LMFDB - Embedded newform 135.4.e.c.91.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [135,4,Mod(46,135)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(135, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

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

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

chi := DirichletCharacter("135.46");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 135 = 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	135.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$7.96525785077$
Analytic rank:	$0$
Dimension:	$14$
Relative dimension:	$7$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} + 48 x^{12} - 60 x^{11} + 1605 x^{10} - 1800 x^{9} + 23232 x^{8} - 2346 x^{7} + \cdots + 82944 $ x^14 - 2x^13 + 48x^12 - 60x^11 + 1605x^10 - 1800x^9 + 23232x^8 - 2346x^7 + 209529x^6 - 55412x^5 + 765088x^4 + 276096x^3 + 1572480x^2 - 345600*x + 82944
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{13} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			91.5
Root			$-0.785104 + 1.35984i$ of defining polynomial
Character	$\chi$	$=$	135.91
Dual form			135.4.e.c.46.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.785104 + 1.35984i) q^{2} +(2.76722 - 4.79297i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(-17.1199 - 29.6525i) q^{7} +21.2519 q^{8} +O(q^{10})$	\(q+(0.785104 + 1.35984i) q^{2} +(2.76722 - 4.79297i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(-17.1199 - 29.6525i) q^{7} +21.2519 q^{8} -7.85104 q^{10} +(-13.4480 - 23.2926i) q^{11} +(9.74401 - 16.8771i) q^{13} +(26.8818 - 46.5607i) q^{14} +(-5.45281 - 9.44454i) q^{16} -29.1232 q^{17} +47.1006 q^{19} +(13.8361 + 23.9648i) q^{20} +(21.1162 - 36.5743i) q^{22} +(56.6083 - 98.0485i) q^{23} +(-12.5000 - 21.6506i) q^{25} +30.6003 q^{26} -189.498 q^{28} +(40.6219 + 70.3593i) q^{29} +(5.41811 - 9.38445i) q^{31} +(93.5697 - 162.067i) q^{32} +(-22.8648 - 39.6029i) q^{34} +171.199 q^{35} +410.002 q^{37} +(36.9789 + 64.0493i) q^{38} +(-53.1298 + 92.0234i) q^{40} +(-221.755 + 384.091i) q^{41} +(-169.748 - 294.012i) q^{43} -148.854 q^{44} +177.774 q^{46} +(118.121 + 204.591i) q^{47} +(-414.681 + 718.249i) q^{49} +(19.6276 - 33.9960i) q^{50} +(-53.9277 - 93.4055i) q^{52} -609.634 q^{53} +134.480 q^{55} +(-363.830 - 630.173i) q^{56} +(-63.7849 + 110.479i) q^{58} +(7.77272 - 13.4627i) q^{59} +(-30.5255 - 52.8717i) q^{61} +17.0151 q^{62} +206.603 q^{64} +(48.7201 + 84.3856i) q^{65} +(-108.445 + 187.832i) q^{67} +(-80.5903 + 139.587i) q^{68} +(134.409 + 232.803i) q^{70} -65.4315 q^{71} +711.811 q^{73} +(321.895 + 557.538i) q^{74} +(130.338 - 225.752i) q^{76} +(-460.457 + 797.534i) q^{77} +(478.876 + 829.438i) q^{79} +54.5281 q^{80} -696.404 q^{82} +(-261.220 - 452.446i) q^{83} +(72.8080 - 126.107i) q^{85} +(266.540 - 461.660i) q^{86} +(-285.796 - 495.012i) q^{88} +1602.24 q^{89} -667.266 q^{91} +(-313.296 - 542.644i) q^{92} +(-185.474 + 321.251i) q^{94} +(-117.751 + 203.952i) q^{95} +(-400.969 - 694.499i) q^{97} -1302.27 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 2 q^{2} - 36 q^{4} - 35 q^{5} - 22 q^{7} + 36 q^{8}+O(q^{10}) $ 14 * q - 2 * q^2 - 36 * q^4 - 35 * q^5 - 22 * q^7 + 36 * q^8	$ 14 q - 2 q^{2} - 36 q^{4} - 35 q^{5} - 22 q^{7} + 36 q^{8} + 20 q^{10} - 23 q^{11} - 96 q^{13} + 21 q^{14} - 324 q^{16} + 322 q^{17} + 558 q^{19} - 180 q^{20} - 311 q^{22} - 96 q^{23} - 175 q^{25} - 716 q^{26} + 674 q^{28} + 296 q^{29} - 244 q^{31} + 314 q^{32} - 125 q^{34} + 220 q^{35} + 808 q^{37} - 305 q^{38} - 90 q^{40} + 47 q^{41} - 525 q^{43} + 110 q^{44} + 1434 q^{46} - 164 q^{47} - 1225 q^{49} - 50 q^{50} - 1682 q^{52} + 1012 q^{53} + 230 q^{55} + 981 q^{56} - 1183 q^{58} + 85 q^{59} - 828 q^{61} - 1572 q^{62} + 4472 q^{64} - 480 q^{65} - 1093 q^{67} - 2473 q^{68} + 105 q^{70} + 656 q^{71} + 4170 q^{73} + 1316 q^{74} - 2789 q^{76} - 24 q^{77} - 2110 q^{79} + 3240 q^{80} - 124 q^{82} - 1290 q^{83} - 805 q^{85} + 2569 q^{86} - 2271 q^{88} - 6096 q^{89} + 6676 q^{91} - 2763 q^{92} + 517 q^{94} - 1395 q^{95} - 1787 q^{97} + 2558 q^{98}+O(q^{100}) $ 14 * q - 2 * q^2 - 36 * q^4 - 35 * q^5 - 22 * q^7 + 36 * q^8 + 20 * q^10 - 23 * q^11 - 96 * q^13 + 21 * q^14 - 324 * q^16 + 322 * q^17 + 558 * q^19 - 180 * q^20 - 311 * q^22 - 96 * q^23 - 175 * q^25 - 716 * q^26 + 674 * q^28 + 296 * q^29 - 244 * q^31 + 314 * q^32 - 125 * q^34 + 220 * q^35 + 808 * q^37 - 305 * q^38 - 90 * q^40 + 47 * q^41 - 525 * q^43 + 110 * q^44 + 1434 * q^46 - 164 * q^47 - 1225 * q^49 - 50 * q^50 - 1682 * q^52 + 1012 * q^53 + 230 * q^55 + 981 * q^56 - 1183 * q^58 + 85 * q^59 - 828 * q^61 - 1572 * q^62 + 4472 * q^64 - 480 * q^65 - 1093 * q^67 - 2473 * q^68 + 105 * q^70 + 656 * q^71 + 4170 * q^73 + 1316 * q^74 - 2789 * q^76 - 24 * q^77 - 2110 * q^79 + 3240 * q^80 - 124 * q^82 - 1290 * q^83 - 805 * q^85 + 2569 * q^86 - 2271 * q^88 - 6096 * q^89 + 6676 * q^91 - 2763 * q^92 + 517 * q^94 - 1395 * q^95 - 1787 * q^97 + 2558 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/135\mathbb{Z}\right)^\times$.

$n$	$56$	$82$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$

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 $

$2$	0.785104	+	1.35984i	0.277576	+	0.480776i	0.970782	−	0.239964i	$-0.0771355\pi$
$2$	0.785104	+	1.35984i	0.277576	+	0.480776i	−0.693206	+	0.720740i	$0.743802\pi$
$3$		0			0
$3$		0			0
$4$	2.76722	−	4.79297i	0.345903	−	0.599121i
$5$	−2.50000	+	4.33013i	−0.223607	+	0.387298i
$5$	−2.50000	+	4.33013i	−0.223607	+	0.387298i
$6$		0			0
$7$	−17.1199	−	29.6525i	−0.924387	−	1.60109i	−0.792544	−	0.609815i	$-0.791244\pi$
$7$	−17.1199	−	29.6525i	−0.924387	−	1.60109i	−0.131844	−	0.991271i	$-0.542090\pi$
$8$	21.2519			0.939210
$9$		0			0
$10$	−7.85104			−0.248272
$11$	−13.4480	−	23.2926i	−0.368611	−	0.638453i	0.620737	−	0.784019i	$-0.286833\pi$
$11$	−13.4480	−	23.2926i	−0.368611	−	0.638453i	−0.989349	+	0.145565i	$0.953500\pi$
$12$		0			0
$13$	9.74401	−	16.8771i	0.207885	−	0.360067i	−0.743163	−	0.669110i	$-0.766675\pi$
$13$	9.74401	−	16.8771i	0.207885	−	0.360067i	0.951048	+	0.309043i	$0.100009\pi$
$14$	26.8818	−	46.5607i	0.513176	−	0.888847i
$15$		0			0
$16$	−5.45281	−	9.44454i	−0.0852001	−	0.147571i
$17$	−29.1232			−0.415495			−0.207747	−	0.978182i	$-0.566613\pi$
$17$	−29.1232			−0.415495			−0.207747	+	0.978182i	$0.566613\pi$
$18$		0			0
$19$	47.1006			0.568717			0.284358	−	0.958718i	$-0.408219\pi$
$19$	47.1006			0.568717			0.284358	+	0.958718i	$0.408219\pi$
$20$	13.8361	+	23.9648i	0.154692	+	0.267935i
$21$		0			0
$22$	21.1162	−	36.5743i	0.204636	−	0.354439i
$23$	56.6083	−	98.0485i	0.513202	−	0.888892i	−0.486680	−	0.873580i	$-0.661792\pi$
$23$	56.6083	−	98.0485i	0.513202	−	0.888892i	0.999883	−	0.0153124i	$-0.00487428\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.100000	−	0.173205i
$26$	30.6003			0.230816
$27$		0			0
$28$	−189.498			−1.27899
$29$	40.6219	+	70.3593i	0.260114	+	0.450531i	0.966272	−	0.257524i	$-0.0829065\pi$
$29$	40.6219	+	70.3593i	0.260114	+	0.450531i	−0.706158	+	0.708054i	$0.749573\pi$
$30$		0			0
$31$	5.41811	−	9.38445i	0.0313910	−	0.0543708i	−0.849903	−	0.526939i	$-0.823340\pi$
$31$	5.41811	−	9.38445i	0.0313910	−	0.0543708i	0.881294	+	0.472568i	$0.156673\pi$
$32$	93.5697	−	162.067i	0.516904	−	0.895304i
$33$		0			0
$34$	−22.8648	−	39.6029i	−0.115332	−	0.199760i
$35$	171.199			0.826797
$36$		0			0
$37$	410.002			1.82173			0.910864	−	0.412706i	$-0.135416\pi$
$37$	410.002			1.82173			0.910864	+	0.412706i	$0.135416\pi$
$38$	36.9789	+	64.0493i	0.157862	+	0.273426i
$39$		0			0
$40$	−53.1298	+	92.0234i	−0.210014	+	0.363755i
