LMFDB - Embedded newform 81.5.f.a.17.6

Error: no document with id 276274424 found in table mf_hecke_traces.

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [81,5,Mod(8,81)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("81.8"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	81.f (of order $18$, degree $6$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$8.37296700979$
Analytic rank:	$0$
Dimension:	$66$
Relative dimension:	$11$ over $\Q(\zeta_{18})$
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			17.6
Character	$\chi$	$=$	81.17
Dual form			81.5.f.a.62.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.254601 - 0.699511i) q^{2} +(11.8322 + 9.92841i) q^{4} +(32.8927 + 5.79987i) q^{5} +(-64.8213 + 54.3915i) q^{7} +(20.2723 - 11.7042i) q^{8} +(12.4316 - 21.5322i) q^{10} +(-67.4291 + 11.8896i) q^{11} +(75.3208 - 27.4145i) q^{13} +(21.5439 + 59.1913i) q^{14} +(39.8885 + 226.219i) q^{16} +(196.337 + 113.355i) q^{17} +(156.848 + 271.668i) q^{19} +(331.610 + 395.198i) q^{20} +(-8.85064 + 50.1945i) q^{22} +(96.4022 - 114.888i) q^{23} +(460.984 + 167.784i) q^{25} -59.6675i q^{26} -1307.00 q^{28} +(540.798 - 1485.83i) q^{29} +(-249.929 - 209.716i) q^{31} +(537.243 + 94.7304i) q^{32} +(129.281 - 108.480i) q^{34} +(-2447.61 + 1413.13i) q^{35} +(374.335 - 648.368i) q^{37} +(229.969 - 40.5497i) q^{38} +(734.693 - 267.407i) q^{40} +(655.793 + 1801.78i) q^{41} +(145.099 + 822.898i) q^{43} +(-915.880 - 528.784i) q^{44} +(-55.8211 - 96.6849i) q^{46} +(-2633.58 - 3138.58i) q^{47} +(826.432 - 4686.93i) q^{49} +(234.734 - 279.745i) q^{50} +(1163.39 + 423.441i) q^{52} -2026.25i q^{53} -2286.88 q^{55} +(-677.466 + 1861.32i) q^{56} +(-901.667 - 756.588i) q^{58} +(-550.354 - 97.0423i) q^{59} +(-1370.05 + 1149.60i) q^{61} +(-210.331 + 121.435i) q^{62} +(-1634.62 + 2831.25i) q^{64} +(2636.51 - 464.887i) q^{65} +(3291.06 - 1197.85i) q^{67} +(1197.67 + 3290.56i) q^{68} +(365.334 + 2071.91i) q^{70} +(359.082 + 207.316i) q^{71} +(-3822.91 - 6621.48i) q^{73} +(-358.234 - 426.927i) q^{74} +(-841.377 + 4771.69i) q^{76} +(3724.15 - 4438.27i) q^{77} +(4707.92 + 1713.54i) q^{79} +7672.29i q^{80} +1427.33 q^{82} +(2025.31 - 5564.49i) q^{83} +(5800.61 + 4867.29i) q^{85} +(612.569 + 108.012i) q^{86} +(-1227.78 + 1030.23i) q^{88} +(-1948.72 + 1125.09i) q^{89} +(-3391.27 + 5873.86i) q^{91} +(2281.30 - 402.255i) q^{92} +(-2865.98 + 1043.13i) q^{94} +(3583.51 + 9845.60i) q^{95} +(-405.930 - 2302.15i) q^{97} +(-3068.15 - 1771.40i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 66 q + 6 q^{2} - 6 q^{4} - 3 q^{5} - 6 q^{7} + 9 q^{8} - 3 q^{10} + 492 q^{11} - 6 q^{13} - 1137 q^{14} - 54 q^{16} + 9 q^{17} - 3 q^{19} + 2487 q^{20} + 1002 q^{22} + 2724 q^{23} + 435 q^{25} - 12 q^{28}+ \cdots + 259938 q^{98}+O(q^{100}) $ 66 * q + 6 * q^2 - 6 * q^4 - 3 * q^5 - 6 * q^7 + 9 * q^8 - 3 * q^10 + 492 * q^11 - 6 * q^13 - 1137 * q^14 - 54 * q^16 + 9 * q^17 - 3 * q^19 + 2487 * q^20 + 1002 * q^22 + 2724 * q^23 + 435 * q^25 - 12 * q^28 - 5016 * q^29 - 1671 * q^31 - 10224 * q^32 - 342 * q^34 + 2682 * q^35 - 3 * q^37 + 6564 * q^38 - 1113 * q^40 - 5754 * q^41 - 1266 * q^43 + 4545 * q^44 - 3 * q^46 + 9159 * q^47 + 5898 * q^49 + 34977 * q^50 + 2871 * q^52 - 12 * q^55 - 39243 * q^56 - 12291 * q^58 - 24762 * q^59 - 8358 * q^61 - 38304 * q^62 + 6141 * q^64 - 23727 * q^65 + 15996 * q^67 + 43533 * q^68 - 7251 * q^70 + 19773 * q^71 + 6108 * q^73 + 74847 * q^74 - 28614 * q^76 + 33909 * q^77 - 5658 * q^79 - 12 * q^82 - 18813 * q^83 + 24219 * q^85 - 96474 * q^86 + 36042 * q^88 - 110232 * q^89 - 6042 * q^91 - 73545 * q^92 + 2631 * q^94 + 65163 * q^95 - 3696 * q^97 + 259938 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/81\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$e\left(\frac{11}{18}\right)$

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 $

$2$	0.254601	−	0.699511i	0.0636503	−	0.174878i	−0.903791	−	0.427975i	$-0.859227\pi$
$2$	0.254601	−	0.699511i	0.0636503	−	0.174878i	0.967441	+	0.253097i	$0.0814493\pi$
$3$		0			0
$3$		0			0
$4$	11.8322	+	9.92841i	0.739514	+	0.620526i
$5$	32.8927	+	5.79987i	1.31571	+	0.231995i	0.787076	−	0.616856i	$-0.211594\pi$
$5$	32.8927	+	5.79987i	1.31571	+	0.231995i	0.528632	+	0.848851i	$0.322705\pi$
$6$		0			0
$7$	−64.8213	+	54.3915i	−1.32288	+	1.11003i	−0.337198	+	0.941434i	$0.609479\pi$
$7$	−64.8213	+	54.3915i	−1.32288	+	1.11003i	−0.985685	+	0.168597i	$0.946076\pi$
$8$	20.2723	−	11.7042i	0.316755	−	0.182878i
$9$		0			0
$10$	12.4316	−	21.5322i	0.124316	−	0.215322i
$11$	−67.4291	+	11.8896i	−0.557265	+	0.0982609i	−0.445185	−	0.895438i	$-0.646862\pi$
$11$	−67.4291	+	11.8896i	−0.557265	+	0.0982609i	−0.112080	+	0.993699i	$0.535751\pi$
$12$		0			0
$13$	75.3208	−	27.4145i	0.445685	−	0.162216i	−0.109421	−	0.993995i	$-0.534900\pi$
$13$	75.3208	−	27.4145i	0.445685	−	0.162216i	0.555106	+	0.831779i	$0.312678\pi$
$14$	21.5439	+	59.1913i	0.109918	+	0.301997i
$15$		0			0
$16$	39.8885	+	226.219i	0.155814	+	0.883667i
$17$	196.337	+	113.355i	0.679367	+	0.392233i	0.799617	−	0.600511i	$-0.205036\pi$
$17$	196.337	+	113.355i	0.679367	+	0.392233i	−0.120249	+	0.992744i	$0.538369\pi$
$18$		0			0
$19$	156.848	+	271.668i	0.434481	+	0.752544i	0.997253	−	0.0740685i	$-0.0235984\pi$
$19$	156.848	+	271.668i	0.434481	+	0.752544i	−0.562772	+	0.826612i	$0.690265\pi$
$20$	331.610	+	395.198i	0.829025	+	0.987994i
$21$		0			0
$22$	−8.85064	+	50.1945i	−0.0182864	+	0.103708i
$23$	96.4022	−	114.888i	0.182235	−	0.217179i	−0.667192	−	0.744886i	$-0.732504\pi$
$23$	96.4022	−	114.888i	0.182235	−	0.217179i	0.849426	+	0.527707i	$0.176948\pi$
$24$		0			0
$25$	460.984	+	167.784i	0.737574	+	0.268455i
$26$		−	59.6675i		−	0.0882655i
$27$		0			0
$28$	−1307.00			−1.66709
$29$	540.798	−	1485.83i	0.643042	−	1.76674i	0.00108784	−	0.999999i	$-0.499654\pi$
$29$	540.798	−	1485.83i	0.643042	−	1.76674i	0.641954	−	0.766743i	$-0.278124\pi$
$30$		0			0
$31$	−249.929	−	209.716i	−0.260072	−	0.218227i	0.503423	−	0.864040i	$-0.332074\pi$
$31$	−249.929	−	209.716i	−0.260072	−	0.218227i	−0.763495	+	0.645814i	$0.776518\pi$
$32$	537.243	+	94.7304i	0.524651	+	0.0925102i
$33$		0			0
$34$	129.281	−	108.480i	0.111835	−	0.0938405i
$35$	−2447.61	+	1413.13i	−1.99805	+	1.15357i
$36$		0			0
$37$	374.335	−	648.368i	0.273437	−	0.473607i	−0.696302	−	0.717748i	$-0.745173\pi$
$37$	374.335	−	648.368i	0.273437	−	0.473607i	0.969740	+	0.244141i	$0.0785061\pi$
$38$	229.969	−	40.5497i	0.159258	−	0.0280815i
$39$		0			0
$40$	734.693	−	267.407i	0.459183	−	0.167129i
$41$	655.793	+	1801.78i	0.390121	+	1.07185i	0.966946	+	0.254981i	$0.0820692\pi$
$41$	655.793	+	1801.78i	0.390121	+	1.07185i	−0.576825	+	0.816868i	$0.695709\pi$
$42$		0			0
$43$	145.099	+	822.898i	0.0784744	+	0.445050i	0.998575	+	0.0533680i	$0.0169957\pi$
$43$	145.099	+	822.898i	0.0784744	+	0.445050i	−0.920101	+	0.391682i	$0.871893\pi$
$44$	−915.880	−	528.784i	−0.473078	−	0.273132i
$45$		0			0
$46$	−55.8211	−	96.6849i	−0.0263805	−	0.0456923i
$47$	−2633.58	−	3138.58i	−1.19221	−	1.42082i	−0.882712	−	0.469914i	$-0.844285\pi$
$47$	−2633.58	−	3138.58i	−1.19221	−	1.42082i	−0.309493	−	0.950902i	$-0.600159\pi$
$48$		0			0
$49$	826.432	−	4686.93i	0.344203	−	1.95207i
$50$	234.734	−	279.745i	0.0938936	−	0.111898i
$51$		0			0
$52$	1163.39	+	423.441i	0.430250	+	0.156598i
$53$		−	2026.25i		−	0.721344i	−0.932693	−	0.360672i	$-0.882547\pi$
$53$		−	2026.25i		−	0.721344i	0.932693	−	0.360672i	$-0.117453\pi$
$54$		0			0
$55$	−2286.88			−0.755994
$56$	−677.466	+	1861.32i	−0.216029	+	0.593534i
$57$		0			0
$58$	−901.667	−	756.588i	−0.268034	−	0.224907i
$59$	−550.354	−	97.0423i	−0.158102	−	0.0278777i	0.0940367	−	0.995569i	$-0.470023\pi$
$59$	−550.354	−	97.0423i	−0.158102	−	0.0278777i	−0.252139	+	0.967691i	$0.581134\pi$
$60$		0			0
