374i	$-0.937083\pi$
$997$	−20787.6	−	36005.1i	−0.660330	−	1.14372i	0.320199	−	0.947350i	$-0.396250\pi$
$998$	−32842.3			−1.04169
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.4.e.bd.271.2		4
3.2	odd	2		810.4.e.z.271.2		4
9.2	odd	6		810.4.e.z.541.2		4
9.4	even	3		270.4.a.m.1.1	✓	2
9.5	odd	6		270.4.a.n.1.1	yes	2
9.7	even	3	inner	810.4.e.bd.541.2		4
36.23	even	6		2160.4.a.bb.1.2		2
36.31	odd	6		2160.4.a.w.1.2		2
45.4	even	6		1350.4.a.bm.1.2		2
45.13	odd	12		1350.4.c.u.649.4		4
45.14	odd	6		1350.4.a.bf.1.2		2
45.22	odd	12		1350.4.c.u.649.1		4
45.23	even	12		1350.4.c.bb.649.2		4
45.32	even	12		1350.4.c.bb.649.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.4.a.m.1.1	✓	2	9.4	even	3
270.4.a.n.1.1	yes	2	9.5	odd	6
810.4.e.z.271.2		4	3.2	odd	2
810.4.e.z.541.2		4	9.2	odd	6
810.4.e.bd.271.2		4	1.1	even	1	trivial
810.4.e.bd.541.2		4	9.7	even	3	inner
1350.4.a.bf.1.2		2	45.14	odd	6
1350.4.a.bm.1.2		2	45.4	even	6
1350.4.c.u.649.1		4	45.22	odd	12
1350.4.c.u.649.4		4	45.13	odd	12
1350.4.c.bb.649.2		4	45.23	even	12
1350.4.c.bb.649.3		4	45.32	even	12
2160.4.a.w.1.2		2	36.31	odd	6
2160.4.a.bb.1.2		2	36.23	even	6

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:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	810.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(2.50000 - 4.33013i) q^{5} +(14.7687 + 25.5802i) q^{7} -8.00000 q^{8} +10.0000 q^{10} +(23.2687 + 40.3026i) q^{11} +(-46.0375 + 79.7392i) q^{13} +(-29.5375 + 51.1604i) q^{14} +(-8.00000 - 13.8564i) q^{16} +4.53748 q^{17} +87.5375 q^{19} +(10.0000 + 17.3205i) q^{20} +(-46.5375 + 80.6053i) q^{22} +(80.3437 - 139.159i) q^{23} +(-12.5000 - 21.6506i) q^{25} -184.150 q^{26} -118.150 q^{28} +(-120.881 - 209.372i) q^{29} +(1.34369 - 2.32734i) q^{31} +(16.0000 - 27.7128i) q^{32} +(4.53748 + 7.85914i) q^{34} +147.687 q^{35} -20.6124 q^{37} +(87.5375 + 151.619i) q^{38} +(-20.0000 + 34.6410i) q^{40} +(-250.612 + 434.073i) q^{41} +(-147.344 - 255.207i) q^{43} -186.150 q^{44} +321.375 q^{46} +(239.494 + 414.815i) q^{47} +(-264.731 + 458.528i) q^{49} +(25.0000 - 43.3013i) q^{50} +(-184.150 - 318.957i) q^{52} +243.075 q^{53} +232.687 q^{55} +(-118.150 - 204.642i) q^{56} +(241.762 - 418.745i) q^{58} +(-191.850 + 332.294i) q^{59} +(-66.3062 - 114.846i) q^{61} +5.37477 q^{62} +64.0000 q^{64} +(230.187 + 398.696i) q^{65} +(-291.306 + 504.557i) q^{67} +(-9.07495 + 15.7183i) q^{68} +(147.687 + 255.802i) q^{70} +566.775 q^{71} -839.388 q^{73} +(-20.6124 - 35.7018i) q^{74} +(-175.075 + 303.239i) q^{76} +(-687.300 + 1190.44i) q^{77} +(-225.888 - 391.249i) q^{79} -80.0000 q^{80} -1002.45 q^{82} +(-150.525 - 260.716i) q^{83} +(11.3437 - 19.6478i) q^{85} +(294.687 - 510.414i) q^{86} +(-186.150 - 322.421i) q^{88} -739.349 q^{89} -2719.66 q^{91} +(321.375 + 556.637i) q^{92} +(-478.987 + 829.630i) q^{94} +(218.844 - 379.048i) q^{95} +(573.069 + 992.584i) q^{97} -1058.93 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 8 q^{4} + 10 q^{5} - q^{7} - 32 q^{8} + 40 q^{10} + 33 q^{11} - 64 q^{13} + 2 q^{14} - 32 q^{16} - 102 q^{17} + 230 q^{19} + 40 q^{20} - 66 q^{22} + 21 q^{23} - 50 q^{25} - 256 q^{26} + 8 q^{28}+ \cdots - 4476 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 - 8 * q^4 + 10 * q^5 - q^7 - 32 * q^8 + 40 * q^10 + 33 * q^11 - 64 * q^13 + 2 * q^14 - 32 * q^16 - 102 * q^17 + 230 * q^19 + 40 * q^20 - 66 * q^22 + 21 * q^23 - 50 * q^25 - 256 * q^26 + 8 * q^28 - 63 * q^29 - 295 * q^31 + 64 * q^32 - 102 * q^34 - 10 * q^35 + 278 * q^37 + 230 * q^38 - 80 * q^40 - 642 * q^41 - 289 * q^43 - 264 * q^44 + 84 * q^46 + 177 * q^47 - 1119 * q^49 + 100 * q^50 - 256 * q^52 + 732 * q^53 + 330 * q^55 + 8 * q^56 + 126 * q^58 - 1248 * q^59 - 85 * q^61 - 1180 * q^62 + 256 * q^64 + 320 * q^65 - 985 * q^67 + 204 * q^68 - 10 * q^70 + 2988 * q^71 - 3718 * q^73 + 278 * q^74 - 460 * q^76 - 1788 * q^77 - 1264 * q^79 - 320 * q^80 - 2568 * q^82 + 1080 * q^83 - 255 * q^85 + 578 * q^86 - 264 * q^88 + 1368 * q^89 - 7154 * q^91 + 84 * q^92 - 354 * q^94 + 575 * q^95 + 1271 * q^97 - 4476 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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