LMFDB - Embedded newform 1008.3.cg.q.145.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,3,Mod(145,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1008.145");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1008.cg (of order $6$, 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:	$27.4660106475$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 81 x^{14} - 118 x^{13} + 1960 x^{12} - 366 x^{11} + 37625 x^{10} - 83714 x^{9} + \cdots + 1148023744 $ x^16 - 2x^15 + 81x^14 - 118x^13 + 1960x^12 - 366x^11 + 37625x^10 - 83714x^9 + 623931x^8 + 289492x^7 + 241286x^6 - 4777608x^5 + 98362220x^4 - 239083824x^3 + 523977616x^2 - 403737152*x + 1148023744
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			145.4
Root			$2.29298 + 1.67052i$ of defining polynomial
Character	$\chi$	$=$	1008.145
Dual form			1008.3.cg.q.577.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.27883 - 0.738332i) q^{5} +(4.08597 + 5.68374i) q^{7} +O(q^{10})$	$q+(-1.27883 - 0.738332i) q^{5} +(4.08597 + 5.68374i) q^{7} +(-7.76437 - 13.4483i) q^{11} +18.6695i q^{13} +(11.8009 - 6.81327i) q^{17} +(-1.20669 - 0.696684i) q^{19} +(-20.0154 + 34.6678i) q^{23} +(-11.4097 - 19.7622i) q^{25} +52.1017 q^{29} +(-41.5945 + 24.0146i) q^{31} +(-1.02877 - 10.2853i) q^{35} +(-7.93414 + 13.7423i) q^{37} +20.6334i q^{41} +16.0549 q^{43} +(-65.1271 - 37.6011i) q^{47} +(-15.6097 + 46.4471i) q^{49} +(9.13459 + 15.8216i) q^{53} +22.9307i q^{55} +(-41.7253 + 24.0901i) q^{59} +(23.2757 + 13.4382i) q^{61} +(13.7843 - 23.8751i) q^{65} +(0.310414 + 0.537652i) q^{67} +18.7693 q^{71} +(-89.5780 + 51.7179i) q^{73} +(44.7115 - 99.0798i) q^{77} +(-55.1116 + 95.4561i) q^{79} +23.8805i q^{83} -20.1218 q^{85} +(114.503 + 66.1084i) q^{89} +(-106.113 + 76.2831i) q^{91} +(1.02877 + 1.78188i) q^{95} +72.0450i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{7}+O(q^{10}) $ 16 * q - 4 * q^7	$ 16 q - 4 q^{7} + 24 q^{19} + 36 q^{25} + 84 q^{31} - 68 q^{37} - 80 q^{43} - 184 q^{49} + 216 q^{61} - 56 q^{67} + 156 q^{73} - 28 q^{79} + 448 q^{85} - 48 q^{91}+O(q^{100}) $ 16 * q - 4 * q^7 + 24 * q^19 + 36 * q^25 + 84 * q^31 - 68 * q^37 - 80 * q^43 - 184 * q^49 + 216 * q^61 - 56 * q^67 + 156 * q^73 - 28 * q^79 + 448 * q^85 - 48 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$1$	$e\left(\frac{5}{6}\right)$	$1$	$1$

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			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.27883	−	0.738332i	−0.255766	−	0.147666i	0.366636	−	0.930365i	$-0.380510\pi$
$5$	−1.27883	−	0.738332i	−0.255766	−	0.147666i	−0.622401	+	0.782698i	$0.713843\pi$
$6$		0			0
$7$	4.08597	+	5.68374i	0.583710	+	0.811962i
$7$	4.08597	+	5.68374i	0.583710	+	0.811962i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−7.76437	−	13.4483i	−0.705852	−	1.22257i	−0.966383	−	0.257105i	$-0.917231\pi$
$11$	−7.76437	−	13.4483i	−0.705852	−	1.22257i	0.260532	−	0.965465i	$-0.416102\pi$
$12$		0			0
$13$			18.6695i			1.43612i	0.695982	+	0.718059i	$0.254969\pi$
$13$			18.6695i			1.43612i	−0.695982	+	0.718059i	$0.745031\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	11.8009	−	6.81327i	0.694173	−	0.400781i	−0.111001	−	0.993820i	$-0.535406\pi$
$17$	11.8009	−	6.81327i	0.694173	−	0.400781i	0.805173	+	0.593040i	$0.202072\pi$
$18$		0			0
$19$	−1.20669	−	0.696684i	−0.0635101	−	0.0366676i	0.467909	−	0.883777i	$-0.345008\pi$
$19$	−1.20669	−	0.696684i	−0.0635101	−	0.0366676i	−0.531419	+	0.847109i	$0.678341\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−20.0154	+	34.6678i	−0.870237	+	1.50729i	−0.00848505	+	0.999964i	$0.502701\pi$
$23$	−20.0154	+	34.6678i	−0.870237	+	1.50729i	−0.861752	+	0.507330i	$0.830632\pi$
$24$		0			0
$25$	−11.4097	−	19.7622i	−0.456389	−	0.790489i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	52.1017			1.79661			0.898305	−	0.439373i	$-0.144799\pi$
$29$	52.1017			1.79661			0.898305	+	0.439373i	$0.144799\pi$
$30$		0			0
$31$	−41.5945	+	24.0146i	−1.34176	+	0.774664i	−0.987065	−	0.160319i	$-0.948748\pi$
$31$	−41.5945	+	24.0146i	−1.34176	+	0.774664i	−0.354692	+	0.934983i	$0.615414\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.02877	−	10.2853i	−0.0293934	−	0.293866i
$36$		0			0
$37$	−7.93414	+	13.7423i	−0.214436	+	0.371414i	−0.953098	−	0.302662i	$-0.902125\pi$
$37$	−7.93414	+	13.7423i	−0.214436	+	0.371414i	0.738662	+	0.674076i	$0.235458\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			20.6334i			0.503253i	0.967824	+	0.251626i	$0.0809654\pi$
$41$			20.6334i			0.503253i	−0.967824	+	0.251626i	$0.919035\pi$
$42$		0			0
$43$	16.0549			0.373370			0.186685	−	0.982420i	$-0.440226\pi$
$43$	16.0549			0.373370			0.186685	+	0.982420i	$0.440226\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−65.1271	−	37.6011i	−1.38568	−	0.800024i	−0.392857	−	0.919599i	$-0.628513\pi$
$47$	−65.1271	−	37.6011i	−1.38568	−	0.800024i	−0.992825	+	0.119575i	$0.961847\pi$
$48$		0			0
$49$	−15.6097	+	46.4471i	−0.318566	+	0.947901i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	9.13459	+	15.8216i	0.172351	+	0.298520i	0.939241	−	0.343258i	$-0.111530\pi$
$53$	9.13459	+	15.8216i	0.172351	+	0.298520i	−0.766890	+	0.641778i	$0.778197\pi$
$54$		0			0
$55$			22.9307i			0.416922i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−41.7253	+	24.0901i	−0.707209	+	0.408307i	−0.810027	−	0.586393i	$-0.800548\pi$
$59$	−41.7253	+	24.0901i	−0.707209	+	0.408307i	0.102818	+	0.994700i	$0.467214\pi$
$60$		0			0
$61$	23.2757	+	13.4382i	0.381568	+	0.220298i	0.678500	−	0.734600i	$-0.262630\pi$
$61$	23.2757	+	13.4382i	0.381568	+	0.220298i	−0.296932	+	0.954899i	$0.595964\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	13.7843	−	23.8751i	0.212066	−	0.367310i
$66$		0			0
$67$	0.310414	+	0.537652i	0.00463304	+	0.00802466i	0.868333	−	0.495982i	$-0.165192\pi$
$67$	0.310414	+	0.537652i	0.00463304	+	0.00802466i	−0.863700	+	0.504007i	$0.831859\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	18.7693			0.264356			0.132178	−	0.991226i	$-0.457803\pi$
