LMFDB - Embedded newform 54.4.c.a.37.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [54,4,Mod(19,54)] 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("54.19"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(54, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 54 = 2 \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	54.c (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$3.18610314031$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			37.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	54.37
Dual form			54.4.c.a.19.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-4.50000 + 7.79423i) q^{5} +(15.5000 + 26.8468i) q^{7} -8.00000 q^{8} -18.0000 q^{10} +(-7.50000 - 12.9904i) q^{11} +(18.5000 - 32.0429i) q^{13} +(-31.0000 + 53.6936i) q^{14} +(-8.00000 - 13.8564i) q^{16} +42.0000 q^{17} -28.0000 q^{19} +(-18.0000 - 31.1769i) q^{20} +(15.0000 - 25.9808i) q^{22} +(97.5000 - 168.875i) q^{23} +(22.0000 + 38.1051i) q^{25} +74.0000 q^{26} -124.000 q^{28} +(55.5000 + 96.1288i) q^{29} +(102.500 - 177.535i) q^{31} +(16.0000 - 27.7128i) q^{32} +(42.0000 + 72.7461i) q^{34} -279.000 q^{35} -166.000 q^{37} +(-28.0000 - 48.4974i) q^{38} +(36.0000 - 62.3538i) q^{40} +(-130.500 + 226.033i) q^{41} +(21.5000 + 37.2391i) q^{43} +60.0000 q^{44} +390.000 q^{46} +(88.5000 + 153.286i) q^{47} +(-309.000 + 535.204i) q^{49} +(-44.0000 + 76.2102i) q^{50} +(74.0000 + 128.172i) q^{52} -114.000 q^{53} +135.000 q^{55} +(-124.000 - 214.774i) q^{56} +(-111.000 + 192.258i) q^{58} +(79.5000 - 137.698i) q^{59} +(-95.5000 - 165.411i) q^{61} +410.000 q^{62} +64.0000 q^{64} +(166.500 + 288.386i) q^{65} +(210.500 - 364.597i) q^{67} +(-84.0000 + 145.492i) q^{68} +(-279.000 - 483.242i) q^{70} -156.000 q^{71} +182.000 q^{73} +(-166.000 - 287.520i) q^{74} +(56.0000 - 96.9948i) q^{76} +(232.500 - 402.702i) q^{77} +(-566.500 - 981.207i) q^{79} +144.000 q^{80} -522.000 q^{82} +(-541.500 - 937.906i) q^{83} +(-189.000 + 327.358i) q^{85} +(-43.0000 + 74.4782i) q^{86} +(60.0000 + 103.923i) q^{88} +1050.00 q^{89} +1147.00 q^{91} +(390.000 + 675.500i) q^{92} +(-177.000 + 306.573i) q^{94} +(126.000 - 218.238i) q^{95} +(450.500 + 780.289i) q^{97} -1236.00 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 4 q^{4} - 9 q^{5} + 31 q^{7} - 16 q^{8} - 36 q^{10} - 15 q^{11} + 37 q^{13} - 62 q^{14} - 16 q^{16} + 84 q^{17} - 56 q^{19} - 36 q^{20} + 30 q^{22} + 195 q^{23} + 44 q^{25} + 148 q^{26}+ \cdots - 2472 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 4 * q^4 - 9 * q^5 + 31 * q^7 - 16 * q^8 - 36 * q^10 - 15 * q^11 + 37 * q^13 - 62 * q^14 - 16 * q^16 + 84 * q^17 - 56 * q^19 - 36 * q^20 + 30 * q^22 + 195 * q^23 + 44 * q^25 + 148 * q^26 - 248 * q^28 + 111 * q^29 + 205 * q^31 + 32 * q^32 + 84 * q^34 - 558 * q^35 - 332 * q^37 - 56 * q^38 + 72 * q^40 - 261 * q^41 + 43 * q^43 + 120 * q^44 + 780 * q^46 + 177 * q^47 - 618 * q^49 - 88 * q^50 + 148 * q^52 - 228 * q^53 + 270 * q^55 - 248 * q^56 - 222 * q^58 + 159 * q^59 - 191 * q^61 + 820 * q^62 + 128 * q^64 + 333 * q^65 + 421 * q^67 - 168 * q^68 - 558 * q^70 - 312 * q^71 + 364 * q^73 - 332 * q^74 + 112 * q^76 + 465 * q^77 - 1133 * q^79 + 288 * q^80 - 1044 * q^82 - 1083 * q^83 - 378 * q^85 - 86 * q^86 + 120 * q^88 + 2100 * q^89 + 2294 * q^91 + 780 * q^92 - 354 * q^94 + 252 * q^95 + 901 * q^97 - 2472 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/54\mathbb{Z}\right)^\times$.

$n$	$29$
$\chi(n)$	$e\left(\frac{1}{3}\right)$

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 $

$2$	1.00000	+	1.73205i	0.353553	+	0.612372i
$2$	1.00000	+	1.73205i	0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−2.00000	+	3.46410i	−0.250000	+	0.433013i
$5$	−4.50000	+	7.79423i	−0.402492	+	0.697137i	−0.994026	−	0.109143i	$-0.965189\pi$
$5$	−4.50000	+	7.79423i	−0.402492	+	0.697137i	0.591534	+	0.806280i	$0.298523\pi$
$6$		0			0
$7$	15.5000	+	26.8468i	0.836921	+	1.44959i	0.892456	+	0.451134i	$0.148980\pi$
$7$	15.5000	+	26.8468i	0.836921	+	1.44959i	−0.0555351	+	0.998457i	$0.517686\pi$
$8$	−8.00000			−0.353553
$9$		0			0
$10$	−18.0000			−0.569210
$11$	−7.50000	−	12.9904i	−0.205576	−	0.356068i	0.744740	−	0.667355i	$-0.232573\pi$
$11$	−7.50000	−	12.9904i	−0.205576	−	0.356068i	−0.950316	+	0.311287i	$0.899240\pi$
$12$		0			0
$13$	18.5000	−	32.0429i	0.394691	−	0.683624i	−0.598371	−	0.801219i	$-0.704185\pi$
$13$	18.5000	−	32.0429i	0.394691	−	0.683624i	0.993062	+	0.117595i	$0.0375185\pi$
$14$	−31.0000	+	53.6936i	−0.591793	+	1.02502i
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.125000	−	0.216506i
$17$	42.0000			0.599206			0.299603	−	0.954064i	$-0.403146\pi$
$17$	42.0000			0.599206			0.299603	+	0.954064i	$0.403146\pi$
$18$		0			0
$19$	−28.0000			−0.338086			−0.169043	−	0.985609i	$-0.554068\pi$
$19$	−28.0000			−0.338086			−0.169043	+	0.985609i	$0.554068\pi$
$20$	−18.0000	−	31.1769i	−0.201246	−	0.348569i
$21$		0			0
$22$	15.0000	−	25.9808i	0.145364	−	0.251778i
$23$	97.5000	−	168.875i	0.883920	−	1.53099i	0.0369731	−	0.999316i	$-0.488228\pi$
$23$	97.5000	−	168.875i	0.883920	−	1.53099i	0.846947	−	0.531678i	$-0.178438\pi$
$24$		0			0
$25$	22.0000	+	38.1051i	0.176000	+	0.304841i
$26$	74.0000			0.558177
$27$		0			0
$28$	−124.000			−0.836921
$29$	55.5000	+	96.1288i	0.355382	+	0.615540i	0.987183	−	0.159590i	$-0.0510173\pi$
$29$	55.5000	+	96.1288i	0.355382	+	0.615540i	−0.631801	+	0.775131i	$0.717684\pi$
$30$		0			0
$31$	102.500	−	177.535i	0.593856	−	1.02859i	−0.399851	−	0.916580i	$-0.630938\pi$
$31$	102.500	−	177.535i	0.593856	−	1.02859i	0.993707	−	0.112009i	$-0.0357285\pi$
$32$	16.0000	−	27.7128i	0.0883883	−	0.153093i
$33$		0			0
$34$	42.0000	+	72.7461i	0.211851	+	0.366937i
$35$	−279.000			−1.34742
$36$		0			0
$37$	−166.000			−0.737574			−0.368787	−	0.929514i	$-0.620227\pi$
$37$	−166.000			−0.737574			−0.368787	+	0.929514i	$0.620227\pi$
$38$	−28.0000	−	48.4974i	−0.119532	−	0.207035i
$39$		0			0
$40$	36.0000	−	62.3538i	0.142302	−	0.246475i
$41$	−130.500	+	226.033i	−0.497090	+	0.860985i	−0.999994	−	0.00335732i	$-0.998931\pi$
$41$	−130.500	+	226.033i	−0.497090	+	0.860985i	0.502905	+	0.864342i	$0.332265\pi$
$42$		0			0
$43$	21.5000	+	37.2391i	0.0762493	+	0.132068i	0.901629	−	0.432511i	$-0.142372\pi$
$43$	21.5000	+	37.2391i	0.0762493	+	0.132068i	−0.825380	+	0.564578i	$0.809039\pi$
$44$	60.0000			0.205576
$45$		0			0
$46$	390.000			1.25005
$47$	88.5000	+	153.286i	0.274661	+	0.475726i	0.970049	−	0.242907i	$-0.0781011\pi$
$47$	88.5000	+	153.286i	0.274661	+	0.475726i	−0.695389	+	0.718634i	$0.744768\pi$
$48$		0			0
$49$	−309.000	+	535.204i	−0.900875	+	1.56036i
$50$	−44.0000	+	76.2102i	−0.124451	+	0.215555i
$51$		0			0
$52$	74.0000	+	128.172i	0.197345	+	0.341812i
$53$	−114.000			−0.295455			−0.147727	−	0.989028i	$-0.547196\pi$
$53$	−114.000			−0.295455			−0.147727	+	0.989028i	$0.547196\pi$
$54$		0			0
$55$	135.000			0.330971
$56$	−124.000	−	214.774i	−0.295896	−	0.512508i
$57$		0			0
$58$	−111.000	+	192.258i	−0.251293	+	0.435253i
$59$	79.5000	−	137.698i	0.175424	−	0.303843i	−0.764884	−	0.644168i	$-0.777204\pi$
$59$	79.5000	−	137.698i	0.175424	−	0.303843i	0.940308	+	0.340325i	$0.110537\pi$
$60$		0			0
$61$	−95.5000	−	165.411i	−0.200451	−	0.347192i	0.748223	−	0.663448i	$-0.230907\pi$
