LMFDB - Embedded newform 225.5.g.m.118.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [225,5,Mod(82,225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 5, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("225.82"); S:= CuspForms(chi, 5); N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	225.g (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,-20,180,0,0,288,0,340,0,0,620] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(16)] == traces)

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

Self dual:	no
Analytic conductor:	$23.2582416939$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 60x^{5} + 1973x^{4} - 3300x^{3} + 1800x^{2} + 31560x + 276676 $ x^8 - 60x^5 + 1973x^4 - 3300x^3 + 1800x^2 + 31560*x + 276676
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			118.1
Root			$-5.02811 + 5.02811i$ of defining polynomial
Character	$\chi$	$=$	225.118
Dual form			225.5.g.m.82.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-5.02811 + 5.02811i) q^{2} -34.5637i q^{4} +(38.5593 - 38.5593i) q^{7} +(93.3405 + 93.3405i) q^{8} +40.4333 q^{11} +(-20.8688 - 20.8688i) q^{13} +387.760i q^{14} -385.633 q^{16} +(15.8713 - 15.8713i) q^{17} +314.926i q^{19} +(-203.303 + 203.303i) q^{22} +(-572.869 - 572.869i) q^{23} +209.862 q^{26} +(-1332.75 - 1332.75i) q^{28} -824.433i q^{29} -1347.19 q^{31} +(445.555 - 445.555i) q^{32} +159.606i q^{34} +(589.843 - 589.843i) q^{37} +(-1583.48 - 1583.48i) q^{38} -1856.55 q^{41} +(671.078 + 671.078i) q^{43} -1397.53i q^{44} +5760.90 q^{46} +(504.865 - 504.865i) q^{47} -572.636i q^{49} +(-721.305 + 721.305i) q^{52} +(2251.32 + 2251.32i) q^{53} +7198.29 q^{56} +(4145.34 + 4145.34i) q^{58} -2585.03i q^{59} -3276.74 q^{61} +(6773.81 - 6773.81i) q^{62} -1689.53i q^{64} +(3428.94 - 3428.94i) q^{67} +(-548.573 - 548.573i) q^{68} +5679.51 q^{71} +(-4450.06 - 4450.06i) q^{73} +5931.59i q^{74} +10885.0 q^{76} +(1559.08 - 1559.08i) q^{77} -6465.77i q^{79} +(9334.92 - 9334.92i) q^{82} +(-621.380 - 621.380i) q^{83} -6748.50 q^{86} +(3774.07 + 3774.07i) q^{88} -1856.76i q^{89} -1609.37 q^{91} +(-19800.5 + 19800.5i) q^{92} +5077.03i q^{94} +(-12383.5 + 12383.5i) q^{97} +(2879.27 + 2879.27i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 20 q^{7} + 180 q^{8} + 288 q^{11} + 340 q^{13} + 620 q^{16} + 900 q^{17} + 1100 q^{22} - 1560 q^{23} + 3024 q^{26} - 3580 q^{28} - 512 q^{31} + 4980 q^{32} + 3820 q^{37} - 7680 q^{38} + 2712 q^{41}+ \cdots + 46440 q^{98}+O(q^{100}) $ 8 * q - 20 * q^7 + 180 * q^8 + 288 * q^11 + 340 * q^13 + 620 * q^16 + 900 * q^17 + 1100 * q^22 - 1560 * q^23 + 3024 * q^26 - 3580 * q^28 - 512 * q^31 + 4980 * q^32 + 3820 * q^37 - 7680 * q^38 + 2712 * q^41 + 1240 * q^43 + 13528 * q^46 + 4800 * q^47 + 1240 * q^52 + 1020 * q^53 + 30720 * q^56 - 2340 * q^58 - 4760 * q^61 + 28680 * q^62 + 8920 * q^67 - 1920 * q^68 - 7536 * q^71 - 11600 * q^73 + 4344 * q^76 - 360 * q^77 + 27200 * q^82 - 32400 * q^83 - 14592 * q^86 + 14340 * q^88 + 16528 * q^91 - 31800 * q^92 - 58640 * q^97 + 46440 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	−5.02811	+	5.02811i	−1.25703	+	1.25703i	−0.304522	+	0.952505i	$0.598497\pi$
$2$	−5.02811	+	5.02811i	−1.25703	+	1.25703i	−0.952505	+	0.304522i	$0.901503\pi$
$3$		0			0
$3$		0			0
$4$		−	34.5637i		−	2.16023i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	38.5593	−	38.5593i	0.786924	−	0.786924i	−0.194065	−	0.980989i	$-0.562167\pi$
$7$	38.5593	−	38.5593i	0.786924	−	0.786924i	0.980989	+	0.194065i	$0.0621672\pi$
$8$	93.3405	+	93.3405i	1.45845	+	1.45845i
$9$		0			0
$10$		0			0
$11$	40.4333			0.334160			0.167080	−	0.985943i	$-0.446566\pi$
$11$	40.4333			0.334160			0.167080	+	0.985943i	$0.446566\pi$
$12$		0			0
$13$	−20.8688	−	20.8688i	−0.123484	−	0.123484i	0.642664	−	0.766148i	$-0.277829\pi$
$13$	−20.8688	−	20.8688i	−0.123484	−	0.123484i	−0.766148	+	0.642664i	$0.777829\pi$
$14$			387.760i			1.97837i
$15$		0			0
$16$	−385.633			−1.50638
$17$	15.8713	−	15.8713i	0.0549181	−	0.0549181i	−0.679114	−	0.734032i	$-0.737636\pi$
$17$	15.8713	−	15.8713i	0.0549181	−	0.0549181i	0.734032	+	0.679114i	$0.237636\pi$
$18$		0			0
$19$			314.926i			0.872371i	0.899857	+	0.436185i	$0.143671\pi$
$19$			314.926i			0.872371i	−0.899857	+	0.436185i	$0.856329\pi$
$20$		0			0
$21$		0			0
$22$	−203.303	+	203.303i	−0.420048	+	0.420048i
$23$	−572.869	−	572.869i	−1.08293	−	1.08293i	−0.996235	−	0.0866935i	$-0.972370\pi$
$23$	−572.869	−	572.869i	−1.08293	−	1.08293i	−0.0866935	−	0.996235i	$-0.527630\pi$
$24$		0			0
$25$		0			0
$26$	209.862			0.310446
$27$		0			0
$28$	−1332.75	−	1332.75i	−1.69994	−	1.69994i
$29$		−	824.433i		−	0.980301i	−0.871638	−	0.490151i	$-0.836942\pi$
$29$		−	824.433i		−	0.980301i	0.871638	−	0.490151i	$-0.163058\pi$
$30$		0			0
$31$	−1347.19			−1.40186			−0.700931	−	0.713229i	$-0.747232\pi$
$31$	−1347.19			−1.40186			−0.700931	+	0.713229i	$0.747232\pi$
$32$	445.555	−	445.555i	0.435112	−	0.435112i
$33$		0			0
$34$			159.606i			0.138067i
$35$		0			0
$36$		0			0
$37$	589.843	−	589.843i	0.430857	−	0.430857i	−0.458063	−	0.888920i	$-0.651457\pi$
$37$	589.843	−	589.843i	0.430857	−	0.430857i	0.888920	+	0.458063i	$0.151457\pi$
$38$	−1583.48	−	1583.48i	−1.09659	−	1.09659i
$39$		0			0
$40$		0			0
$41$	−1856.55			−1.10443			−0.552215	−	0.833702i	$-0.686217\pi$
$41$	−1856.55			−1.10443			−0.552215	+	0.833702i	$0.686217\pi$
$42$		0			0
$43$	671.078	+	671.078i	0.362941	+	0.362941i	0.864894	−	0.501954i	$-0.167385\pi$
$43$	671.078	+	671.078i	0.362941	+	0.362941i	−0.501954	+	0.864894i	$0.667385\pi$
$44$		−	1397.53i		−	0.721863i
$45$		0			0
$46$	5760.90			2.72254
$47$	504.865	−	504.865i	0.228549	−	0.228549i	−0.583537	−	0.812086i	$-0.698332\pi$
$47$	504.865	−	504.865i	0.228549	−	0.228549i	0.812086	+	0.583537i	$0.198332\pi$
$48$		0			0
$49$		−	572.636i		−	0.238499i
$50$		0			0
$51$		0			0
$52$	−721.305	+	721.305i	−0.266755	+	0.266755i
$53$	2251.32	+	2251.32i	0.801468	+	0.801468i	0.983325	−	0.181857i	$-0.0582107\pi$
$53$	2251.32	+	2251.32i	0.801468	+	0.801468i	−0.181857	+	0.983325i	$0.558211\pi$
$54$		0			0
$55$		0			0
$56$	7198.29			2.29537
$57$		0			0
$58$	4145.34	+	4145.34i	1.23227	+	1.23227i
$59$		−	2585.03i		−	0.742610i	−0.928511	−	0.371305i	$-0.878910\pi$
$59$		−	2585.03i		−	0.742610i	0.928511	−	0.371305i	$-0.121090\pi$
$60$		0			0
$61$	−3276.74			−0.880607			−0.440304	−	0.897849i	$-0.645129\pi$
