LMFDB - Embedded newform 108.4.i.a.25.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,4,Mod(13,108)]

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, 8]))

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

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

chi := DirichletCharacter("108.13");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	108.i (of order $9$, 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:	$6.37220628062$
Analytic rank:	$0$
Dimension:	$54$
Relative dimension:	$9$ over $\Q(\zeta_{9})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			25.6
Character	$\chi$	$=$	108.25
Dual form			108.4.i.a.13.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.34095 + 4.63896i) q^{3} +(-5.19451 + 4.35871i) q^{5} +(-17.8009 + 6.47901i) q^{7} +(-16.0399 + 21.7191i) q^{9} +O(q^{10})$	\(q+(2.34095 + 4.63896i) q^{3} +(-5.19451 + 4.35871i) q^{5} +(-17.8009 + 6.47901i) q^{7} +(-16.0399 + 21.7191i) q^{9} +(-0.134044 - 0.112476i) q^{11} +(-1.24242 - 7.04614i) q^{13} +(-32.3799 - 13.8936i) q^{15} +(-26.6029 - 46.0776i) q^{17} +(-65.4831 + 113.420i) q^{19} +(-71.7270 - 67.4109i) q^{21} +(129.122 + 46.9965i) q^{23} +(-13.7215 + 77.8183i) q^{25} +(-138.303 - 23.5653i) q^{27} +(-9.32414 + 52.8798i) q^{29} +(139.135 + 50.6411i) q^{31} +(0.207983 - 0.885126i) q^{33} +(64.2270 - 111.244i) q^{35} +(58.6266 + 101.544i) q^{37} +(29.7783 - 22.2582i) q^{39} +(-27.9173 - 158.327i) q^{41} +(51.8941 + 43.5443i) q^{43} +(-11.3478 - 182.734i) q^{45} +(597.576 - 217.500i) q^{47} +(12.1427 - 10.1889i) q^{49} +(151.476 - 231.275i) q^{51} +36.5908 q^{53} +1.18654 q^{55} +(-679.444 - 38.2632i) q^{57} +(574.476 - 482.043i) q^{59} +(-64.9040 + 23.6231i) q^{61} +(144.807 - 490.544i) q^{63} +(37.1659 + 31.1859i) q^{65} +(109.760 + 622.479i) q^{67} +(84.2524 + 709.008i) q^{69} +(-66.9800 - 116.013i) q^{71} +(-435.761 + 754.761i) q^{73} +(-393.117 + 118.515i) q^{75} +(3.11484 + 1.13371i) q^{77} +(55.7069 - 315.930i) q^{79} +(-214.441 - 696.747i) q^{81} +(-197.919 + 1122.46i) q^{83} +(339.028 + 123.396i) q^{85} +(-267.135 + 80.5346i) q^{87} +(-259.505 + 449.475i) q^{89} +(67.7684 + 117.378i) q^{91} +(90.7863 + 763.992i) q^{93} +(-154.213 - 874.583i) q^{95} +(-1153.80 - 968.155i) q^{97} +(4.59294 - 1.10721i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 54 q + 12 q^{5} - 48 q^{9}+O(q^{10}) $ 54 * q + 12 * q^5 - 48 * q^9	$ 54 q + 12 q^{5} - 48 q^{9} - 87 q^{11} + 234 q^{15} + 204 q^{17} - 12 q^{21} + 96 q^{23} - 216 q^{25} + 27 q^{27} + 318 q^{29} - 54 q^{31} + 63 q^{33} + 6 q^{35} + 66 q^{39} + 867 q^{41} - 513 q^{43} - 306 q^{45} - 1548 q^{47} + 594 q^{49} - 1368 q^{51} - 1068 q^{53} - 1269 q^{57} - 1218 q^{59} - 54 q^{61} + 30 q^{63} + 96 q^{65} - 2997 q^{67} + 1476 q^{69} - 120 q^{71} - 216 q^{73} + 732 q^{75} + 3480 q^{77} + 2808 q^{79} + 3348 q^{81} + 4464 q^{83} + 2160 q^{85} + 4824 q^{87} + 4029 q^{89} + 270 q^{91} + 1164 q^{93} - 1650 q^{95} - 3483 q^{97} - 5076 q^{99}+O(q^{100}) $ 54 * q + 12 * q^5 - 48 * q^9 - 87 * q^11 + 234 * q^15 + 204 * q^17 - 12 * q^21 + 96 * q^23 - 216 * q^25 + 27 * q^27 + 318 * q^29 - 54 * q^31 + 63 * q^33 + 6 * q^35 + 66 * q^39 + 867 * q^41 - 513 * q^43 - 306 * q^45 - 1548 * q^47 + 594 * q^49 - 1368 * q^51 - 1068 * q^53 - 1269 * q^57 - 1218 * q^59 - 54 * q^61 + 30 * q^63 + 96 * q^65 - 2997 * q^67 + 1476 * q^69 - 120 * q^71 - 216 * q^73 + 732 * q^75 + 3480 * q^77 + 2808 * q^79 + 3348 * q^81 + 4464 * q^83 + 2160 * q^85 + 4824 * q^87 + 4029 * q^89 + 270 * q^91 + 1164 * q^93 - 1650 * q^95 - 3483 * q^97 - 5076 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{5}{9}\right)$	$1$

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
$3$	2.34095	+	4.63896i	0.450516	+	0.892769i
$3$	2.34095	+	4.63896i	0.450516	+	0.892769i
$4$		0			0
$5$	−5.19451	+	4.35871i	−0.464611	+	0.389855i	−0.844824	−	0.535044i	$-0.820295\pi$
$5$	−5.19451	+	4.35871i	−0.464611	+	0.389855i	0.380213	+	0.924899i	$0.375851\pi$
$6$		0			0
$7$	−17.8009	+	6.47901i	−0.961160	+	0.349834i	−0.774488	−	0.632588i	$-0.781992\pi$
$7$	−17.8009	+	6.47901i	−0.961160	+	0.349834i	−0.186672	+	0.982422i	$0.559770\pi$
$8$		0			0
$9$	−16.0399	+	21.7191i	−0.594072	+	0.804412i
$10$		0			0
$11$	−0.134044	−	0.112476i	−0.00367416	−	0.00308299i	0.640949	−	0.767584i	$-0.278541\pi$
$11$	−0.134044	−	0.112476i	−0.00367416	−	0.00308299i	−0.644623	+	0.764501i	$0.722986\pi$
$12$		0			0
$13$	−1.24242	−	7.04614i	−0.0265067	−	0.150327i	0.968682	−	0.248305i	$-0.0798735\pi$
$13$	−1.24242	−	7.04614i	−0.0265067	−	0.150327i	−0.995189	+	0.0979783i	$0.968762\pi$
$14$		0			0
$15$	−32.3799	−	13.8936i	−0.557364	−	0.239154i
$16$		0			0
$17$	−26.6029	−	46.0776i	−0.379538	−	0.657380i	0.611457	−	0.791278i	$-0.290584\pi$
$17$	−26.6029	−	46.0776i	−0.379538	−	0.657380i	−0.990995	+	0.133898i	$0.957251\pi$
$18$		0			0
$19$	−65.4831	+	113.420i	−0.790677	+	1.36949i	0.134872	+	0.990863i	$0.456938\pi$
$19$	−65.4831	+	113.420i	−0.790677	+	1.36949i	−0.925548	+	0.378629i	$0.876396\pi$
$20$		0			0
$21$	−71.7270	−	67.4109i	−0.745338	−	0.700488i
$22$		0			0
$23$	129.122	+	46.9965i	1.17060	+	0.426063i	0.852872	−	0.522119i	$-0.174858\pi$
$23$	129.122	+	46.9965i	1.17060	+	0.426063i	0.317726	+	0.948182i	$0.397081\pi$
$24$		0			0
$25$	−13.7215	+	77.8183i	−0.109772	+	0.622546i
$26$		0			0
$27$	−138.303	−	23.5653i	−0.985792	−	0.167968i
$28$		0			0
$29$	−9.32414	+	52.8798i	−0.0597052	+	0.338605i	−0.999998	−	0.00176047i	$-0.999440\pi$
$29$	−9.32414	+	52.8798i	−0.0597052	+	0.338605i	0.940293	+	0.340365i	$0.110551\pi$
$30$		0			0
$31$	139.135	+	50.6411i	0.806111	+	0.293401i	0.712017	−	0.702163i	$-0.247782\pi$
$31$	139.135	+	50.6411i	0.806111	+	0.293401i	0.0940949	+	0.995563i	$0.470004\pi$
$32$		0			0
$33$	0.207983	−	0.885126i	0.00109713	−	0.00466911i
$34$		0			0
$35$	64.2270	−	111.244i	0.310181	−	0.537249i
$36$		0			0
$37$	58.6266	+	101.544i	0.260491	+	0.451183i	0.966372	−	0.257147i	$-0.0827824\pi$
$37$	58.6266	+	101.544i	0.260491	+	0.451183i	−0.705882	+	0.708330i	$0.749449\pi$
$38$		0			0
$39$	29.7783	−	22.2582i	0.122265	−	0.0913889i
$40$		0			0
$41$	−27.9173	−	158.327i	−0.106340	−	0.603085i	−0.990677	−	0.136235i	$-0.956500\pi$
$41$	−27.9173	−	158.327i	−0.106340	−	0.603085i	0.884336	−	0.466850i	$-0.154611\pi$
$42$		0			0
$43$	51.8941	+	43.5443i	0.184041	+	0.154429i	0.730154	−	0.683283i	$-0.239448\pi$
$43$	51.8941	+	43.5443i	0.184041	+	0.154429i	−0.546113	+	0.837712i	$0.683893\pi$
$44$		0			0
$45$	−11.3478	−	182.734i	−0.0375919	−	0.605340i
$46$		0			0
$47$	597.576	−	217.500i	1.85458	−	0.675013i	0.871909	−	0.489668i	$-0.162882\pi$
$47$	597.576	−	217.500i	1.85458	−	0.675013i	0.982674	−	0.185345i	$-0.0593402\pi$
$48$		0			0
$49$	12.1427	−	10.1889i	0.0354013	−	0.0297052i
$50$		0			0
$51$	151.476	−	231.275i	0.415900	−	0.635000i
$52$		0			0
$53$	36.5908			0.0948328			0.0474164	−	0.998875i	$-0.484901\pi$
$53$	36.5908			0.0948328			0.0474164	+	0.998875i	$0.484901\pi$
$54$		0			0
$55$	1.18654			0.00290897
$56$		0			0
$57$	−679.444	−	38.2632i	−1.57885	−	0.0889138i
$58$		0			0
$59$	574.476	−	482.043i	1.26763	−	1.06367i	0.272809	−	0.962068i	$-0.412047\pi$