$61$	−95.5000	−	165.411i	−0.200451	−	0.347192i	−0.948674	+	0.316256i	$0.897574\pi$
$62$	410.000			0.839840
$63$		0			0
$64$	64.0000			0.125000
$65$	166.500	+	288.386i	0.317720	+	0.550307i
$66$		0			0
$67$	210.500	−	364.597i	0.383831	−	0.664815i	−0.607775	−	0.794109i	$-0.707938\pi$
$67$	210.500	−	364.597i	0.383831	−	0.664815i	0.991606	+	0.129294i	$0.0412712\pi$
$68$	−84.0000	+	145.492i	−0.149801	+	0.259464i
$69$		0			0
$70$	−279.000	−	483.242i	−0.476384	−	0.825121i
$71$	−156.000			−0.260758			−0.130379	−	0.991464i	$-0.541619\pi$
$71$	−156.000			−0.260758			−0.130379	+	0.991464i	$0.541619\pi$
$72$		0			0
$73$	182.000			0.291801			0.145901	−	0.989299i	$-0.453392\pi$
$73$	182.000			0.291801			0.145901	+	0.989299i	$0.453392\pi$
$74$	−166.000	−	287.520i	−0.260772	−	0.451670i
$75$		0			0
$76$	56.0000	−	96.9948i	0.0845216	−	0.146396i
$77$	232.500	−	402.702i	0.344102	−	0.596002i
$78$		0			0
$79$	−566.500	−	981.207i	−0.806788	−	1.39740i	−0.915078	−	0.403278i	$-0.867871\pi$
$79$	−566.500	−	981.207i	−0.806788	−	1.39740i	0.108290	−	0.994119i	$-0.465463\pi$
$80$	144.000			0.201246
$81$		0			0
$82$	−522.000			−0.702991
$83$	−541.500	−	937.906i	−0.716113	−	1.24034i	−0.962529	−	0.271179i	$-0.912586\pi$
$83$	−541.500	−	937.906i	−0.716113	−	1.24034i	0.246416	−	0.969164i	$-0.420747\pi$
$84$		0			0
$85$	−189.000	+	327.358i	−0.241176	+	0.417728i
$86$	−43.0000	+	74.4782i	−0.0539164	+	0.0933859i
$87$		0			0
$88$	60.0000	+	103.923i	0.0726821	+	0.125889i
$89$	1050.00			1.25056			0.625280	−	0.780401i	$-0.284985\pi$
$89$	1050.00			1.25056			0.625280	+	0.780401i	$0.284985\pi$
$90$		0			0
$91$	1147.00			1.32130
$92$	390.000	+	675.500i	0.441960	+	0.765497i
$93$		0			0
$94$	−177.000	+	306.573i	−0.194214	+	0.336389i
$95$	126.000	−	218.238i	0.136077	−	0.235693i
$96$		0			0
$97$	450.500	+	780.289i	0.471560	+	0.816766i	0.999471	−	0.0325338i	$-0.0103576\pi$
$97$	450.500	+	780.289i	0.471560	+	0.816766i	−0.527910	+	0.849300i	$0.677024\pi$
$98$	−1236.00			−1.27403
$99$		0			0
$100$	−176.000			−0.176000
$101$	193.500	+	335.152i	0.190633	+	0.330187i	0.945460	−	0.325737i	$-0.105613\pi$
$101$	193.500	+	335.152i	0.190633	+	0.330187i	−0.754827	+	0.655924i	$0.772279\pi$
$102$		0			0
$103$	−275.500	+	477.180i	−0.263552	+	0.456485i	−0.967183	−	0.254080i	$-0.918227\pi$
$103$	−275.500	+	477.180i	−0.263552	+	0.456485i	0.703631	+	0.710565i	$0.251561\pi$
$104$	−148.000	+	256.344i	−0.139544	+	0.241698i
$105$		0			0
$106$	−114.000	−	197.454i	−0.104459	−	0.180928i
$107$	12.0000			0.0108419			0.00542095	−	0.999985i	$-0.498274\pi$
$107$	12.0000			0.0108419			0.00542095	+	0.999985i	$0.498274\pi$
$108$		0			0
$109$	−502.000			−0.441127			−0.220564	−	0.975373i	$-0.570790\pi$
$109$	−502.000			−0.441127			−0.220564	+	0.975373i	$0.570790\pi$
$110$	135.000	+	233.827i	0.117016	+	0.202677i
$111$		0			0
$112$	248.000	−	429.549i	0.209230	−	0.362398i
$113$	−700.500	+	1213.30i	−0.583164	+	1.01007i	0.411938	+	0.911212i	$0.364852\pi$
$113$	−700.500	+	1213.30i	−0.583164	+	1.01007i	−0.995102	+	0.0988572i	$0.968481\pi$
$114$		0			0
$115$	877.500	+	1519.87i	0.711542	+	1.23243i
$116$	−444.000			−0.355382
$117$		0			0
$118$	318.000			0.248087
$119$	651.000	+	1127.57i	0.501488	+	0.868603i
$120$		0			0
$121$	553.000	−	957.824i	0.415477	−	0.719627i
$122$	191.000	−	330.822i	0.141740	−	0.245502i
$123$		0			0
$124$	410.000	+	710.141i	0.296928	+	0.514295i
$125$	−1521.00			−1.08834
$126$		0			0
$127$	−880.000			−0.614861			−0.307431	−	0.951571i	$-0.599469\pi$
$127$	−880.000			−0.614861			−0.307431	+	0.951571i	$0.599469\pi$
$128$	64.0000	+	110.851i	0.0441942	+	0.0765466i
$129$		0			0
$130$	−333.000	+	576.773i	−0.224662	+	0.389126i
$131$	751.500	−	1301.64i	0.501213	−	0.868126i	−0.498786	−	0.866725i	$-0.666221\pi$
$131$	751.500	−	1301.64i	0.501213	−	0.868126i	0.999999	−	0.00140084i	$-0.000445901\pi$
$132$		0			0
$133$	−434.000	−	751.710i	−0.282952	−	0.490087i
$134$	842.000			0.542819
$135$		0			0
$136$	−336.000			−0.211851
$137$	−1330.50	−	2304.49i	−0.829725	−	1.43713i	−0.898254	−	0.439476i	$-0.855164\pi$
$137$	−1330.50	−	2304.49i	−0.829725	−	1.43713i	0.0685295	−	0.997649i	$-0.478169\pi$
$138$		0			0
$139$	60.5000	−	104.789i	0.0369176	−	0.0639431i	−0.846976	−	0.531631i	$-0.821579\pi$
$139$	60.5000	−	104.789i	0.0369176	−	0.0639431i	0.883894	+	0.467688i	$0.154913\pi$
$140$	558.000	−	966.484i	0.336854	−	0.583449i
$141$		0			0
$142$	−156.000	−	270.200i	−0.0921918	−	0.159681i
$143$	−555.000			−0.324555
$144$		0			0
$145$	−999.000			−0.572155
$146$	182.000	+	315.233i	0.103167	+	0.178691i
$147$		0			0
$148$	332.000	−	575.041i	0.184393	−	0.319379i
$149$	−1414.50	+	2449.99i	−0.777721	+	1.34705i	0.155532	+	0.987831i	$0.450291\pi$
$149$	−1414.50	+	2449.99i	−0.777721	+	1.34705i	−0.933253	+	0.359221i	$0.883043\pi$
$150$		0			0
$151$	−230.500	−	399.238i	−0.124224	−	0.215162i	0.797205	−	0.603708i	$-0.206311\pi$
$151$	−230.500	−	399.238i	−0.124224	−	0.215162i	−0.921429	+	0.388546i	$0.872977\pi$
$152$	224.000			0.119532
$153$		0			0
$154$	930.000			0.486633
$155$	922.500	+	1597.82i	0.478045	+	0.827998i
$156$		0			0
$157$	1488.50	−	2578.16i	0.756658	−	1.31057i	−0.187889	−	0.982190i	$-0.560164\pi$
$157$	1488.50	−	2578.16i	0.756658	−	1.31057i	0.944546	−	0.328379i	$-0.106502\pi$
$158$	1133.00	−	1962.41i	0.570485	−	0.988109i
$159$		0			0
$160$	144.000	+	249.415i	0.0711512	+	0.123238i
$161$	6045.00			2.95909
$162$		0			0
$163$	−3316.00			−1.59343			−0.796715	−	0.604355i	$-0.793431\pi$
$163$	−3316.00			−1.59343			−0.796715	+	0.604355i	$0.793431\pi$
$164$	−522.000	−	904.131i	−0.248545	−	0.430492i
$165$		0			0
$166$	1083.00	−	1875.81i	0.506368	−	0.877055i
$167$	−340.500	+	589.763i	−0.157777	+	0.273277i	−0.934067	−	0.357099i	$-0.883766\pi$
$167$	−340.500	+	589.763i	−0.157777	+	0.273277i	0.776290	+	0.630376i	$0.217099\pi$
$168$		0			0
$169$	414.000	+	717.069i	0.188439	+	0.326386i
$170$	−756.000			−0.341074
$171$		0			0
$172$	−172.000			−0.0762493
$173$	−1990.50	−	3447.65i	−0.874768	−	1.51514i	−0.857009	−	0.515301i	$-0.827680\pi$
$173$	−1990.50	−	3447.65i	−0.874768	−	1.51514i	−0.0177589	−	0.999842i	$-0.505653\pi$
$174$		0			0
$175$	−682.000	+	1181.26i	−0.294596	+	0.510256i
$176$	−120.000	+	207.846i	−0.0513940	+	0.0890170i
$177$		0			0
$178$	1050.00	+	1818.65i	0.442139	+	0.765808i
$179$	−2004.00			−0.836793			−0.418397	−	0.908264i	$-0.637408\pi$
$179$	−2004.00			−0.836793			−0.418397	+	0.908264i	$0.637408\pi$
$180$		0			0
$181$	1274.00			0.523181			0.261590	−	0.965179i	$-0.415753\pi$
$181$	1274.00			0.523181			0.261590	+	0.965179i	$0.415753\pi$
$182$	1147.00	+	1986.66i	0.467150	+	0.809128i
$183$		0			0
$184$	−780.000	+	1351.00i	−0.312513	+	0.541288i
$185$	747.000	−	1293.84i	0.296868	−	0.514190i
$186$		0			0
$187$	−315.000	−	545.596i	−0.123182	−	0.213358i
$188$	−708.000			−0.274661
$189$		0			0
$190$	504.000			0.192442
$191$	580.500	+	1005.46i	0.219914	+	0.380902i	0.954781	−	0.297309i	$-0.0960891\pi$
$191$	580.500	+	1005.46i	0.219914	+	0.380902i	−0.734868	+	0.678210i	$0.762756\pi$
$192$		0			0
$193$	−1805.50	+	3127.22i	−0.673382	+	1.16633i	0.303557	+	0.952813i	$0.401826\pi$
$193$	−1805.50	+	3127.22i	−0.673382	+	1.16633i	−0.976939	+	0.213519i	$0.931508\pi$
$194$	−901.000	+	1560.58i	−0.333443	+	0.577541i
$195$		0			0