$61$	−3276.74			−0.880607			−0.440304	+	0.897849i	$0.645129\pi$
$62$	6773.81	−	6773.81i	1.76218	−	1.76218i
$63$		0			0
$64$		−	1689.53i		−	0.412483i
$65$		0			0
$66$		0			0
$67$	3428.94	−	3428.94i	0.763854	−	0.763854i	−0.213163	−	0.977017i	$-0.568376\pi$
$67$	3428.94	−	3428.94i	0.763854	−	0.763854i	0.977017	+	0.213163i	$0.0683764\pi$
$68$	−548.573	−	548.573i	−0.118636	−	0.118636i
$69$		0			0
$70$		0			0
$71$	5679.51			1.12666			0.563331	−	0.826231i	$-0.309519\pi$
$71$	5679.51			1.12666			0.563331	+	0.826231i	$0.309519\pi$
$72$		0			0
$73$	−4450.06	−	4450.06i	−0.835065	−	0.835065i	0.153140	−	0.988205i	$-0.451061\pi$
$73$	−4450.06	−	4450.06i	−0.835065	−	0.835065i	−0.988205	+	0.153140i	$0.951061\pi$
$74$			5931.59i			1.08320i
$75$		0			0
$76$	10885.0			1.88453
$77$	1559.08	−	1559.08i	0.262958	−	0.262958i
$78$		0			0
$79$		−	6465.77i		−	1.03602i	−0.855376	−	0.518008i	$-0.826674\pi$
$79$		−	6465.77i		−	1.03602i	0.855376	−	0.518008i	$-0.173326\pi$
$80$		0			0
$81$		0			0
$82$	9334.92	−	9334.92i	1.38830	−	1.38830i
$83$	−621.380	−	621.380i	−0.0901988	−	0.0901988i	0.660568	−	0.750766i	$-0.270316\pi$
$83$	−621.380	−	621.380i	−0.0901988	−	0.0901988i	−0.750766	+	0.660568i	$0.770316\pi$
$84$		0			0
$85$		0			0
$86$	−6748.50			−0.912453
$87$		0			0
$88$	3774.07	+	3774.07i	0.487354	+	0.487354i
$89$		−	1856.76i		−	0.234410i	−0.993108	−	0.117205i	$-0.962606\pi$
$89$		−	1856.76i		−	0.234410i	0.993108	−	0.117205i	$-0.0373935\pi$
$90$		0			0
$91$	−1609.37			−0.194345
$92$	−19800.5	+	19800.5i	−2.33938	+	2.33938i
$93$		0			0
$94$			5077.03i			0.574585i
$95$		0			0
$96$		0			0
$97$	−12383.5	+	12383.5i	−1.31614	+	1.31614i	−0.399332	+	0.916806i	$0.630758\pi$
$97$	−12383.5	+	12383.5i	−1.31614	+	1.31614i	−0.916806	+	0.399332i	$0.869242\pi$
$98$	2879.27	+	2879.27i	0.299800	+	0.299800i
$99$		0			0
$100$		0			0
$101$	189.344			0.0185613			0.00928067	−	0.999957i	$-0.497046\pi$
$101$	189.344			0.0185613			0.00928067	+	0.999957i	$0.497046\pi$
$102$		0			0
$103$	−14330.9	−	14330.9i	−1.35083	−	1.35083i	−0.884736	−	0.466092i	$-0.845662\pi$
$103$	−14330.9	−	14330.9i	−1.35083	−	1.35083i	−0.466092	−	0.884736i	$-0.654338\pi$
$104$		−	3895.82i		−	0.360190i
$105$		0			0
$106$	−22639.8			−2.01493
$107$	−384.065	+	384.065i	−0.0335457	+	0.0335457i	−0.723681	−	0.690135i	$-0.757551\pi$
$107$	−384.065	+	384.065i	−0.0335457	+	0.0335457i	0.690135	+	0.723681i	$0.257551\pi$
$108$		0			0
$109$		−	14104.7i		−	1.18717i	−0.804773	−	0.593583i	$-0.797713\pi$
$109$		−	14104.7i		−	1.18717i	0.804773	−	0.593583i	$-0.202287\pi$
$110$		0			0
$111$		0			0
$112$	−14869.7	+	14869.7i	−1.18540	+	1.18540i
$113$	−12599.1	−	12599.1i	−0.986693	−	0.986693i	0.0132199	−	0.999913i	$-0.495792\pi$
$113$	−12599.1	−	12599.1i	−0.986693	−	0.986693i	−0.999913	+	0.0132199i	$0.995792\pi$
$114$		0			0
$115$		0			0
$116$	−28495.5			−2.11768
$117$		0			0
$118$	12997.8	+	12997.8i	0.933481	+	0.933481i
$119$		−	1223.97i		−	0.0864328i
$120$		0			0
$121$	−13006.1			−0.888337
$122$	16475.8	−	16475.8i	1.10695	−	1.10695i
$123$		0			0
$124$			46563.9i			3.02835i
$125$		0			0
$126$		0			0
$127$	10957.5	−	10957.5i	0.679367	−	0.679367i	−0.280490	−	0.959857i	$-0.590497\pi$
$127$	10957.5	−	10957.5i	0.679367	−	0.679367i	0.959857	+	0.280490i	$0.0904970\pi$
$128$	15624.0	+	15624.0i	0.953614	+	0.953614i
$129$		0			0
$130$		0			0
$131$	−10872.8			−0.633579			−0.316789	−	0.948496i	$-0.602605\pi$
$131$	−10872.8			−0.633579			−0.316789	+	0.948496i	$0.602605\pi$
$132$		0			0
$133$	12143.3	+	12143.3i	0.686489	+	0.686489i
$134$			34482.2i			1.92037i
$135$		0			0
$136$	2962.88			0.160190
$137$	−3387.27	+	3387.27i	−0.180471	+	0.180471i	−0.791561	−	0.611090i	$-0.790731\pi$
$137$	−3387.27	+	3387.27i	−0.180471	+	0.180471i	0.611090	+	0.791561i	$0.290731\pi$
$138$		0			0
$139$			6096.09i			0.315516i	0.987478	+	0.157758i	$0.0504266\pi$
$139$			6096.09i			0.315516i	−0.987478	+	0.157758i	$0.949573\pi$
$140$		0			0
$141$		0			0
$142$	−28557.2	+	28557.2i	−1.41625	+	1.41625i
$143$	−843.796	−	843.796i	−0.0412634	−	0.0412634i
$144$		0			0
$145$		0			0
$146$	44750.8			2.09940
$147$		0			0
$148$	−20387.2	−	20387.2i	−0.930752	−	0.930752i
$149$		−	41060.3i		−	1.84948i	−0.380600	−	0.924740i	$-0.624282\pi$
$149$		−	41060.3i		−	1.84948i	0.380600	−	0.924740i	$-0.375718\pi$
$150$		0			0
$151$	12096.5			0.530523			0.265262	−	0.964176i	$-0.414542\pi$
$151$	12096.5			0.530523			0.265262	+	0.964176i	$0.414542\pi$
$152$	−29395.3	+	29395.3i	−1.27231	+	1.27231i
$153$		0			0
$154$			15678.4i			0.661091i
$155$		0			0
$156$		0			0
$157$	21184.3	−	21184.3i	0.859440	−	0.859440i	−0.131832	−	0.991272i	$-0.542086\pi$
$157$	21184.3	−	21184.3i	0.859440	−	0.859440i	0.991272	+	0.131832i	$0.0420859\pi$
$158$	32510.6	+	32510.6i	1.30230	+	1.30230i
$159$		0			0
$160$		0			0
$161$	−44178.8			−1.70436
$162$		0			0
$163$	−3953.29	−	3953.29i	−0.148793	−	0.148793i	0.628786	−	0.777579i	$-0.283552\pi$
$163$	−3953.29	−	3953.29i	−0.148793	−	0.148793i	−0.777579	+	0.628786i	$0.783552\pi$
$164$			64169.2i			2.38583i
$165$		0			0
$166$	6248.73			0.226765
$167$	−7863.94	+	7863.94i	−0.281973	+	0.281973i	−0.833895	−	0.551922i	$-0.813894\pi$
$167$	−7863.94	+	7863.94i	−0.281973	+	0.281973i	0.551922	+	0.833895i	$0.313894\pi$
$168$		0			0
$169$		−	27690.0i		−	0.969503i
$170$		0			0
$171$		0			0
$172$	23195.0	−	23195.0i	0.784037	−	0.784037i
$173$	−20563.0	−	20563.0i	−0.687059	−	0.687059i	0.274522	−	0.961581i	$-0.411480\pi$
$173$	−20563.0	−	20563.0i	−0.687059	−	0.687059i	−0.961581	+	0.274522i	$0.911480\pi$
$174$		0			0
$175$		0			0
$176$	−15592.4			−0.503371
$177$		0			0
$178$	9336.01	+	9336.01i	0.294660	+	0.294660i
$179$			11578.3i			0.361360i	0.983542	+	0.180680i	$0.0578298\pi$
$179$			11578.3i			0.361360i	−0.983542	+	0.180680i	$0.942170\pi$
$180$		0			0
$181$	15410.3			0.470385			0.235193	−	0.971949i	$-0.424428\pi$
$181$	15410.3			0.470385			0.235193	+	0.971949i	$0.424428\pi$
$182$	8092.11	−	8092.11i	0.244298	−	0.244298i
$183$		0			0
$184$		−	106944.i		−	3.15879i
$185$		0			0
$186$		0			0
$187$	641.731	−	641.731i	0.0183514	−	0.0183514i
$188$	−17450.0	−	17450.0i	−0.493720	−	0.493720i
$189$		0			0
$190$		0			0
$191$	8129.00			0.222828			0.111414	−	0.993774i	$-0.464462\pi$
$191$	8129.00			0.222828			0.111414	+	0.993774i	$0.464462\pi$
$192$		0			0
$193$	28495.5	+	28495.5i	0.765001	+	0.765001i	0.977222	−	0.212220i	$-0.0680695\pi$
$193$	28495.5	+	28495.5i	0.765001	+	0.765001i	−0.212220	+	0.977222i	$0.568069\pi$