$59$	574.476	−	482.043i	1.26763	−	1.06367i	0.994825	−	0.101603i	$-0.0323973\pi$
$60$		0			0
$61$	−64.9040	+	23.6231i	−0.136231	+	0.0495841i	−0.409236	−	0.912429i	$-0.634205\pi$
$61$	−64.9040	+	23.6231i	−0.136231	+	0.0495841i	0.273005	+	0.962013i	$0.411982\pi$
$62$		0			0
$63$	144.807	−	490.544i	0.289587	−	0.980996i
$64$		0			0
$65$	37.1659	+	31.1859i	0.0709209	+	0.0595097i
$66$		0			0
$67$	109.760	+	622.479i	0.200139	+	1.13504i	0.904908	+	0.425607i	$0.139939\pi$
$67$	109.760	+	622.479i	0.200139	+	1.13504i	−0.704769	+	0.709437i	$0.748949\pi$
$68$		0			0
$69$	84.2524	+	709.008i	0.146997	+	1.23702i
$70$		0			0
$71$	−66.9800	−	116.013i	−0.111959	−	0.193918i	0.804601	−	0.593816i	$-0.202379\pi$
$71$	−66.9800	−	116.013i	−0.111959	−	0.193918i	−0.916560	+	0.399897i	$0.869046\pi$
$72$		0			0
$73$	−435.761	+	754.761i	−0.698658	+	1.21011i	0.270274	+	0.962783i	$0.412886\pi$
$73$	−435.761	+	754.761i	−0.698658	+	1.21011i	−0.968932	+	0.247327i	$0.920448\pi$
$74$		0			0
$75$	−393.117	+	118.515i	−0.605244	+	0.182466i
$76$		0			0
$77$	3.11484	+	1.13371i	0.00460999	+	0.00167790i
$78$		0			0
$79$	55.7069	−	315.930i	0.0793357	−	0.449935i	−0.919100	−	0.394024i	$-0.871083\pi$
$79$	55.7069	−	315.930i	0.0793357	−	0.449935i	0.998436	−	0.0559108i	$-0.0178062\pi$
$80$		0			0
$81$	−214.441	−	696.747i	−0.294158	−	0.955757i
$82$		0			0
$83$	−197.919	+	1122.46i	−0.261741	+	1.48440i	0.516418	+	0.856337i	$0.327265\pi$
$83$	−197.919	+	1122.46i	−0.261741	+	1.48440i	−0.778159	+	0.628068i	$0.783846\pi$
$84$		0			0
$85$	339.028	+	123.396i	0.432620	+	0.157461i
$86$		0			0
$87$	−267.135	+	80.5346i	−0.329194	+	0.0992438i
$88$		0			0
$89$	−259.505	+	449.475i	−0.309073	+	0.535329i	−0.978160	−	0.207855i	$-0.933352\pi$
$89$	−259.505	+	449.475i	−0.309073	+	0.535329i	0.669087	+	0.743184i	$0.266685\pi$
$90$		0			0
$91$	67.7684	+	117.378i	0.0780665	+	0.135215i
$92$		0			0
$93$	90.7863	+	763.992i	0.101227	+	0.851853i
$94$		0			0
$95$	−154.213	−	874.583i	−0.166546	−	0.944530i
$96$		0			0
$97$	−1153.80	−	968.155i	−1.20774	−	1.01341i	−0.999375	−	0.0353635i	$-0.988741\pi$
$97$	−1153.80	−	968.155i	−1.20774	−	1.01341i	−0.208366	−	0.978051i	$-0.566814\pi$
$98$		0			0
$99$	4.59294	−	1.10721i	0.00466271	−	0.00112403i
$100$		0			0
$101$	−956.621	+	348.182i	−0.942449	+	0.343023i	−0.767132	−	0.641489i	$-0.778317\pi$
$101$	−956.621	+	348.182i	−0.942449	+	0.343023i	−0.175317	+	0.984512i	$0.556095\pi$
$102$		0			0
$103$	749.339	−	628.770i	0.716840	−	0.601500i	−0.209669	−	0.977772i	$-0.567239\pi$
$103$	749.339	−	628.770i	0.716840	−	0.601500i	0.926509	+	0.376272i	$0.122794\pi$
$104$		0			0
$105$	666.410	+	37.5292i	0.619381	+	0.0348807i
$106$		0			0
$107$	−1665.48			−1.50475			−0.752374	−	0.658736i	$-0.771091\pi$
$107$	−1665.48			−1.50475			−0.752374	+	0.658736i	$0.771091\pi$
$108$		0			0
$109$	531.528			0.467075			0.233538	−	0.972348i	$-0.424970\pi$
$109$	531.528			0.467075			0.233538	+	0.972348i	$0.424970\pi$
$110$		0			0
$111$	−333.818	+	509.676i	−0.285447	+	0.435823i
$112$		0			0
$113$	−277.167	+	232.571i	−0.230740	+	0.193614i	−0.750826	−	0.660500i	$-0.770344\pi$
$113$	−277.167	+	232.571i	−0.230740	+	0.193614i	0.520086	+	0.854114i	$0.325900\pi$
$114$		0			0
$115$	−875.568	+	318.681i	−0.709975	+	0.258410i
$116$		0			0
$117$	172.964	+	86.0352i	0.136672	+	0.0679826i
$118$		0			0
$119$	772.094	+	647.864i	0.594771	+	0.499072i
$120$		0			0
$121$	−231.120	−	1310.75i	−0.173644	−	0.984785i
$122$		0			0
$123$	669.119	−	500.142i	0.490507	−	0.366636i
$124$		0			0
$125$	−691.720	−	1198.09i	−0.494955	−	0.857287i
$126$		0			0
$127$	−947.516	+	1641.15i	−0.662035	+	1.14668i	0.318045	+	0.948076i	$0.396974\pi$
$127$	−947.516	+	1641.15i	−0.662035	+	1.14668i	−0.980080	+	0.198603i	$0.936360\pi$
$128$		0			0
$129$	−80.5190	+	342.669i	−0.0549559	+	0.233879i
$130$		0			0
$131$	2543.01	+	925.581i	1.69606	+	0.617316i	0.995367	−	0.0961447i	$-0.0306512\pi$
$131$	2543.01	+	925.581i	1.69606	+	0.617316i	0.700695	+	0.713461i	$0.252873\pi$
$132$		0			0
$133$	430.811	−	2443.25i	0.280873	−	1.59291i
$134$		0			0
$135$	821.129	−	480.412i	0.523493	−	0.306276i
$136$		0			0
$137$	−255.797	+	1450.70i	−0.159520	+	0.904681i	0.795017	+	0.606587i	$0.207462\pi$
$137$	−255.797	+	1450.70i	−0.159520	+	0.904681i	−0.954537	+	0.298094i	$0.903649\pi$
$138$		0			0
$139$	2088.85	+	760.279i	1.27463	+	0.463928i	0.888654	−	0.458579i	$-0.151641\pi$
$139$	2088.85	+	760.279i	1.27463	+	0.463928i	0.385979	+	0.922507i	$0.373864\pi$
$140$		0			0
$141$	2407.87	+	2262.98i	1.43815	+	1.35161i
$142$		0			0
$143$	−0.625984	+	1.08424i	−0.000366066	+	0.000634045i
$144$		0			0
$145$	−182.053	−	315.326i	−0.104267	−	0.180596i
$146$		0			0
$147$	75.6912	+	32.4776i	0.0424687	+	0.0182225i
$148$		0			0
$149$	−387.086	−	2195.27i	−0.212828	−	1.20701i	−0.884637	−	0.466281i	$-0.845594\pi$
$149$	−387.086	−	2195.27i	−0.212828	−	1.20701i	0.671809	−	0.740724i	$-0.265518\pi$
$150$		0			0
$151$	−1652.75	−	1386.82i	−0.890720	−	0.747403i	0.0776345	−	0.996982i	$-0.475263\pi$
$151$	−1652.75	−	1386.82i	−0.890720	−	0.747403i	−0.968354	+	0.249579i	$0.919708\pi$
$152$		0			0
$153$	1427.47	+	161.289i	0.754277	+	0.0852253i
$154$		0			0
$155$	−943.470	+	343.395i	−0.488912	+	0.177949i
$156$		0			0
$157$	2419.53	−	2030.22i	1.22993	−	1.03204i	0.231688	−	0.972790i	$-0.425575\pi$
$157$	2419.53	−	2030.22i	1.22993	−	1.03204i	0.998243	−	0.0592452i	$-0.0188694\pi$
$158$		0			0
$159$	85.6572	+	169.744i	0.0427237	+	0.0846637i
$160$		0			0
$161$	−2602.98			−1.27418
$162$		0			0
$163$	−1919.61			−0.922428			−0.461214	−	0.887289i	$-0.652586\pi$
$163$	−1919.61			−0.922428			−0.461214	+	0.887289i	$0.652586\pi$
$164$		0			0
$165$	2.77764	+	5.50433i	0.00131054	+	0.00259704i
$166$		0			0
$167$	−531.834	+	446.262i	−0.246435	+	0.206783i	−0.757635	−	0.652678i	$-0.773645\pi$
$167$	−531.834	+	446.262i	−0.246435	+	0.206783i	0.511201	+	0.859461i	$0.329201\pi$
$168$		0			0
$169$	2016.40	−	733.910i	0.917797	−	0.334051i
$170$		0			0
$171$	−1413.04	−	3241.49i	−0.631918	−	1.44961i
$172$		0			0
$173$	2035.08	+	1707.63i	0.894358	+	0.750456i	0.969080	−	0.246749i	$-0.0793622\pi$
$173$	2035.08	+	1707.63i	0.894358	+	0.750456i	−0.0747211	+	0.997204i	$0.523807\pi$
$174$		0			0
$175$	−259.931	−	1474.14i	−0.112280	−	0.636769i
$176$		0			0
$177$	3581.00	+	1536.54i	1.52070	+	0.652504i
$178$		0			0
$179$	−554.255	−	959.998i	−0.231436	−	0.400858i	0.726795	−	0.686854i	$-0.241009\pi$
$179$	−554.255	−	959.998i	−0.231436	−	0.400858i	−0.958231	+	0.285996i	$0.907676\pi$
$180$		0			0
$181$	1019.98	−	1766.65i	0.418864	−	0.725493i	−0.576962	−	0.816771i	$-0.695762\pi$
$181$	1019.98	−	1766.65i	0.418864	−	0.725493i	0.995825	+	0.0912779i	$0.0290952\pi$
$182$		0			0
$183$	−261.523	−	245.787i	−0.105641	−	0.0992845i
$184$		0			0
$185$	−747.138	−	271.936i	−0.296922	−	0.108071i
$186$		0			0
$187$	−1.61667	+	9.16862i	−0.000632208	+	0.00358543i
$188$		0			0
$189$	2614.60	−	476.582i	1.00627	−	0.183419i
$190$		0			0
$191$	187.456	−	1063.12i	0.0710149	−	0.402745i	−0.928492	−	0.371352i	$-0.878894\pi$
$191$	187.456	−	1063.12i	0.0710149	−	0.402745i	0.999507	−	0.0313936i	$-0.00999454\pi$
