LMFDB - Embedded newform 270.2.i.b.199.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [270,2,Mod(19,270)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(270, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 3])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("270.19"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 270 = 2 \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	270.i (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$2.15596085457$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			199.4
Root			$-0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	270.199
Dual form			270.2.i.b.19.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(1.30701 - 1.81431i) q^{5} +(-0.389270 + 0.224745i) q^{7} -1.00000i q^{8} +(0.224745 - 2.22474i) q^{10} +(-1.72474 - 2.98735i) q^{11} +(2.12132 + 1.22474i) q^{13} +(-0.224745 + 0.389270i) q^{14} +(-0.500000 - 0.866025i) q^{16} +5.89898i q^{17} +5.44949 q^{19} +(-0.917738 - 2.03906i) q^{20} +(-2.98735 - 1.72474i) q^{22} +(-5.97469 - 3.44949i) q^{23} +(-1.58346 - 4.74264i) q^{25} +2.44949 q^{26} +0.449490i q^{28} +(3.00000 + 5.19615i) q^{29} +(-0.775255 + 1.34278i) q^{31} +(-0.866025 - 0.500000i) q^{32} +(2.94949 + 5.10867i) q^{34} +(-0.101021 + 1.00000i) q^{35} +8.00000i q^{37} +(4.71940 - 2.72474i) q^{38} +(-1.81431 - 1.30701i) q^{40} +(-0.500000 + 0.866025i) q^{41} +(-2.20881 + 1.27526i) q^{43} -3.44949 q^{44} -6.89898 q^{46} +(3.85337 - 2.22474i) q^{47} +(-3.39898 + 5.88721i) q^{49} +(-3.74264 - 3.31552i) q^{50} +(2.12132 - 1.22474i) q^{52} +3.55051i q^{53} +(-7.67423 - 0.775255i) q^{55} +(0.224745 + 0.389270i) q^{56} +(5.19615 + 3.00000i) q^{58} +(-6.62372 + 11.4726i) q^{59} +(-2.22474 - 3.85337i) q^{61} +1.55051i q^{62} -1.00000 q^{64} +(4.99465 - 2.24799i) q^{65} +(-3.94086 - 2.27526i) q^{67} +(5.10867 + 2.94949i) q^{68} +(0.412514 + 0.916536i) q^{70} +2.44949 q^{71} -14.7980i q^{73} +(4.00000 + 6.92820i) q^{74} +(2.72474 - 4.71940i) q^{76} +(1.34278 + 0.775255i) q^{77} +(3.67423 + 6.36396i) q^{79} +(-2.22474 - 0.224745i) q^{80} +1.00000i q^{82} +(-3.46410 + 2.00000i) q^{83} +(10.7026 + 7.71001i) q^{85} +(-1.27526 + 2.20881i) q^{86} +(-2.98735 + 1.72474i) q^{88} +3.10102 q^{89} -1.10102 q^{91} +(-5.97469 + 3.44949i) q^{92} +(2.22474 - 3.85337i) q^{94} +(7.12252 - 9.88708i) q^{95} +(11.2583 - 6.50000i) q^{97} +6.79796i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{4} + 4 q^{5} - 8 q^{10} - 4 q^{11} + 8 q^{14} - 4 q^{16} + 24 q^{19} - 4 q^{20} + 24 q^{29} - 16 q^{31} + 4 q^{34} - 40 q^{35} - 4 q^{40} - 4 q^{41} - 8 q^{44} - 16 q^{46} + 12 q^{49} + 4 q^{50}+ \cdots + 24 q^{95}+O(q^{100}) $ 8 * q + 4 * q^4 + 4 * q^5 - 8 * q^10 - 4 * q^11 + 8 * q^14 - 4 * q^16 + 24 * q^19 - 4 * q^20 + 24 * q^29 - 16 * q^31 + 4 * q^34 - 40 * q^35 - 4 * q^40 - 4 * q^41 - 8 * q^44 - 16 * q^46 + 12 * q^49 + 4 * q^50 - 32 * q^55 - 8 * q^56 - 4 * q^59 - 8 * q^61 - 8 * q^64 + 12 * q^65 + 4 * q^70 + 32 * q^74 + 12 * q^76 - 8 * q^80 - 20 * q^85 - 20 * q^86 + 64 * q^89 - 48 * q^91 + 8 * q^94 + 24 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/270\mathbb{Z}\right)^\times$.

$n$	$191$	$217$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$-1$

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 $

$2$	0.866025	−	0.500000i	0.612372	−	0.353553i
$2$	0.866025	−	0.500000i	0.612372	−	0.353553i
$3$		0			0
$3$		0			0
$4$	0.500000	−	0.866025i	0.250000	−	0.433013i
$5$	1.30701	−	1.81431i	0.584511	−	0.811386i
$5$	1.30701	−	1.81431i	0.584511	−	0.811386i
$6$		0			0
$7$	−0.389270	+	0.224745i	−0.147130	+	0.0849456i	−0.571758	−	0.820422i	$-0.693738\pi$
$7$	−0.389270	+	0.224745i	−0.147130	+	0.0849456i	0.424628	+	0.905368i	$0.360405\pi$
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	0.224745	−	2.22474i	0.0710706	−	0.703526i
$11$	−1.72474	−	2.98735i	−0.520030	−	0.900719i	−0.999729	−	0.0232854i	$-0.992587\pi$
$11$	−1.72474	−	2.98735i	−0.520030	−	0.900719i	0.479699	−	0.877433i	$-0.340746\pi$
$12$		0			0
$13$	2.12132	+	1.22474i	0.588348	+	0.339683i	0.764444	−	0.644690i	$-0.223014\pi$
$13$	2.12132	+	1.22474i	0.588348	+	0.339683i	−0.176096	+	0.984373i	$0.556347\pi$
$14$	−0.224745	+	0.389270i	−0.0600656	+	0.104037i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$			5.89898i			1.43071i	0.698760	+	0.715356i	$0.253736\pi$
$17$			5.89898i			1.43071i	−0.698760	+	0.715356i	$0.746264\pi$
$18$		0			0
$19$	5.44949			1.25020			0.625099	−	0.780545i	$-0.285058\pi$
$19$	5.44949			1.25020			0.625099	+	0.780545i	$0.285058\pi$
$20$	−0.917738	−	2.03906i	−0.205212	−	0.455947i
$21$		0			0
$22$	−2.98735	−	1.72474i	−0.636904	−	0.367717i
$23$	−5.97469	−	3.44949i	−1.24581	−	0.719268i	−0.275538	−	0.961290i	$-0.588856\pi$
$23$	−5.97469	−	3.44949i	−1.24581	−	0.719268i	−0.970271	+	0.242022i	$0.922189\pi$
$24$		0			0
$25$	−1.58346	−	4.74264i	−0.316693	−	0.948528i
$26$	2.44949			0.480384
$27$		0			0
$28$			0.449490i			0.0849456i
$29$	3.00000	+	5.19615i	0.557086	+	0.964901i	0.997738	+	0.0672232i	$0.0214140\pi$
$29$	3.00000	+	5.19615i	0.557086	+	0.964901i	−0.440652	+	0.897678i	$0.645253\pi$
$30$		0			0
$31$	−0.775255	+	1.34278i	−0.139240	+	0.241171i	−0.927209	−	0.374544i	$-0.877799\pi$
$31$	−0.775255	+	1.34278i	−0.139240	+	0.241171i	0.787969	+	0.615715i	$0.211133\pi$
$32$	−0.866025	−	0.500000i	−0.153093	−	0.0883883i
$33$		0			0
$34$	2.94949	+	5.10867i	0.505833	+	0.876129i
$35$	−0.101021	+	1.00000i	−0.0170756	+	0.169031i
$36$		0			0
$37$			8.00000i			1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$			8.00000i			1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	4.71940	−	2.72474i	0.765587	−	0.442012i
$39$		0			0
$40$	−1.81431	−	1.30701i	−0.286868	−	0.206656i
$41$	−0.500000	+	0.866025i	−0.0780869	+	0.135250i	−0.902424	−	0.430848i	$-0.858214\pi$
$41$	−0.500000	+	0.866025i	−0.0780869	+	0.135250i	0.824338	+	0.566099i	$0.191548\pi$
$42$		0			0
$43$	−2.20881	+	1.27526i	−0.336840	+	0.194475i	−0.658874	−	0.752254i	$-0.728967\pi$
$43$	−2.20881	+	1.27526i	−0.336840	+	0.194475i	0.322034	+	0.946728i	$0.395634\pi$
$44$	−3.44949			−0.520030
$45$		0			0
$46$	−6.89898			−1.01720
$47$	3.85337	−	2.22474i	0.562072	−	0.324512i	−0.191905	−	0.981414i	$-0.561466\pi$
$47$	3.85337	−	2.22474i	0.562072	−	0.324512i	0.753977	+	0.656901i	$0.228133\pi$
$48$		0			0
$49$	−3.39898	+	5.88721i	−0.485568	+	0.841029i
$50$	−3.74264	−	3.31552i	−0.529289	−	0.468885i
$51$		0			0
$52$	2.12132	−	1.22474i	0.294174	−	0.169842i
$53$			3.55051i			0.487700i	0.969813	+	0.243850i	$0.0784105\pi$
$53$			3.55051i			0.487700i	−0.969813	+	0.243850i	$0.921590\pi$
$54$		0			0
$55$	−7.67423	−	0.775255i	−1.03479	−	0.104535i
$56$	0.224745	+	0.389270i	0.0300328	+	0.0520183i
$57$		0			0
$58$	5.19615	+	3.00000i	0.682288	+	0.393919i
$59$	−6.62372	+	11.4726i	−0.862335	+	1.49361i	0.00733331	+	0.999973i	$0.497666\pi$
$59$	−6.62372	+	11.4726i	−0.862335	+	1.49361i	−0.869669	+	0.493636i	$0.835668\pi$
$60$		0			0
$61$	−2.22474	−	3.85337i	−0.284849	−	0.493374i	0.687723	−	0.725973i	$-0.258610\pi$
$61$	−2.22474	−	3.85337i	−0.284849	−	0.493374i	−0.972573	+	0.232599i	$0.925277\pi$
$62$			1.55051i			0.196915i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	4.99465	−	2.24799i	0.619510	−	0.278829i
$66$		0			0
$67$	−3.94086	−	2.27526i	−0.481452	−	0.277967i	0.239569	−	0.970879i	$-0.422994\pi$
$67$	−3.94086	−	2.27526i	−0.481452	−	0.277967i	−0.721022	+	0.692913i	$0.756327\pi$
$68$	5.10867	+	2.94949i	0.619517	+	0.357678i
$69$		0			0
$70$	0.412514	+	0.916536i	0.0493048	+	0.109547i