$61$	−1370.05	+	1149.60i	−0.368193	+	0.308950i	−0.808046	−	0.589119i	$-0.799475\pi$
$61$	−1370.05	+	1149.60i	−0.368193	+	0.308950i	0.439853	+	0.898070i	$0.355030\pi$
$62$	−210.331	+	121.435i	−0.0547166	+	0.0315907i
$63$		0			0
$64$	−1634.62	+	2831.25i	−0.399077	+	0.691222i
$65$	2636.51	−	464.887i	0.624025	−	0.110032i
$66$		0			0
$67$	3291.06	−	1197.85i	0.733139	−	0.266841i	0.0516461	−	0.998665i	$-0.483553\pi$
$67$	3291.06	−	1197.85i	0.733139	−	0.266841i	0.681493	+	0.731825i	$0.261331\pi$
$68$	1197.67	+	3290.56i	0.259011	+	0.711626i
$69$		0			0
$70$	365.334	+	2071.91i	0.0745581	+	0.422840i
$71$	359.082	+	207.316i	0.0712324	+	0.0411260i	0.535193	−	0.844730i	$-0.320239\pi$
$71$	359.082	+	207.316i	0.0712324	+	0.0411260i	−0.463961	+	0.885856i	$0.653572\pi$
$72$		0			0
$73$	−3822.91	−	6621.48i	−0.717379	−	1.24254i	−0.962035	−	0.272927i	$-0.912008\pi$
$73$	−3822.91	−	6621.48i	−0.717379	−	1.24254i	0.244655	−	0.969610i	$-0.421325\pi$
$74$	−358.234	−	426.927i	−0.0654190	−	0.0779633i
$75$		0			0
$76$	−841.377	+	4771.69i	−0.145668	+	0.826123i
$77$	3724.15	−	4438.27i	0.628124	−	0.748569i
$78$		0			0
$79$	4707.92	+	1713.54i	0.754354	+	0.274562i	0.690437	−	0.723393i	$-0.257418\pi$
$79$	4707.92	+	1713.54i	0.754354	+	0.274562i	0.0639174	+	0.997955i	$0.479641\pi$
$80$			7672.29i			1.19880i
$81$		0			0
$82$	1427.33			0.212274
$83$	2025.31	−	5564.49i	0.293992	−	0.807736i	−0.701481	−	0.712688i	$-0.747477\pi$
$83$	2025.31	−	5564.49i	0.293992	−	0.807736i	0.995473	−	0.0950476i	$-0.0303003\pi$
$84$		0			0
$85$	5800.61	+	4867.29i	0.802853	+	0.673674i
$86$	612.569	+	108.012i	0.0828243	+	0.0146042i
$87$		0			0
$88$	−1227.78	+	1030.23i	−0.158546	+	0.133036i
$89$	−1948.72	+	1125.09i	−0.246019	+	0.142039i	−0.617940	−	0.786225i	$-0.712033\pi$
$89$	−1948.72	+	1125.09i	−0.246019	+	0.142039i	0.371921	+	0.928264i	$0.378699\pi$
$90$		0			0
$91$	−3391.27	+	5873.86i	−0.409525	+	0.709317i
$92$	2281.30	−	402.255i	0.269530	−	0.0475254i
$93$		0			0
$94$	−2865.98	+	1043.13i	−0.324353	+	0.118055i
$95$	3583.51	+	9845.60i	0.397064	+	1.09093i
$96$		0			0
$97$	−405.930	−	2302.15i	−0.0431428	−	0.244675i	0.955608	−	0.294641i	$-0.0951999\pi$
$97$	−405.930	−	2302.15i	−0.0431428	−	0.244675i	−0.998751	+	0.0499657i	$0.984089\pi$
$98$	−3068.15	−	1771.40i	−0.319466	−	0.184444i
$99$		0			0
$100$	3788.63	+	6562.09i	0.378863	+	0.656209i
$101$	5027.20	+	5991.18i	0.492814	+	0.587313i	0.953931	−	0.300027i	$-0.0969955\pi$
$101$	5027.20	+	5991.18i	0.492814	+	0.587313i	−0.461117	+	0.887339i	$0.652551\pi$
$102$		0			0
$103$	−71.8400	+	407.425i	−0.00677161	+	0.0384037i	−0.988006	−	0.154413i	$-0.950651\pi$
$103$	−71.8400	+	407.425i	−0.00677161	+	0.0384037i	0.981235	+	0.192817i	$0.0617624\pi$
$104$	1206.06	−	1437.33i	0.111507	−	0.132889i
$105$		0			0
$106$	−1417.39	−	515.887i	−0.126147	−	0.0459137i
$107$		−	2540.42i		−	0.221890i	−0.993827	−	0.110945i	$-0.964612\pi$
$107$		−	2540.42i		−	0.221890i	0.993827	−	0.110945i	$-0.0353878\pi$
$108$		0			0
$109$	11640.7			0.979778			0.489889	−	0.871785i	$-0.337037\pi$
$109$	11640.7			0.979778			0.489889	+	0.871785i	$0.337037\pi$
$110$	−582.243	+	1599.70i	−0.0481192	+	0.132207i
$111$		0			0
$112$	−14890.0	−	12494.2i	−1.18702	−	0.996029i
$113$	−17181.1	−	3029.49i	−1.34553	−	0.237254i	−0.545954	−	0.837815i	$-0.683833\pi$
$113$	−17181.1	−	3029.49i	−1.34553	−	0.237254i	−0.799579	+	0.600562i	$0.794944\pi$
$114$		0			0
$115$	3837.26	−	3219.84i	0.290152	−	0.243467i
$116$	21150.8	−	12211.4i	1.57185	−	0.907506i
$117$		0			0
$118$	−208.003	+	360.272i	−0.0149385	+	0.0258742i
$119$	−18892.4	+	3331.24i	−1.33411	+	0.235240i
$120$		0			0
$121$	−9352.72	+	3404.11i	−0.638803	+	0.232505i
$122$	455.346	+	1251.05i	0.0305930	+	0.0840535i
$123$		0			0
$124$	−875.076	−	4962.80i	−0.0569118	−	0.322763i
$125$	−3888.48	−	2245.02i	−0.248863	−	0.143681i
$126$		0			0
$127$	1750.66	+	3032.24i	0.108541	+	0.187999i	0.915180	−	0.403046i	$-0.132049\pi$
$127$	1750.66	+	3032.24i	0.108541	+	0.187999i	−0.806638	+	0.591046i	$0.798715\pi$
$128$	7174.88	+	8550.69i	0.437920	+	0.521892i
$129$		0			0
$130$	346.064	−	1962.63i	0.0204771	−	0.116132i
$131$	18137.8	−	21615.8i	1.05692	−	1.25959i	0.0923590	−	0.995726i	$-0.470559\pi$
$131$	18137.8	−	21615.8i	1.05692	−	1.25959i	0.964560	−	0.263862i	$-0.0849963\pi$
$132$		0			0
$133$	−24943.5	−	9078.70i	−1.41011	−	0.513240i
$134$		−	2607.11i		−	0.145194i
$135$		0			0
$136$	5306.94			0.286924
$137$	−2812.61	+	7727.59i	−0.149854	+	0.411721i	−0.991793	−	0.127852i	$-0.959192\pi$
$137$	−2812.61	+	7727.59i	−0.149854	+	0.411721i	0.841939	+	0.539572i	$0.181414\pi$
$138$		0			0
$139$	17872.4	+	14996.8i	0.925027	+	0.776190i	0.974918	−	0.222565i	$-0.0714429\pi$
$139$	17872.4	+	14996.8i	0.925027	+	0.776190i	−0.0498909	+	0.998755i	$0.515887\pi$
$140$	−42990.8	−	7580.43i	−2.19341	−	0.386757i
$141$		0			0
$142$	236.443	−	198.399i	0.0117260	−	0.00983927i
$143$	−4752.86	+	2744.07i	−0.232425	+	0.134191i
$144$		0			0
$145$	26405.9	−	45736.4i	1.25593	−	2.17534i
$146$	−5605.12	+	988.333i	−0.262954	+	0.0463658i
$147$		0			0
$148$	10866.5	−	3955.08i	0.496096	−	0.180564i
$149$	416.981	+	1145.65i	0.0187821	+	0.0516034i	0.948729	−	0.316089i	$-0.102370\pi$
$149$	416.981	+	1145.65i	0.0187821	+	0.0516034i	−0.929947	+	0.367693i	$0.880148\pi$
$150$		0			0
$151$	−4810.34	−	27280.8i	−0.210971	−	1.19647i	−0.887763	−	0.460300i	$-0.847742\pi$
$151$	−4810.34	−	27280.8i	−0.210971	−	1.19647i	0.676792	−	0.736174i	$-0.263369\pi$
$152$	6359.33	+	3671.56i	0.275248	+	0.158914i
$153$		0			0
$154$	−2156.44	−	3735.07i	−0.0909278	−	0.157492i
$155$	−7004.53	−	8347.67i	−0.291552	−	0.347458i
$156$		0			0
$157$	−4822.84	+	27351.7i	−0.195661	+	1.10965i	0.715814	+	0.698291i	$0.246056\pi$
$157$	−4822.84	+	27351.7i	−0.195661	+	1.10965i	−0.911475	+	0.411356i	$0.865055\pi$
$158$	2397.29	−	2856.97i	0.0960297	−	0.114444i
$159$		0			0
$160$	17121.9	+	6231.88i	0.668826	+	0.243433i
$161$			12690.6i			0.489588i
$162$		0			0
$163$	−20535.9			−0.772925			−0.386463	−	0.922305i	$-0.626303\pi$
$163$	−20535.9			−0.772925			−0.386463	+	0.922305i	$0.626303\pi$
$164$	−10129.3	+	27830.0i	−0.376610	+	1.03473i
$165$		0			0
$166$	−3376.78	−	2833.45i	−0.122542	−	0.102825i
$167$	−18828.8	−	3320.02i	−0.675132	−	0.119044i	−0.174438	−	0.984668i	$-0.555811\pi$
$167$	−18828.8	−	3320.02i	−0.675132	−	0.119044i	−0.500694	+	0.865624i	$0.666922\pi$
$168$		0			0
$169$	−16957.3	+	14228.9i	−0.593723	+	0.498193i
$170$	4881.57	−	2818.37i	0.168912	−	0.0975216i
$171$		0			0
$172$	−6453.22	+	11177.3i	−0.218132	+	0.377816i
$173$	39817.9	−	7020.98i	1.33041	−	0.234588i	0.537160	−	0.843480i	$-0.319497\pi$
$173$	39817.9	−	7020.98i	1.33041	−	0.234588i	0.793253	+	0.608892i	$0.208386\pi$
$174$		0			0
$175$	−39007.6	+	14197.6i	−1.27372	+	0.463595i
$176$	−5379.28	−	14779.5i	−0.173660	−	0.477126i
$177$		0			0
$178$	290.869	+	1649.60i	0.00918030	+	0.0520641i
$179$	37606.9	+	21712.4i	1.17371	+	0.677643i	0.954552	−	0.298046i	$-0.0963348\pi$
$179$	37606.9	+	21712.4i	1.17371	+	0.677643i	0.219160	+	0.975689i	$0.429668\pi$
$180$		0			0
$181$	23540.6	+	40773.4i	0.718554	+	1.24457i	0.961573	+	0.274551i	$0.0885292\pi$
$181$	23540.6	+	40773.4i	0.718554	+	1.24457i	−0.243018	+	0.970022i	$0.578137\pi$
$182$	3245.41	+	3867.72i	0.0979774	+	0.116765i
$183$		0			0
$184$	609.624	−	3457.35i	0.0180064	−	0.102119i
$185$	16073.4	−	19155.5i	0.469638	−	0.559693i
$186$		0			0
$187$	−14586.6	−	5309.08i	−0.417129	−	0.151822i
$188$		−	63283.7i		−	1.79051i
$189$		0			0
$190$	7799.47			0.216052
$191$	−4278.66	+	11755.5i	−0.117285	+	0.322237i	−0.984419	−	0.175837i	$-0.943737\pi$
$191$	−4278.66	+	11755.5i	−0.117285	+	0.322237i	0.867135	+	0.498074i	$0.165959\pi$
$192$		0			0
$193$	7137.74	+	5989.28i	0.191622	+	0.160790i	0.733551	−	0.679634i	$-0.237861\pi$
$193$	7137.74	+	5989.28i	0.191622	+	0.160790i	−0.541929	+	0.840424i	$0.682306\pi$
