LMFDB - Embedded newform 108.4.i.a.97.3

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			97.3
Character	$\chi$	$=$	108.97
Dual form			108.4.i.a.49.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.72085 + 3.62702i) q^{3} +(-6.50317 - 2.36696i) q^{5} +(1.34685 - 7.63837i) q^{7} +(0.689498 - 26.9912i) q^{9} +O(q^{10})$	\(q+(-3.72085 + 3.62702i) q^{3} +(-6.50317 - 2.36696i) q^{5} +(1.34685 - 7.63837i) q^{7} +(0.689498 - 26.9912i) q^{9} +(13.7454 - 5.00292i) q^{11} +(40.8838 - 34.3056i) q^{13} +(32.7823 - 14.7800i) q^{15} +(-29.6006 - 51.2697i) q^{17} +(46.8884 - 81.2131i) q^{19} +(22.6931 + 33.3063i) q^{21} +(4.51998 + 25.6341i) q^{23} +(-59.0669 - 49.5630i) q^{25} +(95.3320 + 102.931i) q^{27} +(-172.381 - 144.645i) q^{29} +(4.27764 + 24.2597i) q^{31} +(-32.9990 + 68.4700i) q^{33} +(-26.8385 + 46.4856i) q^{35} +(-38.3554 - 66.4335i) q^{37} +(-27.6957 + 275.932i) q^{39} +(-305.098 + 256.008i) q^{41} +(282.678 - 102.886i) q^{43} +(-68.3710 + 173.896i) q^{45} +(-3.08562 + 17.4994i) q^{47} +(265.784 + 96.7374i) q^{49} +(296.096 + 83.4053i) q^{51} -242.682 q^{53} -101.230 q^{55} +(120.096 + 472.247i) q^{57} +(-211.368 - 76.9317i) q^{59} +(81.1159 - 460.031i) q^{61} +(-205.240 - 41.6197i) q^{63} +(-347.074 + 126.325i) q^{65} +(333.309 - 279.680i) q^{67} +(-109.793 - 78.9866i) q^{69} +(277.317 + 480.327i) q^{71} +(-226.923 + 393.043i) q^{73} +(399.545 - 29.8199i) q^{75} +(-19.7011 - 111.731i) q^{77} +(-537.292 - 450.842i) q^{79} +(-728.049 - 37.2208i) q^{81} +(-427.701 - 358.883i) q^{83} +(71.1442 + 403.479i) q^{85} +(1166.03 - 87.0265i) q^{87} +(578.094 - 1001.29i) q^{89} +(-206.974 - 358.490i) q^{91} +(-103.907 - 74.7517i) q^{93} +(-497.151 + 417.160i) q^{95} +(1388.22 - 505.270i) q^{97} +(-125.557 - 374.455i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{2}{9}\right)$	$1$

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
$3$	−3.72085	+	3.62702i	−0.716079	+	0.698020i
$3$	−3.72085	+	3.62702i	−0.716079	+	0.698020i
$4$		0			0
$5$	−6.50317	−	2.36696i	−0.581661	−	0.211707i	0.0343968	−	0.999408i	$-0.489049\pi$
$5$	−6.50317	−	2.36696i	−0.581661	−	0.211707i	−0.616058	+	0.787701i	$0.711271\pi$
$6$		0			0
$7$	1.34685	−	7.63837i	0.0727231	−	0.412433i	−0.926614	−	0.376015i	$-0.877294\pi$
$7$	1.34685	−	7.63837i	0.0727231	−	0.412433i	0.999337	−	0.0364180i	$-0.0115948\pi$
$8$		0			0
$9$	0.689498	−	26.9912i	0.0255370	−	0.999674i
$10$		0			0
$11$	13.7454	−	5.00292i	0.376763	−	0.137131i	−0.146695	−	0.989182i	$-0.546864\pi$
$11$	13.7454	−	5.00292i	0.376763	−	0.137131i	0.523459	+	0.852051i	$0.324641\pi$
$12$		0			0
$13$	40.8838	−	34.3056i	0.872241	−	0.731897i	−0.0923280	−	0.995729i	$-0.529431\pi$
$13$	40.8838	−	34.3056i	0.872241	−	0.731897i	0.964569	+	0.263832i	$0.0849864\pi$
$14$		0			0
$15$	32.7823	−	14.7800i	0.564291	−	0.254412i
$16$		0			0
$17$	−29.6006	−	51.2697i	−0.422306	−	0.731455i	0.573859	−	0.818954i	$-0.305446\pi$
$17$	−29.6006	−	51.2697i	−0.422306	−	0.731455i	−0.996165	+	0.0874994i	$0.972112\pi$
$18$		0			0
$19$	46.8884	−	81.2131i	0.566155	−	0.980609i	−0.430786	−	0.902454i	$-0.641764\pi$
$19$	46.8884	−	81.2131i	0.566155	−	0.980609i	0.996941	−	0.0781550i	$-0.0249029\pi$
$20$		0			0
$21$	22.6931	+	33.3063i	0.235811	+	0.346097i
$22$		0			0
$23$	4.51998	+	25.6341i	0.0409774	+	0.232395i	0.998417	−	0.0562379i	$-0.0179105\pi$
$23$	4.51998	+	25.6341i	0.0409774	+	0.232395i	−0.957440	+	0.288633i	$0.906799\pi$
$24$		0			0
$25$	−59.0669	−	49.5630i	−0.472535	−	0.396504i
$26$		0			0
$27$	95.3320	+	102.931i	0.679506	+	0.733670i
$28$		0			0
$29$	−172.381	−	144.645i	−1.10380	−	0.926201i	−0.106129	−	0.994352i	$-0.533846\pi$
$29$	−172.381	−	144.645i	−1.10380	−	0.926201i	−0.997675	+	0.0681509i	$0.978290\pi$
$30$		0			0
$31$	4.27764	+	24.2597i	0.0247834	+	0.140554i	0.994689	−	0.102929i	$-0.0328213\pi$
$31$	4.27764	+	24.2597i	0.0247834	+	0.140554i	−0.969905	+	0.243482i	$0.921710\pi$
$32$		0			0
$33$	−32.9990	+	68.4700i	−0.174072	+	0.361185i
$34$		0			0
$35$	−26.8385	+	46.4856i	−0.129615	+	0.224500i
$36$		0			0
$37$	−38.3554	−	66.4335i	−0.170421	−	0.295178i	0.768146	−	0.640275i	$-0.221180\pi$
$37$	−38.3554	−	66.4335i	−0.170421	−	0.295178i	−0.938567	+	0.345096i	$0.887846\pi$
$38$		0			0
$39$	−27.6957	+	275.932i	−0.113714	+	1.13294i
$40$		0			0
$41$	−305.098	+	256.008i	−1.16215	+	0.975164i	−0.999933	−	0.0115905i	$-0.996311\pi$
$41$	−305.098	+	256.008i	−1.16215	+	0.975164i	−0.162222	+	0.986754i	$0.551866\pi$
$42$		0			0
$43$	282.678	−	102.886i	1.00251	−	0.364885i	0.211960	−	0.977278i	$-0.432016\pi$
$43$	282.678	−	102.886i	1.00251	−	0.364885i	0.790553	+	0.612394i	$0.209793\pi$
$44$		0			0
$45$	−68.3710	+	173.896i	−0.226492	+	0.576065i
$46$		0			0
$47$	−3.08562	+	17.4994i	−0.00957624	+	0.0543096i	−0.989221	−	0.146427i	$-0.953223\pi$
$47$	−3.08562	+	17.4994i	−0.00957624	+	0.0543096i	0.979645	+	0.200737i	$0.0643336\pi$
$48$		0			0
$49$	265.784	+	96.7374i	0.774880	+	0.282033i
$50$		0			0
$51$	296.096	+	83.4053i	0.812974	+	0.229001i
$52$		0			0
$53$	−242.682			−0.628960			−0.314480	−	0.949264i	$-0.601830\pi$
$53$	−242.682			−0.628960			−0.314480	+	0.949264i	$0.601830\pi$
$54$		0			0
$55$	−101.230			−0.248180
$56$		0			0
$57$	120.096	+	472.247i	0.279073	+	1.09738i
$58$		0			0
$59$	−211.368	−	76.9317i	−0.466403	−	0.169757i	0.0981193	−	0.995175i	$-0.468717\pi$
$59$	−211.368	−	76.9317i	−0.466403	−	0.169757i	−0.564522	+	0.825418i	$0.690940\pi$
$60$		0			0
$61$	81.1159	−	460.031i	0.170259	−	0.965589i	−0.773215	−	0.634144i	$-0.781353\pi$
$61$	81.1159	−	460.031i	0.170259	−	0.965589i	0.943475	−	0.331445i	$-0.107536\pi$
$62$		0			0
$63$	−205.240	−	41.6197i	−0.410441	−	0.0832316i
$64$		0			0
$65$	−347.074	+	126.325i	−0.662296	+	0.241056i
$66$		0			0
$67$	333.309	−	279.680i	0.607764	−	0.509975i	−0.286166	−	0.958180i	$-0.592381\pi$
$67$	333.309	−	279.680i	0.607764	−	0.509975i	0.893931	+	0.448205i	$0.147937\pi$
$68$		0			0
$69$	−109.793	−	78.9866i	−0.191559	−	0.137810i
$70$		0			0
$71$	277.317	+	480.327i	0.463541	+	0.802877i	0.999134	−	0.0415991i	$-0.0132452\pi$
$71$	277.317	+	480.327i	0.463541	+	0.802877i	−0.535593	+	0.844476i	$0.679912\pi$
$72$		0			0
$73$	−226.923	+	393.043i	−0.363827	+	0.630166i	−0.988587	−	0.150650i	$-0.951863\pi$
$73$	−226.923	+	393.043i	−0.363827	+	0.630166i	0.624760	+	0.780817i	$0.285197\pi$
$74$		0			0
$75$	399.545	−	29.8199i	0.615140	−	0.0459108i
$76$		0			0
$77$	−19.7011	−	111.731i	−0.0291578	−	0.165362i
$78$		0			0
$79$	−537.292	−	450.842i	−0.765191	−	0.642072i	0.174281	−	0.984696i	$-0.444240\pi$
$79$	−537.292	−	450.842i	−0.765191	−	0.642072i	−0.939473	+	0.342624i	$0.888684\pi$
$80$		0			0
$81$	−728.049	−	37.2208i	−0.998696	−	0.0510573i
$82$		0			0
