LMFDB - Embedded newform 108.5.k.a.65.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,5,Mod(5,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 5]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("108.5");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	108.k (of order $18$, degree $6$, 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:	$11.1639560131$
Analytic rank:	$0$
Dimension:	$72$
Relative dimension:	$12$ over $\Q(\zeta_{18})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			65.3
Character	$\chi$	$=$	108.65
Dual form			108.5.k.a.5.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-7.74246 - 4.58850i) q^{3} +(-2.39469 + 6.57935i) q^{5} +(-4.62159 + 26.2104i) q^{7} +(38.8914 + 71.0525i) q^{9} +O(q^{10})$	\(q+(-7.74246 - 4.58850i) q^{3} +(-2.39469 + 6.57935i) q^{5} +(-4.62159 + 26.2104i) q^{7} +(38.8914 + 71.0525i) q^{9} +(-16.9285 - 46.5107i) q^{11} +(94.6619 - 79.4308i) q^{13} +(48.7301 - 39.9523i) q^{15} +(285.051 - 164.574i) q^{17} +(116.459 - 201.713i) q^{19} +(156.049 - 181.727i) q^{21} +(392.366 - 69.1848i) q^{23} +(441.224 + 370.231i) q^{25} +(24.9088 - 728.574i) q^{27} +(555.504 - 662.024i) q^{29} +(-8.30835 - 47.1190i) q^{31} +(-82.3458 + 437.784i) q^{33} +(-161.380 - 93.1727i) q^{35} +(822.095 + 1423.91i) q^{37} +(-1097.38 + 180.634i) q^{39} +(242.838 + 289.403i) q^{41} +(-807.328 + 293.843i) q^{43} +(-560.612 + 85.7317i) q^{45} +(-579.809 - 102.236i) q^{47} +(1590.58 + 578.923i) q^{49} +(-2962.14 - 33.7453i) q^{51} -1451.91i q^{53} +346.549 q^{55} +(-1827.24 + 1027.38i) q^{57} +(1907.08 - 5239.66i) q^{59} +(-744.330 + 4221.31i) q^{61} +(-2042.05 + 690.982i) q^{63} +(295.917 + 813.026i) q^{65} +(-2803.64 + 2352.53i) q^{67} +(-3355.34 - 1264.71i) q^{69} +(68.5308 - 39.5663i) q^{71} +(-1304.49 + 2259.43i) q^{73} +(-1717.36 - 4891.06i) q^{75} +(1297.30 - 228.749i) q^{77} +(4203.68 + 3527.30i) q^{79} +(-3535.92 + 5526.66i) q^{81} +(2372.59 - 2827.55i) q^{83} +(400.183 + 2269.55i) q^{85} +(-7338.66 + 2576.77i) q^{87} +(-3207.30 - 1851.74i) q^{89} +(1644.42 + 2848.22i) q^{91} +(-151.878 + 402.940i) q^{93} +(1048.25 + 1249.26i) q^{95} +(-1634.86 + 595.041i) q^{97} +(2646.33 - 3011.68i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q + 9 q^{5} - 102 q^{9}+O(q^{10}) $ 72 * q + 9 * q^5 - 102 * q^9	$ 72 q + 9 q^{5} - 102 q^{9} + 18 q^{11} - 225 q^{15} - 282 q^{21} - 1278 q^{23} + 441 q^{25} + 54 q^{27} + 1854 q^{29} - 1665 q^{31} - 45 q^{33} - 2673 q^{35} + 6951 q^{39} - 5472 q^{41} + 1260 q^{43} + 5553 q^{45} + 5103 q^{47} - 5904 q^{49} + 1899 q^{51} + 1107 q^{57} - 10944 q^{59} + 8352 q^{61} - 11985 q^{63} + 8757 q^{65} + 378 q^{67} + 5607 q^{69} - 19764 q^{71} + 6111 q^{73} - 3453 q^{75} - 5679 q^{77} - 5652 q^{79} - 20466 q^{81} - 20061 q^{83} + 26100 q^{85} + 40545 q^{87} + 15633 q^{89} - 6039 q^{91} + 40179 q^{93} + 48024 q^{95} - 37530 q^{97} + 12177 q^{99}+O(q^{100}) $ 72 * q + 9 * q^5 - 102 * q^9 + 18 * q^11 - 225 * q^15 - 282 * q^21 - 1278 * q^23 + 441 * q^25 + 54 * q^27 + 1854 * q^29 - 1665 * q^31 - 45 * q^33 - 2673 * q^35 + 6951 * q^39 - 5472 * q^41 + 1260 * q^43 + 5553 * q^45 + 5103 * q^47 - 5904 * q^49 + 1899 * q^51 + 1107 * q^57 - 10944 * q^59 + 8352 * q^61 - 11985 * q^63 + 8757 * q^65 + 378 * q^67 + 5607 * q^69 - 19764 * q^71 + 6111 * q^73 - 3453 * q^75 - 5679 * q^77 - 5652 * q^79 - 20466 * q^81 - 20061 * q^83 + 26100 * q^85 + 40545 * q^87 + 15633 * q^89 - 6039 * q^91 + 40179 * q^93 + 48024 * q^95 - 37530 * q^97 + 12177 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{13}{18}\right)$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−7.74246	−	4.58850i	−0.860273	−	0.509833i
$3$	−7.74246	−	4.58850i	−0.860273	−	0.509833i
$4$		0			0
$5$	−2.39469	+	6.57935i	−0.0957875	+	0.263174i	−0.978328	−	0.207063i	$-0.933609\pi$
$5$	−2.39469	+	6.57935i	−0.0957875	+	0.263174i	0.882540	+	0.470237i	$0.155832\pi$
$6$		0			0
$7$	−4.62159	+	26.2104i	−0.0943182	+	0.534905i	0.900636	+	0.434574i	$0.143101\pi$
$7$	−4.62159	+	26.2104i	−0.0943182	+	0.534905i	−0.994954	+	0.100331i	$0.968010\pi$
$8$		0			0
$9$	38.8914	+	71.0525i	0.480141	+	0.877191i
$10$		0			0
$11$	−16.9285	−	46.5107i	−0.139905	−	0.384386i	0.849876	−	0.526983i	$-0.176677\pi$
$11$	−16.9285	−	46.5107i	−0.139905	−	0.384386i	−0.989781	+	0.142597i	$0.954455\pi$
$12$		0			0
$13$	94.6619	−	79.4308i	0.560130	−	0.470005i	−0.318224	−	0.948015i	$-0.603086\pi$
$13$	94.6619	−	79.4308i	0.560130	−	0.470005i	0.878354	+	0.478011i	$0.158642\pi$
$14$		0			0
$15$	48.7301	−	39.9523i	0.216578	−	0.177566i
$16$		0			0
$17$	285.051	−	164.574i	0.986335	−	0.569461i	0.0821585	−	0.996619i	$-0.473819\pi$
$17$	285.051	−	164.574i	0.986335	−	0.569461i	0.904177	+	0.427158i	$0.140485\pi$
$18$		0			0
$19$	116.459	−	201.713i	0.322601	−	0.558761i	−0.658423	−	0.752648i	$-0.728776\pi$
$19$	116.459	−	201.713i	0.322601	−	0.558761i	0.981024	+	0.193887i	$0.0621096\pi$
$20$		0			0
$21$	156.049	−	181.727i	0.353852	−	0.412078i
$22$		0			0
$23$	392.366	−	69.1848i	0.741713	−	0.130784i	0.209992	−	0.977703i	$-0.432656\pi$
$23$	392.366	−	69.1848i	0.741713	−	0.130784i	0.531722	+	0.846919i	$0.321545\pi$
$24$		0			0
$25$	441.224	+	370.231i	0.705959	+	0.592370i
$26$		0			0
$27$	24.9088	−	728.574i	0.0341684	−	0.999416i
$28$		0			0
$29$	555.504	−	662.024i	0.660528	−	0.787187i	−0.326933	−	0.945047i	$-0.606015\pi$
$29$	555.504	−	662.024i	0.660528	−	0.787187i	0.987461	+	0.157861i	$0.0504597\pi$
$30$		0			0
$31$	−8.30835	−	47.1190i	−0.00864552	−	0.0490312i	0.980180	−	0.198110i	$-0.0634803\pi$
$31$	−8.30835	−	47.1190i	−0.00864552	−	0.0490312i	−0.988825	+	0.149079i	$0.952369\pi$
$32$		0			0
$33$	−82.3458	+	437.784i	−0.0756160	+	0.402005i
$34$		0			0
$35$	−161.380	−	93.1727i	−0.131739	−	0.0760593i
$36$		0			0
$37$	822.095	+	1423.91i	0.600508	+	1.04011i	0.992744	+	0.120246i	$0.0383682\pi$
$37$	822.095	+	1423.91i	0.600508	+	1.04011i	−0.392236	+	0.919864i	$0.628298\pi$
$38$		0			0
$39$	−1097.38	+	180.634i	−0.721488	+	0.118760i
$40$		0			0
$41$	242.838	+	289.403i	0.144460	+	0.172161i	0.833423	−	0.552636i	$-0.186378\pi$
$41$	242.838	+	289.403i	0.144460	+	0.172161i	−0.688962	+	0.724797i	$0.741934\pi$
$42$		0			0
$43$	−807.328	+	293.843i	−0.436629	+	0.158920i	−0.550976	−	0.834521i	$-0.685745\pi$
$43$	−807.328	+	293.843i	−0.436629	+	0.158920i	0.114347	+	0.993441i	$0.463522\pi$
$44$		0			0
$45$	−560.612	+	85.7317i	−0.276845	+	0.0423366i
$46$		0			0
$47$	−579.809	−	102.236i	−0.262476	−	0.0462815i	0.0408616	−	0.999165i	$-0.486990\pi$
$47$	−579.809	−	102.236i	−0.262476	−	0.0462815i	−0.303337	+	0.952883i	$0.598101\pi$
$48$		0			0
$49$	1590.58	+	578.923i	0.662465	+	0.241118i
$50$		0			0
$51$	−2962.14	−	33.7453i	−1.13885	−	0.0129740i
$52$		0			0
$53$		−	1451.91i		−	0.516880i	−0.966027	−	0.258440i	$-0.916792\pi$
$53$		−	1451.91i		−	0.516880i	0.966027	−	0.258440i	$-0.0832084\pi$
$54$		0			0
$55$	346.549			0.114562
$56$		0			0
$57$	−1827.24	+	1027.38i	−0.562399	+	0.316215i
$58$		0			0
$59$	1907.08	−	5239.66i	0.547854	−	1.50522i	−0.288747	−	0.957405i	$-0.593239\pi$
$59$	1907.08	−	5239.66i	0.547854	−	1.50522i	0.836601	−	0.547812i	$-0.184539\pi$