$41$	−221.755	+	384.091i	−0.844691	+	1.46305i	0.0411978	+	0.999151i	$0.486883\pi$
$41$	−221.755	+	384.091i	−0.844691	+	1.46305i	−0.885889	+	0.463897i	$0.846451\pi$
$42$		0			0
$43$	−169.748	−	294.012i	−0.602007	−	1.04271i	−0.992517	−	0.122108i	$-0.961035\pi$
$43$	−169.748	−	294.012i	−0.602007	−	1.04271i	0.390510	−	0.920599i	$-0.372299\pi$
$44$	−148.854			−0.510015
$45$		0			0
$46$	177.774			0.569811
$47$	118.121	+	204.591i	0.366589	+	0.634950i	0.989030	−	0.147716i	$-0.0471923\pi$
$47$	118.121	+	204.591i	0.366589	+	0.634950i	−0.622441	+	0.782667i	$0.713859\pi$
$48$		0			0
$49$	−414.681	+	718.249i	−1.20898	+	2.09402i
$50$	19.6276	−	33.9960i	0.0555153	−	0.0961553i
$51$		0			0
$52$	−53.9277	−	93.4055i	−0.143816	−	0.249096i
$53$	−609.634			−1.57999			−0.789997	−	0.613111i	$-0.789918\pi$
$53$	−609.634			−1.57999			−0.789997	+	0.613111i	$0.789918\pi$
$54$		0			0
$55$	134.480			0.329696
$56$	−363.830	−	630.173i	−0.868194	−	1.50376i
$57$		0			0
$58$	−63.7849	+	110.479i	−0.144403	+	0.250113i
$59$	7.77272	−	13.4627i	0.0171512	−	0.0297068i	−0.857322	−	0.514780i	$-0.827874\pi$
$59$	7.77272	−	13.4627i	0.0171512	−	0.0297068i	0.874474	+	0.485073i	$0.161207\pi$
$60$		0			0
$61$	−30.5255	−	52.8717i	−0.0640719	−	0.110976i	0.832210	−	0.554461i	$-0.187075\pi$
$61$	−30.5255	−	52.8717i	−0.0640719	−	0.110976i	−0.896282	+	0.443485i	$0.853742\pi$
$62$	17.0151			0.0348536
$63$		0			0
$64$	206.603			0.403521
$65$	48.7201	+	84.3856i	0.0929689	+	0.161027i
$66$		0			0
$67$	−108.445	+	187.832i	−0.197741	+	0.342497i	−0.947796	−	0.318879i	$-0.896694\pi$
$67$	−108.445	+	187.832i	−0.197741	+	0.342497i	0.750055	+	0.661376i	$0.230027\pi$
$68$	−80.5903	+	139.587i	−0.143721	+	0.248932i
$69$		0			0
$70$	134.409	+	232.803i	0.229499	+	0.397504i
$71$	−65.4315			−0.109370			−0.0546852	−	0.998504i	$-0.517416\pi$
$71$	−65.4315			−0.109370			−0.0546852	+	0.998504i	$0.517416\pi$
$72$		0			0
$73$	711.811			1.14125			0.570624	−	0.821211i	$-0.306701\pi$
$73$	711.811			1.14125			0.570624	+	0.821211i	$0.306701\pi$
$74$	321.895	+	557.538i	0.505669	+	0.875844i
$75$		0			0
$76$	130.338	−	225.752i	0.196721	−	0.340730i
$77$	−460.457	+	797.534i	−0.681479	+	1.18036i
$78$		0			0
$79$	478.876	+	829.438i	0.681997	+	1.18125i	0.974370	+	0.224949i	$0.0722217\pi$
$79$	478.876	+	829.438i	0.681997	+	1.18125i	−0.292373	+	0.956304i	$0.594445\pi$
$80$	54.5281			0.0762053
$81$		0			0
$82$	−696.404			−0.937865
$83$	−261.220	−	452.446i	−0.345453	−	0.598343i	0.639983	−	0.768389i	$-0.278942\pi$
$83$	−261.220	−	452.446i	−0.345453	−	0.598343i	−0.985436	+	0.170047i	$0.945608\pi$
$84$		0			0
$85$	72.8080	−	126.107i	0.0929075	−	0.160920i
$86$	266.540	−	461.660i	0.334206	−	0.578862i
$87$		0			0
$88$	−285.796	−	495.012i	−0.346204	−	0.599642i
$89$	1602.24			1.90828			0.954141	−	0.299359i	$-0.0967728\pi$
$89$	1602.24			1.90828			0.954141	+	0.299359i	$0.0967728\pi$
$90$		0			0
$91$	−667.266			−0.768664
$92$	−313.296	−	542.644i	−0.355036	−	0.614941i
$93$		0			0
$94$	−185.474	+	321.251i	−0.203513	+	0.352494i
$95$	−117.751	+	203.952i	−0.127169	+	0.220263i
$96$		0			0
$97$	−400.969	−	694.499i	−0.419714	−	0.726966i	0.576197	−	0.817311i	$-0.304536\pi$
$97$	−400.969	−	694.499i	−0.419714	−	0.726966i	−0.995910	+	0.0903455i	$0.971203\pi$
$98$	−1302.27			−1.34234
$99$		0			0
$100$	−138.361			−0.138361
$101$	293.065	+	507.604i	0.288724	+	0.500084i	0.973505	−	0.228664i	$-0.0734358\pi$
$101$	293.065	+	507.604i	0.288724	+	0.500084i	−0.684782	+	0.728748i	$0.740102\pi$
$102$		0			0
$103$	410.337	−	710.725i	0.392541	−	0.679901i	−0.600243	−	0.799818i	$-0.704929\pi$
$103$	410.337	−	710.725i	0.392541	−	0.679901i	0.992784	+	0.119917i	$0.0382627\pi$
$104$	207.079	−	358.671i	0.195248	−	0.338179i
$105$		0			0
$106$	−478.626	−	829.005i	−0.438569	−	0.759623i
$107$	1358.33			1.22724			0.613621	−	0.789600i	$-0.289712\pi$
$107$	1358.33			1.22724			0.613621	+	0.789600i	$0.289712\pi$
$108$		0			0
$109$	489.495			0.430139			0.215069	−	0.976599i	$-0.431002\pi$
$109$	489.495			0.430139			0.215069	+	0.976599i	$0.431002\pi$
$110$	105.581	+	182.871i	0.0915158	+	0.158510i
$111$		0			0
$112$	−186.703	+	323.379i	−0.157516	+	0.272825i
$113$	398.832	−	690.797i	0.332026	−	0.575086i	−0.650883	−	0.759178i	$-0.725601\pi$
$113$	398.832	−	690.797i	0.332026	−	0.575086i	0.982909	+	0.184092i	$0.0589345\pi$
$114$		0			0
$115$	283.042	+	490.243i	0.229511	+	0.397525i
$116$	449.640			0.359897
$117$		0			0
$118$	24.4096			0.0190431
$119$	498.586	+	863.576i	0.384078	+	0.665243i
$120$		0			0
$121$	303.803	−	526.202i	0.228251	−	0.395343i
$122$	47.9314	−	83.0196i	0.0355697	−	0.0616085i
$123$		0			0
$124$	−29.9862	−	51.9377i	−0.0217165	−	0.0376141i
$125$	125.000			0.0894427
$126$		0			0
$127$	1728.22			1.20752			0.603759	−	0.797167i	$-0.293669\pi$
$127$	1728.22			1.20752			0.603759	+	0.797167i	$0.293669\pi$
$128$	−586.352	−	1015.59i	−0.404896	−	0.701301i
$129$		0			0
$130$	−76.5007	+	132.503i	−0.0516120	+	0.0893945i
$131$	319.551	−	553.478i	0.213124	−	0.369142i	−0.739566	−	0.673084i	$-0.764969\pi$
$131$	319.551	−	553.478i	0.213124	−	0.369142i	0.952691	+	0.303942i	$0.0983028\pi$
$132$		0			0
$133$	−806.357	−	1396.65i	−0.525715	−	0.910564i
$134$	−340.562			−0.219553
$135$		0			0
$136$	−618.923			−0.390237
$137$	−1417.97	−	2456.00i	−0.884274	−	1.53161i	−0.846544	−	0.532319i	$-0.821321\pi$
$137$	−1417.97	−	2456.00i	−0.884274	−	1.53161i	−0.0377300	−	0.999288i	$-0.512013\pi$
$138$		0			0
$139$	−87.4981	+	151.551i	−0.0533920	+	0.0924777i	−0.891486	−	0.453048i	$-0.850337\pi$
$139$	−87.4981	+	151.551i	−0.0533920	+	0.0924777i	0.838094	+	0.545526i	$0.183670\pi$
$140$	473.745	−	820.551i	0.285991	−	0.495352i
$141$		0			0
$142$	−51.3706	−	88.9764i	−0.0303586	−	0.0525827i
$143$	−524.150			−0.306515
$144$		0			0
$145$	−406.219			−0.232653
$146$	558.846	+	967.949i	0.316784	+	0.548685i
$147$		0			0
$148$	1134.57	−	1965.13i	0.630141	−	1.09144i
$149$	511.766	−	886.405i	0.281379	−	0.487363i	−0.690345	−	0.723480i	$-0.742541\pi$
$149$	511.766	−	886.405i	0.281379	−	0.487363i	0.971725	+	0.236117i	$0.0758748\pi$
$150$		0			0
$151$	436.998	+	756.902i	0.235512	+	0.407920i	0.959421	−	0.281976i	$-0.0909898\pi$
$151$	436.998	+	756.902i	0.235512	+	0.407920i	−0.723909	+	0.689895i	$0.757656\pi$
$152$	1000.98			0.534145
$153$		0			0
$154$	−1446.03			−0.756650
$155$	27.0906	+	46.9222i	0.0140385	+	0.0243154i
$156$		0			0
$157$	−61.7762	+	107.000i	−0.0314031	+	0.0543917i	−0.881300	−	0.472558i	$-0.843331\pi$
$157$	−61.7762	+	107.000i	−0.0314031	+	0.0543917i	0.849897	+	0.526949i	$0.176664\pi$
$158$	−751.936	+	1302.39i	−0.378613	+	0.655776i
$159$		0			0
$160$	467.848	+	810.337i	0.231167	+	0.400392i
$161$	−3876.51			−1.89759
$162$		0			0
$163$	−1410.89			−0.677972			−0.338986	−	0.940791i	$-0.610084\pi$
$163$	−1410.89			−0.677972			−0.338986	+	0.940791i	$0.610084\pi$
$164$	1227.29	+	2125.73i	0.584362	+	1.01214i
$165$		0			0
$166$	410.170	−	710.435i	0.191779	−	0.332172i
$167$	−1013.85	+	1756.04i	−0.469786	+	0.813692i	−0.999403	−	0.0345441i	$-0.989002\pi$
$167$	−1013.85	+	1756.04i	−0.469786	+	0.813692i	0.529618	+	0.848236i	$0.322335\pi$
$168$		0			0
$169$	908.608	+	1573.76i	0.413568	+	0.716320i
$170$	228.648			0.103156
$171$		0			0
$172$	−1878.92			−0.832944
$173$	1351.99	+	2341.71i	0.594160	+	1.02912i	0.993665	+	0.112384i	$0.0358488\pi$
$173$	1351.99	+	2341.71i	0.594160	+	1.02912i	−0.399505	+	0.916731i	$0.630818\pi$
$174$		0			0
$175$	−427.997	+	741.313i	−0.184877	+	0.320217i
$176$	−146.659	+	254.020i	−0.0628115	+	0.108793i
$177$		0			0
$178$	1257.93	+	2178.79i	0.529694	+	0.917456i
$179$	750.694			0.313461			0.156730	−	0.987641i	$-0.449905\pi$