$71$	18.7693			0.264356			0.132178	+	0.991226i	$0.457803\pi$
$72$		0			0
$73$	−89.5780	+	51.7179i	−1.22710	+	0.708464i	−0.966421	−	0.256963i	$-0.917278\pi$
$73$	−89.5780	+	51.7179i	−1.22710	+	0.708464i	−0.260674	+	0.965427i	$0.583945\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	44.7115	−	99.0798i	0.580669	−	1.28675i
$78$		0			0
$79$	−55.1116	+	95.4561i	−0.697615	+	1.20830i	0.271676	+	0.962389i	$0.412422\pi$
$79$	−55.1116	+	95.4561i	−0.697615	+	1.20830i	−0.969291	+	0.245916i	$0.920911\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			23.8805i			0.287717i	0.989598	+	0.143858i	$0.0459510\pi$
$83$			23.8805i			0.287717i	−0.989598	+	0.143858i	$0.954049\pi$
$84$		0			0
$85$	−20.1218			−0.236727
$86$		0			0
$87$		0			0
$88$		0			0
$89$	114.503	+	66.1084i	1.28655	+	0.742790i	0.978037	−	0.208429i	$-0.0668352\pi$
$89$	114.503	+	66.1084i	1.28655	+	0.742790i	0.308513	+	0.951220i	$0.400169\pi$
$90$		0			0
$91$	−106.113	+	76.2831i	−1.16607	+	0.838276i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.02877	+	1.78188i	0.0108291	+	0.0187566i
$96$		0			0
$97$			72.0450i			0.742732i	0.928486	+	0.371366i	$0.121111\pi$
$97$			72.0450i			0.742732i	−0.928486	+	0.371366i	$0.878889\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−145.743	+	84.1449i	−1.44300	+	0.833118i	−0.998049	−	0.0624354i	$-0.980113\pi$
$101$	−145.743	+	84.1449i	−1.44300	+	0.833118i	−0.444954	+	0.895553i	$0.646780\pi$
$102$		0			0
$103$	74.2263	+	42.8546i	0.720644	+	0.416064i	0.814990	−	0.579476i	$-0.196743\pi$
$103$	74.2263	+	42.8546i	0.720644	+	0.416064i	−0.0943459	+	0.995539i	$0.530076\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.54612	+	2.67796i	−0.0144497	+	0.0250277i	−0.873160	−	0.487434i	$-0.837933\pi$
$107$	−1.54612	+	2.67796i	−0.0144497	+	0.0250277i	0.858710	+	0.512462i	$0.171266\pi$
$108$		0			0
$109$	46.1317	+	79.9024i	0.423227	+	0.733050i	0.996253	−	0.0864871i	$-0.0275641\pi$
$109$	46.1317	+	79.9024i	0.423227	+	0.733050i	−0.573026	+	0.819537i	$0.694231\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−131.406			−1.16289			−0.581444	−	0.813586i	$-0.697512\pi$
$113$	−131.406			−1.16289			−0.581444	+	0.813586i	$0.697512\pi$
$114$		0			0
$115$	51.1926	−	29.5561i	0.445153	−	0.257009i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	86.9431	+	39.2346i	0.730614	+	0.329702i
$120$		0			0
$121$	−60.0708	+	104.046i	−0.496453	+	0.859881i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			70.6133i			0.564906i
$126$		0			0
$127$	−88.0157			−0.693037			−0.346519	−	0.938043i	$-0.612636\pi$
$127$	−88.0157			−0.693037			−0.346519	+	0.938043i	$0.612636\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	28.0479	+	16.1935i	0.214106	+	0.123614i	0.603218	−	0.797576i	$-0.293885\pi$
$131$	28.0479	+	16.1935i	0.214106	+	0.123614i	−0.389112	+	0.921190i	$0.627218\pi$
$132$		0			0
$133$	−0.970737	−	9.70515i	−0.00729877	−	0.0729710i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−57.5739	−	99.7209i	−0.420247	−	0.727890i	0.575716	−	0.817650i	$-0.304723\pi$
$137$	−57.5739	−	99.7209i	−0.420247	−	0.727890i	−0.995963	+	0.0897598i	$0.971390\pi$
$138$		0			0
$139$			206.700i			1.48705i	0.668706	+	0.743527i	$0.266848\pi$
$139$			206.700i			1.48705i	−0.668706	+	0.743527i	$0.733152\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	251.073	−	144.957i	1.75576	−	1.01369i
$144$		0			0
$145$	−66.6291	−	38.4683i	−0.459511	−	0.265299i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	75.5206	−	130.806i	0.506850	−	0.877890i	−0.493119	−	0.869962i	$-0.664143\pi$
$149$	75.5206	−	130.806i	0.506850	−	0.877890i	0.999969	−	0.00792779i	$-0.00252352\pi$
$150$		0			0
$151$	−31.1207	−	53.9027i	−0.206098	−	0.356971i	0.744384	−	0.667751i	$-0.232743\pi$
$151$	−31.1207	−	53.9027i	−0.206098	−	0.356971i	−0.950482	+	0.310780i	$0.899410\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	70.9229			0.457567
$156$		0			0
$157$	209.429	−	120.914i	1.33394	−	0.770153i	0.348043	−	0.937479i	$-0.386846\pi$
$157$	209.429	−	120.914i	1.33394	−	0.770153i	0.985902	+	0.167325i	$0.0535130\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−278.825	+	27.8889i	−1.73183	+	0.173223i
$162$		0			0
$163$	0.640226	−	1.10890i	0.00392776	−	0.00680309i	−0.864055	−	0.503398i	$-0.832083\pi$
$163$	0.640226	−	1.10890i	0.00392776	−	0.00680309i	0.867983	+	0.496595i	$0.165416\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	158.859i		−	0.951254i	−0.879647	−	0.475627i	$-0.842221\pi$
$167$		−	158.859i		−	0.951254i	0.879647	−	0.475627i	$-0.157779\pi$
$168$		0			0
$169$	−179.551			−1.06243
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−19.5532	−	11.2890i	−0.113024	−	0.0652546i	0.442422	−	0.896807i	$-0.354119\pi$
$173$	−19.5532	−	11.2890i	−0.113024	−	0.0652546i	−0.555446	+	0.831552i	$0.687453\pi$
$174$		0			0
$175$	65.7035	−	145.598i	0.375449	−	0.831987i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−53.3082	−	92.3326i	−0.297811	−	0.515824i	0.677824	−	0.735225i	$-0.262923\pi$
$179$	−53.3082	−	92.3326i	−0.297811	−	0.515824i	−0.975635	+	0.219400i	$0.929590\pi$
$180$		0			0
$181$			141.955i			0.784279i	0.919906	+	0.392140i	$0.128265\pi$
$181$			141.955i			0.784279i	−0.919906	+	0.392140i	$0.871735\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	20.2928	−	11.7161i	0.109691	−	0.0633300i
$186$		0			0
$187$	−183.254	−	105.801i	−0.979966	−	0.565783i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−165.284	+	286.281i	−0.865363	+	1.49885i	0.00132256	+	0.999999i	$0.499579\pi$
$191$	−165.284	+	286.281i	−0.865363	+	1.49885i	−0.866686	+	0.498854i	$0.833754\pi$
$192$		0			0
$193$	−85.3048	−	147.752i	−0.441994	−	0.765555i	0.555844	−	0.831287i	$-0.312395\pi$
$193$	−85.3048	−	147.752i	−0.441994	−	0.765555i	−0.997837	+	0.0657314i	$0.979062\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−291.740			−1.48091			−0.740457	−	0.672104i	$-0.765391\pi$
$197$	−291.740			−1.48091			−0.740457	+	0.672104i	$0.765391\pi$
