LMFDB - Embedded newform 45.4.f.a.17.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,4,Mod(8,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3]))

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

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

chi := DirichletCharacter("45.8");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	45.f (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$2.65508595026$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 16x^{10} - 14x^{8} - 512x^{6} + 3889x^{4} + 126224x^{2} + 506944 $ x^12 - 16x^10 - 14x^8 - 512x^6 + 3889x^4 + 126224*x^2 + 506944
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			17.3
Root			$2.02004 - 2.30794i$ of defining polynomial
Character	$\chi$	$=$	45.17
Dual form			45.4.f.a.8.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.287902 + 0.287902i) q^{2} +7.83422i q^{4} +(-9.93887 + 5.12043i) q^{5} +(15.0557 + 15.0557i) q^{7} +(-4.55871 - 4.55871i) q^{8} +O(q^{10})$	\(q+(-0.287902 + 0.287902i) q^{2} +7.83422i q^{4} +(-9.93887 + 5.12043i) q^{5} +(15.0557 + 15.0557i) q^{7} +(-4.55871 - 4.55871i) q^{8} +(1.38724 - 4.33560i) q^{10} +27.9992i q^{11} +(2.49862 - 2.49862i) q^{13} -8.66913 q^{14} -60.0489 q^{16} +(67.9104 - 67.9104i) q^{17} -95.2200i q^{19} +(-40.1146 - 77.8634i) q^{20} +(-8.06104 - 8.06104i) q^{22} +(121.994 + 121.994i) q^{23} +(72.5624 - 101.783i) q^{25} +1.43872i q^{26} +(-117.950 + 117.950i) q^{28} -99.0852 q^{29} -28.7800 q^{31} +(53.7578 - 53.7578i) q^{32} +39.1031i q^{34} +(-226.728 - 72.5450i) q^{35} +(271.064 + 271.064i) q^{37} +(27.4140 + 27.4140i) q^{38} +(68.6509 + 21.9659i) q^{40} +453.748i q^{41} +(30.5679 - 30.5679i) q^{43} -219.352 q^{44} -70.2445 q^{46} +(254.600 - 254.600i) q^{47} +110.348i q^{49} +(8.41255 + 50.1943i) q^{50} +(19.5748 + 19.5748i) q^{52} +(-224.021 - 224.021i) q^{53} +(-143.368 - 278.281i) q^{55} -137.269i q^{56} +(28.5268 - 28.5268i) q^{58} +483.263 q^{59} -264.372 q^{61} +(8.28582 - 8.28582i) q^{62} -449.437i q^{64} +(-12.0395 + 37.6275i) q^{65} +(-498.817 - 498.817i) q^{67} +(532.025 + 532.025i) q^{68} +(86.1614 - 44.3897i) q^{70} -609.904i q^{71} +(-74.6843 + 74.6843i) q^{73} -156.080 q^{74} +745.975 q^{76} +(-421.548 + 421.548i) q^{77} +406.574i q^{79} +(596.818 - 307.476i) q^{80} +(-130.635 - 130.635i) q^{82} +(-652.278 - 652.278i) q^{83} +(-327.223 + 1022.68i) q^{85} +17.6011i q^{86} +(127.640 - 127.640i) q^{88} -139.860 q^{89} +75.2369 q^{91} +(-955.726 + 955.726i) q^{92} +146.600i q^{94} +(487.567 + 946.380i) q^{95} +(557.633 + 557.633i) q^{97} +(-31.7693 - 31.7693i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 24 q^{7}+O(q^{10}) $ 12 * q + 24 * q^7	$ 12 q + 24 q^{7} + 192 q^{10} - 108 q^{13} - 648 q^{16} + 1056 q^{22} - 144 q^{25} - 576 q^{28} - 1248 q^{31} + 828 q^{37} + 2568 q^{40} - 96 q^{43} + 672 q^{46} - 312 q^{52} - 1512 q^{55} - 3864 q^{58} + 96 q^{61} + 1632 q^{67} - 1536 q^{70} + 3972 q^{73} - 480 q^{76} - 7848 q^{82} - 1752 q^{85} + 7968 q^{88} + 4752 q^{91} + 2772 q^{97}+O(q^{100}) $ 12 * q + 24 * q^7 + 192 * q^10 - 108 * q^13 - 648 * q^16 + 1056 * q^22 - 144 * q^25 - 576 * q^28 - 1248 * q^31 + 828 * q^37 + 2568 * q^40 - 96 * q^43 + 672 * q^46 - 312 * q^52 - 1512 * q^55 - 3864 * q^58 + 96 * q^61 + 1632 * q^67 - 1536 * q^70 + 3972 * q^73 - 480 * q^76 - 7848 * q^82 - 1752 * q^85 + 7968 * q^88 + 4752 * q^91 + 2772 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$

Coefficient data

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

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

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

$2$	−0.287902	+	0.287902i	−0.101789	+	0.101789i	−0.756167	−	0.654378i	$-0.772930\pi$
$2$	−0.287902	+	0.287902i	−0.101789	+	0.101789i	0.654378	+	0.756167i	$0.272930\pi$
$3$		0			0
$3$		0			0
$4$			7.83422i			0.979278i
$5$	−9.93887	+	5.12043i	−0.888960	+	0.457985i
$5$	−9.93887	+	5.12043i	−0.888960	+	0.457985i
$6$		0			0
$7$	15.0557	+	15.0557i	0.812931	+	0.812931i	0.985072	−	0.172141i	$-0.0550686\pi$
$7$	15.0557	+	15.0557i	0.812931	+	0.812931i	−0.172141	+	0.985072i	$0.555069\pi$
$8$	−4.55871	−	4.55871i	−0.201468	−	0.201468i
$9$		0			0
$10$	1.38724	−	4.33560i	0.0438684	−	0.137104i
$11$			27.9992i			0.767462i	0.923445	+	0.383731i	$0.125361\pi$
$11$			27.9992i			0.767462i	−0.923445	+	0.383731i	$0.874639\pi$
$12$		0			0
$13$	2.49862	−	2.49862i	0.0533071	−	0.0533071i	−0.679951	−	0.733258i	$-0.737999\pi$
$13$	2.49862	−	2.49862i	0.0533071	−	0.0533071i	0.733258	+	0.679951i	$0.237999\pi$
$14$	−8.66913			−0.165494
$15$		0			0
$16$	−60.0489			−0.938264
$17$	67.9104	−	67.9104i	0.968864	−	0.968864i	−0.0306653	−	0.999530i	$-0.509763\pi$
$17$	67.9104	−	67.9104i	0.968864	−	0.968864i	0.999530	+	0.0306653i	$0.00976260\pi$
$18$		0			0
$19$		−	95.2200i		−	1.14974i	−0.818246	−	0.574868i	$-0.805054\pi$
$19$		−	95.2200i		−	1.14974i	0.818246	−	0.574868i	$-0.194946\pi$
$20$	−40.1146	−	77.8634i	−0.448495	−	0.870539i
$21$		0			0
$22$	−8.06104	−	8.06104i	−0.0781191	−	0.0781191i
$23$	121.994	+	121.994i	1.10598	+	1.10598i	0.993674	+	0.112302i	$0.0358224\pi$
$23$	121.994	+	121.994i	1.10598	+	1.10598i	0.112302	+	0.993674i	$0.464178\pi$
$24$		0			0
$25$	72.5624	−	101.783i	0.580499	−	0.814261i
$26$			1.43872i			0.0108521i
$27$		0			0
$28$	−117.950	+	117.950i	−0.796085	+	0.796085i
$29$	−99.0852			−0.634471			−0.317235	−	0.948347i	$-0.602755\pi$
$29$	−99.0852			−0.634471			−0.317235	+	0.948347i	$0.602755\pi$
$30$		0			0
$31$	−28.7800			−0.166743			−0.0833716	−	0.996519i	$-0.526569\pi$
$31$	−28.7800			−0.166743			−0.0833716	+	0.996519i	$0.526569\pi$
$32$	53.7578	−	53.7578i	0.296973	−	0.296973i
$33$		0			0
$34$			39.1031i			0.197239i
$35$	−226.728	−	72.5450i	−1.09497	−	0.350353i
$36$		0			0
$37$	271.064	+	271.064i	1.20439	+	1.20439i	0.972816	+	0.231578i	$0.0743888\pi$
$37$	271.064	+	271.064i	1.20439	+	1.20439i	0.231578	+	0.972816i	$0.425611\pi$
$38$	27.4140	+	27.4140i	0.117030	+	0.117030i
$39$		0			0
$40$	68.6509	+	21.9659i	0.271367	+	0.0868277i
$41$			453.748i			1.72838i	0.503166	+	0.864190i	$0.332168\pi$
$41$			453.748i			1.72838i	−0.503166	+	0.864190i	$0.667832\pi$
$42$		0			0
$43$	30.5679	−	30.5679i	0.108409	−	0.108409i	−0.650822	−	0.759230i	$-0.725576\pi$
$43$	30.5679	−	30.5679i	0.108409	−	0.108409i	0.759230	+	0.650822i	$0.225576\pi$
$44$	−219.352			−0.751559
$45$		0			0
$46$	−70.2445			−0.225152
$47$	254.600	−	254.600i	0.790153	−	0.790153i	−0.191365	−	0.981519i	$-0.561292\pi$
$47$	254.600	−	254.600i	0.790153	−	0.790153i	0.981519	+	0.191365i	$0.0612915\pi$
$48$		0			0
$49$			110.348i			0.321713i
$50$	8.41255	+	50.1943i	0.0237943	+	0.141971i
$51$		0			0
$52$	19.5748	+	19.5748i	0.0522025	+	0.0522025i
$53$	−224.021	−	224.021i	−0.580598	−	0.580598i	0.354470	−	0.935068i	$-0.384661\pi$
$53$	−224.021	−	224.021i	−0.580598	−	0.580598i	−0.935068	+	0.354470i	$0.884661\pi$
$54$		0			0
$55$	−143.368	−	278.281i	−0.351486	−	0.682243i
$56$		−	137.269i		−	0.327560i
$57$		0			0
$58$	28.5268	−	28.5268i	0.0645820	−	0.0645820i
$59$	483.263			1.06636			0.533182	−	0.846001i	$-0.320996\pi$
$59$	483.263			1.06636			0.533182	+	0.846001i	$0.320996\pi$
$60$		0			0
