LMFDB - Embedded newform 45.4.f.a.8.4

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			8.4
Root			$-2.02004 - 2.30794i$ of defining polynomial
Character	$\chi$	$=$	45.8
Dual form			45.4.f.a.17.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.287902 + 0.287902i) q^{2} -7.83422i q^{4} +(9.93887 + 5.12043i) q^{5} +(15.0557 - 15.0557i) q^{7} +(4.55871 - 4.55871i) q^{8} +O(q^{10})$	\(q+(0.287902 + 0.287902i) q^{2} -7.83422i q^{4} +(9.93887 + 5.12043i) q^{5} +(15.0557 - 15.0557i) q^{7} +(4.55871 - 4.55871i) q^{8} +(1.38724 + 4.33560i) q^{10} +27.9992i q^{11} +(2.49862 + 2.49862i) q^{13} +8.66913 q^{14} -60.0489 q^{16} +(-67.9104 - 67.9104i) q^{17} +95.2200i q^{19} +(40.1146 - 77.8634i) q^{20} +(-8.06104 + 8.06104i) q^{22} +(-121.994 + 121.994i) q^{23} +(72.5624 + 101.783i) q^{25} +1.43872i q^{26} +(-117.950 - 117.950i) q^{28} +99.0852 q^{29} -28.7800 q^{31} +(-53.7578 - 53.7578i) q^{32} -39.1031i q^{34} +(226.728 - 72.5450i) q^{35} +(271.064 - 271.064i) q^{37} +(-27.4140 + 27.4140i) q^{38} +(68.6509 - 21.9659i) q^{40} +453.748i q^{41} +(30.5679 + 30.5679i) q^{43} +219.352 q^{44} -70.2445 q^{46} +(-254.600 - 254.600i) q^{47} -110.348i q^{49} +(-8.41255 + 50.1943i) q^{50} +(19.5748 - 19.5748i) q^{52} +(224.021 - 224.021i) q^{53} +(-143.368 + 278.281i) q^{55} -137.269i q^{56} +(28.5268 + 28.5268i) q^{58} -483.263 q^{59} -264.372 q^{61} +(-8.28582 - 8.28582i) q^{62} +449.437i q^{64} +(12.0395 + 37.6275i) q^{65} +(-498.817 + 498.817i) q^{67} +(-532.025 + 532.025i) q^{68} +(86.1614 + 44.3897i) q^{70} -609.904i q^{71} +(-74.6843 - 74.6843i) q^{73} +156.080 q^{74} +745.975 q^{76} +(421.548 + 421.548i) q^{77} -406.574i q^{79} +(-596.818 - 307.476i) q^{80} +(-130.635 + 130.635i) q^{82} +(652.278 - 652.278i) q^{83} +(-327.223 - 1022.68i) q^{85} +17.6011i q^{86} +(127.640 + 127.640i) q^{88} +139.860 q^{89} +75.2369 q^{91} +(955.726 + 955.726i) q^{92} -146.600i q^{94} +(-487.567 + 946.380i) q^{95} +(557.633 - 557.633i) q^{97} +(31.7693 - 31.7693i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$

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	$-0.227070\pi$
$2$	0.287902	+	0.287902i	0.101789	+	0.101789i	−0.654378	+	0.756167i	$0.727070\pi$
$3$		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	−0.172141	−	0.985072i	$-0.555069\pi$
$7$	15.0557	−	15.0557i	0.812931	−	0.812931i	0.985072	+	0.172141i	$0.0550686\pi$
$8$	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	0.733258	−	0.679951i	$-0.237999\pi$
$13$	2.49862	+	2.49862i	0.0533071	+	0.0533071i	−0.679951	+	0.733258i	$0.737999\pi$
$14$	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	$-0.490237\pi$
$17$	−67.9104	−	67.9104i	−0.968864	−	0.968864i	−0.999530	+	0.0306653i	$0.990237\pi$
$18$		0			0
$19$			95.2200i			1.14974i	0.818246	+	0.574868i	$0.194946\pi$
$19$			95.2200i			1.14974i	−0.818246	+	0.574868i	$0.805054\pi$
$20$	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	−0.112302	+	0.993674i	$0.535822\pi$
$23$	−121.994	+	121.994i	−1.10598	+	1.10598i	−0.993674	+	0.112302i	$0.964178\pi$
$24$		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	$-0.397245\pi$
$29$	99.0852			0.634471			0.317235	+	0.948347i	$0.397245\pi$
$30$		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	0.231578	−	0.972816i	$-0.425611\pi$
$37$	271.064	−	271.064i	1.20439	−	1.20439i	0.972816	−	0.231578i	$-0.0743888\pi$