$83$	−427.701	−	358.883i	−0.565617	−	0.474609i	0.314571	−	0.949234i	$-0.398139\pi$
$83$	−427.701	−	358.883i	−0.565617	−	0.474609i	−0.880188	+	0.474625i	$0.842584\pi$
$84$		0			0
$85$	71.1442	+	403.479i	0.0907844	+	0.514864i
$86$		0			0
$87$	1166.03	−	87.0265i	1.43692	−	0.107244i
$88$		0			0
$89$	578.094	−	1001.29i	0.688515	−	1.19254i	−0.283804	−	0.958882i	$-0.591596\pi$
$89$	578.094	−	1001.29i	0.688515	−	1.19254i	0.972318	−	0.233660i	$-0.0750702\pi$
$90$		0			0
$91$	−206.974	−	358.490i	−0.238426	−	0.412967i
$92$		0			0
$93$	−103.907	−	74.7517i	−0.115856	−	0.0833483i
$94$		0			0
$95$	−497.151	+	417.160i	−0.536912	+	0.450523i
$96$		0			0
$97$	1388.22	−	505.270i	1.45312	−	0.528891i	0.509656	−	0.860378i	$-0.329773\pi$
$97$	1388.22	−	505.270i	1.45312	−	0.528891i	0.943460	+	0.331488i	$0.107551\pi$
$98$		0			0
$99$	−125.557	−	374.455i	−0.127465	−	0.380142i
$100$		0			0
$101$	285.927	−	1621.57i	0.281691	−	1.59755i	−0.435178	−	0.900344i	$-0.643315\pi$
$101$	285.927	−	1621.57i	0.281691	−	1.59755i	0.716870	−	0.697207i	$-0.245574\pi$
$102$		0			0
$103$	495.107	+	180.204i	0.473634	+	0.172389i	0.567798	−	0.823168i	$-0.307796\pi$
$103$	495.107	+	180.204i	0.473634	+	0.172389i	−0.0941639	+	0.995557i	$0.530018\pi$
$104$		0			0
$105$	−68.7421	−	270.310i	−0.0638909	−	0.251234i
$106$		0			0
$107$	1997.93			1.80512			0.902559	−	0.430566i	$-0.141686\pi$
$107$	1997.93			1.80512			0.902559	+	0.430566i	$0.141686\pi$
$108$		0			0
$109$	−1268.88			−1.11502			−0.557509	−	0.830171i	$-0.688243\pi$
$109$	−1268.88			−1.11502			−0.557509	+	0.830171i	$0.688243\pi$
$110$		0			0
$111$	383.670	+	108.074i	0.328075	+	0.0924135i
$112$		0			0
$113$	216.017	+	78.6237i	0.179833	+	0.0654539i	0.430367	−	0.902654i	$-0.358384\pi$
$113$	216.017	+	78.6237i	0.179833	+	0.0654539i	−0.250534	+	0.968108i	$0.580606\pi$
$114$		0			0
$115$	31.2806	−	177.401i	0.0253647	−	0.143850i
$116$		0			0
$117$	−897.760	−	1127.16i	−0.709384	−	0.890647i
$118$		0			0
$119$	−431.484	+	157.047i	−0.332387	+	0.120979i
$120$		0			0
$121$	−855.698	+	718.016i	−0.642899	+	0.539456i
$122$		0			0
$123$	206.681	−	2059.16i	0.151511	−	1.50950i
$124$		0			0
$125$	699.341	+	1211.29i	0.500408	+	0.866732i
$126$		0			0
$127$	−923.845	+	1600.15i	−0.645496	+	1.11803i	0.338691	+	0.940898i	$0.390016\pi$
$127$	−923.845	+	1600.15i	−0.645496	+	1.11803i	−0.984187	+	0.177134i	$0.943317\pi$
$128$		0			0
$129$	−678.633	+	1408.10i	−0.463181	+	0.961059i
$130$		0			0
$131$	462.901	+	2625.24i	0.308732	+	1.75091i	0.605397	+	0.795924i	$0.293014\pi$
$131$	462.901	+	2625.24i	0.308732	+	1.75091i	−0.296665	+	0.954981i	$0.595875\pi$
$132$		0			0
$133$	−557.184	−	467.533i	−0.363263	−	0.304814i
$134$		0			0
$135$	−376.326	−	895.025i	−0.239919	−	0.570604i
$136$		0			0
$137$	−150.629	−	126.392i	−0.0939348	−	0.0788206i	0.594611	−	0.804014i	$-0.297306\pi$
$137$	−150.629	−	126.392i	−0.0939348	−	0.0788206i	−0.688546	+	0.725193i	$0.741751\pi$
$138$		0			0
$139$	236.445	+	1340.95i	0.144281	+	0.818257i	0.967942	+	0.251174i	$0.0808168\pi$
$139$	236.445	+	1340.95i	0.144281	+	0.818257i	−0.823661	+	0.567082i	$0.808072\pi$
$140$		0			0
$141$	−51.9895	−	76.3043i	−0.0310518	−	0.0455743i
$142$		0			0
$143$	390.337	−	676.083i	0.228263	−	0.395363i
$144$		0			0
$145$	778.653	+	1348.67i	0.445956	+	0.772419i
$146$		0			0
$147$	−1339.81	+	604.057i	−0.751740	+	0.338924i
$148$		0			0
$149$	−582.798	+	489.026i	−0.320434	+	0.268876i	−0.788789	−	0.614664i	$-0.789291\pi$
$149$	−582.798	+	489.026i	−0.320434	+	0.268876i	0.468355	+	0.883541i	$0.344847\pi$
$150$		0			0
$151$	−2128.86	+	774.842i	−1.14731	+	0.417588i	−0.844549	−	0.535478i	$-0.820132\pi$
$151$	−2128.86	+	774.842i	−1.14731	+	0.417588i	−0.302764	+	0.953066i	$0.597909\pi$
$152$		0			0
$153$	−1404.24	+	763.605i	−0.742001	+	0.403489i
$154$		0			0
$155$	29.6035	−	167.890i	0.0153407	−	0.0870016i
$156$		0			0
$157$	3067.14	+	1116.35i	1.55914	+	0.567480i	0.970539	−	0.240946i	$-0.0774576\pi$
$157$	3067.14	+	1116.35i	1.55914	+	0.567480i	0.588599	+	0.808425i	$0.299680\pi$
$158$		0			0
$159$	902.982	−	880.210i	0.450385	−	0.439026i
$160$		0			0
$161$	201.890			0.0988272
$162$		0			0
$163$	−3101.93			−1.49057			−0.745283	−	0.666749i	$-0.767685\pi$
$163$	−3101.93			−1.49057			−0.745283	+	0.666749i	$0.767685\pi$
$164$		0			0
$165$	376.664	−	367.164i	0.177716	−	0.173235i
$166$		0			0
$167$	875.969	+	318.826i	0.405895	+	0.147734i	0.536896	−	0.843649i	$-0.319597\pi$
$167$	875.969	+	318.826i	0.405895	+	0.147734i	−0.131001	+	0.991382i	$0.541819\pi$
$168$		0			0
$169$	113.107	−	641.464i	0.0514827	−	0.291973i
$170$		0			0
$171$	−2159.71	−	1321.57i	−0.965831	−	0.591012i
$172$		0			0
$173$	2953.58	−	1075.01i	1.29801	−	0.472439i	0.401665	−	0.915787i	$-0.368432\pi$
$173$	2953.58	−	1075.01i	1.29801	−	0.472439i	0.896349	+	0.443348i	$0.146210\pi$
$174$		0			0
$175$	−458.134	+	384.420i	−0.197895	+	0.166054i
$176$		0			0
$177$	1065.50	−	480.384i	0.452475	−	0.203999i
$178$		0			0
$179$	1491.37	+	2583.12i	0.622737	+	1.07861i	0.988974	+	0.148090i	$0.0473125\pi$
$179$	1491.37	+	2583.12i	0.622737	+	1.07861i	−0.366237	+	0.930522i	$0.619354\pi$
$180$		0			0
$181$	−475.603	+	823.768i	−0.195311	+	0.338289i	−0.947002	−	0.321226i	$-0.895905\pi$
$181$	−475.603	+	823.768i	−0.195311	+	0.338289i	0.751691	+	0.659515i	$0.229238\pi$
$182$		0			0
$183$	1366.72	+	2005.92i	0.552081	+	0.810282i
$184$		0			0
$185$	92.1862	+	522.814i	0.0366360	+	0.207773i
$186$		0			0
$187$	−663.370	−	556.634i	−0.259414	−	0.217674i
$188$		0			0
$189$	914.623	−	589.548i	0.352006	−	0.226896i
$190$		0			0
$191$	−2073.86	−	1740.18i	−0.785651	−	0.659240i	0.159014	−	0.987276i	$-0.449169\pi$
$191$	−2073.86	−	1740.18i	−0.785651	−	0.659240i	−0.944665	+	0.328037i	$0.893613\pi$
$192$		0			0
$193$	742.402	+	4210.37i	0.276887	+	1.57031i	0.732903	+	0.680334i	$0.238165\pi$
$193$	742.402	+	4210.37i	0.276887	+	1.57031i	−0.456015	+	0.889972i	$0.650724\pi$
$194$		0			0
$195$	833.231	−	1728.88i	0.305994	−	0.634911i
$196$		0			0
$197$	2280.07	−	3949.20i	0.824610	−	1.42827i	−0.0776069	−	0.996984i	$-0.524728\pi$
$197$	2280.07	−	3949.20i	0.824610	−	1.42827i	0.902217	−	0.431282i	$-0.141939\pi$
$198$		0			0
$199$	−1556.41	−	2695.78i	−0.554428	−	0.960297i	−0.997948	−	0.0640327i	$-0.979604\pi$
$199$	−1556.41	−	2695.78i	−0.554428	−	0.960297i	0.443520	−	0.896264i	$-0.353730\pi$
$200$		0			0
$201$	−225.792	+	2249.57i	−0.0792345	+	0.789414i
$202$		0			0
$203$	−1337.02	+	1121.89i	−0.462268	+	0.387889i
$204$		0			0
$205$	2590.07	−	942.707i	0.882429	−	0.321178i
$206$		0			0
$207$	695.011	−	104.325i	0.233365	−	0.0350294i
$208$		0			0
$209$	238.198	−	1350.89i	0.0788349	−	0.447095i
$210$		0			0
$211$	−1845.57	−	671.733i	−0.602153	−	0.219166i	0.0229132	−	0.999737i	$-0.492706\pi$
