LMFDB - Embedded newform 144.5.q.a.113.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,5,Mod(65,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("144.65");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	144.q (of order $6$, degree $2$, not 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:	$14.8852746841$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{6})$
Coefficient field:	6.0.39400128.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 11x^{4} + 14x^{3} + 98x^{2} + 20x + 4 $ x^6 - x^5 + 11x^4 + 14x^3 + 98x^2 + 20x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	no (minimal twist has level 9)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			113.1
Root			$-0.102534 + 0.177594i$ of defining polynomial
Character	$\chi$	$=$	144.113
Dual form			144.5.q.a.65.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-8.32172 + 3.42768i) q^{3} +(-30.0804 - 17.3669i) q^{5} +(-15.6054 - 27.0294i) q^{7} +(57.5020 - 57.0484i) q^{9} +(-49.9968 + 28.8657i) q^{11} +(36.6478 - 63.4758i) q^{13} +(309.848 + 41.4165i) q^{15} +386.985i q^{17} -115.791 q^{19} +(222.512 + 171.441i) q^{21} +(474.852 + 274.156i) q^{23} +(290.719 + 503.539i) q^{25} +(-282.971 + 671.839i) q^{27} +(-680.082 + 392.645i) q^{29} +(272.367 - 471.753i) q^{31} +(317.117 - 411.585i) q^{33} +1084.07i q^{35} +898.827 q^{37} +(-87.3975 + 653.845i) q^{39} +(2242.01 + 1294.43i) q^{41} +(1000.05 + 1732.14i) q^{43} +(-2720.43 + 717.406i) q^{45} +(702.646 - 405.673i) q^{47} +(713.441 - 1235.72i) q^{49} +(-1326.46 - 3220.38i) q^{51} -2221.00i q^{53} +2005.23 q^{55} +(963.579 - 396.895i) q^{57} +(-1309.65 - 756.128i) q^{59} +(951.281 + 1647.67i) q^{61} +(-2439.33 - 663.979i) q^{63} +(-2204.76 + 1272.92i) q^{65} +(2253.55 - 3903.26i) q^{67} +(-4891.30 - 653.806i) q^{69} +3993.54i q^{71} -3436.70 q^{73} +(-4145.25 - 3193.82i) q^{75} +(1560.44 + 900.923i) q^{77} +(601.388 + 1041.63i) q^{79} +(51.9555 - 6560.79i) q^{81} +(-8016.22 + 4628.17i) q^{83} +(6720.72 - 11640.6i) q^{85} +(4313.58 - 5598.59i) q^{87} +8929.99i q^{89} -2287.62 q^{91} +(-649.539 + 4859.38i) q^{93} +(3483.03 + 2010.93i) q^{95} +(-3335.14 - 5776.64i) q^{97} +(-1228.18 + 4512.07i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{3} - 12 q^{5} - 12 q^{7} + 99 q^{9} - 483 q^{11} - 6 q^{13} + 1026 q^{15} + 258 q^{19} + 480 q^{21} + 282 q^{23} - 273 q^{25} - 54 q^{27} - 1056 q^{29} - 1290 q^{31} + 279 q^{33} + 12 q^{37}+ \cdots + 9126 q^{99}+O(q^{100}) $ 6 * q + 3 * q^3 - 12 * q^5 - 12 * q^7 + 99 * q^9 - 483 * q^11 - 6 * q^13 + 1026 * q^15 + 258 * q^19 + 480 * q^21 + 282 * q^23 - 273 * q^25 - 54 * q^27 - 1056 * q^29 - 1290 * q^31 + 279 * q^33 + 12 * q^37 - 1974 * q^39 + 7629 * q^41 + 285 * q^43 - 4212 * q^45 + 9642 * q^47 - 1863 * q^49 - 2457 * q^51 - 2016 * q^55 + 5367 * q^57 - 6225 * q^59 + 3630 * q^61 + 7578 * q^63 - 7158 * q^65 + 5055 * q^67 - 13878 * q^69 - 14622 * q^73 - 21021 * q^75 + 2580 * q^77 - 4764 * q^79 + 18387 * q^81 + 1866 * q^83 + 12366 * q^85 + 21564 * q^87 - 34836 * q^91 + 19254 * q^93 + 13362 * q^95 - 28959 * q^97 + 9126 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$1$	$e\left(\frac{5}{6}\right)$	$1$

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
$3$	−8.32172	+	3.42768i	−0.924635	+	0.380854i
$3$	−8.32172	+	3.42768i	−0.924635	+	0.380854i
$4$		0			0
$5$	−30.0804	−	17.3669i	−1.20321	−	0.694676i	−0.241946	−	0.970290i	$-0.577786\pi$
$5$	−30.0804	−	17.3669i	−1.20321	−	0.694676i	−0.961268	+	0.275614i	$0.911119\pi$
$6$		0			0
$7$	−15.6054	−	27.0294i	−0.318478	−	0.551620i	0.661693	−	0.749775i	$-0.269838\pi$
$7$	−15.6054	−	27.0294i	−0.318478	−	0.551620i	−0.980171	+	0.198155i	$0.936505\pi$
$8$		0			0
$9$	57.5020	−	57.0484i	0.709901	−	0.704301i
$10$		0			0
$11$	−49.9968	+	28.8657i	−0.413197	+	0.238559i	−0.692162	−	0.721742i	$-0.743342\pi$
$11$	−49.9968	+	28.8657i	−0.413197	+	0.238559i	0.278965	+	0.960301i	$0.410009\pi$
$12$		0			0
$13$	36.6478	−	63.4758i	0.216851	−	0.375597i	−0.736993	−	0.675901i	$-0.763755\pi$
$13$	36.6478	−	63.4758i	0.216851	−	0.375597i	0.953844	+	0.300304i	$0.0970882\pi$
$14$		0			0
$15$	309.848	+	41.4165i	1.37710	+	0.184074i
$16$		0			0
$17$			386.985i			1.33905i	0.742791	+	0.669524i	$0.233502\pi$
$17$			386.985i			1.33905i	−0.742791	+	0.669524i	$0.766498\pi$
$18$		0			0
$19$	−115.791			−0.320750			−0.160375	−	0.987056i	$-0.551270\pi$
$19$	−115.791			−0.320750			−0.160375	+	0.987056i	$0.551270\pi$
$20$		0			0
$21$	222.512	+	171.441i	0.504563	+	0.388754i
$22$		0			0
$23$	474.852	+	274.156i	0.897641	+	0.518253i	0.876434	−	0.481522i	$-0.159916\pi$
$23$	474.852	+	274.156i	0.897641	+	0.518253i	0.0212066	+	0.999775i	$0.493249\pi$
$24$		0			0
$25$	290.719	+	503.539i	0.465150	+	0.805663i
$26$		0			0
$27$	−282.971	+	671.839i	−0.388164	+	0.921590i
$28$		0			0
$29$	−680.082	+	392.645i	−0.808658	+	0.466879i	−0.846490	−	0.532405i	$-0.821288\pi$
$29$	−680.082	+	392.645i	−0.808658	+	0.466879i	0.0378313	+	0.999284i	$0.487955\pi$
$30$		0			0
$31$	272.367	−	471.753i	0.283420	−	0.490898i	−0.688805	−	0.724947i	$-0.741864\pi$
$31$	272.367	−	471.753i	0.283420	−	0.490898i	0.972225	+	0.234049i	$0.0751976\pi$
$32$		0			0
$33$	317.117	−	411.585i	0.291200	−	0.377948i
$34$		0			0
$35$			1084.07i			0.884957i
$36$		0			0
$37$	898.827			0.656557			0.328279	−	0.944581i	$-0.393532\pi$
$37$	898.827			0.656557			0.328279	+	0.944581i	$0.393532\pi$
$38$		0			0
$39$	−87.3975	+	653.845i	−0.0574606	+	0.429878i
$40$		0			0
$41$	2242.01	+	1294.43i	1.33374	+	0.770034i	0.985870	−	0.167510i	$-0.0535727\pi$
$41$	2242.01	+	1294.43i	1.33374	+	0.770034i	0.347867	+	0.937544i	$0.386906\pi$
$42$		0			0
$43$	1000.05	+	1732.14i	0.540862	+	0.936801i	0.998855	+	0.0478447i	$0.0152352\pi$
$43$	1000.05	+	1732.14i	0.540862	+	0.936801i	−0.457993	+	0.888956i	$0.651431\pi$
$44$		0			0
$45$	−2720.43	+	717.406i	−1.34342	+	0.354274i
$46$		0			0
$47$	702.646	−	405.673i	0.318083	−	0.183645i	−0.332455	−	0.943119i	$-0.607877\pi$
$47$	702.646	−	405.673i	0.318083	−	0.183645i	0.650538	+	0.759474i	$0.274544\pi$
$48$		0			0
$49$	713.441	−	1235.72i	0.297143	−	0.514667i
$50$		0			0
$51$	−1326.46	−	3220.38i	−0.509981	−	1.23813i
$52$		0			0
$53$		−	2221.00i		−	0.790672i	−0.918537	−	0.395336i	$-0.870628\pi$
$53$		−	2221.00i		−	0.790672i	0.918537	−	0.395336i	$-0.129372\pi$
$54$		0			0
$55$	2005.23			0.662886
$56$		0			0
$57$	963.579	−	396.895i	0.296577	−	0.122159i
$58$		0			0
$59$	−1309.65	−	756.128i	−0.376229	−	0.217216i	0.299947	−	0.953956i	$-0.403031\pi$
$59$	−1309.65	−	756.128i	−0.376229	−	0.217216i	−0.676176	+	0.736740i	$0.736364\pi$
$60$		0			0
$61$	951.281	+	1647.67i	0.255652	+	0.442802i	0.965072	−	0.261983i	$-0.0843765\pi$
$61$	951.281	+	1647.67i	0.255652	+	0.442802i	−0.709420	+	0.704786i	$0.751043\pi$