$71$	2.44949			0.290701			0.145350	−	0.989380i	$-0.453569\pi$
$71$	2.44949			0.290701			0.145350	+	0.989380i	$0.453569\pi$
$72$		0			0
$73$		−	14.7980i		−	1.73197i	−0.500070	−	0.865985i	$-0.666692\pi$
$73$		−	14.7980i		−	1.73197i	0.500070	−	0.865985i	$-0.333308\pi$
$74$	4.00000	+	6.92820i	0.464991	+	0.805387i
$75$		0			0
$76$	2.72474	−	4.71940i	0.312550	−	0.541352i
$77$	1.34278	+	0.775255i	0.153024	+	0.0883485i
$78$		0			0
$79$	3.67423	+	6.36396i	0.413384	+	0.716002i	0.995257	−	0.0972777i	$-0.0310135\pi$
$79$	3.67423	+	6.36396i	0.413384	+	0.716002i	−0.581874	+	0.813279i	$0.697680\pi$
$80$	−2.22474	−	0.224745i	−0.248734	−	0.0251272i
$81$		0			0
$82$			1.00000i			0.110432i
$83$	−3.46410	+	2.00000i	−0.380235	+	0.219529i	−0.677920	−	0.735135i	$-0.737119\pi$
$83$	−3.46410	+	2.00000i	−0.380235	+	0.219529i	0.297686	+	0.954664i	$0.403785\pi$
$84$		0			0
$85$	10.7026	+	7.71001i	1.16086	+	0.836268i
$86$	−1.27526	+	2.20881i	−0.137514	+	0.238182i
$87$		0			0
$88$	−2.98735	+	1.72474i	−0.318452	+	0.183858i
$89$	3.10102			0.328708			0.164354	−	0.986401i	$-0.447446\pi$
$89$	3.10102			0.328708			0.164354	+	0.986401i	$0.447446\pi$
$90$		0			0
$91$	−1.10102			−0.115418
$92$	−5.97469	+	3.44949i	−0.622905	+	0.359634i
$93$		0			0
$94$	2.22474	−	3.85337i	0.229465	−	0.397445i
$95$	7.12252	−	9.88708i	0.730755	−	1.01439i
$96$		0			0
$97$	11.2583	−	6.50000i	1.14311	−	0.659975i	0.195911	−	0.980622i	$-0.437234\pi$
$97$	11.2583	−	6.50000i	1.14311	−	0.659975i	0.947199	+	0.320647i	$0.103900\pi$
$98$			6.79796i			0.686698i
$99$		0			0
$100$	−4.89898	−	1.00000i	−0.489898	−	0.100000i
$101$	−4.00000	−	6.92820i	−0.398015	−	0.689382i	0.595466	−	0.803380i	$-0.296967\pi$
$101$	−4.00000	−	6.92820i	−0.398015	−	0.689382i	−0.993481	+	0.113998i	$0.963634\pi$
$102$		0			0
$103$	−12.3387	−	7.12372i	−1.21576	−	0.701921i	−0.251755	−	0.967791i	$-0.581008\pi$
$103$	−12.3387	−	7.12372i	−1.21576	−	0.701921i	−0.964009	+	0.265870i	$0.914341\pi$
$104$	1.22474	−	2.12132i	0.120096	−	0.208013i
$105$		0			0
$106$	1.77526	+	3.07483i	0.172428	+	0.298654i
$107$		−	16.3485i		−	1.58047i	−0.612806	−	0.790233i	$-0.709959\pi$
$107$		−	16.3485i		−	1.58047i	0.612806	−	0.790233i	$-0.290041\pi$
$108$		0			0
$109$	8.00000			0.766261			0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000			0.766261			0.383131	+	0.923694i	$0.374846\pi$
$110$	−7.03371	+	3.16573i	−0.670638	+	0.301840i
$111$		0			0
$112$	0.389270	+	0.224745i	0.0367825	+	0.0212364i
$113$	4.24264	+	2.44949i	0.399114	+	0.230429i	0.686102	−	0.727506i	$-0.259321\pi$
$113$	4.24264	+	2.44949i	0.399114	+	0.230429i	−0.286988	+	0.957934i	$0.592654\pi$
$114$		0			0
$115$	−14.0674	+	6.33145i	−1.31179	+	0.590411i
$116$	6.00000			0.557086
$117$		0			0
$118$			13.2474i			1.21953i
$119$	−1.32577	−	2.29629i	−0.121533	−	0.210501i
$120$		0			0
$121$	−0.449490	+	0.778539i	−0.0408627	+	0.0707763i
$122$	−3.85337	−	2.22474i	−0.348868	−	0.201419i
$123$		0			0
$124$	0.775255	+	1.34278i	0.0696200	+	0.120585i
$125$	−10.6742	−	3.32577i	−0.954733	−	0.297465i
$126$		0			0
$127$			6.89898i			0.612185i	0.952002	+	0.306093i	$0.0990218\pi$
$127$			6.89898i			0.612185i	−0.952002	+	0.306093i	$0.900978\pi$
$128$	−0.866025	+	0.500000i	−0.0765466	+	0.0441942i
$129$		0			0
$130$	3.20150	−	4.44414i	0.280790	−	0.389777i
$131$	2.44949	−	4.24264i	0.214013	−	0.370681i	−0.738954	−	0.673756i	$-0.764680\pi$
$131$	2.44949	−	4.24264i	0.214013	−	0.370681i	0.952967	+	0.303075i	$0.0980132\pi$
$132$		0			0
$133$	−2.12132	+	1.22474i	−0.183942	+	0.106199i
$134$	−4.55051			−0.393104
$135$		0			0
$136$	5.89898			0.505833
$137$	−2.59808	+	1.50000i	−0.221969	+	0.128154i	−0.606861	−	0.794808i	$-0.707572\pi$
$137$	−2.59808	+	1.50000i	−0.221969	+	0.128154i	0.384893	+	0.922961i	$0.374238\pi$
$138$		0			0
$139$	6.62372	−	11.4726i	0.561817	−	0.973096i	−0.435521	−	0.900179i	$-0.643436\pi$
$139$	6.62372	−	11.4726i	0.561817	−	0.973096i	0.997338	−	0.0729170i	$-0.0232308\pi$
$140$	0.815515	+	0.587486i	0.0689236	+	0.0496517i
$141$		0			0
$142$	2.12132	−	1.22474i	0.178017	−	0.102778i
$143$		−	8.44949i		−	0.706582i
$144$		0			0
$145$	13.3485	+	1.34847i	1.10853	+	0.111984i
$146$	−7.39898	−	12.8154i	−0.612344	−	1.06061i
$147$		0			0
$148$	6.92820	+	4.00000i	0.569495	+	0.328798i
$149$	4.12372	−	7.14250i	0.337829	−	0.585136i	−0.646195	−	0.763172i	$-0.723641\pi$
$149$	4.12372	−	7.14250i	0.337829	−	0.585136i	0.984024	+	0.178036i	$0.0569742\pi$
$150$		0			0
$151$	−1.44949	−	2.51059i	−0.117958	−	0.204309i	0.801000	−	0.598664i	$-0.204301\pi$
$151$	−1.44949	−	2.51059i	−0.117958	−	0.204309i	−0.918958	+	0.394355i	$0.870968\pi$
$152$		−	5.44949i		−	0.442012i
$153$		0			0
$154$	1.55051			0.124944
$155$	1.42296	+	3.16158i	0.114295	+	0.253944i
$156$		0			0
$157$	−13.8564	−	8.00000i	−1.10586	−	0.638470i	−0.168107	−	0.985769i	$-0.553765\pi$
$157$	−13.8564	−	8.00000i	−1.10586	−	0.638470i	−0.937754	+	0.347299i	$0.887099\pi$
$158$	6.36396	+	3.67423i	0.506290	+	0.292306i
$159$		0			0
$160$	−2.03906	+	0.917738i	−0.161202	+	0.0725535i
$161$	3.10102			0.244395
$162$		0			0
$163$		−	8.89898i		−	0.697022i	−0.937305	−	0.348511i	$-0.886687\pi$
$163$		−	8.89898i		−	0.697022i	0.937305	−	0.348511i	$-0.113313\pi$
$164$	0.500000	+	0.866025i	0.0390434	+	0.0676252i
$165$		0			0
$166$	−2.00000	+	3.46410i	−0.155230	+	0.268866i
$167$	0.214297	+	0.123724i	0.0165828	+	0.00957408i	0.508268	−	0.861199i	$-0.330286\pi$
$167$	0.214297	+	0.123724i	0.0165828	+	0.00957408i	−0.491686	+	0.870773i	$0.663619\pi$
$168$		0			0
$169$	−3.50000	−	6.06218i	−0.269231	−	0.466321i
$170$	13.1237	+	1.32577i	1.00654	+	0.101682i
$171$		0			0
$172$			2.55051i			0.194475i
$173$	−10.2173	+	5.89898i	−0.776809	+	0.448491i	−0.835298	−	0.549797i	$-0.814705\pi$
$173$	−10.2173	+	5.89898i	−0.776809	+	0.448491i	0.0584890	+	0.998288i	$0.481372\pi$
$174$		0			0
$175$	1.68228	+	1.49029i	0.127168	+	0.112655i
$176$	−1.72474	+	2.98735i	−0.130008	+	0.225180i
$177$		0			0
$178$	2.68556	−	1.55051i	0.201291	−	0.116216i
$179$	−0.898979			−0.0671929			−0.0335964	−	0.999435i	$-0.510696\pi$
$179$	−0.898979			−0.0671929			−0.0335964	+	0.999435i	$0.510696\pi$
$180$		0			0
$181$	5.55051			0.412566			0.206283	−	0.978492i	$-0.433863\pi$
$181$	5.55051			0.412566			0.206283	+	0.978492i	$0.433863\pi$
$182$	−0.953512	+	0.550510i	−0.0706790	+	0.0408065i
$183$		0			0
$184$	−3.44949	+	5.97469i	−0.254300	+	0.440460i
$185$	14.5145	+	10.4561i	1.06713	+	0.768745i
$186$		0			0
$187$	17.6223	−	10.1742i	1.28867	−	0.744014i
$188$		−	4.44949i		−	0.324512i
$189$		0			0
$190$	1.22474	−	12.1237i	0.0888523	−	0.879547i
$191$	9.12372	+	15.8028i	0.660170	+	1.14345i	0.980571	+	0.196165i	$0.0628489\pi$
$191$	9.12372	+	15.8028i	0.660170	+	1.14345i	−0.320401	+	0.947282i	$0.603818\pi$
$192$		0			0
$193$	11.8619	+	6.84847i	0.853838	+	0.492964i	0.861944	−	0.507004i	$-0.169247\pi$
$193$	11.8619	+	6.84847i	0.853838	+	0.492964i	−0.00810596	+	0.999967i	$0.502580\pi$
$194$	6.50000	−	11.2583i	0.466673	−	0.808301i
$195$		0			0
$196$	3.39898	+	5.88721i	0.242784	+	0.420515i
$197$			8.00000i			0.569976i	0.958531	+	0.284988i	$0.0919897\pi$
$197$			8.00000i			0.569976i	−0.958531	+	0.284988i	$0.908010\pi$
$198$		0			0
$199$	−15.5505			−1.10235			−0.551173	−	0.834391i	$-0.685820\pi$
$199$	−15.5505			−1.10235			−0.551173	+	0.834391i	$0.685820\pi$
$200$	−4.74264	+	1.58346i	−0.335355	+	0.111968i
$201$		0			0
$202$	−6.92820	−	4.00000i	−0.487467	−	0.281439i
$203$	−2.33562	−	1.34847i	−0.163928	−	0.0946440i
$204$		0			0
$205$	0.917738	+	2.03906i	0.0640976	+	0.142414i
$206$	−14.2474			−0.992667