$211$	−1845.57	−	671.733i	−0.602153	−	0.219166i	−0.625066	+	0.780572i	$0.714928\pi$
$212$		0			0
$213$	−2774.01	−	781.393i	−0.892356	−	0.251362i
$214$		0			0
$215$	−2081.83			−0.660371
$216$		0			0
$217$	191.066			0.0597714
$218$		0			0
$219$	−581.224	−	2285.51i	−0.179340	−	0.705207i
$220$		0			0
$221$	−2969.02	−	1080.64i	−0.903702	−	0.328920i
$222$		0			0
$223$	737.919	−	4184.95i	0.221591	−	1.25670i	−0.647506	−	0.762061i	$-0.724188\pi$
$223$	737.919	−	4184.95i	0.221591	−	1.25670i	0.869096	−	0.494643i	$-0.164701\pi$
$224$		0			0
$225$	−1378.49	+	1560.11i	−0.408442	+	0.462255i
$226$		0			0
$227$	−1470.13	+	535.082i	−0.429849	+	0.156452i	−0.547879	−	0.836558i	$-0.684564\pi$
$227$	−1470.13	+	535.082i	−0.429849	+	0.156452i	0.118030	+	0.993010i	$0.462342\pi$
$228$		0			0
$229$	3409.84	−	2861.19i	0.983967	−	0.825647i	−0.000715825	−	1.00000i	$-0.500228\pi$
$229$	3409.84	−	2861.19i	0.983967	−	0.825647i	0.984683	+	0.174353i	$0.0557834\pi$
$230$		0			0
$231$	478.554	+	344.277i	0.136305	+	0.0980596i
$232$		0			0
$233$	1513.22	+	2620.97i	0.425469	+	0.736934i	0.996464	−	0.0840197i	$-0.0267759\pi$
$233$	1513.22	+	2620.97i	0.425469	+	0.736934i	−0.570995	+	0.820953i	$0.693443\pi$
$234$		0			0
$235$	61.4866	−	106.498i	0.0170679	−	0.0295624i
$236$		0			0
$237$	3634.40	−	271.252i	0.996116	−	0.0743448i
$238$		0			0
$239$	352.245	+	1997.68i	0.0953340	+	0.540666i	0.994645	+	0.103355i	$0.0329579\pi$
$239$	352.245	+	1997.68i	0.0953340	+	0.540666i	−0.899311	+	0.437311i	$0.855931\pi$
$240$		0			0
$241$	1684.85	+	1413.75i	0.450334	+	0.377875i	0.839560	−	0.543267i	$-0.182813\pi$
$241$	1684.85	+	1413.75i	0.450334	+	0.377875i	−0.389226	+	0.921142i	$0.627257\pi$
$242$		0			0
$243$	2843.96	−	2502.15i	0.750784	−	0.660548i
$244$		0			0
$245$	−1499.46	−	1258.20i	−0.391009	−	0.328096i
$246$		0			0
$247$	−869.087	−	4928.84i	−0.223881	−	1.26969i
$248$		0			0
$249$	2893.09	−	215.925i	0.736313	−	0.0549545i
$250$		0			0
$251$	1848.46	−	3201.63i	0.464836	−	0.805119i	−0.534359	−	0.845258i	$-0.679447\pi$
$251$	1848.46	−	3201.63i	0.464836	−	0.805119i	0.999194	+	0.0401391i	$0.0127801\pi$
$252$		0			0
$253$	190.374	+	329.738i	0.0473072	+	0.0819385i
$254$		0			0
$255$	−1728.14	−	1243.24i	−0.424394	−	0.305314i
$256$		0			0
$257$	2233.86	−	1874.44i	0.542197	−	0.454957i	−0.330091	−	0.943949i	$-0.607080\pi$
$257$	2233.86	−	1874.44i	0.542197	−	0.454957i	0.872289	+	0.488992i	$0.162635\pi$
$258$		0			0
$259$	−559.102	+	203.497i	−0.134135	+	0.0488211i
$260$		0			0
$261$	−4022.99	+	4553.03i	−0.954087	+	1.07979i
$262$		0			0
$263$	−270.803	+	1535.80i	−0.0634920	+	0.360081i	0.936465	+	0.350762i	$0.114077\pi$
$263$	−270.803	+	1535.80i	−0.0634920	+	0.360081i	−0.999957	+	0.00931899i	$0.997034\pi$
$264$		0			0
$265$	1578.20	+	574.417i	0.365841	+	0.133155i
$266$		0			0
$267$	1480.68	+	5822.40i	0.339387	+	1.33455i
$268$		0			0
$269$	−7131.45			−1.61640			−0.808201	−	0.588906i	$-0.799559\pi$
$269$	−7131.45			−1.61640			−0.808201	+	0.588906i	$0.799559\pi$
$270$		0			0
$271$	5551.33			1.24435			0.622175	−	0.782878i	$-0.286249\pi$
$271$	5551.33			1.24435			0.622175	+	0.782878i	$0.286249\pi$
$272$		0			0
$273$	2070.37	+	583.189i	0.458991	+	0.129290i
$274$		0			0
$275$	−1059.86	−	385.757i	−0.232407	−	0.0845891i
$276$		0			0
$277$	680.783	−	3860.91i	0.147669	−	0.837471i	−0.817517	−	0.575905i	$-0.804650\pi$
$277$	680.783	−	3860.91i	0.147669	−	0.837471i	0.965185	−	0.261567i	$-0.0842391\pi$
$278$		0			0
$279$	657.748	−	98.7316i	0.141141	−	0.0211860i
$280$		0			0
$281$	−5176.79	+	1884.20i	−1.09901	+	0.400006i	−0.826950	−	0.562275i	$-0.809926\pi$
$281$	−5176.79	+	1884.20i	−1.09901	+	0.400006i	−0.272057	+	0.962281i	$0.587704\pi$
$282$		0			0
$283$	−962.891	+	807.961i	−0.202254	+	0.169711i	−0.738289	−	0.674484i	$-0.764366\pi$
$283$	−962.891	+	807.961i	−0.202254	+	0.169711i	0.536035	+	0.844196i	$0.319922\pi$
$284$		0			0
$285$	336.783	−	3355.37i	0.0699975	−	0.697385i
$286$		0			0
$287$	1544.56	+	2675.26i	0.317674	+	0.550228i
$288$		0			0
$289$	704.111	−	1219.56i	0.143316	−	0.248230i
$290$		0			0
$291$	−3332.73	+	6915.13i	−0.671369	+	1.39303i
$292$		0			0
$293$	−1207.87	−	6850.17i	−0.240834	−	1.36584i	−0.829971	−	0.557806i	$-0.811643\pi$
$293$	−1207.87	−	6850.17i	−0.240834	−	1.36584i	0.589137	−	0.808033i	$-0.299468\pi$
$294$		0			0
$295$	1192.47	+	1000.60i	0.235350	+	0.197482i
$296$		0			0
$297$	1825.33	+	937.892i	0.356621	+	0.183239i
$298$		0			0
$299$	1064.19	+	892.959i	0.205831	+	0.172713i
$300$		0			0
$301$	−405.159	−	2297.77i	−0.0775847	−	0.440005i
$302$		0			0
$303$	4817.58	+	7070.70i	0.913409	+	1.34060i
$304$		0			0
$305$	−1616.38	+	2799.66i	−0.303455	+	0.525600i
$306$		0			0
$307$	1319.29	+	2285.08i	0.245263	+	0.424808i	0.962206	−	0.272324i	$-0.0877923\pi$
$307$	1319.29	+	2285.08i	0.245263	+	0.424808i	−0.716942	+	0.697132i	$0.754459\pi$
$308$		0			0
$309$	−2495.82	+	1125.25i	−0.459490	+	0.207162i
$310$		0			0
$311$	−4431.20	+	3718.22i	−0.807943	+	0.677945i	−0.950116	−	0.311897i	$-0.899036\pi$
$311$	−4431.20	+	3718.22i	−0.807943	+	0.677945i	0.142173	+	0.989842i	$0.454591\pi$
$312$		0			0
$313$	−1731.49	+	630.210i	−0.312682	+	0.113807i	−0.493594	−	0.869692i	$-0.664317\pi$
$313$	−1731.49	+	630.210i	−0.312682	+	0.113807i	0.180912	+	0.983499i	$0.442095\pi$
$314$		0			0
$315$	1236.20	+	756.455i	0.221117	+	0.135306i
$316$		0			0
$317$	−71.8687	+	407.588i	−0.0127336	+	0.0722158i	−0.990512	−	0.137424i	$-0.956118\pi$
$317$	−71.8687	+	407.588i	−0.0127336	+	0.0722158i	0.977779	+	0.209639i	$0.0672290\pi$
$318$		0			0
$319$	−3093.09	−	1125.79i	−0.542883	−	0.197593i
$320$		0			0
$321$	−7434.02	+	7246.54i	−1.29261	+	1.26001i
$322$		0			0
$323$	−5551.70			−0.956361
$324$		0			0
$325$	−4115.17			−0.702364
$326$		0			0
$327$	4721.33	−	4602.26i	0.798441	−	0.778305i
$328$		0			0
$329$	129.511	+	47.1381i	0.0217026	+	0.00789912i
$330$		0			0
$331$	−1439.06	+	8161.32i	−0.238967	+	1.35525i	0.595131	+	0.803629i	$0.297100\pi$
$331$	−1439.06	+	8161.32i	−0.238967	+	1.35525i	−0.834097	+	0.551618i	$0.814011\pi$
$332$		0			0
$333$	−1819.57	+	989.452i	−0.299434	+	0.162828i
$334$		0			0
$335$	−2829.56	+	1029.87i	−0.461478	+	0.167964i
$336$		0			0
$337$	−992.442	+	832.758i	−0.160421	+	0.134609i	−0.719465	−	0.694529i	$-0.755613\pi$
$337$	−992.442	+	832.758i	−0.160421	+	0.134609i	0.559044	+	0.829138i	$0.311168\pi$
$338$		0			0
$339$	−1088.94	+	490.949i	−0.174463	+	0.0786570i
$340$		0			0
$341$	180.167	+	312.059i	0.0286117	+	0.0495570i
$342$		0			0
$343$	2427.08	−	4203.82i	0.382069	−	0.661763i
$344$		0			0
$345$	527.047	+	773.540i	0.0822471	+	0.120713i
$346$		0			0
$347$	229.552	+	1301.85i	0.0355129	+	0.201404i	0.997402	−	0.0720360i	$-0.0229497\pi$