$198$		0			0
$199$	113.785	−	65.6941i	0.571786	−	0.330121i	−0.186076	−	0.982535i	$-0.559577\pi$
$199$	113.785	−	65.6941i	0.571786	−	0.330121i	0.757862	+	0.652414i	$0.226244\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	212.886	+	296.132i	1.04870	+	1.45878i
$204$		0			0
$205$	15.2343	−	26.3865i	0.0743135	−	0.128715i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			21.6372i			0.103527i
$210$		0			0
$211$	30.5925			0.144988			0.0724942	−	0.997369i	$-0.476904\pi$
$211$	30.5925			0.144988			0.0724942	+	0.997369i	$0.476904\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−20.5315	−	11.8538i	−0.0954952	−	0.0551342i
$216$		0			0
$217$	−306.446	−	138.289i	−1.41219	−	0.637278i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	127.201	+	220.318i	0.575568	+	0.996913i
$222$		0			0
$223$		−	60.1683i		−	0.269813i	−0.990858	−	0.134906i	$-0.956927\pi$
$223$		−	60.1683i		−	0.269813i	0.990858	−	0.134906i	$-0.0430734\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	293.987	−	169.734i	1.29510	−	0.747725i	0.315545	−	0.948911i	$-0.397813\pi$
$227$	293.987	−	169.734i	1.29510	−	0.747725i	0.979553	+	0.201185i	$0.0644793\pi$
$228$		0			0
$229$	253.061	+	146.105i	1.10507	+	0.638011i	0.937548	−	0.347857i	$-0.113091\pi$
$229$	253.061	+	146.105i	1.10507	+	0.638011i	0.167521	+	0.985869i	$0.446424\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	54.1324	−	93.7601i	0.232328	−	0.402404i	−0.726165	−	0.687521i	$-0.758699\pi$
$233$	54.1324	−	93.7601i	0.232328	−	0.402404i	0.958493	+	0.285117i	$0.0920324\pi$
$234$		0			0
$235$	55.5242	+	96.1708i	0.236273	+	0.409237i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−13.6245			−0.0570064			−0.0285032	−	0.999594i	$-0.509074\pi$
$239$	−13.6245			−0.0570064			−0.0285032	+	0.999594i	$0.509074\pi$
$240$		0			0
$241$	161.485	−	93.2334i	0.670062	−	0.386860i	−0.126038	−	0.992025i	$-0.540226\pi$
$241$	161.485	−	93.2334i	0.670062	−	0.386860i	0.796100	+	0.605165i	$0.206893\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	54.2556	−	47.8727i	0.221451	−	0.195399i
$246$		0			0
$247$	13.0068	−	22.5284i	0.0526589	−	0.0912080i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	338.180i		−	1.34733i	−0.739037	−	0.673665i	$-0.764719\pi$
$251$		−	338.180i		−	1.34733i	0.739037	−	0.673665i	$-0.235281\pi$
$252$		0			0
$253$	621.629			2.45703
$254$		0			0
$255$		0			0
$256$		0			0
$257$	399.200	+	230.478i	1.55331	+	0.896803i	0.997869	+	0.0652458i	$0.0207832\pi$
$257$	399.200	+	230.478i	1.55331	+	0.896803i	0.555439	+	0.831557i	$0.312550\pi$
$258$		0			0
$259$	−110.526	+	11.0552i	−0.426743	+	0.0426841i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	62.4410	+	108.151i	0.237418	+	0.411221i	0.959973	−	0.280093i	$-0.0903654\pi$
$263$	62.4410	+	108.151i	0.237418	+	0.411221i	−0.722554	+	0.691314i	$0.757032\pi$
$264$		0			0
$265$		−	26.9774i		−	0.101802i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−368.369	+	212.678i	−1.36940	+	0.790625i	−0.990852	−	0.134955i	$-0.956911\pi$
$269$	−368.369	+	212.678i	−1.36940	+	0.790625i	−0.378551	+	0.925580i	$0.623578\pi$
$270$		0			0
$271$	353.970	+	204.365i	1.30616	+	0.754114i	0.981454	−	0.191700i	$-0.0614000\pi$
$271$	353.970	+	204.365i	1.30616	+	0.754114i	0.324710	+	0.945814i	$0.394733\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−177.179	+	306.882i	−0.644286	+	1.11594i
$276$		0			0
$277$	−179.920	−	311.630i	−0.649530	−	1.12502i	−0.983235	−	0.182341i	$-0.941632\pi$
$277$	−179.920	−	311.630i	−0.649530	−	1.12502i	0.333706	−	0.942677i	$-0.391701\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	188.703			0.671540			0.335770	−	0.941944i	$-0.391004\pi$
$281$	188.703			0.671540			0.335770	+	0.941944i	$0.391004\pi$
$282$		0			0
$283$	320.934	−	185.291i	1.13404	−	0.654739i	0.189093	−	0.981959i	$-0.439445\pi$
$283$	320.934	−	185.291i	1.13404	−	0.654739i	0.944948	+	0.327220i	$0.106112\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−117.275	+	84.3073i	−0.408622	+	0.293754i
$288$		0			0
$289$	−51.6586	+	89.4754i	−0.178750	+	0.309603i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	325.697i		−	1.11159i	−0.831318	−	0.555796i	$-0.812413\pi$
$293$		−	325.697i		−	1.11159i	0.831318	−	0.555796i	$-0.187587\pi$
$294$		0			0
$295$	71.1461			0.241173
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−647.231	−	373.679i	−2.16465	−	1.24976i
$300$		0			0
$301$	65.5998	+	91.2519i	0.217940	+	0.303162i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−19.8437	−	34.3703i	−0.0650614	−	0.112690i
$306$		0			0
$307$		−	185.193i		−	0.603236i	−0.953429	−	0.301618i	$-0.902473\pi$
$307$		−	185.193i		−	0.603236i	0.953429	−	0.301618i	$-0.0975267\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−307.643	+	177.618i	−0.989206	+	0.571119i	−0.905037	−	0.425333i	$-0.860157\pi$
$311$	−307.643	+	177.618i	−0.989206	+	0.571119i	−0.0841694	+	0.996451i	$0.526824\pi$
$312$		0			0
$313$	−191.007	−	110.278i	−0.610248	−	0.352327i	0.162815	−	0.986657i	$-0.447943\pi$
$313$	−191.007	−	110.278i	−0.610248	−	0.352327i	−0.773062	+	0.634330i	$0.781276\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	228.767	−	396.235i	0.721661	−	1.24995i	−0.238673	−	0.971100i	$-0.576712\pi$
$317$	228.767	−	396.235i	0.721661	−	1.24995i	0.960334	−	0.278853i	$-0.0899543\pi$
$318$		0			0
$319$	−404.537	−	700.678i	−1.26814	−	2.19648i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−18.9868			−0.0587826
$324$		0			0
$325$	368.952	−	213.014i	1.13524	−	0.655429i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−52.3922	−	523.802i	−0.159247	−	1.59210i
$330$		0			0
$331$	159.231	−	275.796i	0.481061	−	0.833221i	−0.518703	−	0.854954i	$-0.673585\pi$
$331$	159.231	−	275.796i	0.481061	−	0.833221i	0.999764	+	0.0217331i	$0.00691840\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	0.916753i		−	0.00273658i
$336$		0			0
$337$	−358.473			−1.06372			−0.531859	−	0.846833i	$-0.678506\pi$