$192$		0			0
$193$	−231.941	−	84.4197i	−0.0865051	−	0.0314853i	0.298405	−	0.954439i	$-0.403545\pi$
$193$	−231.941	−	84.4197i	−0.0865051	−	0.0314853i	−0.384910	+	0.922954i	$0.625768\pi$
$194$		0			0
$195$	−57.6667	+	245.415i	−0.0211774	+	0.0901260i
$196$		0			0
$197$	−2154.60	+	3731.87i	−0.779232	+	1.34967i	0.153154	+	0.988202i	$0.451057\pi$
$197$	−2154.60	+	3731.87i	−0.779232	+	1.34967i	−0.932385	+	0.361466i	$0.882276\pi$
$198$		0			0
$199$	1279.40	+	2215.99i	0.455751	+	0.789383i	0.998731	−	0.0503621i	$-0.0160376\pi$
$199$	1279.40	+	2215.99i	0.455751	+	0.789383i	−0.542980	+	0.839745i	$0.682704\pi$
$200$		0			0
$201$	−2630.72	+	1966.36i	−0.923166	+	0.690033i
$202$		0			0
$203$	−176.631	−	1001.72i	−0.0610692	−	0.346340i
$204$		0			0
$205$	835.117	+	700.746i	0.284522	+	0.238743i
$206$		0			0
$207$	−3091.83	+	2050.59i	−1.03815	+	0.688532i
$208$		0			0
$209$	21.5347	−	7.83798i	0.00712720	−	0.00259409i
$210$		0			0
$211$	−1331.28	+	1117.08i	−0.434357	+	0.364468i	−0.833593	−	0.552380i	$-0.813720\pi$
$211$	−1331.28	+	1117.08i	−0.434357	+	0.364468i	0.399236	+	0.916848i	$0.369275\pi$
$212$		0			0
$213$	381.382	−	582.298i	0.122685	−	0.187316i
$214$		0			0
$215$	−459.361			−0.145712
$216$		0			0
$217$	−2804.85			−0.877444
$218$		0			0
$219$	−4521.40	−	254.625i	−1.39511	−	0.0785660i
$220$		0			0
$221$	−291.617	+	244.696i	−0.0887615	+	0.0744797i
$222$		0			0
$223$	−3665.27	+	1334.05i	−1.10065	+	0.400603i	−0.827555	−	0.561385i	$-0.810269\pi$
$223$	−3665.27	+	1334.05i	−1.10065	+	0.400603i	−0.273093	+	0.961988i	$0.588047\pi$
$224$		0			0
$225$	−1470.05	−	1546.22i	−0.435572	−	0.458139i
$226$		0			0
$227$	2227.65	+	1869.22i	0.651342	+	0.546541i	0.907478	−	0.420100i	$-0.138005\pi$
$227$	2227.65	+	1869.22i	0.651342	+	0.546541i	−0.256136	+	0.966641i	$0.582449\pi$
$228$		0			0
$229$	862.951	+	4894.04i	0.249019	+	1.41226i	0.810970	+	0.585088i	$0.198940\pi$
$229$	862.951	+	4894.04i	0.249019	+	1.41226i	−0.561950	+	0.827171i	$0.689949\pi$
$230$		0			0
$231$	2.03245	+	17.1036i	0.000578897	+	0.00487158i
$232$		0			0
$233$	−254.515	−	440.832i	−0.0715614	−	0.123948i	0.828024	−	0.560692i	$-0.189465\pi$
$233$	−254.515	−	440.832i	−0.0715614	−	0.123948i	−0.899586	+	0.436744i	$0.856132\pi$
$234$		0			0
$235$	−2156.09	+	3734.46i	−0.598502	+	1.03664i
$236$		0			0
$237$	1595.99	−	481.152i	0.437430	−	0.131874i
$238$		0			0
$239$	4164.81	+	1515.87i	1.12719	+	0.410265i	0.837272	−	0.546786i	$-0.184149\pi$
$239$	4164.81	+	1515.87i	1.12719	+	0.410265i	0.289921	+	0.957051i	$0.406371\pi$
$240$		0			0
$241$	90.6192	−	513.927i	0.0242211	−	0.137365i	−0.970299	−	0.241908i	$-0.922227\pi$
$241$	90.6192	−	513.927i	0.0242211	−	0.137365i	0.994520	+	0.104543i	$0.0333380\pi$
$242$		0			0
$243$	2730.19	−	2625.83i	0.720747	−	0.693198i
$244$		0			0
$245$	−18.6647	+	105.853i	−0.00486711	+	0.0276027i
$246$		0			0
$247$	880.532	+	320.487i	0.226830	+	0.0825592i
$248$		0			0
$249$	−5670.35	+	1709.47i	−1.44315	+	0.435073i
$250$		0			0
$251$	3252.48	−	5633.46i	0.817907	−	1.41666i	−0.0893143	−	0.996003i	$-0.528468\pi$
$251$	3252.48	−	5633.46i	0.817907	−	1.41666i	0.907221	−	0.420653i	$-0.138199\pi$
$252$		0			0
$253$	−12.0220	−	20.8227i	−0.00298742	−	0.00517437i
$254$		0			0
$255$	221.217	+	1861.60i	0.0543260	+	0.457168i
$256$		0			0
$257$	−993.934	−	5636.88i	−0.241245	−	1.36817i	−0.829054	−	0.559168i	$-0.811121\pi$
$257$	−993.934	−	5636.88i	−0.241245	−	1.36817i	0.587810	−	0.808999i	$-0.299990\pi$
$258$		0			0
$259$	−1701.51	−	1427.74i	−0.408212	−	0.342531i
$260$		0			0
$261$	−998.946	−	1050.70i	−0.236909	−	0.249183i
$262$		0			0
$263$	−2466.09	+	897.584i	−0.578197	+	0.210446i	−0.614530	−	0.788893i	$-0.710654\pi$
$263$	−2466.09	+	897.584i	−0.578197	+	0.210446i	0.0363333	+	0.999340i	$0.488432\pi$
$264$		0			0
$265$	−190.071	+	159.489i	−0.0440603	+	0.0369710i
$266$		0			0
$267$	−2692.59	−	151.634i	−0.617167	−	0.0347561i
$268$		0			0
$269$	2872.70			0.651122			0.325561	−	0.945521i	$-0.394447\pi$
$269$	2872.70			0.651122			0.325561	+	0.945521i	$0.394447\pi$
$270$		0			0
$271$	3701.45			0.829695			0.414847	−	0.909891i	$-0.363835\pi$
$271$	3701.45			0.829695			0.414847	+	0.909891i	$0.363835\pi$
$272$		0			0
$273$	−385.871	+	589.151i	−0.0855457	+	0.130612i
$274$		0			0
$275$	10.5920	−	8.88773i	0.00232262	−	0.00194891i
$276$		0			0
$277$	1213.14	−	441.548i	0.263143	−	0.0957763i	−0.207079	−	0.978324i	$-0.566396\pi$
$277$	1213.14	−	441.548i	0.263143	−	0.0957763i	0.470223	+	0.882548i	$0.344174\pi$
$278$		0			0
$279$	−3331.60	+	2209.62i	−0.714903	+	0.474145i
$280$		0			0
$281$	−3031.24	−	2543.52i	−0.643519	−	0.539977i	0.261578	−	0.965182i	$-0.415757\pi$
$281$	−3031.24	−	2543.52i	−0.643519	−	0.539977i	−0.905097	+	0.425206i	$0.860202\pi$
$282$		0			0
$283$	−174.948	−	992.178i	−0.0367476	−	0.208406i	0.960906	−	0.276876i	$-0.0892992\pi$
$283$	−174.948	−	992.178i	−0.0367476	−	0.208406i	−0.997653	+	0.0684704i	$0.978188\pi$
$284$		0			0
$285$	3696.15	−	2762.74i	0.768215	−	0.574212i
$286$		0			0
$287$	1522.75	+	2637.49i	0.313189	+	0.542460i
$288$		0			0
$289$	1041.07	−	1803.19i	0.211901	−	0.367024i
$290$		0			0
$291$	1790.24	−	7618.84i	0.360639	−	1.53479i
$292$		0			0
$293$	7284.28	+	2651.26i	1.45240	+	0.528629i	0.943259	−	0.332057i	$-0.107743\pi$
$293$	7284.28	+	2651.26i	1.45240	+	0.528629i	0.509137	+	0.860686i	$0.329965\pi$
$294$		0			0
$295$	−883.036	+	5007.95i	−0.174279	+	0.988387i
$296$		0			0
$297$	15.8881	+	18.7146i	0.00310412	+	0.00365633i
$298$		0			0
$299$	170.720	−	968.201i	0.0330200	−	0.187266i
$300$		0			0
$301$	−1205.89	−	438.907i	−0.230918	−	0.0840471i
$302$		0			0
$303$	−3854.60	−	3622.65i	−0.730828	−	0.686851i
$304$		0			0
$305$	234.178	−	405.608i	0.0439639	−	0.0761477i
$306$		0			0
$307$	4498.36	+	7791.39i	0.836271	+	1.44846i	0.892991	+	0.450074i	$0.148602\pi$
$307$	4498.36	+	7791.39i	0.836271	+	1.44846i	−0.0567205	+	0.998390i	$0.518064\pi$
$308$		0			0
$309$	4671.00	+	2004.24i	0.859948	+	0.368987i
$310$		0			0
$311$	−933.478	−	5294.02i	−0.170202	−	0.965261i	−0.943538	−	0.331265i	$-0.892525\pi$
$311$	−933.478	−	5294.02i	−0.170202	−	0.965261i	0.773336	−	0.633996i	$-0.218587\pi$
$312$		0			0
$313$	2795.71	+	2345.88i	0.504865	+	0.423632i	0.859318	−	0.511442i	$-0.170888\pi$
$313$	2795.71	+	2345.88i	0.504865	+	0.423632i	−0.354453	+	0.935074i	$0.615333\pi$
$314$		0			0
$315$	1385.93	+	3179.31i	0.247900	+	0.568678i
$316$		0			0
$317$	9666.93	−	3518.47i	1.71277	−	0.623398i	0.715597	−	0.698513i	$-0.246155\pi$
$317$	9666.93	−	3518.47i	1.71277	−	0.623398i	0.997175	+	0.0751153i	$0.0239325\pi$
$318$		0			0
$319$	7.19757	−	6.03948i	0.00126328	−	0.00106002i
$320$		0			0
$321$	−3898.80	−	7726.10i	−0.677912	−	1.34339i
$322$		0			0
$323$	6968.17			1.20037
$324$		0			0
$325$	565.367			0.0964951
$326$		0			0
$327$	1244.28	+	2465.74i	0.210425	+	0.416990i
$328$		0			0
$329$	−9228.23	+	7743.40i	−1.54641	+	1.29759i
$330$		0			0
$331$	807.479	−	293.898i	0.134088	−	0.0488039i	−0.274105	−	0.961700i	$-0.588381\pi$
$331$	807.479	−	293.898i	0.134088	−	0.0488039i	0.408192	+	0.912896i	$0.366159\pi$
$332$		0			0