$62$		0			0
$63$	−2439.33	−	663.979i	−0.614595	−	0.167291i
$64$		0			0
$65$	−2204.76	+	1272.92i	−0.521836	+	0.301282i
$66$		0			0
$67$	2253.55	−	3903.26i	0.502015	−	0.869516i	−0.497982	−	0.867187i	$-0.665925\pi$
$67$	2253.55	−	3903.26i	0.502015	−	0.869516i	0.999997	−	0.00232883i	$-0.000741290\pi$
$68$		0			0
$69$	−4891.30	−	653.806i	−1.02737	−	0.137325i
$70$		0			0
$71$			3993.54i			0.792213i	0.918205	+	0.396106i	$0.129639\pi$
$71$			3993.54i			0.792213i	−0.918205	+	0.396106i	$0.870361\pi$
$72$		0			0
$73$	−3436.70			−0.644905			−0.322452	−	0.946586i	$-0.604507\pi$
$73$	−3436.70			−0.644905			−0.322452	+	0.946586i	$0.604507\pi$
$74$		0			0
$75$	−4145.25	−	3193.82i	−0.736934	−	0.567790i
$76$		0			0
$77$	1560.44	+	900.923i	0.263188	+	0.151952i
$78$		0			0
$79$	601.388	+	1041.63i	0.0963608	+	0.166902i	0.910176	−	0.414222i	$-0.135946\pi$
$79$	601.388	+	1041.63i	0.0963608	+	0.166902i	−0.813815	+	0.581124i	$0.802613\pi$
$80$		0			0
$81$	51.9555	−	6560.79i	0.00791883	−	0.999969i
$82$		0			0
$83$	−8016.22	+	4628.17i	−1.16363	+	0.671820i	−0.952170	−	0.305567i	$-0.901154\pi$
$83$	−8016.22	+	4628.17i	−1.16363	+	0.671820i	−0.211456	+	0.977387i	$0.567821\pi$
$84$		0			0
$85$	6720.72	−	11640.6i	0.930204	−	1.61116i
$86$		0			0
$87$	4313.58	−	5598.59i	0.569902	−	0.739673i
$88$		0			0
$89$			8929.99i			1.12738i	0.825986	+	0.563691i	$0.190619\pi$
$89$			8929.99i			1.12738i	−0.825986	+	0.563691i	$0.809381\pi$
$90$		0			0
$91$	−2287.62			−0.276249
$92$		0			0
$93$	−649.539	+	4859.38i	−0.0751000	+	0.561843i
$94$		0			0
$95$	3483.03	+	2010.93i	0.385932	+	0.222818i
$96$		0			0
$97$	−3335.14	−	5776.64i	−0.354463	−	0.613948i	0.632563	−	0.774509i	$-0.282003\pi$
$97$	−3335.14	−	5776.64i	−0.354463	−	0.613948i	−0.987026	+	0.160561i	$0.948670\pi$
$98$		0			0
$99$	−1228.18	+	4512.07i	−0.125311	+	0.460369i
$100$		0			0
$101$	−7924.19	+	4575.03i	−0.776805	+	0.448489i	−0.835297	−	0.549799i	$-0.814704\pi$
$101$	−7924.19	+	4575.03i	−0.776805	+	0.448489i	0.0584918	+	0.998288i	$0.481371\pi$
$102$		0			0
$103$	7656.20	−	13260.9i	0.721670	−	1.24997i	−0.238660	−	0.971103i	$-0.576708\pi$
$103$	7656.20	−	13260.9i	0.721670	−	1.24997i	0.960330	−	0.278866i	$-0.0899586\pi$
$104$		0			0
$105$	−3715.85	−	9021.34i	−0.337039	−	0.818262i
$106$		0			0
$107$			6099.28i			0.532735i	0.963872	+	0.266367i	$0.0858234\pi$
$107$			6099.28i			0.532735i	−0.963872	+	0.266367i	$0.914177\pi$
$108$		0			0
$109$	15169.5			1.27679			0.638393	−	0.769710i	$-0.279599\pi$
$109$	15169.5			1.27679			0.638393	+	0.769710i	$0.279599\pi$
$110$		0			0
$111$	−7479.78	+	3080.89i	−0.607076	+	0.250052i
$112$		0			0
$113$	−1189.11	−	686.532i	−0.0931246	−	0.0537655i	0.452714	−	0.891656i	$-0.350456\pi$
$113$	−1189.11	−	686.532i	−0.0931246	−	0.0537655i	−0.545839	+	0.837890i	$0.683789\pi$
$114$		0			0
$115$	−9522.48	−	16493.4i	−0.720036	−	1.24714i
$116$		0			0
$117$	−1513.88	−	5740.68i	−0.110591	−	0.419365i
$118$		0			0
$119$	10460.0	−	6039.06i	0.738646	−	0.426457i
$120$		0			0
$121$	−5654.04	+	9793.09i	−0.386179	+	0.668881i
$122$		0			0
$123$	−23094.3	−	3086.94i	−1.52649	−	0.204042i
$124$		0			0
$125$			1513.10i			0.0968386i
$126$		0			0
$127$	19152.4			1.18745			0.593726	−	0.804667i	$-0.297656\pi$
$127$	19152.4			1.18745			0.593726	+	0.804667i	$0.297656\pi$
$128$		0			0
$129$	−14259.4	−	10986.5i	−0.856884	−	0.660210i
$130$		0			0
$131$	2615.39	+	1510.00i	0.152403	+	0.0879902i	0.574262	−	0.818671i	$-0.305289\pi$
$131$	2615.39	+	1510.00i	0.152403	+	0.0879902i	−0.421859	+	0.906661i	$0.638622\pi$
$132$		0			0
$133$	1806.97	+	3129.76i	0.102152	+	0.176932i
$134$		0			0
$135$	20179.6	−	15294.8i	1.10725	−	0.839223i
$136$		0			0
$137$	−8788.30	+	5073.93i	−0.468235	+	0.270336i	−0.715501	−	0.698612i	$-0.753801\pi$
$137$	−8788.30	+	5073.93i	−0.468235	+	0.270336i	0.247266	+	0.968948i	$0.420468\pi$
$138$		0			0
$139$	−17563.2	+	30420.3i	−0.909021	+	1.57447i	−0.0935938	+	0.995610i	$0.529835\pi$
$139$	−17563.2	+	30420.3i	−0.909021	+	1.57447i	−0.815427	+	0.578860i	$0.803498\pi$
$140$		0			0
$141$	−4456.70	+	5784.34i	−0.224169	+	0.290948i
$142$		0			0
$143$			4231.45i			0.206927i
$144$		0			0
$145$	27276.1			1.29732
$146$		0			0
$147$	−1701.41	+	12728.7i	−0.0787363	+	0.589048i
$148$		0			0
$149$	32373.7	+	18691.0i	1.45821	+	0.841899i	0.998923	−	0.0463890i	$-0.0147714\pi$
$149$	32373.7	+	18691.0i	1.45821	+	0.841899i	0.459288	+	0.888288i	$0.348105\pi$
$150$		0			0
$151$	16635.0	+	28812.7i	0.729573	+	1.26366i	0.957064	+	0.289878i	$0.0936146\pi$
$151$	16635.0	+	28812.7i	0.729573	+	1.26366i	−0.227490	+	0.973780i	$0.573052\pi$
$152$		0			0
$153$	22076.9	+	22252.4i	0.943093	+	0.950591i
$154$		0			0
$155$	−16385.8	+	9460.33i	−0.682030	+	0.393770i
$156$		0			0
$157$	−4040.16	+	6997.77i	−0.163908	+	0.283897i	−0.936267	−	0.351289i	$-0.885743\pi$
$157$	−4040.16	+	6997.77i	−0.163908	+	0.283897i	0.772359	+	0.635186i	$0.219077\pi$
$158$		0			0
$159$	7612.87	+	18482.5i	0.301130	+	0.731083i
$160$		0			0
$161$		−	17113.3i		−	0.660209i
$162$		0			0
$163$	25427.1			0.957022			0.478511	−	0.878081i	$-0.341177\pi$
$163$	25427.1			0.957022			0.478511	+	0.878081i	$0.341177\pi$
$164$		0			0
$165$	−16687.0	+	6873.29i	−0.612928	+	0.252463i
$166$		0			0
$167$	30356.6	+	17526.4i	1.08848	+	0.628434i	0.933171	−	0.359432i	$-0.117029\pi$
$167$	30356.6	+	17526.4i	1.08848	+	0.628434i	0.155309	+	0.987866i	$0.450363\pi$
$168$		0			0
$169$	11594.4	+	20082.1i	0.405951	+	0.703129i
$170$		0			0
$171$	−6658.21	+	6605.69i	−0.227701	+	0.225905i
$172$		0			0
$173$	12731.0	−	7350.23i	0.425372	−	0.245589i	−0.272001	−	0.962297i	$-0.587685\pi$
$173$	12731.0	−	7350.23i	0.425372	−	0.245589i	0.697373	+	0.716708i	$0.254352\pi$
$174$		0			0
$175$	9073.58	−	15715.9i	0.296280	−	0.513172i
$176$		0			0
$177$	13490.3	+	1803.21i	0.430602	+	0.0575573i
$178$		0			0
$179$		−	37052.5i		−	1.15641i	−0.815892	−	0.578205i	$-0.803753\pi$
$179$		−	37052.5i		−	1.15641i	0.815892	−	0.578205i	$-0.196247\pi$
$180$		0			0
$181$	−39664.7			−1.21073			−0.605365	−	0.795948i	$-0.706973\pi$
$181$	−39664.7			−1.21073			−0.605365	+	0.795948i	$0.706973\pi$
$182$		0			0
$183$	−13564.0	−	10450.7i	−0.405028	−	0.312065i
$184$		0			0
$185$	−27037.0	−	15609.8i	−0.789979	−	0.456094i
$186$		0			0
$187$	−11170.6	−	19348.0i	−0.319442	−	0.553290i
$188$		0			0
$189$	22575.3	−	2835.80i	0.631990	−	0.0793874i
$190$		0			0
$191$	−49915.1	+	28818.5i	−1.36825	+	0.789959i	−0.990704	−	0.136032i	$-0.956565\pi$
$191$	−49915.1	+	28818.5i	−1.36825	+	0.789959i	−0.377545	+	0.925991i	$0.623232\pi$
$192$		0			0
$193$	−2089.81	+	3619.67i	−0.0561039	+	0.0971748i	−0.892713	−	0.450625i	$-0.851201\pi$
$193$	−2089.81	+	3619.67i	−0.0561039	+	0.0971748i	0.836609	+	0.547800i	$0.184534\pi$
$194$		0			0
