LMFDB - Embedded newform 864.3.q.a.737.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,3,Mod(449,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([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("864.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	864.q (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:	$23.5422948407$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 288)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			737.8
Character	$\chi$	$=$	864.737
Dual form			864.3.q.a.449.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.83987 - 1.06225i) q^{5} +(3.42470 - 5.93176i) q^{7} +(-4.54662 - 2.62499i) q^{11} +(-10.0161 - 17.3484i) q^{13} +0.0666728i q^{17} -27.2587 q^{19} +(-18.5685 + 10.7205i) q^{23} +(-10.2432 + 17.7418i) q^{25} +(37.2921 + 21.5306i) q^{29} +(-8.06516 - 13.9693i) q^{31} -14.5516i q^{35} -22.3835 q^{37} +(2.56602 - 1.48149i) q^{41} +(14.6708 - 25.4106i) q^{43} +(-38.3723 - 22.1542i) q^{47} +(1.04281 + 1.80621i) q^{49} +12.8946i q^{53} -11.1536 q^{55} +(-75.0094 + 43.3067i) q^{59} +(6.45271 - 11.1764i) q^{61} +(-36.8568 - 21.2793i) q^{65} +(-61.4191 - 106.381i) q^{67} +125.360i q^{71} +90.7813 q^{73} +(-31.1416 + 17.9796i) q^{77} +(24.2523 - 42.0063i) q^{79} +(-101.140 - 58.3934i) q^{83} +(0.0708233 + 0.122669i) q^{85} -116.172i q^{89} -137.209 q^{91} +(-50.1525 + 28.9556i) q^{95} +(-23.2781 + 40.3188i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 60 q^{25} - 72 q^{29} - 252 q^{41} - 36 q^{49} - 96 q^{61} + 288 q^{65} + 24 q^{73} + 720 q^{77} + 96 q^{85} - 132 q^{97}+O(q^{100}) $ 24 * q + 60 * q^25 - 72 * q^29 - 252 * q^41 - 36 * q^49 - 96 * q^61 + 288 * q^65 + 24 * q^73 + 720 * q^77 + 96 * q^85 - 132 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$.

$n$	$325$	$353$	$703$
$\chi(n)$	$1$	$e\left(\frac{1}{6}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.83987	−	1.06225i	0.367974	−	0.212450i	−0.304599	−	0.952481i	$-0.598522\pi$
$5$	1.83987	−	1.06225i	0.367974	−	0.212450i	0.672573	+	0.740031i	$0.265189\pi$
$6$		0			0
$7$	3.42470	−	5.93176i	0.489243	−	0.847394i	−0.510680	−	0.859771i	$-0.670606\pi$
$7$	3.42470	−	5.93176i	0.489243	−	0.847394i	0.999923	+	0.0123765i	$0.00393967\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.54662	−	2.62499i	−0.413329	−	0.238636i	0.278890	−	0.960323i	$-0.410033\pi$
$11$	−4.54662	−	2.62499i	−0.413329	−	0.238636i	−0.692219	+	0.721687i	$0.743367\pi$
$12$		0			0
$13$	−10.0161	−	17.3484i	−0.770471	−	1.33449i	−0.937305	−	0.348510i	$-0.886688\pi$
$13$	−10.0161	−	17.3484i	−0.770471	−	1.33449i	0.166834	−	0.985985i	$-0.446646\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			0.0666728i			0.00392193i	0.999998	+	0.00196097i	$0.000624195\pi$
$17$			0.0666728i			0.00392193i	−0.999998	+	0.00196097i	$0.999376\pi$
$18$		0			0
$19$	−27.2587			−1.43467			−0.717334	−	0.696730i	$-0.754638\pi$
$19$	−27.2587			−1.43467			−0.717334	+	0.696730i	$0.754638\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−18.5685	+	10.7205i	−0.807326	+	0.466110i	−0.846026	−	0.533141i	$-0.821012\pi$
$23$	−18.5685	+	10.7205i	−0.807326	+	0.466110i	0.0387003	+	0.999251i	$0.487678\pi$
$24$		0			0
$25$	−10.2432	+	17.7418i	−0.409730	+	0.709673i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	37.2921	+	21.5306i	1.28594	+	0.742435i	0.977927	−	0.208948i	$-0.0670041\pi$
$29$	37.2921	+	21.5306i	1.28594	+	0.742435i	0.308009	+	0.951384i	$0.400337\pi$
$30$		0			0
$31$	−8.06516	−	13.9693i	−0.260167	−	0.450622i	0.706119	−	0.708093i	$-0.250444\pi$
$31$	−8.06516	−	13.9693i	−0.260167	−	0.450622i	−0.966286	+	0.257471i	$0.917111\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	14.5516i		−	0.415759i
$36$		0			0
$37$	−22.3835			−0.604961			−0.302480	−	0.953156i	$-0.597815\pi$
$37$	−22.3835			−0.604961			−0.302480	+	0.953156i	$0.597815\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.56602	−	1.48149i	0.0625859	−	0.0361340i	−0.468381	−	0.883527i	$-0.655162\pi$
$41$	2.56602	−	1.48149i	0.0625859	−	0.0361340i	0.530966	+	0.847393i	$0.321829\pi$
$42$		0			0
$43$	14.6708	−	25.4106i	0.341182	−	0.590944i	−0.643471	−	0.765471i	$-0.722506\pi$
$43$	14.6708	−	25.4106i	0.341182	−	0.590944i	0.984652	+	0.174527i	$0.0558396\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−38.3723	−	22.1542i	−0.816431	−	0.471367i	0.0327529	−	0.999463i	$-0.489573\pi$
$47$	−38.3723	−	22.1542i	−0.816431	−	0.471367i	−0.849184	+	0.528097i	$0.822906\pi$
$48$		0			0
$49$	1.04281	+	1.80621i	0.0212819	+	0.0368614i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			12.8946i			0.243294i	0.992573	+	0.121647i	$0.0388176\pi$
$53$			12.8946i			0.243294i	−0.992573	+	0.121647i	$0.961182\pi$
$54$		0			0
$55$	−11.1536			−0.202793
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−75.0094	+	43.3067i	−1.27135	+	0.734012i	−0.975241	−	0.221144i	$-0.929021\pi$
$59$	−75.0094	+	43.3067i	−1.27135	+	0.734012i	−0.296105	+	0.955155i	$0.595688\pi$
$60$		0			0
$61$	6.45271	−	11.1764i	0.105782	−	0.183220i	−0.808275	−	0.588805i	$-0.799599\pi$
$61$	6.45271	−	11.1764i	0.105782	−	0.183220i	0.914057	+	0.405585i	$0.132932\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−36.8568	−	21.2793i	−0.567027	−	0.327373i
$66$		0			0
$67$	−61.4191	−	106.381i	−0.916704	−	1.58778i	−0.804388	−	0.594105i	$-0.797506\pi$
$67$	−61.4191	−	106.381i	−0.916704	−	1.58778i	−0.112316	−	0.993673i	$-0.535827\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			125.360i			1.76563i	0.469717	+	0.882817i	$0.344356\pi$
$71$			125.360i			1.76563i	−0.469717	+	0.882817i	$0.655644\pi$
$72$		0			0
$73$	90.7813			1.24358			0.621790	−	0.783184i	$-0.286406\pi$
$73$	90.7813			1.24358			0.621790	+	0.783184i	$0.286406\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−31.1416	+	17.9796i	−0.404437	+	0.233502i
$78$		0			0
$79$	24.2523	−	42.0063i	0.306992	−	0.531725i	−0.670711	−	0.741719i	$-0.734011\pi$