$179$	750.694			0.313461			0.156730	+	0.987641i	$0.449905\pi$
$180$		0			0
$181$	−1134.08			−0.465722			−0.232861	−	0.972510i	$-0.574809\pi$
$181$	−1134.08			−0.465722			−0.232861	+	0.972510i	$0.574809\pi$
$182$	−523.873	−	907.375i	−0.213363	−	0.369556i
$183$		0			0
$184$	1203.03	−	2083.72i	0.482005	−	0.834857i
$185$	−1025.01	+	1775.36i	−0.407351	+	0.705552i
$186$		0			0
$187$	391.649	+	678.355i	0.153156	+	0.265274i
$188$	1307.46			0.507216
$189$		0			0
$190$	−369.789			−0.141196
$191$	−267.627	−	463.544i	−0.101387	−	0.175607i	0.810870	−	0.585227i	$-0.198995\pi$
$191$	−267.627	−	463.544i	−0.101387	−	0.175607i	−0.912256	+	0.409620i	$0.865661\pi$
$192$		0			0
$193$	−1311.77	+	2272.06i	−0.489241	+	0.847391i	−0.999923	−	0.0123786i	$-0.996060\pi$
$193$	−1311.77	+	2272.06i	−0.489241	+	0.847391i	0.510682	+	0.859770i	$0.329393\pi$
$194$	629.605	−	1090.51i	0.233005	−	0.403577i
$195$		0			0
$196$	2295.03	+	3975.11i	0.836382	+	1.44866i
$197$	−2954.08			−1.06837			−0.534186	−	0.845367i	$-0.679382\pi$
$197$	−2954.08			−1.06837			−0.534186	+	0.845367i	$0.679382\pi$
$198$		0			0
$199$	2639.35			0.940192			0.470096	−	0.882615i	$-0.344219\pi$
$199$	2639.35			0.940192			0.470096	+	0.882615i	$0.344219\pi$
$200$	−265.649	−	460.117i	−0.0939210	−	0.162676i
$201$		0			0
$202$	−460.174	+	797.044i	−0.160286	+	0.277623i
$203$	1390.89	−	2409.09i	0.480892	−	0.832929i
$204$		0			0
$205$	−1108.78	−	1920.46i	−0.377757	−	0.654295i
$206$	1288.63			0.435841
$207$		0			0
$208$	−212.529			−0.0708473
$209$	−633.409	−	1097.10i	−0.209635	−	0.363099i
$210$		0			0
$211$	995.327	−	1723.96i	0.324745	−	0.562474i	−0.656716	−	0.754138i	$-0.728055\pi$
$211$	995.327	−	1723.96i	0.324745	−	0.562474i	0.981461	+	0.191664i	$0.0613883\pi$
$212$	−1686.99	+	2921.96i	−0.546524	+	0.946607i
$213$		0			0
$214$	1066.43	+	1847.12i	0.340654	+	0.590029i
$215$	1697.48			0.538452
$216$		0			0
$217$	−371.030			−0.116070
$218$	384.305	+	665.635i	0.119396	+	0.206800i
$219$		0			0
$220$	372.136	−	644.558i	0.114043	−	0.197528i
$221$	−283.777	+	491.516i	−0.0863751	+	0.149606i
$222$		0			0
$223$	337.450	+	584.481i	0.101333	+	0.175515i	0.912234	−	0.409669i	$-0.134356\pi$
$223$	337.450	+	584.481i	0.101333	+	0.175515i	−0.810901	+	0.585184i	$0.801022\pi$
$224$	−6407.61			−1.91128
$225$		0			0
$226$	1252.50			0.368650
$227$	945.926	+	1638.39i	0.276579	+	0.479048i	0.970532	−	0.240972i	$-0.0774661\pi$
$227$	945.926	+	1638.39i	0.276579	+	0.479048i	−0.693954	+	0.720020i	$0.744133\pi$
$228$		0			0
$229$	−1704.26	+	2951.86i	−0.491792	+	0.851810i	−0.999955	−	0.00945146i	$-0.996991\pi$
$229$	−1704.26	+	2951.86i	−0.491792	+	0.851810i	0.508163	+	0.861261i	$0.330325\pi$
$230$	−444.435	+	769.783i	−0.127414	+	0.220687i
$231$		0			0
$232$	863.294	+	1495.27i	0.244302	+	0.423143i
$233$	2841.86			0.799041			0.399521	−	0.916724i	$-0.369177\pi$
$233$	2841.86			0.799041			0.399521	+	0.916724i	$0.369177\pi$
$234$		0			0
$235$	−1181.21			−0.327887
$236$	−43.0177	−	74.5088i	−0.0118653	−	0.0205513i
$237$		0			0
$238$	−782.884	+	1356.00i	−0.213222	+	0.369311i
$239$	809.803	−	1402.62i	0.219171	−	0.379615i	−0.735384	−	0.677651i	$-0.762998\pi$
$239$	809.803	−	1402.62i	0.219171	−	0.379615i	0.954555	+	0.298036i	$0.0963315\pi$
$240$		0			0
$241$	−13.3504	−	23.1235i	−0.00356835	−	0.00618057i	0.864236	−	0.503087i	$-0.167803\pi$
$241$	−13.3504	−	23.1235i	−0.00356835	−	0.00618057i	−0.867804	+	0.496907i	$0.834469\pi$
$242$	954.068			0.253429
$243$		0			0
$244$	−337.883			−0.0886506
$245$	−2073.41	−	3591.25i	−0.540674	−	0.936475i
$246$		0			0
$247$	458.949	−	794.923i	0.118228	−	0.204776i
$248$	115.145	−	199.437i	0.0294828	−	0.0510657i
$249$		0			0
$250$	98.1381	+	169.980i	0.0248272	+	0.0430019i
$251$	427.354			0.107467			0.0537337	−	0.998555i	$-0.482888\pi$
$251$	427.354			0.107467			0.0537337	+	0.998555i	$0.482888\pi$
$252$		0			0
$253$	−3045.07			−0.756689
$254$	1356.83	+	2350.10i	0.335178	+	0.580546i
$255$		0			0
$256$	1747.11	−	3026.08i	0.426540	−	0.738789i
$257$	1920.77	−	3326.88i	0.466204	−	0.807490i	−0.533051	−	0.846083i	$-0.678954\pi$
$257$	1920.77	−	3326.88i	0.466204	−	0.807490i	0.999255	+	0.0385936i	$0.0122878\pi$
$258$		0			0
$259$	−7019.19	−	12157.6i	−1.68398	−	2.91674i
$260$	539.277			0.128633
$261$		0			0
$262$	1003.52			0.236633
$263$	−399.437	−	691.845i	−0.0936514	−	0.162209i	0.815394	−	0.578907i	$-0.196521\pi$
$263$	−399.437	−	691.845i	−0.0936514	−	0.162209i	−0.909045	+	0.416698i	$0.863187\pi$
$264$		0			0
$265$	1524.08	−	2639.79i	0.353297	−	0.611929i
$266$	1266.15	−	2193.03i	0.291852	−	0.505502i
$267$		0			0
$268$	600.181	+	1039.54i	0.136798	+	0.236941i
$269$	−7934.42			−1.79840			−0.899201	−	0.437536i	$-0.855851\pi$
$269$	−7934.42			−1.79840			−0.899201	+	0.437536i	$0.855851\pi$
$270$		0			0
$271$	−2309.09			−0.517592			−0.258796	−	0.965932i	$-0.583326\pi$
$271$	−2309.09			−0.517592			−0.258796	+	0.965932i	$0.583326\pi$
$272$	158.803	+	275.055i	0.0354002	+	0.0613150i
$273$		0			0
$274$	2226.51	−	3856.43i	0.490907	−	0.850276i
$275$	−336.200	+	582.315i	−0.0737223	+	0.127691i
$276$		0			0
$277$	−380.651	−	659.307i	−0.0825672	−	0.143011i	0.821785	−	0.569798i	$-0.192979\pi$
$277$	−380.651	−	659.307i	−0.0825672	−	0.143011i	−0.904352	+	0.426787i	$0.859645\pi$
$278$	−274.781			−0.0592815
$279$		0			0
$280$	3638.30			0.776536
$281$	−643.638	−	1114.81i	−0.136641	−	0.236670i	0.789582	−	0.613645i	$-0.210297\pi$
$281$	−643.638	−	1114.81i	−0.136641	−	0.236670i	−0.926223	+	0.376975i	$0.876964\pi$
$282$		0			0
$283$	1363.25	−	2361.22i	0.286350	−	0.495972i	−0.686586	−	0.727049i	$-0.740891\pi$
$283$	1363.25	−	2361.22i	0.286350	−	0.495972i	0.972936	+	0.231076i	$0.0742248\pi$
$284$	−181.064	+	313.611i	−0.0378315	+	0.0655261i
$285$		0			0
$286$	−411.512	−	712.760i	−0.0850813	−	0.147365i
$287$	15185.7			3.12329
$288$		0			0
$289$	−4064.84			−0.827364
$290$	−318.925	−	552.394i	−0.0645790	−	0.111854i
$291$		0			0
$292$	1969.74	−	3411.69i	0.394761	−	0.683746i
$293$	−48.1869	+	83.4622i	−0.00960789	+	0.0166414i	−0.870789	−	0.491656i	$-0.836392\pi$
$293$	−48.1869	+	83.4622i	−0.00960789	+	0.0166414i	0.861181	+	0.508298i	$0.169725\pi$
$294$		0			0
$295$	38.8636	+	67.3137i	0.00767026	+	0.0132853i
$296$	8713.33			1.71099
$297$		0			0
$298$	1607.16			0.312417
$299$	−1103.18	−	1910.77i	−0.213374	−	0.369575i
$300$		0			0
$301$	−5812.13	+	10066.9i	−1.11298	+	1.92773i
$302$	−686.178	+	1188.50i	−0.130745	+	0.226458i
$303$		0			0
$304$	−256.831	−	444.844i	−0.0484547	−	0.0839261i
$305$	305.255			0.0573076
$306$		0			0
$307$	5258.02			0.977495			0.488747	−	0.872425i	$-0.337454\pi$
$307$	5258.02			0.977495			0.488747	+	0.872425i	$0.337454\pi$
$308$	2548.37	+	4413.91i	0.471451	+	0.816577i
$309$		0			0
$310$	−42.5378	+	73.6777i	−0.00779351	+	0.0134988i
$311$	−2615.66	+	4530.45i	−0.476914	+	0.826039i	−0.999650	−	0.0264555i	$-0.991578\pi$
$311$	−2615.66	+	4530.45i	−0.476914	+	0.826039i	0.522736	+	0.852495i	$0.324911\pi$
$312$		0			0
$313$	1996.33	+	3457.74i	0.360509	+	0.624419i	0.988045	−	0.154168i	$-0.0492698\pi$
$313$	1996.33	+	3457.74i	0.360509	+	0.624419i	−0.627536	+	0.778588i	$0.715936\pi$
$314$	−194.003			−0.0348670
$315$		0			0
$316$	5300.63			0.943619
$317$	−2366.79	−	4099.40i	−0.419344	−	0.726325i	0.576529	−	0.817076i	$-0.304407\pi$
$317$	−2366.79	−	4099.40i	−0.419344	−	0.726325i	−0.995874	+	0.0907509i	$0.971073\pi$
$318$		0			0
$319$	1092.57	−	1892.38i	0.191762	−	0.332141i
$320$	−516.507	+	894.617i	−0.0902301	+	0.156283i
$321$		0			0
$322$	−3043.47	−	5271.44i	−0.526726	−	0.912317i
$323$	−1371.72			−0.236299
$324$		0			0
$325$	−487.201			−0.0831539