$61$	−264.372			−0.554908			−0.277454	−	0.960739i	$-0.589491\pi$
$61$	−264.372			−0.554908			−0.277454	+	0.960739i	$0.589491\pi$
$62$	8.28582	−	8.28582i	0.0169726	−	0.0169726i
$63$		0			0
$64$		−	449.437i		−	0.877807i
$65$	−12.0395	+	37.6275i	−0.0229740	+	0.0718018i
$66$		0			0
$67$	−498.817	−	498.817i	−0.909556	−	0.909556i	0.0866805	−	0.996236i	$-0.472374\pi$
$67$	−498.817	−	498.817i	−0.909556	−	0.909556i	−0.996236	+	0.0866805i	$0.972374\pi$
$68$	532.025	+	532.025i	0.948788	+	0.948788i
$69$		0			0
$70$	86.1614	−	44.3897i	0.147118	−	0.0757940i
$71$		−	609.904i		−	1.01947i	−0.860332	−	0.509735i	$-0.829744\pi$
$71$		−	609.904i		−	1.01947i	0.860332	−	0.509735i	$-0.170256\pi$
$72$		0			0
$73$	−74.6843	+	74.6843i	−0.119742	+	0.119742i	−0.764438	−	0.644697i	$-0.776984\pi$
$73$	−74.6843	+	74.6843i	−0.119742	+	0.119742i	0.644697	+	0.764438i	$0.276984\pi$
$74$	−156.080			−0.245188
$75$		0			0
$76$	745.975			1.12591
$77$	−421.548	+	421.548i	−0.623894	+	0.623894i
$78$		0			0
$79$			406.574i			0.579027i	0.957174	+	0.289513i	$0.0934935\pi$
$79$			406.574i			0.579027i	−0.957174	+	0.289513i	$0.906507\pi$
$80$	596.818	−	307.476i	0.834079	−	0.429711i
$81$		0			0
$82$	−130.635	−	130.635i	−0.175930	−	0.175930i
$83$	−652.278	−	652.278i	−0.862613	−	0.862613i	0.129028	−	0.991641i	$-0.458814\pi$
$83$	−652.278	−	652.278i	−0.862613	−	0.862613i	−0.991641	+	0.129028i	$0.958814\pi$
$84$		0			0
$85$	−327.223	+	1022.68i	−0.417556	+	1.30501i
$86$			17.6011i			0.0220695i
$87$		0			0
$88$	127.640	−	127.640i	0.154619	−	0.154619i
$89$	−139.860			−0.166575			−0.0832873	−	0.996526i	$-0.526542\pi$
$89$	−139.860			−0.166575			−0.0832873	+	0.996526i	$0.526542\pi$
$90$		0			0
$91$	75.2369			0.0866700
$92$	−955.726	+	955.726i	−1.08306	+	1.08306i
$93$		0			0
$94$			146.600i			0.160857i
$95$	487.567	+	946.380i	0.526561	+	1.02207i
$96$		0			0
$97$	557.633	+	557.633i	0.583701	+	0.583701i	0.935918	−	0.352217i	$-0.114572\pi$
$97$	557.633	+	557.633i	0.583701	+	0.583701i	−0.352217	+	0.935918i	$0.614572\pi$
$98$	−31.7693	−	31.7693i	−0.0327468	−	0.0327468i
$99$		0			0
$100$	797.388	+	568.470i	0.797388	+	0.568470i
$101$		−	299.833i		−	0.295391i	−0.989033	−	0.147695i	$-0.952814\pi$
$101$		−	299.833i		−	0.295391i	0.989033	−	0.147695i	$-0.0471855\pi$
$102$		0			0
$103$	577.974	−	577.974i	0.552907	−	0.552907i	−0.374372	−	0.927279i	$-0.622142\pi$
$103$	577.974	−	577.974i	0.552907	−	0.552907i	0.927279	+	0.374372i	$0.122142\pi$
$104$	−22.7810			−0.0214794
$105$		0			0
$106$	128.992			0.118197
$107$	−216.994	+	216.994i	−0.196053	+	0.196053i	−0.798305	−	0.602253i	$-0.794270\pi$
$107$	−216.994	+	216.994i	−0.196053	+	0.196053i	0.602253	+	0.798305i	$0.294270\pi$
$108$		0			0
$109$			936.415i			0.822865i	0.911440	+	0.411432i	$0.134971\pi$
$109$			936.415i			0.822865i	−0.911440	+	0.411432i	$0.865029\pi$
$110$	121.394	+	38.8417i	0.105222	+	0.0336673i
$111$		0			0
$112$	−904.077	−	904.077i	−0.762743	−	0.762743i
$113$	−1084.00	−	1084.00i	−0.902428	−	0.902428i	0.0932173	−	0.995646i	$-0.470285\pi$
$113$	−1084.00	−	1084.00i	−0.902428	−	0.902428i	−0.995646	+	0.0932173i	$0.970285\pi$
$114$		0			0
$115$	−1837.14	−	587.820i	−1.48969	−	0.476648i
$116$		−	776.256i		−	0.621323i
$117$		0			0
$118$	−139.132	+	139.132i	−0.108544	+	0.108544i
$119$	2044.88			1.57524
$120$		0			0
$121$	547.043			0.411001
$122$	76.1133	−	76.1133i	0.0564834	−	0.0564834i
$123$		0			0
$124$		−	225.469i		−	0.163288i
$125$	−200.018	+	1383.15i	−0.143121	+	0.989705i
$126$		0			0
$127$	−915.574	−	915.574i	−0.639717	−	0.639717i	0.310769	−	0.950486i	$-0.399414\pi$
$127$	−915.574	−	915.574i	−0.639717	−	0.639717i	−0.950486	+	0.310769i	$0.899414\pi$
$128$	559.457	+	559.457i	0.386324	+	0.386324i
$129$		0			0
$130$	−7.36684	−	14.2992i	−0.00497011	−	0.00964711i
$131$		−	629.329i		−	0.419731i	−0.977730	−	0.209865i	$-0.932697\pi$
$131$		−	629.329i		−	0.419731i	0.977730	−	0.209865i	$-0.0673026\pi$
$132$		0			0
$133$	1433.60	−	1433.60i	0.934655	−	0.934655i
$134$	287.221			0.185165
$135$		0			0
$136$	−619.167			−0.390391
$137$	660.931	−	660.931i	0.412169	−	0.412169i	−0.470325	−	0.882494i	$-0.655863\pi$
$137$	660.931	−	660.931i	0.412169	−	0.412169i	0.882494	+	0.470325i	$0.155863\pi$
$138$		0			0
$139$		−	1378.35i		−	0.841081i	−0.907274	−	0.420540i	$-0.861840\pi$
$139$		−	1378.35i		−	0.841081i	0.907274	−	0.420540i	$-0.138160\pi$
$140$	568.334	−	1776.24i	0.343093	−	1.07228i
$141$		0			0
$142$	175.593	+	175.593i	0.103771	+	0.103771i
$143$	69.9595	+	69.9595i	0.0409112	+	0.0409112i
$144$		0			0
$145$	984.795	−	507.359i	0.564019	−	0.290578i
$146$		−	43.0035i		−	0.0243767i
$147$		0			0
$148$	−2123.57	+	2123.57i	−1.17944	+	1.17944i
$149$	−924.719			−0.508429			−0.254215	−	0.967148i	$-0.581817\pi$
$149$	−924.719			−0.508429			−0.254215	+	0.967148i	$0.581817\pi$
$150$		0			0
$151$	−401.768			−0.216526			−0.108263	−	0.994122i	$-0.534529\pi$
$151$	−401.768			−0.216526			−0.108263	+	0.994122i	$0.534529\pi$
$152$	−434.080	+	434.080i	−0.231635	+	0.231635i
$153$		0			0
$154$		−	242.729i		−	0.127011i
$155$	286.041	−	147.366i	0.148228	−	0.0763659i
$156$		0			0
$157$	1305.55	+	1305.55i	0.663659	+	0.663659i	0.956240	−	0.292582i	$-0.0945144\pi$
$157$	1305.55	+	1305.55i	0.663659	+	0.663659i	−0.292582	+	0.956240i	$0.594514\pi$
$158$	−117.053	−	117.053i	−0.0589384	−	0.0589384i
$159$		0			0
$160$	−259.029	+	809.556i	−0.127988	+	0.400006i
$161$			3673.40i			1.79816i
$162$		0			0
$163$	−78.8393	+	78.8393i	−0.0378845	+	0.0378845i	−0.725795	−	0.687911i	$-0.758528\pi$
$163$	−78.8393	+	78.8393i	−0.0378845	+	0.0378845i	0.687911	+	0.725795i	$0.258528\pi$
$164$	−3554.77			−1.69256
$165$		0			0
$166$	375.585			0.175609
$167$	428.116	−	428.116i	0.198375	−	0.198375i	−0.600928	−	0.799303i	$-0.705202\pi$
$167$	428.116	−	428.116i	0.198375	−	0.198375i	0.799303	+	0.600928i	$0.205202\pi$
$168$		0			0
$169$			2184.51i			0.994317i
$170$	−200.225	−	388.641i	−0.0903325	−	0.175338i
$171$		0			0
$172$	239.476	+	239.476i	0.106162	+	0.106162i
$173$	1390.57	+	1390.57i	0.611114	+	0.611114i	0.943236	−	0.332122i	$-0.107765\pi$
$173$	1390.57	+	1390.57i	0.611114	+	0.611114i	−0.332122	+	0.943236i	$0.607765\pi$
$174$		0			0
$175$	2624.88	−	439.930i	1.13384	−	0.190032i
$176$		−	1681.32i		−	0.720082i
$177$		0			0
$178$	40.2660	−	40.2660i	0.0169554	−	0.0169554i
$179$	110.674			0.0462131			0.0231065	−	0.999733i	$-0.492644\pi$
$179$	110.674			0.0462131			0.0231065	+	0.999733i	$0.492644\pi$
$180$		0			0
$181$	700.593			0.287706			0.143853	−	0.989599i	$-0.454051\pi$
$181$	700.593			0.287706			0.143853	+	0.989599i	$0.454051\pi$
$182$	−21.6609	+	21.6609i	−0.00882203	+	0.00882203i
$183$		0			0
$184$		−	1112.27i		−	0.445638i
$185$	−4082.03	−	1306.11i	−1.62225	−	0.519064i
$186$		0			0
$187$	1901.44	+	1901.44i	0.743567	+	0.743567i
$188$	1994.59	+	1994.59i	0.773780	+	0.773780i
$189$		0			0
$190$	−412.836	−	132.093i	−0.157633	−	0.0504370i
$191$			2214.32i			0.838862i	0.907787	+	0.419431i	$0.137770\pi$
$191$			2214.32i			0.838862i	−0.907787	+	0.419431i	$0.862230\pi$
$192$		0			0
$193$	−1964.16	+	1964.16i	−0.732556	+	0.732556i	−0.971125	−	0.238569i	$-0.923322\pi$