$79$	24.2523	−	42.0063i	0.306992	−	0.531725i	0.977703	+	0.209994i	$0.0673443\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−101.140	−	58.3934i	−1.21856	−	0.703534i	−0.253948	−	0.967218i	$-0.581729\pi$
$83$	−101.140	−	58.3934i	−1.21856	−	0.703534i	−0.964609	+	0.263683i	$0.915063\pi$
$84$		0			0
$85$	0.0708233	+	0.122669i	0.000833215	+	0.00144317i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	116.172i		−	1.30531i	−0.757656	−	0.652654i	$-0.773655\pi$
$89$		−	116.172i		−	1.30531i	0.757656	−	0.652654i	$-0.226345\pi$
$90$		0			0
$91$	−137.209			−1.50779
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−50.1525	+	28.9556i	−0.527921	+	0.304795i
$96$		0			0
$97$	−23.2781	+	40.3188i	−0.239980	+	0.415658i	−0.960708	−	0.277560i	$-0.910474\pi$
$97$	−23.2781	+	40.3188i	−0.239980	+	0.415658i	0.720728	+	0.693218i	$0.243808\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−117.017	−	67.5600i	−1.15859	−	0.668911i	−0.207622	−	0.978209i	$-0.566572\pi$
$101$	−117.017	−	67.5600i	−1.15859	−	0.668911i	−0.950965	+	0.309298i	$0.899906\pi$
$102$		0			0
$103$	−7.91649	−	13.7118i	−0.0768592	−	0.133124i	0.825034	−	0.565083i	$-0.191156\pi$
$103$	−7.91649	−	13.7118i	−0.0768592	−	0.133124i	−0.901893	+	0.431959i	$0.857823\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	191.129i		−	1.78625i	−0.449810	−	0.893124i	$-0.648508\pi$
$107$		−	191.129i		−	1.78625i	0.449810	−	0.893124i	$-0.351492\pi$
$108$		0			0
$109$	144.818			1.32861			0.664304	−	0.747463i	$-0.268728\pi$
$109$	144.818			1.32861			0.664304	+	0.747463i	$0.268728\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−83.4871	+	48.2013i	−0.738824	+	0.426560i	−0.821641	−	0.570005i	$-0.806941\pi$
$113$	−83.4871	+	48.2013i	−0.738824	+	0.426560i	0.0828178	+	0.996565i	$0.473608\pi$
$114$		0			0
$115$	−22.7758	+	39.4488i	−0.198050	+	0.343033i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.395487	+	0.228335i	0.00332342	+	0.00191878i
$120$		0			0
$121$	−46.7188	−	80.9194i	−0.386106	−	0.668755i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			96.6361i			0.773089i
$126$		0			0
$127$	127.727			1.00572			0.502861	−	0.864367i	$-0.332281\pi$
$127$	127.727			1.00572			0.502861	+	0.864367i	$0.332281\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	28.3110	−	16.3454i	0.216115	−	0.124774i	−0.388035	−	0.921645i	$-0.626846\pi$
$131$	28.3110	−	16.3454i	0.216115	−	0.124774i	0.604150	+	0.796871i	$0.293513\pi$
$132$		0			0
$133$	−93.3529	+	161.692i	−0.701901	+	1.21573i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	159.718	+	92.2133i	1.16583	+	0.673089i	0.952693	−	0.303934i	$-0.0983002\pi$
$137$	159.718	+	92.2133i	1.16583	+	0.673089i	0.213132	+	0.977023i	$0.431634\pi$
$138$		0			0
$139$	−119.353	−	206.725i	−0.858652	−	1.48723i	−0.873215	−	0.487336i	$-0.837969\pi$
$139$	−119.353	−	206.725i	−0.858652	−	1.48723i	0.0145624	−	0.999894i	$-0.495364\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			105.169i			0.735447i
$144$		0			0
$145$	91.4837			0.630922
$146$		0			0
$147$		0			0
$148$		0			0
$149$	222.274	−	128.330i	1.49177	−	0.861274i	0.491814	−	0.870700i	$-0.336334\pi$
$149$	222.274	−	128.330i	1.49177	−	0.861274i	0.999956	+	0.00942657i	$0.00300061\pi$
$150$		0			0
$151$	−41.6371	+	72.1175i	−0.275742	+	0.477600i	−0.970322	−	0.241816i	$-0.922257\pi$
$151$	−41.6371	+	72.1175i	−0.275742	+	0.477600i	0.694580	+	0.719416i	$0.255590\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−29.6777	−	17.1345i	−0.191469	−	0.110545i
$156$		0			0
$157$	−79.7821	−	138.187i	−0.508166	−	0.880170i	−0.999955	−	0.00945526i	$-0.996990\pi$
$157$	−79.7821	−	138.187i	−0.508166	−	0.880170i	0.491789	−	0.870714i	$-0.336343\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			146.859i			0.912165i
$162$		0			0
$163$	−18.4140			−0.112970			−0.0564848	−	0.998403i	$-0.517989\pi$
$163$	−18.4140			−0.112970			−0.0564848	+	0.998403i	$0.517989\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−175.064	+	101.073i	−1.04829	+	0.605229i	−0.922170	−	0.386786i	$-0.873585\pi$
$167$	−175.064	+	101.073i	−1.04829	+	0.605229i	−0.126118	+	0.992015i	$0.540252\pi$
$168$		0			0
$169$	−116.145	+	201.170i	−0.687251	+	1.19035i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−87.4383	−	50.4825i	−0.505424	−	0.291807i	0.225527	−	0.974237i	$-0.427590\pi$
$173$	−87.4383	−	50.4825i	−0.505424	−	0.291807i	−0.730951	+	0.682430i	$0.760923\pi$
$174$		0			0
$175$	70.1602	+	121.521i	0.400915	+	0.694405i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			45.8305i			0.256036i	0.991772	+	0.128018i	$0.0408616\pi$
$179$			45.8305i			0.256036i	−0.991772	+	0.128018i	$0.959138\pi$
$180$		0			0
$181$	55.7708			0.308126			0.154063	−	0.988061i	$-0.450764\pi$
$181$	55.7708			0.308126			0.154063	+	0.988061i	$0.450764\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−41.1829	+	23.7769i	−0.222610	+	0.128524i
$186$		0			0
$187$	0.175016	−	0.303136i	0.000935912	−	0.00162105i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	274.007	+	158.198i	1.43459	+	0.828262i	0.997466	−	0.0711380i	$-0.0226631\pi$
$191$	274.007	+	158.198i	1.43459	+	0.828262i	0.437126	+	0.899400i	$0.355996\pi$
$192$		0			0
$193$	−187.647	−	325.013i	−0.972262	−	1.68401i	−0.688690	−	0.725056i	$-0.741814\pi$
$193$	−187.647	−	325.013i	−0.972262	−	1.68401i	−0.283571	−	0.958951i	$-0.591519\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	46.6367i		−	0.236734i	−0.992970	−	0.118367i	$-0.962234\pi$
$197$		−	46.6367i		−	0.236734i	0.992970	−	0.118367i	$-0.0377660\pi$
$198$		0			0
$199$	192.041			0.965033			0.482516	−	0.875887i	$-0.339723\pi$
$199$	192.041			0.965033			0.482516	+	0.875887i	$0.339723\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	255.429	−	147.472i	1.25827	−	0.726463i
$204$		0			0
$205$	3.14744	−	5.45152i	0.0153533	−	0.0265928i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	123.935	+	71.5538i	0.592990	+	0.342363i
$210$		0			0
