LMFDB - Embedded newform 750.3.f.b.193.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [750,3,Mod(193,750)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(750, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("750.193"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 750 = 2 \cdot 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	750.f (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,-16,0,0,0,0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$20.4360198270$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	16.0.6879707136000000000000.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 21x^{12} + 86x^{8} + 36x^{4} + 1 $ x^16 + 21x^12 + 86x^8 + 36*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{8}\cdot 5^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			193.1
Root			$-1.40647 - 1.40647i$ of defining polynomial
Character	$\chi$	$=$	750.193
Dual form			750.3.f.b.307.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.00000 + 1.00000i) q^{2} +(-1.22474 - 1.22474i) q^{3} -2.00000i q^{4} +2.44949 q^{6} +(-3.25953 + 3.25953i) q^{7} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} +10.3780 q^{11} +(-2.44949 + 2.44949i) q^{12} +(-3.45217 - 3.45217i) q^{13} -6.51907i q^{14} -4.00000 q^{16} +(-3.95806 + 3.95806i) q^{17} +(-3.00000 - 3.00000i) q^{18} -3.80019i q^{19} +7.98420 q^{21} +(-10.3780 + 10.3780i) q^{22} +(-7.19505 - 7.19505i) q^{23} -4.89898i q^{24} +6.90433 q^{26} +(3.67423 - 3.67423i) q^{27} +(6.51907 + 6.51907i) q^{28} -21.2791i q^{29} +22.4937 q^{31} +(4.00000 - 4.00000i) q^{32} +(-12.7104 - 12.7104i) q^{33} -7.91612i q^{34} +6.00000 q^{36} +(-28.5269 + 28.5269i) q^{37} +(3.80019 + 3.80019i) q^{38} +8.45604i q^{39} -26.1336 q^{41} +(-7.98420 + 7.98420i) q^{42} +(3.75315 + 3.75315i) q^{43} -20.7560i q^{44} +14.3901 q^{46} +(-61.3876 + 61.3876i) q^{47} +(4.89898 + 4.89898i) q^{48} +27.7509i q^{49} +9.69522 q^{51} +(-6.90433 + 6.90433i) q^{52} +(19.1975 + 19.1975i) q^{53} +7.34847i q^{54} -13.0381 q^{56} +(-4.65427 + 4.65427i) q^{57} +(21.2791 + 21.2791i) q^{58} +39.1066i q^{59} +82.6505 q^{61} +(-22.4937 + 22.4937i) q^{62} +(-9.77860 - 9.77860i) q^{63} +8.00000i q^{64} +25.4208 q^{66} +(-43.8132 + 43.8132i) q^{67} +(7.91612 + 7.91612i) q^{68} +17.6242i q^{69} -59.6858 q^{71} +(-6.00000 + 6.00000i) q^{72} +(66.8754 + 66.8754i) q^{73} -57.0539i q^{74} -7.60039 q^{76} +(-33.8275 + 33.8275i) q^{77} +(-8.45604 - 8.45604i) q^{78} +120.001i q^{79} -9.00000 q^{81} +(26.1336 - 26.1336i) q^{82} +(-88.5686 - 88.5686i) q^{83} -15.9684i q^{84} -7.50629 q^{86} +(-26.0615 + 26.0615i) q^{87} +(20.7560 + 20.7560i) q^{88} +88.3609i q^{89} +22.5049 q^{91} +(-14.3901 + 14.3901i) q^{92} +(-27.5490 - 27.5490i) q^{93} -122.775i q^{94} -9.79796 q^{96} +(-71.9212 + 71.9212i) q^{97} +(-27.7509 - 27.7509i) q^{98} +31.1340i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{2} + 24 q^{7} + 32 q^{8} - 24 q^{11} - 48 q^{13} - 64 q^{16} - 16 q^{17} - 48 q^{18} - 48 q^{21} + 24 q^{22} + 104 q^{23} + 96 q^{26} - 48 q^{28} + 200 q^{31} + 64 q^{32} - 48 q^{33} + 96 q^{36}+ \cdots - 224 q^{98}+O(q^{100}) $ 16 * q - 16 * q^2 + 24 * q^7 + 32 * q^8 - 24 * q^11 - 48 * q^13 - 64 * q^16 - 16 * q^17 - 48 * q^18 - 48 * q^21 + 24 * q^22 + 104 * q^23 + 96 * q^26 - 48 * q^28 + 200 * q^31 + 64 * q^32 - 48 * q^33 + 96 * q^36 - 128 * q^37 + 56 * q^38 - 192 * q^41 + 48 * q^42 + 88 * q^43 - 208 * q^46 + 16 * q^47 - 48 * q^51 - 96 * q^52 - 160 * q^53 + 96 * q^56 + 48 * q^57 + 24 * q^58 - 24 * q^61 - 200 * q^62 + 72 * q^63 + 96 * q^66 + 264 * q^67 + 32 * q^68 + 176 * q^71 - 96 * q^72 + 32 * q^73 - 112 * q^76 + 32 * q^77 + 96 * q^78 - 144 * q^81 + 192 * q^82 - 288 * q^83 - 176 * q^86 - 168 * q^87 - 48 * q^88 + 32 * q^91 + 208 * q^92 + 144 * q^93 - 48 * q^97 - 224 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/750\mathbb{Z}\right)^\times$.

$n$	$127$	$251$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$

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 $

$2$	−1.00000	+	1.00000i	−0.500000	+	0.500000i
$2$	−1.00000	+	1.00000i	−0.500000	+	0.500000i
$3$	−1.22474	−	1.22474i	−0.408248	−	0.408248i
$3$	−1.22474	−	1.22474i	−0.408248	−	0.408248i
$4$		−	2.00000i		−	0.500000i
$5$		0			0
$5$		0			0
$6$	2.44949			0.408248
$7$	−3.25953	+	3.25953i	−0.465648	+	0.465648i	−0.900501	−	0.434853i	$-0.856800\pi$
$7$	−3.25953	+	3.25953i	−0.465648	+	0.465648i	0.434853	+	0.900501i	$0.356800\pi$
$8$	2.00000	+	2.00000i	0.250000	+	0.250000i
$9$			3.00000i			0.333333i
$10$		0			0
$11$	10.3780			0.943455			0.471727	−	0.881744i	$-0.343631\pi$
$11$	10.3780			0.943455			0.471727	+	0.881744i	$0.343631\pi$
$12$	−2.44949	+	2.44949i	−0.204124	+	0.204124i
$13$	−3.45217	−	3.45217i	−0.265551	−	0.265551i	0.561753	−	0.827305i	$-0.310127\pi$
$13$	−3.45217	−	3.45217i	−0.265551	−	0.265551i	−0.827305	+	0.561753i	$0.810127\pi$
$14$		−	6.51907i		−	0.465648i
$15$		0			0
$16$	−4.00000			−0.250000
$17$	−3.95806	+	3.95806i	−0.232827	+	0.232827i	−0.813872	−	0.581045i	$-0.802644\pi$
$17$	−3.95806	+	3.95806i	−0.232827	+	0.232827i	0.581045	+	0.813872i	$0.302644\pi$
$18$	−3.00000	−	3.00000i	−0.166667	−	0.166667i
$19$		−	3.80019i		−	0.200010i	−0.994987	−	0.100005i	$-0.968114\pi$
$19$		−	3.80019i		−	0.200010i	0.994987	−	0.100005i	$-0.0318859\pi$
$20$		0			0
$21$	7.98420			0.380200
$22$	−10.3780	+	10.3780i	−0.471727	+	0.471727i
$23$	−7.19505	−	7.19505i	−0.312828	−	0.312828i	0.533176	−	0.846004i	$-0.320998\pi$
$23$	−7.19505	−	7.19505i	−0.312828	−	0.312828i	−0.846004	+	0.533176i	$0.820998\pi$
$24$		−	4.89898i		−	0.204124i
$25$		0			0
$26$	6.90433			0.265551
$27$	3.67423	−	3.67423i	0.136083	−	0.136083i
$28$	6.51907	+	6.51907i	0.232824	+	0.232824i
$29$		−	21.2791i		−	0.733763i	−0.930268	−	0.366882i	$-0.880425\pi$
$29$		−	21.2791i		−	0.733763i	0.930268	−	0.366882i	$-0.119575\pi$
$30$		0			0
$31$	22.4937			0.725603			0.362802	−	0.931866i	$-0.381820\pi$
$31$	22.4937			0.725603			0.362802	+	0.931866i	$0.381820\pi$
$32$	4.00000	−	4.00000i	0.125000	−	0.125000i
$33$	−12.7104	−	12.7104i	−0.385164	−	0.385164i
$34$		−	7.91612i		−	0.232827i
$35$		0			0
$36$	6.00000			0.166667
$37$	−28.5269	+	28.5269i	−0.770998	+	0.770998i	−0.978281	−	0.207283i	$-0.933538\pi$
$37$	−28.5269	+	28.5269i	−0.770998	+	0.770998i	0.207283	+	0.978281i	$0.433538\pi$
$38$	3.80019	+	3.80019i	0.100005	+	0.100005i
$39$			8.45604i			0.216822i
$40$		0			0
$41$	−26.1336			−0.637404			−0.318702	−	0.947855i	$-0.603247\pi$
$41$	−26.1336			−0.637404			−0.318702	+	0.947855i	$0.603247\pi$
$42$	−7.98420	+	7.98420i	−0.190100	+	0.190100i
$43$	3.75315	+	3.75315i	0.0872825	+	0.0872825i	0.749400	−	0.662118i	$-0.230342\pi$
$43$	3.75315	+	3.75315i	0.0872825	+	0.0872825i	−0.662118	+	0.749400i	$0.730342\pi$
$44$		−	20.7560i		−	0.471727i
$45$		0			0
$46$	14.3901			0.312828
$47$	−61.3876	+	61.3876i	−1.30612	+	1.30612i	−0.381925	+	0.924193i	$0.624739\pi$
$47$	−61.3876	+	61.3876i	−1.30612	+	1.30612i	−0.924193	+	0.381925i	$0.875261\pi$
$48$	4.89898	+	4.89898i	0.102062	+	0.102062i
$49$			27.7509i			0.566344i
$50$		0			0
$51$	9.69522			0.190102
$52$	−6.90433	+	6.90433i	−0.132776	+	0.132776i
$53$	19.1975	+	19.1975i	0.362217	+	0.362217i	0.864629	−	0.502412i	$-0.167554\pi$
$53$	19.1975	+	19.1975i	0.362217	+	0.362217i	−0.502412	+	0.864629i	$0.667554\pi$
$54$			7.34847i			0.136083i
$55$		0			0
$56$	−13.0381			−0.232824
$57$	−4.65427	+	4.65427i	−0.0816538	+	0.0816538i
$58$	21.2791	+	21.2791i	0.366882	+	0.366882i