$347$	229.552	+	1301.85i	0.0355129	+	0.201404i	−0.961889	+	0.273440i	$0.911839\pi$
$348$		0			0
$349$	1971.21	+	1654.04i	0.302339	+	0.253693i	0.781317	−	0.624134i	$-0.214548\pi$
$349$	1971.21	+	1654.04i	0.302339	+	0.253693i	−0.478978	+	0.877827i	$0.658993\pi$
$350$		0			0
$351$	7428.65	+	937.795i	1.12966	+	0.142609i
$352$		0			0
$353$	7466.29	+	6264.96i	1.12575	+	0.944618i	0.998881	−	0.0473007i	$-0.0150619\pi$
$353$	7466.29	+	6264.96i	1.12575	+	0.944618i	0.126872	+	0.991919i	$0.459506\pi$
$354$		0			0
$355$	−666.523	−	3780.04i	−0.0996490	−	0.565137i
$356$		0			0
$357$	1035.88	−	2149.35i	0.153570	−	0.318644i
$358$		0			0
$359$	5633.32	−	9757.19i	0.828176	−	1.43444i	−0.0712921	−	0.997455i	$-0.522712\pi$
$359$	5633.32	−	9757.19i	0.828176	−	1.43444i	0.899468	−	0.436987i	$-0.143954\pi$
$360$		0			0
$361$	−967.548	−	1675.84i	−0.141063	−	0.244328i
$362$		0			0
$363$	579.671	−	5775.26i	0.0838150	−	0.835049i
$364$		0			0
$365$	2406.04	−	2018.90i	0.345035	−	0.289518i
$366$		0			0
$367$	−721.883	+	262.744i	−0.102676	+	0.0373709i	−0.392847	−	0.919604i	$-0.628510\pi$
$367$	−721.883	+	262.744i	−0.102676	+	0.0373709i	0.290171	+	0.956975i	$0.406288\pi$
$368$		0			0
$369$	6699.59	+	8411.48i	0.945168	+	1.18668i
$370$		0			0
$371$	−326.856	+	1853.69i	−0.0457399	+	0.259404i
$372$		0			0
$373$	−3384.75	−	1231.95i	−0.469854	−	0.171013i	0.0962323	−	0.995359i	$-0.469321\pi$
$373$	−3384.75	−	1231.95i	−0.469854	−	0.171013i	−0.566086	+	0.824346i	$0.691543\pi$
$374$		0			0
$375$	−6995.53	−	1970.53i	−0.963327	−	0.271353i
$376$		0			0
$377$	−12009.7			−1.64067
$378$		0			0
$379$	11322.2			1.53451			0.767257	−	0.641340i	$-0.221621\pi$
$379$	11322.2			1.53451			0.767257	+	0.641340i	$0.221621\pi$
$380$		0			0
$381$	−2366.27	−	9304.71i	−0.318182	−	1.25117i
$382$		0			0
$383$	6956.61	+	2532.00i	0.928110	+	0.337805i	0.761460	−	0.648212i	$-0.224483\pi$
$383$	6956.61	+	2532.00i	0.928110	+	0.337805i	0.166650	+	0.986016i	$0.446705\pi$
$384$		0			0
$385$	−136.342	+	773.235i	−0.0180484	+	0.102358i
$386$		0			0
$387$	−2582.12	−	7700.76i	−0.339165	−	1.01150i
$388$		0			0
$389$	6863.75	−	2498.20i	0.894617	−	0.325614i	0.146523	−	0.989207i	$-0.453192\pi$
$389$	6863.75	−	2498.20i	0.894617	−	0.325614i	0.748094	+	0.663593i	$0.230969\pi$
$390$		0			0
$391$	1180.46	−	990.522i	0.152681	−	0.128115i
$392$		0			0
$393$	−11244.2	−	8089.19i	−1.44324	−	1.03828i
$394$		0			0
$395$	2426.98	+	4203.65i	0.309151	+	0.535465i
$396$		0			0
$397$	4660.31	−	8071.90i	0.589155	−	1.02045i	−0.405189	−	0.914233i	$-0.632794\pi$
$397$	4660.31	−	8071.90i	0.589155	−	1.02045i	0.994343	−	0.106213i	$-0.0338725\pi$
$398$		0			0
$399$	3768.95	−	281.294i	0.472891	−	0.0352941i
$400$		0			0
$401$	−1648.21	−	9347.46i	−0.205256	−	1.16406i	−0.897037	−	0.441956i	$-0.854285\pi$
$401$	−1648.21	−	9347.46i	−0.205256	−	1.16406i	0.691781	−	0.722108i	$-0.256827\pi$
$402$		0			0
$403$	1007.13	+	845.082i	0.124488	+	0.104458i
$404$		0			0
$405$	4646.53	+	1965.32i	0.570093	+	0.241129i
$406$		0			0
$407$	−859.572	−	721.267i	−0.104687	−	0.0878424i
$408$		0			0
$409$	727.417	+	4125.39i	0.0879424	+	0.498746i	0.996683	+	0.0813844i	$0.0259341\pi$
$409$	727.417	+	4125.39i	0.0879424	+	0.498746i	−0.908740	+	0.417362i	$0.862955\pi$
$410$		0			0
$411$	1018.89	−	76.0448i	0.122283	−	0.00912656i
$412$		0			0
$413$	−872.313	+	1510.89i	−0.103932	+	0.180015i
$414$		0			0
$415$	1931.95	+	3346.23i	0.228519	+	0.395807i
$416$		0			0
$417$	−5743.42	−	4131.88i	−0.674476	−	0.485225i
$418$		0			0
$419$	12673.7	−	10634.5i	1.47769	−	1.23993i	0.569077	−	0.822284i	$-0.307301\pi$
$419$	12673.7	−	10634.5i	1.47769	−	1.23993i	0.908611	−	0.417643i	$-0.137144\pi$
$420$		0			0
$421$	−6440.45	+	2344.13i	−0.745578	+	0.271368i	−0.686744	−	0.726900i	$-0.740961\pi$
$421$	−6440.45	+	2344.13i	−0.745578	+	0.271368i	−0.0588342	+	0.998268i	$0.518738\pi$
$422$		0			0
$423$	470.202	+	95.3503i	0.0540473	+	0.0109600i
$424$		0			0
$425$	−792.666	+	4495.43i	−0.0904705	+	0.513084i
$426$		0			0
$427$	−3404.63	−	1239.19i	−0.385859	−	0.140441i
$428$		0			0
$429$	999.779	+	3931.36i	0.112517	+	0.442443i
$430$		0			0
$431$	−16248.1			−1.81588			−0.907938	−	0.419104i	$-0.862344\pi$
$431$	−16248.1			−1.81588			−0.907938	+	0.419104i	$0.862344\pi$
$432$		0			0
$433$	−6251.26			−0.693803			−0.346901	−	0.937902i	$-0.612766\pi$
$433$	−6251.26			−0.693803			−0.346901	+	0.937902i	$0.612766\pi$
$434$		0			0
$435$	−7788.89	−	2194.00i	−0.858503	−	0.241826i
$436$		0			0
$437$	2293.76	+	834.860i	0.251088	+	0.0913885i
$438$		0			0
$439$	−2321.71	+	13167.1i	−0.252412	+	1.43150i	0.550216	+	0.835022i	$0.314545\pi$
$439$	−2321.71	+	13167.1i	−0.252412	+	1.43150i	−0.802629	+	0.596479i	$0.796566\pi$
$440$		0			0
$441$	2794.32	−	7107.13i	0.301729	−	0.767425i
$442$		0			0
$443$	13255.0	−	4824.43i	1.42159	−	0.517417i	0.487081	−	0.873357i	$-0.338061\pi$
$443$	13255.0	−	4824.43i	1.42159	−	0.517417i	0.934509	+	0.355940i	$0.115839\pi$
$444$		0			0
$445$	−6129.45	+	5143.22i	−0.652952	+	0.547892i
$446$		0			0
$447$	394.802	−	3933.41i	0.0417752	−	0.416206i
$448$		0			0
$449$	6125.14	+	10609.0i	0.643793	+	1.11508i	0.984579	+	0.174941i	$0.0559736\pi$
$449$	6125.14	+	10609.0i	0.643793	+	1.11508i	−0.340786	+	0.940141i	$0.610693\pi$
$450$		0			0
$451$	−2912.91	+	5045.32i	−0.304133	+	0.526773i
$452$		0			0
$453$	5110.82	−	10604.5i	0.530082	−	1.09987i
$454$		0			0
$455$	497.457	+	2821.22i	0.0512553	+	0.290683i
$456$		0			0
$457$	−3161.89	−	2653.14i	−0.323648	−	0.271573i	0.466458	−	0.884544i	$-0.345530\pi$
$457$	−3161.89	−	2653.14i	−0.323648	−	0.271573i	−0.790106	+	0.612971i	$0.789974\pi$
$458$		0			0
$459$	2455.37	−	7934.46i	0.249688	−	0.806861i
$460$		0			0
$461$	11312.5	+	9492.29i	1.14289	+	0.959002i	0.999530	−	0.0306683i	$-0.00976354\pi$
$461$	11312.5	+	9492.29i	1.14289	+	0.959002i	0.143364	+	0.989670i	$0.454208\pi$
$462$		0			0
$463$	−1803.16	−	10226.2i	−0.180993	−	1.02646i	−0.930996	−	0.365029i	$-0.881059\pi$
$463$	−1803.16	−	10226.2i	−0.180993	−	1.02646i	0.750003	−	0.661434i	$-0.230052\pi$
$464$		0			0
$465$	498.789	+	732.066i	0.0497436	+	0.0730081i
$466$		0			0
$467$	9089.22	−	15743.0i	0.900640	−	1.55995i	0.0739745	−	0.997260i	$-0.476432\pi$
$467$	9089.22	−	15743.0i	0.900640	−	1.55995i	0.826665	−	0.562694i	$-0.190235\pi$
$468$		0			0
$469$	−1687.38	−	2922.63i	−0.166132	−	0.287749i
$470$		0			0
$471$	−15461.4	+	6970.81i	−1.51258	+	0.681949i
$472$		0			0
$473$	3370.80	−	2828.43i	0.327673	−	0.274950i
$474$		0			0
$475$	−6794.72	+	2473.07i	−0.656343	+	0.238889i
$476$		0			0
$477$	−167.329	+	6550.26i	−0.0160617	+	0.628755i
$478$		0			0
$479$	−878.825	+	4984.07i	−0.0838300	+	0.475423i	0.913773	+	0.406225i	$0.133155\pi$
$479$	−878.825	+	4984.07i	−0.0838300	+	0.475423i	−0.997603	+	0.0691981i	$0.977956\pi$