$193$	−1964.16	+	1964.16i	−0.732556	+	0.732556i	0.238569	+	0.971125i	$0.423322\pi$
$194$	−321.087			−0.118828
$195$		0			0
$196$	−864.488			−0.315047
$197$	2700.70	−	2700.70i	0.976735	−	0.976735i	−0.0230004	−	0.999735i	$-0.507322\pi$
$197$	2700.70	−	2700.70i	0.976735	−	0.976735i	0.999735	+	0.0230004i	$0.00732189\pi$
$198$		0			0
$199$		−	841.195i		−	0.299652i	−0.988712	−	0.149826i	$-0.952129\pi$
$199$		−	841.195i		−	0.299652i	0.988712	−	0.149826i	$-0.0478713\pi$
$200$	−794.788	+	133.206i	−0.281000	+	0.0470955i
$201$		0			0
$202$	86.3224	+	86.3224i	0.0300674	+	0.0300674i
$203$	−1491.80	−	1491.80i	−0.515781	−	0.515781i
$204$		0			0
$205$	−2323.39	−	4509.75i	−0.791572	−	1.53646i
$206$			332.800i			0.112559i
$207$		0			0
$208$	−150.039	+	150.039i	−0.0500161	+	0.0500161i
$209$	2666.09			0.882379
$210$		0			0
$211$	−3970.06			−1.29531			−0.647654	−	0.761934i	$-0.724250\pi$
$211$	−3970.06			−1.29531			−0.647654	+	0.761934i	$0.724250\pi$
$212$	1755.03	−	1755.03i	0.568567	−	0.568567i
$213$		0			0
$214$		−	124.946i		−	0.0399119i
$215$	−147.290	+	460.332i	−0.0467213	+	0.146020i
$216$		0			0
$217$	−433.303	−	433.303i	−0.135551	−	0.135551i
$218$	−269.596	−	269.596i	−0.0837584	−	0.0837584i
$219$		0			0
$220$	2180.11	−	1123.18i	0.668106	−	0.344203i
$221$		−	339.365i		−	0.103295i
$222$		0			0
$223$	3600.34	−	3600.34i	1.08115	−	1.08115i	0.0847496	−	0.996402i	$-0.472991\pi$
$223$	3600.34	−	3600.34i	1.08115	−	1.08115i	0.996402	−	0.0847496i	$-0.0270090\pi$
$224$	1618.72			0.482837
$225$		0			0
$226$	624.173			0.183714
$227$	1734.81	−	1734.81i	0.507239	−	0.507239i	−0.406439	−	0.913678i	$-0.633230\pi$
$227$	1734.81	−	1734.81i	0.507239	−	0.507239i	0.913678	+	0.406439i	$0.133230\pi$
$228$		0			0
$229$		−	3467.74i		−	1.00067i	−0.865831	−	0.500337i	$-0.833209\pi$
$229$		−	3467.74i		−	1.00067i	0.865831	−	0.500337i	$-0.166791\pi$
$230$	698.151	−	359.682i	0.200151	−	0.103116i
$231$		0			0
$232$	451.700	+	451.700i	0.127826	+	0.127826i
$233$	1070.17	+	1070.17i	0.300899	+	0.300899i	0.841365	−	0.540467i	$-0.181752\pi$
$233$	1070.17	+	1070.17i	0.300899	+	0.300899i	−0.540467	+	0.841365i	$0.681752\pi$
$234$		0			0
$235$	−1226.78	+	3834.10i	−0.340536	+	1.06429i
$236$			3785.99i			1.04427i
$237$		0			0
$238$	−588.724	+	588.724i	−0.160342	+	0.160342i
$239$	−3909.44			−1.05808			−0.529038	−	0.848598i	$-0.677447\pi$
$239$	−3909.44			−1.05808			−0.529038	+	0.848598i	$0.677447\pi$
$240$		0			0
$241$	−4787.09			−1.27952			−0.639759	−	0.768576i	$-0.720966\pi$
$241$	−4787.09			−1.27952			−0.639759	+	0.768576i	$0.720966\pi$
$242$	−157.495	+	157.495i	−0.0418353	+	0.0418353i
$243$		0			0
$244$		−	2071.15i		−	0.543409i
$245$	−565.027	−	1096.73i	−0.147340	−	0.285990i
$246$		0			0
$247$	−237.919	−	237.919i	−0.0612891	−	0.0612891i
$248$	131.200	+	131.200i	0.0335935	+	0.0335935i
$249$		0			0
$250$	−340.628	−	455.799i	−0.0861727	−	0.115309i
$251$		−	2337.42i		−	0.587794i	−0.955837	−	0.293897i	$-0.905048\pi$
$251$		−	2337.42i		−	0.587794i	0.955837	−	0.293897i	$-0.0949523\pi$
$252$		0			0
$253$	−3415.73	+	3415.73i	−0.848795	+	0.848795i
$254$	527.191			0.130232
$255$		0			0
$256$	3273.36			0.799160
$257$	−977.764	+	977.764i	−0.237320	+	0.237320i	−0.815740	−	0.578419i	$-0.803670\pi$
$257$	−977.764	+	977.764i	−0.237320	+	0.237320i	0.578419	+	0.815740i	$0.303670\pi$
$258$		0			0
$259$			8162.10i			1.95818i
$260$	−294.782	−	94.3199i	−0.0703139	−	0.0224980i
$261$		0			0
$262$	181.185	+	181.185i	0.0427239	+	0.0427239i
$263$	−790.690	−	790.690i	−0.185384	−	0.185384i	0.608313	−	0.793697i	$-0.291847\pi$
$263$	−790.690	−	790.690i	−0.185384	−	0.185384i	−0.793697	+	0.608313i	$0.791847\pi$
$264$		0			0
$265$	3373.60	+	1079.43i	0.782034	+	0.250223i
$266$			825.474i			0.190275i
$267$		0			0
$268$	3907.85	−	3907.85i	0.890708	−	0.890708i
$269$	−3954.49			−0.896317			−0.448159	−	0.893954i	$-0.647920\pi$
$269$	−3954.49			−0.896317			−0.448159	+	0.893954i	$0.647920\pi$
$270$		0			0
$271$	7863.30			1.76259			0.881295	−	0.472567i	$-0.156673\pi$
$271$	7863.30			1.76259			0.881295	+	0.472567i	$0.156673\pi$
$272$	−4077.94	+	4077.94i	−0.909050	+	0.909050i
$273$		0			0
$274$			380.567i			0.0839083i
$275$	2849.83	+	2031.69i	0.624915	+	0.445511i
$276$		0			0
$277$	908.009	+	908.009i	0.196957	+	0.196957i	0.798694	−	0.601737i	$-0.205525\pi$
$277$	908.009	+	908.009i	0.196957	+	0.196957i	−0.601737	+	0.798694i	$0.705525\pi$
$278$	396.830	+	396.830i	0.0856126	+	0.0856126i
$279$		0			0
$280$	702.876	+	1364.30i	0.150017	+	0.291187i
$281$		−	7102.70i		−	1.50787i	−0.656949	−	0.753935i	$-0.728153\pi$
$281$		−	7102.70i		−	1.50787i	0.656949	−	0.753935i	$-0.271847\pi$
$282$		0			0
$283$	−4721.93	+	4721.93i	−0.991837	+	0.991837i	−0.999967	−	0.00813014i	$-0.997412\pi$
$283$	−4721.93	+	4721.93i	−0.991837	+	0.991837i	0.00813014	+	0.999967i	$0.497412\pi$
$284$	4778.13			0.998344
$285$		0			0
$286$	−40.2829			−0.00832860
$287$	−6831.49	+	6831.49i	−1.40505	+	1.40505i
$288$		0			0
$289$		−	4310.65i		−	0.877396i
$290$	−137.455	+	429.594i	−0.0278332	+	0.0869884i
$291$		0			0
$292$	−585.094	−	585.094i	−0.117260	−	0.117260i
$293$	2078.57	+	2078.57i	0.414442	+	0.414442i	0.883283	−	0.468841i	$-0.155328\pi$
$293$	2078.57	+	2078.57i	0.414442	+	0.414442i	−0.468841	+	0.883283i	$0.655328\pi$
$294$		0			0
$295$	−4803.09	+	2474.51i	−0.947955	+	0.488379i
$296$		−	2471.40i		−	0.485294i
$297$		0			0
$298$	266.229	−	266.229i	0.0517524	−	0.0517524i
$299$	609.632			0.117913
$300$		0			0
$301$	920.443			0.176257
$302$	115.670	−	115.670i	0.0220399	−	0.0220399i
$303$		0			0
$304$			5717.85i			1.07875i
$305$	2627.56	−	1353.70i	0.493291	−	0.254140i
$306$		0			0
$307$	−5325.58	−	5325.58i	−0.990056	−	0.990056i	0.00989542	−	0.999951i	$-0.496850\pi$
$307$	−5325.58	−	5325.58i	−0.990056	−	0.990056i	−0.999951	+	0.00989542i	$0.996850\pi$
$308$	−3302.50	−	3302.50i	−0.610966	−	0.610966i
$309$		0			0
$310$	−39.9248	+	124.779i	−0.00731476	+	0.0228611i
$311$			2005.56i			0.365676i	0.983143	+	0.182838i	$0.0585283\pi$
$311$			2005.56i			0.365676i	−0.983143	+	0.182838i	$0.941472\pi$
$312$		0			0
$313$	5026.49	−	5026.49i	0.907713	−	0.907713i	−0.0883746	−	0.996087i	$-0.528167\pi$
$313$	5026.49	−	5026.49i	0.907713	−	0.907713i	0.996087	+	0.0883746i	$0.0281673\pi$
$314$	−751.742			−0.135106
$315$		0			0
$316$	−3185.19			−0.567028
$317$	3137.15	−	3137.15i	0.555836	−	0.555836i	−0.372283	−	0.928119i	$-0.621425\pi$
$317$	3137.15	−	3137.15i	0.555836	−	0.555836i	0.928119	+	0.372283i	$0.121425\pi$
$318$		0			0
$319$		−	2774.31i		−	0.486933i
$320$	2301.31	+	4466.90i	0.402022	+	0.780335i
$321$		0			0
$322$	−1057.58	−	1057.58i	−0.183033	−	0.183033i
$323$	−6466.43	−	6466.43i	−1.11394	−	1.11394i
$324$		0			0
$325$	−73.0101	−	435.622i	−0.0124611	−	0.0743506i
$326$		−	45.3960i		−	0.00771243i
$327$		0			0
$328$	2068.50	−	2068.50i	0.348214	−	0.348214i
$329$	7666.35			1.28468
$330$		0			0
$331$	2978.02			0.494523			0.247261	−	0.968949i	$-0.420469\pi$
$331$	2978.02			0.494523			0.247261	+	0.968949i	$0.420469\pi$
$332$	5110.10	−	5110.10i	0.844738	−	0.844738i
$333$		0			0
$334$			246.511i			0.0403847i
$335$	7511.84	+	2403.52i	1.22512	+	0.391996i
$336$		0			0