$60$		0			0
$61$	−744.330	+	4221.31i	−0.200035	+	1.13445i	0.705029	+	0.709179i	$0.250934\pi$
$61$	−744.330	+	4221.31i	−0.200035	+	1.13445i	−0.905064	+	0.425276i	$0.860177\pi$
$62$		0			0
$63$	−2042.05	+	690.982i	−0.514500	+	0.174095i
$64$		0			0
$65$	295.917	+	813.026i	0.0700396	+	0.192432i
$66$		0			0
$67$	−2803.64	+	2352.53i	−0.624557	+	0.524066i	−0.899232	−	0.437471i	$-0.855874\pi$
$67$	−2803.64	+	2352.53i	−0.624557	+	0.524066i	0.274675	+	0.961537i	$0.411430\pi$
$68$		0			0
$69$	−3355.34	−	1264.71i	−0.704754	−	0.265640i
$70$		0			0
$71$	68.5308	−	39.5663i	0.0135947	−	0.00784889i	−0.493187	−	0.869923i	$-0.664168\pi$
$71$	68.5308	−	39.5663i	0.0135947	−	0.00784889i	0.506782	+	0.862074i	$0.330835\pi$
$72$		0			0
$73$	−1304.49	+	2259.43i	−0.244790	+	0.423988i	−0.962073	−	0.272794i	$-0.912052\pi$
$73$	−1304.49	+	2259.43i	−0.244790	+	0.423988i	0.717283	+	0.696782i	$0.245386\pi$
$74$		0			0
$75$	−1717.36	−	4891.06i	−0.305308	−	0.869521i
$76$		0			0
$77$	1297.30	−	228.749i	0.218806	−	0.0385814i
$78$		0			0
$79$	4203.68	+	3527.30i	0.673558	+	0.565182i	0.914116	−	0.405452i	$-0.132886\pi$
$79$	4203.68	+	3527.30i	0.673558	+	0.565182i	−0.240558	+	0.970635i	$0.577330\pi$
$80$		0			0
$81$	−3535.92	+	5526.66i	−0.538929	+	0.842351i
$82$		0			0
$83$	2372.59	−	2827.55i	0.344403	−	0.410444i	−0.565842	−	0.824514i	$-0.691449\pi$
$83$	2372.59	−	2827.55i	0.344403	−	0.410444i	0.910245	+	0.414070i	$0.135893\pi$
$84$		0			0
$85$	400.183	+	2269.55i	0.0553887	+	0.314125i
$86$		0			0
$87$	−7338.66	+	2576.77i	−0.969569	+	0.340437i
$88$		0			0
$89$	−3207.30	−	1851.74i	−0.404911	−	0.233776i	0.283690	−	0.958916i	$-0.408441\pi$
$89$	−3207.30	−	1851.74i	−0.404911	−	0.233776i	−0.688601	+	0.725140i	$0.741775\pi$
$90$		0			0
$91$	1644.42	+	2848.22i	0.198577	+	0.343946i
$92$		0			0
$93$	−151.878	+	402.940i	−0.0175602	+	0.0465880i
$94$		0			0
$95$	1048.25	+	1249.26i	0.116150	+	0.138422i
$96$		0			0
$97$	−1634.86	+	595.041i	−0.173755	+	0.0632417i	−0.427433	−	0.904047i	$-0.640582\pi$
$97$	−1634.86	+	595.041i	−0.173755	+	0.0632417i	0.253678	+	0.967289i	$0.418360\pi$
$98$		0			0
$99$	2646.33	−	3011.68i	0.270006	−	0.307283i
$100$		0			0
$101$	11040.4	+	1946.73i	1.08229	+	0.190837i	0.686227	−	0.727387i	$-0.259265\pi$
$101$	11040.4	+	1946.73i	1.08229	+	0.190837i	0.396062	+	0.918224i	$0.370377\pi$
$102$		0			0
$103$	−3421.96	−	1245.49i	−0.322553	−	0.117400i	0.175669	−	0.984449i	$-0.443791\pi$
$103$	−3421.96	−	1245.49i	−0.322553	−	0.117400i	−0.498222	+	0.867050i	$0.666013\pi$
$104$		0			0
$105$	821.954	+	1461.88i	0.0745537	+	0.132596i
$106$		0			0
$107$		−	12263.4i		−	1.07113i	−0.844493	−	0.535566i	$-0.820098\pi$
$107$		−	12263.4i		−	1.07113i	0.844493	−	0.535566i	$-0.179902\pi$
$108$		0			0
$109$	20211.4			1.70115			0.850575	−	0.525854i	$-0.176254\pi$
$109$	20211.4			1.70115			0.850575	+	0.525854i	$0.176254\pi$
$110$		0			0
$111$	168.567	−	14796.8i	0.0136813	−	1.20094i
$112$		0			0
$113$	5651.59	−	15527.6i	0.442602	−	1.21604i	−0.495172	−	0.868795i	$-0.664895\pi$
$113$	5651.59	−	15527.6i	0.442602	−	1.21604i	0.937774	−	0.347245i	$-0.112883\pi$
$114$		0			0
$115$	−484.404	+	2747.19i	−0.0366279	+	0.207727i
$116$		0			0
$117$	9325.29	+	3636.79i	0.681225	+	0.265672i
$118$		0			0
$119$	2996.16	+	8231.88i	0.211578	+	0.581306i
$120$		0			0
$121$	9338.98	−	7836.34i	0.637865	−	0.535232i
$122$		0			0
$123$	−552.239	−	3354.95i	−0.0365020	−	0.221756i
$124$		0			0
$125$	−7282.20	+	4204.38i	−0.466061	+	0.269080i
$126$		0			0
$127$	3442.74	−	5963.01i	0.213451	−	0.369707i	−0.739342	−	0.673331i	$-0.764863\pi$
$127$	3442.74	−	5963.01i	0.213451	−	0.369707i	0.952792	+	0.303623i	$0.0981964\pi$
$128$		0			0
$129$	7599.00	+	1429.35i	0.456643	+	0.0858932i
$130$		0			0
$131$	15623.6	−	2754.87i	0.910415	−	0.160531i	0.301223	−	0.953554i	$-0.402605\pi$
$131$	15623.6	−	2754.87i	0.910415	−	0.160531i	0.609192	+	0.793023i	$0.291494\pi$
$132$		0			0
$133$	4748.73	+	3984.66i	0.268457	+	0.225262i
$134$		0			0
$135$	4733.90	+	1908.59i	0.259747	+	0.104724i
$136$		0			0
$137$	−16763.9	+	19978.5i	−0.893170	+	1.06444i	0.104384	+	0.994537i	$0.466713\pi$
$137$	−16763.9	+	19978.5i	−0.893170	+	1.06444i	−0.997554	+	0.0699017i	$0.977731\pi$
$138$		0			0
$139$	−3919.05	−	22226.1i	−0.202839	−	1.15036i	−0.900804	−	0.434226i	$-0.857022\pi$
$139$	−3919.05	−	22226.1i	−0.202839	−	1.15036i	0.697965	−	0.716132i	$-0.254089\pi$
$140$		0			0
$141$	4020.04	+	3452.01i	0.202205	+	0.173633i
$142$		0			0
$143$	−5296.87	−	3058.15i	−0.259028	−	0.149550i
$144$		0			0
$145$	3025.43	+	5240.20i	0.143897	+	0.249236i
$146$		0			0
$147$	−9658.60	−	11780.7i	−0.446971	−	0.545173i
$148$		0			0
$149$	−2520.68	−	3004.02i	−0.113539	−	0.135310i	0.706282	−	0.707931i	$-0.250371\pi$
$149$	−2520.68	−	3004.02i	−0.113539	−	0.135310i	−0.819820	+	0.572621i	$0.805927\pi$
$150$		0			0
$151$	−35103.6	+	12776.7i	−1.53957	+	0.560356i	−0.965941	−	0.258762i	$-0.916686\pi$
$151$	−35103.6	+	12776.7i	−1.53957	+	0.560356i	−0.573625	+	0.819118i	$0.694463\pi$
$152$		0			0
$153$	22779.4	+	13853.1i	0.973106	+	0.591783i
$154$		0			0
$155$	329.908	+	58.1717i	0.0137319	+	0.00242130i
$156$		0			0
$157$	−35398.4	−	12884.0i	−1.43610	−	0.522697i	−0.497425	−	0.867507i	$-0.665721\pi$
$157$	−35398.4	−	12884.0i	−1.43610	−	0.522697i	−0.938672	+	0.344810i	$0.887943\pi$
$158$		0			0
$159$	−6662.10	+	11241.4i	−0.263522	+	0.444658i
$160$		0			0
$161$			10603.8i			0.409082i
$162$		0			0
$163$	5722.24			0.215373			0.107686	−	0.994185i	$-0.465656\pi$
$163$	5722.24			0.215373			0.107686	+	0.994185i	$0.465656\pi$
$164$		0			0
$165$	−2683.14	−	1590.14i	−0.0985543	−	0.0584072i
$166$		0			0
$167$	−3814.37	+	10479.9i	−0.136770	+	0.375772i	−0.989103	−	0.147228i	$-0.952965\pi$
$167$	−3814.37	+	10479.9i	−0.136770	+	0.375772i	0.852333	+	0.523000i	$0.175187\pi$
$168$		0			0
$169$	−2307.94	+	13089.0i	−0.0808073	+	0.458281i
$170$		0			0
$171$	18861.4	+	429.802i	0.645034	+	0.0146986i
$172$		0			0
$173$	9145.34	+	25126.6i	0.305568	+	0.839541i	0.993507	+	0.113772i	$0.0362934\pi$
$173$	9145.34	+	25126.6i	0.305568	+	0.839541i	−0.687939	+	0.725769i	$0.741484\pi$
$174$		0			0
$175$	−11743.1	+	9853.59i	−0.383447	+	0.321750i
$176$		0			0
$177$	−38807.7	+	31817.2i	−1.23871	+	1.01558i
$178$		0			0
$179$	−40225.0	+	23223.9i	−1.25542	+	0.724819i	−0.972181	−	0.234230i	$-0.924743\pi$
$179$	−40225.0	+	23223.9i	−1.25542	+	0.724819i	−0.283242	+	0.959049i	$0.591410\pi$
$180$		0			0
$181$	24531.3	−	42489.5i	0.748797	−	1.29695i	−0.199603	−	0.979877i	$-0.563965\pi$
$181$	24531.3	−	42489.5i	0.748797	−	1.29695i	0.948400	−	0.317077i	$-0.102701\pi$
$182$		0			0
$183$	25132.4	−	29267.9i	0.750467	−	0.873957i
$184$		0			0
$185$	−11337.1	+	1999.03i	−0.331251	+	0.0584085i
$186$		0			0
$187$	−12480.0	−	10471.9i	−0.356886	−	0.299463i
$188$		0			0
$189$	18981.1	+	4020.04i	0.531370	+	0.112540i
$190$		0			0
$191$	−22885.6	+	27273.9i	−0.627328	+	0.747621i	−0.982312	−	0.187252i	$-0.940042\pi$
$191$	−22885.6	+	27273.9i	−0.627328	+	0.747621i	0.354984	+	0.934872i	$0.384486\pi$
$192$		0			0
$193$	−9753.90	−	55317.1i	−0.261857	−	1.48506i	−0.777839	−	0.628464i	$-0.783684\pi$