$207$		0			0
$208$		−	2.44949i		−	0.169842i
$209$	−9.39898	−	16.2795i	−0.650141	−	1.12608i
$210$		0			0
$211$	−1.89898	+	3.28913i	−0.130731	+	0.226433i	−0.923959	−	0.382492i	$-0.875066\pi$
$211$	−1.89898	+	3.28913i	−0.130731	+	0.226433i	0.793227	+	0.608925i	$0.208399\pi$
$212$	3.07483	+	1.77526i	0.211180	+	0.121925i
$213$		0			0
$214$	−8.17423	−	14.1582i	−0.558779	−	0.967834i
$215$	−0.573214	+	5.67423i	−0.0390929	+	0.386980i
$216$		0			0
$217$		−	0.696938i		−	0.0473113i
$218$	6.92820	−	4.00000i	0.469237	−	0.270914i
$219$		0			0
$220$	−4.50851	+	6.25845i	−0.303964	+	0.421945i
$221$	−7.22474	+	12.5136i	−0.485989	+	0.841758i
$222$		0			0
$223$	−7.88171	+	4.55051i	−0.527799	+	0.304725i	−0.740120	−	0.672475i	$-0.765231\pi$
$223$	−7.88171	+	4.55051i	−0.527799	+	0.304725i	0.212321	+	0.977200i	$0.431898\pi$
$224$	0.449490			0.0300328
$225$		0			0
$226$	4.89898			0.325875
$227$	2.98735	−	1.72474i	0.198277	−	0.114475i	−0.397574	−	0.917570i	$-0.630148\pi$
$227$	2.98735	−	1.72474i	0.198277	−	0.114475i	0.595852	+	0.803095i	$0.296815\pi$
$228$		0			0
$229$	−9.22474	+	15.9777i	−0.609588	+	1.05584i	0.381720	+	0.924278i	$0.375332\pi$
$229$	−9.22474	+	15.9777i	−0.609588	+	1.05584i	−0.991308	+	0.131560i	$0.958001\pi$
$230$	−9.01702	+	12.5169i	−0.594564	+	0.825341i
$231$		0			0
$232$	5.19615	−	3.00000i	0.341144	−	0.196960i
$233$		−	13.6969i		−	0.897316i	−0.893703	−	0.448658i	$-0.851902\pi$
$233$		−	13.6969i		−	0.897316i	0.893703	−	0.448658i	$-0.148098\pi$
$234$		0			0
$235$	1.00000	−	9.89898i	0.0652328	−	0.645738i
$236$	6.62372	+	11.4726i	0.431168	+	0.746804i
$237$		0			0
$238$	−2.29629	−	1.32577i	−0.148847	−	0.0859366i
$239$	0.348469	−	0.603566i	0.0225406	−	0.0390415i	−0.854535	−	0.519394i	$-0.826158\pi$
$239$	0.348469	−	0.603566i	0.0225406	−	0.0390415i	0.877076	+	0.480352i	$0.159491\pi$
$240$		0			0
$241$	−0.500000	−	0.866025i	−0.0322078	−	0.0557856i	0.849472	−	0.527633i	$-0.176921\pi$
$241$	−0.500000	−	0.866025i	−0.0322078	−	0.0557856i	−0.881680	+	0.471848i	$0.843587\pi$
$242$			0.898979i			0.0577886i
$243$		0			0
$244$	−4.44949			−0.284849
$245$	6.23874	+	13.8614i	0.398579	+	0.885574i
$246$		0			0
$247$	11.5601	+	6.67423i	0.735552	+	0.424671i
$248$	1.34278	+	0.775255i	0.0852667	+	0.0492287i
$249$		0			0
$250$	−10.9070	+	2.45692i	−0.689822	+	0.155389i
$251$	−6.55051			−0.413465			−0.206732	−	0.978398i	$-0.566283\pi$
$251$	−6.55051			−0.413465			−0.206732	+	0.978398i	$0.566283\pi$
$252$		0			0
$253$			23.7980i			1.49616i
$254$	3.44949	+	5.97469i	0.216440	+	0.374885i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	8.74774	+	5.05051i	0.545669	+	0.315042i	0.747373	−	0.664404i	$-0.231315\pi$
$257$	8.74774	+	5.05051i	0.545669	+	0.315042i	−0.201704	+	0.979446i	$0.564648\pi$
$258$		0			0
$259$	−1.79796	−	3.11416i	−0.111720	−	0.193504i
$260$	0.550510	−	5.44949i	0.0341412	−	0.337963i
$261$		0			0
$262$		−	4.89898i		−	0.302660i
$263$	−10.7816	+	6.22474i	−0.664820	+	0.383834i	−0.794111	−	0.607773i	$-0.792063\pi$
$263$	−10.7816	+	6.22474i	−0.664820	+	0.383834i	0.129291	+	0.991607i	$0.458730\pi$
$264$		0			0
$265$	6.44174	+	4.64054i	0.395713	+	0.285066i
$266$	−1.22474	+	2.12132i	−0.0750939	+	0.130066i
$267$		0			0
$268$	−3.94086	+	2.27526i	−0.240726	+	0.138983i
$269$	16.0454			0.978306			0.489153	−	0.872198i	$-0.337306\pi$
$269$	16.0454			0.978306			0.489153	+	0.872198i	$0.337306\pi$
$270$		0			0
$271$	−15.5959			−0.947385			−0.473692	−	0.880690i	$-0.657079\pi$
$271$	−15.5959			−0.947385			−0.473692	+	0.880690i	$0.657079\pi$
$272$	5.10867	−	2.94949i	0.309758	−	0.178839i
$273$		0			0
$274$	−1.50000	+	2.59808i	−0.0906183	+	0.156956i
$275$	−11.4368	+	12.9102i	−0.689667	+	0.778514i
$276$		0			0
$277$	−25.6308	+	14.7980i	−1.54001	+	0.889123i	−0.541169	+	0.840914i	$0.682018\pi$
$277$	−25.6308	+	14.7980i	−1.54001	+	0.889123i	−0.998837	+	0.0482095i	$0.984648\pi$
$278$		−	13.2474i		−	0.794529i
$279$		0			0
$280$	1.00000	+	0.101021i	0.0597614	+	0.00603713i
$281$	6.00000	+	10.3923i	0.357930	+	0.619953i	0.987615	−	0.156898i	$-0.0501493\pi$
$281$	6.00000	+	10.3923i	0.357930	+	0.619953i	−0.629685	+	0.776851i	$0.716816\pi$
$282$		0			0
$283$	3.46410	+	2.00000i	0.205919	+	0.118888i	0.599414	−	0.800439i	$-0.295400\pi$
$283$	3.46410	+	2.00000i	0.205919	+	0.118888i	−0.393494	+	0.919327i	$0.628734\pi$
$284$	1.22474	−	2.12132i	0.0726752	−	0.125877i
$285$		0			0
$286$	−4.22474	−	7.31747i	−0.249814	−	0.432691i
$287$		−	0.449490i		−	0.0265325i
$288$		0			0
$289$	−17.7980			−1.04694
$290$	12.2343	−	5.50643i	0.718426	−	0.323348i
$291$		0			0
$292$	−12.8154	−	7.39898i	−0.749965	−	0.432993i
$293$	−15.5885	−	9.00000i	−0.910687	−	0.525786i	−0.0300351	−	0.999549i	$-0.509562\pi$
$293$	−15.5885	−	9.00000i	−0.910687	−	0.525786i	−0.880652	+	0.473763i	$0.842895\pi$
$294$		0			0
$295$	12.1577	+	27.0123i	0.707848	+	1.57272i
$296$	8.00000			0.464991
$297$		0			0
$298$		−	8.24745i		−	0.477762i
$299$	−8.44949	−	14.6349i	−0.488647	−	0.846361i
$300$		0			0
$301$	0.573214	−	0.992836i	0.0330395	−	0.0572261i
$302$	−2.51059	−	1.44949i	−0.144468	−	0.0834088i
$303$		0			0
$304$	−2.72474	−	4.71940i	−0.156275	−	0.270676i
$305$	−9.89898	−	1.00000i	−0.566814	−	0.0572598i
$306$		0			0
$307$			29.9444i			1.70902i	0.519438	+	0.854508i	$0.326141\pi$
$307$			29.9444i			1.70902i	−0.519438	+	0.854508i	$0.673859\pi$
$308$	1.34278	−	0.775255i	0.0765121	−	0.0441743i
$309$		0			0
$310$	2.81311	+	2.02653i	0.159774	+	0.115099i
$311$	6.55051	−	11.3458i	0.371445	−	0.643362i	−0.618343	−	0.785909i	$-0.712196\pi$
$311$	6.55051	−	11.3458i	0.371445	−	0.643362i	0.989788	+	0.142546i	$0.0455290\pi$
$312$		0			0
$313$	18.7901	−	10.8485i	1.06208	−	0.613192i	0.136073	−	0.990699i	$-0.456552\pi$
$313$	18.7901	−	10.8485i	1.06208	−	0.613192i	0.926007	+	0.377507i	$0.123219\pi$
$314$	−16.0000			−0.902932
$315$		0			0
$316$	7.34847			0.413384
$317$	−19.8704	+	11.4722i	−1.11603	+	0.644343i	−0.940386	−	0.340110i	$-0.889536\pi$
$317$	−19.8704	+	11.4722i	−1.11603	+	0.644343i	−0.175649	+	0.984453i	$0.556202\pi$
$318$		0			0
$319$	10.3485	−	17.9241i	0.579403	−	1.00356i
$320$	−1.30701	+	1.81431i	−0.0730639	+	0.101423i
$321$		0			0
$322$	2.68556	−	1.55051i	0.149661	−	0.0864066i
$323$			32.1464i			1.78868i
$324$		0			0
$325$	2.44949	−	12.0000i	0.135873	−	0.665640i
$326$	−4.44949	−	7.70674i	−0.246434	−	0.426837i
$327$		0			0
$328$	0.866025	+	0.500000i	0.0478183	+	0.0276079i
$329$	−1.00000	+	1.73205i	−0.0551318	+	0.0954911i
$330$		0			0
$331$	12.6969	+	21.9917i	0.697887	+	1.20878i	0.969198	+	0.246284i	$0.0792095\pi$
$331$	12.6969	+	21.9917i	0.697887	+	1.20878i	−0.271311	+	0.962492i	$0.587457\pi$
$332$			4.00000i			0.219529i
$333$		0			0
$334$	0.247449			0.0135398
$335$	−9.27875	+	4.17617i	−0.506953	+	0.228169i
$336$		0			0
$337$	−16.1045	−	9.29796i	−0.877270	−	0.506492i	−0.00751272	−	0.999972i	$-0.502391\pi$
$337$	−16.1045	−	9.29796i	−0.877270	−	0.506492i	−0.869757	+	0.493480i	$0.835725\pi$
$338$	−6.06218	−	3.50000i	−0.329739	−	0.190375i
$339$		0			0
$340$	12.0284	−	5.41372i	0.652329	−	0.293600i
$341$	5.34847			0.289636
$342$		0			0
$343$		−	6.20204i		−	0.334879i
$344$	1.27526	+	2.20881i	0.0687571	+	0.119091i
$345$		0			0
$346$	−5.89898	+	10.2173i	−0.317131	+	0.549287i
$347$	8.00853	+	4.62372i	0.429920	+	0.248215i	0.699313	−	0.714816i	$-0.253490\pi$
$347$	8.00853	+	4.62372i	0.429920	+	0.248215i	−0.269392	+	0.963030i	$0.586823\pi$
$348$		0			0
$349$	−13.7980	−	23.8988i	−0.738588	−	1.27927i	−0.953131	−	0.302557i	$-0.902160\pi$
$349$	−13.7980	−	23.8988i	−0.738588	−	1.27927i	0.214543	−	0.976714i	$-0.431174\pi$