$337$	−5985.73	−	5985.73i	−0.967547	−	0.967547i	0.0319428	−	0.999490i	$-0.489831\pi$
$337$	−5985.73	−	5985.73i	−0.967547	−	0.967547i	−0.999490	+	0.0319428i	$0.989831\pi$
$338$	−628.926	−	628.926i	−0.101210	−	0.101210i
$339$		0			0
$340$	−8011.93	−	2563.54i	−1.27796	−	0.408904i
$341$		−	805.818i		−	0.127969i
$342$		0			0
$343$	3502.74	−	3502.74i	0.551400	−	0.551400i
$344$	−278.701			−0.0436818
$345$		0			0
$346$	−800.693			−0.124409
$347$	−8183.37	+	8183.37i	−1.26601	+	1.26601i	−0.317881	+	0.948130i	$0.602971\pi$
$347$	−8183.37	+	8183.37i	−1.26601	+	1.26601i	−0.948130	+	0.317881i	$0.897029\pi$
$348$		0			0
$349$		−	1421.22i		−	0.217983i	−0.994043	−	0.108991i	$-0.965238\pi$
$349$		−	1421.22i		−	0.217983i	0.994043	−	0.108991i	$-0.0347621\pi$
$350$	−629.053	+	882.366i	−0.0960694	+	0.134756i
$351$		0			0
$352$	1505.18	+	1505.18i	0.227916	+	0.227916i
$353$	1399.56	+	1399.56i	0.211023	+	0.211023i	0.804702	−	0.593679i	$-0.202325\pi$
$353$	1399.56	+	1399.56i	0.211023	+	0.211023i	−0.593679	+	0.804702i	$0.702325\pi$
$354$		0			0
$355$	3122.97	+	6061.76i	0.466902	+	0.906268i
$356$		−	1095.70i		−	0.163123i
$357$		0			0
$358$	−31.8632	+	31.8632i	−0.00470397	+	0.00470397i
$359$	4313.98			0.634215			0.317108	−	0.948390i	$-0.397288\pi$
$359$	4313.98			0.634215			0.317108	+	0.948390i	$0.397288\pi$
$360$		0			0
$361$	−2207.85			−0.321891
$362$	−201.702	+	201.702i	−0.0292852	+	0.0292852i
$363$		0			0
$364$			589.423i			0.0848740i
$365$	359.862	−	1124.69i	0.0516056	−	0.161285i
$366$		0			0
$367$	4533.34	+	4533.34i	0.644792	+	0.644792i	0.951730	−	0.306938i	$-0.0993043\pi$
$367$	4533.34	+	4533.34i	0.644792	+	0.644792i	−0.306938	+	0.951730i	$0.599304\pi$
$368$	−7325.59	−	7325.59i	−1.03770	−	1.03770i
$369$		0			0
$370$	1551.26	−	799.194i	0.217962	−	0.112292i
$371$		−	6745.59i		−	0.943972i
$372$		0			0
$373$	−2787.95	+	2787.95i	−0.387010	+	0.387010i	−0.873620	−	0.486609i	$-0.838234\pi$
$373$	−2787.95	+	2787.95i	−0.387010	+	0.387010i	0.486609	+	0.873620i	$0.338234\pi$
$374$	−1094.86			−0.151374
$375$		0			0
$376$	−2321.29			−0.318382
$377$	−247.576	+	247.576i	−0.0338218	+	0.0338218i
$378$		0			0
$379$			8409.09i			1.13970i	0.821749	+	0.569849i	$0.192998\pi$
$379$			8409.09i			1.13970i	−0.821749	+	0.569849i	$0.807002\pi$
$380$	−7414.15	+	3819.71i	−1.00089	+	0.515650i
$381$		0			0
$382$	−637.508	−	637.508i	−0.0853868	−	0.0853868i
$383$	8507.38	+	8507.38i	1.13500	+	1.13500i	0.989333	+	0.145672i	$0.0465343\pi$
$383$	8507.38	+	8507.38i	1.13500	+	1.13500i	0.145672	+	0.989333i	$0.453466\pi$
$384$		0			0
$385$	2031.20	−	6348.22i	0.268883	−	0.840351i
$386$		−	1130.97i		−	0.149132i
$387$		0			0
$388$	−4368.62	+	4368.62i	−0.571606	+	0.571606i
$389$	−8641.23			−1.12629			−0.563146	−	0.826357i	$-0.690409\pi$
$389$	−8641.23			−1.12629			−0.563146	+	0.826357i	$0.690409\pi$
$390$		0			0
$391$	16569.3			2.14308
$392$	503.042	−	503.042i	0.0648150	−	0.0648150i
$393$		0			0
$394$			1555.07i			0.198841i
$395$	−2081.83	−	4040.89i	−0.265186	−	0.514732i
$396$		0			0
$397$	1902.21	+	1902.21i	0.240476	+	0.240476i	0.817047	−	0.576571i	$-0.195610\pi$
$397$	1902.21	+	1902.21i	0.240476	+	0.240476i	−0.576571	+	0.817047i	$0.695610\pi$
$398$	242.182	+	242.182i	0.0305012	+	0.0305012i
$399$		0			0
$400$	−4357.29	+	6111.93i	−0.544661	+	0.763991i
$401$			13640.1i			1.69864i	0.527882	+	0.849318i	$0.322986\pi$
$401$			13640.1i			1.69864i	−0.527882	+	0.849318i	$0.677014\pi$
$402$		0			0
$403$	−71.9103	+	71.9103i	−0.00888860	+	0.00888860i
$404$	2348.96			0.289270
$405$		0			0
$406$	858.982			0.105001
$407$	−7589.57	+	7589.57i	−0.924327	+	0.924327i
$408$		0			0
$409$		−	6395.02i		−	0.773138i	−0.922260	−	0.386569i	$-0.873660\pi$
$409$		−	6395.02i		−	0.773138i	0.922260	−	0.386569i	$-0.126340\pi$
$410$	1967.27	+	629.458i	0.236968	+	0.0758212i
$411$		0			0
$412$	4527.98	+	4527.98i	0.541450	+	0.541450i
$413$	7275.86	+	7275.86i	0.866880	+	0.866880i
$414$		0			0
$415$	9822.86	+	3142.97i	1.16189	+	0.371764i
$416$		−	268.641i		−	0.0316615i
$417$		0			0
$418$	−767.572	+	767.572i	−0.0898162	+	0.0898162i
$419$	−8474.94			−0.988133			−0.494067	−	0.869424i	$-0.664490\pi$
$419$	−8474.94			−0.988133			−0.494067	+	0.869424i	$0.664490\pi$
$420$		0			0
$421$	−9069.21			−1.04990			−0.524948	−	0.851134i	$-0.675915\pi$
$421$	−9069.21			−1.04990			−0.524948	+	0.851134i	$0.675915\pi$
$422$	1142.99	−	1142.99i	0.131848	−	0.131848i
$423$		0			0
$424$			2042.49i			0.233944i
$425$	−1984.35	−	11839.8i	−0.226483	−	1.35133i
$426$		0			0
$427$	−3980.30	−	3980.30i	−0.451102	−	0.451102i
$428$	−1699.98	−	1699.98i	−0.191990	−	0.191990i
$429$		0			0
$430$	−90.1254	−	174.936i	−0.0101075	−	0.0196189i
$431$		−	8388.11i		−	0.937450i	−0.883344	−	0.468725i	$-0.844713\pi$
$431$		−	8388.11i		−	0.937450i	0.883344	−	0.468725i	$-0.155287\pi$
$432$		0			0
$433$	−4132.17	+	4132.17i	−0.458613	+	0.458613i	−0.898200	−	0.439587i	$-0.855125\pi$
$433$	−4132.17	+	4132.17i	−0.458613	+	0.458613i	0.439587	+	0.898200i	$0.355125\pi$
$434$	249.498			0.0275951
$435$		0			0
$436$	−7336.08			−0.805814
$437$	11616.2	−	11616.2i	1.27158	−	1.27158i
$438$		0			0
$439$		−	10498.9i		−	1.14143i	−0.821149	−	0.570714i	$-0.806667\pi$
$439$		−	10498.9i		−	1.14143i	0.821149	−	0.570714i	$-0.193333\pi$
$440$	−615.028	+	1922.17i	−0.0666370	+	0.208264i
$441$		0			0
$442$	97.7038	+	97.7038i	0.0105142	+	0.0105142i
$443$	1465.93	+	1465.93i	0.157220	+	0.157220i	0.781333	−	0.624114i	$-0.214540\pi$
$443$	1465.93	+	1465.93i	0.157220	+	0.157220i	−0.624114	+	0.781333i	$0.714540\pi$
$444$		0			0
$445$	1390.05	−	716.144i	0.148078	−	0.0762887i
$446$			2073.09i			0.220098i
$447$		0			0
$448$	6766.58	−	6766.58i	0.713596	−	0.713596i
$449$	−6295.42			−0.661691			−0.330846	−	0.943685i	$-0.607334\pi$
$449$	−6295.42			−0.661691			−0.330846	+	0.943685i	$0.607334\pi$
$450$		0			0
$451$	−12704.6			−1.32647
$452$	8492.32	−	8492.32i	0.883728	−	0.883728i
$453$		0			0
$454$			998.909i			0.103262i
$455$	−747.770	+	385.245i	−0.0770462	+	0.0396936i
$456$		0			0
$457$	9065.09	+	9065.09i	0.927893	+	0.927893i	0.997570	−	0.0696767i	$-0.0221968\pi$
$457$	9065.09	+	9065.09i	0.927893	+	0.927893i	−0.0696767	+	0.997570i	$0.522197\pi$
$458$	998.368	+	998.368i	0.101857	+	0.101857i
$459$		0			0
$460$	4605.11	−	14392.6i	0.466771	−	1.45882i
$461$			13305.5i			1.34424i	0.740440	+	0.672122i	$0.234617\pi$
$461$			13305.5i			1.34424i	−0.740440	+	0.672122i	$0.765383\pi$
$462$		0			0
$463$	−3124.86	+	3124.86i	−0.313660	+	0.313660i	−0.846326	−	0.532666i	$-0.821190\pi$
$463$	−3124.86	+	3124.86i	−0.313660	+	0.313660i	0.532666	+	0.846326i	$0.321190\pi$
$464$	5949.95			0.595301
$465$		0			0
$466$	−616.210			−0.0612562
$467$	4255.07	−	4255.07i	0.421629	−	0.421629i	−0.464135	−	0.885764i	$-0.653635\pi$
$467$	4255.07	−	4255.07i	0.421629	−	0.421629i	0.885764	+	0.464135i	$0.153635\pi$
$468$		0			0
$469$		−	15020.1i		−	1.47881i
$470$	−750.653	−	1457.04i	−0.0736703	−	0.142996i
$471$		0			0
$472$	−2203.05	−	2203.05i	−0.214838	−	0.214838i
$473$	855.879	+	855.879i	0.0831995	+	0.0831995i
$474$		0			0
$475$	−9691.74	−	6909.39i	−0.936184	−	0.667420i
$476$			16020.0i			1.54260i
$477$		0			0
$478$	1125.53	−	1125.53i	0.107700	−	0.107700i