$333$	−3145.82	−	355.444i	−0.517687	−	0.0584931i
$334$		0			0
$335$	−3283.35	−	2755.06i	−0.535489	−	0.449329i
$336$		0			0
$337$	475.226	+	2695.14i	0.0768166	+	0.435648i	0.998824	+	0.0484770i	$0.0154368\pi$
$337$	475.226	+	2695.14i	0.0768166	+	0.435648i	−0.922008	+	0.387171i	$0.873452\pi$
$338$		0			0
$339$	−1727.72	−	741.331i	−0.276805	−	0.118772i
$340$		0			0
$341$	−12.9543	−	22.4376i	−0.00205723	−	0.00356323i
$342$		0			0
$343$	3098.65	−	5367.02i	0.487788	−	0.844874i
$344$		0			0
$345$	−3528.01	−	3315.71i	−0.550555	−	0.517426i
$346$		0			0
$347$	9139.24	+	3326.41i	1.41389	+	0.514614i	0.932269	−	0.361765i	$-0.117826\pi$
$347$	9139.24	+	3326.41i	1.41389	+	0.514614i	0.481622	+	0.876379i	$0.340048\pi$
$348$		0			0
$349$	−244.747	+	1388.03i	−0.0375387	+	0.212892i	−0.997807	−	0.0661847i	$-0.978917\pi$
$349$	−244.747	+	1388.03i	−0.0375387	+	0.212892i	0.960269	+	0.279077i	$0.0900285\pi$
$350$		0			0
$351$	5.78654	+	1003.78i	0.000879951	+	0.152643i
$352$		0			0
$353$	145.344	−	824.287i	0.0219147	−	0.124284i	−0.971888	−	0.235444i	$-0.924345\pi$
$353$	145.344	−	824.287i	0.0219147	−	0.124284i	0.993803	+	0.111160i	$0.0354566\pi$
$354$		0			0
$355$	853.594	+	310.683i	0.127617	+	0.0464488i
$356$		0			0
$357$	−1197.98	+	5098.33i	−0.177602	+	0.755833i
$358$		0			0
$359$	2517.07	−	4359.69i	0.370044	−	0.640935i	−0.619528	−	0.784975i	$-0.712676\pi$
$359$	2517.07	−	4359.69i	0.370044	−	0.640935i	0.989572	+	0.144040i	$0.0460093\pi$
$360$		0			0
$361$	−5146.58	−	8914.13i	−0.750339	−	1.29963i
$362$		0			0
$363$	5539.47	−	4140.55i	0.800956	−	0.598685i
$364$		0			0
$365$	−1026.22	−	5819.97i	−0.147163	−	0.834606i
$366$		0			0
$367$	4798.61	+	4026.51i	0.682521	+	0.572703i	0.916742	−	0.399480i	$-0.130809\pi$
$367$	4798.61	+	4026.51i	0.682521	+	0.572703i	−0.234221	+	0.972183i	$0.575254\pi$
$368$		0			0
$369$	3886.51	+	1933.21i	0.548303	+	0.272734i
$370$		0			0
$371$	−651.351	+	237.073i	−0.0911495	+	0.0331757i
$372$		0			0
$373$	−9425.34	+	7908.80i	−1.30838	+	1.09786i	−0.319749	+	0.947502i	$0.603598\pi$
$373$	−9425.34	+	7908.80i	−1.30838	+	1.09786i	−0.988632	+	0.150359i	$0.951957\pi$
$374$		0			0
$375$	3938.63	−	6013.54i	0.542374	−	0.828101i
$376$		0			0
$377$	384.183			0.0524840
$378$		0			0
$379$	−4996.58			−0.677195			−0.338598	−	0.940931i	$-0.609953\pi$
$379$	−4996.58			−0.677195			−0.338598	+	0.940931i	$0.609953\pi$
$380$		0			0
$381$	−9831.30	−	553.655i	−1.32198	−	0.0744477i
$382$		0			0
$383$	−10520.9	+	8828.08i	−1.40364	+	1.17779i	−0.444177	+	0.895939i	$0.646504\pi$
$383$	−10520.9	+	8828.08i	−1.40364	+	1.17779i	−0.959458	+	0.281851i	$0.909052\pi$
$384$		0			0
$385$	−21.1216	+	7.68763i	−0.00279599	+	0.00101766i
$386$		0			0
$387$	−1778.12	+	428.647i	−0.233558	+	0.0563032i
$388$		0			0
$389$	−6156.52	−	5165.93i	−0.802437	−	0.673324i	0.146353	−	0.989232i	$-0.453246\pi$
$389$	−6156.52	−	5165.93i	−0.802437	−	0.673324i	−0.948790	+	0.315908i	$0.897691\pi$
$390$		0			0
$391$	−1269.53	−	7199.87i	−0.164202	−	0.931235i
$392$		0			0
$393$	1659.33	+	13963.7i	0.212982	+	1.79230i
$394$		0			0
$395$	1087.68	+	1883.91i	0.138549	+	0.239974i
$396$		0			0
$397$	3689.69	−	6390.72i	0.466448	−	0.807912i	−0.532817	−	0.846230i	$-0.678867\pi$
$397$	3689.69	−	6390.72i	0.466448	−	0.807912i	0.999266	+	0.0383180i	$0.0122000\pi$
$398$		0			0
$399$	12342.6	−	3721.00i	1.54863	−	0.466875i
$400$		0			0
$401$	−7940.72	−	2890.19i	−0.988880	−	0.359923i	−0.203593	−	0.979056i	$-0.565262\pi$
$401$	−7940.72	−	2890.19i	−0.988880	−	0.359923i	−0.785286	+	0.619133i	$0.787484\pi$
$402$		0			0
$403$	183.959	−	1043.29i	0.0227386	−	0.128957i
$404$		0			0
$405$	4150.83	+	2684.57i	0.509275	+	0.329376i
$406$		0			0
$407$	3.56277	−	20.2055i	0.000433907	−	0.00246081i
$408$		0			0
$409$	−12312.8	−	4481.48i	−1.48857	−	0.541796i	−0.535500	−	0.844535i	$-0.679877\pi$
$409$	−12312.8	−	4481.48i	−1.48857	−	0.541796i	−0.953073	+	0.302739i	$0.902099\pi$
$410$		0			0
$411$	−7328.53	+	2209.37i	−0.879537	+	0.265159i
$412$		0			0
$413$	−7103.06	+	12302.9i	−0.846292	+	1.46582i
$414$		0			0
$415$	−3864.37	−	6693.28i	−0.457095	−	0.791711i
$416$		0			0
$417$	1362.98	+	11469.9i	0.160061	+	1.34696i
$418$		0			0
$419$	978.770	+	5550.88i	0.114119	+	0.647204i	0.987182	+	0.159596i	$0.0510192\pi$
$419$	978.770	+	5550.88i	0.114119	+	0.647204i	−0.873063	+	0.487607i	$0.837870\pi$
$420$		0			0
$421$	8502.48	+	7134.42i	0.984288	+	0.825916i	0.984731	−	0.174084i	$-0.0556964\pi$
$421$	8502.48	+	7134.42i	0.984288	+	0.825916i	−0.000442623		1.00000i	$0.500141\pi$
$422$		0			0
$423$	−4861.17	+	16467.5i	−0.558766	+	1.89285i
$424$		0			0
$425$	3950.71	−	1437.94i	0.450912	−	0.164119i
$426$		0			0
$427$	1002.30	−	841.027i	0.113594	−	0.0953165i
$428$		0			0
$429$	−6.49513	−	0.365776i	−0.000730973	−	4.11651e-5i
$430$		0			0
$431$	−3944.54			−0.440839			−0.220420	−	0.975405i	$-0.570743\pi$
$431$	−3944.54			−0.440839			−0.220420	+	0.975405i	$0.570743\pi$
$432$		0			0
$433$	1909.68			0.211948			0.105974	−	0.994369i	$-0.466204\pi$
$433$	1909.68			0.211948			0.105974	+	0.994369i	$0.466204\pi$
$434$		0			0
$435$	1036.61	−	1582.70i	0.114256	−	0.174448i
$436$		0			0
$437$	−13785.7	+	11567.5i	−1.50906	+	1.26625i
$438$		0			0
$439$	7600.00	−	2766.17i	0.826260	−	0.300734i	0.105937	−	0.994373i	$-0.466216\pi$
$439$	7600.00	−	2766.17i	0.826260	−	0.300734i	0.720323	+	0.693639i	$0.243994\pi$
$440$		0			0
$441$	26.5266	+	427.157i	0.00286434	+	0.0461243i
$442$		0			0
$443$	−4879.88	−	4094.71i	−0.523364	−	0.439155i	0.342439	−	0.939540i	$-0.388747\pi$
$443$	−4879.88	−	4094.71i	−0.523364	−	0.439155i	−0.865803	+	0.500386i	$0.833192\pi$
$444$		0			0
$445$	−611.133	−	3465.91i	−0.0651022	−	0.369213i
$446$		0			0
$447$	9277.64	−	6934.70i	0.981694	−	0.733780i
$448$		0			0
$449$	6820.97	+	11814.3i	0.716929	+	1.24176i	0.962211	+	0.272306i	$0.0877864\pi$
$449$	6820.97	+	11814.3i	0.716929	+	1.24176i	−0.245281	+	0.969452i	$0.578880\pi$
$450$		0			0
$451$	−14.0659	+	24.3628i	−0.00146859	+	0.00254368i
$452$		0			0
$453$	2564.41	−	10913.5i	0.265975	−	1.13192i
$454$		0			0
$455$	−863.641	−	314.340i	−0.0889849	−	0.0323878i
$456$		0			0
$457$	−2344.18	+	13294.5i	−0.239947	+	1.36081i	0.591993	+	0.805943i	$0.298341\pi$
$457$	−2344.18	+	13294.5i	−0.239947	+	1.36081i	−0.831940	+	0.554865i	$0.812770\pi$
$458$		0			0
$459$	2593.43	+	6999.57i	0.263727	+	0.711790i
$460$		0			0
$461$	2081.24	−	11803.3i	0.210267	−	1.19248i	−0.678666	−	0.734447i	$-0.737442\pi$
$461$	2081.24	−	11803.3i	0.210267	−	1.19248i	0.888933	−	0.458037i	$-0.151447\pi$
$462$		0			0
$463$	−6171.60	−	2246.28i	−0.619479	−	0.225472i	0.0131668	−	0.999913i	$-0.495809\pi$
$463$	−6171.60	−	2246.28i	−0.619479	−	0.225472i	−0.632646	+	0.774442i	$0.718031\pi$
$464$		0			0
$465$	−3801.61	−	3572.85i	−0.379130	−	0.356316i
$466$		0			0
$467$	2152.08	−	3727.52i	0.213247	−	0.369355i	−0.739482	−	0.673177i	$-0.764929\pi$
$467$	2152.08	−	3727.52i	0.213247	−	0.369355i	0.952729	+	0.303822i	$0.0982627\pi$
$468$		0			0
$469$	−5986.88	−	10369.6i	−0.589442	−	1.02094i
$470$		0			0
$471$	15082.1	+	6471.45i	1.47547	+	0.633097i
$472$		0			0
$473$	−2.05839	−	11.6737i	−0.000200095	−	0.00113479i