$59$			39.1066i			0.662824i	0.943486	+	0.331412i	$0.107525\pi$
$59$			39.1066i			0.662824i	−0.943486	+	0.331412i	$0.892475\pi$
$60$		0			0
$61$	82.6505			1.35493			0.677463	−	0.735557i	$-0.263079\pi$
$61$	82.6505			1.35493			0.677463	+	0.735557i	$0.263079\pi$
$62$	−22.4937	+	22.4937i	−0.362802	+	0.362802i
$63$	−9.77860	−	9.77860i	−0.155216	−	0.155216i
$64$			8.00000i			0.125000i
$65$		0			0
$66$	25.4208			0.385164
$67$	−43.8132	+	43.8132i	−0.653928	+	0.653928i	−0.953936	−	0.300009i	$-0.903010\pi$
$67$	−43.8132	+	43.8132i	−0.653928	+	0.653928i	0.300009	+	0.953936i	$0.403010\pi$
$68$	7.91612	+	7.91612i	0.116413	+	0.116413i
$69$			17.6242i			0.255423i
$70$		0			0
$71$	−59.6858			−0.840645			−0.420323	−	0.907375i	$-0.638083\pi$
$71$	−59.6858			−0.840645			−0.420323	+	0.907375i	$0.638083\pi$
$72$	−6.00000	+	6.00000i	−0.0833333	+	0.0833333i
$73$	66.8754	+	66.8754i	0.916101	+	0.916101i	0.996743	−	0.0806423i	$-0.0256971\pi$
$73$	66.8754	+	66.8754i	0.916101	+	0.916101i	−0.0806423	+	0.996743i	$0.525697\pi$
$74$		−	57.0539i		−	0.770998i
$75$		0			0
$76$	−7.60039			−0.100005
$77$	−33.8275	+	33.8275i	−0.439318	+	0.439318i
$78$	−8.45604	−	8.45604i	−0.108411	−	0.108411i
$79$			120.001i			1.51901i	0.650504	+	0.759503i	$0.274558\pi$
$79$			120.001i			1.51901i	−0.650504	+	0.759503i	$0.725442\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$	26.1336	−	26.1336i	0.318702	−	0.318702i
$83$	−88.5686	−	88.5686i	−1.06709	−	1.06709i	−0.997581	−	0.0695105i	$-0.977856\pi$
$83$	−88.5686	−	88.5686i	−1.06709	−	1.06709i	−0.0695105	−	0.997581i	$-0.522144\pi$
$84$		−	15.9684i		−	0.190100i
$85$		0			0
$86$	−7.50629			−0.0872825
$87$	−26.0615	+	26.0615i	−0.299558	+	0.299558i
$88$	20.7560	+	20.7560i	0.235864	+	0.235864i
$89$			88.3609i			0.992819i	0.868089	+	0.496409i	$0.165348\pi$
$89$			88.3609i			0.992819i	−0.868089	+	0.496409i	$0.834652\pi$
$90$		0			0
$91$	22.5049			0.247307
$92$	−14.3901	+	14.3901i	−0.156414	+	0.156414i
$93$	−27.5490	−	27.5490i	−0.296226	−	0.296226i
$94$		−	122.775i		−	1.30612i
$95$		0			0
$96$	−9.79796			−0.102062
$97$	−71.9212	+	71.9212i	−0.741456	+	0.741456i	−0.972858	−	0.231402i	$-0.925669\pi$
$97$	−71.9212	+	71.9212i	−0.741456	+	0.741456i	0.231402	+	0.972858i	$0.425669\pi$
$98$	−27.7509	−	27.7509i	−0.283172	−	0.283172i
$99$			31.1340i			0.314485i
$100$		0			0
$101$	−157.358			−1.55800			−0.779000	−	0.627023i	$-0.784273\pi$
$101$	−157.358			−1.55800			−0.779000	+	0.627023i	$0.784273\pi$
$102$	−9.69522	+	9.69522i	−0.0950512	+	0.0950512i
$103$	43.2518	+	43.2518i	0.419920	+	0.419920i	0.885176	−	0.465256i	$-0.154038\pi$
$103$	43.2518	+	43.2518i	0.419920	+	0.419920i	−0.465256	+	0.885176i	$0.654038\pi$
$104$		−	13.8087i		−	0.132776i
$105$		0			0
$106$	−38.3950			−0.362217
$107$	0.412082	−	0.412082i	0.00385123	−	0.00385123i	−0.705179	−	0.709030i	$-0.749133\pi$
$107$	0.412082	−	0.412082i	0.00385123	−	0.00385123i	0.709030	+	0.705179i	$0.249133\pi$
$108$	−7.34847	−	7.34847i	−0.0680414	−	0.0680414i
$109$			22.5380i			0.206771i	0.994641	+	0.103385i	$0.0329675\pi$
$109$			22.5380i			0.206771i	−0.994641	+	0.103385i	$0.967032\pi$
$110$		0			0
$111$	69.8764			0.629517
$112$	13.0381	−	13.0381i	0.116412	−	0.116412i
$113$	−29.7957	−	29.7957i	−0.263679	−	0.263679i	0.562868	−	0.826547i	$-0.309698\pi$
$113$	−29.7957	−	29.7957i	−0.263679	−	0.263679i	−0.826547	+	0.562868i	$0.809698\pi$
$114$		−	9.30854i		−	0.0816538i
$115$		0			0
$116$	−42.5583			−0.366882
$117$	10.3565	−	10.3565i	0.0885171	−	0.0885171i
$118$	−39.1066	−	39.1066i	−0.331412	−	0.331412i
$119$		−	25.8029i		−	0.216831i
$120$		0			0
$121$	−13.2970			−0.109893
$122$	−82.6505	+	82.6505i	−0.677463	+	0.677463i
$123$	32.0069	+	32.0069i	0.260219	+	0.260219i
$124$		−	44.9874i		−	0.362802i
$125$		0			0
$126$	19.5572			0.155216
$127$	−101.603	+	101.603i	−0.800021	+	0.800021i	−0.983099	−	0.183077i	$-0.941394\pi$
$127$	−101.603	+	101.603i	−0.800021	+	0.800021i	0.183077	+	0.983099i	$0.441394\pi$
$128$	−8.00000	−	8.00000i	−0.0625000	−	0.0625000i
$129$		−	9.19330i		−	0.0712659i
$130$		0			0
$131$	−230.730			−1.76130			−0.880648	−	0.473771i	$-0.842892\pi$
$131$	−230.730			−1.76130			−0.880648	+	0.473771i	$0.842892\pi$
$132$	−25.4208	+	25.4208i	−0.192582	+	0.192582i
$133$	12.3869	+	12.3869i	0.0931343	+	0.0931343i
$134$		−	87.6263i		−	0.653928i
$135$		0			0
$136$	−15.8322			−0.116413
$137$	107.410	−	107.410i	0.784012	−	0.784012i	−0.196493	−	0.980505i	$-0.562955\pi$
$137$	107.410	−	107.410i	0.784012	−	0.784012i	0.980505	+	0.196493i	$0.0629554\pi$
$138$	−17.6242	−	17.6242i	−0.127712	−	0.127712i
$139$			25.1870i			0.181201i	0.995887	+	0.0906007i	$0.0288787\pi$
$139$			25.1870i			0.181201i	−0.995887	+	0.0906007i	$0.971121\pi$
$140$		0			0
$141$	150.368			1.06644
$142$	59.6858	−	59.6858i	0.420323	−	0.420323i
$143$	−35.8266	−	35.8266i	−0.250536	−	0.250536i
$144$		−	12.0000i		−	0.0833333i
$145$		0			0
$146$	−133.751			−0.916101
$147$	33.9877	−	33.9877i	0.231209	−	0.231209i
$148$	57.0539	+	57.0539i	0.385499	+	0.385499i
$149$			236.105i			1.58460i	0.610132	+	0.792300i	$0.291116\pi$
$149$			236.105i			1.58460i	−0.610132	+	0.792300i	$0.708884\pi$
$150$		0			0
$151$	−228.749			−1.51489			−0.757447	−	0.652896i	$-0.773554\pi$
$151$	−228.749			−1.51489			−0.757447	+	0.652896i	$0.773554\pi$
$152$	7.60039	−	7.60039i	0.0500025	−	0.0500025i
$153$	−11.8742	−	11.8742i	−0.0776090	−	0.0776090i
$154$		−	67.6549i		−	0.439318i
$155$		0			0
$156$	16.9121			0.108411
$157$	179.990	−	179.990i	1.14643	−	1.14643i	0.159183	−	0.987249i	$-0.449114\pi$
$157$	179.990	−	179.990i	1.14643	−	1.14643i	0.987249	−	0.159183i	$-0.0508861\pi$
$158$	−120.001	−	120.001i	−0.759503	−	0.759503i
$159$		−	47.0241i		−	0.295749i
$160$		0			0
$161$	46.9050			0.291335
$162$	9.00000	−	9.00000i	0.0555556	−	0.0555556i
$163$	116.681	+	116.681i	0.715831	+	0.715831i	0.967749	−	0.251917i	$-0.0810611\pi$
$163$	116.681	+	116.681i	0.715831	+	0.715831i	−0.251917	+	0.967749i	$0.581061\pi$
$164$			52.2671i			0.318702i
$165$		0			0
$166$	177.137			1.06709
$167$	111.143	−	111.143i	0.665528	−	0.665528i	−0.291150	−	0.956677i	$-0.594038\pi$
$167$	111.143	−	111.143i	0.665528	−	0.665528i	0.956677	+	0.291150i	$0.0940378\pi$
$168$	15.9684	+	15.9684i	0.0950499	+	0.0950499i
$169$		−	145.165i		−	0.858965i
$170$		0			0
$171$	11.4006			0.0666701
$172$	7.50629	−	7.50629i	0.0436412	−	0.0436412i
$173$	130.368	+	130.368i	0.753572	+	0.753572i	0.975144	−	0.221572i	$-0.0711187\pi$
$173$	130.368	+	130.368i	0.753572	+	0.753572i	−0.221572	+	0.975144i	$0.571119\pi$
$174$		−	52.1230i		−	0.299558i
$175$		0			0
$176$	−41.5120			−0.235864
$177$	47.8957	−	47.8957i	0.270597	−	0.270597i
$178$	−88.3609	−	88.3609i	−0.496409	−	0.496409i
$179$		−	125.012i		−	0.698389i	−0.937050	−	0.349194i	$-0.886455\pi$
$179$		−	125.012i		−	0.698389i	0.937050	−	0.349194i	$-0.113545\pi$
$180$		0			0
$181$	261.923			1.44709			0.723544	−	0.690279i	$-0.242512\pi$
$181$	261.923			1.44709			0.723544	+	0.690279i	$0.242512\pi$
$182$	−22.5049	+	22.5049i	−0.123653	+	0.123653i
$183$	−101.226	−	101.226i	−0.553146	−	0.553146i
$184$		−	28.7802i		−	0.156414i
$185$		0			0
$186$	55.0981			0.296226
$187$	−41.0767	+	41.0767i	−0.219662	+	0.219662i
$188$	122.775	+	122.775i	0.653059	+	0.653059i
$189$			23.9526i			0.126733i
$190$		0			0
$191$	−207.160			−1.08460			−0.542302	−	0.840183i	$-0.682447\pi$
$191$	−207.160			−1.08460			−0.542302	+	0.840183i	$0.682447\pi$
