LMFDB - Embedded newform 135.4.e.c.91.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [135,4,Mod(46,135)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(135, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

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

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

chi := DirichletCharacter("135.46");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 135 = 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	135.e (of order $3$, 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:	$7.96525785077$
Analytic rank:	$0$
Dimension:	$14$
Relative dimension:	$7$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} + 48 x^{12} - 60 x^{11} + 1605 x^{10} - 1800 x^{9} + 23232 x^{8} - 2346 x^{7} + \cdots + 82944 $ x^14 - 2x^13 + 48x^12 - 60x^11 + 1605x^10 - 1800x^9 + 23232x^8 - 2346x^7 + 209529x^6 - 55412x^5 + 765088x^4 + 276096x^3 + 1572480x^2 - 345600*x + 82944
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{13} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			91.4
Root			$0.112625 - 0.195072i$ of defining polynomial
Character	$\chi$	$=$	135.91
Dual form			135.4.e.c.46.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.112625 - 0.195072i) q^{2} +(3.97463 - 6.88426i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(15.5970 + 27.0148i) q^{7} -3.59257 q^{8} +O(q^{10})$	\(q+(-0.112625 - 0.195072i) q^{2} +(3.97463 - 6.88426i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(15.5970 + 27.0148i) q^{7} -3.59257 q^{8} +1.12625 q^{10} +(9.06424 + 15.6997i) q^{11} +(25.0781 - 43.4366i) q^{13} +(3.51322 - 6.08508i) q^{14} +(-31.3924 - 54.3733i) q^{16} +131.631 q^{17} +23.2428 q^{19} +(19.8732 + 34.4213i) q^{20} +(2.04172 - 3.53636i) q^{22} +(-16.4928 + 28.5664i) q^{23} +(-12.5000 - 21.6506i) q^{25} -11.2977 q^{26} +247.969 q^{28} +(62.9087 + 108.961i) q^{29} +(-62.5563 + 108.351i) q^{31} +(-21.4414 + 37.1376i) q^{32} +(-14.8249 - 25.6775i) q^{34} -155.970 q^{35} +99.9894 q^{37} +(-2.61772 - 4.53402i) q^{38} +(8.98142 - 15.5563i) q^{40} +(122.663 - 212.458i) q^{41} +(-69.5882 - 120.530i) q^{43} +144.108 q^{44} +7.43001 q^{46} +(-236.480 - 409.596i) q^{47} +(-315.033 + 545.654i) q^{49} +(-2.81562 + 4.87680i) q^{50} +(-199.352 - 345.289i) q^{52} -421.529 q^{53} -90.6424 q^{55} +(-56.0333 - 97.0526i) q^{56} +(14.1702 - 24.5435i) q^{58} +(-371.207 + 642.949i) q^{59} +(-4.48868 - 7.77462i) q^{61} +28.1816 q^{62} -492.620 q^{64} +(125.391 + 217.183i) q^{65} +(-294.453 + 510.008i) q^{67} +(523.185 - 906.182i) q^{68} +(17.5661 + 30.4254i) q^{70} +48.5526 q^{71} +409.800 q^{73} +(-11.2613 - 19.5051i) q^{74} +(92.3816 - 160.010i) q^{76} +(-282.750 + 489.737i) q^{77} +(-265.263 - 459.449i) q^{79} +313.924 q^{80} -55.2595 q^{82} +(-147.295 - 255.122i) q^{83} +(-329.077 + 569.979i) q^{85} +(-15.6747 + 27.1494i) q^{86} +(-32.5639 - 56.4023i) q^{88} -852.817 q^{89} +1564.57 q^{91} +(131.106 + 227.082i) q^{92} +(-53.2672 + 92.2615i) q^{94} +(-58.1070 + 100.644i) q^{95} +(194.045 + 336.096i) q^{97} +141.922 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 2 q^{2} - 36 q^{4} - 35 q^{5} - 22 q^{7} + 36 q^{8}+O(q^{10}) $ 14 * q - 2 * q^2 - 36 * q^4 - 35 * q^5 - 22 * q^7 + 36 * q^8	$ 14 q - 2 q^{2} - 36 q^{4} - 35 q^{5} - 22 q^{7} + 36 q^{8} + 20 q^{10} - 23 q^{11} - 96 q^{13} + 21 q^{14} - 324 q^{16} + 322 q^{17} + 558 q^{19} - 180 q^{20} - 311 q^{22} - 96 q^{23} - 175 q^{25} - 716 q^{26} + 674 q^{28} + 296 q^{29} - 244 q^{31} + 314 q^{32} - 125 q^{34} + 220 q^{35} + 808 q^{37} - 305 q^{38} - 90 q^{40} + 47 q^{41} - 525 q^{43} + 110 q^{44} + 1434 q^{46} - 164 q^{47} - 1225 q^{49} - 50 q^{50} - 1682 q^{52} + 1012 q^{53} + 230 q^{55} + 981 q^{56} - 1183 q^{58} + 85 q^{59} - 828 q^{61} - 1572 q^{62} + 4472 q^{64} - 480 q^{65} - 1093 q^{67} - 2473 q^{68} + 105 q^{70} + 656 q^{71} + 4170 q^{73} + 1316 q^{74} - 2789 q^{76} - 24 q^{77} - 2110 q^{79} + 3240 q^{80} - 124 q^{82} - 1290 q^{83} - 805 q^{85} + 2569 q^{86} - 2271 q^{88} - 6096 q^{89} + 6676 q^{91} - 2763 q^{92} + 517 q^{94} - 1395 q^{95} - 1787 q^{97} + 2558 q^{98}+O(q^{100}) $ 14 * q - 2 * q^2 - 36 * q^4 - 35 * q^5 - 22 * q^7 + 36 * q^8 + 20 * q^10 - 23 * q^11 - 96 * q^13 + 21 * q^14 - 324 * q^16 + 322 * q^17 + 558 * q^19 - 180 * q^20 - 311 * q^22 - 96 * q^23 - 175 * q^25 - 716 * q^26 + 674 * q^28 + 296 * q^29 - 244 * q^31 + 314 * q^32 - 125 * q^34 + 220 * q^35 + 808 * q^37 - 305 * q^38 - 90 * q^40 + 47 * q^41 - 525 * q^43 + 110 * q^44 + 1434 * q^46 - 164 * q^47 - 1225 * q^49 - 50 * q^50 - 1682 * q^52 + 1012 * q^53 + 230 * q^55 + 981 * q^56 - 1183 * q^58 + 85 * q^59 - 828 * q^61 - 1572 * q^62 + 4472 * q^64 - 480 * q^65 - 1093 * q^67 - 2473 * q^68 + 105 * q^70 + 656 * q^71 + 4170 * q^73 + 1316 * q^74 - 2789 * q^76 - 24 * q^77 - 2110 * q^79 + 3240 * q^80 - 124 * q^82 - 1290 * q^83 - 805 * q^85 + 2569 * q^86 - 2271 * q^88 - 6096 * q^89 + 6676 * q^91 - 2763 * q^92 + 517 * q^94 - 1395 * q^95 - 1787 * q^97 + 2558 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/135\mathbb{Z}\right)^\times$.

$n$	$56$	$82$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$

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 $

$2$	−0.112625	−	0.195072i	−0.0398189	−	0.0689684i	0.845429	−	0.534088i	$-0.179345\pi$
$2$	−0.112625	−	0.195072i	−0.0398189	−	0.0689684i	−0.885248	+	0.465119i	$0.846011\pi$
$3$		0			0
$3$		0			0
$4$	3.97463	−	6.88426i	0.496829	−	0.860533i
$5$	−2.50000	+	4.33013i	−0.223607	+	0.387298i
$5$	−2.50000	+	4.33013i	−0.223607	+	0.387298i
$6$		0			0
$7$	15.5970	+	27.0148i	0.842159	+	1.45866i	0.888066	+	0.459717i	$0.152049\pi$
$7$	15.5970	+	27.0148i	0.842159	+	1.45866i	−0.0459062	+	0.998946i	$0.514618\pi$
$8$	−3.59257			−0.158771
$9$		0			0
$10$	1.12625			0.0356151
$11$	9.06424	+	15.6997i	0.248452	+	0.430331i	0.963096	−	0.269156i	$-0.0867449\pi$
$11$	9.06424	+	15.6997i	0.248452	+	0.430331i	−0.714645	+	0.699488i	$0.753412\pi$
$12$		0			0
$13$	25.0781	−	43.4366i	0.535032	−	0.926702i	−0.464130	−	0.885767i	$-0.653633\pi$
$13$	25.0781	−	43.4366i	0.535032	−	0.926702i	0.999162	−	0.0409353i	$-0.0130337\pi$
$14$	3.51322	−	6.08508i	0.0670678	−	0.116165i
$15$		0			0
$16$	−31.3924	−	54.3733i	−0.490507	−	0.849583i
$17$	131.631			1.87795			0.938977	−	0.343981i	$-0.111776\pi$
$17$	131.631			1.87795			0.938977	+	0.343981i	$0.111776\pi$
$18$		0			0
$19$	23.2428			0.280646			0.140323	−	0.990106i	$-0.455186\pi$
$19$	23.2428			0.280646			0.140323	+	0.990106i	$0.455186\pi$
$20$	19.8732	+	34.4213i	0.222189	+	0.384842i
$21$		0			0
$22$	2.04172	−	3.53636i	0.0197862	−	0.0342707i
$23$	−16.4928	+	28.5664i	−0.149521	+	0.258978i	−0.931051	−	0.364890i	$-0.881107\pi$
$23$	−16.4928	+	28.5664i	−0.149521	+	0.258978i	0.781529	+	0.623868i	$0.214440\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.100000	−	0.173205i
$26$	−11.2977			−0.0852176
$27$		0			0
$28$	247.969			1.67364
$29$	62.9087	+	108.961i	0.402823	+	0.697709i	0.994065	−	0.108784i	$-0.0346958\pi$
$29$	62.9087	+	108.961i	0.402823	+	0.697709i	−0.591243	+	0.806494i	$0.701363\pi$
$30$		0			0
$31$	−62.5563	+	108.351i	−0.362434	+	0.627754i	−0.988361	−	0.152128i	$-0.951387\pi$
$31$	−62.5563	+	108.351i	−0.362434	+	0.627754i	0.625927	+	0.779882i	$0.284721\pi$
$32$	−21.4414	+	37.1376i	−0.118448	+	0.205158i
$33$		0			0
$34$	−14.8249	−	25.6775i	−0.0747781	−	0.129519i
$35$	−155.970			−0.753250
$36$		0			0
$37$	99.9894			0.444274			0.222137	−	0.975015i	$-0.428697\pi$
$37$	99.9894			0.444274			0.222137	+	0.975015i	$0.428697\pi$
$38$	−2.61772	−	4.53402i	−0.0111750	−	0.0193557i
$39$		0			0
$40$	8.98142	−	15.5563i	0.0355022	−	0.0614916i