$211$	157.590	+	272.954i	0.746872	+	1.29362i	0.949315	+	0.314326i	$0.101778\pi$
$211$	157.590	+	272.954i	0.746872	+	1.29362i	−0.202443	+	0.979294i	$0.564888\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	62.3363i		−	0.289936i
$216$		0			0
$217$	−110.483			−0.509139
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.15667	−	0.667803i	0.00523380	−	0.00302173i
$222$		0			0
$223$	92.1859	−	159.671i	0.413390	−	0.716012i	−0.581868	−	0.813283i	$-0.697678\pi$
$223$	92.1859	−	159.671i	0.413390	−	0.716012i	0.995258	+	0.0972713i	$0.0310114\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−114.806	−	66.2830i	−0.505752	−	0.291996i	0.225334	−	0.974282i	$-0.427653\pi$
$227$	−114.806	−	66.2830i	−0.505752	−	0.291996i	−0.731086	+	0.682286i	$0.760986\pi$
$228$		0			0
$229$	12.0352	+	20.8456i	0.0525556	+	0.0910290i	0.891106	−	0.453794i	$-0.149930\pi$
$229$	12.0352	+	20.8456i	0.0525556	+	0.0910290i	−0.838551	+	0.544824i	$0.816597\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	135.751i		−	0.582621i	−0.956629	−	0.291311i	$-0.905909\pi$
$233$		−	135.751i		−	0.582621i	0.956629	−	0.291311i	$-0.0940913\pi$
$234$		0			0
$235$	−94.1335			−0.400568
$236$		0			0
$237$		0			0
$238$		0			0
$239$	371.772	−	214.642i	1.55553	−	0.898085i	0.557854	−	0.829939i	$-0.311625\pi$
$239$	371.772	−	214.642i	1.55553	−	0.898085i	0.997675	−	0.0681461i	$-0.0217084\pi$
$240$		0			0
$241$	91.2013	−	157.965i	0.378429	−	0.655458i	−0.612405	−	0.790544i	$-0.709798\pi$
$241$	91.2013	−	157.965i	0.378429	−	0.655458i	0.990834	+	0.135086i	$0.0431312\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3.83729	+	2.21546i	0.0156624	+	0.00904269i
$246$		0			0
$247$	273.026	+	472.895i	1.10537	+	1.91456i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	69.0483i		−	0.275093i	−0.990495	−	0.137546i	$-0.956078\pi$
$251$		−	69.0483i		−	0.275093i	0.990495	−	0.137546i	$-0.0439216\pi$
$252$		0			0
$253$	112.565			0.444922
$254$		0			0
$255$		0			0
$256$		0			0
$257$	92.0921	−	53.1694i	0.358335	−	0.206885i	−0.310015	−	0.950732i	$-0.600334\pi$
$257$	92.0921	−	53.1694i	0.358335	−	0.206885i	0.668350	+	0.743847i	$0.267001\pi$
$258$		0			0
$259$	−76.6570	+	132.774i	−0.295973	+	0.512640i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	209.762	+	121.106i	0.797575	+	0.460480i	0.842622	−	0.538505i	$-0.181011\pi$
$263$	209.762	+	121.106i	0.797575	+	0.460480i	−0.0450477	+	0.998985i	$0.514344\pi$
$264$		0			0
$265$	13.6973	+	23.7244i	0.0516879	+	0.0895260i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			143.438i			0.533228i	0.963803	+	0.266614i	$0.0859049\pi$
$269$			143.438i			0.533228i	−0.963803	+	0.266614i	$0.914095\pi$
$270$		0			0
$271$	−59.8857			−0.220980			−0.110490	−	0.993877i	$-0.535242\pi$
$271$	−59.8857			−0.220980			−0.110490	+	0.993877i	$0.535242\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	93.1443	−	53.7769i	0.338706	−	0.195552i
$276$		0			0
$277$	−157.458	+	272.725i	−0.568441	+	0.984568i	0.428280	+	0.903646i	$0.359120\pi$
$277$	−157.458	+	272.725i	−0.568441	+	0.984568i	−0.996720	+	0.0809221i	$0.974214\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	35.1739	+	20.3077i	0.125174	+	0.0722693i	0.561280	−	0.827626i	$-0.310309\pi$
$281$	35.1739	+	20.3077i	0.125174	+	0.0722693i	−0.436106	+	0.899896i	$0.643643\pi$
$282$		0			0
$283$	50.9484	+	88.2451i	0.180030	+	0.311820i	0.941890	−	0.335920i	$-0.109047\pi$
$283$	50.9484	+	88.2451i	0.180030	+	0.311820i	−0.761861	+	0.647741i	$0.775714\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	20.2947i		−	0.0707133i
$288$		0			0
$289$	288.996			0.999985
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−116.145	+	67.0565i	−0.396400	+	0.228862i	−0.684930	−	0.728609i	$-0.740167\pi$
$293$	−116.145	+	67.0565i	−0.396400	+	0.228862i	0.288529	+	0.957471i	$0.406834\pi$
$294$		0			0
$295$	−92.0052	+	159.358i	−0.311882	+	0.540195i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	371.969	+	214.756i	1.24404	+	0.718248i
$300$		0			0
$301$	−100.486	−	174.047i	−0.333842	−	0.578231i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	27.4176i		−	0.0898937i
$306$		0			0
$307$	129.799			0.422799			0.211400	−	0.977400i	$-0.432198\pi$
$307$	129.799			0.422799			0.211400	+	0.977400i	$0.432198\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−175.014	+	101.044i	−0.562746	+	0.324902i	−0.754247	−	0.656591i	$-0.771998\pi$
$311$	−175.014	+	101.044i	−0.562746	+	0.324902i	0.191501	+	0.981492i	$0.438665\pi$
$312$		0			0
$313$	−168.502	+	291.854i	−0.538346	+	0.932442i	0.460648	+	0.887583i	$0.347617\pi$
$313$	−168.502	+	291.854i	−0.538346	+	0.932442i	−0.998993	+	0.0448588i	$0.985716\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−268.759	−	155.168i	−0.847821	−	0.489490i	0.0120937	−	0.999927i	$-0.496150\pi$
$317$	−268.759	−	155.168i	−0.847821	−	0.489490i	−0.859915	+	0.510437i	$0.829484\pi$
$318$		0			0
$319$	−113.035	−	195.783i	−0.354343	−	0.613740i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	1.81741i		−	0.00562667i
$324$		0			0
$325$	410.390			1.26274
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−262.827	+	151.743i	−0.798867	+	0.461226i
$330$		0			0
$331$	82.2294	−	142.426i	0.248427	−	0.430289i	−0.714662	−	0.699470i	$-0.753420\pi$
$331$	82.2294	−	142.426i	0.248427	−	0.430289i	0.963090	+	0.269181i	$0.0867530\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−226.007	−	130.485i	−0.674647	−	0.389508i
$336$		0			0
$337$	94.5387	+	163.746i	0.280530	+	0.485892i	0.971515	−	0.236976i	$-0.0761564\pi$
$337$	94.5387	+	163.746i	0.280530	+	0.485892i	−0.690985	+	0.722869i	$0.742823\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			84.6840i			0.248340i
$342$		0			0
$343$	349.906			1.02013
$344$		0			0
$345$		0			0
$346$		0			0
$347$	373.319	−	215.536i	1.07585	−	0.621141i	0.146075	−	0.989274i	$-0.453336\pi$
$347$	373.319	−	215.536i	1.07585	−	0.621141i	0.929773	+	0.368132i	$0.120003\pi$
$348$		0			0