$192$	9.79796	−	9.79796i	0.0510310	−	0.0510310i
$193$	−141.876	−	141.876i	−0.735111	−	0.735111i	0.236516	−	0.971627i	$-0.423994\pi$
$193$	−141.876	−	141.876i	−0.735111	−	0.735111i	−0.971627	+	0.236516i	$0.923994\pi$
$194$		−	143.842i		−	0.741456i
$195$		0			0
$196$	55.5018			0.283172
$197$	104.952	−	104.952i	0.532749	−	0.532749i	−0.388641	−	0.921389i	$-0.627055\pi$
$197$	104.952	−	104.952i	0.532749	−	0.532749i	0.921389	+	0.388641i	$0.127055\pi$
$198$	−31.1340	−	31.1340i	−0.157242	−	0.157242i
$199$			259.207i			1.30255i	0.758842	+	0.651274i	$0.225765\pi$
$199$			259.207i			1.30255i	−0.758842	+	0.651274i	$0.774235\pi$
$200$		0			0
$201$	107.320			0.533930
$202$	157.358	−	157.358i	0.779000	−	0.779000i
$203$	69.3601	+	69.3601i	0.341675	+	0.341675i
$204$		−	19.3904i		−	0.0950512i
$205$		0			0
$206$	−86.5035			−0.419920
$207$	21.5851	−	21.5851i	0.104276	−	0.104276i
$208$	13.8087	+	13.8087i	0.0663878	+	0.0663878i
$209$		−	39.4384i		−	0.188701i
$210$		0			0
$211$	174.522			0.827117			0.413559	−	0.910477i	$-0.364286\pi$
$211$	174.522			0.827117			0.413559	+	0.910477i	$0.364286\pi$
$212$	38.3950	−	38.3950i	0.181108	−	0.181108i
$213$	73.0999	+	73.0999i	0.343192	+	0.343192i
$214$			0.824163i			0.00385123i
$215$		0			0
$216$	14.6969			0.0680414
$217$	−73.3190	+	73.3190i	−0.337875	+	0.337875i
$218$	−22.5380	−	22.5380i	−0.103385	−	0.103385i
$219$		−	163.811i		−	0.747993i
$220$		0			0
$221$	27.3277			0.123655
$222$	−69.8764	+	69.8764i	−0.314759	+	0.314759i
$223$	48.0152	+	48.0152i	0.215315	+	0.215315i	0.806521	−	0.591206i	$-0.201348\pi$
$223$	48.0152	+	48.0152i	0.215315	+	0.215315i	−0.591206	+	0.806521i	$0.701348\pi$
$224$			26.0763i			0.116412i
$225$		0			0
$226$	59.5914			0.263679
$227$	−15.9885	+	15.9885i	−0.0704340	+	0.0704340i	−0.741446	−	0.671012i	$-0.765860\pi$
$227$	−15.9885	+	15.9885i	−0.0704340	+	0.0704340i	0.671012	+	0.741446i	$0.265860\pi$
$228$	9.30854	+	9.30854i	0.0408269	+	0.0408269i
$229$		−	66.5652i		−	0.290678i	−0.989382	−	0.145339i	$-0.953573\pi$
$229$		−	66.5652i		−	0.290678i	0.989382	−	0.145339i	$-0.0464273\pi$
$230$		0			0
$231$	82.8600			0.358701
$232$	42.5583	−	42.5583i	0.183441	−	0.183441i
$233$	68.7971	+	68.7971i	0.295267	+	0.295267i	0.839157	−	0.543890i	$-0.183049\pi$
$233$	68.7971	+	68.7971i	0.295267	+	0.295267i	−0.543890	+	0.839157i	$0.683049\pi$
$234$			20.7130i			0.0885171i
$235$		0			0
$236$	78.2133			0.331412
$237$	146.971	−	146.971i	0.620131	−	0.620131i
$238$	25.8029	+	25.8029i	0.108415	+	0.108415i
$239$			396.575i			1.65931i	0.558278	+	0.829654i	$0.311462\pi$
$239$			396.575i			1.65931i	−0.558278	+	0.829654i	$0.688538\pi$
$240$		0			0
$241$	359.811			1.49299			0.746497	−	0.665389i	$-0.231734\pi$
$241$	359.811			1.49299			0.746497	+	0.665389i	$0.231734\pi$
$242$	13.2970	−	13.2970i	0.0549464	−	0.0549464i
$243$	11.0227	+	11.0227i	0.0453609	+	0.0453609i
$244$		−	165.301i		−	0.677463i
$245$		0			0
$246$	−64.0139			−0.260219
$247$	−13.1189	+	13.1189i	−0.0531130	+	0.0531130i
$248$	44.9874	+	44.9874i	0.181401	+	0.181401i
$249$			216.948i			0.871277i
$250$		0			0
$251$	−433.572			−1.72738			−0.863689	−	0.504024i	$-0.831852\pi$
$251$	−433.572			−1.72738			−0.863689	+	0.504024i	$0.831852\pi$
$252$	−19.5572	+	19.5572i	−0.0776080	+	0.0776080i
$253$	−74.6702	−	74.6702i	−0.295139	−	0.295139i
$254$		−	203.205i		−	0.800021i
$255$		0			0
$256$	16.0000			0.0625000
$257$	215.029	−	215.029i	0.836689	−	0.836689i	−0.151733	−	0.988422i	$-0.548485\pi$
$257$	215.029	−	215.029i	0.836689	−	0.836689i	0.988422	+	0.151733i	$0.0484853\pi$
$258$	9.19330	+	9.19330i	0.0356329	+	0.0356329i
$259$		−	185.969i		−	0.718027i
$260$		0			0
$261$	63.8374			0.244588
$262$	230.730	−	230.730i	0.880648	−	0.880648i
$263$	−26.3760	−	26.3760i	−0.100289	−	0.100289i	0.655182	−	0.755471i	$-0.272592\pi$
$263$	−26.3760	−	26.3760i	−0.100289	−	0.100289i	−0.755471	+	0.655182i	$0.772592\pi$
$264$		−	50.8416i		−	0.192582i
$265$		0			0
$266$	−24.7737			−0.0931343
$267$	108.220	−	108.220i	0.405317	−	0.405317i
$268$	87.6263	+	87.6263i	0.326964	+	0.326964i
$269$			24.2923i			0.0903059i	0.998980	+	0.0451530i	$0.0143775\pi$
$269$			24.2923i			0.0903059i	−0.998980	+	0.0451530i	$0.985622\pi$
$270$		0			0
$271$	5.91723			0.0218348			0.0109174	−	0.999940i	$-0.496525\pi$
$271$	5.91723			0.0218348			0.0109174	+	0.999940i	$0.496525\pi$
$272$	15.8322	−	15.8322i	0.0582067	−	0.0582067i
$273$	−27.5628	−	27.5628i	−0.100963	−	0.100963i
$274$			214.819i			0.784012i
$275$		0			0
$276$	35.2484			0.127712
$277$	−22.3759	+	22.3759i	−0.0807792	+	0.0807792i	−0.746342	−	0.665563i	$-0.768192\pi$
$277$	−22.3759	+	22.3759i	−0.0807792	+	0.0807792i	0.665563	+	0.746342i	$0.268192\pi$
$278$	−25.1870	−	25.1870i	−0.0906007	−	0.0906007i
$279$			67.4811i			0.241868i
$280$		0			0
$281$	26.5064			0.0943288			0.0471644	−	0.998887i	$-0.484982\pi$
$281$	26.5064			0.0943288			0.0471644	+	0.998887i	$0.484982\pi$
$282$	−150.368	+	150.368i	−0.533221	+	0.533221i
$283$	85.0173	+	85.0173i	0.300414	+	0.300414i	0.841176	−	0.540762i	$-0.181864\pi$
$283$	85.0173	+	85.0173i	0.300414	+	0.300414i	−0.540762	+	0.841176i	$0.681864\pi$
$284$			119.372i			0.420323i
$285$		0			0
$286$	71.6532			0.250536
$287$	85.1832	−	85.1832i	0.296806	−	0.296806i
$288$	12.0000	+	12.0000i	0.0416667	+	0.0416667i
$289$			257.668i			0.891583i
$290$		0			0
$291$	176.170			0.605396
$292$	133.751	−	133.751i	0.458050	−	0.458050i
$293$	−103.562	−	103.562i	−0.353454	−	0.353454i	0.507939	−	0.861393i	$-0.330408\pi$
$293$	−103.562	−	103.562i	−0.353454	−	0.353454i	−0.861393	+	0.507939i	$0.830408\pi$
$294$			67.9755i			0.231209i
$295$		0			0
$296$	−114.108			−0.385499
$297$	38.1312	−	38.1312i	0.128388	−	0.128388i
$298$	−236.105	−	236.105i	−0.792300	−	0.792300i
$299$			49.6770i			0.166144i
$300$		0			0
$301$	−24.4670			−0.0812858
$302$	228.749	−	228.749i	0.757447	−	0.757447i
$303$	192.724	+	192.724i	0.636051	+	0.636051i
$304$			15.2008i			0.0500025i
$305$		0			0
$306$	23.7483			0.0776090
$307$	292.105	−	292.105i	0.951482	−	0.951482i	−0.0473939	−	0.998876i	$-0.515092\pi$
$307$	292.105	−	292.105i	0.951482	−	0.951482i	0.998876	+	0.0473939i	$0.0150916\pi$
$308$	67.6549	+	67.6549i	0.219659	+	0.219659i
$309$		−	105.945i		−	0.342863i
$310$		0			0
$311$	88.6420			0.285023			0.142511	−	0.989793i	$-0.454482\pi$
$311$	88.6420			0.285023			0.142511	+	0.989793i	$0.454482\pi$
$312$	−16.9121	+	16.9121i	−0.0542054	+	0.0542054i
$313$	−308.953	−	308.953i	−0.987070	−	0.987070i	0.0128476	−	0.999917i	$-0.495910\pi$
$313$	−308.953	−	308.953i	−0.987070	−	0.987070i	−0.999917	+	0.0128476i	$0.995910\pi$
$314$			359.980i			1.14643i
$315$		0			0
$316$	240.003			0.759503
$317$	−63.0305	+	63.0305i	−0.198835	+	0.198835i	−0.799500	−	0.600666i	$-0.794902\pi$
$317$	−63.0305	+	63.0305i	−0.198835	+	0.198835i	0.600666	+	0.799500i	$0.294902\pi$
$318$	47.0241	+	47.0241i	0.147874	+	0.147874i
$319$		−	220.835i		−	0.692272i
$320$		0			0
$321$	−1.00939			−0.00314452
$322$	−46.9050	+	46.9050i	−0.145668	+	0.145668i
$323$	15.0414	+	15.0414i	0.0465678	+	0.0465678i
$324$			18.0000i			0.0555556i
$325$		0			0
$326$	−233.361			−0.715831
$327$	27.6033	−	27.6033i	0.0844139	−	0.0844139i
$328$	−52.2671	−	52.2671i	−0.159351	−	0.159351i
$329$		−	400.190i		−	1.21638i
$330$		0			0
$331$	−228.413			−0.690071			−0.345035	−	0.938590i	$-0.612133\pi$
$331$	−228.413			−0.690071			−0.345035	+	0.938590i	$0.612133\pi$
$332$	−177.137	+	177.137i	−0.533546	+	0.533546i
$333$	−85.5808	−	85.5808i	−0.256999	−	0.256999i