$479$	10625.2			1.01352			0.506761	−	0.862087i	$-0.330843\pi$
$479$	10625.2			1.01352			0.506761	+	0.862087i	$0.330843\pi$
$480$		0			0
$481$	1354.57			0.128406
$482$	1378.21	−	1378.21i	0.130241	−	0.130241i
$483$		0			0
$484$			4285.66i			0.402485i
$485$	−8397.56	−	2686.92i	−0.786213	−	0.251560i
$486$		0			0
$487$	−3269.03	−	3269.03i	−0.304177	−	0.304177i	0.538469	−	0.842645i	$-0.319003\pi$
$487$	−3269.03	−	3269.03i	−0.304177	−	0.304177i	−0.842645	+	0.538469i	$0.819003\pi$
$488$	1205.19	+	1205.19i	0.111796	+	0.111796i
$489$		0			0
$490$	478.424	+	153.079i	0.0441081	+	0.0141130i
$491$		−	18791.1i		−	1.72715i	−0.504220	−	0.863575i	$-0.668220\pi$
$491$		−	18791.1i		−	1.72715i	0.504220	−	0.863575i	$-0.331780\pi$
$492$		0			0
$493$	−6728.92	+	6728.92i	−0.614716	+	0.614716i
$494$	136.995			0.0124771
$495$		0			0
$496$	1728.21			0.156449
$497$	9182.53	−	9182.53i	0.828758	−	0.828758i
$498$		0			0
$499$			7220.67i			0.647778i	0.946095	+	0.323889i	$0.104990\pi$
$499$			7220.67i			0.647778i	−0.946095	+	0.323889i	$0.895010\pi$
$500$	−10835.9	−	1566.99i	−0.969197	−	0.140156i
$501$		0			0
$502$	672.947	+	672.947i	0.0598308	+	0.0598308i
$503$	2081.38	+	2081.38i	0.184501	+	0.184501i	0.793314	−	0.608813i	$-0.208354\pi$
$503$	2081.38	+	2081.38i	0.184501	+	0.184501i	−0.608813	+	0.793314i	$0.708354\pi$
$504$		0			0
$505$	1535.27	+	2980.00i	0.135285	+	0.262590i
$506$		−	1966.79i		−	0.172796i
$507$		0			0
$508$	7172.81	−	7172.81i	0.626461	−	0.626461i
$509$	−12624.1			−1.09932			−0.549659	−	0.835389i	$-0.685242\pi$
$509$	−12624.1			−1.09932			−0.549659	+	0.835389i	$0.685242\pi$
$510$		0			0
$511$	−2248.85			−0.194683
$512$	−5418.06	+	5418.06i	−0.467669	+	0.467669i
$513$		0			0
$514$		−	563.001i		−	0.0483130i
$515$	−2784.93	+	8703.88i	−0.238289	+	0.744735i
$516$		0			0
$517$	7128.60	+	7128.60i	0.606413	+	0.606413i
$518$	−2349.89	−	2349.89i	−0.199321	−	0.199321i
$519$		0			0
$520$	226.417	−	116.648i	0.0190943	−	0.00983724i
$521$			11837.5i			0.995409i	0.867347	+	0.497705i	$0.165824\pi$
$521$			11837.5i			0.995409i	−0.867347	+	0.497705i	$0.834176\pi$
$522$		0			0
$523$	−5538.72	+	5538.72i	−0.463081	+	0.463081i	−0.899664	−	0.436583i	$-0.856189\pi$
$523$	−5538.72	+	5538.72i	−0.463081	+	0.463081i	0.436583	+	0.899664i	$0.356189\pi$
$524$	4930.31			0.411033
$525$		0			0
$526$	455.282			0.0377400
$527$	−1954.46	+	1954.46i	−0.161552	+	0.161552i
$528$		0			0
$529$			17597.9i			1.44637i
$530$	−1282.04	+	660.496i	−0.105072	+	0.0541323i
$531$		0			0
$532$	11231.2	+	11231.2i	0.915287	+	0.915287i
$533$	1133.74	+	1133.74i	0.0921349	+	0.0921349i
$534$		0			0
$535$	1045.57	−	3267.78i	0.0844937	−	0.264072i
$536$			4547.92i			0.366493i
$537$		0			0
$538$	1138.51	−	1138.51i	0.0912350	−	0.0912350i
$539$	−3089.65			−0.246903
$540$		0			0
$541$	5207.24			0.413820			0.206910	−	0.978360i	$-0.433659\pi$
$541$	5207.24			0.413820			0.206910	+	0.978360i	$0.433659\pi$
$542$	−2263.86	+	2263.86i	−0.179412	+	0.179412i
$543$		0			0
$544$		−	7301.44i		−	0.575453i
$545$	−4794.85	−	9306.91i	−0.376860	−	0.731494i
$546$		0			0
$547$	−3940.59	−	3940.59i	−0.308021	−	0.308021i	0.536121	−	0.844141i	$-0.319889\pi$
$547$	−3940.59	−	3940.59i	−0.308021	−	0.308021i	−0.844141	+	0.536121i	$0.819889\pi$
$548$	5177.88	+	5177.88i	0.403628	+	0.403628i
$549$		0			0
$550$	−1405.40	+	235.545i	−0.108957	+	0.0182612i
$551$			9434.89i			0.729473i
$552$		0			0
$553$	−6121.25	+	6121.25i	−0.470709	+	0.470709i
$554$	−522.836			−0.0400959
$555$		0			0
$556$	10798.3			0.823652
$557$	−2850.64	+	2850.64i	−0.216850	+	0.216850i	−0.807170	−	0.590320i	$-0.799002\pi$
$557$	−2850.64	+	2850.64i	−0.216850	+	0.216850i	0.590320	+	0.807170i	$0.299002\pi$
$558$		0			0
$559$		−	152.755i		−	0.0115579i
$560$	13614.8	+	4356.25i	1.02737	+	0.328723i
$561$		0			0
$562$	2044.88	+	2044.88i	0.153484	+	0.153484i
$563$	−12845.3	−	12845.3i	−0.961569	−	0.961569i	0.0377198	−	0.999288i	$-0.487991\pi$
$563$	−12845.3	−	12845.3i	−0.961569	−	0.961569i	−0.999288	+	0.0377198i	$0.987991\pi$
$564$		0			0
$565$	16324.3	+	5223.21i	1.21552	+	0.388924i
$566$		−	2718.91i		−	0.201916i
$567$		0			0
$568$	−2780.38	+	2780.38i	−0.205391	+	0.205391i
$569$	−25072.7			−1.84728			−0.923639	−	0.383263i	$-0.874800\pi$
$569$	−25072.7			−1.84728			−0.923639	+	0.383263i	$0.874800\pi$
$570$		0			0
$571$	−3587.24			−0.262909			−0.131455	−	0.991322i	$-0.541965\pi$
$571$	−3587.24			−0.262909			−0.131455	+	0.991322i	$0.541965\pi$
$572$	−548.078	+	548.078i	−0.0400635	+	0.0400635i
$573$		0			0
$574$		−	3933.60i		−	0.286037i
$575$	21269.0	−	3564.68i	1.54257	−	0.258534i
$576$		0			0
$577$	760.083	+	760.083i	0.0548400	+	0.0548400i	0.733995	−	0.679155i	$-0.237654\pi$
$577$	760.083	+	760.083i	0.0548400	+	0.0548400i	−0.679155	+	0.733995i	$0.737654\pi$
$578$	1241.04	+	1241.04i	0.0893091	+	0.0893091i
$579$		0			0
$580$	3974.76	+	7715.11i	0.284557	+	0.552332i
$581$		−	19641.0i		−	1.40249i
$582$		0			0
$583$	6272.43	−	6272.43i	0.445587	−	0.445587i
$584$	680.928			0.0482483
$585$		0			0
$586$	−1196.85			−0.0843711
$587$	−8165.43	+	8165.43i	−0.574146	+	0.574146i	−0.933284	−	0.359139i	$-0.883071\pi$
$587$	−8165.43	+	8165.43i	−0.574146	+	0.574146i	0.359139	+	0.933284i	$0.383071\pi$
$588$		0			0
$589$			2740.43i			0.191711i
$590$	670.402	−	2095.24i	0.0467797	−	0.146203i
$591$		0			0
$592$	−16277.1	−	16277.1i	−1.13004	−	1.13004i
$593$	9861.32	+	9861.32i	0.682893	+	0.682893i	0.960651	−	0.277758i	$-0.0895913\pi$
$593$	9861.32	+	9861.32i	0.682893	+	0.682893i	−0.277758	+	0.960651i	$0.589591\pi$
$594$		0			0
$595$	−20323.8	+	10470.6i	−1.40032	+	0.721436i
$596$		−	7244.46i		−	0.497893i
$597$		0			0
$598$	−175.514	+	175.514i	−0.0120022	+	0.0120022i
$599$	20769.5			1.41673			0.708363	−	0.705848i	$-0.249434\pi$
$599$	20769.5			1.41673			0.708363	+	0.705848i	$0.249434\pi$
$600$		0			0
$601$	−11642.9			−0.790222			−0.395111	−	0.918633i	$-0.629294\pi$
$601$	−11642.9			−0.790222			−0.395111	+	0.918633i	$0.629294\pi$
$602$	−264.997	+	264.997i	−0.0179410	+	0.0179410i
$603$		0			0
$604$		−	3147.54i		−	0.212039i
$605$	−5436.99	+	2801.09i	−0.365364	+	0.188232i
$606$		0			0
$607$	−17174.6	−	17174.6i	−1.14843	−	1.14843i	−0.986862	−	0.161565i	$-0.948346\pi$
$607$	−17174.6	−	17174.6i	−1.14843	−	1.14843i	−0.161565	−	0.986862i	$-0.551654\pi$
$608$	−5118.82	−	5118.82i	−0.341440	−	0.341440i
$609$		0			0
$610$	−366.748	+	1146.21i	−0.0243429	+	0.0760800i
$611$		−	1272.30i		−	0.0842416i
$612$		0			0
$613$	11694.1	−	11694.1i	0.770504	−	0.770504i	−0.207691	−	0.978195i	$-0.566595\pi$
$613$	11694.1	−	11694.1i	0.770504	−	0.770504i	0.978195	+	0.207691i	$0.0665947\pi$
$614$	3066.49			0.201553
$615$		0			0
$616$	3843.43			0.251390
$617$	−5586.69	+	5586.69i	−0.364525	+	0.364525i	−0.865476	−	0.500951i	$-0.832984\pi$
$617$	−5586.69	+	5586.69i	−0.364525	+	0.364525i	0.500951	+	0.865476i	$0.332984\pi$
$618$		0			0
$619$		−	16137.1i		−	1.04783i	−0.851771	−	0.523914i	$-0.824471\pi$
$619$		−	16137.1i		−	1.04783i	0.851771	−	0.523914i	$-0.175529\pi$
$620$	1154.50	+	2240.91i	0.0747835	+	0.145156i
$621$		0			0
$622$	−577.406	−	577.406i	−0.0372217	−	0.0372217i
$623$	−2105.69	−	2105.69i	−0.135414	−	0.135414i
$624$		0			0
