LMFDB - Embedded newform 864.3.t.b.559.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,3,Mod(559,864)]

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([3, 3, 4]))

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

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

chi := DirichletCharacter("864.559");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	864.t (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:	$40$
Relative dimension:	$20$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			559.8
Character	$\chi$	$=$	864.559
Dual form			864.3.t.b.847.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.70411 + 0.983869i) q^{5} +(-8.69613 - 5.02071i) q^{7} +O(q^{10})$	$q+(-1.70411 + 0.983869i) q^{5} +(-8.69613 - 5.02071i) q^{7} +(6.08627 - 10.5417i) q^{11} +(-4.28857 + 2.47601i) q^{13} -4.71370 q^{17} +20.5288 q^{19} +(3.33400 - 1.92488i) q^{23} +(-10.5640 + 18.2974i) q^{25} +(40.7608 + 23.5333i) q^{29} +(-49.9323 + 28.8285i) q^{31} +19.7589 q^{35} +7.93672i q^{37} +(11.3426 + 19.6460i) q^{41} +(-30.7806 + 53.3136i) q^{43} +(-44.7133 - 25.8152i) q^{47} +(25.9151 + 44.8862i) q^{49} +51.0133i q^{53} +23.9524i q^{55} +(16.7094 + 28.9415i) q^{59} +(-39.7329 - 22.9398i) q^{61} +(4.87213 - 8.43878i) q^{65} +(-26.9209 - 46.6284i) q^{67} +132.571i q^{71} +24.6995 q^{73} +(-105.854 + 61.1148i) q^{77} +(-84.3214 - 48.6830i) q^{79} +(0.187458 - 0.324687i) q^{83} +(8.03267 - 4.63767i) q^{85} +134.821 q^{89} +49.7252 q^{91} +(-34.9834 + 20.1977i) q^{95} +(10.8826 - 18.8492i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q+O(q^{10}) $ 40 * q	$ 40 q - 16 q^{11} + 4 q^{17} + 76 q^{19} + 118 q^{25} - 108 q^{35} - 20 q^{41} + 16 q^{43} + 166 q^{49} - 64 q^{59} + 102 q^{65} + 64 q^{67} - 292 q^{73} + 554 q^{83} + 688 q^{89} + 204 q^{91} + 92 q^{97}+O(q^{100}) $ 40 * q - 16 * q^11 + 4 * q^17 + 76 * q^19 + 118 * q^25 - 108 * q^35 - 20 * q^41 + 16 * q^43 + 166 * q^49 - 64 * q^59 + 102 * q^65 + 64 * q^67 - 292 * q^73 + 554 * q^83 + 688 * q^89 + 204 * q^91 + 92 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$325$	$353$	$703$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.70411	+	0.983869i	−0.340822	+	0.196774i	−0.660636	−	0.750707i	$-0.729713\pi$
$5$	−1.70411	+	0.983869i	−0.340822	+	0.196774i	0.319813	+	0.947481i	$0.396380\pi$
$6$		0			0
$7$	−8.69613	−	5.02071i	−1.24230	−	0.717244i	−0.272741	−	0.962087i	$-0.587930\pi$
$7$	−8.69613	−	5.02071i	−1.24230	−	0.717244i	−0.969563	+	0.244843i	$0.921264\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	6.08627	−	10.5417i	0.553298	−	0.958339i	−0.444736	−	0.895662i	$-0.646703\pi$
$11$	6.08627	−	10.5417i	0.553298	−	0.958339i	0.998034	−	0.0626779i	$-0.0199641\pi$
$12$		0			0
$13$	−4.28857	+	2.47601i	−0.329890	+	0.190462i	−0.655792	−	0.754941i	$-0.727665\pi$
$13$	−4.28857	+	2.47601i	−0.329890	+	0.190462i	0.325902	+	0.945403i	$0.394332\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.71370			−0.277277			−0.138638	−	0.990343i	$-0.544273\pi$
$17$	−4.71370			−0.277277			−0.138638	+	0.990343i	$0.544273\pi$
$18$		0			0
$19$	20.5288			1.08046			0.540232	−	0.841516i	$-0.318337\pi$
$19$	20.5288			1.08046			0.540232	+	0.841516i	$0.318337\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.33400	−	1.92488i	0.144956	−	0.0836906i	−0.425768	−	0.904832i	$-0.639996\pi$
$23$	3.33400	−	1.92488i	0.144956	−	0.0836906i	0.570724	+	0.821142i	$0.306663\pi$
$24$		0			0
$25$	−10.5640	+	18.2974i	−0.422560	+	0.731896i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	40.7608	+	23.5333i	1.40555	+	0.811492i	0.994955	−	0.100327i	$-0.0319889\pi$
$29$	40.7608	+	23.5333i	1.40555	+	0.811492i	0.410592	+	0.911819i	$0.365322\pi$
$30$		0			0
$31$	−49.9323	+	28.8285i	−1.61072	+	0.929950i	−0.621517	+	0.783400i	$0.713483\pi$
$31$	−49.9323	+	28.8285i	−1.61072	+	0.929950i	−0.989203	+	0.146550i	$0.953183\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	19.7589			0.564540
$36$		0			0
$37$			7.93672i			0.214506i	0.994232	+	0.107253i	$0.0342055\pi$
$37$			7.93672i			0.214506i	−0.994232	+	0.107253i	$0.965795\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	11.3426	+	19.6460i	0.276649	+	0.479171i	0.970550	−	0.240900i	$-0.0774426\pi$
$41$	11.3426	+	19.6460i	0.276649	+	0.479171i	−0.693901	+	0.720071i	$0.744109\pi$
$42$		0			0
$43$	−30.7806	+	53.3136i	−0.715828	+	1.23985i	0.246811	+	0.969064i	$0.420617\pi$
$43$	−30.7806	+	53.3136i	−0.715828	+	1.23985i	−0.962639	+	0.270787i	$0.912716\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−44.7133	−	25.8152i	−0.951346	−	0.549260i	−0.0578471	−	0.998325i	$-0.518424\pi$