$195$	13984.2	−	18150.1i	0.367763	−	0.477319i
$196$		0			0
$197$			22191.5i			0.571812i	0.958258	+	0.285906i	$0.0922945\pi$
$197$			22191.5i			0.571812i	−0.958258	+	0.285906i	$0.907705\pi$
$198$		0			0
$199$	−50608.7			−1.27797			−0.638983	−	0.769221i	$-0.720645\pi$
$199$	−50608.7			−1.27797			−0.638983	+	0.769221i	$0.720645\pi$
$200$		0			0
$201$	−5374.25	+	40206.3i	−0.133023	+	0.995180i
$202$		0			0
$203$	21225.9	+	12254.8i	0.515080	+	0.297382i
$204$		0			0
$205$	−44960.4	−	77873.6i	−1.06985	−	1.85303i
$206$		0			0
$207$	42945.1	−	11325.0i	1.00224	−	0.264301i
$208$		0			0
$209$	5789.18	−	3342.38i	0.132533	−	0.0765180i
$210$		0			0
$211$	−4459.02	+	7723.26i	−0.100156	+	0.173474i	−0.911749	−	0.410749i	$-0.865267\pi$
$211$	−4459.02	+	7723.26i	−0.100156	+	0.173474i	0.811593	+	0.584223i	$0.198601\pi$
$212$		0			0
$213$	−13688.6	−	33233.1i	−0.301717	−	0.732508i
$214$		0			0
$215$		−	69471.4i		−	1.50290i
$216$		0			0
$217$	−17001.6			−0.361052
$218$		0			0
$219$	28599.2	−	11779.9i	0.596302	−	0.245614i
$220$		0			0
$221$	24564.2	+	14182.1i	0.502941	+	0.290373i
$222$		0			0
$223$	−4248.76	−	7359.07i	−0.0854384	−	0.147984i	0.820140	−	0.572164i	$-0.193896\pi$
$223$	−4248.76	−	7359.07i	−0.0854384	−	0.147984i	−0.905578	+	0.424180i	$0.860562\pi$
$224$		0			0
$225$	45443.0	+	12369.5i	0.897640	+	0.244335i
$226$		0			0
$227$	−16543.5	+	9551.40i	−0.321052	+	0.185360i	−0.651862	−	0.758338i	$-0.726012\pi$
$227$	−16543.5	+	9551.40i	−0.321052	+	0.185360i	0.330809	+	0.943698i	$0.392678\pi$
$228$		0			0
$229$	5361.51	−	9286.40i	0.102239	−	0.177083i	−0.810368	−	0.585921i	$-0.800733\pi$
$229$	5361.51	−	9286.40i	0.102239	−	0.177083i	0.912607	+	0.408839i	$0.134066\pi$
$230$		0			0
$231$	−16073.7	−	2148.52i	−0.301225	−	0.0402638i
$232$		0			0
$233$		−	102200.i		−	1.88251i	−0.337694	−	0.941256i	$-0.609647\pi$
$233$		−	102200.i		−	1.88251i	0.337694	−	0.941256i	$-0.390353\pi$
$234$		0			0
$235$	−28181.1			−0.510296
$236$		0			0
$237$	−8574.97	−	6606.82i	−0.152664	−	0.117624i
$238$		0			0
$239$	−13440.9	−	7760.09i	−0.235305	−	0.135854i	0.377712	−	0.925923i	$-0.376711\pi$
$239$	−13440.9	−	7760.09i	−0.235305	−	0.135854i	−0.613017	+	0.790070i	$0.710044\pi$
$240$		0			0
$241$	35978.9	+	62317.3i	0.619461	+	1.07294i	0.989584	+	0.143955i	$0.0459820\pi$
$241$	35978.9	+	62317.3i	0.619461	+	1.07294i	−0.370123	+	0.928983i	$0.620685\pi$
$242$		0			0
$243$	22056.0	+	54775.2i	0.373520	+	0.927622i
$244$		0			0
$245$	−42921.1	+	24780.5i	−0.715054	+	0.412837i
$246$		0			0
$247$	−4243.48	+	7349.92i	−0.0695550	+	0.120473i
$248$		0			0
$249$	50844.9	−	65991.4i	0.820065	−	1.06436i
$250$		0			0
$251$		−	41487.8i		−	0.658526i	−0.944238	−	0.329263i	$-0.893200\pi$
$251$		−	41487.8i		−	0.658526i	0.944238	−	0.329263i	$-0.106800\pi$
$252$		0			0
$253$	−31654.8			−0.494537
$254$		0			0
$255$	−16027.6	+	119907.i	−0.246483	+	1.84401i
$256$		0			0
$257$	−62812.2	−	36264.6i	−0.950994	−	0.549057i	−0.0576045	−	0.998339i	$-0.518346\pi$
$257$	−62812.2	−	36264.6i	−0.950994	−	0.549057i	−0.893390	+	0.449283i	$0.851680\pi$
$258$		0			0
$259$	−14026.6	−	24294.7i	−0.209099	−	0.362170i
$260$		0			0
$261$	−16706.2	+	61375.5i	−0.245244	+	0.900977i
$262$		0			0
$263$	74383.1	−	42945.1i	1.07538	−	0.620872i	0.145735	−	0.989324i	$-0.453445\pi$
$263$	74383.1	−	42945.1i	1.07538	−	0.620872i	0.929647	+	0.368451i	$0.120112\pi$
$264$		0			0
$265$	−38571.8	+	66808.4i	−0.549261	+	0.951347i
$266$		0			0
$267$	−30609.2	−	74312.8i	−0.429367	−	1.04242i
$268$		0			0
$269$		−	88967.6i		−	1.22950i	−0.788724	−	0.614748i	$-0.789258\pi$
$269$		−	88967.6i		−	1.22950i	0.788724	−	0.614748i	$-0.210742\pi$
$270$		0			0
$271$	−96541.6			−1.31455			−0.657273	−	0.753652i	$-0.728290\pi$
$271$	−96541.6			−1.31455			−0.657273	+	0.753652i	$0.728290\pi$
$272$		0			0
$273$	19036.9	−	7841.23i	0.255429	−	0.105210i
$274$		0			0
$275$	−29070.0	−	16783.6i	−0.384397	−	0.221932i
$276$		0			0
$277$	11770.8	+	20387.6i	0.153407	+	0.265710i	0.932478	−	0.361227i	$-0.117642\pi$
$277$	11770.8	+	20387.6i	0.153407	+	0.265710i	−0.779071	+	0.626936i	$0.784309\pi$
$278$		0			0
$279$	−11251.1	−	42664.8i	−0.144540	−	0.548102i
$280$		0			0
$281$	−50584.4	+	29204.9i	−0.640626	+	0.369865i	−0.784855	−	0.619679i	$-0.787263\pi$
$281$	−50584.4	+	29204.9i	−0.640626	+	0.369865i	0.144230	+	0.989544i	$0.453930\pi$
$282$		0			0
$283$	−38058.6	+	65919.5i	−0.475204	+	0.823078i	−0.999597	−	0.0283989i	$-0.990959\pi$
$283$	−38058.6	+	65919.5i	−0.475204	+	0.823078i	0.524393	+	0.851477i	$0.324292\pi$
$284$		0			0
$285$	−35877.6	−	4795.66i	−0.441707	−	0.0590417i
$286$		0			0
$287$		−	80800.3i		−	0.980956i
$288$		0			0
$289$	−66236.1			−0.793047
$290$		0			0
$291$	47554.6	+	36639.7i	0.561573	+	0.432680i
$292$		0			0
$293$	121166.	+	69955.5i	1.41139	+	0.814866i	0.995519	−	0.0945573i	$-0.0301436\pi$
$293$	121166.	+	69955.5i	1.41139	+	0.814866i	0.415871	+	0.909424i	$0.363477\pi$
$294$		0			0
$295$	26263.2	+	45489.2i	0.301789	+	0.522714i
$296$		0			0
$297$	−5245.43	−	41758.0i	−0.0594660	−	0.473399i
$298$		0			0
$299$	34804.5	−	20094.4i	0.389308	−	0.224767i
$300$		0			0
$301$	31212.5	−	54061.7i	0.344505	−	0.596701i
$302$		0			0
$303$	50261.1	−	65233.7i	0.547453	−	0.710537i
$304$		0			0
$305$		−	66083.2i		−	0.710381i
$306$		0			0
$307$	81796.8			0.867880			0.433940	−	0.900942i	$-0.357123\pi$
$307$	81796.8			0.867880			0.433940	+	0.900942i	$0.357123\pi$
$308$		0			0
$309$	−18258.5	+	136597.i	−0.191226	+	1.43062i
$310$		0			0
$311$	1474.59	+	851.354i	0.0152458	+	0.00880216i	0.507604	−	0.861591i	$-0.330531\pi$
$311$	1474.59	+	851.354i	0.0152458	+	0.00880216i	−0.492358	+	0.870393i	$0.663865\pi$
$312$		0			0
$313$	34980.1	+	60587.3i	0.357052	+	0.618433i	0.987467	−	0.157826i	$-0.0504484\pi$
$313$	34980.1	+	60587.3i	0.357052	+	0.618433i	−0.630415	+	0.776259i	$0.717115\pi$
$314$		0			0
$315$	61844.6	+	62336.3i	0.623276	+	0.628232i
$316$		0			0
$317$	101004.	−	58314.7i	1.00512	−	0.580309i	0.0953639	−	0.995442i	$-0.469599\pi$
$317$	101004.	−	58314.7i	1.00512	−	0.580309i	0.909760	+	0.415134i	$0.136265\pi$
$318$		0			0
$319$	22668.0	−	39262.1i	0.222757	−	0.385826i
$320$		0			0
$321$	−20906.4	−	50756.5i	−0.202894	−	0.492585i
$322$		0			0
$323$		−	44809.3i		−	0.429500i
$324$		0			0
$325$	42616.8			0.403472
$326$		0			0
$327$	−126236.	+	51996.2i	−1.18056	+	0.486269i
$328$		0			0
$329$	−21930.2	−	12661.4i	−0.202605	−	0.116974i
$330$		0			0
$331$	50418.1	+	87326.8i	0.460183	+	0.797061i	0.998970	−	0.0453817i	$-0.0144504\pi$
$331$	50418.1	+	87326.8i	0.460183	+	0.797061i	−0.538787	+	0.842442i	$0.681117\pi$
$332$		0			0
$333$	51684.3	−	51276.6i	0.466090	−	0.462414i
$334$		0			0
$335$	−135575.	+	78274.3i	−1.20806	+	0.697476i
$336$		0			0
$337$	−20047.2	+	34722.8i	−0.176520	+	0.305742i	−0.940686	−	0.339277i	$-0.889817\pi$