$194$		−	124532.i		−	3.30884i
$195$		0			0
$196$	−19792.4			−0.515213
$197$	−14908.5	+	14908.5i	−0.384150	+	0.384150i	−0.872595	−	0.488445i	$-0.837564\pi$
$197$	−14908.5	+	14908.5i	−0.384150	+	0.384150i	0.488445	+	0.872595i	$0.337564\pi$
$198$		0			0
$199$			3870.42i			0.0977353i	0.998805	+	0.0488677i	$0.0155613\pi$
$199$			3870.42i			0.0977353i	−0.998805	+	0.0488677i	$0.984439\pi$
$200$		0			0
$201$		0			0
$202$	−952.043	+	952.043i	−0.0233321	+	0.0233321i
$203$	−31789.6	−	31789.6i	−0.771423	−	0.771423i
$204$		0			0
$205$		0			0
$206$	144115.			3.39606
$207$		0			0
$208$	8047.71	+	8047.71i	0.186014	+	0.186014i
$209$			12733.5i			0.291511i
$210$		0			0
$211$	−17666.9			−0.396822			−0.198411	−	0.980119i	$-0.563578\pi$
$211$	−17666.9			−0.396822			−0.198411	+	0.980119i	$0.563578\pi$
$212$	77814.2	−	77814.2i	1.73136	−	1.73136i
$213$		0			0
$214$		−	3862.24i		−	0.0843358i
$215$		0			0
$216$		0			0
$217$	−51946.6	+	51946.6i	−1.10316	+	1.10316i
$218$	70920.0	+	70920.0i	1.49230	+	1.49230i
$219$		0			0
$220$		0			0
$221$	−662.433			−0.0135630
$222$		0			0
$223$	−10730.0	−	10730.0i	−0.215768	−	0.215768i	0.590944	−	0.806713i	$-0.298755\pi$
$223$	−10730.0	−	10730.0i	−0.215768	−	0.215768i	−0.806713	+	0.590944i	$0.798755\pi$
$224$		−	34360.5i		−	0.684800i
$225$		0			0
$226$	126699.			2.48060
$227$	−34199.9	+	34199.9i	−0.663703	+	0.663703i	−0.956251	−	0.292548i	$-0.905497\pi$
$227$	−34199.9	+	34199.9i	−0.663703	+	0.663703i	0.292548	+	0.956251i	$0.405497\pi$
$228$		0			0
$229$			73430.0i			1.40024i	0.714026	+	0.700120i	$0.246870\pi$
$229$			73430.0i			1.40024i	−0.714026	+	0.700120i	$0.753130\pi$
$230$		0			0
$231$		0			0
$232$	76953.1	−	76953.1i	1.42972	−	1.42972i
$233$	18203.1	+	18203.1i	0.335300	+	0.335300i	0.854595	−	0.519295i	$-0.173805\pi$
$233$	18203.1	+	18203.1i	0.335300	+	0.335300i	−0.519295	+	0.854595i	$0.673805\pi$
$234$		0			0
$235$		0			0
$236$	−89348.2			−1.60421
$237$		0			0
$238$	6154.28	+	6154.28i	0.108648	+	0.108648i
$239$			102775.i			1.79925i	0.436661	+	0.899626i	$0.356161\pi$
$239$			102775.i			1.79925i	−0.436661	+	0.899626i	$0.643839\pi$
$240$		0			0
$241$	69403.6			1.19495			0.597473	−	0.801889i	$-0.296172\pi$
$241$	69403.6			1.19495			0.597473	+	0.801889i	$0.296172\pi$
$242$	65396.3	−	65396.3i	1.11666	−	1.11666i
$243$		0			0
$244$			113256.i			1.90232i
$245$		0			0
$246$		0			0
$247$	6572.14	−	6572.14i	0.107724	−	0.107724i
$248$	−125747.	−	125747.i	−2.04454	−	2.04454i
$249$		0			0
$250$		0			0
$251$	40858.0			0.648530			0.324265	−	0.945966i	$-0.394883\pi$
$251$	40858.0			0.648530			0.324265	+	0.945966i	$0.394883\pi$
$252$		0			0
$253$	−23163.0	−	23163.0i	−0.361871	−	0.361871i
$254$			110191.i			1.70796i
$255$		0			0
$256$	−130086.			−1.98495
$257$	53271.7	−	53271.7i	0.806548	−	0.806548i	−0.177562	−	0.984110i	$-0.556821\pi$
$257$	53271.7	−	53271.7i	0.806548	−	0.806548i	0.984110	+	0.177562i	$0.0568211\pi$
$258$		0			0
$259$		−	45487.9i		−	0.678103i
$260$		0			0
$261$		0			0
$262$	54669.8	−	54669.8i	0.796425	−	0.796425i
$263$	47788.6	+	47788.6i	0.690896	+	0.690896i	0.962429	−	0.271533i	$-0.0875306\pi$
$263$	47788.6	+	47788.6i	0.690896	+	0.690896i	−0.271533	+	0.962429i	$0.587531\pi$
$264$		0			0
$265$		0			0
$266$	−122116.			−1.72587
$267$		0			0
$268$	−118517.	−	118517.i	−1.65010	−	1.65010i
$269$		−	130832.i		−	1.80805i	−0.427483	−	0.904023i	$-0.640600\pi$
$269$		−	130832.i		−	1.80805i	0.427483	−	0.904023i	$-0.359400\pi$
$270$		0			0
$271$	−116029.			−1.57989			−0.789945	−	0.613178i	$-0.789891\pi$
$271$	−116029.			−1.57989			−0.789945	+	0.613178i	$0.789891\pi$
$272$	−6120.51	+	6120.51i	−0.0827274	+	0.0827274i
$273$		0			0
$274$		−	34063.1i		−	0.453715i
$275$		0			0
$276$		0			0
$277$	−41111.1	+	41111.1i	−0.535796	+	0.535796i	−0.922291	−	0.386495i	$-0.873686\pi$
$277$	−41111.1	+	41111.1i	−0.535796	+	0.535796i	0.386495	+	0.922291i	$0.373686\pi$
$278$	−30651.8	−	30651.8i	−0.396613	−	0.396613i
$279$		0			0
$280$		0			0
$281$	61086.4			0.773627			0.386814	−	0.922158i	$-0.373576\pi$
$281$	61086.4			0.773627			0.386814	+	0.922158i	$0.373576\pi$
$282$		0			0
$283$	−54739.9	−	54739.9i	−0.683489	−	0.683489i	0.277296	−	0.960785i	$-0.410562\pi$
$283$	−54739.9	−	54739.9i	−0.683489	−	0.683489i	−0.960785	+	0.277296i	$0.910562\pi$
$284$		−	196305.i		−	2.43386i
$285$		0			0
$286$	8485.40			0.103739
$287$	−71587.1	+	71587.1i	−0.869103	+	0.869103i
$288$		0			0
$289$			83017.2i			0.993968i
$290$		0			0
$291$		0			0
$292$	−153811.	+	153811.i	−1.80394	+	1.80394i
$293$	−33601.9	−	33601.9i	−0.391406	−	0.391406i	0.483782	−	0.875189i	$-0.339263\pi$
$293$	−33601.9	−	33601.9i	−0.391406	−	0.391406i	−0.875189	+	0.483782i	$0.839263\pi$
$294$		0			0
$295$		0			0
$296$	110113.			1.25676
$297$		0			0
$298$	206456.	+	206456.i	2.32485	+	2.32485i
$299$			23910.2i			0.267449i
$300$		0			0
$301$	51752.5			0.571214
$302$	−60822.3	+	60822.3i	−0.666882	+	0.666882i
$303$		0			0
$304$		−	121446.i		−	1.31412i
$305$		0			0
$306$		0			0
$307$	−84437.4	+	84437.4i	−0.895897	+	0.895897i	−0.995070	−	0.0991732i	$-0.968380\pi$
$307$	−84437.4	+	84437.4i	−0.895897	+	0.895897i	0.0991732	+	0.995070i	$0.468380\pi$
$308$	−53887.6	−	53887.6i	−0.568051	−	0.568051i
$309$		0			0
$310$		0			0
$311$	−93477.0			−0.966460			−0.483230	−	0.875493i	$-0.660537\pi$
$311$	−93477.0			−0.966460			−0.483230	+	0.875493i	$0.660537\pi$
$312$		0			0
$313$	6979.59	+	6979.59i	0.0712429	+	0.0712429i	0.741830	−	0.670588i	$-0.233958\pi$
$313$	6979.59	+	6979.59i	0.0712429	+	0.0712429i	−0.670588	+	0.741830i	$0.733958\pi$
$314$			213034.i			2.16068i
$315$		0			0
$316$	−223481.			−2.23804
$317$	41396.6	−	41396.6i	0.411951	−	0.411951i	−0.470467	−	0.882418i	$-0.655914\pi$
$317$	41396.6	−	41396.6i	0.411951	−	0.411951i	0.882418	+	0.470467i	$0.155914\pi$
$318$		0			0
$319$		−	33334.6i		−	0.327577i
$320$		0			0
$321$		0			0
$322$	222136.	−	222136.i	2.14243	−	2.14243i
$323$	4998.29	+	4998.29i	0.0479090	+	0.0479090i
$324$		0			0
$325$		0			0
$326$	39755.1			0.374074
$327$		0			0
$328$	−173291.	−	173291.i	−1.61075	−	1.61075i
$329$		−	38934.5i		−	0.359702i
$330$		0			0
$331$	65879.3			0.601303			0.300651	−	0.953734i	$-0.402796\pi$
$331$	65879.3			0.601303			0.300651	+	0.953734i	$0.402796\pi$
$332$	−21477.2	+	21477.2i	−0.194851	+	0.194851i
$333$		0			0
$334$		−	79081.5i		−	0.708895i
$335$		0			0
$336$		0			0