$337$	−358.473			−1.06372			−0.531859	+	0.846833i	$0.678506\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	645.909	+	372.916i	1.89416	+	1.09360i
$342$		0			0
$343$	−327.774	+	101.060i	−0.955610	+	0.294635i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−17.3454	−	30.0431i	−0.0499867	−	0.0865795i	0.839949	−	0.542665i	$-0.182585\pi$
$347$	−17.3454	−	30.0431i	−0.0499867	−	0.0865795i	−0.889936	+	0.456085i	$0.849251\pi$
$348$		0			0
$349$		−	367.693i		−	1.05356i	−0.850001	−	0.526781i	$-0.823399\pi$
$349$		−	367.693i		−	1.05356i	0.850001	−	0.526781i	$-0.176601\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	15.8743	−	9.16504i	0.0449697	−	0.0259633i	−0.477347	−	0.878715i	$-0.658401\pi$
$353$	15.8743	−	9.16504i	0.0449697	−	0.0259633i	0.522316	+	0.852752i	$0.325068\pi$
$354$		0			0
$355$	−24.0027	−	13.8580i	−0.0676133	−	0.0390365i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	82.0379	−	142.094i	0.228518	−	0.395805i	−0.728851	−	0.684672i	$-0.759945\pi$
$359$	82.0379	−	142.094i	0.228518	−	0.395805i	0.957369	+	0.288868i	$0.0932787\pi$
$360$		0			0
$361$	−179.529	−	310.954i	−0.497311	−	0.861368i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	152.740			0.418465
$366$		0			0
$367$	303.246	−	175.079i	0.826284	−	0.477055i	−0.0262946	−	0.999654i	$-0.508371\pi$
$367$	303.246	−	175.079i	0.826284	−	0.477055i	0.852579	+	0.522599i	$0.175037\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−52.6020	+	116.565i	−0.141784	+	0.314192i
$372$		0			0
$373$	353.395	−	612.098i	0.947440	−	1.64101i	0.196650	−	0.980474i	$-0.436994\pi$
$373$	353.395	−	612.098i	0.947440	−	1.64101i	0.750790	−	0.660541i	$-0.229673\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			972.714i			2.58014i
$378$		0			0
$379$	534.245			1.40962			0.704809	−	0.709397i	$-0.251033\pi$
$379$	534.245			1.40962			0.704809	+	0.709397i	$0.251033\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−379.794	−	219.274i	−0.991630	−	0.572518i	−0.0858685	−	0.996306i	$-0.527367\pi$
$383$	−379.794	−	219.274i	−0.991630	−	0.572518i	−0.905761	+	0.423789i	$0.860700\pi$
$384$		0			0
$385$	−130.332	+	93.6942i	−0.338525	+	0.243361i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	156.927	+	271.805i	0.403410	+	0.698727i	0.994135	−	0.108146i	$-0.0344913\pi$
$389$	156.927	+	271.805i	0.403410	+	0.698727i	−0.590725	+	0.806873i	$0.701158\pi$
$390$		0			0
$391$			545.483i			1.39510i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	140.956	−	81.3813i	0.356852	−	0.206029i
$396$		0			0
$397$	487.980	+	281.736i	1.22917	+	0.709661i	0.966856	−	0.255321i	$-0.0821813\pi$
$397$	487.980	+	281.736i	1.22917	+	0.709661i	0.262313	+	0.964983i	$0.415515\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	300.808	−	521.014i	0.750144	−	1.29929i	−0.197609	−	0.980281i	$-0.563318\pi$
$401$	300.808	−	521.014i	0.750144	−	1.29929i	0.947753	−	0.319006i	$-0.103349\pi$
$402$		0			0
$403$	−448.341	−	776.549i	−1.11251	−	1.92692i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	246.414			0.605441
$408$		0			0
$409$	−134.300	+	77.5382i	−0.328362	+	0.189580i	−0.655114	−	0.755530i	$-0.727379\pi$
$409$	−134.300	+	77.5382i	−0.328362	+	0.189580i	0.326752	+	0.945110i	$0.394046\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−307.410	−	138.724i	−0.744335	−	0.335894i
$414$		0			0
$415$	17.6317	−	30.5391i	0.0424861	−	0.0735881i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			539.266i			1.28703i	0.765433	+	0.643516i	$0.222525\pi$
$419$			539.266i			1.28703i	−0.765433	+	0.643516i	$0.777475\pi$
$420$		0			0
$421$	259.595			0.616615			0.308307	−	0.951287i	$-0.400238\pi$
$421$	259.595			0.616615			0.308307	+	0.951287i	$0.400238\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−269.291	−	155.475i	−0.633626	−	0.365824i
$426$		0			0
$427$	18.7244	+	187.201i	0.0438510	+	0.438409i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−248.827	−	430.981i	−0.577324	−	0.999955i	−0.995785	−	0.0917199i	$-0.970764\pi$
$431$	−248.827	−	430.981i	−0.577324	−	0.999955i	0.418461	−	0.908235i	$-0.362570\pi$
$432$		0			0
$433$			416.661i			0.962266i	0.876648	+	0.481133i	$0.159775\pi$
$433$			416.661i			0.962266i	−0.876648	+	0.481133i	$0.840225\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	48.3049	−	27.8889i	0.110538	−	0.0638189i
$438$		0			0
$439$	141.093	+	81.4599i	0.321396	+	0.185558i	0.652015	−	0.758206i	$-0.273924\pi$
$439$	141.093	+	81.4599i	0.321396	+	0.185558i	−0.330619	+	0.943764i	$0.607257\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−321.003	+	555.993i	−0.724611	+	1.25506i	0.234522	+	0.972111i	$0.424647\pi$
$443$	−321.003	+	555.993i	−0.724611	+	1.25506i	−0.959134	+	0.282953i	$0.908686\pi$
$444$		0			0
$445$	−97.6198	−	169.082i	−0.219370	−	0.379961i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	238.462			0.531097			0.265548	−	0.964098i	$-0.414447\pi$
$449$	238.462			0.531097			0.265548	+	0.964098i	$0.414447\pi$
$450$		0			0
$451$	277.483	−	160.205i	0.615262	−	0.355222i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	192.022	−	19.2066i	0.422027	−	0.0422123i
$456$		0			0
$457$	−87.9517	+	152.337i	−0.192454	+	0.333341i	−0.946063	−	0.323982i	$-0.894978\pi$
$457$	−87.9517	+	152.337i	−0.192454	+	0.333341i	0.753609	+	0.657323i	$0.228311\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			161.802i			0.350980i	0.984481	+	0.175490i	$0.0561509\pi$
$461$			161.802i			0.350980i	−0.984481	+	0.175490i	$0.943849\pi$
$462$		0			0
$463$	88.3545			0.190831			0.0954153	−	0.995438i	$-0.469582\pi$
$463$	88.3545			0.190831			0.0954153	+	0.995438i	$0.469582\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−5.32384	−	3.07372i	−0.0114001	−	0.00658184i	0.494289	−	0.869298i	$-0.335428\pi$
$467$	−5.32384	−	3.07372i	−0.0114001	−	0.00658184i	−0.505689	+	0.862716i	$0.668762\pi$
$468$		0			0
$469$	−1.78753	+	3.96114i	−0.00381137	+	0.00844593i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−124.656	−	215.911i	−0.263544	−	0.456471i
$474$		0			0