$41$	122.663	−	212.458i	0.467237	−	0.809278i	−0.532063	−	0.846705i	$-0.678583\pi$
$41$	122.663	−	212.458i	0.467237	−	0.809278i	0.999299	+	0.0374272i	$0.0119162\pi$
$42$		0			0
$43$	−69.5882	−	120.530i	−0.246793	−	0.427458i	0.715841	−	0.698263i	$-0.246043\pi$
$43$	−69.5882	−	120.530i	−0.246793	−	0.427458i	−0.962634	+	0.270805i	$0.912710\pi$
$44$	144.108			0.493752
$45$		0			0
$46$	7.43001			0.0238151
$47$	−236.480	−	409.596i	−0.733919	−	1.27119i	−0.955196	−	0.295975i	$-0.904356\pi$
$47$	−236.480	−	409.596i	−0.733919	−	1.27119i	0.221276	−	0.975211i	$-0.428978\pi$
$48$		0			0
$49$	−315.033	+	545.654i	−0.918465	+	1.59083i
$50$	−2.81562	+	4.87680i	−0.00796378	+	0.0137937i
$51$		0			0
$52$	−199.352	−	345.289i	−0.531639	−	0.920825i
$53$	−421.529			−1.09248			−0.546240	−	0.837628i	$-0.683941\pi$
$53$	−421.529			−1.09248			−0.546240	+	0.837628i	$0.683941\pi$
$54$		0			0
$55$	−90.6424			−0.222222
$56$	−56.0333	−	97.0526i	−0.133710	−	0.231593i
$57$		0			0
$58$	14.1702	−	24.5435i	0.0320799	−	0.0555641i
$59$	−371.207	+	642.949i	−0.819101	+	1.41873i	0.0872437	+	0.996187i	$0.472194\pi$
$59$	−371.207	+	642.949i	−0.819101	+	1.41873i	−0.906345	+	0.422538i	$0.861139\pi$
$60$		0			0
$61$	−4.48868	−	7.77462i	−0.00942158	−	0.0163187i	0.861276	−	0.508137i	$-0.169666\pi$
$61$	−4.48868	−	7.77462i	−0.00942158	−	0.0163187i	−0.870698	+	0.491818i	$0.836332\pi$
$62$	28.1816			0.0577269
$63$		0			0
$64$	−492.620			−0.962148
$65$	125.391	+	217.183i	0.239274	+	0.414434i
$66$		0			0
$67$	−294.453	+	510.008i	−0.536913	+	0.929960i	0.462155	+	0.886799i	$0.347076\pi$
$67$	−294.453	+	510.008i	−0.536913	+	0.929960i	−0.999068	+	0.0431613i	$0.986257\pi$
$68$	523.185	−	906.182i	0.933021	−	1.61604i
$69$		0			0
$70$	17.5661	+	30.4254i	0.0299936	+	0.0519505i
$71$	48.5526			0.0811568			0.0405784	−	0.999176i	$-0.487080\pi$
$71$	48.5526			0.0811568			0.0405784	+	0.999176i	$0.487080\pi$
$72$		0			0
$73$	409.800			0.657034			0.328517	−	0.944498i	$-0.393451\pi$
$73$	409.800			0.657034			0.328517	+	0.944498i	$0.393451\pi$
$74$	−11.2613	−	19.5051i	−0.0176905	−	0.0306409i
$75$		0			0
$76$	92.3816	−	160.010i	0.139433	−	0.241505i
$77$	−282.750	+	489.737i	−0.418472	+	0.724815i
$78$		0			0
$79$	−265.263	−	459.449i	−0.377778	−	0.654330i	0.612961	−	0.790113i	$-0.289978\pi$
$79$	−265.263	−	459.449i	−0.377778	−	0.654330i	−0.990739	+	0.135783i	$0.956645\pi$
$80$	313.924			0.438723
$81$		0			0
$82$	−55.2595			−0.0744195
$83$	−147.295	−	255.122i	−0.194792	−	0.337389i	0.752040	−	0.659117i	$-0.229070\pi$
$83$	−147.295	−	255.122i	−0.194792	−	0.337389i	−0.946832	+	0.321728i	$0.895736\pi$
$84$		0			0
$85$	−329.077	+	569.979i	−0.419923	+	0.727328i
$86$	−15.6747	+	27.1494i	−0.0196541	+	0.0340418i
$87$		0			0
$88$	−32.5639	−	56.4023i	−0.0394469	−	0.0683240i
$89$	−852.817			−1.01571			−0.507856	−	0.861442i	$-0.669562\pi$
$89$	−852.817			−1.01571			−0.507856	+	0.861442i	$0.669562\pi$
$90$		0			0
$91$	1564.57			1.80233
$92$	131.106	+	227.082i	0.148573	+	0.257336i
$93$		0			0
$94$	−53.2672	+	92.2615i	−0.0584478	+	0.101234i
$95$	−58.1070	+	100.644i	−0.0627543	+	0.108694i
$96$		0			0
$97$	194.045	+	336.096i	0.203117	+	0.351808i	0.949531	−	0.313673i	$-0.101560\pi$
$97$	194.045	+	336.096i	0.203117	+	0.351808i	−0.746414	+	0.665481i	$0.768226\pi$
$98$	141.922			0.146289
$99$		0			0
$100$	−198.732			−0.198732
$101$	−539.997	−	935.302i	−0.531997	−	0.921446i	−0.999302	−	0.0373500i	$-0.988108\pi$
$101$	−539.997	−	935.302i	−0.531997	−	0.921446i	0.467305	−	0.884096i	$-0.345225\pi$
$102$		0			0
$103$	297.498	−	515.282i	0.284596	−	0.492935i	−0.687915	−	0.725791i	$-0.741474\pi$
$103$	297.498	−	515.282i	0.284596	−	0.492935i	0.972511	+	0.232856i	$0.0748072\pi$
$104$	−90.0948	+	156.049i	−0.0849473	+	0.147133i
$105$		0			0
$106$	47.4747	+	82.2286i	0.0435014	+	0.0753467i
$107$	498.693			0.450565			0.225282	−	0.974293i	$-0.427670\pi$
$107$	498.693			0.450565			0.225282	+	0.974293i	$0.427670\pi$
$108$		0			0
$109$	−959.301			−0.842976			−0.421488	−	0.906834i	$-0.638492\pi$
$109$	−959.301			−0.842976			−0.421488	+	0.906834i	$0.638492\pi$
$110$	10.2086	+	17.6818i	0.00884864	+	0.0153263i
$111$		0			0
$112$	979.256	−	1696.12i	0.826170	−	1.43097i
$113$	864.396	−	1497.18i	0.719607	−	1.24640i	−0.241549	−	0.970389i	$-0.577655\pi$
$113$	864.396	−	1497.18i	0.719607	−	1.24640i	0.961156	−	0.276007i	$-0.0890113\pi$
$114$		0			0
$115$	−82.4641	−	142.832i	−0.0668679	−	0.115819i
$116$	1000.16			0.800536
$117$		0			0
$118$	167.228			0.130463
$119$	2053.05	+	3555.99i	1.58154	+	2.73930i
$120$		0			0
$121$	501.179	−	868.068i	0.376543	−	0.652192i
$122$	−1.01107	+	1.75123i	−0.000750315	+	0.00129958i
$123$		0			0
$124$	497.277	+	861.309i	0.360135	+	0.623773i
$125$	125.000			0.0894427
$126$		0			0
$127$	1019.95			0.712647			0.356324	−	0.934363i	$-0.384030\pi$
$127$	1019.95			0.712647			0.356324	+	0.934363i	$0.384030\pi$
$128$	227.013	+	393.197i	0.156760	+	0.271516i
$129$		0			0
$130$	28.2442	−	48.9204i	0.0190552	−	0.0330046i
$131$	−851.196	+	1474.31i	−0.567705	+	0.983294i	0.429088	+	0.903263i	$0.358835\pi$
$131$	−851.196	+	1474.31i	−0.567705	+	0.983294i	−0.996792	+	0.0800308i	$0.974498\pi$
$132$		0			0
$133$	362.518	+	627.900i	0.236348	+	0.409367i
$134$	132.651			0.0855172
$135$		0			0
$136$	−472.893			−0.298164
$137$	−524.839	−	909.049i	−0.327300	−	0.566900i	0.654675	−	0.755910i	$-0.272805\pi$
$137$	−524.839	−	909.049i	−0.327300	−	0.566900i	−0.981975	+	0.189010i	$0.939472\pi$
$138$		0			0
$139$	−717.411	+	1242.59i	−0.437770	+	0.758240i	−0.997517	−	0.0704236i	$-0.977565\pi$
$139$	−717.411	+	1242.59i	−0.437770	+	0.758240i	0.559747	+	0.828663i	$0.310898\pi$
$140$	−619.924	+	1073.74i	−0.374236	+	0.648197i
$141$		0			0
$142$	−5.46823	−	9.47126i	−0.00323158	−	0.00559726i
$143$	909.256			0.531719
$144$		0			0
$145$	−629.087			−0.360295
$146$	−46.1537	−	79.9406i	−0.0261624	−	0.0453146i
$147$		0			0
$148$	397.421	−	688.353i	0.220728	−	0.382313i
$149$	609.231	−	1055.22i	0.334968	−	0.580181i	−0.648511	−	0.761205i	$-0.724608\pi$
$149$	609.231	−	1055.22i	0.334968	−	0.580181i	0.983479	+	0.181024i	$0.0579413\pi$
$150$		0			0
$151$	−804.751	−	1393.87i	−0.433707	−	0.751202i	0.563482	−	0.826128i	$-0.309461\pi$
$151$	−804.751	−	1393.87i	−0.433707	−	0.751202i	−0.997189	+	0.0749258i	$0.976128\pi$
$152$	−83.5014			−0.0445583
$153$		0			0
$154$	127.379			0.0666524
$155$	−312.782	−	541.754i	−0.162085	−	0.280740i
$156$		0			0
$157$	−643.168	+	1114.00i	−0.326945	+	0.566286i	−0.981904	−	0.189379i	$-0.939353\pi$
$157$	−643.168	+	1114.00i	−0.326945	+	0.566286i	0.654959	+	0.755664i	$0.272686\pi$
$158$	−59.7505	+	103.491i	−0.0300854	+	0.0521095i
$159$		0			0
$160$	−107.207	−	185.688i	−0.0529717	−	0.0917496i
$161$	−1028.95			−0.503683
$162$		0			0
$163$	1416.84			0.680830			0.340415	−	0.940275i	$-0.389432\pi$
$163$	1416.84			0.680830			0.340415	+	0.940275i	$0.389432\pi$
$164$	−975.079	−	1688.89i	−0.464273	−	0.804145i
$165$		0			0
$166$	−33.1782	+	57.4663i	−0.0155128	+	0.0268690i
$167$	447.487	−	775.071i	0.207351	−	0.359142i	−0.743528	−	0.668704i	$-0.766849\pi$
$167$	447.487	−	775.071i	0.207351	−	0.359142i	0.950879	+	0.309562i	$0.100182\pi$
$168$		0			0
$169$	−159.323	−	275.955i	−0.0725183	−	0.125605i
$170$	148.249			0.0668835
$171$		0			0
$172$	−1106.35			−0.490456
$173$	−867.548	−	1502.64i	−0.381263	−	0.660367i	0.609980	−	0.792417i	$-0.291177\pi$
$173$	−867.548	−	1502.64i	−0.381263	−	0.660367i	−0.991243	+	0.132050i	$0.957844\pi$
$174$		0			0
$175$	389.925	−	675.370i	0.168432	−	0.291733i
$176$	569.097	−	985.705i	0.243735	−	0.422161i
$177$		0			0
$178$	96.0485	+	166.361i	0.0404446	+	0.0700521i
$179$	−2133.37			−0.890815			−0.445408	−	0.895328i	$-0.646941\pi$