$480$		0			0
$481$	−3847.16	−	1400.25i	−0.364689	−	0.132736i
$482$		0			0
$483$	−751.204	+	732.259i	−0.0707681	+	0.0689834i
$484$		0			0
$485$	−10223.8			−0.957191
$486$		0			0
$487$	16549.8			1.53993			0.769963	−	0.638089i	$-0.220275\pi$
$487$	16549.8			1.53993			0.769963	+	0.638089i	$0.220275\pi$
$488$		0			0
$489$	11541.8	−	11250.8i	1.06736	−	1.04044i
$490$		0			0
$491$	1518.20	+	552.579i	0.139542	+	0.0507893i	0.410847	−	0.911704i	$-0.365233\pi$
$491$	1518.20	+	552.579i	0.139542	+	0.0507893i	−0.271305	+	0.962493i	$0.587455\pi$
$492$		0			0
$493$	−2313.32	+	13119.5i	−0.211332	+	1.19852i
$494$		0			0
$495$	−69.7982	+	2732.33i	−0.00633777	+	0.248099i
$496$		0			0
$497$	4042.41	−	1471.32i	0.364843	−	0.132792i
$498$		0			0
$499$	−4436.72	+	3722.85i	−0.398026	+	0.333983i	−0.819730	−	0.572750i	$-0.805876\pi$
$499$	−4436.72	+	3722.85i	−0.398026	+	0.333983i	0.421704	+	0.906733i	$0.361432\pi$
$500$		0			0
$501$	−4415.74	+	1990.85i	−0.393774	+	0.177534i
$502$		0			0
$503$	2966.70	+	5138.47i	0.262979	+	0.455493i	0.967032	−	0.254654i	$-0.0819617\pi$
$503$	2966.70	+	5138.47i	0.262979	+	0.455493i	−0.704053	+	0.710147i	$0.748628\pi$
$504$		0			0
$505$	−5697.64	+	9868.59i	−0.502062	+	0.869597i
$506$		0			0
$507$	1905.74	+	2797.04i	0.166937	+	0.245011i
$508$		0			0
$509$	−1327.68	−	7529.66i	−0.115616	−	0.655690i	−0.986443	−	0.164102i	$-0.947528\pi$
$509$	−1327.68	−	7529.66i	−0.115616	−	0.655690i	0.870828	−	0.491589i	$-0.163584\pi$
$510$		0			0
$511$	2696.57	+	2262.69i	0.233443	+	0.195882i
$512$		0			0
$513$	12829.3	−	2915.93i	1.10415	−	0.250958i
$514$		0			0
$515$	−2793.23	−	2343.79i	−0.238998	−	0.200544i
$516$		0			0
$517$	45.1350	+	255.974i	0.00383953	+	0.0217751i
$518$		0			0
$519$	−7090.74	+	14712.7i	−0.599709	+	1.24434i
$520$		0			0
$521$	294.488	−	510.068i	0.0247635	−	0.0428916i	−0.853378	−	0.521292i	$-0.825450\pi$
$521$	294.488	−	510.068i	0.0247635	−	0.0428916i	0.878142	+	0.478401i	$0.158783\pi$
$522$		0			0
$523$	4872.57	+	8439.54i	0.407386	+	0.705613i	0.994596	−	0.103822i	$-0.0331071\pi$
$523$	4872.57	+	8439.54i	0.407386	+	0.705613i	−0.587210	+	0.809434i	$0.699774\pi$
$524$		0			0
$525$	310.352	−	3092.03i	0.0257997	−	0.257043i
$526$		0			0
$527$	1117.17	−	937.415i	0.0923426	−	0.0774847i
$528$		0			0
$529$	10796.6	−	3929.63i	0.887365	−	0.322974i
$530$		0			0
$531$	−2222.22	+	5652.03i	−0.181612	+	0.461916i
$532$		0			0
$533$	−3691.08	+	20933.2i	−0.299960	+	1.70115i
$534$		0			0
$535$	−12992.9	−	4729.03i	−1.04997	−	0.382157i
$536$		0			0
$537$	−14918.2	−	4202.20i	−1.19882	−	0.337688i
$538$		0			0
$539$	4137.28			0.330622
$540$		0			0
$541$	−4636.36			−0.368452			−0.184226	−	0.982884i	$-0.558978\pi$
$541$	−4636.36			−0.368452			−0.184226	+	0.982884i	$0.558978\pi$
$542$		0			0
$543$	−1218.17	−	4790.14i	−0.0962741	−	0.378572i
$544$		0			0
$545$	8251.77	+	3003.40i	0.648563	+	0.236058i
$546$		0			0
$547$	−3576.84	+	20285.3i	−0.279588	+	1.58562i	0.444411	+	0.895823i	$0.353413\pi$
$547$	−3576.84	+	20285.3i	−0.279588	+	1.58562i	−0.723999	+	0.689801i	$0.757698\pi$
$548$		0			0
$549$	−12360.9	−	2506.60i	−0.960926	−	0.194862i
$550$		0			0
$551$	−19829.7	+	7217.42i	−1.53317	+	0.558027i
$552$		0			0
$553$	−4167.35	+	3496.82i	−0.320459	+	0.268897i
$554$		0			0
$555$	−2239.27	−	1610.95i	−0.171264	−	0.123209i
$556$		0			0
$557$	2376.10	+	4115.53i	0.180752	+	0.313071i	0.942137	−	0.335229i	$-0.108814\pi$
$557$	2376.10	+	4115.53i	0.180752	+	0.313071i	−0.761385	+	0.648300i	$0.775480\pi$
$558$		0			0
$559$	8027.38	−	13903.8i	0.607374	−	1.05200i
$560$		0			0
$561$	4487.22	−	334.903i	0.337702	−	0.0252043i
$562$		0			0
$563$	1868.80	+	10598.5i	0.139894	+	0.793379i	0.971326	+	0.237752i	$0.0764106\pi$
$563$	1868.80	+	10598.5i	0.139894	+	0.793379i	−0.831432	+	0.555627i	$0.812478\pi$
$564$		0			0
$565$	−1218.69	−	1022.61i	−0.0907449	−	0.0761440i
$566$		0			0
$567$	−1264.88	+	5510.98i	−0.0936859	+	0.408182i
$568$		0			0
$569$	2744.52	+	2302.93i	0.202208	+	0.169673i	0.738269	−	0.674507i	$-0.235644\pi$
$569$	2744.52	+	2302.93i	0.202208	+	0.169673i	−0.536061	+	0.844180i	$0.680088\pi$
$570$		0			0
$571$	588.801	+	3339.25i	0.0431533	+	0.244735i	0.998752	−	0.0499351i	$-0.0159015\pi$
$571$	588.801	+	3339.25i	0.0431533	+	0.244735i	−0.955599	+	0.294670i	$0.904790\pi$
$572$		0			0
$573$	14028.2	−	1046.99i	1.02275	−	0.0763327i
$574$		0			0
$575$	1003.52	−	1738.15i	0.0727821	−	0.126062i
$576$		0			0
$577$	−1802.55	−	3122.12i	−0.130054	−	0.225261i	0.793643	−	0.608384i	$-0.208182\pi$
$577$	−1802.55	−	3122.12i	−0.130054	−	0.225261i	−0.923697	+	0.383123i	$0.874848\pi$
$578$		0			0
$579$	−18033.4	−	12973.5i	−1.29438	−	0.931189i
$580$		0			0
$581$	−3317.33	+	2783.57i	−0.236878	+	0.198764i
$582$		0			0
$583$	−3335.76	+	1214.12i	−0.236969	+	0.0862497i
$584$		0			0
$585$	3170.35	+	9455.05i	0.224064	+	0.668236i
$586$		0			0
$587$	3392.97	−	19242.5i	0.238574	−	1.35302i	−0.596381	−	0.802702i	$-0.703395\pi$
$587$	3392.97	−	19242.5i	0.238574	−	1.35302i	0.834955	−	0.550318i	$-0.185494\pi$
$588$		0			0
$589$	2170.78	+	790.099i	0.151860	+	0.0552724i
$590$		0			0
$591$	5840.00	+	22964.2i	0.406473	+	1.59835i
$592$		0			0
$593$	−9718.65			−0.673013			−0.336507	−	0.941681i	$-0.609245\pi$
$593$	−9718.65			−0.673013			−0.336507	+	0.941681i	$0.609245\pi$
$594$		0			0
$595$	3177.74			0.218949
$596$		0			0
$597$	15568.8	+	4385.49i	1.06732	+	0.300647i
$598$		0			0
$599$	−23413.0	−	8521.62i	−1.59704	−	0.581275i	−0.618222	−	0.786003i	$-0.712147\pi$
$599$	−23413.0	−	8521.62i	−1.59704	−	0.581275i	−0.978819	+	0.204728i	$0.934369\pi$
$600$		0			0
$601$	4759.65	−	26993.3i	0.323045	−	1.83208i	−0.200024	−	0.979791i	$-0.564102\pi$
$601$	4759.65	−	26993.3i	0.323045	−	1.83208i	0.523069	−	0.852290i	$-0.324787\pi$
$602$		0			0
$603$	−7319.07	−	9189.26i	−0.494288	−	0.620589i
$604$		0			0
$605$	7264.26	−	2643.98i	0.488156	−	0.177674i
$606$		0			0
$607$	19958.2	−	16746.9i	1.33456	−	1.11983i	0.351574	−	0.936160i	$-0.385646\pi$
$607$	19958.2	−	16746.9i	1.33456	−	1.11983i	0.982988	−	0.183671i	$-0.0587979\pi$
$608$		0			0
$609$	905.731	−	9023.79i	0.0602661	−	0.600431i
$610$		0			0
$611$	474.175	+	821.296i	0.0313962	+	0.0543798i
$612$		0			0
$613$	−694.965	+	1203.72i	−0.0457902	+	0.0793109i	−0.888012	−	0.459820i	$-0.847914\pi$
$613$	−694.965	+	1203.72i	−0.0457902	+	0.0793109i	0.842222	+	0.539131i	$0.181247\pi$
$614$		0			0
$615$	−6218.04	+	12901.9i	−0.407700	+	0.845942i
$616$		0			0
$617$	−1604.15	−	9097.57i	−0.104669	−	0.593605i	−0.991352	−	0.131228i	$-0.958108\pi$
$617$	−1604.15	−	9097.57i	−0.104669	−	0.593605i	0.886684	−	0.462377i	$-0.153003\pi$
$618$		0			0
$619$	5940.36	+	4984.55i	0.385724	+	0.323661i	0.814945	−	0.579539i	$-0.196767\pi$