$475$			31.7959i			0.0669387i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	518.538	−	299.378i	1.08254	−	0.625006i	0.150961	−	0.988540i	$-0.451763\pi$
$479$	518.538	−	299.378i	1.08254	−	0.625006i	0.931581	+	0.363534i	$0.118430\pi$
$480$		0			0
$481$	−256.563	−	148.127i	−0.533395	−	0.307956i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	53.1931	−	92.1332i	0.109677	−	0.189965i
$486$		0			0
$487$	209.892	+	363.544i	0.430990	+	0.746496i	0.996959	−	0.0779302i	$-0.0248311\pi$
$487$	209.892	+	363.544i	0.430990	+	0.746496i	−0.565969	+	0.824427i	$0.691498\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−39.5914			−0.0806341			−0.0403171	−	0.999187i	$-0.512837\pi$
$491$	−39.5914			−0.0806341			−0.0403171	+	0.999187i	$0.512837\pi$
$492$		0			0
$493$	614.849	−	354.983i	1.24716	−	0.720047i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	76.6908	+	106.680i	0.154307	+	0.214647i
$498$		0			0
$499$	−334.884	+	580.035i	−0.671109	+	1.16240i	0.306480	+	0.951877i	$0.400849\pi$
$499$	−334.884	+	580.035i	−0.671109	+	1.16240i	−0.977590	+	0.210519i	$0.932485\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	213.355i		−	0.424166i	−0.977252	−	0.212083i	$-0.931975\pi$
$503$		−	213.355i		−	0.424166i	0.977252	−	0.212083i	$-0.0680247\pi$
$504$		0			0
$505$	248.508			0.492094
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−391.019	−	225.755i	−0.768210	−	0.443526i	0.0640257	−	0.997948i	$-0.479606\pi$
$509$	−391.019	−	225.755i	−0.768210	−	0.443526i	−0.832236	+	0.554422i	$0.812939\pi$
$510$		0			0
$511$	−659.964	−	297.820i	−1.29151	−	0.582818i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−63.2818	−	109.607i	−0.122877	−	0.212830i
$516$		0			0
$517$			1167.80i			2.25879i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−633.929	+	365.999i	−1.21675	+	0.702494i	−0.964222	−	0.265095i	$-0.914597\pi$
$521$	−633.929	+	365.999i	−1.21675	+	0.702494i	−0.252532	+	0.967588i	$0.581263\pi$
$522$		0			0
$523$	365.430	+	210.981i	0.698720	+	0.403406i	0.806870	−	0.590728i	$-0.201160\pi$
$523$	365.430	+	210.981i	0.698720	+	0.403406i	−0.108151	+	0.994135i	$0.534493\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−327.236	+	566.789i	−0.620941	+	1.07550i
$528$		0			0
$529$	−536.736	−	929.654i	−1.01462	−	1.75738i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−385.215			−0.722730
$534$		0			0
$535$	3.95445	−	2.28310i	0.00739149	−	0.00426748i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	745.834	−	150.709i	1.38374	−	0.279608i
$540$		0			0
$541$	−448.892	+	777.504i	−0.829745	+	1.43716i	0.0684934	+	0.997652i	$0.478181\pi$
$541$	−448.892	+	777.504i	−0.829745	+	1.43716i	−0.898238	+	0.439509i	$0.855153\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	136.242i		−	0.249985i
$546$		0			0
$547$	−545.813			−0.997829			−0.498915	−	0.866651i	$-0.666268\pi$
$547$	−545.813			−0.997829			−0.498915	+	0.866651i	$0.666268\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−62.8707	−	36.2984i	−0.114103	−	0.0658773i
$552$		0			0
$553$	−767.731	+	76.7907i	−1.38830	+	0.138862i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−105.374	−	182.513i	−0.189181	−	0.327671i	0.755796	−	0.654807i	$-0.227250\pi$
$557$	−105.374	−	182.513i	−0.189181	−	0.327671i	−0.944977	+	0.327135i	$0.893917\pi$
$558$		0			0
$559$			299.738i			0.536203i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	258.646	−	149.329i	0.459407	−	0.265239i	−0.252388	−	0.967626i	$-0.581216\pi$
$563$	258.646	−	149.329i	0.459407	−	0.265239i	0.711795	+	0.702387i	$0.247883\pi$
$564$		0			0
$565$	168.046	+	97.0215i	0.297427	+	0.171719i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−234.818	+	406.717i	−0.412686	+	0.714793i	−0.995182	−	0.0980404i	$-0.968743\pi$
$569$	−234.818	+	406.717i	−0.412686	+	0.714793i	0.582497	+	0.812833i	$0.302076\pi$
$570$		0			0
$571$	109.586	+	189.808i	0.191919	+	0.332414i	0.945886	−	0.324498i	$-0.105195\pi$
$571$	109.586	+	189.808i	0.191919	+	0.332414i	−0.753967	+	0.656912i	$0.771862\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	913.483			1.58867
$576$		0			0
$577$	230.500	−	133.079i	0.399480	−	0.230640i	−0.286780	−	0.957997i	$-0.592585\pi$
$577$	230.500	−	133.079i	0.399480	−	0.230640i	0.686260	+	0.727357i	$0.259251\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−135.731	+	97.5750i	−0.233615	+	0.167943i
$582$		0			0
$583$	141.849	−	245.689i	0.243308	−	0.421422i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	209.316i		−	0.356586i	−0.983977	−	0.178293i	$-0.942943\pi$
$587$		−	209.316i		−	0.356586i	0.983977	−	0.178293i	$-0.0570575\pi$
$588$		0			0
$589$	66.9223			0.113620
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−363.251	−	209.723i	−0.612565	−	0.353665i	0.161404	−	0.986888i	$-0.448398\pi$
$593$	−363.251	−	209.723i	−0.612565	−	0.353665i	−0.773969	+	0.633224i	$0.781731\pi$
$594$		0			0
$595$	−82.2171	−	114.367i	−0.138180	−	0.192214i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	350.383	+	606.881i	0.584947	+	1.01316i	0.994882	+	0.101043i	$0.0322180\pi$
$599$	350.383	+	606.881i	0.584947	+	1.01316i	−0.409935	+	0.912115i	$0.634449\pi$
$600$		0			0
$601$			192.210i			0.319817i	0.987132	+	0.159909i	$0.0511200\pi$
$601$			192.210i			0.319817i	−0.987132	+	0.159909i	$0.948880\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	153.640	−	88.7043i	0.253951	−	0.146619i
$606$		0			0
$607$	−326.241	−	188.356i	−0.537465	−	0.310306i	0.206586	−	0.978428i	$-0.433765\pi$
$607$	−326.241	−	188.356i	−0.537465	−	0.310306i	−0.744051	+	0.668123i	$0.767098\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	701.995	−	1215.89i	1.14893	−	1.99000i
$612$		0			0
$613$	222.681	+	385.694i	0.363264	+	0.629191i	0.988496	−	0.151248i	$-0.0483291\pi$
$613$	222.681	+	385.694i	0.363264	+	0.629191i	−0.625232	+	0.780439i	$0.714996\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−562.357			−0.911437			−0.455719	−	0.890124i	$-0.650618\pi$
$617$	−562.357			−0.911437			−0.455719	+	0.890124i	$0.650618\pi$
$618$		0			0