$350$	2.20204	+	0.449490i	0.117704	+	0.0240262i
$351$		0			0
$352$			3.44949i			0.183858i
$353$	28.2289	−	16.2980i	1.50247	−	0.867453i	0.502476	−	0.864591i	$-0.332422\pi$
$353$	28.2289	−	16.2980i	1.50247	−	0.867453i	0.999996	−	0.00286194i	$-0.000910986\pi$
$354$		0			0
$355$	3.20150	−	4.44414i	0.169918	−	0.235871i
$356$	1.55051	−	2.68556i	0.0821769	−	0.142335i
$357$		0			0
$358$	−0.778539	+	0.449490i	−0.0411471	+	0.0237563i
$359$	3.55051			0.187389			0.0936944	−	0.995601i	$-0.470132\pi$
$359$	3.55051			0.187389			0.0936944	+	0.995601i	$0.470132\pi$
$360$		0			0
$361$	10.6969			0.562997
$362$	4.80688	−	2.77526i	0.252644	−	0.145864i
$363$		0			0
$364$	−0.550510	+	0.953512i	−0.0288546	+	0.0499776i
$365$	−26.8481	−	19.3410i	−1.40530	−	1.01236i
$366$		0			0
$367$	23.8988	−	13.7980i	1.24751	−	0.720248i	0.276894	−	0.960900i	$-0.410695\pi$
$367$	23.8988	−	13.7980i	1.24751	−	0.720248i	0.970611	+	0.240653i	$0.0773615\pi$
$368$			6.89898i			0.359634i
$369$		0			0
$370$	17.7980	+	1.79796i	0.925272	+	0.0934714i
$371$	−0.797959	−	1.38211i	−0.0414280	−	0.0717553i
$372$		0			0
$373$	−11.7744	−	6.79796i	−0.609656	−	0.351985i	0.163175	−	0.986597i	$-0.447827\pi$
$373$	−11.7744	−	6.79796i	−0.609656	−	0.351985i	−0.772831	+	0.634612i	$0.781160\pi$
$374$	10.1742	−	17.6223i	0.526097	−	0.911227i
$375$		0			0
$376$	−2.22474	−	3.85337i	−0.114732	−	0.198722i
$377$			14.6969i			0.756931i
$378$		0			0
$379$	−4.14643			−0.212988			−0.106494	−	0.994313i	$-0.533962\pi$
$379$	−4.14643			−0.212988			−0.106494	+	0.994313i	$0.533962\pi$
$380$	−5.00120	−	11.1118i	−0.256556	−	0.570025i
$381$		0			0
$382$	15.8028	+	9.12372i	0.808539	+	0.466810i
$383$	15.4135	+	8.89898i	0.787592	+	0.454717i	0.839114	−	0.543955i	$-0.183074\pi$
$383$	15.4135	+	8.89898i	0.787592	+	0.454717i	−0.0515220	+	0.998672i	$0.516407\pi$
$384$		0			0
$385$	3.16158	−	1.42296i	0.161129	−	0.0725208i
$386$	13.6969			0.697156
$387$		0			0
$388$		−	13.0000i		−	0.659975i
$389$	−8.77526	−	15.1992i	−0.444923	−	0.770629i	0.553124	−	0.833099i	$-0.313436\pi$
$389$	−8.77526	−	15.1992i	−0.444923	−	0.770629i	−0.998047	+	0.0624697i	$0.980102\pi$
$390$		0			0
$391$	20.3485	−	35.2446i	1.02907	−	1.78240i
$392$	5.88721	+	3.39898i	0.297349	+	0.171674i
$393$		0			0
$394$	4.00000	+	6.92820i	0.201517	+	0.349038i
$395$	16.3485	+	1.65153i	0.822581	+	0.0830975i
$396$		0			0
$397$		−	1.79796i		−	0.0902370i	−0.998982	−	0.0451185i	$-0.985633\pi$
$397$		−	1.79796i		−	0.0902370i	0.998982	−	0.0451185i	$-0.0143665\pi$
$398$	−13.4671	+	7.77526i	−0.675047	+	0.389738i
$399$		0			0
$400$	−3.31552	+	3.74264i	−0.165776	+	0.187132i
$401$	4.60102	−	7.96920i	0.229764	−	0.397963i	−0.727974	−	0.685605i	$-0.759538\pi$
$401$	4.60102	−	7.96920i	0.229764	−	0.397963i	0.957738	+	0.287642i	$0.0928712\pi$
$402$		0			0
$403$	−3.28913	+	1.89898i	−0.163843	+	0.0945949i
$404$	−8.00000			−0.398015
$405$		0			0
$406$	−2.69694			−0.133847
$407$	23.8988	−	13.7980i	1.18462	−	0.683939i
$408$		0			0
$409$	7.94949	−	13.7689i	0.393077	−	0.680829i	−0.599777	−	0.800167i	$-0.704744\pi$
$409$	7.94949	−	13.7689i	0.393077	−	0.680829i	0.992854	+	0.119338i	$0.0380773\pi$
$410$	1.81431	+	1.30701i	0.0896025	+	0.0645485i
$411$		0			0
$412$	−12.3387	+	7.12372i	−0.607882	+	0.350961i
$413$		−	5.95459i		−	0.293006i
$414$		0			0
$415$	−0.898979	+	8.89898i	−0.0441292	+	0.436834i
$416$	−1.22474	−	2.12132i	−0.0600481	−	0.104006i
$417$		0			0
$418$	−16.2795	−	9.39898i	−0.796257	−	0.459719i
$419$	4.44949	−	7.70674i	0.217372	−	0.376499i	−0.736632	−	0.676294i	$-0.763585\pi$
$419$	4.44949	−	7.70674i	0.217372	−	0.376499i	0.954004	+	0.299795i	$0.0969183\pi$
$420$		0			0
$421$	5.77526	+	10.0030i	0.281469	+	0.487518i	0.971747	−	0.236026i	$-0.0758451\pi$
$421$	5.77526	+	10.0030i	0.281469	+	0.487518i	−0.690278	+	0.723544i	$0.742512\pi$
$422$			3.79796i			0.184882i
$423$		0			0
$424$	3.55051			0.172428
$425$	27.9767	−	9.34082i	1.35707	−	0.453097i
$426$		0			0
$427$	1.73205	+	1.00000i	0.0838198	+	0.0483934i
$428$	−14.1582	−	8.17423i	−0.684362	−	0.395117i
$429$		0			0
$430$	2.34070	+	5.20064i	0.112879	+	0.250797i
$431$	−38.2474			−1.84231			−0.921157	−	0.389190i	$-0.872755\pi$
$431$	−38.2474			−1.84231			−0.921157	+	0.389190i	$0.872755\pi$
$432$		0			0
$433$		−	23.0000i		−	1.10531i	−0.833410	−	0.552655i	$-0.813615\pi$
$433$		−	23.0000i		−	1.10531i	0.833410	−	0.552655i	$-0.186385\pi$
$434$	−0.348469	−	0.603566i	−0.0167271	−	0.0289721i
$435$		0			0
$436$	4.00000	−	6.92820i	0.191565	−	0.331801i
$437$	−32.5590	−	18.7980i	−1.55751	−	0.899228i
$438$		0			0
$439$	11.0227	+	19.0919i	0.526085	+	0.911206i	0.999538	+	0.0303869i	$0.00967395\pi$
$439$	11.0227	+	19.0919i	0.526085	+	0.911206i	−0.473453	+	0.880819i	$0.656993\pi$
$440$	−0.775255	+	7.67423i	−0.0369588	+	0.365855i
$441$		0			0
$442$			14.4495i			0.687292i
$443$	18.4008	−	10.6237i	0.874250	−	0.504748i	0.00549166	−	0.999985i	$-0.498252\pi$
$443$	18.4008	−	10.6237i	0.874250	−	0.504748i	0.868758	+	0.495237i	$0.164919\pi$
$444$		0			0
$445$	4.05306	−	5.62622i	0.192133	−	0.266709i
$446$	−4.55051	+	7.88171i	−0.215473	+	0.373210i
$447$		0			0
$448$	0.389270	−	0.224745i	0.0183913	−	0.0106182i
$449$	18.7980			0.887131			0.443565	−	0.896242i	$-0.353713\pi$
$449$	18.7980			0.887131			0.443565	+	0.896242i	$0.353713\pi$
$450$		0			0
$451$	3.44949			0.162430
$452$	4.24264	−	2.44949i	0.199557	−	0.115214i
$453$		0			0
$454$	1.72474	−	2.98735i	0.0809463	−	0.140203i
$455$	−1.43904	+	1.99760i	−0.0674633	+	0.0936487i
$456$		0			0
$457$	−15.5010	+	8.94949i	−0.725105	+	0.418639i	−0.816629	−	0.577163i	$-0.804160\pi$
$457$	−15.5010	+	8.94949i	−0.725105	+	0.418639i	0.0915238	+	0.995803i	$0.470826\pi$
$458$			18.4495i			0.862088i
$459$		0			0
$460$	−1.55051	+	15.3485i	−0.0722929	+	0.715626i
$461$	−1.22474	−	2.12132i	−0.0570421	−	0.0987997i	0.836094	−	0.548586i	$-0.184834\pi$
$461$	−1.22474	−	2.12132i	−0.0570421	−	0.0987997i	−0.893136	+	0.449786i	$0.851500\pi$
$462$		0			0
$463$	20.7846	+	12.0000i	0.965943	+	0.557687i	0.897997	−	0.440002i	$-0.145022\pi$
$463$	20.7846	+	12.0000i	0.965943	+	0.557687i	0.0679458	+	0.997689i	$0.478356\pi$
$464$	3.00000	−	5.19615i	0.139272	−	0.241225i
$465$		0			0
$466$	−6.84847	−	11.8619i	−0.317249	−	0.549492i
$467$			10.3485i			0.478870i	0.970912	+	0.239435i	$0.0769622\pi$
$467$			10.3485i			0.478870i	−0.970912	+	0.239435i	$0.923038\pi$
$468$		0			0
$469$	2.04541			0.0944482
$470$	−4.08346	−	9.07277i	−0.188356	−	0.418496i
$471$		0			0
$472$	11.4726	+	6.62372i	0.528070	+	0.304882i
$473$	7.61926	+	4.39898i	0.350334	+	0.202265i
$474$		0			0
$475$	−8.62907	−	25.8450i	−0.395929	−	1.18585i
$476$	−2.65153			−0.121533
$477$		0			0
$478$		−	0.696938i		−	0.0318772i
$479$	−8.34847	−	14.4600i	−0.381451	−	0.660693i	0.609819	−	0.792541i	$-0.291242\pi$
$479$	−8.34847	−	14.4600i	−0.381451	−	0.660693i	−0.991270	+	0.131848i	$0.957909\pi$
$480$		0			0
$481$	−9.79796	+	16.9706i	−0.446748	+	0.773791i
$482$	−0.866025	−	0.500000i	−0.0394464	−	0.0227744i
$483$		0			0
$484$	0.449490	+	0.778539i	0.0204314	+	0.0353881i
$485$	2.92168	−	28.9217i	0.132667	−	1.31327i
$486$		0			0
$487$		−	25.1010i		−	1.13744i	−0.822533	−	0.568718i	$-0.807440\pi$
$487$		−	25.1010i		−	1.13744i	0.822533	−	0.568718i	$-0.192560\pi$
$488$	−3.85337	+	2.22474i	−0.174434	+	0.100709i
$489$		0			0
$490$	12.3336	+	8.88498i	0.557176	+	0.401383i
$491$	9.27526	−	16.0652i	0.418586	−	0.725013i	−0.577211	−	0.816595i	$-0.695859\pi$
$491$	9.27526	−	16.0652i	0.418586	−	0.725013i	0.995798	+	0.0915820i	$0.0291924\pi$