$194$	−1713.73	−	302.176i	−0.0455342	−	0.00802891i
$195$		0			0
$196$	56312.3	−	47251.6i	1.46585	−	1.23000i
$197$	−5465.01	+	3155.23i	−0.140818	+	0.0813014i	−0.568754	−	0.822508i	$-0.692574\pi$
$197$	−5465.01	+	3155.23i	−0.140818	+	0.0813014i	0.427936	+	0.903809i	$0.359241\pi$
$198$		0			0
$199$	−19132.7	+	33138.8i	−0.483136	+	0.836816i	−0.999812	−	0.0193646i	$-0.993836\pi$
$199$	−19132.7	+	33138.8i	−0.483136	+	0.836816i	0.516677	+	0.856181i	$0.327169\pi$
$200$	11309.0	−	1994.08i	0.282724	−	0.0498519i
$201$		0			0
$202$	5470.83	−	1991.22i	0.134076	−	0.0487996i
$203$	45761.3	+	125728.i	1.11047	+	3.05099i
$204$		0			0
$205$	11120.7	+	63068.8i	0.264622	+	1.50075i
$206$	266.708	+	153.984i	0.00628494	+	0.00362861i
$207$		0			0
$208$	9206.11	+	15945.4i	0.212789	+	0.368562i
$209$	−13806.1	−	16453.5i	−0.316067	−	0.376674i
$210$		0			0
$211$	2970.23	−	16845.0i	0.0667153	−	0.378361i	−0.933109	−	0.359595i	$-0.882915\pi$
$211$	2970.23	−	16845.0i	0.0667153	−	0.378361i	0.999824	−	0.0187666i	$-0.00597395\pi$
$212$	20117.5	−	23975.1i	0.447612	−	0.533444i
$213$		0			0
$214$	−1777.05	−	646.795i	−0.0388037	−	0.0141234i
$215$			27908.9i			0.603762i
$216$		0			0
$217$	27607.5			0.586283
$218$	2963.75	−	8142.82i	0.0623631	−	0.171341i
$219$		0			0
$220$	−27058.9	−	22705.1i	−0.559068	−	0.469114i
$221$	17895.9	+	3155.52i	0.366410	+	0.0646080i
$222$		0			0
$223$	−55489.0	+	46560.8i	−1.11583	+	0.936291i	−0.998386	−	0.0567890i	$-0.981914\pi$
$223$	−55489.0	+	46560.8i	−1.11583	+	0.936291i	−0.117442	+	0.993080i	$0.537469\pi$
$224$	−39977.3	+	23080.9i	−0.796741	+	0.459999i
$225$		0			0
$226$	−6493.49	+	11247.1i	−0.127134	+	0.220202i
$227$	−50165.4	+	8845.52i	−0.973538	+	0.171661i	−0.637722	−	0.770267i	$-0.720123\pi$
$227$	−50165.4	+	8845.52i	−0.973538	+	0.171661i	−0.335816	+	0.941928i	$0.609012\pi$
$228$		0			0
$229$	85834.2	−	31241.1i	1.63678	−	0.595738i	0.650306	−	0.759673i	$-0.274641\pi$
$229$	85834.2	−	31241.1i	1.63678	−	0.595738i	0.986471	+	0.163935i	$0.0524186\pi$
$230$	−1275.35	−	3503.98i	−0.0241086	−	0.0662379i
$231$		0			0
$232$	−6427.26	−	36450.8i	−0.119413	−	0.677222i
$233$	−45583.3	−	26317.5i	−0.839642	−	0.484768i	0.0175004	−	0.999847i	$-0.494429\pi$
$233$	−45583.3	−	26317.5i	−0.839642	−	0.484768i	−0.857143	+	0.515079i	$0.827762\pi$
$234$		0			0
$235$	−68422.3	−	118511.i	−1.23897	−	2.14596i
$236$	−5548.44	−	6612.37i	−0.0996200	−	0.118722i
$237$		0			0
$238$	−2479.79	+	14063.6i	−0.0437785	+	0.248280i
$239$	4247.15	−	5061.55i	0.0743535	−	0.0886110i	−0.727585	−	0.686017i	$-0.759357\pi$
$239$	4247.15	−	5061.55i	0.0743535	−	0.0886110i	0.801939	+	0.597406i	$0.203802\pi$
$240$		0			0
$241$	16925.2	+	6160.29i	0.291408	+	0.106064i	0.483588	−	0.875296i	$-0.339333\pi$
$241$	16925.2	+	6160.29i	0.291408	+	0.106064i	−0.192180	+	0.981360i	$0.561556\pi$
$242$			7409.02i			0.126512i
$243$		0			0
$244$	−27624.4			−0.463995
$245$	54367.2	−	149373.i	0.905742	−	2.48851i
$246$		0			0
$247$	19261.6	+	16162.4i	0.315717	+	0.264918i
$248$	−7521.20	−	1326.19i	−0.122288	−	0.0215627i
$249$		0			0
$250$	−2560.43	+	2148.45i	−0.0409668	+	0.0343752i
$251$	−21455.0	+	12387.0i	−0.340550	+	0.196617i	−0.660515	−	0.750813i	$-0.729662\pi$
$251$	−21455.0	+	12387.0i	−0.340550	+	0.196617i	0.319965	+	0.947429i	$0.396329\pi$
$252$		0			0
$253$	−5134.35	+	8892.95i	−0.0802129	+	0.138933i
$254$	2566.81	−	452.597i	0.0397856	−	0.00701527i
$255$		0			0
$256$	−41345.3	+	15048.4i	−0.630879	+	0.229621i
$257$	−17438.2	−	47911.0i	−0.264019	−	0.725385i	−0.998887	−	0.0471749i	$-0.984978\pi$
$257$	−17438.2	−	47911.0i	−0.264019	−	0.725385i	0.734868	−	0.678210i	$-0.237244\pi$
$258$		0			0
$259$	11000.8	+	62388.7i	0.163993	+	0.930050i
$260$	35811.3	+	20675.7i	0.529753	+	0.305853i
$261$		0			0
$262$	−10502.6	−	18191.0i	−0.153001	−	0.265005i
$263$	43299.3	+	51602.1i	0.625993	+	0.746029i	0.982088	−	0.188420i	$-0.0603367\pi$
$263$	43299.3	+	51602.1i	0.625993	+	0.746029i	−0.356096	+	0.934449i	$0.615892\pi$
$264$		0			0
$265$	11752.0	−	66649.0i	0.167348	−	0.949078i
$266$	−12701.3	+	15136.8i	−0.179508	+	0.213930i
$267$		0			0
$268$	50833.3	+	18501.8i	0.707748	+	0.257599i
$269$			53854.7i			0.744250i	0.928183	+	0.372125i	$0.121371\pi$
$269$			53854.7i			0.744250i	−0.928183	+	0.372125i	$0.878629\pi$
$270$		0			0
$271$	−55212.6			−0.751795			−0.375898	−	0.926661i	$-0.622666\pi$
$271$	−55212.6			−0.751795			−0.375898	+	0.926661i	$0.622666\pi$
$272$	−17811.5	+	48936.7i	−0.240748	+	0.661450i
$273$		0			0
$274$	4689.44	+	3934.91i	0.0624625	+	0.0524123i
$275$	−33078.6	−	5832.64i	−0.437403	−	0.0771259i
$276$		0			0
$277$	75052.6	−	62976.6i	0.978152	−	0.820767i	−0.00565790	−	0.999984i	$-0.501801\pi$
$277$	75052.6	−	62976.6i	0.978152	−	0.820767i	0.983810	+	0.179217i	$0.0573565\pi$
$278$	15040.7	−	8683.78i	0.194617	−	0.112362i
$279$		0			0
$280$	−33079.1	+	57294.7i	−0.421928	+	0.730800i
$281$	−56691.2	+	9996.19i	−0.717965	+	0.126597i	−0.520683	−	0.853750i	$-0.674323\pi$
$281$	−56691.2	+	9996.19i	−0.717965	+	0.126597i	−0.197282	+	0.980347i	$0.563211\pi$
$282$		0			0
$283$	42767.7	−	15566.2i	0.534002	−	0.194361i	−0.0609224	−	0.998143i	$-0.519404\pi$
$283$	42767.7	−	15566.2i	0.534002	−	0.194361i	0.594924	+	0.803782i	$0.297182\pi$
$284$	2190.42	+	6018.13i	0.0271576	+	0.0746148i
$285$		0			0
$286$	709.421	+	4023.32i	0.00867305	+	0.0491873i
$287$	−140511.	−	81123.9i	−1.70587	−	0.984884i
$288$		0			0
$289$	−16061.7	−	27819.6i	−0.192307	−	0.333085i
$290$	−25270.1	−	30115.8i	−0.300477	−	0.358095i
$291$		0			0
$292$	20507.2	−	116302.i	0.240514	−	1.36403i
$293$	25799.3	−	30746.4i	0.300520	−	0.358146i	−0.594560	−	0.804051i	$-0.702674\pi$
$293$	25799.3	−	30746.4i	0.300520	−	0.358146i	0.895080	+	0.445905i	$0.147118\pi$
$294$		0			0
$295$	−17539.8	−	6383.97i	−0.201549	−	0.0733579i
$296$		−	17525.2i		−	0.200023i
$297$		0			0
$298$	907.556			0.0102198
$299$	4111.50	−	11296.2i	0.0459894	−	0.126355i
$300$		0			0
$301$	−54164.2	−	45449.1i	−0.597832	−	0.501641i
$302$	−20307.9	−	3580.84i	−0.222665	−	0.0392619i
$303$		0			0
$304$	−55200.0	+	46318.3i	−0.597300	+	0.501194i
$305$	−51732.0	+	29867.5i	−0.556109	+	0.321070i
$306$		0			0
$307$	44563.2	−	77185.8i	0.472825	−	0.818956i	−0.526692	−	0.850056i	$-0.676568\pi$
$307$	44563.2	−	77185.8i	0.472825	−	0.818956i	0.999516	+	0.0311002i	$0.00990110\pi$
$308$	88129.8	−	15539.7i	0.929012	−	0.163810i
$309$		0			0
$310$	−7622.65	+	2774.42i	−0.0793200	+	0.0288701i
$311$	23554.2	+	64714.7i	0.243528	+	0.669087i	0.999889	+	0.0149321i	$0.00475320\pi$
$311$	23554.2	+	64714.7i	0.243528	+	0.669087i	−0.756361	+	0.654155i	$0.773025\pi$
$312$		0			0
$313$	−4982.98	−	28259.9i	−0.0508628	−	0.288457i	0.948758	−	0.316004i	$-0.102341\pi$
$313$	−4982.98	−	28259.9i	−0.0508628	−	0.288457i	−0.999621	+	0.0275472i	$0.991230\pi$
$314$	17904.9	+	10337.4i	0.181599	+	0.104846i
$315$		0			0
$316$	38692.4	+	67017.2i	0.387482	+	0.671139i
$317$	−98084.4	−	116892.i	−0.976071	−	1.16324i	−0.986578	−	0.163291i	$-0.947789\pi$
$317$	−98084.4	−	116892.i	−0.976071	−	1.16324i	0.0105069	−	0.999945i	$-0.496655\pi$
$318$		0			0
$319$	−18799.6	+	106618.i	−0.184743	+	1.04773i
$320$	−70187.9	+	83646.7i	−0.685429	+	0.816863i
$321$		0			0
$322$	8877.23	+	3231.05i	0.0856181	+	0.0311624i
$323$			71118.1i			0.681671i
$324$		0			0
$325$	39321.4			0.372273
$326$	−5228.45	+	14365.1i	−0.0491969	+	0.135167i
$327$		0			0
$328$	34382.8	+	28850.6i	0.319590	+	0.268168i
$329$	341424.	+	60202.3i	3.15430	+	0.556188i
$330$		0			0
$331$	−81778.9	+	68620.6i	−0.746423	+	0.626324i	−0.934554	−	0.355820i	$-0.884202\pi$
$331$	−81778.9	+	68620.6i	−0.746423	+	0.626324i	0.188131	+	0.982144i	$0.439757\pi$
$332$	79210.5	−	45732.2i	0.718632	−	0.414902i
$333$		0			0
$334$	−7116.22	+	12325.6i	−0.0637905	+	0.110488i
$335$	115199.	−	20312.7i	1.02650	−	0.181000i
$336$		0			0
