LMFDB - Embedded newform 513.2.f.h.163.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [513,2,Mod(163,513)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(513, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("513.163"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 513 = 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	513.f (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,1,0,-3,5,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$4.09632562369$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + 8x^{10} - x^{9} + 41x^{8} - 7x^{7} + 91x^{6} + 9x^{5} + 135x^{4} - 12x^{3} + 45x^{2} + 9x + 9 $ x^12 - x^11 + 8x^10 - x^9 + 41x^8 - 7x^7 + 91x^6 + 9x^5 + 135x^4 - 12x^3 + 45x^2 + 9*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			163.1
Root			$-0.963952 - 1.66961i$ of defining polynomial
Character	$\chi$	$=$	513.163
Dual form			513.2.f.h.406.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.963952 + 1.66961i) q^{2} +(-0.858406 - 1.48680i) q^{4} +(0.100478 - 0.174033i) q^{5} +3.71133 q^{7} -0.545959 q^{8} +(0.193712 + 0.335518i) q^{10} +1.10424 q^{11} +(0.126543 + 0.219179i) q^{13} +(-3.57755 + 6.19649i) q^{14} +(2.24309 - 3.88515i) q^{16} +(1.77914 - 3.08156i) q^{17} +(4.04150 - 1.63288i) q^{19} -0.345003 q^{20} +(-1.06443 + 1.84365i) q^{22} +(3.67802 + 6.37053i) q^{23} +(2.47981 + 4.29515i) q^{25} -0.487925 q^{26} +(-3.18583 - 5.51802i) q^{28} +(0.934935 + 1.61935i) q^{29} -4.12338 q^{31} +(3.77850 + 6.54456i) q^{32} +(3.43001 + 5.94095i) q^{34} +(0.372907 - 0.645893i) q^{35} -7.85784 q^{37} +(-1.16954 + 8.32176i) q^{38} +(-0.0548567 + 0.0950146i) q^{40} +(-1.39952 + 2.42404i) q^{41} +(-1.21088 + 2.09730i) q^{43} +(-0.947882 - 1.64178i) q^{44} -14.1818 q^{46} +(1.26067 + 2.18354i) q^{47} +6.77400 q^{49} -9.56166 q^{50} +(0.217250 - 0.376289i) q^{52} +(-2.85436 - 4.94389i) q^{53} +(0.110951 - 0.192173i) q^{55} -2.02624 q^{56} -3.60493 q^{58} +(7.21720 - 12.5005i) q^{59} +(2.47619 + 4.28889i) q^{61} +(3.97474 - 6.88445i) q^{62} -5.59682 q^{64} +0.0508590 q^{65} +(2.59984 + 4.50306i) q^{67} -6.10889 q^{68} +(0.718928 + 1.24522i) q^{70} +(0.350284 - 0.606709i) q^{71} +(-0.444681 + 0.770211i) q^{73} +(7.57458 - 13.1196i) q^{74} +(-5.89701 - 4.60724i) q^{76} +4.09819 q^{77} +(-6.09274 + 10.5529i) q^{79} +(-0.450762 - 0.780742i) q^{80} +(-2.69814 - 4.67332i) q^{82} -16.5557 q^{83} +(-0.357528 - 0.619256i) q^{85} +(-2.33446 - 4.04340i) q^{86} -0.602867 q^{88} +(-0.534540 - 0.925851i) q^{89} +(0.469643 + 0.813445i) q^{91} +(6.31448 - 10.9370i) q^{92} -4.86089 q^{94} +(0.121907 - 0.867421i) q^{95} +(3.19691 - 5.53720i) q^{97} +(-6.52981 + 11.3100i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + q^{2} - 3 q^{4} + 5 q^{5} + 2 q^{7} - 12 q^{8} + q^{10} - 4 q^{11} - 5 q^{13} + 2 q^{14} + 3 q^{16} + 10 q^{17} - 9 q^{19} - 2 q^{20} - 4 q^{22} + 3 q^{23} - 5 q^{25} - 2 q^{26} - 2 q^{28} - 6 q^{29}+ \cdots - 59 q^{98}+O(q^{100}) $ 12 * q + q^2 - 3 * q^4 + 5 * q^5 + 2 * q^7 - 12 * q^8 + q^10 - 4 * q^11 - 5 * q^13 + 2 * q^14 + 3 * q^16 + 10 * q^17 - 9 * q^19 - 2 * q^20 - 4 * q^22 + 3 * q^23 - 5 * q^25 - 2 * q^26 - 2 * q^28 - 6 * q^29 - 4 * q^31 + 8 * q^32 + 4 * q^34 + 3 * q^35 + 26 * q^37 - 10 * q^38 - 6 * q^40 - 13 * q^41 - 3 * q^43 - 20 * q^44 - 10 * q^46 + 18 * q^47 + 26 * q^49 + 6 * q^50 - 22 * q^52 - q^53 - 6 * q^55 + 48 * q^56 + 28 * q^58 + 11 * q^59 - 13 * q^61 + 22 * q^62 - 72 * q^65 + 9 * q^67 - 10 * q^68 - 30 * q^70 - 6 * q^71 - 15 * q^73 + 43 * q^74 - 33 * q^76 - 58 * q^77 - 13 * q^79 + q^80 + 4 * q^82 - 16 * q^83 - 3 * q^85 - 7 * q^86 - 30 * q^88 + 29 * q^89 + 22 * q^91 + 21 * q^92 + 36 * q^94 + 22 * q^95 + 20 * q^97 - 59 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/513\mathbb{Z}\right)^\times$.

$n$	$191$	$325$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	−0.963952	+	1.66961i	−0.681617	+	1.18060i	0.292870	+	0.956152i	$0.405390\pi$
$2$	−0.963952	+	1.66961i	−0.681617	+	1.18060i	−0.974487	+	0.224443i	$0.927944\pi$
$3$		0			0
$3$		0			0
$4$	−0.858406	−	1.48680i	−0.429203	−	0.743402i
$5$	0.100478	−	0.174033i	0.0449350	−	0.0778298i	−0.842683	−	0.538410i	$-0.819025\pi$
$5$	0.100478	−	0.174033i	0.0449350	−	0.0778298i	0.887618	+	0.460580i	$0.152359\pi$
$6$		0			0
$7$	3.71133			1.40275			0.701376	−	0.712791i	$-0.252569\pi$
$7$	3.71133			1.40275			0.701376	+	0.712791i	$0.252569\pi$
$8$	−0.545959			−0.193026
$9$		0			0
$10$	0.193712	+	0.335518i	0.0612570	+	0.106100i
$11$	1.10424			0.332939			0.166470	−	0.986047i	$-0.446763\pi$
$11$	1.10424			0.332939			0.166470	+	0.986047i	$0.446763\pi$
$12$		0			0
$13$	0.126543	+	0.219179i	0.0350967	+	0.0607892i	0.883040	−	0.469297i	$-0.155493\pi$
$13$	0.126543	+	0.219179i	0.0350967	+	0.0607892i	−0.847944	+	0.530087i	$0.822159\pi$
$14$	−3.57755	+	6.19649i	−0.956140	+	1.65608i
$15$		0			0
$16$	2.24309	−	3.88515i	0.560773	−	0.971287i
$17$	1.77914	−	3.08156i	0.431504	−	0.747388i	−0.565499	−	0.824749i	$-0.691316\pi$
$17$	1.77914	−	3.08156i	0.431504	−	0.747388i	0.997003	+	0.0773616i	$0.0246496\pi$
$18$		0			0
$19$	4.04150	−	1.63288i	0.927183	−	0.374608i
$19$	4.04150	−	1.63288i	0.927183	−	0.374608i
$20$	−0.345003			−0.0771450
$21$		0			0
$22$	−1.06443	+	1.84365i	−0.226937	+	0.393067i
$23$	3.67802	+	6.37053i	0.766921	+	1.32835i	0.939225	+	0.343303i	$0.111546\pi$
$23$	3.67802	+	6.37053i	0.766921	+	1.32835i	−0.172304	+	0.985044i	$0.555121\pi$
$24$		0			0
$25$	2.47981	+	4.29515i	0.495962	+	0.859031i
$26$	−0.487925			−0.0956899
$27$		0			0
$28$	−3.18583	−	5.51802i	−0.602066	−	1.04281i
$29$	0.934935	+	1.61935i	0.173613	+	0.300707i	0.939680	−	0.342054i	$-0.111122\pi$
$29$	0.934935	+	1.61935i	0.173613	+	0.300707i	−0.766067	+	0.642760i	$0.777789\pi$
$30$		0			0
$31$	−4.12338			−0.740581			−0.370291	−	0.928916i	$-0.620742\pi$
$31$	−4.12338			−0.740581			−0.370291	+	0.928916i	$0.620742\pi$
$32$	3.77850	+	6.54456i	0.667951	+	1.15693i
$33$		0			0
$34$	3.43001	+	5.94095i	0.588241	+	1.01886i
$35$	0.372907	−	0.645893i	0.0630327	−	0.109176i
$36$		0			0
$37$	−7.85784			−1.29182			−0.645911	−	0.763413i	$-0.723522\pi$
$37$	−7.85784			−1.29182			−0.645911	+	0.763413i	$0.723522\pi$
$38$	−1.16954	+	8.32176i	−0.189724	+	1.34997i
$39$		0			0
$40$	−0.0548567	+	0.0950146i	−0.00867361	+	0.0150231i
$41$	−1.39952	+	2.42404i	−0.218569	+	0.378572i	−0.954371	−	0.298625i	$-0.903472\pi$
$41$	−1.39952	+	2.42404i	−0.218569	+	0.378572i	0.735802	+	0.677197i	$0.236805\pi$
$42$		0			0
$43$	−1.21088	+	2.09730i	−0.184657	+	0.319836i	−0.943461	−	0.331484i	$-0.892451\pi$
$43$	−1.21088	+	2.09730i	−0.184657	+	0.319836i	0.758804	+	0.651319i	$0.225784\pi$
$44$	−0.947882	−	1.64178i	−0.142899	−	0.247508i
$45$		0			0
$46$	−14.1818			−2.09099
$47$	1.26067	+	2.18354i	0.183887	+	0.318502i	0.943201	−	0.332223i	$-0.107799\pi$
$47$	1.26067	+	2.18354i	0.183887	+	0.318502i	−0.759314	+	0.650725i	$0.774465\pi$
$48$		0			0
$49$	6.77400			0.967714
$50$	−9.56166			−1.35222
$51$		0			0
$52$	0.217250	−	0.376289i	0.0301272	−	0.0521818i
$53$	−2.85436	−	4.94389i	−0.392076	−	0.679096i	0.600647	−	0.799514i	$-0.294910\pi$
$53$	−2.85436	−	4.94389i	−0.392076	−	0.679096i	−0.992723	+	0.120419i	$0.961576\pi$
$54$		0			0
$55$	0.110951	−	0.192173i	0.0149606	−	0.0259126i
$56$	−2.02624			−0.270767
$57$		0			0
$58$	−3.60493			−0.473350
$59$	7.21720	−	12.5005i	0.939599	−	1.62743i	0.173378	−	0.984855i	$-0.444532\pi$