$47$	−44.7133	−	25.8152i	−0.951346	−	0.549260i	−0.893499	+	0.449066i	$0.851757\pi$
$48$		0			0
$49$	25.9151	+	44.8862i	0.528879	+	0.916045i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			51.0133i			0.962515i	0.876579	+	0.481258i	$0.159820\pi$
$53$			51.0133i			0.962515i	−0.876579	+	0.481258i	$0.840180\pi$
$54$		0			0
$55$			23.9524i			0.435498i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	16.7094	+	28.9415i	0.283209	+	0.490533i	0.972173	−	0.234262i	$-0.0752675\pi$
$59$	16.7094	+	28.9415i	0.283209	+	0.490533i	−0.688964	+	0.724796i	$0.741934\pi$
$60$		0			0
$61$	−39.7329	−	22.9398i	−0.651359	−	0.376062i	0.137618	−	0.990485i	$-0.456055\pi$
$61$	−39.7329	−	22.9398i	−0.651359	−	0.376062i	−0.788977	+	0.614423i	$0.789389\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.87213	−	8.43878i	0.0749559	−	0.129827i
$66$		0			0
$67$	−26.9209	−	46.6284i	−0.401805	−	0.695946i	0.592139	−	0.805836i	$-0.298284\pi$
$67$	−26.9209	−	46.6284i	−0.401805	−	0.695946i	−0.993944	+	0.109889i	$0.964950\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			132.571i			1.86720i	0.358313	+	0.933601i	$0.383352\pi$
$71$			132.571i			1.86720i	−0.358313	+	0.933601i	$0.616648\pi$
$72$		0			0
$73$	24.6995			0.338349			0.169175	−	0.985586i	$-0.445890\pi$
$73$	24.6995			0.338349			0.169175	+	0.985586i	$0.445890\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−105.854	+	61.1148i	−1.37473	+	0.793699i
$78$		0			0
$79$	−84.3214	−	48.6830i	−1.06736	−	0.616241i	−0.139901	−	0.990166i	$-0.544678\pi$
$79$	−84.3214	−	48.6830i	−1.06736	−	0.616241i	−0.927459	+	0.373925i	$0.878012\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	0.187458	−	0.324687i	0.00225853	−	0.00391189i	−0.864894	−	0.501955i	$-0.832614\pi$
$83$	0.187458	−	0.324687i	0.00225853	−	0.00391189i	0.867152	+	0.498043i	$0.165948\pi$
$84$		0			0
$85$	8.03267	−	4.63767i	0.0945021	−	0.0545608i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	134.821			1.51485			0.757423	−	0.652925i	$-0.226458\pi$
$89$	134.821			1.51485			0.757423	+	0.652925i	$0.226458\pi$
$90$		0			0
$91$	49.7252			0.546431
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−34.9834	+	20.1977i	−0.368246	+	0.212607i
$96$		0			0
$97$	10.8826	−	18.8492i	0.112192	−	0.194322i	−0.804462	−	0.594004i	$-0.797546\pi$
$97$	10.8826	−	18.8492i	0.112192	−	0.194322i	0.916654	+	0.399682i	$0.130880\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−40.3177	−	23.2774i	−0.399185	−	0.230470i	0.286947	−	0.957946i	$-0.407360\pi$
$101$	−40.3177	−	23.2774i	−0.399185	−	0.230470i	−0.686132	+	0.727477i	$0.740693\pi$
$102$		0			0
$103$	87.3356	−	50.4232i	0.847918	−	0.489546i	−0.0120298	−	0.999928i	$-0.503829\pi$
$103$	87.3356	−	50.4232i	0.847918	−	0.489546i	0.859948	+	0.510382i	$0.170496\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−12.9467			−0.120997			−0.0604986	−	0.998168i	$-0.519269\pi$
$107$	−12.9467			−0.120997			−0.0604986	+	0.998168i	$0.519269\pi$
$108$		0			0
$109$			117.693i			1.07975i	0.841745	+	0.539875i	$0.181529\pi$
$109$			117.693i			1.07975i	−0.841745	+	0.539875i	$0.818471\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−14.6904	−	25.4446i	−0.130004	−	0.225173i	0.793674	−	0.608343i	$-0.208166\pi$
$113$	−14.6904	−	25.4446i	−0.130004	−	0.225173i	−0.923678	+	0.383170i	$0.874832\pi$
$114$		0			0
$115$	−3.78766	+	6.56043i	−0.0329362	+	0.0570472i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	40.9910	+	23.6661i	0.344462	+	0.198875i
$120$		0			0
$121$	−13.5854	−	23.5307i	−0.112276	−	0.194468i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	90.7678i		−	0.726143i
$126$		0			0
$127$			112.463i			0.885532i	0.896637	+	0.442766i	$0.146003\pi$
$127$			112.463i			0.885532i	−0.896637	+	0.442766i	$0.853997\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	89.1863	+	154.475i	0.680812	+	1.17920i	0.974733	+	0.223371i	$0.0717062\pi$
$131$	89.1863	+	154.475i	0.680812	+	1.17920i	−0.293922	+	0.955829i	$0.594961\pi$
$132$		0			0
$133$	−178.521	−	103.069i	−1.34226	−	0.774956i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−64.0732	+	110.978i	−0.467688	+	0.810059i	−0.999318	−	0.0369173i	$-0.988246\pi$
$137$	−64.0732	+	110.978i	−0.467688	+	0.810059i	0.531631	+	0.846976i	$0.321580\pi$
$138$		0			0
$139$	96.3171	+	166.826i	0.692929	+	1.20019i	0.970874	+	0.239591i	$0.0770134\pi$
$139$	96.3171	+	166.826i	0.692929	+	1.20019i	−0.277945	+	0.960597i	$0.589653\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			60.2786i			0.421529i
