LMFDB - Embedded newform 27.5.d.a.17.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [27,5,Mod(8,27)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(27, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("27.8");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 27 = 3^{3} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	27.d (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:	$2.79098900326$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	no (minimal twist has level 9)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.2
Root			$-0.102534 + 0.177594i$ of defining polynomial
Character	$\chi$	$=$	27.17
Dual form			27.5.d.a.8.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.307601 + 0.177594i) q^{2} +(-7.93692 + 13.7472i) q^{4} +(30.0804 + 17.3669i) q^{5} +(15.6054 + 27.0294i) q^{7} -11.3212i q^{8} -12.3370 q^{10} +(-49.9968 + 28.8657i) q^{11} +(36.6478 - 63.4758i) q^{13} +(-9.60051 - 5.54285i) q^{14} +(-124.980 - 216.472i) q^{16} -386.985i q^{17} +115.791 q^{19} +(-477.491 + 275.679i) q^{20} +(10.2527 - 17.7583i) q^{22} +(474.852 + 274.156i) q^{23} +(290.719 + 503.539i) q^{25} +26.0337i q^{26} -495.436 q^{28} +(680.082 - 392.645i) q^{29} +(-272.367 + 471.753i) q^{31} +(233.759 + 134.961i) q^{32} +(68.7261 + 119.037i) q^{34} +1084.07i q^{35} +898.827 q^{37} +(-35.6175 + 20.5637i) q^{38} +(196.614 - 340.546i) q^{40} +(-2242.01 - 1294.43i) q^{41} +(-1000.05 - 1732.14i) q^{43} -916.419i q^{44} -194.753 q^{46} +(702.646 - 405.673i) q^{47} +(713.441 - 1235.72i) q^{49} +(-178.851 - 103.260i) q^{50} +(581.741 + 1007.61i) q^{52} +2221.00i q^{53} -2005.23 q^{55} +(306.005 - 176.672i) q^{56} +(-139.463 + 241.557i) q^{58} +(-1309.65 - 756.128i) q^{59} +(951.281 + 1647.67i) q^{61} -193.483i q^{62} +3903.49 q^{64} +(2204.76 - 1272.92i) q^{65} +(-2253.55 + 3903.26i) q^{67} +(5319.94 + 3071.47i) q^{68} +(-192.524 - 333.462i) q^{70} +3993.54i q^{71} -3436.70 q^{73} +(-276.480 + 159.626i) q^{74} +(-919.023 + 1591.80i) q^{76} +(-1560.44 - 900.923i) q^{77} +(-601.388 - 1041.63i) q^{79} -8682.07i q^{80} +919.529 q^{82} +(-8016.22 + 4628.17i) q^{83} +(6720.72 - 11640.6i) q^{85} +(615.236 + 355.207i) q^{86} +(326.794 + 566.024i) q^{88} -8929.99i q^{89} +2287.62 q^{91} +(-7537.72 + 4351.91i) q^{92} +(-144.090 + 249.571i) q^{94} +(3483.03 + 2010.93i) q^{95} +(-3335.14 - 5776.64i) q^{97} +506.811i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{2} + 15 q^{4} + 12 q^{5} + 12 q^{7} - 36 q^{10} - 483 q^{11} - 6 q^{13} + 1146 q^{14} + 15 q^{16} - 258 q^{19} - 1614 q^{20} - 369 q^{22} + 282 q^{23} - 273 q^{25} + 1308 q^{28} + 1056 q^{29}+ \cdots - 28959 q^{97}+O(q^{100}) $ 6 * q + 3 * q^2 + 15 * q^4 + 12 * q^5 + 12 * q^7 - 36 * q^10 - 483 * q^11 - 6 * q^13 + 1146 * q^14 + 15 * q^16 - 258 * q^19 - 1614 * q^20 - 369 * q^22 + 282 * q^23 - 273 * q^25 + 1308 * q^28 + 1056 * q^29 + 1290 * q^31 + 1161 * q^32 + 513 * q^34 + 12 * q^37 + 789 * q^38 - 1314 * q^40 - 7629 * q^41 - 285 * q^43 - 5760 * q^46 + 9642 * q^47 - 1863 * q^49 - 3027 * q^50 - 240 * q^52 + 2016 * q^55 + 462 * q^56 + 6462 * q^58 - 6225 * q^59 + 3630 * q^61 + 15450 * q^64 + 7158 * q^65 - 5055 * q^67 + 10503 * q^68 - 9684 * q^70 - 14622 * q^73 - 26454 * q^74 - 4047 * q^76 - 2580 * q^77 + 4764 * q^79 - 9702 * q^82 + 1866 * q^83 + 12366 * q^85 + 37731 * q^86 + 14787 * q^88 + 34836 * q^91 - 33636 * q^92 - 12708 * q^94 + 13362 * q^95 - 28959 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/27\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$e\left(\frac{5}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−0.307601	+	0.177594i	−0.0769004	+	0.0443984i	−0.537957	−	0.842972i	$-0.680804\pi$
$2$	−0.307601	+	0.177594i	−0.0769004	+	0.0443984i	0.461057	+	0.887371i	$0.347470\pi$
$3$		0			0
$3$		0			0
$4$	−7.93692	+	13.7472i	−0.496058	+	0.859197i
$5$	30.0804	+	17.3669i	1.20321	+	0.694676i	0.961268	−	0.275614i	$-0.0888811\pi$
$5$	30.0804	+	17.3669i	1.20321	+	0.694676i	0.241946	+	0.970290i	$0.422214\pi$
$6$		0			0
$7$	15.6054	+	27.0294i	0.318478	+	0.551620i	0.980171	−	0.198155i	$-0.0634949\pi$
$7$	15.6054	+	27.0294i	0.318478	+	0.551620i	−0.661693	+	0.749775i	$0.730162\pi$
$8$		−	11.3212i		−	0.176894i
$9$		0			0
$10$	−12.3370			−0.123370
$11$	−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	$0.0970882\pi$
$14$	−9.60051	−	5.54285i	−0.0489822	−	0.0282799i
$15$		0			0
$16$	−124.980	−	216.472i	−0.488204	−	0.845594i
$17$		−	386.985i		−	1.33905i	−0.742791	−	0.669524i	$-0.766498\pi$
$17$		−	386.985i		−	1.33905i	0.742791	−	0.669524i	$-0.233502\pi$
$18$		0			0
$19$	115.791			0.320750			0.160375	−	0.987056i	$-0.448730\pi$
$19$	115.791			0.320750			0.160375	+	0.987056i	$0.448730\pi$
$20$	−477.491	+	275.679i	−1.19373	+	0.689199i
$21$		0			0
$22$	10.2527	−	17.7583i	0.0211833	−	0.0366906i
$23$	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
$26$			26.0337i			0.0385113i
$27$		0			0
$28$	−495.436			−0.631934
$29$	680.082	−	392.645i	0.808658	−	0.466879i	−0.0378313	−	0.999284i	$-0.512045\pi$
$29$	680.082	−	392.645i	0.808658	−	0.466879i	0.846490	+	0.532405i	$0.178712\pi$
$30$		0			0
$31$	−272.367	+	471.753i	−0.283420	+	0.490898i	−0.972225	−	0.234049i	$-0.924802\pi$
$31$	−272.367	+	471.753i	−0.283420	+	0.490898i	0.688805	+	0.724947i	$0.258136\pi$
$32$	233.759	+	134.961i	0.228280	+	0.131798i
$33$		0			0
$34$	68.7261	+	119.037i	0.0594516	+	0.102973i
$35$			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	$0.393532\pi$
$38$	−35.6175	+	20.5637i	−0.0246658	+	0.0142408i
$39$		0			0
$40$	196.614	−	340.546i	0.122884	−	0.212841i
$41$	−2242.01	−	1294.43i	−1.33374	−	0.770034i	−0.347867	−	0.937544i	$-0.613094\pi$
$41$	−2242.01	−	1294.43i	−1.33374	−	0.770034i	−0.985870	+	0.167510i	$0.946427\pi$
$42$		0			0
$43$	−1000.05	−	1732.14i	−0.540862	−	0.936801i	−0.998855	−	0.0478447i	$-0.984765\pi$
$43$	−1000.05	−	1732.14i	−0.540862	−	0.936801i	0.457993	−	0.888956i	$-0.348569\pi$