$59$	7.21720	−	12.5005i	0.939599	−	1.62743i	0.766221	−	0.642577i	$-0.222135\pi$
$60$		0			0
$61$	2.47619	+	4.28889i	0.317044	+	0.549136i	0.979870	−	0.199637i	$-0.0639765\pi$
$61$	2.47619	+	4.28889i	0.317044	+	0.549136i	−0.662826	+	0.748773i	$0.730643\pi$
$62$	3.97474	−	6.88445i	0.504793	−	0.874326i
$63$		0			0
$64$	−5.59682			−0.699602
$65$	0.0508590			0.00630828
$66$		0			0
$67$	2.59984	+	4.50306i	0.317621	+	0.550136i	0.979991	−	0.199041i	$-0.0637826\pi$
$67$	2.59984	+	4.50306i	0.317621	+	0.550136i	−0.662370	+	0.749177i	$0.730449\pi$
$68$	−6.10889			−0.740812
$69$		0			0
$70$	0.718928	+	1.24522i	0.0859284	+	0.148832i
$71$	0.350284	−	0.606709i	0.0415710	−	0.0720031i	−0.844491	−	0.535569i	$-0.820097\pi$
$71$	0.350284	−	0.606709i	0.0415710	−	0.0720031i	0.886062	+	0.463566i	$0.153430\pi$
$72$		0			0
$73$	−0.444681	+	0.770211i	−0.0520460	+	0.0901463i	−0.890875	−	0.454249i	$-0.849908\pi$
$73$	−0.444681	+	0.770211i	−0.0520460	+	0.0901463i	0.838829	+	0.544396i	$0.183241\pi$
$74$	7.57458	−	13.1196i	0.880527	−	1.52512i
$75$		0			0
$76$	−5.89701	−	4.60724i	−0.676434	−	0.528487i
$77$	4.09819			0.467032
$78$		0			0
$79$	−6.09274	+	10.5529i	−0.685487	+	1.18730i	0.287797	+	0.957691i	$0.407077\pi$
$79$	−6.09274	+	10.5529i	−0.685487	+	1.18730i	−0.973284	+	0.229606i	$0.926256\pi$
$80$	−0.450762	−	0.780742i	−0.0503967	−	0.0872896i
$81$		0			0
$82$	−2.69814	−	4.67332i	−0.297960	−	0.516082i
$83$	−16.5557			−1.81723			−0.908615	−	0.417636i	$-0.862859\pi$
$83$	−16.5557			−1.81723			−0.908615	+	0.417636i	$0.862859\pi$
$84$		0			0
$85$	−0.357528	−	0.619256i	−0.0387793	−	0.0671678i
$86$	−2.33446	−	4.04340i	−0.251731	−	0.436011i
$87$		0			0
$88$	−0.602867			−0.0642658
$89$	−0.534540	−	0.925851i	−0.0566612	−	0.0981400i	0.836304	−	0.548267i	$-0.184712\pi$
$89$	−0.534540	−	0.925851i	−0.0566612	−	0.0981400i	−0.892965	+	0.450127i	$0.851379\pi$
$90$		0			0
$91$	0.469643	+	0.813445i	0.0492319	+	0.0852722i
$92$	6.31448	−	10.9370i	0.658330	−	1.14026i
$93$		0			0
$94$	−4.86089			−0.501363
$95$	0.121907	−	0.867421i	0.0125074	−	0.0889955i
$96$		0			0
$97$	3.19691	−	5.53720i	0.324597	−	0.562218i	−0.656834	−	0.754035i	$-0.728105\pi$
$97$	3.19691	−	5.53720i	0.324597	−	0.562218i	0.981431	+	0.191817i	$0.0614381\pi$
$98$	−6.52981	+	11.3100i	−0.659610	+	1.14248i
$99$		0			0
$100$	4.25737	−	7.37397i	0.425737	−	0.737397i
$101$	2.03917	+	3.53195i	0.202905	+	0.351442i	0.949463	−	0.313878i	$-0.101628\pi$
$101$	2.03917	+	3.53195i	0.202905	+	0.351442i	−0.746558	+	0.665320i	$0.768295\pi$
$102$		0			0
$103$	14.6274			1.44128			0.720641	−	0.693308i	$-0.243848\pi$
$103$	14.6274			1.44128			0.720641	+	0.693308i	$0.243848\pi$
$104$	−0.0690872	−	0.119662i	−0.00677455	−	0.0117339i
$105$		0			0
$106$	11.0058			1.06898
$107$	10.6993			1.03434			0.517170	−	0.855883i	$-0.326985\pi$
$107$	10.6993			1.03434			0.517170	+	0.855883i	$0.326985\pi$
$108$		0			0
$109$	4.93290	−	8.54403i	0.472486	−	0.818370i	−0.527018	−	0.849854i	$-0.676690\pi$
$109$	4.93290	−	8.54403i	0.472486	−	0.818370i	0.999504	+	0.0314842i	$0.0100234\pi$
$110$	0.213903	+	0.370491i	0.0203949	+	0.0353249i
$111$		0			0
$112$	8.32486	−	14.4191i	0.786625	−	1.36247i
$113$	15.0356			1.41443			0.707213	−	0.707001i	$-0.249952\pi$
$113$	15.0356			1.41443			0.707213	+	0.707001i	$0.249952\pi$
$114$		0			0
$115$	1.47824			0.137847
$116$	1.60511	−	2.78013i	0.149031	−	0.258128i
$117$		0			0
$118$	13.9141	+	24.0999i	1.28089	+	2.21857i
$119$	6.60298	−	11.4367i	0.605294	−	1.04840i
$120$		0			0
$121$	−9.78066			−0.889151
$122$	−9.54772			−0.864410
$123$		0			0
$124$	3.53954	+	6.13066i	0.317860	+	0.550549i
$125$	2.00144			0.179014
$126$		0			0
$127$	−8.25515	−	14.2983i	−0.732526	−	1.26877i	−0.955800	−	0.294017i	$-0.905008\pi$
$127$	−8.25515	−	14.2983i	−0.732526	−	1.26877i	0.223274	−	0.974756i	$-0.428325\pi$
$128$	−2.16194	+	3.74460i	−0.191091	+	0.330979i
$129$		0			0
$130$	−0.0490256	+	0.0849149i	−0.00429983	+	0.00744753i
$131$	3.81895	−	6.61462i	0.333663	−	0.577922i	−0.649564	−	0.760307i	$-0.725048\pi$
$131$	3.81895	−	6.61462i	0.333663	−	0.577922i	0.983227	+	0.182385i	$0.0583818\pi$
$132$		0			0
$133$	14.9994	−	6.06015i	1.30061	−	0.525482i
$134$	−10.0245			−0.865984
$135$		0			0
$136$	−0.971336	+	1.68240i	−0.0832914	+	0.144265i
$137$	1.09192	+	1.89126i	0.0932888	+	0.161581i	0.908893	−	0.417029i	$-0.136929\pi$
$137$	1.09192	+	1.89126i	0.0932888	+	0.161581i	−0.815604	+	0.578610i	$0.803595\pi$
$138$		0			0
$139$	−8.45796	−	14.6496i	−0.717395	−	1.24256i	−0.962029	−	0.272948i	$-0.912001\pi$
$139$	−8.45796	−	14.6496i	−0.717395	−	1.24256i	0.244634	−	0.969615i	$-0.421332\pi$
$140$	−1.28042			−0.108215
$141$		0			0
$142$	0.675313	+	1.16968i	0.0566710	+	0.0981571i
$143$	0.139733	+	0.242025i	0.0116851	+	0.0202391i
$144$		0			0
$145$	0.375761			0.0312052
$146$	−0.857303	−	1.48489i	−0.0709509	−	0.122891i
$147$		0			0
$148$	6.74522	+	11.6831i	0.554454	+	0.960342i
$149$	10.6794	−	18.4973i	0.874890	−	1.51535i	0.0180106	−	0.999838i	$-0.494267\pi$
$149$	10.6794	−	18.4973i	0.874890	−	1.51535i	0.856880	−	0.515517i	$-0.172400\pi$
$150$		0			0
$151$	−16.0232			−1.30395			−0.651975	−	0.758240i	$-0.726059\pi$
$151$	−16.0232			−1.30395			−0.651975	+	0.758240i	$0.726059\pi$
$152$	−2.20649	+	0.891483i	−0.178970	+	0.0723088i
$153$		0			0
$154$	−3.95045	+	6.84239i	−0.318337	+	0.551375i
$155$	−0.414308	+	0.717603i	−0.0332780	+	0.0576393i
$156$		0			0
$157$	−10.5440	+	18.2627i	−0.841502	+	1.45752i	0.0471233	+	0.998889i	$0.484995\pi$
$157$	−10.5440	+	18.2627i	−0.841502	+	1.45752i	−0.888625	+	0.458635i	$0.848339\pi$
$158$	−11.7462	−	20.3450i	−0.934478	−	1.61856i
$159$		0			0
$160$	1.51862			0.120058
$161$	13.6504	+	23.6432i	1.07580	+	1.86334i
$162$		0			0
$163$	16.6930			1.30749			0.653747	−	0.756713i	$-0.273196\pi$
$163$	16.6930			1.30749			0.653747	+	0.756713i	$0.273196\pi$
$164$	4.80543			0.375241
$165$		0			0
$166$	15.9589	−	27.6417i	1.23865	−	2.14541i
$167$	−7.56635	−	13.1053i	−0.585502	−	1.01412i	−0.994813	−	0.101724i	$-0.967564\pi$
$167$	−7.56635	−	13.1053i	−0.585502	−	1.01412i	0.409311	−	0.912395i	$-0.365769\pi$
$168$		0			0
$169$	6.46797	−	11.2029i	0.497536	−	0.861758i
$170$	1.37856			0.105731
$171$		0			0
$172$	4.15770			0.317022
$173$	2.50992	−	4.34730i	0.190825	−	0.330519i	−0.754699	−	0.656072i	$-0.772217\pi$
$173$	2.50992	−	4.34730i	0.190825	−	0.330519i	0.945524	+	0.325552i	$0.105550\pi$
$174$		0			0
$175$	9.20340	+	15.9408i	0.695711	+	1.20501i
$176$	2.47690	−	4.29012i	0.186703	−	0.323380i
$177$		0			0
$178$	2.06108			0.154485
$179$	−24.1826			−1.80749			−0.903747	−	0.428066i	$-0.859195\pi$
$179$	−24.1826			−1.80749			−0.903747	+	0.428066i	$0.859195\pi$
$180$		0			0
$181$	−5.09176	−	8.81918i	−0.378467	−	0.655524i	0.612372	−	0.790570i	$-0.290215\pi$
$181$	−5.09176	−	8.81918i	−0.378467	−	0.655524i	−0.990839	+	0.135045i	$0.956882\pi$
$182$	−1.81085			−0.134229
$183$		0			0
$184$	−2.00805	−	3.47804i	−0.148035	−	0.256405i
$185$	−0.789539	+	1.36752i	−0.0580481	+	0.100542i
$186$		0			0
$187$	1.96459	−	3.40277i	0.143665	−	0.248835i
$188$	2.16433	−	3.74873i	0.157850	−	0.273404i
$189$		0			0
$190$	1.33074	+	1.03969i	0.0965424	+	0.0754270i
$191$	−17.5661			−1.27104			−0.635521	−	0.772084i	$-0.719215\pi$
$191$	−17.5661			−1.27104			−0.635521	+	0.772084i	$0.719215\pi$
$192$		0			0