$38$	−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	0.759230	−	0.650822i	$-0.225576\pi$
$43$	30.5679	+	30.5679i	0.108409	+	0.108409i	−0.650822	+	0.759230i	$0.725576\pi$
$44$	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	$-0.438708\pi$
$47$	−254.600	−	254.600i	−0.790153	−	0.790153i	−0.981519	+	0.191365i	$0.938708\pi$
$48$		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	$-0.615339\pi$
$53$	224.021	−	224.021i	0.580598	−	0.580598i	0.935068	+	0.354470i	$0.115339\pi$
$54$		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	$-0.679004\pi$
$59$	−483.263			−1.06636			−0.533182	+	0.846001i	$0.679004\pi$
$60$		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	−0.996236	−	0.0866805i	$-0.972374\pi$
$67$	−498.817	+	498.817i	−0.909556	+	0.909556i	0.0866805	+	0.996236i	$0.472374\pi$
$68$	−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	0.644697	−	0.764438i	$-0.276984\pi$
$73$	−74.6843	−	74.6843i	−0.119742	−	0.119742i	−0.764438	+	0.644697i	$0.776984\pi$
$74$	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	$-0.906507\pi$
$79$		−	406.574i		−	0.579027i	0.957174	−	0.289513i	$-0.0934935\pi$
$80$	−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	$-0.541186\pi$
$83$	652.278	−	652.278i	0.862613	−	0.862613i	0.991641	+	0.129028i	$0.0411858\pi$
$84$		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	$-0.473458\pi$
$89$	139.860			0.166575			0.0832873	+	0.996526i	$0.473458\pi$
$90$		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	−0.352217	−	0.935918i	$-0.614572\pi$
$97$	557.633	−	557.633i	0.583701	−	0.583701i	0.935918	+	0.352217i	$0.114572\pi$
$98$	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	0.927279	−	0.374372i	$-0.122142\pi$
$103$	577.974	+	577.974i	0.552907	+	0.552907i	−0.374372	+	0.927279i	$0.622142\pi$
$104$	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	$-0.205730\pi$
$107$	216.994	+	216.994i	0.196053	+	0.196053i	−0.602253	+	0.798305i	$0.705730\pi$
$108$		0			0
$109$		−	936.415i		−	0.822865i	−0.911440	−	0.411432i	$-0.865029\pi$
$109$		−	936.415i		−	0.822865i	0.911440	−	0.411432i	$-0.134971\pi$
$110$	−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	$-0.529715\pi$
$113$	1084.00	−	1084.00i	0.902428	−	0.902428i	0.995646	+	0.0932173i	$0.0297151\pi$
$114$		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	−0.950486	−	0.310769i	$-0.899414\pi$
$127$	−915.574	+	915.574i	−0.639717	+	0.639717i	0.310769	+	0.950486i	$0.399414\pi$
$128$	−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	$-0.344137\pi$
$137$	−660.931	−	660.931i	−0.412169	−	0.412169i	−0.882494	+	0.470325i	$0.844137\pi$
$138$		0			0
$139$			1378.35i			0.841081i	0.907274	+	0.420540i	$0.138160\pi$
$139$			1378.35i			0.841081i	−0.907274	+	0.420540i	$0.861840\pi$
$140$	−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	$-0.418183\pi$
$149$	924.719			0.508429			0.254215	+	0.967148i	$0.418183\pi$
$150$		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	−0.292582	−	0.956240i	$-0.594514\pi$
$157$	1305.55	−	1305.55i	0.663659	−	0.663659i	0.956240	+	0.292582i	$0.0945144\pi$
$158$	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	0.687911	−	0.725795i	$-0.258528\pi$
$163$	−78.8393	−	78.8393i	−0.0378845	−	0.0378845i	−0.725795	+	0.687911i	$0.758528\pi$
$164$	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	$-0.294798\pi$
$167$	−428.116	−	428.116i	−0.198375	−	0.198375i	−0.799303	+	0.600928i	$0.794798\pi$