$492$		0			0
$493$	−30.6520	+	17.6969i	−1.38050	+	0.797030i
$494$	13.3485			0.600576
$495$		0			0
$496$	1.55051			0.0696200
$497$	−0.953512	+	0.550510i	−0.0427708	+	0.0246938i
$498$		0			0
$499$	−10.6237	+	18.4008i	−0.475583	+	0.823734i	−0.999609	−	0.0279682i	$-0.991096\pi$
$499$	−10.6237	+	18.4008i	−0.475583	+	0.823734i	0.524026	+	0.851703i	$0.324430\pi$
$500$	−8.21731	+	7.58128i	−0.367489	+	0.339045i
$501$		0			0
$502$	−5.67291	+	3.27526i	−0.253194	+	0.146182i
$503$			14.4495i			0.644271i	0.946694	+	0.322135i	$0.104401\pi$
$503$			14.4495i			0.644271i	−0.946694	+	0.322135i	$0.895599\pi$
$504$		0			0
$505$	−17.7980	−	1.79796i	−0.791999	−	0.0800081i
$506$	11.8990	+	20.6096i	0.528974	+	0.916210i
$507$		0			0
$508$	5.97469	+	3.44949i	0.265084	+	0.153046i
$509$	−15.7980	+	27.3629i	−0.700232	+	1.21284i	0.268153	+	0.963376i	$0.413587\pi$
$509$	−15.7980	+	27.3629i	−0.700232	+	1.21284i	−0.968385	+	0.249461i	$0.919746\pi$
$510$		0			0
$511$	3.32577	+	5.76039i	0.147123	+	0.254825i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	10.1010			0.445537
$515$	−29.0514	+	13.0754i	−1.28016	+	0.576172i
$516$		0			0
$517$	−13.2922	−	7.67423i	−0.584589	−	0.337512i
$518$	−3.11416	−	1.79796i	−0.136828	−	0.0789978i
$519$		0			0
$520$	−2.24799	−	4.99465i	−0.0985808	−	0.219030i
$521$	−21.6969			−0.950560			−0.475280	−	0.879835i	$-0.657653\pi$
$521$	−21.6969			−0.950560			−0.475280	+	0.879835i	$0.657653\pi$
$522$		0			0
$523$			10.2020i			0.446104i	0.974807	+	0.223052i	$0.0716020\pi$
$523$			10.2020i			0.446104i	−0.974807	+	0.223052i	$0.928398\pi$
$524$	−2.44949	−	4.24264i	−0.107006	−	0.185341i
$525$		0			0
$526$	−6.22474	+	10.7816i	−0.271412	+	0.470099i
$527$	−7.92104	−	4.57321i	−0.345046	−	0.199212i
$528$		0			0
$529$	12.2980	+	21.3007i	0.534694	+	0.926117i
$530$	7.89898	+	0.797959i	0.343110	+	0.0346611i
$531$		0			0
$532$			2.44949i			0.106199i
$533$	−2.12132	+	1.22474i	−0.0918846	+	0.0530496i
$534$		0			0
$535$	−29.6612	−	21.3676i	−1.28237	−	0.923800i
$536$	−2.27526	+	3.94086i	−0.0982761	+	0.170219i
$537$		0			0
$538$	13.8957	−	8.02270i	0.599087	−	0.345883i
$539$	23.4495			1.01004
$540$		0			0
$541$	−0.404082			−0.0173728			−0.00868642	−	0.999962i	$-0.502765\pi$
$541$	−0.404082			−0.0173728			−0.00868642	+	0.999962i	$0.502765\pi$
$542$	−13.5065	+	7.79796i	−0.580152	+	0.334951i
$543$		0			0
$544$	2.94949	−	5.10867i	0.126458	−	0.219032i
$545$	10.4561	−	14.5145i	0.447888	−	0.621733i
$546$		0			0
$547$	14.9367	−	8.62372i	0.638648	−	0.368724i	−0.145445	−	0.989366i	$-0.546461\pi$
$547$	14.9367	−	8.62372i	0.638648	−	0.368724i	0.784094	+	0.620642i	$0.213128\pi$
$548$			3.00000i			0.128154i
$549$		0			0
$550$	−3.44949	+	16.8990i	−0.147087	+	0.720575i
$551$	16.3485	+	28.3164i	0.696468	+	1.20632i
$552$		0			0
$553$	−2.86054	−	1.65153i	−0.121642	−	0.0702302i
$554$	−14.7980	+	25.6308i	−0.628705	+	1.08895i
$555$		0			0
$556$	−6.62372	−	11.4726i	−0.280908	−	0.486548i
$557$		−	14.9444i		−	0.633214i	−0.948557	−	0.316607i	$-0.897456\pi$
$557$		−	14.9444i		−	0.633214i	0.948557	−	0.316607i	$-0.102544\pi$
$558$		0			0
$559$	−6.24745			−0.264239
$560$	0.916536	−	0.412514i	0.0387307	−	0.0174319i
$561$		0			0
$562$	10.3923	+	6.00000i	0.438373	+	0.253095i
$563$	−1.60524	−	0.926786i	−0.0676528	−	0.0390594i	0.465792	−	0.884894i	$-0.345769\pi$
$563$	−1.60524	−	0.926786i	−0.0676528	−	0.0390594i	−0.533445	+	0.845835i	$0.679103\pi$
$564$		0			0
$565$	9.98930	−	4.49598i	0.420253	−	0.189147i
$566$	4.00000			0.168133
$567$		0			0
$568$		−	2.44949i		−	0.102778i
$569$	15.7474	+	27.2754i	0.660167	+	1.14344i	0.980571	+	0.196162i	$0.0628480\pi$
$569$	15.7474	+	27.2754i	0.660167	+	1.14344i	−0.320404	+	0.947281i	$0.603819\pi$
$570$		0			0
$571$	−18.6237	+	32.2572i	−0.779379	+	1.34992i	0.152922	+	0.988238i	$0.451132\pi$
$571$	−18.6237	+	32.2572i	−0.779379	+	1.34992i	−0.932300	+	0.361685i	$0.882202\pi$
$572$	−7.31747	−	4.22474i	−0.305959	−	0.176645i
$573$		0			0
$574$	−0.224745	−	0.389270i	−0.00938067	−	0.0162478i
$575$	−6.89898	+	33.7980i	−0.287707	+	1.40947i
$576$		0			0
$577$		−	15.6969i		−	0.653472i	−0.945116	−	0.326736i	$-0.894051\pi$
$577$		−	15.6969i		−	0.653472i	0.945116	−	0.326736i	$-0.105949\pi$
$578$	−15.4135	+	8.89898i	−0.641116	+	0.370149i
$579$		0			0
$580$	7.84204	−	10.8859i	0.325623	−	0.452012i
$581$	0.898979	−	1.55708i	0.0372960	−	0.0645985i
$582$		0			0
$583$	10.6066	−	6.12372i	0.439281	−	0.253619i
$584$	−14.7980			−0.612344
$585$		0			0
$586$	−18.0000			−0.743573
$587$	20.7364	−	11.9722i	0.855885	−	0.494145i	−0.00674727	−	0.999977i	$-0.502148\pi$
$587$	20.7364	−	11.9722i	0.855885	−	0.494145i	0.862632	+	0.505832i	$0.168814\pi$
$588$		0			0
$589$	−4.22474	+	7.31747i	−0.174078	+	0.301511i
$590$	24.0350	+	17.3145i	0.989506	+	0.712827i
$591$		0			0
$592$	6.92820	−	4.00000i	0.284747	−	0.164399i
$593$		−	17.3939i		−	0.714281i	−0.934051	−	0.357140i	$-0.883752\pi$
$593$		−	17.3939i		−	0.714281i	0.934051	−	0.357140i	$-0.116248\pi$
$594$		0			0
$595$	−5.89898	−	0.595918i	−0.241835	−	0.0244303i
$596$	−4.12372	−	7.14250i	−0.168914	−	0.292568i
$597$		0			0
$598$	−14.6349	−	8.44949i	−0.598467	−	0.345525i
$599$	−16.8990	+	29.2699i	−0.690474	+	1.19594i	0.281209	+	0.959646i	$0.409264\pi$
$599$	−16.8990	+	29.2699i	−0.690474	+	1.19594i	−0.971683	+	0.236289i	$0.924069\pi$
$600$		0			0
$601$	−19.3990	−	33.6000i	−0.791301	−	1.37057i	−0.925162	−	0.379573i	$-0.876071\pi$
$601$	−19.3990	−	33.6000i	−0.791301	−	1.37057i	0.133861	−	0.991000i	$-0.457262\pi$
$602$		−	1.14643i		−	0.0467249i
$603$		0			0
$604$	−2.89898			−0.117958
$605$	0.825027	+	1.83307i	0.0335421	+	0.0745249i
$606$		0			0
$607$	23.8988	+	13.7980i	0.970021	+	0.560042i	0.899243	−	0.437450i	$-0.144118\pi$
$607$	23.8988	+	13.7980i	0.970021	+	0.560042i	0.0707783	+	0.997492i	$0.477452\pi$
$608$	−4.71940	−	2.72474i	−0.191397	−	0.110503i
$609$		0			0
$610$	−9.07277	+	4.08346i	−0.367346	+	0.165335i
$611$	10.8990			0.440926
$612$		0			0
$613$			36.9444i			1.49217i	0.665851	+	0.746085i	$0.268069\pi$
$613$			36.9444i			1.49217i	−0.665851	+	0.746085i	$0.731931\pi$
$614$	14.9722	+	25.9326i	0.604229	+	1.04655i
$615$		0			0
$616$	0.775255	−	1.34278i	0.0312359	−	0.0541022i
$617$	7.19066	+	4.15153i	0.289485	+	0.167134i	0.637710	−	0.770277i	$-0.279882\pi$
$617$	7.19066	+	4.15153i	0.289485	+	0.167134i	−0.348224	+	0.937411i	$0.613215\pi$
$618$		0			0
$619$	−14.2753	−	24.7255i	−0.573771	−	0.993800i	−0.996174	−	0.0873923i	$-0.972147\pi$
$619$	−14.2753	−	24.7255i	−0.573771	−	0.993800i	0.422403	−	0.906408i	$-0.361187\pi$
$620$	3.44949	+	0.348469i	0.138535	+	0.0139949i
$621$		0			0
$622$		−	13.1010i		−	0.525303i
$623$	−1.20713	+	0.696938i	−0.0483628	+	0.0279222i
$624$		0			0
$625$	−19.9853	+	15.0196i	−0.799411	+	0.600784i
$626$	10.8485	−	18.7901i	0.433592	−	0.751003i
$627$		0			0
$628$	−13.8564	+	8.00000i	−0.552931	+	0.319235i
$629$	−47.1918			−1.88166
$630$		0			0
$631$	−11.3485			−0.451775			−0.225888	−	0.974153i	$-0.572528\pi$
$631$	−11.3485			−0.451775			−0.225888	+	0.974153i	$0.572528\pi$
$632$	6.36396	−	3.67423i	0.253145	−	0.146153i
$633$		0			0
$634$	−11.4722	+	19.8704i	−0.455619	+	0.789155i
$635$	12.5169	+	9.01702i	0.496718	+	0.357829i
$636$		0			0
$637$	−14.4206	+	8.32577i	−0.571367	+	0.329879i
$638$		−	20.6969i		−	0.819400i
$639$		0			0
$640$	−0.224745	+	2.22474i	−0.00888382	+	0.0879408i
$641$	−18.5000	−	32.0429i	−0.730706	−	1.26562i	−0.956582	−	0.291464i	$-0.905858\pi$
$641$	−18.5000	−	32.0429i	−0.730706	−	1.26562i	0.225876	−	0.974156i	$-0.427476\pi$