$193$	−9.82521	+	17.0178i	−0.707234	+	1.22497i	0.258645	+	0.965972i	$0.416724\pi$
$193$	−9.82521	+	17.0178i	−0.707234	+	1.22497i	−0.965879	+	0.258993i	$0.916609\pi$
$194$	6.16333	+	10.6752i	0.442501	+	0.766434i
$195$		0			0
$196$	−5.81484	−	10.0716i	−0.415346	−	0.719400i
$197$	11.7871			0.839796			0.419898	−	0.907571i	$-0.362066\pi$
$197$	11.7871			0.839796			0.419898	+	0.907571i	$0.362066\pi$
$198$		0			0
$199$	−8.67627	−	15.0277i	−0.615045	−	1.06529i	−0.990377	−	0.138398i	$-0.955805\pi$
$199$	−8.67627	−	15.0277i	−0.615045	−	1.06529i	0.375332	−	0.926890i	$-0.377529\pi$
$200$	−1.35387	−	2.34498i	−0.0957333	−	0.165815i
$201$		0			0
$202$	−7.86265			−0.553214
$203$	3.46986	+	6.00997i	0.243536	+	0.421817i
$204$		0			0
$205$	0.281242	+	0.487125i	0.0196428	+	0.0340223i
$206$	−14.1001	+	24.4221i	−0.982402	+	1.70157i
$207$		0			0
$208$	1.13539			0.0787250
$209$	4.46277	−	1.80308i	0.308696	−	0.124722i
$210$		0			0
$211$	−6.43806	+	11.1510i	−0.443214	+	0.767669i	−0.997926	−	0.0643732i	$-0.979495\pi$
$211$	−6.43806	+	11.1510i	−0.443214	+	0.767669i	0.554712	+	0.832043i	$0.312829\pi$
$212$	−4.90039	+	8.48773i	−0.336560	+	0.582940i
$213$		0			0
$214$	−10.3136	+	17.8637i	−0.705024	+	1.22114i
$215$	0.243333	+	0.421465i	0.0165952	+	0.0287437i
$216$		0			0
$217$	−15.3032			−1.03885
$218$	9.51015	+	16.4721i	0.644109	+	1.11563i
$219$		0			0
$220$	−0.380965			−0.0256846
$221$	0.900549			0.0605775
$222$		0			0
$223$	−0.452663	+	0.784036i	−0.0303126	+	0.0525029i	−0.880784	−	0.473519i	$-0.842984\pi$
$223$	−0.452663	+	0.784036i	−0.0303126	+	0.0525029i	0.850471	+	0.526022i	$0.176317\pi$
$224$	14.0233	+	24.2890i	0.936970	+	1.62288i
$225$		0			0
$226$	−14.4935	+	25.1036i	−0.964096	+	1.66986i
$227$	2.26554			0.150369			0.0751845	−	0.997170i	$-0.476045\pi$
$227$	2.26554			0.150369			0.0751845	+	0.997170i	$0.476045\pi$
$228$		0			0
$229$	−11.1098			−0.734156			−0.367078	−	0.930190i	$-0.619642\pi$
$229$	−11.1098			−0.734156			−0.367078	+	0.930190i	$0.619642\pi$
$230$	−1.42495	+	2.46809i	−0.0939585	+	0.162741i
$231$		0			0
$232$	−0.510436	−	0.884101i	−0.0335118	−	0.0580441i
$233$	−10.0569	+	17.4190i	−0.658847	+	1.14116i	0.322068	+	0.946717i	$0.395622\pi$
$233$	−10.0569	+	17.4190i	−0.658847	+	1.14116i	−0.980915	+	0.194440i	$0.937711\pi$
$234$		0			0
$235$	0.506676			0.0330519
$236$	−24.7811			−1.61311
$237$		0			0
$238$	12.7299	+	22.0488i	0.825157	+	1.42921i
$239$	−10.2465			−0.662793			−0.331396	−	0.943492i	$-0.607520\pi$
$239$	−10.2465			−0.662793			−0.331396	+	0.943492i	$0.607520\pi$
$240$		0			0
$241$	−5.28060	−	9.14627i	−0.340153	−	0.589163i	0.644308	−	0.764766i	$-0.277146\pi$
$241$	−5.28060	−	9.14627i	−0.340153	−	0.589163i	−0.984461	+	0.175603i	$0.943812\pi$
$242$	9.42809	−	16.3299i	0.606061	−	1.04973i
$243$		0			0
$244$	4.25116	−	7.36322i	0.272152	−	0.471382i
$245$	0.680637	−	1.17890i	0.0434843	−	0.0753170i
$246$		0			0
$247$	0.869314	+	0.679181i	0.0553132	+	0.0432153i
$248$	2.25120			0.142951
$249$		0			0
$250$	−1.92929	+	3.34163i	−0.122019	+	0.211343i
$251$	−12.2800	−	21.2697i	−0.775109	−	1.34253i	−0.934733	−	0.355351i	$-0.884362\pi$
$251$	−12.2800	−	21.2697i	−0.775109	−	1.34253i	0.159624	−	0.987178i	$-0.448972\pi$
$252$		0			0
$253$	4.06141	+	7.03456i	0.255338	+	0.442259i
$254$	31.8303			1.99721
$255$		0			0
$256$	−9.76484	−	16.9132i	−0.610302	−	1.05707i
$257$	8.09227	+	14.0162i	0.504782	+	0.874308i	0.999985	+	0.00553074i	$0.00176050\pi$
$257$	8.09227	+	14.0162i	0.504782	+	0.874308i	−0.495203	+	0.868778i	$0.664906\pi$
$258$		0			0
$259$	−29.1631			−1.81211
$260$	−0.0436577	−	0.0756173i	−0.00270753	−	0.00468959i
$261$		0			0
$262$	7.36257	+	12.7523i	0.454861	+	0.787842i
$263$	−6.42369	+	11.1262i	−0.396101	+	0.686068i	−0.993241	−	0.116069i	$-0.962970\pi$
$263$	−6.42369	+	11.1262i	−0.396101	+	0.686068i	0.597140	+	0.802137i	$0.296304\pi$
$264$		0			0
$265$	−1.14720			−0.0704718
$266$	−4.34054	+	30.8848i	−0.266136	+	1.89367i
$267$		0			0
$268$	4.46344	−	7.73091i	0.272648	−	0.472240i
$269$	−4.99697	+	8.65500i	−0.304671	+	0.527705i	−0.977188	−	0.212376i	$-0.931880\pi$
$269$	−4.99697	+	8.65500i	−0.304671	+	0.527705i	0.672517	+	0.740081i	$0.265213\pi$
$270$		0			0
$271$	8.40716	−	14.5616i	0.510699	−	0.884556i	−0.489224	−	0.872158i	$-0.662720\pi$
$271$	8.40716	−	14.5616i	0.510699	−	0.884556i	0.999923	−	0.0123982i	$-0.00394656\pi$
$272$	−7.98154	−	13.8244i	−0.483952	−	0.838229i
$273$		0			0
$274$	−4.21022			−0.254349
$275$	2.73829	+	4.74286i	0.165125	+	0.286005i
$276$		0			0
$277$	−3.77985			−0.227109			−0.113555	−	0.993532i	$-0.536224\pi$
$277$	−3.77985			−0.227109			−0.113555	+	0.993532i	$0.536224\pi$
$278$	32.6122			1.95595
$279$		0			0
$280$	−0.203592	+	0.352631i	−0.0121669	+	0.0210737i
$281$	11.5655	+	20.0320i	0.689939	+	1.19501i	0.971857	+	0.235571i	$0.0756959\pi$
$281$	11.5655	+	20.0320i	0.689939	+	1.19501i	−0.281918	+	0.959438i	$0.590971\pi$
$282$		0			0
$283$	10.0506	−	17.4082i	0.597449	−	1.03481i	−0.395748	−	0.918359i	$-0.629514\pi$
$283$	10.0506	−	17.4082i	0.597449	−	1.03481i	0.993196	−	0.116452i	$-0.0371522\pi$
$284$	−1.20274			−0.0713697
$285$		0			0
$286$	−0.538784			−0.0318590
$287$	−5.19409	+	8.99644i	−0.306598	+	0.531043i
$288$		0			0
$289$	2.16933	+	3.75740i	0.127608	+	0.221023i
$290$	−0.362215	+	0.627375i	−0.0212700	+	0.0368407i
$291$		0			0
$292$	1.52687			0.0893532
$293$	6.78228			0.396225			0.198113	−	0.980179i	$-0.436519\pi$
$293$	6.78228			0.396225			0.198113	+	0.980179i	$0.436519\pi$
$294$		0			0
$295$	−1.45034	−	2.51206i	−0.0844418	−	0.146258i
$296$	4.29006			0.249355
$297$		0			0
$298$	20.5888	+	35.6609i	1.19268	+	2.06578i
$299$	−0.930855	+	1.61229i	−0.0538328	+	0.0932411i
$300$		0			0
$301$	−4.49397	+	7.78379i	−0.259028	+	0.448650i
$302$	15.4456	−	26.7526i	0.888794	−	1.53944i
$303$		0			0
$304$	2.72148	−	19.3645i	0.156088	−	1.11063i
$305$	0.995209			0.0569855
$306$		0			0
$307$	−6.64501	+	11.5095i	−0.379251	+	0.656881i	−0.990953	−	0.134206i	$-0.957152\pi$
$307$	−6.64501	+	11.5095i	−0.379251	+	0.656881i	0.611703	+	0.791088i	$0.290485\pi$
$308$	−3.51791	−	6.09320i	−0.200451	−	0.347192i
$309$		0			0
$310$	−0.798746	−	1.38347i	−0.0453658	−	0.0785758i
$311$	−28.1106			−1.59400			−0.797002	−	0.603976i	$-0.793582\pi$
$311$	−28.1106			−1.59400			−0.797002	+	0.603976i	$0.793582\pi$
$312$		0			0
$313$	−2.93523	−	5.08397i	−0.165909	−	0.287363i	0.771069	−	0.636752i	$-0.219723\pi$
$313$	−2.93523	−	5.08397i	−0.165909	−	0.287363i	−0.936978	+	0.349389i	$0.886389\pi$
$314$	−20.3278	−	35.2088i	−1.14716	−	1.98695i
$315$		0			0
$316$	20.9202			1.17685
$317$	9.21984	+	15.9692i	0.517838	+	0.896922i	0.999785	+	0.0207215i	$0.00659635\pi$
$317$	9.21984	+	15.9692i	0.517838	+	0.896922i	−0.481947	+	0.876200i	$0.660070\pi$
$318$		0			0
$319$	1.03239	+	1.78815i	0.0578026	+	0.100117i
$320$	−0.562356	+	0.974029i	−0.0314367	+	0.0544499i
$321$		0			0
$322$	−52.6332			−2.93314
$323$	2.15858	−	15.3592i	0.120107	−	0.854610i
$324$		0			0
$325$	−0.627604	+	1.08704i	−0.0348132	+	0.0602982i
$326$	−16.0912	+	27.8708i	−0.891210	+	1.54362i
$327$		0			0
$328$	0.764081	−	1.32343i	0.0421893	−	0.0730741i
$329$	4.67876	+	8.10385i	0.257948	+	0.446780i
$330$		0			0
$331$	−32.6088			−1.79234			−0.896171	−	0.443708i	$-0.853663\pi$
$331$	−32.6088			−1.79234			−0.896171	+	0.443708i	$0.853663\pi$
$332$	14.2116	+	24.6151i	0.779960	+	1.35093i
$333$		0			0