$179$	−2133.37			−0.890815			−0.445408	+	0.895328i	$0.646941\pi$
$180$		0			0
$181$	−3611.98			−1.48330			−0.741648	−	0.670789i	$-0.765955\pi$
$181$	−3611.98			−1.48330			−0.741648	+	0.670789i	$0.765955\pi$
$182$	−176.210	−	305.205i	−0.0717668	−	0.124304i
$183$		0			0
$184$	59.2516	−	102.627i	0.0237396	−	0.0411182i
$185$	−249.974	+	432.967i	−0.0993428	+	0.172067i
$186$		0			0
$187$	1193.13	+	2066.57i	0.466581	+	0.808142i
$188$	−3759.69			−1.45853
$189$		0			0
$190$	26.1772			0.00999523
$191$	−298.495	−	517.008i	−0.113080	−	0.195861i	0.803930	−	0.594723i	$-0.202738\pi$
$191$	−298.495	−	517.008i	−0.113080	−	0.195861i	−0.917011	+	0.398863i	$0.869405\pi$
$192$		0			0
$193$	603.872	−	1045.94i	0.225221	−	0.390094i	−0.731165	−	0.682201i	$-0.761023\pi$
$193$	603.872	−	1045.94i	0.225221	−	0.390094i	0.956386	+	0.292107i	$0.0943563\pi$
$194$	43.7087	−	75.7056i	0.0161758	−	0.0280173i
$195$		0			0
$196$	2504.28	+	4337.54i	0.912639	+	1.58074i
$197$	3268.56			1.18211			0.591054	−	0.806632i	$-0.298712\pi$
$197$	3268.56			1.18211			0.591054	+	0.806632i	$0.298712\pi$
$198$		0			0
$199$	−2109.88			−0.751585			−0.375793	−	0.926704i	$-0.622629\pi$
$199$	−2109.88			−0.751585			−0.375793	+	0.926704i	$0.622629\pi$
$200$	44.9071	+	77.7814i	0.0158771	+	0.0274999i
$201$		0			0
$202$	−121.634	+	210.677i	−0.0423671	+	0.0733820i
$203$	−1962.38	+	3398.93i	−0.678482	+	1.17516i
$204$		0			0
$205$	613.314	+	1062.29i	0.208955	+	0.361920i
$206$	−134.023			−0.0453292
$207$		0			0
$208$	−3149.05			−1.04975
$209$	210.678	+	364.906i	0.0697269	+	0.120771i
$210$		0			0
$211$	−329.321	+	570.400i	−0.107447	+	0.186104i	−0.914735	−	0.404053i	$-0.867601\pi$
$211$	−329.321	+	570.400i	−0.107447	+	0.186104i	0.807288	+	0.590157i	$0.200934\pi$
$212$	−1675.42	+	2901.92i	−0.542776	+	0.940116i
$213$		0			0
$214$	−56.1652	−	97.2810i	−0.0179410	−	0.0310747i
$215$	695.882			0.220738
$216$		0			0
$217$	−3902.77			−1.22091
$218$	108.041	+	187.133i	0.0335664	+	0.0581387i
$219$		0			0
$220$	−360.270	+	624.006i	−0.110406	+	0.191229i
$221$	3301.06	−	5717.60i	1.00476	−	1.74030i
$222$		0			0
$223$	−609.397	−	1055.51i	−0.182997	−	0.316959i	0.759903	−	0.650036i	$-0.225246\pi$
$223$	−609.397	−	1055.51i	−0.182997	−	0.316959i	−0.942900	+	0.333077i	$0.891913\pi$
$224$	−1337.69			−0.399009
$225$		0			0
$226$	−389.410			−0.114616
$227$	−762.785	−	1321.18i	−0.223030	−	0.386299i	0.732697	−	0.680555i	$-0.238261\pi$
$227$	−762.785	−	1321.18i	−0.223030	−	0.386299i	−0.955727	+	0.294256i	$0.904928\pi$
$228$		0			0
$229$	3091.33	−	5354.33i	0.892055	−	1.54508i	0.0546469	−	0.998506i	$-0.482597\pi$
$229$	3091.33	−	5354.33i	0.892055	−	1.54508i	0.837408	−	0.546578i	$-0.184070\pi$
$230$	−18.5750	+	32.1729i	−0.00532522	+	0.00922355i
$231$		0			0
$232$	−226.004	−	391.450i	−0.0639564	−	0.110776i
$233$	2013.82			0.566222			0.283111	−	0.959087i	$-0.408633\pi$
$233$	2013.82			0.566222			0.283111	+	0.959087i	$0.408633\pi$
$234$		0			0
$235$	2364.80			0.656438
$236$	2950.82	+	5110.97i	0.813906	+	1.40973i
$237$		0			0
$238$	462.449	−	800.985i	0.125950	−	0.218152i
$239$	2543.15	−	4404.87i	0.688297	−	1.19216i	−0.284092	−	0.958797i	$-0.591692\pi$
$239$	2543.15	−	4404.87i	0.688297	−	1.19216i	0.972389	−	0.233368i	$-0.0749746\pi$
$240$		0			0
$241$	−1643.41	−	2846.47i	−0.439259	−	0.760820i	0.558373	−	0.829590i	$-0.311426\pi$
$241$	−1643.41	−	2846.47i	−0.439259	−	0.760820i	−0.997633	+	0.0687703i	$0.978092\pi$
$242$	−225.781			−0.0599742
$243$		0			0
$244$	−71.3634			−0.0187237
$245$	−1575.17	−	2728.27i	−0.410750	−	0.711440i
$246$		0			0
$247$	582.886	−	1009.59i	0.150154	−	0.260075i
$248$	224.738	−	389.258i	0.0575439	−	0.0996689i
$249$		0			0
$250$	−14.0781	−	24.3840i	−0.00356151	−	0.00616872i
$251$	−3480.55			−0.875260			−0.437630	−	0.899155i	$-0.644182\pi$
$251$	−3480.55			−0.875260			−0.437630	+	0.899155i	$0.644182\pi$
$252$		0			0
$253$	−597.979			−0.148595
$254$	−114.872	−	198.964i	−0.0283768	−	0.0491501i
$255$		0			0
$256$	−1919.34	+	3324.40i	−0.468590	+	0.811621i
$257$	−3397.49	+	5884.62i	−0.824628	+	1.42830i	0.0775746	+	0.996987i	$0.475282\pi$
$257$	−3397.49	+	5884.62i	−0.824628	+	1.42830i	−0.902203	+	0.431312i	$0.858051\pi$
$258$		0			0
$259$	1559.54	+	2701.19i	0.374150	+	0.648047i
$260$	1993.52			0.475512
$261$		0			0
$262$	383.464			0.0904216
$263$	2682.12	+	4645.56i	0.628846	+	1.08919i	0.987784	+	0.155832i	$0.0498057\pi$
$263$	2682.12	+	4645.56i	0.628846	+	1.08919i	−0.358938	+	0.933362i	$0.616861\pi$
$264$		0			0
$265$	1053.82	−	1825.27i	0.244286	−	0.423116i
$266$	81.6572	−	141.434i	0.0188223	−	0.0326011i
$267$		0			0
$268$	2340.68	+	4054.18i	0.533508	+	0.924062i
$269$	−48.4985			−0.0109926			−0.00549629	−	0.999985i	$-0.501750\pi$
$269$	−48.4985			−0.0109926			−0.00549629	+	0.999985i	$0.501750\pi$
$270$		0			0
$271$	7643.16			1.71324			0.856622	−	0.515945i	$-0.172559\pi$
$271$	7643.16			1.71324			0.856622	+	0.515945i	$0.172559\pi$
$272$	−4132.22	−	7157.21i	−0.921149	−	1.59548i
$273$		0			0
$274$	−118.220	+	204.763i	−0.0260654	+	0.0451467i
$275$	226.606	−	392.493i	0.0496904	−	0.0860663i
$276$		0			0
$277$	−2573.01	−	4456.59i	−0.558113	−	0.966680i	−0.997654	−	0.0684576i	$-0.978192\pi$
$277$	−2573.01	−	4456.59i	−0.558113	−	0.966680i	0.439541	−	0.898223i	$-0.355141\pi$
$278$	323.193			0.0697261
$279$		0			0
$280$	560.333			0.119594
$281$	2927.37	+	5070.35i	0.621467	+	1.07641i	0.989213	+	0.146486i	$0.0467963\pi$
$281$	2927.37	+	5070.35i	0.621467	+	1.07641i	−0.367746	+	0.929926i	$0.619870\pi$
$282$		0			0
$283$	−4696.92	+	8135.31i	−0.986583	+	1.70881i	−0.351904	+	0.936036i	$0.614466\pi$
$283$	−4696.92	+	8135.31i	−0.986583	+	1.70881i	−0.634679	+	0.772776i	$0.718868\pi$
$284$	192.979	−	334.249i	0.0403211	−	0.0698381i
$285$		0			0
$286$	−102.405	−	177.370i	−0.0211725	−	0.0366718i
$287$	7652.69			1.57395
$288$		0			0
$289$	12413.7			2.52671
$290$	70.8509	+	122.717i	0.0143466	+	0.0248490i
$291$		0			0
$292$	1628.81	−	2821.17i	0.326434	−	0.565400i
$293$	−890.291	+	1542.03i	−0.177513	+	0.307462i	−0.941028	−	0.338328i	$-0.890139\pi$
$293$	−890.291	+	1542.03i	−0.177513	+	0.307462i	0.763515	+	0.645790i	$0.223472\pi$
$294$		0			0
$295$	−1856.03	−	3214.74i	−0.366313	−	0.634473i
$296$	−359.219			−0.0705377
$297$		0			0
$298$	−274.458			−0.0533522
$299$	827.217	+	1432.78i	0.159997	+	0.277123i
$300$		0			0
$301$	2170.74	−	3759.82i	0.415678	−	0.719976i
$302$	−181.270	+	313.969i	−0.0345395	+	0.0598241i
$303$		0			0
$304$	−729.649	−	1263.79i	−0.137659	−	0.238432i
$305$	44.8868			0.00842692
$306$		0			0
$307$	8480.18			1.57651			0.788256	−	0.615347i	$-0.210984\pi$
$307$	8480.18			1.57651			0.788256	+	0.615347i	$0.210984\pi$
$308$	2247.65	+	3893.05i	0.415818	+	0.720218i
$309$		0			0
$310$	−70.4540	+	122.030i	−0.0129081	+	0.0223575i
$311$	−3405.40	+	5898.32i	−0.620908	+	1.07544i	0.368409	+	0.929664i	$0.379903\pi$
$311$	−3405.40	+	5898.32i	−0.620908	+	1.07544i	−0.989317	+	0.145780i	$0.953431\pi$
$312$		0			0
$313$	−4023.41	−	6968.74i	−0.726570	−	1.25846i	−0.958325	−	0.285682i	$-0.907780\pi$
$313$	−4023.41	−	6968.74i	−0.726570	−	1.25846i	0.231755	−	0.972774i	$-0.425553\pi$
$314$	289.747			0.0520744
$315$		0			0
$316$	−4217.30			−0.750764
$317$	783.112	+	1356.39i	0.138751	+	0.240323i	0.927024	−	0.375002i	$-0.122358\pi$
$317$	783.112	+	1356.39i	0.138751	+	0.240323i	−0.788273	+	0.615325i	$0.789025\pi$
$318$		0			0
$319$	−1140.44	+	1975.30i	−0.200164	+	0.346694i
$320$	1231.55	−	2133.11i	0.215143	−	0.372638i
$321$		0			0
$322$	115.886	+	200.720i	0.0200561	+	0.0347382i
$323$	3059.47			0.527039
$324$		0			0
$325$	−1253.91			−0.214013
$326$	−159.571	−	276.385i	−0.0271099	−	0.0469557i