$337$	−20047.2	+	34722.8i	−0.176520	+	0.305742i	0.764166	+	0.645020i	$0.223151\pi$
$338$		0			0
$339$	12248.6	+	1637.24i	0.106583	+	0.0142466i
$340$		0			0
$341$			31448.2i			0.270450i
$342$		0			0
$343$	−119471.			−1.01549
$344$		0			0
$345$	135778.	+	104613.i	1.14075	+	0.878920i
$346$		0			0
$347$	196035.	+	113181.i	1.62808	+	0.939971i	0.984666	+	0.174451i	$0.0558150\pi$
$347$	196035.	+	113181.i	1.62808	+	0.939971i	0.643412	+	0.765520i	$0.277518\pi$
$348$		0			0
$349$	−39799.1	−	68934.1i	−0.326755	−	0.565957i	0.655111	−	0.755533i	$-0.272622\pi$
$349$	−39799.1	−	68934.1i	−0.326755	−	0.565957i	−0.981866	+	0.189576i	$0.939289\pi$
$350$		0			0
$351$	32275.3	+	42583.3i	0.261973	+	0.345641i
$352$		0			0
$353$	−121173.	+	69959.4i	−0.972427	+	0.561431i	−0.899975	−	0.435941i	$-0.856416\pi$
$353$	−121173.	+	69959.4i	−0.972427	+	0.561431i	−0.0724520	+	0.997372i	$0.523082\pi$
$354$		0			0
$355$	69355.5	−	120127.i	0.550331	−	0.953202i
$356$		0			0
$357$	−66344.9	+	86108.8i	−0.520560	+	0.675633i
$358$		0			0
$359$			211616.i			1.64195i	0.570963	+	0.820976i	$0.306570\pi$
$359$			211616.i			1.64195i	−0.570963	+	0.820976i	$0.693430\pi$
$360$		0			0
$361$	−116913.			−0.897119
$362$		0			0
$363$	13483.7	−	100876.i	0.102329	−	0.765549i
$364$		0			0
$365$	103377.	+	59684.8i	0.775959	+	0.448000i
$366$		0			0
$367$	−15996.8	−	27707.3i	−0.118769	−	0.205713i	0.800511	−	0.599318i	$-0.204561\pi$
$367$	−15996.8	−	27707.3i	−0.118769	−	0.205713i	−0.919280	+	0.393604i	$0.871228\pi$
$368$		0			0
$369$	202765.	−	53471.2i	1.48916	−	0.392706i
$370$		0			0
$371$	−60032.2	+	34659.6i	−0.436151	+	0.251812i
$372$		0			0
$373$	61573.9	−	106649.i	0.442567	−	0.766548i	−0.555312	−	0.831642i	$-0.687401\pi$
$373$	61573.9	−	106649.i	0.442567	−	0.766548i	0.997879	+	0.0650938i	$0.0207347\pi$
$374$		0			0
$375$	−5186.44	−	12591.6i	−0.0368813	−	0.0895404i
$376$		0			0
$377$			57558.3i			0.404972i
$378$		0			0
$379$	116524.			0.811218			0.405609	−	0.914047i	$-0.367059\pi$
$379$	116524.			0.811218			0.405609	+	0.914047i	$0.367059\pi$
$380$		0			0
$381$	−159381.	+	65648.4i	−1.09796	+	0.452246i
$382$		0			0
$383$	−156848.	−	90556.2i	−1.06925	−	0.617334i	−0.141277	−	0.989970i	$-0.545121\pi$
$383$	−156848.	−	90556.2i	−1.06925	−	0.617334i	−0.927978	+	0.372636i	$0.878454\pi$
$384$		0			0
$385$	−31292.5	−	54200.2i	−0.211115	−	0.365661i
$386$		0			0
$387$	156321.	+	42550.2i	1.04375	+	0.284106i
$388$		0			0
$389$	−44017.9	+	25413.7i	−0.290891	+	0.167946i	−0.638343	−	0.769752i	$-0.720380\pi$
$389$	−44017.9	+	25413.7i	−0.290891	+	0.167946i	0.347453	+	0.937697i	$0.387047\pi$
$390$		0			0
$391$	−106094.	+	183760.i	−0.693965	+	1.20198i
$392$		0			0
$393$	−26940.4	−	3601.04i	−0.174429	−	0.0233154i
$394$		0			0
$395$		−	41777.0i		−	0.267758i
$396$		0			0
$397$	228710.			1.45112			0.725561	−	0.688158i	$-0.241580\pi$
$397$	228710.			1.45112			0.725561	+	0.688158i	$0.241580\pi$
$398$		0			0
$399$	−25764.9	−	19851.3i	−0.161839	−	0.124693i
$400$		0			0
$401$	163618.	+	94464.9i	1.01752	+	0.587465i	0.913384	−	0.407099i	$-0.133459\pi$
$401$	163618.	+	94464.9i	1.01752	+	0.587465i	0.104134	+	0.994563i	$0.466793\pi$
$402$		0			0
$403$	−19963.3	−	34577.4i	−0.122920	−	0.212903i
$404$		0			0
$405$	−115504.	+	196449.i	−0.704182	+	1.19768i
$406$		0			0
$407$	−44938.5	+	25945.2i	−0.271287	+	0.156628i
$408$		0			0
$409$	−138713.	+	240258.i	−0.829223	+	1.43626i	0.0694256	+	0.997587i	$0.477883\pi$
$409$	−138713.	+	240258.i	−0.829223	+	1.43626i	−0.898649	+	0.438669i	$0.855450\pi$
$410$		0			0
$411$	55741.9	−	72347.3i	0.329988	−	0.428291i
$412$		0			0
$413$			47198.8i			0.276714i
$414$		0			0
$415$	321508.			1.86679
$416$		0			0
$417$	41884.6	−	313351.i	0.240870	−	1.80201i
$418$		0			0
$419$	88699.9	+	51210.9i	0.505237	+	0.291699i	0.730874	−	0.682513i	$-0.239113\pi$
$419$	88699.9	+	51210.9i	0.505237	+	0.291699i	−0.225637	+	0.974212i	$0.572446\pi$
$420$		0			0
$421$	−23567.9	−	40820.8i	−0.132971	−	0.230312i	0.791850	−	0.610716i	$-0.209118\pi$
$421$	−23567.9	−	40820.8i	−0.132971	−	0.230312i	−0.924821	+	0.380404i	$0.875785\pi$
$422$		0			0
$423$	17260.5	−	63411.8i	0.0964658	−	0.354397i
$424$		0			0
$425$	−194862.	+	112504.i	−1.07882	+	0.622857i
$426$		0			0
$427$	29690.3	−	51425.1i	0.162839	−	0.282046i
$428$		0			0
$429$	−14504.1	−	35213.0i	−0.0788090	−	0.191332i
$430$		0			0
$431$			45556.1i			0.245240i	0.992454	+	0.122620i	$0.0391297\pi$
$431$			45556.1i			0.245240i	−0.992454	+	0.122620i	$0.960870\pi$
$432$		0			0
$433$	209599.			1.11793			0.558965	−	0.829192i	$-0.311199\pi$
$433$	209599.			1.11793			0.558965	+	0.829192i	$0.311199\pi$
$434$		0			0
$435$	−226984.	+	93493.9i	−1.19955	+	0.494089i
$436$		0			0
$437$	−54983.5	−	31744.8i	−0.287919	−	0.166230i
$438$		0			0
$439$	−91842.1	−	159075.i	−0.476555	−	0.825417i	0.523084	−	0.852281i	$-0.324781\pi$
$439$	−91842.1	−	159075.i	−0.476555	−	0.825417i	−0.999639	+	0.0268637i	$0.991448\pi$
$440$		0			0
$441$	−29471.4	−	111757.i	−0.151539	−	0.574641i
$442$		0			0
$443$	100410.	−	57971.5i	0.511644	−	0.295398i	−0.221865	−	0.975077i	$-0.571215\pi$
$443$	100410.	−	57971.5i	0.511644	−	0.295398i	0.733509	+	0.679680i	$0.237881\pi$
$444$		0			0
$445$	155086.	−	268617.i	0.783165	−	1.35648i
$446$		0			0
$447$	−333472.	−	44574.2i	−1.66895	−	0.223084i
$448$		0			0
$449$		−	328940.i		−	1.63164i	−0.578305	−	0.815820i	$-0.696286\pi$
$449$		−	328940.i		−	1.63164i	0.578305	−	0.815820i	$-0.303714\pi$
$450$		0			0
$451$	−149458.			−0.734795
$452$		0			0
$453$	−237192.	−	182751.i	−1.15586	−	0.890562i
$454$		0			0
$455$	68812.3	+	39728.8i	0.332387	+	0.191904i
$456$		0			0
$457$	106369.	+	184236.i	0.509308	+	0.882148i	0.999942	+	0.0107819i	$0.00343204\pi$
$457$	106369.	+	184236.i	0.509308	+	0.882148i	−0.490634	+	0.871366i	$0.663235\pi$
$458$		0			0
$459$	−259992.	−	109506.i	−1.23405	−	0.519770i
$460$		0			0
$461$	−41957.2	+	24224.0i	−0.197426	+	0.113984i	−0.595454	−	0.803389i	$-0.703028\pi$
$461$	−41957.2	+	24224.0i	−0.197426	+	0.113984i	0.398028	+	0.917373i	$0.369695\pi$
$462$		0			0
$463$	84550.6	−	146446.i	0.394416	−	0.683149i	−0.598610	−	0.801040i	$-0.704280\pi$
$463$	84550.6	−	146446.i	0.394416	−	0.683149i	0.993026	+	0.117892i	$0.0376135\pi$
$464$		0			0
$465$	103931.	−	134891.i	0.480660	−	0.623848i
$466$		0			0
$467$		−	54646.4i		−	0.250569i	−0.992121	−	0.125285i	$-0.960016\pi$
$467$		−	54646.4i		−	0.250569i	0.992121	−	0.125285i	$-0.0399844\pi$
$468$		0			0
$469$	−140670.			−0.639524
$470$		0			0
$471$	9634.97	−	72081.9i	0.0434319	−	0.324926i
$472$		0			0
$473$	−99999.1	−	57734.5i	−0.446965	−	0.258055i
$474$		0			0
$475$	−33662.6	−	58305.3i	−0.149197	−	0.258417i
$476$		0			0
$477$	−126704.	−	127712.i	−0.556871	−	0.561299i