$642$		0			0
$643$	13.2047	+	7.62372i	0.520742	+	0.300650i	0.737238	−	0.675633i	$-0.236130\pi$
$643$	13.2047	+	7.62372i	0.520742	+	0.300650i	−0.216496	+	0.976283i	$0.569463\pi$
$644$	1.55051	−	2.68556i	0.0610987	−	0.105826i
$645$		0			0
$646$	16.0732	+	27.8396i	0.632392	+	1.09534i
$647$		−	34.8990i		−	1.37202i	−0.727592	−	0.686010i	$-0.759361\pi$
$647$		−	34.8990i		−	1.37202i	0.727592	−	0.686010i	$-0.240639\pi$
$648$		0			0
$649$	45.6969			1.79376
$650$	−3.87868	−	11.6170i	−0.152134	−	0.455658i
$651$		0			0
$652$	−7.70674	−	4.44949i	−0.301819	−	0.174255i
$653$	29.4449	+	17.0000i	1.15227	+	0.665261i	0.949439	−	0.313953i	$-0.101653\pi$
$653$	29.4449	+	17.0000i	1.15227	+	0.665261i	0.202828	+	0.979214i	$0.434987\pi$
$654$		0			0
$655$	−4.49598	−	9.98930i	−0.175672	−	0.390314i
$656$	1.00000			0.0390434
$657$		0			0
$658$			2.00000i			0.0779681i
$659$	17.8990	+	31.0019i	0.697245	+	1.20766i	0.969418	+	0.245416i	$0.0789245\pi$
$659$	17.8990	+	31.0019i	0.697245	+	1.20766i	−0.272173	+	0.962248i	$0.587742\pi$
$660$		0			0
$661$	−2.89898	+	5.02118i	−0.112757	+	0.195301i	−0.916881	−	0.399161i	$-0.869302\pi$
$661$	−2.89898	+	5.02118i	−0.112757	+	0.195301i	0.804124	+	0.594462i	$0.202635\pi$
$662$	21.9917	+	12.6969i	0.854733	+	0.493481i
$663$		0			0
$664$	2.00000	+	3.46410i	0.0776151	+	0.134433i
$665$	−0.550510	+	5.44949i	−0.0213479	+	0.211322i
$666$		0			0
$667$		−	41.3939i		−	1.60278i
$668$	0.214297	−	0.123724i	0.00829139	−	0.00478704i
$669$		0			0
$670$	−5.94755	+	8.25605i	−0.229774	+	0.318959i
$671$	−7.67423	+	13.2922i	−0.296261	+	0.513138i
$672$		0			0
$673$	25.0273	−	14.4495i	0.964730	−	0.556987i	0.0671042	−	0.997746i	$-0.478624\pi$
$673$	25.0273	−	14.4495i	0.964730	−	0.556987i	0.897625	+	0.440759i	$0.145291\pi$
$674$	−18.5959			−0.716288
$675$		0			0
$676$	−7.00000			−0.269231
$677$	−34.2911	+	19.7980i	−1.31791	+	0.760897i	−0.983393	−	0.181491i	$-0.941908\pi$
$677$	−34.2911	+	19.7980i	−1.31791	+	0.760897i	−0.334520	+	0.942389i	$0.608574\pi$
$678$		0			0
$679$	−2.92168	+	5.06050i	−0.112124	+	0.194204i
$680$	7.71001	−	10.7026i	0.295665	−	0.410426i
$681$		0			0
$682$	4.63191	−	2.67423i	0.177365	−	0.102402i
$683$		−	45.4495i		−	1.73908i	−0.493866	−	0.869538i	$-0.664417\pi$
$683$		−	45.4495i		−	1.73908i	0.493866	−	0.869538i	$-0.335583\pi$
$684$		0			0
$685$	−0.674235	+	6.67423i	−0.0257612	+	0.255009i
$686$	−3.10102	−	5.37113i	−0.118398	−	0.205071i
$687$		0			0
$688$	2.20881	+	1.27526i	0.0842100	+	0.0486186i
$689$	−4.34847	+	7.53177i	−0.165663	+	0.286938i
$690$		0			0
$691$	−8.79796	−	15.2385i	−0.334690	−	0.579700i	0.648735	−	0.761014i	$-0.275298\pi$
$691$	−8.79796	−	15.2385i	−0.334690	−	0.579700i	−0.983425	+	0.181314i	$0.941965\pi$
$692$			11.7980i			0.448491i
$693$		0			0
$694$	9.24745			0.351028
$695$	−12.1577	−	27.0123i	−0.461167	−	1.02464i
$696$		0			0
$697$	−5.10867	−	2.94949i	−0.193505	−	0.111720i
$698$	−23.8988	−	13.7980i	−0.904582	−	0.522260i
$699$		0			0
$700$	2.13177	−	0.711751i	0.0805733	−	0.0269017i
$701$	39.3939			1.48789			0.743943	−	0.668243i	$-0.232953\pi$
$701$	39.3939			1.48789			0.743943	+	0.668243i	$0.232953\pi$
$702$		0			0
$703$			43.5959i			1.64425i
$704$	1.72474	+	2.98735i	0.0650038	+	0.112590i
$705$		0			0
$706$	16.2980	−	28.2289i	0.613382	−	1.06241i
$707$	3.11416	+	1.79796i	0.117120	+	0.0676192i
$708$		0			0
$709$	18.6742	+	32.3447i	0.701326	+	1.21473i	0.968001	+	0.250945i	$0.0807414\pi$
$709$	18.6742	+	32.3447i	0.701326	+	1.21473i	−0.266676	+	0.963786i	$0.585925\pi$
$710$	0.550510	−	5.44949i	0.0206603	−	0.204516i
$711$		0			0
$712$		−	3.10102i		−	0.116216i
$713$	9.26382	−	5.34847i	0.346933	−	0.200302i
$714$		0			0
$715$	−15.3300	−	11.0435i	−0.573310	−	0.413005i
$716$	−0.449490	+	0.778539i	−0.0167982	+	0.0290954i
$717$		0			0
$718$	3.07483	−	1.77526i	0.114752	−	0.0662519i
$719$	41.7980			1.55880			0.779400	−	0.626526i	$-0.215524\pi$
$719$	41.7980			1.55880			0.779400	+	0.626526i	$0.215524\pi$
$720$		0			0
$721$	6.40408			0.238500
$722$	9.26382	−	5.34847i	0.344764	−	0.199049i
$723$		0			0
$724$	2.77526	−	4.80688i	0.103142	−	0.178646i
$725$	19.8931	−	22.4558i	0.738811	−	0.833989i
$726$		0			0
$727$	−21.9524	+	12.6742i	−0.814170	+	0.470061i	−0.848402	−	0.529353i	$-0.822435\pi$
$727$	−21.9524	+	12.6742i	−0.814170	+	0.470061i	0.0342318	+	0.999414i	$0.489102\pi$
$728$			1.10102i			0.0408065i
$729$		0			0
$730$	−32.9217	−	3.32577i	−1.21849	−	0.123092i
$731$	−7.52270	−	13.0297i	−0.278237	−	0.481921i
$732$		0			0
$733$	18.3133	+	10.5732i	0.676419	+	0.390531i	0.798504	−	0.601989i	$-0.205625\pi$
$733$	18.3133	+	10.5732i	0.676419	+	0.390531i	−0.122086	+	0.992520i	$0.538958\pi$
$734$	13.7980	−	23.8988i	0.509292	−	0.882120i
$735$		0			0
$736$	3.44949	+	5.97469i	0.127150	+	0.220230i
$737$			15.6969i			0.578204i
$738$		0			0
$739$	17.2474			0.634458			0.317229	−	0.948349i	$-0.397248\pi$
$739$	17.2474			0.634458			0.317229	+	0.948349i	$0.397248\pi$
$740$	16.3125	−	7.34190i	0.599658	−	0.269894i
$741$		0			0
$742$	−1.38211	−	0.797959i	−0.0507387	−	0.0292940i
$743$	19.8311	+	11.4495i	0.727532	+	0.420041i	0.817519	−	0.575902i	$-0.195349\pi$
$743$	19.8311	+	11.4495i	0.727532	+	0.420041i	−0.0899863	+	0.995943i	$0.528682\pi$
$744$		0			0
$745$	−7.56899	−	16.8170i	−0.277306	−	0.616128i
$746$	−13.5959			−0.497782
$747$		0			0
$748$		−	20.3485i		−	0.744014i
$749$	3.67423	+	6.36396i	0.134254	+	0.232534i
$750$		0			0
$751$	26.4949	−	45.8905i	0.966813	−	1.67457i	0.262148	−	0.965028i	$-0.415569\pi$
$751$	26.4949	−	45.8905i	0.966813	−	1.67457i	0.704664	−	0.709541i	$-0.251098\pi$
$752$	−3.85337	−	2.22474i	−0.140518	−	0.0811281i
$753$		0			0
$754$	7.34847	+	12.7279i	0.267615	+	0.463524i
$755$	−6.44949	−	0.651531i	−0.234721	−	0.0237116i
$756$		0			0
$757$		−	10.0000i		−	0.363456i	−0.983349	−	0.181728i	$-0.941831\pi$
$757$		−	10.0000i		−	0.363456i	0.983349	−	0.181728i	$-0.0581691\pi$
$758$	−3.59091	+	2.07321i	−0.130428	+	0.0753025i
$759$		0			0
$760$	−9.88708	−	7.12252i	−0.358642	−	0.258361i
$761$	0.247449	−	0.428594i	0.00897001	−	0.0155365i	−0.861506	−	0.507748i	$-0.830478\pi$
$761$	0.247449	−	0.428594i	0.00897001	−	0.0155365i	0.870476	+	0.492212i	$0.163811\pi$
$762$		0			0
$763$	−3.11416	+	1.79796i	−0.112740	+	0.0650905i
$764$	18.2474			0.660170
$765$		0			0
$766$	17.7980			0.643066
$767$	−28.1021	+	16.2247i	−1.01471	+	0.585842i
$768$		0			0
$769$	12.2474	−	21.2132i	0.441654	−	0.764968i	−0.556158	−	0.831076i	$-0.687725\pi$
$769$	12.2474	−	21.2132i	0.441654	−	0.764968i	0.997812	+	0.0661088i	$0.0210584\pi$
$770$	2.02653	−	2.81311i	0.0730310	−	0.101377i
$771$		0			0
$772$	11.8619	−	6.84847i	0.426919	−	0.246482i
$773$			35.3939i			1.27303i	0.771265	+	0.636515i	$0.219625\pi$
$773$			35.3939i			1.27303i	−0.771265	+	0.636515i	$0.780375\pi$
$774$		0			0
$775$	7.59592	+	1.55051i	0.272853	+	0.0556960i
$776$	−6.50000	−	11.2583i	−0.233336	−	0.404151i
$777$		0			0
$778$	−15.1992	−	8.77526i	−0.544917	−	0.314608i
$779$	−2.72474	+	4.71940i	−0.0976241	+	0.169090i
$780$		0			0
$781$	−4.22474	−	7.31747i	−0.151173	−	0.261840i
$782$		−	40.6969i		−	1.45532i
$783$		0			0
$784$	6.79796			0.242784
$785$	−32.6249	+	14.6838i	−1.16443	+	0.524087i
$786$		0			0
$787$	44.5084	+	25.6969i	1.58655	+	0.915997i	0.993869	+	0.110562i	$0.0352650\pi$
$787$	44.5084	+	25.6969i	1.58655	+	0.915997i	0.592684	+	0.805435i	$0.298068\pi$
$788$	6.92820	+	4.00000i	0.246807	+	0.142494i
$789$		0			0
$790$	14.9840	−	6.74397i	0.533105	−	0.239940i
$791$	−2.20204			−0.0782956
$792$		0			0
$793$		−	10.8990i		−	0.387034i