$619$	351.292	−	202.819i	0.567515	−	0.327655i	−0.188641	−	0.982046i	$-0.560408\pi$
$619$	351.292	−	202.819i	0.567515	−	0.327655i	0.756156	+	0.654391i	$0.227075\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	92.1132	+	920.922i	0.147854	+	1.47820i
$624$		0			0
$625$	−233.107	+	403.754i	−0.372972	+	0.646006i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			216.230i			0.343768i
$630$		0			0
$631$	369.672			0.585851			0.292926	−	0.956135i	$-0.405371\pi$
$631$	369.672			0.585851			0.292926	+	0.956135i	$0.405371\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	112.557	+	64.9848i	0.177255	+	0.102338i
$636$		0			0
$637$	−867.146	−	291.426i	−1.36130	−	0.457498i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	353.414	+	612.132i	0.551349	+	0.954964i	0.998178	+	0.0603445i	$0.0192199\pi$
$641$	353.414	+	612.132i	0.551349	+	0.954964i	−0.446829	+	0.894619i	$0.647447\pi$
$642$		0			0
$643$			445.022i			0.692102i	0.938216	+	0.346051i	$0.112478\pi$
$643$			445.022i			0.692102i	−0.938216	+	0.346051i	$0.887522\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−321.002	+	185.331i	−0.496140	+	0.286446i	−0.727118	−	0.686513i	$-0.759141\pi$
$647$	−321.002	+	185.331i	−0.496140	+	0.286446i	0.230978	+	0.972959i	$0.425807\pi$
$648$		0			0
$649$	647.942	+	374.089i	0.998369	+	0.576409i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−107.834	+	186.774i	−0.165136	+	0.286025i	−0.936704	−	0.350123i	$-0.886140\pi$
$653$	−107.834	+	186.774i	−0.165136	+	0.286025i	0.771567	+	0.636148i	$0.219473\pi$
$654$		0			0
$655$	−23.9123	−	41.4174i	−0.0365074	−	0.0632326i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−146.496			−0.222301			−0.111151	−	0.993804i	$-0.535454\pi$
$659$	−146.496			−0.222301			−0.111151	+	0.993804i	$0.535454\pi$
$660$		0			0
$661$	792.061	−	457.297i	1.19828	−	0.691826i	0.238106	−	0.971239i	$-0.423473\pi$
$661$	792.061	−	457.297i	1.19828	−	0.691826i	0.960171	+	0.279414i	$0.0901401\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−5.92421	+	13.1279i	−0.00890859	+	0.0197413i
$666$		0			0
$667$	−1042.84	+	1806.25i	−1.56348	+	2.70802i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	417.357i		−	0.621992i
$672$		0			0
$673$	−1092.50			−1.62333			−0.811666	−	0.584122i	$-0.801439\pi$
$673$	−1092.50			−1.62333			−0.811666	+	0.584122i	$0.801439\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	151.935	+	87.7196i	0.224424	+	0.129571i	0.607997	−	0.793939i	$-0.291973\pi$
$677$	151.935	+	87.7196i	0.224424	+	0.129571i	−0.383573	+	0.923510i	$0.625307\pi$
$678$		0			0
$679$	−409.485	+	294.374i	−0.603071	+	0.433540i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	494.398	+	856.322i	0.723862	+	1.25377i	0.959441	+	0.281910i	$0.0909681\pi$
$683$	494.398	+	856.322i	0.723862	+	1.25377i	−0.235579	+	0.971855i	$0.575699\pi$
$684$		0			0
$685$			170.035i			0.248226i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−295.381	+	170.539i	−0.428710	+	0.247516i
$690$		0			0
$691$	−158.527	−	91.5256i	−0.229417	−	0.132454i	0.380886	−	0.924622i	$-0.375619\pi$
$691$	−158.527	−	91.5256i	−0.229417	−	0.132454i	−0.610303	+	0.792168i	$0.708952\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	152.614	−	264.334i	0.219588	−	0.380337i
$696$		0			0
$697$	140.581	+	243.493i	0.201694	+	0.349344i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−119.325			−0.170222			−0.0851109	−	0.996371i	$-0.527124\pi$
$701$	−119.325			−0.170222			−0.0851109	+	0.996371i	$0.527124\pi$
$702$		0			0
$703$	19.1481	−	11.0552i	0.0272377	−	0.0157257i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1073.76	−	484.553i	−1.51876	−	0.685365i
$708$		0			0
$709$	−101.857	+	176.421i	−0.143662	+	0.248830i	−0.928873	−	0.370398i	$-0.879221\pi$
$709$	−101.857	+	176.421i	−0.143662	+	0.248830i	0.785211	+	0.619229i	$0.212555\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	1922.65i		−	2.69656i
$714$		0			0
$715$	−428.106			−0.598749
$716$		0			0
$717$		0			0
$718$		0			0
$719$	149.327	+	86.2140i	0.207687	+	0.119908i	0.600236	−	0.799823i	$-0.295073\pi$
$719$	149.327	+	86.2140i	0.207687	+	0.119908i	−0.392549	+	0.919731i	$0.628407\pi$
$720$		0			0
$721$	59.7122	+	596.985i	0.0828186	+	0.827996i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−594.466	−	1029.65i	−0.819953	−	1.42020i
$726$		0			0
$727$			1002.21i			1.37855i	0.724499	+	0.689276i	$0.242071\pi$
$727$			1002.21i			1.37855i	−0.724499	+	0.689276i	$0.757929\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	189.463	−	109.386i	0.259183	−	0.149639i
$732$		0			0
$733$	−284.326	−	164.156i	−0.387894	−	0.223951i	0.293353	−	0.956004i	$-0.405229\pi$
$733$	−284.326	−	164.156i	−0.387894	−	0.223951i	−0.681247	+	0.732053i	$0.738562\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4.82033	−	8.34906i	0.00654048	−	0.0113284i
$738$		0			0
$739$	−612.564	−	1060.99i	−0.828910	−	1.43571i	−0.898894	−	0.438166i	$-0.855628\pi$
$739$	−612.564	−	1060.99i	−0.828910	−	1.43571i	0.0699840	−	0.997548i	$-0.477705\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−743.413			−1.00056			−0.500278	−	0.865865i	$-0.666769\pi$
$743$	−743.413			−1.00056			−0.500278	+	0.865865i	$0.666769\pi$
$744$		0			0
$745$	−193.156	+	111.519i	−0.259270	+	0.149689i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−21.5382	+	2.15432i	−0.0287560	+	0.00287626i
$750$		0			0
$751$	555.950	−	962.934i	0.740280	−	1.28220i	−0.212088	−	0.977251i	$-0.568026\pi$
$751$	555.950	−	962.934i	0.740280	−	1.28220i	0.952368	−	0.304952i	$-0.0986404\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			91.9097i			0.121735i
$756$		0			0
$757$	−716.530			−0.946539			−0.473270	−	0.880918i	$-0.656926\pi$
$757$	−716.530			−0.946539			−0.473270	+	0.880918i	$0.656926\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1022.88	+	590.561i	1.34413	+	0.776032i	0.987410	−	0.158180i	$-0.0505627\pi$
$761$	1022.88	+	590.561i	1.34413	+	0.776032i	0.356717	+	0.934212i	$0.383896\pi$
$762$		0			0