$196$	−1236.00	−	2140.81i	−0.450437	−	0.780180i
$197$	−2046.00			−0.739957			−0.369978	−	0.929040i	$-0.620635\pi$
$197$	−2046.00			−0.739957			−0.369978	+	0.929040i	$0.620635\pi$
$198$		0			0
$199$	2996.00			1.06724			0.533620	−	0.845724i	$-0.320831\pi$
$199$	2996.00			1.06724			0.533620	+	0.845724i	$0.320831\pi$
$200$	−176.000	−	304.841i	−0.0622254	−	0.107778i
$201$		0			0
$202$	−387.000	+	670.304i	−0.134798	+	0.233477i
$203$	−1720.50	+	2979.99i	−0.594854	+	1.03032i
$204$		0			0
$205$	−1174.50	−	2034.29i	−0.400149	−	0.693079i
$206$	−1102.00			−0.372718
$207$		0			0
$208$	−592.000			−0.197345
$209$	210.000	+	363.731i	0.0695024	+	0.120382i
$210$		0			0
$211$	−377.500	+	653.849i	−0.123167	+	0.213331i	−0.921015	−	0.389528i	$-0.872638\pi$
$211$	−377.500	+	653.849i	−0.123167	+	0.213331i	0.797848	+	0.602858i	$0.205972\pi$
$212$	228.000	−	394.908i	0.0738637	−	0.127936i
$213$		0			0
$214$	12.0000	+	20.7846i	0.00383319	+	0.00663928i
$215$	−387.000			−0.122759
$216$		0			0
$217$	6355.00			1.98804
$218$	−502.000	−	869.490i	−0.155962	−	0.270134i
$219$		0			0
$220$	−270.000	+	467.654i	−0.0827427	+	0.143315i
$221$	777.000	−	1345.80i	0.236501	−	0.409631i
$222$		0			0
$223$	1731.50	+	2999.05i	0.519954	+	0.900587i	0.999731	+	0.0231966i	$0.00738438\pi$
$223$	1731.50	+	2999.05i	0.519954	+	0.900587i	−0.479777	+	0.877391i	$0.659282\pi$
$224$	992.000			0.295896
$225$		0			0
$226$	−2802.00			−0.824718
$227$	3112.50	+	5391.01i	0.910061	+	1.57627i	0.813976	+	0.580899i	$0.197299\pi$
$227$	3112.50	+	5391.01i	0.910061	+	1.57627i	0.0960856	+	0.995373i	$0.469368\pi$
$228$		0			0
$229$	732.500	−	1268.73i	0.211375	−	0.366113i	−0.740770	−	0.671759i	$-0.765539\pi$
$229$	732.500	−	1268.73i	0.211375	−	0.366113i	0.952145	+	0.305646i	$0.0988724\pi$
$230$	−1755.00	+	3039.75i	−0.503136	+	0.871457i
$231$		0			0
$232$	−444.000	−	769.031i	−0.125647	−	0.217626i
$233$	−2634.00			−0.740597			−0.370298	−	0.928913i	$-0.620745\pi$
$233$	−2634.00			−0.740597			−0.370298	+	0.928913i	$0.620745\pi$
$234$		0			0
$235$	−1593.00			−0.442195
$236$	318.000	+	550.792i	0.0877120	+	0.151922i
$237$		0			0
$238$	−1302.00	+	2255.13i	−0.354606	+	0.614195i
$239$	3457.50	−	5988.57i	0.935762	−	1.62079i	0.162492	−	0.986710i	$-0.448047\pi$
$239$	3457.50	−	5988.57i	0.935762	−	1.62079i	0.773270	−	0.634077i	$-0.218620\pi$
$240$		0			0
$241$	744.500	+	1289.51i	0.198994	+	0.344667i	0.948202	−	0.317667i	$-0.102899\pi$
$241$	744.500	+	1289.51i	0.198994	+	0.344667i	−0.749209	+	0.662334i	$0.769566\pi$
$242$	2212.00			0.587573
$243$		0			0
$244$	764.000			0.200451
$245$	−2781.00	−	4816.83i	−0.725190	−	1.25607i
$246$		0			0
$247$	−518.000	+	897.202i	−0.133439	+	0.231124i
$248$	−820.000	+	1420.28i	−0.209960	+	0.363661i
$249$		0			0
$250$	−1521.00	−	2634.45i	−0.384786	−	0.666469i
$251$	4620.00			1.16180			0.580900	−	0.813975i	$-0.302701\pi$
$251$	4620.00			1.16180			0.580900	+	0.813975i	$0.302701\pi$
$252$		0			0
$253$	−2925.00			−0.726850
$254$	−880.000	−	1524.20i	−0.217386	−	0.376524i
$255$		0			0
$256$	−128.000	+	221.703i	−0.0312500	+	0.0541266i
$257$	1675.50	−	2902.05i	0.406672	−	0.704377i	−0.587842	−	0.808976i	$-0.700022\pi$
$257$	1675.50	−	2902.05i	0.406672	−	0.704377i	0.994515	+	0.104598i	$0.0333557\pi$
$258$		0			0
$259$	−2573.00	−	4456.57i	−0.617291	−	1.06918i
$260$	−1332.00			−0.317720
$261$		0			0
$262$	3006.00			0.708822
$263$	−301.500	−	522.213i	−0.0706893	−	0.122437i	0.828514	−	0.559968i	$-0.189187\pi$
$263$	−301.500	−	522.213i	−0.0706893	−	0.122437i	−0.899204	+	0.437530i	$0.855853\pi$
$264$		0			0
$265$	513.000	−	888.542i	0.118918	−	0.205972i
$266$	868.000	−	1503.42i	0.200077	−	0.346544i
$267$		0			0
$268$	842.000	+	1458.39i	0.191915	+	0.332407i
$269$	1470.00			0.333188			0.166594	−	0.986026i	$-0.446723\pi$
$269$	1470.00			0.333188			0.166594	+	0.986026i	$0.446723\pi$
$270$		0			0
$271$	2072.00			0.464447			0.232223	−	0.972662i	$-0.425400\pi$
$271$	2072.00			0.464447			0.232223	+	0.972662i	$0.425400\pi$
$272$	−336.000	−	581.969i	−0.0749007	−	0.129732i
$273$		0			0
$274$	2661.00	−	4608.99i	0.586704	−	1.01620i
$275$	330.000	−	571.577i	0.0723627	−	0.125336i
$276$		0			0
$277$	−3569.50	−	6182.56i	−0.774262	−	1.34106i	−0.935209	−	0.354097i	$-0.884788\pi$
$277$	−3569.50	−	6182.56i	−0.774262	−	1.34106i	0.160947	−	0.986963i	$-0.448545\pi$
$278$	242.000			0.0522093
$279$		0			0
$280$	2232.00			0.476384
$281$	4213.50	+	7298.00i	0.894507	+	1.54933i	0.834414	+	0.551138i	$0.185806\pi$
$281$	4213.50	+	7298.00i	0.894507	+	1.54933i	0.0600924	+	0.998193i	$0.480860\pi$
$282$		0			0
$283$	228.500	−	395.774i	0.0479962	−	0.0831318i	−0.841029	−	0.540990i	$-0.818050\pi$
$283$	228.500	−	395.774i	0.0479962	−	0.0831318i	0.889025	+	0.457858i	$0.151383\pi$
$284$	312.000	−	540.400i	0.0651894	−	0.112911i
$285$		0			0
$286$	−555.000	−	961.288i	−0.114748	−	0.198749i
$287$	−8091.00			−1.66410
$288$		0			0
$289$	−3149.00			−0.640953
$290$	−999.000	−	1730.32i	−0.202287	−	0.350372i
$291$		0			0
$292$	−364.000	+	630.466i	−0.0729503	+	0.126354i
$293$	−2944.50	+	5100.02i	−0.587097	+	1.01688i	0.407513	+	0.913199i	$0.366396\pi$
$293$	−2944.50	+	5100.02i	−0.587097	+	1.01688i	−0.994610	+	0.103683i	$0.966937\pi$
$294$		0			0
$295$	715.500	+	1239.28i	0.141214	+	0.244589i
$296$	1328.00			0.260772
$297$		0			0
$298$	−5658.00			−1.09986
$299$	−3607.50	−	6248.37i	−0.697750	−	1.20854i
$300$		0			0
$301$	−666.500	+	1154.41i	−0.127629	+	0.221060i
$302$	461.000	−	798.475i	0.0878396	−	0.152143i
$303$		0			0
$304$	224.000	+	387.979i	0.0422608	+	0.0731978i
$305$	1719.00			0.322720
$306$		0			0
$307$	−1204.00			−0.223830			−0.111915	−	0.993718i	$-0.535698\pi$
$307$	−1204.00			−0.223830			−0.111915	+	0.993718i	$0.535698\pi$
$308$	930.000	+	1610.81i	0.172051	+	0.298001i
$309$		0			0
$310$	−1845.00	+	3195.63i	−0.338029	+	0.585483i
$311$	−1642.50	+	2844.89i	−0.299478	+	0.518711i	−0.976017	−	0.217696i	$-0.930146\pi$
$311$	−1642.50	+	2844.89i	−0.299478	+	0.518711i	0.676539	+	0.736407i	$0.263479\pi$
$312$		0			0
$313$	5028.50	+	8709.62i	0.908075	+	1.57283i	0.816735	+	0.577013i	$0.195782\pi$
$313$	5028.50	+	8709.62i	0.908075	+	1.57283i	0.0913406	+	0.995820i	$0.470885\pi$
$314$	5954.00			1.07008
$315$		0			0
$316$	4532.00			0.806788
$317$	1147.50	+	1987.53i	0.203312	+	0.352147i	0.949594	−	0.313483i	$-0.101496\pi$
$317$	1147.50	+	1987.53i	0.203312	+	0.352147i	−0.746281	+	0.665631i	$0.768163\pi$
$318$		0			0
$319$	832.500	−	1441.93i	0.146116	−	0.253081i
$320$	−288.000	+	498.831i	−0.0503115	+	0.0871421i
$321$		0			0
$322$	6045.00	+	10470.2i	1.04619	+	1.81206i
$323$	−1176.00			−0.202583
$324$		0			0
$325$	1628.00			0.277862
$326$	−3316.00	−	5743.48i	−0.563363	−	0.975773i
$327$		0			0
$328$	1044.00	−	1808.26i	0.175748	−	0.304404i
$329$	−2743.50	+	4751.88i	−0.459739	+	0.796291i
$330$		0			0
$331$	3339.50	+	5784.18i	0.554548	+	0.960506i	0.997939	+	0.0641773i	$0.0204423\pi$
$331$	3339.50	+	5784.18i	0.554548	+	0.960506i	−0.443390	+	0.896329i	$0.646224\pi$
$332$	4332.00			0.716113
$333$		0			0
$334$	−1362.00			−0.223130
$335$	1894.50	+	3281.37i	0.308978	+	0.535165i
$336$		0			0
$337$	−1091.50	+	1890.53i	−0.176433	+	0.305590i	−0.940656	−	0.339361i	$-0.889789\pi$
$337$	−1091.50	+	1890.53i	−0.176433	+	0.305590i	0.764224	+	0.644951i	$0.223122\pi$