$349$	−183.325	+	317.529i	−0.525287	+	0.909824i	0.474279	+	0.880375i	$0.342709\pi$
$349$	−183.325	+	317.529i	−0.525287	+	0.909824i	−0.999566	+	0.0294495i	$0.990625\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−50.6831	−	29.2619i	−0.143578	−	0.0828949i	0.426490	−	0.904492i	$-0.359750\pi$
$353$	−50.6831	−	29.2619i	−0.143578	−	0.0828949i	−0.570068	+	0.821597i	$0.693083\pi$
$354$		0			0
$355$	133.164	+	230.646i	0.375109	+	0.649708i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			48.0665i			0.133890i	0.997757	+	0.0669449i	$0.0213252\pi$
$359$			48.0665i			0.133890i	−0.997757	+	0.0669449i	$0.978675\pi$
$360$		0			0
$361$	382.036			1.05827
$362$		0			0
$363$		0			0
$364$		0			0
$365$	167.026	−	96.4325i	0.457605	−	0.264199i
$366$		0			0
$367$	−57.1737	+	99.0277i	−0.155787	+	0.269830i	−0.933345	−	0.358980i	$-0.883125\pi$
$367$	−57.1737	+	99.0277i	−0.155787	+	0.269830i	0.777559	+	0.628810i	$0.216458\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	76.4876	+	44.1601i	0.206166	+	0.119030i
$372$		0			0
$373$	−172.978	−	299.607i	−0.463748	−	0.803235i	0.535396	−	0.844601i	$-0.320162\pi$
$373$	−172.978	−	299.607i	−0.463748	−	0.803235i	−0.999144	+	0.0413660i	$0.986829\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	862.613i		−	2.28810i
$378$		0			0
$379$	323.425			0.853364			0.426682	−	0.904402i	$-0.359682\pi$
$379$	323.425			0.853364			0.426682	+	0.904402i	$0.359682\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	449.868	−	259.732i	1.17459	−	0.678151i	0.219834	−	0.975537i	$-0.429448\pi$
$383$	449.868	−	259.732i	1.17459	−	0.678151i	0.954757	+	0.297387i	$0.0961151\pi$
$384$		0			0
$385$	−38.1978	+	66.1605i	−0.0992150	+	0.171845i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	148.157	+	85.5386i	0.380867	+	0.219893i	0.678195	−	0.734882i	$-0.262762\pi$
$389$	148.157	+	85.5386i	0.380867	+	0.219893i	−0.297329	+	0.954775i	$0.596096\pi$
$390$		0			0
$391$	−0.714768	−	1.23801i	−0.00182805	−	0.00316628i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	103.048i		−	0.260882i
$396$		0			0
$397$	159.942			0.402878			0.201439	−	0.979501i	$-0.435438\pi$
$397$	159.942			0.402878			0.201439	+	0.979501i	$0.435438\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−568.013	+	327.942i	−1.41649	+	0.817811i	−0.995989	−	0.0894809i	$-0.971479\pi$
$401$	−568.013	+	327.942i	−1.41649	+	0.817811i	−0.420502	+	0.907292i	$0.638146\pi$
$402$		0			0
$403$	−161.563	+	279.836i	−0.400902	+	0.694382i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	101.769	+	58.7566i	0.250048	+	0.144365i
$408$		0			0
$409$	−33.1589	−	57.4330i	−0.0810732	−	0.140423i	0.822638	−	0.568566i	$-0.192501\pi$
$409$	−33.1589	−	57.4330i	−0.0810732	−	0.140423i	−0.903711	+	0.428143i	$0.859168\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			593.250i			1.43644i
$414$		0			0
$415$	−248.114			−0.597864
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−644.388	+	372.038i	−1.53792	+	0.887918i	−0.538959	+	0.842332i	$0.681182\pi$
$419$	−644.388	+	372.038i	−1.53792	+	0.887918i	−0.998960	+	0.0455866i	$0.985484\pi$
$420$		0			0
$421$	340.558	−	589.864i	0.808927	−	1.40110i	−0.104680	−	0.994506i	$-0.533382\pi$
$421$	340.558	−	589.864i	0.808927	−	1.40110i	0.913608	−	0.406597i	$-0.133285\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.18290	−	0.682946i	−0.00278329	−	0.00160693i
$426$		0			0
$427$	−44.1972	−	76.5518i	−0.103506	−	0.179278i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	251.031i		−	0.582439i	−0.956656	−	0.291220i	$-0.905939\pi$
$431$		−	251.031i		−	0.582439i	0.956656	−	0.291220i	$-0.0940610\pi$
$432$		0			0
$433$	−87.8113			−0.202797			−0.101399	−	0.994846i	$-0.532332\pi$
$433$	−87.8113			−0.202797			−0.101399	+	0.994846i	$0.532332\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	506.153	−	292.227i	1.15824	−	0.668713i
$438$		0			0
$439$	−196.634	+	340.580i	−0.447913	+	0.775809i	−0.998250	−	0.0591342i	$-0.981166\pi$
$439$	−196.634	+	340.580i	−0.447913	+	0.775809i	0.550337	+	0.834943i	$0.314499\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−43.0034	−	24.8280i	−0.0970731	−	0.0560452i	0.450678	−	0.892687i	$-0.351182\pi$
$443$	−43.0034	−	24.8280i	−0.0970731	−	0.0560452i	−0.547751	+	0.836642i	$0.684516\pi$
$444$		0			0
$445$	−123.404	−	213.743i	−0.277313	−	0.480320i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	276.997i		−	0.616919i	−0.951237	−	0.308459i	$-0.900187\pi$
$449$		−	276.997i		−	0.616919i	0.951237	−	0.308459i	$-0.0998134\pi$
$450$		0			0
$451$	−15.5556			−0.0344914
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−252.447	+	145.750i	−0.554829	+	0.320331i
$456$		0			0
$457$	3.56704	−	6.17829i	0.00780534	−	0.0135192i	−0.862096	−	0.506744i	$-0.830849\pi$
$457$	3.56704	−	6.17829i	0.00780534	−	0.0135192i	0.869902	+	0.493225i	$0.164182\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−298.324	−	172.238i	−0.647124	−	0.373617i	0.140229	−	0.990119i	$-0.455216\pi$
$461$	−298.324	−	172.238i	−0.647124	−	0.373617i	−0.787354	+	0.616502i	$0.788549\pi$
$462$		0			0
$463$	−309.452	−	535.986i	−0.668362	−	1.15764i	−0.978362	−	0.206900i	$-0.933662\pi$
$463$	−309.452	−	535.986i	−0.668362	−	1.15764i	0.310000	−	0.950737i	$-0.399671\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			682.172i			1.46075i	0.683044	+	0.730377i	$0.260656\pi$
$467$			682.172i			1.46075i	−0.683044	+	0.730377i	$0.739344\pi$
$468$		0			0
$469$	−841.369			−1.79396
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−133.405	+	77.0215i	−0.282041	+	0.162836i
$474$		0			0
$475$	279.217	−	483.619i	0.587826	−	1.01814i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−120.167	−	69.3787i	−0.250871	−	0.144841i	0.369292	−	0.929313i	$-0.379600\pi$
$479$	−120.167	−	69.3787i	−0.250871	−	0.144841i	−0.620163	+	0.784473i	$0.712934\pi$
$480$		0			0
$481$	224.196	+	388.319i	0.466105	+	0.807317i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			98.9087i			0.203935i