$337$	−18022.9	+	18022.9i	−0.158696	+	0.158696i	−0.781989	−	0.623293i	$-0.785795\pi$
$337$	−18022.9	+	18022.9i	−0.158696	+	0.158696i	0.623293	+	0.781989i	$0.285795\pi$
$338$	139228.	+	139228.i	1.21869	+	1.21869i
$339$		0			0
$340$		0			0
$341$	−54471.3			−0.468445
$342$		0			0
$343$	70500.4	+	70500.4i	0.599244	+	0.599244i
$344$			125277.i			1.05866i
$345$		0			0
$346$	206786.			1.72730
$347$	−32583.8	+	32583.8i	−0.270610	+	0.270610i	−0.829346	−	0.558736i	$-0.811287\pi$
$347$	−32583.8	+	32583.8i	−0.270610	+	0.270610i	0.558736	+	0.829346i	$0.311287\pi$
$348$		0			0
$349$			54340.9i			0.446145i	0.974802	+	0.223072i	$0.0716086\pi$
$349$			54340.9i			0.446145i	−0.974802	+	0.223072i	$0.928391\pi$
$350$		0			0
$351$		0			0
$352$	18015.2	−	18015.2i	0.145397	−	0.145397i
$353$	−11345.9	−	11345.9i	−0.0910524	−	0.0910524i	0.660114	−	0.751166i	$-0.270508\pi$
$353$	−11345.9	−	11345.9i	−0.0910524	−	0.0910524i	−0.751166	+	0.660114i	$0.770508\pi$
$354$		0			0
$355$		0			0
$356$	−64176.7			−0.506381
$357$		0			0
$358$	−58217.1	−	58217.1i	−0.454239	−	0.454239i
$359$		−	26830.8i		−	0.208183i	−0.994568	−	0.104091i	$-0.966807\pi$
$359$		−	26830.8i		−	0.208183i	0.994568	−	0.104091i	$-0.0331934\pi$
$360$		0			0
$361$	31142.7			0.238969
$362$	−77484.6	+	77484.6i	−0.591287	+	0.591287i
$363$		0			0
$364$			55626.0i			0.419832i
$365$		0			0
$366$		0			0
$367$	112367.	−	112367.i	0.834267	−	0.834267i	−0.153830	−	0.988097i	$-0.549161\pi$
$367$	112367.	−	112367.i	0.834267	−	0.834267i	0.988097	+	0.153830i	$0.0491609\pi$
$368$	220917.	+	220917.i	1.63130	+	1.63130i
$369$		0			0
$370$		0			0
$371$	173619.			1.26139
$372$		0			0
$373$	16877.0	+	16877.0i	0.121305	+	0.121305i	0.765153	−	0.643848i	$-0.222663\pi$
$373$	16877.0	+	16877.0i	0.121305	+	0.121305i	−0.643848	+	0.765153i	$0.722663\pi$
$374$			6453.38i			0.0461364i
$375$		0			0
$376$	94248.7			0.666653
$377$	−17205.0	+	17205.0i	−0.121052	+	0.121052i
$378$		0			0
$379$			50523.2i			0.351733i	0.984414	+	0.175866i	$0.0562726\pi$
$379$			50523.2i			0.351733i	−0.984414	+	0.175866i	$0.943727\pi$
$380$		0			0
$381$		0			0
$382$	−40873.5	+	40873.5i	−0.280101	+	0.280101i
$383$	17592.0	+	17592.0i	0.119927	+	0.119927i	0.764523	−	0.644596i	$-0.222974\pi$
$383$	17592.0	+	17592.0i	0.119927	+	0.119927i	−0.644596	+	0.764523i	$0.722974\pi$
$384$		0			0
$385$		0			0
$386$	−286557.			−1.92325
$387$		0			0
$388$	428022.	+	428022.i	2.84317	+	2.84317i
$389$		−	6343.53i		−	0.0419210i	−0.999780	−	0.0209605i	$-0.993328\pi$
$389$		−	6343.53i		−	0.0419210i	0.999780	−	0.0209605i	$-0.00667242\pi$
$390$		0			0
$391$	−18184.4			−0.118945
$392$	53450.1	−	53450.1i	0.347838	−	0.347838i
$393$		0			0
$394$		−	149923.i		−	0.965773i
$395$		0			0
$396$		0			0
$397$	145003.	−	145003.i	0.920018	−	0.920018i	−0.0770121	−	0.997030i	$-0.524538\pi$
$397$	145003.	−	145003.i	0.920018	−	0.920018i	0.997030	+	0.0770121i	$0.0245380\pi$
$398$	−19460.9	−	19460.9i	−0.122856	−	0.122856i
$399$		0			0
$400$		0			0
$401$	299021.			1.85957			0.929786	−	0.368100i	$-0.119992\pi$
$401$	299021.			1.85957			0.929786	+	0.368100i	$0.119992\pi$
$402$		0			0
$403$	28114.3	+	28114.3i	0.173108	+	0.173108i
$404$		−	6544.44i		−	0.0400968i
$405$		0			0
$406$	319683.			1.93940
$407$	23849.3	−	23849.3i	0.143975	−	0.143975i
$408$		0			0
$409$			17363.2i			0.103796i	0.998652	+	0.0518982i	$0.0165271\pi$
$409$			17363.2i			0.103796i	−0.998652	+	0.0518982i	$0.983473\pi$
$410$		0			0
$411$		0			0
$412$	−495331.	+	495331.i	−2.91811	+	2.91811i
$413$	−99676.8	−	99676.8i	−0.584378	−	0.584378i
$414$		0			0
$415$		0			0
$416$	−18596.4			−0.107459
$417$		0			0
$418$	−64025.4	−	64025.4i	−0.366437	−	0.366437i
$419$			36236.3i			0.206403i	0.994660	+	0.103201i	$0.0329086\pi$
$419$			36236.3i			0.206403i	−0.994660	+	0.103201i	$0.967091\pi$
$420$		0			0
$421$	−121899.			−0.687761			−0.343881	−	0.939013i	$-0.611742\pi$
$421$	−121899.			−0.687761			−0.343881	+	0.939013i	$0.611742\pi$
$422$	88831.2	−	88831.2i	0.498817	−	0.498817i
$423$		0			0
$424$			420280.i			2.33780i
$425$		0			0
$426$		0			0
$427$	−126349.	+	126349.i	−0.692971	+	0.692971i
$428$	13274.7	+	13274.7i	0.0724666	+	0.0724666i
$429$		0			0
$430$		0			0
$431$	−330169.			−1.77739			−0.888693	−	0.458504i	$-0.848386\pi$
$431$	−330169.			−1.77739			−0.888693	+	0.458504i	$0.848386\pi$
$432$		0			0
$433$	−140458.	−	140458.i	−0.749151	−	0.749151i	0.225169	−	0.974320i	$-0.427707\pi$
$433$	−140458.	−	140458.i	−0.749151	−	0.749151i	−0.974320	+	0.225169i	$0.927707\pi$
$434$		−	522387.i		−	2.77340i
$435$		0			0
$436$	−487512.			−2.56455
$437$	180411.	−	180411.i	0.944715	−	0.944715i
$438$		0			0
$439$		−	25797.9i		−	0.133861i	−0.997758	−	0.0669306i	$-0.978679\pi$
$439$		−	25797.9i		−	0.133861i	0.997758	−	0.0669306i	$-0.0213206\pi$
$440$		0			0
$441$		0			0
$442$	3330.78	−	3330.78i	0.0170491	−	0.0170491i
$443$	131776.	+	131776.i	0.671473	+	0.671473i	0.958056	−	0.286583i	$-0.0925193\pi$
$443$	131776.	+	131776.i	0.671473	+	0.671473i	−0.286583	+	0.958056i	$0.592519\pi$
$444$		0			0
$445$		0			0
$446$	107903.			0.542454
$447$		0			0
$448$	−65147.1	−	65147.1i	−0.324593	−	0.324593i
$449$			133600.i			0.662697i	0.943508	+	0.331349i	$0.107504\pi$
$449$			133600.i			0.662697i	−0.943508	+	0.331349i	$0.892496\pi$
$450$		0			0
$451$	−75066.4			−0.369056
$452$	−435471.	+	435471.i	−2.13149	+	2.13149i
$453$		0			0
$454$		−	343922.i		−	1.66858i
$455$		0			0
$456$		0			0
$457$	147483.	−	147483.i	0.706170	−	0.706170i	−0.259558	−	0.965728i	$-0.583577\pi$
$457$	147483.	−	147483.i	0.706170	−	0.706170i	0.965728	+	0.259558i	$0.0835768\pi$
$458$	−369214.	−	369214.i	−1.76014	−	1.76014i
$459$		0			0
$460$		0			0
$461$	−206578.			−0.972034			−0.486017	−	0.873949i	$-0.661551\pi$
$461$	−206578.			−0.972034			−0.486017	+	0.873949i	$0.661551\pi$
$462$		0			0
$463$	156702.	+	156702.i	0.730990	+	0.730990i	0.970816	−	0.239826i	$-0.0770904\pi$
$463$	156702.	+	156702.i	0.730990	+	0.730990i	−0.239826	+	0.970816i	$0.577090\pi$
$464$			317928.i			1.47670i
$465$		0			0
$466$	−183055.			−0.842963
$467$	−100008.	+	100008.i	−0.458566	+	0.458566i	−0.898184	−	0.439619i	$-0.855114\pi$
$467$	−100008.	+	100008.i	−0.458566	+	0.458566i	0.439619	+	0.898184i	$0.355114\pi$
$468$		0			0
$469$		−	264435.i		−	1.20219i
$470$		0			0
$471$		0			0
$472$	241288.	−	241288.i	1.08306	−	1.08306i
$473$	27133.9	+	27133.9i	0.121280	+	0.121280i
$474$		0			0
$475$		0			0