$474$		0			0
$475$	−7927.63	−	6652.07i	−0.765778	−	0.642564i
$476$		0			0
$477$	−586.915	+	794.721i	−0.0563375	+	0.0762847i
$478$		0			0
$479$	−14016.5	+	5101.57i	−1.33701	+	0.486632i	−0.908870	−	0.417080i	$-0.863053\pi$
$479$	−14016.5	+	5101.57i	−1.33701	+	0.486632i	−0.428141	+	0.903712i	$0.640831\pi$
$480$		0			0
$481$	642.656	−	539.252i	0.0609201	−	0.0511181i
$482$		0			0
$483$	−6093.44	−	12075.1i	−0.574040	−	1.13755i
$484$		0			0
$485$	10213.3			0.956214
$486$		0			0
$487$	21294.9			1.98144			0.990722	−	0.135901i	$-0.0433931\pi$
$487$	21294.9			1.98144			0.990722	+	0.135901i	$0.0433931\pi$
$488$		0			0
$489$	−4493.71	−	8905.01i	−0.415568	−	0.823514i
$490$		0			0
$491$	−8618.26	+	7231.58i	−0.792132	+	0.664677i	−0.946272	−	0.323372i	$-0.895184\pi$
$491$	−8618.26	+	7231.58i	−0.792132	+	0.664677i	0.154140	+	0.988049i	$0.450739\pi$
$492$		0			0
$493$	2684.63	−	977.124i	0.245252	−	0.0892646i
$494$		0			0
$495$	−19.0321	+	25.7707i	−0.00172814	+	0.00234001i
$496$		0			0
$497$	1943.96	+	1631.17i	0.175449	+	0.147220i
$498$		0			0
$499$	137.225	+	778.243i	0.0123107	+	0.0698175i	0.990344	−	0.138630i	$-0.0442700\pi$
$499$	137.225	+	778.243i	0.0123107	+	0.0698175i	−0.978033	+	0.208448i	$0.933159\pi$
$500$		0			0
$501$	−3315.19	−	1422.48i	−0.295632	−	0.126850i
$502$		0			0
$503$	−5970.64	−	10341.5i	−0.529260	−	0.916705i	−0.999418	−	0.0341229i	$-0.989136\pi$
$503$	−5970.64	−	10341.5i	−0.529260	−	0.916705i	0.470158	−	0.882583i	$-0.344197\pi$
$504$		0			0
$505$	3451.55	−	5978.26i	0.304143	−	0.526791i
$506$		0			0
$507$	8124.87	+	7635.96i	0.711712	+	0.668885i
$508$		0			0
$509$	10169.7	+	3701.49i	0.885592	+	0.322329i	0.744464	−	0.667662i	$-0.232705\pi$
$509$	10169.7	+	3701.49i	0.885592	+	0.322329i	0.141128	+	0.989991i	$0.454927\pi$
$510$		0			0
$511$	2866.86	−	16258.8i	0.248185	−	1.40752i
$512$		0			0
$513$	11729.3	−	14143.2i	1.00947	−	1.21723i
$514$		0			0
$515$	−1151.82	+	6532.30i	−0.0985539	+	0.558927i
$516$		0			0
$517$	−104.565	−	38.0585i	−0.00889509	−	0.00323755i
$518$		0			0
$519$	−3157.63	+	13438.1i	−0.267061	+	1.13655i
$520$		0			0
$521$	6195.35	−	10730.7i	0.520966	−	0.902339i	−0.478737	−	0.877958i	$-0.658905\pi$
$521$	6195.35	−	10730.7i	0.520966	−	0.902339i	0.999703	−	0.0243806i	$-0.00776137\pi$
$522$		0			0
$523$	−4693.00	−	8128.52i	−0.392372	−	0.679609i	0.600390	−	0.799708i	$-0.295012\pi$
$523$	−4693.00	−	8128.52i	−0.392372	−	0.679609i	−0.992762	+	0.120099i	$0.961679\pi$
$524$		0			0
$525$	6230.00	−	4656.69i	0.517904	−	0.387114i
$526$		0			0
$527$	−1367.98	−	7758.23i	−0.113075	−	0.641278i
$528$		0			0
$529$	5143.32	+	4315.76i	0.422727	+	0.354710i
$530$		0			0
$531$	1254.99	+	20209.1i	0.102565	+	1.65160i
$532$		0			0
$533$	−1080.91	+	393.418i	−0.0878411	+	0.0319715i
$534$		0			0
$535$	8651.35	−	7259.34i	0.699122	−	0.586633i
$536$		0			0
$537$	3155.91	−	4818.47i	0.253608	−	0.387211i
$538$		0			0
$539$	−2.77366			−0.000221651
$540$		0			0
$541$	1596.07			0.126840			0.0634202	−	0.997987i	$-0.479799\pi$
$541$	1596.07			0.126840			0.0634202	+	0.997987i	$0.479799\pi$
$542$		0			0
$543$	10583.1	+	595.995i	0.836402	+	0.0471024i
$544$		0			0
$545$	−2761.03	+	2316.78i	−0.217008	+	0.182091i
$546$		0			0
$547$	−9720.40	+	3537.94i	−0.759807	+	0.276547i	−0.692726	−	0.721201i	$-0.743591\pi$
$547$	−9720.40	+	3537.94i	−0.759807	+	0.276547i	−0.0670804	+	0.997748i	$0.521368\pi$
$548$		0			0
$549$	527.982	−	1788.57i	0.0410450	−	0.139043i
$550$		0			0
$551$	−5387.06	−	4520.28i	−0.416509	−	0.349493i
$552$		0			0
$553$	1055.28	+	5984.77i	0.0811481	+	0.460214i
$554$		0			0
$555$	−487.510	−	4102.53i	−0.0372858	−	0.313771i
$556$		0			0
$557$	4716.05	+	8168.44i	0.358753	+	0.621379i	0.987753	−	0.156027i	$-0.0498687\pi$
$557$	4716.05	+	8168.44i	0.358753	+	0.621379i	−0.629000	+	0.777406i	$0.716535\pi$
$558$		0			0
$559$	242.345	−	419.753i	0.0183365	−	0.0317597i
$560$		0			0
$561$	−46.3174	+	13.9636i	−0.00348578	+	0.00105088i
$562$		0			0
$563$	20619.8	+	7504.98i	1.54355	+	0.561807i	0.966894	−	0.255179i	$-0.0821343\pi$
$563$	20619.8	+	7504.98i	1.54355	+	0.561807i	0.576658	+	0.816986i	$0.304356\pi$
$564$		0			0
$565$	426.038	−	2416.18i	0.0317231	−	0.179911i
$566$		0			0
$567$	8331.49	+	11013.4i	0.617089	+	0.815729i
$568$		0			0
$569$	1838.39	−	10426.1i	0.135447	−	0.768160i	−0.839100	−	0.543977i	$-0.816918\pi$
$569$	1838.39	−	10426.1i	0.135447	−	0.768160i	0.974547	−	0.224182i	$-0.0719711\pi$
$570$		0			0
$571$	−2257.85	−	821.791i	−0.165478	−	0.0602292i	0.257953	−	0.966158i	$-0.416952\pi$
$571$	−2257.85	−	821.791i	−0.165478	−	0.0602292i	−0.423431	+	0.905928i	$0.639174\pi$
$572$		0			0
$573$	5370.58	−	1619.10i	0.391552	−	0.118043i
$574$		0			0
$575$	−5428.93	+	9403.18i	−0.393743	+	0.681982i
$576$		0			0
$577$	7640.34	+	13233.5i	0.551251	+	0.954794i	0.998185	+	0.0602273i	$0.0191825\pi$
$577$	7640.34	+	13233.5i	0.551251	+	0.954794i	−0.446934	+	0.894567i	$0.647484\pi$
$578$		0			0
$579$	−151.342	−	1273.59i	−0.0108628	−	0.0914137i
$580$		0			0
$581$	−3749.26	−	21263.1i	−0.267720	−	1.51832i
$582$		0			0
$583$	−4.90478	−	4.11560i	−0.000348431	−	0.000292368i
$584$		0			0
$585$	−1273.47	+	306.991i	−0.0900024	+	0.0216966i
$586$		0			0
$587$	−12260.7	+	4462.53i	−0.862101	+	0.313779i	−0.734964	−	0.678106i	$-0.762801\pi$
$587$	−12260.7	+	4462.53i	−0.862101	+	0.313779i	−0.127137	+	0.991885i	$0.540579\pi$
$588$		0			0
$589$	−14854.7	+	12464.6i	−1.03918	+	0.871978i
$590$		0			0
$591$	−22355.8	−	1258.98i	−1.55600	−	0.0876268i
$592$		0			0
$593$	16271.5			1.12680			0.563400	−	0.826185i	$-0.309493\pi$
$593$	16271.5			1.12680			0.563400	+	0.826185i	$0.309493\pi$
$594$		0			0
$595$	−6834.50			−0.470903
$596$		0			0
$597$	−7284.87	+	11122.6i	−0.499414	+	0.762509i
$598$		0			0
$599$	−12021.9	+	10087.6i	−0.820038	+	0.688094i	−0.952981	−	0.303030i	$-0.902002\pi$
$599$	−12021.9	+	10087.6i	−0.820038	+	0.688094i	0.132943	+	0.991124i	$0.457557\pi$
$600$		0			0
$601$	9308.20	−	3387.91i	0.631763	−	0.229943i	−0.00623571	−	0.999981i	$-0.501985\pi$
$601$	9308.20	−	3387.91i	0.631763	−	0.229943i	0.637998	+	0.770038i	$0.279763\pi$
$602$		0			0
$603$	−15280.3	−	7600.64i	−1.03194	−	0.513303i
$604$		0			0
$605$	6913.73	+	5801.31i	0.464600	+	0.389846i
$606$		0			0
$607$	−2791.61	−	15832.0i	−0.186669	−	1.05865i	−0.923792	−	0.382894i	$-0.874927\pi$
$607$	−2791.61	−	15832.0i	−0.186669	−	1.05865i	0.737123	−	0.675758i	$-0.236184\pi$
$608$		0			0
$609$	4233.47	−	3164.36i	0.281689	−	0.210552i
$610$		0			0
$611$	−2274.98	−	3940.38i	−0.150631	−	0.260901i
$612$		0			0
$613$	−2802.75	+	4854.51i	−0.184669	+	0.319856i	−0.943465	−	0.331472i	$-0.892455\pi$
$613$	−2802.75	+	4854.51i	−0.184669	+	0.319856i	0.758796	+	0.651328i	$0.225788\pi$
$614$		0			0
$615$	−1295.77	+	5514.48i	−0.0849602	+	0.361570i
$616$		0			0
$617$	3393.93	+	1235.29i	0.221450	+	0.0806012i	0.450362	−	0.892846i	$-0.351295\pi$
$617$	3393.93	+	1235.29i	0.221450	+	0.0806012i	−0.228913	+	0.973447i	$0.573517\pi$
$618$		0			0
$619$	−3759.85	+	21323.2i	−0.244138	+	1.38457i	0.578349	+	0.815789i	$0.303697\pi$
$619$	−3759.85	+	21323.2i	−0.244138	+	1.38457i	−0.822487	+	0.568784i	$0.807414\pi$