$486$		0			0
$487$	−630.925			−1.29553			−0.647766	−	0.761839i	$-0.724297\pi$
$487$	−630.925			−1.29553			−0.647766	+	0.761839i	$0.724297\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−351.529	+	202.955i	−0.715945	+	0.413351i	−0.813259	−	0.581903i	$-0.802309\pi$
$491$	−351.529	+	202.955i	−0.715945	+	0.413351i	0.0973131	+	0.995254i	$0.468975\pi$
$492$		0			0
$493$	−1.43551	+	2.48637i	−0.00291178	+	0.00504335i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	743.605	+	429.321i	1.49619	+	0.863825i
$498$		0			0
$499$	52.0887	+	90.2203i	0.104386	+	0.180802i	0.913487	−	0.406867i	$-0.133379\pi$
$499$	52.0887	+	90.2203i	0.104386	+	0.180802i	−0.809101	+	0.587670i	$0.800046\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	275.389i		−	0.547493i	−0.961802	−	0.273746i	$-0.911737\pi$
$503$		−	275.389i		−	0.547493i	0.961802	−	0.273746i	$-0.0882629\pi$
$504$		0			0
$505$	−287.063			−0.568441
$506$		0			0
$507$		0			0
$508$		0			0
$509$	661.144	−	381.712i	1.29891	−	0.749925i	0.318693	−	0.947858i	$-0.396756\pi$
$509$	661.144	−	381.712i	1.29891	−	0.749925i	0.980216	+	0.197933i	$0.0634228\pi$
$510$		0			0
$511$	310.899	−	538.493i	0.608413	−	1.05380i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−29.1307	−	16.8186i	−0.0565644	−	0.0326575i
$516$		0			0
$517$	116.309	+	201.454i	0.224970	+	0.389659i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			727.642i			1.39663i	0.715793	+	0.698313i	$0.246065\pi$
$521$			727.642i			1.39663i	−0.715793	+	0.698313i	$0.753935\pi$
$522$		0			0
$523$	585.050			1.11864			0.559321	−	0.828951i	$-0.311062\pi$
$523$	585.050			1.11864			0.559321	+	0.828951i	$0.311062\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	0.931371	−	0.537727i	0.00176731	−	0.00102036i
$528$		0			0
$529$	−34.6405	+	59.9991i	−0.0654830	+	0.113420i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−51.4032	−	29.6777i	−0.0964413	−	0.0556804i
$534$		0			0
$535$	−203.027	−	351.652i	−0.379489	−	0.657294i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	10.9495i		−	0.0203145i
$540$		0			0
$541$	−359.365			−0.664260			−0.332130	−	0.943234i	$-0.607767\pi$
$541$	−359.365			−0.664260			−0.332130	+	0.943234i	$0.607767\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	266.447	−	153.833i	0.488894	−	0.282263i
$546$		0			0
$547$	−48.9820	+	84.8393i	−0.0895467	+	0.155099i	−0.907320	−	0.420442i	$-0.861875\pi$
$547$	−48.9820	+	84.8393i	−0.0895467	+	0.155099i	0.817773	+	0.575541i	$0.195208\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1016.53	−	586.896i	−1.84489	−	1.06515i
$552$		0			0
$553$	−166.114	−	287.718i	−0.300387	−	0.520286i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			550.731i			0.988745i	0.869250	+	0.494373i	$0.164602\pi$
$557$			550.731i			0.988745i	−0.869250	+	0.494373i	$0.835398\pi$
$558$		0			0
$559$	−587.779			−1.05148
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−464.850	+	268.382i	−0.825667	+	0.476699i	−0.852367	−	0.522944i	$-0.824833\pi$
$563$	−464.850	+	268.382i	−0.825667	+	0.476699i	0.0266998	+	0.999643i	$0.491500\pi$
$564$		0			0
$565$	−102.404	+	177.368i	−0.181245	+	0.313926i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	477.998	+	275.972i	0.840067	+	0.485013i	0.857287	−	0.514839i	$-0.172148\pi$
$569$	477.998	+	275.972i	0.840067	+	0.485013i	−0.0172202	+	0.999852i	$0.505482\pi$
$570$		0			0
$571$	−409.760	−	709.726i	−0.717619	−	1.24295i	−0.961941	−	0.273258i	$-0.911899\pi$
$571$	−409.760	−	709.726i	−0.717619	−	1.24295i	0.244322	−	0.969694i	$-0.421435\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	439.252i		−	0.763917i
$576$		0			0
$577$	−828.718			−1.43625			−0.718126	−	0.695913i	$-0.755000\pi$
$577$	−828.718			−1.43625			−0.718126	+	0.695913i	$0.755000\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−692.751	+	399.960i	−1.19234	+	0.688399i
$582$		0			0
$583$	33.8482	−	58.6268i	0.0580587	−	0.100561i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	66.3759	+	38.3222i	0.113077	+	0.0652848i	0.555472	−	0.831535i	$-0.312538\pi$
$587$	66.3759	+	38.3222i	0.113077	+	0.0652848i	−0.442395	+	0.896820i	$0.645871\pi$
$588$		0			0
$589$	219.846	+	380.784i	0.373252	+	0.646492i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	943.303i		−	1.59073i	−0.606130	−	0.795366i	$-0.707279\pi$
$593$		−	943.303i		−	1.59073i	0.606130	−	0.795366i	$-0.292721\pi$
$594$		0			0
$595$	0.970195			0.00163058
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−285.323	+	164.731i	−0.476332	+	0.275010i	−0.718887	−	0.695127i	$-0.755348\pi$
$599$	−285.323	+	164.731i	−0.476332	+	0.275010i	0.242555	+	0.970138i	$0.422015\pi$
$600$		0			0
$601$	381.291	−	660.415i	0.634427	−	1.09886i	−0.352209	−	0.935921i	$-0.614569\pi$
$601$	381.291	−	660.415i	0.634427	−	1.09886i	0.986636	−	0.162939i	$-0.0520972\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−171.913	−	99.2542i	−0.284154	−	0.164057i
$606$		0			0
$607$	−479.687	−	830.843i	−0.790259	−	1.36877i	−0.925806	−	0.377998i	$-0.876613\pi$
$607$	−479.687	−	830.843i	−0.790259	−	1.36877i	0.135547	−	0.990771i	$-0.456721\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			887.599i			1.45270i
$612$		0			0
$613$	573.016			0.934773			0.467386	−	0.884053i	$-0.345196\pi$
$613$	573.016			0.934773			0.467386	+	0.884053i	$0.345196\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−97.1976	+	56.1171i	−0.157533	+	0.0909515i	−0.576694	−	0.816960i	$-0.695657\pi$
$617$	−97.1976	+	56.1171i	−0.157533	+	0.0909515i	0.419161	+	0.907912i	$0.362324\pi$
$618$		0			0
$619$	−22.5587	+	39.0728i	−0.0364438	+	0.0631225i	−0.883672	−	0.468107i	$-0.844936\pi$
$619$	−22.5587	+	39.0728i	−0.0364438	+	0.0631225i	0.847228	+	0.531229i	$0.178270\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−689.107	−	397.856i	−1.10611	−	0.638613i
$624$		0			0
$625$	−153.429	−	265.747i	−0.245487	−	0.425196i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	1.49237i		−	0.00237261i