$334$	29.1744			1.59635
$335$	1.04491			0.0570893
$336$		0			0
$337$	11.0880	−	19.2050i	0.604003	−	1.04616i	−0.388206	−	0.921573i	$-0.626905\pi$
$337$	11.0880	−	19.2050i	0.604003	−	1.04616i	0.992208	−	0.124590i	$-0.0397617\pi$
$338$	12.4696	+	21.5980i	0.678258	+	1.17478i
$339$		0			0
$340$	−0.613808	+	1.06315i	−0.0332884	+	0.0576572i
$341$	−4.55318			−0.246569
$342$		0			0
$343$	−0.838759			−0.0452887
$344$	0.661090	−	1.14504i	0.0356436	−	0.0617365i
$345$		0			0
$346$	4.83888	+	8.38118i	0.260140	+	0.450575i
$347$	4.91160	−	8.50715i	0.263669	−	0.456688i	−0.703545	−	0.710650i	$-0.748401\pi$
$347$	4.91160	−	8.50715i	0.263669	−	0.456688i	0.967214	+	0.253963i	$0.0817341\pi$
$348$		0			0
$349$	24.9982			1.33812			0.669061	−	0.743208i	$-0.266697\pi$
$349$	24.9982			1.33812			0.669061	+	0.743208i	$0.266697\pi$
$350$	−35.4865			−1.89683
$351$		0			0
$352$	4.17236	+	7.22673i	0.222387	+	0.385186i
$353$	−29.6800			−1.57971			−0.789853	−	0.613296i	$-0.789843\pi$
$353$	−29.6800			−1.57971			−0.789853	+	0.613296i	$0.789843\pi$
$354$		0			0
$355$	−0.0703915	−	0.121922i	−0.00373599	−	0.00647093i
$356$	−0.917705	+	1.58951i	−0.0486383	+	0.0842440i
$357$		0			0
$358$	23.3109	−	40.3756i	1.23202	−	2.13392i
$359$	−0.606086	+	1.04977i	−0.0319880	+	0.0554049i	−0.881576	−	0.472042i	$-0.843517\pi$
$359$	−0.606086	+	1.04977i	−0.0319880	+	0.0554049i	0.849588	+	0.527447i	$0.176851\pi$
$360$		0			0
$361$	13.6674	−	13.1985i	0.719338	−	0.694660i
$362$	19.6328			1.03188
$363$		0			0
$364$	0.806288	−	1.39653i	0.0422610	−	0.0731982i
$365$	0.0893612	+	0.154778i	0.00467738	+	0.00810146i
$366$		0			0
$367$	16.5923	+	28.7387i	0.866110	+	1.50015i	0.865941	+	0.500146i	$0.166720\pi$
$367$	16.5923	+	28.7387i	0.866110	+	1.50015i	0.000168352		1.00000i	$0.499946\pi$
$368$	33.0006			1.72027
$369$		0			0
$370$	−1.52215	−	2.63645i	−0.0791331	−	0.137062i
$371$	−10.5935	−	18.3484i	−0.549986	−	0.952603i
$372$		0			0
$373$	−2.50301			−0.129601			−0.0648005	−	0.997898i	$-0.520641\pi$
$373$	−2.50301			−0.129601			−0.0648005	+	0.997898i	$0.520641\pi$
$374$	3.78753	+	6.56020i	0.195849	+	0.339220i
$375$		0			0
$376$	−0.688272	−	1.19212i	−0.0354949	−	0.0614790i
$377$	−0.236619	+	0.409835i	−0.0121865	+	0.0211076i
$378$		0			0
$379$	19.9040			1.02240			0.511200	−	0.859462i	$-0.329201\pi$
$379$	19.9040			1.02240			0.511200	+	0.859462i	$0.329201\pi$
$380$	−1.39433	+	0.563347i	−0.0715276	+	0.0288991i
$381$		0			0
$382$	16.9329	−	29.3287i	0.866363	−	1.50059i
$383$	−5.61569	+	9.72666i	−0.286948	+	0.497009i	−0.973080	−	0.230469i	$-0.925974\pi$
$383$	−5.61569	+	9.72666i	−0.286948	+	0.497009i	0.686132	+	0.727477i	$0.259307\pi$
$384$		0			0
$385$	0.411777	−	0.713218i	0.0209861	−	0.0363490i
$386$	−18.9421	−	32.8086i	−0.964125	−	1.66991i
$387$		0			0
$388$	−10.9770			−0.557271
$389$	−6.85978	−	11.8815i	−0.347805	−	0.602416i	0.638054	−	0.769991i	$-0.279739\pi$
$389$	−6.85978	−	11.8815i	−0.347805	−	0.602416i	−0.985859	+	0.167576i	$0.946406\pi$
$390$		0			0
$391$	26.1749			1.32372
$392$	−3.69832			−0.186794
$393$		0			0
$394$	−11.3622	+	19.6799i	−0.572419	+	0.991459i
$395$	1.22437	+	2.12067i	0.0616047	+	0.106703i
$396$		0			0
$397$	11.4421	−	19.8182i	0.574261	−	0.994649i	−0.421861	−	0.906661i	$-0.638623\pi$
$397$	11.4421	−	19.8182i	0.574261	−	0.994649i	0.996122	−	0.0879883i	$-0.0280438\pi$
$398$	33.4540			1.67690
$399$		0			0
$400$	22.2497			1.11249
$401$	−15.8845	+	27.5128i	−0.793235	+	1.37392i	0.130719	+	0.991419i	$0.458271\pi$
$401$	−15.8845	+	27.5128i	−0.793235	+	1.37392i	−0.923954	+	0.382504i	$0.875062\pi$
$402$		0			0
$403$	−0.521784	−	0.903757i	−0.0259919	−	0.0450193i
$404$	3.50087	−	6.06369i	0.174175	−	0.301680i
$405$		0			0
$406$	−13.3791			−0.663993
$407$	−8.67691			−0.430098
$408$		0			0
$409$	2.44119	+	4.22827i	0.120709	+	0.209075i	0.920048	−	0.391807i	$-0.128150\pi$
$409$	2.44119	+	4.22827i	0.120709	+	0.209075i	−0.799338	+	0.600881i	$0.794816\pi$
$410$	−1.08441			−0.0535554
$411$		0			0
$412$	−12.5563	−	21.7481i	−0.618603	−	1.07145i
$413$	26.7854	−	46.3937i	1.31802	−	2.28289i
$414$		0			0
$415$	−1.66349	+	2.88124i	−0.0816573	+	0.141435i
$416$	−0.956285	+	1.65633i	−0.0468857	+	0.0812085i
$417$		0			0
$418$	−1.29144	+	9.18918i	−0.0631666	+	0.449457i
$419$	−36.6816			−1.79201			−0.896006	−	0.444041i	$-0.853544\pi$
$419$	−36.6816			−1.79201			−0.896006	+	0.444041i	$0.853544\pi$
$420$		0			0
$421$	−4.25458	+	7.36914i	−0.207355	+	0.359150i	−0.950881	−	0.309558i	$-0.899819\pi$
$421$	−4.25458	+	7.36914i	−0.207355	+	0.359150i	0.743525	+	0.668708i	$0.233152\pi$
$422$	−12.4120	−	21.4981i	−0.604204	−	1.04651i
$423$		0			0
$424$	1.55836	+	2.69916i	0.0756807	+	0.131083i
$425$	17.6477			0.856039
$426$		0			0
$427$	9.18997	+	15.9175i	0.444734	+	0.770302i
$428$	−9.18434	−	15.9077i	−0.443942	−	0.768930i
$429$		0			0
$430$	−0.938245			−0.0452462
$431$	−9.82913	−	17.0246i	−0.473453	−	0.820044i	0.526086	−	0.850432i	$-0.323659\pi$
$431$	−9.82913	−	17.0246i	−0.473453	−	0.820044i	−0.999538	+	0.0303877i	$0.990326\pi$
$432$		0			0
$433$	0.214463	+	0.371461i	0.0103064	+	0.0178513i	0.871133	−	0.491048i	$-0.163386\pi$
$433$	0.214463	+	0.371461i	0.0103064	+	0.0178513i	−0.860826	+	0.508899i	$0.830053\pi$
$434$	14.7516	−	25.5505i	0.708099	−	1.22646i
$435$		0			0
$436$	−16.9377			−0.811170
$437$	25.2670	+	19.7407i	1.20869	+	0.944326i
$438$		0			0
$439$	9.03327	−	15.6461i	0.431134	−	0.746746i	−0.565837	−	0.824517i	$-0.691447\pi$
$439$	9.03327	−	15.6461i	0.431134	−	0.746746i	0.996971	+	0.0777706i	$0.0247802\pi$
$440$	−0.0605747	+	0.104919i	−0.00288779	+	0.00500179i
$441$		0			0
$442$	−0.868086	+	1.50357i	−0.0412906	+	0.0715175i
$443$	−12.9787	−	22.4798i	−0.616636	−	1.06805i	−0.990095	−	0.140398i	$-0.955162\pi$
$443$	−12.9787	−	22.4798i	−0.616636	−	1.06805i	0.373459	−	0.927647i	$-0.378172\pi$
$444$		0			0
$445$	−0.214838			−0.0101843
$446$	−0.872692	−	1.51155i	−0.0413231	−	0.0715738i
$447$		0			0
$448$	−20.7717			−0.981369
$449$	22.5625			1.06479			0.532395	−	0.846496i	$-0.321292\pi$
$449$	22.5625			1.06479			0.532395	+	0.846496i	$0.321292\pi$
$450$		0			0
$451$	−1.54540	+	2.67671i	−0.0727701	+	0.126042i
$452$	−12.9066	−	22.3549i	−0.607076	−	1.05149i
$453$		0			0
$454$	−2.18387	+	3.78257i	−0.102494	+	0.177525i
$455$	0.188755			0.00884896
$456$		0			0
$457$	−9.51926			−0.445292			−0.222646	−	0.974899i	$-0.571469\pi$
$457$	−9.51926			−0.445292			−0.222646	+	0.974899i	$0.571469\pi$
$458$	10.7093	−	18.5491i	0.500413	−	0.866741i
$459$		0			0
$460$	−1.26893	−	2.19785i	−0.0591642	−	0.102475i
$461$	−6.19201	+	10.7249i	−0.288391	+	0.499507i	−0.973426	−	0.229003i	$-0.926453\pi$
$461$	−6.19201	+	10.7249i	−0.288391	+	0.499507i	0.685035	+	0.728510i	$0.259787\pi$
$462$		0			0
$463$	−6.35948			−0.295550			−0.147775	−	0.989021i	$-0.547211\pi$
$463$	−6.35948			−0.295550			−0.147775	+	0.989021i	$0.547211\pi$
$464$	8.38857			0.389430
$465$		0			0
$466$	−19.3887	−	33.5821i	−0.898162	−	1.55566i
$467$	−8.32816			−0.385381			−0.192691	−	0.981260i	$-0.561721\pi$
$467$	−8.32816			−0.385381			−0.192691	+	0.981260i	$0.561721\pi$
$468$		0			0
$469$	9.64888	+	16.7124i	0.445544	+	0.771705i
$470$	−0.488412	+	0.845954i	−0.0225287	+	0.0390209i
$471$		0			0
$472$	−3.94029	+	6.82478i	−0.181367	+	0.314136i
$473$	−1.33709	+	2.31592i	−0.0614797	+	0.106486i
$474$		0			0
$475$	17.0356	+	13.3096i	0.781647	+	0.610688i