$338$	−828.000	+	1434.14i	−0.133246	+	0.230789i
$339$		0			0
$340$	−756.000	−	1309.43i	−0.120588	−	0.208864i
$341$	−3075.00			−0.488330
$342$		0			0
$343$	−8525.00			−1.34200
$344$	−172.000	−	297.913i	−0.0269582	−	0.0466930i
$345$		0			0
$346$	3981.00	−	6895.29i	0.618555	−	1.07137i
$347$	1945.50	−	3369.70i	0.300980	−	0.521312i	−0.675379	−	0.737471i	$-0.736020\pi$
$347$	1945.50	−	3369.70i	0.300980	−	0.521312i	0.976358	+	0.216159i	$0.0693531\pi$
$348$		0			0
$349$	−1397.50	−	2420.54i	−0.214345	−	0.371257i	0.738725	−	0.674007i	$-0.235428\pi$
$349$	−1397.50	−	2420.54i	−0.214345	−	0.371257i	−0.953070	+	0.302751i	$0.902095\pi$
$350$	−2728.00			−0.416622
$351$		0			0
$352$	−480.000			−0.0726821
$353$	2377.50	+	4117.95i	0.358475	+	0.620896i	0.987706	−	0.156322i	$-0.0499636\pi$
$353$	2377.50	+	4117.95i	0.358475	+	0.620896i	−0.629232	+	0.777218i	$0.716630\pi$
$354$		0			0
$355$	702.000	−	1215.90i	0.104953	−	0.181784i
$356$	−2100.00	+	3637.31i	−0.312640	+	0.541508i
$357$		0			0
$358$	−2004.00	−	3471.03i	−0.295851	−	0.512429i
$359$	−4608.00			−0.677440			−0.338720	−	0.940887i	$-0.609994\pi$
$359$	−4608.00			−0.677440			−0.338720	+	0.940887i	$0.609994\pi$
$360$		0			0
$361$	−6075.00			−0.885698
$362$	1274.00	+	2206.63i	0.184972	+	0.320381i
$363$		0			0
$364$	−2294.00	+	3973.32i	−0.330325	+	0.572140i
$365$	−819.000	+	1418.55i	−0.117448	+	0.203425i
$366$		0			0
$367$	−1922.50	−	3329.87i	−0.273443	−	0.473618i	0.696298	−	0.717753i	$-0.254829\pi$
$367$	−1922.50	−	3329.87i	−0.273443	−	0.473618i	−0.969741	+	0.244135i	$0.921496\pi$
$368$	−3120.00			−0.441960
$369$		0			0
$370$	2988.00			0.419834
$371$	−1767.00	−	3060.53i	−0.247272	−	0.428288i
$372$		0			0
$373$	4158.50	−	7202.73i	0.577263	−	0.999848i	−0.418529	−	0.908203i	$-0.637454\pi$
$373$	4158.50	−	7202.73i	0.577263	−	0.999848i	0.995792	−	0.0916449i	$-0.0292125\pi$
$374$	630.000	−	1091.19i	0.0871030	−	0.150867i
$375$		0			0
$376$	−708.000	−	1226.29i	−0.0971072	−	0.168195i
$377$	4107.00			0.561064
$378$		0			0
$379$	12560.0			1.70228			0.851140	−	0.524939i	$-0.175912\pi$
$379$	12560.0			1.70228			0.851140	+	0.524939i	$0.175912\pi$
$380$	504.000	+	872.954i	0.0680386	+	0.117846i
$381$		0			0
$382$	−1161.00	+	2010.91i	−0.155502	+	0.269338i
$383$	6043.50	−	10467.6i	0.806288	−	1.39653i	−0.109130	−	0.994028i	$-0.534806\pi$
$383$	6043.50	−	10467.6i	0.806288	−	1.39653i	0.915418	−	0.402505i	$-0.131860\pi$
$384$		0			0
$385$	2092.50	+	3624.32i	0.276997	+	0.479772i
$386$	−7222.00			−0.952306
$387$		0			0
$388$	−3604.00			−0.471560
$389$	−4270.50	−	7396.72i	−0.556614	−	0.964084i	−0.997776	−	0.0666565i	$-0.978767\pi$
$389$	−4270.50	−	7396.72i	−0.556614	−	0.964084i	0.441162	−	0.897428i	$-0.354567\pi$
$390$		0			0
$391$	4095.00	−	7092.75i	0.529650	−	0.917380i
$392$	2472.00	−	4281.63i	0.318507	−	0.551671i
$393$		0			0
$394$	−2046.00	−	3543.78i	−0.261614	−	0.453129i
$395$	10197.0			1.29890
$396$		0			0
$397$	−13174.0			−1.66545			−0.832726	−	0.553686i	$-0.813221\pi$
$397$	−13174.0			−1.66545			−0.832726	+	0.553686i	$0.813221\pi$
$398$	2996.00	+	5189.22i	0.377326	+	0.653549i
$399$		0			0
$400$	352.000	−	609.682i	0.0440000	−	0.0762102i
$401$	4801.50	−	8316.44i	0.597944	−	1.03567i	−0.395180	−	0.918604i	$-0.629318\pi$
$401$	4801.50	−	8316.44i	0.597944	−	1.03567i	0.993124	−	0.117066i	$-0.0373488\pi$
$402$		0			0
$403$	−3792.50	−	6568.80i	−0.468779	−	0.811949i
$404$	−1548.00			−0.190633
$405$		0			0
$406$	−6882.00			−0.841251
$407$	1245.00	+	2156.40i	0.151627	+	0.262626i
$408$		0			0
$409$	−5735.50	+	9934.18i	−0.693404	+	1.20101i	0.277312	+	0.960780i	$0.410557\pi$
$409$	−5735.50	+	9934.18i	−0.693404	+	1.20101i	−0.970716	+	0.240231i	$0.922777\pi$
$410$	2349.00	−	4068.59i	0.282948	−	0.490081i
$411$		0			0
$412$	−1102.00	−	1908.72i	−0.131776	−	0.228242i
$413$	4929.00			0.587264
$414$		0			0
$415$	9747.00			1.15292
$416$	−592.000	−	1025.37i	−0.0697721	−	0.120849i
$417$		0			0
$418$	−420.000	+	727.461i	−0.0491456	+	0.0851227i
$419$	−2986.50	+	5172.77i	−0.348210	+	0.603118i	−0.985932	−	0.167149i	$-0.946544\pi$
$419$	−2986.50	+	5172.77i	−0.348210	+	0.603118i	0.637721	+	0.770267i	$0.279877\pi$
$420$		0			0
$421$	4452.50	+	7711.96i	0.515443	+	0.892774i	0.999839	+	0.0179250i	$0.00570601\pi$
$421$	4452.50	+	7711.96i	0.515443	+	0.892774i	−0.484396	+	0.874849i	$0.660961\pi$
$422$	−1510.00			−0.174184
$423$		0			0
$424$	912.000			0.104459
$425$	924.000	+	1600.41i	0.105460	+	0.182662i
$426$		0			0
$427$	2960.50	−	5127.74i	0.335524	−	0.581144i
$428$	−24.0000	+	41.5692i	−0.00271048	+	0.00469468i
$429$		0			0
$430$	−387.000	−	670.304i	−0.0434019	−	0.0751742i
$431$	−1416.00			−0.158251			−0.0791257	−	0.996865i	$-0.525213\pi$
$431$	−1416.00			−0.158251			−0.0791257	+	0.996865i	$0.525213\pi$
$432$		0			0
$433$	10766.0			1.19488			0.597438	−	0.801915i	$-0.296186\pi$
$433$	10766.0			1.19488			0.597438	+	0.801915i	$0.296186\pi$
$434$	6355.00	+	11007.2i	0.702880	+	1.21742i
$435$		0			0
$436$	1004.00	−	1738.98i	0.110282	−	0.191014i
$437$	−2730.00	+	4728.50i	−0.298841	+	0.517608i
$438$		0			0
$439$	−2174.50	−	3766.34i	−0.236408	−	0.409471i	0.723273	−	0.690562i	$-0.242637\pi$
$439$	−2174.50	−	3766.34i	−0.236408	−	0.409471i	−0.959681	+	0.281091i	$0.909304\pi$
$440$	−1080.00			−0.117016
$441$		0			0
$442$	3108.00			0.334463
$443$	−7273.50	−	12598.1i	−0.780078	−	1.35113i	−0.931896	−	0.362726i	$-0.881846\pi$
$443$	−7273.50	−	12598.1i	−0.780078	−	1.35113i	0.151818	−	0.988408i	$-0.451487\pi$
$444$		0			0
$445$	−4725.00	+	8183.94i	−0.503340	+	0.871811i
$446$	−3463.00	+	5998.09i	−0.367663	+	0.636811i
$447$		0			0
$448$	992.000	+	1718.19i	0.104615	+	0.181199i
$449$	3330.00			0.350005			0.175003	−	0.984568i	$-0.444007\pi$
$449$	3330.00			0.350005			0.175003	+	0.984568i	$0.444007\pi$
$450$		0			0
$451$	3915.00			0.408759
$452$	−2802.00	−	4853.21i	−0.291582	−	0.505035i
$453$		0			0
$454$	−6225.00	+	10782.0i	−0.643510	+	1.11459i
$455$	−5161.50	+	8939.98i	−0.531813	+	0.921127i
$456$		0			0
$457$	−4073.50	−	7055.51i	−0.416959	−	0.722194i	0.578673	−	0.815560i	$-0.303571\pi$
$457$	−4073.50	−	7055.51i	−0.416959	−	0.722194i	−0.995632	+	0.0933655i	$0.970237\pi$
$458$	2930.00			0.298930
$459$		0			0
$460$	−7020.00			−0.711542
$461$	4015.50	+	6955.05i	0.405684	+	0.702666i	0.994401	−	0.105674i	$-0.0336999\pi$
$461$	4015.50	+	6955.05i	0.405684	+	0.702666i	−0.588717	+	0.808340i	$0.700367\pi$
$462$		0			0
$463$	−2141.50	+	3709.19i	−0.214955	+	0.372312i	−0.953259	−	0.302156i	$-0.902294\pi$
$463$	−2141.50	+	3709.19i	−0.214955	+	0.372312i	0.738304	+	0.674468i	$0.235627\pi$
$464$	888.000	−	1538.06i	0.0888456	−	0.153885i
$465$		0			0
$466$	−2634.00	−	4562.22i	−0.261841	−	0.453521i
$467$	−5460.00			−0.541025			−0.270512	−	0.962716i	$-0.587193\pi$
$467$	−5460.00			−0.541025			−0.270512	+	0.962716i	$0.587193\pi$
$468$		0			0
$469$	13051.0			1.28494
$470$	−1593.00	−	2759.16i	−0.156340	−	0.270788i
$471$		0			0
$472$	−636.000	+	1101.58i	−0.0620218	+	0.107425i
$473$	322.500	−	558.586i	0.0313500	−	0.0542999i
$474$		0			0
$475$	−616.000	−	1066.94i	−0.0595032	−	0.103063i
$476$	−5208.00			−0.501488
$477$		0			0
$478$	13830.0			1.32337
$479$	214.500	+	371.525i	0.0204609	+	0.0354393i	0.876075	−	0.482176i	$-0.160153\pi$
$479$	214.500	+	371.525i	0.0204609	+	0.0354393i	−0.855614	+	0.517615i	$0.826820\pi$