$337$	−6305.13	+	2294.88i	−0.0555180	+	0.0202069i	−0.369630	−	0.929179i	$-0.620516\pi$
$337$	−6305.13	+	2294.88i	−0.0555180	+	0.0202069i	0.314112	+	0.949386i	$0.398293\pi$
$338$	5635.91	+	15484.5i	0.0493322	+	0.135539i
$339$		0			0
$340$	20309.6	+	115182.i	0.175689	+	0.996382i
$341$	19345.9	+	11169.4i	0.166372	+	0.0960551i
$342$		0			0
$343$	99774.7	+	172815.i	0.848071	+	1.46890i
$344$	12572.9	+	14983.8i	0.106247	+	0.126620i
$345$		0			0
$346$	5226.44	−	29640.6i	0.0436570	−	0.247591i
$347$	−23538.4	+	28052.0i	−0.195487	+	0.232973i	−0.854880	−	0.518826i	$-0.826369\pi$
$347$	−23538.4	+	28052.0i	−0.195487	+	0.232973i	0.659392	+	0.751799i	$0.270814\pi$
$348$		0			0
$349$	31908.3	+	11613.7i	0.261971	+	0.0953496i	0.469667	−	0.882844i	$-0.344374\pi$
$349$	31908.3	+	11613.7i	0.261971	+	0.0953496i	−0.207696	+	0.978193i	$0.566596\pi$
$350$			30901.0i			0.252253i
$351$		0			0
$352$	−37352.1			−0.301460
$353$	−18275.2	+	50210.7i	−0.146660	+	0.402946i	−0.991171	−	0.132593i	$-0.957670\pi$
$353$	−18275.2	+	50210.7i	−0.146660	+	0.402946i	0.844510	+	0.535540i	$0.179892\pi$
$354$		0			0
$355$	10608.8	+	8901.82i	0.0841800	+	0.0706354i
$356$	−34228.0	−	6035.32i	−0.270073	−	0.0476212i
$357$		0			0
$358$	24762.8	−	20778.5i	0.193212	−	0.162124i
$359$	94896.6	−	54788.6i	0.736312	−	0.425110i	−0.0844152	−	0.996431i	$-0.526902\pi$
$359$	94896.6	−	54788.6i	0.736312	−	0.425110i	0.820727	+	0.571321i	$0.193569\pi$
$360$		0			0
$361$	15958.1	−	27640.2i	0.122452	−	0.212093i
$362$	34514.9	−	6085.91i	0.263384	−	0.0464418i
$363$		0			0
$364$	−98444.3	+	35830.8i	−0.742998	+	0.270429i
$365$	−87342.2	−	239971.i	−0.655599	−	1.80124i
$366$		0			0
$367$	−31256.6	−	177265.i	−0.232065	−	1.31610i	−0.848708	−	0.528862i	$-0.822619\pi$
$367$	−31256.6	−	177265.i	−0.232065	−	1.31610i	0.616643	−	0.787243i	$-0.288492\pi$
$368$	29835.1	+	17225.3i	0.220309	+	0.127195i
$369$		0			0
$370$	−9307.17	−	16120.5i	−0.0679852	−	0.117754i
$371$	110211.	+	131344.i	0.800714	+	0.954254i
$372$		0			0
$373$	−35563.1	+	201689.i	−0.255613	+	1.44965i	0.538882	+	0.842381i	$0.318847\pi$
$373$	−35563.1	+	201689.i	−0.255613	+	1.44965i	−0.794495	+	0.607271i	$0.792264\pi$
$374$	−7427.52	+	8851.77i	−0.0531007	+	0.0632830i
$375$		0			0
$376$	−90123.4	−	32802.2i	−0.637473	−	0.232021i
$377$		−	126740.i		−	0.891723i
$378$		0			0
$379$	−102840.			−0.715949			−0.357975	−	0.933731i	$-0.616533\pi$
$379$	−102840.			−0.715949			−0.357975	+	0.933731i	$0.616533\pi$
$380$	−55350.3	+	152074.i	−0.383313	+	1.05314i
$381$		0			0
$382$	7133.76	+	5985.93i	0.0488868	+	0.0410209i
$383$	−223273.	−	39369.1i	−1.52208	−	0.268385i	−0.650833	−	0.759221i	$-0.725580\pi$
$383$	−223273.	−	39369.1i	−1.52208	−	0.268385i	−0.871252	+	0.490836i	$0.836691\pi$
$384$		0			0
$385$	148239.	−	124387.i	1.00009	−	0.839177i
$386$	6006.84	−	3468.05i	0.0403155	−	0.0232761i
$387$		0			0
$388$	18053.6	−	31269.7i	0.119922	−	0.207712i
$389$	−194117.	+	34228.1i	−1.28282	+	0.226195i	−0.773176	−	0.634192i	$-0.781333\pi$
$389$	−194117.	+	34228.1i	−1.28282	+	0.226195i	−0.509641	+	0.860387i	$0.670222\pi$
$390$		0			0
$391$	31950.5	−	11629.0i	0.208989	−	0.0760658i
$392$	−38103.2	−	104688.i	−0.247964	−	0.681276i
$393$		0			0
$394$	815.717	+	4626.16i	0.00525469	+	0.0298008i
$395$	144918.	+	83668.5i	0.928813	+	0.536250i
$396$		0			0
$397$	−11400.4	−	19746.1i	−0.0723335	−	0.125285i	0.827590	−	0.561333i	$-0.189711\pi$
$397$	−11400.4	−	19746.1i	−0.0723335	−	0.125285i	−0.899924	+	0.436048i	$0.856378\pi$
$398$	18309.7	+	21820.7i	0.115589	+	0.137753i
$399$		0			0
$400$	−19568.0	+	110976.i	−0.122300	+	0.693598i
$401$	−136140.	+	162246.i	−0.846640	+	1.00899i	0.153144	+	0.988204i	$0.451060\pi$
$401$	−136140.	+	162246.i	−0.846640	+	1.00899i	−0.999784	+	0.0207820i	$0.993384\pi$
$402$		0			0
$403$	−24574.1	−	8944.26i	−0.151310	−	0.0550724i
$404$			120801.i			0.740130i
$405$		0			0
$406$	99599.2			0.604232
$407$	−17532.3	+	48169.5i	−0.105840	+	0.290793i
$408$		0			0
$409$	31824.6	+	26704.0i	0.190247	+	0.159636i	0.732936	−	0.680298i	$-0.238149\pi$
$409$	31824.6	+	26704.0i	0.190247	+	0.159636i	−0.542689	+	0.839933i	$0.682594\pi$
$410$	46948.7	+	8278.32i	0.279290	+	0.0492464i
$411$		0			0
$412$	−4895.11	+	4107.49i	−0.0288382	+	0.0241981i
$413$	40952.9	−	23644.2i	0.240096	−	0.138620i
$414$		0			0
$415$	98891.2	−	171285.i	0.574198	−	0.994540i
$416$	43062.6	−	7593.09i	0.248836	−	0.0438765i
$417$		0			0
$418$	−15024.5	+	5468.45i	−0.0859896	+	0.0312977i
$419$	−115543.	−	317452.i	−0.658136	−	1.80821i	−0.585217	−	0.810877i	$-0.698991\pi$
$419$	−115543.	−	317452.i	−0.658136	−	1.80821i	−0.0729191	−	0.997338i	$-0.523231\pi$
$420$		0			0
$421$	−24231.5	−	137423.i	−0.136715	−	0.775348i	−0.973650	−	0.228046i	$-0.926766\pi$
$421$	−24231.5	−	137423.i	−0.136715	−	0.775348i	0.836936	−	0.547302i	$-0.184345\pi$
$422$	−11027.1	−	6366.47i	−0.0619205	−	0.0357498i
$423$		0			0
$424$	−23715.7	−	41076.8i	−0.131918	−	0.228489i
$425$	71488.9	+	85197.2i	0.395787	+	0.471680i
$426$		0			0
$427$	26279.4	−	149038.i	0.144132	−	0.817411i
$428$	25222.4	−	30058.8i	0.137689	−	0.164091i
$429$		0			0
$430$	19522.6	+	7105.64i	0.105585	+	0.0384296i
$431$		−	344917.i		−	1.85678i	−0.371610	−	0.928389i	$-0.621194\pi$
$431$		−	344917.i		−	1.85678i	0.371610	−	0.928389i	$-0.378806\pi$
$432$		0			0
$433$	205873.			1.09805			0.549027	−	0.835805i	$-0.314999\pi$
$433$	205873.			1.09805			0.549027	+	0.835805i	$0.314999\pi$
$434$	7028.90	−	19311.7i	0.0373171	−	0.102528i
$435$		0			0
$436$	137736.	+	115574.i	0.724559	+	0.607977i
$437$	46331.8	+	8169.55i	0.242614	+	0.0427794i
$438$		0			0
$439$	−218946.	+	183718.i	−1.13608	+	0.953282i	−0.999303	−	0.0373199i	$-0.988118\pi$
$439$	−218946.	+	183718.i	−1.13608	+	0.953282i	−0.136774	+	0.990602i	$0.543674\pi$
$440$	−46360.3	+	26766.2i	−0.239465	+	0.138255i
$441$		0			0
$442$	6763.63	−	11714.9i	0.0346206	−	0.0599647i
$443$	−252713.	+	44560.2i	−1.28772	+	0.227059i	−0.775254	−	0.631650i	$-0.782378\pi$
$443$	−252713.	+	44560.2i	−1.28772	+	0.227059i	−0.512463	+	0.858709i	$0.671267\pi$
$444$		0			0
$445$	−70624.0	+	25705.0i	−0.356642	+	0.129807i
$446$	18442.2	+	50669.6i	0.0927136	+	0.254729i
$447$		0			0
$448$	−48037.5	−	272434.i	−0.239345	−	1.35739i
$449$	195404.	+	112817.i	0.969261	+	0.559603i	0.899011	−	0.437926i	$-0.144287\pi$
$449$	195404.	+	112817.i	0.969261	+	0.559603i	0.0702500	+	0.997529i	$0.477620\pi$
$450$		0			0
$451$	−65641.9	−	113695.i	−0.322722	−	0.558970i
$452$	−173213.	−	206427.i	−0.847818	−	1.01039i
$453$		0			0
$454$	−6584.64	+	37343.4i	−0.0319463	+	0.181176i
$455$	−145616.	+	173538.i	−0.703373	+	0.838247i
$456$		0			0
$457$	−182922.	−	66578.1i	−0.875856	−	0.318786i	−0.135320	−	0.990802i	$-0.543206\pi$
$457$	−182922.	−	66578.1i	−0.875856	−	0.318786i	−0.740536	+	0.672016i	$0.765428\pi$
$458$		−	67996.0i		−	0.324155i
$459$		0			0
$460$	77371.3			0.365649
$461$	77924.7	−	214096.i	0.366668	−	1.00741i	−0.609952	−	0.792439i	$-0.708811\pi$
$461$	77924.7	−	214096.i	0.366668	−	1.00741i	0.976620	−	0.214974i	$-0.0689667\pi$
$462$		0			0
$463$	−271314.	−	227659.i	−1.26564	−	1.06200i	−0.995058	−	0.0992998i	$-0.968340\pi$
$463$	−271314.	−	227659.i	−1.26564	−	1.06200i	−0.270581	−	0.962697i	$-0.587216\pi$
$464$	357694.	+	63071.1i	1.66141	+	0.292951i
$465$		0			0
$466$	−30015.0	+	25185.6i	−0.138219	+	0.115979i
$467$	−127308.	+	73501.4i	−0.583744	+	0.337025i	−0.762620	−	0.646847i	$-0.776087\pi$
$467$	−127308.	+	73501.4i	−0.583744	+	0.337025i	0.178876	+	0.983872i	$0.442754\pi$
$468$		0			0
$469$	−148178.	+	256652.i	−0.673656	+	1.16681i
$470$	−100320.	+	17689.1i	−0.454142	+	0.0800775i
$471$		0			0
$472$	−12292.7	+	4474.19i	−0.0551779	+	0.0200831i
$473$	−19567.8	−	53762.1i	−0.0874621	−	0.240300i
$474$		0			0
$475$	26722.6	+	151551.i	0.118438	+	0.671695i
$476$	−256613.	−	148155.i	−1.13257	−	0.653888i