$630$		0			0
$631$	−1118.03			−1.77185			−0.885923	−	0.463832i	$-0.846474\pi$
$631$	−1118.03			−1.77185			−0.885923	+	0.463832i	$0.846474\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	235.001	−	135.678i	0.370080	−	0.213666i
$636$		0			0
$637$	20.8899	−	36.1824i	0.0327942	−	0.0568012i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−583.312	−	336.775i	−0.910003	−	0.525390i	−0.0295708	−	0.999563i	$-0.509414\pi$
$641$	−583.312	−	336.775i	−0.910003	−	0.525390i	−0.880432	+	0.474172i	$0.842747\pi$
$642$		0			0
$643$	285.246	+	494.060i	0.443617	+	0.768368i	0.997955	−	0.0639243i	$-0.0203616\pi$
$643$	285.246	+	494.060i	0.443617	+	0.768368i	−0.554337	+	0.832292i	$0.687028\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			376.087i			0.581278i	0.956833	+	0.290639i	$0.0938678\pi$
$647$			376.087i			0.581278i	−0.956833	+	0.290639i	$0.906132\pi$
$648$		0			0
$649$	454.719			0.700646
$650$		0			0
$651$		0			0
$652$		0			0
$653$	557.218	−	321.710i	0.853321	−	0.492665i	−0.00844912	−	0.999964i	$-0.502689\pi$
$653$	557.218	−	321.710i	0.853321	−	0.492665i	0.861770	+	0.507299i	$0.169356\pi$
$654$		0			0
$655$	34.7258	−	60.1468i	0.0530164	−	0.0918272i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	140.020	+	80.8403i	0.212473	+	0.122671i	0.602460	−	0.798149i	$-0.294187\pi$
$659$	140.020	+	80.8403i	0.212473	+	0.122671i	−0.389987	+	0.920820i	$0.627521\pi$
$660$		0			0
$661$	358.633	+	621.171i	0.542561	+	0.939744i	0.998756	+	0.0498638i	$0.0158787\pi$
$661$	358.633	+	621.171i	0.542561	+	0.939744i	−0.456195	+	0.889880i	$0.650788\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			396.657i			0.596476i
$666$		0			0
$667$	−923.278			−1.38423
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−58.6760	+	33.8766i	−0.0874456	+	0.0504868i
$672$		0			0
$673$	82.4149	−	142.747i	0.122459	−	0.212105i	−0.798278	−	0.602289i	$-0.794255\pi$
$673$	82.4149	−	142.747i	0.122459	−	0.212105i	0.920737	+	0.390184i	$0.127589\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−703.172	−	405.977i	−1.03866	−	0.599670i	−0.119206	−	0.992870i	$-0.538035\pi$
$677$	−703.172	−	405.977i	−1.03866	−	0.599670i	−0.919453	+	0.393200i	$0.871368\pi$
$678$		0			0
$679$	159.441	+	276.160i	0.234818	+	0.406716i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	546.230i		−	0.799751i	−0.916569	−	0.399876i	$-0.869053\pi$
$683$		−	546.230i		−	0.799751i	0.916569	−	0.399876i	$-0.130947\pi$
$684$		0			0
$685$	391.814			0.571992
$686$		0			0
$687$		0			0
$688$		0			0
$689$	223.701	−	129.154i	0.324675	−	0.187451i
$690$		0			0
$691$	11.8911	−	20.5960i	0.0172085	−	0.0298061i	−0.857293	−	0.514829i	$-0.827855\pi$
$691$	11.8911	−	20.5960i	0.0172085	−	0.0298061i	0.874501	+	0.485023i	$0.161189\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−439.187	−	253.565i	−0.631924	−	0.364842i
$696$		0			0
$697$	0.0987754	+	0.171084i	0.000141715	+	0.000245458i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	959.326i		−	1.36851i	−0.729242	−	0.684255i	$-0.760127\pi$
$701$		−	959.326i		−	1.36851i	0.729242	−	0.684255i	$-0.239873\pi$
$702$		0			0
$703$	610.146			0.867917
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−801.499	+	462.746i	−1.13366	+	0.654520i
$708$		0			0
$709$	218.458	−	378.381i	0.308121	−	0.533682i	−0.669830	−	0.742515i	$-0.733633\pi$
$709$	218.458	−	378.381i	0.308121	−	0.533682i	0.977951	+	0.208833i	$0.0669664\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	299.516	+	172.926i	0.420079	+	0.242532i
$714$		0			0
$715$	111.716	+	193.497i	0.156246	+	0.270626i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	643.581i		−	0.895106i	−0.894257	−	0.447553i	$-0.852296\pi$
$719$		−	643.581i		−	0.895106i	0.894257	−	0.447553i	$-0.147704\pi$
$720$		0			0
$721$	−108.447			−0.150411
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−763.985	+	441.087i	−1.05377	+	0.608396i
$726$		0			0
$727$	365.499	−	633.062i	0.502749	−	0.870787i	−0.497246	−	0.867610i	$-0.665655\pi$
$727$	365.499	−	633.062i	0.502749	−	0.870787i	0.999995	−	0.00317756i	$-0.00101145\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.69420	+	0.978144i	0.00231764	+	0.00133809i
$732$		0			0
$733$	471.405	+	816.497i	0.643117	+	1.11391i	0.984733	+	0.174072i	$0.0556924\pi$
$733$	471.405	+	816.497i	0.643117	+	1.11391i	−0.341616	+	0.939840i	$0.610974\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			644.899i			0.875033i
$738$		0			0
$739$	354.141			0.479216			0.239608	−	0.970870i	$-0.422981\pi$
$739$	354.141			0.479216			0.239608	+	0.970870i	$0.422981\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	108.989	−	62.9246i	0.146687	−	0.0846899i	−0.424860	−	0.905259i	$-0.639677\pi$
$743$	108.989	−	62.9246i	0.146687	−	0.0846899i	0.571547	+	0.820569i	$0.306343\pi$
$744$		0			0
$745$	272.637	−	472.221i	0.365955	−	0.633853i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1133.73	−	654.559i	−1.51366	−	0.873910i
$750$		0			0
$751$	−275.702	−	477.530i	−0.367113	−	0.635858i	0.622000	−	0.783017i	$-0.286320\pi$
$751$	−275.702	−	477.530i	−0.367113	−	0.635858i	−0.989113	+	0.147159i	$0.952987\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			176.916i			0.234326i
$756$		0			0
$757$	−816.186			−1.07818			−0.539092	−	0.842247i	$-0.681233\pi$
$757$	−816.186			−1.07818			−0.539092	+	0.842247i	$0.681233\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−816.829	+	471.596i	−1.07336	+	0.619706i	−0.929098	−	0.369833i	$-0.879415\pi$
$761$	−816.829	+	471.596i	−1.07336	+	0.619706i	−0.144264	+	0.989539i	$0.546081\pi$
$762$		0			0
$763$	495.960	−	859.027i	0.650012	−	1.12585i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1502.61	+	867.531i	1.95907	+	1.13107i
$768$		0			0
$769$	272.289	+	471.618i	0.354081	+	0.613287i	0.986960	−	0.160964i	$-0.0514603\pi$
$769$	272.289	+	471.618i	0.354081	+	0.613287i	−0.632879	+	0.774251i	$0.718127\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			24.6789i			0.0319262i	0.999873	+	0.0159631i	$0.00508143\pi$