$480$		0			0
$481$	−3071.00	+	5319.13i	−0.291113	+	0.504223i
$482$	−1489.00	+	2579.02i	−0.140710	+	0.243716i
$483$		0			0
$484$	2212.00	+	3831.30i	0.207739	+	0.359814i
$485$	−8109.00			−0.759197
$486$		0			0
$487$	−11296.0			−1.05107			−0.525535	−	0.850772i	$-0.676135\pi$
$487$	−11296.0			−1.05107			−0.525535	+	0.850772i	$0.676135\pi$
$488$	764.000	+	1323.29i	0.0708702	+	0.122751i
$489$		0			0
$490$	5562.00	−	9633.67i	0.512787	−	0.888173i
$491$	−7336.50	+	12707.2i	−0.674321	+	1.16796i	0.302346	+	0.953198i	$0.402230\pi$
$491$	−7336.50	+	12707.2i	−0.674321	+	1.16796i	−0.976667	+	0.214760i	$0.931103\pi$
$492$		0			0
$493$	2331.00	+	4037.41i	0.212947	+	0.368835i
$494$	−2072.00			−0.188712
$495$		0			0
$496$	−3280.00			−0.296928
$497$	−2418.00	−	4188.10i	−0.218234	−	0.377992i
$498$		0			0
$499$	−6719.50	+	11638.5i	−0.602818	+	1.04411i	0.389574	+	0.920995i	$0.372622\pi$
$499$	−6719.50	+	11638.5i	−0.602818	+	1.04411i	−0.992392	+	0.123116i	$0.960711\pi$
$500$	3042.00	−	5268.90i	0.272085	−	0.471265i
$501$		0			0
$502$	4620.00	+	8002.07i	0.410758	+	0.711454i
$503$	17388.0			1.54134			0.770669	−	0.637236i	$-0.219922\pi$
$503$	17388.0			1.54134			0.770669	+	0.637236i	$0.219922\pi$
$504$		0			0
$505$	−3483.00			−0.306914
$506$	−2925.00	−	5066.25i	−0.256980	−	0.445103i
$507$		0			0
$508$	1760.00	−	3048.41i	0.153715	−	0.266243i
$509$	−1894.50	+	3281.37i	−0.164975	+	0.285745i	−0.936646	−	0.350276i	$-0.886088\pi$
$509$	−1894.50	+	3281.37i	−0.164975	+	0.285745i	0.771671	+	0.636021i	$0.219421\pi$
$510$		0			0
$511$	2821.00	+	4886.12i	0.244215	+	0.422992i
$512$	−512.000			−0.0441942
$513$		0			0
$514$	6702.00			0.575122
$515$	−2479.50	−	4294.62i	−0.212155	−	0.367463i
$516$		0			0
$517$	1327.50	−	2299.30i	0.112927	−	0.195596i
$518$	5146.00	−	8913.13i	0.436491	−	0.756024i
$519$		0			0
$520$	−1332.00	−	2307.09i	−0.112331	−	0.194563i
$521$	−9786.00			−0.822903			−0.411451	−	0.911432i	$-0.634978\pi$
$521$	−9786.00			−0.822903			−0.411451	+	0.911432i	$0.634978\pi$
$522$		0			0
$523$	−8008.00			−0.669532			−0.334766	−	0.942301i	$-0.608657\pi$
$523$	−8008.00			−0.669532			−0.334766	+	0.942301i	$0.608657\pi$
$524$	3006.00	+	5206.54i	0.250606	+	0.434063i
$525$		0			0
$526$	603.000	−	1044.43i	0.0499849	−	0.0865764i
$527$	4305.00	−	7456.48i	0.355842	−	0.616336i
$528$		0			0
$529$	−12929.0	−	22393.7i	−1.06263	−	1.84053i
$530$	2052.00			0.168176
$531$		0			0
$532$	3472.00			0.282952
$533$	4828.50	+	8363.21i	0.392393	+	0.679645i
$534$		0			0
$535$	−54.0000	+	93.5307i	−0.00436378	+	0.00755829i
$536$	−1684.00	+	2916.77i	−0.135705	+	0.235047i
$537$		0			0
$538$	1470.00	+	2546.11i	0.117800	+	0.204035i
$539$	9270.00			0.740793
$540$		0			0
$541$	−2938.00			−0.233483			−0.116742	−	0.993162i	$-0.537245\pi$
$541$	−2938.00			−0.233483			−0.116742	+	0.993162i	$0.537245\pi$
$542$	2072.00	+	3588.81i	0.164207	+	0.284414i
$543$		0			0
$544$	672.000	−	1163.94i	0.0529628	−	0.0917343i
$545$	2259.00	−	3912.70i	0.177550	−	0.307526i
$546$		0			0
$547$	5187.50	+	8985.01i	0.405487	+	0.702324i	0.994378	−	0.105888i	$-0.0337686\pi$
$547$	5187.50	+	8985.01i	0.405487	+	0.702324i	−0.588891	+	0.808213i	$0.700435\pi$
$548$	10644.0			0.829725
$549$		0			0
$550$	1320.00			0.102336
$551$	−1554.00	−	2691.61i	−0.120150	−	0.208106i
$552$		0			0
$553$	17561.5	−	30417.4i	1.35044	−	2.33902i
$554$	7139.00	−	12365.1i	0.547486	−	0.948273i
$555$		0			0
$556$	242.000	+	419.156i	0.0184588	+	0.0319716i
$557$	−3306.00			−0.251490			−0.125745	−	0.992063i	$-0.540132\pi$
$557$	−3306.00			−0.251490			−0.125745	+	0.992063i	$0.540132\pi$
$558$		0			0
$559$	1591.00			0.120379
$560$	2232.00	+	3865.94i	0.168427	+	0.291724i
$561$		0			0
$562$	−8427.00	+	14596.0i	−0.632512	+	1.09554i
$563$	−10546.5	+	18267.1i	−0.789488	+	1.36743i	0.136792	+	0.990600i	$0.456321\pi$
$563$	−10546.5	+	18267.1i	−0.789488	+	1.36743i	−0.926281	+	0.376834i	$0.877013\pi$
$564$		0			0
$565$	−6304.50	−	10919.7i	−0.469438	−	0.813090i
$566$	914.000			0.0678768
$567$		0			0
$568$	1248.00			0.0921918
$569$	643.500	+	1114.57i	0.0474111	+	0.0821185i	0.888757	−	0.458379i	$-0.151570\pi$
$569$	643.500	+	1114.57i	0.0474111	+	0.0821185i	−0.841346	+	0.540497i	$0.818236\pi$
$570$		0			0
$571$	−7517.50	+	13020.7i	−0.550959	+	0.954289i	0.447247	+	0.894411i	$0.352405\pi$
$571$	−7517.50	+	13020.7i	−0.550959	+	0.954289i	−0.998206	+	0.0598783i	$0.980929\pi$
$572$	1110.00	−	1922.58i	0.0811389	−	0.140537i
$573$		0			0
$574$	−8091.00	−	14014.0i	−0.588348	−	1.01905i
$575$	8580.00			0.622280
$576$		0			0
$577$	1190.00			0.0858585			0.0429292	−	0.999078i	$-0.486331\pi$
$577$	1190.00			0.0858585			0.0429292	+	0.999078i	$0.486331\pi$
$578$	−3149.00	−	5454.23i	−0.226611	−	0.392502i
$579$		0			0
$580$	1998.00	−	3460.64i	0.143039	−	0.247750i
$581$	16786.5	−	29075.1i	1.19866	−	2.07614i
$582$		0			0
$583$	855.000	+	1480.90i	0.0607384	+	0.105202i
$584$	−1456.00			−0.103167
$585$		0			0
$586$	−11778.0			−0.830281
$587$	−8941.50	−	15487.1i	−0.628714	−	1.08896i	−0.987810	−	0.155664i	$-0.950248\pi$
$587$	−8941.50	−	15487.1i	−0.628714	−	1.08896i	0.359096	−	0.933301i	$-0.383085\pi$
$588$		0			0
$589$	−2870.00	+	4970.99i	−0.200775	+	0.347752i
$590$	−1431.00	+	2478.56i	−0.0998531	+	0.172951i
$591$		0			0
$592$	1328.00	+	2300.16i	0.0921967	+	0.159689i
$593$	−20118.0			−1.39317			−0.696583	−	0.717476i	$-0.745297\pi$
$593$	−20118.0			−1.39317			−0.696583	+	0.717476i	$0.745297\pi$
$594$		0			0
$595$	−11718.0			−0.807380
$596$	−5658.00	−	9799.94i	−0.388860	−	0.673526i
$597$		0			0
$598$	7215.00	−	12496.7i	0.493383	−	0.854565i
$599$	−532.500	+	922.317i	−0.0363228	+	0.0629129i	−0.883615	−	0.468214i	$-0.844898\pi$
$599$	−532.500	+	922.317i	−0.0363228	+	0.0629129i	0.847293	+	0.531127i	$0.178231\pi$
$600$		0			0
$601$	10362.5	+	17948.4i	0.703320	+	1.21819i	0.967294	+	0.253656i	$0.0816331\pi$
$601$	10362.5	+	17948.4i	0.703320	+	1.21819i	−0.263975	+	0.964530i	$0.585034\pi$
$602$	−2666.00			−0.180495
$603$		0			0
$604$	1844.00			0.124224
$605$	4977.00	+	8620.42i	0.334453	+	0.579289i
$606$		0			0
$607$	7872.50	−	13635.6i	0.526417	−	0.911780i	−0.473110	−	0.881004i	$-0.656869\pi$
$607$	7872.50	−	13635.6i	0.526417	−	0.911780i	0.999526	−	0.0307768i	$-0.00979812\pi$
$608$	−448.000	+	775.959i	−0.0298829	+	0.0517587i
$609$		0			0
$610$	1719.00	+	2977.40i	0.114099	+	0.197625i
$611$	6549.00			0.433624
$612$		0			0
$613$	5042.00			0.332210			0.166105	−	0.986108i	$-0.446881\pi$
$613$	5042.00			0.332210			0.166105	+	0.986108i	$0.446881\pi$
$614$	−1204.00	−	2085.39i	−0.0791360	−	0.137068i
$615$		0			0
$616$	−1860.00	+	3221.61i	−0.121658	+	0.210718i
$617$	−5026.50	+	8706.15i	−0.327973	+	0.568066i	−0.982110	−	0.188311i	$-0.939699\pi$
$617$	−5026.50	+	8706.15i	−0.327973	+	0.568066i	0.654137	+	0.756376i	$0.273032\pi$
$618$		0			0
$619$	2991.50	+	5181.43i	0.194246	+	0.336445i	0.946653	−	0.322254i	$-0.104441\pi$
$619$	2991.50	+	5181.43i	0.194246	+	0.336445i	−0.752407	+	0.658699i	$0.771107\pi$
$620$	−7380.00			−0.478045
$621$		0			0
$622$	−6570.00			−0.423526
$623$	16275.0	+	28189.1i	1.04662	+	1.81280i
$624$		0			0
$625$	4094.50	−	7091.88i	0.262048	−	0.453880i
$626$	−10057.0	+	17419.2i	−0.642106	+	1.11216i
$627$		0			0
$628$	5954.00	+	10312.6i	0.378329	+	0.655285i