$326$	−1107.70	−	1918.59i	−0.188189	−	0.325953i
$327$		0			0
$328$	−4712.72	+	8162.67i	−0.793343	+	1.37411i
$329$	4044.43	−	7005.15i	0.677740	−	1.17388i
$330$		0			0
$331$	−4684.92	−	8114.52i	−0.777965	−	1.34747i	−0.933113	−	0.359583i	$-0.882919\pi$
$331$	−4684.92	−	8114.52i	−0.777965	−	1.34747i	0.155148	−	0.987891i	$-0.450415\pi$
$332$	−2891.42			−0.477973
$333$		0			0
$334$	−3183.92			−0.521605
$335$	−542.223	−	939.159i	−0.0884323	−	0.153169i
$336$		0			0
$337$	−2724.41	+	4718.83i	−0.440381	+	0.762762i	−0.997718	−	0.0675246i	$-0.978490\pi$
$337$	−2724.41	+	4718.83i	−0.440381	+	0.762762i	0.557337	+	0.830287i	$0.311823\pi$
$338$	−1426.71	+	2471.13i	−0.229593	+	0.397667i
$339$		0			0
$340$	−402.952	−	697.933i	−0.0642739	−	0.111326i
$341$	−291.451			−0.0462843
$342$		0			0
$343$	16653.0			2.62150
$344$	−3607.47	−	6248.31i	−0.565411	−	0.979321i
$345$		0			0
$346$	−2122.90	+	3676.98i	−0.329850	+	0.571316i
$347$	−1948.77	+	3375.38i	−0.301486	+	0.522189i	−0.976473	−	0.215641i	$-0.930816\pi$
$347$	−1948.77	+	3375.38i	−0.301486	+	0.522189i	0.674987	+	0.737830i	$0.264149\pi$
$348$		0			0
$349$	3129.53	+	5420.51i	0.480000	+	0.831384i	0.999737	−	0.0229422i	$-0.00730337\pi$
$349$	3129.53	+	5420.51i	0.480000	+	0.831384i	−0.519737	+	0.854326i	$0.673970\pi$
$350$	−1344.09			−0.205270
$351$		0			0
$352$	−5033.30			−0.762147
$353$	171.433	+	296.931i	0.0258483	+	0.0447706i	0.878660	−	0.477448i	$-0.158438\pi$
$353$	171.433	+	296.931i	0.0258483	+	0.0447706i	−0.852812	+	0.522218i	$0.825105\pi$
$354$		0			0
$355$	163.579	−	283.327i	0.0244559	−	0.0423589i
$356$	4433.75	−	7679.48i	0.660080	−	1.14329i
$357$		0			0
$358$	589.373	+	1020.82i	0.0870093	+	0.150705i
$359$	−10448.9			−1.53613			−0.768066	−	0.640371i	$-0.778781\pi$
$359$	−10448.9			−1.53613			−0.768066	+	0.640371i	$0.778781\pi$
$360$		0			0
$361$	−4640.53			−0.676561
$362$	−890.373	−	1542.17i	−0.129273	−	0.223908i
$363$		0			0
$364$	−1846.47	+	3198.18i	−0.265883	+	0.460523i
$365$	−1779.53	+	3082.23i	−0.255191	+	0.442004i
$366$		0			0
$367$	2586.87	+	4480.58i	0.367938	+	0.637288i	0.989243	−	0.146281i	$-0.0467305\pi$
$367$	2586.87	+	4480.58i	0.367938	+	0.637288i	−0.621305	+	0.783569i	$0.713397\pi$
$368$	−1234.70			−0.174900
$369$		0			0
$370$	−3218.95			−0.452284
$371$	10436.9	+	18077.2i	1.46053	+	2.52970i
$372$		0			0
$373$	−1050.53	+	1819.58i	−0.145830	+	0.252585i	−0.929682	−	0.368362i	$-0.879919\pi$
$373$	−1050.53	+	1819.58i	−0.145830	+	0.252585i	0.783852	+	0.620947i	$0.213252\pi$
$374$	−614.970	+	1065.16i	−0.0850250	+	0.147268i
$375$		0			0
$376$	2510.29	+	4347.95i	0.344304	+	0.596352i
$377$	1583.28			0.216295
$378$		0			0
$379$	11242.6			1.52374			0.761868	−	0.647732i	$-0.224282\pi$
$379$	11242.6			1.52374			0.761868	+	0.647732i	$0.224282\pi$
$380$	651.689	+	1128.76i	0.0879762	+	0.152379i
$381$		0			0
$382$	420.231	−	727.862i	0.0562851	−	0.0974886i
$383$	3397.44	−	5884.53i	0.453266	−	0.785080i	−0.545321	−	0.838227i	$-0.683592\pi$
$383$	3397.44	−	5884.53i	0.453266	−	0.785080i	0.998587	+	0.0531478i	$0.0169255\pi$
$384$		0			0
$385$	−2302.28	−	3987.67i	−0.304767	−	0.527871i
$386$	−4119.52			−0.543207
$387$		0			0
$388$	−4438.28			−0.580721
$389$	3524.70	+	6104.96i	0.459407	+	0.795717i	0.998930	−	0.0462544i	$-0.0147285\pi$
$389$	3524.70	+	6104.96i	0.459407	+	0.795717i	−0.539522	+	0.841971i	$0.681395\pi$
$390$		0			0
$391$	−1648.62	+	2855.49i	−0.213233	+	0.369330i
$392$	−8812.77	+	15264.2i	−1.13549	+	1.96673i
$393$		0			0
$394$	−2319.26	−	4017.08i	−0.296555	−	0.513648i
$395$	−4788.76			−0.609997
$396$		0			0
$397$	3011.36			0.380694			0.190347	−	0.981717i	$-0.439039\pi$
$397$	3011.36			0.380694			0.190347	+	0.981717i	$0.439039\pi$
$398$	2072.16	+	3589.09i	0.260975	+	0.452022i
$399$		0			0
$400$	−136.320	+	236.114i	−0.0170400	+	0.0295142i
$401$	−2084.07	+	3609.72i	−0.259535	+	0.449528i	−0.966118	−	0.258103i	$-0.916903\pi$
$401$	−2084.07	+	3609.72i	−0.259535	+	0.449528i	0.706582	+	0.707631i	$0.250236\pi$
$402$		0			0
$403$	−105.588	−	182.884i	−0.0130514	−	0.0226058i
$404$	3243.91			0.399481
$405$		0			0
$406$	4367.96			0.533937
$407$	−5513.71	−	9550.03i	−0.671510	−	1.16309i
$408$		0			0
$409$	6302.68	−	10916.6i	0.761974	−	1.31978i	−0.179858	−	0.983693i	$-0.557564\pi$
$409$	6302.68	−	10916.6i	0.761974	−	1.31978i	0.941832	−	0.336085i	$-0.109103\pi$
$410$	1741.01	−	3015.52i	0.209713	−	0.363234i
$411$		0			0
$412$	−2270.99	−	3933.47i	−0.271562	−	0.470359i
$413$	−532.272			−0.0634175
$414$		0			0
$415$	2612.20			0.308983
$416$	−1823.49	−	3158.37i	−0.214913	−	0.372240i
$417$		0			0
$418$	994.584	−	1722.67i	0.116380	−	0.201575i
$419$	−2572.24	+	4455.24i	−0.299909	+	0.519458i	−0.976115	−	0.217255i	$-0.930290\pi$
$419$	−2572.24	+	4455.24i	−0.299909	+	0.519458i	0.676206	+	0.736713i	$0.263623\pi$
$420$		0			0
$421$	1685.03	+	2918.56i	0.195068	+	0.337867i	0.946923	−	0.321461i	$-0.104174\pi$
$421$	1685.03	+	2918.56i	0.195068	+	0.337867i	−0.751855	+	0.659329i	$0.770841\pi$
$422$	3125.74			0.360566
$423$		0			0
$424$	−12955.9			−1.48395
$425$	364.040	+	630.536i	0.0415495	+	0.0719658i
$426$		0			0
$427$	−1045.19	+	1810.31i	−0.118454	+	0.205169i
$428$	3758.81	−	6510.45i	0.424507	−	0.735267i
$429$		0			0
$430$	1332.70	+	2308.30i	0.149461	+	0.258875i
$431$	−14948.3			−1.67061			−0.835306	−	0.549785i	$-0.814710\pi$
$431$	−14948.3			−1.67061			−0.835306	+	0.549785i	$0.814710\pi$
$432$		0			0
$433$	−14011.1			−1.55504			−0.777520	−	0.628858i	$-0.783523\pi$
$433$	−14011.1			−1.55504			−0.777520	+	0.628858i	$0.783523\pi$
$434$	−291.297	−	504.542i	−0.0322182	−	0.0558036i
$435$		0			0
$436$	1354.54	−	2346.13i	0.148786	−	0.257705i
$437$	2666.29	−	4618.14i	0.291867	−	0.505528i
$438$		0			0
$439$	2391.10	+	4141.50i	0.259956	+	0.450257i	0.966230	−	0.257681i	$-0.0829585\pi$
$439$	2391.10	+	4141.50i	0.259956	+	0.450257i	−0.706274	+	0.707939i	$0.749625\pi$
$440$	2857.96			0.309654
$441$		0			0
$442$	−891.178			−0.0959027
$443$	2387.01	+	4134.42i	0.256005	+	0.443413i	0.965168	−	0.261631i	$-0.0842604\pi$
$443$	2387.01	+	4134.42i	0.256005	+	0.443413i	−0.709163	+	0.705044i	$0.750927\pi$
$444$		0			0
$445$	−4005.60	+	6937.90i	−0.426705	+	0.739074i
$446$	−529.867	+	917.757i	−0.0562555	+	0.0974374i
$447$		0			0
$448$	−3537.02	−	6126.30i	−0.373010	−	0.646072i
$449$	855.812			0.0899516			0.0449758	−	0.998988i	$-0.485679\pi$
$449$	855.812			0.0899516			0.0449758	+	0.998988i	$0.485679\pi$
$450$		0			0
$451$	11928.7			1.24545
$452$	−2207.31	−	3823.18i	−0.229697	−	0.397848i
$453$		0			0
$454$	−1485.30	+	2572.62i	−0.153543	+	0.265945i
$455$	1668.16	−	2889.35i	0.171879	−	0.297702i
$456$		0			0
$457$	6708.99	+	11620.3i	0.686725	+	1.18944i	0.972891	+	0.231263i	$0.0742857\pi$
$457$	6708.99	+	11620.3i	0.686725	+	1.18944i	−0.286166	+	0.958180i	$0.592381\pi$
$458$	−5352.08			−0.546040
$459$		0			0
$460$	3132.96			0.317554
$461$	−7942.99	−	13757.7i	−0.802477	−	1.38993i	−0.917981	−	0.396625i	$-0.870181\pi$
$461$	−7942.99	−	13757.7i	−0.802477	−	1.38993i	0.115503	−	0.993307i	$-0.463152\pi$
$462$		0			0
$463$	−3280.69	+	5682.32i	−0.329302	+	0.570367i	−0.982373	−	0.186929i	$-0.940147\pi$
$463$	−3280.69	+	5682.32i	−0.329302	+	0.570367i	0.653072	+	0.757296i	$0.273480\pi$
$464$	443.007	−	767.311i	0.0443235	−	0.0767705i
$465$		0			0
$466$	2231.16	+	3864.48i	0.221795	+	0.384160i
$467$	6792.37			0.673048			0.336524	−	0.941675i	$-0.390749\pi$
$467$	6792.37			0.673048			0.336524	+	0.941675i	$0.390749\pi$
$468$		0			0
$469$	7426.25			0.731156
$470$	−927.371	−	1606.25i	−0.0910137	−	0.157640i
$471$		0			0
$472$	165.185	−	286.109i	0.0161086	−	0.0279009i