$327$		0			0
$328$	−440.674	+	763.271i	−0.0741835	+	0.128490i
$329$	7376.77	−	12776.9i	1.23615	−	2.14108i
$330$		0			0
$331$	−1221.28	−	2115.32i	−0.202802	−	0.351264i	0.746628	−	0.665242i	$-0.231672\pi$
$331$	−1221.28	−	2115.32i	−0.202802	−	0.351264i	−0.949430	+	0.313978i	$0.898338\pi$
$332$	−2341.77			−0.387113
$333$		0			0
$334$	−201.593			−0.0330260
$335$	−1472.27	−	2550.04i	−0.240115	−	0.415891i
$336$		0			0
$337$	−4736.49	+	8203.84i	−0.765617	+	1.32609i	0.174302	+	0.984692i	$0.444233\pi$
$337$	−4736.49	+	8203.84i	−0.765617	+	1.32609i	−0.939920	+	0.341396i	$0.889100\pi$
$338$	−35.8874	+	62.1588i	−0.00577520	+	0.0100029i
$339$		0			0
$340$	2615.92	+	4530.91i	0.417260	+	0.722715i
$341$	−2268.10			−0.360190
$342$		0			0
$343$	−8954.76			−1.40966
$344$	250.000	+	433.013i	0.0391835	+	0.0678678i
$345$		0			0
$346$	−195.415	+	338.469i	−0.0303630	+	0.0525902i
$347$	−1656.23	+	2868.68i	−0.256229	+	0.443801i	−0.965229	−	0.261408i	$-0.915813\pi$
$347$	−1656.23	+	2868.68i	−0.256229	+	0.443801i	0.709000	+	0.705209i	$0.249147\pi$
$348$		0			0
$349$	−2834.48	−	4909.46i	−0.434745	−	0.753001i	0.562530	−	0.826777i	$-0.309828\pi$
$349$	−2834.48	−	4909.46i	−0.434745	−	0.753001i	−0.997275	+	0.0737765i	$0.976495\pi$
$350$	−175.661			−0.0268271
$351$		0			0
$352$	−777.400			−0.117715
$353$	−869.541	−	1506.09i	−0.131108	−	0.227085i	0.792996	−	0.609227i	$-0.208520\pi$
$353$	−869.541	−	1506.09i	−0.131108	−	0.227085i	−0.924104	+	0.382142i	$0.875187\pi$
$354$		0			0
$355$	−121.382	+	210.239i	−0.0181472	+	0.0314319i
$356$	−3389.63	+	5871.02i	−0.504636	+	0.874054i
$357$		0			0
$358$	240.271	+	416.162i	0.0354713	+	0.0614381i
$359$	−8624.58			−1.26793			−0.633966	−	0.773361i	$-0.718574\pi$
$359$	−8624.58			−1.26793			−0.633966	+	0.773361i	$0.718574\pi$
$360$		0			0
$361$	−6318.77			−0.921238
$362$	406.799	+	704.597i	0.0590632	+	0.102301i
$363$		0			0
$364$	6218.60	−	10770.9i	0.895449	−	1.55096i
$365$	−1024.50	+	1774.49i	−0.146917	+	0.254468i
$366$		0			0
$367$	3054.59	+	5290.70i	0.434463	+	0.752513i	0.997252	−	0.0740883i	$-0.0236047\pi$
$367$	3054.59	+	5290.70i	0.434463	+	0.752513i	−0.562788	+	0.826601i	$0.690271\pi$
$368$	2071.00			0.293365
$369$		0			0
$370$	112.613			0.0158229
$371$	−6574.59	−	11387.5i	−0.920043	−	1.59356i
$372$		0			0
$373$	−3277.67	+	5677.08i	−0.454990	+	0.788065i	−0.998688	−	0.0512159i	$-0.983690\pi$
$373$	−3277.67	+	5677.08i	−0.454990	+	0.788065i	0.543698	+	0.839281i	$0.317024\pi$
$374$	268.753	−	465.494i	0.0371575	−	0.0643587i
$375$		0			0
$376$	849.572	+	1471.50i	0.116525	+	0.201827i
$377$	6310.52			0.862092
$378$		0			0
$379$	5032.40			0.682050			0.341025	−	0.940054i	$-0.389226\pi$
$379$	5032.40			0.682050			0.341025	+	0.940054i	$0.389226\pi$
$380$	461.908	+	800.048i	0.0623563	+	0.108004i
$381$		0			0
$382$	−67.2359	+	116.456i	−0.00900547	+	0.0155979i
$383$	1831.49	−	3172.24i	0.244347	−	0.423222i	−0.717601	−	0.696455i	$-0.754760\pi$
$383$	1831.49	−	3172.24i	0.244347	−	0.423222i	0.961948	+	0.273233i	$0.0880930\pi$
$384$		0			0
$385$	−1413.75	−	2448.69i	−0.187146	−	0.324147i
$386$	−272.044			−0.0358722
$387$		0			0
$388$	3085.03			0.403657
$389$	−2435.48	−	4218.37i	−0.317439	−	0.549820i	0.662514	−	0.749049i	$-0.269489\pi$
$389$	−2435.48	−	4218.37i	−0.317439	−	0.549820i	−0.979953	+	0.199229i	$0.936156\pi$
$390$		0			0
$391$	−2170.97	+	3760.22i	−0.280794	+	0.486349i
$392$	1131.78	−	1960.30i	0.145825	−	0.252577i
$393$		0			0
$394$	−368.121	−	637.605i	−0.0470703	−	0.0815281i
$395$	2652.63			0.337895
$396$		0			0
$397$	−3744.62			−0.473393			−0.236696	−	0.971584i	$-0.576065\pi$
$397$	−3744.62			−0.473393			−0.236696	+	0.971584i	$0.576065\pi$
$398$	237.625	+	411.579i	0.0299273	+	0.0518356i
$399$		0			0
$400$	−784.811	+	1359.33i	−0.0981014	+	0.169917i
$401$	−900.435	+	1559.60i	−0.112134	+	0.194221i	−0.916630	−	0.399736i	$-0.869102\pi$
$401$	−900.435	+	1559.60i	−0.112134	+	0.194221i	0.804497	+	0.593957i	$0.202435\pi$
$402$		0			0
$403$	3137.59	+	5434.46i	0.387827	+	0.671737i
$404$	−8585.16			−1.05725
$405$		0			0
$406$	884.049			0.108066
$407$	906.328	+	1569.81i	0.110381	+	0.191185i
$408$		0			0
$409$	−7965.99	+	13797.5i	−0.963064	+	1.66808i	−0.248334	+	0.968675i	$0.579883\pi$
$409$	−7965.99	+	13797.5i	−0.963064	+	1.66808i	−0.714730	+	0.699401i	$0.753450\pi$
$410$	138.149	−	239.281i	0.0166407	−	0.0288225i
$411$		0			0
$412$	−2364.89	−	4096.12i	−0.282791	−	0.489809i
$413$	−23158.8			−2.75926
$414$		0			0
$415$	1472.95			0.174227
$416$	1075.42	+	1862.68i	0.126747	+	0.219532i
$417$		0			0
$418$	47.4553	−	82.1949i	0.00555290	−	0.00961791i
$419$	−1723.87	+	2985.84i	−0.200995	+	0.348133i	−0.948849	−	0.315730i	$-0.897751\pi$
$419$	−1723.87	+	2985.84i	−0.200995	+	0.348133i	0.747854	+	0.663863i	$0.231084\pi$
$420$		0			0
$421$	−2807.99	−	4863.58i	−0.325066	−	0.563032i	0.656459	−	0.754361i	$-0.272053\pi$
$421$	−2807.99	−	4863.58i	−0.325066	−	0.563032i	−0.981526	+	0.191330i	$0.938720\pi$
$422$	148.359			0.0171137
$423$		0			0
$424$	1514.37			0.173454
$425$	−1645.39	−	2849.89i	−0.187795	−	0.325271i
$426$		0			0
$427$	140.020	−	242.522i	0.0158689	−	0.0274858i
$428$	1982.12	−	3433.13i	0.223854	−	0.387726i
$429$		0			0
$430$	−78.3737	−	135.747i	−0.00878957	−	0.0152240i
$431$	3534.04			0.394962			0.197481	−	0.980307i	$-0.436724\pi$
$431$	3534.04			0.394962			0.197481	+	0.980307i	$0.436724\pi$
$432$		0			0
$433$	6674.80			0.740809			0.370405	−	0.928871i	$-0.379219\pi$
$433$	6674.80			0.740809			0.370405	+	0.928871i	$0.379219\pi$
$434$	439.549	+	761.321i	0.0486153	+	0.0842041i
$435$		0			0
$436$	−3812.87	+	6604.08i	−0.418815	+	0.725408i
$437$	−383.339	+	663.963i	−0.0419625	+	0.0726812i
$438$		0			0
$439$	3271.76	+	5666.85i	0.355700	+	0.616091i	0.987238	−	0.159254i	$-0.0509090\pi$
$439$	3271.76	+	5666.85i	0.355700	+	0.616091i	−0.631537	+	0.775346i	$0.717576\pi$
$440$	325.639			0.0352823
$441$		0			0
$442$	−1487.12			−0.160035
$443$	7154.86	+	12392.6i	0.767353	+	1.32909i	0.938993	+	0.343935i	$0.111760\pi$
$443$	7154.86	+	12392.6i	0.767353	+	1.32909i	−0.171640	+	0.985160i	$0.554907\pi$
$444$		0			0
$445$	2132.04	−	3692.81i	0.227120	−	0.393384i
$446$	−137.267	+	237.753i	−0.0145734	+	0.0252420i
$447$		0			0
$448$	−7683.39	−	13308.0i	−0.810282	−	1.40345i
$449$	14587.4			1.53324			0.766618	−	0.642104i	$-0.221938\pi$
$449$	14587.4			1.53324			0.766618	+	0.642104i	$0.221938\pi$
$450$		0			0
$451$	4447.38			0.464343
$452$	−6871.31	−	11901.5i	−0.715043	−	1.23849i
$453$		0			0
$454$	−171.817	+	297.596i	−0.0177616	+	0.0307640i
$455$	−3911.43	+	6774.80i	−0.403013	+	0.698039i
$456$		0			0
$457$	−1012.89	−	1754.37i	−0.103678	−	0.179576i	0.809519	−	0.587093i	$-0.199728\pi$
$457$	−1012.89	−	1754.37i	−0.103678	−	0.179576i	−0.913197	+	0.407518i	$0.866394\pi$
$458$	−1392.64			−0.142083
$459$		0			0
$460$	−1311.06			−0.132888
$461$	−1778.09	−	3079.75i	−0.179640	−	0.311146i	0.762117	−	0.647439i	$-0.224160\pi$
$461$	−1778.09	−	3079.75i	−0.179640	−	0.311146i	−0.941757	+	0.336293i	$0.890827\pi$
$462$		0			0
$463$	5141.34	−	8905.07i	0.516066	−	0.893852i	−0.483760	−	0.875201i	$-0.660729\pi$
$463$	5141.34	−	8905.07i	0.516066	−	0.893852i	0.999826	−	0.0186517i	$-0.00593735\pi$
$464$	3949.72	−	6841.11i	0.395174	−	0.684462i
$465$		0			0
$466$	−226.806	−	392.840i	−0.0225464	−	0.0390514i
$467$	−8217.44			−0.814256			−0.407128	−	0.913371i	$-0.633470\pi$
$467$	−8217.44			−0.814256			−0.407128	+	0.913371i	$0.633470\pi$
$468$		0			0
$469$	−18370.3			−1.80866
$470$	−266.336	−	461.307i	−0.0261386	−	0.0452734i
$471$		0			0
$472$	1333.59	−	2309.84i	0.130049	−	0.225252i
$473$	1261.53	−	2185.03i	0.122632	−	0.212406i