$763$	−265.652	+	588.679i	−0.348167	+	0.771532i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−449.751	−	778.992i	−0.586377	−	1.01564i
$768$		0			0
$769$			1046.71i			1.36113i	0.732687	+	0.680566i	$0.238266\pi$
$769$			1046.71i			1.36113i	−0.732687	+	0.680566i	$0.761734\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	516.667	−	298.298i	0.668391	−	0.385896i	−0.127075	−	0.991893i	$-0.540559\pi$
$773$	516.667	−	298.298i	0.668391	−	0.385896i	0.795467	+	0.605997i	$0.207226\pi$
$774$		0			0
$775$	949.164	+	548.000i	1.22473	+	0.707097i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	14.3749	−	24.8981i	0.0184531	−	0.0319616i
$780$		0			0
$781$	−145.732	−	252.415i	−0.186596	−	0.323194i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−357.099			−0.454903
$786$		0			0
$787$	858.587	−	495.705i	1.09096	−	0.629867i	0.157129	−	0.987578i	$-0.449776\pi$
$787$	858.587	−	495.705i	1.09096	−	0.629867i	0.933832	+	0.357711i	$0.116443\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−536.922	−	746.879i	−0.678789	−	0.944222i
$792$		0			0
$793$	−250.885	+	434.546i	−0.316375	+	0.547977i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			952.073i			1.19457i	0.802029	+	0.597285i	$0.203754\pi$
$797$			952.073i			1.19457i	−0.802029	+	0.597285i	$0.796246\pi$
$798$		0			0
$799$	−1024.75			−1.28254
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1391.03	+	803.113i	1.73229	+	1.00014i
$804$		0			0
$805$	377.160	+	170.200i	0.468522	+	0.211429i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	409.068	+	708.527i	0.505647	+	0.875806i	0.999979	+	0.00653275i	$0.00207945\pi$
$809$	409.068	+	708.527i	0.505647	+	0.875806i	−0.494332	+	0.869273i	$0.664587\pi$
$810$		0			0
$811$			1453.98i			1.79282i	0.443222	+	0.896412i	$0.353835\pi$
$811$			1453.98i			1.79282i	−0.443222	+	0.896412i	$0.646165\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.63748	+	0.945398i	−0.00200917	+	0.00116000i
$816$		0			0
$817$	−19.3733	−	11.1852i	−0.0237128	−	0.0136906i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−305.538	+	529.207i	−0.372153	+	0.644588i	−0.989897	−	0.141792i	$-0.954714\pi$
$821$	−305.538	+	529.207i	−0.372153	+	0.644588i	0.617744	+	0.786380i	$0.288047\pi$
$822$		0			0
$823$	120.570	+	208.833i	0.146500	+	0.253746i	0.929932	−	0.367732i	$-0.119866\pi$
$823$	120.570	+	208.833i	0.146500	+	0.253746i	−0.783431	+	0.621478i	$0.786532\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−868.740			−1.05047			−0.525236	−	0.850957i	$-0.676023\pi$
$827$	−868.740			−1.05047			−0.525236	+	0.850957i	$0.676023\pi$
$828$		0			0
$829$	−689.094	+	397.849i	−0.831236	+	0.479914i	−0.854276	−	0.519820i	$-0.825999\pi$
$829$	−689.094	+	397.849i	−0.831236	+	0.479914i	0.0230398	+	0.999735i	$0.492666\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	132.248	+	654.473i	0.158761	+	0.785682i
$834$		0			0
$835$	−117.291	+	203.154i	−0.140468	+	0.243298i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			271.638i			0.323763i	0.986810	+	0.161882i	$0.0517563\pi$
$839$			271.638i			0.323763i	−0.986810	+	0.161882i	$0.948244\pi$
$840$		0			0
$841$	1873.58			2.22781
$842$		0			0
$843$		0			0
$844$		0			0
$845$	229.615	+	132.568i	0.271734	+	0.156886i
$846$		0			0
$847$	−836.815	+	83.7007i	−0.987976	+	0.0988202i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−317.611	−	550.118i	−0.373221	−	0.646437i
$852$		0			0
$853$			61.5771i			0.0721889i	0.999348	+	0.0360945i	$0.0114917\pi$
$853$			61.5771i			0.0721889i	−0.999348	+	0.0360945i	$0.988508\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	935.801	−	540.285i	1.09195	−	0.630438i	0.157855	−	0.987462i	$-0.449542\pi$
$857$	935.801	−	540.285i	1.09195	−	0.630438i	0.934095	+	0.357025i	$0.116209\pi$
$858$		0			0
$859$	824.546	+	476.052i	0.959891	+	0.554193i	0.896140	−	0.443772i	$-0.146360\pi$
$859$	824.546	+	476.052i	0.959891	+	0.554193i	0.0637516	+	0.997966i	$0.479693\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−204.942	+	354.970i	−0.237476	+	0.411320i	−0.959989	−	0.280036i	$-0.909653\pi$
$863$	−204.942	+	354.970i	−0.237476	+	0.411320i	0.722513	+	0.691357i	$0.242987\pi$
$864$		0			0
$865$	16.6701	+	28.8735i	0.0192718	+	0.0333797i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1711.63			1.96965
$870$		0			0
$871$	−10.0377	+	5.79528i	−0.0115244	+	0.00665359i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−401.347	+	288.524i	−0.458683	+	0.329741i
$876$		0			0
$877$	−169.305	+	293.245i	−0.193051	+	0.334373i	−0.946260	−	0.323408i	$-0.895171\pi$
$877$	−169.305	+	293.245i	−0.193051	+	0.334373i	0.753209	+	0.657781i	$0.228505\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	446.960i		−	0.507333i	−0.967292	−	0.253666i	$-0.918363\pi$
$881$		−	446.960i		−	0.507333i	0.967292	−	0.253666i	$-0.0816365\pi$
$882$		0			0
$883$	884.678			1.00190			0.500950	−	0.865476i	$-0.332984\pi$
$883$	884.678			1.00190			0.500950	+	0.865476i	$0.332984\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	775.206	+	447.565i	0.873963	+	0.504583i	0.868663	−	0.495403i	$-0.164980\pi$
$887$	775.206	+	447.565i	0.873963	+	0.504583i	0.00530011	+	0.999986i	$0.498313\pi$
$888$		0			0
$889$	−359.629	−	500.258i	−0.404533	−	0.562720i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	52.3922	+	90.7460i	0.0586699	+	0.101619i
$894$		0			0
$895$			157.437i			0.175907i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2167.14	+	1251.20i	−2.41061	+	1.39177i
$900$		0			0
$901$	215.593	+	124.473i	0.239282	+	0.138150i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	104.810	−	181.535i	0.115812	−	0.200592i
$906$		0			0
$907$	−702.978	−	1217.59i	−0.775059	−	1.34244i	−0.934761	−	0.355276i	$-0.884387\pi$
$907$	−702.978	−	1217.59i	−0.775059	−	1.34244i	0.159703	−	0.987165i	$-0.448946\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−709.400			−0.778704			−0.389352	−	0.921089i	$-0.627301\pi$
$911$	−709.400			−0.778704			−0.389352	+	0.921089i	$0.627301\pi$
$912$		0			0