$619$	5940.36	+	4984.55i	0.385724	+	0.323661i	−0.429221	+	0.903200i	$0.641212\pi$
$620$		0			0
$621$	−2207.65	+	2909.00i	−0.142657	+	0.187977i
$622$		0			0
$623$	−6869.59	−	5764.27i	−0.441773	−	0.370691i
$624$		0			0
$625$	−7.17646	−	40.6997i	−0.000459293	−	0.00260478i
$626$		0			0
$627$	4013.39	+	5890.40i	0.255629	+	0.375183i
$628$		0			0
$629$	−2270.68	+	3932.94i	−0.143940	+	0.249311i
$630$		0			0
$631$	−1569.14	−	2717.83i	−0.0989958	−	0.171466i	0.812273	−	0.583277i	$-0.198230\pi$
$631$	−1569.14	−	2717.83i	−0.0989958	−	0.171466i	−0.911269	+	0.411811i	$0.864896\pi$
$632$		0			0
$633$	9303.48	−	4194.50i	0.584171	−	0.263375i
$634$		0			0
$635$	9795.40	−	8219.32i	0.612155	−	0.513659i
$636$		0			0
$637$	14184.9	−	5162.88i	0.882301	−	0.321131i
$638$		0			0
$639$	13155.8	−	7153.92i	0.814453	−	0.442887i
$640$		0			0
$641$	1044.51	−	5923.70i	0.0643613	−	0.365011i	−0.935568	−	0.353146i	$-0.885112\pi$
$641$	1044.51	−	5923.70i	0.0643613	−	0.365011i	0.999930	−	0.0118653i	$-0.00377692\pi$
$642$		0			0
$643$	−3903.59	−	1420.79i	−0.239413	−	0.0871393i	0.219527	−	0.975606i	$-0.429549\pi$
$643$	−3903.59	−	1420.79i	−0.239413	−	0.0871393i	−0.458940	+	0.888467i	$0.651771\pi$
$644$		0			0
$645$	7746.19	−	7550.84i	0.472878	−	0.460952i
$646$		0			0
$647$	−3878.14			−0.235650			−0.117825	−	0.993034i	$-0.537592\pi$
$647$	−3878.14			−0.235650			−0.117825	+	0.993034i	$0.537592\pi$
$648$		0			0
$649$	−3290.22			−0.199002
$650$		0			0
$651$	−710.928	+	692.999i	−0.0428010	+	0.0417216i
$652$		0			0
$653$	3094.96	+	1126.48i	0.185475	+	0.0675075i	0.433088	−	0.901352i	$-0.357424\pi$
$653$	3094.96	+	1126.48i	0.185475	+	0.0675075i	−0.247613	+	0.968859i	$0.579646\pi$
$654$		0			0
$655$	3203.52	−	18168.1i	0.191102	−	1.08379i
$656$		0			0
$657$	10452.2	+	6395.93i	0.620670	+	0.379801i
$658$		0			0
$659$	−14236.8	+	5181.76i	−0.841556	+	0.306301i	−0.726593	−	0.687068i	$-0.758897\pi$
$659$	−14236.8	+	5181.76i	−0.841556	+	0.306301i	−0.114963	+	0.993370i	$0.536675\pi$
$660$		0			0
$661$	3706.41	−	3110.04i	0.218098	−	0.183006i	−0.527193	−	0.849746i	$-0.676755\pi$
$661$	3706.41	−	3110.04i	0.218098	−	0.183006i	0.745290	+	0.666740i	$0.232311\pi$
$662$		0			0
$663$	14966.8	−	6747.81i	0.876714	−	0.395269i
$664$		0			0
$665$	2516.83	+	4359.28i	0.146765	+	0.254204i
$666$		0			0
$667$	2928.68	−	5072.62i	0.170013	−	0.294471i
$668$		0			0
$669$	12433.2	+	18248.0i	0.718527	+	1.05457i
$670$		0			0
$671$	−1186.53	−	6729.13i	−0.0682643	−	0.387146i
$672$		0			0
$673$	−15159.9	−	12720.6i	−0.868305	−	0.728595i	0.0954352	−	0.995436i	$-0.469576\pi$
$673$	−15159.9	−	12720.6i	−0.868305	−	0.728595i	−0.963741	+	0.266841i	$0.914020\pi$
$674$		0			0
$675$	−529.390	−	10804.8i	−0.0301870	−	0.616111i
$676$		0			0
$677$	6219.26	+	5218.58i	0.353066	+	0.296257i	0.802020	−	0.597298i	$-0.203759\pi$
$677$	6219.26	+	5218.58i	0.353066	+	0.296257i	−0.448954	+	0.893555i	$0.648203\pi$
$678$		0			0
$679$	−1989.72	−	11284.2i	−0.112457	−	0.637775i
$680$		0			0
$681$	3529.37	−	7323.13i	0.198599	−	0.412075i
$682$		0			0
$683$	9569.27	−	16574.5i	0.536102	−	0.928557i	−0.463007	−	0.886355i	$-0.653229\pi$
$683$	9569.27	−	16574.5i	0.536102	−	0.928557i	0.999109	−	0.0422019i	$-0.0134373\pi$
$684$		0			0
$685$	680.397	+	1178.48i	0.0379513	+	0.0657336i
$686$		0			0
$687$	−2309.91	+	23013.6i	−0.128280	+	1.27806i
$688$		0			0
$689$	−9921.75	+	8325.33i	−0.548604	+	0.460334i
$690$		0			0
$691$	101.637	−	36.9930i	0.00559547	−	0.00203658i	−0.339221	−	0.940707i	$-0.610163\pi$
$691$	101.637	−	36.9930i	0.00559547	−	0.00203658i	0.344816	+	0.938670i	$0.387941\pi$
$692$		0			0
$693$	−3029.33	+	454.719i	−0.166053	+	0.0249255i
$694$		0			0
$695$	1636.33	−	9280.06i	0.0893085	−	0.506493i
$696$		0			0
$697$	22156.5	+	8064.32i	1.20407	+	0.438247i
$698$		0			0
$699$	−15136.8	−	4263.78i	−0.819063	−	0.230717i
$700$		0			0
$701$	−27394.8			−1.47601			−0.738007	−	0.674793i	$-0.764233\pi$
$701$	−27394.8			−1.47601			−0.738007	+	0.674793i	$0.764233\pi$
$702$		0			0
$703$	−7193.70			−0.385939
$704$		0			0
$705$	157.487	+	619.277i	0.00841321	+	0.0330827i
$706$		0			0
$707$	−12001.1	−	4368.04i	−0.638398	−	0.232358i
$708$		0			0
$709$	2265.35	−	12847.4i	0.119996	−	0.680529i	−0.864159	−	0.503218i	$-0.832149\pi$
$709$	2265.35	−	12847.4i	0.119996	−	0.680529i	0.984155	−	0.177310i	$-0.0567397\pi$
$710$		0			0
$711$	−12539.2	+	14191.3i	−0.661403	+	0.748545i
$712$		0			0
$713$	−602.540	+	219.307i	−0.0316484	+	0.0115191i
$714$		0			0
$715$	−4138.69	+	3472.77i	−0.216473	+	0.181642i
$716$		0			0
$717$	−8556.27	−	6155.47i	−0.445662	−	0.320614i
$718$		0			0
$719$	−7365.93	−	12758.2i	−0.382063	−	0.661752i	0.609294	−	0.792944i	$-0.291453\pi$
$719$	−7365.93	−	12758.2i	−0.382063	−	0.661752i	−0.991357	+	0.131192i	$0.958119\pi$
$720$		0			0
$721$	2043.30	−	3539.10i	0.105543	−	0.182806i
$722$		0			0
$723$	−11396.8	+	850.594i	−0.586238	+	0.0437537i
$724$		0			0
$725$	3012.97	+	17087.4i	0.154343	+	0.875325i
$726$		0			0
$727$	616.890	+	517.632i	0.0314707	+	0.0264070i	0.658387	−	0.752679i	$-0.271239\pi$
$727$	616.890	+	517.632i	0.0314707	+	0.0264070i	−0.626917	+	0.779086i	$0.715683\pi$
$728$		0			0
$729$	−1506.62	+	19625.3i	−0.0765443	+	0.997066i
$730$		0			0
$731$	−13642.4	−	11447.3i	−0.690263	−	0.579200i
$732$		0			0
$733$	314.266	+	1782.29i	0.0158358	+	0.0898095i	0.991701	−	0.128563i	$-0.0410364\pi$
$733$	314.266	+	1782.29i	0.0158358	+	0.0898095i	−0.975866	+	0.218372i	$0.929925\pi$
$734$		0			0
$735$	10142.8	−	757.005i	0.509010	−	0.0379898i
$736$		0			0
$737$	3182.26	−	5511.83i	0.159050	−	0.275483i
$738$		0			0
$739$	−2005.53	−	3473.68i	−0.0998303	−	0.172911i	0.811784	−	0.583958i	$-0.198497\pi$
$739$	−2005.53	−	3473.68i	−0.0998303	−	0.172911i	−0.911614	+	0.411047i	$0.865163\pi$
$740$		0			0
$741$	21110.7	+	15187.3i	1.04659	+	0.752927i
$742$		0			0
$743$	−20551.0	+	17244.3i	−1.01473	+	0.851456i	−0.988956	−	0.148211i	$-0.952648\pi$
$743$	−20551.0	+	17244.3i	−1.01473	+	0.851456i	−0.0257705	+	0.999668i	$0.508204\pi$
$744$		0			0
$745$	4947.54	−	1800.76i	0.243307	−	0.0885566i
$746$		0			0
$747$	−9981.59	+	11296.7i	−0.488899	+	0.553313i
$748$		0			0
$749$	2690.92	−	15261.0i	0.131274	−	0.744490i
$750$		0			0
$751$	33872.2	+	12328.5i	1.64582	+	0.599031i	0.988043	−	0.154179i	$-0.0492734\pi$
$751$	33872.2	+	12328.5i	1.64582	+	0.599031i	0.657780	+	0.753210i	$0.271496\pi$
$752$		0			0
$753$	4734.50	+	18617.2i	0.229130	+	0.900993i
$754$		0			0
$755$	15678.4			0.755754
$756$		0			0
$757$	−20176.1			−0.968708			−0.484354	−	0.874872i	$-0.660945\pi$
$757$	−20176.1			−0.968708			−0.484354	+	0.874872i	$0.660945\pi$
$758$		0			0
$759$	−1904.32	−	536.416i	−0.0910704	−	0.0256530i
$760$		0			0