$478$		0			0
$479$	120011.	−	69288.6i	0.523060	−	0.301989i	−0.215126	−	0.976586i	$-0.569016\pi$
$479$	120011.	−	69288.6i	0.523060	−	0.301989i	0.738186	+	0.674597i	$0.235683\pi$
$480$		0			0
$481$	32940.0	−	57053.7i	0.142375	−	0.246601i
$482$		0			0
$483$	58658.9	+	142412.i	0.251443	+	0.610453i
$484$		0			0
$485$			231684.i			0.984948i
$486$		0			0
$487$	−23464.1			−0.0989340			−0.0494670	−	0.998776i	$-0.515752\pi$
$487$	−23464.1			−0.0989340			−0.0494670	+	0.998776i	$0.515752\pi$
$488$		0			0
$489$	−211597.	+	87156.1i	−0.884897	+	0.364485i
$490$		0			0
$491$	−64368.8	−	37163.4i	−0.267001	−	0.154153i	0.360523	−	0.932750i	$-0.382598\pi$
$491$	−64368.8	−	37163.4i	−0.267001	−	0.154153i	−0.627524	+	0.778597i	$0.715932\pi$
$492$		0			0
$493$	−151948.	−	263181.i	−0.625173	−	1.08283i
$494$		0			0
$495$	115305.	−	114395.i	0.470583	−	0.466872i
$496$		0			0
$497$	107943.	−	62321.0i	0.437001	−	0.252302i
$498$		0			0
$499$	−9486.01	+	16430.2i	−0.0380963	+	0.0659847i	−0.884445	−	0.466645i	$-0.845463\pi$
$499$	−9486.01	+	16430.2i	−0.0380963	+	0.0659847i	0.846349	+	0.532629i	$0.178796\pi$
$500$		0			0
$501$	−312694.	−	41796.9i	−1.24579	−	0.166521i
$502$		0			0
$503$		−	117856.i		−	0.465818i	−0.972498	−	0.232909i	$-0.925175\pi$
$503$		−	117856.i		−	0.465818i	0.972498	−	0.232909i	$-0.0748245\pi$
$504$		0			0
$505$	317816.			1.24622
$506$		0			0
$507$	−165320.	−	127375.i	−0.643146	−	0.495529i
$508$		0			0
$509$	158881.	+	91730.2i	0.613250	+	0.354060i	0.774236	−	0.632897i	$-0.218134\pi$
$509$	158881.	+	91730.2i	0.613250	+	0.354060i	−0.160986	+	0.986957i	$0.551468\pi$
$510$		0			0
$511$	53631.2	+	92891.9i	0.205388	+	0.355743i
$512$		0			0
$513$	32765.5	−	77792.9i	0.124504	−	0.295601i
$514$		0			0
$515$	−460602.	+	265929.i	−1.73665	+	1.00265i
$516$		0			0
$517$	−23420.0	+	40564.7i	−0.0876207	+	0.151763i
$518$		0			0
$519$	−80749.2	+	104804.i	−0.299781	+	0.389085i
$520$		0			0
$521$			409498.i			1.50861i	0.656526	+	0.754303i	$0.272025\pi$
$521$			409498.i			1.50861i	−0.656526	+	0.754303i	$0.727975\pi$
$522$		0			0
$523$	211852.			0.774513			0.387256	−	0.921972i	$-0.373423\pi$
$523$	211852.			0.774513			0.387256	+	0.921972i	$0.373423\pi$
$524$		0			0
$525$	−21638.6	+	161885.i	−0.0785075	+	0.587336i
$526$		0			0
$527$	182561.	+	105402.i	0.657336	+	0.379513i
$528$		0			0
$529$	10402.4	+	18017.4i	0.0371724	+	0.0643845i
$530$		0			0
$531$	−118444.	+	31234.7i	−0.420071	+	0.110777i
$532$		0			0
$533$	164330.	−	94875.7i	0.578444	−	0.333965i
$534$		0			0
$535$	105926.	−	183468.i	0.370078	−	0.640994i
$536$		0			0
$537$	127004.	+	308341.i	0.440423	+	1.06926i
$538$		0			0
$539$			82375.9i			0.283545i
$540$		0			0
$541$	44016.5			0.150391			0.0751954	−	0.997169i	$-0.476042\pi$
$541$	44016.5			0.150391			0.0751954	+	0.997169i	$0.476042\pi$
$542$		0			0
$543$	330078.	−	135958.i	1.11948	−	0.461111i
$544$		0			0
$545$	−456304.	−	263447.i	−1.53625	−	0.886953i
$546$		0			0
$547$	−21429.6	−	37117.1i	−0.0716207	−	0.124051i	0.827991	−	0.560741i	$-0.189484\pi$
$547$	−21429.6	−	37117.1i	−0.0716207	−	0.124051i	−0.899612	+	0.436691i	$0.856150\pi$
$548$		0			0
$549$	148697.	+	40475.0i	0.493354	+	0.134290i
$550$		0			0
$551$	78747.3	−	45464.8i	0.259378	−	0.149752i
$552$		0			0
$553$	18769.8	−	32510.3i	0.0613776	−	0.106309i
$554$		0			0
$555$	278500.	+	37226.3i	0.904148	+	0.120855i
$556$		0			0
$557$			311007.i			1.00244i	0.865319	+	0.501222i	$0.167116\pi$
$557$			311007.i			1.00244i	−0.865319	+	0.501222i	$0.832884\pi$
$558$		0			0
$559$	146599.			0.469145
$560$		0			0
$561$	159277.	+	122719.i	0.506090	+	0.389931i
$562$		0			0
$563$	247482.	+	142884.i	0.780777	+	0.450782i	0.836705	−	0.547653i	$-0.184479\pi$
$563$	247482.	+	142884.i	0.780777	+	0.450782i	−0.0559288	+	0.998435i	$0.517812\pi$
$564$		0			0
$565$	23845.9	+	41302.2i	0.0746992	+	0.129383i
$566$		0			0
$567$	−178145.	+	100980.i	−0.554125	+	0.314100i
$568$		0			0
$569$	−118371.	+	68341.3i	−0.365611	+	0.211086i	−0.671539	−	0.740969i	$-0.734367\pi$
$569$	−118371.	+	68341.3i	−0.365611	+	0.211086i	0.305928	+	0.952055i	$0.401033\pi$
$570$		0			0
$571$	−14255.0	+	24690.4i	−0.0437215	+	0.0757279i	−0.887058	−	0.461658i	$-0.847255\pi$
$571$	−14255.0	+	24690.4i	−0.0437215	+	0.0757279i	0.843337	+	0.537386i	$0.180588\pi$
$572$		0			0
$573$	316599.	−	410912.i	0.964273	−	1.25153i
$574$		0			0
$575$			318809.i			0.964261i
$576$		0			0
$577$	−293742.			−0.882297			−0.441149	−	0.897434i	$-0.645429\pi$
$577$	−293742.			−0.882297			−0.441149	+	0.897434i	$0.645429\pi$
$578$		0			0
$579$	4983.78	−	37285.1i	0.0148663	−	0.111219i
$580$		0			0
$581$	250193.	+	144449.i	0.741179	+	0.427920i
$582$		0			0
$583$	64110.6	+	111043.i	0.188622	+	0.326703i
$584$		0			0
$585$	−54160.0	+	198973.i	−0.158258	+	0.581410i
$586$		0			0
$587$	−334334.	+	193028.i	−0.970295	+	0.560200i	−0.899326	−	0.437279i	$-0.855942\pi$
$587$	−334334.	+	193028.i	−0.970295	+	0.560200i	−0.0709687	+	0.997479i	$0.522609\pi$
$588$		0			0
$589$	−31537.6	+	54624.7i	−0.0909072	+	0.157456i
$590$		0			0
$591$	−76065.3	−	184671.i	−0.217777	−	0.528718i
$592$		0			0
$593$		−	305581.i		−	0.868995i	−0.900673	−	0.434497i	$-0.856926\pi$
$593$		−	305581.i		−	0.868995i	0.900673	−	0.434497i	$-0.143074\pi$
$594$		0			0
$595$	−419519.			−1.18500
$596$		0			0
$597$	421152.	−	173471.i	1.18165	−	0.486718i
$598$		0			0
$599$	−229270.	−	132369.i	−0.638988	−	0.368920i	0.145236	−	0.989397i	$-0.453606\pi$
$599$	−229270.	−	132369.i	−0.638988	−	0.368920i	−0.784225	+	0.620477i	$0.786939\pi$
$600$		0			0
$601$	66101.5	+	114491.i	0.183005	+	0.316973i	0.942902	−	0.333069i	$-0.108084\pi$
$601$	66101.5	+	114491.i	0.183005	+	0.316973i	−0.759898	+	0.650043i	$0.774751\pi$
$602$		0			0
$603$	−93091.3	−	353006.i	−0.256020	−	0.970841i
$604$		0			0
$605$	340151.	−	196386.i	0.929312	−	0.536538i
$606$		0			0
$607$	−127073.	+	220097.i	−0.344886	+	0.597360i	−0.985333	−	0.170642i	$-0.945416\pi$
$607$	−127073.	+	220097.i	−0.344886	+	0.597360i	0.640447	+	0.768002i	$0.278749\pi$
$608$		0			0
$609$	−218642.	−	29225.2i	−0.589520	−	0.0787994i
$610$		0			0
$611$		−	59468.0i		−	0.159295i
$612$		0			0
$613$	−492878.			−1.31165			−0.655826	−	0.754912i	$-0.727680\pi$
$613$	−492878.			−1.31165			−0.655826	+	0.754912i	$0.727680\pi$
$614$		0			0
$615$	641074.	+	493933.i	1.69495	+	1.30592i
$616$		0			0
$617$	−105029.	−	60638.8i	−0.275893	−	0.159287i	0.355670	−	0.934612i	$-0.384253\pi$
$617$	−105029.	−	60638.8i	−0.275893	−	0.159287i	−0.631563	+	0.775325i	$0.717586\pi$
$618$		0			0
$619$	153682.	+	266185.i	0.401090	+	0.694709i	0.993858	−	0.110665i	$-0.0352981\pi$
$619$	153682.	+	266185.i	0.401090	+	0.694709i	−0.592768	+	0.805374i	$0.701965\pi$
$620$		0			0
$621$	−318558.	+	241446.i	−0.826049	+	0.626090i
$622$		0			0