$193$	−9753.90	−	55317.1i	−0.261857	−	1.48506i	0.515982	−	0.856599i	$-0.327427\pi$
$194$		0			0
$195$	1439.44	−	7652.63i	0.0378550	−	0.201253i
$196$		0			0
$197$	−54922.0	−	31709.2i	−1.41519	−	0.817059i	−0.419317	−	0.907840i	$-0.637730\pi$
$197$	−54922.0	−	31709.2i	−1.41519	−	0.817059i	−0.995871	+	0.0907812i	$0.971064\pi$
$198$		0			0
$199$	1739.17	+	3012.33i	0.0439172	+	0.0760669i	0.887148	−	0.461484i	$-0.152683\pi$
$199$	1739.17	+	3012.33i	0.0439172	+	0.0760669i	−0.843231	+	0.537551i	$0.819350\pi$
$200$		0			0
$201$	32501.6	−	5349.90i	0.804476	−	0.132420i
$202$		0			0
$203$	14784.6	+	17619.6i	0.358770	+	0.427566i
$204$		0			0
$205$	−2485.60	+	904.686i	−0.0591458	+	0.0215273i
$206$		0			0
$207$	20175.4	+	25187.9i	0.470850	+	0.587830i
$208$		0			0
$209$	−11353.3	−	2001.89i	−0.259913	−	0.0458297i
$210$		0			0
$211$	−41197.3	−	14994.6i	−0.925346	−	0.336799i	−0.164983	−	0.986296i	$-0.552757\pi$
$211$	−41197.3	−	14994.6i	−0.925346	−	0.336799i	−0.760363	+	0.649498i	$0.774979\pi$
$212$		0			0
$213$	−712.146	−	8.11291i	−0.0156968	−	0.000178820i
$214$		0			0
$215$		−	6015.35i		−	0.130132i
$216$		0			0
$217$	1273.40			0.0270425
$218$		0			0
$219$	20467.3	−	11508.0i	0.426749	−	0.239944i
$220$		0			0
$221$	13911.2	−	38220.7i	0.284826	−	0.782554i
$222$		0			0
$223$	−9275.03	+	52601.3i	−0.186512	+	1.05776i	0.737486	+	0.675362i	$0.236013\pi$
$223$	−9275.03	+	52601.3i	−0.186512	+	1.05776i	−0.923998	+	0.382397i	$0.875099\pi$
$224$		0			0
$225$	−9146.01	+	45748.9i	−0.180662	+	0.903682i
$226$		0			0
$227$	31965.3	+	87823.9i	0.620336	+	1.70436i	0.706163	+	0.708049i	$0.250425\pi$
$227$	31965.3	+	87823.9i	0.620336	+	1.70436i	−0.0858271	+	0.996310i	$0.527353\pi$
$228$		0			0
$229$	45556.5	−	38226.5i	0.868720	−	0.728942i	−0.0951084	−	0.995467i	$-0.530320\pi$
$229$	45556.5	−	38226.5i	0.868720	−	0.728942i	0.963828	+	0.266525i	$0.0858753\pi$
$230$		0			0
$231$	−11093.9	−	4181.57i	−0.207903	−	0.0783638i
$232$		0			0
$233$	−90002.3	+	51962.9i	−1.65784	+	0.957153i	−0.684127	+	0.729363i	$0.739817\pi$
$233$	−90002.3	+	51962.9i	−1.65784	+	0.957153i	−0.973710	+	0.227790i	$0.926850\pi$
$234$		0			0
$235$	2061.11	−	3569.94i	0.0373220	−	0.0646435i
$236$		0			0
$237$	−16361.8	−	46598.6i	−0.291296	−	0.829614i
$238$		0			0
$239$	94510.7	−	16664.8i	1.65457	−	0.291745i	0.733079	−	0.680144i	$-0.238083\pi$
$239$	94510.7	−	16664.8i	1.65457	−	0.291745i	0.921491	+	0.388399i	$0.126972\pi$
$240$		0			0
$241$	4706.66	+	3949.36i	0.0810362	+	0.0679975i	0.682406	−	0.730973i	$-0.260934\pi$
$241$	4706.66	+	3949.36i	0.0810362	+	0.0679975i	−0.601370	+	0.798971i	$0.705378\pi$
$242$		0			0
$243$	52735.8	−	26565.5i	0.893085	−	0.449888i
$244$		0			0
$245$	−7617.87	+	9078.63i	−0.126912	+	0.151247i
$246$		0			0
$247$	−4997.97	−	28344.9i	−0.0819219	−	0.464602i
$248$		0			0
$249$	−31343.9	+	11005.5i	−0.505539	+	0.177506i
$250$		0			0
$251$	56184.4	+	32438.1i	0.891802	+	0.514882i	0.874531	−	0.484969i	$-0.161169\pi$
$251$	56184.4	+	32438.1i	0.891802	+	0.514882i	0.0172704	+	0.999851i	$0.494502\pi$
$252$		0			0
$253$	−9860.01	−	17078.0i	−0.154041	−	0.266807i
$254$		0			0
$255$	7315.43	−	19408.2i	0.112502	−	0.298472i
$256$		0			0
$257$	−54498.0	−	64948.2i	−0.825115	−	0.983334i	0.174884	−	0.984589i	$-0.444045\pi$
$257$	−54498.0	−	64948.2i	−0.825115	−	0.983334i	−0.999999	+	0.00125545i	$0.999600\pi$
$258$		0			0
$259$	−41120.6	+	14966.7i	−0.612999	+	0.223113i
$260$		0			0
$261$	68642.8	+	13722.9i	1.00766	+	0.201449i
$262$		0			0
$263$	58941.3	+	10392.9i	0.852134	+	0.150254i	0.582622	−	0.812743i	$-0.302027\pi$
$263$	58941.3	+	10392.9i	0.852134	+	0.150254i	0.269512	+	0.962997i	$0.413138\pi$
$264$		0			0
$265$	9552.65	+	3476.88i	0.136029	+	0.0495106i
$266$		0			0
$267$	16335.7	+	29053.7i	0.229148	+	0.407548i
$268$		0			0
$269$		−	112684.i		−	1.55725i	−0.627491	−	0.778624i	$-0.715918\pi$
$269$		−	112684.i		−	1.55725i	0.627491	−	0.778624i	$-0.284082\pi$
$270$		0			0
$271$	124765.			1.69885			0.849423	−	0.527712i	$-0.176950\pi$
$271$	124765.			1.69885			0.849423	+	0.527712i	$0.176950\pi$
$272$		0			0
$273$	337.182	−	29597.6i	0.00452417	−	0.397129i
$274$		0			0
$275$	9750.45	−	26789.1i	0.128932	−	0.354236i
$276$		0			0
$277$	2480.76	−	14069.1i	0.0323314	−	0.183361i	−0.964365	−	0.264574i	$-0.914769\pi$
$277$	2480.76	−	14069.1i	0.0323314	−	0.183361i	0.996697	+	0.0812137i	$0.0258796\pi$
$278$		0			0
$279$	3024.80	−	2422.85i	0.0388587	−	0.0311257i
$280$		0			0
$281$	2564.76	+	7046.62i	0.0324813	+	0.0892417i	0.954873	−	0.297013i	$-0.0959906\pi$
$281$	2564.76	+	7046.62i	0.0324813	+	0.0892417i	−0.922392	+	0.386255i	$0.873768\pi$
$282$		0			0
$283$	−35218.1	+	29551.5i	−0.439737	+	0.368983i	−0.835611	−	0.549322i	$-0.814886\pi$
$283$	−35218.1	+	29551.5i	−0.439737	+	0.368983i	0.395874	+	0.918305i	$0.370442\pi$
$284$		0			0
$285$	−2383.84	−	14482.3i	−0.0293486	−	0.178298i
$286$		0			0
$287$	−8707.65	+	5027.37i	−0.105715	+	0.0610347i
$288$		0			0
$289$	12408.8	−	21492.7i	0.148572	−	0.257333i
$290$		0			0
$291$	15388.2	+	2894.48i	0.181720	+	0.0341809i
$292$		0			0
$293$	76504.6	−	13489.8i	0.891153	−	0.157134i	0.290719	−	0.956808i	$-0.406106\pi$
$293$	76504.6	−	13489.8i	0.891153	−	0.157134i	0.600434	+	0.799674i	$0.294994\pi$
$294$		0			0
$295$	29906.7	+	25094.7i	0.343656	+	0.288362i
$296$		0			0
$297$	−34308.2	+	11175.2i	−0.388942	+	0.126690i
$298$		0			0
$299$	31646.8	−	37715.1i	0.353987	−	0.421865i
$300$		0			0
$301$	−3970.60	−	22518.4i	−0.0438251	−	0.248544i
$302$		0			0
$303$	−76547.6	−	65731.4i	−0.833770	−	0.715958i
$304$		0			0
$305$	−25991.0	−	15005.9i	−0.279398	−	0.161311i
$306$		0			0
$307$	39030.5	+	67602.8i	0.414121	+	0.717279i	0.995336	−	0.0964716i	$-0.0307557\pi$
$307$	39030.5	+	67602.8i	0.414121	+	0.717279i	−0.581215	+	0.813750i	$0.697422\pi$
$308$		0			0
$309$	20779.5	+	25344.9i	0.217630	+	0.265444i
$310$		0			0
$311$	22796.0	+	27167.3i	0.235689	+	0.280883i	0.870905	−	0.491451i	$-0.163533\pi$
$311$	22796.0	+	27167.3i	0.235689	+	0.280883i	−0.635216	+	0.772334i	$0.719089\pi$
$312$		0			0
$313$	39308.2	−	14307.0i	0.401231	−	0.146036i	−0.133519	−	0.991046i	$-0.542628\pi$
$313$	39308.2	−	14307.0i	0.401231	−	0.146036i	0.534750	+	0.845010i	$0.320406\pi$
$314$		0			0
$315$	343.862	−	15090.1i	0.00346548	−	0.152079i
$316$		0			0
$317$	−28243.0	−	4980.00i	−0.281055	−	0.0495576i	0.0313435	−	0.999509i	$-0.490021\pi$
$317$	−28243.0	−	4980.00i	−0.281055	−	0.0495576i	−0.312399	+	0.949951i	$0.601133\pi$
$318$		0			0
$319$	−40195.1	−	14629.8i	−0.394995	−	0.143766i
$320$		0			0
$321$	−56270.5	+	94948.8i	−0.546098	+	0.921466i
$322$		0			0
$323$		−	76664.5i		−	0.734834i
$324$		0			0
$325$	71174.9			0.673845
$326$		0			0
$327$	−156486.	−	92739.8i	−1.46345	−	0.867302i
$328$		0			0
$329$	5359.28	−	14724.5i	0.0495125	−	0.136034i
$330$		0			0
$331$	−30145.6	+	170964.i	−0.275149	+	1.56045i	0.463340	+	0.886181i	$0.346651\pi$
$331$	−30145.6	+	170964.i	−0.275149	+	1.56045i	−0.738489	+	0.674266i	$0.764460\pi$
$332$		0			0
$333$	−69200.0	+	113790.i	−0.624047	+	1.02616i
$334$		0			0