$761$	−3112.67	−	1132.92i	−0.148271	−	0.0539662i	0.266819	−	0.963747i	$-0.414028\pi$
$761$	−3112.67	−	1132.92i	−0.148271	−	0.0539662i	−0.415090	+	0.909781i	$0.636250\pi$
$762$		0			0
$763$	−1709.00	+	9692.20i	−0.0810876	+	0.459871i
$764$		0			0
$765$	10939.4	−	1642.07i	0.517014	−	0.0776067i
$766$		0			0
$767$	−11280.7	+	4105.84i	−0.531060	+	0.193290i
$768$		0			0
$769$	−872.027	+	731.718i	−0.0408922	+	0.0343126i	−0.663005	−	0.748615i	$-0.730719\pi$
$769$	−872.027	+	731.718i	−0.0408922	+	0.0343126i	0.622113	+	0.782928i	$0.286275\pi$
$770$		0			0
$771$	−1513.28	+	15076.8i	−0.0706865	+	0.704250i
$772$		0			0
$773$	20299.3	+	35159.4i	0.944521	+	1.63596i	0.756707	+	0.653754i	$0.226807\pi$
$773$	20299.3	+	35159.4i	0.944521	+	1.63596i	0.187814	+	0.982205i	$0.439860\pi$
$774$		0			0
$775$	949.716	−	1644.96i	0.0440191	−	0.0762434i
$776$		0			0
$777$	1342.25	−	2785.05i	0.0619730	−	0.128589i
$778$		0			0
$779$	6485.62	+	36781.8i	0.298295	+	1.69171i
$780$		0			0
$781$	6214.87	+	5214.89i	0.284744	+	0.238929i
$782$		0			0
$783$	−1544.97	−	31532.6i	−0.0705144	−	1.43919i
$784$		0			0
$785$	−17303.8	−	14519.6i	−0.786750	−	0.660161i
$786$		0			0
$787$	4660.94	+	26433.5i	0.211111	+	1.19727i	0.887529	+	0.460752i	$0.152420\pi$
$787$	4660.94	+	26433.5i	0.211111	+	1.19727i	−0.676418	+	0.736518i	$0.736469\pi$
$788$		0			0
$789$	−4562.75	−	6696.69i	−0.205879	−	0.302165i
$790$		0			0
$791$	891.499	−	1544.12i	0.0400734	−	0.0694091i
$792$		0			0
$793$	−12465.3	−	21590.5i	−0.558204	−	0.966838i
$794$		0			0
$795$	−7955.67	+	3586.83i	−0.354916	+	0.160015i
$796$		0			0
$797$	−23016.9	+	19313.5i	−1.02296	+	0.858368i	−0.989997	−	0.141088i	$-0.954940\pi$
$797$	−23016.9	+	19313.5i	−1.02296	+	0.858368i	−0.0329662	+	0.999456i	$0.510495\pi$
$798$		0			0
$799$	988.525	−	359.794i	0.0437691	−	0.0159306i
$800$		0			0
$801$	−26627.3	−	16293.8i	−1.17457	−	0.718744i
$802$		0			0
$803$	−1152.79	+	6537.81i	−0.0506615	+	0.287315i
$804$		0			0
$805$	−1312.93	−	477.866i	−0.0574839	−	0.0209224i
$806$		0			0
$807$	26535.1	−	25865.9i	1.15747	−	1.12828i
$808$		0			0
$809$	21235.4			0.922865			0.461432	−	0.887175i	$-0.347336\pi$
$809$	21235.4			0.922865			0.461432	+	0.887175i	$0.347336\pi$
$810$		0			0
$811$	16153.8			0.699427			0.349714	−	0.936857i	$-0.386279\pi$
$811$	16153.8			0.699427			0.349714	+	0.936857i	$0.386279\pi$
$812$		0			0
$813$	−20655.7	+	20134.7i	−0.891053	+	0.868581i
$814$		0			0
$815$	20172.4	+	7342.15i	0.867004	+	0.315564i
$816$		0			0
$817$	4898.61	−	27781.4i	0.209768	−	1.18965i
$818$		0			0
$819$	−9818.78	+	5339.31i	−0.418921	+	0.227803i
$820$		0			0
$821$	20735.6	−	7547.15i	0.881459	−	0.320825i	0.138661	−	0.990340i	$-0.455720\pi$
$821$	20735.6	−	7547.15i	0.881459	−	0.320825i	0.742798	+	0.669515i	$0.233498\pi$
$822$		0			0
$823$	−12610.1	+	10581.1i	−0.534094	+	0.448158i	−0.869512	−	0.493911i	$-0.835567\pi$
$823$	−12610.1	+	10581.1i	−0.534094	+	0.448158i	0.335418	+	0.942069i	$0.391122\pi$
$824$		0			0
$825$	5342.72	−	2408.78i	0.225466	−	0.101652i
$826$		0			0
$827$	3411.08	+	5908.16i	0.143428	+	0.248424i	0.928785	−	0.370618i	$-0.120854\pi$
$827$	3411.08	+	5908.16i	0.143428	+	0.248424i	−0.785357	+	0.619043i	$0.787521\pi$
$828$		0			0
$829$	−16850.1	+	29185.1i	−0.705943	+	1.22273i	0.260408	+	0.965499i	$0.416143\pi$
$829$	−16850.1	+	29185.1i	−0.705943	+	1.22273i	−0.966350	+	0.257230i	$0.917190\pi$
$830$		0			0
$831$	11470.5	+	16835.1i	0.478829	+	0.702771i
$832$		0			0
$833$	−2907.66	−	16490.2i	−0.120942	−	0.685894i
$834$		0			0
$835$	−4941.92	−	4146.76i	−0.204817	−	0.171862i
$836$		0			0
$837$	−2089.28	+	2753.03i	−0.0862797	+	0.113690i
$838$		0			0
$839$	−17763.1	−	14905.0i	−0.730931	−	0.613324i	0.199454	−	0.979907i	$-0.436083\pi$
$839$	−17763.1	−	14905.0i	−0.730931	−	0.613324i	−0.930385	+	0.366583i	$0.880527\pi$
$840$		0			0
$841$	4557.96	+	25849.5i	0.186886	+	1.05988i
$842$		0			0
$843$	12428.0	−	25787.1i	0.507764	−	1.05356i
$844$		0			0
$845$	−2253.88	+	3903.83i	−0.0917582	+	0.158930i
$846$		0			0
$847$	4331.97	+	7503.20i	0.175736	+	0.304384i
$848$		0			0
$849$	652.286	−	6498.73i	0.0263680	−	0.262704i
$850$		0			0
$851$	1529.60	−	1283.48i	0.0616144	−	0.0517007i
$852$		0			0
$853$	−27893.1	+	10152.3i	−1.11963	+	0.407511i	−0.834516	−	0.550984i	$-0.814253\pi$
$853$	−27893.1	+	10152.3i	−1.11963	+	0.407511i	−0.285111	+	0.958495i	$0.592030\pi$
$854$		0			0
$855$	10916.9	+	13706.3i	0.436665	+	0.548242i
$856$		0			0
$857$	−1789.08	+	10146.4i	−0.0713114	+	0.404427i	0.928168	+	0.372162i	$0.121383\pi$
$857$	−1789.08	+	10146.4i	−0.0713114	+	0.404427i	−0.999479	+	0.0322654i	$0.989728\pi$
$858$		0			0
$859$	−955.259	−	347.686i	−0.0379430	−	0.0138101i	0.322979	−	0.946406i	$-0.395316\pi$
$859$	−955.259	−	347.686i	−0.0379430	−	0.0138101i	−0.360922	+	0.932596i	$0.617538\pi$
$860$		0			0
$861$	−15450.3	−	4352.09i	−0.611550	−	0.172264i
$862$		0			0
$863$	43831.8			1.72891			0.864456	−	0.502709i	$-0.167663\pi$
$863$	43831.8			1.72891			0.864456	+	0.502709i	$0.167663\pi$
$864$		0			0
$865$	−21752.1			−0.855023
$866$		0			0
$867$	1803.46	+	7091.61i	0.0706443	+	0.277790i
$868$		0			0
$869$	−9640.83	−	3508.97i	−0.376344	−	0.136978i
$870$		0			0
$871$	4032.38	−	22868.7i	0.156868	−	0.889642i
$872$		0			0
$873$	−12680.7	−	37818.0i	−0.491610	−	1.46615i
$874$		0			0
$875$	10194.2	−	3710.39i	0.393860	−	0.143353i
$876$		0			0
$877$	6970.64	−	5849.06i	0.268394	−	0.225210i	−0.498650	−	0.866803i	$-0.666171\pi$
$877$	6970.64	−	5849.06i	0.268394	−	0.225210i	0.767045	+	0.641594i	$0.221726\pi$
$878$		0			0
$879$	29340.0	+	21107.5i	1.12584	+	0.809941i
$880$		0			0
$881$	−17753.9	−	30750.6i	−0.678937	−	1.17595i	−0.975301	−	0.220878i	$-0.929108\pi$
$881$	−17753.9	−	30750.6i	−0.678937	−	1.17595i	0.296365	−	0.955075i	$-0.404226\pi$
$882$		0			0
$883$	−23165.8	+	40124.3i	−0.882889	+	1.52921i	−0.0347750	+	0.999395i	$0.511071\pi$
$883$	−23165.8	+	40124.3i	−0.882889	+	1.52921i	−0.848114	+	0.529814i	$0.822262\pi$
$884$		0			0
$885$	−8066.19	+	602.017i	−0.306375	+	0.0228662i
$886$		0			0
$887$	3331.76	+	18895.3i	0.126121	+	0.715268i	0.980636	+	0.195841i	$0.0627436\pi$
$887$	3331.76	+	18895.3i	0.126121	+	0.715268i	−0.854515	+	0.519427i	$0.826145\pi$
$888$		0			0
$889$	10978.2	+	9211.82i	0.414171	+	0.347530i
$890$		0			0
$891$	−10193.5	+	3130.76i	−0.383273	+	0.117715i
$892$		0			0
$893$	1276.50	+	1071.11i	0.0478348	+	0.0401382i
$894$		0			0
$895$	−3584.46	−	20328.5i	−0.133872	−	0.759224i
$896$		0			0
$897$	−7198.46	+	537.255i	−0.267948	+	0.0199982i
$898$		0			0
$899$	2771.65	−	4800.64i	0.102825	−	0.178098i
$900$		0			0
$901$	7183.51	+	12442.2i	0.265613	+	0.460056i
$902$		0			0
$903$	9841.60	+	7080.16i	0.362689	+	0.260922i
$904$		0			0
$905$	5042.75	−	4231.37i	0.185223	−	0.155420i