$168$		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	$-0.892235\pi$
$173$	−1390.57	+	1390.57i	−0.611114	+	0.611114i	0.332122	+	0.943236i	$0.392235\pi$
$174$		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	$-0.507356\pi$
$179$	−110.674			−0.0462131			−0.0231065	+	0.999733i	$0.507356\pi$
$180$		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	0.238569	−	0.971125i	$-0.423322\pi$
$193$	−1964.16	−	1964.16i	−0.732556	−	0.732556i	−0.971125	+	0.238569i	$0.923322\pi$
$194$	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	$-0.492678\pi$
$197$	−2700.70	−	2700.70i	−0.976735	−	0.976735i	−0.999735	+	0.0230004i	$0.992678\pi$
$198$		0			0
$199$			841.195i			0.299652i	0.988712	+	0.149826i	$0.0478713\pi$
$199$			841.195i			0.299652i	−0.988712	+	0.149826i	$0.952129\pi$
$200$	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	0.996402	+	0.0847496i	$0.0270090\pi$
$223$	3600.34	+	3600.34i	1.08115	+	1.08115i	0.0847496	+	0.996402i	$0.472991\pi$
$224$	−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	$-0.366770\pi$
$227$	−1734.81	−	1734.81i	−0.507239	−	0.507239i	−0.913678	+	0.406439i	$0.866770\pi$
$228$		0			0
$229$			3467.74i			1.00067i	0.865831	+	0.500337i	$0.166791\pi$
$229$			3467.74i			1.00067i	−0.865831	+	0.500337i	$0.833209\pi$
$230$	−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	$-0.818248\pi$
$233$	−1070.17	+	1070.17i	−0.300899	+	0.300899i	0.540467	+	0.841365i	$0.318248\pi$
$234$		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	$-0.322553\pi$
$239$	3909.44			1.05808			0.529038	+	0.848598i	$0.322553\pi$
$240$		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	$-0.196330\pi$
$257$	977.764	+	977.764i	0.237320	+	0.237320i	−0.578419	+	0.815740i	$0.696330\pi$
$258$		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	$-0.708153\pi$
$263$	790.690	−	790.690i	0.185384	−	0.185384i	0.793697	+	0.608313i	$0.208153\pi$
$264$		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	$-0.352080\pi$
$269$	3954.49			0.896317			0.448159	+	0.893954i	$0.352080\pi$
$270$		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	−0.601737	−	0.798694i	$-0.705525\pi$
$277$	908.009	−	908.009i	0.196957	−	0.196957i	0.798694	+	0.601737i	$0.205525\pi$
$278$	−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	0.00813014	−	0.999967i	$-0.497412\pi$
$283$	−4721.93	−	4721.93i	−0.991837	−	0.991837i	−0.999967	+	0.00813014i	$0.997412\pi$
$284$	−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	$-0.844672\pi$
$293$	−2078.57	+	2078.57i	−0.414442	+	0.414442i	0.468841	+	0.883283i	$0.344672\pi$
$294$		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	−0.999951	−	0.00989542i	$-0.996850\pi$
$307$	−5325.58	+	5325.58i	−0.990056	+	0.990056i	0.00989542	+	0.999951i	$0.496850\pi$
$308$	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	0.996087	−	0.0883746i	$-0.0281673\pi$
$313$	5026.49	+	5026.49i	0.907713	+	0.907713i	−0.0883746	+	0.996087i	$0.528167\pi$
$314$	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	$-0.378575\pi$
$317$	−3137.15	−	3137.15i	−0.555836	−	0.555836i	−0.928119	+	0.372283i	$0.878575\pi$
$318$		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	−0.999490	−	0.0319428i	$-0.989831\pi$
$337$	−5985.73	+	5985.73i	−0.967547	+	0.967547i	0.0319428	+	0.999490i	$0.489831\pi$
$338$	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	0.948130	+	0.317881i	$0.102971\pi$