$473$	−4565.54	+	7907.74i	−0.443813	+	0.768707i
$474$		0			0
$475$	−588.757	−	1019.76i	−0.0568717	−	0.0985046i
$476$	5518.79			0.531415
$477$		0			0
$478$	2543.12			0.243346
$479$	8826.24	+	15287.5i	0.841923	+	1.45825i	0.888267	+	0.459328i	$0.151910\pi$
$479$	8826.24	+	15287.5i	0.841923	+	1.45825i	−0.0463438	+	0.998926i	$0.514757\pi$
$480$		0			0
$481$	3995.07	−	6919.66i	0.378710	−	0.655945i
$482$	20.9629	−	36.3088i	0.00198098	−	0.00343116i
$483$		0			0
$484$	−1681.38	−	2912.23i	−0.157906	−	0.273501i
$485$	4009.69			0.375403
$486$		0			0
$487$	−5318.16			−0.494844			−0.247422	−	0.968908i	$-0.579583\pi$
$487$	−5318.16			−0.494844			−0.247422	+	0.968908i	$0.579583\pi$
$488$	−648.724	−	1123.62i	−0.0601770	−	0.104230i
$489$		0			0
$490$	3255.68	−	5639.01i	0.300157	−	0.519886i
$491$	5434.98	−	9413.66i	0.499546	−	0.865239i	−0.500454	−	0.865763i	$-0.666833\pi$
$491$	5434.98	−	9413.66i	0.499546	−	0.865239i	1.00000		0.000523887i	$0.000166759\pi$
$492$		0			0
$493$	−1183.04	−	2049.09i	−0.108076	−	0.187193i
$494$	1441.29			0.131269
$495$		0			0
$496$	−118.176			−0.0106981
$497$	1120.18	+	1940.21i	0.101101	+	0.175111i
$498$		0			0
$499$	−5185.25	+	8981.12i	−0.465178	+	0.805711i	−0.999210	−	0.0397529i	$-0.987343\pi$
$499$	−5185.25	+	8981.12i	−0.465178	+	0.805711i	0.534032	+	0.845464i	$0.320676\pi$
$500$	345.903	−	599.121i	0.0309385	−	0.0535870i
$501$		0			0
$502$	335.517	+	581.133i	0.0298304	+	0.0516678i
$503$	12396.4			1.09886			0.549432	−	0.835538i	$-0.314844\pi$
$503$	12396.4			1.09886			0.549432	+	0.835538i	$0.314844\pi$
$504$		0			0
$505$	−2930.65			−0.258242
$506$	−2390.70	−	4140.82i	−0.210039	−	0.363798i
$507$		0			0
$508$	4782.37	−	8283.30i	0.417684	−	0.723449i
$509$	2780.52	−	4816.00i	0.242130	−	0.419382i	−0.719190	−	0.694813i	$-0.755487\pi$
$509$	2780.52	−	4816.00i	0.242130	−	0.419382i	0.961321	+	0.275431i	$0.0888205\pi$
$510$		0			0
$511$	−12186.1	−	21107.0i	−1.05496	−	1.82724i
$512$	−3894.99			−0.336203
$513$		0			0
$514$	6032.03			0.517629
$515$	2051.69	+	3553.63i	0.175550	+	0.304061i
$516$		0			0
$517$	3176.97	−	5502.68i	0.270258	−	0.468100i
$518$	11021.6	−	19090.0i	0.934868	−	1.61924i
$519$		0			0
$520$	1035.39	+	1793.36i	0.0873174	+	0.151238i
$521$	2613.39			0.219760			0.109880	−	0.993945i	$-0.464953\pi$
$521$	2613.39			0.219760			0.109880	+	0.993945i	$0.464953\pi$
$522$		0			0
$523$	−2927.73			−0.244781			−0.122391	−	0.992482i	$-0.539056\pi$
$523$	−2927.73			−0.244781			−0.122391	+	0.992482i	$0.539056\pi$
$524$	−1768.53	−	3063.19i	−0.147440	−	0.255374i
$525$		0			0
$526$	627.199	−	1086.34i	0.0519908	−	0.0900508i
$527$	−157.793	+	273.305i	−0.0130428	+	0.0225908i
$528$		0			0
$529$	−325.506	−	563.793i	−0.0267532	−	0.0463379i
$530$	4786.26			0.392268
$531$		0			0
$532$	−8925.48			−0.727384
$533$	4321.57	+	7485.18i	0.351197	+	0.608291i
$534$		0			0
$535$	−3395.83	+	5881.75i	−0.274420	+	0.475309i
$536$	−2304.66	+	3991.78i	−0.185720	+	0.321677i
$537$		0			0
$538$	−6229.35	−	10789.5i	−0.499194	−	0.864629i
$539$	22306.5			1.78258
$540$		0			0
$541$	−4023.02			−0.319710			−0.159855	−	0.987140i	$-0.551103\pi$
$541$	−4023.02			−0.319710			−0.159855	+	0.987140i	$0.551103\pi$
$542$	−1812.88	−	3140.00i	−0.143671	−	0.248846i
$543$		0			0
$544$	−2725.05	+	4719.92i	−0.214771	+	0.371994i
$545$	−1223.74	+	2119.57i	−0.0961819	+	0.166592i
$546$		0			0
$547$	−1652.77	−	2862.68i	−0.129191	−	0.223765i	0.794172	−	0.607692i	$-0.207905\pi$
$547$	−1652.77	−	2862.68i	−0.129191	−	0.223765i	−0.923363	+	0.383927i	$0.874571\pi$
$548$	−15695.4			−1.22349
$549$		0			0
$550$	−1055.81			−0.0818542
$551$	1913.32	+	3313.96i	0.147931	+	0.256224i
$552$		0			0
$553$	16396.6	−	28399.8i	1.26086	−	2.18387i
$554$	597.702	−	1035.25i	0.0458374	−	0.0793927i
$555$		0			0
$556$	484.253	+	838.752i	0.0369369	+	0.0639766i
$557$	−6472.87			−0.492396			−0.246198	−	0.969220i	$-0.579181\pi$
$557$	−6472.87			−0.492396			−0.246198	+	0.969220i	$0.579181\pi$
$558$		0			0
$559$	−6616.10			−0.500593
$560$	−933.515	−	1616.90i	−0.0704432	−	0.122011i
$561$		0			0
$562$	1010.65	−	1750.49i	0.0758569	−	0.131388i
$563$	−10277.6	+	17801.4i	−0.769362	+	1.33257i	0.168547	+	0.985694i	$0.446093\pi$
$563$	−10277.6	+	17801.4i	−0.769362	+	1.33257i	−0.937909	+	0.346881i	$0.887241\pi$
$564$		0			0
$565$	1994.16	+	3453.98i	0.148487	+	0.257186i
$566$	4281.18			0.317936
$567$		0			0
$568$	−1390.54			−0.102722
$569$	635.311	+	1100.39i	0.0468078	+	0.0810735i	0.888480	−	0.458915i	$-0.151762\pi$
$569$	635.311	+	1100.39i	0.0468078	+	0.0810735i	−0.841672	+	0.539989i	$0.818429\pi$
$570$		0			0
$571$	8376.66	−	14508.8i	0.613927	−	1.06335i	−0.376645	−	0.926358i	$-0.622922\pi$
$571$	8376.66	−	14508.8i	0.613927	−	1.06335i	0.990572	−	0.136995i	$-0.0437444\pi$
$572$	−1450.44	+	2512.23i	−0.106024	+	0.183639i
$573$		0			0
$574$	11922.4	+	20650.1i	0.866951	+	1.50160i
$575$	−2830.42			−0.205281
$576$		0			0
$577$	−7800.01			−0.562770			−0.281385	−	0.959595i	$-0.590794\pi$
$577$	−7800.01			−0.562770			−0.281385	+	0.959595i	$0.590794\pi$
$578$	−3191.32	−	5527.53i	−0.229657	−	0.397777i
$579$		0			0
$580$	−1124.10	+	1947.00i	−0.0804753	+	0.139387i
$581$	−8944.12	+	15491.7i	−0.638665	+	1.10620i
$582$		0			0
$583$	8198.35	+	14200.0i	0.582403	+	1.00875i
$584$	15127.3			1.07187
$585$		0			0
$586$	−151.327			−0.0106677
$587$	−1184.62	−	2051.83i	−0.0832958	−	0.144273i	0.821368	−	0.570399i	$-0.193211\pi$
$587$	−1184.62	−	2051.83i	−0.0832958	−	0.144273i	−0.904664	+	0.426126i	$0.859878\pi$
$588$		0			0
$589$	255.196	−	442.013i	0.0178526	−	0.0309216i
$590$	−61.0240	+	105.697i	−0.00425816	+	0.00737536i
$591$		0			0
$592$	−2235.66	−	3872.28i	−0.155212	−	0.268834i
$593$	−11974.3			−0.829218			−0.414609	−	0.910000i	$-0.636082\pi$
$593$	−11974.3			−0.829218			−0.414609	+	0.910000i	$0.636082\pi$
$594$		0			0
$595$	−4985.86			−0.343530
$596$	−2832.34	−	4905.76i	−0.194660	−	0.337160i
$597$		0			0
$598$	1732.23	−	3000.31i	0.118455	−	0.205170i
$599$	11099.7	−	19225.3i	0.757134	−	1.31139i	−0.187173	−	0.982327i	$-0.559932\pi$
$599$	11099.7	−	19225.3i	0.757134	−	1.31139i	0.944307	−	0.329067i	$-0.106734\pi$
$600$		0			0
$601$	5749.47	+	9958.38i	0.390226	+	0.675891i	0.992479	−	0.122414i	$-0.0390635\pi$
$601$	5749.47	+	9958.38i	0.390226	+	0.675891i	−0.602253	+	0.798305i	$0.705730\pi$
$602$	−18252.5			−1.23574
$603$		0			0
$604$	4837.08			0.325858
$605$	1519.01	+	2631.01i	0.102077	+	0.176803i
$606$		0			0
$607$	7636.51	−	13226.8i	0.510636	−	0.884448i	−0.489288	−	0.872122i	$-0.662743\pi$
$607$	7636.51	−	13226.8i	0.510636	−	0.884448i	0.999924	−	0.0123257i	$-0.00392350\pi$
$608$	4407.19	−	7633.47i	0.293972	−	0.509175i
$609$		0			0
$610$	239.657	+	415.098i	0.0159072	+	0.0275522i
$611$	4603.88			0.304833
$612$		0			0
$613$	−16440.8			−1.08326			−0.541630	−	0.840617i	$-0.682192\pi$
$613$	−16440.8			−1.08326			−0.541630	+	0.840617i	$0.682192\pi$
$614$	4128.09	+	7150.07i	0.271329	+	0.469956i
$615$		0			0
$616$	−9785.58	+	16949.1i	−0.640052	+	1.10860i
$617$	4679.57	−	8105.25i	0.305336	−	0.528857i	−0.672000	−	0.740551i	$-0.734565\pi$
$617$	4679.57	−	8105.25i	0.305336	−	0.528857i	0.977336	+	0.211694i	$0.0678979\pi$
$618$		0			0
$619$	12742.3	+	22070.3i	0.827392	+	1.43308i	0.900078	+	0.435729i	$0.143510\pi$
$619$	12742.3	+	22070.3i	0.827392	+	1.43308i	−0.0726861	+	0.997355i	$0.523157\pi$
$620$	299.862			0.0194238
$621$		0			0
$622$	−8214.25			−0.529520
$623$	−27430.2	−	47510.4i	−1.76399	−	3.05532i
$624$		0			0
$625$	−312.500	+	541.266i	−0.0200000	+	0.0346410i
$626$	−3134.65	+	5429.38i	−0.200137	+	0.346648i
$627$		0			0
$628$	341.897	+	592.183i	0.0217248	+	0.0376285i