$477$		0			0
$478$	−2459.28	−	4259.60i	−0.0107635	−	0.0186429i
$479$	210335.	+	250668.i	0.916729	+	1.09251i	0.995418	+	0.0956156i	$0.0304820\pi$
$479$	210335.	+	250668.i	0.916729	+	1.09251i	−0.0786896	+	0.996899i	$0.525074\pi$
$480$		0			0
$481$	10420.5	−	59097.8i	0.0450402	−	0.255436i
$482$	8618.38	−	10271.0i	0.0370964	−	0.0442097i
$483$		0			0
$484$	−144461.	−	52579.5i	−0.616679	−	0.224453i
$485$		−	78078.1i		−	0.331930i
$486$		0			0
$487$	241088.			1.01652			0.508261	−	0.861203i	$-0.330288\pi$
$487$	241088.			1.01652			0.508261	+	0.861203i	$0.330288\pi$
$488$	−14318.7	+	39340.4i	−0.0601264	+	0.165196i
$489$		0			0
$490$	−90645.8	−	76060.9i	−0.377534	−	0.316788i
$491$	−91179.6	−	16077.4i	−0.378211	−	0.0666889i	−0.0186891	−	0.999825i	$-0.505949\pi$
$491$	−91179.6	−	16077.4i	−0.378211	−	0.0666889i	−0.359522	+	0.933136i	$0.617060\pi$
$492$		0			0
$493$	274606.	−	230421.i	1.12984	−	0.948045i
$494$	16209.8	−	9358.71i	0.0664237	−	0.0383497i
$495$		0			0
$496$	37472.3	−	64903.9i	0.152317	−	0.263820i
$497$	−34552.4	+	6092.52i	−0.139883	+	0.0246652i
$498$		0			0
$499$	12674.5	−	4613.13i	0.0509012	−	0.0185265i	−0.316444	−	0.948611i	$-0.602489\pi$
$499$	12674.5	−	4613.13i	0.0509012	−	0.0185265i	0.367345	+	0.930085i	$0.380267\pi$
$500$	−23719.9	−	65170.0i	−0.0948797	−	0.260680i
$501$		0			0
$502$	3202.41	+	18161.8i	0.0127078	+	0.0720693i
$503$	157137.	+	90723.2i	0.621074	+	0.358577i	0.777287	−	0.629146i	$-0.216595\pi$
$503$	157137.	+	90723.2i	0.621074	+	0.358577i	−0.156213	+	0.987723i	$0.549929\pi$
$504$		0			0
$505$	130610.	+	226223.i	0.512146	+	0.887063i
$506$	4913.50	+	5855.69i	0.0191907	+	0.0228706i
$507$		0			0
$508$	−9391.07	+	53259.4i	−0.0363905	+	0.206381i
$509$	−209919.	+	250172.i	−0.810245	+	0.965613i	−0.999868	−	0.0162444i	$-0.994829\pi$
$509$	−209919.	+	250172.i	−0.810245	+	0.965613i	0.189623	+	0.981857i	$0.439273\pi$
$510$		0			0
$511$	607958.	+	221279.i	2.32826	+	0.847419i
$512$			211347.i			0.806224i
$513$		0			0
$514$	−37954.0			−0.143659
$515$	−4726.03	+	12984.7i	−0.0178189	+	0.0489571i
$516$		0			0
$517$	214896.	+	180319.i	0.803985	+	0.674624i
$518$	46442.4	+	8189.05i	0.173083	+	0.0305193i
$519$		0			0
$520$	48006.9	−	40282.5i	0.177540	−	0.148974i
$521$	39759.4	−	22955.1i	0.146475	−	0.0845676i	−0.424971	−	0.905207i	$-0.639716\pi$
$521$	39759.4	−	22955.1i	0.146475	−	0.0845676i	0.571447	+	0.820639i	$0.306382\pi$
$522$		0			0
$523$	−231013.	+	400127.i	−0.844566	+	1.46283i	0.0414310	+	0.999141i	$0.486808\pi$
$523$	−231013.	+	400127.i	−0.844566	+	1.46283i	−0.885997	+	0.463690i	$0.846525\pi$
$524$	429221.	−	75683.2i	1.56321	−	0.275637i
$525$		0			0
$526$	47120.3	−	17150.4i	0.170309	−	0.0619872i
$527$	−25298.0	−	69505.8i	−0.0910890	−	0.250265i
$528$		0			0
$529$	44688.1	+	253439.i	0.159691	+	0.905653i
$530$	−43629.6	−	25189.6i	−0.155321	−	0.0896745i
$531$		0			0
$532$	−205000.	−	355071.i	−0.724321	−	1.25456i
$533$	98789.8	+	117733.i	0.347742	+	0.414423i
$534$		0			0
$535$	14734.1	−	83561.4i	0.0514774	−	0.291943i
$536$	52697.5	−	62802.4i	0.183426	−	0.218598i
$537$		0			0
$538$	37672.0	+	13711.5i	0.130153	+	0.0473718i
$539$			325861.i			1.12164i
$540$		0			0
$541$	−391014.			−1.33597			−0.667986	−	0.744174i	$-0.732843\pi$
$541$	−391014.			−1.33597			−0.667986	+	0.744174i	$0.732843\pi$
$542$	−14057.2	+	38621.8i	−0.0478520	+	0.131472i
$543$		0			0
$544$	94742.5	+	79498.4i	0.320145	+	0.268634i
$545$	382895.	+	67514.8i	1.28910	+	0.227303i
$546$		0			0
$547$	−163659.	+	137326.i	−0.546972	+	0.458964i	−0.873914	−	0.486080i	$-0.838426\pi$
$547$	−163659.	+	137326.i	−0.546972	+	0.458964i	0.326942	+	0.945044i	$0.393982\pi$
$548$	−110002.	+	63509.7i	−0.366302	+	0.211485i
$549$		0			0
$550$	−12501.8	+	21653.8i	−0.0413284	+	0.0715829i
$551$	488476.	−	86131.5i	1.60894	−	0.283700i
$552$		0			0
$553$	−398376.	+	144997.i	−1.30270	+	0.474142i
$554$	−24944.3	−	68534.0i	−0.0812742	−	0.223299i
$555$		0			0
$556$	62576.7	+	354890.i	0.202424	+	1.14801i
$557$	−55557.4	−	32076.1i	−0.179074	−	0.103388i	0.407784	−	0.913079i	$-0.366302\pi$
$557$	−55557.4	−	32076.1i	−0.179074	−	0.103388i	−0.586857	+	0.809690i	$0.699635\pi$
$558$		0			0
$559$	33488.3	+	58003.5i	0.107169	+	0.185623i
$560$	−417308.	−	497328.i	−1.33070	−	1.58587i
$561$		0			0
$562$	−7441.20	+	42201.2i	−0.0235597	+	0.133614i
$563$	210551.	−	250925.i	0.664264	−	0.791640i	−0.323727	−	0.946151i	$-0.604936\pi$
$563$	210551.	−	250925.i	0.664264	−	0.791640i	0.987991	+	0.154511i	$0.0493803\pi$
$564$		0			0
$565$	−547562.	−	199296.i	−1.71529	−	0.624313i
$566$		−	33879.6i		−	0.105756i
$567$		0			0
$568$	9705.90			0.0300842
$569$	−7988.21	+	21947.4i	−0.0246732	+	0.0677889i	−0.951418	−	0.307903i	$-0.900373\pi$
$569$	−7988.21	+	21947.4i	−0.0246732	+	0.0677889i	0.926745	+	0.375692i	$0.122595\pi$
$570$		0			0
$571$	44186.9	+	37077.2i	0.135526	+	0.113720i	0.708031	−	0.706182i	$-0.249584\pi$
$571$	44186.9	+	37077.2i	0.135526	+	0.113720i	−0.572505	+	0.819901i	$0.694028\pi$
$572$	−83481.2	−	14720.0i	−0.255150	−	0.0449899i
$573$		0			0
$574$	−92521.3	+	77634.6i	−0.280813	+	0.235630i
$575$	63716.2	−	36786.5i	0.192714	−	0.111264i
$576$		0			0
$577$	220787.	−	382414.i	0.663165	−	1.14864i	−0.316614	−	0.948554i	$-0.602546\pi$
$577$	220787.	−	382414.i	0.663165	−	1.14864i	0.979779	−	0.200081i	$-0.0641206\pi$
$578$	−23549.4	+	4152.40i	−0.0704895	+	0.0124292i
$579$		0			0
$580$	766531.	−	278994.i	2.27863	−	0.829353i
$581$	171378.	+	470857.i	0.507695	+	1.39488i
$582$		0			0
$583$	24091.3	+	136628.i	0.0708799	+	0.401980i
$584$	−154998.	−	89488.4i	−0.454466	−	0.262386i
$585$		0			0
$586$	−14938.9	−	25875.0i	−0.0435035	−	0.0753503i
$587$	−154877.	−	184575.i	−0.449480	−	0.535670i	0.492956	−	0.870054i	$-0.335916\pi$
$587$	−154877.	−	184575.i	−0.449480	−	0.535670i	−0.942437	+	0.334384i	$0.891472\pi$
$588$		0			0
$589$	17772.2	−	100791.i	0.0512285	−	0.290531i
$590$	−8931.31	+	10643.9i	−0.0256573	+	0.0305772i
$591$		0			0
$592$	161605.	+	58819.3i	0.461116	+	0.167833i
$593$			435330.i			1.23797i	0.785404	+	0.618984i	$0.212455\pi$
$593$			435330.i			1.23797i	−0.785404	+	0.618984i	$0.787545\pi$
$594$		0			0
$595$	−640742.			−1.80988
$596$	−6440.64	+	17695.5i	−0.0181316	+	0.0498162i
$597$		0			0
$598$	−6855.06	−	5752.08i	−0.0191694	−	0.0160850i
$599$	137450.	+	24236.2i	0.383082	+	0.0675477i	0.361873	−	0.932227i	$-0.382137\pi$
$599$	137450.	+	24236.2i	0.383082	+	0.0675477i	0.0212090	+	0.999775i	$0.493248\pi$
$600$		0			0
$601$	−30948.8	+	25969.1i	−0.0856830	+	0.0718966i	−0.684623	−	0.728897i	$-0.740033\pi$
$601$	−30948.8	+	25969.1i	−0.0856830	+	0.0718966i	0.598940	+	0.800794i	$0.295589\pi$
$602$	−45582.4	+	26317.0i	−0.125778	+	0.0726179i
$603$		0			0
$604$	213938.	−	370552.i	0.586427	−	1.01572i
$605$	−327380.	+	57725.9i	−0.894419	+	0.157710i
$606$		0			0
$607$	292764.	−	106557.i	0.794585	−	0.289205i	0.0873443	−	0.996178i	$-0.472162\pi$
$607$	292764.	−	106557.i	0.794585	−	0.289205i	0.707241	+	0.706973i	$0.249940\pi$
$608$	58530.1	+	160810.i	0.158333	+	0.435017i
$609$		0			0
$610$	7721.61	+	43791.4i	0.0207514	+	0.117687i
$611$	−284406.	−	164202.i	−0.761828	−	0.439841i
$612$		0			0
$613$	51413.6	+	89051.0i	0.136822	+	0.236983i	0.926292	−	0.376806i	$-0.122978\pi$
$613$	51413.6	+	89051.0i	0.136822	+	0.236983i	−0.789470	+	0.613789i	$0.789644\pi$
$614$	−42646.5	−	50824.1i	−0.113122	−	0.134813i
$615$		0			0
$616$	23550.6	−	133562.i	0.0620641	−	0.351983i
$617$	23587.4	−	28110.4i	0.0619597	−	0.0738407i	−0.734174	−	0.678961i	$-0.762431\pi$
$617$	23587.4	−	28110.4i	0.0619597	−	0.0738407i	0.796134	+	0.605120i	$0.206875\pi$
$618$		0			0
$619$	475026.	+	172895.i	1.23976	+	0.451234i	0.876928	−	0.480622i	$-0.159589\pi$
$619$	475026.	+	172895.i	1.23976	+	0.451234i	0.362828	+	0.931856i	$0.381811\pi$
$620$		−	168315.i		−	0.437865i