$44$		−	916.419i		−	0.473357i
$45$		0			0
$46$	−194.753			−0.0920385
$47$	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
$50$	−178.851	−	103.260i	−0.0715404	−	0.0413039i
$51$		0			0
$52$	581.741	+	1007.61i	0.215141	+	0.372635i
$53$			2221.00i			0.790672i	0.918537	+	0.395336i	$0.129372\pi$
$53$			2221.00i			0.790672i	−0.918537	+	0.395336i	$0.870628\pi$
$54$		0			0
$55$	−2005.23			−0.662886
$56$	306.005	−	176.672i	0.0975781	−	0.0563368i
$57$		0			0
$58$	−139.463	+	241.557i	−0.0414574	+	0.0718064i
$59$	−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	$0.751043\pi$
$62$		−	193.483i		−	0.0503337i
$63$		0			0
$64$	3903.49			0.953001
$65$	2204.76	−	1272.92i	0.521836	−	0.301282i
$66$		0			0
$67$	−2253.55	+	3903.26i	−0.502015	+	0.869516i	0.497982	+	0.867187i	$0.334075\pi$
$67$	−2253.55	+	3903.26i	−0.502015	+	0.869516i	−0.999997	+	0.00232883i	$0.999259\pi$
$68$	5319.94	+	3071.47i	1.15051	+	0.664244i
$69$		0			0
$70$	−192.524	−	333.462i	−0.0392907	−	0.0680535i
$71$			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	$0.604507\pi$
$74$	−276.480	+	159.626i	−0.0504895	+	0.0291501i
$75$		0			0
$76$	−919.023	+	1591.80i	−0.159111	+	0.275588i
$77$	−1560.44	−	900.923i	−0.263188	−	0.151952i
$78$		0			0
$79$	−601.388	−	1041.63i	−0.0963608	−	0.166902i	0.813815	−	0.581124i	$-0.197387\pi$
$79$	−601.388	−	1041.63i	−0.0963608	−	0.166902i	−0.910176	+	0.414222i	$0.864054\pi$
$80$		−	8682.07i		−	1.35657i
$81$		0			0
$82$	919.529			0.136753
$83$	−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
$86$	615.236	+	355.207i	0.0831850	+	0.0480269i
$87$		0			0
$88$	326.794	+	566.024i	0.0421996	+	0.0730919i
$89$		−	8929.99i		−	1.12738i	−0.825986	−	0.563691i	$-0.809381\pi$
$89$		−	8929.99i		−	1.12738i	0.825986	−	0.563691i	$-0.190619\pi$
$90$		0			0
$91$	2287.62			0.276249
$92$	−7537.72	+	4351.91i	−0.890563	+	0.514167i
$93$		0			0
$94$	−144.090	+	249.571i	−0.0163071	+	0.0282448i
$95$	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	$0.948670\pi$
$98$			506.811i			0.0527708i
$99$		0			0
$100$	−9229.64			−0.922964
$101$	7924.19	−	4575.03i	0.776805	−	0.448489i	−0.0584918	−	0.998288i	$-0.518629\pi$
$101$	7924.19	−	4575.03i	0.776805	−	0.448489i	0.835297	+	0.549799i	$0.185296\pi$
$102$		0			0
$103$	−7656.20	+	13260.9i	−0.721670	+	1.24997i	0.238660	+	0.971103i	$0.423292\pi$
$103$	−7656.20	+	13260.9i	−0.721670	+	1.24997i	−0.960330	+	0.278866i	$0.910041\pi$
$104$	−718.622	−	414.897i	−0.0664406	−	0.0383595i
$105$		0			0
$106$	−394.435	−	683.182i	−0.0351046	−	0.0608029i
$107$			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	$0.279599\pi$
$110$	616.812	−	356.116i	0.0509762	−	0.0294311i
$111$		0			0
$112$	3900.74	−	6756.28i	0.310964	−	0.538606i
$113$	1189.11	+	686.532i	0.0931246	+	0.0537655i	0.545839	−	0.837890i	$-0.316211\pi$
$113$	1189.11	+	686.532i	0.0931246	+	0.0537655i	−0.452714	+	0.891656i	$0.649544\pi$
$114$		0			0
$115$	9522.48	+	16493.4i	0.720036	+	1.24714i
$116$			12465.6i			0.926396i
$117$		0			0
$118$	537.135			0.0385762
$119$	10460.0	−	6039.06i	0.738646	−	0.426457i
$120$		0			0
$121$	−5654.04	+	9793.09i	−0.386179	+	0.668881i
$122$	−585.231	−	337.883i	−0.0393195	−	0.0227011i
$123$		0			0
$124$	−4323.51	−	7488.53i	−0.281185	−	0.487027i
$125$		−	1513.10i		−	0.0968386i
$126$		0			0
$127$	−19152.4			−1.18745			−0.593726	−	0.804667i	$-0.702344\pi$
$127$	−19152.4			−1.18745			−0.593726	+	0.804667i	$0.702344\pi$
$128$	−4940.87	+	2852.61i	−0.301567	+	0.174110i
$129$		0			0
$130$	−452.124	+	783.102i	−0.0267529	+	0.0463374i
$131$	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
$134$		−	1600.86i		−	0.0891548i
$135$		0			0
$136$	−4381.13			−0.236869
$137$	8788.30	−	5073.93i	0.468235	−	0.270336i	−0.247266	−	0.968948i	$-0.579532\pi$
$137$	8788.30	−	5073.93i	0.468235	−	0.270336i	0.715501	+	0.698612i	$0.246199\pi$
$138$		0			0
$139$	17563.2	−	30420.3i	0.909021	−	1.57447i	0.0935938	−	0.995610i	$-0.470165\pi$
$139$	17563.2	−	30420.3i	0.909021	−	1.57447i	0.815427	−	0.578860i	$-0.196502\pi$
$140$	−14902.9	−	8604.19i	−0.760352	−	0.438989i
$141$		0			0
$142$	−709.229	−	1228.42i	−0.0351730	−	0.0609214i
$143$			4231.45i			0.206927i
$144$		0			0
$145$	27276.1			1.29732
$146$	1057.13	−	610.336i	0.0495934	−	0.0286328i
$147$		0			0
$148$	−7133.92	+	12356.3i	−0.325690	+	0.564112i
$149$	−32373.7	−	18691.0i	−1.45821	−	0.841899i	−0.459288	−	0.888288i	$-0.651895\pi$
$149$	−32373.7	−	18691.0i	−1.45821	−	0.841899i	−0.998923	+	0.0463890i	$0.985229\pi$
$150$		0			0
$151$	−16635.0	−	28812.7i	−0.729573	−	1.26366i	−0.957064	−	0.289878i	$-0.906385\pi$
$151$	−16635.0	−	28812.7i	−0.729573	−	1.26366i	0.227490	−	0.973780i	$-0.426948\pi$
$152$		−	1310.89i		−	0.0567387i
$153$		0			0
$154$	639.993			0.0269857
$155$	−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	$0.219077\pi$
$158$	369.975	+	213.605i	0.0148204	+	0.00855654i
$159$		0			0
$160$	4687.71	+	8119.35i	0.183114	+	0.317162i
$161$			17113.3i			0.660209i
$162$		0			0
$163$	−25427.1			−0.957022			−0.478511	−	0.878081i	$-0.658823\pi$
$163$	−25427.1			−0.957022			−0.478511	+	0.878081i	$0.658823\pi$
$164$	35589.4	−	20547.5i	1.32322	−	0.763962i
$165$		0			0
$166$	1643.87	−	2847.26i	0.0596555	−	0.103326i
$167$	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
$170$			4774.24i			0.165198i
$171$		0			0
$172$	31749.4			1.07319
$173$	−12731.0	+	7350.23i	−0.425372	+	0.245589i	−0.697373	−	0.716708i	$-0.745648\pi$
$173$	−12731.0	+	7350.23i	−0.425372	+	0.245589i	0.272001	+	0.962297i	$0.412315\pi$
$174$		0			0
$175$	−9073.58	+	15715.9i	−0.296280	+	0.513172i
$176$	12497.2	+	7215.28i	0.403449	+	0.232931i
$177$		0			0
$178$	1585.91	+	2746.88i	0.0500540	+	0.0866960i
$179$		−	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	$0.706973\pi$
$182$	−703.674	+	406.267i	−0.0212436	+	0.0122650i
$183$		0			0
$184$	3103.77	−	5375.89i	0.0916757	−	0.158787i
$185$	27037.0	+	15609.8i	0.789979	+	0.456094i