$334$			222.286i			0.665528i
$335$		0			0
$336$	−31.9368			−0.0950499
$337$	26.8074	−	26.8074i	0.0795473	−	0.0795473i	−0.666214	−	0.745761i	$-0.732086\pi$
$337$	26.8074	−	26.8074i	0.0795473	−	0.0795473i	0.745761	+	0.666214i	$0.232086\pi$
$338$	145.165	+	145.165i	0.429483	+	0.429483i
$339$			72.9842i			0.215293i
$340$		0			0
$341$	233.440			0.684574
$342$	−11.4006	+	11.4006i	−0.0333350	+	0.0333350i
$343$	−250.172	−	250.172i	−0.729365	−	0.729365i
$344$			15.0126i			0.0436412i
$345$		0			0
$346$	−260.736			−0.753572
$347$	345.508	−	345.508i	0.995700	−	0.995700i	−0.00429061	−	0.999991i	$-0.501366\pi$
$347$	345.508	−	345.508i	0.995700	−	0.995700i	0.999991	+	0.00429061i	$0.00136575\pi$
$348$	52.1230	+	52.1230i	0.149779	+	0.149779i
$349$			195.192i			0.559289i	0.960104	+	0.279644i	$0.0902165\pi$
$349$			195.192i			0.559289i	−0.960104	+	0.279644i	$0.909783\pi$
$350$		0			0
$351$	−25.3681			−0.0722739
$352$	41.5120	−	41.5120i	0.117932	−	0.117932i
$353$	49.4995	+	49.4995i	0.140225	+	0.140225i	0.773735	−	0.633510i	$-0.218386\pi$
$353$	49.4995	+	49.4995i	0.140225	+	0.140225i	−0.633510	+	0.773735i	$0.718386\pi$
$354$			95.7913i			0.270597i
$355$		0			0
$356$	176.722			0.496409
$357$	−31.6019	+	31.6019i	−0.0885208	+	0.0885208i
$358$	125.012	+	125.012i	0.349194	+	0.349194i
$359$		−	452.011i		−	1.25908i	−0.776966	−	0.629542i	$-0.783243\pi$
$359$		−	452.011i		−	1.25908i	0.776966	−	0.629542i	$-0.216757\pi$
$360$		0			0
$361$	346.559			0.959996
$362$	−261.923	+	261.923i	−0.723544	+	0.723544i
$363$	16.2855	+	16.2855i	0.0448635	+	0.0448635i
$364$		−	45.0098i		−	0.123653i
$365$		0			0
$366$	202.451			0.553146
$367$	−476.321	+	476.321i	−1.29788	+	1.29788i	−0.368084	+	0.929793i	$0.619986\pi$
$367$	−476.321	+	476.321i	−1.29788	+	1.29788i	−0.929793	+	0.368084i	$0.880014\pi$
$368$	28.7802	+	28.7802i	0.0782070	+	0.0782070i
$369$		−	78.4007i		−	0.212468i
$370$		0			0
$371$	−125.150			−0.337331
$372$	−55.0981	+	55.0981i	−0.148113	+	0.148113i
$373$	−170.014	−	170.014i	−0.455802	−	0.455802i	0.441473	−	0.897275i	$-0.354456\pi$
$373$	−170.014	−	170.014i	−0.455802	−	0.455802i	−0.897275	+	0.441473i	$0.854456\pi$
$374$		−	82.1535i		−	0.219662i
$375$		0			0
$376$	−245.550			−0.653059
$377$	−73.4591	+	73.4591i	−0.194852	+	0.194852i
$378$	−23.9526	−	23.9526i	−0.0633666	−	0.0633666i
$379$			124.175i			0.327638i	0.986490	+	0.163819i	$0.0523814\pi$
$379$			124.175i			0.327638i	−0.986490	+	0.163819i	$0.947619\pi$
$380$		0			0
$381$	248.875			0.653215
$382$	207.160	−	207.160i	0.542302	−	0.542302i
$383$	−136.998	−	136.998i	−0.357697	−	0.357697i	0.505267	−	0.862963i	$-0.331394\pi$
$383$	−136.998	−	136.998i	−0.357697	−	0.357697i	−0.862963	+	0.505267i	$0.831394\pi$
$384$			19.5959i			0.0510310i
$385$		0			0
$386$	283.753			0.735111
$387$	−11.2594	+	11.2594i	−0.0290942	+	0.0290942i
$388$	143.842	+	143.842i	0.370728	+	0.370728i
$389$			72.9663i			0.187574i	0.995592	+	0.0937871i	$0.0298973\pi$
$389$			72.9663i			0.187574i	−0.995592	+	0.0937871i	$0.970103\pi$
$390$		0			0
$391$	56.9568			0.145670
$392$	−55.5018	+	55.5018i	−0.141586	+	0.141586i
$393$	282.585	+	282.585i	0.719046	+	0.719046i
$394$			209.903i			0.532749i
$395$		0			0
$396$	62.2680			0.157242
$397$	−178.254	+	178.254i	−0.449002	+	0.449002i	−0.895023	−	0.446020i	$-0.852841\pi$
$397$	−178.254	+	178.254i	−0.449002	+	0.449002i	0.446020	+	0.895023i	$0.352841\pi$
$398$	−259.207	−	259.207i	−0.651274	−	0.651274i
$399$		−	30.3415i		−	0.0760438i
$400$		0			0
$401$	−604.681			−1.50793			−0.753967	−	0.656912i	$-0.771862\pi$
$401$	−604.681			−1.50793			−0.753967	+	0.656912i	$0.771862\pi$
$402$	−107.320	+	107.320i	−0.266965	+	0.266965i
$403$	−77.6520	−	77.6520i	−0.192685	−	0.192685i
$404$			314.716i			0.779000i
$405$		0			0
$406$	−138.720			−0.341675
$407$	−296.053	+	296.053i	−0.727402	+	0.727402i
$408$	19.3904	+	19.3904i	0.0475256	+	0.0475256i
$409$		−	596.739i		−	1.45902i	−0.683970	−	0.729510i	$-0.739748\pi$
$409$		−	596.739i		−	1.45902i	0.683970	−	0.729510i	$-0.260252\pi$
$410$		0			0
$411$	−263.099			−0.640143
$412$	86.5035	−	86.5035i	0.209960	−	0.209960i
$413$	−127.469	−	127.469i	−0.308643	−	0.308643i
$414$			43.1703i			0.104276i
$415$		0			0
$416$	−27.6173			−0.0663878
$417$	30.8476	−	30.8476i	0.0739752	−	0.0739752i
$418$	39.4384	+	39.4384i	0.0943503	+	0.0943503i
$419$		−	475.207i		−	1.13415i	−0.823668	−	0.567073i	$-0.808076\pi$
$419$		−	475.207i		−	1.13415i	0.823668	−	0.567073i	$-0.191924\pi$
$420$		0			0
$421$	−512.865			−1.21821			−0.609103	−	0.793091i	$-0.708471\pi$
$421$	−512.865			−1.21821			−0.609103	+	0.793091i	$0.708471\pi$
$422$	−174.522	+	174.522i	−0.413559	+	0.413559i
$423$	−184.163	−	184.163i	−0.435373	−	0.435373i
$424$			76.7900i			0.181108i
$425$		0			0
$426$	−146.200			−0.343192
$427$	−269.402	+	269.402i	−0.630918	+	0.630918i
$428$	−0.824163	−	0.824163i	−0.00192561	−	0.00192561i
$429$			87.7569i			0.204561i
$430$		0			0
$431$	−614.993			−1.42690			−0.713449	−	0.700707i	$-0.752868\pi$
$431$	−614.993			−1.42690			−0.713449	+	0.700707i	$0.752868\pi$
$432$	−14.6969	+	14.6969i	−0.0340207	+	0.0340207i
$433$	318.276	+	318.276i	0.735049	+	0.735049i	0.971615	−	0.236566i	$-0.0760221\pi$
$433$	318.276	+	318.276i	0.735049	+	0.735049i	−0.236566	+	0.971615i	$0.576022\pi$
$434$		−	146.638i		−	0.337875i
$435$		0			0
$436$	45.0761			0.103385
$437$	−27.3426	+	27.3426i	−0.0625688	+	0.0625688i
$438$	163.811	+	163.811i	0.373997	+	0.373997i
$439$		−	22.3789i		−	0.0509771i	−0.999675	−	0.0254885i	$-0.991886\pi$
$439$		−	22.3789i		−	0.0509771i	0.999675	−	0.0254885i	$-0.00811413\pi$
$440$		0			0
$441$	−83.2526			−0.188781
$442$	−27.3277	+	27.3277i	−0.0618275	+	0.0618275i
$443$	284.497	+	284.497i	0.642206	+	0.642206i	0.951097	−	0.308891i	$-0.0999579\pi$
$443$	284.497	+	284.497i	0.642206	+	0.642206i	−0.308891	+	0.951097i	$0.599958\pi$
$444$		−	139.753i		−	0.314759i
$445$		0			0
$446$	−96.0305			−0.215315
$447$	289.169	−	289.169i	0.646910	−	0.646910i
$448$	−26.0763	−	26.0763i	−0.0582060	−	0.0582060i
$449$		−	389.187i		−	0.866786i	−0.901205	−	0.433393i	$-0.857316\pi$
$449$		−	389.187i		−	0.866786i	0.901205	−	0.433393i	$-0.142684\pi$
$450$		0			0
$451$	−271.214			−0.601362
$452$	−59.5914	+	59.5914i	−0.131839	+	0.131839i
$453$	280.159	+	280.159i	0.618453	+	0.618453i
$454$		−	31.9770i		−	0.0704340i
$455$		0			0
$456$	−18.6171			−0.0408269
$457$	−283.679	+	283.679i	−0.620741	+	0.620741i	−0.945721	−	0.324980i	$-0.894642\pi$
$457$	−283.679	+	283.679i	−0.620741	+	0.620741i	0.324980	+	0.945721i	$0.394642\pi$
$458$	66.5652	+	66.5652i	0.145339	+	0.145339i
$459$			29.0857i			0.0633675i
$460$		0			0
$461$	871.374			1.89018			0.945091	−	0.326806i	$-0.105973\pi$
$461$	871.374			1.89018			0.945091	+	0.326806i	$0.105973\pi$
$462$	−82.8600	+	82.8600i	−0.179351	+	0.179351i
$463$	−240.388	−	240.388i	−0.519196	−	0.519196i	0.398132	−	0.917328i	$-0.369658\pi$
$463$	−240.388	−	240.388i	−0.519196	−	0.519196i	−0.917328	+	0.398132i	$0.869658\pi$
$464$			85.1165i			0.183441i
$465$		0			0
$466$	−137.594			−0.295267
$467$	−54.7139	+	54.7139i	−0.117160	+	0.117160i	−0.763256	−	0.646096i	$-0.776401\pi$
$467$	−54.7139	+	54.7139i	−0.117160	+	0.117160i	0.646096	+	0.763256i	$0.276401\pi$
$468$	−20.7130	−	20.7130i	−0.0442585	−	0.0442585i
$469$		−	285.621i		−	0.609000i
$470$		0			0
$471$	−440.883			−0.936058
$472$	−78.2133	+	78.2133i	−0.165706	+	0.165706i
$473$	38.9502	+	38.9502i	0.0823471	+	0.0823471i