$474$		0			0
$475$	−290.535	−	503.222i	−0.0280646	−	0.0486093i
$476$	32640.5			3.14301
$477$		0			0
$478$	−1145.69			−0.109629
$479$	6842.59	+	11851.7i	0.652705	+	1.13052i	0.982464	+	0.186453i	$0.0596993\pi$
$479$	6842.59	+	11851.7i	0.652705	+	1.13052i	−0.329759	+	0.944065i	$0.606967\pi$
$480$		0			0
$481$	2507.54	−	4343.19i	0.237701	−	0.411710i
$482$	−370.178	+	641.168i	−0.0349817	+	0.0605900i
$483$		0			0
$484$	−3984.00	−	6900.50i	−0.374155	−	0.648056i
$485$	−1940.45			−0.181673
$486$		0			0
$487$	−3239.59			−0.301437			−0.150718	−	0.988577i	$-0.548159\pi$
$487$	−3239.59			−0.301437			−0.150718	+	0.988577i	$0.548159\pi$
$488$	16.1259	+	27.9309i	0.00149587	+	0.00259092i
$489$		0			0
$490$	−354.806	+	614.542i	−0.0327112	+	0.0566575i
$491$	5564.04	−	9637.20i	0.511409	−	0.885786i	−0.488504	−	0.872562i	$-0.662457\pi$
$491$	5564.04	−	9637.20i	0.511409	−	0.885786i	0.999913	−	0.0132239i	$-0.00420943\pi$
$492$		0			0
$493$	8280.74	+	14342.7i	0.756482	+	1.31027i
$494$	−262.590			−0.0239159
$495$		0			0
$496$	7855.18			0.711105
$497$	757.276	+	1311.64i	0.0683470	+	0.118380i
$498$		0			0
$499$	2713.61	−	4700.11i	0.243443	−	0.421655i	−0.718250	−	0.695785i	$-0.755057\pi$
$499$	2713.61	−	4700.11i	0.243443	−	0.421655i	0.961693	+	0.274130i	$0.0883899\pi$
$500$	496.829	−	860.533i	0.0444377	−	0.0769684i
$501$		0			0
$502$	391.997	+	678.958i	0.0348519	+	0.0603653i
$503$	9600.22			0.850999			0.425500	−	0.904959i	$-0.360098\pi$
$503$	9600.22			0.850999			0.425500	+	0.904959i	$0.360098\pi$
$504$		0			0
$505$	5399.97			0.475833
$506$	67.3474	+	116.649i	0.00591691	+	0.0102484i
$507$		0			0
$508$	4053.94	−	7021.62i	0.354064	−	0.613256i
$509$	−9469.94	+	16402.4i	−0.824652	+	1.42834i	0.0775333	+	0.996990i	$0.475296\pi$
$509$	−9469.94	+	16402.4i	−0.824652	+	1.42834i	−0.902185	+	0.431349i	$0.858038\pi$
$510$		0			0
$511$	6391.66	+	11070.7i	0.553328	+	0.958392i
$512$	4496.87			0.388155
$513$		0			0
$514$	1530.57			0.131343
$515$	1487.49	+	2576.41i	0.127275	+	0.220447i
$516$		0			0
$517$	4287.03	−	7425.35i	0.364687	−	0.631657i
$518$	351.285	−	608.444i	0.0297965	−	0.0516090i
$519$		0			0
$520$	−450.474	−	780.244i	−0.0379896	−	0.0657999i
$521$	−19292.6			−1.62231			−0.811155	−	0.584831i	$-0.801161\pi$
$521$	−19292.6			−1.62231			−0.811155	+	0.584831i	$0.801161\pi$
$522$		0			0
$523$	17967.5			1.50223			0.751114	−	0.660172i	$-0.229517\pi$
$523$	17967.5			1.50223			0.751114	+	0.660172i	$0.229517\pi$
$524$	6766.38	+	11719.7i	0.564104	+	0.977057i
$525$		0			0
$526$	604.147	−	1046.41i	0.0500799	−	0.0867410i
$527$	−8234.35	+	14262.3i	−0.680634	+	1.17889i
$528$		0			0
$529$	5539.47	+	9594.65i	0.455287	+	0.788580i
$530$	−474.747			−0.0389088
$531$		0			0
$532$	5763.51			0.469699
$533$	−6152.30	−	10656.1i	−0.499973	−	0.865979i
$534$		0			0
$535$	−1246.73	+	2159.40i	−0.100749	+	0.174503i
$536$	1057.84	−	1832.24i	0.0852460	−	0.147650i
$537$		0			0
$538$	5.46214	+	9.46070i	0.000437713	+	0.000758141i
$539$	−11422.1			−0.912777
$540$		0			0
$541$	−8299.36			−0.659552			−0.329776	−	0.944059i	$-0.606973\pi$
$541$	−8299.36			−0.659552			−0.329776	+	0.944059i	$0.606973\pi$
$542$	−860.811	−	1490.97i	−0.0682195	−	0.118160i
$543$		0			0
$544$	−2822.35	+	4888.46i	−0.222440	+	0.385278i
$545$	2398.25	−	4153.89i	0.188495	−	0.326483i
$546$		0			0
$547$	−42.3605	−	73.3706i	−0.00331116	−	0.00573510i	0.864365	−	0.502865i	$-0.167721\pi$
$547$	−42.3605	−	73.3706i	−0.00331116	−	0.00573510i	−0.867676	+	0.497130i	$0.834387\pi$
$548$	−8344.17			−0.650448
$549$		0			0
$550$	−102.086			−0.00791447
$551$	1462.18	+	2532.56i	0.113050	+	0.195809i
$552$		0			0
$553$	8274.63	−	14332.1i	0.636298	−	1.10210i
$554$	−579.571	+	1003.85i	−0.0444469	+	0.0769843i
$555$		0			0
$556$	5702.89	+	9877.69i	0.434994	+	0.753431i
$557$	20914.0			1.59094			0.795469	−	0.605994i	$-0.207224\pi$
$557$	20914.0			1.59094			0.795469	+	0.605994i	$0.207224\pi$
$558$		0			0
$559$	−6980.56			−0.528169
$560$	4896.28	+	8480.61i	0.369474	+	0.639948i
$561$		0			0
$562$	659.389	−	1142.10i	0.0494923	−	0.0857231i
$563$	5520.48	−	9561.76i	0.413252	−	0.715773i	−0.581991	−	0.813195i	$-0.697726\pi$
$563$	5520.48	−	9561.76i	0.413252	−	0.715773i	0.995243	+	0.0974220i	$0.0310596\pi$
$564$		0			0
$565$	4321.98	+	7485.89i	0.321818	+	0.557405i
$566$	2115.96			0.157139
$567$		0			0
$568$	−174.429			−0.0128853
$569$	−3007.31	−	5208.81i	−0.221569	−	0.383770i	0.733715	−	0.679457i	$-0.237785\pi$
$569$	−3007.31	−	5208.81i	−0.221569	−	0.383770i	−0.955285	+	0.295688i	$0.904451\pi$
$570$		0			0
$571$	10968.7	−	18998.3i	0.803896	−	1.39239i	−0.113137	−	0.993579i	$-0.536090\pi$
$571$	10968.7	−	18998.3i	0.803896	−	1.39239i	0.917034	−	0.398810i	$-0.130577\pi$
$572$	3613.96	−	6259.56i	0.264173	−	0.457561i
$573$		0			0
$574$	−861.884	−	1492.83i	−0.0626730	−	0.108553i
$575$	824.641			0.0598085
$576$		0			0
$577$	473.507			0.0341635			0.0170818	−	0.999854i	$-0.494562\pi$
$577$	473.507			0.0341635			0.0170818	+	0.999854i	$0.494562\pi$
$578$	−1398.09	−	2421.57i	−0.100611	−	0.174263i
$579$		0			0
$580$	−2500.39	+	4330.80i	−0.179005	+	0.310046i
$581$	4594.72	−	7958.29i	0.328092	−	0.568271i
$582$		0			0
$583$	−3820.84	−	6617.89i	−0.271429	−	0.470129i
$584$	−1472.24			−0.104318
$585$		0			0
$586$	401.076			0.0282735
$587$	6510.17	+	11275.9i	0.457757	+	0.792859i	0.998842	−	0.0481093i	$-0.0153196\pi$
$587$	6510.17	+	11275.9i	0.457757	+	0.792859i	−0.541085	+	0.840968i	$0.681986\pi$
$588$		0			0
$589$	−1453.99	+	2518.38i	−0.101715	+	0.176176i
$590$	−418.071	+	724.120i	−0.0291724	+	0.0505281i
$591$		0			0
$592$	−3138.91	−	5436.75i	−0.217920	−	0.377448i
$593$	12887.3			0.892441			0.446220	−	0.894923i	$-0.352770\pi$
$593$	12887.3			0.892441			0.446220	+	0.894923i	$0.352770\pi$
$594$		0			0
$595$	−20530.5			−1.41457
$596$	−4842.94	−	8388.22i	−0.332843	−	0.576501i
$597$		0			0
$598$	186.330	−	322.734i	0.0127418	−	0.0220695i
$599$	−4337.24	+	7512.32i	−0.295851	+	0.512429i	−0.975182	−	0.221403i	$-0.928937\pi$
$599$	−4337.24	+	7512.32i	−0.295851	+	0.512429i	0.679332	+	0.733832i	$0.262270\pi$
$600$		0			0
$601$	−5467.86	−	9470.61i	−0.371112	−	0.642786i	0.618625	−	0.785687i	$-0.287690\pi$
$601$	−5467.86	−	9470.61i	−0.371112	−	0.642786i	−0.989737	+	0.142901i	$0.954357\pi$
$602$	−977.916			−0.0662074
$603$		0			0
$604$	−12794.4			−0.861913
$605$	2505.90	+	4340.34i	0.168395	+	0.291669i
$606$		0			0
$607$	−10835.2	+	18767.1i	−0.724527	+	1.25492i	0.234641	+	0.972082i	$0.424608\pi$
$607$	−10835.2	+	18767.1i	−0.724527	+	1.25492i	−0.959168	+	0.282836i	$0.908725\pi$
$608$	−498.359	+	863.183i	−0.0332420	+	0.0575768i
$609$		0			0
$610$	−5.05537	−	8.75616i	−0.000335551	−	0.000581191i
$611$	−23721.9			−1.57068
$612$		0			0
$613$	−15571.2			−1.02596			−0.512982	−	0.858399i	$-0.671459\pi$
$613$	−15571.2			−1.02596			−0.512982	+	0.858399i	$0.671459\pi$
$614$	−955.079	−	1654.25i	−0.0627750	−	0.108730i
$615$		0			0
$616$	1015.80	−	1759.42i	0.0664411	−	0.115079i
$617$	4863.27	−	8423.43i	0.317322	−	0.549618i	−0.662606	−	0.748968i	$-0.730550\pi$
$617$	4863.27	−	8423.43i	0.317322	−	0.549618i	0.979928	+	0.199350i	$0.0638829\pi$
$618$		0			0
$619$	−3148.79	−	5453.87i	−0.204460	−	0.354135i	0.745501	−	0.666505i	$-0.232210\pi$
$619$	−3148.79	−	5453.87i	−0.204460	−	0.354135i	−0.949961	+	0.312370i	$0.898877\pi$
$620$	−4972.77			−0.322115
$621$		0			0
$622$	1534.13			0.0988955
$623$	−13301.4	−	23038.7i	−0.855392	−	1.48158i
$624$		0			0
$625$	−312.500	+	541.266i	−0.0200000	+	0.0346410i
$626$	−906.272	+	1569.71i	−0.0578625	+	0.100221i
$627$		0			0
$628$	5112.71	+	8855.47i	0.324872	+	0.562694i
$629$	13161.7			0.834327
$630$		0			0