$144$		0			0
$145$	−92.6146			−0.638722
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−31.9061	+	18.4210i	−0.214135	+	0.123631i	−0.603231	−	0.797566i	$-0.706120\pi$
$149$	−31.9061	+	18.4210i	−0.214135	+	0.123631i	0.389097	+	0.921197i	$0.372787\pi$
$150$		0			0
$151$	−1.53781	−	0.887854i	−0.0101842	−	0.00587983i	0.494899	−	0.868950i	$-0.335205\pi$
$151$	−1.53781	−	0.887854i	−0.0101842	−	0.00587983i	−0.505083	+	0.863071i	$0.668538\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	56.7268	−	98.2538i	0.365980	−	0.633895i
$156$		0			0
$157$	−28.4676	+	16.4358i	−0.181322	+	0.104686i	−0.587914	−	0.808924i	$-0.700051\pi$
$157$	−28.4676	+	16.4358i	−0.181322	+	0.104686i	0.406592	+	0.913610i	$0.366717\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−38.6571			−0.240106
$162$		0			0
$163$	−26.1023			−0.160137			−0.0800686	−	0.996789i	$-0.525514\pi$
$163$	−26.1023			−0.160137			−0.0800686	+	0.996789i	$0.525514\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	113.333	−	65.4331i	0.678643	−	0.391815i	−0.120700	−	0.992689i	$-0.538514\pi$
$167$	113.333	−	65.4331i	0.678643	−	0.391815i	0.799344	+	0.600874i	$0.205181\pi$
$168$		0			0
$169$	−72.2388	+	125.121i	−0.427448	+	0.740362i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	159.107	+	91.8606i	0.919695	+	0.530986i	0.883538	−	0.468360i	$-0.155155\pi$
$173$	159.107	+	91.8606i	0.919695	+	0.530986i	0.0361571	+	0.999346i	$0.488488\pi$
$174$		0			0
$175$	183.732	−	106.078i	1.04990	−	0.606158i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−24.4312			−0.136487			−0.0682435	−	0.997669i	$-0.521739\pi$
$179$	−24.4312			−0.136487			−0.0682435	+	0.997669i	$0.521739\pi$
$180$		0			0
$181$			101.654i			0.561622i	0.959763	+	0.280811i	$0.0906034\pi$
$181$			101.654i			0.561622i	−0.959763	+	0.280811i	$0.909397\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−7.80869	−	13.5250i	−0.0422091	−	0.0731084i
$186$		0			0
$187$	−28.6889	+	49.6906i	−0.153417	+	0.265725i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	25.0116	+	14.4405i	0.130951	+	0.0756046i	0.564044	−	0.825744i	$-0.309245\pi$
$191$	25.0116	+	14.4405i	0.130951	+	0.0756046i	−0.433093	+	0.901349i	$0.642578\pi$
$192$		0			0
$193$	−142.662	−	247.098i	−0.739181	−	1.28030i	−0.952864	−	0.303396i	$-0.901879\pi$
$193$	−142.662	−	247.098i	−0.739181	−	1.28030i	0.213683	−	0.976903i	$-0.431454\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	191.958i		−	0.974405i	−0.873289	−	0.487202i	$-0.838017\pi$
$197$		−	191.958i		−	0.974405i	0.873289	−	0.487202i	$-0.161983\pi$
$198$		0			0
$199$			33.7936i			0.169817i	0.996389	+	0.0849085i	$0.0270598\pi$
$199$			33.7936i			0.169817i	−0.996389	+	0.0849085i	$0.972940\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−236.308	−	409.297i	−1.16408	−	2.01624i
$204$		0			0
$205$	−38.6582	−	22.3193i	−0.188576	−	0.108875i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	124.944	−	216.409i	0.597818	−	1.03545i
$210$		0			0
$211$	−129.688	−	224.627i	−0.614636	−	1.06458i	−0.990448	−	0.137885i	$-0.955969\pi$
$211$	−129.688	−	224.627i	−0.614636	−	1.06458i	0.375812	−	0.926696i	$-0.377364\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	121.136i		−	0.563425i
$216$		0			0
$217$	578.957			2.66801
$218$		0			0
$219$		0			0
$220$		0			0
$221$	20.2150	−	11.6712i	0.0914708	−	0.0528107i
$222$		0			0
$223$	−107.192	−	61.8873i	−0.480681	−	0.277521i	0.240019	−	0.970768i	$-0.422846\pi$
$223$	−107.192	−	61.8873i	−0.480681	−	0.277521i	−0.720700	+	0.693247i	$0.756180\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−127.486	+	220.813i	−0.561614	+	0.972745i	0.435741	+	0.900072i	$0.356486\pi$
$227$	−127.486	+	220.813i	−0.561614	+	0.972745i	−0.997356	+	0.0726729i	$0.976847\pi$
$228$		0			0
$229$	216.736	−	125.132i	0.946444	−	0.546430i	0.0544698	−	0.998515i	$-0.482653\pi$
$229$	216.736	−	125.132i	0.946444	−	0.546430i	0.891975	+	0.452085i	$0.149320\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−231.979			−0.995619			−0.497810	−	0.867286i	$-0.665862\pi$
$233$	−231.979			−0.995619			−0.497810	+	0.867286i	$0.665862\pi$
$234$		0			0
$235$	101.595			0.432320
$236$		0			0
$237$		0			0
$238$		0			0
$239$	209.313	−	120.847i	0.875786	−	0.505635i	0.00651903	−	0.999979i	$-0.497925\pi$
$239$	209.313	−	120.847i	0.875786	−	0.505635i	0.869267	+	0.494344i	$0.164592\pi$
$240$		0			0
$241$	−115.503	+	200.057i	−0.479266	+	0.830113i	−0.999717	−	0.0237786i	$-0.992430\pi$