$474$			293.942i			0.620131i
$475$		0			0
$476$	−51.6057			−0.108415
$477$	−57.5925	+	57.5925i	−0.120739	+	0.120739i
$478$	−396.575	−	396.575i	−0.829654	−	0.829654i
$479$		−	125.855i		−	0.262746i	−0.991333	−	0.131373i	$-0.958061\pi$
$479$		−	125.855i		−	0.262746i	0.991333	−	0.131373i	$-0.0419386\pi$
$480$		0			0
$481$	196.959			0.409479
$482$	−359.811	+	359.811i	−0.746497	+	0.746497i
$483$	−57.4466	−	57.4466i	−0.118937	−	0.118937i
$484$			26.5941i			0.0549464i
$485$		0			0
$486$	−22.0454			−0.0453609
$487$	−333.136	+	333.136i	−0.684058	+	0.684058i	−0.960912	−	0.276854i	$-0.910708\pi$
$487$	−333.136	+	333.136i	−0.684058	+	0.684058i	0.276854	+	0.960912i	$0.410708\pi$
$488$	165.301	+	165.301i	0.338731	+	0.338731i
$489$		−	285.808i		−	0.584474i
$490$		0			0
$491$	−89.4786			−0.182237			−0.0911187	−	0.995840i	$-0.529044\pi$
$491$	−89.4786			−0.182237			−0.0911187	+	0.995840i	$0.529044\pi$
$492$	64.0139	−	64.0139i	0.130109	−	0.130109i
$493$	84.2240	+	84.2240i	0.170840	+	0.170840i
$494$		−	26.2378i		−	0.0531130i
$495$		0			0
$496$	−89.9748			−0.181401
$497$	194.548	−	194.548i	0.391444	−	0.391444i
$498$	−216.948	−	216.948i	−0.435638	−	0.435638i
$499$			122.886i			0.246264i	0.992390	+	0.123132i	$0.0392938\pi$
$499$			122.886i			0.246264i	−0.992390	+	0.123132i	$0.960706\pi$
$500$		0			0
$501$	−272.244			−0.543401
$502$	433.572	−	433.572i	0.863689	−	0.863689i
$503$	213.403	+	213.403i	0.424261	+	0.424261i	0.886668	−	0.462407i	$-0.153014\pi$
$503$	213.403	+	213.403i	0.424261	+	0.424261i	−0.462407	+	0.886668i	$0.653014\pi$
$504$		−	39.1144i		−	0.0776080i
$505$		0			0
$506$	149.340			0.295139
$507$	−177.790	+	177.790i	−0.350671	+	0.350671i
$508$	203.205	+	203.205i	0.400011	+	0.400011i
$509$			147.919i			0.290608i	0.989387	+	0.145304i	$0.0464160\pi$
$509$			147.919i			0.290608i	−0.989387	+	0.145304i	$0.953584\pi$
$510$		0			0
$511$	−435.965			−0.853161
$512$	−16.0000	+	16.0000i	−0.0312500	+	0.0312500i
$513$	−13.9628	−	13.9628i	−0.0272179	−	0.0272179i
$514$			430.058i			0.836689i
$515$		0			0
$516$	−18.3866			−0.0356329
$517$	−637.080	+	637.080i	−1.23226	+	1.23226i
$518$	185.969	+	185.969i	0.359014	+	0.359014i
$519$		−	319.335i		−	0.615289i
$520$		0			0
$521$	663.440			1.27340			0.636698	−	0.771113i	$-0.280300\pi$
$521$	663.440			1.27340			0.636698	+	0.771113i	$0.280300\pi$
$522$	−63.8374	+	63.8374i	−0.122294	+	0.122294i
$523$	486.528	+	486.528i	0.930264	+	0.930264i	0.997722	−	0.0674581i	$-0.0214889\pi$
$523$	486.528	+	486.528i	0.930264	+	0.930264i	−0.0674581	+	0.997722i	$0.521489\pi$
$524$			461.460i			0.880648i
$525$		0			0
$526$	52.7521			0.100289
$527$	−89.0314	+	89.0314i	−0.168940	+	0.168940i
$528$	50.8416	+	50.8416i	0.0962910	+	0.0962910i
$529$		−	425.463i		−	0.804277i
$530$		0			0
$531$	−117.320			−0.220941
$532$	24.7737	−	24.7737i	0.0465671	−	0.0465671i
$533$	90.2173	+	90.2173i	0.169263	+	0.169263i
$534$			216.439i			0.405317i
$535$		0			0
$536$	−175.253			−0.326964
$537$	−153.107	+	153.107i	−0.285116	+	0.285116i
$538$	−24.2923	−	24.2923i	−0.0451530	−	0.0451530i
$539$			287.999i			0.534320i
$540$		0			0
$541$	572.097			1.05748			0.528741	−	0.848783i	$-0.322664\pi$
$541$	572.097			1.05748			0.528741	+	0.848783i	$0.322664\pi$
$542$	−5.91723	+	5.91723i	−0.0109174	+	0.0109174i
$543$	−320.789	−	320.789i	−0.590771	−	0.590771i
$544$			31.6645i			0.0582067i
$545$		0			0
$546$	55.1255			0.100963
$547$	−82.4854	+	82.4854i	−0.150796	+	0.150796i	−0.778473	−	0.627678i	$-0.784006\pi$
$547$	−82.4854	+	82.4854i	−0.150796	+	0.150796i	0.627678	+	0.778473i	$0.284006\pi$
$548$	−214.819	−	214.819i	−0.392006	−	0.392006i
$549$			247.951i			0.451642i
$550$		0			0
$551$	−80.8648			−0.146760
$552$	−35.2484	+	35.2484i	−0.0638558	+	0.0638558i
$553$	−391.149	−	391.149i	−0.707321	−	0.707321i
$554$		−	44.7517i		−	0.0807792i
$555$		0			0
$556$	50.3740			0.0906007
$557$	−451.347	+	451.347i	−0.810318	+	0.810318i	−0.984681	−	0.174364i	$-0.944213\pi$
$557$	−451.347	+	451.347i	−0.810318	+	0.810318i	0.174364	+	0.984681i	$0.444213\pi$
$558$	−67.4811	−	67.4811i	−0.120934	−	0.120934i
$559$		−	25.9130i		−	0.0463559i
$560$		0			0
$561$	100.617			0.179353
$562$	−26.5064	+	26.5064i	−0.0471644	+	0.0471644i
$563$	−41.9310	−	41.9310i	−0.0744778	−	0.0744778i	0.668887	−	0.743364i	$-0.266771\pi$
$563$	−41.9310	−	41.9310i	−0.0744778	−	0.0744778i	−0.743364	+	0.668887i	$0.766771\pi$
$564$		−	300.736i		−	0.533221i
$565$		0			0
$566$	−170.035			−0.300414
$567$	29.3358	−	29.3358i	0.0517386	−	0.0517386i
$568$	−119.372	−	119.372i	−0.210161	−	0.210161i
$569$			97.7105i			0.171723i	0.996307	+	0.0858616i	$0.0273643\pi$
$569$			97.7105i			0.171723i	−0.996307	+	0.0858616i	$0.972636\pi$
$570$		0			0
$571$	−452.840			−0.793065			−0.396532	−	0.918021i	$-0.629787\pi$
$571$	−452.840			−0.793065			−0.396532	+	0.918021i	$0.629787\pi$
$572$	−71.6532	+	71.6532i	−0.125268	+	0.125268i
$573$	253.718	+	253.718i	0.442788	+	0.442788i
$574$			170.366i			0.296806i
$575$		0			0
$576$	−24.0000			−0.0416667
$577$	−577.336	+	577.336i	−1.00058	+	1.00058i	−0.000582984		1.00000i	$0.500186\pi$
$577$	−577.336	+	577.336i	−1.00058	+	1.00058i	−1.00000		0.000582984i	$0.999814\pi$
$578$	−257.668	−	257.668i	−0.445792	−	0.445792i
$579$			347.525i			0.600216i
$580$		0			0
$581$	577.385			0.993778
$582$	−176.170	+	176.170i	−0.302698	+	0.302698i
$583$	199.232	+	199.232i	0.341735	+	0.341735i
$584$			267.501i			0.458050i
$585$		0			0
$586$	207.124			0.353454
$587$	−286.660	+	286.660i	−0.488347	+	0.488347i	−0.907784	−	0.419437i	$-0.862227\pi$
$587$	−286.660	+	286.660i	−0.488347	+	0.488347i	0.419437	+	0.907784i	$0.362227\pi$
$588$	−67.9755	−	67.9755i	−0.115605	−	0.115605i
$589$		−	85.4804i		−	0.145128i
$590$		0			0
$591$	−257.078			−0.434988
$592$	114.108	−	114.108i	0.192750	−	0.192750i
$593$	158.533	+	158.533i	0.267341	+	0.267341i	0.828028	−	0.560687i	$-0.189463\pi$
$593$	158.533	+	158.533i	0.267341	+	0.267341i	−0.560687	+	0.828028i	$0.689463\pi$
$594$			76.2624i			0.128388i
$595$		0			0
$596$	472.211			0.792300
$597$	317.463	−	317.463i	0.531763	−	0.531763i
$598$	−49.6770	−	49.6770i	−0.0830719	−	0.0830719i
$599$		−	434.969i		−	0.726159i	−0.931758	−	0.363080i	$-0.881725\pi$
$599$		−	434.969i		−	0.726159i	0.931758	−	0.363080i	$-0.118275\pi$
$600$		0			0
$601$	649.549			1.08078			0.540390	−	0.841415i	$-0.318277\pi$
$601$	649.549			1.08078			0.540390	+	0.841415i	$0.318277\pi$
$602$	24.4670	−	24.4670i	0.0406429	−	0.0406429i
$603$	−131.440	−	131.440i	−0.217976	−	0.217976i
$604$			457.498i			0.757447i
$605$		0			0
$606$	−385.447			−0.636051
$607$	189.146	−	189.146i	0.311607	−	0.311607i	−0.533925	−	0.845532i	$-0.679283\pi$
$607$	189.146	−	189.146i	0.311607	−	0.311607i	0.845532	+	0.533925i	$0.179283\pi$
$608$	−15.2008	−	15.2008i	−0.0250013	−	0.0250013i
$609$		−	169.897i		−	0.278977i
$610$		0			0
$611$	423.840			0.693683
$612$	−23.7483	+	23.7483i	−0.0388045	+	0.0388045i
$613$	−715.958	−	715.958i	−1.16796	−	1.16796i	−0.982688	−	0.185269i	$-0.940684\pi$
$613$	−715.958	−	715.958i	−1.16796	−	1.16796i	−0.185269	−	0.982688i	$-0.559316\pi$
$614$			584.210i			0.951482i
$615$		0			0
$616$	−135.310			−0.219659
$617$	487.963	−	487.963i	0.790864	−	0.790864i	−0.190771	−	0.981635i	$-0.561099\pi$
$617$	487.963	−	487.963i	0.790864	−	0.790864i	0.981635	+	0.190771i	$0.0610987\pi$
$618$	105.945	+	105.945i	0.171432	+	0.171432i
$619$		−	742.728i		−	1.19988i	−0.800044	−	0.599941i	$-0.795191\pi$