$347$	8183.37	+	8183.37i	1.26601	+	1.26601i	0.317881	+	0.948130i	$0.397029\pi$
$348$		0			0
$349$			1421.22i			0.217983i	0.994043	+	0.108991i	$0.0347621\pi$
$349$			1421.22i			0.217983i	−0.994043	+	0.108991i	$0.965238\pi$
$350$	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	$-0.797675\pi$
$353$	−1399.56	+	1399.56i	−0.211023	+	0.211023i	0.593679	+	0.804702i	$0.297675\pi$
$354$		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	$-0.602712\pi$
$359$	−4313.98			−0.634215			−0.317108	+	0.948390i	$0.602712\pi$
$360$		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	−0.306938	−	0.951730i	$-0.599304\pi$
$367$	4533.34	−	4533.34i	0.644792	−	0.644792i	0.951730	+	0.306938i	$0.0993043\pi$
$368$	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	0.486609	−	0.873620i	$-0.338234\pi$
$373$	−2787.95	−	2787.95i	−0.387010	−	0.387010i	−0.873620	+	0.486609i	$0.838234\pi$
$374$	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	$-0.807002\pi$
$379$		−	8409.09i		−	1.13970i	0.821749	−	0.569849i	$-0.192998\pi$
$380$	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	−0.145672	+	0.989333i	$0.546534\pi$
$383$	−8507.38	+	8507.38i	−1.13500	+	1.13500i	−0.989333	+	0.145672i	$0.953466\pi$
$384$		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	$-0.309591\pi$
$389$	8641.23			1.12629			0.563146	+	0.826357i	$0.309591\pi$
$390$		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	−0.576571	−	0.817047i	$-0.695610\pi$
$397$	1902.21	−	1902.21i	0.240476	−	0.240476i	0.817047	+	0.576571i	$0.195610\pi$
$398$	−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	$0.126340\pi$
$409$			6395.02i			0.773138i	−0.922260	+	0.386569i	$0.873660\pi$
$410$	−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	$-0.335510\pi$
$419$	8474.94			0.988133			0.494067	+	0.869424i	$0.335510\pi$
$420$		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	0.439587	−	0.898200i	$-0.355125\pi$
$433$	−4132.17	−	4132.17i	−0.458613	−	0.458613i	−0.898200	+	0.439587i	$0.855125\pi$
$434$	−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	$0.193333\pi$
$439$			10498.9i			1.14143i	−0.821149	+	0.570714i	$0.806667\pi$
$440$	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	$-0.785460\pi$
$443$	−1465.93	+	1465.93i	−0.157220	+	0.157220i	0.624114	+	0.781333i	$0.285460\pi$
$444$		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	$-0.392666\pi$
$449$	6295.42			0.661691			0.330846	+	0.943685i	$0.392666\pi$
$450$		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	−0.0696767	−	0.997570i	$-0.522197\pi$
$457$	9065.09	−	9065.09i	0.927893	−	0.927893i	0.997570	+	0.0696767i	$0.0221968\pi$
$458$	−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	0.532666	−	0.846326i	$-0.321190\pi$
$463$	−3124.86	−	3124.86i	−0.313660	−	0.313660i	−0.846326	+	0.532666i	$0.821190\pi$
$464$	−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	$-0.346365\pi$
$467$	−4255.07	−	4255.07i	−0.421629	−	0.421629i	−0.885764	+	0.464135i	$0.846365\pi$
$468$		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	$-0.669157\pi$
$479$	−10625.2			−1.01352			−0.506761	+	0.862087i	$0.669157\pi$
$480$		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	−0.842645	−	0.538469i	$-0.819003\pi$
$487$	−3269.03	+	3269.03i	−0.304177	+	0.304177i	0.538469	+	0.842645i	$0.319003\pi$