$476$	−22.6721			−1.03918
$477$		0			0
$478$	9.87716	−	17.1077i	0.451771	−	0.782490i
$479$	15.1768	+	26.2870i	0.693445	+	1.20108i	0.970702	+	0.240286i	$0.0772413\pi$
$479$	15.1768	+	26.2870i	0.693445	+	1.20108i	−0.277257	+	0.960796i	$0.589425\pi$
$480$		0			0
$481$	−0.994354	−	1.72227i	−0.0453386	−	0.0785288i
$482$	20.3610			0.927417
$483$		0			0
$484$	8.39578	+	14.5419i	0.381626	+	0.660996i
$485$	−0.642436	−	1.11273i	−0.0291715	−	0.0505266i
$486$		0			0
$487$	−23.9955			−1.08734			−0.543669	−	0.839300i	$-0.682965\pi$
$487$	−23.9955			−1.08734			−0.543669	+	0.839300i	$0.682965\pi$
$488$	−1.35190	−	2.34156i	−0.0611975	−	0.105997i
$489$		0			0
$490$	1.31220	+	2.27280i	0.0592792	+	0.102675i
$491$	16.7317	−	28.9802i	0.755092	−	1.30786i	−0.190237	−	0.981738i	$-0.560926\pi$
$491$	16.7317	−	28.9802i	0.755092	−	1.30786i	0.945329	−	0.326119i	$-0.105741\pi$
$492$		0			0
$493$	6.65351			0.299659
$494$	−1.97195	+	0.796721i	−0.0887221	+	0.0358462i
$495$		0			0
$496$	−9.24912	+	16.0199i	−0.415298	+	0.719316i
$497$	1.30002	−	2.25170i	0.0583139	−	0.101003i
$498$		0			0
$499$	8.52440	−	14.7647i	0.381605	−	0.660958i	−0.609687	−	0.792642i	$-0.708705\pi$
$499$	8.52440	−	14.7647i	0.381605	−	0.660958i	0.991292	+	0.131684i	$0.0420383\pi$
$500$	−1.71805	−	2.97575i	−0.0768335	−	0.133080i
$501$		0			0
$502$	47.3495			2.11331
$503$	4.74902	+	8.22554i	0.211748	+	0.366759i	0.952262	−	0.305283i	$-0.0987510\pi$
$503$	4.74902	+	8.22554i	0.211748	+	0.366759i	−0.740514	+	0.672041i	$0.765418\pi$
$504$		0			0
$505$	0.819565			0.0364702
$506$	−15.6600			−0.696172
$507$		0			0
$508$	−14.1725	+	24.5476i	−0.628805	+	1.08912i
$509$	6.51224	+	11.2795i	0.288650	+	0.499956i	0.973488	−	0.228739i	$-0.0734603\pi$
$509$	6.51224	+	11.2795i	0.288650	+	0.499956i	−0.684838	+	0.728696i	$0.740127\pi$
$510$		0			0
$511$	−1.65036	+	2.85851i	−0.0730077	+	0.126453i
$512$	29.0036			1.28179
$513$		0			0
$514$	−31.2022			−1.37627
$515$	1.46973	−	2.54565i	0.0647641	−	0.112175i
$516$		0			0
$517$	1.39207	+	2.41114i	0.0612233	+	0.106042i
$518$	28.1118	−	48.6911i	1.23516	−	2.13936i
$519$		0			0
$520$	−0.0277669			−0.00121766
$521$	−21.7645			−0.953522			−0.476761	−	0.879033i	$-0.658189\pi$
$521$	−21.7645			−0.953522			−0.476761	+	0.879033i	$0.658189\pi$
$522$		0			0
$523$	9.84381	+	17.0500i	0.430440	+	0.745544i	0.996911	−	0.0785379i	$-0.0250251\pi$
$523$	9.84381	+	17.0500i	0.430440	+	0.745544i	−0.566471	+	0.824081i	$0.691692\pi$
$524$	−13.1128			−0.572837
$525$		0			0
$526$	−12.3842	−	21.4501i	−0.539979	−	0.935271i
$527$	−7.33607	+	12.7064i	−0.319564	+	0.553501i
$528$		0			0
$529$	−15.5557	+	26.9433i	−0.676336	+	1.17145i
$530$	1.10584	−	1.91538i	0.0480348	−	0.0831987i
$531$		0			0
$532$	−21.8858	−	17.0990i	−0.948869	−	0.741336i
$533$	−0.708398			−0.0306841
$534$		0			0
$535$	1.07504	−	1.86203i	0.0464781	−	0.0805025i
$536$	−1.41941	−	2.45848i	−0.0613090	−	0.106190i
$537$		0			0
$538$	−9.63368	−	16.6860i	−0.415337	−	0.719385i
$539$	7.48009			0.322190
$540$		0			0
$541$	−8.27531	−	14.3333i	−0.355783	−	0.616235i	0.631468	−	0.775402i	$-0.282453\pi$
$541$	−8.27531	−	14.3333i	−0.355783	−	0.616235i	−0.987252	+	0.159167i	$0.949119\pi$
$542$	16.2082	+	28.0734i	0.696202	+	1.20586i
$543$		0			0
$544$	26.8899			1.15290
$545$	−0.991294	−	1.71697i	−0.0424624	−	0.0735470i
$546$		0			0
$547$	−1.42152	−	2.46214i	−0.0607797	−	0.105274i	0.834034	−	0.551712i	$-0.186025\pi$
$547$	−1.42152	−	2.46214i	−0.0607797	−	0.105274i	−0.894814	+	0.446439i	$0.852692\pi$
$548$	1.87462	−	3.24693i	0.0800797	−	0.138702i
$549$		0			0
$550$	−10.5583			−0.450209
$551$	6.42274	+	5.01799i	0.273618	+	0.213773i
$552$		0			0
$553$	−22.6122	+	39.1655i	−0.961568	+	1.66548i
$554$	3.64360	−	6.31090i	0.154802	−	0.268124i
$555$		0			0
$556$	−14.5207	+	25.1506i	−0.615816	+	1.06662i
$557$	−0.233985	−	0.405274i	−0.00991427	−	0.0171720i	0.861026	−	0.508561i	$-0.169823\pi$
$557$	−0.233985	−	0.405274i	−0.00991427	−	0.0171720i	−0.870940	+	0.491389i	$0.836489\pi$
$558$		0			0
$559$	−0.612912			−0.0259234
$560$	−1.67293	−	2.89759i	−0.0706941	−	0.122446i
$561$		0			0
$562$	−44.5943			−1.88110
$563$	−27.4971			−1.15887			−0.579433	−	0.815020i	$-0.696726\pi$
$563$	−27.4971			−1.15887			−0.579433	+	0.815020i	$0.696726\pi$
$564$		0			0
$565$	1.51074	−	2.61668i	0.0635573	−	0.110084i
$566$	19.3767	+	33.5614i	0.814462	+	1.41069i
$567$		0			0
$568$	−0.191240	+	0.331238i	−0.00802427	+	0.0138984i
$569$	27.8501			1.16754			0.583769	−	0.811920i	$-0.301577\pi$
$569$	27.8501			1.16754			0.583769	+	0.811920i	$0.301577\pi$
$570$		0			0
$571$	−20.9629			−0.877269			−0.438635	−	0.898665i	$-0.644538\pi$
$571$	−20.9629			−0.877269			−0.438635	+	0.898665i	$0.644538\pi$
$572$	0.239895	−	0.415511i	0.0100305	−	0.0173734i
$573$		0			0
$574$	−10.0137	−	17.3443i	−0.417964	−	0.723935i
$575$	−18.2416	+	31.5954i	−0.760727	+	1.31762i
$576$		0			0
$577$	−3.53779			−0.147280			−0.0736401	−	0.997285i	$-0.523462\pi$
$577$	−3.53779			−0.147280			−0.0736401	+	0.997285i	$0.523462\pi$
$578$	−8.36453			−0.347919
$579$		0			0
$580$	−0.322555	−	0.558682i	−0.0133934	−	0.0231980i
$581$	−61.4439			−2.54912
$582$		0			0
$583$	−3.15188	−	5.45922i	−0.130538	−	0.226098i
$584$	0.242778	−	0.420503i	0.0100462	−	0.0174005i
$585$		0			0
$586$	−6.53779	+	11.3238i	−0.270074	+	0.467781i
$587$	20.9145	−	36.2250i	0.863234	−	1.49516i	−0.00555648	−	0.999985i	$-0.501769\pi$
$587$	20.9145	−	36.2250i	0.863234	−	1.49516i	0.868790	−	0.495180i	$-0.164898\pi$
$588$		0			0
$589$	−16.6646	+	6.73297i	−0.686655	+	0.277427i
$590$	5.59222			0.230228
$591$		0			0
$592$	−17.6259	+	30.5289i	−0.724418	+	1.25473i
$593$	−10.4661	−	18.1279i	−0.429793	−	0.744422i	0.567062	−	0.823675i	$-0.308080\pi$
$593$	−10.4661	−	18.1279i	−0.429793	−	0.744422i	−0.996855	+	0.0792526i	$0.974747\pi$
$594$		0			0
$595$	−1.32691	−	2.29827i	−0.0543978	−	0.0942198i
$596$	−36.6690			−1.50202
$597$		0			0
$598$	−1.79460	−	3.10834i	−0.0733866	−	0.127109i
$599$	14.8465	+	25.7149i	0.606611	+	1.05068i	0.991795	+	0.127841i	$0.0408047\pi$
$599$	14.8465	+	25.7149i	0.606611	+	1.05068i	−0.385184	+	0.922840i	$0.625862\pi$
$600$		0			0
$601$	26.5584			1.08334			0.541669	−	0.840592i	$-0.317793\pi$
$601$	26.5584			1.08334			0.541669	+	0.840592i	$0.317793\pi$
$602$	−8.66395	−	15.0064i	−0.353116	−	0.611615i
$603$		0			0
$604$	13.7544	+	23.8234i	0.559659	+	0.969358i
$605$	−0.982740	+	1.70216i	−0.0399541	+	0.0692024i
$606$		0			0
$607$	39.6964			1.61123			0.805613	−	0.592442i	$-0.201836\pi$
$607$	39.6964			1.61123			0.805613	+	0.592442i	$0.201836\pi$
$608$	25.9573	+	20.2800i	1.05271	+	0.822463i
$609$		0			0
$610$	−0.959334	+	1.66161i	−0.0388423	+	0.0672768i
$611$	−0.319057	+	0.552623i	−0.0129077	+	0.0223567i
$612$		0			0
$613$	9.71124	−	16.8204i	0.392233	−	0.679368i	−0.600511	−	0.799617i	$-0.705036\pi$
$613$	9.71124	−	16.8204i	0.392233	−	0.679368i	0.992744	+	0.120249i	$0.0383693\pi$
$614$	−12.8109	−	22.1892i	−0.517007	−	0.895483i
$615$		0			0
$616$	−2.23744			−0.0901490
$617$	−24.0124	−	41.5908i	−0.966704	−	1.67438i	−0.704964	−	0.709243i	$-0.749037\pi$
$617$	−24.0124	−	41.5908i	−0.966704	−	1.67438i	−0.261740	−	0.965138i	$-0.584296\pi$
$618$		0			0
$619$	−1.57935			−0.0634794			−0.0317397	−	0.999496i	$-0.510105\pi$
$619$	−1.57935			−0.0634794			−0.0317397	+	0.999496i	$0.510105\pi$
$620$	1.42258			0.0571322