$629$	−6972.00			−0.441958
$630$		0			0
$631$	−19696.0			−1.24261			−0.621304	−	0.783570i	$-0.713397\pi$
$631$	−19696.0			−1.24261			−0.621304	+	0.783570i	$0.713397\pi$
$632$	4532.00	+	7849.65i	0.285243	+	0.494055i
$633$		0			0
$634$	−2295.00	+	3975.06i	−0.143764	+	0.249006i
$635$	3960.00	−	6858.92i	0.247477	−	0.428642i
$636$		0			0
$637$	11433.0	+	19802.5i	0.711133	+	1.23172i
$638$	3330.00			0.206639
$639$		0			0
$640$	−1152.00			−0.0711512
$641$	−5488.50	−	9506.36i	−0.338195	−	0.585770i	0.645899	−	0.763423i	$-0.276483\pi$
$641$	−5488.50	−	9506.36i	−0.338195	−	0.585770i	−0.984093	+	0.177653i	$0.943150\pi$
$642$		0			0
$643$	7914.50	−	13708.3i	0.485408	−	0.840752i	−0.514451	−	0.857520i	$-0.672004\pi$
$643$	7914.50	−	13708.3i	0.485408	−	0.840752i	0.999859	+	0.0167681i	$0.00533770\pi$
$644$	−12090.0	+	20940.5i	−0.739771	+	1.28132i
$645$		0			0
$646$	−1176.00	−	2036.89i	−0.0716240	−	0.124056i
$647$	−28224.0			−1.71499			−0.857496	−	0.514490i	$-0.827981\pi$
$647$	−28224.0			−1.71499			−0.857496	+	0.514490i	$0.827981\pi$
$648$		0			0
$649$	−2385.00			−0.144252
$650$	1628.00	+	2819.78i	0.0982391	+	0.170155i
$651$		0			0
$652$	6632.00	−	11487.0i	0.398358	−	0.689976i
$653$	14083.5	−	24393.3i	0.843997	−	1.46185i	−0.0424927	−	0.999097i	$-0.513530\pi$
$653$	14083.5	−	24393.3i	0.843997	−	1.46185i	0.886490	−	0.462749i	$-0.153137\pi$
$654$		0			0
$655$	6763.50	+	11714.7i	0.403468	+	0.698828i
$656$	4176.00			0.248545
$657$		0			0
$658$	−10974.0			−0.650169
$659$	5368.50	+	9298.51i	0.317340	+	0.549649i	0.979932	−	0.199331i	$-0.0638770\pi$
$659$	5368.50	+	9298.51i	0.317340	+	0.549649i	−0.662592	+	0.748980i	$0.730544\pi$
$660$		0			0
$661$	−5063.50	+	8770.24i	−0.297954	+	0.516071i	−0.975668	−	0.219255i	$-0.929637\pi$
$661$	−5063.50	+	8770.24i	−0.297954	+	0.516071i	0.677714	+	0.735326i	$0.262971\pi$
$662$	−6679.00	+	11568.4i	−0.392125	+	0.679180i
$663$		0			0
$664$	4332.00	+	7503.24i	0.253184	+	0.438528i
$665$	7812.00			0.455543
$666$		0			0
$667$	21645.0			1.25652
$668$	−1362.00	−	2359.05i	−0.0788883	−	0.136638i
$669$		0			0
$670$	−3789.00	+	6562.74i	−0.218480	+	0.378419i
$671$	−1432.50	+	2481.16i	−0.0824159	+	0.142748i
$672$		0			0
$673$	−125.500	−	217.372i	−0.00718822	−	0.0124504i	0.862409	−	0.506212i	$-0.168955\pi$
$673$	−125.500	−	217.372i	−0.00718822	−	0.0124504i	−0.869597	+	0.493762i	$0.835621\pi$
$674$	−4366.00			−0.249513
$675$		0			0
$676$	−3312.00			−0.188439
$677$	4225.50	+	7318.78i	0.239881	+	0.415485i	0.960680	−	0.277659i	$-0.0895584\pi$
$677$	4225.50	+	7318.78i	0.239881	+	0.415485i	−0.720799	+	0.693144i	$0.756225\pi$
$678$		0			0
$679$	−13965.5	+	24189.0i	−0.789318	+	1.36714i
$680$	1512.00	−	2618.86i	0.0852685	−	0.147689i
$681$		0			0
$682$	−3075.00	−	5326.06i	−0.172651	−	0.299040i
$683$	25884.0			1.45011			0.725054	−	0.688692i	$-0.241815\pi$
$683$	25884.0			1.45011			0.725054	+	0.688692i	$0.241815\pi$
$684$		0			0
$685$	23949.0			1.33583
$686$	−8525.00	−	14765.7i	−0.474469	−	0.821805i
$687$		0			0
$688$	344.000	−	595.825i	0.0190623	−	0.0330169i
$689$	−2109.00	+	3652.90i	−0.116613	+	0.201980i
$690$		0			0
$691$	−3182.50	−	5512.25i	−0.175207	−	0.303467i	0.765026	−	0.643999i	$-0.222726\pi$
$691$	−3182.50	−	5512.25i	−0.175207	−	0.303467i	−0.940233	+	0.340532i	$0.889393\pi$
$692$	15924.0			0.874768
$693$		0			0
$694$	7782.00			0.425649
$695$	544.500	+	943.102i	0.0297181	+	0.0514732i
$696$		0			0
$697$	−5481.00	+	9493.37i	−0.297859	+	0.515907i
$698$	2795.00	−	4841.08i	0.151565	−	0.262518i
$699$		0			0
$700$	−2728.00	−	4725.03i	−0.147298	−	0.255128i
$701$	−1122.00			−0.0604527			−0.0302264	−	0.999543i	$-0.509623\pi$
$701$	−1122.00			−0.0604527			−0.0302264	+	0.999543i	$0.509623\pi$
$702$		0			0
$703$	4648.00			0.249364
$704$	−480.000	−	831.384i	−0.0256970	−	0.0445085i
$705$		0			0
$706$	−4755.00	+	8235.90i	−0.253480	+	0.439040i
$707$	−5998.50	+	10389.7i	−0.319090	+	0.552681i
$708$		0			0
$709$	−2141.50	−	3709.19i	−0.113435	−	0.196476i	0.803718	−	0.595011i	$-0.202852\pi$
$709$	−2141.50	−	3709.19i	−0.113435	−	0.196476i	−0.917153	+	0.398535i	$0.869519\pi$
$710$	2808.00			0.148426
$711$		0			0
$712$	−8400.00			−0.442139
$713$	−19987.5	−	34619.4i	−1.04984	−	1.81838i
$714$		0			0
$715$	2497.50	−	4325.80i	0.130631	−	0.226260i
$716$	4008.00	−	6942.06i	0.209198	−	0.362342i
$717$		0			0
$718$	−4608.00	−	7981.29i	−0.239511	−	0.414846i
$719$	−4032.00			−0.209135			−0.104568	−	0.994518i	$-0.533346\pi$
$719$	−4032.00			−0.209135			−0.104568	+	0.994518i	$0.533346\pi$
$720$		0			0
$721$	−17081.0			−0.882288
$722$	−6075.00	−	10522.2i	−0.313141	−	0.542377i
$723$		0			0
$724$	−2548.00	+	4413.27i	−0.130795	+	0.226544i
$725$	−2442.00	+	4229.67i	−0.125095	+	0.216670i
$726$		0			0
$727$	−12002.5	−	20788.9i	−0.612308	−	1.06055i	−0.990850	−	0.134965i	$-0.956908\pi$
$727$	−12002.5	−	20788.9i	−0.612308	−	1.06055i	0.378542	−	0.925584i	$-0.376426\pi$
$728$	−9176.00			−0.467150
$729$		0			0
$730$	−3276.00			−0.166096
$731$	903.000	+	1564.04i	0.0456890	+	0.0791357i
$732$		0			0
$733$	18750.5	−	32476.8i	0.944837	−	1.63651i	0.188760	−	0.982023i	$-0.439553\pi$
$733$	18750.5	−	32476.8i	0.944837	−	1.63651i	0.756077	−	0.654482i	$-0.227113\pi$
$734$	3845.00	−	6659.74i	0.193354	−	0.334898i
$735$		0			0
$736$	−3120.00	−	5404.00i	−0.156256	−	0.270644i
$737$	−6315.00			−0.315626
$738$		0			0
$739$	−880.000			−0.0438042			−0.0219021	−	0.999760i	$-0.506972\pi$
$739$	−880.000			−0.0438042			−0.0219021	+	0.999760i	$0.506972\pi$
$740$	2988.00	+	5175.37i	0.148434	+	0.257095i
$741$		0			0
$742$	3534.00	−	6121.07i	0.174848	−	0.302846i
$743$	811.500	−	1405.56i	0.0400687	−	0.0694010i	−0.845296	−	0.534299i	$-0.820576\pi$
$743$	811.500	−	1405.56i	0.0400687	−	0.0694010i	0.885364	+	0.464898i	$0.153909\pi$
$744$		0			0
$745$	−12730.5	−	22049.9i	−0.626053	−	1.08436i
$746$	16634.0			0.816373
$747$		0			0
$748$	2520.00			0.123182
$749$	186.000	+	322.161i	0.00907382	+	0.0157163i
$750$		0			0
$751$	3444.50	−	5966.05i	0.167366	−	0.289886i	−0.770127	−	0.637890i	$-0.779807\pi$
$751$	3444.50	−	5966.05i	0.167366	−	0.289886i	0.937493	+	0.348005i	$0.113141\pi$
$752$	1416.00	−	2452.58i	0.0686652	−	0.118932i
$753$		0			0
$754$	4107.00	+	7113.53i	0.198366	+	0.343580i
$755$	4149.00			0.199997
$756$		0			0
$757$	−12850.0			−0.616963			−0.308482	−	0.951230i	$-0.599821\pi$
$757$	−12850.0			−0.616963			−0.308482	+	0.951230i	$0.599821\pi$
$758$	12560.0	+	21754.6i	0.601847	+	1.04243i
$759$		0			0
$760$	−1008.00	+	1745.91i	−0.0481105	+	0.0833299i
$761$	2305.50	−	3993.24i	0.109822	−	0.190217i	−0.805876	−	0.592084i	$-0.798305\pi$
$761$	2305.50	−	3993.24i	0.109822	−	0.190217i	0.915698	+	0.401867i	$0.131639\pi$
$762$		0			0
$763$	−7781.00	−	13477.1i	−0.369189	−	0.639454i
$764$	−4644.00			−0.219914
$765$		0			0
$766$	24174.0			1.14026
$767$	−2941.50	−	5094.83i	−0.138476	−	0.239848i
$768$		0			0
$769$	1152.50	−	1996.19i	0.0540445	−	0.0936078i	−0.837737	−	0.546073i	$-0.816122\pi$
$769$	1152.50	−	1996.19i	0.0540445	−	0.0936078i	0.891782	+	0.452465i	$0.149455\pi$
$770$	−4185.00	+	7248.63i	−0.195866	+	0.339250i
$771$		0			0
$772$	−7222.00	−	12508.9i	−0.336691	−	0.583166i
$773$	34902.0			1.62398			0.811991	−	0.583670i	$-0.198384\pi$
$773$	34902.0			1.62398			0.811991	+	0.583670i	$0.198384\pi$