$621$		0			0
$622$	51265.6			0.132509
$623$	65122.9	−	178924.i	0.167787	−	0.460990i
$624$		0			0
$625$	−349755.	−	293479.i	−0.895373	−	0.751307i
$626$	−21036.8	−	3709.35i	−0.0536822	−	0.00946561i
$627$		0			0
$628$	−328624.	+	275748.i	−0.833258	+	0.699187i
$629$	146992.	−	84865.8i	0.371529	−	0.214502i
$630$		0			0
$631$	−244451.	+	423401.i	−0.613949	+	1.06339i	0.376619	+	0.926368i	$0.377087\pi$
$631$	−244451.	+	423401.i	−0.613949	+	1.06339i	−0.990568	+	0.137023i	$0.956247\pi$
$632$	115496.	−	20365.1i	0.289157	−	0.0509861i
$633$		0			0
$634$	−106740.	+	38850.2i	−0.265551	+	0.0966528i
$635$	39997.5	+	109892.i	0.0991940	+	0.272533i
$636$		0			0
$637$	−66242.4	−	375680.i	−0.163252	−	0.925846i
$638$	69794.1	+	40295.6i	0.171466	+	0.0989957i
$639$		0			0
$640$	186408.	+	322869.i	0.455098	+	0.788253i
$641$	160001.	+	190682.i	0.389410	+	0.464081i	0.924761	−	0.380549i	$-0.124265\pi$
$641$	160001.	+	190682.i	0.389410	+	0.464081i	−0.535351	+	0.844630i	$0.679821\pi$
$642$		0			0
$643$	101222.	−	574056.i	0.244822	−	1.38846i	−0.576083	−	0.817391i	$-0.695419\pi$
$643$	101222.	−	574056.i	0.244822	−	1.38846i	0.820905	−	0.571065i	$-0.193469\pi$
$644$	−125998.	+	150158.i	−0.303802	+	0.362057i
$645$		0			0
$646$	49747.9	+	18106.8i	0.119209	+	0.0433886i
$647$		−	512607.i		−	1.22455i	−0.790645	−	0.612274i	$-0.790255\pi$
$647$		−	512607.i		−	1.22455i	0.790645	−	0.612274i	$-0.209745\pi$
$648$		0			0
$649$	38263.7			0.0908442
$650$	10011.3	−	27505.7i	0.0236953	−	0.0651023i
$651$		0			0
$652$	−242985.	−	203888.i	−0.571589	−	0.479620i
$653$	−376391.	−	66367.9i	−0.882700	−	0.155644i	−0.286117	−	0.958195i	$-0.592364\pi$
$653$	−376391.	−	66367.9i	−0.882700	−	0.155644i	−0.596584	+	0.802551i	$0.703476\pi$
$654$		0			0
$655$	721970.	−	605805.i	1.68282	−	1.41205i
$656$	−381437.	+	220223.i	−0.886371	+	0.511746i
$657$		0			0
$658$	129039.	−	223502.i	0.298037	−	0.516215i
$659$	−751732.	+	132551.i	−1.73098	+	0.305219i	−0.948340	−	0.317256i	$-0.897239\pi$
$659$	−751732.	+	132551.i	−1.73098	+	0.305219i	−0.782640	+	0.622474i	$0.786128\pi$
$660$		0			0
$661$	−190739.	+	69423.3i	−0.436553	+	0.158892i	−0.550941	−	0.834544i	$-0.685731\pi$
$661$	−190739.	+	69423.3i	−0.436553	+	0.158892i	0.114389	+	0.993436i	$0.463509\pi$
$662$	27179.9	+	74676.1i	0.0620200	+	0.170399i
$663$		0			0
$664$	−24070.3	−	136510.i	−0.0545941	−	0.309619i
$665$	−767805.	−	443292.i	−1.73623	−	1.00241i
$666$		0			0
$667$	−118569.	−	205368.i	−0.266515	−	0.461617i
$668$	−189824.	−	226223.i	−0.425400	−	0.506972i
$669$		0			0
$670$	15120.9	−	85754.8i	0.0336843	−	0.191033i
$671$	78712.6	−	93806.0i	0.174823	−	0.208346i
$672$		0			0
$673$	−63092.7	−	22963.9i	−0.139299	−	0.0507008i	0.271430	−	0.962458i	$-0.412504\pi$
$673$	−63092.7	−	22963.9i	−0.139299	−	0.0507008i	−0.410729	+	0.911757i	$0.634726\pi$
$674$			4994.79i			0.0109950i
$675$		0			0
$676$	−341913.			−0.748208
$677$	208638.	−	573229.i	0.455215	−	1.25069i	−0.473794	−	0.880636i	$-0.657116\pi$
$677$	208638.	−	573229.i	0.455215	−	1.25069i	0.929009	−	0.370058i	$-0.120662\pi$
$678$		0			0
$679$	151530.	+	127149.i	0.328669	+	0.275786i
$680$	174560.	+	30779.6i	0.377508	+	0.0665648i
$681$		0			0
$682$	12738.6	−	10689.0i	0.0273875	−	0.0229809i
$683$	171841.	−	99212.2i	0.368370	−	0.212679i	−0.304376	−	0.952552i	$-0.598448\pi$
$683$	171841.	−	99212.2i	0.368370	−	0.212679i	0.672746	+	0.739873i	$0.265115\pi$
$684$		0			0
$685$	−137333.	+	237868.i	−0.292681	+	0.506939i
$686$	146289.	−	25794.6i	0.310858	−	0.0548127i
$687$		0			0
$688$	−180367.	+	65648.3i	−0.381049	+	0.138690i
$689$	−55548.8	−	152619.i	−0.117014	−	0.321492i
$690$		0			0
$691$	50674.2	+	287388.i	0.106128	+	0.601883i	0.990764	+	0.135600i	$0.0432962\pi$
$691$	50674.2	+	287388.i	0.106128	+	0.601883i	−0.884635	+	0.466283i	$0.845593\pi$
$692$	540842.	+	312255.i	1.12943	+	0.652075i
$693$		0			0
$694$	13629.8	+	23607.5i	0.0282989	+	0.0490152i
$695$	500894.	+	596942.i	1.03699	+	1.23584i
$696$		0			0
$697$	−75484.4	+	428093.i	−0.155379	+	0.881197i
$698$	16247.8	−	19363.4i	0.0333490	−	0.0397438i
$699$		0			0
$700$	−602506.	−	219294.i	−1.22960	−	0.447539i
$701$			661865.i			1.34689i	0.739235	+	0.673447i	$0.235187\pi$
$701$			661865.i			1.34689i	−0.739235	+	0.673447i	$0.764813\pi$
$702$		0			0
$703$	234855.			0.475213
$704$	76558.6	−	210343.i	0.154472	−	0.424408i
$705$		0			0
$706$	30470.1	+	25567.4i	0.0611313	+	0.0512953i
$707$	−651739.	−	114919.i	−1.30387	−	0.229908i
$708$		0			0
$709$	122517.	−	102804.i	0.243728	−	0.204512i	−0.512738	−	0.858545i	$-0.671369\pi$
$709$	122517.	−	102804.i	0.243728	−	0.204512i	0.756466	+	0.654033i	$0.226924\pi$
$710$	8927.93	−	5154.54i	0.0177106	−	0.0102252i
$711$		0			0
$712$	−26336.6	+	45616.4i	−0.0519518	+	0.0899831i
$713$	−48187.5	+	8496.75i	−0.0947884	+	0.0167138i
$714$		0			0
$715$	−172250.	+	62693.8i	−0.336935	+	0.122634i
$716$	229404.	+	630282.i	0.447481	+	1.22944i
$717$		0			0
$718$	−14164.4	−	80330.4i	−0.0274758	−	0.155823i
$719$	271500.	+	156751.i	0.525185	+	0.303216i	0.739053	−	0.673647i	$-0.235273\pi$
$719$	271500.	+	156751.i	0.525185	+	0.303216i	−0.213869	+	0.976862i	$0.568606\pi$
$720$		0			0
$721$	−17503.7	−	30317.3i	−0.0336713	−	0.0583203i
$722$	−15271.7	−	18200.0i	−0.0292962	−	0.0349139i
$723$		0			0
$724$	−126278.	+	716161.i	−0.240908	+	1.36626i
$725$	498598.	−	594206.i	0.948581	−	1.13048i
$726$		0			0
$727$	174359.	+	63461.4i	0.329895	+	0.120072i	0.501657	−	0.865067i	$-0.332724\pi$
$727$	174359.	+	63461.4i	0.329895	+	0.120072i	−0.171762	+	0.985138i	$0.554946\pi$
$728$			158769.i			0.299573i
$729$		0			0
$730$	−190100.			−0.356727
$731$	−64791.5	+	178013.i	−0.121250	+	0.333133i
$732$		0			0
$733$	566483.	+	475336.i	1.05434	+	0.884693i	0.993543	−	0.113457i	$-0.0361925\pi$
$733$	566483.	+	475336.i	1.05434	+	0.884693i	0.0607933	+	0.998150i	$0.480637\pi$
$734$	−131957.	−	23267.5i	−0.244928	−	0.0431875i
$735$		0			0
$736$	62674.7	−	52590.4i	0.115701	−	0.0970846i
$737$	−207671.	+	119899.i	−0.382333	+	0.220740i
$738$		0			0
$739$	212883.	−	368724.i	0.389809	−	0.675170i	−0.602614	−	0.798033i	$-0.705874\pi$
$739$	212883.	−	368724.i	0.389809	−	0.675170i	0.992424	+	0.122863i	$0.0392076\pi$
$740$	380367.	−	67068.9i	0.694607	−	0.122478i
$741$		0			0
$742$	119937.	−	43653.4i	0.217843	−	0.0792885i
$743$	46087.3	+	126624.i	0.0834841	+	0.229371i	0.974410	−	0.224778i	$-0.0721657\pi$
$743$	46087.3	+	126624.i	0.0834841	+	0.229371i	−0.890926	+	0.454149i	$0.849943\pi$
$744$		0			0
$745$	7071.04	+	40101.9i	0.0127400	+	0.0722523i
$746$	132029.	+	76227.0i	0.237242	+	0.136972i
$747$		0			0
$748$	−119881.	−	207640.i	−0.214263	−	0.371114i
$749$	138177.	+	164673.i	0.246305	+	0.293535i
$750$		0			0
$751$	47356.0	−	268569.i	0.0839644	−	0.476186i	−0.913611	−	0.406590i	$-0.866718\pi$
$751$	47356.0	−	268569.i	0.0839644	−	0.476186i	0.997575	−	0.0695961i	$-0.0221710\pi$
$752$	604956.	−	720959.i	1.06976	−	1.27490i
$753$		0			0
$754$	−88655.8	−	32268.1i	−0.155942	−	0.0567584i
$755$		−	925239.i		−	1.62316i
$756$		0			0
$757$	914027.			1.59502			0.797512	−	0.603303i	$-0.206149\pi$
$757$	914027.			1.59502			0.797512	+	0.603303i	$0.206149\pi$
$758$	−26183.1	+	71937.5i	−0.0455704	+	0.125204i
$759$		0			0
$760$	187881.	+	157651.i	0.325279	+	0.272941i
$761$	−717737.	−	126556.i	−1.23936	−	0.218532i	−0.484717	−	0.874671i	$-0.661077\pi$
$761$	−717737.	−	126556.i	−1.23936	−	0.218532i	−0.754640	+	0.656139i	$0.772188\pi$
$762$		0			0
$763$	−754567.	+	633157.i	−1.29613	+	1.08758i
$764$	−167340.	+	96613.5i	−0.286690	+	0.165520i
$765$		0			0
$766$	−84384.7	+	146159.i	−0.143816	+	0.249096i
$767$	−44113.5	+	7778.40i	−0.0749861	+	0.0132221i
$768$		0			0
$769$	−253227.	+	92167.2i	−0.428211	+	0.155856i	−0.547130	−	0.837048i	$-0.684280\pi$
$769$	−253227.	+	92167.2i	−0.428211	+	0.155856i	0.118918	+	0.992904i	$0.462057\pi$