$488$	−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	$-0.895010\pi$
$499$		−	7220.67i		−	0.647778i	0.946095	−	0.323889i	$-0.104990\pi$
$500$	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	$-0.791646\pi$
$503$	−2081.38	+	2081.38i	−0.184501	+	0.184501i	0.608813	+	0.793314i	$0.291646\pi$
$504$		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	$-0.314758\pi$
$509$	12624.1			1.09932			0.549659	+	0.835389i	$0.314758\pi$
$510$		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	0.436583	−	0.899664i	$-0.356189\pi$
$523$	−5538.72	−	5538.72i	−0.463081	−	0.463081i	−0.899664	+	0.436583i	$0.856189\pi$
$524$	−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	−0.844141	−	0.536121i	$-0.819889\pi$
$547$	−3940.59	+	3940.59i	−0.308021	+	0.308021i	0.536121	+	0.844141i	$0.319889\pi$
$548$	−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	$-0.200998\pi$
$557$	2850.64	+	2850.64i	0.216850	+	0.216850i	−0.590320	+	0.807170i	$0.700998\pi$
$558$		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	$-0.512009\pi$
$563$	12845.3	−	12845.3i	0.961569	−	0.961569i	0.999288	+	0.0377198i	$0.0120094\pi$
$564$		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	$-0.125200\pi$
$569$	25072.7			1.84728			0.923639	+	0.383263i	$0.125200\pi$
$570$		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	−0.679155	−	0.733995i	$-0.737654\pi$
$577$	760.083	−	760.083i	0.0548400	−	0.0548400i	0.733995	+	0.679155i	$0.237654\pi$
$578$	−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	$-0.116929\pi$
$587$	8165.43	+	8165.43i	0.574146	+	0.574146i	−0.359139	+	0.933284i	$0.616929\pi$
$588$		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	$-0.910409\pi$
$593$	−9861.32	+	9861.32i	−0.682893	+	0.682893i	0.277758	+	0.960651i	$0.410409\pi$
$594$		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	$-0.750566\pi$
$599$	−20769.5			−1.41673			−0.708363	+	0.705848i	$0.750566\pi$
$600$		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	−0.161565	+	0.986862i	$0.551654\pi$
$607$	−17174.6	+	17174.6i	−1.14843	+	1.14843i	−0.986862	+	0.161565i	$0.948346\pi$
$608$	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	0.978195	−	0.207691i	$-0.0665947\pi$
$613$	11694.1	+	11694.1i	0.770504	+	0.770504i	−0.207691	+	0.978195i	$0.566595\pi$
$614$	−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	$-0.167016\pi$
$617$	5586.69	+	5586.69i	0.364525	+	0.364525i	−0.500951	+	0.865476i	$0.667016\pi$
$618$		0			0
$619$			16137.1i			1.04783i	0.851771	+	0.523914i	$0.175529\pi$
$619$			16137.1i			1.04783i	−0.851771	+	0.523914i	$0.824471\pi$
$620$	−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	0.997133	−	0.0756629i	$-0.0241073\pi$
$643$	15024.4	+	15024.4i	0.921471	+	0.921471i	−0.0756629	+	0.997133i	$0.524107\pi$
$644$	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	$-0.217163\pi$
$647$	2396.69	+	2396.69i	0.145631	+	0.145631i	−0.630532	+	0.776163i	$0.717163\pi$
$648$		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	$-0.597899\pi$
$653$	10852.1	−	10852.1i	0.650343	−	0.650343i	0.953076	+	0.302732i	$0.0978988\pi$
$654$		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	$-0.715121\pi$
$659$	−21164.7			−1.25108			−0.625538	+	0.780193i	$0.715121\pi$
$660$		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	0.659241	−	0.751932i	$-0.270878\pi$
$673$	−1618.29	−	1618.29i	−0.0926903	−	0.0926903i	−0.751932	+	0.659241i	$0.770878\pi$