$241$	−115.503	+	200.057i	−0.479266	+	0.830113i	0.520451	+	0.853891i	$0.325764\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−88.3243	−	50.9941i	−0.360507	−	0.208139i
$246$		0			0
$247$	−88.0392	+	50.8294i	−0.356434	+	0.205787i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−78.5229			−0.312840			−0.156420	−	0.987691i	$-0.549995\pi$
$251$	−78.5229			−0.312840			−0.156420	+	0.987691i	$0.549995\pi$
$252$		0			0
$253$		−	46.8615i		−	0.185223i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−130.836	−	226.615i	−0.509090	−	0.881770i	−0.999945	−	0.0105283i	$-0.996649\pi$
$257$	−130.836	−	226.615i	−0.509090	−	0.881770i	0.490855	−	0.871242i	$-0.336685\pi$
$258$		0			0
$259$	39.8480	−	69.0187i	0.153853	−	0.266481i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−275.353	−	158.975i	−1.04697	−	0.604469i	−0.125171	−	0.992135i	$-0.539948\pi$
$263$	−275.353	−	158.975i	−1.04697	−	0.604469i	−0.921800	+	0.387667i	$0.873281\pi$
$264$		0			0
$265$	−50.1904	−	86.9323i	−0.189398	−	0.328046i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	135.043i		−	0.502020i	−0.967985	−	0.251010i	$-0.919237\pi$
$269$		−	135.043i		−	0.502020i	0.967985	−	0.251010i	$-0.0807627\pi$
$270$		0			0
$271$			161.415i			0.595628i	0.954624	+	0.297814i	$0.0962575\pi$
$271$			161.415i			0.595628i	−0.954624	+	0.297814i	$0.903742\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	128.591	+	222.726i	0.467603	+	0.809912i
$276$		0			0
$277$	367.934	+	212.427i	1.32828	+	0.766884i	0.985034	−	0.172361i	$-0.0551397\pi$
$277$	367.934	+	212.427i	1.32828	+	0.766884i	0.343248	+	0.939245i	$0.388473\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−80.5450	+	139.508i	−0.286637	+	0.496470i	−0.973005	−	0.230785i	$-0.925871\pi$
$281$	−80.5450	+	139.508i	−0.286637	+	0.496470i	0.686368	+	0.727255i	$0.259204\pi$
$282$		0			0
$283$	−182.383	−	315.897i	−0.644464	−	1.11625i	−0.984425	−	0.175805i	$-0.943747\pi$
$283$	−182.383	−	315.897i	−0.644464	−	1.11625i	0.339960	−	0.940440i	$-0.389586\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	227.792i		−	0.793701i
$288$		0			0
$289$	−266.781			−0.923118
$290$		0			0
$291$		0			0
$292$		0			0
$293$	437.041	−	252.326i	1.49161	−	0.861180i	0.491654	−	0.870790i	$-0.336392\pi$
$293$	437.041	−	252.326i	1.49161	−	0.861180i	0.999954	+	0.00961009i	$0.00305903\pi$
$294$		0			0
$295$	−56.9492	−	32.8796i	−0.193048	−	0.111456i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−9.53204	+	16.5100i	−0.0318797	+	0.0552173i
$300$		0			0
$301$	535.344	−	309.081i	1.77855	−	1.02685i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	90.2790			0.295997
$306$		0			0
$307$	−42.6361			−0.138880			−0.0694400	−	0.997586i	$-0.522121\pi$
$307$	−42.6361			−0.138880			−0.0694400	+	0.997586i	$0.522121\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−170.776	+	98.5973i	−0.549117	+	0.317033i	−0.748766	−	0.662834i	$-0.769353\pi$
$311$	−170.776	+	98.5973i	−0.549117	+	0.317033i	0.199649	+	0.979868i	$0.436020\pi$
$312$		0			0
$313$	73.0920	−	126.599i	0.233521	−	0.404470i	−0.725321	−	0.688411i	$-0.758309\pi$
$313$	73.0920	−	126.599i	0.233521	−	0.404470i	0.958842	+	0.283941i	$0.0916420\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−61.0328	−	35.2373i	−0.192533	−	0.111159i	0.400635	−	0.916238i	$-0.368790\pi$
$317$	−61.0328	−	35.2373i	−0.192533	−	0.111159i	−0.593168	+	0.805079i	$0.702123\pi$
$318$		0			0
$319$	496.163	−	286.460i	1.55537	−	0.897993i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−96.7667			−0.299587
$324$		0			0
$325$		−	104.626i		−	0.321927i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	259.221	+	448.985i	0.787907	+	1.36470i
$330$		0			0
$331$	−54.0633	+	93.6405i	−0.163333	+	0.282902i	−0.936062	−	0.351835i	$-0.885558\pi$
$331$	−54.0633	+	93.6405i	−0.163333	+	0.282902i	0.772729	+	0.634736i	$0.218891\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	91.7525	+	52.9733i	0.273888	+	0.158129i
$336$		0			0
$337$	−168.695	−	292.189i	−0.500580	−	0.867030i	−1.00000		0.000669822i	$-0.999787\pi$
$337$	−168.695	−	292.189i	−0.500580	−	0.867030i	0.499420	−	0.866360i	$-0.333547\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			701.831i			2.05816i
$342$		0			0
$343$		−	28.4187i		−	0.0828534i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	12.3560	+	21.4012i	0.0356081	+	0.0616750i	0.883280	−	0.468845i	$-0.155330\pi$
$347$	12.3560	+	21.4012i	0.0356081	+	0.0616750i	−0.847672	+	0.530520i	$0.821997\pi$
$348$		0			0
$349$	12.1159	+	6.99512i	0.0347160	+	0.0200433i	0.517258	−	0.855830i	$-0.326953\pi$