$774$		0			0
$775$	9020.00			0.418075
$776$	−3604.00	−	6242.31i	−0.166722	−	0.288771i
$777$		0			0
$778$	8541.00	−	14793.4i	0.393586	−	0.681710i
$779$	3654.00	−	6328.91i	0.168059	−	0.291087i
$780$		0			0
$781$	1170.00	+	2026.50i	0.0536055	+	0.0928474i
$782$	16380.0			0.749038
$783$		0			0
$784$	9888.00			0.450437
$785$	13396.5	+	23203.4i	0.609098	+	1.05499i
$786$		0			0
$787$	−13127.5	+	22737.5i	−0.594593	+	1.02987i	0.399011	+	0.916946i	$0.369354\pi$
$787$	−13127.5	+	22737.5i	−0.594593	+	1.02987i	−0.993604	+	0.112919i	$0.963980\pi$
$788$	4092.00	−	7087.55i	0.184989	−	0.320411i
$789$		0			0
$790$	10197.0	+	17661.7i	0.459232	+	0.795412i
$791$	−43431.0			−1.95225
$792$		0			0
$793$	−7067.00			−0.316465
$794$	−13174.0	−	22818.0i	−0.588826	−	1.01988i
$795$		0			0
$796$	−5992.00	+	10378.4i	−0.266810	+	0.462129i
$797$	−718.500	+	1244.48i	−0.0319330	+	0.0553095i	−0.881550	−	0.472090i	$-0.843500\pi$
$797$	−718.500	+	1244.48i	−0.0319330	+	0.0553095i	0.849617	+	0.527400i	$0.176833\pi$
$798$		0			0
$799$	3717.00	+	6438.03i	0.164578	+	0.285058i
$800$	1408.00			0.0622254
$801$		0			0
$802$	19206.0			0.845620
$803$	−1365.00	−	2364.25i	−0.0599873	−	0.103901i
$804$		0			0
$805$	−27202.5	+	47116.1i	−1.19101	+	2.06289i
$806$	7585.00	−	13137.6i	0.331477	−	0.574135i
$807$		0			0
$808$	−1548.00	−	2681.21i	−0.0673991	−	0.116739i
$809$	−25734.0			−1.11837			−0.559184	−	0.829044i	$-0.688885\pi$
$809$	−25734.0			−1.11837			−0.559184	+	0.829044i	$0.688885\pi$
$810$		0			0
$811$	−3220.00			−0.139420			−0.0697099	−	0.997567i	$-0.522207\pi$
$811$	−3220.00			−0.139420			−0.0697099	+	0.997567i	$0.522207\pi$
$812$	−6882.00	−	11920.0i	−0.297427	−	0.515159i
$813$		0			0
$814$	−2490.00	+	4312.81i	−0.107217	+	0.185705i
$815$	14922.0	−	25845.7i	0.641343	−	1.11084i
$816$		0			0
$817$	−602.000	−	1042.69i	−0.0257788	−	0.0446503i
$818$	−22942.0			−0.980621
$819$		0			0
$820$	9396.00			0.400149
$821$	22315.5	+	38651.6i	0.948619	+	1.64306i	0.748337	+	0.663318i	$0.230852\pi$
$821$	22315.5	+	38651.6i	0.948619	+	1.64306i	0.200282	+	0.979738i	$0.435814\pi$
$822$		0			0
$823$	−12473.5	+	21604.7i	−0.528310	+	0.915059i	0.471146	+	0.882055i	$0.343841\pi$
$823$	−12473.5	+	21604.7i	−0.528310	+	0.915059i	−0.999455	+	0.0330037i	$0.989493\pi$
$824$	2204.00	−	3817.44i	0.0931796	−	0.161392i
$825$		0			0
$826$	4929.00	+	8537.28i	0.207629	+	0.359625i
$827$	14964.0			0.629201			0.314601	−	0.949224i	$-0.398129\pi$
$827$	14964.0			0.629201			0.314601	+	0.949224i	$0.398129\pi$
$828$		0			0
$829$	−7462.00			−0.312625			−0.156312	−	0.987708i	$-0.549961\pi$
$829$	−7462.00			−0.312625			−0.156312	+	0.987708i	$0.549961\pi$
$830$	9747.00	+	16882.3i	0.407618	+	0.706016i
$831$		0			0
$832$	1184.00	−	2050.75i	0.0493363	−	0.0854530i
$833$	−12978.0	+	22478.6i	−0.539809	+	0.934977i
$834$		0			0
$835$	−3064.50	−	5307.87i	−0.127008	−	0.219984i
$836$	−1680.00			−0.0695024
$837$		0			0
$838$	−11946.0			−0.492444
$839$	1348.50	+	2335.67i	0.0554891	+	0.0961100i	0.892436	−	0.451175i	$-0.148995\pi$
$839$	1348.50	+	2335.67i	0.0554891	+	0.0961100i	−0.836947	+	0.547285i	$0.815662\pi$
$840$		0			0
$841$	6034.00	−	10451.2i	0.247407	−	0.428521i
$842$	−8905.00	+	15423.9i	−0.364473	+	0.631286i
$843$		0			0
$844$	−1510.00	−	2615.40i	−0.0615833	−	0.106665i
$845$	−7452.00			−0.303381
$846$		0			0
$847$	34286.0			1.39089
$848$	912.000	+	1579.63i	0.0369318	+	0.0639678i
$849$		0			0
$850$	−1848.00	+	3200.83i	−0.0745716	+	0.129162i
$851$	−16185.0	+	28033.2i	−0.651956	+	1.12922i
$852$		0			0
$853$	19812.5	+	34316.3i	0.795272	+	1.37745i	0.922666	+	0.385599i	$0.126005\pi$
$853$	19812.5	+	34316.3i	0.795272	+	1.37745i	−0.127394	+	0.991852i	$0.540661\pi$
$854$	11842.0			0.474502
$855$		0			0
$856$	−96.0000			−0.00383319
$857$	−1486.50	−	2574.69i	−0.0592507	−	0.102625i	0.834879	−	0.550434i	$-0.185538\pi$
$857$	−1486.50	−	2574.69i	−0.0592507	−	0.102625i	−0.894129	+	0.447809i	$0.852204\pi$
$858$		0			0
$859$	22614.5	−	39169.5i	0.898250	−	1.55581i	0.0685200	−	0.997650i	$-0.478172\pi$
$859$	22614.5	−	39169.5i	0.898250	−	1.55581i	0.829730	−	0.558165i	$-0.188494\pi$
$860$	774.000	−	1340.61i	0.0306897	−	0.0531562i
$861$		0			0
$862$	−1416.00	−	2452.58i	−0.0559503	−	0.0969087i
$863$	−1416.00			−0.0558531			−0.0279265	−	0.999610i	$-0.508890\pi$
$863$	−1416.00			−0.0558531			−0.0279265	+	0.999610i	$0.508890\pi$
$864$		0			0
$865$	35829.0			1.40835
$866$	10766.0	+	18647.3i	0.422452	+	0.731709i
$867$		0			0
$868$	−12710.0	+	22014.4i	−0.497011	+	0.860848i
$869$	−8497.50	+	14718.1i	−0.331712	+	0.574542i
$870$		0			0
$871$	−7788.50	−	13490.1i	−0.302989	−	0.524792i
$872$	4016.00			0.155962
$873$		0			0
$874$	−10920.0			−0.422625
$875$	−23575.5	−	40834.0i	−0.910854	−	1.57765i
$876$		0			0
$877$	−8777.50	+	15203.1i	−0.337965	+	0.585372i	−0.984050	−	0.177893i	$-0.943072\pi$
$877$	−8777.50	+	15203.1i	−0.337965	+	0.585372i	0.646085	+	0.763265i	$0.276405\pi$
$878$	4349.00	−	7532.69i	0.167166	−	0.289540i
$879$		0			0
$880$	−1080.00	−	1870.61i	−0.0413714	−	0.0716573i
$881$	−30030.0			−1.14840			−0.574198	−	0.818717i	$-0.694686\pi$
$881$	−30030.0			−1.14840			−0.574198	+	0.818717i	$0.694686\pi$
$882$		0			0
$883$	−35740.0			−1.36211			−0.681057	−	0.732230i	$-0.738479\pi$
$883$	−35740.0			−1.36211			−0.681057	+	0.732230i	$0.738479\pi$
$884$	3108.00	+	5383.21i	0.118250	+	0.204816i
$885$		0			0
$886$	14547.0	−	25196.1i	0.551598	−	0.955396i
$887$	373.500	−	646.921i	0.0141386	−	0.0244887i	−0.858870	−	0.512194i	$-0.828833\pi$
$887$	373.500	−	646.921i	0.0141386	−	0.0244887i	0.873008	+	0.487706i	$0.162166\pi$
$888$		0			0
$889$	−13640.0	−	23625.2i	−0.514590	−	0.891297i
$890$	−18900.0			−0.711831
$891$		0			0
$892$	−13852.0			−0.519954
$893$	−2478.00	−	4292.02i	−0.0928590	−	0.160837i
$894$		0			0
$895$	9018.00	−	15619.6i	0.336803	−	0.583360i
$896$	−1984.00	+	3436.39i	−0.0739741	+	0.128127i
$897$		0			0
$898$	3330.00	+	5767.73i	0.123746	+	0.214334i
$899$	22755.0			0.844184
$900$		0			0
$901$	−4788.00			−0.177038
$902$	3915.00	+	6780.98i	0.144518	+	0.250313i
$903$		0			0
$904$	5604.00	−	9706.41i	0.206179	−	0.357113i
$905$	−5733.00	+	9929.85i	−0.210576	+	0.364728i
$906$		0			0
$907$	10269.5	+	17787.3i	0.375957	+	0.651177i	0.990470	−	0.137730i	$-0.0439806\pi$
$907$	10269.5	+	17787.3i	0.375957	+	0.651177i	−0.614513	+	0.788907i	$0.710647\pi$
$908$	−24900.0			−0.910061
$909$		0			0
$910$	−20646.0			−0.752097
$911$	−12061.5	−	20891.1i	−0.438656	−	0.759774i	0.558930	−	0.829215i	$-0.311212\pi$
$911$	−12061.5	−	20891.1i	−0.438656	−	0.759774i	−0.997586	+	0.0694406i	$0.977879\pi$
$912$		0			0
$913$	−8122.50	+	14068.6i	−0.294431	+	0.509969i
$914$	8147.00	−	14111.0i	0.294835	−	0.510668i
$915$		0			0
$916$	2930.00	+	5074.91i	0.105688	+	0.183056i
$917$	46593.0			1.67790
$918$		0			0
$919$	23312.0			0.836770			0.418385	−	0.908270i	$-0.362596\pi$
$919$	23312.0			0.836770			0.418385	+	0.908270i	$0.362596\pi$
$920$	−7020.00	−	12159.0i	−0.251568	−	0.435729i
$921$		0			0
$922$	−8031.00	+	13910.1i	−0.286862	+	0.496860i
$923$	−2886.00	+	4998.70i	−0.102919	+	0.178260i
$924$		0			0
$925$	−3652.00	−	6325.45i	−0.129813	−	0.224843i
$926$	−8566.00			−0.303992
$927$		0			0
$928$	3552.00			0.125647
$929$	9349.50	+	16193.8i	0.330191	+	0.571907i	0.982549	−	0.186004i	$-0.0595536\pi$