$770$	−49268.3	−	135364.i	−0.0830972	−	0.228308i
$771$		0			0
$772$	24991.3	+	141733.i	0.0419329	+	0.237813i
$773$	798882.	+	461235.i	1.33698	+	0.771904i	0.986358	−	0.164615i	$-0.0526382\pi$
$773$	798882.	+	461235.i	1.33698	+	0.771904i	0.350618	+	0.936518i	$0.385971\pi$
$774$		0			0
$775$	−80026.3	−	138610.i	−0.133238	−	0.230776i
$776$	−35173.9	−	41918.7i	−0.0584114	−	0.0696120i
$777$		0			0
$778$	−25479.5	+	144502.i	−0.0420952	+	0.238734i
$779$	−386626.	+	460763.i	−0.637113	+	0.759281i
$780$		0			0
$781$	−26677.5	−	9709.81i	−0.0437364	−	0.0159187i
$782$		−	25310.5i		−	0.0413891i
$783$		0			0
$784$	1.09324e6			1.77861
$785$	−317273.	+	871699.i	−0.514865	+	1.41458i
$786$		0			0
$787$	−58281.0	−	48903.6i	−0.0940974	−	0.0789571i	0.594525	−	0.804077i	$-0.297340\pi$
$787$	−58281.0	−	48903.6i	−0.0940974	−	0.0789571i	−0.688623	+	0.725120i	$0.741784\pi$
$788$	−95989.6	−	16925.6i	−0.154587	−	0.0272578i
$789$		0			0
$790$	95423.3	−	80069.7i	0.152897	−	0.128296i
$791$	1.27848e6	−	738131.i	2.04334	−	1.17972i
$792$		0			0
$793$	−71677.0	+	124148.i	−0.113981	+	0.197421i
$794$	−16715.2	+	2947.34i	−0.0265137	+	0.00467508i
$795$		0			0
$796$	−555397.	+	202148.i	−0.876551	+	0.319039i
$797$	16522.6	+	45395.5i	0.0260113	+	0.0714655i	0.952019	−	0.306040i	$-0.0990041\pi$
$797$	16522.6	+	45395.5i	0.0260113	+	0.0714655i	−0.926007	+	0.377506i	$0.876782\pi$
$798$		0			0
$799$	−161295.	−	914750.i	−0.252655	−	1.43288i
$800$	231766.	+	133810.i	0.362134	+	0.209078i
$801$		0			0
$802$	78831.3	+	136540.i	0.122560	+	0.212281i
$803$	336502.	+	401028.i	0.521863	+	0.621932i
$804$		0			0
$805$	−73604.0	+	417429.i	−0.113582	+	0.644156i
$806$	−12513.2	+	14912.7i	−0.0192619	+	0.0229554i
$807$		0			0
$808$	172035.	+	62615.6i	0.263508	+	0.0959090i
$809$		−	427329.i		−	0.652928i	−0.945210	−	0.326464i	$-0.894143\pi$
$809$		−	427329.i		−	0.652928i	0.945210	−	0.326464i	$-0.105857\pi$
$810$		0			0
$811$	−31386.9			−0.0477207			−0.0238604	−	0.999715i	$-0.507596\pi$
$811$	−31386.9			−0.0477207			−0.0238604	+	0.999715i	$0.507596\pi$
$812$	−706823.	+	1.94198e6i	−1.07201	+	2.94532i
$813$		0			0
$814$	29231.4	+	24528.0i	0.0441165	+	0.0370181i
$815$	−675480.	−	119105.i	−1.01694	−	0.179315i
$816$		0			0
$817$	−200797.	+	168489.i	−0.300824	+	0.252421i
$818$	26782.4	−	15462.8i	0.0400260	−	0.0231090i
$819$		0			0
$820$	−494590.	+	856656.i	−0.735560	+	1.27403i
$821$	−240577.	+	42420.2i	−0.356917	+	0.0629341i	−0.349232	−	0.937036i	$-0.613557\pi$
$821$	−240577.	+	42420.2i	−0.356917	+	0.0629341i	−0.00768530	+	0.999970i	$0.502446\pi$
$822$		0			0
$823$	761740.	−	277251.i	1.12462	−	0.409329i	0.288286	−	0.957544i	$-0.406915\pi$
$823$	761740.	−	277251.i	1.12462	−	0.409329i	0.836337	+	0.548215i	$0.184692\pi$
$824$	3312.23	+	9100.27i	0.00487827	+	0.0134029i
$825$		0			0
$826$	−6112.70	−	34666.9i	−0.00895928	−	0.0508106i
$827$	425347.	+	245574.i	0.621917	+	0.359064i	0.777615	−	0.628741i	$-0.216429\pi$
$827$	425347.	+	245574.i	0.621917	+	0.359064i	−0.155698	+	0.987805i	$0.549763\pi$
$828$		0			0
$829$	397952.	+	689273.i	0.579057	+	1.00296i	0.995588	+	0.0938348i	$0.0299126\pi$
$829$	397952.	+	689273.i	0.579057	+	1.00296i	−0.416531	+	0.909122i	$0.636754\pi$
$830$	−94637.7	−	112785.i	−0.137375	−	0.163717i
$831$		0			0
$832$	−45503.6	+	258064.i	−0.0657354	+	0.372804i
$833$	693548.	−	826538.i	0.999508	−	1.19117i
$834$		0			0
$835$	−600073.	−	218409.i	−0.860659	−	0.313254i
$836$		−	331754.i		−	0.474683i
$837$		0			0
$838$	−251479.			−0.358107
$839$	237489.	−	652495.i	0.337380	−	0.926944i	−0.648755	−	0.760997i	$-0.724710\pi$
$839$	237489.	−	652495.i	0.337380	−	0.926944i	0.986135	−	0.165946i	$-0.0530678\pi$
$840$		0			0
$841$	−1.37342e6	−	1.15244e6i	−1.94183	−	1.62939i
$842$	−102299.	−	18038.0i	−0.144293	−	0.0254428i
$843$		0			0
$844$	202389.	−	169824.i	0.284120	−	0.238405i
$845$	−640298.	+	369676.i	−0.896745	+	0.517736i
$846$		0			0
$847$	421101.	−	729368.i	0.586974	−	1.01667i
$848$	458377.	−	80824.2i	0.637428	−	0.112396i
$849$		0			0
$850$	77797.6	−	28316.0i	0.107678	−	0.0391917i
$851$	−38402.7	−	105511.i	−0.0530277	−	0.145692i
$852$		0			0
$853$	83911.0	+	475883.i	0.115324	+	0.654037i	0.986589	+	0.163224i	$0.0521892\pi$
$853$	83911.0	+	475883.i	0.115324	+	0.654037i	−0.871265	+	0.490813i	$0.836700\pi$
$854$	−97562.7	−	56327.9i	−0.133773	−	0.0772338i
$855$		0			0
$856$	−29733.6	−	51500.2i	−0.0405789	−	0.0702848i
$857$	−695915.	−	829359.i	−0.947533	−	1.12923i	−0.991489	−	0.130192i	$-0.958441\pi$
$857$	−695915.	−	829359.i	−0.947533	−	1.12923i	0.0439559	−	0.999033i	$-0.486004\pi$
$858$		0			0
$859$	55630.3	−	315495.i	0.0753920	−	0.427569i	−0.923627	−	0.383292i	$-0.874790\pi$
$859$	55630.3	−	315495.i	0.0753920	−	0.427569i	0.999019	−	0.0442774i	$-0.0140986\pi$
$860$	−277091.	+	330224.i	−0.374650	+	0.446490i
$861$		0			0
$862$	−241273.	−	87816.2i	−0.324709	−	0.118184i
$863$		−	360673.i		−	0.484274i	−0.970242	−	0.242137i	$-0.922152\pi$
$863$		−	360673.i		−	0.484274i	0.970242	−	0.242137i	$-0.0778484\pi$
$864$		0			0
$865$	1.35044e6			1.80486
$866$	52415.5	−	144010.i	0.0698914	−	0.192025i
$867$		0			0
$868$	326658.	+	274099.i	0.433564	+	0.363804i
$869$	−337824.	−	59567.5i	−0.447354	−	0.0788806i
$870$		0			0
$871$	215047.	−	180446.i	0.283463	−	0.237854i
$872$	235984.	−	136246.i	0.310349	−	0.179180i
$873$		0			0
$874$	17510.8	−	30329.6i	0.0229236	−	0.0397049i
$875$	374166.	−	65975.6i	0.488707	−	0.0861722i
$876$		0			0
$877$	−1.11500e6	+	405826.i	−1.44969	+	0.527644i	−0.942504	−	0.334194i	$-0.891536\pi$
$877$	−1.11500e6	+	405826.i	−1.44969	+	0.527644i	−0.507184	+	0.861838i	$0.669314\pi$
$878$	72768.5	+	199930.i	0.0943962	+	0.259351i
$879$		0			0
$880$	−91220.2	−	517336.i	−0.117795	−	0.668047i
$881$	−669688.	−	386644.i	−0.862821	−	0.498150i	0.00213514	−	0.999998i	$-0.499320\pi$
$881$	−669688.	−	386644.i	−0.862821	−	0.498150i	−0.864956	+	0.501848i	$0.832654\pi$
$882$		0			0
$883$	−537395.	−	930796.i	−0.689243	−	1.19380i	−0.972083	−	0.234637i	$-0.924610\pi$
$883$	−537395.	−	930796.i	−0.689243	−	1.19380i	0.282840	−	0.959167i	$-0.408724\pi$
$884$	180418.	+	215014.i	0.230875	+	0.275146i
$885$		0			0
$886$	−33170.8	+	188121.i	−0.0422560	+	0.239645i
$887$	400533.	−	477336.i	0.509086	−	0.606705i	−0.448878	−	0.893593i	$-0.648176\pi$
$887$	400533.	−	477336.i	0.509086	−	0.606705i	0.957964	+	0.286888i	$0.0926208\pi$
$888$		0			0
$889$	−278408.	−	101332.i	−0.352273	−	0.128217i
$890$			55946.8i			0.0706309i
$891$		0			0
$892$	−1.11883e6			−1.40616
$893$	439581.	−	1.20774e6i	0.551235	−	1.51450i
$894$		0			0
$895$	1.11106e6	+	932293.i	1.38705	+	1.16388i
$896$	−930169.	−	164014.i	−1.15863	−	0.204298i
$897$		0			0
$898$	128666.	−	107964.i	0.159556	−	0.133883i
$899$	−446763.	+	257939.i	−0.552787	+	0.319152i
$900$		0			0
$901$	229687.	−	397829.i	0.282935	−	0.490057i
$902$	−96243.5	+	16970.3i	−0.118293	+	0.0208582i
$903$		0			0
$904$	−383758.	+	139677.i	−0.469592	+	0.170918i
$905$	537832.	+	1.47768e6i	0.656673	+	1.80420i
$906$		0			0
$907$	−150017.	−	850789.i	−0.182358	−	1.03421i	−0.929303	−	0.369319i	$-0.879591\pi$
$907$	−150017.	−	850789.i	−0.182358	−	1.03421i	0.746944	−	0.664887i	$-0.231520\pi$
$908$	−681390.	−	393401.i	−0.826465	−	0.477160i
$909$		0			0
$910$	84317.8	+	146043.i	0.101821	+	0.176359i
$911$	−187578.	−	223547.i	−0.226020	−	0.269360i	0.641103	−	0.767455i	$-0.278477\pi$
$911$	−187578.	−	223547.i	−0.226020	−	0.269360i	−0.867122	+	0.498096i	$0.834033\pi$
$912$		0			0
$913$	−70405.4	+	399289.i	−0.0844625	+	0.479011i
$914$	−93144.2	+	111005.i	−0.111497	+	0.132877i
$915$		0			0
$916$	1.32578e6	+	482546.i	1.58009	+	0.575106i
$917$			2.38770e6i			2.83950i
$918$		0			0
$919$	−690029.			−0.817026			−0.408513	−	0.912752i	$-0.633953\pi$
$919$	−690029.			−0.817026			−0.408513	+	0.912752i	$0.633953\pi$