$620$		0			0
$621$	−16750.4	−	9542.55i	−1.08240	−	0.616633i
$622$		0			0
$623$	1707.27	−	9682.42i	0.109792	−	0.622661i
$624$		0			0
$625$	−466.380	−	169.748i	−0.0298483	−	0.0108639i
$626$		0			0
$627$	86.7717	+	81.5503i	0.00552684	+	0.00519426i
$628$		0			0
$629$	3119.28	−	5402.74i	0.197732	−	0.342482i
$630$		0			0
$631$	4449.63	+	7706.98i	0.280724	+	0.486228i	0.971563	−	0.236780i	$-0.0760921\pi$
$631$	4449.63	+	7706.98i	0.280724	+	0.486228i	−0.690839	+	0.723008i	$0.742759\pi$
$632$		0			0
$633$	−8298.55	−	3560.75i	−0.521070	−	0.223581i
$634$		0			0
$635$	−2231.40	−	12654.9i	−0.139449	−	0.790857i
$636$		0			0
$637$	−86.8787	−	72.8999i	−0.00540386	−	0.00453438i
$638$		0			0
$639$	3594.05	+	406.090i	0.222502	+	0.0251403i
$640$		0			0
$641$	20334.9	−	7401.30i	1.25301	−	0.456059i	0.371593	−	0.928396i	$-0.378812\pi$
$641$	20334.9	−	7401.30i	1.25301	−	0.456059i	0.881418	+	0.472337i	$0.156589\pi$
$642$		0			0
$643$	−6779.27	+	5688.49i	−0.415783	+	0.348883i	−0.826556	−	0.562854i	$-0.809703\pi$
$643$	−6779.27	+	5688.49i	−0.415783	+	0.348883i	0.410773	+	0.911738i	$0.365259\pi$
$644$		0			0
$645$	−1075.34	−	2130.96i	−0.0656457	−	0.130087i
$646$		0			0
$647$	−6445.91			−0.391677			−0.195838	−	0.980636i	$-0.562743\pi$
$647$	−6445.91			−0.391677			−0.195838	+	0.980636i	$0.562743\pi$
$648$		0			0
$649$	−131.223			−0.00793678
$650$		0			0
$651$	−6566.00	−	13011.6i	−0.395302	−	0.783354i
$652$		0			0
$653$	2338.71	−	1962.41i	0.140154	−	0.117603i	−0.570015	−	0.821634i	$-0.693063\pi$
$653$	2338.71	−	1962.41i	0.140154	−	0.117603i	0.710170	+	0.704031i	$0.248618\pi$
$654$		0			0
$655$	−17244.0	+	6276.32i	−1.02867	+	0.374406i
$656$		0			0
$657$	−9403.17	−	21570.7i	−0.558375	−	1.28090i
$658$		0			0
$659$	11991.4	+	10062.0i	0.708828	+	0.594777i	0.924270	−	0.381739i	$-0.124675\pi$
$659$	11991.4	+	10062.0i	0.708828	+	0.594777i	−0.215442	+	0.976517i	$0.569119\pi$
$660$		0			0
$661$	5685.36	+	32243.3i	0.334546	+	1.89730i	0.431671	+	0.902031i	$0.357924\pi$
$661$	5685.36	+	32243.3i	0.334546	+	1.89730i	−0.0971257	+	0.995272i	$0.530965\pi$
$662$		0			0
$663$	−1817.79	−	779.981i	−0.106482	−	0.0456892i
$664$		0			0
$665$	8411.56	+	14569.3i	0.490506	+	0.849581i
$666$		0			0
$667$	−3689.12	+	6389.74i	−0.214158	+	0.370932i
$668$		0			0
$669$	−14768.8	−	13880.1i	−0.853505	−	0.802146i
$670$		0			0
$671$	11.3570	+	4.13362i	0.000653402	+	0.000237819i
$672$		0			0
$673$	−959.177	+	5439.76i	−0.0549384	+	0.311571i	−0.999877	−	0.0156762i	$-0.995010\pi$
$673$	−959.177	+	5439.76i	−0.0549384	+	0.311571i	0.944939	+	0.327247i	$0.106121\pi$
$674$		0			0
$675$	3731.53	−	10439.1i	0.212780	−	0.595263i
$676$		0			0
$677$	−621.476	+	3524.57i	−0.0352810	+	0.200089i	−0.997353	−	0.0727055i	$-0.976837\pi$
$677$	−621.476	+	3524.57i	−0.0352810	+	0.200089i	0.962072	+	0.272794i	$0.0879478\pi$
$678$		0			0
$679$	26811.4	+	9758.57i	1.51536	+	0.551546i
$680$		0			0
$681$	−3456.44	+	14709.8i	−0.194495	+	0.827723i
$682$		0			0
$683$	813.215	−	1408.53i	0.0455590	−	0.0789105i	−0.842347	−	0.538936i	$-0.818826\pi$
$683$	813.215	−	1408.53i	0.0455590	−	0.0789105i	0.887906	+	0.460026i	$0.152160\pi$
$684$		0			0
$685$	−4994.42	−	8650.59i	−0.278580	−	0.482514i
$686$		0			0
$687$	−20683.1	+	15459.9i	−1.14863	+	0.858561i
$688$		0			0
$689$	−45.4614	−	257.824i	−0.00251370	−	0.0142559i
$690$		0			0
$691$	−771.115	−	647.042i	−0.0424524	−	0.0356218i	0.621315	−	0.783561i	$-0.286599\pi$
$691$	−771.115	−	647.042i	−0.0424524	−	0.0356218i	−0.663767	+	0.747939i	$0.731043\pi$
$692$		0			0
$693$	−74.5851	+	49.4671i	−0.00408839	+	0.00271154i
$694$		0			0
$695$	−14164.4	+	5155.41i	−0.773073	+	0.281376i
$696$		0			0
$697$	−6552.63	+	5498.31i	−0.356096	+	0.298800i
$698$		0			0
$699$	1449.20	−	2212.65i	0.0784174	−	0.119728i
$700$		0			0
$701$	−30593.6			−1.64837			−0.824183	−	0.566324i	$-0.808365\pi$
$701$	−30593.6			−1.64837			−0.824183	+	0.566324i	$0.808365\pi$
$702$		0			0
$703$	−15356.2			−0.823855
$704$		0			0
$705$	−22371.3	−	1259.85i	−1.19511	−	0.0673032i
$706$		0			0
$707$	14772.9	−	12395.9i	0.785843	−	0.659401i
$708$		0			0
$709$	15119.8	−	5503.15i	0.800896	−	0.291502i	0.0910385	−	0.995847i	$-0.470981\pi$
$709$	15119.8	−	5503.15i	0.800896	−	0.291502i	0.709858	+	0.704345i	$0.248759\pi$
$710$		0			0
$711$	5968.18	+	6277.40i	0.314802	+	0.331112i
$712$		0			0
$713$	15585.5	+	13077.8i	0.818626	+	0.686909i
$714$		0			0
$715$	−1.47419	−	8.36055i	−7.71072e−5	−	0.000437296i
$716$		0			0
$717$	2717.55	+	22869.0i	0.141547	+	1.19115i
$718$		0			0
$719$	−13641.9	−	23628.5i	−0.707591	−	1.22558i	−0.965748	−	0.259481i	$-0.916449\pi$
$719$	−13641.9	−	23628.5i	−0.707591	−	1.22558i	0.258157	−	0.966103i	$-0.416885\pi$
$720$		0			0
$721$	−9265.12	+	16047.7i	−0.478573	+	0.828913i
$722$		0			0
$723$	2596.22	−	782.697i	0.133547	−	0.0402611i
$724$		0			0
$725$	−3987.08	−	1451.18i	−0.204243	−	0.0743385i
$726$		0			0
$727$	3276.60	−	18582.5i	0.167156	−	0.947989i	−0.779657	−	0.626206i	$-0.784607\pi$
$727$	3276.60	−	18582.5i	0.167156	−	0.947989i	0.946814	−	0.321783i	$-0.104282\pi$
$728$		0			0
$729$	18572.4	+	6518.29i	0.943573	+	0.331164i
$730$		0			0
$731$	625.883	−	3549.56i	0.0316677	−	0.179597i
$732$		0			0
$733$	−7560.06	−	2751.64i	−0.380951	−	0.138655i	0.144445	−	0.989513i	$-0.453860\pi$
$733$	−7560.06	−	2751.64i	−0.380951	−	0.138655i	−0.525396	+	0.850858i	$0.676083\pi$
$734$		0			0
$735$	−534.739	+	161.211i	−0.0268356	+	0.00809026i
$736$		0			0
$737$	55.3015	−	95.7850i	0.00276398	−	0.00478736i
$738$		0			0
$739$	13107.9	+	22703.6i	0.652480	+	1.13013i	0.982519	+	0.186161i	$0.0596045\pi$
$739$	13107.9	+	22703.6i	0.652480	+	1.13013i	−0.330040	+	0.943967i	$0.607062\pi$
$740$		0			0
$741$	574.550	+	4835.00i	0.0284840	+	0.239700i
$742$		0			0
$743$	4176.05	+	23683.6i	0.206197	+	1.16940i	0.895545	+	0.444971i	$0.146786\pi$
$743$	4176.05	+	23683.6i	0.206197	+	1.16940i	−0.689348	+	0.724431i	$0.742103\pi$
$744$		0			0
$745$	11579.3	+	9716.17i	0.569439	+	0.477816i
$746$		0			0
$747$	−21204.2	−	22302.8i	−1.03858	−	1.09239i
$748$		0			0
$749$	29647.1	−	10790.7i	1.44630	−	0.526412i
$750$		0			0
$751$	29364.6	−	24639.8i	1.42680	−	1.19723i	0.479228	−	0.877690i	$-0.340917\pi$
$751$	29364.6	−	24639.8i	1.42680	−	1.19723i	0.947573	−	0.319538i	$-0.103528\pi$
$752$		0			0
$753$	33747.3	+	1900.50i	1.63323	+	0.0919760i
$754$		0			0
$755$	14630.0			0.705217
$756$		0			0
$757$	−13077.8			−0.627900			−0.313950	−	0.949440i	$-0.601652\pi$
$757$	−13077.8			−0.627900			−0.313950	+	0.949440i	$0.601652\pi$
$758$		0			0
$759$	68.4530	−	104.515i	0.00327363	−	0.00499821i
$760$		0			0
$761$	23600.6	−	19803.3i	1.12421	−	0.943322i	0.125398	−	0.992106i	$-0.459979\pi$
$761$	23600.6	−	19803.3i	1.12421	−	0.943322i	0.998809	+	0.0487844i	$0.0155347\pi$
$762$		0			0
$763$	−9461.70	+	3443.78i	−0.448934	+	0.163399i
$764$		0			0
$765$	−8118.04	+	5384.12i	−0.383671	+	0.254462i
$766$		0			0
$767$	−4110.29	−	3448.94i	−0.193499	−	0.162365i
$768$		0			0
$769$	−2655.63	−	15060.8i	−0.124531	−	0.706251i	−0.981585	−	0.191024i	$-0.938819\pi$
$769$	−2655.63	−	15060.8i	−0.124531	−	0.706251i	0.857054	−	0.515226i	$-0.172292\pi$