$913$	321.152	−	185.417i	0.351754	−	0.203085i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	22.5635	+	225.583i	0.0246058	+	0.246001i
$918$		0			0
$919$	−11.5829	+	20.0622i	−0.0126039	+	0.0218305i	−0.872259	−	0.489045i	$-0.837345\pi$
$919$	−11.5829	+	20.0622i	−0.0126039	+	0.0218305i	0.859655	+	0.510876i	$0.170679\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			350.414i			0.379647i
$924$		0			0
$925$	362.106			0.391466
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1051.65	−	607.168i	−1.13202	−	0.653572i	−0.187577	−	0.982250i	$-0.560064\pi$
$929$	−1051.65	−	607.168i	−1.13202	−	0.653572i	−0.944442	+	0.328678i	$0.893397\pi$
$930$		0			0
$931$	51.1951	−	45.1723i	0.0549894	−	0.0485202i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	156.233	+	270.604i	0.167094	+	0.289416i
$936$		0			0
$937$		−	465.914i		−	0.497240i	−0.968601	−	0.248620i	$-0.920023\pi$
$937$		−	465.914i		−	0.497240i	0.968601	−	0.248620i	$-0.0799770\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1354.48	−	782.009i	1.43940	−	0.831041i	0.441597	−	0.897214i	$-0.354412\pi$
$941$	1354.48	−	782.009i	1.43940	−	0.831041i	0.997808	+	0.0661729i	$0.0210789\pi$
$942$		0			0
$943$	−715.313	−	412.986i	−0.758550	−	0.437949i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−26.1548	+	45.3015i	−0.0276186	+	0.0478369i	−0.879504	−	0.475891i	$-0.842126\pi$
$947$	−26.1548	+	45.3015i	−0.0276186	+	0.0478369i	0.851886	+	0.523728i	$0.175459\pi$
$948$		0			0
$949$	−965.548	−	1672.38i	−1.01744	−	1.76225i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	248.105			0.260341			0.130170	−	0.991492i	$-0.458448\pi$
$953$	248.105			0.260341			0.130170	+	0.991492i	$0.458448\pi$
$954$		0			0
$955$	422.741	−	244.069i	0.442660	−	0.255570i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	331.542	−	734.691i	0.345717	−	0.766102i
$960$		0			0
$961$	672.900	−	1165.50i	0.700208	−	1.21280i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			251.933i			0.261070i
$966$		0			0
$967$	944.970			0.977218			0.488609	−	0.872503i	$-0.337505\pi$
$967$	944.970			0.977218			0.488609	+	0.872503i	$0.337505\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−896.377	−	517.523i	−0.923148	−	0.532980i	−0.0385102	−	0.999258i	$-0.512261\pi$
$971$	−896.377	−	517.523i	−0.923148	−	0.532980i	−0.884638	+	0.466278i	$0.845595\pi$
$972$		0			0
$973$	−1174.83	+	844.572i	−1.20743	+	0.868008i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	128.007	+	221.715i	0.131020	+	0.226934i	0.924070	−	0.382223i	$-0.124841\pi$
$977$	128.007	+	221.715i	0.131020	+	0.226934i	−0.793050	+	0.609157i	$0.791508\pi$
$978$		0			0
$979$		−	2053.16i		−	2.09720i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−547.281	+	315.973i	−0.556746	+	0.321437i	−0.751838	−	0.659348i	$-0.770833\pi$
$983$	−547.281	+	315.973i	−0.556746	+	0.321437i	0.195093	+	0.980785i	$0.437499\pi$
$984$		0			0
$985$	373.085	+	215.401i	0.378767	+	0.218681i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−321.346	+	556.588i	−0.324920	+	0.562778i
$990$		0			0
$991$	−130.255	−	225.608i	−0.131438	−	0.227657i	0.792793	−	0.609491i	$-0.208626\pi$
$991$	−130.255	−	225.608i	−0.131438	−	0.227657i	−0.924231	+	0.381834i	$0.875293\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−194.016			−0.194991
$996$		0			0
$997$	−592.313	+	341.972i	−0.594096	+	0.343001i	−0.766715	−	0.641987i	$-0.778110\pi$
$997$	−592.313	+	341.972i	−0.594096	+	0.343001i	0.172620	+	0.984989i	$0.444777\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.3.cg.q.145.4		16
3.2	odd	2	inner	1008.3.cg.q.145.5		16
4.3	odd	2		504.3.by.d.145.4	yes	16
7.3	odd	6	inner	1008.3.cg.q.577.4		16
12.11	even	2		504.3.by.d.145.5	yes	16
21.17	even	6	inner	1008.3.cg.q.577.5		16
28.3	even	6		504.3.by.d.73.4	✓	16
28.19	even	6		3528.3.f.i.2449.7		16
28.23	odd	6		3528.3.f.i.2449.9		16
84.23	even	6		3528.3.f.i.2449.8		16
84.47	odd	6		3528.3.f.i.2449.10		16
84.59	odd	6		504.3.by.d.73.5	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.3.by.d.73.4	✓	16	28.3	even	6
504.3.by.d.73.5	yes	16	84.59	odd	6
504.3.by.d.145.4	yes	16	4.3	odd	2
504.3.by.d.145.5	yes	16	12.11	even	2
1008.3.cg.q.145.4		16	1.1	even	1	trivial
1008.3.cg.q.145.5		16	3.2	odd	2	inner
1008.3.cg.q.577.4		16	7.3	odd	6	inner
1008.3.cg.q.577.5		16	21.17	even	6	inner
3528.3.f.i.2449.7		16	28.19	even	6
3528.3.f.i.2449.8		16	84.23	even	6
3528.3.f.i.2449.9		16	28.23	odd	6
3528.3.f.i.2449.10		16	84.47	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:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1008.cg (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.27883 - 0.738332i) q^{5} +(4.08597 + 5.68374i) q^{7} +O(q^{10})\)	\(q+(-1.27883 - 0.738332i) q^{5} +(4.08597 + 5.68374i) q^{7} +(-7.76437 - 13.4483i) q^{11} +18.6695i q^{13} +(11.8009 - 6.81327i) q^{17} +(-1.20669 - 0.696684i) q^{19} +(-20.0154 + 34.6678i) q^{23} +(-11.4097 - 19.7622i) q^{25} +52.1017 q^{29} +(-41.5945 + 24.0146i) q^{31} +(-1.02877 - 10.2853i) q^{35} +(-7.93414 + 13.7423i) q^{37} +20.6334i q^{41} +16.0549 q^{43} +(-65.1271 - 37.6011i) q^{47} +(-15.6097 + 46.4471i) q^{49} +(9.13459 + 15.8216i) q^{53} +22.9307i q^{55} +(-41.7253 + 24.0901i) q^{59} +(23.2757 + 13.4382i) q^{61} +(13.7843 - 23.8751i) q^{65} +(0.310414 + 0.537652i) q^{67} +18.7693 q^{71} +(-89.5780 + 51.7179i) q^{73} +(44.7115 - 99.0798i) q^{77} +(-55.1116 + 95.4561i) q^{79} +23.8805i q^{83} -20.1218 q^{85} +(114.503 + 66.1084i) q^{89} +(-106.113 + 76.2831i) q^{91} +(1.02877 + 1.78188i) q^{95} +72.0450i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{7}+O(q^{10}) \) 16 * q - 4 * q^7	\( 16 q - 4 q^{7} + 24 q^{19} + 36 q^{25} + 84 q^{31} - 68 q^{37} - 80 q^{43} - 184 q^{49} + 216 q^{61} - 56 q^{67} + 156 q^{73} - 28 q^{79} + 448 q^{85} - 48 q^{91}+O(q^{100}) \) 16 * q - 4 * q^7 + 24 * q^19 + 36 * q^25 + 84 * q^31 - 68 * q^37 - 80 * q^43 - 184 * q^49 + 216 * q^61 - 56 * q^67 + 156 * q^73 - 28 * q^79 + 448 * q^85 - 48 * q^91

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