$674$	−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	$-0.0259874\pi$
$677$	16119.8	+	16119.8i	0.915118	+	0.915118i	−0.0815511	+	0.996669i	$0.525987\pi$
$678$		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	0.411351	−	0.911477i	$-0.365057\pi$
$683$	23612.1	−	23612.1i	1.32283	−	1.32283i	0.911477	−	0.411351i	$-0.134943\pi$
$684$		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	$-0.957832\pi$
$709$		−	4987.22i		−	0.264173i	0.991238	−	0.132087i	$-0.0421677\pi$
$710$	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	$-0.162905\pi$
$719$	33618.4			1.74375			0.871873	+	0.489732i	$0.162905\pi$
$720$		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	−0.915477	−	0.402371i	$-0.868186\pi$
$727$	−10057.9	+	10057.9i	−0.513106	+	0.513106i	0.402371	+	0.915477i	$0.368186\pi$
$728$	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	0.593048	−	0.805167i	$-0.297924\pi$
$733$	−4209.55	−	4209.55i	−0.212119	−	0.212119i	−0.805167	+	0.593048i	$0.797924\pi$
$734$	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	$-0.962278\pi$
$739$		−	4750.38i		−	0.236462i	0.992986	−	0.118231i	$-0.0377223\pi$
$740$	−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	$-0.942251\pi$
$743$	−16266.1	+	16266.1i	−0.803159	+	0.803159i	0.180429	+	0.983588i	$0.442251\pi$
$744$		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	−0.849836	−	0.527048i	$-0.823299\pi$
$757$	−6722.97	+	6722.97i	−0.322788	+	0.322788i	0.527048	+	0.849836i	$0.323299\pi$
$758$	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	$0.200991\pi$
$769$			25176.4i			1.18060i	−0.807183	+	0.590301i	$0.799009\pi$
$770$	−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	$-0.827446\pi$
$773$	−7322.13	+	7322.13i	−0.340697	+	0.340697i	0.515932	+	0.856629i	$0.327446\pi$
$774$		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	−0.215763	−	0.976446i	$-0.569224\pi$
$787$	16794.5	−	16794.5i	0.760683	−	0.760683i	0.976446	+	0.215763i	$0.0692237\pi$
$788$	−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	$-0.453070\pi$
$797$	−18950.9	−	18950.9i	−0.842251	−	0.842251i	−0.989151	+	0.146900i	$0.953070\pi$
$798$		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	$-0.527475\pi$
$809$	−3967.32			−0.172415			−0.0862074	+	0.996277i	$0.527475\pi$
$810$		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	0.748628	−	0.662990i	$-0.230713\pi$
$823$	2021.92	+	2021.92i	0.0856375	+	0.0856375i	−0.662990	+	0.748628i	$0.730713\pi$
$824$	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	$-0.430039\pi$
$827$	−18025.3	−	18025.3i	−0.757922	−	0.757922i	−0.975944	+	0.218022i	$0.930039\pi$
$828$		0			0
$829$		−	42400.5i		−	1.77640i	−0.459462	−	0.888198i	$-0.651958\pi$
$829$		−	42400.5i		−	1.77640i	0.459462	−	0.888198i	$-0.348042\pi$
$830$	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	$-0.213510\pi$
$839$	38073.9			1.56670			0.783348	+	0.621584i	$0.213510\pi$
$840$		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	−0.997973	−	0.0636372i	$-0.979730\pi$
$853$	−26447.7	−	26447.7i	−1.06161	−	1.06161i	−0.0636372	−	0.997973i	$-0.520270\pi$
$854$	−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	$-0.429854\pi$
$857$	−18997.5	−	18997.5i	−0.757227	−	0.757227i	−0.975817	+	0.218590i	$0.929854\pi$
$858$		0			0
$859$		−	29326.5i		−	1.16485i	−0.812885	−	0.582425i	$-0.802104\pi$
$859$		−	29326.5i		−	1.16485i	0.812885	−	0.582425i	$-0.197896\pi$