$623$	241372.	−	139356.i	0.621886	−	0.359046i
$624$		0			0
$625$	207977.	−	360227.i	0.532421	−	0.922181i
$626$		0			0
$627$	−36719.3	+	47657.9i	−0.0934026	+	0.121227i
$628$		0			0
$629$			347832.i			0.879161i
$630$		0			0
$631$	254196.			0.638425			0.319212	−	0.947683i	$-0.396582\pi$
$631$	254196.			0.638425			0.319212	+	0.947683i	$0.396582\pi$
$632$		0			0
$633$	10633.9	−	79554.9i	0.0265390	−	0.198545i
$634$		0			0
$635$	−576112.	−	332618.i	−1.42876	−	0.824895i
$636$		0			0
$637$	−52292.1	−	90572.5i	−0.128872	−	0.223212i
$638$		0			0
$639$	227825.	+	229637.i	0.557957	+	0.562393i
$640$		0			0
$641$	316954.	−	182993.i	0.771400	−	0.445368i	−0.0619737	−	0.998078i	$-0.519739\pi$
$641$	316954.	−	182993.i	0.771400	−	0.445368i	0.833374	+	0.552710i	$0.186406\pi$
$642$		0			0
$643$	115391.	−	199864.i	0.279094	−	0.483406i	−0.692066	−	0.721835i	$-0.743299\pi$
$643$	115391.	−	199864.i	0.279094	−	0.483406i	0.971160	+	0.238429i	$0.0766324\pi$
$644$		0			0
$645$	238126.	+	578121.i	0.572383	+	1.38963i
$646$		0			0
$647$			278596.i			0.665529i	0.943010	+	0.332764i	$0.107981\pi$
$647$			278596.i			0.665529i	−0.943010	+	0.332764i	$0.892019\pi$
$648$		0			0
$649$	87304.7			0.207276
$650$		0			0
$651$	141483.	−	58276.1i	0.333842	−	0.137508i
$652$		0			0
$653$	−67988.8	−	39253.4i	−0.159445	−	0.0920557i	0.418154	−	0.908376i	$-0.362677\pi$
$653$	−67988.8	−	39253.4i	−0.159445	−	0.0920557i	−0.577600	+	0.816320i	$0.696010\pi$
$654$		0			0
$655$	−52448.0	−	90842.6i	−0.122249	−	0.211742i
$656$		0			0
$657$	−197617.	+	196058.i	−0.457819	+	0.454208i
$658$		0			0
$659$	742420.	−	428637.i	1.70954	−	0.987003i	0.774418	−	0.632675i	$-0.218043\pi$
$659$	742420.	−	428637.i	1.70954	−	0.987003i	0.935121	−	0.354328i	$-0.115290\pi$
$660$		0			0
$661$	−11172.0	+	19350.4i	−0.0255698	+	0.0442882i	−0.878527	−	0.477692i	$-0.841473\pi$
$661$	−11172.0	+	19350.4i	−0.0255698	+	0.0442882i	0.852957	+	0.521981i	$0.174807\pi$
$662$		0			0
$663$	−253028.	−	33821.5i	−0.575627	−	0.0769424i
$664$		0			0
$665$		−	125526.i		−	0.283850i
$666$		0			0
$667$	−430584.			−0.967846
$668$		0			0
$669$	60581.6	+	46676.7i	0.135359	+	0.104291i
$670$		0			0
$671$	−95122.1	−	54918.8i	−0.211269	−	0.121976i
$672$		0			0
$673$	331739.	+	574588.i	0.732430	+	1.26861i	0.955842	+	0.293882i	$0.0949472\pi$
$673$	331739.	+	574588.i	0.732430	+	1.26861i	−0.223412	+	0.974724i	$0.571719\pi$
$674$		0			0
$675$	−420563.	+	52829.0i	−0.923046	+	0.115948i
$676$		0			0
$677$	−197558.	+	114060.i	−0.431040	+	0.248861i	−0.699790	−	0.714349i	$-0.746723\pi$
$677$	−197558.	+	114060.i	−0.431040	+	0.248861i	0.268749	+	0.963210i	$0.413390\pi$
$678$		0			0
$679$	−104093.	+	180294.i	−0.225777	+	0.391058i
$680$		0			0
$681$	104931.	−	136190.i	0.226261	−	0.293664i
$682$		0			0
$683$			53606.6i			0.114915i	0.998348	+	0.0574575i	$0.0182994\pi$
$683$			53606.6i			0.114915i	−0.998348	+	0.0574575i	$0.981701\pi$
$684$		0			0
$685$	352474.			0.751182
$686$		0			0
$687$	−12786.1	+	95656.4i	−0.0270910	+	0.202675i
$688$		0			0
$689$	−140980.	−	81394.6i	−0.296974	−	0.171458i
$690$		0			0
$691$	378663.	+	655863.i	0.793043	+	1.37359i	0.924074	+	0.382213i	$0.124838\pi$
$691$	378663.	+	655863.i	0.793043	+	1.37359i	−0.131031	+	0.991378i	$0.541829\pi$
$692$		0			0
$693$	141125.	−	37216.0i	0.293858	−	0.0774932i
$694$		0			0
$695$	1.05661e6	−	610036.i	2.18749	−	1.26295i
$696$		0			0
$697$	−500923.	+	867624.i	−1.03111	+	1.78594i
$698$		0			0
$699$	350308.	+	850477.i	0.716962	+	1.74064i
$700$		0			0
$701$		−	506359.i		−	1.03044i	−0.857058	−	0.515220i	$-0.827710\pi$
$701$		−	506359.i		−	1.03044i	0.857058	−	0.515220i	$-0.172290\pi$
$702$		0			0
$703$	−104076.			−0.210591
$704$		0			0
$705$	234515.	−	96595.9i	0.471838	−	0.194348i
$706$		0			0
$707$	247321.	+	142791.i	0.494791	+	0.285668i
$708$		0			0
$709$	325622.	+	563993.i	0.647770	+	1.12197i	0.983654	+	0.180067i	$0.0576314\pi$
$709$	325622.	+	563993.i	0.647770	+	1.12197i	−0.335885	+	0.941903i	$0.609035\pi$
$710$		0			0
$711$	94004.6	+	25587.8i	0.185956	+	0.0506167i
$712$		0			0
$713$	258668.	−	149342.i	0.508819	−	0.293767i
$714$		0			0
$715$	73487.2	−	127284.i	0.143747	−	0.248978i
$716$		0			0
$717$	138450.	+	18506.3i	0.269312	+	0.0359981i
$718$		0			0
$719$		−	69724.6i		−	0.134874i	−0.997724	−	0.0674370i	$-0.978518\pi$
$719$		−	69724.6i		−	0.134874i	0.997724	−	0.0674370i	$-0.0214822\pi$
$720$		0			0
$721$	−477913.			−0.919345
$722$		0			0
$723$	−513010.	−	395263.i	−0.981407	−	0.756152i
$724$		0			0
$725$	−395425.	−	228299.i	−0.752294	−	0.434337i
$726$		0			0
$727$	−373864.	−	647552.i	−0.707367	−	1.22520i	−0.965830	−	0.259175i	$-0.916549\pi$
$727$	−373864.	−	647552.i	−0.707367	−	1.22520i	0.258463	−	0.966021i	$-0.416784\pi$
$728$		0			0
$729$	−371295.	−	380223.i	−0.698658	−	0.715456i
$730$		0			0
$731$	−670313.	+	387006.i	−1.25442	+	0.724240i
$732$		0			0
$733$	−32869.3	+	56931.2i	−0.0611761	+	0.105960i	−0.894991	−	0.446083i	$-0.852818\pi$
$733$	−32869.3	+	56931.2i	−0.0611761	+	0.105960i	0.833815	+	0.552044i	$0.186152\pi$
$734$		0			0
$735$	272238.	−	353337.i	0.503934	−	0.654054i
$736$		0			0
$737$			260201.i			0.479042i
$738$		0			0
$739$	−977921.			−1.79067			−0.895334	−	0.445395i	$-0.853063\pi$
$739$	−977921.			−1.79067			−0.895334	+	0.445395i	$0.853063\pi$
$740$		0			0
$741$	10119.8	−	75709.3i	0.0184305	−	0.137884i
$742$		0			0
$743$	−437233.	−	252437.i	−0.792019	−	0.457272i	0.0486539	−	0.998816i	$-0.484507\pi$
$743$	−437233.	−	252437.i	−0.792019	−	0.457272i	−0.840673	+	0.541543i	$0.817840\pi$
$744$		0			0
$745$	−649209.	−	1.12446e6i	−1.16969	−	2.02597i
$746$		0			0
$747$	−196919.	+	723442.i	−0.352896	+	1.29647i
$748$		0			0
$749$	164860.	−	95181.9i	0.293867	−	0.169664i
$750$		0			0
$751$	−266395.	+	461411.i	−0.472332	+	0.818102i	−0.999499	−	0.0316593i	$-0.989921\pi$
$751$	−266395.	+	461411.i	−0.472332	+	0.818102i	0.527167	+	0.849762i	$0.323254\pi$
$752$		0			0
$753$	142207.	+	345250.i	0.250802	+	0.608896i
$754$		0			0
$755$		−	1.15559e6i		−	2.02727i
$756$		0			0
$757$	−293571.			−0.512297			−0.256148	−	0.966637i	$-0.582454\pi$
$757$	−293571.			−0.512297			−0.256148	+	0.966637i	$0.582454\pi$
$758$		0			0
$759$	263422.	−	108503.i	0.457266	−	0.188346i
$760$		0			0
$761$	−22249.2	−	12845.6i	−0.0384190	−	0.0221812i	0.480667	−	0.876903i	$-0.340394\pi$
$761$	−22249.2	−	12845.6i	−0.0384190	−	0.0221812i	−0.519086	+	0.854722i	$0.673728\pi$
$762$		0			0
$763$	−236727.	−	410022.i	−0.406629	−	0.704301i
$764$		0			0
$765$	−277625.	−	1.05277e6i	−0.474390	−	1.79891i
$766$		0			0
$767$	−95991.7	+	55420.9i	−0.163171	+	0.0942069i
$768$		0			0
$769$	323572.	−	560443.i	0.547165	−	0.947717i	−0.451302	−	0.892371i	$-0.649040\pi$
$769$	323572.	−	560443.i	0.547165	−	0.947717i	0.998467	−	0.0553463i	$-0.0176263\pi$