$625$	−5094.39	−	14771.2i	−0.326041	−	0.945356i
$626$			2894.27i			0.184790i
$627$		0			0
$628$	−10228.0	+	10228.0i	−0.649906	+	0.649906i
$629$	36816.1			2.33379
$630$		0			0
$631$	25292.3			1.59567			0.797837	−	0.602874i	$-0.205978\pi$
$631$	25292.3			1.59567			0.797837	+	0.602874i	$0.205978\pi$
$632$	1853.45	−	1853.45i	0.116656	−	0.116656i
$633$		0			0
$634$			1806.38i			0.113156i
$635$	13787.9	+	4411.64i	0.861663	+	0.275702i
$636$		0			0
$637$	275.717	+	275.717i	0.0171496	+	0.0171496i
$638$	798.729	+	798.729i	0.0495643	+	0.0495643i
$639$		0			0
$640$	−8425.03	−	2695.71i	−0.520357	−	0.166496i
$641$		−	8057.35i		−	0.496484i	−0.968698	−	0.248242i	$-0.920147\pi$
$641$		−	8057.35i		−	0.496484i	0.968698	−	0.248242i	$-0.0798528\pi$
$642$		0			0
$643$	15024.4	−	15024.4i	0.921471	−	0.921471i	−0.0756629	−	0.997133i	$-0.524107\pi$
$643$	15024.4	−	15024.4i	0.921471	−	0.921471i	0.997133	+	0.0756629i	$0.0241073\pi$
$644$	−28778.2			−1.76090
$645$		0			0
$646$	3723.40			0.226773
$647$	−2396.69	+	2396.69i	−0.145631	+	0.145631i	−0.776163	−	0.630532i	$-0.782837\pi$
$647$	−2396.69	+	2396.69i	−0.145631	+	0.145631i	0.630532	+	0.776163i	$0.282837\pi$
$648$		0			0
$649$			13531.0i			0.818394i
$650$	146.436	+	104.397i	0.00883646	+	0.00629965i
$651$		0			0
$652$	−617.645	−	617.645i	−0.0370995	−	0.0370995i
$653$	−10852.1	−	10852.1i	−0.650343	−	0.650343i	0.302732	−	0.953076i	$-0.402101\pi$
$653$	−10852.1	−	10852.1i	−0.650343	−	0.650343i	−0.953076	+	0.302732i	$0.902101\pi$
$654$		0			0
$655$	3222.44	+	6254.82i	0.192230	+	0.373124i
$656$		−	27247.1i		−	1.62168i
$657$		0			0
$658$	−2207.16	+	2207.16i	−0.130766	+	0.130766i
$659$	21164.7			1.25108			0.625538	−	0.780193i	$-0.284879\pi$
$659$	21164.7			1.25108			0.625538	+	0.780193i	$0.284879\pi$
$660$		0			0
$661$	−6399.33			−0.376558			−0.188279	−	0.982116i	$-0.560291\pi$
$661$	−6399.33			−0.376558			−0.188279	+	0.982116i	$0.560291\pi$
$662$	−857.380	+	857.380i	−0.0503369	+	0.0503369i
$663$		0			0
$664$			5947.09i			0.347578i
$665$	−6907.74	+	21589.1i	−0.402813	+	1.25893i
$666$		0			0
$667$	−12087.8	−	12087.8i	−0.701710	−	0.701710i
$668$	3353.96	+	3353.96i	0.194264	+	0.194264i
$669$		0			0
$670$	−2854.65	+	1470.70i	−0.164604	+	0.0848028i
$671$		−	7402.22i		−	0.425871i
$672$		0			0
$673$	−1618.29	+	1618.29i	−0.0926903	+	0.0926903i	−0.751932	−	0.659241i	$-0.770878\pi$
$673$	−1618.29	+	1618.29i	−0.0926903	+	0.0926903i	0.659241	+	0.751932i	$0.270878\pi$
$674$	3446.61			0.196971
$675$		0			0
$676$	−17114.0			−0.973713
$677$	−16119.8	+	16119.8i	−0.915118	+	0.915118i	−0.996669	−	0.0815511i	$-0.974013\pi$
$677$	−16119.8	+	16119.8i	−0.915118	+	0.915118i	0.0815511	+	0.996669i	$0.474013\pi$
$678$		0			0
$679$			16791.1i			0.949017i
$680$	6153.83	−	3170.40i	0.347042	−	0.178793i
$681$		0			0
$682$	231.997	+	231.997i	0.0130258	+	0.0130258i
$683$	−23612.1	−	23612.1i	−1.32283	−	1.32283i	−0.911477	−	0.411351i	$-0.865057\pi$
$683$	−23612.1	−	23612.1i	−1.32283	−	1.32283i	−0.411351	−	0.911477i	$-0.634943\pi$
$684$		0			0
$685$	−3184.66	+	9953.16i	−0.177634	+	0.555169i
$686$			2016.89i			0.112253i
$687$		0			0
$688$	−1835.57	+	1835.57i	−0.101716	+	0.101716i
$689$	−1119.49			−0.0619000
$690$		0			0
$691$	12036.1			0.662624			0.331312	−	0.943521i	$-0.392509\pi$
$691$	12036.1			0.662624			0.331312	+	0.943521i	$0.392509\pi$
$692$	−10894.0	+	10894.0i	−0.598451	+	0.598451i
$693$		0			0
$694$		−	4712.02i		−	0.257732i
$695$	7057.75	+	13699.3i	0.385202	+	0.747687i
$696$		0			0
$697$	30814.2	+	30814.2i	1.67457	+	1.67457i
$698$	409.171	+	409.171i	0.0221882	+	0.0221882i
$699$		0			0
$700$	3446.51	+	20563.9i	0.186094	+	1.11035i
$701$			33366.3i			1.79776i	0.438200	+	0.898878i	$0.355616\pi$
$701$			33366.3i			1.79776i	−0.438200	+	0.898878i	$0.644384\pi$
$702$		0			0
$703$	25810.7	−	25810.7i	1.38473	−	1.38473i
$704$	12583.9			0.673684
$705$		0			0
$706$	−805.875			−0.0429596
$707$	4514.19	−	4514.19i	0.240132	−	0.240132i
$708$		0			0
$709$			4987.22i			0.264173i	0.991238	+	0.132087i	$0.0421677\pi$
$709$			4987.22i			0.264173i	−0.991238	+	0.132087i	$0.957832\pi$
$710$	−2644.30	−	846.084i	−0.139773	−	0.0447225i
$711$		0			0
$712$	637.581	+	637.581i	0.0335595	+	0.0335595i
$713$	−3510.98	−	3510.98i	−0.184414	−	0.184414i
$714$		0			0
$715$	−1053.54	−	337.096i	−0.0551052	−	0.0176317i
$716$			867.043i			0.0452555i
$717$		0			0
$718$	−1242.00	+	1242.00i	−0.0645560	+	0.0645560i
$719$	−33618.4			−1.74375			−0.871873	−	0.489732i	$-0.837095\pi$
$719$	−33618.4			−1.74375			−0.871873	+	0.489732i	$0.837095\pi$
$720$		0			0
$721$	17403.6			0.898951
$722$	635.644	−	635.644i	0.0327649	−	0.0327649i
$723$		0			0
$724$			5488.61i			0.281744i
$725$	−7189.86	+	10085.1i	−0.368310	+	0.516625i
$726$		0			0
$727$	−10057.9	−	10057.9i	−0.513106	−	0.513106i	0.402371	−	0.915477i	$-0.368186\pi$
$727$	−10057.9	−	10057.9i	−0.513106	−	0.513106i	−0.915477	+	0.402371i	$0.868186\pi$
$728$	−342.983	−	342.983i	−0.0174613	−	0.0174613i
$729$		0			0
$730$	220.197	+	427.407i	0.0111642	+	0.0216699i
$731$		−	4151.76i		−	0.210066i
$732$		0			0
$733$	−4209.55	+	4209.55i	−0.212119	+	0.212119i	−0.805167	−	0.593048i	$-0.797924\pi$
$733$	−4209.55	+	4209.55i	−0.212119	+	0.212119i	0.593048	+	0.805167i	$0.297924\pi$
$734$	−2610.32			−0.131265
$735$		0			0
$736$	13116.2			0.656890
$737$	13966.5	−	13966.5i	0.698050	−	0.698050i
$738$		0			0
$739$			4750.38i			0.236462i	0.992986	+	0.118231i	$0.0377223\pi$
$739$			4750.38i			0.236462i	−0.992986	+	0.118231i	$0.962278\pi$
$740$	10232.3	−	31979.5i	0.508308	−	1.58864i
$741$		0			0
$742$	1942.07	+	1942.07i	0.0960857	+	0.0960857i
$743$	16266.1	+	16266.1i	0.803159	+	0.803159i	0.983588	−	0.180429i	$-0.0577487\pi$
$743$	16266.1	+	16266.1i	0.803159	+	0.803159i	−0.180429	+	0.983588i	$0.557749\pi$
$744$		0			0
$745$	9190.66	−	4734.96i	0.451973	−	0.232853i
$746$		−	1605.32i		−	0.0787866i
$747$		0			0
$748$	−14896.3	+	14896.3i	−0.728159	+	0.728159i
$749$	−6534.00			−0.318754
$750$		0			0
$751$	−32690.2			−1.58839			−0.794197	−	0.607660i	$-0.792108\pi$
$751$	−32690.2			−1.58839			−0.794197	+	0.607660i	$0.792108\pi$
$752$	−15288.4	+	15288.4i	−0.741372	+	0.741372i
$753$		0			0
$754$		−	142.555i		−	0.00688536i
$755$	3993.12	−	2057.22i	0.192483	−	0.0991656i
$756$		0			0
$757$	−6722.97	−	6722.97i	−0.322788	−	0.322788i	0.527048	−	0.849836i	$-0.323299\pi$
$757$	−6722.97	−	6722.97i	−0.322788	−	0.322788i	−0.849836	+	0.527048i	$0.823299\pi$
$758$	−2420.99	−	2420.99i	−0.116008	−	0.116008i
$759$		0			0
$760$	2091.59	−	6536.94i	0.0998289	−	0.312000i
$761$			7001.94i			0.333535i	0.985996	+	0.166767i	$0.0533329\pi$
$761$			7001.94i			0.333535i	−0.985996	+	0.166767i	$0.946667\pi$
$762$		0			0
$763$	−14098.4	+	14098.4i	−0.668932	+	0.668932i
$764$	−17347.5			−0.821480
$765$		0			0
$766$	−4898.58			−0.231061
$767$	1207.49	−	1207.49i	0.0568448	−	0.0568448i
$768$		0			0
$769$		−	25176.4i		−	1.18060i	−0.807183	−	0.590301i	$-0.799009\pi$
$769$		−	25176.4i		−	1.18060i	0.807183	−	0.590301i	$-0.200991\pi$
$770$	1242.88	+	2412.45i	0.0581690	+	0.112907i
$771$		0			0
$772$	−15387.7	−	15387.7i	−0.717376	−	0.717376i
$773$	7322.13	+	7322.13i	0.340697	+	0.340697i	0.856629	−	0.515932i	$-0.172554\pi$