$335$	−8764.29	−	24079.7i	−0.0780957	−	0.214566i
$336$		0			0
$337$	−82889.8	+	69552.8i	−0.729863	+	0.612428i	−0.930094	−	0.367321i	$-0.880275\pi$
$337$	−82889.8	+	69552.8i	−0.729863	+	0.612428i	0.200231	+	0.979749i	$0.435831\pi$
$338$		0			0
$339$	−115006.	+	94289.6i	−1.00074	+	0.820474i
$340$		0			0
$341$	−2050.89	+	1184.08i	−0.0176374	+	0.0101829i
$342$		0			0
$343$	−54475.7	+	94354.7i	−0.463036	+	0.802002i
$344$		0			0
$345$	16356.0	−	19047.3i	0.137416	−	0.160028i
$346$		0			0
$347$	−65651.3	+	11576.1i	−0.545236	+	0.0961398i	−0.439480	−	0.898252i	$-0.644837\pi$
$347$	−65651.3	+	11576.1i	−0.545236	+	0.0961398i	−0.105756	+	0.994392i	$0.533726\pi$
$348$		0			0
$349$	17393.2	+	14594.6i	0.142800	+	0.119823i	0.711389	−	0.702798i	$-0.248066\pi$
$349$	17393.2	+	14594.6i	0.142800	+	0.119823i	−0.568589	+	0.822621i	$0.692511\pi$
$350$		0			0
$351$	−55513.3	−	70946.8i	−0.450591	−	0.575862i
$352$		0			0
$353$	−46885.1	+	55875.5i	−0.376258	+	0.448407i	−0.920629	−	0.390437i	$-0.872324\pi$
$353$	−46885.1	+	55875.5i	−0.376258	+	0.448407i	0.544371	+	0.838844i	$0.316768\pi$
$354$		0			0
$355$	96.2104	+	545.637i	0.000763424	+	0.00432959i
$356$		0			0
$357$	14574.3	−	77482.9i	0.114354	−	0.607952i
$358$		0			0
$359$	90021.6	+	51974.0i	0.698487	+	0.403271i	0.806783	−	0.590847i	$-0.201206\pi$
$359$	90021.6	+	51974.0i	0.698487	+	0.403271i	−0.108297	+	0.994119i	$0.534540\pi$
$360$		0			0
$361$	38035.2	+	65878.9i	0.291858	+	0.505512i
$362$		0			0
$363$	−108264.	+	17820.7i	−0.821618	+	0.135242i
$364$		0			0
$365$	−11741.8	−	13993.3i	−0.0881349	−	0.105035i
$366$		0			0
$367$	−142673.	+	51928.7i	−1.05928	+	0.385545i	−0.812156	−	0.583440i	$-0.801706\pi$
$367$	−142673.	+	51928.7i	−1.05928	+	0.385545i	−0.247120	+	0.968985i	$0.579484\pi$
$368$		0			0
$369$	−11118.5	+	28509.5i	−0.0816570	+	0.209381i
$370$		0			0
$371$	38055.2	+	6710.16i	0.276482	+	0.0487512i
$372$		0			0
$373$	128474.	+	46760.5i	0.923413	+	0.336095i	0.759596	−	0.650396i	$-0.225397\pi$
$373$	128474.	+	46760.5i	0.923413	+	0.336095i	0.163818	+	0.986491i	$0.447619\pi$
$374$		0			0
$375$	75673.9	+	862.092i	0.538126	+	0.00613043i
$376$		0			0
$377$		−	106793.i		−	0.751378i
$378$		0			0
$379$	−102503.			−0.713606			−0.356803	−	0.934180i	$-0.616133\pi$
$379$	−102503.			−0.713606			−0.356803	+	0.934180i	$0.616133\pi$
$380$		0			0
$381$	−54016.5	+	30371.3i	−0.372115	+	0.209225i
$382$		0			0
$383$	32174.5	−	88398.6i	0.219338	−	0.602626i	−0.780406	−	0.625274i	$-0.784987\pi$
$383$	32174.5	−	88398.6i	0.219338	−	0.602626i	0.999744	+	0.0226476i	$0.00720957\pi$
$384$		0			0
$385$	−1601.61	+	9083.16i	−0.0108052	+	0.0612796i
$386$		0			0
$387$	−52276.4	−	45934.7i	−0.349047	−	0.306703i
$388$		0			0
$389$	−36200.3	−	99459.6i	−0.239229	−	0.657276i	−0.999966	−	0.00821594i	$-0.997385\pi$
$389$	−36200.3	−	99459.6i	−0.239229	−	0.657276i	0.760737	−	0.649060i	$-0.224837\pi$
$390$		0			0
$391$	100458.	−	84294.6i	0.657102	−	0.551374i
$392$		0			0
$393$	−133606.	−	50359.5i	−0.865050	−	0.326059i
$394$		0			0
$395$	−33273.8	+	19210.7i	−0.213260	+	0.123126i
$396$		0			0
$397$	31794.5	−	55069.7i	0.201730	−	0.349407i	−0.747356	−	0.664424i	$-0.768677\pi$
$397$	31794.5	−	55069.7i	0.201730	−	0.349407i	0.949086	+	0.315017i	$0.102010\pi$
$398$		0			0
$399$	−18483.3	−	52640.6i	−0.116100	−	0.330655i
$400$		0			0
$401$	−46597.3	+	8216.37i	−0.289783	+	0.0510965i	−0.316650	−	0.948543i	$-0.602558\pi$
$401$	−46597.3	+	8216.37i	−0.289783	+	0.0510965i	0.0268672	+	0.999639i	$0.491447\pi$
$402$		0			0
$403$	−4529.18	−	3800.43i	−0.0278875	−	0.0234004i
$404$		0			0
$405$	−27894.4	−	36498.7i	−0.170062	−	0.222519i
$406$		0			0
$407$	52310.3	−	62340.9i	0.315790	−	0.376344i
$408$		0			0
$409$	39403.7	+	223470.i	0.235554	+	1.33589i	0.841444	+	0.540345i	$0.181706\pi$
$409$	39403.7	+	223470.i	0.235554	+	1.33589i	−0.605890	+	0.795549i	$0.707183\pi$
$410$		0			0
$411$	221465.	−	77761.3i	1.31106	−	0.460341i
$412$		0			0
$413$	128520.	+	74200.9i	0.753476	+	0.435020i
$414$		0			0
$415$	12921.8	+	22381.2i	0.0750286	+	0.129953i
$416$		0			0
$417$	−71641.0	+	190067.i	−0.411993	+	1.09304i
$418$		0			0
$419$	55767.8	+	66461.5i	0.317655	+	0.378566i	0.901118	−	0.433573i	$-0.142747\pi$
$419$	55767.8	+	66461.5i	0.317655	+	0.378566i	−0.583464	+	0.812139i	$0.698303\pi$
$420$		0			0
$421$	283952.	−	103350.i	1.60207	−	0.583105i	0.622218	−	0.782844i	$-0.286232\pi$
$421$	283952.	−	103350.i	1.60207	−	0.583105i	0.979849	+	0.199739i	$0.0640095\pi$
$422$		0			0
$423$	−15285.5	−	45173.0i	−0.0854275	−	0.252463i
$424$		0			0
$425$	186702.	+	32920.6i	1.03364	+	0.182259i
$426$		0			0
$427$	−107202.	−	39018.3i	−0.587959	−	0.213999i
$428$		0			0
$429$	26978.5	+	47982.2i	0.146590	+	0.260715i
$430$		0			0
$431$		−	203554.i		−	1.09579i	−0.836549	−	0.547893i	$-0.815430\pi$
$431$		−	203554.i		−	1.09579i	0.836549	−	0.547893i	$-0.184570\pi$
$432$		0			0
$433$	−172649.			−0.920846			−0.460423	−	0.887700i	$-0.652302\pi$
$433$	−172649.			−0.920846			−0.460423	+	0.887700i	$0.652302\pi$
$434$		0			0
$435$	620.352	−	54454.2i	0.00327838	−	0.287775i
$436$		0			0
$437$	31739.1	−	87202.4i	0.166200	−	0.456631i
$438$		0			0
$439$	20306.1	−	115162.i	0.105365	−	0.597557i	−0.885708	−	0.464242i	$-0.846327\pi$
$439$	20306.1	−	115162.i	0.105365	−	0.597557i	0.991074	−	0.133315i	$-0.0425622\pi$
$440$		0			0
$441$	20725.9	+	135530.i	0.106570	+	0.696879i
$442$		0			0
$443$	−27746.3	−	76232.3i	−0.141383	−	0.388447i	0.848710	−	0.528858i	$-0.177380\pi$
$443$	−27746.3	−	76232.3i	−0.141383	−	0.388447i	−0.990093	+	0.140411i	$0.955157\pi$
$444$		0			0
$445$	19863.7	−	16667.6i	0.100309	−	0.0841693i
$446$		0			0
$447$	5732.28	+	34824.7i	0.0286888	+	0.174290i
$448$		0			0
$449$	−284164.	+	164062.i	−1.40954	+	0.813796i	−0.995343	−	0.0963929i	$-0.969269\pi$
$449$	−284164.	+	164062.i	−1.40954	+	0.813796i	−0.414193	+	0.910189i	$0.635936\pi$
$450$		0			0
$451$	9349.45	−	16193.7i	0.0459656	−	0.0796148i
$452$		0			0
$453$	330414.	+	62150.0i	1.61014	+	0.302862i
$454$		0			0
$455$	−22677.3	+	3998.62i	−0.109539	+	0.0193147i
$456$		0			0
$457$	−77985.4	−	65437.5i	−0.373406	−	0.313325i	0.436701	−	0.899607i	$-0.356147\pi$
$457$	−77985.4	−	65437.5i	−0.373406	−	0.313325i	−0.810107	+	0.586282i	$0.800591\pi$
$458$		0			0
$459$	−112804.	−	211780.i	−0.535427	−	1.00522i
$460$		0			0
$461$	78911.8	−	94043.5i	0.371313	−	0.442514i	−0.547739	−	0.836649i	$-0.684511\pi$
$461$	78911.8	−	94043.5i	0.371313	−	0.442514i	0.919052	+	0.394135i	$0.128956\pi$
$462$		0			0
$463$	7079.25	+	40148.4i	0.0330237	+	0.187287i	0.996857	−	0.0792181i	$-0.0252423\pi$
$463$	7079.25	+	40148.4i	0.0330237	+	0.187287i	−0.963834	+	0.266505i	$0.914131\pi$
$464$		0			0
$465$	−2287.38	−	1964.17i	−0.0105787	−	0.00908393i
$466$		0			0
$467$	333060.	+	192292.i	1.52718	+	0.881715i	0.999479	+	0.0322828i	$0.0102777\pi$
$467$	333060.	+	192292.i	1.52718	+	0.881715i	0.527697	+	0.849433i	$0.323056\pi$
$468$		0			0
$469$	−48703.4	−	84356.8i	−0.221418	−	0.383508i
$470$		0			0
$471$	214953.	+	262179.i	0.968949	+	1.18183i
$472$		0			0
$473$	27333.7	+	32575.1i	0.122173	+	0.145600i
$474$		0			0
$475$	126065.	−	45883.8i	0.558736	−	0.203363i