$621$		0			0
$622$	27.0972	−	46.9338i	1.08650	−	1.88187i
$623$	−1.98386	−	3.43614i	−0.0794816	−	0.137666i
$624$		0			0
$625$	−12.1979	+	21.1275i	−0.487918	+	0.845098i
$626$	11.3177			0.452346
$627$		0			0
$628$	36.2041			1.44470
$629$	−13.9802	+	24.2144i	−0.557427	+	0.965491i
$630$		0			0
$631$	4.28754	+	7.42624i	0.170684	+	0.295634i	0.938659	−	0.344846i	$-0.112069\pi$
$631$	4.28754	+	7.42624i	0.170684	+	0.295634i	−0.767975	+	0.640480i	$0.778735\pi$
$632$	3.32638	−	5.76147i	0.132316	−	0.229179i
$633$		0			0
$634$	−35.5499			−1.41187
$635$	−3.31784			−0.131664
$636$		0			0
$637$	0.857201	+	1.48472i	0.0339636	+	0.0588266i
$638$	−3.98069			−0.157597
$639$		0			0
$640$	0.434455	+	0.752497i	0.0171733	+	0.0297451i
$641$	4.72915	−	8.19112i	0.186790	−	0.323530i	−0.757388	−	0.652965i	$-0.773525\pi$
$641$	4.72915	−	8.19112i	0.186790	−	0.323530i	0.944178	+	0.329435i	$0.106858\pi$
$642$		0			0
$643$	−7.42218	+	12.8556i	−0.292702	+	0.506975i	−0.974448	−	0.224614i	$-0.927888\pi$
$643$	−7.42218	+	12.8556i	−0.292702	+	0.506975i	0.681746	+	0.731589i	$0.261221\pi$
$644$	23.4351	−	40.5909i	0.923474	−	1.59950i
$645$		0			0
$646$	23.5632	+	18.4096i	0.927082	+	0.724314i
$647$	4.84046			0.190298			0.0951491	−	0.995463i	$-0.469667\pi$
$647$	4.84046			0.190298			0.0951491	+	0.995463i	$0.469667\pi$
$648$		0			0
$649$	7.96948	−	13.8035i	0.312830	−	0.541837i
$650$	−1.20996	−	2.09571i	−0.0474585	−	0.0822006i
$651$		0			0
$652$	−14.3294	−	24.8192i	−0.561181	−	0.971993i
$653$	−22.1655			−0.867404			−0.433702	−	0.901056i	$-0.642793\pi$
$653$	−22.1655			−0.867404			−0.433702	+	0.901056i	$0.642793\pi$
$654$		0			0
$655$	−0.767440	−	1.32924i	−0.0299863	−	0.0519379i
$656$	6.27851	+	10.8747i	0.245135	+	0.424586i
$657$		0			0
$658$	−18.0404			−0.703288
$659$	9.10369	+	15.7681i	0.354629	+	0.614236i	0.987054	−	0.160385i	$-0.0512737\pi$
$659$	9.10369	+	15.7681i	0.354629	+	0.614236i	−0.632425	+	0.774622i	$0.717940\pi$
$660$		0			0
$661$	−18.4435	−	31.9451i	−0.717369	−	1.24252i	−0.962039	−	0.272913i	$-0.912013\pi$
$661$	−18.4435	−	31.9451i	−0.717369	−	1.24252i	0.244670	−	0.969606i	$-0.421320\pi$
$662$	31.4333	−	54.4441i	1.22169	−	2.11603i
$663$		0			0
$664$	9.03875			0.350772
$665$	0.452438	−	3.21929i	0.0175448	−	0.124839i
$666$		0			0
$667$	−6.87743	+	11.9121i	−0.266295	+	0.461237i
$668$	−12.9900	+	22.4994i	−0.502598	+	0.870526i
$669$		0			0
$670$	−1.00724	+	1.74459i	−0.0389130	+	0.0673993i
$671$	2.73430	+	4.73594i	0.105556	+	0.182829i
$672$		0			0
$673$	26.9092			1.03727			0.518636	−	0.854995i	$-0.326440\pi$
$673$	26.9092			1.03727			0.518636	+	0.854995i	$0.326440\pi$
$674$	21.3766	+	37.0254i	0.823397	+	1.42616i
$675$		0			0
$676$	−22.2086			−0.854177
$677$	−8.78946			−0.337806			−0.168903	−	0.985633i	$-0.554023\pi$
$677$	−8.78946			−0.337806			−0.168903	+	0.985633i	$0.554023\pi$
$678$		0			0
$679$	11.8648	−	20.5504i	0.455329	−	0.788652i
$680$	0.195195	+	0.338088i	0.00748540	+	0.0129651i
$681$		0			0
$682$	4.38905	−	7.60206i	0.168065	−	0.291098i
$683$	1.11970			0.0428442			0.0214221	−	0.999771i	$-0.493181\pi$
$683$	1.11970			0.0428442			0.0214221	+	0.999771i	$0.493181\pi$
$684$		0			0
$685$	0.438854			0.0167677
$686$	0.808523	−	1.40040i	0.0308696	−	0.0534676i
$687$		0			0
$688$	5.43222	+	9.40888i	0.207101	+	0.358710i
$689$	0.722397	−	1.25123i	0.0275211	−	0.0476680i
$690$		0			0
$691$	15.4107			0.586252			0.293126	−	0.956074i	$-0.405304\pi$
$691$	15.4107			0.586252			0.293126	+	0.956074i	$0.405304\pi$
$692$	−8.61811			−0.327611
$693$		0			0
$694$	9.46910	+	16.4010i	0.359442	+	0.622572i
$695$	−3.39935			−0.128945
$696$		0			0
$697$	4.97989	+	8.62542i	0.188627	+	0.326711i
$698$	−24.0970	+	41.7373i	−0.912086	+	1.57978i
$699$		0			0
$700$	15.8005	−	27.3673i	0.597203	−	1.03439i
$701$	−14.4419	+	25.0140i	−0.545462	+	0.944767i	0.453116	+	0.891451i	$0.350312\pi$
$701$	−14.4419	+	25.0140i	−0.545462	+	0.944767i	−0.998578	+	0.0533156i	$0.983021\pi$
$702$		0			0
$703$	−31.7575	+	12.8309i	−1.19776	+	0.483926i
$704$	−6.18020			−0.232925
$705$		0			0
$706$	28.6101	−	49.5541i	1.07675	−	1.86499i
$707$	7.56804	+	13.1082i	0.284626	+	0.492986i
$708$		0			0
$709$	0.939283	+	1.62689i	0.0352755	+	0.0610990i	0.883124	−	0.469139i	$-0.155436\pi$
$709$	0.939283	+	1.62689i	0.0352755	+	0.0610990i	−0.847849	+	0.530238i	$0.822102\pi$
$710$	0.271416			0.0101861
$711$		0			0
$712$	0.291837	+	0.505476i	0.0109371	+	0.0189435i
$713$	−15.1659	−	26.2681i	−0.567967	−	0.983748i
$714$		0			0
$715$	0.0561603			0.00210028
$716$	20.7585	+	35.9548i	0.775782	+	1.34369i
$717$		0			0
$718$	−1.16848	−	2.02386i	−0.0436071	−	0.0755298i
$719$	−20.6247	+	35.7230i	−0.769171	+	1.33224i	0.168842	+	0.985643i	$0.445997\pi$
$719$	−20.6247	+	35.7230i	−0.769171	+	1.33224i	−0.938013	+	0.346600i	$0.887336\pi$
$720$		0			0
$721$	54.2872			2.02176
$722$	8.86172	+	35.5421i	0.329799	+	1.32274i
$723$		0			0
$724$	−8.74159	+	15.1409i	−0.324879	+	0.562706i
$725$	−4.63692	+	8.03138i	−0.172211	+	0.298278i
$726$		0			0
$727$	−8.82911	+	15.2925i	−0.327454	+	0.567166i	−0.982006	−	0.188850i	$-0.939524\pi$
$727$	−8.82911	+	15.2925i	−0.327454	+	0.567166i	0.654552	+	0.756017i	$0.272857\pi$
$728$	−0.256406	−	0.444107i	−0.00950302	−	0.0164597i
$729$		0			0
$730$	−0.344560			−0.0127527
$731$	4.30864	+	7.46278i	0.159361	+	0.276021i
$732$		0			0
$733$	27.2602			1.00688			0.503439	−	0.864031i	$-0.332068\pi$
$733$	27.2602			1.00688			0.503439	+	0.864031i	$0.332068\pi$
$734$	−63.9766			−2.36142
$735$		0			0
$736$	−27.7949	+	48.1421i	−1.02453	+	1.77454i
$737$	2.87084	+	4.97244i	0.105749	+	0.183162i
$738$		0			0
$739$	3.80969	−	6.59858i	0.140142	−	0.242733i	−0.787408	−	0.616432i	$-0.788577\pi$
$739$	3.80969	−	6.59858i	0.140142	−	0.242733i	0.927550	+	0.373699i	$0.121911\pi$
$740$	2.71098			0.0996576
$741$		0			0
$742$	40.8464			1.49952
$743$	15.8724	−	27.4918i	0.582302	−	1.00858i	−0.412904	−	0.910775i	$-0.635485\pi$
$743$	15.8724	−	27.4918i	0.582302	−	1.00858i	0.995206	−	0.0978020i	$-0.0311812\pi$
$744$		0			0
$745$	−2.14608	−	3.71713i	−0.0786265	−	0.136185i
$746$	2.41278	−	4.17906i	0.0883382	−	0.153006i
$747$		0			0
$748$	−6.74566			−0.246646
$749$	39.7087			1.45092
$750$		0			0
$751$	−20.4677	−	35.4511i	−0.746877	−	1.29363i	−0.949313	−	0.314333i	$-0.898219\pi$
$751$	−20.4677	−	35.4511i	−0.746877	−	1.29363i	0.202436	−	0.979296i	$-0.435114\pi$
$752$	11.3112			0.412476
$753$		0			0
$754$	−0.456178	−	0.790123i	−0.0166130	−	0.0287746i
$755$	−1.60998	+	2.78856i	−0.0585931	+	0.101486i
$756$		0			0
$757$	−17.6665	+	30.5994i	−0.642102	+	1.11215i	0.342861	+	0.939386i	$0.388604\pi$
$757$	−17.6665	+	30.5994i	−0.642102	+	1.11215i	−0.984963	+	0.172767i	$0.944729\pi$
$758$	−19.1865	+	33.2320i	−0.696885	+	1.20704i
$759$		0			0
$760$	−0.0665562	+	0.473576i	−0.00241425	+	0.0171784i
$761$	16.1891			0.586855			0.293428	−	0.955981i	$-0.405204\pi$
$761$	16.1891			0.586855			0.293428	+	0.955981i	$0.405204\pi$
$762$		0			0
$763$	18.3076	−	31.7098i	0.662781	−	1.14797i
$764$	15.0789	+	26.1174i	0.545535	+	0.944894i
$765$		0			0
$766$	−10.8265	−	18.7521i	−0.391177	−	0.677539i
$767$	3.65314			0.131907
$768$		0			0
$769$	−2.45805	−	4.25746i	−0.0886395	−	0.153528i	0.818297	−	0.574796i	$-0.194919\pi$
$769$	−2.45805	−	4.25746i	−0.0886395	−	0.153528i	−0.906936	+	0.421268i	$0.861585\pi$
$770$	0.793866	+	1.37502i	0.0286089	+	0.0495521i