$349$	12.1159	+	6.99512i	0.0347160	+	0.0200433i	−0.482542	+	0.875873i	$0.660286\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−157.831	+	273.371i	−0.447113	+	0.774422i	−0.998197	−	0.0600276i	$-0.980881\pi$
$353$	−157.831	+	273.371i	−0.447113	+	0.774422i	0.551084	+	0.834450i	$0.314214\pi$
$354$		0			0
$355$	−130.433	−	225.916i	−0.367417	−	0.636384i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	629.848i		−	1.75445i	−0.480078	−	0.877226i	$-0.659392\pi$
$359$		−	629.848i		−	1.75445i	0.480078	−	0.877226i	$-0.340608\pi$
$360$		0			0
$361$	60.4319			0.167401
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−42.0907	+	24.3011i	−0.115317	+	0.0665783i
$366$		0			0
$367$	224.072	+	129.368i	0.610550	+	0.352501i	0.773181	−	0.634186i	$-0.218665\pi$
$367$	224.072	+	129.368i	0.610550	+	0.352501i	−0.162631	+	0.986687i	$0.551998\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	256.123	−	443.618i	0.690359	−	1.19574i
$372$		0			0
$373$	107.907	−	62.3003i	0.289296	−	0.167025i	−0.348329	−	0.937373i	$-0.613251\pi$
$373$	107.907	−	62.3003i	0.289296	−	0.167025i	0.637624	+	0.770348i	$0.279917\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−233.074			−0.618234
$378$		0			0
$379$	154.148			0.406724			0.203362	−	0.979104i	$-0.434813\pi$
$379$	154.148			0.406724			0.203362	+	0.979104i	$0.434813\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−344.678	+	199.000i	−0.899944	+	0.519583i	−0.877182	−	0.480158i	$-0.840579\pi$
$383$	−344.678	+	199.000i	−0.899944	+	0.519583i	−0.0227617	+	0.999741i	$0.507246\pi$
$384$		0			0
$385$	120.258	−	208.293i	0.312358	−	0.541020i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−279.310	−	161.260i	−0.718020	−	0.414549i	0.0960032	−	0.995381i	$-0.469394\pi$
$389$	−279.310	−	161.260i	−0.718020	−	0.414549i	−0.814024	+	0.580832i	$0.802727\pi$
$390$		0			0
$391$	−15.7155	+	9.07333i	−0.0401930	+	0.0232054i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	191.591			0.485040
$396$		0			0
$397$			286.878i			0.722615i	0.932447	+	0.361308i	$0.117670\pi$
$397$			286.878i			0.722615i	−0.932447	+	0.361308i	$0.882330\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−296.251	−	513.122i	−0.738781	−	1.27961i	−0.953044	−	0.302831i	$-0.902068\pi$
$401$	−296.251	−	513.122i	−0.738781	−	1.27961i	0.214263	−	0.976776i	$-0.431265\pi$
$402$		0			0
$403$	142.759	−	247.266i	0.354240	−	0.613562i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	83.6668	+	48.3050i	0.205569	+	0.118686i
$408$		0			0
$409$	258.131	+	447.095i	0.631126	+	1.09314i	0.987322	+	0.158731i	$0.0507403\pi$
$409$	258.131	+	447.095i	0.631126	+	1.09314i	−0.356196	+	0.934411i	$0.615926\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	335.571i		−	0.812522i
$414$		0			0
$415$			0.737736i			0.00177768i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−155.891	−	270.011i	−0.372054	−	0.644417i	0.617827	−	0.786314i	$-0.288013\pi$
$419$	−155.891	−	270.011i	−0.372054	−	0.644417i	−0.989881	+	0.141897i	$0.954680\pi$
$420$		0			0
$421$	−567.466	−	327.627i	−1.34790	−	0.778211i	−0.359949	−	0.932972i	$-0.617206\pi$
$421$	−567.466	−	327.627i	−1.34790	−	0.778211i	−0.987952	+	0.154761i	$0.950539\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	49.7956	−	86.2485i	0.117166	−	0.202938i
$426$		0			0
$427$	230.348	+	398.975i	0.539457	+	0.934367i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			204.753i			0.475064i	0.971380	+	0.237532i	$0.0763384\pi$
$431$			204.753i			0.475064i	−0.971380	+	0.237532i	$0.923662\pi$
$432$		0			0
$433$	−561.451			−1.29665			−0.648327	−	0.761362i	$-0.724531\pi$
$433$	−561.451			−1.29665			−0.648327	+	0.761362i	$0.724531\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	68.4429	−	39.5155i	0.156620	−	0.0904246i
$438$		0			0
$439$	−261.544	−	151.002i	−0.595771	−	0.343969i	0.171605	−	0.985166i	$-0.445105\pi$
$439$	−261.544	−	151.002i	−0.595771	−	0.343969i	−0.767376	+	0.641197i	$0.778438\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−228.326	+	395.472i	−0.515408	+	0.892713i	0.484432	+	0.874829i	$0.339026\pi$
$443$	−228.326	+	395.472i	−0.515408	+	0.892713i	−0.999840	+	0.0178842i	$0.994307\pi$
$444$		0			0
$445$	−229.750	+	132.646i	−0.516293	+	0.298082i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−418.640			−0.932384			−0.466192	−	0.884684i	$-0.654374\pi$
$449$	−418.640			−0.932384			−0.466192	+	0.884684i	$0.654374\pi$
$450$		0			0
$451$	276.137			0.612278
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−84.7373	+	48.9231i	−0.186236	+	0.107523i
$456$		0			0