$186$		0			0
$187$	11170.6	+	19348.0i	0.319442	+	0.553290i
$188$			12879.2i			0.364395i
$189$		0			0
$190$	−1428.51			−0.0395710
$191$	−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	$0.184534\pi$
$194$	2051.79	+	1184.60i	0.0545167	+	0.0314752i
$195$		0			0
$196$	11325.1	+	19615.6i	0.294800	+	0.510609i
$197$		−	22191.5i		−	0.571812i	−0.958258	−	0.285906i	$-0.907705\pi$
$197$		−	22191.5i		−	0.571812i	0.958258	−	0.285906i	$-0.0922945\pi$
$198$		0			0
$199$	50608.7			1.27797			0.638983	−	0.769221i	$-0.279355\pi$
$199$	50608.7			1.27797			0.638983	+	0.769221i	$0.279355\pi$
$200$	5700.67	−	3291.28i	0.142517	−	0.0822820i
$201$		0			0
$202$	−1624.99	+	2814.57i	−0.0398244	+	0.0689779i
$203$	21225.9	+	12254.8i	0.515080	+	0.297382i
$204$		0			0
$205$	−44960.4	−	77873.6i	−1.06985	−	1.85303i
$206$		−	5438.77i		−	0.128164i
$207$		0			0
$208$	−18321.0			−0.423469
$209$	−5789.18	+	3342.38i	−0.132533	+	0.0765180i
$210$		0			0
$211$	4459.02	−	7723.26i	0.100156	−	0.173474i	−0.811593	−	0.584223i	$-0.801399\pi$
$211$	4459.02	−	7723.26i	0.100156	−	0.173474i	0.911749	+	0.410749i	$0.134733\pi$
$212$	−30532.4	−	17627.9i	−0.679343	−	0.392219i
$213$		0			0
$214$	−1083.19	−	1876.15i	−0.0236526	−	0.0409675i
$215$		−	69471.4i		−	1.50290i
$216$		0			0
$217$	−17001.6			−0.361052
$218$	−4666.16	+	2694.01i	−0.0981853	+	0.0566873i
$219$		0			0
$220$	15915.4	−	27566.2i	0.328830	−	0.569550i
$221$	−24564.2	−	14182.1i	−0.502941	−	0.290373i
$222$		0			0
$223$	4248.76	+	7359.07i	0.0854384	+	0.147984i	0.905578	−	0.424180i	$-0.139438\pi$
$223$	4248.76	+	7359.07i	0.0854384	+	0.147984i	−0.820140	+	0.572164i	$0.806104\pi$
$224$			8424.49i			0.167899i
$225$		0			0
$226$	−487.695			−0.00954842
$227$	−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	$0.134066\pi$
$230$	−5858.25	−	3382.26i	−0.110742	−	0.0639370i
$231$		0			0
$232$	−4445.21	−	7699.34i	−0.0825880	−	0.143047i
$233$			102200.i			1.88251i	0.337694	+	0.941256i	$0.390353\pi$
$233$			102200.i			1.88251i	−0.337694	+	0.941256i	$0.609647\pi$
$234$		0			0
$235$	28181.1			0.510296
$236$	20789.2	−	12002.7i	0.373262	−	0.215503i
$237$		0			0
$238$	−2145.00	+	3715.25i	−0.0378681	+	0.0655894i
$239$	−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	$0.620685\pi$
$242$		−	4016.49i		−	0.0685830i
$243$		0			0
$244$	−30201.0			−0.507272
$245$	42921.1	−	24780.5i	0.715054	−	0.412837i
$246$		0			0
$247$	4243.48	−	7349.92i	0.0695550	−	0.120473i
$248$	5340.81	+	3083.52i	0.0868368	+	0.0501352i
$249$		0			0
$250$	268.718	+	465.433i	0.00429948	+	0.00744692i
$251$		−	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
$254$	5891.31	−	3401.35i	0.0913155	−	0.0527210i
$255$		0			0
$256$	−30214.7	+	52333.4i	−0.461040	+	0.798545i
$257$	62812.2	+	36264.6i	0.950994	+	0.549057i	0.893390	−	0.449283i	$-0.148320\pi$
$257$	62812.2	+	36264.6i	0.950994	+	0.549057i	0.0576045	+	0.998339i	$0.481654\pi$
$258$		0			0
$259$	14026.6	+	24294.7i	0.209099	+	0.362170i
$260$			40412.2i			0.597813i
$261$		0			0
$262$	−1072.67			−0.0156265
$263$	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
$266$	−1111.65	−	641.812i	−0.0157111	−	0.00907078i
$267$		0			0
$268$	−35772.5	−	61959.7i	−0.498057	−	0.862660i
$269$			88967.6i			1.22950i	0.788724	+	0.614748i	$0.210742\pi$
$269$			88967.6i			1.22950i	−0.788724	+	0.614748i	$0.789258\pi$
$270$		0			0
$271$	96541.6			1.31455			0.657273	−	0.753652i	$-0.271710\pi$
$271$	96541.6			1.31455			0.657273	+	0.753652i	$0.271710\pi$
$272$	−83771.3	+	48365.4i	−1.13229	+	0.653728i
$273$		0			0
$274$	−1802.20	+	3121.49i	−0.0240050	+	0.0415778i
$275$	−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	$0.784309\pi$
$278$			12476.5i			0.161436i
$279$		0			0
$280$	12273.0			0.156543
$281$	50584.4	−	29204.9i	0.640626	−	0.369865i	−0.144230	−	0.989544i	$-0.546070\pi$
$281$	50584.4	−	29204.9i	0.640626	−	0.369865i	0.784855	+	0.619679i	$0.212737\pi$
$282$		0			0
$283$	38058.6	−	65919.5i	0.475204	−	0.823078i	−0.524393	−	0.851477i	$-0.675708\pi$
$283$	38058.6	−	65919.5i	0.475204	−	0.823078i	0.999597	+	0.0283989i	$0.00904087\pi$
$284$	−54899.9	−	31696.4i	−0.680667	−	0.392983i
$285$		0			0
$286$	−751.480	−	1301.60i	−0.00918724	−	0.0159128i
$287$		−	80800.3i		−	0.980956i
$288$		0			0
$289$	−66236.1			−0.793047
$290$	−8390.18	+	4844.07i	−0.0997643	+	0.0575990i
$291$		0			0
$292$	27276.8	−	47244.8i	0.319910	−	0.554100i
$293$	−121166.	−	69955.5i	−1.41139	−	0.814866i	−0.415871	−	0.909424i	$-0.636523\pi$
$293$	−121166.	−	69955.5i	−1.41139	−	0.814866i	−0.995519	+	0.0945573i	$0.969856\pi$
$294$		0			0
$295$	−26263.2	−	45489.2i	−0.301789	−	0.522714i
$296$		−	10175.8i		−	0.116141i
$297$		0			0
$298$	13277.6			0.149516
$299$	34804.5	−	20094.4i	0.389308	−	0.224767i
$300$		0			0
$301$	31212.5	−	54061.7i	0.344505	−	0.596701i
$302$	10233.9	+	5908.55i	0.112209	+	0.0647838i
$303$		0			0
$304$	−14471.6	−	25065.5i	−0.156592	−	0.271225i
$305$			66083.2i			0.710381i
$306$		0			0
$307$	−81796.8			−0.867880			−0.433940	−	0.900942i	$-0.642877\pi$
$307$	−81796.8			−0.867880			−0.433940	+	0.900942i	$0.642877\pi$
$308$	24770.2	−	14301.1i	0.261113	−	0.150754i
$309$		0			0
$310$	3360.19	−	5820.03i	0.0349656	−	0.0605622i
$311$	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	$0.717115\pi$
$314$		−	2870.03i		−	0.0291090i
$315$		0			0
$316$	19092.7			0.191202
$317$	−101004.	+	58314.7i	−1.00512	+	0.580309i	−0.909760	−	0.415134i	$-0.863735\pi$
$317$	−101004.	+	58314.7i	−1.00512	+	0.580309i	−0.0953639	+	0.995442i	$0.530401\pi$
$318$		0			0
$319$	−22668.0	+	39262.1i	−0.222757	+	0.385826i
$320$	117418.	+	67791.6i	1.14666	+	0.662027i
$321$		0			0
$322$	−3039.21	−	5264.07i	−0.0293123	−	0.0507703i
$323$		−	44809.3i		−	0.429500i
$324$		0			0
$325$	42616.8			0.403472
$326$	7821.42	−	4515.70i	0.0735954	−	0.0424903i
$327$		0			0
$328$	−14654.5	+	25382.3i	−0.136214	+	0.235930i
$329$	21930.2	+	12661.4i	0.202605	+	0.116974i
$330$		0			0