$619$		−	742.728i		−	1.19988i	0.800044	−	0.599941i	$-0.204809\pi$
$620$		0			0
$621$	−52.8726			−0.0851410
$622$	−88.6420	+	88.6420i	−0.142511	+	0.142511i
$623$	−288.015	−	288.015i	−0.462304	−	0.462304i
$624$		−	33.8242i		−	0.0542054i
$625$		0			0
$626$	617.906			0.987070
$627$	−48.3020	+	48.3020i	−0.0770367	+	0.0770367i
$628$	−359.980	−	359.980i	−0.573216	−	0.573216i
$629$		−	225.823i		−	0.359018i
$630$		0			0
$631$	−335.875			−0.532289			−0.266145	−	0.963933i	$-0.585750\pi$
$631$	−335.875			−0.532289			−0.266145	+	0.963933i	$0.585750\pi$
$632$	−240.003	+	240.003i	−0.379751	+	0.379751i
$633$	−213.745	−	213.745i	−0.337669	−	0.337669i
$634$		−	126.061i		−	0.198835i
$635$		0			0
$636$	−94.0481			−0.147874
$637$	95.8006	−	95.8006i	0.150393	−	0.150393i
$638$	220.835	+	220.835i	0.346136	+	0.346136i
$639$		−	179.057i		−	0.280215i
$640$		0			0
$641$	589.403			0.919506			0.459753	−	0.888047i	$-0.347938\pi$
$641$	589.403			0.919506			0.459753	+	0.888047i	$0.347938\pi$
$642$	1.00939	−	1.00939i	0.00157226	−	0.00157226i
$643$	403.195	+	403.195i	0.627053	+	0.627053i	0.947325	−	0.320273i	$-0.103775\pi$
$643$	403.195	+	403.195i	0.627053	+	0.627053i	−0.320273	+	0.947325i	$0.603775\pi$
$644$		−	93.8100i		−	0.145668i
$645$		0			0
$646$	−30.0828			−0.0465678
$647$	394.144	−	394.144i	0.609188	−	0.609188i	−0.333546	−	0.942734i	$-0.608245\pi$
$647$	394.144	−	394.144i	0.609188	−	0.609188i	0.942734	+	0.333546i	$0.108245\pi$
$648$	−18.0000	−	18.0000i	−0.0277778	−	0.0277778i
$649$			405.849i			0.625345i
$650$		0			0
$651$	179.594			0.275874
$652$	233.361	−	233.361i	0.357916	−	0.357916i
$653$	480.918	+	480.918i	0.736474	+	0.736474i	0.971894	−	0.235420i	$-0.0756465\pi$
$653$	480.918	+	480.918i	0.736474	+	0.736474i	−0.235420	+	0.971894i	$0.575646\pi$
$654$			55.2067i			0.0844139i
$655$		0			0
$656$	104.534			0.159351
$657$	−200.626	+	200.626i	−0.305367	+	0.305367i
$658$	400.190	+	400.190i	0.608191	+	0.608191i
$659$			951.208i			1.44341i	0.692200	+	0.721706i	$0.256642\pi$
$659$			951.208i			1.44341i	−0.692200	+	0.721706i	$0.743358\pi$
$660$		0			0
$661$	14.7771			0.0223557			0.0111778	−	0.999938i	$-0.496442\pi$
$661$	14.7771			0.0223557			0.0111778	+	0.999938i	$0.496442\pi$
$662$	228.413	−	228.413i	0.345035	−	0.345035i
$663$	−33.4695	−	33.4695i	−0.0504819	−	0.0504819i
$664$		−	354.274i		−	0.533546i
$665$		0			0
$666$	171.162			0.256999
$667$	−153.104	+	153.104i	−0.229542	+	0.229542i
$668$	−222.286	−	222.286i	−0.332764	−	0.332764i
$669$		−	117.613i		−	0.175804i
$670$		0			0
$671$	857.747			1.27831
$672$	31.9368	−	31.9368i	0.0475250	−	0.0475250i
$673$	785.103	+	785.103i	1.16657	+	1.16657i	0.983008	+	0.183565i	$0.0587637\pi$
$673$	785.103	+	785.103i	1.16657	+	1.16657i	0.183565	+	0.983008i	$0.441236\pi$
$674$			53.6149i			0.0795473i
$675$		0			0
$676$	−290.330			−0.429483
$677$	−876.553	+	876.553i	−1.29476	+	1.29476i	−0.362954	+	0.931807i	$0.618232\pi$
$677$	−876.553	+	876.553i	−1.29476	+	1.29476i	−0.931807	+	0.362954i	$0.881768\pi$
$678$	−72.9842	−	72.9842i	−0.107646	−	0.107646i
$679$		−	468.859i		−	0.690514i
$680$		0			0
$681$	39.1637			0.0575091
$682$	−233.440	+	233.440i	−0.342287	+	0.342287i
$683$	−494.646	−	494.646i	−0.724225	−	0.724225i	0.245238	−	0.969463i	$-0.421134\pi$
$683$	−494.646	−	494.646i	−0.724225	−	0.724225i	−0.969463	+	0.245238i	$0.921134\pi$
$684$		−	22.8012i		−	0.0333350i
$685$		0			0
$686$	500.344			0.729365
$687$	−81.5254	+	81.5254i	−0.118669	+	0.118669i
$688$	−15.0126	−	15.0126i	−0.0218206	−	0.0218206i
$689$		−	132.546i		−	0.192374i
$690$		0			0
$691$	−1053.96			−1.52527			−0.762635	−	0.646829i	$-0.776095\pi$
$691$	−1053.96			−1.52527			−0.762635	+	0.646829i	$0.776095\pi$
$692$	260.736	−	260.736i	0.376786	−	0.376786i
$693$	−101.482	−	101.482i	−0.146439	−	0.146439i
$694$			691.016i			0.995700i
$695$		0			0
$696$	−104.246			−0.149779
$697$	103.438	−	103.438i	0.148405	−	0.148405i
$698$	−195.192	−	195.192i	−0.279644	−	0.279644i
$699$		−	168.518i		−	0.241084i
$700$		0			0
$701$	773.811			1.10387			0.551933	−	0.833888i	$-0.313890\pi$
$701$	773.811			1.10387			0.551933	+	0.833888i	$0.313890\pi$
$702$	25.3681	−	25.3681i	0.0361369	−	0.0361369i
$703$	108.408	+	108.408i	0.154208	+	0.154208i
$704$			83.0240i			0.117932i
$705$		0			0
$706$	−98.9990			−0.140225
$707$	512.914	−	512.914i	0.725480	−	0.725480i
$708$	−95.7913	−	95.7913i	−0.135298	−	0.135298i
$709$			1127.93i			1.59088i	0.606033	+	0.795439i	$0.292760\pi$
$709$			1127.93i			1.59088i	−0.606033	+	0.795439i	$0.707240\pi$
$710$		0			0
$711$	−360.004			−0.506335
$712$	−176.722	+	176.722i	−0.248205	+	0.248205i
$713$	−161.843	−	161.843i	−0.226989	−	0.226989i
$714$		−	63.2038i		−	0.0885208i
$715$		0			0
$716$	−250.023			−0.349194
$717$	485.703	−	485.703i	0.677410	−	0.677410i
$718$	452.011	+	452.011i	0.629542	+	0.629542i
$719$			370.533i			0.515345i	0.966232	+	0.257673i	$0.0829556\pi$
$719$			370.533i			0.515345i	−0.966232	+	0.257673i	$0.917044\pi$
$720$		0			0
$721$	−281.961			−0.391070
$722$	−346.559	+	346.559i	−0.479998	+	0.479998i
$723$	−440.677	−	440.677i	−0.609512	−	0.609512i
$724$		−	523.846i		−	0.723544i
$725$		0			0
$726$	−32.5709			−0.0448635
$727$	814.509	−	814.509i	1.12037	−	1.12037i	0.128685	−	0.991686i	$-0.458924\pi$
$727$	814.509	−	814.509i	1.12037	−	1.12037i	0.991686	−	0.128685i	$-0.0410755\pi$
$728$	45.0098	+	45.0098i	0.0618267	+	0.0618267i
$729$		−	27.0000i		−	0.0370370i
$730$		0			0
$731$	−29.7104			−0.0406434
$732$	−202.451	+	202.451i	−0.276573	+	0.276573i
$733$	419.033	+	419.033i	0.571669	+	0.571669i	0.932595	−	0.360926i	$-0.117539\pi$
$733$	419.033	+	419.033i	0.571669	+	0.571669i	−0.360926	+	0.932595i	$0.617539\pi$
$734$		−	952.641i		−	1.29788i
$735$		0			0
$736$	−57.5604			−0.0782070
$737$	−454.693	+	454.693i	−0.616952	+	0.616952i
$738$	78.4007	+	78.4007i	0.106234	+	0.106234i
$739$			695.444i			0.941062i	0.882384	+	0.470531i	$0.155938\pi$
$739$			695.444i			0.941062i	−0.882384	+	0.470531i	$0.844062\pi$
$740$		0			0
$741$	32.1346			0.0433665
$742$	125.150	−	125.150i	0.168665	−	0.168665i
$743$	−378.997	−	378.997i	−0.510090	−	0.510090i	0.404464	−	0.914554i	$-0.367458\pi$
$743$	−378.997	−	378.997i	−0.510090	−	0.510090i	−0.914554	+	0.404464i	$0.867458\pi$
$744$		−	110.196i		−	0.148113i
$745$		0			0
$746$	340.028			0.455802
$747$	265.706	−	265.706i	0.355697	−	0.355697i
$748$	82.1535	+	82.1535i	0.109831	+	0.109831i
$749$			2.68639i			0.00358663i
$750$		0			0
$751$	−262.131			−0.349042			−0.174521	−	0.984653i	$-0.555838\pi$
$751$	−262.131			−0.349042			−0.174521	+	0.984653i	$0.555838\pi$
$752$	245.550	−	245.550i	0.326530	−	0.326530i
$753$	531.015	+	531.015i	0.705200	+	0.705200i
$754$		−	146.918i		−	0.194852i
$755$		0			0
$756$	47.9052			0.0633666
$757$	682.177	−	682.177i	0.901159	−	0.901159i	−0.0943779	−	0.995536i	$-0.530086\pi$
$757$	682.177	−	682.177i	0.901159	−	0.901159i	0.995536	+	0.0943779i	$0.0300862\pi$
$758$	−124.175	−	124.175i	−0.163819	−	0.163819i
$759$			182.904i			0.240980i
$760$		0			0
$761$	984.696			1.29395			0.646975	−	0.762511i	$-0.276034\pi$
$761$	984.696			1.29395			0.646975	+	0.762511i	$0.276034\pi$
$762$	−248.875	+	248.875i	−0.326607	+	0.326607i
$763$	−73.4635	−	73.4635i	−0.0962824	−	0.0962824i
$764$			414.319i			0.542302i
$765$		0			0
$766$	273.996			0.357697
$767$	135.003	−	135.003i	0.176014	−	0.176014i
$768$	−19.5959	−	19.5959i	−0.0255155	−	0.0255155i
$769$		−	675.108i		−	0.877904i	−0.898511	−	0.438952i	$-0.855350\pi$