$773$	7322.13	+	7322.13i	0.340697	+	0.340697i	−0.515932	+	0.856629i	$0.672554\pi$
$774$		0			0
$775$	−2088.35	+	2929.30i	−0.0967943	+	0.135772i
$776$		−	5084.17i		−	0.235195i
$777$		0			0
$778$	2487.83	−	2487.83i	0.114644	−	0.114644i
$779$	43205.9			1.98718
$780$		0			0
$781$	17076.9			0.782405
$782$	−4770.33	+	4770.33i	−0.218142	+	0.218142i
$783$		0			0
$784$		−	6626.25i		−	0.301852i
$785$	−19660.7	−	6290.73i	−0.893911	−	0.286020i
$786$		0			0
$787$	16794.5	+	16794.5i	0.760683	+	0.760683i	0.976446	−	0.215763i	$-0.0692237\pi$
$787$	16794.5	+	16794.5i	0.760683	+	0.760683i	−0.215763	+	0.976446i	$0.569224\pi$
$788$	21157.9	+	21157.9i	0.956495	+	0.956495i
$789$		0			0
$790$	1762.74	+	564.016i	0.0793868	+	0.0254010i
$791$		−	32640.8i		−	1.46722i
$792$		0			0
$793$	−660.565	+	660.565i	−0.0295805	+	0.0295805i
$794$	−1095.30			−0.0489556
$795$		0			0
$796$	6590.11			0.293443
$797$	18950.9	−	18950.9i	0.842251	−	0.842251i	−0.146900	−	0.989151i	$-0.546930\pi$
$797$	18950.9	−	18950.9i	0.842251	−	0.842251i	0.989151	+	0.146900i	$0.0469296\pi$
$798$		0			0
$799$		−	34580.0i		−	1.53110i
$800$	−1570.81	−	9372.41i	−0.0694208	−	0.414206i
$801$		0			0
$802$	−3927.01	−	3927.01i	−0.172902	−	0.172902i
$803$	−2091.10	−	2091.10i	−0.0918972	−	0.0918972i
$804$		0			0
$805$	−18809.4	−	36509.5i	−0.823532	−	1.59850i
$806$		−	41.4062i		−	0.00180952i
$807$		0			0
$808$	−1366.85	+	1366.85i	−0.0595118	+	0.0595118i
$809$	3967.32			0.172415			0.0862074	−	0.996277i	$-0.472525\pi$
$809$	3967.32			0.172415			0.0862074	+	0.996277i	$0.472525\pi$
$810$		0			0
$811$	−23204.4			−1.00471			−0.502353	−	0.864663i	$-0.667532\pi$
$811$	−23204.4			−1.00471			−0.502353	+	0.864663i	$0.667532\pi$
$812$	11687.1	−	11687.1i	0.505093	−	0.505093i
$813$		0			0
$814$		−	4370.11i		−	0.188172i
$815$	379.883	−	1187.27i	0.0163273	−	0.0510283i
$816$		0			0
$817$	−2910.68	−	2910.68i	−0.124641	−	0.124641i
$818$	1841.14	+	1841.14i	0.0786968	+	0.0786968i
$819$		0			0
$820$	35330.4	−	18201.9i	1.50462	−	0.775169i
$821$			5783.29i			0.245844i	0.992416	+	0.122922i	$0.0392265\pi$
$821$			5783.29i			0.245844i	−0.992416	+	0.122922i	$0.960773\pi$
$822$		0			0
$823$	2021.92	−	2021.92i	0.0856375	−	0.0856375i	−0.662990	−	0.748628i	$-0.730713\pi$
$823$	2021.92	−	2021.92i	0.0856375	−	0.0856375i	0.748628	+	0.662990i	$0.230713\pi$
$824$	−5269.62			−0.222786
$825$		0			0
$826$	−4189.47			−0.176477
$827$	18025.3	−	18025.3i	0.757922	−	0.757922i	−0.218022	−	0.975944i	$-0.569961\pi$
$827$	18025.3	−	18025.3i	0.757922	−	0.757922i	0.975944	+	0.218022i	$0.0699605\pi$
$828$		0			0
$829$			42400.5i			1.77640i	0.459462	+	0.888198i	$0.348042\pi$
$829$			42400.5i			1.77640i	−0.459462	+	0.888198i	$0.651958\pi$
$830$	−3732.89	+	1923.15i	−0.156109	+	0.0804261i
$831$		0			0
$832$	−1122.97	−	1122.97i	−0.0467933	−	0.0467933i
$833$	7493.75	+	7493.75i	0.311696	+	0.311696i
$834$		0			0
$835$	−2062.85	+	6447.13i	−0.0854947	+	0.267200i
$836$			20886.7i			0.864094i
$837$		0			0
$838$	2439.95	−	2439.95i	0.100581	−	0.100581i
$839$	−38073.9			−1.56670			−0.783348	−	0.621584i	$-0.786490\pi$
$839$	−38073.9			−1.56670			−0.783348	+	0.621584i	$0.786490\pi$
$840$		0			0
$841$	−14571.1			−0.597447
$842$	2611.04	−	2611.04i	0.106868	−	0.106868i
$843$		0			0
$844$		−	31102.3i		−	1.26847i
$845$	−11185.6	−	21711.6i	−0.455382	−	0.883908i
$846$		0			0
$847$	8236.11	+	8236.11i	0.334116	+	0.334116i
$848$	13452.2	+	13452.2i	0.544754	+	0.544754i
$849$		0			0
$850$	3980.01	+	2837.42i	0.160604	+	0.114497i
$851$			66136.1i			2.66406i
$852$		0			0
$853$	−26447.7	+	26447.7i	−1.06161	+	1.06161i	−0.0636372	+	0.997973i	$0.520270\pi$
$853$	−26447.7	+	26447.7i	−1.06161	+	1.06161i	−0.997973	+	0.0636372i	$0.979730\pi$
$854$	2291.88			0.0918342
$855$		0			0
$856$	1978.43			0.0789967
$857$	18997.5	−	18997.5i	0.757227	−	0.757227i	−0.218590	−	0.975817i	$-0.570146\pi$
$857$	18997.5	−	18997.5i	0.757227	−	0.757227i	0.975817	+	0.218590i	$0.0701457\pi$
$858$		0			0
$859$			29326.5i			1.16485i	0.812885	+	0.582425i	$0.197896\pi$
$859$			29326.5i			1.16485i	−0.812885	+	0.582425i	$0.802104\pi$
$860$	−3606.34	−	1153.90i	−0.142995	−	0.0457532i
$861$		0			0
$862$	2414.95	+	2414.95i	0.0954219	+	0.0954219i
$863$	−23070.8	−	23070.8i	−0.910011	−	0.910011i	0.0862616	−	0.996273i	$-0.472508\pi$
$863$	−23070.8	−	23070.8i	−0.910011	−	0.910011i	−0.996273	+	0.0862616i	$0.972508\pi$
$864$		0			0
$865$	−20940.9	−	6700.36i	−0.823137	−	0.263375i
$866$		−	2379.32i		−	0.0933633i
$867$		0			0
$868$	3394.59	−	3394.59i	0.132742	−	0.132742i
$869$	−11383.8			−0.444381
$870$		0			0
$871$	−2492.71			−0.0969716
$872$	4268.84	−	4268.84i	0.165781	−	0.165781i
$873$		0			0
$874$			6688.68i			0.258865i
$875$	−23835.8	+	17812.9i	−0.920910	+	0.688214i
$876$		0			0
$877$	20252.6	+	20252.6i	0.779795	+	0.779795i	0.979796	−	0.200001i	$-0.0640944\pi$
$877$	20252.6	+	20252.6i	0.779795	+	0.779795i	−0.200001	+	0.979796i	$0.564094\pi$
$878$	3022.67	+	3022.67i	0.116185	+	0.116185i
$879$		0			0
$880$	8609.09	+	16710.5i	0.329787	+	0.640124i
$881$		−	15547.7i		−	0.594569i	−0.954789	−	0.297284i	$-0.903919\pi$
$881$		−	15547.7i		−	0.594569i	0.954789	−	0.297284i	$-0.0960809\pi$
$882$		0			0
$883$	17599.2	−	17599.2i	0.670737	−	0.670737i	−0.287149	−	0.957886i	$-0.592707\pi$
$883$	17599.2	−	17599.2i	0.670737	−	0.670737i	0.957886	+	0.287149i	$0.0927075\pi$
$884$	2658.66			0.101154
$885$		0			0
$886$	−844.087			−0.0320064
$887$	−9099.97	+	9099.97i	−0.344472	+	0.344472i	−0.858046	−	0.513573i	$-0.828321\pi$
$887$	−9099.97	+	9099.97i	−0.344472	+	0.344472i	0.513573	+	0.858046i	$0.328321\pi$
$888$		0			0
$889$		−	27569.2i		−	1.04009i
$890$	−194.020	+	606.378i	−0.00730736	+	0.0228380i
$891$		0			0
$892$	28205.9	+	28205.9i	1.05875	+	1.05875i
$893$	−24243.0	−	24243.0i	−0.908467	−	0.908467i
$894$		0			0
$895$	−1099.97	+	566.697i	−0.0410816	+	0.0211649i
$896$			16846.0i			0.628109i
$897$		0			0
$898$	1812.47	−	1812.47i	0.0673528	−	0.0673528i
$899$	2851.67			0.105794
$900$		0			0
$901$	−30426.8			−1.12504
$902$	3657.68	−	3657.68i	0.135019	−	0.135019i
$903$		0			0
$904$			9883.30i			0.363621i
$905$	−6963.11	+	3587.34i	−0.255759	+	0.131765i
$906$		0			0
$907$	28827.2	+	28827.2i	1.05534	+	1.05534i	0.998376	+	0.0569624i	$0.0181415\pi$
$907$	28827.2	+	28827.2i	1.05534	+	1.05534i	0.0569624	+	0.998376i	$0.481858\pi$
$908$	13590.9	+	13590.9i	0.496728	+	0.496728i
$909$		0			0
$910$	104.372	−	326.197i	0.00380207	−	0.0118828i
$911$			32922.1i			1.19732i	0.801003	+	0.598660i	$0.204300\pi$
$911$			32922.1i			1.19732i	−0.801003	+	0.598660i	$0.795700\pi$
$912$		0			0
$913$	18263.3	−	18263.3i	0.662023	−	0.662023i
$914$	−5219.72			−0.188898
$915$		0			0
$916$	27167.0			0.979938
$917$	9474.99	−	9474.99i	0.341212	−	0.341212i
$918$		0			0
$919$			28643.9i			1.02815i	0.857744	+	0.514077i	$0.171865\pi$
$919$			28643.9i			1.02815i	−0.857744	+	0.514077i	$0.828135\pi$
$920$	5695.28	+	11054.7i	0.204096	+	0.396154i
$921$		0			0
$922$	−3830.67	−	3830.67i	−0.136829	−	0.136829i
$923$	−1523.92	−	1523.92i	−0.0543450	−	0.0543450i
$924$		0			0
$925$	47258.6	−	7920.53i	1.67984	−	0.281541i
$926$		−	1799.31i		−	0.0638541i
$927$		0			0