$476$	−42305.1			−0.186715
$477$		0			0
$478$	−516764.	−	516764.i	−2.26171	−	2.26171i
$479$		−	124818.i		−	0.544010i	−0.962296	−	0.272005i	$-0.912313\pi$
$479$		−	124818.i		−	0.544010i	0.962296	−	0.272005i	$-0.0876867\pi$
$480$		0			0
$481$	−24618.7			−0.106408
$482$	−348969.	+	348969.i	−1.50208	+	1.50208i
$483$		0			0
$484$			449541.i			1.91902i
$485$		0			0
$486$		0			0
$487$	106165.	−	106165.i	0.447635	−	0.447635i	−0.446932	−	0.894568i	$-0.647484\pi$
$487$	106165.	−	106165.i	0.447635	−	0.447635i	0.894568	+	0.446932i	$0.147484\pi$
$488$	−305853.	−	305853.i	−1.28432	−	1.28432i
$489$		0			0
$490$		0			0
$491$	−185092.			−0.767758			−0.383879	−	0.923383i	$-0.625412\pi$
$491$	−185092.			−0.767758			−0.383879	+	0.923383i	$0.625412\pi$
$492$		0			0
$493$	−13084.9	−	13084.9i	−0.0538363	−	0.0538363i
$494$			66090.8i			0.270824i
$495$		0			0
$496$	519520.			2.11173
$497$	218998.	−	218998.i	0.886598	−	0.886598i
$498$		0			0
$499$		−	482480.i		−	1.93766i	−0.247723	−	0.968831i	$-0.579682\pi$
$499$		−	482480.i		−	1.93766i	0.247723	−	0.968831i	$-0.420318\pi$
$500$		0			0
$501$		0			0
$502$	−205439.	+	205439.i	−0.815220	+	0.815220i
$503$	224311.	+	224311.i	0.886572	+	0.886572i	0.994192	−	0.107620i	$-0.0343230\pi$
$503$	224311.	+	224311.i	0.886572	+	0.886572i	−0.107620	+	0.994192i	$0.534323\pi$
$504$		0			0
$505$		0			0
$506$	232932.			0.909763
$507$		0			0
$508$	−378732.	−	378732.i	−1.46759	−	1.46759i
$509$			390482.i			1.50718i	0.657344	+	0.753591i	$0.271680\pi$
$509$			390482.i			1.50718i	−0.657344	+	0.753591i	$0.728320\pi$
$510$		0			0
$511$	−343182.			−1.31426
$512$	404102.	−	404102.i	1.54153	−	1.54153i
$513$		0			0
$514$			535711.i			2.02770i
$515$		0			0
$516$		0			0
$517$	20413.4	−	20413.4i	0.0763719	−	0.0763719i
$518$	228718.	+	228718.i	0.852394	+	0.852394i
$519$		0			0
$520$		0			0
$521$	71234.7			0.262432			0.131216	−	0.991354i	$-0.458112\pi$
$521$	71234.7			0.262432			0.131216	+	0.991354i	$0.458112\pi$
$522$		0			0
$523$	26895.3	+	26895.3i	0.0983272	+	0.0983272i	0.754559	−	0.656232i	$-0.227851\pi$
$523$	26895.3	+	26895.3i	0.0983272	+	0.0983272i	−0.656232	+	0.754559i	$0.727851\pi$
$524$			375806.i			1.36868i
$525$		0			0
$526$	−480573.			−1.73695
$527$	−21381.7	+	21381.7i	−0.0769876	+	0.0769876i
$528$		0			0
$529$			376517.i			1.34547i
$530$		0			0
$531$		0			0
$532$	419718.	−	419718.i	1.48298	−	1.48298i
$533$	38744.0	+	38744.0i	0.136380	+	0.136380i
$534$		0			0
$535$		0			0
$536$	640119.			2.22808
$537$		0			0
$538$	657838.	+	657838.i	2.27276	+	2.27276i
$539$		−	23153.6i		−	0.0796967i
$540$		0			0
$541$	293490.			1.00276			0.501382	−	0.865226i	$-0.332825\pi$
$541$	293490.			1.00276			0.501382	+	0.865226i	$0.332825\pi$
$542$	583405.	−	583405.i	1.98596	−	1.98596i
$543$		0			0
$544$		−	14143.1i		−	0.0477911i
$545$		0			0
$546$		0			0
$547$	−178128.	+	178128.i	−0.595330	+	0.595330i	−0.939066	−	0.343736i	$-0.888307\pi$
$547$	−178128.	+	178128.i	−0.595330	+	0.595330i	0.343736	+	0.939066i	$0.388307\pi$
$548$	117077.	+	117077.i	0.389860	+	0.389860i
$549$		0			0
$550$		0			0
$551$	259635.			0.855186
$552$		0			0
$553$	−249316.	−	249316.i	−0.815266	−	0.815266i
$554$		−	413422.i		−	1.34702i
$555$		0			0
$556$	210704.			0.681589
$557$	122805.	−	122805.i	0.395827	−	0.395827i	−0.480931	−	0.876758i	$-0.659701\pi$
$557$	122805.	−	122805.i	0.395827	−	0.395827i	0.876758	+	0.480931i	$0.159701\pi$
$558$		0			0
$559$		−	28009.2i		−	0.0896349i
$560$		0			0
$561$		0			0
$562$	−307149.	+	307149.i	−0.972470	+	0.972470i
$563$	−86662.0	−	86662.0i	−0.273408	−	0.273408i	0.557062	−	0.830471i	$-0.311928\pi$
$563$	−86662.0	−	86662.0i	−0.273408	−	0.273408i	−0.830471	+	0.557062i	$0.811928\pi$
$564$		0			0
$565$		0			0
$566$	550477.			1.71833
$567$		0			0
$568$	530128.	+	530128.i	1.64318	+	1.64318i
$569$		−	358179.i		−	1.10631i	−0.833080	−	0.553153i	$-0.813425\pi$
$569$		−	358179.i		−	1.10631i	0.833080	−	0.553153i	$-0.186575\pi$
$570$		0			0
$571$	420094.			1.28847			0.644234	−	0.764828i	$-0.277176\pi$
$571$	420094.			1.28847			0.644234	+	0.764828i	$0.277176\pi$
$572$	−29164.8	+	29164.8i	−0.0891387	+	0.0891387i
$573$		0			0
$574$		−	719896.i		−	2.18497i
$575$		0			0
$576$		0			0
$577$	167153.	−	167153.i	0.502068	−	0.502068i	−0.410012	−	0.912080i	$-0.634476\pi$
$577$	167153.	−	167153.i	0.502068	−	0.502068i	0.912080	+	0.410012i	$0.134476\pi$
$578$	−417419.	−	417419.i	−1.24944	−	1.24944i
$579$		0			0
$580$		0			0
$581$	−47919.9			−0.141959
$582$		0			0
$583$	91028.5	+	91028.5i	0.267818	+	0.267818i
$584$		−	830742.i		−	2.43579i
$585$		0			0
$586$	337908.			0.984017
$587$	−43094.6	+	43094.6i	−0.125068	+	0.125068i	−0.766870	−	0.641802i	$-0.778187\pi$
$587$	−43094.6	+	43094.6i	−0.125068	+	0.125068i	0.641802	+	0.766870i	$0.278187\pi$
$588$		0			0
$589$		−	424265.i		−	1.22294i
$590$		0			0
$591$		0			0
$592$	−227463.	+	227463.i	−0.649033	+	0.649033i
$593$	−114499.	−	114499.i	−0.325605	−	0.325605i	0.525307	−	0.850912i	$-0.323950\pi$
$593$	−114499.	−	114499.i	−0.325605	−	0.325605i	−0.850912	+	0.525307i	$0.823950\pi$
$594$		0			0
$595$		0			0
$596$	−1.41920e6			−3.99531
$597$		0			0
$598$	−120223.	−	120223.i	−0.336191	−	0.336191i
$599$			282109.i			0.786254i	0.919484	+	0.393127i	$0.128607\pi$
$599$			282109.i			0.786254i	−0.919484	+	0.393127i	$0.871393\pi$
$600$		0			0
$601$	−155225.			−0.429747			−0.214874	−	0.976642i	$-0.568934\pi$
$601$	−155225.			−0.429747			−0.214874	+	0.976642i	$0.568934\pi$
$602$	−260217.	+	260217.i	−0.718031	+	0.718031i
$603$		0			0
$604$		−	418099.i		−	1.14605i
$605$		0			0
$606$		0			0
$607$	262539.	−	262539.i	0.712553	−	0.712553i	−0.254516	−	0.967069i	$-0.581916\pi$
$607$	262539.	−	262539.i	0.712553	−	0.712553i	0.967069	+	0.254516i	$0.0819160\pi$
$608$	140317.	+	140317.i	0.379579	+	0.379579i
$609$		0			0
$610$		0			0
$611$	−21071.9			−0.0564444
$612$		0			0
$613$	98919.2	+	98919.2i	0.263245	+	0.263245i	0.826371	−	0.563126i	$-0.190402\pi$
$613$	98919.2	+	98919.2i	0.263245	+	0.263245i	−0.563126	+	0.826371i	$0.690402\pi$
$614$		−	849121.i		−	2.25233i
$615$		0			0
$616$	291051.			0.767021
$617$	−440524.	+	440524.i	−1.15717	+	1.15717i	−0.172095	+	0.985080i	$0.555053\pi$
$617$	−440524.	+	440524.i	−1.15717	+	1.15717i	−0.985080	+	0.172095i	$0.944947\pi$
$618$		0			0
$619$			468975.i			1.22396i	0.790871	+	0.611982i	$0.209628\pi$
$619$			468975.i			1.22396i	−0.790871	+	0.611982i	$0.790372\pi$
$620$		0			0