$331$	−50418.1	−	87326.8i	−0.460183	−	0.797061i	0.538787	−	0.842442i	$-0.318883\pi$
$331$	−50418.1	−	87326.8i	−0.460183	−	0.797061i	−0.998970	+	0.0453817i	$0.985550\pi$
$332$		−	146934.i		−	1.33305i
$333$		0			0
$334$	−12450.3			−0.111606
$335$	−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	$0.223151\pi$
$338$	−7132.90	−	4118.18i	−0.0624356	−	0.0360472i
$339$		0			0
$340$	106684.	+	184782.i	0.922869	+	1.59846i
$341$		−	31448.2i		−	0.270450i
$342$		0			0
$343$	119471.			1.01549
$344$	−19609.9	+	11321.8i	−0.165714	+	0.0956751i
$345$		0			0
$346$	2610.71	−	4521.88i	0.0218075	−	0.0377717i
$347$	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	$0.939289\pi$
$350$		−	6445.64i		−	0.0526175i
$351$		0			0
$352$	−15583.0			−0.125766
$353$	121173.	−	69959.4i	0.972427	−	0.561431i	0.0724520	−	0.997372i	$-0.476918\pi$
$353$	121173.	−	69959.4i	0.972427	−	0.561431i	0.899975	+	0.435941i	$0.143584\pi$
$354$		0			0
$355$	−69355.5	+	120127.i	−0.550331	+	0.953202i
$356$	122762.	+	70876.6i	0.968642	+	0.559246i
$357$		0			0
$358$	6580.30	+	11397.4i	0.0513428	+	0.0889283i
$359$			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
$362$	12200.9	−	7044.20i	0.0931055	−	0.0537545i
$363$		0			0
$364$	−18156.6	+	31448.2i	−0.137035	+	0.237352i
$365$	−103377.	−	59684.8i	−0.775959	−	0.448000i
$366$		0			0
$367$	15996.8	+	27707.3i	0.118769	+	0.205713i	0.919280	−	0.393604i	$-0.128772\pi$
$367$	15996.8	+	27707.3i	0.118769	+	0.205713i	−0.800511	+	0.599318i	$0.795439\pi$
$368$		−	137056.i		−	1.01205i
$369$		0			0
$370$	−11088.8			−0.0809995
$371$	−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	$0.0207347\pi$
$374$	−6872.17	−	3967.65i	−0.0491305	−	0.0283655i
$375$		0			0
$376$	−4592.70	−	7954.79i	−0.0324857	−	0.0562669i
$377$		−	57558.3i		−	0.404972i
$378$		0			0
$379$	−116524.			−0.811218			−0.405609	−	0.914047i	$-0.632941\pi$
$379$	−116524.			−0.811218			−0.405609	+	0.914047i	$0.632941\pi$
$380$	−55289.1	+	31921.2i	−0.382889	+	0.221061i
$381$		0			0
$382$	10236.0	−	17729.2i	0.0701459	−	0.121496i
$383$	−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
$386$		−	1484.55i		−	0.00996371i
$387$		0			0
$388$	105883.			0.703336
$389$	44017.9	−	25413.7i	0.290891	−	0.167946i	−0.347453	−	0.937697i	$-0.612953\pi$
$389$	44017.9	−	25413.7i	0.290891	−	0.167946i	0.638343	+	0.769752i	$0.279620\pi$
$390$		0			0
$391$	106094.	−	183760.i	0.693965	−	1.20198i
$392$	−13989.8	−	8077.01i	−0.0910414	−	0.0525628i
$393$		0			0
$394$	3941.06	+	6826.12i	0.0253876	+	0.0439726i
$395$		−	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	$0.241580\pi$
$398$	−15567.3	+	8987.80i	−0.0982761	+	0.0567397i
$399$		0			0
$400$	72668.1	−	125865.i	0.454176	−	0.786655i
$401$	−163618.	−	94464.9i	−1.01752	−	0.587465i	−0.104134	−	0.994563i	$-0.533207\pi$
$401$	−163618.	−	94464.9i	−1.01752	−	0.587465i	−0.913384	+	0.407099i	$0.866541\pi$
$402$		0			0
$403$	19963.3	+	34577.4i	0.122920	+	0.212903i
$404$			145247.i			0.889904i
$405$		0			0
$406$	−8705.50			−0.0528131
$407$	−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	$0.855450\pi$
$410$	27659.7	+	15969.4i	0.164543	+	0.0949992i
$411$		0			0
$412$	−121533.	−	210502.i	−0.715980	−	1.24011i
$413$		−	47198.8i		−	0.276714i
$414$		0			0
$415$	−321508.			−1.86679
$416$	17133.5	−	9892.04i	0.0990056	−	0.0571609i
$417$		0			0
$418$	1187.17	−	2056.24i	0.00679456	−	0.0117685i
$419$	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	$0.875785\pi$
$422$			3167.58i			0.0177870i
$423$		0			0
$424$	25144.3			0.139865
$425$	194862.	−	112504.i	1.07882	−	0.622857i
$426$		0			0
$427$	−29690.3	+	51425.1i	−0.162839	+	0.282046i
$428$	−83847.7	−	48409.5i	−0.457724	−	0.264267i
$429$		0			0
$430$	12337.7	+	21369.5i	0.0667262	+	0.115573i
$431$			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	$0.311199\pi$
$434$	5229.72	−	3019.38i	0.0277651	−	0.0160302i
$435$		0			0
$436$	−120399.	+	208537.i	−0.633360	+	1.09701i
$437$	54983.5	+	31744.8i	0.287919	+	0.166230i
$438$		0			0
$439$	91842.1	+	159075.i	0.476555	+	0.825417i	0.999639	−	0.0268637i	$-0.00855202\pi$
$439$	91842.1	+	159075.i	0.476555	+	0.825417i	−0.523084	+	0.852281i	$0.675219\pi$
$440$			22701.6i			0.117260i
$441$		0			0
$442$	10074.6			0.0515685
$443$	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
$446$	−2613.85	−	1509.11i	−0.0131405	−	0.00758666i
$447$		0			0
$448$	60915.7	+	105509.i	0.303510	+	0.525695i
$449$			328940.i			1.63164i	0.578305	+	0.815820i	$0.303714\pi$
$449$			328940.i			1.63164i	−0.578305	+	0.815820i	$0.696286\pi$
$450$		0			0
$451$	149458.			0.734795
$452$	−18875.7	+	10897.9i	−0.0923903	+	0.0533416i
$453$		0			0
$454$	3392.54	−	5876.05i	0.0164594	−	0.0285084i
$455$	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	$0.663235\pi$
$458$			3808.68i			0.0181570i
$459$		0			0
$460$	−302317.			−1.42872
$461$	41957.2	−	24224.0i	0.197426	−	0.113984i	−0.398028	−	0.917373i	$-0.630305\pi$
$461$	41957.2	−	24224.0i	0.197426	−	0.113984i	0.595454	+	0.803389i	$0.296972\pi$
$462$		0			0
$463$	−84550.6	+	146446.i	−0.394416	+	0.683149i	−0.993026	−	0.117892i	$-0.962386\pi$
$463$	−84550.6	+	146446.i	−0.394416	+	0.683149i	0.598610	+	0.801040i	$0.295720\pi$
$464$	−169993.	−	98145.8i	−0.789580	−	0.455864i
$465$		0			0
$466$	−18150.0	−	31436.8i	−0.0835806	−	0.144766i
$467$		−	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
$470$	−8668.55	+	5004.79i	−0.0392420	+	0.0226564i
$471$		0			0
$472$	−8560.28	+	14826.8i	−0.0384241	+	0.0665525i
$473$	99999.1	+	57734.5i	0.446965	+	0.258055i
$474$		0			0
$475$	33662.6	+	58305.3i	0.149197	+	0.258417i
$476$			191726.i			0.846189i
$477$		0			0
$478$	5512.58			0.0241268
$479$	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
$482$	−22134.3	−	12779.3i	−0.0952735	−	0.0550062i
$483$		0			0
$484$	−89751.4	−	155454.i	−0.383134	−	0.663607i
$485$		−	231684.i		−	0.984948i
$486$		0			0
$487$	23464.1			0.0989340			0.0494670	−	0.998776i	$-0.484248\pi$