$771$		0			0
$772$	33.7361			1.21419
$773$	27.0916	+	46.9240i	0.974416	+	1.68774i	0.681847	+	0.731495i	$0.261177\pi$
$773$	27.0916	+	46.9240i	0.974416	+	1.68774i	0.292569	+	0.956244i	$0.405490\pi$
$774$		0			0
$775$	−10.2252	−	17.7106i	−0.367300	−	0.636182i
$776$	−1.74538	+	3.02308i	−0.0626554	+	0.108522i
$777$		0			0
$778$	26.4500			0.948278
$779$	−1.69800	+	12.0820i	−0.0608373	+	0.432883i
$780$		0			0
$781$	0.386796	−	0.669950i	0.0138406	−	0.0239727i
$782$	−25.2313	+	43.7019i	−0.902270	+	1.56278i
$783$		0			0
$784$	15.1947	−	26.3180i	0.542668	−	0.939928i
$785$	2.11887	+	3.67000i	0.0756258	+	0.130988i
$786$		0			0
$787$	26.2374			0.935261			0.467631	−	0.883924i	$-0.345108\pi$
$787$	26.2374			0.935261			0.467631	+	0.883924i	$0.345108\pi$
$788$	−10.1181	−	17.5251i	−0.360443	−	0.624306i
$789$		0			0
$790$	−4.72093			−0.167963
$791$	55.8019			1.98409
$792$		0			0
$793$	−0.626689	+	1.08546i	−0.0222544	+	0.0385457i
$794$	22.0592	+	38.2077i	0.782852	+	1.35594i
$795$		0			0
$796$	−14.8955	+	25.7998i	−0.527958	+	0.914450i
$797$	33.2236			1.17684			0.588420	−	0.808555i	$-0.299750\pi$
$797$	33.2236			1.17684			0.588420	+	0.808555i	$0.299750\pi$
$798$		0			0
$799$	8.97161			0.317393
$800$	−18.7399	+	32.4585i	−0.662556	+	1.14758i
$801$		0			0
$802$	−30.6238	−	53.0420i	−1.08136	−	1.87298i
$803$	−0.491033	+	0.850494i	−0.0173282	+	0.0300133i
$804$		0			0
$805$	5.48624			0.193365
$806$	2.01190			0.0708662
$807$		0			0
$808$	−1.11330	−	1.92830i	−0.0391659	−	0.0678372i
$809$	−6.95935			−0.244678			−0.122339	−	0.992488i	$-0.539039\pi$
$809$	−6.95935			−0.244678			−0.122339	+	0.992488i	$0.539039\pi$
$810$		0			0
$811$	−2.83833	−	4.91613i	−0.0996672	−	0.172629i	0.811880	−	0.583825i	$-0.198444\pi$
$811$	−2.83833	−	4.91613i	−0.0996672	−	0.172629i	−0.911547	+	0.411196i	$0.865111\pi$
$812$	5.95709	−	10.3180i	0.209053	−	0.362090i
$813$		0			0
$814$	8.36412	−	14.4871i	0.293162	−	0.507772i
$815$	1.67727	−	2.90512i	0.0587523	−	0.101762i
$816$		0			0
$817$	−1.46913	+	10.4535i	−0.0513982	+	0.365720i
$818$	−9.41278			−0.329110
$819$		0			0
$820$	0.482839	−	0.836302i	0.0168615	−	0.0292049i
$821$	5.35004	+	9.26654i	0.186718	+	0.323404i	0.944154	−	0.329505i	$-0.106882\pi$
$821$	5.35004	+	9.26654i	0.186718	+	0.323404i	−0.757436	+	0.652909i	$0.773548\pi$
$822$		0			0
$823$	21.2280	+	36.7680i	0.739962	+	1.28165i	0.952512	+	0.304502i	$0.0984900\pi$
$823$	21.2280	+	36.7680i	0.739962	+	1.28165i	−0.212549	+	0.977150i	$0.568177\pi$
$824$	−7.98597			−0.278204
$825$		0			0
$826$	51.6397	+	89.4426i	1.79678	+	3.11211i
$827$	−12.5928	−	21.8114i	−0.437896	−	0.758458i	0.559631	−	0.828742i	$-0.310943\pi$
$827$	−12.5928	−	21.8114i	−0.437896	−	0.758458i	−0.997527	+	0.0702836i	$0.977610\pi$
$828$		0			0
$829$	−15.4727			−0.537389			−0.268694	−	0.963225i	$-0.586592\pi$
$829$	−15.4727			−0.537389			−0.268694	+	0.963225i	$0.586592\pi$
$830$	−3.20704	−	5.55475i	−0.111318	−	0.192808i
$831$		0			0
$832$	−0.708237	−	1.22670i	−0.0245537	−	0.0425283i
$833$	12.0519	−	20.8745i	0.417573	−	0.723258i
$834$		0			0
$835$	−3.04100			−0.105238
$836$	−6.51169	−	5.08748i	−0.225212	−	0.175954i
$837$		0			0
$838$	35.3593	−	61.2441i	1.22147	−	2.11564i
$839$	17.5746	−	30.4401i	0.606743	−	1.05091i	−0.385031	−	0.922904i	$-0.625809\pi$
$839$	17.5746	−	30.4401i	0.606743	−	1.05091i	0.991773	−	0.128006i	$-0.0408576\pi$
$840$		0			0
$841$	12.7518	−	22.0868i	0.439717	−	0.761612i
$842$	−8.20241	−	14.2070i	−0.282674	−	0.489605i
$843$		0			0
$844$	22.1059			0.760915
$845$	−1.29978	−	2.25128i	−0.0447136	−	0.0774463i
$846$		0			0
$847$	−36.2993			−1.24726
$848$	−25.6103			−0.879462
$849$		0			0
$850$	−17.0115	+	29.4648i	−0.583490	+	1.01063i
$851$	−28.9013	−	50.0586i	−0.990725	−	1.71599i
$852$		0			0
$853$	1.67252	−	2.89689i	0.0572660	−	0.0991876i	−0.835971	−	0.548773i	$-0.815095\pi$
$853$	1.67252	−	2.89689i	0.0572660	−	0.0991876i	0.893237	+	0.449586i	$0.148428\pi$
$854$	−35.4348			−1.21255
$855$		0			0
$856$	−5.84137			−0.199654
$857$	−13.8220	+	23.9405i	−0.472152	+	0.817791i	−0.999492	−	0.0318629i	$-0.989856\pi$
$857$	−13.8220	+	23.9405i	−0.472152	+	0.817791i	0.527340	+	0.849654i	$0.323189\pi$
$858$		0			0
$859$	−19.9394	−	34.5360i	−0.680322	−	1.17835i	−0.974882	−	0.222720i	$-0.928506\pi$
$859$	−19.9394	−	34.5360i	−0.680322	−	1.17835i	0.294560	−	0.955633i	$-0.404827\pi$
$860$	0.417757	−	0.723576i	0.0142454	−	0.0246737i
$861$		0			0
$862$	37.8992			1.29085
$863$	−13.4302			−0.457170			−0.228585	−	0.973524i	$-0.573410\pi$
$863$	−13.4302			−0.457170			−0.228585	+	0.973524i	$0.573410\pi$
$864$		0			0
$865$	−0.504382	−	0.873615i	−0.0171495	−	0.0297038i
$866$	−0.826928			−0.0281002
$867$		0			0
$868$	13.1364	+	22.7529i	0.445878	+	0.772284i
$869$	−6.72782	+	11.6529i	−0.228226	+	0.395298i
$870$		0			0
$871$	−0.657983	+	1.13966i	−0.0222949	+	0.0386159i
$872$	−2.69316	+	4.66469i	−0.0912019	+	0.157966i
$873$		0			0
$874$	−57.3155	+	23.1571i	−1.93873	+	0.783299i
$875$	7.42802			0.251113
$876$		0			0
$877$	−10.0707	+	17.4429i	−0.340062	+	0.589005i	−0.984444	−	0.175699i	$-0.943782\pi$
$877$	−10.0707	+	17.4429i	−0.340062	+	0.589005i	0.644382	+	0.764704i	$0.277115\pi$
$878$	17.4153	+	30.1641i	0.587737	+	1.01799i
$879$		0			0
$880$	−0.497747	−	0.862123i	−0.0167790	−	0.0290622i
$881$	47.8690			1.61275			0.806374	−	0.591406i	$-0.201427\pi$
$881$	47.8690			1.61275			0.806374	+	0.591406i	$0.201427\pi$
$882$		0			0
$883$	−13.8313	−	23.9565i	−0.465460	−	0.806200i	0.533762	−	0.845634i	$-0.320778\pi$
$883$	−13.8313	−	23.9565i	−0.465460	−	0.806200i	−0.999222	+	0.0394347i	$0.987444\pi$
$884$	−0.773037	−	1.33894i	−0.0260000	−	0.0450334i
$885$		0			0
$886$	50.0433			1.68124
$887$	18.2879	+	31.6755i	0.614047	+	1.06356i	0.990551	+	0.137145i	$0.0437928\pi$
$887$	18.2879	+	31.6755i	0.614047	+	1.06356i	−0.376504	+	0.926415i	$0.622874\pi$
$888$		0			0
$889$	−30.6376	−	53.0659i	−1.02755	−	1.77977i
$890$	0.207093	−	0.358696i	0.00694178	−	0.0120235i
$891$		0			0
$892$	1.55428			0.0520410
$893$	8.66044	+	6.76626i	0.289810	+	0.226424i
$894$		0			0
$895$	−2.42982	+	4.20857i	−0.0812199	+	0.140677i
$896$	−8.02369	+	13.8974i	−0.268053	+	0.464281i
$897$		0			0
$898$	−21.7492	+	37.6706i	−0.725779	+	1.25709i
$899$	−3.85509	−	6.67722i	−0.128575	−	0.222698i
$900$		0			0
$901$	−20.3132			−0.676730
$902$	−2.97939	−	5.16045i	−0.0992027	−	0.171824i
$903$		0			0
$904$	−8.20879			−0.273020
$905$	−2.04643			−0.0680258
$906$		0			0
$907$	−22.3142	+	38.6493i	−0.740931	+	1.28333i	0.211142	+	0.977455i	$0.432282\pi$
$907$	−22.3142	+	38.6493i	−0.740931	+	1.28333i	−0.952072	+	0.305874i	$0.901051\pi$
$908$	−1.94475	−	3.36841i	−0.0645388	−	0.111785i
$909$		0			0
$910$	−0.181950	+	0.315147i	−0.00603160	+	0.0104470i
$911$	−53.4274			−1.77013			−0.885065	−	0.465467i	$-0.845886\pi$
$911$	−53.4274			−1.77013			−0.885065	+	0.465467i	$0.845886\pi$
$912$		0			0
$913$	−18.2814			−0.605027
$914$	9.17610	−	15.8935i	0.303519	−	0.525709i
$915$		0			0
$916$	9.53671	+	16.5181i	0.315102	+	0.545772i
$917$	14.1734	−	24.5491i	0.468047	−	0.810681i
$918$		0			0
$919$	−4.65987			−0.153715			−0.0768574	−	0.997042i	$-0.524489\pi$
$919$	−4.65987			−0.153715			−0.0768574	+	0.997042i	$0.524489\pi$
$920$	−0.807058			−0.0266079
$921$		0			0
$922$	−11.9376	−	20.6765i	−0.393144	−	0.680945i
$923$	0.177304			0.00583602
$924$		0			0
$925$	−19.4859	−	33.7507i	−0.640694	−	1.10971i