$476$		0			0
$477$	103162.	−	56467.0i	0.453402	−	0.248175i
$478$		0			0
$479$	−403511.	−	71149.9i	−1.75867	−	0.310101i	−0.801150	−	0.598464i	$-0.795778\pi$
$479$	−403511.	−	71149.9i	−1.75867	−	0.310101i	−0.957521	+	0.288363i	$0.906889\pi$
$480$		0			0
$481$	190923.	+	69490.5i	0.825219	+	0.300355i
$482$		0			0
$483$	48655.5	−	82099.6i	0.208563	−	0.351922i
$484$		0			0
$485$		−	12181.3i		−	0.0517856i
$486$		0			0
$487$	−10720.1			−0.0452001			−0.0226001	−	0.999745i	$-0.507194\pi$
$487$	−10720.1			−0.0452001			−0.0226001	+	0.999745i	$0.507194\pi$
$488$		0			0
$489$	−44304.2	−	26256.5i	−0.185279	−	0.109804i
$490$		0			0
$491$	−142217.	+	390738.i	−0.589913	+	1.62077i	0.180749	+	0.983529i	$0.442148\pi$
$491$	−142217.	+	390738.i	−0.589913	+	1.62077i	−0.770662	+	0.637244i	$0.780074\pi$
$492$		0			0
$493$	49394.9	−	280132.i	0.203230	−	1.15258i
$494$		0			0
$495$	13477.8	+	24623.1i	0.0550057	+	0.100492i
$496$		0			0
$497$	720.324	+	1979.07i	0.00291619	+	0.00801216i
$498$		0			0
$499$	26084.9	−	21887.9i	0.104758	−	0.0879027i	−0.588904	−	0.808203i	$-0.700440\pi$
$499$	26084.9	−	21887.9i	0.104758	−	0.0879027i	0.693662	+	0.720300i	$0.255996\pi$
$500$		0			0
$501$	77619.5	−	63637.9i	0.309240	−	0.253537i
$502$		0			0
$503$	92901.4	−	53636.6i	0.367186	−	0.211995i	−0.305042	−	0.952339i	$-0.598671\pi$
$503$	92901.4	−	53636.6i	0.367186	−	0.211995i	0.672228	+	0.740344i	$0.265337\pi$
$504$		0			0
$505$	−39246.6	+	67977.0i	−0.153893	+	0.266551i
$506$		0			0
$507$	77927.7	−	90750.8i	0.303163	−	0.353049i
$508$		0			0
$509$	−188545.	+	33245.5i	−0.727744	+	0.128321i	−0.525232	−	0.850959i	$-0.676022\pi$
$509$	−188545.	+	33245.5i	−0.727744	+	0.128321i	−0.202512	+	0.979280i	$0.564910\pi$
$510$		0			0
$511$	−53191.8	−	44633.2i	−0.203706	−	0.170929i
$512$		0			0
$513$	−144062.	−	89873.3i	−0.547412	−	0.341504i
$514$		0			0
$515$	16389.1	−	19531.7i	0.0617931	−	0.0736421i
$516$		0			0
$517$	5060.23	+	28698.0i	0.0189317	+	0.107367i
$518$		0			0
$519$	44485.9	−	236505.i	0.165154	−	0.878023i
$520$		0			0
$521$	412944.	+	238413.i	1.52130	+	0.878324i	0.999684	+	0.0251453i	$0.00800483\pi$
$521$	412944.	+	238413.i	1.52130	+	0.878324i	0.521618	+	0.853179i	$0.325329\pi$
$522$		0			0
$523$	−190822.	−	330513.i	−0.697629	−	1.20833i	−0.969286	−	0.245934i	$-0.920905\pi$
$523$	−190822.	−	330513.i	−0.697629	−	1.20833i	0.271658	−	0.962394i	$-0.412428\pi$
$524$		0			0
$525$	136133.	−	22408.1i	0.493908	−	0.0812993i
$526$		0			0
$527$	−10122.9	−	12064.0i	−0.0364487	−	0.0434379i
$528$		0			0
$529$	−113800.	+	41419.7i	−0.406658	+	0.148011i
$530$		0			0
$531$	446460.	−	68275.0i	1.58341	−	0.242143i
$532$		0			0
$533$	45975.0	+	8106.63i	0.161833	+	0.0285355i
$534$		0			0
$535$	80685.1	+	29367.0i	0.281894	+	0.102601i
$536$		0			0
$537$	418003.	+	4761.97i	1.44954	+	0.0165135i
$538$		0			0
$539$		−	83779.2i		−	0.288376i
$540$		0			0
$541$	−360726.			−1.23249			−0.616244	−	0.787555i	$-0.711346\pi$
$541$	−360726.			−1.23249			−0.616244	+	0.787555i	$0.711346\pi$
$542$		0			0
$543$	−384896.	+	216411.i	−1.30540	+	0.733974i
$544$		0			0
$545$	−48399.9	+	132978.i	−0.162949	+	0.447698i
$546$		0			0
$547$	51311.5	−	291002.i	0.171490	−	0.972571i	−0.770627	−	0.637287i	$-0.780057\pi$
$547$	51311.5	−	291002.i	0.171490	−	0.972571i	0.942117	−	0.335284i	$-0.108832\pi$
$548$		0			0
$549$	−328882.	+	111286.i	−1.09118	+	0.369229i
$550$		0			0
$551$	−68845.2	−	189151.i	−0.226762	−	0.623024i
$552$		0			0
$553$	−111880.	+	93878.1i	−0.365848	+	0.306983i
$554$		0			0
$555$	96949.3	+	36542.7i	0.314745	+	0.118635i
$556$		0			0
$557$	−504904.	+	291507.i	−1.62742	+	0.939589i	−0.642556	+	0.766239i	$0.722126\pi$
$557$	−504904.	+	291507.i	−1.62742	+	0.939589i	−0.984860	+	0.173350i	$0.944541\pi$
$558$		0			0
$559$	−53083.0	+	91942.4i	−0.169876	+	0.294234i
$560$		0			0
$561$	48575.2	+	138343.i	0.154344	+	0.439572i
$562$		0			0
$563$	−445514.	+	78556.2i	−1.40555	+	0.247836i	−0.824420	−	0.565978i	$-0.808499\pi$
$563$	−445514.	+	78556.2i	−1.40555	+	0.247836i	−0.581125	+	0.813814i	$0.697387\pi$
$564$		0			0
$565$	88627.8	+	74367.5i	0.277634	+	0.232963i
$566$		0			0
$567$	−128514.	−	118220.i	−0.399747	−	0.367725i
$568$		0			0
$569$	−84329.0	+	100499.i	−0.260467	+	0.310412i	−0.880390	−	0.474250i	$-0.842719\pi$
$569$	−84329.0	+	100499.i	−0.260467	+	0.310412i	0.619923	+	0.784662i	$0.287164\pi$
$570$		0			0
$571$	−22878.3	−	129749.i	−0.0701700	−	0.397954i	−0.999582	−	0.0289101i	$-0.990796\pi$
$571$	−22878.3	−	129749.i	−0.0701700	−	0.397954i	0.929412	−	0.369044i	$-0.120315\pi$
$572$		0			0
$573$	302337.	−	106157.i	0.920835	−	0.323326i
$574$		0			0
$575$	198736.	+	114740.i	0.601092	+	0.347041i
$576$		0			0
$577$	−195982.	−	339451.i	−0.588661	−	1.01959i	−0.994408	−	0.105605i	$-0.966322\pi$
$577$	−195982.	−	339451.i	−0.588661	−	1.01959i	0.405747	−	0.913985i	$-0.367011\pi$
$578$		0			0
$579$	−178303.	+	473046.i	−0.531866	+	1.41106i
$580$		0			0
$581$	63145.8	+	75254.3i	0.187065	+	0.222935i
$582$		0			0
$583$	−67529.6	+	24578.8i	−0.198681	+	0.0723141i
$584$		0			0
$585$	−46258.9	+	52645.4i	−0.135171	+	0.153833i
$586$		0			0
$587$	−483459.	−	85246.9i	−1.40308	−	0.247401i	−0.579674	−	0.814849i	$-0.696820\pi$
$587$	−483459.	−	85246.9i	−1.40308	−	0.247401i	−0.823410	+	0.567447i	$0.807931\pi$
$588$		0			0
$589$	−10472.1	−	3811.52i	−0.0301858	−	0.0109867i
$590$		0			0
$591$	279734.	+	497517.i	0.800885	+	1.42440i
$592$		0			0
$593$		−	126196.i		−	0.358870i	−0.983770	−	0.179435i	$-0.942573\pi$
$593$		−	126196.i		−	0.358870i	0.983770	−	0.179435i	$-0.0574269\pi$
$594$		0			0
$595$	−61335.3			−0.173251
$596$		0			0
$597$	356.609	−	31303.0i	0.00100056	−	0.0878288i
$598$		0			0
$599$	−214012.	+	587994.i	−0.596465	+	1.63878i	0.161796	+	0.986824i	$0.448271\pi$
$599$	−214012.	+	587994.i	−0.596465	+	1.63878i	−0.758261	+	0.651951i	$0.773951\pi$
$600$		0			0
$601$	64146.8	−	363795.i	0.177593	−	1.00718i	−0.757515	−	0.652818i	$-0.773587\pi$
$601$	64146.8	−	363795.i	0.177593	−	1.00718i	0.935108	−	0.354363i	$-0.115302\pi$
$602$		0			0
$603$	−276191.	−	107712.i	−0.759581	−	0.296231i
$604$		0			0
$605$	29194.1	+	80210.0i	0.0797597	+	0.219138i
$606$		0			0
$607$	16151.0	−	13552.3i	0.0438350	−	0.0367819i	−0.620607	−	0.784122i	$-0.713114\pi$
$607$	16151.0	−	13552.3i	0.0438350	−	0.0367819i	0.664442	+	0.747340i	$0.268669\pi$
$608$		0			0
$609$	−33621.7	−	204258.i	−0.0906535	−	0.550737i
$610$		0			0
$611$	−63006.5	+	36376.8i	−0.168773	+	0.0974411i
$612$		0			0
$613$	−234797.	+	406680.i	−0.624844	+	1.08226i	0.363727	+	0.931506i	$0.381504\pi$
$613$	−234797.	+	406680.i	−0.624844	+	1.08226i	−0.988571	+	0.150756i	$0.951829\pi$
$614$		0			0
$615$	23395.8	+	4400.69i	0.0618569	+	0.0116351i
$616$		0			0
$617$	−329956.	+	58180.1i	−0.866734	+	0.152829i	−0.589298	−	0.807916i	$-0.700596\pi$
$617$	−329956.	+	58180.1i	−0.866734	+	0.152829i	−0.277436	+	0.960744i	$0.589485\pi$
$618$		0			0
$619$	−391964.	−	328896.i	−1.02297	−	0.858377i	−0.0329751	−	0.999456i	$-0.510498\pi$
$619$	−391964.	−	328896.i	−1.02297	−	0.858377i	−0.989998	+	0.141080i	$0.954943\pi$
$620$		0			0