$621$		0			0
$622$	470012.	−	470012.i	1.21487	−	1.21487i
$623$	−71595.5	−	71595.5i	−0.184463	−	0.184463i
$624$		0			0
$625$		0			0
$626$	−70188.3			−0.179108
$627$		0			0
$628$	−732210.	−	732210.i	−1.85659	−	1.85659i
$629$		−	18723.2i		−	0.0473237i
$630$		0			0
$631$	327190.			0.821752			0.410876	−	0.911691i	$-0.365223\pi$
$631$	327190.			0.821752			0.410876	+	0.911691i	$0.365223\pi$
$632$	603519.	−	603519.i	1.51097	−	1.51097i
$633$		0			0
$634$			416293.i			1.03567i
$635$		0			0
$636$		0			0
$637$	−11950.2	+	11950.2i	−0.0294509	+	0.0294509i
$638$	167610.	+	167610.i	0.411773	+	0.411773i
$639$		0			0
$640$		0			0
$641$	584738.			1.42313			0.711567	−	0.702619i	$-0.247986\pi$
$641$	584738.			1.42313			0.711567	+	0.702619i	$0.247986\pi$
$642$		0			0
$643$	202062.	+	202062.i	0.488723	+	0.488723i	0.907903	−	0.419180i	$-0.137682\pi$
$643$	202062.	+	202062.i	0.488723	+	0.488723i	−0.419180	+	0.907903i	$0.637682\pi$
$644$			1.52699e6i			3.68183i
$645$		0			0
$646$	−50263.9			−0.120446
$647$	119543.	−	119543.i	0.285573	−	0.285573i	−0.549754	−	0.835327i	$-0.685278\pi$
$647$	119543.	−	119543.i	0.285573	−	0.285573i	0.835327	+	0.549754i	$0.185278\pi$
$648$		0			0
$649$		−	104521.i		−	0.248150i
$650$		0			0
$651$		0			0
$652$	−136640.	+	136640.i	−0.321428	+	0.321428i
$653$	−279985.	−	279985.i	−0.656611	−	0.656611i	0.297965	−	0.954577i	$-0.403692\pi$
$653$	−279985.	−	279985.i	−0.656611	−	0.656611i	−0.954577	+	0.297965i	$0.903692\pi$
$654$		0			0
$655$		0			0
$656$	715946.			1.66369
$657$		0			0
$658$	195767.	+	195767.i	0.452155	+	0.452155i
$659$		−	171782.i		−	0.395555i	−0.980247	−	0.197778i	$-0.936628\pi$
$659$		−	171782.i		−	0.395555i	0.980247	−	0.197778i	$-0.0633724\pi$
$660$		0			0
$661$	566188.			1.29586			0.647929	−	0.761701i	$-0.275635\pi$
$661$	566188.			1.29586			0.647929	+	0.761701i	$0.275635\pi$
$662$	−331248.	+	331248.i	−0.755854	+	0.755854i
$663$		0			0
$664$		−	116000.i		−	0.263100i
$665$		0			0
$666$		0			0
$667$	−472292.	+	472292.i	−1.06160	+	1.06160i
$668$	271807.	+	271807.i	0.609128	+	0.609128i
$669$		0			0
$670$		0			0
$671$	−132489.			−0.294263
$672$		0			0
$673$	115466.	+	115466.i	0.254932	+	0.254932i	0.822989	−	0.568057i	$-0.192305\pi$
$673$	115466.	+	115466.i	0.254932	+	0.254932i	−0.568057	+	0.822989i	$0.692305\pi$
$674$		−	181243.i		−	0.398970i
$675$		0			0
$676$	−957070.			−2.09435
$677$	255557.	−	255557.i	0.557584	−	0.557584i	−0.371035	−	0.928619i	$-0.620997\pi$
$677$	255557.	−	255557.i	0.557584	−	0.557584i	0.928619	+	0.371035i	$0.120997\pi$
$678$		0			0
$679$			955001.i			2.07140i
$680$		0			0
$681$		0			0
$682$	273888.	−	273888.i	0.588849	−	0.588849i
$683$	97131.8	+	97131.8i	0.208219	+	0.208219i	0.803510	−	0.595291i	$-0.202963\pi$
$683$	97131.8	+	97131.8i	0.208219	+	0.208219i	−0.595291	+	0.803510i	$0.702963\pi$
$684$		0			0
$685$		0			0
$686$	−708967.			−1.50653
$687$		0			0
$688$	−258789.	−	258789.i	−0.546726	−	0.546726i
$689$		−	93965.1i		−	0.197937i
$690$		0			0
$691$	−40145.6			−0.0840778			−0.0420389	−	0.999116i	$-0.513385\pi$
$691$	−40145.6			−0.0840778			−0.0420389	+	0.999116i	$0.513385\pi$
$692$	−710734.	+	710734.i	−1.48421	+	1.48421i
$693$		0			0
$694$		−	327670.i		−	0.680327i
$695$		0			0
$696$		0			0
$697$	−29465.9	+	29465.9i	−0.0606532	+	0.0606532i
$698$	−273232.	−	273232.i	−0.560816	−	0.560816i
$699$		0			0
$700$		0			0
$701$	−225909.			−0.459724			−0.229862	−	0.973223i	$-0.573828\pi$
$701$	−225909.			−0.459724			−0.229862	+	0.973223i	$0.573828\pi$
$702$		0			0
$703$	185757.	+	185757.i	0.375867	+	0.375867i
$704$		−	68313.3i		−	0.137835i
$705$		0			0
$706$	114097.			0.228911
$707$	7300.97	−	7300.97i	0.0146064	−	0.0146064i
$708$		0			0
$709$		−	459779.i		−	0.914654i	−0.889299	−	0.457327i	$-0.848807\pi$
$709$		−	459779.i		−	0.914654i	0.889299	−	0.457327i	$-0.151193\pi$
$710$		0			0
$711$		0			0
$712$	173311.	−	173311.i	0.341875	−	0.341875i
$713$	771763.	+	771763.i	1.51812	+	1.51812i
$714$		0			0
$715$		0			0
$716$	400191.			0.780622
$717$		0			0
$718$	134908.	+	134908.i	0.261691	+	0.261691i
$719$			633365.i			1.22517i	0.790405	+	0.612585i	$0.209870\pi$
$719$			633365.i			1.22517i	−0.790405	+	0.612585i	$0.790130\pi$
$720$		0			0
$721$	−1.10518e6			−2.12600
$722$	−156589.	+	156589.i	−0.300391	+	0.300391i
$723$		0			0
$724$		−	532637.i		−	1.01614i
$725$		0			0
$726$		0			0
$727$	−483916.	+	483916.i	−0.915590	+	0.915590i	−0.996705	−	0.0811152i	$-0.974152\pi$
$727$	−483916.	+	483916.i	−0.915590	+	0.915590i	0.0811152	+	0.996705i	$0.474152\pi$
$728$	−150220.	−	150220.i	−0.283442	−	0.283442i
$729$		0			0
$730$		0			0
$731$	21301.8			0.0398641
$732$		0			0
$733$	79446.4	+	79446.4i	0.147865	+	0.147865i	0.777164	−	0.629298i	$-0.216658\pi$
$733$	79446.4	+	79446.4i	0.147865	+	0.147865i	−0.629298	+	0.777164i	$0.716658\pi$
$734$			1.12998e6i			2.09739i
$735$		0			0
$736$	−510489.			−0.942390
$737$	138643.	−	138643.i	0.255249	−	0.255249i
$738$		0			0
$739$			885328.i			1.62112i	0.585655	+	0.810561i	$0.300837\pi$
$739$			885328.i			1.62112i	−0.585655	+	0.810561i	$0.699163\pi$
$740$		0			0
$741$		0			0
$742$	−872975.	+	872975.i	−1.58560	+	1.58560i
$743$	693877.	+	693877.i	1.25691	+	1.25691i	0.952560	+	0.304352i	$0.0984400\pi$
$743$	693877.	+	693877.i	1.25691	+	1.25691i	0.304352	+	0.952560i	$0.401560\pi$
$744$		0			0
$745$		0			0
$746$	−169719.			−0.304967
$747$		0			0
$748$	−22180.6	−	22180.6i	−0.0396434	−	0.0396434i
$749$			29618.5i			0.0527959i
$750$		0			0
$751$	−1.03267e6			−1.83097			−0.915487	−	0.402347i	$-0.868194\pi$
$751$	−1.03267e6			−1.83097			−0.915487	+	0.402347i	$0.868194\pi$
$752$	−194692.	+	194692.i	−0.344281	+	0.344281i
$753$		0			0
$754$		−	173017.i		−	0.304331i
$755$		0			0
$756$		0			0
$757$	366937.	−	366937.i	0.640323	−	0.640323i	−0.310312	−	0.950635i	$-0.600433\pi$
$757$	366937.	−	366937.i	0.640323	−	0.640323i	0.950635	+	0.310312i	$0.100433\pi$
$758$	−254036.	−	254036.i	−0.442137	−	0.442137i
$759$		0			0
$760$		0			0
$761$	721432.			1.24574			0.622868	−	0.782327i	$-0.285967\pi$
$761$	721432.			1.24574			0.622868	+	0.782327i	$0.285967\pi$
$762$		0			0
$763$	−543867.	−	543867.i	−0.934209	−	0.934209i
$764$		−	280969.i		−	0.481361i
$765$		0			0
$766$	−176909.			−0.301504
$767$	−53946.5	+	53946.5i	−0.0917007	+	0.0917007i
$768$		0			0
$769$			543034.i			0.918279i	0.888364	+	0.459139i	$0.151842\pi$
$769$			543034.i			0.918279i	−0.888364	+	0.459139i	$0.848158\pi$