$457$	117.305	−	203.178i	0.256684	−	0.444590i	−0.708668	−	0.705543i	$-0.750703\pi$
$457$	117.305	−	203.178i	0.256684	−	0.444590i	0.965352	+	0.260953i	$0.0840367\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−104.947	−	60.5913i	−0.227651	−	0.131435i	0.381837	−	0.924230i	$-0.375292\pi$
$461$	−104.947	−	60.5913i	−0.227651	−	0.131435i	−0.609488	+	0.792795i	$0.708625\pi$
$462$		0			0
$463$	−213.979	+	123.541i	−0.462157	+	0.266827i	−0.712951	−	0.701214i	$-0.752642\pi$
$463$	−213.979	+	123.541i	−0.462157	+	0.266827i	0.250794	+	0.968041i	$0.419308\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−276.509			−0.592097			−0.296049	−	0.955173i	$-0.595669\pi$
$467$	−276.509			−0.592097			−0.296049	+	0.955173i	$0.595669\pi$
$468$		0			0
$469$			540.649i			1.15277i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	374.678	+	648.962i	0.792132	+	1.37201i
$474$		0			0
$475$	−216.866	+	375.624i	−0.456561	+	0.790787i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	613.932	+	354.454i	1.28169	+	0.739987i	0.977158	−	0.212514i	$-0.0681652\pi$
$479$	613.932	+	354.454i	1.28169	+	0.739987i	0.304536	+	0.952501i	$0.401499\pi$
$480$		0			0
$481$	−19.6514	−	34.0372i	−0.0408552	−	0.0707633i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			42.8283i			0.0883057i
$486$		0			0
$487$			204.394i			0.419699i	0.977734	+	0.209850i	$0.0672975\pi$
$487$			204.394i			0.419699i	−0.977734	+	0.209850i	$0.932703\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	113.951	+	197.369i	0.232080	+	0.401974i	0.958420	−	0.285361i	$-0.0921136\pi$
$491$	113.951	+	197.369i	0.232080	+	0.401974i	−0.726340	+	0.687335i	$0.758780\pi$
$492$		0			0
$493$	−192.135	−	110.929i	−0.389725	−	0.225008i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	665.603	−	1152.86i	1.33924	−	2.31963i
$498$		0			0
$499$	190.463	+	329.891i	0.381689	+	0.661105i	0.991304	−	0.131593i	$-0.0420092\pi$
$499$	190.463	+	329.891i	0.381689	+	0.661105i	−0.609615	+	0.792698i	$0.708676\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			697.917i			1.38751i	0.720212	+	0.693754i	$0.244045\pi$
$503$			697.917i			1.38751i	−0.720212	+	0.693754i	$0.755955\pi$
$504$		0			0
$505$	91.6077			0.181401
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−64.0084	+	36.9552i	−0.125753	+	0.0726036i	−0.561557	−	0.827438i	$-0.689797\pi$
$509$	−64.0084	+	36.9552i	−0.125753	+	0.0726036i	0.435804	+	0.900042i	$0.356464\pi$
$510$		0			0
$511$	−214.790	−	124.009i	−0.420333	−	0.242679i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−99.2196	+	171.853i	−0.192659	+	0.333696i
$516$		0			0
$517$	−544.274	+	314.237i	−1.05275	+	0.607808i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	994.276			1.90840			0.954200	−	0.299171i	$-0.0967100\pi$
$521$	994.276			1.90840			0.954200	+	0.299171i	$0.0967100\pi$
$522$		0			0
$523$	−342.943			−0.655722			−0.327861	−	0.944726i	$-0.606328\pi$
$523$	−342.943			−0.655722			−0.327861	+	0.944726i	$0.606328\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	235.366	−	135.889i	0.446615	−	0.257854i
$528$		0			0
$529$	−257.090	+	445.292i	−0.485992	+	0.841762i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−97.2872	−	56.1688i	−0.182528	−	0.105382i
$534$		0			0
$535$	22.0626	−	12.7379i	0.0412385	−	0.0238091i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	630.905			1.17051
$540$		0			0
$541$		−	571.225i		−	1.05587i	−0.849285	−	0.527934i	$-0.822967\pi$
$541$		−	571.225i		−	1.05587i	0.849285	−	0.527934i	$-0.177033\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−115.794	−	200.562i	−0.212467	−	0.368003i
$546$		0			0
$547$	−390.615	+	676.565i	−0.714104	+	1.23686i	0.249200	+	0.968452i	$0.419832\pi$
$547$	−390.615	+	676.565i	−0.714104	+	1.23686i	−0.963304	+	0.268413i	$0.913501\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	836.771	+	483.110i	1.51864	+	0.876788i
$552$		0			0
$553$	488.847	+	846.707i	0.883990	+	1.53112i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			439.321i			0.788727i	0.918954	+	0.394364i	$0.129035\pi$
$557$			439.321i			0.788727i	−0.918954	+	0.394364i	$0.870965\pi$
$558$		0			0
$559$		−	304.852i		−	0.545352i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	16.1013	+	27.8883i	0.0285992	+	0.0495352i	0.879971	−	0.475028i	$-0.157562\pi$
$563$	16.1013	+	27.8883i	0.0285992	+	0.0495352i	−0.851372	+	0.524563i	$0.824229\pi$
$564$		0			0
$565$	50.0682	+	28.9069i	0.0886163	+	0.0511626i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−402.885	+	697.818i	−0.708058	+	1.22639i	0.257518	+	0.966274i	$0.417095\pi$