$621$	−40632.9	−	287591.i	−0.105365	−	0.745749i
$622$		0			0
$623$	63357.5	−	75506.6i	0.163238	−	0.194540i
$624$		0			0
$625$	52287.4	+	296537.i	0.133856	+	0.759134i
$626$		0			0
$627$	78716.6	+	67594.0i	0.200231	+	0.171938i
$628$		0			0
$629$	468678.	+	270591.i	1.18460	+	0.683932i
$630$		0			0
$631$	213064.	+	369037.i	0.535120	+	0.926854i	0.999158	+	0.0410389i	$0.0130668\pi$
$631$	213064.	+	369037.i	0.535120	+	0.926854i	−0.464038	+	0.885815i	$0.653600\pi$
$632$		0			0
$633$	250166.	+	305129.i	0.624340	+	0.761511i
$634$		0			0
$635$	30988.4	+	36930.5i	0.0768514	+	0.0915879i
$636$		0			0
$637$	196552.	−	71538.9i	0.484393	−	0.176304i
$638$		0			0
$639$	5476.54	+	3330.49i	0.0134123	+	0.00815656i
$640$		0			0
$641$	161675.	+	28507.7i	0.393484	+	0.0693819i	0.366890	−	0.930264i	$-0.380423\pi$
$641$	161675.	+	28507.7i	0.393484	+	0.0693819i	0.0265944	+	0.999646i	$0.491534\pi$
$642$		0			0
$643$	135755.	+	49410.8i	0.328348	+	0.119509i	0.500934	−	0.865486i	$-0.332990\pi$
$643$	135755.	+	49410.8i	0.328348	+	0.119509i	−0.172586	+	0.984994i	$0.555212\pi$
$644$		0			0
$645$	−27601.4	+	46573.6i	−0.0663456	+	0.111949i
$646$		0			0
$647$		−	517527.i		−	1.23630i	−0.786059	−	0.618151i	$-0.787882\pi$
$647$		−	517527.i		−	1.23630i	0.786059	−	0.618151i	$-0.212118\pi$
$648$		0			0
$649$	−275984.			−0.655232
$650$		0			0
$651$	−9859.27	−	5843.00i	−0.0232639	−	0.0137871i
$652$		0			0
$653$	56327.0	−	154757.i	0.132096	−	0.362931i	−0.855956	−	0.517048i	$-0.827031\pi$
$653$	56327.0	−	154757.i	0.132096	−	0.362931i	0.988053	+	0.154116i	$0.0492531\pi$
$654$		0			0
$655$	−19288.5	+	109390.i	−0.0449589	+	0.254974i
$656$		0			0
$657$	−211272.	−	4814.32i	−0.489453	−	0.0111533i
$658$		0			0
$659$	−159820.	−	439101.i	−0.368010	−	1.01110i	−0.976117	−	0.217244i	$-0.930293\pi$
$659$	−159820.	−	439101.i	−0.368010	−	1.01110i	0.608107	−	0.793855i	$-0.291929\pi$
$660$		0			0
$661$	−68881.5	+	57798.4i	−0.157652	+	0.132286i	−0.718201	−	0.695835i	$-0.755034\pi$
$661$	−68881.5	+	57798.4i	−0.157652	+	0.132286i	0.560550	+	0.828121i	$0.310590\pi$
$662$		0			0
$663$	−283083.	+	232091.i	−0.644000	+	0.527997i
$664$		0			0
$665$	−37588.2	+	21701.6i	−0.0849979	+	0.0490736i
$666$		0			0
$667$	172159.	−	298188.i	0.386971	−	0.670254i
$668$		0			0
$669$	313172.	−	364705.i	0.699731	−	0.814873i
$670$		0			0
$671$	208936.	−	36841.1i	0.464054	−	0.0818253i
$672$		0			0
$673$	−12569.7	−	10547.2i	−0.0277521	−	0.0232867i	0.628806	−	0.777562i	$-0.283544\pi$
$673$	−12569.7	−	10547.2i	−0.0277521	−	0.0232867i	−0.656558	+	0.754275i	$0.727988\pi$
$674$		0			0
$675$	280731.	−	312243.i	0.616146	−	0.685307i
$676$		0			0
$677$	282969.	−	337229.i	0.617393	−	0.735780i	−0.363227	−	0.931701i	$-0.618325\pi$
$677$	282969.	−	337229.i	0.617393	−	0.735780i	0.980620	+	0.195920i	$0.0627694\pi$
$678$		0			0
$679$	−8040.58	−	45600.4i	−0.0174400	−	0.0989074i
$680$		0			0
$681$	155490.	−	826646.i	0.335280	−	1.78248i
$682$		0			0
$683$	586826.	+	338804.i	1.25796	+	0.726285i	0.972678	−	0.232158i	$-0.0745785\pi$
$683$	586826.	+	338804.i	1.25796	+	0.726285i	0.285285	+	0.958443i	$0.407912\pi$
$684$		0			0
$685$	−91300.9	−	158138.i	−0.194578	−	0.337019i
$686$		0			0
$687$	−528122.	+	86931.0i	−1.11898	+	0.184188i
$688$		0			0
$689$	−115327.	−	137441.i	−0.242936	−	0.289520i
$690$		0			0
$691$	405109.	−	147448.i	0.848430	−	0.308803i	0.119030	−	0.992891i	$-0.462022\pi$
$691$	405109.	−	147448.i	0.848430	−	0.308803i	0.729400	+	0.684087i	$0.239799\pi$
$692$		0			0
$693$	66707.0	+	83280.0i	0.138901	+	0.173410i
$694$		0			0
$695$	155618.	+	27439.6i	0.322173	+	0.0568079i
$696$		0			0
$697$	116849.	+	42529.7i	0.240525	+	0.0875441i
$698$		0			0
$699$	935271.	+	10654.8i	1.91418	+	0.0218067i
$700$		0			0
$701$		−	267426.i		−	0.544211i	−0.962267	−	0.272105i	$-0.912280\pi$
$701$		−	267426.i		−	0.544211i	0.962267	−	0.272105i	$-0.0877199\pi$
$702$		0			0
$703$	382961.			0.774897
$704$		0			0
$705$	−32338.7	+	18182.7i	−0.0650645	+	0.0365832i
$706$		0			0
$707$	−102049.	+	280377.i	−0.204159	+	0.560923i
$708$		0			0
$709$	80759.1	−	458008.i	0.160657	−	0.911130i	−0.792774	−	0.609516i	$-0.791364\pi$
$709$	80759.1	−	458008.i	0.160657	−	0.911130i	0.953430	−	0.301613i	$-0.0975252\pi$
$710$		0			0
$711$	−87136.8	+	435864.i	−0.172370	+	0.862207i
$712$		0			0
$713$	−6519.83	−	17913.1i	−0.0128250	−	0.0352364i
$714$		0			0
$715$	32805.0	−	27526.6i	0.0641693	−	0.0538445i
$716$		0			0
$717$	−808212.	−	304635.i	−1.57212	−	0.592573i
$718$		0			0
$719$	166203.	−	95957.6i	0.321501	−	0.185619i	−0.330560	−	0.943785i	$-0.607238\pi$
$719$	166203.	−	95957.6i	0.321501	−	0.185619i	0.652061	+	0.758166i	$0.273904\pi$
$720$		0			0
$721$	48459.7	−	83934.7i	0.0932203	−	0.161462i
$722$		0			0
$723$	−18319.5	−	52174.3i	−0.0350460	−	0.0998113i
$724$		0			0
$725$	490204.	−	86436.2i	0.932612	−	0.164445i
$726$		0			0
$727$	646158.	+	542191.i	1.22256	+	1.02585i	0.998687	+	0.0512314i	$0.0163146\pi$
$727$	646158.	+	542191.i	1.22256	+	1.02585i	0.223873	+	0.974618i	$0.428130\pi$
$728$		0			0
$729$	−530200.	−	36295.8i	−0.997665	−	0.0682970i
$730$		0			0
$731$	−181770.	+	216626.i	−0.340164	+	0.405392i
$732$		0			0
$733$	169815.	+	963071.i	0.316060	+	1.79246i	0.566214	+	0.824258i	$0.308408\pi$
$733$	169815.	+	963071.i	0.316060	+	1.79246i	−0.250154	+	0.968206i	$0.580481\pi$
$734$		0			0
$735$	100638.	−	35336.4i	0.186290	−	0.0654104i
$736$		0			0
$737$	156879.	+	90574.3i	0.288822	+	0.166752i
$738$		0			0
$739$	101513.	+	175825.i	0.185880	+	0.321953i	0.943873	−	0.330310i	$-0.107153\pi$
$739$	101513.	+	175825.i	0.185880	+	0.321953i	−0.757993	+	0.652263i	$0.773820\pi$
$740$		0			0
$741$	−91363.9	+	242393.i	−0.166394	+	0.441451i
$742$		0			0
$743$	−231095.	−	275408.i	−0.418613	−	0.498883i	0.514988	−	0.857197i	$-0.327796\pi$
$743$	−231095.	−	275408.i	−0.418613	−	0.498883i	−0.933601	+	0.358314i	$0.883352\pi$
$744$		0			0
$745$	25800.8	−	9390.71i	0.0464857	−	0.0169194i
$746$		0			0
$747$	293178.	+	58611.4i	0.525400	+	0.105037i
$748$		0			0
$749$	321428.	+	56676.4i	0.572954	+	0.101027i
$750$		0			0
$751$	679048.	+	247153.i	1.20398	+	0.438214i	0.864613	−	0.502439i	$-0.167564\pi$
$751$	679048.	+	247153.i	1.20398	+	0.438214i	0.339370	+	0.940653i	$0.389786\pi$
$752$		0			0
$753$	−286164.	−	508952.i	−0.504690	−	0.897609i
$754$		0			0
$755$		−	261555.i		−	0.458849i
$756$		0			0
$757$	−916562.			−1.59945			−0.799724	−	0.600367i	$-0.795021\pi$
$757$	−916562.			−1.59945			−0.799724	+	0.600367i	$0.795021\pi$
$758$		0			0
$759$	−2021.76	+	177469.i	−0.00350950	+	0.308062i
$760$		0			0
$761$	231376.	−	635700.i	0.399529	−	1.09770i	−0.562985	−	0.826467i	$-0.690347\pi$
$761$	231376.	−	635700.i	0.399529	−	1.09770i	0.962514	−	0.271231i	$-0.0874306\pi$
$762$		0			0
$763$	−93408.7	+	529747.i	−0.160449	+	0.909954i
$764$		0			0
$765$	−145694.	+	116700.i	−0.248953	+	0.199411i
$766$		0			0
$767$	−235662.	−	647477.i	−0.400590	−	1.10061i
$768$		0			0
$769$	−179399.	+	150534.i	−0.303366	+	0.254555i	−0.781744	−	0.623600i	$-0.785670\pi$
$769$	−179399.	+	150534.i	−0.303366	+	0.254555i	0.478377	+	0.878154i	$0.341225\pi$
$770$		0			0