$487$	23464.1			0.0989340			0.0494670	+	0.998776i	$0.484248\pi$
$488$	18653.6	−	10769.6i	0.0783289	−	0.0452232i
$489$		0			0
$490$	−8801.74	+	15245.1i	−0.0366586	+	0.0634946i
$491$	−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
$494$			3014.46i			0.0123525i
$495$		0			0
$496$	136162.			0.553467
$497$	−107943.	+	62321.0i	−0.437001	+	0.252302i
$498$		0			0
$499$	9486.01	−	16430.2i	0.0380963	−	0.0659847i	−0.846349	−	0.532629i	$-0.821204\pi$
$499$	9486.01	−	16430.2i	0.0380963	−	0.0659847i	0.884445	+	0.466645i	$0.154537\pi$
$500$	20800.9	+	12009.4i	0.0832034	+	0.0480375i
$501$		0			0
$502$	7367.97	+	12761.7i	0.0292375	+	0.0506409i
$503$		−	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
$506$	9737.06	−	5621.69i	0.0380300	−	0.0219567i
$507$		0			0
$508$	152011.	−	263291.i	0.589045	−	1.02026i
$509$	−158881.	−	91730.2i	−0.613250	−	0.354060i	0.160986	−	0.986957i	$-0.448532\pi$
$509$	−158881.	−	91730.2i	−0.613250	−	0.354060i	−0.774236	+	0.632897i	$0.781866\pi$
$510$		0			0
$511$	−53631.2	−	92891.9i	−0.205388	−	0.355743i
$512$		−	112747.i		−	0.430097i
$513$		0			0
$514$	−25761.5			−0.0975091
$515$	−460602.	+	265929.i	−1.73665	+	1.00265i
$516$		0			0
$517$	−23420.0	+	40564.7i	−0.0876207	+	0.151763i
$518$	−8629.19	−	4982.06i	−0.0321596	−	0.0185673i
$519$		0			0
$520$	−14410.9	−	24960.5i	−0.0532949	−	0.0923094i
$521$		−	409498.i		−	1.50861i	−0.656526	−	0.754303i	$-0.727975\pi$
$521$		−	409498.i		−	1.50861i	0.656526	−	0.754303i	$-0.272025\pi$
$522$		0			0
$523$	−211852.			−0.774513			−0.387256	−	0.921972i	$-0.626577\pi$
$523$	−211852.			−0.774513			−0.387256	+	0.921972i	$0.626577\pi$
$524$	−41516.4	+	23969.5i	−0.151202	+	0.0872964i
$525$		0			0
$526$	−15253.6	+	26420.0i	−0.0551315	+	0.0954906i
$527$	182561.	+	105402.i	0.657336	+	0.379513i
$528$		0			0
$529$	10402.4	+	18017.4i	0.0371724	+	0.0643845i
$530$		−	27400.5i		−	0.0975453i
$531$		0			0
$532$	−57367.0			−0.202693
$533$	−164330.	+	94875.7i	−0.578444	+	0.333965i
$534$		0			0
$535$	−105926.	+	183468.i	−0.370078	+	0.640994i
$536$	44189.5	+	25512.8i	0.153812	+	0.0888033i
$537$		0			0
$538$	−15800.1	−	27366.6i	−0.0545877	−	0.0945487i
$539$			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	$0.476042\pi$
$542$	−29696.3	+	17145.2i	−0.101089	+	0.0583638i
$543$		0			0
$544$	52227.8	−	90461.2i	0.176483	−	0.305678i
$545$	456304.	+	263447.i	1.53625	+	0.886953i
$546$		0			0
$547$	21429.6	+	37117.1i	0.0716207	+	0.124051i	0.899612	−	0.436691i	$-0.143850\pi$
$547$	21429.6	+	37117.1i	0.0716207	+	0.124051i	−0.827991	+	0.560741i	$0.810516\pi$
$548$			161085.i			0.536408i
$549$		0			0
$550$	11922.6			0.0394137
$551$	78747.3	−	45464.8i	0.259378	−	0.149752i
$552$		0			0
$553$	18769.8	−	32510.3i	0.0613776	−	0.106309i
$554$	−7241.43	−	4180.84i	−0.0235942	−	0.0136221i
$555$		0			0
$556$	278795.	+	482888.i	0.901853	+	1.56206i
$557$		−	311007.i		−	1.00244i	−0.865319	−	0.501222i	$-0.832884\pi$
$557$		−	311007.i		−	1.00244i	0.865319	−	0.501222i	$-0.167116\pi$
$558$		0			0
$559$	−146599.			−0.469145
$560$	234671.	−	135487.i	0.748314	−	0.432039i
$561$		0			0
$562$	−10373.2	+	17967.0i	−0.0328429	+	0.0568856i
$563$	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
$566$			27035.9i			0.0843933i
$567$		0			0
$568$	45211.7			0.140137
$569$	118371.	−	68341.3i	0.365611	−	0.211086i	−0.305928	−	0.952055i	$-0.598967\pi$
$569$	118371.	−	68341.3i	0.365611	−	0.211086i	0.671539	+	0.740969i	$0.265633\pi$
$570$		0			0
$571$	14255.0	−	24690.4i	0.0437215	−	0.0757279i	−0.843337	−	0.537386i	$-0.819412\pi$
$571$	14255.0	−	24690.4i	0.0437215	−	0.0757279i	0.887058	+	0.461658i	$0.152745\pi$
$572$	−58170.4	−	33584.7i	−0.177791	−	0.102648i
$573$		0			0
$574$	14349.6	+	24854.3i	0.0435529	+	0.0754358i
$575$			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	$0.645429\pi$
$578$	20374.3	−	11763.1i	0.0609856	−	0.0352101i
$579$		0			0
$580$	−216489.	+	374969.i	−0.643545	+	1.11465i
$581$	−250193.	−	144449.i	−0.741179	−	0.427920i
$582$		0			0
$583$	−64110.6	−	111043.i	−0.188622	−	0.326703i
$584$			38907.5i			0.114080i
$585$		0			0
$586$	49694.6			0.144715
$587$	−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
$590$	16157.2	+	9328.37i	0.0464154	+	0.0267980i
$591$		0			0
$592$	−112335.	−	194571.i	−0.320534	−	0.555180i
$593$			305581.i			0.868995i	0.900673	+	0.434497i	$0.143074\pi$
$593$			305581.i			0.868995i	−0.900673	+	0.434497i	$0.856926\pi$
$594$		0			0
$595$	419519.			1.18500
$596$	513896.	−	296698.i	1.44671	−	0.835260i
$597$		0			0
$598$	−7137.28	+	12362.1i	−0.0199586	+	0.0345693i
$599$	−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	$0.774751\pi$
$602$			22172.6i			0.0611820i
$603$		0			0
$604$	528123.			1.44764
$605$	−340151.	+	196386.i	−0.929312	+	0.536538i
$606$		0			0
$607$	127073.	−	220097.i	0.344886	−	0.597360i	−0.640447	−	0.768002i	$-0.721251\pi$
$607$	127073.	−	220097.i	0.344886	−	0.597360i	0.985333	+	0.170642i	$0.0545843\pi$
$608$	27067.2	+	15627.3i	0.0732211	+	0.0422742i
$609$		0			0
$610$	−11736.0	−	20327.3i	−0.0315398	−	0.0546286i
$611$		−	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	$0.727680\pi$
$614$	25160.8	−	14526.6i	0.0667403	−	0.0385325i
$615$		0			0
$616$	−10199.5	+	17666.1i	−0.0268793	+	0.0465564i
$617$	105029.	+	60638.8i	0.275893	+	0.159287i	0.631563	−	0.775325i	$-0.282414\pi$
$617$	105029.	+	60638.8i	0.275893	+	0.159287i	−0.355670	+	0.934612i	$0.615747\pi$
$618$		0			0
$619$	−153682.	−	266185.i	−0.401090	−	0.694709i	0.592768	−	0.805374i	$-0.298035\pi$
$619$	−153682.	−	266185.i	−0.401090	−	0.694709i	−0.993858	+	0.110665i	$0.964702\pi$
$620$		−	300344.i		−	0.781331i
$621$		0			0
$622$	−604.780			−0.00156321
$623$	241372.	−	139356.i	0.621886	−	0.359046i
$624$		0			0
$625$	207977.	−	360227.i	0.532421	−	0.922181i
$626$	−21519.8	−	12424.5i	−0.0549149	−	0.0317051i
$627$		0			0
$628$	−64132.9	−	111081.i	−0.162615	−	0.281658i
$629$		−	347832.i		−	0.879161i