$629$	−11940.6			−0.756919
$630$		0			0
$631$	−21733.8			−1.37117			−0.685585	−	0.727993i	$-0.740454\pi$
$631$	−21733.8			−1.37117			−0.685585	+	0.727993i	$0.740454\pi$
$632$	10177.0	+	17627.1i	0.640539	+	1.10945i
$633$		0			0
$634$	3716.35	−	6436.91i	0.232800	−	0.403222i
$635$	−4320.55	+	7483.41i	−0.270009	+	0.467669i
$636$		0			0
$637$	8081.32	+	13997.3i	0.502659	+	0.870631i
$638$	3431.12			0.212914
$639$		0			0
$640$	5863.52			0.362150
$641$	−10162.4	−	17601.8i	−0.626195	−	1.08460i	−0.988308	−	0.152468i	$-0.951278\pi$
$641$	−10162.4	−	17601.8i	−0.626195	−	1.08460i	0.362113	−	0.932134i	$-0.382055\pi$
$642$		0			0
$643$	−3363.72	+	5826.13i	−0.206302	+	0.357325i	−0.950547	−	0.310581i	$-0.899476\pi$
$643$	−3363.72	+	5826.13i	−0.206302	+	0.357325i	0.744245	+	0.667907i	$0.232810\pi$
$644$	−10727.2	+	18580.0i	−0.656382	+	1.13689i
$645$		0			0
$646$	−1076.94	−	1865.32i	−0.0655910	−	0.113607i
$647$	−6724.69			−0.408616			−0.204308	−	0.978907i	$-0.565494\pi$
$647$	−6724.69			−0.408616			−0.204308	+	0.978907i	$0.565494\pi$
$648$		0			0
$649$	−418.110			−0.0252885
$650$	−382.503	−	662.515i	−0.0230816	−	0.0399784i
$651$		0			0
$652$	−3904.25	+	6762.35i	−0.234512	+	0.406187i
$653$	−15398.2	+	26670.4i	−0.922782	+	1.59831i	−0.127693	+	0.991814i	$0.540757\pi$
$653$	−15398.2	+	26670.4i	−0.922782	+	1.59831i	−0.795089	+	0.606492i	$0.792576\pi$
$654$		0			0
$655$	1597.75	+	2767.39i	0.0953120	+	0.165085i
$656$	4836.75			0.287871
$657$		0			0
$658$	12701.2			0.752498
$659$	8267.16	+	14319.1i	0.488684	+	0.846426i	0.999915	−	0.0130174i	$-0.00414369\pi$
$659$	8267.16	+	14319.1i	0.488684	+	0.846426i	−0.511231	+	0.859443i	$0.670810\pi$
$660$		0			0
$661$	−968.527	+	1677.54i	−0.0569914	+	0.0987121i	−0.893114	−	0.449831i	$-0.851484\pi$
$661$	−968.527	+	1677.54i	−0.0569914	+	0.0987121i	0.836122	+	0.548543i	$0.184817\pi$
$662$	7356.30	−	12741.5i	0.431889	−	0.748054i
$663$		0			0
$664$	−5551.42	−	9615.35i	−0.324453	−	0.561970i
$665$	8063.57			0.470213
$666$		0			0
$667$	9198.16			0.533964
$668$	5611.10	+	9718.72i	0.325000	+	0.562917i
$669$		0			0
$670$	851.404	−	1474.68i	0.0490935	−	0.0850324i
$671$	−821.013	+	1422.04i	−0.0472352	+	0.0818138i
$672$		0			0
$673$	230.294	+	398.881i	0.0131905	+	0.0228465i	0.872545	−	0.488533i	$-0.162468\pi$
$673$	230.294	+	398.881i	0.0131905	+	0.0228465i	−0.859355	+	0.511380i	$0.829135\pi$
$674$	−8555.80			−0.488957
$675$		0			0
$676$	10057.3			0.572217
$677$	14529.8	+	25166.4i	0.824854	+	1.42869i	0.902031	+	0.431671i	$0.142076\pi$
$677$	14529.8	+	25166.4i	0.824854	+	1.42869i	−0.0771775	+	0.997017i	$0.524591\pi$
$678$		0			0
$679$	−13729.1	+	23779.5i	−0.775956	+	1.34400i
$680$	1547.31	−	2680.02i	0.0872597	−	0.151138i
$681$		0			0
$682$	−228.820	−	396.327i	−0.0128474	−	0.0222524i
$683$	−14366.9			−0.804881			−0.402440	−	0.915446i	$-0.631838\pi$
$683$	−14366.9			−0.804881			−0.402440	+	0.915446i	$0.631838\pi$
$684$		0			0
$685$	14179.7			0.790918
$686$	13074.3	+	22645.4i	0.727667	+	1.26036i
$687$		0			0
$688$	−1851.21	+	3206.38i	−0.102582	+	0.177678i
$689$	−5940.28	+	10288.9i	−0.328457	+	0.568904i
$690$		0			0
$691$	6353.97	+	11005.4i	0.349807	+	0.605883i	0.986215	−	0.165469i	$-0.0529139\pi$
$691$	6353.97	+	11005.4i	0.349807	+	0.605883i	−0.636408	+	0.771352i	$0.719581\pi$
$692$	14965.0			0.822087
$693$		0			0
$694$	−6119.97			−0.334742
$695$	−437.491	−	757.756i	−0.0238776	−	0.0413573i
$696$		0			0
$697$	6458.22	−	11186.0i	0.350965	−	0.607889i
$698$	−4914.02	+	8511.33i	−0.266473	+	0.461545i
$699$		0			0
$700$	2368.73	+	4102.76i	0.127899	+	0.221528i
$701$	−5959.06			−0.321071			−0.160535	−	0.987030i	$-0.551322\pi$
$701$	−5959.06			−0.321071			−0.160535	+	0.987030i	$0.551322\pi$
$702$		0			0
$703$	19311.4			1.03605
$704$	−2778.40	−	4812.32i	−0.148743	−	0.257630i
$705$		0			0
$706$	−269.186	+	466.243i	−0.0143498	+	0.0248545i
$707$	10034.5	−	17380.2i	0.533785	−	0.924542i
$708$		0			0
$709$	−225.912	−	391.292i	−0.0119666	−	0.0207268i	0.859980	−	0.510328i	$-0.170476\pi$
$709$	−225.912	−	391.292i	−0.0119666	−	0.0207268i	−0.871947	+	0.489601i	$0.837143\pi$
$710$	513.706			0.0271536
$711$		0			0
$712$	34050.6			1.79228
$713$	−613.421	−	1062.48i	−0.0322199	−	0.0558065i
$714$		0			0
$715$	1310.37	−	2269.64i	0.0685388	−	0.118713i
$716$	2077.34	−	3598.05i	0.108427	−	0.187801i
$717$		0			0
$718$	−8203.47	−	14208.8i	−0.426394	−	0.738536i
$719$	13488.4			0.699627			0.349814	−	0.936819i	$-0.386245\pi$
$719$	13488.4			0.699627			0.349814	+	0.936819i	$0.386245\pi$
$720$		0			0
$721$	−28099.7			−1.45144
$722$	−3643.30	−	6310.39i	−0.187797	−	0.325275i
$723$		0			0
$724$	−3138.26	+	5435.62i	−0.161094	+	0.279024i
$725$	1015.55	−	1758.98i	0.0520228	−	0.0901061i
$726$		0			0
$727$	−14591.6	−	25273.3i	−0.744389	−	1.28932i	−0.950480	−	0.310787i	$-0.899407\pi$
$727$	−14591.6	−	25273.3i	−0.744389	−	1.28932i	0.206090	−	0.978533i	$-0.433926\pi$
$728$	−14180.7			−0.721938
$729$		0			0
$730$	−5588.46			−0.283340
$731$	4943.60	+	8562.57i	0.250131	+	0.433239i
$732$		0			0
$733$	−6256.24	+	10836.1i	−0.315252	+	0.546032i	−0.979491	−	0.201488i	$-0.935422\pi$
$733$	−6256.24	+	10836.1i	−0.315252	+	0.546032i	0.664239	+	0.747520i	$0.268756\pi$
$734$	−4061.92	+	7035.45i	−0.204262	+	0.353792i
$735$		0			0
$736$	−10593.6	−	18348.7i	−0.530553	−	0.918945i
$737$	5833.46			0.291558
$738$		0			0
$739$	−32248.2			−1.60524			−0.802619	−	0.596492i	$-0.796561\pi$
$739$	−32248.2			−1.60524			−0.802619	+	0.596492i	$0.796561\pi$
$740$	5672.84	+	9825.64i	0.281808	+	0.488105i
$741$		0			0
$742$	−16388.1	+	28384.9i	−0.810815	+	1.40437i
$743$	9244.50	−	16011.9i	0.456457	−	0.790607i	−0.542313	−	0.840176i	$-0.682451\pi$
$743$	9244.50	−	16011.9i	0.456457	−	0.790607i	0.998771	+	0.0495691i	$0.0157848\pi$
$744$		0			0
$745$	2558.83	+	4432.02i	0.125837	+	0.217955i
$746$	−3299.12			−0.161916
$747$		0			0
$748$	4335.12			0.211908
$749$	−23254.5	−	40278.0i	−1.13445	−	1.96492i
$750$		0			0
$751$	1026.10	−	1777.26i	0.0498575	−	0.0863558i	−0.840020	−	0.542556i	$-0.817457\pi$
$751$	1026.10	−	1777.26i	0.0498575	−	0.0863558i	0.889877	+	0.456200i	$0.150790\pi$
$752$	1288.18	−	2231.19i	0.0624668	−	0.108196i
$753$		0			0
$754$	1243.04	+	2153.01i	0.0600384	+	0.103990i
$755$	−4369.98			−0.210649
$756$		0			0
$757$	10118.9			0.485834			0.242917	−	0.970047i	$-0.421896\pi$
$757$	10118.9			0.485834			0.242917	+	0.970047i	$0.421896\pi$
$758$	8826.65	+	15288.2i	0.422953	+	0.732576i
$759$		0			0
$760$	−2502.44	+	4334.36i	−0.119438	+	0.206873i
$761$	13652.0	−	23646.0i	0.650309	−	1.12637i	−0.332739	−	0.943019i	$-0.607973\pi$
$761$	13652.0	−	23646.0i	0.650309	−	1.12637i	0.983048	−	0.183349i	$-0.0586938\pi$
$762$		0			0
$763$	−8380.10	−	14514.8i	−0.397615	−	0.688689i
$764$	−2962.34			−0.140280
$765$		0			0
$766$	10669.4			0.503264
$767$	−151.475	−	262.362i	−0.00713096	−	0.0123512i
$768$		0			0
$769$	11236.9	−	19462.9i	0.526935	−	0.912679i	−0.472572	−	0.881292i	$-0.656674\pi$
$769$	11236.9	−	19462.9i	0.526935	−	0.912679i	0.999507	−	0.0313867i	$-0.00999234\pi$
$770$	3615.06	−	6261.48i	0.169192	−	0.293049i
$771$		0			0
$772$	7259.94	+	12574.6i	0.338460	+	0.586230i
$773$	35405.6			1.64741			0.823707	−	0.567015i	$-0.191902\pi$
$773$	35405.6			1.64741			0.823707	+	0.567015i	$0.191902\pi$
$774$		0			0
$775$	−270.906			−0.0125564
$776$	−8521.35	−	14759.4i	−0.394200	−	0.682774i
$777$		0			0
$778$	−5534.52	+	9586.07i	−0.255041	+	0.441744i
$779$	−10444.8	+	18090.9i	−0.480390	+	0.832060i
$780$		0			0
$781$	879.923	+	1524.07i	0.0403151	+	0.0698278i
$782$	−5177.34			−0.236754
$783$		0			0
$784$	9044.71			0.412022
$785$	−308.881	−	534.998i	−0.0140439	−	0.0243247i