$770$		0			0
$771$	647009.	+	86483.8i	1.08843	+	0.145488i
$772$		0			0
$773$		−	610238.i		−	1.02127i	−0.859798	−	0.510634i	$-0.829411\pi$
$773$		−	610238.i		−	1.02127i	0.859798	−	0.510634i	$-0.170589\pi$
$774$		0			0
$775$	316728.			0.527331
$776$		0			0
$777$	200000.	+	154095.i	0.331274	+	0.255239i
$778$		0			0
$779$	−259605.	−	149883.i	−0.427797	−	0.246989i
$780$		0			0
$781$	−115276.	−	199665.i	−0.188990	−	0.327340i
$782$		0			0
$783$	−71351.0	−	568013.i	−0.116379	−	0.926477i
$784$		0			0
$785$	243059.	−	140330.i	0.394432	−	0.227726i
$786$		0			0
$787$	−290160.	+	502572.i	−0.468477	+	0.811427i	−0.999351	−	0.0360244i	$-0.988531\pi$
$787$	−290160.	+	502572.i	−0.468477	+	0.811427i	0.530874	+	0.847451i	$0.321864\pi$
$788$		0			0
$789$	−471793.	+	612339.i	−0.757875	+	0.983644i
$790$		0			0
$791$			42854.5i			0.0684926i
$792$		0			0
$793$	139449.			0.221753
$794$		0			0
$795$	91986.0	−	688172.i	0.145542	−	1.08884i
$796$		0			0
$797$	−915508.	−	528569.i	−1.44127	−	0.832118i	−0.443336	−	0.896355i	$-0.646205\pi$
$797$	−915508.	−	528569.i	−1.44127	−	0.832118i	−0.997935	+	0.0642373i	$0.979539\pi$
$798$		0			0
$799$	156989.	+	271913.i	0.245910	+	0.425928i
$800$		0			0
$801$	509442.	+	513492.i	0.794016	+	0.800329i
$802$		0			0
$803$	171824.	−	99202.7i	0.266473	−	0.153848i
$804$		0			0
$805$	−297205.	+	514774.i	−0.458631	+	0.794373i
$806$		0			0
$807$	304953.	+	740363.i	0.468258	+	1.13684i
$808$		0			0
$809$			355969.i			0.543895i	0.962312	+	0.271947i	$0.0876677\pi$
$809$			355969.i			0.543895i	−0.962312	+	0.271947i	$0.912332\pi$
$810$		0			0
$811$	136417.			0.207409			0.103704	−	0.994608i	$-0.466930\pi$
$811$	136417.			0.207409			0.103704	+	0.994608i	$0.466930\pi$
$812$		0			0
$813$	803392.	−	330914.i	1.21548	−	0.500650i
$814$		0			0
$815$	−764857.	−	441590.i	−1.15150	−	0.664820i
$816$		0			0
$817$	−115797.	−	200567.i	−0.173482	−	0.300479i
$818$		0			0
$819$	−131543.	+	130505.i	−0.196109	+	0.194563i
$820$		0			0
$821$	−394889.	+	227989.i	−0.585854	+	0.338243i	−0.763456	−	0.645860i	$-0.776499\pi$
$821$	−394889.	+	227989.i	−0.585854	+	0.338243i	0.177603	+	0.984102i	$0.443166\pi$
$822$		0			0
$823$	251795.	−	436122.i	0.371747	−	0.643885i	−0.618087	−	0.786109i	$-0.712092\pi$
$823$	251795.	−	436122.i	0.371747	−	0.643885i	0.989834	+	0.142225i	$0.0454255\pi$
$824$		0			0
$825$	299441.	+	40025.4i	0.439951	+	0.0588069i
$826$		0			0
$827$			551861.i			0.806898i	0.915002	+	0.403449i	$0.132189\pi$
$827$			551861.i			0.806898i	−0.915002	+	0.403449i	$0.867811\pi$
$828$		0			0
$829$	−182308.			−0.265275			−0.132638	−	0.991165i	$-0.542345\pi$
$829$	−182308.			−0.265275			−0.132638	+	0.991165i	$0.542345\pi$
$830$		0			0
$831$	−167836.	−	129314.i	−0.243042	−	0.187259i
$832$		0			0
$833$	478203.	+	276091.i	0.689164	+	0.397889i
$834$		0			0
$835$	−608759.	−	1.05440e6i	−0.873117	−	1.51228i
$836$		0			0
$837$	239870.	+	316479.i	0.342394	+	0.451746i
$838$		0			0
$839$	359993.	−	207842.i	0.511410	−	0.295263i	−0.222003	−	0.975046i	$-0.571259\pi$
$839$	359993.	−	207842.i	0.511410	−	0.295263i	0.733413	+	0.679783i	$0.237926\pi$
$840$		0			0
$841$	−45299.7	+	78461.4i	−0.0640477	+	0.110934i
$842$		0			0
$843$	320844.	−	416423.i	0.451480	−	0.585975i
$844$		0			0
$845$		−	805434.i		−	1.12802i
$846$		0			0
$847$	352935.			0.491958
$848$		0			0
$849$	90762.1	−	679016.i	0.125918	−	0.942030i
$850$		0			0
$851$	426809.	+	246419.i	0.589352	+	0.340263i
$852$		0			0
$853$	−75135.3	−	130138.i	−0.103263	−	0.178857i	0.809764	−	0.586756i	$-0.199595\pi$
$853$	−75135.3	−	130138.i	−0.103263	−	0.178857i	−0.913027	+	0.407898i	$0.866262\pi$
$854$		0			0
$855$	315002.	−	83069.1i	0.430904	−	0.113634i
$856$		0			0
$857$	125373.	−	72383.9i	0.170703	−	0.0985554i	−0.412214	−	0.911087i	$-0.635245\pi$
$857$	125373.	−	72383.9i	0.170703	−	0.0985554i	0.582917	+	0.812532i	$0.301911\pi$
$858$		0			0
$859$	166948.	−	289163.i	0.226254	−	0.391883i	−0.730441	−	0.682976i	$-0.760685\pi$
$859$	166948.	−	289163.i	0.226254	−	0.391883i	0.956695	+	0.291093i	$0.0940188\pi$
$860$		0			0
$861$	276958.	+	672398.i	0.373601	+	0.907026i
$862$		0			0
$863$			1.13280e6i			1.52101i	0.649330	+	0.760507i	$0.275049\pi$
$863$			1.13280e6i			1.52101i	−0.649330	+	0.760507i	$0.724951\pi$
$864$		0			0
$865$	−510603.			−0.682418
$866$		0			0
$867$	551198.	−	227036.i	0.733279	−	0.302035i
$868$		0			0
$869$	−60135.0	−	34718.9i	−0.0796320	−	0.0459755i
$870$		0			0
$871$	−165175.	−	286092.i	−0.217725	−	0.377111i
$872$		0			0
$873$	−521325.	−	141903.i	−0.684038	−	0.186193i
$874$		0			0
$875$	40898.3	−	23612.6i	0.0534181	−	0.0308410i
$876$		0			0
$877$	−248271.	+	430017.i	−0.322795	+	0.559097i	−0.981064	−	0.193686i	$-0.937956\pi$
$877$	−248271.	+	430017.i	−0.322795	+	0.559097i	0.658269	+	0.752783i	$0.271289\pi$
$878$		0			0
$879$	−1.24810e6	−	166830.i	−1.61537	−	0.215921i
$880$		0			0
$881$			1.21533e6i			1.56583i	0.622131	+	0.782913i	$0.286267\pi$
$881$			1.21533e6i			1.56583i	−0.622131	+	0.782913i	$0.713733\pi$
$882$		0			0
$883$	999070.			1.28137			0.640685	−	0.767804i	$-0.278651\pi$
$883$	999070.			1.28137			0.640685	+	0.767804i	$0.278651\pi$
$884$		0			0
$885$	−374478.	−	288527.i	−0.478123	−	0.368383i
$886$		0			0
$887$	−938048.	−	541582.i	−1.19228	−	0.688362i	−0.233456	−	0.972367i	$-0.575003\pi$
$887$	−938048.	−	541582.i	−1.19228	−	0.688362i	−0.958823	+	0.284005i	$0.908337\pi$
$888$		0			0
$889$	−298882.	−	517678.i	−0.378178	−	0.655023i
$890$		0			0
$891$	186784.	+	329519.i	0.235280	+	0.415073i
$892$		0			0
$893$	−81360.0	+	46973.2i	−0.102025	+	0.0589044i
$894$		0			0
$895$	−643488.	+	1.11455e6i	−0.803330	+	1.39141i
$896$		0			0
$897$	−220756.	+	286519.i	−0.274365	+	0.356097i
$898$		0			0
$899$			427774.i			0.529292i
$900$		0			0
$901$	859491.			1.05875
$902$		0			0
$903$	−74435.6	+	556873.i	−0.0912862	+	0.682937i
$904$		0			0
$905$	1.19313e6	+	688853.i	1.45677	+	0.841065i
$906$		0			0
$907$	638426.	+	1.10579e6i	0.776061	+	1.34418i	0.934196	+	0.356760i	$0.116118\pi$
$907$	638426.	+	1.10579e6i	0.776061	+	1.34418i	−0.158135	+	0.987417i	$0.550548\pi$
$908$		0			0
$909$	−194658.	+	715136.i	−0.235583	+	0.865487i
$910$		0			0
$911$	−236846.	+	136743.i	−0.285383	+	0.164766i	−0.635858	−	0.771806i	$-0.719354\pi$
$911$	−236846.	+	136743.i	−0.285383	+	0.164766i	0.350475	+	0.936572i	$0.386020\pi$
$912$		0			0
$913$	267191.	−	462788.i	0.320538	−	0.555188i
$914$		0			0
$915$	226512.	+	549926.i	0.270551	+	0.656844i
$916$		0			0
$917$		−	94256.7i		−	0.112092i
$918$		0			0
$919$	−1.15882e6			−1.37210			−0.686051	−	0.727553i	$-0.740657\pi$
$919$	−1.15882e6			−1.37210			−0.686051	+	0.727553i	$0.740657\pi$
$920$		0			0
$921$	−680690.	+	280373.i	−0.802472	+	0.330535i
$922$		0			0
$923$	253493.	+	146355.i	0.297552	+	0.171792i