$630$		0			0
$631$	−254196.			−0.638425			−0.319212	−	0.947683i	$-0.603418\pi$
$631$	−254196.			−0.638425			−0.319212	+	0.947683i	$0.603418\pi$
$632$	−11792.5	+	6808.43i	−0.0295239	+	0.0170456i
$633$		0			0
$634$	20712.6	−	35875.3i	0.0515296	−	0.0892519i
$635$	−576112.	−	332618.i	−1.42876	−	0.824895i
$636$		0			0
$637$	−52292.1	−	90572.5i	−0.128872	−	0.223212i
$638$		−	16102.8i		−	0.0395602i
$639$		0			0
$640$	−198164.			−0.483799
$641$	−316954.	+	182993.i	−0.771400	+	0.445368i	−0.833374	−	0.552710i	$-0.813594\pi$
$641$	−316954.	+	182993.i	−0.771400	+	0.445368i	0.0619737	+	0.998078i	$0.480261\pi$
$642$		0			0
$643$	−115391.	+	199864.i	−0.279094	+	0.483406i	−0.971160	−	0.238429i	$-0.923368\pi$
$643$	−115391.	+	199864.i	−0.279094	+	0.483406i	0.692066	+	0.721835i	$0.256701\pi$
$644$	−235259.	−	135827.i	−0.567250	−	0.327502i
$645$		0			0
$646$	7957.85	+	13783.4i	0.0190691	+	0.0330287i
$647$			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
$650$	−13109.0	+	7568.47i	−0.0310272	+	0.0179135i
$651$		0			0
$652$	201813.	−	349551.i	0.474738	−	0.822271i
$653$	67988.8	+	39253.4i	0.159445	+	0.0920557i	0.577600	−	0.816320i	$-0.303990\pi$
$653$	67988.8	+	39253.4i	0.159445	+	0.0920557i	−0.418154	+	0.908376i	$0.637323\pi$
$654$		0			0
$655$	52448.0	+	90842.6i	0.122249	+	0.211742i
$656$			647111.i			1.50373i
$657$		0			0
$658$	−8994.34			−0.0207739
$659$	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	$0.174807\pi$
$662$	31017.4	+	17907.9i	0.0707765	+	0.0408628i
$663$		0			0
$664$	52396.4	+	90753.2i	0.118841	+	0.205838i
$665$			125526.i			0.283850i
$666$		0			0
$667$	430584.			0.967846
$668$	−481876.	+	278211.i	−1.07990	+	0.623479i
$669$		0			0
$670$	27802.0	−	48154.6i	0.0619337	−	0.107272i
$671$	−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	$0.571719\pi$
$674$		−	14241.1i		−	0.0313489i
$675$		0			0
$676$	−368095.			−0.805501
$677$	197558.	−	114060.i	0.431040	−	0.248861i	−0.268749	−	0.963210i	$-0.586610\pi$
$677$	197558.	−	114060.i	0.431040	−	0.248861i	0.699790	+	0.714349i	$0.253277\pi$
$678$		0			0
$679$	104093.	−	180294.i	0.225777	−	0.391058i
$680$	−131786.	−	76086.6i	−0.285004	−	0.164547i
$681$		0			0
$682$	5585.01	+	9673.52i	0.0120076	+	0.0207977i
$683$			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
$686$	−36749.6	+	21217.4i	−0.0780916	+	0.0450862i
$687$		0			0
$688$	−249974.	+	432967.i	−0.528102	+	0.914699i
$689$	140980.	+	81394.6i	0.296974	+	0.171458i
$690$		0			0
$691$	−378663.	−	655863.i	−0.793043	−	1.37359i	−0.924074	−	0.382213i	$-0.875162\pi$
$691$	−378663.	−	655863.i	−0.793043	−	1.37359i	0.131031	−	0.991378i	$-0.458171\pi$
$692$		−	233353.i		−	0.487305i
$693$		0			0
$694$	−80400.9			−0.166933
$695$	1.05661e6	−	610036.i	2.18749	−	1.26295i
$696$		0			0
$697$	−500923.	+	867624.i	−1.03111	+	1.78594i
$698$	24484.5	+	14136.2i	0.0502552	+	0.0290149i
$699$		0			0
$700$	−144033.	−	249472.i	−0.293944	−	0.509126i
$701$			506359.i			1.03044i	0.857058	+	0.515220i	$0.172290\pi$
$701$			506359.i			1.03044i	−0.857058	+	0.515220i	$0.827710\pi$
$702$		0			0
$703$	104076.			0.210591
$704$	−195162.	+	112677.i	−0.393777	+	0.227347i
$705$		0			0
$706$	−24848.7	+	43039.2i	−0.0498534	+	0.0863485i
$707$	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	$0.609035\pi$
$710$		−	49268.4i		−	0.0977354i
$711$		0			0
$712$	−101098.			−0.199427
$713$	−258668.	+	149342.i	−0.508819	+	0.293767i
$714$		0			0
$715$	−73487.2	+	127284.i	−0.143747	+	0.248978i
$716$	509367.	+	294083.i	0.993584	+	0.573646i
$717$		0			0
$718$	−37581.8	−	65093.5i	−0.0729001	−	0.126267i
$719$		−	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
$722$	35962.8	−	20763.1i	0.0689888	−	0.0398307i
$723$		0			0
$724$	314816.	−	545277.i	0.600591	−	1.04025i
$725$	395425.	+	228299.i	0.752294	+	0.434337i
$726$		0			0
$727$	373864.	+	647552.i	0.707367	+	1.22520i	0.965830	+	0.259175i	$0.0834506\pi$
$727$	373864.	+	647552.i	0.707367	+	1.22520i	−0.258463	+	0.966021i	$0.583216\pi$
$728$		−	25898.6i		−	0.0488667i
$729$		0			0
$730$	42398.6			0.0795620
$731$	−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	$0.186152\pi$
$734$	−9841.29	−	5681.87i	−0.0182667	−	0.0105463i
$735$		0			0
$736$	74000.7	+	128173.i	0.136609	+	0.236614i
$737$		−	260201.i		−	0.479042i
$738$		0			0
$739$	977921.			1.79067			0.895334	−	0.445395i	$-0.146937\pi$
$739$	977921.			1.79067			0.895334	+	0.445395i	$0.146937\pi$
$740$	−429181.	+	247788.i	−0.783750	+	0.452498i
$741$		0			0
$742$	12310.7	−	21322.7i	0.0223601	−	0.0387288i
$743$	−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
$746$			43740.5i			0.0785971i
$747$		0			0
$748$	−354640.			−0.633847
$749$	−164860.	+	95181.9i	−0.293867	+	0.169664i
$750$		0			0
$751$	266395.	−	461411.i	0.472332	−	0.818102i	−0.527167	−	0.849762i	$-0.676746\pi$
$751$	266395.	−	461411.i	0.472332	−	0.818102i	0.999499	+	0.0316593i	$0.0100792\pi$
$752$	−175634.	−	101402.i	−0.310579	−	0.179313i
$753$		0			0
$754$	10222.0	+	17705.0i	0.0179801	+	0.0311425i
$755$		−	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	$0.582454\pi$
$758$	35843.0	−	20694.0i	0.0623830	−	0.0360168i
$759$		0			0
$760$	22766.1	−	39432.1i	0.0394150	−	0.0682688i
$761$	22249.2	+	12845.6i	0.0384190	+	0.0221812i	0.519086	−	0.854722i	$-0.326272\pi$
$761$	22249.2	+	12845.6i	0.0384190	+	0.0221812i	−0.480667	+	0.876903i	$0.659606\pi$
$762$		0			0
$763$	236727.	+	410022.i	0.406629	+	0.704301i
$764$		−	914920.i		−	1.56746i
$765$		0			0
$766$	64328.9			0.109635
$767$	−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	$-0.0176263\pi$
$770$	19251.2	+	11114.7i	0.0324696	+	0.0187463i
$771$		0			0
$772$	−33173.4	−	57458.0i	−0.0556615	−	0.0964086i
$773$			610238.i			1.02127i	0.859798	+	0.510634i	$0.170589\pi$
$773$			610238.i			1.02127i	−0.859798	+	0.510634i	$0.829411\pi$
$774$		0			0
$775$	−316728.			−0.527331
$776$	−65398.4	+	37757.8i	−0.108604	+	0.0627023i
$777$		0			0