$920$	40104.3	−	110186.i	0.0473823	−	0.130182i
$921$		0			0
$922$	−129923.	−	109018.i	−0.152836	−	0.128244i
$923$	32729.8	+	5771.16i	0.0384185	+	0.00677422i
$924$		0			0
$925$	281348.	−	236079.i	0.328822	−	0.275915i
$926$	−228327.	+	131825.i	−0.266278	+	0.153736i
$927$		0			0
$928$	431293.	−	747022.i	0.500814	−	0.867436i
$929$	−748149.	+	131919.i	−0.866875	+	0.152854i	−0.589363	−	0.807868i	$-0.700621\pi$
$929$	−748149.	+	131919.i	−0.866875	+	0.152854i	−0.277513	+	0.960722i	$0.589510\pi$
$930$		0			0
$931$	1.40291e6	−	510619.i	1.61857	−	0.589112i
$932$	−278060.	−	763965.i	−0.320116	−	0.879512i
$933$		0			0
$934$	19002.2	+	107767.i	0.0217826	+	0.123535i
$935$	−449000.	−	259230.i	−0.513598	−	0.296526i
$936$		0			0
$937$	106034.	+	183656.i	0.120772	+	0.209183i	0.920072	−	0.391749i	$-0.128130\pi$
$937$	106034.	+	183656.i	0.120772	+	0.209183i	−0.799300	+	0.600932i	$0.794796\pi$
$938$	141804.	+	168996.i	0.161170	+	0.192075i
$939$		0			0
$940$	367037.	−	2.08157e6i	0.415388	−	2.35578i
$941$	151335.	−	180355.i	0.170908	−	0.203680i	−0.673791	−	0.738922i	$-0.735335\pi$
$941$	151335.	−	180355.i	0.170908	−	0.203680i	0.844699	+	0.535242i	$0.179780\pi$
$942$		0			0
$943$	270222.	+	98352.7i	0.303876	+	0.110602i
$944$		−	128371.i		−	0.144054i
$945$		0			0
$946$	−42589.2			−0.0475901
$947$	−110408.	+	303342.i	−0.123112	+	0.338246i	−0.985904	−	0.167312i	$-0.946491\pi$
$947$	−110408.	+	303342.i	−0.123112	+	0.338246i	0.862792	+	0.505558i	$0.168713\pi$
$948$		0			0
$949$	−469470.	−	393932.i	−0.521285	−	0.437410i
$950$	112815.	+	19892.4i	0.125003	+	0.0220414i
$951$		0			0
$952$	−344003.	+	288652.i	−0.379566	+	0.318494i
$953$	−866462.	+	500252.i	−0.954034	+	0.550812i	−0.894332	−	0.447405i	$-0.852348\pi$
$953$	−866462.	+	500252.i	−0.954034	+	0.550812i	−0.0597020	+	0.998216i	$0.519015\pi$
$954$		0			0
$955$	−208917.	+	361855.i	−0.229069	+	0.396760i
$956$	100506.	−	17722.0i	0.109971	−	0.0193908i
$957$		0			0
$958$	228896.	−	83311.5i	0.249407	−	0.0907766i
$959$	−237998.	−	653894.i	−0.258783	−	0.711001i
$960$		0			0
$961$	−141884.	−	804662.i	−0.153633	−	0.871298i
$962$	−38686.5	−	22335.7i	−0.0418032	−	0.0241351i
$963$		0			0
$964$	139101.	+	240931.i	0.149685	+	0.259261i
$965$	200043.	+	238402.i	0.214817	+	0.256009i
$966$		0			0
$967$	−270413.	+	1.53359e6i	−0.289184	+	1.64004i	0.400761	+	0.916182i	$0.368746\pi$
$967$	−270413.	+	1.53359e6i	−0.289184	+	1.64004i	−0.689945	+	0.723861i	$0.742365\pi$
$968$	−149759.	+	178475.i	−0.159824	+	0.190470i
$969$		0			0
$970$	−54616.5	−	19878.8i	−0.0580471	−	0.0211274i
$971$		−	1.32763e6i		−	1.40811i	−0.710143	−	0.704057i	$-0.751370\pi$
$971$		−	1.32763e6i		−	1.40811i	0.710143	−	0.704057i	$-0.248630\pi$
$972$		0			0
$973$	−1.97421e6			−2.08530
$974$	61381.2	−	168643.i	0.0647020	−	0.177767i
$975$		0			0
$976$	−314711.	−	264074.i	−0.330379	−	0.277221i
$977$	1.76176e6	+	310645.i	1.84568	+	0.325443i	0.983465	−	0.181098i	$-0.0579650\pi$
$977$	1.76176e6	+	310645.i	1.84568	+	0.325443i	0.862216	+	0.506541i	$0.169076\pi$
$978$		0			0
$979$	118023.	−	99033.3i	0.123141	−	0.103328i
$980$	2.12632e6	−	1.22763e6i	2.21399	−	1.27825i
$981$		0			0
$982$	−34460.8	+	59687.8i	−0.0357357	+	0.0618960i
$983$	−390849.	+	68917.3i	−0.404485	+	0.0713216i	−0.372190	−	0.928157i	$-0.621393\pi$
$983$	−390849.	+	68917.3i	−0.404485	+	0.0713216i	−0.0322954	+	0.999478i	$0.510282\pi$
$984$		0			0
$985$	−198059.	+	72087.5i	−0.204137	+	0.0742998i
$986$	−91267.4	−	250755.i	−0.0938776	−	0.257927i
$987$		0			0
$988$	67440.4	+	382473.i	0.0690885	+	0.391821i
$989$	108529.	+	62659.1i	0.110956	+	0.0640607i
$990$		0			0
$991$	451332.	+	781730.i	0.459567	+	0.795994i	0.998938	−	0.0460748i	$-0.0146713\pi$
$991$	451332.	+	781730.i	0.459567	+	0.795994i	−0.539371	+	0.842068i	$0.681338\pi$
$992$	−114406.	−	136344.i	−0.116259	−	0.138552i
$993$		0			0
$994$	−4535.30	+	25721.0i	−0.00459022	+	0.0260324i
$995$	−821526.	+	979056.i	−0.829803	+	0.988921i
$996$		0			0
$997$	−281527.	−	102467.i	−0.283224	−	0.103085i	0.196502	−	0.980503i	$-0.437042\pi$
$997$	−281527.	−	102467.i	−0.283224	−	0.103085i	−0.479726	+	0.877418i	$0.659264\pi$
$998$		−	10040.4i		−	0.0100807i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.5.f.a.17.6		66
3.2	odd	2		27.5.f.a.23.6	yes	66
27.7	even	9		27.5.f.a.20.6	✓	66
27.20	odd	18	inner	81.5.f.a.62.6		66

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.5.f.a.20.6	✓	66	27.7	even	9
27.5.f.a.23.6	yes	66	3.2	odd	2
81.5.f.a.17.6		66	1.1	even	1	trivial
81.5.f.a.62.6		66	27.20	odd	18	inner

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:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	81.f (of order \(18\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.254601 - 0.699511i) q^{2} +(11.8322 + 9.92841i) q^{4} +(32.8927 + 5.79987i) q^{5} +(-64.8213 + 54.3915i) q^{7} +(20.2723 - 11.7042i) q^{8} +(12.4316 - 21.5322i) q^{10} +(-67.4291 + 11.8896i) q^{11} +(75.3208 - 27.4145i) q^{13} +(21.5439 + 59.1913i) q^{14} +(39.8885 + 226.219i) q^{16} +(196.337 + 113.355i) q^{17} +(156.848 + 271.668i) q^{19} +(331.610 + 395.198i) q^{20} +(-8.85064 + 50.1945i) q^{22} +(96.4022 - 114.888i) q^{23} +(460.984 + 167.784i) q^{25} -59.6675i q^{26} -1307.00 q^{28} +(540.798 - 1485.83i) q^{29} +(-249.929 - 209.716i) q^{31} +(537.243 + 94.7304i) q^{32} +(129.281 - 108.480i) q^{34} +(-2447.61 + 1413.13i) q^{35} +(374.335 - 648.368i) q^{37} +(229.969 - 40.5497i) q^{38} +(734.693 - 267.407i) q^{40} +(655.793 + 1801.78i) q^{41} +(145.099 + 822.898i) q^{43} +(-915.880 - 528.784i) q^{44} +(-55.8211 - 96.6849i) q^{46} +(-2633.58 - 3138.58i) q^{47} +(826.432 - 4686.93i) q^{49} +(234.734 - 279.745i) q^{50} +(1163.39 + 423.441i) q^{52} -2026.25i q^{53} -2286.88 q^{55} +(-677.466 + 1861.32i) q^{56} +(-901.667 - 756.588i) q^{58} +(-550.354 - 97.0423i) q^{59} +(-1370.05 + 1149.60i) q^{61} +(-210.331 + 121.435i) q^{62} +(-1634.62 + 2831.25i) q^{64} +(2636.51 - 464.887i) q^{65} +(3291.06 - 1197.85i) q^{67} +(1197.67 + 3290.56i) q^{68} +(365.334 + 2071.91i) q^{70} +(359.082 + 207.316i) q^{71} +(-3822.91 - 6621.48i) q^{73} +(-358.234 - 426.927i) q^{74} +(-841.377 + 4771.69i) q^{76} +(3724.15 - 4438.27i) q^{77} +(4707.92 + 1713.54i) q^{79} +7672.29i q^{80} +1427.33 q^{82} +(2025.31 - 5564.49i) q^{83} +(5800.61 + 4867.29i) q^{85} +(612.569 + 108.012i) q^{86} +(-1227.78 + 1030.23i) q^{88} +(-1948.72 + 1125.09i) q^{89} +(-3391.27 + 5873.86i) q^{91} +(2281.30 - 402.255i) q^{92} +(-2865.98 + 1043.13i) q^{94} +(3583.51 + 9845.60i) q^{95} +(-405.930 - 2302.15i) q^{97} +(-3068.15 - 1771.40i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q + 6 q^{2} - 6 q^{4} - 3 q^{5} - 6 q^{7} + 9 q^{8} - 3 q^{10} + 492 q^{11} - 6 q^{13} - 1137 q^{14} - 54 q^{16} + 9 q^{17} - 3 q^{19} + 2487 q^{20} + 1002 q^{22} + 2724 q^{23} + 435 q^{25} - 12 q^{28}+ \cdots + 259938 q^{98}+O(q^{100}) \) 66 * q + 6 * q^2 - 6 * q^4 - 3 * q^5 - 6 * q^7 + 9 * q^8 - 3 * q^10 + 492 * q^11 - 6 * q^13 - 1137 * q^14 - 54 * q^16 + 9 * q^17 - 3 * q^19 + 2487 * q^20 + 1002 * q^22 + 2724 * q^23 + 435 * q^25 - 12 * q^28 - 5016 * q^29 - 1671 * q^31 - 10224 * q^32 - 342 * q^34 + 2682 * q^35 - 3 * q^37 + 6564 * q^38 - 1113 * q^40 - 5754 * q^41 - 1266 * q^43 + 4545 * q^44 - 3 * q^46 + 9159 * q^47 + 5898 * q^49 + 34977 * q^50 + 2871 * q^52 - 12 * q^55 - 39243 * q^56 - 12291 * q^58 - 24762 * q^59 - 8358 * q^61 - 38304 * q^62 + 6141 * q^64 - 23727 * q^65 + 15996 * q^67 + 43533 * q^68 - 7251 * q^70 + 19773 * q^71 + 6108 * q^73 + 74847 * q^74 - 28614 * q^76 + 33909 * q^77 - 5658 * q^79 - 12 * q^82 - 18813 * q^83 + 24219 * q^85 - 96474 * q^86 + 36042 * q^88 - 110232 * q^89 - 6042 * q^91 - 73545 * q^92 + 2631 * q^94 + 65163 * q^95 - 3696 * q^97 + 259938 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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