$569$	−402.885	+	697.818i	−0.708058	+	1.22639i	−0.965576	+	0.260120i	$0.916238\pi$
$570$		0			0
$571$	−250.021	−	433.050i	−0.437866	−	0.758406i	0.559659	−	0.828723i	$-0.310932\pi$
$571$	−250.021	−	433.050i	−0.437866	−	0.758406i	−0.997525	+	0.0703173i	$0.977599\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			81.3379i			0.141457i
$576$		0			0
$577$	408.301			0.707628			0.353814	−	0.935316i	$-0.384885\pi$
$577$	408.301			0.707628			0.353814	+	0.935316i	$0.384885\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3.26032	+	1.88235i	−0.00561156	+	0.00323984i
$582$		0			0
$583$	537.769	+	310.481i	0.922416	+	0.532557i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−231.965	+	401.775i	−0.395171	+	0.684456i	−0.993123	−	0.117076i	$-0.962648\pi$
$587$	−231.965	+	401.775i	−0.395171	+	0.684456i	0.597952	+	0.801532i	$0.295981\pi$
$588$		0			0
$589$	−1025.05	+	591.814i	−1.74032	+	1.00478i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	80.8523			0.136344			0.0681722	−	0.997674i	$-0.478283\pi$
$593$	80.8523			0.136344			0.0681722	+	0.997674i	$0.478283\pi$
$594$		0			0
$595$	−93.1375			−0.156534
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−153.902	+	88.8554i	−0.256932	+	0.148339i	−0.622934	−	0.782274i	$-0.714059\pi$
$599$	−153.902	+	88.8554i	−0.256932	+	0.148339i	0.366002	+	0.930614i	$0.380726\pi$
$600$		0			0
$601$	99.6250	−	172.555i	0.165765	−	0.287114i	−0.771161	−	0.636640i	$-0.780324\pi$
$601$	99.6250	−	172.555i	0.165765	−	0.287114i	0.936927	+	0.349526i	$0.113657\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	46.3022	+	26.7326i	0.0765325	+	0.0441861i
$606$		0			0
$607$	−967.595	+	558.641i	−1.59406	+	0.920332i	−0.601461	+	0.798902i	$0.705415\pi$
$607$	−967.595	+	558.641i	−1.59406	+	0.920332i	−0.992600	+	0.121430i	$0.961252\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	255.674			0.418453
$612$		0			0
$613$		−	131.929i		−	0.215218i	−0.994193	−	0.107609i	$-0.965681\pi$
$613$		−	131.929i		−	0.215218i	0.994193	−	0.107609i	$-0.0343194\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	57.7694	+	100.060i	0.0936295	+	0.162171i	0.909036	−	0.416718i	$-0.136820\pi$
$617$	57.7694	+	100.060i	0.0936295	+	0.162171i	−0.815406	+	0.578889i	$0.803486\pi$
$618$		0			0
$619$	337.518	−	584.599i	0.545264	−	0.944424i	−0.453327	−	0.891344i	$-0.649763\pi$
$619$	337.518	−	584.599i	0.545264	−	0.944424i	0.998590	−	0.0530799i	$-0.0169038\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1172.42	−	676.899i	−1.88190	−	1.08651i
$624$		0			0
$625$	−174.796	−	302.756i	−0.279674	−	0.484410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	37.4113i		−	0.0594775i
$630$		0			0
$631$		−	214.708i		−	0.340267i	−0.985421	−	0.170133i	$-0.945580\pi$
$631$		−	214.708i		−	0.340267i	0.985421	−	0.170133i	$-0.0544199\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−110.648	−	191.649i	−0.174250	−	0.301809i
$636$		0			0
$637$	−222.277	−	128.332i	−0.348944	−	0.201463i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−90.7182	+	157.129i	−0.141526	+	0.245130i	−0.928072	−	0.372402i	$-0.878534\pi$
$641$	−90.7182	+	157.129i	−0.141526	+	0.245130i	0.786545	+	0.617532i	$0.211868\pi$
$642$		0			0
$643$	544.280	+	942.721i	0.846470	+	1.46613i	0.884338	+	0.466847i	$0.154610\pi$
$643$	544.280	+	942.721i	0.846470	+	1.46613i	−0.0378680	+	0.999283i	$0.512057\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			162.166i			0.250643i	0.992116	+	0.125321i	$0.0399962\pi$
$647$			162.166i			0.250643i	−0.992116	+	0.125321i	$0.960004\pi$
$648$		0			0
$649$	406.791			0.626796
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−441.606	+	254.961i	−0.676272	+	0.390446i	−0.798449	−	0.602062i	$-0.794346\pi$
$653$	−441.606	+	254.961i	−0.676272	+	0.390446i	0.122177	+	0.992508i	$0.461013\pi$
$654$		0			0
$655$	−303.967	−	175.495i	−0.464071	−	0.267932i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	392.592	−	679.989i	0.595739	−	1.03185i	−0.397703	−	0.917514i	$-0.630193\pi$
$659$	392.592	−	679.989i	0.595739	−	1.03185i	0.993442	−	0.114336i	$-0.0364740\pi$
$660$		0			0
$661$	−219.088	+	126.491i	−0.331450	+	0.191363i	−0.656485	−	0.754339i	$-0.727957\pi$
$661$	−219.088	+	126.491i	−0.331450	+	0.191363i	0.325035	+	0.945702i	$0.394624\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	405.626			0.609964
$666$		0			0
$667$	181.195			0.271657
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−483.650	+	279.236i	−0.720790	+	0.416149i
$672$		0			0
$673$	−54.8126	+	94.9383i	−0.0814452	+	0.141067i	−0.903871	−	0.427805i	$-0.859287\pi$