$778$	−9026.64	+	15634.6i	−0.0149131	+	0.0258302i
$779$	−259605.	−	149883.i	−0.427797	−	0.246989i
$780$		0			0
$781$	−115276.	−	199665.i	−0.188990	−	0.327340i
$782$			75366.6i			0.123244i
$783$		0			0
$784$	−356664.			−0.580266
$785$	−243059.	+	140330.i	−0.394432	+	0.227726i
$786$		0			0
$787$	290160.	−	502572.i	0.468477	−	0.811427i	−0.530874	−	0.847451i	$-0.678136\pi$
$787$	290160.	−	502572.i	0.468477	−	0.811427i	0.999351	+	0.0360244i	$0.0114694\pi$
$788$	305069.	+	176132.i	0.491299	+	0.283652i
$789$		0			0
$790$	7419.33	+	12850.7i	0.0118880	+	0.0205907i
$791$			42854.5i			0.0684926i
$792$		0			0
$793$	139449.			0.221753
$794$	−70351.5	+	40617.4i	−0.111592	+	0.0644276i
$795$		0			0
$796$	−401678.	+	695726.i	−0.633945	+	1.09802i
$797$	915508.	+	528569.i	1.44127	+	0.832118i	0.997935	−	0.0642373i	$-0.0204615\pi$
$797$	915508.	+	528569.i	1.44127	+	0.832118i	0.443336	+	0.896355i	$0.353795\pi$
$798$		0			0
$799$	−156989.	−	271913.i	−0.245910	−	0.425928i
$800$			156943.i			0.245223i
$801$		0			0
$802$	67105.5			0.104330
$803$	171824.	−	99202.7i	0.266473	−	0.153848i
$804$		0			0
$805$	−297205.	+	514774.i	−0.458631	+	0.794373i
$806$	−12281.5	−	7090.71i	−0.0189051	−	0.0109149i
$807$		0			0
$808$	−51794.8	−	89711.2i	−0.0793348	−	0.137412i
$809$		−	355969.i		−	0.543895i	−0.962312	−	0.271947i	$-0.912332\pi$
$809$		−	355969.i		−	0.543895i	0.962312	−	0.271947i	$-0.0876677\pi$
$810$		0			0
$811$	−136417.			−0.207409			−0.103704	−	0.994608i	$-0.533070\pi$
$811$	−136417.			−0.207409			−0.103704	+	0.994608i	$0.533070\pi$
$812$	−336937.	+	194531.i	−0.511019	+	0.295037i
$813$		0			0
$814$	9215.43	−	15961.6i	0.0139081	−	0.0240895i
$815$	−764857.	−	441590.i	−1.15150	−	0.664820i
$816$		0			0
$817$	−115797.	−	200567.i	−0.173482	−	0.300479i
$818$		−	98538.4i		−	0.147265i
$819$		0			0
$820$	1.42739e6			2.12282
$821$	394889.	−	227989.i	0.585854	−	0.338243i	−0.177603	−	0.984102i	$-0.556834\pi$
$821$	394889.	−	227989.i	0.585854	−	0.338243i	0.763456	+	0.645860i	$0.223501\pi$
$822$		0			0
$823$	−251795.	+	436122.i	−0.371747	+	0.643885i	−0.989834	−	0.142225i	$-0.954574\pi$
$823$	−251795.	+	436122.i	−0.371747	+	0.643885i	0.618087	+	0.786109i	$0.287908\pi$
$824$	150129.	+	86677.3i	0.221112	+	0.127659i
$825$		0			0
$826$	8382.22	+	14518.4i	0.0122857	+	0.0212794i
$827$			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	$0.542345\pi$
$830$	98896.3	−	57097.8i	0.143557	−	0.0828826i
$831$		0			0
$832$	143054.	−	247777.i	0.206659	−	0.357944i
$833$	−478203.	−	276091.i	−0.689164	−	0.397889i
$834$		0			0
$835$	608759.	+	1.05440e6i	0.873117	+	1.51228i
$836$		−	106113.i		−	0.151829i
$837$		0			0
$838$	−36379.0			−0.0518039
$839$	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
$842$	14499.0	+	8371.03i	0.0204510	+	0.0118074i
$843$		0			0
$844$	70781.8	+	122598.i	0.0993658	+	0.172107i
$845$			805434.i			1.12802i
$846$		0			0
$847$	−352935.			−0.491958
$848$	480784.	−	277581.i	0.668587	−	0.386009i
$849$		0			0
$850$	−39959.9	+	69212.6i	−0.0553078	+	0.0957959i
$851$	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	$0.866262\pi$
$854$		−	21091.3i		−	0.0289192i
$855$		0			0
$856$	69051.1			0.0942374
$857$	−125373.	+	72383.9i	−0.170703	+	0.0985554i	−0.582917	−	0.812532i	$-0.698089\pi$
$857$	−125373.	+	72383.9i	−0.170703	+	0.0985554i	0.412214	+	0.911087i	$0.364755\pi$
$858$		0			0
$859$	−166948.	+	289163.i	−0.226254	+	0.391883i	−0.956695	−	0.291093i	$-0.905981\pi$
$859$	−166948.	+	289163.i	−0.226254	+	0.391883i	0.730441	+	0.682976i	$0.239315\pi$
$860$	955033.	+	551389.i	1.29128	+	0.745523i
$861$		0			0
$862$	−8090.47	−	14013.1i	−0.0108883	−	0.0188591i
$863$			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
$866$	−64473.1	+	37223.6i	−0.0859692	+	0.0496343i
$867$		0			0
$868$	134940.	−	233724.i	0.179103	−	0.310215i
$869$	60135.0	+	34718.9i	0.0796320	+	0.0459755i
$870$		0			0
$871$	165175.	+	286092.i	0.217725	+	0.377111i
$872$		−	171737.i		−	0.225855i
$873$		0			0
$874$	−22550.7			−0.0295214
$875$	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	$0.271289\pi$
$878$	−56501.5	−	32621.2i	−0.0732945	−	0.0423166i
$879$		0			0
$880$	250614.	+	434076.i	0.323623	+	0.560532i
$881$		−	1.21533e6i		−	1.56583i	−0.622131	−	0.782913i	$-0.713733\pi$
$881$		−	1.21533e6i		−	1.56583i	0.622131	−	0.782913i	$-0.286267\pi$
$882$		0			0
$883$	−999070.			−1.28137			−0.640685	−	0.767804i	$-0.721349\pi$
$883$	−999070.			−1.28137			−0.640685	+	0.767804i	$0.721349\pi$
$884$	389928.	−	225125.i	0.498976	−	0.288084i
$885$		0			0
$886$	−20590.8	+	35664.2i	−0.0262304	+	0.0454324i
$887$	−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
$890$			110169.i			0.139085i
$891$		0			0
$892$	−134888.			−0.169529
$893$	81360.0	−	46973.2i	0.102025	−	0.0589044i
$894$		0			0
$895$	643488.	−	1.11455e6i	0.803330	−	1.39141i
$896$	−154209.	−	89032.4i	−0.192085	−	0.110900i
$897$		0			0
$898$	−58417.8	−	101183.i	−0.0724423	−	0.125474i
$899$			427774.i			0.529292i
$900$		0			0
$901$	859491.			1.05875
$902$	−45973.5	+	26542.8i	−0.0565060	+	0.0326238i
$903$		0			0
$904$	7772.36	−	13462.1i	0.00951078	−	0.0164731i
$905$	−1.19313e6	−	688853.i	−1.45677	−	0.841065i
$906$		0			0
$907$	−638426.	−	1.10579e6i	−0.776061	−	1.34418i	−0.934196	−	0.356760i	$-0.883882\pi$
$907$	−638426.	−	1.10579e6i	−0.776061	−	1.34418i	0.158135	−	0.987417i	$-0.449452\pi$
$908$		−	303235.i		−	0.367796i
$909$		0			0
$910$	−28222.4			−0.0340809
$911$	−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
$914$	−65438.2	−	37780.8i	−0.0783320	−	0.0452250i
$915$		0			0
$916$	85107.7	+	147411.i	0.101433	+	0.175687i
$917$			94256.7i			0.112092i
$918$		0			0
$919$	1.15882e6			1.37210			0.686051	−	0.727553i	$-0.259343\pi$
$919$	1.15882e6			1.37210			0.686051	+	0.727553i	$0.259343\pi$
$920$	186725.	−	107806.i	0.220611	−	0.127370i
$921$		0			0
$922$	−8604.07	+	14902.7i	−0.0101214	+	0.0175308i
$923$	253493.	+	146355.i	0.297552	+	0.171792i
$924$		0			0