$794$	−0.898979	−	1.55708i	−0.0319036	−	0.0552586i
$795$		0			0
$796$	−7.77526	+	13.4671i	−0.275587	+	0.477330i
$797$	3.11416	+	1.79796i	0.110309	+	0.0636870i	0.554139	−	0.832424i	$-0.313047\pi$
$797$	3.11416	+	1.79796i	0.110309	+	0.0636870i	−0.443830	+	0.896111i	$0.646381\pi$
$798$		0			0
$799$	13.1237	+	22.7310i	0.464284	+	0.804163i
$800$	−1.00000	+	4.89898i	−0.0353553	+	0.173205i
$801$		0			0
$802$		−	9.20204i		−	0.324935i
$803$	−44.2066	+	25.5227i	−1.56002	+	0.900677i
$804$		0			0
$805$	4.05306	−	5.62622i	0.142851	−	0.198298i
$806$	−1.89898	+	3.28913i	−0.0668887	+	0.115855i
$807$		0			0
$808$	−6.92820	+	4.00000i	−0.243733	+	0.140720i
$809$	−41.0908			−1.44468			−0.722338	−	0.691540i	$-0.756933\pi$
$809$	−41.0908			−1.44468			−0.722338	+	0.691540i	$0.756933\pi$
$810$		0			0
$811$	7.24745			0.254492			0.127246	−	0.991871i	$-0.459386\pi$
$811$	7.24745			0.254492			0.127246	+	0.991871i	$0.459386\pi$
$812$	−2.33562	+	1.34847i	−0.0819641	+	0.0473220i
$813$		0			0
$814$	13.7980	−	23.8988i	0.483618	−	0.837651i
$815$	−16.1455	−	11.6310i	−0.565553	−	0.407417i
$816$		0			0
$817$	−12.0369	+	6.94949i	−0.421117	+	0.243132i
$818$		−	15.8990i		−	0.555895i
$819$		0			0
$820$	2.22474	+	0.224745i	0.0776915	+	0.00784843i
$821$	1.02270	+	1.77138i	0.0356926	+	0.0618214i	0.883320	−	0.468771i	$-0.155303\pi$
$821$	1.02270	+	1.77138i	0.0356926	+	0.0618214i	−0.847627	+	0.530592i	$0.821970\pi$
$822$		0			0
$823$	−27.3629	−	15.7980i	−0.953810	−	0.550682i	−0.0595473	−	0.998225i	$-0.518966\pi$
$823$	−27.3629	−	15.7980i	−0.953810	−	0.550682i	−0.894262	+	0.447543i	$0.852299\pi$
$824$	−7.12372	+	12.3387i	−0.248167	+	0.429837i
$825$		0			0
$826$	−2.97730	−	5.15683i	−0.103593	−	0.179429i
$827$		−	5.79796i		−	0.201615i	−0.994906	−	0.100807i	$-0.967857\pi$
$827$		−	5.79796i		−	0.201615i	0.994906	−	0.100807i	$-0.0321426\pi$
$828$		0			0
$829$	26.4495			0.918629			0.459314	−	0.888274i	$-0.348095\pi$
$829$	26.4495			0.918629			0.459314	+	0.888274i	$0.348095\pi$
$830$	3.67095	+	8.15623i	0.127421	+	0.283107i
$831$		0			0
$832$	−2.12132	−	1.22474i	−0.0735436	−	0.0424604i
$833$	−34.7285	−	20.0505i	−1.20327	−	0.694709i
$834$		0			0
$835$	0.504562	−	0.227093i	0.0174611	−	0.00785888i
$836$	−18.7980			−0.650141
$837$		0			0
$838$		−	8.89898i		−	0.307410i
$839$	−20.1237	−	34.8553i	−0.694748	−	1.20334i	−0.970266	−	0.242043i	$-0.922183\pi$
$839$	−20.1237	−	34.8553i	−0.694748	−	1.20334i	0.275517	−	0.961296i	$-0.411151\pi$
$840$		0			0
$841$	−3.50000	+	6.06218i	−0.120690	+	0.209041i
$842$	10.0030	+	5.77526i	0.344727	+	0.199028i
$843$		0			0
$844$	1.89898	+	3.28913i	0.0653656	+	0.113216i
$845$	−15.5732	−	1.57321i	−0.535735	−	0.0541202i
$846$		0			0
$847$		−	0.404082i		−	0.0138844i
$848$	3.07483	−	1.77526i	0.105590	−	0.0609625i
$849$		0			0
$850$	19.5582	−	22.0778i	0.670839	−	0.757261i
$851$	27.5959	−	47.7975i	0.945976	−	1.63848i
$852$		0			0
$853$	−7.92104	+	4.57321i	−0.271211	+	0.156584i	−0.629438	−	0.777051i	$-0.716715\pi$
$853$	−7.92104	+	4.57321i	−0.271211	+	0.156584i	0.358227	+	0.933635i	$0.383381\pi$
$854$	2.00000			0.0684386
$855$		0			0
$856$	−16.3485			−0.558779
$857$	−4.67123	+	2.69694i	−0.159566	+	0.0921257i	−0.577657	−	0.816280i	$-0.696033\pi$
$857$	−4.67123	+	2.69694i	−0.159566	+	0.0921257i	0.418091	+	0.908405i	$0.362699\pi$
$858$		0			0
$859$	−18.8712	+	32.6858i	−0.643876	+	1.11523i	0.340684	+	0.940178i	$0.389341\pi$
$859$	−18.8712	+	32.6858i	−0.643876	+	1.11523i	−0.984560	+	0.175048i	$0.943992\pi$
$860$	4.62742	+	3.33354i	0.157794	+	0.113673i
$861$		0			0
$862$	−33.1233	+	19.1237i	−1.12818	+	0.651357i
$863$			26.4495i			0.900351i	0.892940	+	0.450176i	$0.148639\pi$
$863$			26.4495i			0.900351i	−0.892940	+	0.450176i	$0.851361\pi$
$864$		0			0
$865$	−2.65153	+	26.2474i	−0.0901548	+	0.892440i
$866$	−11.5000	−	19.9186i	−0.390786	−	0.676861i
$867$		0			0
$868$	−0.603566	−	0.348469i	−0.0204864	−	0.0118278i
$869$	12.6742	−	21.9524i	0.429944	−	0.744685i
$870$		0			0
$871$	−5.57321	−	9.65309i	−0.188841	−	0.327082i
$872$		−	8.00000i		−	0.270914i
$873$		0			0
$874$	−37.5959			−1.27170
$875$	4.90260	−	1.10436i	0.165738	−	0.0373342i
$876$		0			0
$877$	18.0597	+	10.4268i	0.609834	+	0.352088i	0.772900	−	0.634527i	$-0.218805\pi$
$877$	18.0597	+	10.4268i	0.609834	+	0.352088i	−0.163067	+	0.986615i	$0.552139\pi$
$878$	19.0919	+	11.0227i	0.644320	+	0.371998i
$879$		0			0
$880$	3.16573	+	7.03371i	0.106717	+	0.237106i
$881$	30.0000			1.01073			0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000			1.01073			0.505363	+	0.862907i	$0.331359\pi$
$882$		0			0
$883$			6.55051i			0.220442i	0.993907	+	0.110221i	$0.0351559\pi$
$883$			6.55051i			0.220442i	−0.993907	+	0.110221i	$0.964844\pi$
$884$	7.22474	+	12.5136i	0.242994	+	0.420879i
$885$		0			0
$886$	10.6237	−	18.4008i	0.356911	−	0.618188i
$887$	16.7563	+	9.67423i	0.562620	+	0.324829i	0.754197	−	0.656649i	$-0.228027\pi$
$887$	16.7563	+	9.67423i	0.562620	+	0.324829i	−0.191576	+	0.981478i	$0.561360\pi$
$888$		0			0
$889$	−1.55051	−	2.68556i	−0.0520024	−	0.0900709i
$890$	0.696938	−	6.89898i	0.0233614	−	0.231254i
$891$		0			0
$892$			9.10102i			0.304725i
$893$	20.9989	−	12.1237i	0.702702	−	0.405705i
$894$		0			0
$895$	−1.17497	+	1.63103i	−0.0392750	+	0.0545193i
$896$	0.224745	−	0.389270i	0.00750820	−	0.0130046i
$897$		0			0
$898$	16.2795	−	9.39898i	0.543254	−	0.313648i
$899$	−9.30306			−0.310274
$900$		0			0
$901$	−20.9444			−0.697759
$902$	2.98735	−	1.72474i	0.0994677	−	0.0574277i
$903$		0			0
$904$	2.44949	−	4.24264i	0.0814688	−	0.141108i
$905$	7.25456	−	10.0704i	0.241150	−	0.334750i
$906$		0			0
$907$	−34.4179	+	19.8712i	−1.14283	+	0.659811i	−0.947129	−	0.320853i	$-0.896031\pi$
$907$	−34.4179	+	19.8712i	−1.14283	+	0.659811i	−0.195698	+	0.980664i	$0.562697\pi$
$908$		−	3.44949i		−	0.114475i
$909$		0			0
$910$	−0.247449	+	2.44949i	−0.00820284	+	0.0811998i
$911$	12.1237	+	20.9989i	0.401677	+	0.695725i	0.993928	−	0.110028i	$-0.0350942\pi$
$911$	12.1237	+	20.9989i	0.401677	+	0.695725i	−0.592252	+	0.805753i	$0.701761\pi$
$912$		0			0
$913$	11.9494	+	6.89898i	0.395467	+	0.228323i
$914$	−8.94949	+	15.5010i	−0.296023	+	0.512727i
$915$		0			0
$916$	9.22474	+	15.9777i	0.304794	+	0.527919i
$917$			2.20204i			0.0727178i
$918$		0			0
$919$	−1.10102			−0.0363193			−0.0181597	−	0.999835i	$-0.505781\pi$
$919$	−1.10102			−0.0363193			−0.0181597	+	0.999835i	$0.505781\pi$
$920$	6.33145	+	14.0674i	0.208742	+	0.463789i
$921$		0			0
$922$	−2.12132	−	1.22474i	−0.0698620	−	0.0403348i
$923$	5.19615	+	3.00000i	0.171033	+	0.0987462i
$924$		0			0
$925$	37.9411	−	12.6677i	1.24750	−	0.416512i
$926$	24.0000			0.788689
$927$		0			0
$928$		−	6.00000i		−	0.196960i
$929$	8.20204	+	14.2064i	0.269100	+	0.466095i	0.968630	−	0.248508i	$-0.0799404\pi$
$929$	8.20204	+	14.2064i	0.269100	+	0.466095i	−0.699529	+	0.714604i	$0.746607\pi$
$930$		0			0
$931$	−18.5227	+	32.0823i	−0.607057	+	1.05145i
$932$	−11.8619	−	6.84847i	−0.388549	−	0.224329i
$933$		0			0
$934$	5.17423	+	8.96204i	0.169306	+	0.293247i
$935$	4.57321	−	45.2702i	0.149560	−	1.48049i
$936$		0			0
$937$		−	0.404082i		−	0.0132008i	−0.999978	−	0.00660039i	$-0.997899\pi$
$937$		−	0.404082i		−	0.0132008i	0.999978	−	0.00660039i	$-0.00210099\pi$
$938$	1.77138	−	1.02270i	0.0578374	−	0.0333925i
$939$		0			0
$940$	−8.07277	−	5.81552i	−0.263305	−	0.189681i
$941$	−15.1010	+	26.1557i	−0.492279	+	0.852653i	−0.999960	−	0.00889239i	$-0.997169\pi$
$941$	−15.1010	+	26.1557i	−0.492279	+	0.852653i	0.507681	+	0.861545i	$0.330503\pi$
$942$		0			0
$943$	5.97469	−	3.44949i	0.194563	−	0.112331i