$926$	6.13023	−	10.6179i	0.201452	−	0.348925i
$927$		0			0
$928$	−7.06531	+	12.2375i	−0.231930	+	0.401715i
$929$	21.0762	−	36.5050i	0.691486	−	1.19769i	−0.279865	−	0.960039i	$-0.590290\pi$
$929$	21.0762	−	36.5050i	0.691486	−	1.19769i	0.971351	−	0.237650i	$-0.0763771\pi$
$930$		0			0
$931$	27.3771	−	11.0611i	0.897249	−	0.362513i
$932$	34.5315			1.13112
$933$		0			0
$934$	8.02794	−	13.9048i	0.262682	−	0.454979i
$935$	−0.394795	−	0.683805i	−0.0129112	−	0.0223628i
$936$		0			0
$937$	25.9049	+	44.8687i	0.846278	+	1.46580i	0.884507	+	0.466527i	$0.154495\pi$
$937$	25.9049	+	44.8687i	0.846278	+	1.46580i	−0.0382292	+	0.999269i	$0.512172\pi$
$938$	−37.2042			−1.21476
$939$		0			0
$940$	−0.434934	−	0.753328i	−0.0141860	−	0.0245708i
$941$	−11.0890	−	19.2067i	−0.361490	−	0.626120i	0.626716	−	0.779248i	$-0.284399\pi$
$941$	−11.0890	−	19.2067i	−0.361490	−	0.626120i	−0.988206	+	0.153128i	$0.951065\pi$
$942$		0			0
$943$	−20.5899			−0.670500
$944$	−32.3776	−	56.0797i	−1.05380	−	1.82524i
$945$		0			0
$946$	−2.57779	−	4.46486i	−0.0838112	−	0.145165i
$947$	15.5028	−	26.8517i	0.503774	−	0.872563i	−0.496216	−	0.868199i	$-0.665278\pi$
$947$	15.5028	−	26.8517i	0.503774	−	0.872563i	0.999990	−	0.00436376i	$-0.00138903\pi$
$948$		0			0
$949$	−0.225085			−0.00730657
$950$	−38.6435	+	15.6130i	−1.25376	+	0.506553i
$951$		0			0
$952$	−3.60495	+	6.24396i	−0.116837	+	0.202368i
$953$	10.9581	−	18.9799i	0.354966	−	0.614820i	−0.632146	−	0.774849i	$-0.717826\pi$
$953$	10.9581	−	18.9799i	0.354966	−	0.614820i	0.987112	+	0.160030i	$0.0511590\pi$
$954$		0			0
$955$	−1.76501	+	3.05708i	−0.0571143	+	0.0989249i
$956$	8.79568	+	15.2346i	0.284473	+	0.492721i
$957$		0			0
$958$	−58.5188			−1.89066
$959$	4.05247	+	7.01908i	0.130861	+	0.226658i
$960$		0			0
$961$	−13.9977			−0.451540
$962$	3.83404			0.123614
$963$		0			0
$964$	−9.06580	+	15.7024i	−0.291990	+	0.505741i
$965$	1.97443	+	3.41981i	0.0635592	+	0.110088i
$966$		0			0
$967$	−7.86066	+	13.6151i	−0.252782	+	0.437831i	−0.964291	−	0.264846i	$-0.914679\pi$
$967$	−7.86066	+	13.6151i	−0.252782	+	0.437831i	0.711509	+	0.702677i	$0.248012\pi$
$968$	5.33984			0.171629
$969$		0			0
$970$	2.47711			0.0795352
$971$	13.5978	−	23.5521i	0.436374	−	0.755822i	−0.561033	−	0.827794i	$-0.689596\pi$
$971$	13.5978	−	23.5521i	0.436374	−	0.755822i	0.997407	+	0.0719720i	$0.0229292\pi$
$972$		0			0
$973$	−31.3903	−	54.3696i	−1.00633	−	1.74301i
$974$	23.1305	−	40.0631i	0.741148	−	1.28371i
$975$		0			0
$976$	22.2173			0.711158
$977$	−6.11737			−0.195712			−0.0978560	−	0.995201i	$-0.531198\pi$
$977$	−6.11737			−0.195712			−0.0978560	+	0.995201i	$0.531198\pi$
$978$		0			0
$979$	−0.590258	−	1.02236i	−0.0188647	−	0.0326747i
$980$	−2.33705			−0.0746544
$981$		0			0
$982$	32.2571	+	55.8710i	1.02937	+	1.78291i
$983$	−25.2388	+	43.7148i	−0.804991	+	1.39429i	0.111306	+	0.993786i	$0.464497\pi$
$983$	−25.2388	+	43.7148i	−0.804991	+	1.39429i	−0.916297	+	0.400499i	$0.868837\pi$
$984$		0			0
$985$	1.18434	−	2.05134i	0.0377363	−	0.0653611i
$986$	−6.41367	+	11.1088i	−0.204253	+	0.353776i
$987$		0			0
$988$	0.263584	−	1.87551i	0.00838573	−	0.0596680i
$989$	−17.8146			−0.566470
$990$		0			0
$991$	−8.20353	+	14.2089i	−0.260594	+	0.451362i	−0.966400	−	0.257044i	$-0.917252\pi$
$991$	−8.20353	+	14.2089i	−0.260594	+	0.451362i	0.705806	+	0.708405i	$0.250585\pi$
$992$	−15.5802	−	26.9857i	−0.494672	−	0.856797i
$993$		0			0
$994$	2.50631	+	4.34106i	0.0794954	+	0.137690i
$995$	−3.48709			−0.110548
$996$		0			0
$997$	21.8758	+	37.8900i	0.692814	+	1.19999i	0.970912	+	0.239436i	$0.0769625\pi$
$997$	21.8758	+	37.8900i	0.692814	+	1.19999i	−0.278099	+	0.960553i	$0.589704\pi$
$998$	16.4342	+	28.4649i	0.520216	+	0.901041i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	513.2.f.h.163.1	yes	12
3.2	odd	2		513.2.f.f.163.6	✓	12
19.7	even	3	inner	513.2.f.h.406.1	yes	12
19.8	odd	6		9747.2.a.br.1.1		6
19.11	even	3		9747.2.a.bl.1.6		6
57.8	even	6		9747.2.a.bm.1.6		6
57.11	odd	6		9747.2.a.bs.1.1		6
57.26	odd	6		513.2.f.f.406.6	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
513.2.f.f.163.6	✓	12	3.2	odd	2
513.2.f.f.406.6	yes	12	57.26	odd	6
513.2.f.h.163.1	yes	12	1.1	even	1	trivial
513.2.f.h.406.1	yes	12	19.7	even	3	inner
9747.2.a.bl.1.6		6	19.11	even	3
9747.2.a.bm.1.6		6	57.8	even	6
9747.2.a.br.1.1		6	19.8	odd	6
9747.2.a.bs.1.1		6	57.11	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:	\( N \)	\(=\)	\( 513 = 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	513.f (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.963952 + 1.66961i) q^{2} +(-0.858406 - 1.48680i) q^{4} +(0.100478 - 0.174033i) q^{5} +3.71133 q^{7} -0.545959 q^{8} +(0.193712 + 0.335518i) q^{10} +1.10424 q^{11} +(0.126543 + 0.219179i) q^{13} +(-3.57755 + 6.19649i) q^{14} +(2.24309 - 3.88515i) q^{16} +(1.77914 - 3.08156i) q^{17} +(4.04150 - 1.63288i) q^{19} -0.345003 q^{20} +(-1.06443 + 1.84365i) q^{22} +(3.67802 + 6.37053i) q^{23} +(2.47981 + 4.29515i) q^{25} -0.487925 q^{26} +(-3.18583 - 5.51802i) q^{28} +(0.934935 + 1.61935i) q^{29} -4.12338 q^{31} +(3.77850 + 6.54456i) q^{32} +(3.43001 + 5.94095i) q^{34} +(0.372907 - 0.645893i) q^{35} -7.85784 q^{37} +(-1.16954 + 8.32176i) q^{38} +(-0.0548567 + 0.0950146i) q^{40} +(-1.39952 + 2.42404i) q^{41} +(-1.21088 + 2.09730i) q^{43} +(-0.947882 - 1.64178i) q^{44} -14.1818 q^{46} +(1.26067 + 2.18354i) q^{47} +6.77400 q^{49} -9.56166 q^{50} +(0.217250 - 0.376289i) q^{52} +(-2.85436 - 4.94389i) q^{53} +(0.110951 - 0.192173i) q^{55} -2.02624 q^{56} -3.60493 q^{58} +(7.21720 - 12.5005i) q^{59} +(2.47619 + 4.28889i) q^{61} +(3.97474 - 6.88445i) q^{62} -5.59682 q^{64} +0.0508590 q^{65} +(2.59984 + 4.50306i) q^{67} -6.10889 q^{68} +(0.718928 + 1.24522i) q^{70} +(0.350284 - 0.606709i) q^{71} +(-0.444681 + 0.770211i) q^{73} +(7.57458 - 13.1196i) q^{74} +(-5.89701 - 4.60724i) q^{76} +4.09819 q^{77} +(-6.09274 + 10.5529i) q^{79} +(-0.450762 - 0.780742i) q^{80} +(-2.69814 - 4.67332i) q^{82} -16.5557 q^{83} +(-0.357528 - 0.619256i) q^{85} +(-2.33446 - 4.04340i) q^{86} -0.602867 q^{88} +(-0.534540 - 0.925851i) q^{89} +(0.469643 + 0.813445i) q^{91} +(6.31448 - 10.9370i) q^{92} -4.86089 q^{94} +(0.121907 - 0.867421i) q^{95} +(3.19691 - 5.53720i) q^{97} +(-6.52981 + 11.3100i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + q^{2} - 3 q^{4} + 5 q^{5} + 2 q^{7} - 12 q^{8} + q^{10} - 4 q^{11} - 5 q^{13} + 2 q^{14} + 3 q^{16} + 10 q^{17} - 9 q^{19} - 2 q^{20} - 4 q^{22} + 3 q^{23} - 5 q^{25} - 2 q^{26} - 2 q^{28} - 6 q^{29}+ \cdots - 59 q^{98}+O(q^{100}) \) 12 * q + q^2 - 3 * q^4 + 5 * q^5 + 2 * q^7 - 12 * q^8 + q^10 - 4 * q^11 - 5 * q^13 + 2 * q^14 + 3 * q^16 + 10 * q^17 - 9 * q^19 - 2 * q^20 - 4 * q^22 + 3 * q^23 - 5 * q^25 - 2 * q^26 - 2 * q^28 - 6 * q^29 - 4 * q^31 + 8 * q^32 + 4 * q^34 + 3 * q^35 + 26 * q^37 - 10 * q^38 - 6 * q^40 - 13 * q^41 - 3 * q^43 - 20 * q^44 - 10 * q^46 + 18 * q^47 + 26 * q^49 + 6 * q^50 - 22 * q^52 - q^53 - 6 * q^55 + 48 * q^56 + 28 * q^58 + 11 * q^59 - 13 * q^61 + 22 * q^62 - 72 * q^65 + 9 * q^67 - 10 * q^68 - 30 * q^70 - 6 * q^71 - 15 * q^73 + 43 * q^74 - 33 * q^76 - 58 * q^77 - 13 * q^79 + q^80 + 4 * q^82 - 16 * q^83 - 3 * q^85 - 7 * q^86 - 30 * q^88 + 29 * q^89 + 22 * q^91 + 21 * q^92 + 36 * q^94 + 22 * q^95 + 20 * q^97 - 59 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more