$773$			24.6789i			0.0319262i	−0.999873	+	0.0159631i	$0.994919\pi$
$774$		0			0
$775$	330.454			0.426392
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−69.9464	+	40.3836i	−0.0897900	+	0.0518403i
$780$		0			0
$781$	329.069	−	569.964i	0.421343	−	0.729788i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−293.578	−	169.497i	−0.373984	−	0.215920i
$786$		0			0
$787$	358.205	+	620.429i	0.455152	+	0.788346i	0.998697	−	0.0510338i	$-0.0162516\pi$
$787$	358.205	+	620.429i	0.455152	+	0.788346i	−0.543545	+	0.839380i	$0.682918\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			660.300i			0.834767i
$792$		0			0
$793$	−258.525			−0.326008
$794$		0			0
$795$		0			0
$796$		0			0
$797$	833.455	−	481.195i	1.04574	−	0.603758i	0.124286	−	0.992246i	$-0.460336\pi$
$797$	833.455	−	481.195i	1.04574	−	0.603758i	0.921454	+	0.388488i	$0.127003\pi$
$798$		0			0
$799$	1.47709	−	2.55839i	0.00184867	−	0.00320199i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−412.748	−	238.300i	−0.514007	−	0.296762i
$804$		0			0
$805$	156.001	+	270.201i	0.193790	+	0.335653i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	674.254i		−	0.833441i	−0.909035	−	0.416721i	$-0.863179\pi$
$809$		−	674.254i		−	0.833441i	0.909035	−	0.416721i	$-0.136821\pi$
$810$		0			0
$811$	36.7884			0.0453618			0.0226809	−	0.999743i	$-0.492780\pi$
$811$	36.7884			0.0453618			0.0226809	+	0.999743i	$0.492780\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−33.8795	+	19.5603i	−0.0415699	+	0.0240004i
$816$		0			0
$817$	−399.907	+	692.659i	−0.489482	+	0.847808i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1312.18	+	757.588i	1.59827	+	0.922762i	0.991820	+	0.127642i	$0.0407408\pi$
$821$	1312.18	+	757.588i	1.59827	+	0.922762i	0.606451	+	0.795121i	$0.292593\pi$
$822$		0			0
$823$	381.212	+	660.278i	0.463198	+	0.802282i	0.999118	−	0.0419867i	$-0.0133687\pi$
$823$	381.212	+	660.278i	0.463198	+	0.802282i	−0.535921	+	0.844268i	$0.680035\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			648.915i			0.784662i	0.919824	+	0.392331i	$0.128331\pi$
$827$			648.915i			0.784662i	−0.919824	+	0.392331i	$0.871669\pi$
$828$		0			0
$829$	−958.477			−1.15618			−0.578092	−	0.815971i	$-0.696203\pi$
$829$	−958.477			−1.15618			−0.578092	+	0.815971i	$0.696203\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.120425	+	0.0695273i	−0.000144568	+	8.34662e-5i
$834$		0			0
$835$	−214.730	+	371.924i	−0.257162	+	0.445418i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−249.394	−	143.988i	−0.297251	−	0.171618i	0.343956	−	0.938986i	$-0.388233\pi$
$839$	−249.394	−	143.988i	−0.297251	−	0.171618i	−0.641207	+	0.767368i	$0.721566\pi$
$840$		0			0
$841$	506.635	+	877.517i	0.602420	+	1.04342i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			493.502i			0.584027i
$846$		0			0
$847$	−639.993			−0.755599
$848$		0			0
$849$		0			0
$850$		0			0
$851$	415.629	−	239.963i	0.488400	−	0.281978i
$852$		0			0
$853$	333.286	−	577.268i	0.390722	−	0.676751i	−0.601823	−	0.798630i	$-0.705559\pi$
$853$	333.286	−	577.268i	0.390722	−	0.676751i	0.992545	+	0.121879i	$0.0388920\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	633.567	+	365.790i	0.739285	+	0.426826i	0.821809	−	0.569763i	$-0.192965\pi$
$857$	633.567	+	365.790i	0.739285	+	0.426826i	−0.0825243	+	0.996589i	$0.526298\pi$
$858$		0			0
$859$	−388.156	−	672.307i	−0.451870	−	0.782662i	0.546632	−	0.837373i	$-0.315910\pi$
$859$	−388.156	−	672.307i	−0.451870	−	0.782662i	−0.998502	+	0.0547109i	$0.982576\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	925.254i		−	1.07214i	−0.844175	−	0.536068i	$-0.819909\pi$
$863$		−	925.254i		−	1.07214i	0.844175	−	0.536068i	$-0.180091\pi$
$864$		0			0
$865$	−214.501			−0.247977
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−220.532	+	127.324i	−0.253777	+	0.146518i
$870$		0			0
$871$	−1230.36	+	2131.05i	−1.41259	+	2.44667i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	573.222	+	330.950i	0.655111	+	0.378229i
$876$		0			0
$877$	348.075	+	602.884i	0.396893	+	0.687439i	0.993341	−	0.115213i	$-0.0367550\pi$
$877$	348.075	+	602.884i	0.396893	+	0.687439i	−0.596448	+	0.802652i	$0.703422\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			183.830i			0.208660i	0.994543	+	0.104330i	$0.0332699\pi$
$881$			183.830i			0.208660i	−0.994543	+	0.104330i	$0.966730\pi$
$882$		0			0
$883$	823.282			0.932369			0.466184	−	0.884688i	$-0.345628\pi$
$883$	823.282			0.932369			0.466184	+	0.884688i	$0.345628\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−316.421	+	182.686i	−0.356732	+	0.205959i	−0.667646	−	0.744479i	$-0.732698\pi$
$887$	−316.421	+	182.686i	−0.356732	+	0.205959i	0.310914	+	0.950438i	$0.399365\pi$
$888$		0			0
$889$	437.426	−	757.644i	0.492043	−	0.852243i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1045.98	+	603.895i	1.17131	+	0.676255i
$894$		0			0
$895$	48.6835	+	84.3223i	0.0543950	+	0.0942148i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	694.592i		−	0.772627i
$900$		0			0
$901$	−0.859719			−0.000954183
$902$		0			0
$903$		0			0
$904$		0			0
$905$	102.611	−	59.2426i	0.113383	−	0.0654615i
$906$		0			0
$907$	−153.202	+	265.354i	−0.168911	+	0.292562i	−0.938037	−	0.346535i	$-0.887358\pi$
$907$	−153.202	+	265.354i	−0.168911	+	0.292562i	0.769126	+	0.639097i	$0.220692\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	682.548	+	394.069i	0.749229	+	0.432568i	0.825415	−	0.564526i	$-0.190941\pi$
$911$	682.548	+	394.069i	0.749229	+	0.432568i	−0.0761860	+	0.997094i	$0.524274\pi$
$912$		0			0
$913$	306.564	+	530.985i	0.335777	+	0.581582i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	223.912i		−	0.244179i
$918$		0			0
$919$	1319.46			1.43575			0.717876	−	0.696171i	$-0.245114\pi$
$919$	1319.46			1.43575			0.717876	+	0.696171i	$0.245114\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	2174.80	−	1255.62i	2.35623	−	1.36037i