$860$	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	$-0.527492\pi$
$863$	23070.8	−	23070.8i	0.910011	−	0.910011i	0.996273	+	0.0862616i	$0.0274921\pi$
$864$		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	−0.200001	−	0.979796i	$-0.564094\pi$
$877$	20252.6	−	20252.6i	0.779795	−	0.779795i	0.979796	+	0.200001i	$0.0640944\pi$
$878$	−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	0.957886	−	0.287149i	$-0.0927075\pi$
$883$	17599.2	+	17599.2i	0.670737	+	0.670737i	−0.287149	+	0.957886i	$0.592707\pi$
$884$	−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	$-0.171679\pi$
$887$	9099.97	+	9099.97i	0.344472	+	0.344472i	−0.513573	+	0.858046i	$0.671679\pi$
$888$		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	0.0569624	−	0.998376i	$-0.481858\pi$
$907$	28827.2	−	28827.2i	1.05534	−	1.05534i	0.998376	−	0.0569624i	$-0.0181415\pi$
$908$	−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	$-0.828135\pi$
$919$		−	28643.9i		−	1.02815i	0.857744	−	0.514077i	$-0.171865\pi$
$920$	−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	$-0.568429\pi$
$929$	−12080.7			−0.426648			−0.213324	+	0.976981i	$0.568429\pi$
$930$		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	−0.948944	−	0.315444i	$-0.897846\pi$
$937$	−18170.0	+	18170.0i	−0.633500	+	0.633500i	0.315444	+	0.948944i	$0.397846\pi$
$938$	−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	$-0.173636\pi$
$947$	9792.71	+	9792.71i	0.336030	+	0.336030i	−0.518841	+	0.854871i	$0.673636\pi$
$948$		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	$-0.932975\pi$
$953$	−22620.8	+	22620.8i	−0.768899	+	0.768899i	0.209014	+	0.977913i	$0.432975\pi$
$954$		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	−0.776866	−	0.629666i	$-0.783192\pi$
$967$	−4426.36	+	4426.36i	−0.147200	+	0.147200i	0.629666	+	0.776866i	$0.283192\pi$
$968$	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	$-0.361254\pi$
$977$	−14789.2	−	14789.2i	−0.484287	−	0.484287i	−0.906498	+	0.422211i	$0.861254\pi$
$978$		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	$-0.839808\pi$
$983$	−12134.8	+	12134.8i	−0.393733	+	0.393733i	0.482283	+	0.876015i	$0.339808\pi$
$984$		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	−0.677912	+	0.735143i	$0.737115\pi$
$997$	−44483.8	+	44483.8i	−1.41306	+	1.41306i	−0.735143	+	0.677912i	$0.762885\pi$
$998$	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.8.4	yes	12
3.2	odd	2	inner	45.4.f.a.8.3	✓	12
4.3	odd	2		720.4.w.d.593.6		12
5.2	odd	4	inner	45.4.f.a.17.3	yes	12
5.3	odd	4		225.4.f.c.107.4		12
5.4	even	2		225.4.f.c.143.3		12
12.11	even	2		720.4.w.d.593.1		12
15.2	even	4	inner	45.4.f.a.17.4	yes	12
15.8	even	4		225.4.f.c.107.3		12
15.14	odd	2		225.4.f.c.143.4		12
20.7	even	4		720.4.w.d.17.1		12
60.47	odd	4		720.4.w.d.17.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.f.a.8.3	✓	12	3.2	odd	2	inner
45.4.f.a.8.4	yes	12	1.1	even	1	trivial
45.4.f.a.17.3	yes	12	5.2	odd	4	inner
45.4.f.a.17.4	yes	12	15.2	even	4	inner
225.4.f.c.107.3		12	15.8	even	4
225.4.f.c.107.4		12	5.3	odd	4
225.4.f.c.143.3		12	5.4	even	2
225.4.f.c.143.4		12	15.14	odd	2
720.4.w.d.17.1		12	20.7	even	4
720.4.w.d.17.6		12	60.47	odd	4
720.4.w.d.593.1		12	12.11	even	2
720.4.w.d.593.6		12	4.3	odd	2

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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