$631$	−5670.98			−0.357778			−0.178889	−	0.983869i	$-0.557250\pi$
$631$	−5670.98			−0.357778			−0.178889	+	0.983869i	$0.557250\pi$
$632$	952.977	+	1650.60i	0.0599800	+	0.103888i
$633$		0			0
$634$	176.396	−	305.527i	0.0110498	−	0.0191388i
$635$	−2549.88	+	4416.53i	−0.159353	+	0.276007i
$636$		0			0
$637$	15800.9	+	27367.9i	0.982816	+	1.70229i
$638$	513.767			0.0318813
$639$		0			0
$640$	−2270.13			−0.140210
$641$	10832.3	+	18762.0i	0.667470	+	1.15609i	0.978609	+	0.205727i	$0.0659560\pi$
$641$	10832.3	+	18762.0i	0.667470	+	1.15609i	−0.311140	+	0.950364i	$0.600711\pi$
$642$		0			0
$643$	−3111.86	+	5389.90i	−0.190855	+	0.330570i	−0.945534	−	0.325524i	$-0.894459\pi$
$643$	−3111.86	+	5389.90i	−0.190855	+	0.330570i	0.754679	+	0.656094i	$0.227793\pi$
$644$	−4089.71	+	7083.59i	−0.250244	+	0.433436i
$645$		0			0
$646$	−344.573	−	596.818i	−0.0209861	−	0.0363491i
$647$	−20451.9			−1.24273			−0.621365	−	0.783521i	$-0.713422\pi$
$647$	−20451.9			−1.24273			−0.621365	+	0.783521i	$0.713422\pi$
$648$		0			0
$649$	−13458.8			−0.814029
$650$	141.221	+	244.602i	0.00852176	+	0.0147601i
$651$		0			0
$652$	5631.41	−	9753.88i	0.338256	−	0.585876i
$653$	3724.22	−	6450.54i	0.223185	−	0.386568i	−0.732588	−	0.680672i	$-0.761688\pi$
$653$	3724.22	−	6450.54i	0.223185	−	0.386568i	0.955773	+	0.294104i	$0.0950211\pi$
$654$		0			0
$655$	−4255.98	−	7371.57i	−0.253885	−	0.439742i
$656$	−15402.7			−0.916731
$657$		0			0
$658$	−3323.23			−0.196889
$659$	413.185	+	715.658i	0.0244240	+	0.0423036i	0.877979	−	0.478699i	$-0.158891\pi$
$659$	413.185	+	715.658i	0.0244240	+	0.0423036i	−0.853555	+	0.521003i	$0.825558\pi$
$660$		0			0
$661$	3129.11	−	5419.77i	0.184127	−	0.318918i	−0.759155	−	0.650910i	$-0.774387\pi$
$661$	3129.11	−	5419.77i	0.184127	−	0.318918i	0.943282	+	0.331992i	$0.107721\pi$
$662$	−275.093	+	476.475i	−0.0161507	+	0.0279739i
$663$		0			0
$664$	529.167	+	916.545i	0.0309272	+	0.0535675i
$665$	−3625.18			−0.211396
$666$		0			0
$667$	−4150.17			−0.240922
$668$	−3557.19	−	6161.24i	−0.206036	−	0.356865i
$669$		0			0
$670$	−331.627	+	574.396i	−0.0191222	+	0.0331207i
$671$	81.3729	−	140.942i	0.00468162	−	0.00810880i
$672$		0			0
$673$	5039.90	+	8729.36i	0.288668	+	0.499988i	0.973492	−	0.228720i	$-0.0734542\pi$
$673$	5039.90	+	8729.36i	0.288668	+	0.499988i	−0.684824	+	0.728709i	$0.740121\pi$
$674$	2133.79			0.121944
$675$		0			0
$676$	−2532.99			−0.144117
$677$	−12855.2	−	22265.9i	−0.729788	−	1.26403i	−0.956973	−	0.290178i	$-0.906285\pi$
$677$	−12855.2	−	22265.9i	−0.729788	−	1.26403i	0.227185	−	0.973852i	$-0.427048\pi$
$678$		0			0
$679$	−6053.05	+	10484.2i	−0.342113	+	0.592557i
$680$	1182.23	−	2047.69i	0.0666714	−	0.115478i
$681$		0			0
$682$	255.445	+	442.443i	0.0143424	+	0.0248417i
$683$	32091.0			1.79784			0.898922	−	0.438109i	$-0.144352\pi$
$683$	32091.0			1.79784			0.898922	+	0.438109i	$0.144352\pi$
$684$		0			0
$685$	5248.39			0.292746
$686$	1008.53	+	1746.82i	0.0561310	+	0.0972217i
$687$		0			0
$688$	−4369.09	+	7567.48i	−0.242107	+	0.419342i
$689$	−10571.2	+	18309.8i	−0.584512	+	1.01240i
$690$		0			0
$691$	−5282.87	−	9150.20i	−0.290839	−	0.503748i	0.683169	−	0.730260i	$-0.260601\pi$
$691$	−5282.87	−	9150.20i	−0.290839	−	0.503748i	−0.974008	+	0.226512i	$0.927268\pi$
$692$	−13792.7			−0.757690
$693$		0			0
$694$	746.133			0.0408110
$695$	−3587.06	−	6212.96i	−0.195777	−	0.339095i
$696$		0			0
$697$	16146.2	−	27966.1i	0.877449	−	1.51979i
$698$	−638.465	+	1105.85i	−0.0346222	+	0.0599673i
$699$		0			0
$700$	−3099.62	−	5368.70i	−0.167364	−	0.289882i
$701$	−13081.8			−0.704837			−0.352419	−	0.935842i	$-0.614641\pi$
$701$	−13081.8			−0.704837			−0.352419	+	0.935842i	$0.614641\pi$
$702$		0			0
$703$	2324.04			0.124684
$704$	−4465.22	−	7733.99i	−0.239047	−	0.414042i
$705$		0			0
$706$	−195.864	+	339.246i	−0.0104411	+	0.0180846i
$707$	16844.7	−	29175.8i	0.896053	−	1.55201i
$708$		0			0
$709$	14110.6	+	24440.3i	0.747440	+	1.29460i	0.949046	+	0.315137i	$0.102051\pi$
$709$	14110.6	+	24440.3i	0.747440	+	1.29460i	−0.201606	+	0.979467i	$0.564616\pi$
$710$	54.6823			0.00289041
$711$		0			0
$712$	3063.80			0.161265
$713$	−2063.46	−	3574.02i	−0.108383	−	0.187725i
$714$		0			0
$715$	−2273.14	+	3937.19i	−0.118896	+	0.205934i
$716$	−8479.38	+	14686.7i	−0.442583	+	0.766576i
$717$		0			0
$718$	971.342	+	1682.41i	0.0504877	+	0.0874473i
$719$	11471.4			0.595010			0.297505	−	0.954720i	$-0.403846\pi$
$719$	11471.4			0.595010			0.297505	+	0.954720i	$0.403846\pi$
$720$		0			0
$721$	18560.3			0.958701
$722$	711.651	+	1232.62i	0.0366827	+	0.0635363i
$723$		0			0
$724$	−14356.3	+	24865.8i	−0.736944	+	1.27642i
$725$	1572.72	−	2724.03i	0.0805645	−	0.139542i
$726$		0			0
$727$	−10829.6	−	18757.5i	−0.552475	−	0.956914i	−0.998095	−	0.0616922i	$-0.980350\pi$
$727$	−10829.6	−	18757.5i	−0.552475	−	0.956914i	0.445621	−	0.895222i	$-0.352983\pi$
$728$	−5620.84			−0.286157
$729$		0			0
$730$	461.537			0.0234004
$731$	−9159.96	−	15865.5i	−0.463466	−	0.802746i
$732$		0			0
$733$	−2840.86	+	4920.51i	−0.143151	+	0.247944i	−0.928682	−	0.370878i	$-0.879057\pi$
$733$	−2840.86	+	4920.51i	−0.143151	+	0.247944i	0.785531	+	0.618823i	$0.212390\pi$
$734$	688.045	−	1191.73i	0.0345997	−	0.0599285i
$735$		0			0
$736$	−707.259	−	1225.01i	−0.0354211	−	0.0613511i
$737$	−10676.0			−0.533588
$738$		0			0
$739$	261.324			0.0130080			0.00650402	−	0.999979i	$-0.497930\pi$
$739$	261.324			0.0130080			0.00650402	+	0.999979i	$0.497930\pi$
$740$	1987.11	+	3441.77i	0.0987127	+	0.170975i
$741$		0			0
$742$	−1480.93	+	2565.04i	−0.0732702	+	0.126908i
$743$	−8202.97	+	14208.0i	−0.405031	+	0.701533i	−0.994325	−	0.106385i	$-0.966072\pi$
$743$	−8202.97	+	14208.0i	−0.405031	+	0.701533i	0.589294	+	0.807918i	$0.299406\pi$
$744$		0			0
$745$	3046.16	+	5276.10i	0.149802	+	0.259465i
$746$	1476.59			0.0724688
$747$		0			0
$748$	18969.1			0.927244
$749$	7778.11	+	13472.1i	0.379447	+	0.657222i
$750$		0			0
$751$	−10737.3	+	18597.5i	−0.521716	+	0.903639i	0.477965	+	0.878379i	$0.341375\pi$
$751$	−10737.3	+	18597.5i	−0.521716	+	0.903639i	−0.999681	+	0.0252601i	$0.991959\pi$
$752$	−14847.4	+	25716.4i	−0.719985	+	1.24705i
$753$		0			0
$754$	−710.722	−	1231.01i	−0.0343276	−	0.0594571i
$755$	8047.51			0.387919
$756$		0			0
$757$	−13643.2			−0.655046			−0.327523	−	0.944843i	$-0.606214\pi$
$757$	−13643.2			−0.655046			−0.327523	+	0.944843i	$0.606214\pi$
$758$	−566.773	−	981.680i	−0.0271585	−	0.0470399i
$759$		0			0
$760$	208.754	−	361.572i	0.00996353	−	0.0172573i
$761$	13469.0	−	23328.9i	0.641589	−	1.11127i	−0.343489	−	0.939157i	$-0.611609\pi$
$761$	13469.0	−	23328.9i	0.641589	−	1.11127i	0.985078	−	0.172109i	$-0.0550580\pi$
$762$		0			0
$763$	−14962.2	−	25915.3i	−0.709920	−	1.22962i
$764$	−4745.63			−0.224726
$765$		0			0
$766$	−825.088			−0.0389186
$767$	18618.3	+	32247.9i	0.876491	+	1.51813i
$768$		0			0
$769$	14442.7	−	25015.5i	0.677265	−	1.17306i	−0.298536	−	0.954398i	$-0.596498\pi$
$769$	14442.7	−	25015.5i	0.677265	−	1.17306i	0.975801	−	0.218659i	$-0.0701683\pi$
$770$	−318.447	+	551.566i	−0.0149039	+	0.0258144i
$771$		0			0
$772$	−4800.34	−	8314.43i	−0.223793	−	0.387620i
$773$	−3031.34			−0.141048			−0.0705238	−	0.997510i	$-0.522467\pi$
$773$	−3031.34			−0.141048			−0.0705238	+	0.997510i	$0.522467\pi$
$774$		0			0
$775$	3127.82			0.144974
$776$	−697.121	−	1207.45i	−0.0322490	−	0.0558568i
$777$		0			0
$778$	−548.591	+	950.188i	−0.0252801	+	0.0437865i
$779$	2851.03	−	4938.13i	0.131128	−	0.227120i
$780$		0			0
$781$	440.092	+	762.262i	0.0201636	+	0.0349243i
$782$	978.019			0.0447236
$783$		0			0
$784$	39558.7			1.80205
$785$	−3215.84	−	5570.00i	−0.146214	−	0.253251i
$786$		0			0