$928$	−5326.61	+	5326.61i	−0.188421	+	0.188421i
$929$	12080.7			0.426648			0.213324	−	0.976981i	$-0.431571\pi$
$929$	12080.7			0.426648			0.213324	+	0.976981i	$0.431571\pi$
$930$		0			0
$931$	10507.3			0.369885
$932$	−8383.98	+	8383.98i	−0.294664	+	0.294664i
$933$		0			0
$934$			2450.08i			0.0858343i
$935$	−28634.4	−	9161.98i	−1.00154	−	0.320459i
$936$		0			0
$937$	−18170.0	−	18170.0i	−0.633500	−	0.633500i	0.315444	−	0.948944i	$-0.397846\pi$
$937$	−18170.0	−	18170.0i	−0.633500	−	0.633500i	−0.948944	+	0.315444i	$0.897846\pi$
$938$	4324.31	+	4324.31i	0.150526	+	0.150526i
$939$		0			0
$940$	−30037.2	−	9610.84i	−1.04224	−	0.333480i
$941$		−	7693.26i		−	0.266518i	−0.991081	−	0.133259i	$-0.957456\pi$
$941$		−	7693.26i		−	0.266518i	0.991081	−	0.133259i	$-0.0425442\pi$
$942$		0			0
$943$	−55354.4	+	55354.4i	−1.91155	+	1.91155i
$944$	−29019.4			−1.00053
$945$		0			0
$946$	−492.819			−0.0169375
$947$	−9792.71	+	9792.71i	−0.336030	+	0.336030i	−0.854871	−	0.518841i	$-0.826364\pi$
$947$	−9792.71	+	9792.71i	−0.336030	+	0.336030i	0.518841	+	0.854871i	$0.326364\pi$
$948$		0			0
$949$			373.216i			0.0127662i
$950$	4779.50	−	801.043i	0.163229	−	0.0273571i
$951$		0			0
$952$	−9321.99	−	9321.99i	−0.317361	−	0.317361i
$953$	22620.8	+	22620.8i	0.768899	+	0.768899i	0.977913	−	0.209014i	$-0.0670254\pi$
$953$	22620.8	+	22620.8i	0.768899	+	0.768899i	−0.209014	+	0.977913i	$0.567025\pi$
$954$		0			0
$955$	−11338.3	−	22007.9i	−0.384186	−	0.745715i
$956$		−	30627.4i		−	1.03615i
$957$		0			0
$958$	−3059.01	+	3059.01i	−0.103165	+	0.103165i
$959$	19901.5			0.670130
$960$		0			0
$961$	−28962.7			−0.972197
$962$	−389.984	+	389.984i	−0.0130702	+	0.0130702i
$963$		0			0
$964$		−	37503.2i		−	1.25300i
$965$	9464.20	−	29578.9i	0.315713	−	0.986713i
$966$		0			0
$967$	−4426.36	−	4426.36i	−0.147200	−	0.147200i	0.629666	−	0.776866i	$-0.283192\pi$
$967$	−4426.36	−	4426.36i	−0.147200	−	0.147200i	−0.776866	+	0.629666i	$0.783192\pi$
$968$	−2493.81	−	2493.81i	−0.0828037	−	0.0828037i
$969$		0			0
$970$	3191.24	−	1644.10i	0.105634	−	0.0544216i
$971$		−	36943.3i		−	1.22097i	−0.792026	−	0.610487i	$-0.790974\pi$
$971$		−	36943.3i		−	1.22097i	0.792026	−	0.610487i	$-0.209026\pi$
$972$		0			0
$973$	20752.0	−	20752.0i	0.683741	−	0.683741i
$974$	1882.32			0.0619236
$975$		0			0
$976$	15875.2			0.520650
$977$	14789.2	−	14789.2i	0.484287	−	0.484287i	−0.422211	−	0.906498i	$-0.638746\pi$
$977$	14789.2	−	14789.2i	0.484287	−	0.484287i	0.906498	+	0.422211i	$0.138746\pi$
$978$		0			0
$979$		−	3915.98i		−	0.127840i
$980$	8592.04	−	4426.55i	0.280064	−	0.144287i
$981$		0			0
$982$	5410.00	+	5410.00i	0.175805	+	0.175805i
$983$	12134.8	+	12134.8i	0.393733	+	0.393733i	0.876015	−	0.482283i	$-0.160192\pi$
$983$	12134.8	+	12134.8i	0.393733	+	0.393733i	−0.482283	+	0.876015i	$0.660192\pi$
$984$		0			0
$985$	−13013.2	+	40670.6i	−0.420948	+	1.31561i
$986$		−	3874.54i		−	0.125142i
$987$		0			0
$988$	1863.91	−	1863.91i	0.0600190	−	0.0600190i
$989$	7458.19			0.239794
$990$		0			0
$991$	29327.6			0.940084			0.470042	−	0.882644i	$-0.344239\pi$
$991$	29327.6			0.940084			0.470042	+	0.882644i	$0.344239\pi$
$992$	−1547.15	+	1547.15i	−0.0495182	+	0.0495182i
$993$		0			0
$994$			5287.34i			0.168717i
$995$	4307.28	+	8360.53i	0.137236	+	0.266379i
$996$		0			0
$997$	−44483.8	−	44483.8i	−1.41306	−	1.41306i	−0.735143	−	0.677912i	$-0.762885\pi$
$997$	−44483.8	−	44483.8i	−1.41306	−	1.41306i	−0.677912	−	0.735143i	$-0.737115\pi$
$998$	−2078.84	−	2078.84i	−0.0659365	−	0.0659365i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.4.f.a.17.3	yes	12
3.2	odd	2	inner	45.4.f.a.17.4	yes	12
4.3	odd	2		720.4.w.d.17.1		12
5.2	odd	4		225.4.f.c.143.3		12
5.3	odd	4	inner	45.4.f.a.8.4	yes	12
5.4	even	2		225.4.f.c.107.4		12
12.11	even	2		720.4.w.d.17.6		12
15.2	even	4		225.4.f.c.143.4		12
15.8	even	4	inner	45.4.f.a.8.3	✓	12
15.14	odd	2		225.4.f.c.107.3		12
20.3	even	4		720.4.w.d.593.6		12
60.23	odd	4		720.4.w.d.593.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.f.a.8.3	✓	12	15.8	even	4	inner
45.4.f.a.8.4	yes	12	5.3	odd	4	inner
45.4.f.a.17.3	yes	12	1.1	even	1	trivial
45.4.f.a.17.4	yes	12	3.2	odd	2	inner
225.4.f.c.107.3		12	15.14	odd	2
225.4.f.c.107.4		12	5.4	even	2
225.4.f.c.143.3		12	5.2	odd	4
225.4.f.c.143.4		12	15.2	even	4
720.4.w.d.17.1		12	4.3	odd	2
720.4.w.d.17.6		12	12.11	even	2
720.4.w.d.593.1		12	60.23	odd	4
720.4.w.d.593.6		12	20.3	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	45.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.287902 + 0.287902i) q^{2} +7.83422i q^{4} +(-9.93887 + 5.12043i) q^{5} +(15.0557 + 15.0557i) q^{7} +(-4.55871 - 4.55871i) q^{8} +O(q^{10})\)	\(q+(-0.287902 + 0.287902i) q^{2} +7.83422i q^{4} +(-9.93887 + 5.12043i) q^{5} +(15.0557 + 15.0557i) q^{7} +(-4.55871 - 4.55871i) q^{8} +(1.38724 - 4.33560i) q^{10} +27.9992i q^{11} +(2.49862 - 2.49862i) q^{13} -8.66913 q^{14} -60.0489 q^{16} +(67.9104 - 67.9104i) q^{17} -95.2200i q^{19} +(-40.1146 - 77.8634i) q^{20} +(-8.06104 - 8.06104i) q^{22} +(121.994 + 121.994i) q^{23} +(72.5624 - 101.783i) q^{25} +1.43872i q^{26} +(-117.950 + 117.950i) q^{28} -99.0852 q^{29} -28.7800 q^{31} +(53.7578 - 53.7578i) q^{32} +39.1031i q^{34} +(-226.728 - 72.5450i) q^{35} +(271.064 + 271.064i) q^{37} +(27.4140 + 27.4140i) q^{38} +(68.6509 + 21.9659i) q^{40} +453.748i q^{41} +(30.5679 - 30.5679i) q^{43} -219.352 q^{44} -70.2445 q^{46} +(254.600 - 254.600i) q^{47} +110.348i q^{49} +(8.41255 + 50.1943i) q^{50} +(19.5748 + 19.5748i) q^{52} +(-224.021 - 224.021i) q^{53} +(-143.368 - 278.281i) q^{55} -137.269i q^{56} +(28.5268 - 28.5268i) q^{58} +483.263 q^{59} -264.372 q^{61} +(8.28582 - 8.28582i) q^{62} -449.437i q^{64} +(-12.0395 + 37.6275i) q^{65} +(-498.817 - 498.817i) q^{67} +(532.025 + 532.025i) q^{68} +(86.1614 - 44.3897i) q^{70} -609.904i q^{71} +(-74.6843 + 74.6843i) q^{73} -156.080 q^{74} +745.975 q^{76} +(-421.548 + 421.548i) q^{77} +406.574i q^{79} +(596.818 - 307.476i) q^{80} +(-130.635 - 130.635i) q^{82} +(-652.278 - 652.278i) q^{83} +(-327.223 + 1022.68i) q^{85} +17.6011i q^{86} +(127.640 - 127.640i) q^{88} -139.860 q^{89} +75.2369 q^{91} +(-955.726 + 955.726i) q^{92} +146.600i q^{94} +(487.567 + 946.380i) q^{95} +(557.633 + 557.633i) q^{97} +(-31.7693 - 31.7693i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 24 q^{7}+O(q^{10}) \) 12 * q + 24 * q^7	\( 12 q + 24 q^{7} + 192 q^{10} - 108 q^{13} - 648 q^{16} + 1056 q^{22} - 144 q^{25} - 576 q^{28} - 1248 q^{31} + 828 q^{37} + 2568 q^{40} - 96 q^{43} + 672 q^{46} - 312 q^{52} - 1512 q^{55} - 3864 q^{58} + 96 q^{61} + 1632 q^{67} - 1536 q^{70} + 3972 q^{73} - 480 q^{76} - 7848 q^{82} - 1752 q^{85} + 7968 q^{88} + 4752 q^{91} + 2772 q^{97}+O(q^{100}) \) 12 * q + 24 * q^7 + 192 * q^10 - 108 * q^13 - 648 * q^16 + 1056 * q^22 - 144 * q^25 - 576 * q^28 - 1248 * q^31 + 828 * q^37 + 2568 * q^40 - 96 * q^43 + 672 * q^46 - 312 * q^52 - 1512 * q^55 - 3864 * q^58 + 96 * q^61 + 1632 * q^67 - 1536 * q^70 + 3972 * q^73 - 480 * q^76 - 7848 * q^82 - 1752 * q^85 + 7968 * q^88 + 4752 * q^91 + 2772 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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