$786$		0			0
$787$	6090.02	−	10548.2i	0.275840	−	0.477769i	−0.694507	−	0.719486i	$-0.744377\pi$
$787$	6090.02	−	10548.2i	0.275840	−	0.477769i	0.970347	+	0.241718i	$0.0777107\pi$
$788$	−8174.59	+	14158.8i	−0.369553	+	0.640085i
$789$		0			0
$790$	−3759.68	−	6511.95i	−0.169321	−	0.293272i
$791$	−27311.8			−1.22768
$792$		0			0
$793$	−1189.76			−0.0532783
$794$	2364.23	+	4094.97i	0.105672	+	0.183029i
$795$		0			0
$796$	7303.65	−	12650.3i	0.325215	−	0.563289i
$797$	1146.54	−	1985.86i	0.0509567	−	0.0882596i	−0.839422	−	0.543480i	$-0.817106\pi$
$797$	1146.54	−	1985.86i	0.0509567	−	0.0882596i	0.890379	+	0.455221i	$0.150440\pi$
$798$		0			0
$799$	−3440.05	−	5958.35i	−0.152316	−	0.263819i
$800$	−4678.48			−0.206762
$801$		0			0
$802$	−6544.86			−0.288164
$803$	−9572.43	−	16579.9i	−0.420677	−	0.728634i
$804$		0			0
$805$	9691.29	−	16785.8i	0.424314	−	0.734934i
$806$	165.796	−	287.167i	0.00724554	−	0.0125496i
$807$		0			0
$808$	6228.19	+	10787.5i	0.271172	+	0.469684i
$809$	25264.0			1.09794			0.548972	−	0.835841i	$-0.315019\pi$
$809$	25264.0			1.09794			0.548972	+	0.835841i	$0.315019\pi$
$810$		0			0
$811$	18781.2			0.813190			0.406595	−	0.913608i	$-0.366716\pi$
$811$	18781.2			0.813190			0.406595	+	0.913608i	$0.366716\pi$
$812$	−7697.78	−	13333.0i	−0.332684	−	0.576225i
$813$		0			0
$814$	8657.68	−	14995.5i	0.372790	−	0.645692i
$815$	3527.23	−	6109.33i	0.151599	−	0.262577i
$816$		0			0
$817$	−7995.23	−	13848.1i	−0.342372	−	0.593005i
$818$	19793.0			0.846023
$819$		0			0
$820$	−12272.9			−0.522669
$821$	15787.4	+	27344.6i	0.671114	+	1.16240i	0.977588	+	0.210525i	$0.0675174\pi$
$821$	15787.4	+	27344.6i	0.671114	+	1.16240i	−0.306474	+	0.951879i	$0.599149\pi$
$822$		0			0
$823$	473.129	−	819.484i	0.0200392	−	0.0347089i	−0.855832	−	0.517254i	$-0.826954\pi$
$823$	473.129	−	819.484i	0.0200392	−	0.0347089i	0.875871	+	0.482545i	$0.160288\pi$
$824$	8720.45	−	15104.3i	0.368679	−	0.638570i
$825$		0			0
$826$	−417.889	−	723.806i	−0.0176032	−	0.0304896i
$827$	−35044.7			−1.47355			−0.736774	−	0.676139i	$-0.763652\pi$
$827$	−35044.7			−1.47355			−0.736774	+	0.676139i	$0.763652\pi$
$828$		0			0
$829$	31029.2			1.29999			0.649993	−	0.759940i	$-0.274772\pi$
$829$	31029.2			1.29999			0.649993	+	0.759940i	$0.274772\pi$
$830$	2050.85	+	3552.18i	0.0857663	+	0.148552i
$831$		0			0
$832$	2013.14	−	3486.86i	0.0838860	−	0.145295i
$833$	12076.8	−	20917.7i	0.502327	−	0.870055i
$834$		0			0
$835$	−5069.26	−	8780.21i	−0.210094	−	0.363894i
$836$	−7011.13			−0.290054
$837$		0			0
$838$	−8077.90			−0.332991
$839$	−10599.9	−	18359.6i	−0.436175	−	0.755477i	0.561216	−	0.827669i	$-0.310334\pi$
$839$	−10599.9	−	18359.6i	−0.436175	−	0.755477i	−0.997391	+	0.0721924i	$0.977000\pi$
$840$		0			0
$841$	8894.22	−	15405.2i	0.364681	−	0.631647i
$842$	−2645.86	+	4582.76i	−0.108292	+	0.187568i
$843$		0			0
$844$	−5508.58	−	9541.14i	−0.224660	−	0.389123i
$845$	−9086.08			−0.369906
$846$		0			0
$847$	−20804.3			−0.843971
$848$	3324.22	+	5757.71i	0.134616	+	0.233161i
$849$		0			0
$850$	−571.619	+	990.073i	−0.0230663	+	0.0399520i
$851$	23209.5	−	40200.1i	0.934915	−	1.61932i
$852$		0			0
$853$	−5489.93	−	9508.84i	−0.220365	−	0.381684i	0.734554	−	0.678551i	$-0.237392\pi$
$853$	−5489.93	−	9508.84i	−0.220365	−	0.381684i	−0.954919	+	0.296867i	$0.904058\pi$
$854$	−3282.32			−0.131521
$855$		0			0
$856$	28867.2			1.15264
$857$	22407.2	+	38810.4i	0.893133	+	1.54695i	0.836098	+	0.548580i	$0.184831\pi$
$857$	22407.2	+	38810.4i	0.893133	+	1.54695i	0.0570348	+	0.998372i	$0.481835\pi$
$858$		0			0
$859$	−18693.1	+	32377.4i	−0.742492	+	1.28603i	0.208866	+	0.977944i	$0.433023\pi$
$859$	−18693.1	+	32377.4i	−0.742492	+	1.28603i	−0.951358	+	0.308089i	$0.900311\pi$
$860$	4697.30	−	8135.96i	0.186252	−	0.322598i
$861$		0			0
$862$	−11736.0	−	20327.3i	−0.463722	−	0.803191i
$863$	30536.3			1.20448			0.602241	−	0.798314i	$-0.294275\pi$
$863$	30536.3			1.20448			0.602241	+	0.798314i	$0.294275\pi$
$864$		0			0
$865$	−13519.9			−0.531433
$866$	−11000.2	−	19052.9i	−0.431643	−	0.747627i
$867$		0			0
$868$	−1026.72	+	1778.34i	−0.0401489	+	0.0695399i
$869$	12879.8	−	22308.6i	0.502784	−	0.870847i
$870$		0			0
$871$	2113.37	+	3660.47i	0.0822146	+	0.142400i
$872$	10402.7			0.403991
$873$		0			0
$874$	8373.25			0.324061
$875$	−2139.99	−	3706.57i	−0.0826797	−	0.143205i
$876$		0			0
$877$	−17163.7	+	29728.4i	−0.660864	+	1.14465i	0.319525	+	0.947578i	$0.396477\pi$
$877$	−17163.7	+	29728.4i	−0.660864	+	1.14465i	−0.980389	+	0.197072i	$0.936857\pi$
$878$	−3754.52	+	6503.02i	−0.144315	+	0.249962i
$879$		0			0
$880$	−733.294	−	1270.10i	−0.0280901	−	0.0486535i
$881$	18009.6			0.688718			0.344359	−	0.938838i	$-0.388096\pi$
$881$	18009.6			0.688718			0.344359	+	0.938838i	$0.388096\pi$
$882$		0			0
$883$	−29074.4			−1.10808			−0.554039	−	0.832491i	$-0.686914\pi$
$883$	−29074.4			−1.10808			−0.554039	+	0.832491i	$0.686914\pi$
$884$	1570.55	+	2720.27i	0.0597548	+	0.103498i
$885$		0			0
$886$	−3748.10	+	6491.90i	−0.142122	+	0.246162i
$887$	−16451.3	+	28494.5i	−0.622752	+	1.07864i	0.366220	+	0.930528i	$0.380652\pi$
$887$	−16451.3	+	28494.5i	−0.622752	+	1.07864i	−0.988971	+	0.148109i	$0.952681\pi$
$888$		0			0
$889$	−29586.9	−	51246.1i	−1.11621	−	1.93334i
$890$	−12579.3			−0.473772
$891$		0			0
$892$	3735.20			0.140206
$893$	5563.56	+	9636.36i	0.208485	+	0.361107i
$894$		0			0
$895$	−1876.74	+	3250.60i	−0.0700920	+	0.121403i
$896$	−20076.6	+	34773.7i	−0.748562	+	1.29655i
$897$		0			0
$898$	671.902	+	1163.77i	0.0249684	+	0.0432466i
$899$	880.377			0.0326610
$900$		0			0
$901$	17754.5			0.656479
$902$	9365.24	+	16221.1i	0.345708	+	0.598783i
$903$		0			0
$904$	8475.93	−	14680.7i	0.311842	−	0.540127i
$905$	2835.21	−	4910.72i	0.104139	−	0.180373i
$906$		0			0
$907$	407.908	+	706.517i	0.0149331	+	0.0258649i	0.873395	−	0.487012i	$-0.161913\pi$
$907$	407.908	+	706.517i	0.0149331	+	0.0258649i	−0.858462	+	0.512877i	$0.828580\pi$
$908$	10470.4			0.382677
$909$		0			0
$910$	5238.73			0.190838
$911$	7921.41	+	13720.3i	0.288088	+	0.498983i	0.973353	−	0.229311i	$-0.0736472\pi$
$911$	7921.41	+	13720.3i	0.288088	+	0.498983i	−0.685265	+	0.728293i	$0.740314\pi$
$912$		0			0
$913$	−7025.77	+	12169.0i	−0.254676	+	0.441112i
$914$	−10534.5	+	18246.3i	−0.381237	+	0.660322i
$915$		0			0
$916$	9432.11	+	16336.9i	0.340225	+	0.589286i
$917$	−21882.7			−0.788037
$918$		0			0
$919$	23848.4			0.856025			0.428013	−	0.903773i	$-0.359214\pi$
$919$	23848.4			0.856025			0.428013	+	0.903773i	$0.359214\pi$
$920$	6015.17	+	10418.6i	0.215559	+	0.373359i
$921$		0			0
$922$	12472.2	−	21602.4i	0.445498	−	0.771624i
$923$	−637.566	+	1104.30i	−0.0227364	+	0.0393807i
$924$		0			0
$925$	−5125.03	−	8876.81i	−0.182173	−	0.315533i
$926$	−10302.7			−0.365625
$927$		0			0
$928$	15203.9			0.537816
$929$	−19796.0	−	34287.7i	−0.699123	−	1.21092i	−0.968771	−	0.247958i	$-0.920240\pi$
$929$	−19796.0	−	34287.7i	−0.699123	−	1.21092i	0.269647	−	0.962959i	$-0.413093\pi$
$930$		0			0
$931$	−19531.7	+	33830.0i	−0.687569	+	1.19090i
$932$	7864.07	−	13621.0i	0.276391	−	0.478723i
$933$		0			0
$934$	5332.72	+	9236.54i	0.186822	+	0.323585i
$935$	−3916.49			−0.136987
$936$		0			0
$937$	10255.0			0.357540			0.178770	−	0.983891i	$-0.442788\pi$
$937$	10255.0			0.357540			0.178770	+	0.983891i	$0.442788\pi$
$938$	5830.38	+	10098.5i	0.202952	+	0.351523i
$939$		0			0
$940$	−3268.66	+	5661.49i	−0.113417	+	0.196444i
$941$	−11947.2	+	20693.1i	−0.413886	+	0.716871i	−0.995311	−	0.0967289i	$-0.969162\pi$
$941$	−11947.2	+	20693.1i	−0.413886	+	0.716871i	0.581425	+	0.813600i	$0.302495\pi$
$942$		0			0