$769$		−	675.108i		−	0.877904i	0.898511	−	0.438952i	$-0.144650\pi$
$770$		0			0
$771$	−526.711			−0.683153
$772$	−283.753	+	283.753i	−0.367556	+	0.367556i
$773$	−520.462	−	520.462i	−0.673302	−	0.673302i	0.285174	−	0.958476i	$-0.407949\pi$
$773$	−520.462	−	520.462i	−0.673302	−	0.673302i	−0.958476	+	0.285174i	$0.907949\pi$
$774$		−	22.5189i		−	0.0290942i
$775$		0			0
$776$	−287.685			−0.370728
$777$	−227.765	+	227.765i	−0.293133	+	0.293133i
$778$	−72.9663	−	72.9663i	−0.0937871	−	0.0937871i
$779$			99.3126i			0.127487i
$780$		0			0
$781$	−619.420			−0.793111
$782$	−56.9568	+	56.9568i	−0.0728348	+	0.0728348i
$783$	−78.1845	−	78.1845i	−0.0998525	−	0.0998525i
$784$		−	111.004i		−	0.141586i
$785$		0			0
$786$	−565.170			−0.719046
$787$	−184.553	+	184.553i	−0.234502	+	0.234502i	−0.814569	−	0.580067i	$-0.803026\pi$
$787$	−184.553	+	184.553i	−0.234502	+	0.234502i	0.580067	+	0.814569i	$0.303026\pi$
$788$	−209.903	−	209.903i	−0.266374	−	0.266374i
$789$			64.6079i			0.0818858i
$790$		0			0
$791$	194.240			0.245563
$792$	−62.2680	+	62.2680i	−0.0786212	+	0.0786212i
$793$	−285.323	−	285.323i	−0.359802	−	0.359802i
$794$		−	356.508i		−	0.449002i
$795$		0			0
$796$	518.414			0.651274
$797$	768.507	−	768.507i	0.964249	−	0.964249i	−0.0351335	−	0.999383i	$-0.511186\pi$
$797$	768.507	−	768.507i	0.964249	−	0.964249i	0.999383	+	0.0351335i	$0.0111856\pi$
$798$	30.3415	+	30.3415i	0.0380219	+	0.0380219i
$799$		−	485.951i		−	0.608199i
$800$		0			0
$801$	−265.083			−0.330940
$802$	604.681	−	604.681i	0.753967	−	0.753967i
$803$	694.033	+	694.033i	0.864300	+	0.864300i
$804$		−	214.640i		−	0.266965i
$805$		0			0
$806$	155.304			0.192685
$807$	29.7519	−	29.7519i	0.0368672	−	0.0368672i
$808$	−314.716	−	314.716i	−0.389500	−	0.389500i
$809$			202.308i			0.250072i	0.992152	+	0.125036i	$0.0399046\pi$
$809$			202.308i			0.250072i	−0.992152	+	0.125036i	$0.960095\pi$
$810$		0			0
$811$	−497.459			−0.613389			−0.306695	−	0.951808i	$-0.599223\pi$
$811$	−497.459			−0.613389			−0.306695	+	0.951808i	$0.599223\pi$
$812$	138.720	−	138.720i	0.170838	−	0.170838i
$813$	−7.24709	−	7.24709i	−0.00891401	−	0.00891401i
$814$		−	592.105i		−	0.727402i
$815$		0			0
$816$	−38.7809			−0.0475256
$817$	14.2627	−	14.2627i	0.0174574	−	0.0174574i
$818$	596.739	+	596.739i	0.729510	+	0.729510i
$819$			67.5147i			0.0824355i
$820$		0			0
$821$	450.382			0.548578			0.274289	−	0.961647i	$-0.411558\pi$
$821$	450.382			0.548578			0.274289	+	0.961647i	$0.411558\pi$
$822$	263.099	−	263.099i	0.320072	−	0.320072i
$823$	1059.14	+	1059.14i	1.28693	+	1.28693i	0.936642	+	0.350289i	$0.113917\pi$
$823$	1059.14	+	1059.14i	1.28693	+	1.28693i	0.350289	+	0.936642i	$0.386083\pi$
$824$			173.007i			0.209960i
$825$		0			0
$826$	254.939			0.308643
$827$	204.105	−	204.105i	0.246802	−	0.246802i	−0.572855	−	0.819657i	$-0.694164\pi$
$827$	204.105	−	204.105i	0.246802	−	0.246802i	0.819657	+	0.572855i	$0.194164\pi$
$828$	−43.1703	−	43.1703i	−0.0521380	−	0.0521380i
$829$			1220.16i			1.47185i	0.677065	+	0.735923i	$0.263252\pi$
$829$			1220.16i			1.47185i	−0.677065	+	0.735923i	$0.736748\pi$
$830$		0			0
$831$	54.8094			0.0659560
$832$	27.6173	−	27.6173i	0.0331939	−	0.0331939i
$833$	−109.840	−	109.840i	−0.131860	−	0.131860i
$834$			61.6953i			0.0739752i
$835$		0			0
$836$	−78.8769			−0.0943503
$837$	82.6471	−	82.6471i	0.0987421	−	0.0987421i
$838$	475.207	+	475.207i	0.567073	+	0.567073i
$839$		−	926.065i		−	1.10377i	−0.833919	−	0.551886i	$-0.813908\pi$
$839$		−	926.065i		−	1.10377i	0.833919	−	0.551886i	$-0.186092\pi$
$840$		0			0
$841$	388.199			0.461592
$842$	512.865	−	512.865i	0.609103	−	0.609103i
$843$	−32.4636	−	32.4636i	−0.0385096	−	0.0385096i
$844$		−	349.044i		−	0.413559i
$845$		0			0
$846$	368.325			0.435373
$847$	43.3421	−	43.3421i	0.0511713	−	0.0511713i
$848$	−76.7900	−	76.7900i	−0.0905542	−	0.0905542i
$849$		−	208.249i		−	0.245287i
$850$		0			0
$851$	410.505			0.482380
$852$	146.200	−	146.200i	0.171596	−	0.171596i
$853$	−546.901	−	546.901i	−0.641150	−	0.641150i	0.309689	−	0.950838i	$-0.399775\pi$
$853$	−546.901	−	546.901i	−0.641150	−	0.641150i	−0.950838	+	0.309689i	$0.899775\pi$
$854$		−	538.804i		−	0.630918i
$855$		0			0
$856$	1.64833			0.00192561
$857$	134.506	−	134.506i	0.156949	−	0.156949i	−0.624264	−	0.781213i	$-0.714601\pi$
$857$	134.506	−	134.506i	0.156949	−	0.156949i	0.781213	+	0.624264i	$0.214601\pi$
$858$	−87.7569	−	87.7569i	−0.102281	−	0.102281i
$859$		−	1268.50i		−	1.47672i	−0.674409	−	0.738358i	$-0.735602\pi$
$859$		−	1268.50i		−	1.47672i	0.674409	−	0.738358i	$-0.264398\pi$
$860$		0			0
$861$	−208.655			−0.242341
$862$	614.993	−	614.993i	0.713449	−	0.713449i
$863$	−1002.77	−	1002.77i	−1.16195	−	1.16195i	−0.984047	−	0.177907i	$-0.943067\pi$
$863$	−1002.77	−	1002.77i	−1.16195	−	1.16195i	−0.177907	−	0.984047i	$-0.556933\pi$
$864$		−	29.3939i		−	0.0340207i
$865$		0			0
$866$	−636.552			−0.735049
$867$	315.577	−	315.577i	0.363987	−	0.363987i
$868$	146.638	+	146.638i	0.168938	+	0.168938i
$869$			1245.38i			1.43311i
$870$		0			0
$871$	302.501			0.347303
$872$	−45.0761	+	45.0761i	−0.0516927	+	0.0516927i
$873$	−215.764	−	215.764i	−0.247152	−	0.247152i
$874$		−	54.6851i		−	0.0625688i
$875$		0			0
$876$	−327.621			−0.373997
$877$	392.449	−	392.449i	0.447490	−	0.447490i	−0.447029	−	0.894519i	$-0.647518\pi$
$877$	392.449	−	392.449i	0.447490	−	0.447490i	0.894519	+	0.447029i	$0.147518\pi$
$878$	22.3789	+	22.3789i	0.0254885	+	0.0254885i
$879$			253.674i			0.288594i
$880$		0			0
$881$	189.075			0.214614			0.107307	−	0.994226i	$-0.465777\pi$
$881$	189.075			0.214614			0.107307	+	0.994226i	$0.465777\pi$
$882$	83.2526	−	83.2526i	0.0943907	−	0.0943907i
$883$	947.776	+	947.776i	1.07336	+	1.07336i	0.997087	+	0.0762723i	$0.0243018\pi$
$883$	947.776	+	947.776i	1.07336	+	1.07336i	0.0762723	+	0.997087i	$0.475698\pi$
$884$		−	54.6555i		−	0.0618275i
$885$		0			0
$886$	−568.995			−0.642206
$887$	515.610	−	515.610i	0.581296	−	0.581296i	−0.353963	−	0.935259i	$-0.615166\pi$
$887$	515.610	−	515.610i	0.581296	−	0.581296i	0.935259	+	0.353963i	$0.115166\pi$
$888$	139.753	+	139.753i	0.157379	+	0.157379i
$889$		−	662.355i		−	0.745056i
$890$		0			0
$891$	−93.4020			−0.104828
$892$	96.0305	−	96.0305i	0.107657	−	0.107657i
$893$	233.285	+	233.285i	0.261237	+	0.261237i
$894$			578.337i			0.646910i
$895$		0			0
$896$	52.1525			0.0582060
$897$	60.8416	−	60.8416i	0.0678279	−	0.0678279i
$898$	389.187	+	389.187i	0.433393	+	0.433393i
$899$		−	478.646i		−	0.532421i
$900$		0			0
$901$	−151.970			−0.168668
$902$	271.214	−	271.214i	0.300681	−	0.300681i
$903$	29.9659	+	29.9659i	0.0331848	+	0.0331848i
$904$		−	119.183i		−	0.131839i
$905$		0			0
$906$	−560.319			−0.618453
$907$	1246.18	−	1246.18i	1.37396	−	1.37396i	0.519464	−	0.854492i	$-0.326132\pi$
$907$	1246.18	−	1246.18i	1.37396	−	1.37396i	0.854492	−	0.519464i	$-0.173868\pi$
$908$	31.9770	+	31.9770i	0.0352170	+	0.0352170i
$909$		−	472.074i		−	0.519334i
$910$		0			0
$911$	−1120.86			−1.23036			−0.615180	−	0.788387i	$-0.710917\pi$
$911$	−1120.86			−1.23036			−0.615180	+	0.788387i	$0.710917\pi$
$912$	18.6171	−	18.6171i	0.0204135	−	0.0204135i
$913$	−919.165	−	919.165i	−1.00675	−	1.00675i
$914$		−	567.357i		−	0.620741i
$915$		0			0
$916$	−133.130			−0.145339
$917$	752.072	−	752.072i	0.820143	−	0.820143i
$918$	−29.0857	−	29.0857i	−0.0316837	−	0.0316837i
$919$			1550.71i			1.68738i	0.536827	+	0.843692i	$0.319623\pi$
$919$			1550.71i			1.68738i	−0.536827	+	0.843692i	$0.680377\pi$