$771$	123934.	+	752923.i	0.208489	+	1.26661i
$772$		0			0
$773$	−208696.	+	120491.i	−0.349266	+	0.201649i	−0.664362	−	0.747411i	$-0.731297\pi$
$773$	−208696.	+	120491.i	−0.349266	+	0.201649i	0.315096	+	0.949060i	$0.397963\pi$
$774$		0			0
$775$	13779.1	−	23866.1i	0.0229412	−	0.0397354i
$776$		0			0
$777$	387049.	+	72802.8i	0.641098	+	0.120589i
$778$		0			0
$779$	86656.8	−	15279.9i	0.142800	−	0.0251795i
$780$		0			0
$781$	−3000.38	−	2517.62i	−0.00491897	−	0.00412750i
$782$		0			0
$783$	−468497.	−	421216.i	−0.764158	−	0.687039i
$784$		0			0
$785$	169536.	−	202045.i	0.275120	−	0.327876i
$786$		0			0
$787$	−79896.4	−	453115.i	−0.128996	−	0.731575i	−0.978854	−	0.204562i	$-0.934423\pi$
$787$	−79896.4	−	453115.i	−0.128996	−	0.731575i	0.849857	−	0.527013i	$-0.176688\pi$
$788$		0			0
$789$	−408663.	−	350919.i	−0.656464	−	0.563706i
$790$		0			0
$791$	380865.	+	219892.i	0.608721	+	0.351445i
$792$		0			0
$793$	264842.	+	458720.i	0.421153	+	0.729459i
$794$		0			0
$795$	−58007.4	−	70751.9i	−0.0917802	−	0.111945i
$796$		0			0
$797$	−259482.	−	309238.i	−0.408498	−	0.486829i	0.522093	−	0.852888i	$-0.325151\pi$
$797$	−259482.	−	309238.i	−0.408498	−	0.486829i	−0.930592	+	0.366059i	$0.880707\pi$
$798$		0			0
$799$	−182100.	+	66279.1i	−0.285244	+	0.103821i
$800$		0			0
$801$	6834.00	−	299904.i	0.0106515	−	0.467430i
$802$		0			0
$803$	127171.	+	22423.7i	0.197223	+	0.0347757i
$804$		0			0
$805$	−69766.1	−	25392.8i	−0.107660	−	0.0391849i
$806$		0			0
$807$	−517050.	+	872451.i	−0.793936	+	1.33966i
$808$		0			0
$809$		−	651482.i		−	0.995417i	−0.867344	−	0.497709i	$-0.834175\pi$
$809$		−	651482.i		−	0.995417i	0.867344	−	0.497709i	$-0.165825\pi$
$810$		0			0
$811$	748653.			1.13825			0.569126	−	0.822250i	$-0.307281\pi$
$811$	748653.			1.13825			0.569126	+	0.822250i	$0.307281\pi$
$812$		0			0
$813$	−965988.	−	572484.i	−1.46147	−	0.866128i
$814$		0			0
$815$	−13703.0	+	37648.6i	−0.0206300	+	0.0566805i
$816$		0			0
$817$	−34748.5	+	197069.i	−0.0520586	+	0.295239i
$818$		0			0
$819$	−138419.	+	227611.i	−0.206362	+	0.339333i
$820$		0			0
$821$	220966.	+	607100.i	0.327823	+	0.900686i	0.988662	+	0.150159i	$0.0479784\pi$
$821$	220966.	+	607100.i	0.327823	+	0.900686i	−0.660839	+	0.750528i	$0.729799\pi$
$822$		0			0
$823$	245147.	−	205703.i	0.361932	−	0.303697i	−0.443628	−	0.896211i	$-0.646309\pi$
$823$	245147.	−	205703.i	0.361932	−	0.303697i	0.805560	+	0.592514i	$0.201864\pi$
$824$		0			0
$825$	−198414.	+	162674.i	−0.291518	+	0.239007i
$826$		0			0
$827$	−949209.	+	548026.i	−1.38788	+	0.801291i	−0.993076	−	0.117476i	$-0.962520\pi$
$827$	−949209.	+	548026.i	−1.38788	+	0.801291i	−0.394800	+	0.918767i	$0.629186\pi$
$828$		0			0
$829$	−233502.	+	404438.i	−0.339768	+	0.588495i	−0.984389	−	0.176007i	$-0.943682\pi$
$829$	−233502.	+	404438.i	−0.339768	+	0.588495i	0.644621	+	0.764502i	$0.277015\pi$
$830$		0			0
$831$	−83763.1	+	97546.4i	−0.121297	+	0.141257i
$832$		0			0
$833$	548672.	−	96745.6i	0.790720	−	0.139425i
$834$		0			0
$835$	−59816.6	−	50192.1i	−0.0857924	−	0.0719884i
$836$		0			0
$837$	−34536.6	+	4879.57i	−0.0492980	+	0.00696515i
$838$		0			0
$839$	239525.	−	285455.i	0.340273	−	0.405521i	−0.568587	−	0.822623i	$-0.692510\pi$
$839$	239525.	−	285455.i	0.340273	−	0.405521i	0.908860	+	0.417102i	$0.136954\pi$
$840$		0			0
$841$	−6872.95	−	38978.4i	−0.00971742	−	0.0551103i
$842$		0			0
$843$	12475.8	−	66326.6i	0.0175555	−	0.0933324i
$844$		0			0
$845$	−80590.0	−	46528.7i	−0.112867	−	0.0651639i
$846$		0			0
$847$	162232.	+	280994.i	0.226136	+	0.391680i
$848$		0			0
$849$	408272.	−	67203.2i	0.566414	−	0.0932341i
$850$		0			0
$851$	421076.	+	501818.i	0.581435	+	0.692927i
$852$		0			0
$853$	−271408.	+	98784.3i	−0.373013	+	0.135766i	−0.521722	−	0.853115i	$-0.674710\pi$
$853$	−271408.	+	98784.3i	−0.373013	+	0.135766i	0.148709	+	0.988881i	$0.452488\pi$
$854$		0			0
$855$	−47995.0	+	123067.i	−0.0656544	+	0.168348i
$856$		0			0
$857$	256053.	+	45149.0i	0.348632	+	0.0614733i	0.345222	−	0.938521i	$-0.387804\pi$
$857$	256053.	+	45149.0i	0.348632	+	0.0614733i	0.00340982	+	0.999994i	$0.498915\pi$
$858$		0			0
$859$	−645430.	−	234917.i	−0.874707	−	0.318367i	−0.134635	−	0.990895i	$-0.542986\pi$
$859$	−645430.	−	234917.i	−0.874707	−	0.318367i	−0.740072	+	0.672528i	$0.765208\pi$
$860$		0			0
$861$	90486.7	+	1030.84i	0.122061	+	0.00139055i
$862$		0			0
$863$		−	1.40914e6i		−	1.89205i	−0.324089	−	0.946027i	$-0.605058\pi$
$863$		−	1.40914e6i		−	1.89205i	0.324089	−	0.946027i	$-0.394942\pi$
$864$		0			0
$865$	−187217.			−0.250215
$866$		0			0
$867$	−194694.	+	109469.i	−0.259009	+	0.145631i
$868$		0			0
$869$	92895.4	−	255228.i	0.123014	−	0.337978i
$870$		0			0
$871$	−78534.3	+	445390.i	−0.103520	+	0.587089i
$872$		0			0
$873$	−105861.	−	93019.1i	−0.138902	−	0.122052i
$874$		0			0
$875$	−76542.9	−	210300.i	−0.0999745	−	0.274678i
$876$		0			0
$877$	−242516.	+	203495.i	−0.315313	+	0.264579i	−0.786684	−	0.617356i	$-0.788204\pi$
$877$	−242516.	+	203495.i	−0.315313	+	0.264579i	0.471371	+	0.881935i	$0.343759\pi$
$878$		0			0
$879$	−654232.	−	246597.i	−0.846748	−	0.319161i
$880$		0			0
$881$	−700666.	+	404530.i	−0.902733	+	0.521193i	−0.878086	−	0.478503i	$-0.841180\pi$
$881$	−700666.	+	404530.i	−0.902733	+	0.521193i	−0.0246470	+	0.999696i	$0.507846\pi$
$882$		0			0
$883$	−218818.	+	379003.i	−0.280647	+	0.486095i	−0.971544	−	0.236858i	$-0.923882\pi$
$883$	−218818.	+	379003.i	−0.280647	+	0.486095i	0.690897	+	0.722953i	$0.257216\pi$
$884$		0			0
$885$	−116405.	−	331521.i	−0.148622	−	0.423277i
$886$		0			0
$887$	−19943.6	+	3516.60i	−0.0253488	+	0.00446967i	−0.186308	−	0.982491i	$-0.559652\pi$
$887$	−19943.6	+	3516.60i	−0.0253488	+	0.00446967i	0.160959	+	0.986961i	$0.448541\pi$
$888$		0			0
$889$	140382.	+	117794.i	0.177626	+	0.149046i
$890$		0			0
$891$	316907.	+	70899.7i	0.399187	+	0.0893077i
$892$		0			0
$893$	−88146.1	+	105048.i	−0.110535	+	0.131731i
$894$		0			0
$895$	−56471.9	−	320268.i	−0.0704996	−	0.399823i
$896$		0			0
$897$	−418080.	+	146797.i	−0.519606	+	0.182445i
$898$		0			0
$899$	−35809.2	−	20674.5i	−0.0443073	−	0.0255808i
$900$		0			0
$901$	−238948.	−	413870.i	−0.294343	−	0.509817i
$902$		0			0
$903$	−72583.2	+	192567.i	−0.0890145	+	0.236160i
$904$		0			0
$905$	220808.	+	263149.i	0.269599	+	0.321296i
$906$		0			0
$907$	−1.12746e6	+	410361.i	−1.37052	+	0.498829i	−0.919291	−	0.393579i	$-0.871237\pi$
$907$	−1.12746e6	+	410361.i	−1.37052	+	0.498829i	−0.451230	+	0.892408i	$0.649015\pi$
$908$		0			0
$909$	291058.	+	860161.i	0.352251	+	1.04100i
$910$		0			0
$911$	−944703.	−	166577.i	−1.13830	−	0.200714i	−0.427441	−	0.904043i	$-0.640585\pi$
$911$	−944703.	−	166577.i	−1.13830	−	0.200714i	−0.710864	+	0.703330i	$0.751696\pi$
$912$		0			0
$913$	−171676.	−	62484.8i	−0.205953	−	0.0749606i
$914$		0			0
$915$	132380.	+	235442.i	0.158117	+	0.281217i
$916$		0			0
$917$			422233.i			0.502127i
$918$		0			0
$919$	1.34835e6			1.59652			0.798258	−	0.602316i	$-0.205755\pi$
$919$	1.34835e6			1.59652			0.798258	+	0.602316i	$0.205755\pi$
$920$		0			0
$921$	8003.05	−	702503.i	0.00943488	−	0.828188i
$922$		0			0