$770$		0			0
$771$	23822.5	−	17806.5i	1.11277	−	0.831756i
$772$		0			0
$773$	−2757.85	−	4776.74i	−0.128322	−	0.222260i	0.794705	−	0.606996i	$-0.207626\pi$
$773$	−2757.85	−	4776.74i	−0.128322	−	0.222260i	−0.923027	+	0.384736i	$0.874292\pi$
$774$		0			0
$775$	−5849.95	+	10132.4i	−0.271144	+	0.469635i
$776$		0			0
$777$	2640.08	−	11235.5i	0.121895	−	0.518754i
$778$		0			0
$779$	19785.5	+	7201.35i	0.910001	+	0.331213i
$780$		0			0
$781$	−4.07042	+	23.0845i	−0.000186493	+	0.00105765i
$782$		0			0
$783$	2535.68	−	7093.71i	0.115732	−	0.323766i
$784$		0			0
$785$	−3719.09	+	21092.0i	−0.169096	+	0.958989i
$786$		0			0
$787$	7795.37	+	2837.28i	0.353081	+	0.128511i	0.512471	−	0.858705i	$-0.328730\pi$
$787$	7795.37	+	2837.28i	0.353081	+	0.128511i	−0.159389	+	0.987216i	$0.550953\pi$
$788$		0			0
$789$	−9936.85	−	9338.91i	−0.448367	−	0.421387i
$790$		0			0
$791$	3427.00	−	5935.74i	0.154046	−	0.266815i
$792$		0			0
$793$	247.090	+	427.973i	0.0110649	+	0.0191649i
$794$		0			0
$795$	−1184.81	−	508.379i	−0.0528564	−	0.0226797i
$796$		0			0
$797$	−6858.09	−	38894.2i	−0.304801	−	1.72861i	−0.624442	−	0.781071i	$-0.714673\pi$
$797$	−6858.09	−	38894.2i	−0.304801	−	1.72861i	0.319641	−	0.947539i	$-0.396438\pi$
$798$		0			0
$799$	−25919.1	−	21748.7i	−1.14763	−	0.962972i
$800$		0			0
$801$	−5599.78	−	12845.8i	−0.247014	−	0.566646i
$802$		0			0
$803$	143.304	−	52.1584i	0.00629774	−	0.00229219i
$804$		0			0
$805$	13521.2	−	11345.6i	0.592000	−	0.496747i
$806$		0			0
$807$	6724.84	+	13326.4i	0.293340	+	0.581301i
$808$		0			0
$809$	−7558.62			−0.328488			−0.164244	−	0.986420i	$-0.552519\pi$
$809$	−7558.62			−0.328488			−0.164244	+	0.986420i	$0.552519\pi$
$810$		0			0
$811$	12908.2			0.558900			0.279450	−	0.960160i	$-0.409848\pi$
$811$	12908.2			0.558900			0.279450	+	0.960160i	$0.409848\pi$
$812$		0			0
$813$	8664.91	+	17170.9i	0.373790	+	0.740725i
$814$		0			0
$815$	9971.44	−	8367.03i	0.428570	−	0.359613i
$816$		0			0
$817$	−8336.98	+	3034.41i	−0.357006	+	0.129940i
$818$		0			0
$819$	−3636.35	−	410.869i	−0.155146	−	0.0175298i
$820$		0			0
$821$	−9881.42	−	8291.49i	−0.420053	−	0.352467i	0.408130	−	0.912924i	$-0.366181\pi$
$821$	−9881.42	−	8291.49i	−0.420053	−	0.352467i	−0.828183	+	0.560457i	$0.810625\pi$
$822$		0			0
$823$	−809.125	−	4588.78i	−0.0342701	−	0.194356i	0.962866	−	0.269978i	$-0.0870166\pi$
$823$	−809.125	−	4588.78i	−0.0342701	−	0.194356i	−0.997136	+	0.0756229i	$0.975905\pi$
$824$		0			0
$825$	66.0252	+	28.3301i	0.00278630	+	0.00119555i
$826$		0			0
$827$	−10441.8	−	18085.7i	−0.439053	−	0.760463i	0.558563	−	0.829462i	$-0.311353\pi$
$827$	−10441.8	−	18085.7i	−0.439053	−	0.760463i	−0.997617	+	0.0689990i	$0.978019\pi$
$828$		0			0
$829$	2621.32	−	4540.26i	0.109822	−	0.190217i	−0.805876	−	0.592084i	$-0.798305\pi$
$829$	2621.32	−	4540.26i	0.109822	−	0.190217i	0.915698	+	0.401867i	$0.131639\pi$
$830$		0			0
$831$	4888.23	+	4594.08i	0.204056	+	0.191777i
$832$		0			0
$833$	−792.510	−	288.450i	−0.0329638	−	0.0119978i
$834$		0			0
$835$	817.491	−	4636.22i	0.0338808	−	0.192147i
$836$		0			0
$837$	−18049.4	−	10282.6i	−0.745377	−	0.424633i
$838$		0			0
$839$	−2975.93	+	16877.3i	−0.122456	+	0.694482i	0.860330	+	0.509737i	$0.170257\pi$
$839$	−2975.93	+	16877.3i	−0.122456	+	0.694482i	−0.982786	+	0.184746i	$0.940854\pi$
$840$		0			0
$841$	20208.8	+	7355.41i	0.828604	+	0.301587i
$842$		0			0
$843$	4703.29	−	20016.1i	0.192159	−	0.817781i
$844$		0			0
$845$	−7275.31	+	12601.2i	−0.296187	+	0.513011i
$846$		0			0
$847$	12606.5	+	21835.1i	0.511411	+	0.885790i
$848$		0			0
$849$	4193.13	−	3134.21i	0.169503	−	0.126697i
$850$		0			0
$851$	2797.75	+	15866.8i	0.112698	+	0.639139i
$852$		0			0
$853$	−21828.5	−	18316.3i	−0.876196	−	0.735215i	0.0891976	−	0.996014i	$-0.471570\pi$
$853$	−21828.5	−	18316.3i	−0.876196	−	0.735215i	−0.965393	+	0.260798i	$0.916014\pi$
$854$		0			0
$855$	21468.7	+	10678.9i	0.858732	+	0.427147i
$856$		0			0
$857$	9601.95	−	3494.82i	0.382726	−	0.139301i	−0.143490	−	0.989652i	$-0.545833\pi$
$857$	9601.95	−	3494.82i	0.382726	−	0.139301i	0.526216	+	0.850351i	$0.323610\pi$
$858$		0			0
$859$	18718.8	−	15707.0i	0.743514	−	0.623883i	−0.190265	−	0.981733i	$-0.560935\pi$
$859$	18718.8	−	15707.0i	0.743514	−	0.623883i	0.933779	+	0.357850i	$0.116490\pi$
$860$		0			0
$861$	−8670.52	+	13238.2i	−0.343195	+	0.523992i
$862$		0			0
$863$	−14442.6			−0.569679			−0.284840	−	0.958575i	$-0.591940\pi$
$863$	−14442.6			−0.569679			−0.284840	+	0.958575i	$0.591940\pi$
$864$		0			0
$865$	−18014.3			−0.708097
$866$		0			0
$867$	10802.0	+	608.320i	0.423132	+	0.0238289i
$868$		0			0
$869$	−43.0018	+	36.0828i	−0.00167864	+	0.00140854i
$870$		0			0
$871$	4249.71	−	1546.77i	0.165323	−	0.0601725i
$872$		0			0
$873$	39534.4	−	9530.44i	1.53269	−	0.369480i
$874$		0			0
$875$	20075.7	+	16845.5i	0.775639	+	0.650838i
$876$		0			0
$877$	1130.95	+	6413.93i	0.0435455	+	0.246959i	0.998809	−	0.0487984i	$-0.0155392\pi$
$877$	1130.95	+	6413.93i	0.0435455	+	0.246959i	−0.955263	+	0.295757i	$0.904428\pi$
$878$		0			0
$879$	4753.02	+	39997.9i	0.182384	+	1.53481i
$880$		0			0
$881$	−4888.97	−	8467.95i	−0.186962	−	0.323828i	0.757274	−	0.653098i	$-0.226531\pi$
$881$	−4888.97	−	8467.95i	−0.186962	−	0.323828i	−0.944236	+	0.329270i	$0.893198\pi$
$882$		0			0
$883$	−6874.70	+	11907.3i	−0.262007	+	0.453809i	−0.966775	−	0.255628i	$-0.917718\pi$
$883$	−6874.70	+	11907.3i	−0.262007	+	0.453809i	0.704768	+	0.709438i	$0.251051\pi$
$884$		0			0
$885$	−25298.8	+	7626.97i	−0.960916	+	0.289692i
$886$		0			0
$887$	−2269.36	−	825.981i	−0.0859051	−	0.0312669i	0.298710	−	0.954344i	$-0.403444\pi$
$887$	−2269.36	−	825.981i	−0.0859051	−	0.0312669i	−0.384615	+	0.923077i	$0.625666\pi$
$888$		0			0
$889$	6233.67	−	35352.9i	0.235175	−	1.33374i
$890$		0			0
$891$	−49.6229	+	117.514i	−0.00186580	+	0.00441849i
$892$		0			0
$893$	−14462.3	+	82019.6i	−0.541950	+	3.07355i
$894$		0			0
$895$	7063.43	+	2570.88i	0.263804	+	0.0960168i
$896$		0			0
$897$	4891.09	−	1474.54i	0.182061	−	0.0548869i
$898$		0			0
$899$	−3975.21	+	6885.27i	−0.147476	+	0.255436i
$900$		0			0
$901$	−973.423	−	1686.02i	−0.0359927	−	0.0623412i
$902$		0			0
$903$	−786.845	−	6621.52i	−0.0289973	−	0.244020i
$904$		0			0
$905$	2402.05	+	13622.7i	0.0882284	+	0.500368i
$906$		0			0
$907$	−18721.1	−	15708.9i	−0.685362	−	0.575087i	0.232206	−	0.972667i	$-0.425406\pi$
$907$	−18721.1	−	15708.9i	−0.685362	−	0.575087i	−0.917568	+	0.397580i	$0.869850\pi$
$908$		0			0
$909$	7781.93	−	26361.8i	0.283950	−	0.961898i
$910$		0			0
$911$	−47391.2	+	17249.0i	−1.72353	+	0.627315i	−0.998137	−	0.0610158i	$-0.980566\pi$
$911$	−47391.2	+	17249.0i	−1.72353	+	0.627315i	−0.725397	+	0.688331i	$0.758344\pi$
$912$		0			0
$913$	152.780	−	128.197i	0.00553808	−	0.00464700i
$914$		0			0
$915$	2429.80	+	136.835i	0.0877886	+	0.00494386i
$916$		0			0
$917$	−51264.9			−1.84615
$918$		0			0
$919$	−360.717			−0.0129477			−0.00647386	−	0.999979i	$-0.502061\pi$
$919$	−360.717			−0.0129477			−0.00647386	+	0.999979i	$0.502061\pi$
$920$		0			0
$921$	−25613.5	+	39107.0i	−0.916390	+	1.39915i
$922$		0			0