$920$		0			0
$921$	−715.508			−0.776882
$922$	−871.374	+	871.374i	−0.945091	+	0.945091i
$923$	206.045	+	206.045i	0.223234	+	0.223234i
$924$		−	165.720i		−	0.179351i
$925$		0			0
$926$	480.775			0.519196
$927$	−129.755	+	129.755i	−0.139973	+	0.139973i
$928$	−85.1165	−	85.1165i	−0.0917204	−	0.0917204i
$929$		−	956.772i		−	1.02989i	−0.857222	−	0.514947i	$-0.827812\pi$
$929$		−	956.772i		−	1.02989i	0.857222	−	0.514947i	$-0.172188\pi$
$930$		0			0
$931$	105.459			0.113275
$932$	137.594	−	137.594i	0.147633	−	0.147633i
$933$	−108.564	−	108.564i	−0.116360	−	0.116360i
$934$		−	109.428i		−	0.117160i
$935$		0			0
$936$	41.4260			0.0442585
$937$	489.671	−	489.671i	0.522595	−	0.522595i	−0.395760	−	0.918354i	$-0.629519\pi$
$937$	489.671	−	489.671i	0.522595	−	0.522595i	0.918354	+	0.395760i	$0.129519\pi$
$938$	285.621	+	285.621i	0.304500	+	0.304500i
$939$			756.777i			0.805939i
$940$		0			0
$941$	−175.441			−0.186441			−0.0932207	−	0.995645i	$-0.529716\pi$
$941$	−175.441			−0.186441			−0.0932207	+	0.995645i	$0.529716\pi$
$942$	440.883	−	440.883i	0.468029	−	0.468029i
$943$	188.032	+	188.032i	0.199398	+	0.199398i
$944$		−	156.427i		−	0.165706i
$945$		0			0
$946$	−77.9004			−0.0823471
$947$	−62.2446	+	62.2446i	−0.0657282	+	0.0657282i	−0.739207	−	0.673479i	$-0.764799\pi$
$947$	−62.2446	+	62.2446i	−0.0657282	+	0.0657282i	0.673479	+	0.739207i	$0.264799\pi$
$948$	−293.942	−	293.942i	−0.310066	−	0.310066i
$949$		−	461.730i		−	0.486543i
$950$		0			0
$951$	154.393			0.162348
$952$	51.6057	−	51.6057i	0.0542077	−	0.0542077i
$953$	296.351	+	296.351i	0.310966	+	0.310966i	0.845284	−	0.534318i	$-0.179431\pi$
$953$	296.351	+	296.351i	0.310966	+	0.310966i	−0.534318	+	0.845284i	$0.679431\pi$
$954$		−	115.185i		−	0.120739i
$955$		0			0
$956$	793.149			0.829654
$957$	−270.466	+	270.466i	−0.282619	+	0.282619i
$958$	125.855	+	125.855i	0.131373	+	0.131373i
$959$			700.211i			0.730147i
$960$		0			0
$961$	−455.034			−0.473500
$962$	−196.959	+	196.959i	−0.204740	+	0.204740i
$963$	1.23624	+	1.23624i	0.00128374	+	0.00128374i
$964$		−	719.623i		−	0.746497i
$965$		0			0
$966$	114.893			0.118937
$967$	−926.473	+	926.473i	−0.958090	+	0.958090i	−0.999156	−	0.0410668i	$-0.986924\pi$
$967$	−926.473	+	926.473i	−0.958090	+	0.958090i	0.0410668	+	0.999156i	$0.486924\pi$
$968$	−26.5941	−	26.5941i	−0.0274732	−	0.0274732i
$969$		−	36.8437i		−	0.0380224i
$970$		0			0
$971$	130.623			0.134524			0.0672622	−	0.997735i	$-0.478574\pi$
$971$	130.623			0.134524			0.0672622	+	0.997735i	$0.478574\pi$
$972$	22.0454	−	22.0454i	0.0226805	−	0.0226805i
$973$	−82.0979	−	82.0979i	−0.0843760	−	0.0843760i
$974$		−	666.272i		−	0.684058i
$975$		0			0
$976$	−330.602			−0.338731
$977$	9.53233	−	9.53233i	0.00975673	−	0.00975673i	−0.702212	−	0.711968i	$-0.747804\pi$
$977$	9.53233	−	9.53233i	0.00975673	−	0.00975673i	0.711968	+	0.702212i	$0.247804\pi$
$978$	285.808	+	285.808i	0.292237	+	0.292237i
$979$			917.009i			0.936680i
$980$		0			0
$981$	−67.6141			−0.0689237
$982$	89.4786	−	89.4786i	0.0911187	−	0.0911187i
$983$	−959.887	−	959.887i	−0.976487	−	0.976487i	0.0232424	−	0.999730i	$-0.492601\pi$
$983$	−959.887	−	959.887i	−0.976487	−	0.976487i	−0.999730	+	0.0232424i	$0.992601\pi$
$984$			128.028i			0.130109i
$985$		0			0
$986$	−168.448			−0.170840
$987$	−490.130	+	490.130i	−0.496586	+	0.496586i
$988$	26.2378	+	26.2378i	0.0265565	+	0.0265565i
$989$		−	54.0081i		−	0.0546088i
$990$		0			0
$991$	−768.037			−0.775012			−0.387506	−	0.921867i	$-0.626663\pi$
$991$	−768.037			−0.775012			−0.387506	+	0.921867i	$0.626663\pi$
$992$	89.9748	−	89.9748i	0.0907004	−	0.0907004i
$993$	279.748	+	279.748i	0.281720	+	0.281720i
$994$			389.096i			0.391444i
$995$		0			0
$996$	433.896			0.435638
$997$	−977.735	+	977.735i	−0.980677	+	0.980677i	−0.999817	−	0.0191398i	$-0.993907\pi$
$997$	−977.735	+	977.735i	−0.980677	+	0.980677i	0.0191398	+	0.999817i	$0.493907\pi$
$998$	−122.886	−	122.886i	−0.123132	−	0.123132i
$999$			209.629i			0.209839i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	750.3.f.b.193.1	✓	16
5.2	odd	4	inner	750.3.f.b.307.1	yes	16
5.3	odd	4		750.3.f.c.307.8	yes	16
5.4	even	2		750.3.f.c.193.8	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
750.3.f.b.193.1	✓	16	1.1	even	1	trivial
750.3.f.b.307.1	yes	16	5.2	odd	4	inner
750.3.f.c.193.8	yes	16	5.4	even	2
750.3.f.c.307.8	yes	16	5.3	odd	4

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:	\( N \)	\(=\)	\( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	750.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.00000i) q^{2} +(-1.22474 - 1.22474i) q^{3} -2.00000i q^{4} +2.44949 q^{6} +(-3.25953 + 3.25953i) q^{7} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} +10.3780 q^{11} +(-2.44949 + 2.44949i) q^{12} +(-3.45217 - 3.45217i) q^{13} -6.51907i q^{14} -4.00000 q^{16} +(-3.95806 + 3.95806i) q^{17} +(-3.00000 - 3.00000i) q^{18} -3.80019i q^{19} +7.98420 q^{21} +(-10.3780 + 10.3780i) q^{22} +(-7.19505 - 7.19505i) q^{23} -4.89898i q^{24} +6.90433 q^{26} +(3.67423 - 3.67423i) q^{27} +(6.51907 + 6.51907i) q^{28} -21.2791i q^{29} +22.4937 q^{31} +(4.00000 - 4.00000i) q^{32} +(-12.7104 - 12.7104i) q^{33} -7.91612i q^{34} +6.00000 q^{36} +(-28.5269 + 28.5269i) q^{37} +(3.80019 + 3.80019i) q^{38} +8.45604i q^{39} -26.1336 q^{41} +(-7.98420 + 7.98420i) q^{42} +(3.75315 + 3.75315i) q^{43} -20.7560i q^{44} +14.3901 q^{46} +(-61.3876 + 61.3876i) q^{47} +(4.89898 + 4.89898i) q^{48} +27.7509i q^{49} +9.69522 q^{51} +(-6.90433 + 6.90433i) q^{52} +(19.1975 + 19.1975i) q^{53} +7.34847i q^{54} -13.0381 q^{56} +(-4.65427 + 4.65427i) q^{57} +(21.2791 + 21.2791i) q^{58} +39.1066i q^{59} +82.6505 q^{61} +(-22.4937 + 22.4937i) q^{62} +(-9.77860 - 9.77860i) q^{63} +8.00000i q^{64} +25.4208 q^{66} +(-43.8132 + 43.8132i) q^{67} +(7.91612 + 7.91612i) q^{68} +17.6242i q^{69} -59.6858 q^{71} +(-6.00000 + 6.00000i) q^{72} +(66.8754 + 66.8754i) q^{73} -57.0539i q^{74} -7.60039 q^{76} +(-33.8275 + 33.8275i) q^{77} +(-8.45604 - 8.45604i) q^{78} +120.001i q^{79} -9.00000 q^{81} +(26.1336 - 26.1336i) q^{82} +(-88.5686 - 88.5686i) q^{83} -15.9684i q^{84} -7.50629 q^{86} +(-26.0615 + 26.0615i) q^{87} +(20.7560 + 20.7560i) q^{88} +88.3609i q^{89} +22.5049 q^{91} +(-14.3901 + 14.3901i) q^{92} +(-27.5490 - 27.5490i) q^{93} -122.775i q^{94} -9.79796 q^{96} +(-71.9212 + 71.9212i) q^{97} +(-27.7509 - 27.7509i) q^{98} +31.1340i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{2} + 24 q^{7} + 32 q^{8} - 24 q^{11} - 48 q^{13} - 64 q^{16} - 16 q^{17} - 48 q^{18} - 48 q^{21} + 24 q^{22} + 104 q^{23} + 96 q^{26} - 48 q^{28} + 200 q^{31} + 64 q^{32} - 48 q^{33} + 96 q^{36}+ \cdots - 224 q^{98}+O(q^{100}) \) 16 * q - 16 * q^2 + 24 * q^7 + 32 * q^8 - 24 * q^11 - 48 * q^13 - 64 * q^16 - 16 * q^17 - 48 * q^18 - 48 * q^21 + 24 * q^22 + 104 * q^23 + 96 * q^26 - 48 * q^28 + 200 * q^31 + 64 * q^32 - 48 * q^33 + 96 * q^36 - 128 * q^37 + 56 * q^38 - 192 * q^41 + 48 * q^42 + 88 * q^43 - 208 * q^46 + 16 * q^47 - 48 * q^51 - 96 * q^52 - 160 * q^53 + 96 * q^56 + 48 * q^57 + 24 * q^58 - 24 * q^61 - 200 * q^62 + 72 * q^63 + 96 * q^66 + 264 * q^67 + 32 * q^68 + 176 * q^71 - 96 * q^72 + 32 * q^73 - 112 * q^76 + 32 * q^77 + 96 * q^78 - 144 * q^81 + 192 * q^82 - 288 * q^83 - 176 * q^86 - 168 * q^87 - 48 * q^88 + 32 * q^91 + 208 * q^92 + 144 * q^93 - 48 * q^97 - 224 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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