$943$	25106.4	+	43485.5i	0.866995	+	1.50168i
$944$	−169.533			−0.00584514
$945$		0			0
$946$	−14337.7			−0.492768
$947$	−18128.6	−	31399.7i	−0.622072	−	1.07746i	−0.989099	−	0.147250i	$-0.952958\pi$
$947$	−18128.6	−	31399.7i	−0.622072	−	1.07746i	0.367028	−	0.930210i	$-0.380375\pi$
$948$		0			0
$949$	6935.89	−	12013.3i	0.237248	−	0.410926i
$950$	924.472	−	1601.23i	0.0315725	−	0.0546851i
$951$		0			0
$952$	10595.9	+	18352.6i	0.360730	+	0.624803i
$953$	46736.7			1.58861			0.794307	−	0.607516i	$-0.207834\pi$
$953$	46736.7			1.58861			0.794307	+	0.607516i	$0.207834\pi$
$954$		0			0
$955$	2676.27			0.0906830
$956$	−4481.81	−	7762.72i	−0.151623	−	0.262620i
$957$		0			0
$958$	−13859.0	+	24004.6i	−0.467396	+	0.809553i
$959$	−48551.0	+	84092.9i	−1.63482	+	2.83160i
$960$		0			0
$961$	14836.8	+	25698.1i	0.498029	+	0.862612i
$962$	12546.2			0.420484
$963$		0			0
$964$	−147.774			−0.00493721
$965$	−6558.87	−	11360.3i	−0.218795	−	0.378965i
$966$		0			0
$967$	−22178.5	+	38414.3i	−0.737552	+	1.27748i	0.216043	+	0.976384i	$0.430685\pi$
$967$	−22178.5	+	38414.3i	−0.737552	+	1.27748i	−0.953595	+	0.301094i	$0.902648\pi$
$968$	6456.39	−	11182.8i	0.214376	−	0.371310i
$969$		0			0
$970$	3148.03	+	5452.54i	0.104203	+	0.180485i
$971$	650.540			0.0215003			0.0107502	−	0.999942i	$-0.496578\pi$
$971$	650.540			0.0215003			0.0107502	+	0.999942i	$0.496578\pi$
$972$		0			0
$973$	5991.83			0.197420
$974$	−4175.31	−	7231.85i	−0.137357	−	0.237909i
$975$		0			0
$976$	−332.899	+	576.598i	−0.0109179	+	0.0189103i
$977$	25837.5	−	44751.9i	0.846075	−	1.46544i	−0.0386096	−	0.999254i	$-0.512293\pi$
$977$	25837.5	−	44751.9i	0.846075	−	1.46544i	0.884684	−	0.466190i	$-0.154374\pi$
$978$		0			0
$979$	−21546.9	−	37320.3i	−0.703414	−	1.21835i
$980$	−22950.3			−0.748082
$981$		0			0
$982$	17068.1			0.554649
$983$	11759.5	+	20368.0i	0.381556	+	0.660874i	0.991285	−	0.131736i	$-0.0420551\pi$
$983$	11759.5	+	20368.0i	0.381556	+	0.660874i	−0.609729	+	0.792610i	$0.708722\pi$
$984$		0			0
$985$	7385.20	−	12791.5i	0.238895	−	0.413779i
$986$	1857.62	−	3217.49i	0.0599987	−	0.103921i
$987$		0			0
$988$	−2540.03	−	4399.46i	−0.0817905	−	0.141665i
$989$	−38436.6			−1.23581
$990$		0			0
$991$	−27732.9			−0.888966			−0.444483	−	0.895787i	$-0.646613\pi$
$991$	−27732.9			−0.888966			−0.444483	+	0.895787i	$0.646613\pi$
$992$	−1013.94	−	1756.20i	−0.0324523	−	0.0562090i
$993$		0			0
$994$	−1758.92	+	3046.53i	−0.0561262	+	0.0972135i
$995$	−6598.36	+	11428.7i	−0.210233	+	0.364135i
$996$		0			0
$997$	17163.9	+	29728.7i	0.545221	+	0.944351i	0.998593	+	0.0530293i	$0.0168877\pi$
$997$	17163.9	+	29728.7i	0.545221	+	0.944351i	−0.453372	+	0.891322i	$0.649779\pi$
$998$	−16283.9			−0.516489
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.e.c.91.5		14
3.2	odd	2		45.4.e.c.31.3	yes	14
9.2	odd	6		45.4.e.c.16.3	✓	14
9.4	even	3		405.4.a.n.1.3		7
9.5	odd	6		405.4.a.m.1.5		7
9.7	even	3	inner	135.4.e.c.46.5		14
15.2	even	4		225.4.k.d.49.9		28
15.8	even	4		225.4.k.d.49.6		28
15.14	odd	2		225.4.e.d.76.5		14
45.2	even	12		225.4.k.d.124.6		28
45.4	even	6		2025.4.a.ba.1.5		7
45.14	odd	6		2025.4.a.bb.1.3		7
45.29	odd	6		225.4.e.d.151.5		14
45.38	even	12		225.4.k.d.124.9		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.e.c.16.3	✓	14	9.2	odd	6
45.4.e.c.31.3	yes	14	3.2	odd	2
135.4.e.c.46.5		14	9.7	even	3	inner
135.4.e.c.91.5		14	1.1	even	1	trivial
225.4.e.d.76.5		14	15.14	odd	2
225.4.e.d.151.5		14	45.29	odd	6
225.4.k.d.49.6		28	15.8	even	4
225.4.k.d.49.9		28	15.2	even	4
225.4.k.d.124.6		28	45.2	even	12
225.4.k.d.124.9		28	45.38	even	12
405.4.a.m.1.5		7	9.5	odd	6
405.4.a.n.1.3		7	9.4	even	3
2025.4.a.ba.1.5		7	45.4	even	6
2025.4.a.bb.1.3		7	45.14	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	135.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.785104 + 1.35984i) q^{2} +(2.76722 - 4.79297i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(-17.1199 - 29.6525i) q^{7} +21.2519 q^{8} +O(q^{10})\)	\(q+(0.785104 + 1.35984i) q^{2} +(2.76722 - 4.79297i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(-17.1199 - 29.6525i) q^{7} +21.2519 q^{8} -7.85104 q^{10} +(-13.4480 - 23.2926i) q^{11} +(9.74401 - 16.8771i) q^{13} +(26.8818 - 46.5607i) q^{14} +(-5.45281 - 9.44454i) q^{16} -29.1232 q^{17} +47.1006 q^{19} +(13.8361 + 23.9648i) q^{20} +(21.1162 - 36.5743i) q^{22} +(56.6083 - 98.0485i) q^{23} +(-12.5000 - 21.6506i) q^{25} +30.6003 q^{26} -189.498 q^{28} +(40.6219 + 70.3593i) q^{29} +(5.41811 - 9.38445i) q^{31} +(93.5697 - 162.067i) q^{32} +(-22.8648 - 39.6029i) q^{34} +171.199 q^{35} +410.002 q^{37} +(36.9789 + 64.0493i) q^{38} +(-53.1298 + 92.0234i) q^{40} +(-221.755 + 384.091i) q^{41} +(-169.748 - 294.012i) q^{43} -148.854 q^{44} +177.774 q^{46} +(118.121 + 204.591i) q^{47} +(-414.681 + 718.249i) q^{49} +(19.6276 - 33.9960i) q^{50} +(-53.9277 - 93.4055i) q^{52} -609.634 q^{53} +134.480 q^{55} +(-363.830 - 630.173i) q^{56} +(-63.7849 + 110.479i) q^{58} +(7.77272 - 13.4627i) q^{59} +(-30.5255 - 52.8717i) q^{61} +17.0151 q^{62} +206.603 q^{64} +(48.7201 + 84.3856i) q^{65} +(-108.445 + 187.832i) q^{67} +(-80.5903 + 139.587i) q^{68} +(134.409 + 232.803i) q^{70} -65.4315 q^{71} +711.811 q^{73} +(321.895 + 557.538i) q^{74} +(130.338 - 225.752i) q^{76} +(-460.457 + 797.534i) q^{77} +(478.876 + 829.438i) q^{79} +54.5281 q^{80} -696.404 q^{82} +(-261.220 - 452.446i) q^{83} +(72.8080 - 126.107i) q^{85} +(266.540 - 461.660i) q^{86} +(-285.796 - 495.012i) q^{88} +1602.24 q^{89} -667.266 q^{91} +(-313.296 - 542.644i) q^{92} +(-185.474 + 321.251i) q^{94} +(-117.751 + 203.952i) q^{95} +(-400.969 - 694.499i) q^{97} -1302.27 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 2 q^{2} - 36 q^{4} - 35 q^{5} - 22 q^{7} + 36 q^{8}+O(q^{10}) \) 14 * q - 2 * q^2 - 36 * q^4 - 35 * q^5 - 22 * q^7 + 36 * q^8	\( 14 q - 2 q^{2} - 36 q^{4} - 35 q^{5} - 22 q^{7} + 36 q^{8} + 20 q^{10} - 23 q^{11} - 96 q^{13} + 21 q^{14} - 324 q^{16} + 322 q^{17} + 558 q^{19} - 180 q^{20} - 311 q^{22} - 96 q^{23} - 175 q^{25} - 716 q^{26} + 674 q^{28} + 296 q^{29} - 244 q^{31} + 314 q^{32} - 125 q^{34} + 220 q^{35} + 808 q^{37} - 305 q^{38} - 90 q^{40} + 47 q^{41} - 525 q^{43} + 110 q^{44} + 1434 q^{46} - 164 q^{47} - 1225 q^{49} - 50 q^{50} - 1682 q^{52} + 1012 q^{53} + 230 q^{55} + 981 q^{56} - 1183 q^{58} + 85 q^{59} - 828 q^{61} - 1572 q^{62} + 4472 q^{64} - 480 q^{65} - 1093 q^{67} - 2473 q^{68} + 105 q^{70} + 656 q^{71} + 4170 q^{73} + 1316 q^{74} - 2789 q^{76} - 24 q^{77} - 2110 q^{79} + 3240 q^{80} - 124 q^{82} - 1290 q^{83} - 805 q^{85} + 2569 q^{86} - 2271 q^{88} - 6096 q^{89} + 6676 q^{91} - 2763 q^{92} + 517 q^{94} - 1395 q^{95} - 1787 q^{97} + 2558 q^{98}+O(q^{100}) \) 14 * q - 2 * q^2 - 36 * q^4 - 35 * q^5 - 22 * q^7 + 36 * q^8 + 20 * q^10 - 23 * q^11 - 96 * q^13 + 21 * q^14 - 324 * q^16 + 322 * q^17 + 558 * q^19 - 180 * q^20 - 311 * q^22 - 96 * q^23 - 175 * q^25 - 716 * q^26 + 674 * q^28 + 296 * q^29 - 244 * q^31 + 314 * q^32 - 125 * q^34 + 220 * q^35 + 808 * q^37 - 305 * q^38 - 90 * q^40 + 47 * q^41 - 525 * q^43 + 110 * q^44 + 1434 * q^46 - 164 * q^47 - 1225 * q^49 - 50 * q^50 - 1682 * q^52 + 1012 * q^53 + 230 * q^55 + 981 * q^56 - 1183 * q^58 + 85 * q^59 - 828 * q^61 - 1572 * q^62 + 4472 * q^64 - 480 * q^65 - 1093 * q^67 - 2473 * q^68 + 105 * q^70 + 656 * q^71 + 4170 * q^73 + 1316 * q^74 - 2789 * q^76 - 24 * q^77 - 2110 * q^79 + 3240 * q^80 - 124 * q^82 - 1290 * q^83 - 805 * q^85 + 2569 * q^86 - 2271 * q^88 - 6096 * q^89 + 6676 * q^91 - 2763 * q^92 + 517 * q^94 - 1395 * q^95 - 1787 * q^97 + 2558 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more