$924$		0			0
$925$	261306.	+	452595.i	0.305397	+	0.528964i
$926$		0			0
$927$	−316268.	−	1.19930e6i	−0.368041	−	1.39563i
$928$		0			0
$929$	738465.	−	426353.i	0.855655	−	0.494012i	−0.00690018	−	0.999976i	$-0.502196\pi$
$929$	738465.	−	426353.i	0.855655	−	0.494012i	0.862555	+	0.505964i	$0.168863\pi$
$930$		0			0
$931$	−82610.0	+	143085.i	−0.0953089	+	0.165080i
$932$		0			0
$933$	−15189.3	−	2030.31i	−0.0174491	−	0.00233237i
$934$		0			0
$935$			775993.i			0.887636i
$936$		0			0
$937$	629989.			0.717552			0.358776	−	0.933424i	$-0.383194\pi$
$937$	629989.			0.717552			0.358776	+	0.933424i	$0.383194\pi$
$938$		0			0
$939$	−498768.	−	384290.i	−0.565676	−	0.435840i
$940$		0			0
$941$	−366547.	−	211626.i	−0.413953	−	0.238996i	0.278534	−	0.960426i	$-0.410152\pi$
$941$	−366547.	−	211626.i	−0.413953	−	0.238996i	−0.692487	+	0.721431i	$0.743485\pi$
$942$		0			0
$943$	709749.	+	1.22932e6i	0.798145	+	1.38243i
$944$		0			0
$945$	−728322.	−	306761.i	−0.815568	−	0.343508i
$946$		0			0
$947$	1.20762e6	−	697219.i	1.34657	−	0.777444i	0.358810	−	0.933411i	$-0.383183\pi$
$947$	1.20762e6	−	697219.i	1.34657	−	0.777444i	0.987762	+	0.155967i	$0.0498493\pi$
$948$		0			0
$949$	−125947.	+	218147.i	−0.139848	+	0.242224i
$950$		0			0
$951$	−640642.	+	831488.i	−0.708361	+	0.919379i
$952$		0			0
$953$		−	24001.0i		−	0.0264267i	−0.999913	−	0.0132134i	$-0.995794\pi$
$953$		−	24001.0i		−	0.0264267i	0.999913	−	0.0132134i	$-0.00420607\pi$
$954$		0			0
$955$	2.00195e6			2.19506
$956$		0			0
$957$	−54058.5	+	404426.i	−0.0590255	+	0.441586i
$958$		0			0
$959$	274290.	+	158362.i	0.298245	+	0.172192i
$960$		0			0
$961$	313393.	+	542813.i	0.339346	+	0.587765i
$962$		0			0
$963$	347954.	+	350721.i	0.375206	+	0.378189i
$964$		0			0
$965$	125725.	−	72587.2i	0.135010	−	0.0779481i
$966$		0			0
$967$	522031.	−	904183.i	0.558268	−	0.966949i	−0.439373	−	0.898305i	$-0.644799\pi$
$967$	522031.	−	904183.i	0.558268	−	0.966949i	0.997641	−	0.0686443i	$-0.0218673\pi$
$968$		0			0
$969$	153592.	+	372890.i	0.163577	+	0.397131i
$970$		0			0
$971$		−	605213.i		−	0.641904i	−0.947096	−	0.320952i	$-0.895997\pi$
$971$		−	605213.i		−	0.641904i	0.947096	−	0.320952i	$-0.104003\pi$
$972$		0			0
$973$	1.09632e6			1.15801
$974$		0			0
$975$	−354645.	+	146077.i	−0.373065	+	0.153664i
$976$		0			0
$977$	41660.2	+	24052.5i	0.0436448	+	0.0251983i	0.521664	−	0.853151i	$-0.325312\pi$
$977$	41660.2	+	24052.5i	0.0436448	+	0.0251983i	−0.478019	+	0.878350i	$0.658645\pi$
$978$		0			0
$979$	−257770.	−	446471.i	−0.268947	−	0.465831i
$980$		0			0
$981$	872276.	−	865396.i	0.906392	−	0.899243i
$982$		0			0
$983$	107286.	−	61941.5i	0.111029	−	0.0641024i	−0.443457	−	0.896295i	$-0.646248\pi$
$983$	107286.	−	61941.5i	0.111029	−	0.0641024i	0.554486	+	0.832193i	$0.312915\pi$
$984$		0			0
$985$	385397.	−	667527.i	0.397224	−	0.688012i
$986$		0			0
$987$	225896.	+	30194.9i	0.231886	+	0.0309955i
$988$		0			0
$989$			1.09668e6i			1.12121i
$990$		0			0
$991$	−851418.			−0.866953			−0.433477	−	0.901165i	$-0.642713\pi$
$991$	−851418.			−0.866953			−0.433477	+	0.901165i	$0.642713\pi$
$992$		0			0
$993$	−718894.	−	553891.i	−0.729065	−	0.561728i
$994$		0			0
$995$	1.52233e6	+	878917.i	1.53767	+	0.887773i
$996$		0			0
$997$	801524.	+	1.38828e6i	0.806355	+	1.39665i	0.915373	+	0.402607i	$0.131896\pi$
$997$	801524.	+	1.38828e6i	0.806355	+	1.39665i	−0.109018	+	0.994040i	$0.534771\pi$
$998$		0			0
$999$	−254342.	+	603867.i	−0.254852	+	0.605077i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.5.q.a.113.1		6
3.2	odd	2		432.5.q.a.17.3		6
4.3	odd	2		9.5.d.a.5.2	yes	6
9.2	odd	6	inner	144.5.q.a.65.1		6
9.4	even	3		1296.5.e.c.161.6		6
9.5	odd	6		1296.5.e.c.161.1		6
9.7	even	3		432.5.q.a.305.3		6
12.11	even	2		27.5.d.a.17.2		6
36.7	odd	6		27.5.d.a.8.2		6
36.11	even	6		9.5.d.a.2.2	✓	6
36.23	even	6		81.5.b.a.80.3		6
36.31	odd	6		81.5.b.a.80.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.5.d.a.2.2	✓	6	36.11	even	6
9.5.d.a.5.2	yes	6	4.3	odd	2
27.5.d.a.8.2		6	36.7	odd	6
27.5.d.a.17.2		6	12.11	even	2
81.5.b.a.80.3		6	36.23	even	6
81.5.b.a.80.4		6	36.31	odd	6
144.5.q.a.65.1		6	9.2	odd	6	inner
144.5.q.a.113.1		6	1.1	even	1	trivial
432.5.q.a.17.3		6	3.2	odd	2
432.5.q.a.305.3		6	9.7	even	3
1296.5.e.c.161.1		6	9.5	odd	6
1296.5.e.c.161.6		6	9.4	even	3

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	144.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-8.32172 + 3.42768i) q^{3} +(-30.0804 - 17.3669i) q^{5} +(-15.6054 - 27.0294i) q^{7} +(57.5020 - 57.0484i) q^{9} +(-49.9968 + 28.8657i) q^{11} +(36.6478 - 63.4758i) q^{13} +(309.848 + 41.4165i) q^{15} +386.985i q^{17} -115.791 q^{19} +(222.512 + 171.441i) q^{21} +(474.852 + 274.156i) q^{23} +(290.719 + 503.539i) q^{25} +(-282.971 + 671.839i) q^{27} +(-680.082 + 392.645i) q^{29} +(272.367 - 471.753i) q^{31} +(317.117 - 411.585i) q^{33} +1084.07i q^{35} +898.827 q^{37} +(-87.3975 + 653.845i) q^{39} +(2242.01 + 1294.43i) q^{41} +(1000.05 + 1732.14i) q^{43} +(-2720.43 + 717.406i) q^{45} +(702.646 - 405.673i) q^{47} +(713.441 - 1235.72i) q^{49} +(-1326.46 - 3220.38i) q^{51} -2221.00i q^{53} +2005.23 q^{55} +(963.579 - 396.895i) q^{57} +(-1309.65 - 756.128i) q^{59} +(951.281 + 1647.67i) q^{61} +(-2439.33 - 663.979i) q^{63} +(-2204.76 + 1272.92i) q^{65} +(2253.55 - 3903.26i) q^{67} +(-4891.30 - 653.806i) q^{69} +3993.54i q^{71} -3436.70 q^{73} +(-4145.25 - 3193.82i) q^{75} +(1560.44 + 900.923i) q^{77} +(601.388 + 1041.63i) q^{79} +(51.9555 - 6560.79i) q^{81} +(-8016.22 + 4628.17i) q^{83} +(6720.72 - 11640.6i) q^{85} +(4313.58 - 5598.59i) q^{87} +8929.99i q^{89} -2287.62 q^{91} +(-649.539 + 4859.38i) q^{93} +(3483.03 + 2010.93i) q^{95} +(-3335.14 - 5776.64i) q^{97} +(-1228.18 + 4512.07i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{3} - 12 q^{5} - 12 q^{7} + 99 q^{9} - 483 q^{11} - 6 q^{13} + 1026 q^{15} + 258 q^{19} + 480 q^{21} + 282 q^{23} - 273 q^{25} - 54 q^{27} - 1056 q^{29} - 1290 q^{31} + 279 q^{33} + 12 q^{37}+ \cdots + 9126 q^{99}+O(q^{100}) \) 6 * q + 3 * q^3 - 12 * q^5 - 12 * q^7 + 99 * q^9 - 483 * q^11 - 6 * q^13 + 1026 * q^15 + 258 * q^19 + 480 * q^21 + 282 * q^23 - 273 * q^25 - 54 * q^27 - 1056 * q^29 - 1290 * q^31 + 279 * q^33 + 12 * q^37 - 1974 * q^39 + 7629 * q^41 + 285 * q^43 - 4212 * q^45 + 9642 * q^47 - 1863 * q^49 - 2457 * q^51 - 2016 * q^55 + 5367 * q^57 - 6225 * q^59 + 3630 * q^61 + 7578 * q^63 - 7158 * q^65 + 5055 * q^67 - 13878 * q^69 - 14622 * q^73 - 21021 * q^75 + 2580 * q^77 - 4764 * q^79 + 18387 * q^81 + 1866 * q^83 + 12366 * q^85 + 21564 * q^87 - 34836 * q^91 + 19254 * q^93 + 13362 * q^95 - 28959 * q^97 + 9126 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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