$924$		0			0
$925$	229.280	−	397.125i	0.247870	−	0.429324i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−406.971	−	234.965i	−0.438075	−	0.252922i	0.264706	−	0.964329i	$-0.414725\pi$
$929$	−406.971	−	234.965i	−0.438075	−	0.252922i	−0.702781	+	0.711407i	$0.748058\pi$
$930$		0			0
$931$	−28.4257	−	49.2348i	−0.0305325	−	0.0528838i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	0.743642i		−	0.000795339i
$936$		0			0
$937$	−1619.12			−1.72798			−0.863991	−	0.503508i	$-0.832042\pi$
$937$	−1619.12			−1.72798			−0.863991	+	0.503508i	$0.832042\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−578.847	+	334.197i	−0.615140	+	0.355151i	−0.774974	−	0.631993i	$-0.782237\pi$
$941$	−578.847	+	334.197i	−0.615140	+	0.355151i	0.159834	+	0.987144i	$0.448904\pi$
$942$		0			0
$943$	−31.7648	+	55.0182i	−0.0336848	+	0.0583438i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−859.549	−	496.261i	−0.907655	−	0.524035i	−0.0279791	−	0.999609i	$-0.508907\pi$
$947$	−859.549	−	496.261i	−0.907655	−	0.524035i	−0.879676	+	0.475574i	$0.842241\pi$
$948$		0			0
$949$	−909.276	−	1574.91i	−0.958142	−	1.65955i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	565.650i		−	0.593546i	−0.954948	−	0.296773i	$-0.904089\pi$
$953$		−	565.650i		−	0.593546i	0.954948	−	0.296773i	$-0.0959105\pi$
$954$		0			0
$955$	672.184			0.703858
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1093.97	−	631.606i	1.14074	−	0.658609i
$960$		0			0
$961$	350.406	−	606.921i	0.364627	−	0.631552i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−690.491	−	398.655i	−0.715535	−	0.413114i
$966$		0			0
$967$	515.193	+	892.340i	0.532775	+	0.922793i	0.999268	+	0.0382677i	$0.0121840\pi$
$967$	515.193	+	892.340i	0.532775	+	0.922793i	−0.466493	+	0.884525i	$0.654483\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			454.101i			0.467663i	0.972277	+	0.233832i	$0.0751265\pi$
$971$			454.101i			0.467663i	−0.972277	+	0.233832i	$0.924874\pi$
$972$		0			0
$973$	−1634.99			−1.68036
$974$		0			0
$975$		0			0
$976$		0			0
$977$	733.302	−	423.372i	0.750565	−	0.433339i	−0.0753331	−	0.997158i	$-0.524002\pi$
$977$	733.302	−	423.372i	0.750565	−	0.433339i	0.825898	+	0.563820i	$0.190669\pi$
$978$		0			0
$979$	−304.952	+	528.192i	−0.311493	+	0.539522i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1241.22	+	716.621i	1.26269	+	0.729014i	0.973594	−	0.228286i	$-0.0733123\pi$
$983$	1241.22	+	716.621i	1.26269	+	0.729014i	0.289095	+	0.957300i	$0.406646\pi$
$984$		0			0
$985$	−49.5398	−	85.8055i	−0.0502942	−	0.0871122i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			629.115i			0.636113i
$990$		0			0
$991$	955.562			0.964240			0.482120	−	0.876105i	$-0.339867\pi$
$991$	955.562			0.964240			0.482120	+	0.876105i	$0.339867\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	353.332	−	203.996i	0.355107	−	0.205021i
$996$		0			0
$997$	294.723	−	510.475i	0.295610	−	0.512011i	−0.679517	−	0.733660i	$-0.737811\pi$
$997$	294.723	−	510.475i	0.295610	−	0.512011i	0.975127	+	0.221649i	$0.0711439\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.3.q.a.737.8		24
3.2	odd	2		288.3.q.b.65.6	✓	24
4.3	odd	2	inner	864.3.q.a.737.7		24
8.3	odd	2		1728.3.q.k.1601.5		24
8.5	even	2		1728.3.q.k.1601.6		24
9.2	odd	6		2592.3.e.i.161.15		24
9.4	even	3		288.3.q.b.257.6	yes	24
9.5	odd	6	inner	864.3.q.a.449.8		24
9.7	even	3		2592.3.e.i.161.16		24
12.11	even	2		288.3.q.b.65.7	yes	24
24.5	odd	2		576.3.q.l.65.7		24
24.11	even	2		576.3.q.l.65.6		24
36.7	odd	6		2592.3.e.i.161.10		24
36.11	even	6		2592.3.e.i.161.9		24
36.23	even	6	inner	864.3.q.a.449.7		24
36.31	odd	6		288.3.q.b.257.7	yes	24
72.5	odd	6		1728.3.q.k.449.6		24
72.13	even	6		576.3.q.l.257.7		24
72.59	even	6		1728.3.q.k.449.5		24
72.67	odd	6		576.3.q.l.257.6		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.3.q.b.65.6	✓	24	3.2	odd	2
288.3.q.b.65.7	yes	24	12.11	even	2
288.3.q.b.257.6	yes	24	9.4	even	3
288.3.q.b.257.7	yes	24	36.31	odd	6
576.3.q.l.65.6		24	24.11	even	2
576.3.q.l.65.7		24	24.5	odd	2
576.3.q.l.257.6		24	72.67	odd	6
576.3.q.l.257.7		24	72.13	even	6
864.3.q.a.449.7		24	36.23	even	6	inner
864.3.q.a.449.8		24	9.5	odd	6	inner
864.3.q.a.737.7		24	4.3	odd	2	inner
864.3.q.a.737.8		24	1.1	even	1	trivial
1728.3.q.k.449.5		24	72.59	even	6
1728.3.q.k.449.6		24	72.5	odd	6
1728.3.q.k.1601.5		24	8.3	odd	2
1728.3.q.k.1601.6		24	8.5	even	2
2592.3.e.i.161.9		24	36.11	even	6
2592.3.e.i.161.10		24	36.7	odd	6
2592.3.e.i.161.15		24	9.2	odd	6
2592.3.e.i.161.16		24	9.7	even	3

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:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	864.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.83987 - 1.06225i) q^{5} +(3.42470 - 5.93176i) q^{7} +(-4.54662 - 2.62499i) q^{11} +(-10.0161 - 17.3484i) q^{13} +0.0666728i q^{17} -27.2587 q^{19} +(-18.5685 + 10.7205i) q^{23} +(-10.2432 + 17.7418i) q^{25} +(37.2921 + 21.5306i) q^{29} +(-8.06516 - 13.9693i) q^{31} -14.5516i q^{35} -22.3835 q^{37} +(2.56602 - 1.48149i) q^{41} +(14.6708 - 25.4106i) q^{43} +(-38.3723 - 22.1542i) q^{47} +(1.04281 + 1.80621i) q^{49} +12.8946i q^{53} -11.1536 q^{55} +(-75.0094 + 43.3067i) q^{59} +(6.45271 - 11.1764i) q^{61} +(-36.8568 - 21.2793i) q^{65} +(-61.4191 - 106.381i) q^{67} +125.360i q^{71} +90.7813 q^{73} +(-31.1416 + 17.9796i) q^{77} +(24.2523 - 42.0063i) q^{79} +(-101.140 - 58.3934i) q^{83} +(0.0708233 + 0.122669i) q^{85} -116.172i q^{89} -137.209 q^{91} +(-50.1525 + 28.9556i) q^{95} +(-23.2781 + 40.3188i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 60 q^{25} - 72 q^{29} - 252 q^{41} - 36 q^{49} - 96 q^{61} + 288 q^{65} + 24 q^{73} + 720 q^{77} + 96 q^{85} - 132 q^{97}+O(q^{100}) \) 24 * q + 60 * q^25 - 72 * q^29 - 252 * q^41 - 36 * q^49 - 96 * q^61 + 288 * q^65 + 24 * q^73 + 720 * q^77 + 96 * q^85 - 132 * q^97

Introduction

L-functions

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