$925$	261306.	+	452595.i	0.305397	+	0.528964i
$926$		−	60062.6i		−	0.0700459i
$927$		0			0
$928$	211967.			0.246135
$929$	−738465.	+	426353.i	−0.855655	+	0.494012i	−0.862555	−	0.505964i	$-0.831137\pi$
$929$	−738465.	+	426353.i	−0.855655	+	0.494012i	0.00690018	+	0.999976i	$0.497804\pi$
$930$		0			0
$931$	82610.0	−	143085.i	0.0953089	−	0.165080i
$932$	−1.40495e6	−	811151.i	−1.61745	−	0.933834i
$933$		0			0
$934$	9704.86	+	16809.3i	0.0111249	+	0.0192689i
$935$			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	$0.383194\pi$
$938$	43270.4	−	24982.2i	0.0491796	−	0.0283939i
$939$		0			0
$940$	−223671.	+	387410.i	−0.253136	+	0.438445i
$941$	366547.	+	211626.i	0.413953	+	0.238996i	0.692487	−	0.721431i	$-0.256515\pi$
$941$	366547.	+	211626.i	0.413953	+	0.238996i	−0.278534	+	0.960426i	$0.589848\pi$
$942$		0			0
$943$	−709749.	−	1.22932e6i	−0.798145	−	1.38243i
$944$			378004.i			0.424182i
$945$		0			0
$946$	−41013.1			−0.0458290
$947$	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
$950$	−20709.3	−	11956.5i	−0.0229466	−	0.0132482i
$951$		0			0
$952$	−68369.4	−	118419.i	−0.0754376	−	0.130662i
$953$			24001.0i			0.0264267i	0.999913	+	0.0132134i	$0.00420607\pi$
$953$			24001.0i			0.0264267i	−0.999913	+	0.0132134i	$0.995794\pi$
$954$		0			0
$955$	−2.00195e6			−2.19506
$956$	213358.	−	123183.i	0.233450	−	0.134782i
$957$		0			0
$958$	−24610.5	+	42626.6i	−0.0268157	+	0.0464461i
$959$	274290.	+	158362.i	0.298245	+	0.172192i
$960$		0			0
$961$	313393.	+	542813.i	0.339346	+	0.587765i
$962$			23399.8i			0.0252849i
$963$		0			0
$964$	−1.14225e6			−1.22915
$965$	−125725.	+	72587.2i	−0.135010	+	0.0779481i
$966$		0			0
$967$	−522031.	+	904183.i	−0.558268	+	0.966949i	0.439373	+	0.898305i	$0.355201\pi$
$967$	−522031.	+	904183.i	−0.558268	+	0.966949i	−0.997641	+	0.0686443i	$0.978133\pi$
$968$	110869.	+	64010.5i	0.118321	+	0.0683126i
$969$		0			0
$970$	41145.7	+	71266.5i	0.0437302	+	0.0757429i
$971$		−	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
$974$	−7217.58	+	4167.07i	−0.00760806	+	0.00439251i
$975$		0			0
$976$	237783.	−	411851.i	0.249621	−	0.432355i
$977$	−41660.2	−	24052.5i	−0.0436448	−	0.0251983i	0.478019	−	0.878350i	$-0.341355\pi$
$977$	−41660.2	−	24052.5i	−0.0436448	−	0.0251983i	−0.521664	+	0.853151i	$0.674688\pi$
$978$		0			0
$979$	257770.	+	446471.i	0.268947	+	0.465831i
$980$			786724.i			0.819163i
$981$		0			0
$982$	26399.9			0.0273766
$983$	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
$986$	93478.7	+	53969.9i	0.0961521	+	0.0555134i
$987$		0			0
$988$	67360.3	+	116672.i	0.0690066	+	0.119523i
$989$		−	1.09668e6i		−	1.12121i
$990$		0			0
$991$	851418.			0.866953			0.433477	−	0.901165i	$-0.357287\pi$
$991$	851418.			0.866953			0.433477	+	0.901165i	$0.357287\pi$
$992$	−127336.	+	73517.8i	−0.129399	+	0.0747083i
$993$		0			0
$994$	22135.6	−	38340.0i	0.0224037	−	0.0388043i
$995$	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	$0.534771\pi$
$998$			6738.62i			0.00676566i
$999$		0			0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	27.d (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.307601 + 0.177594i) q^{2} +(-7.93692 + 13.7472i) q^{4} +(30.0804 + 17.3669i) q^{5} +(15.6054 + 27.0294i) q^{7} -11.3212i q^{8} -12.3370 q^{10} +(-49.9968 + 28.8657i) q^{11} +(36.6478 - 63.4758i) q^{13} +(-9.60051 - 5.54285i) q^{14} +(-124.980 - 216.472i) q^{16} -386.985i q^{17} +115.791 q^{19} +(-477.491 + 275.679i) q^{20} +(10.2527 - 17.7583i) q^{22} +(474.852 + 274.156i) q^{23} +(290.719 + 503.539i) q^{25} +26.0337i q^{26} -495.436 q^{28} +(680.082 - 392.645i) q^{29} +(-272.367 + 471.753i) q^{31} +(233.759 + 134.961i) q^{32} +(68.7261 + 119.037i) q^{34} +1084.07i q^{35} +898.827 q^{37} +(-35.6175 + 20.5637i) q^{38} +(196.614 - 340.546i) q^{40} +(-2242.01 - 1294.43i) q^{41} +(-1000.05 - 1732.14i) q^{43} -916.419i q^{44} -194.753 q^{46} +(702.646 - 405.673i) q^{47} +(713.441 - 1235.72i) q^{49} +(-178.851 - 103.260i) q^{50} +(581.741 + 1007.61i) q^{52} +2221.00i q^{53} -2005.23 q^{55} +(306.005 - 176.672i) q^{56} +(-139.463 + 241.557i) q^{58} +(-1309.65 - 756.128i) q^{59} +(951.281 + 1647.67i) q^{61} -193.483i q^{62} +3903.49 q^{64} +(2204.76 - 1272.92i) q^{65} +(-2253.55 + 3903.26i) q^{67} +(5319.94 + 3071.47i) q^{68} +(-192.524 - 333.462i) q^{70} +3993.54i q^{71} -3436.70 q^{73} +(-276.480 + 159.626i) q^{74} +(-919.023 + 1591.80i) q^{76} +(-1560.44 - 900.923i) q^{77} +(-601.388 - 1041.63i) q^{79} -8682.07i q^{80} +919.529 q^{82} +(-8016.22 + 4628.17i) q^{83} +(6720.72 - 11640.6i) q^{85} +(615.236 + 355.207i) q^{86} +(326.794 + 566.024i) q^{88} -8929.99i q^{89} +2287.62 q^{91} +(-7537.72 + 4351.91i) q^{92} +(-144.090 + 249.571i) q^{94} +(3483.03 + 2010.93i) q^{95} +(-3335.14 - 5776.64i) q^{97} +506.811i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{2} + 15 q^{4} + 12 q^{5} + 12 q^{7} - 36 q^{10} - 483 q^{11} - 6 q^{13} + 1146 q^{14} + 15 q^{16} - 258 q^{19} - 1614 q^{20} - 369 q^{22} + 282 q^{23} - 273 q^{25} + 1308 q^{28} + 1056 q^{29}+ \cdots - 28959 q^{97}+O(q^{100}) \) 6 * q + 3 * q^2 + 15 * q^4 + 12 * q^5 + 12 * q^7 - 36 * q^10 - 483 * q^11 - 6 * q^13 + 1146 * q^14 + 15 * q^16 - 258 * q^19 - 1614 * q^20 - 369 * q^22 + 282 * q^23 - 273 * q^25 + 1308 * q^28 + 1056 * q^29 + 1290 * q^31 + 1161 * q^32 + 513 * q^34 + 12 * q^37 + 789 * q^38 - 1314 * q^40 - 7629 * q^41 - 285 * q^43 - 5760 * q^46 + 9642 * q^47 - 1863 * q^49 - 3027 * q^50 - 240 * q^52 + 2016 * q^55 + 462 * q^56 + 6462 * q^58 - 6225 * q^59 + 3630 * q^61 + 15450 * q^64 + 7158 * q^65 - 5055 * q^67 + 10503 * q^68 - 9684 * q^70 - 14622 * q^73 - 26454 * q^74 - 4047 * q^76 - 2580 * q^77 + 4764 * q^79 - 9702 * q^82 + 1866 * q^83 + 12366 * q^85 + 37731 * q^86 + 14787 * q^88 + 34836 * q^91 - 33636 * q^92 - 12708 * q^94 + 13362 * q^95 - 28959 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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