$944$	13.2474			0.431168
$945$		0			0
$946$	8.79796			0.286046
$947$	−2.81237	+	1.62372i	−0.0913898	+	0.0527640i	−0.544998	−	0.838437i	$-0.683470\pi$
$947$	−2.81237	+	1.62372i	−0.0913898	+	0.0527640i	0.453609	+	0.891201i	$0.350136\pi$
$948$		0			0
$949$	18.1237	−	31.3912i	0.588321	−	1.01900i
$950$	−20.3955	−	18.0679i	−0.661717	−	0.586199i
$951$		0			0
$952$	−2.29629	+	1.32577i	−0.0744233	+	0.0429683i
$953$		−	31.2020i		−	1.01073i	−0.862905	−	0.505367i	$-0.831357\pi$
$953$		−	31.2020i		−	1.01073i	0.862905	−	0.505367i	$-0.168643\pi$
$954$		0			0
$955$	40.5959	+	4.10102i	1.31365	+	0.132706i
$956$	−0.348469	−	0.603566i	−0.0112703	−	0.0195207i
$957$		0			0
$958$	−14.4600	−	8.34847i	−0.467181	−	0.269727i
$959$	0.674235	−	1.16781i	0.0217722	−	0.0377105i
$960$		0			0
$961$	14.2980	+	24.7648i	0.461224	+	0.798864i
$962$			19.5959i			0.631798i
$963$		0			0
$964$	−1.00000			−0.0322078
$965$	27.9289	−	12.5702i	0.899062	−	0.404649i
$966$		0			0
$967$	−0.603566	−	0.348469i	−0.0194094	−	0.0112060i	0.490264	−	0.871574i	$-0.336900\pi$
$967$	−0.603566	−	0.348469i	−0.0194094	−	0.0112060i	−0.509673	+	0.860368i	$0.670234\pi$
$968$	0.778539	+	0.449490i	0.0250232	+	0.0144471i
$969$		0			0
$970$	−11.9306	−	26.5078i	−0.383068	−	0.851113i
$971$	−35.3939			−1.13584			−0.567922	−	0.823083i	$-0.692252\pi$
$971$	−35.3939			−1.13584			−0.567922	+	0.823083i	$0.692252\pi$
$972$		0			0
$973$			5.95459i			0.190895i
$974$	−12.5505	−	21.7381i	−0.402144	−	0.696534i
$975$		0			0
$976$	−2.22474	+	3.85337i	−0.0712123	+	0.123343i
$977$	24.3362	+	14.0505i	0.778584	+	0.449516i	0.835928	−	0.548839i	$-0.184930\pi$
$977$	24.3362	+	14.0505i	0.778584	+	0.449516i	−0.0573443	+	0.998354i	$0.518263\pi$
$978$		0			0
$979$	−5.34847	−	9.26382i	−0.170938	−	0.296073i
$980$	15.1237	+	1.52781i	0.483110	+	0.0488040i
$981$		0			0
$982$		−	18.5505i		−	0.591971i
$983$	18.5276	−	10.6969i	0.590940	−	0.341179i	−0.174529	−	0.984652i	$-0.555840\pi$
$983$	18.5276	−	10.6969i	0.590940	−	0.341179i	0.765469	+	0.643473i	$0.222507\pi$
$984$		0			0
$985$	14.5145	+	10.4561i	0.462471	+	0.333158i
$986$	−17.6969	+	30.6520i	−0.563585	+	0.976158i
$987$		0			0
$988$	11.5601	−	6.67423i	0.367776	−	0.212336i
$989$	17.5959			0.559518
$990$		0			0
$991$	4.00000			0.127064			0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000			0.127064			0.0635321	+	0.997980i	$0.479763\pi$
$992$	1.34278	−	0.775255i	0.0426333	−	0.0246144i
$993$		0			0
$994$	−0.550510	+	0.953512i	−0.0174611	+	0.0302436i
$995$	−20.3246	+	28.2135i	−0.644334	+	0.894428i
$996$		0			0
$997$	18.1384	−	10.4722i	0.574448	−	0.331658i	−0.184476	−	0.982837i	$-0.559059\pi$
$997$	18.1384	−	10.4722i	0.574448	−	0.331658i	0.758924	+	0.651179i	$0.225725\pi$
$998$			21.2474i			0.672576i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.2.i.b.199.4		8
3.2	odd	2		90.2.i.b.49.2	✓	8
4.3	odd	2		2160.2.by.d.1009.3		8
5.2	odd	4		1350.2.e.m.901.1		4
5.3	odd	4		1350.2.e.j.901.2		4
5.4	even	2	inner	270.2.i.b.199.1		8
9.2	odd	6		90.2.i.b.79.3	yes	8
9.4	even	3		810.2.c.e.649.3		4
9.5	odd	6		810.2.c.f.649.2		4
9.7	even	3	inner	270.2.i.b.19.1		8
12.11	even	2		720.2.by.c.49.3		8
15.2	even	4		450.2.e.k.301.2		4
15.8	even	4		450.2.e.n.301.1		4
15.14	odd	2		90.2.i.b.49.3	yes	8
20.19	odd	2		2160.2.by.d.1009.2		8
36.7	odd	6		2160.2.by.d.289.2		8
36.11	even	6		720.2.by.c.529.2		8
45.2	even	12		450.2.e.k.151.2		4
45.4	even	6		810.2.c.e.649.1		4
45.7	odd	12		1350.2.e.m.451.1		4
45.13	odd	12		4050.2.a.bz.1.1		2
45.14	odd	6		810.2.c.f.649.4		4
45.22	odd	12		4050.2.a.bm.1.2		2
45.23	even	12		4050.2.a.bq.1.1		2
45.29	odd	6		90.2.i.b.79.2	yes	8
45.32	even	12		4050.2.a.bs.1.2		2
45.34	even	6	inner	270.2.i.b.19.4		8
45.38	even	12		450.2.e.n.151.1		4
45.43	odd	12		1350.2.e.j.451.2		4
60.59	even	2		720.2.by.c.49.2		8
180.79	odd	6		2160.2.by.d.289.3		8
180.119	even	6		720.2.by.c.529.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.2.i.b.49.2	✓	8	3.2	odd	2
90.2.i.b.49.3	yes	8	15.14	odd	2
90.2.i.b.79.2	yes	8	45.29	odd	6
90.2.i.b.79.3	yes	8	9.2	odd	6
270.2.i.b.19.1		8	9.7	even	3	inner
270.2.i.b.19.4		8	45.34	even	6	inner
270.2.i.b.199.1		8	5.4	even	2	inner
270.2.i.b.199.4		8	1.1	even	1	trivial
450.2.e.k.151.2		4	45.2	even	12
450.2.e.k.301.2		4	15.2	even	4
450.2.e.n.151.1		4	45.38	even	12
450.2.e.n.301.1		4	15.8	even	4
720.2.by.c.49.2		8	60.59	even	2
720.2.by.c.49.3		8	12.11	even	2
720.2.by.c.529.2		8	36.11	even	6
720.2.by.c.529.3		8	180.119	even	6
810.2.c.e.649.1		4	45.4	even	6
810.2.c.e.649.3		4	9.4	even	3
810.2.c.f.649.2		4	9.5	odd	6
810.2.c.f.649.4		4	45.14	odd	6
1350.2.e.j.451.2		4	45.43	odd	12
1350.2.e.j.901.2		4	5.3	odd	4
1350.2.e.m.451.1		4	45.7	odd	12
1350.2.e.m.901.1		4	5.2	odd	4
2160.2.by.d.289.2		8	36.7	odd	6
2160.2.by.d.289.3		8	180.79	odd	6
2160.2.by.d.1009.2		8	20.19	odd	2
2160.2.by.d.1009.3		8	4.3	odd	2
4050.2.a.bm.1.2		2	45.22	odd	12
4050.2.a.bq.1.1		2	45.23	even	12
4050.2.a.bs.1.2		2	45.32	even	12
4050.2.a.bz.1.1		2	45.13	odd	12

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	270.i (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(1.30701 - 1.81431i) q^{5} +(-0.389270 + 0.224745i) q^{7} -1.00000i q^{8} +(0.224745 - 2.22474i) q^{10} +(-1.72474 - 2.98735i) q^{11} +(2.12132 + 1.22474i) q^{13} +(-0.224745 + 0.389270i) q^{14} +(-0.500000 - 0.866025i) q^{16} +5.89898i q^{17} +5.44949 q^{19} +(-0.917738 - 2.03906i) q^{20} +(-2.98735 - 1.72474i) q^{22} +(-5.97469 - 3.44949i) q^{23} +(-1.58346 - 4.74264i) q^{25} +2.44949 q^{26} +0.449490i q^{28} +(3.00000 + 5.19615i) q^{29} +(-0.775255 + 1.34278i) q^{31} +(-0.866025 - 0.500000i) q^{32} +(2.94949 + 5.10867i) q^{34} +(-0.101021 + 1.00000i) q^{35} +8.00000i q^{37} +(4.71940 - 2.72474i) q^{38} +(-1.81431 - 1.30701i) q^{40} +(-0.500000 + 0.866025i) q^{41} +(-2.20881 + 1.27526i) q^{43} -3.44949 q^{44} -6.89898 q^{46} +(3.85337 - 2.22474i) q^{47} +(-3.39898 + 5.88721i) q^{49} +(-3.74264 - 3.31552i) q^{50} +(2.12132 - 1.22474i) q^{52} +3.55051i q^{53} +(-7.67423 - 0.775255i) q^{55} +(0.224745 + 0.389270i) q^{56} +(5.19615 + 3.00000i) q^{58} +(-6.62372 + 11.4726i) q^{59} +(-2.22474 - 3.85337i) q^{61} +1.55051i q^{62} -1.00000 q^{64} +(4.99465 - 2.24799i) q^{65} +(-3.94086 - 2.27526i) q^{67} +(5.10867 + 2.94949i) q^{68} +(0.412514 + 0.916536i) q^{70} +2.44949 q^{71} -14.7980i q^{73} +(4.00000 + 6.92820i) q^{74} +(2.72474 - 4.71940i) q^{76} +(1.34278 + 0.775255i) q^{77} +(3.67423 + 6.36396i) q^{79} +(-2.22474 - 0.224745i) q^{80} +1.00000i q^{82} +(-3.46410 + 2.00000i) q^{83} +(10.7026 + 7.71001i) q^{85} +(-1.27526 + 2.20881i) q^{86} +(-2.98735 + 1.72474i) q^{88} +3.10102 q^{89} -1.10102 q^{91} +(-5.97469 + 3.44949i) q^{92} +(2.22474 - 3.85337i) q^{94} +(7.12252 - 9.88708i) q^{95} +(11.2583 - 6.50000i) q^{97} +6.79796i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{4} + 4 q^{5} - 8 q^{10} - 4 q^{11} + 8 q^{14} - 4 q^{16} + 24 q^{19} - 4 q^{20} + 24 q^{29} - 16 q^{31} + 4 q^{34} - 40 q^{35} - 4 q^{40} - 4 q^{41} - 8 q^{44} - 16 q^{46} + 12 q^{49} + 4 q^{50}+ \cdots + 24 q^{95}+O(q^{100}) \) 8 * q + 4 * q^4 + 4 * q^5 - 8 * q^10 - 4 * q^11 + 8 * q^14 - 4 * q^16 + 24 * q^19 - 4 * q^20 + 24 * q^29 - 16 * q^31 + 4 * q^34 - 40 * q^35 - 4 * q^40 - 4 * q^41 - 8 * q^44 - 16 * q^46 + 12 * q^49 + 4 * q^50 - 32 * q^55 - 8 * q^56 - 4 * q^59 - 8 * q^61 - 8 * q^64 + 12 * q^65 + 4 * q^70 + 32 * q^74 + 12 * q^76 - 8 * q^80 - 20 * q^85 - 20 * q^86 + 64 * q^89 - 48 * q^91 + 8 * q^94 + 24 * q^95

Introduction

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