$770$		0			0
$771$		0			0
$772$	984913.	−	984913.i	1.65258	−	1.65258i
$773$	−501793.	−	501793.i	−0.839780	−	0.839780i	0.149050	−	0.988830i	$-0.452378\pi$
$773$	−501793.	−	501793.i	−0.839780	−	0.839780i	−0.988830	+	0.149050i	$0.952378\pi$
$774$		0			0
$775$		0			0
$776$	−2.31177e6			−3.83903
$777$		0			0
$778$	31895.9	+	31895.9i	0.0526958	+	0.0526958i
$779$		−	584675.i		−	0.963473i
$780$		0			0
$781$	229641.			0.376485
$782$	91433.1	−	91433.1i	0.149517	−	0.149517i
$783$		0			0
$784$			220827.i			0.359269i
$785$		0			0
$786$		0			0
$787$	219605.	−	219605.i	0.354562	−	0.354562i	−0.507242	−	0.861804i	$-0.669335\pi$
$787$	219605.	−	219605.i	0.354562	−	0.354562i	0.861804	+	0.507242i	$0.169335\pi$
$788$	515293.	+	515293.i	0.829854	+	0.829854i
$789$		0			0
$790$		0			0
$791$	−971623.			−1.55290
$792$		0			0
$793$	68381.7	+	68381.7i	0.108741	+	0.108741i
$794$			1.45818e6i			2.31298i
$795$		0			0
$796$	133776.			0.211131
$797$	−311150.	+	311150.i	−0.489839	+	0.489839i	−0.908255	−	0.418416i	$-0.862585\pi$
$797$	−311150.	+	311150.i	−0.489839	+	0.489839i	0.418416	+	0.908255i	$0.362585\pi$
$798$		0			0
$799$		−	16025.8i		−	0.0251030i
$800$		0			0
$801$		0			0
$802$	−1.50351e6	+	1.50351e6i	−2.33753	+	2.33753i
$803$	−179931.	−	179931.i	−0.279045	−	0.279045i
$804$		0			0
$805$		0			0
$806$	−282723.			−0.435202
$807$		0			0
$808$	17673.5	+	17673.5i	0.0270707	+	0.0270707i
$809$			827176.i			1.26387i	0.775023	+	0.631933i	$0.217738\pi$
$809$			827176.i			1.26387i	−0.775023	+	0.631933i	$0.782262\pi$
$810$		0			0
$811$	1.22035e6			1.85542			0.927711	−	0.373300i	$-0.121774\pi$
$811$	1.22035e6			1.85542			0.927711	+	0.373300i	$0.121774\pi$
$812$	−1.09877e6	+	1.09877e6i	−1.66645	+	1.66645i
$813$		0			0
$814$			239834.i			0.361961i
$815$		0			0
$816$		0			0
$817$	−211340.	+	211340.i	−0.316619	+	0.316619i
$818$	−87303.8	−	87303.8i	−0.130475	−	0.130475i
$819$		0			0
$820$		0			0
$821$	1.13223e6			1.67977			0.839884	−	0.542766i	$-0.182623\pi$
$821$	1.13223e6			1.67977			0.839884	+	0.542766i	$0.182623\pi$
$822$		0			0
$823$	−504583.	−	504583.i	−0.744961	−	0.744961i	0.228567	−	0.973528i	$-0.426596\pi$
$823$	−504583.	−	504583.i	−0.744961	−	0.744961i	−0.973528	+	0.228567i	$0.926596\pi$
$824$		−	2.67531e6i		−	3.94022i
$825$		0			0
$826$	1.00237e6			1.46916
$827$	−707633.	+	707633.i	−1.03466	+	1.03466i	−0.0352807	+	0.999377i	$0.511233\pi$
$827$	−707633.	+	707633.i	−1.03466	+	1.03466i	−0.999377	+	0.0352807i	$0.988767\pi$
$828$		0			0
$829$		−	846141.i		−	1.23121i	−0.788053	−	0.615607i	$-0.788911\pi$
$829$		−	846141.i		−	1.23121i	0.788053	−	0.615607i	$-0.211089\pi$
$830$		0			0
$831$		0			0
$832$	−35258.5	+	35258.5i	−0.0509352	+	0.0509352i
$833$	−9088.50	−	9088.50i	−0.0130979	−	0.0130979i
$834$		0			0
$835$		0			0
$836$	440117.			0.629732
$837$		0			0
$838$	−182200.	−	182200.i	−0.259454	−	0.259454i
$839$		−	499671.i		−	0.709839i	−0.934897	−	0.354920i	$-0.884508\pi$
$839$		−	499671.i		−	0.709839i	0.934897	−	0.354920i	$-0.115492\pi$
$840$		0			0
$841$	27590.7			0.0390096
$842$	612924.	−	612924.i	0.864534	−	0.864534i
$843$		0			0
$844$			610635.i			0.857229i
$845$		0			0
$846$		0			0
$847$	−501508.	+	501508.i	−0.699054	+	0.699054i
$848$	−868184.	−	868184.i	−1.20731	−	1.20731i
$849$		0			0
$850$		0			0
$851$	−675806.			−0.933175
$852$		0			0
$853$	175495.	+	175495.i	0.241194	+	0.241194i	0.817344	−	0.576150i	$-0.195446\pi$
$853$	175495.	+	175495.i	0.241194	+	0.241194i	−0.576150	+	0.817344i	$0.695446\pi$
$854$		−	1.27059e6i		−	1.74217i
$855$		0			0
$856$	−71697.7			−0.0978492
$857$	932907.	−	932907.i	1.27021	−	1.27021i	0.324237	−	0.945976i	$-0.394892\pi$
$857$	932907.	−	932907.i	1.27021	−	1.27021i	0.945976	−	0.324237i	$-0.105108\pi$
$858$		0			0
$859$		−	318689.i		−	0.431897i	−0.976405	−	0.215948i	$-0.930716\pi$
$859$		−	318689.i		−	0.431897i	0.976405	−	0.215948i	$-0.0692843\pi$
$860$		0			0
$861$		0			0
$862$	1.66012e6	−	1.66012e6i	2.23422	−	2.23422i
$863$	−834366.	−	834366.i	−1.12030	−	1.12030i	−0.991696	−	0.128606i	$-0.958950\pi$
$863$	−834366.	−	834366.i	−1.12030	−	1.12030i	−0.128606	−	0.991696i	$-0.541050\pi$
$864$		0			0
$865$		0			0
$866$	1.41247e6			1.88341
$867$		0			0
$868$	1.79547e6	+	1.79547e6i	2.38308	+	2.38308i
$869$		−	261433.i		−	0.346195i
$870$		0			0
$871$	−143116.			−0.188648
$872$	1.31654e6	−	1.31654e6i	1.73142	−	1.73142i
$873$		0			0
$874$			1.81426e6i			2.37507i
$875$		0			0
$876$		0			0
$877$	218162.	−	218162.i	0.283648	−	0.283648i	−0.550914	−	0.834562i	$-0.685721\pi$
$877$	218162.	−	218162.i	0.283648	−	0.283648i	0.834562	+	0.550914i	$0.185721\pi$
$878$	129714.	+	129714.i	0.168267	+	0.168267i
$879$		0			0
$880$		0			0
$881$	279844.			0.360549			0.180274	−	0.983616i	$-0.442301\pi$
$881$	279844.			0.360549			0.180274	+	0.983616i	$0.442301\pi$
$882$		0			0
$883$	−269641.	−	269641.i	−0.345831	−	0.345831i	0.512723	−	0.858554i	$-0.328637\pi$
$883$	−269641.	−	269641.i	−0.345831	−	0.345831i	−0.858554	+	0.512723i	$0.828637\pi$
$884$			22896.2i			0.0292994i
$885$		0			0
$886$	−1.32517e6			−1.68812
$887$	566987.	−	566987.i	0.720653	−	0.720653i	−0.248085	−	0.968738i	$-0.579801\pi$
$887$	566987.	−	566987.i	0.720653	−	0.720653i	0.968738	+	0.248085i	$0.0798013\pi$
$888$		0			0
$889$		−	845027.i		−	1.06922i
$890$		0			0
$891$		0			0
$892$	−370867.	+	370867.i	−0.466110	+	0.466110i
$893$	158995.	+	158995.i	0.199380	+	0.199380i
$894$		0			0
$895$		0			0
$896$	1.20490e6			1.50084
$897$		0			0
$898$	−671758.	−	671758.i	−0.833029	−	0.833029i
$899$			1.11067e6i			1.37425i
$900$		0			0
$901$	71463.1			0.0880303
$902$	377442.	−	377442.i	0.463913	−	0.463913i
$903$		0			0
$904$		−	2.35201e6i		−	2.87808i
$905$		0			0
$906$		0			0
$907$	−256644.	+	256644.i	−0.311973	+	0.311973i	−0.845674	−	0.533700i	$-0.820801\pi$
$907$	−256644.	+	256644.i	−0.311973	+	0.311973i	0.533700	+	0.845674i	$0.320801\pi$
$908$	1.18208e6	+	1.18208e6i	1.43375	+	1.43375i
$909$		0			0
$910$		0			0
$911$	143826.			0.173301			0.0866506	−	0.996239i	$-0.472384\pi$
$911$	143826.			0.173301			0.0866506	+	0.996239i	$0.472384\pi$
$912$		0			0
$913$	−25124.4	−	25124.4i	−0.0301408	−	0.0301408i
$914$			1.48312e6i			1.77535i
$915$		0			0
$916$	2.53801e6			3.02485
$917$	−419249.	+	419249.i	−0.498578	+	0.498578i
$918$		0			0
$919$			993411.i			1.17625i	0.808772	+	0.588123i	$0.200133\pi$
$919$			993411.i			1.17625i	−0.808772	+	0.588123i	$0.799867\pi$