$787$	−7577.38	+	13124.4i	−0.343208	+	0.594453i	−0.985026	−	0.172403i	$-0.944847\pi$
$787$	−7577.38	+	13124.4i	−0.343208	+	0.594453i	0.641819	+	0.766856i	$0.278180\pi$
$788$	12991.3	−	22501.6i	0.587306	−	1.01724i
$789$		0			0
$790$	−298.753	−	517.455i	−0.0134546	−	0.0233041i
$791$	53928.0			2.42409
$792$		0			0
$793$	−450.270			−0.0201634
$794$	421.737	+	730.470i	0.0188500	+	0.0326491i
$795$		0			0
$796$	−8386.00	+	14525.0i	−0.373409	+	0.646764i
$797$	14190.3	−	24578.3i	0.630673	−	1.09236i	−0.356741	−	0.934203i	$-0.616112\pi$
$797$	14190.3	−	24578.3i	0.630673	−	1.09236i	0.987414	−	0.158155i	$-0.0505546\pi$
$798$		0			0
$799$	−31128.1	−	53915.5i	−1.37827	−	2.38723i
$800$	1072.07			0.0473793
$801$		0			0
$802$	405.646			0.0178602
$803$	3714.53	+	6433.75i	0.163241	+	0.282742i
$804$		0			0
$805$	2572.39	−	4455.50i	0.112627	−	0.195076i
$806$	706.741	−	1224.11i	0.0308857	−	0.0534957i
$807$		0			0
$808$	1939.98	+	3360.14i	0.0844655	+	0.146299i
$809$	14569.6			0.633175			0.316588	−	0.948563i	$-0.397463\pi$
$809$	14569.6			0.633175			0.316588	+	0.948563i	$0.397463\pi$
$810$		0			0
$811$	27927.7			1.20921			0.604607	−	0.796524i	$-0.293330\pi$
$811$	27927.7			1.20921			0.604607	+	0.796524i	$0.293330\pi$
$812$	15599.4	+	27019.0i	0.674178	+	1.16771i
$813$		0			0
$814$	204.150	−	353.598i	0.00879049	−	0.0152256i
$815$	−3542.09	+	6135.09i	−0.152238	+	0.263684i
$816$		0			0
$817$	−1617.43	−	2801.46i	−0.0692614	−	0.119964i
$818$	3588.68			0.153393
$819$		0			0
$820$	9750.79			0.415259
$821$	9022.03	+	15626.6i	0.383521	+	0.664279i	0.991563	−	0.129627i	$-0.0413779\pi$
$821$	9022.03	+	15626.6i	0.383521	+	0.664279i	−0.608041	+	0.793905i	$0.708045\pi$
$822$		0			0
$823$	16126.4	−	27931.8i	0.683028	−	1.18304i	−0.291025	−	0.956716i	$-0.593996\pi$
$823$	16126.4	−	27931.8i	0.683028	−	1.18304i	0.974052	−	0.226323i	$-0.0726705\pi$
$824$	−1068.78	+	1851.19i	−0.0451855	+	0.0782636i
$825$		0			0
$826$	2608.26	+	4517.64i	0.109871	+	0.190301i
$827$	−13569.5			−0.570566			−0.285283	−	0.958443i	$-0.592088\pi$
$827$	−13569.5			−0.570566			−0.285283	+	0.958443i	$0.592088\pi$
$828$		0			0
$829$	742.559			0.0311099			0.0155550	−	0.999879i	$-0.495049\pi$
$829$	742.559			0.0311099			0.0155550	+	0.999879i	$0.495049\pi$
$830$	−165.891	−	287.331i	−0.00693754	−	0.0120162i
$831$		0			0
$832$	−12354.0	+	21397.7i	−0.514780	+	0.891625i
$833$	−41468.1	+	71824.9i	−1.72483	+	2.98750i
$834$		0			0
$835$	2237.44	+	3875.35i	0.0927302	+	0.160613i
$836$	3349.48			0.138569
$837$		0			0
$838$	776.605			0.0320136
$839$	−17752.3	−	30747.9i	−0.730487	−	1.26524i	−0.956675	−	0.291157i	$-0.905960\pi$
$839$	−17752.3	−	30747.9i	−0.730487	−	1.26524i	0.226189	−	0.974083i	$-0.427373\pi$
$840$		0			0
$841$	4279.49	−	7412.29i	0.175468	−	0.303919i
$842$	−632.499	+	1095.52i	−0.0258876	+	0.0448386i
$843$		0			0
$844$	2617.86	+	4534.26i	0.106766	+	0.184924i
$845$	1593.23			0.0648623
$846$		0			0
$847$	31267.6			1.26844
$848$	13232.8	+	22919.9i	0.535869	+	0.928153i
$849$		0			0
$850$	−370.623	+	641.938i	−0.0149556	+	0.0259039i
$851$	−1649.11	+	2856.34i	−0.0664285	+	0.115058i
$852$		0			0
$853$	−187.524	−	324.801i	−0.00752719	−	0.0130375i	0.862237	−	0.506505i	$-0.169063\pi$
$853$	−187.524	−	324.801i	−0.00752719	−	0.0130375i	−0.869764	+	0.493467i	$0.835729\pi$
$854$	−63.0790			−0.00252754
$855$		0			0
$856$	−1791.59			−0.0715365
$857$	16097.4	+	27881.5i	0.641629	+	1.11133i	0.985069	+	0.172159i	$0.0550743\pi$
$857$	16097.4	+	27881.5i	0.641629	+	1.11133i	−0.343440	+	0.939174i	$0.611592\pi$
$858$		0			0
$859$	14068.8	−	24367.8i	0.558813	−	0.967893i	−0.438783	−	0.898593i	$-0.644590\pi$
$859$	14068.8	−	24367.8i	0.558813	−	0.967893i	0.997596	−	0.0692994i	$-0.0220764\pi$
$860$	2765.87	−	4790.64i	0.109669	−	0.189953i
$861$		0			0
$862$	−398.021	−	689.392i	−0.0157270	−	0.0272399i
$863$	−15724.6			−0.620244			−0.310122	−	0.950697i	$-0.600370\pi$
$863$	−15724.6			−0.620244			−0.310122	+	0.950697i	$0.600370\pi$
$864$		0			0
$865$	8675.48			0.341012
$866$	−751.749	−	1302.07i	−0.0294982	−	0.0510924i
$867$		0			0
$868$	−15512.1	+	26867.7i	−0.606583	+	1.05063i
$869$	4808.82	−	8329.12i	0.187719	−	0.325139i
$870$		0			0
$871$	14768.6	+	25580.0i	0.574531	+	0.995117i
$872$	3446.35			0.133840
$873$		0			0
$874$	172.694			0.00668361
$875$	1949.63	+	3376.85i	0.0753250	+	0.130467i
$876$		0			0
$877$	−7640.23	+	13233.3i	−0.294176	+	0.509528i	−0.974793	−	0.223112i	$-0.928378\pi$
$877$	−7640.23	+	13233.3i	−0.294176	+	0.509528i	0.680617	+	0.732639i	$0.261712\pi$
$878$	736.963	−	1276.46i	0.0283272	−	0.0490642i
$879$		0			0
$880$	2845.49	+	4928.52i	0.109001	+	0.188796i
$881$	24687.6			0.944092			0.472046	−	0.881574i	$-0.343516\pi$
$881$	24687.6			0.944092			0.472046	+	0.881574i	$0.343516\pi$
$882$		0			0
$883$	6562.59			0.250112			0.125056	−	0.992150i	$-0.460089\pi$
$883$	6562.59			0.250112			0.125056	+	0.992150i	$0.460089\pi$
$884$	−26241.0	−	45450.7i	−0.998392	−	1.72927i
$885$		0			0
$886$	1611.63	−	2791.43i	0.0611104	−	0.105846i
$887$	25255.5	−	43743.9i	0.956030	−	1.65589i	0.224035	−	0.974581i	$-0.428077\pi$
$887$	25255.5	−	43743.9i	0.956030	−	1.65589i	0.731994	−	0.681311i	$-0.238590\pi$
$888$		0			0
$889$	15908.2	+	27553.8i	0.600162	+	1.03951i
$890$	−960.485			−0.0361747
$891$		0			0
$892$	−9688.51			−0.363672
$893$	−5496.47	−	9520.17i	−0.205971	−	0.356753i
$894$		0			0
$895$	5333.44	−	9237.78i	0.199192	−	0.345011i
$896$	−7081.43	+	12265.4i	−0.264034	+	0.457320i
$897$		0			0
$898$	−1642.91	−	2845.60i	−0.0610518	−	0.105745i
$899$	−15741.4			−0.583986
$900$		0			0
$901$	−55486.3			−2.05163
$902$	−500.886	−	867.559i	−0.0184897	−	0.0320250i
$903$		0			0
$904$	−3105.40	+	5378.72i	−0.114252	+	0.197891i
$905$	9029.96	−	15640.3i	0.331675	−	0.574478i
$906$		0			0
$907$	3508.72	+	6077.28i	0.128451	+	0.222484i	0.923077	−	0.384616i	$-0.125666\pi$
$907$	3508.72	+	6077.28i	0.128451	+	0.222484i	−0.794626	+	0.607100i	$0.792333\pi$
$908$	−12127.2			−0.443231
$909$		0			0
$910$	1762.10			0.0641902
$911$	−3062.30	−	5304.05i	−0.111370	−	0.192899i	0.804953	−	0.593339i	$-0.202191\pi$
$911$	−3062.30	−	5304.05i	−0.111370	−	0.192899i	−0.916323	+	0.400440i	$0.868857\pi$
$912$		0			0
$913$	2670.23	−	4624.98i	0.0967928	−	0.167650i
$914$	−228.153	+	395.172i	−0.00825669	+	0.0143010i
$915$		0			0
$916$	−24573.8	−	42563.0i	−0.886397	−	1.53528i
$917$	−53104.4			−1.91239
$918$		0			0
$919$	23780.7			0.853594			0.426797	−	0.904347i	$-0.359642\pi$
$919$	23780.7			0.853594			0.426797	+	0.904347i	$0.359642\pi$
$920$	296.258	+	513.134i	0.0106167	+	0.0183886i
$921$		0			0
$922$	−400.515	+	693.713i	−0.0143062	+	0.0247790i
$923$	1217.61	−	2108.96i	0.0434215	−	0.0752082i
$924$		0			0
$925$	−1249.87	−	2164.83i	−0.0444274	−	0.0769506i
$926$	−2316.17			−0.0821967
$927$		0			0
$928$	−5395.41			−0.190854
$929$	5094.54	+	8824.01i	0.179921	+	0.311632i	0.941853	−	0.336024i	$-0.109082\pi$
$929$	5094.54	+	8824.01i	0.179921	+	0.311632i	−0.761932	+	0.647657i	$0.775749\pi$
$930$		0			0
$931$	−7322.26	+	12682.5i	−0.257763	+	0.446459i
$932$	8004.19	−	13863.7i	0.281316	−	0.487253i
$933$		0			0
$934$	925.488	+	1602.99i	0.0324228	+	0.0561579i
$935$	−11931.3			−0.417323
$936$		0			0
$937$	−3880.22			−0.135284			−0.0676421	−	0.997710i	$-0.521548\pi$
$937$	−3880.22			−0.135284			−0.0676421	+	0.997710i	$0.521548\pi$
$938$	2068.96	+	3583.54i	0.0720191	+	0.124741i
$939$		0			0
$940$	9399.22	−	16279.9i	0.326137	−	0.564886i
$941$	−19336.9	+	33492.6i	−0.669890	+	1.16028i	0.308045	+	0.951372i	$0.400325\pi$
$941$	−19336.9	+	33492.6i	−0.669890	+	1.16028i	−0.977935	+	0.208911i	$0.933008\pi$
$942$		0			0
$943$	4046.11	+	7008.07i	0.139724	+	0.242008i
$944$	46612.3			1.60710