$906$		0			0
$907$	−2377.37	+	865.291i	−0.0870333	+	0.0316775i	−0.385170	−	0.922846i	$-0.625857\pi$
$907$	−2377.37	+	865.291i	−0.0870333	+	0.0316775i	0.298137	+	0.954523i	$0.403635\pi$
$908$		0			0
$909$	−43571.1	−	8835.59i	−1.58984	−	0.322396i
$910$		0			0
$911$	2356.17	−	13362.5i	0.0856898	−	0.485971i	−0.911516	−	0.411265i	$-0.865087\pi$
$911$	2356.17	−	13362.5i	0.0856898	−	0.485971i	0.997206	−	0.0747059i	$-0.0238018\pi$
$912$		0			0
$913$	−7674.38	−	2793.25i	−0.278187	−	0.101252i
$914$		0			0
$915$	−4140.08	−	16279.8i	−0.149581	−	0.588189i
$916$		0			0
$917$	20676.0			0.744583
$918$		0			0
$919$	−34243.4			−1.22915			−0.614574	−	0.788859i	$-0.710672\pi$
$919$	−34243.4			−1.22915			−0.614574	+	0.788859i	$0.710672\pi$
$920$		0			0
$921$	−13196.9	−	3717.35i	−0.472152	−	0.132998i
$922$		0			0
$923$	27815.7	+	10124.1i	0.991943	+	0.361038i
$924$		0			0
$925$	−1027.11	+	5825.03i	−0.0365093	+	0.207055i
$926$		0			0
$927$	5205.30	−	13239.3i	0.184428	−	0.469077i
$928$		0			0
$929$	6133.04	−	2232.25i	0.216597	−	0.0788349i	−0.231443	−	0.972849i	$-0.574345\pi$
$929$	6133.04	−	2232.25i	0.216597	−	0.0788349i	0.448040	+	0.894014i	$0.352122\pi$
$930$		0			0
$931$	20318.5	−	17049.3i	0.715267	−	0.600180i
$932$		0			0
$933$	3001.81	−	29907.0i	0.105332	−	1.04942i
$934$		0			0
$935$	2996.48	+	5190.05i	0.104808	+	0.181533i
$936$		0			0
$937$	−4402.77	+	7625.83i	−0.153503	+	0.265875i	−0.932513	−	0.361137i	$-0.882389\pi$
$937$	−4402.77	+	7625.83i	−0.153503	+	0.265875i	0.779010	+	0.627012i	$0.215722\pi$
$938$		0			0
$939$	4156.83	−	8625.06i	0.144465	−	0.299753i
$940$		0			0
$941$	−7494.30	−	42502.3i	−0.259625	−	1.47241i	−0.783916	−	0.620867i	$-0.786781\pi$
$941$	−7494.30	−	42502.3i	−0.259625	−	1.47241i	0.524291	−	0.851539i	$-0.324331\pi$
$942$		0			0
$943$	−7941.57	−	6663.77i	−0.274245	−	0.230119i
$944$		0			0
$945$	−7343.38	+	1669.05i	−0.252783	+	0.0574543i
$946$		0			0
$947$	5003.21	+	4198.19i	0.171682	+	0.144058i	0.724580	−	0.689191i	$-0.242034\pi$
$947$	5003.21	+	4198.19i	0.171682	+	0.144058i	−0.552898	+	0.833249i	$0.686478\pi$
$948$		0			0
$949$	4206.07	+	23853.8i	0.143872	+	0.815941i
$950$		0			0
$951$	−1210.91	−	1777.24i	−0.0412898	−	0.0606005i
$952$		0			0
$953$	573.765	−	993.790i	0.0195027	−	0.0337797i	−0.856109	−	0.516795i	$-0.827125\pi$
$953$	573.765	−	993.790i	0.0195027	−	0.0337797i	0.875612	+	0.483015i	$0.160458\pi$
$954$		0			0
$955$	9367.74	+	16225.4i	0.317417	+	0.549782i
$956$		0			0
$957$	15592.2	−	7029.78i	0.526671	−	0.237451i
$958$		0			0
$959$	−1168.31	+	980.324i	−0.0393395	+	0.0330097i
$960$		0			0
$961$	27424.1	−	9981.57i	0.920551	−	0.335053i
$962$		0			0
$963$	1377.57	−	53926.6i	0.0460973	−	1.80453i
$964$		0			0
$965$	5137.81	−	29138.0i	0.171391	−	0.972004i
$966$		0			0
$967$	−4869.23	−	1772.25i	−0.161927	−	0.0589368i	0.259785	−	0.965667i	$-0.416348\pi$
$967$	−4869.23	−	1772.25i	−0.161927	−	0.0589368i	−0.421712	+	0.906730i	$0.638571\pi$
$968$		0			0
$969$	20657.1	−	20136.1i	0.684830	−	0.667559i
$970$		0			0
$971$	59813.0			1.97682			0.988409	−	0.151812i	$-0.0485108\pi$
$971$	59813.0			1.97682			0.988409	+	0.151812i	$0.0485108\pi$
$972$		0			0
$973$	10561.1			0.347969
$974$		0			0
$975$	15311.9	−	14925.8i	0.502948	−	0.490264i
$976$		0			0
$977$	17492.6	+	6366.79i	0.572813	+	0.208487i	0.612154	−	0.790739i	$-0.290303\pi$
$977$	17492.6	+	6366.79i	0.572813	+	0.208487i	−0.0393403	+	0.999226i	$0.512526\pi$
$978$		0			0
$979$	2936.77	−	16655.3i	0.0958730	−	0.543723i
$980$		0			0
$981$	−874.894	+	34248.7i	−0.0284742	+	1.11466i
$982$		0			0
$983$	2809.40	−	1022.54i	0.0911555	−	0.0331779i	−0.296040	−	0.955176i	$-0.595666\pi$
$983$	2809.40	−	1022.54i	0.0911555	−	0.0331779i	0.387195	+	0.921998i	$0.373444\pi$
$984$		0			0
$985$	−24175.3	+	20285.4i	−0.782018	+	0.656191i
$986$		0			0
$987$	−652.862	+	294.344i	−0.0210545	+	0.00949249i
$988$		0			0
$989$	3915.10	+	6781.15i	0.125878	+	0.218026i
$990$		0			0
$991$	11105.6	−	19235.4i	0.355984	−	0.616583i	−0.631302	−	0.775537i	$-0.717479\pi$
$991$	11105.6	−	19235.4i	0.355984	−	0.616583i	0.987286	+	0.158954i	$0.0508123\pi$
$992$		0			0
$993$	−24246.7	−	35586.6i	−0.774870	−	1.13727i
$994$		0			0
$995$	3740.80	+	21215.1i	0.119187	+	0.675944i
$996$		0			0
$997$	−17248.2	−	14473.0i	−0.547901	−	0.459744i	0.326328	−	0.945256i	$-0.394188\pi$
$997$	−17248.2	−	14473.0i	−0.547901	−	0.459744i	−0.874230	+	0.485513i	$0.838633\pi$
$998$		0			0
$999$	3181.58	−	10281.2i	0.100761	−	0.325608i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.4.i.a.97.3	yes	54
3.2	odd	2		324.4.i.a.289.7		54
27.5	odd	18		324.4.i.a.37.7		54
27.22	even	9	inner	108.4.i.a.49.3	✓	54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.i.a.49.3	✓	54	27.22	even	9	inner
108.4.i.a.97.3	yes	54	1.1	even	1	trivial
324.4.i.a.37.7		54	27.5	odd	18
324.4.i.a.289.7		54	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	108.i (of order \(9\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(-3.72085 + 3.62702i) q^{3} +(-6.50317 - 2.36696i) q^{5} +(1.34685 - 7.63837i) q^{7} +(0.689498 - 26.9912i) q^{9} +O(q^{10})\)	\(q+(-3.72085 + 3.62702i) q^{3} +(-6.50317 - 2.36696i) q^{5} +(1.34685 - 7.63837i) q^{7} +(0.689498 - 26.9912i) q^{9} +(13.7454 - 5.00292i) q^{11} +(40.8838 - 34.3056i) q^{13} +(32.7823 - 14.7800i) q^{15} +(-29.6006 - 51.2697i) q^{17} +(46.8884 - 81.2131i) q^{19} +(22.6931 + 33.3063i) q^{21} +(4.51998 + 25.6341i) q^{23} +(-59.0669 - 49.5630i) q^{25} +(95.3320 + 102.931i) q^{27} +(-172.381 - 144.645i) q^{29} +(4.27764 + 24.2597i) q^{31} +(-32.9990 + 68.4700i) q^{33} +(-26.8385 + 46.4856i) q^{35} +(-38.3554 - 66.4335i) q^{37} +(-27.6957 + 275.932i) q^{39} +(-305.098 + 256.008i) q^{41} +(282.678 - 102.886i) q^{43} +(-68.3710 + 173.896i) q^{45} +(-3.08562 + 17.4994i) q^{47} +(265.784 + 96.7374i) q^{49} +(296.096 + 83.4053i) q^{51} -242.682 q^{53} -101.230 q^{55} +(120.096 + 472.247i) q^{57} +(-211.368 - 76.9317i) q^{59} +(81.1159 - 460.031i) q^{61} +(-205.240 - 41.6197i) q^{63} +(-347.074 + 126.325i) q^{65} +(333.309 - 279.680i) q^{67} +(-109.793 - 78.9866i) q^{69} +(277.317 + 480.327i) q^{71} +(-226.923 + 393.043i) q^{73} +(399.545 - 29.8199i) q^{75} +(-19.7011 - 111.731i) q^{77} +(-537.292 - 450.842i) q^{79} +(-728.049 - 37.2208i) q^{81} +(-427.701 - 358.883i) q^{83} +(71.1442 + 403.479i) q^{85} +(1166.03 - 87.0265i) q^{87} +(578.094 - 1001.29i) q^{89} +(-206.974 - 358.490i) q^{91} +(-103.907 - 74.7517i) q^{93} +(-497.151 + 417.160i) q^{95} +(1388.22 - 505.270i) q^{97} +(-125.557 - 374.455i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 12 q^{5} - 48 q^{9}+O(q^{10}) \) 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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