$923$	3344.47	−	9188.87i	0.00392577	−	0.0107860i
$924$		0			0
$925$	−164448.	+	932630.i	−0.192196	+	1.09000i
$926$		0			0
$927$	−44589.6	−	291578.i	−0.0518889	−	0.339309i
$928$		0			0
$929$	459824.	+	1.26336e6i	0.532795	+	1.46384i	0.855731	+	0.517421i	$0.173108\pi$
$929$	459824.	+	1.26336e6i	0.532795	+	1.46384i	−0.322936	+	0.946421i	$0.604670\pi$
$930$		0			0
$931$	302013.	−	253419.i	0.348439	−	0.292375i
$932$		0			0
$933$	−51840.6	−	314941.i	−0.0595534	−	0.361798i
$934$		0			0
$935$	98784.0	−	57033.0i	0.112996	−	0.0652383i
$936$		0			0
$937$	277362.	−	480406.i	0.315914	−	0.547178i	−0.663718	−	0.747983i	$-0.731022\pi$
$937$	277362.	−	480406.i	0.315914	−	0.547178i	0.979631	+	0.200805i	$0.0643557\pi$
$938$		0			0
$939$	−369990.	−	69594.0i	−0.419622	−	0.0789297i
$940$		0			0
$941$	−447774.	+	78954.7i	−0.505685	+	0.0891659i	−0.420670	−	0.907214i	$-0.638205\pi$
$941$	−447774.	+	78954.7i	−0.505685	+	0.0891659i	−0.0850150	+	0.996380i	$0.527094\pi$
$942$		0			0
$943$	115304.	+	96751.3i	0.129664	+	0.108801i
$944$		0			0
$945$	−71903.0	+	115256.i	−0.0805162	+	0.129063i
$946$		0			0
$947$	−809229.	+	964401.i	−0.902342	+	1.07537i	0.0944652	+	0.995528i	$0.469886\pi$
$947$	−809229.	+	964401.i	−0.902342	+	1.07537i	−0.996808	+	0.0798416i	$0.974559\pi$
$948$		0			0
$949$	55983.6	+	317499.i	0.0621625	+	0.352541i
$950$		0			0
$951$	195819.	+	168150.i	0.216518	+	0.185924i
$952$		0			0
$953$	218839.	+	126347.i	0.240956	+	0.139116i	0.615616	−	0.788046i	$-0.288907\pi$
$953$	218839.	+	126347.i	0.240956	+	0.139116i	−0.374660	+	0.927162i	$0.622241\pi$
$954$		0			0
$955$	−124641.	−	215885.i	−0.136664	−	0.236709i
$956$		0			0
$957$	244080.	+	297706.i	0.266507	+	0.325060i
$958$		0			0
$959$	−446166.	−	531720.i	−0.485132	−	0.578157i
$960$		0			0
$961$	865675.	−	315080.i	0.937363	−	0.341172i
$962$		0			0
$963$	871344.	−	476941.i	0.939588	−	0.514294i
$964$		0			0
$965$	387308.	+	68292.9i	0.415913	+	0.0733366i
$966$		0			0
$967$	−1.27021e6	−	462319.i	−1.35839	−	0.494412i	−0.442831	−	0.896605i	$-0.646026\pi$
$967$	−1.27021e6	−	462319.i	−1.35839	−	0.494412i	−0.915555	+	0.402193i	$0.868248\pi$
$968$		0			0
$969$	−351775.	+	593572.i	−0.374642	+	0.632158i
$970$		0			0
$971$			286859.i			0.304250i	0.988361	+	0.152125i	$0.0486116\pi$
$971$			286859.i			0.304250i	−0.988361	+	0.152125i	$0.951388\pi$
$972$		0			0
$973$	600665.			0.634464
$974$		0			0
$975$	−551069.	−	326586.i	−0.579691	−	0.343548i
$976$		0			0
$977$	492760.	−	1.35385e6i	0.516234	−	1.41834i	−0.358405	−	0.933566i	$-0.616679\pi$
$977$	492760.	−	1.35385e6i	0.516234	−	1.41834i	0.874639	−	0.484775i	$-0.161098\pi$
$978$		0			0
$979$	−31830.7	+	180521.i	−0.0332110	+	0.188349i
$980$		0			0
$981$	786049.	+	1.43607e6i	0.816792	+	1.49223i
$982$		0			0
$983$	−63612.3	−	174773.i	−0.0658316	−	0.180871i	0.902415	−	0.430868i	$-0.141793\pi$
$983$	−63612.3	−	174773.i	−0.0658316	−	0.180871i	−0.968247	+	0.249997i	$0.919570\pi$
$984$		0			0
$985$	340147.	−	285417.i	0.350586	−	0.294176i
$986$		0			0
$987$	−109057.	+	89412.8i	−0.111949	+	0.0917837i
$988$		0			0
$989$	−296439.	+	171149.i	−0.303070	+	0.174977i
$990$		0			0
$991$	624673.	−	1.08197e6i	0.636071	−	1.10171i	−0.350217	−	0.936669i	$-0.613892\pi$
$991$	624673.	−	1.08197e6i	0.636071	−	1.10171i	0.986287	−	0.165038i	$-0.0527747\pi$
$992$		0			0
$993$	1.01787e6	−	1.18536e6i	1.03227	−	1.20213i
$994$		0			0
$995$	−23983.9	+	4229.01i	−0.0242255	+	0.00427162i
$996$		0			0
$997$	−170540.	−	143100.i	−0.171568	−	0.143963i	0.552961	−	0.833207i	$-0.313498\pi$
$997$	−170540.	−	143100.i	−0.171568	−	0.143963i	−0.724529	+	0.689245i	$0.757942\pi$
$998$		0			0
$999$	1.05790e6	−	563490.i	1.06002	−	0.564618i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.5.k.a.65.3	yes	72
3.2	odd	2		324.5.k.a.197.7		72
27.5	odd	18	inner	108.5.k.a.5.3	✓	72
27.22	even	9		324.5.k.a.125.7		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.k.a.5.3	✓	72	27.5	odd	18	inner
108.5.k.a.65.3	yes	72	1.1	even	1	trivial
324.5.k.a.125.7		72	27.22	even	9
324.5.k.a.197.7		72	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	108.k (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(-7.74246 - 4.58850i) q^{3} +(-2.39469 + 6.57935i) q^{5} +(-4.62159 + 26.2104i) q^{7} +(38.8914 + 71.0525i) q^{9} +O(q^{10})\)	\(q+(-7.74246 - 4.58850i) q^{3} +(-2.39469 + 6.57935i) q^{5} +(-4.62159 + 26.2104i) q^{7} +(38.8914 + 71.0525i) q^{9} +(-16.9285 - 46.5107i) q^{11} +(94.6619 - 79.4308i) q^{13} +(48.7301 - 39.9523i) q^{15} +(285.051 - 164.574i) q^{17} +(116.459 - 201.713i) q^{19} +(156.049 - 181.727i) q^{21} +(392.366 - 69.1848i) q^{23} +(441.224 + 370.231i) q^{25} +(24.9088 - 728.574i) q^{27} +(555.504 - 662.024i) q^{29} +(-8.30835 - 47.1190i) q^{31} +(-82.3458 + 437.784i) q^{33} +(-161.380 - 93.1727i) q^{35} +(822.095 + 1423.91i) q^{37} +(-1097.38 + 180.634i) q^{39} +(242.838 + 289.403i) q^{41} +(-807.328 + 293.843i) q^{43} +(-560.612 + 85.7317i) q^{45} +(-579.809 - 102.236i) q^{47} +(1590.58 + 578.923i) q^{49} +(-2962.14 - 33.7453i) q^{51} -1451.91i q^{53} +346.549 q^{55} +(-1827.24 + 1027.38i) q^{57} +(1907.08 - 5239.66i) q^{59} +(-744.330 + 4221.31i) q^{61} +(-2042.05 + 690.982i) q^{63} +(295.917 + 813.026i) q^{65} +(-2803.64 + 2352.53i) q^{67} +(-3355.34 - 1264.71i) q^{69} +(68.5308 - 39.5663i) q^{71} +(-1304.49 + 2259.43i) q^{73} +(-1717.36 - 4891.06i) q^{75} +(1297.30 - 228.749i) q^{77} +(4203.68 + 3527.30i) q^{79} +(-3535.92 + 5526.66i) q^{81} +(2372.59 - 2827.55i) q^{83} +(400.183 + 2269.55i) q^{85} +(-7338.66 + 2576.77i) q^{87} +(-3207.30 - 1851.74i) q^{89} +(1644.42 + 2848.22i) q^{91} +(-151.878 + 402.940i) q^{93} +(1048.25 + 1249.26i) q^{95} +(-1634.86 + 595.041i) q^{97} +(2646.33 - 3011.68i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{5} - 102 q^{9}+O(q^{10}) \) 72 * q + 9 * q^5 - 102 * q^9	\( 72 q + 9 q^{5} - 102 q^{9} + 18 q^{11} - 225 q^{15} - 282 q^{21} - 1278 q^{23} + 441 q^{25} + 54 q^{27} + 1854 q^{29} - 1665 q^{31} - 45 q^{33} - 2673 q^{35} + 6951 q^{39} - 5472 q^{41} + 1260 q^{43} + 5553 q^{45} + 5103 q^{47} - 5904 q^{49} + 1899 q^{51} + 1107 q^{57} - 10944 q^{59} + 8352 q^{61} - 11985 q^{63} + 8757 q^{65} + 378 q^{67} + 5607 q^{69} - 19764 q^{71} + 6111 q^{73} - 3453 q^{75} - 5679 q^{77} - 5652 q^{79} - 20466 q^{81} - 20061 q^{83} + 26100 q^{85} + 40545 q^{87} + 15633 q^{89} - 6039 q^{91} + 40179 q^{93} + 48024 q^{95} - 37530 q^{97} + 12177 q^{99}+O(q^{100}) \) 72 * q + 9 * q^5 - 102 * q^9 + 18 * q^11 - 225 * q^15 - 282 * q^21 - 1278 * q^23 + 441 * q^25 + 54 * q^27 + 1854 * q^29 - 1665 * q^31 - 45 * q^33 - 2673 * q^35 + 6951 * q^39 - 5472 * q^41 + 1260 * q^43 + 5553 * q^45 + 5103 * q^47 - 5904 * q^49 + 1899 * q^51 + 1107 * q^57 - 10944 * q^59 + 8352 * q^61 - 11985 * q^63 + 8757 * q^65 + 378 * q^67 + 5607 * q^69 - 19764 * q^71 + 6111 * q^73 - 3453 * q^75 - 5679 * q^77 - 5652 * q^79 - 20466 * q^81 - 20061 * q^83 + 26100 * q^85 + 40545 * q^87 + 15633 * q^89 - 6039 * q^91 + 40179 * q^93 + 48024 * q^95 - 37530 * q^97 + 12177 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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