$945$		0			0
$946$	−568.318			−0.0195324
$947$	−3877.44	−	6715.92i	−0.133052	−	0.230452i	0.791800	−	0.610781i	$-0.209144\pi$
$947$	−3877.44	−	6715.92i	−0.133052	−	0.230452i	−0.924851	+	0.380328i	$0.875811\pi$
$948$		0			0
$949$	10277.0	−	17800.3i	0.351534	−	0.608875i
$950$	−65.4430	+	113.351i	−0.00223500	+	0.00387114i
$951$		0			0
$952$	−7375.72	−	12775.1i	−0.251101	−	0.434920i
$953$	22527.9			0.765739			0.382870	−	0.923802i	$-0.374936\pi$
$953$	22527.9			0.765739			0.382870	+	0.923802i	$0.374936\pi$
$954$		0			0
$955$	2984.95			0.101142
$956$	−20216.2	−	35015.5i	−0.683931	−	1.18460i
$957$		0			0
$958$	1541.29	−	2669.60i	0.0519800	−	0.0900321i
$959$	16371.9	−	28356.9i	0.551277	−	0.954840i
$960$		0			0
$961$	7068.91	+	12243.7i	0.237283	+	0.410987i
$962$	−1129.65			−0.0378600
$963$		0			0
$964$	−26127.8			−0.872947
$965$	3019.36	+	5229.68i	0.100722	+	0.174455i
$966$		0			0
$967$	2826.20	−	4895.13i	0.0939861	−	0.162789i	−0.815199	−	0.579181i	$-0.803372\pi$
$967$	2826.20	−	4895.13i	0.0939861	−	0.162789i	0.909185	+	0.416392i	$0.136706\pi$
$968$	−1800.52	+	3118.59i	−0.0597840	+	0.103549i
$969$		0			0
$970$	218.543	+	378.528i	0.00723402	+	0.0125297i
$971$	−19612.4			−0.648188			−0.324094	−	0.946025i	$-0.605059\pi$
$971$	−19612.4			−0.648188			−0.324094	+	0.946025i	$0.605059\pi$
$972$		0			0
$973$	−44757.9			−1.47469
$974$	364.858	+	631.953i	0.0120029	+	0.0207896i
$975$		0			0
$976$	−281.821	+	488.129i	−0.00924270	+	0.0160088i
$977$	−13841.8	+	23974.7i	−0.453264	+	0.785076i	−0.998587	−	0.0531500i	$-0.983074\pi$
$977$	−13841.8	+	23974.7i	−0.453264	+	0.785076i	0.545323	+	0.838226i	$0.316407\pi$
$978$		0			0
$979$	−7730.14	−	13389.0i	−0.252356	−	0.437093i
$980$	−25042.8			−0.816290
$981$		0			0
$982$	−2506.60			−0.0814549
$983$	−15308.2	−	26514.6i	−0.496700	−	0.860310i	0.503293	−	0.864116i	$-0.332122\pi$
$983$	−15308.2	−	26514.6i	−0.496700	−	0.860310i	−0.999993	+	0.00380619i	$0.998788\pi$
$984$		0			0
$985$	−8171.40	+	14153.3i	−0.264327	+	0.457829i
$986$	1865.23	−	3230.68i	0.0602446	−	0.104347i
$987$		0			0
$988$	−4633.51	−	8025.48i	−0.149202	−	0.258426i
$989$	4590.82			0.147603
$990$		0			0
$991$	−14462.4			−0.463587			−0.231793	−	0.972765i	$-0.574459\pi$
$991$	−14462.4			−0.463587			−0.231793	+	0.972765i	$0.574459\pi$
$992$	−2682.59	−	4646.39i	−0.0858593	−	0.148713i
$993$		0			0
$994$	170.576	−	295.447i	0.00544301	−	0.00942756i
$995$	5274.70	−	9136.05i	0.168060	−	0.291088i
$996$		0			0
$997$	3448.63	+	5973.20i	0.109548	+	0.189743i	0.915587	−	0.402120i	$-0.131726\pi$
$997$	3448.63	+	5973.20i	0.109548	+	0.189743i	−0.806039	+	0.591862i	$0.798393\pi$
$998$	−1222.48			−0.0387745
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.e.c.91.4		14
3.2	odd	2		45.4.e.c.31.4	yes	14
9.2	odd	6		45.4.e.c.16.4	✓	14
9.4	even	3		405.4.a.n.1.4		7
9.5	odd	6		405.4.a.m.1.4		7
9.7	even	3	inner	135.4.e.c.46.4		14
15.2	even	4		225.4.k.d.49.7		28
15.8	even	4		225.4.k.d.49.8		28
15.14	odd	2		225.4.e.d.76.4		14
45.2	even	12		225.4.k.d.124.8		28
45.4	even	6		2025.4.a.ba.1.4		7
45.14	odd	6		2025.4.a.bb.1.4		7
45.29	odd	6		225.4.e.d.151.4		14
45.38	even	12		225.4.k.d.124.7		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.e.c.16.4	✓	14	9.2	odd	6
45.4.e.c.31.4	yes	14	3.2	odd	2
135.4.e.c.46.4		14	9.7	even	3	inner
135.4.e.c.91.4		14	1.1	even	1	trivial
225.4.e.d.76.4		14	15.14	odd	2
225.4.e.d.151.4		14	45.29	odd	6
225.4.k.d.49.7		28	15.2	even	4
225.4.k.d.49.8		28	15.8	even	4
225.4.k.d.124.7		28	45.38	even	12
225.4.k.d.124.8		28	45.2	even	12
405.4.a.m.1.4		7	9.5	odd	6
405.4.a.n.1.4		7	9.4	even	3
2025.4.a.ba.1.4		7	45.4	even	6
2025.4.a.bb.1.4		7	45.14	odd	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:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	135.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.112625 - 0.195072i) q^{2} +(3.97463 - 6.88426i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(15.5970 + 27.0148i) q^{7} -3.59257 q^{8} +O(q^{10})\)	\(q+(-0.112625 - 0.195072i) q^{2} +(3.97463 - 6.88426i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(15.5970 + 27.0148i) q^{7} -3.59257 q^{8} +1.12625 q^{10} +(9.06424 + 15.6997i) q^{11} +(25.0781 - 43.4366i) q^{13} +(3.51322 - 6.08508i) q^{14} +(-31.3924 - 54.3733i) q^{16} +131.631 q^{17} +23.2428 q^{19} +(19.8732 + 34.4213i) q^{20} +(2.04172 - 3.53636i) q^{22} +(-16.4928 + 28.5664i) q^{23} +(-12.5000 - 21.6506i) q^{25} -11.2977 q^{26} +247.969 q^{28} +(62.9087 + 108.961i) q^{29} +(-62.5563 + 108.351i) q^{31} +(-21.4414 + 37.1376i) q^{32} +(-14.8249 - 25.6775i) q^{34} -155.970 q^{35} +99.9894 q^{37} +(-2.61772 - 4.53402i) q^{38} +(8.98142 - 15.5563i) q^{40} +(122.663 - 212.458i) q^{41} +(-69.5882 - 120.530i) q^{43} +144.108 q^{44} +7.43001 q^{46} +(-236.480 - 409.596i) q^{47} +(-315.033 + 545.654i) q^{49} +(-2.81562 + 4.87680i) q^{50} +(-199.352 - 345.289i) q^{52} -421.529 q^{53} -90.6424 q^{55} +(-56.0333 - 97.0526i) q^{56} +(14.1702 - 24.5435i) q^{58} +(-371.207 + 642.949i) q^{59} +(-4.48868 - 7.77462i) q^{61} +28.1816 q^{62} -492.620 q^{64} +(125.391 + 217.183i) q^{65} +(-294.453 + 510.008i) q^{67} +(523.185 - 906.182i) q^{68} +(17.5661 + 30.4254i) q^{70} +48.5526 q^{71} +409.800 q^{73} +(-11.2613 - 19.5051i) q^{74} +(92.3816 - 160.010i) q^{76} +(-282.750 + 489.737i) q^{77} +(-265.263 - 459.449i) q^{79} +313.924 q^{80} -55.2595 q^{82} +(-147.295 - 255.122i) q^{83} +(-329.077 + 569.979i) q^{85} +(-15.6747 + 27.1494i) q^{86} +(-32.5639 - 56.4023i) q^{88} -852.817 q^{89} +1564.57 q^{91} +(131.106 + 227.082i) q^{92} +(-53.2672 + 92.2615i) q^{94} +(-58.1070 + 100.644i) q^{95} +(194.045 + 336.096i) q^{97} +141.922 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 2 q^{2} - 36 q^{4} - 35 q^{5} - 22 q^{7} + 36 q^{8}+O(q^{10}) \) 14 * q - 2 * q^2 - 36 * q^4 - 35 * q^5 - 22 * q^7 + 36 * q^8	\( 14 q - 2 q^{2} - 36 q^{4} - 35 q^{5} - 22 q^{7} + 36 q^{8} + 20 q^{10} - 23 q^{11} - 96 q^{13} + 21 q^{14} - 324 q^{16} + 322 q^{17} + 558 q^{19} - 180 q^{20} - 311 q^{22} - 96 q^{23} - 175 q^{25} - 716 q^{26} + 674 q^{28} + 296 q^{29} - 244 q^{31} + 314 q^{32} - 125 q^{34} + 220 q^{35} + 808 q^{37} - 305 q^{38} - 90 q^{40} + 47 q^{41} - 525 q^{43} + 110 q^{44} + 1434 q^{46} - 164 q^{47} - 1225 q^{49} - 50 q^{50} - 1682 q^{52} + 1012 q^{53} + 230 q^{55} + 981 q^{56} - 1183 q^{58} + 85 q^{59} - 828 q^{61} - 1572 q^{62} + 4472 q^{64} - 480 q^{65} - 1093 q^{67} - 2473 q^{68} + 105 q^{70} + 656 q^{71} + 4170 q^{73} + 1316 q^{74} - 2789 q^{76} - 24 q^{77} - 2110 q^{79} + 3240 q^{80} - 124 q^{82} - 1290 q^{83} - 805 q^{85} + 2569 q^{86} - 2271 q^{88} - 6096 q^{89} + 6676 q^{91} - 2763 q^{92} + 517 q^{94} - 1395 q^{95} - 1787 q^{97} + 2558 q^{98}+O(q^{100}) \) 14 * q - 2 * q^2 - 36 * q^4 - 35 * q^5 - 22 * q^7 + 36 * q^8 + 20 * q^10 - 23 * q^11 - 96 * q^13 + 21 * q^14 - 324 * q^16 + 322 * q^17 + 558 * q^19 - 180 * q^20 - 311 * q^22 - 96 * q^23 - 175 * q^25 - 716 * q^26 + 674 * q^28 + 296 * q^29 - 244 * q^31 + 314 * q^32 - 125 * q^34 + 220 * q^35 + 808 * q^37 - 305 * q^38 - 90 * q^40 + 47 * q^41 - 525 * q^43 + 110 * q^44 + 1434 * q^46 - 164 * q^47 - 1225 * q^49 - 50 * q^50 - 1682 * q^52 + 1012 * q^53 + 230 * q^55 + 981 * q^56 - 1183 * q^58 + 85 * q^59 - 828 * q^61 - 1572 * q^62 + 4472 * q^64 - 480 * q^65 - 1093 * q^67 - 2473 * q^68 + 105 * q^70 + 656 * q^71 + 4170 * q^73 + 1316 * q^74 - 2789 * q^76 - 24 * q^77 - 2110 * q^79 + 3240 * q^80 - 124 * q^82 - 1290 * q^83 - 805 * q^85 + 2569 * q^86 - 2271 * q^88 - 6096 * q^89 + 6676 * q^91 - 2763 * q^92 + 517 * q^94 - 1395 * q^95 - 1787 * q^97 + 2558 * q^98

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