$923$	−734.225	+	616.088i	−0.0261834	+	0.0219705i
$924$		0			0
$925$	−8706.44	+	3168.88i	−0.309477	+	0.112640i
$926$		0			0
$927$	1636.99	+	26360.4i	0.0579999	+	0.933969i
$928$		0			0
$929$	−40315.5	−	33828.7i	−1.42380	−	1.19471i	−0.949268	−	0.314469i	$-0.898174\pi$
$929$	−40315.5	−	33828.7i	−1.42380	−	1.19471i	−0.474530	−	0.880239i	$-0.657382\pi$
$930$		0			0
$931$	360.487	+	2044.42i	0.0126901	+	0.0719691i
$932$		0			0
$933$	22373.5	−	16723.4i	0.785076	−	0.586816i
$934$		0			0
$935$	−31.5655	−	54.6731i	−0.00110407	−	0.00191230i
$936$		0			0
$937$	8458.48	−	14650.5i	0.294905	−	0.510791i	−0.680057	−	0.733159i	$-0.738045\pi$
$937$	8458.48	−	14650.5i	0.294905	−	0.510791i	0.974963	+	0.222368i	$0.0713785\pi$
$938$		0			0
$939$	−4337.83	+	18460.8i	−0.150756	+	0.641580i
$940$		0			0
$941$	−12795.9	−	4657.31i	−0.443287	−	0.161343i	0.110727	−	0.993851i	$-0.464682\pi$
$941$	−12795.9	−	4657.31i	−0.443287	−	0.161343i	−0.554013	+	0.832508i	$0.686904\pi$
$942$		0			0
$943$	3836.07	−	21755.5i	0.132471	−	0.751278i
$944$		0			0
$945$	−11504.3	+	13871.9i	−0.396015	+	0.477516i
$946$		0			0
$947$	2895.19	−	16419.4i	0.0993463	−	0.563421i	−0.893982	−	0.448102i	$-0.852100\pi$
$947$	2895.19	−	16419.4i	0.0993463	−	0.563421i	0.993329	−	0.115319i	$-0.0367889\pi$
$948$		0			0
$949$	5859.55	+	2132.70i	0.200431	+	0.0729510i
$950$		0			0
$951$	38951.8	+	36607.9i	1.32818	+	1.24826i
$952$		0			0
$953$	−15893.0	+	27527.6i	−0.540216	+	0.935682i	0.458675	+	0.888604i	$0.348324\pi$
$953$	−15893.0	+	27527.6i	−0.540216	+	0.935682i	−0.998891	+	0.0470781i	$0.985009\pi$
$954$		0			0
$955$	3660.07	+	6339.43i	0.124018	+	0.214805i
$956$		0			0
$957$	44.8661	+	19.2512i	0.00151548	+	0.000650263i
$958$		0			0
$959$	−4845.65	−	27481.1i	−0.163164	−	0.925349i
$960$		0			0
$961$	−6027.10	−	5057.33i	−0.202313	−	0.169760i
$962$		0			0
$963$	26714.2	−	36172.8i	0.893928	−	1.21044i
$964$		0			0
$965$	1572.78	−	572.445i	0.0524659	−	0.0190960i
$966$		0			0
$967$	5283.91	−	4433.72i	0.175718	−	0.147445i	−0.550687	−	0.834712i	$-0.685634\pi$
$967$	5283.91	−	4433.72i	0.175718	−	0.147445i	0.726405	+	0.687267i	$0.241190\pi$
$968$		0			0
$969$	16312.1	+	32325.1i	0.540785	+	1.07165i
$970$		0			0
$971$	−5291.80			−0.174894			−0.0874469	−	0.996169i	$-0.527871\pi$
$971$	−5291.80			−0.174894			−0.0874469	+	0.996169i	$0.527871\pi$
$972$		0			0
$973$	−42109.4			−1.38742
$974$		0			0
$975$	1323.49	+	2622.71i	0.0434725	+	0.0861478i
$976$		0			0
$977$	−11643.4	+	9770.01i	−0.381276	+	0.319929i	−0.813203	−	0.581980i	$-0.802278\pi$
$977$	−11643.4	+	9770.01i	−0.381276	+	0.319929i	0.431927	+	0.901909i	$0.357834\pi$
$978$		0			0
$979$	85.3404	−	31.0614i	0.00278600	−	0.00101402i
$980$		0			0
$981$	−8525.68	+	11544.3i	−0.277476	+	0.375721i
$982$		0			0
$983$	27467.9	+	23048.3i	0.891242	+	0.747841i	0.968459	−	0.249173i	$-0.0801589\pi$
$983$	27467.9	+	23048.3i	0.891242	+	0.747841i	−0.0772168	+	0.997014i	$0.524603\pi$
$984$		0			0
$985$	−5074.07	−	28776.5i	−0.164135	−	0.930858i
$986$		0			0
$987$	−57524.1	−	24682.5i	−1.85513	−	0.796000i
$988$		0			0
$989$	4654.23	+	8061.36i	0.149642	+	0.259187i
$990$		0			0
$991$	12217.3	−	21161.0i	0.391621	−	0.678307i	−0.601043	−	0.799217i	$-0.705248\pi$
$991$	12217.3	−	21161.0i	0.391621	−	0.678307i	0.992663	+	0.120910i	$0.0385812\pi$
$992$		0			0
$993$	3253.65	+	3057.86i	0.103979	+	0.0977224i
$994$		0			0
$995$	−16304.7	−	5934.43i	−0.519491	−	0.189079i
$996$		0			0
$997$	3628.99	−	20581.0i	0.115277	−	0.653768i	−0.871336	−	0.490687i	$-0.836746\pi$
$997$	3628.99	−	20581.0i	0.115277	−	0.653768i	0.986613	−	0.163081i	$-0.0521431\pi$
$998$		0			0
$999$	−5715.30	−	15425.4i	−0.181005	−	0.488527i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.4.i.a.25.6	yes	54
3.2	odd	2		324.4.i.a.73.7		54
27.13	even	9	inner	108.4.i.a.13.6	✓	54
27.14	odd	18		324.4.i.a.253.7		54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.i.a.13.6	✓	54	27.13	even	9	inner
108.4.i.a.25.6	yes	54	1.1	even	1	trivial
324.4.i.a.73.7		54	3.2	odd	2
324.4.i.a.253.7		54	27.14	odd	18

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	108.i (of order \(9\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(2.34095 + 4.63896i) q^{3} +(-5.19451 + 4.35871i) q^{5} +(-17.8009 + 6.47901i) q^{7} +(-16.0399 + 21.7191i) q^{9} +O(q^{10})\)	\(q+(2.34095 + 4.63896i) q^{3} +(-5.19451 + 4.35871i) q^{5} +(-17.8009 + 6.47901i) q^{7} +(-16.0399 + 21.7191i) q^{9} +(-0.134044 - 0.112476i) q^{11} +(-1.24242 - 7.04614i) q^{13} +(-32.3799 - 13.8936i) q^{15} +(-26.6029 - 46.0776i) q^{17} +(-65.4831 + 113.420i) q^{19} +(-71.7270 - 67.4109i) q^{21} +(129.122 + 46.9965i) q^{23} +(-13.7215 + 77.8183i) q^{25} +(-138.303 - 23.5653i) q^{27} +(-9.32414 + 52.8798i) q^{29} +(139.135 + 50.6411i) q^{31} +(0.207983 - 0.885126i) q^{33} +(64.2270 - 111.244i) q^{35} +(58.6266 + 101.544i) q^{37} +(29.7783 - 22.2582i) q^{39} +(-27.9173 - 158.327i) q^{41} +(51.8941 + 43.5443i) q^{43} +(-11.3478 - 182.734i) q^{45} +(597.576 - 217.500i) q^{47} +(12.1427 - 10.1889i) q^{49} +(151.476 - 231.275i) q^{51} +36.5908 q^{53} +1.18654 q^{55} +(-679.444 - 38.2632i) q^{57} +(574.476 - 482.043i) q^{59} +(-64.9040 + 23.6231i) q^{61} +(144.807 - 490.544i) q^{63} +(37.1659 + 31.1859i) q^{65} +(109.760 + 622.479i) q^{67} +(84.2524 + 709.008i) q^{69} +(-66.9800 - 116.013i) q^{71} +(-435.761 + 754.761i) q^{73} +(-393.117 + 118.515i) q^{75} +(3.11484 + 1.13371i) q^{77} +(55.7069 - 315.930i) q^{79} +(-214.441 - 696.747i) q^{81} +(-197.919 + 1122.46i) q^{83} +(339.028 + 123.396i) q^{85} +(-267.135 + 80.5346i) q^{87} +(-259.505 + 449.475i) q^{89} +(67.7684 + 117.378i) q^{91} +(90.7863 + 763.992i) q^{93} +(-154.213 - 874.583i) q^{95} +(-1153.80 - 968.155i) q^{97} +(4.59294 - 1.10721i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 12 q^{5} - 48 q^{9}+O(q^{10}) \) 54 * q + 12 * q^5 - 48 * q^9	\( 54 q + 12 q^{5} - 48 q^{9} - 87 q^{11} + 234 q^{15} + 204 q^{17} - 12 q^{21} + 96 q^{23} - 216 q^{25} + 27 q^{27} + 318 q^{29} - 54 q^{31} + 63 q^{33} + 6 q^{35} + 66 q^{39} + 867 q^{41} - 513 q^{43} - 306 q^{45} - 1548 q^{47} + 594 q^{49} - 1368 q^{51} - 1068 q^{53} - 1269 q^{57} - 1218 q^{59} - 54 q^{61} + 30 q^{63} + 96 q^{65} - 2997 q^{67} + 1476 q^{69} - 120 q^{71} - 216 q^{73} + 732 q^{75} + 3480 q^{77} + 2808 q^{79} + 3348 q^{81} + 4464 q^{83} + 2160 q^{85} + 4824 q^{87} + 4029 q^{89} + 270 q^{91} + 1164 q^{93} - 1650 q^{95} - 3483 q^{97} - 5076 q^{99}+O(q^{100}) \) 54 * q + 12 * q^5 - 48 * q^9 - 87 * q^11 + 234 * q^15 + 204 * q^17 - 12 * q^21 + 96 * q^23 - 216 * q^25 + 27 * q^27 + 318 * q^29 - 54 * q^31 + 63 * q^33 + 6 * q^35 + 66 * q^39 + 867 * q^41 - 513 * q^43 - 306 * q^45 - 1548 * q^47 + 594 * q^49 - 1368 * q^51 - 1068 * q^53 - 1269 * q^57 - 1218 * q^59 - 54 * q^61 + 30 * q^63 + 96 * q^65 - 2997 * q^67 + 1476 * q^69 - 120 * q^71 - 216 * q^73 + 732 * q^75 + 3480 * q^77 + 2808 * q^79 + 3348 * q^81 + 4464 * q^83 + 2160 * q^85 + 4824 * q^87 + 4029 * q^89 + 270 * q^91 + 1164 * q^93 - 1650 * q^95 - 3483 * q^97 - 5076 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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