$929$	9349.50	+	16193.8i	0.330191	+	0.571907i	−0.652358	+	0.757911i	$0.726220\pi$
$930$		0			0
$931$	8652.00	−	14985.7i	0.304573	−	0.527537i
$932$	5268.00	−	9124.44i	0.185149	−	0.320688i
$933$		0			0
$934$	−5460.00	−	9457.00i	−0.191281	−	0.331309i
$935$	5670.00			0.198320
$936$		0			0
$937$	−31234.0			−1.08898			−0.544488	−	0.838769i	$-0.683276\pi$
$937$	−31234.0			−1.08898			−0.544488	+	0.838769i	$0.683276\pi$
$938$	13051.0	+	22605.0i	0.454297	+	0.786865i
$939$		0			0
$940$	3186.00	−	5518.31i	0.110549	−	0.191476i
$941$	5875.50	−	10176.7i	0.203545	−	0.352550i	−0.746123	−	0.665808i	$-0.768087\pi$
$941$	5875.50	−	10176.7i	0.203545	−	0.352550i	0.949668	+	0.313258i	$0.101420\pi$
$942$		0			0
$943$	25447.5	+	44076.4i	0.878775	+	1.52208i
$944$	−2544.00			−0.0877120
$945$		0			0
$946$	1290.00			0.0443356
$947$	−12943.5	−	22418.8i	−0.444147	−	0.769285i	0.553845	−	0.832620i	$-0.313160\pi$
$947$	−12943.5	−	22418.8i	−0.444147	−	0.769285i	−0.997992	+	0.0633343i	$0.979827\pi$
$948$		0			0
$949$	3367.00	−	5831.82i	0.115171	−	0.199482i
$950$	1232.00	−	2133.89i	0.0420751	−	0.0728762i
$951$		0			0
$952$	−5208.00	−	9020.52i	−0.177303	−	0.307097i
$953$	41130.0			1.39804			0.699020	−	0.715102i	$-0.253620\pi$
$953$	41130.0			1.39804			0.699020	+	0.715102i	$0.253620\pi$
$954$		0			0
$955$	−10449.0			−0.354054
$956$	13830.0	+	23954.3i	0.467881	+	0.810394i
$957$		0			0
$958$	−429.000	+	743.050i	−0.0144680	+	0.0250593i
$959$	41245.5	−	71439.3i	1.38883	−	2.40552i
$960$		0			0
$961$	−6117.00	−	10595.0i	−0.205330	−	0.355643i
$962$	−12284.0			−0.411697
$963$		0			0
$964$	−5956.00			−0.198994
$965$	−16249.5	−	28145.0i	−0.542062	−	0.938879i
$966$		0			0
$967$	14322.5	−	24807.3i	0.476298	−	0.824973i	−0.523333	−	0.852128i	$-0.675312\pi$
$967$	14322.5	−	24807.3i	0.476298	−	0.824973i	0.999631	+	0.0271553i	$0.00864488\pi$
$968$	−4424.00	+	7662.59i	−0.146893	+	0.254427i
$969$		0			0
$970$	−8109.00	−	14045.2i	−0.268417	−	0.464912i
$971$	32256.0			1.06606			0.533030	−	0.846096i	$-0.321053\pi$
$971$	32256.0			1.06606			0.533030	+	0.846096i	$0.321053\pi$
$972$		0			0
$973$	3751.00			0.123588
$974$	−11296.0	−	19565.2i	−0.371609	−	0.643646i
$975$		0			0
$976$	−1528.00	+	2646.57i	−0.0501128	+	0.0867979i
$977$	4507.50	−	7807.22i	0.147603	−	0.255655i	−0.782738	−	0.622351i	$-0.786178\pi$
$977$	4507.50	−	7807.22i	0.147603	−	0.255655i	0.930341	+	0.366696i	$0.119511\pi$
$978$		0			0
$979$	−7875.00	−	13639.9i	−0.257085	−	0.445284i
$980$	22248.0			0.725190
$981$		0			0
$982$	−29346.0			−0.953634
$983$	7942.50	+	13756.8i	0.257707	+	0.446362i	0.965627	−	0.259930i	$-0.0836996\pi$
$983$	7942.50	+	13756.8i	0.257707	+	0.446362i	−0.707920	+	0.706293i	$0.750366\pi$
$984$		0			0
$985$	9207.00	−	15947.0i	0.297827	−	0.515851i
$986$	−4662.00	+	8074.82i	−0.150576	+	0.260806i
$987$		0			0
$988$	−2072.00	−	3588.81i	−0.0667197	−	0.115562i
$989$	8385.00			0.269593
$990$		0			0
$991$	34904.0			1.11883			0.559416	−	0.828887i	$-0.311026\pi$
$991$	34904.0			1.11883			0.559416	+	0.828887i	$0.311026\pi$
$992$	−3280.00	−	5681.13i	−0.104980	−	0.181831i
$993$		0			0
$994$	4836.00	−	8376.20i	0.154314	−	0.267281i
$995$	−13482.0	+	23351.5i	−0.429556	+	0.744013i
$996$		0			0
$997$	17670.5	+	30606.2i	0.561314	+	0.972225i	0.997382	+	0.0723111i	$0.0230374\pi$
$997$	17670.5	+	30606.2i	0.561314	+	0.972225i	−0.436068	+	0.899914i	$0.643629\pi$
$998$	−26878.0			−0.852513
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	54.4.c.a.37.1		2
3.2	odd	2		18.4.c.a.13.1	yes	2
4.3	odd	2		432.4.i.a.145.1		2
9.2	odd	6		18.4.c.a.7.1	✓	2
9.4	even	3		162.4.a.a.1.1		1
9.5	odd	6		162.4.a.d.1.1		1
9.7	even	3	inner	54.4.c.a.19.1		2
12.11	even	2		144.4.i.a.49.1		2
36.7	odd	6		432.4.i.a.289.1		2
36.11	even	6		144.4.i.a.97.1		2
36.23	even	6		1296.4.a.b.1.1		1
36.31	odd	6		1296.4.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.4.c.a.7.1	✓	2	9.2	odd	6
18.4.c.a.13.1	yes	2	3.2	odd	2
54.4.c.a.19.1		2	9.7	even	3	inner
54.4.c.a.37.1		2	1.1	even	1	trivial
144.4.i.a.49.1		2	12.11	even	2
144.4.i.a.97.1		2	36.11	even	6
162.4.a.a.1.1		1	9.4	even	3
162.4.a.d.1.1		1	9.5	odd	6
432.4.i.a.145.1		2	4.3	odd	2
432.4.i.a.289.1		2	36.7	odd	6
1296.4.a.b.1.1		1	36.23	even	6
1296.4.a.g.1.1		1	36.31	odd	6

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:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	54.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-4.50000 + 7.79423i) q^{5} +(15.5000 + 26.8468i) q^{7} -8.00000 q^{8} -18.0000 q^{10} +(-7.50000 - 12.9904i) q^{11} +(18.5000 - 32.0429i) q^{13} +(-31.0000 + 53.6936i) q^{14} +(-8.00000 - 13.8564i) q^{16} +42.0000 q^{17} -28.0000 q^{19} +(-18.0000 - 31.1769i) q^{20} +(15.0000 - 25.9808i) q^{22} +(97.5000 - 168.875i) q^{23} +(22.0000 + 38.1051i) q^{25} +74.0000 q^{26} -124.000 q^{28} +(55.5000 + 96.1288i) q^{29} +(102.500 - 177.535i) q^{31} +(16.0000 - 27.7128i) q^{32} +(42.0000 + 72.7461i) q^{34} -279.000 q^{35} -166.000 q^{37} +(-28.0000 - 48.4974i) q^{38} +(36.0000 - 62.3538i) q^{40} +(-130.500 + 226.033i) q^{41} +(21.5000 + 37.2391i) q^{43} +60.0000 q^{44} +390.000 q^{46} +(88.5000 + 153.286i) q^{47} +(-309.000 + 535.204i) q^{49} +(-44.0000 + 76.2102i) q^{50} +(74.0000 + 128.172i) q^{52} -114.000 q^{53} +135.000 q^{55} +(-124.000 - 214.774i) q^{56} +(-111.000 + 192.258i) q^{58} +(79.5000 - 137.698i) q^{59} +(-95.5000 - 165.411i) q^{61} +410.000 q^{62} +64.0000 q^{64} +(166.500 + 288.386i) q^{65} +(210.500 - 364.597i) q^{67} +(-84.0000 + 145.492i) q^{68} +(-279.000 - 483.242i) q^{70} -156.000 q^{71} +182.000 q^{73} +(-166.000 - 287.520i) q^{74} +(56.0000 - 96.9948i) q^{76} +(232.500 - 402.702i) q^{77} +(-566.500 - 981.207i) q^{79} +144.000 q^{80} -522.000 q^{82} +(-541.500 - 937.906i) q^{83} +(-189.000 + 327.358i) q^{85} +(-43.0000 + 74.4782i) q^{86} +(60.0000 + 103.923i) q^{88} +1050.00 q^{89} +1147.00 q^{91} +(390.000 + 675.500i) q^{92} +(-177.000 + 306.573i) q^{94} +(126.000 - 218.238i) q^{95} +(450.500 + 780.289i) q^{97} -1236.00 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 4 q^{4} - 9 q^{5} + 31 q^{7} - 16 q^{8} - 36 q^{10} - 15 q^{11} + 37 q^{13} - 62 q^{14} - 16 q^{16} + 84 q^{17} - 56 q^{19} - 36 q^{20} + 30 q^{22} + 195 q^{23} + 44 q^{25} + 148 q^{26}+ \cdots - 2472 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 4 * q^4 - 9 * q^5 + 31 * q^7 - 16 * q^8 - 36 * q^10 - 15 * q^11 + 37 * q^13 - 62 * q^14 - 16 * q^16 + 84 * q^17 - 56 * q^19 - 36 * q^20 + 30 * q^22 + 195 * q^23 + 44 * q^25 + 148 * q^26 - 248 * q^28 + 111 * q^29 + 205 * q^31 + 32 * q^32 + 84 * q^34 - 558 * q^35 - 332 * q^37 - 56 * q^38 + 72 * q^40 - 261 * q^41 + 43 * q^43 + 120 * q^44 + 780 * q^46 + 177 * q^47 - 618 * q^49 - 88 * q^50 + 148 * q^52 - 228 * q^53 + 270 * q^55 - 248 * q^56 - 222 * q^58 + 159 * q^59 - 191 * q^61 + 820 * q^62 + 128 * q^64 + 333 * q^65 + 421 * q^67 - 168 * q^68 - 558 * q^70 - 312 * q^71 + 364 * q^73 - 332 * q^74 + 112 * q^76 + 465 * q^77 - 1133 * q^79 + 288 * q^80 - 1044 * q^82 - 1083 * q^83 - 378 * q^85 - 86 * q^86 + 120 * q^88 + 2100 * q^89 + 2294 * q^91 + 780 * q^92 - 354 * q^94 + 252 * q^95 + 901 * q^97 - 2472 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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