$920$		0			0
$921$		0			0
$922$	1.03869e6	−	1.03869e6i	1.22187	−	1.22187i
$923$	−118525.	−	118525.i	−0.139125	−	0.139125i
$924$		0			0
$925$		0			0
$926$	−1.57582e6			−1.83775
$927$		0			0
$928$	−367330.	−	367330.i	−0.426541	−	0.426541i
$929$		−	1.39909e6i		−	1.62112i	−0.585658	−	0.810558i	$-0.699164\pi$
$929$		−	1.39909e6i		−	1.62112i	0.585658	−	0.810558i	$-0.300836\pi$
$930$		0			0
$931$	180338.			0.208059
$932$	629168.	−	629168.i	0.724327	−	0.724327i
$933$		0			0
$934$		−	1.00570e6i		−	1.15286i
$935$		0			0
$936$		0			0
$937$	−482443.	+	482443.i	−0.549499	+	0.549499i	−0.926296	−	0.376797i	$-0.877025\pi$
$937$	−482443.	+	482443.i	−0.549499	+	0.549499i	0.376797	+	0.926296i	$0.377025\pi$
$938$	1.32961e6	+	1.32961e6i	1.51119	+	1.51119i
$939$		0			0
$940$		0			0
$941$	−494897.			−0.558902			−0.279451	−	0.960160i	$-0.590152\pi$
$941$	−494897.			−0.558902			−0.279451	+	0.960160i	$0.590152\pi$
$942$		0			0
$943$	1.06356e6	+	1.06356e6i	1.19602	+	1.19602i
$944$			996871.i			1.11865i
$945$		0			0
$946$	−272864.			−0.304905
$947$	−1.05460e6	+	1.05460e6i	−1.17595	+	1.17595i	−0.195183	+	0.980767i	$0.562530\pi$
$947$	−1.05460e6	+	1.05460e6i	−1.17595	+	1.17595i	−0.980767	+	0.195183i	$0.937470\pi$
$948$		0			0
$949$			185735.i			0.206235i
$950$		0			0
$951$		0			0
$952$	114246.	−	114246.i	0.126058	−	0.126058i
$953$	−319054.	−	319054.i	−0.351300	−	0.351300i	0.509293	−	0.860593i	$-0.329907\pi$
$953$	−319054.	−	319054.i	−0.351300	−	0.351300i	−0.860593	+	0.509293i	$0.829907\pi$
$954$		0			0
$955$		0			0
$956$	3.55229e6			3.88681
$957$		0			0
$958$	627599.	+	627599.i	0.683835	+	0.683835i
$959$			261221.i			0.284034i
$960$		0			0
$961$	891397.			0.965216
$962$	123785.	−	123785.i	0.133758	−	0.133758i
$963$		0			0
$964$		−	2.39885e6i		−	2.58136i
$965$		0			0
$966$		0			0
$967$	1.01810e6	−	1.01810e6i	1.08877	−	1.08877i	0.0931182	−	0.995655i	$-0.470317\pi$
$967$	1.01810e6	−	1.01810e6i	1.08877	−	1.08877i	0.995655	−	0.0931182i	$-0.0296835\pi$
$968$	−1.21400e6	−	1.21400e6i	−1.29559	−	1.29559i
$969$		0			0
$970$		0			0
$971$	−610675.			−0.647696			−0.323848	−	0.946109i	$-0.604977\pi$
$971$	−610675.			−0.647696			−0.323848	+	0.946109i	$0.604977\pi$
$972$		0			0
$973$	235061.	+	235061.i	0.248287	+	0.248287i
$974$			1.06762e6i			1.12538i
$975$		0			0
$976$	1.26362e6			1.32653
$977$	−90031.7	+	90031.7i	−0.0943206	+	0.0943206i	−0.752693	−	0.658372i	$-0.771245\pi$
$977$	−90031.7	+	90031.7i	−0.0943206	+	0.0943206i	0.658372	+	0.752693i	$0.271245\pi$
$978$		0			0
$979$		−	75075.1i		−	0.0783305i
$980$		0			0
$981$		0			0
$982$	930662.	−	930662.i	0.965093	−	0.965093i
$983$	406452.	+	406452.i	0.420632	+	0.420632i	0.885421	−	0.464790i	$-0.153870\pi$
$983$	406452.	+	406452.i	0.420632	+	0.420632i	−0.464790	+	0.885421i	$0.653870\pi$
$984$		0			0
$985$		0			0
$986$	131584.			0.135347
$987$		0			0
$988$	−227158.	−	227158.i	−0.232709	−	0.232709i
$989$		−	768879.i		−	0.786078i
$990$		0			0
$991$	−563990.			−0.574281			−0.287140	−	0.957889i	$-0.592705\pi$
$991$	−563990.			−0.574281			−0.287140	+	0.957889i	$0.592705\pi$
$992$	−600246.	+	600246.i	−0.609967	+	0.609967i
$993$		0			0
$994$			2.20229e6i			2.22896i
$995$		0			0
$996$		0			0
$997$	−49988.1	+	49988.1i	−0.0502894	+	0.0502894i	−0.731804	−	0.681515i	$-0.761321\pi$
$997$	−49988.1	+	49988.1i	−0.0502894	+	0.0502894i	0.681515	+	0.731804i	$0.261321\pi$
$998$	2.42596e6	+	2.42596e6i	2.43569	+	2.43569i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.5.g.m.118.1		8
3.2	odd	2		75.5.f.e.43.4		8
5.2	odd	4	inner	225.5.g.m.82.1		8
5.3	odd	4		45.5.g.e.37.4		8
5.4	even	2		45.5.g.e.28.4		8
15.2	even	4		75.5.f.e.7.4		8
15.8	even	4		15.5.f.a.7.1	✓	8
15.14	odd	2		15.5.f.a.13.1	yes	8
60.23	odd	4		240.5.bg.c.97.3		8
60.59	even	2		240.5.bg.c.193.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.5.f.a.7.1	✓	8	15.8	even	4
15.5.f.a.13.1	yes	8	15.14	odd	2
45.5.g.e.28.4		8	5.4	even	2
45.5.g.e.37.4		8	5.3	odd	4
75.5.f.e.7.4		8	15.2	even	4
75.5.f.e.43.4		8	3.2	odd	2
225.5.g.m.82.1		8	5.2	odd	4	inner
225.5.g.m.118.1		8	1.1	even	1	trivial
240.5.bg.c.97.3		8	60.23	odd	4
240.5.bg.c.193.3		8	60.59	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	225.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-5.02811 + 5.02811i) q^{2} -34.5637i q^{4} +(38.5593 - 38.5593i) q^{7} +(93.3405 + 93.3405i) q^{8} +40.4333 q^{11} +(-20.8688 - 20.8688i) q^{13} +387.760i q^{14} -385.633 q^{16} +(15.8713 - 15.8713i) q^{17} +314.926i q^{19} +(-203.303 + 203.303i) q^{22} +(-572.869 - 572.869i) q^{23} +209.862 q^{26} +(-1332.75 - 1332.75i) q^{28} -824.433i q^{29} -1347.19 q^{31} +(445.555 - 445.555i) q^{32} +159.606i q^{34} +(589.843 - 589.843i) q^{37} +(-1583.48 - 1583.48i) q^{38} -1856.55 q^{41} +(671.078 + 671.078i) q^{43} -1397.53i q^{44} +5760.90 q^{46} +(504.865 - 504.865i) q^{47} -572.636i q^{49} +(-721.305 + 721.305i) q^{52} +(2251.32 + 2251.32i) q^{53} +7198.29 q^{56} +(4145.34 + 4145.34i) q^{58} -2585.03i q^{59} -3276.74 q^{61} +(6773.81 - 6773.81i) q^{62} -1689.53i q^{64} +(3428.94 - 3428.94i) q^{67} +(-548.573 - 548.573i) q^{68} +5679.51 q^{71} +(-4450.06 - 4450.06i) q^{73} +5931.59i q^{74} +10885.0 q^{76} +(1559.08 - 1559.08i) q^{77} -6465.77i q^{79} +(9334.92 - 9334.92i) q^{82} +(-621.380 - 621.380i) q^{83} -6748.50 q^{86} +(3774.07 + 3774.07i) q^{88} -1856.76i q^{89} -1609.37 q^{91} +(-19800.5 + 19800.5i) q^{92} +5077.03i q^{94} +(-12383.5 + 12383.5i) q^{97} +(2879.27 + 2879.27i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 20 q^{7} + 180 q^{8} + 288 q^{11} + 340 q^{13} + 620 q^{16} + 900 q^{17} + 1100 q^{22} - 1560 q^{23} + 3024 q^{26} - 3580 q^{28} - 512 q^{31} + 4980 q^{32} + 3820 q^{37} - 7680 q^{38} + 2712 q^{41}+ \cdots + 46440 q^{98}+O(q^{100}) \) 8 * q - 20 * q^7 + 180 * q^8 + 288 * q^11 + 340 * q^13 + 620 * q^16 + 900 * q^17 + 1100 * q^22 - 1560 * q^23 + 3024 * q^26 - 3580 * q^28 - 512 * q^31 + 4980 * q^32 + 3820 * q^37 - 7680 * q^38 + 2712 * q^41 + 1240 * q^43 + 13528 * q^46 + 4800 * q^47 + 1240 * q^52 + 1020 * q^53 + 30720 * q^56 - 2340 * q^58 - 4760 * q^61 + 28680 * q^62 + 8920 * q^67 - 1920 * q^68 - 7536 * q^71 - 11600 * q^73 + 4344 * q^76 - 360 * q^77 + 27200 * q^82 - 32400 * q^83 - 14592 * q^86 + 14340 * q^88 + 16528 * q^91 - 31800 * q^92 - 58640 * q^97 + 46440 * q^98

Introduction

L-functions

Modular forms

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