$673$	−54.8126	+	94.9383i	−0.0814452	+	0.141067i	0.822426	+	0.568873i	$0.192620\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	415.870	+	240.103i	0.614284	+	0.354657i	0.774640	−	0.632402i	$-0.217931\pi$
$677$	415.870	+	240.103i	0.614284	+	0.354657i	−0.160356	+	0.987059i	$0.551264\pi$
$678$		0			0
$679$	−189.273	+	109.277i	−0.278753	+	0.160938i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−27.4345			−0.0401677			−0.0200838	−	0.999798i	$-0.506393\pi$
$683$	−27.4345			−0.0401677			−0.0200838	+	0.999798i	$0.506393\pi$
$684$		0			0
$685$		−	252.159i		−	0.368115i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−126.309	−	218.774i	−0.183323	−	0.317524i
$690$		0			0
$691$	−4.69502	+	8.13201i	−0.00679452	+	0.0117685i	−0.869403	−	0.494104i	$-0.835496\pi$
$691$	−4.69502	+	8.13201i	−0.00679452	+	0.0117685i	0.862608	+	0.505873i	$0.168829\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−328.270	−	189.527i	−0.472331	−	0.272701i
$696$		0			0
$697$	−53.4658	−	92.6055i	−0.0767084	−	0.132863i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			821.246i			1.17153i	0.810479	+	0.585767i	$0.199207\pi$
$701$			821.246i			1.17153i	−0.810479	+	0.585767i	$0.800793\pi$
$702$		0			0
$703$			162.931i			0.231766i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	233.738	+	404.847i	0.330606	+	0.572626i
$708$		0			0
$709$	719.455	+	415.378i	1.01475	+	0.585864i	0.912578	−	0.408903i	$-0.134089\pi$
$709$	719.455	+	415.378i	1.01475	+	0.585864i	0.102168	+	0.994767i	$0.467422\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−110.983	+	192.228i	−0.155656	+	0.269604i
$714$		0			0
$715$	−59.3062	−	102.721i	−0.0829458	−	0.143666i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			195.460i			0.271850i	0.990719	+	0.135925i	$0.0434006\pi$
$719$			195.460i			0.271850i	−0.990719	+	0.135925i	$0.956599\pi$
$720$		0			0
$721$	−1012.64			−1.40450
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−861.195	+	497.211i	−1.18786	+	0.685809i
$726$		0			0
$727$	208.982	+	120.656i	0.287457	+	0.165964i	0.636795	−	0.771033i	$-0.280260\pi$
$727$	208.982	+	120.656i	0.287457	+	0.165964i	−0.349337	+	0.936997i	$0.613593\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	145.091	−	251.304i	0.198482	−	0.343782i
$732$		0			0
$733$	907.973	−	524.218i	1.23871	−	0.715168i	0.269878	−	0.962895i	$-0.413017\pi$
$733$	907.973	−	524.218i	1.23871	−	0.715168i	0.968830	+	0.247726i	$0.0796834\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−655.392			−0.889271
$738$		0			0
$739$	166.542			0.225362			0.112681	−	0.993631i	$-0.464056\pi$
$739$	166.542			0.225362			0.112681	+	0.993631i	$0.464056\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−242.538	+	140.029i	−0.326430	+	0.188465i	−0.654255	−	0.756274i	$-0.727018\pi$
$743$	−242.538	+	140.029i	−0.326430	+	0.188465i	0.327825	+	0.944739i	$0.393684\pi$
$744$		0			0
$745$	36.2476	−	62.7828i	0.0486545	−	0.0842722i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	112.586	+	65.0017i	0.150315	+	0.0867846i
$750$		0			0
$751$	−189.292	+	109.288i	−0.252053	+	0.145523i	−0.620704	−	0.784045i	$-0.713153\pi$
$751$	−189.292	+	109.288i	−0.252053	+	0.145523i	0.368651	+	0.929568i	$0.379820\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	3.49413			0.00462799
$756$		0			0
$757$		−	1241.72i		−	1.64031i	−0.572138	−	0.820157i	$-0.693886\pi$
$757$		−	1241.72i		−	1.64031i	0.572138	−	0.820157i	$-0.306114\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	353.625	+	612.496i	0.464684	+	0.804857i	0.999187	−	0.0403097i	$-0.0128345\pi$
$761$	353.625	+	612.496i	0.464684	+	0.804857i	−0.534503	+	0.845167i	$0.679501\pi$
$762$		0			0
$763$	590.902	−	1023.47i	0.774445	−	1.34138i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−143.318	−	82.7449i	−0.186856	−	0.107881i
$768$		0			0
$769$	−469.232	−	812.734i	−0.610185	−	1.05687i	−0.991209	−	0.132307i	$-0.957762\pi$
$769$	−469.232	−	812.734i	−0.610185	−	1.05687i	0.381024	−	0.924565i	$-0.375572\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	419.774i		−	0.543045i	−0.962432	−	0.271522i	$-0.912473\pi$
$773$		−	419.774i		−	0.543045i	0.962432	−	0.271522i	$-0.0875271\pi$
$774$		0			0
$775$		−	1218.18i		−	1.57184i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	232.851	+	403.309i	0.298910	+	0.517727i
$780$		0			0
$781$	1397.53	+	806.866i	1.78941	+	1.03312i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	32.3413	−	56.0168i	0.0411991	−	0.0713589i
$786$		0			0
$787$	−43.1357	−	74.7132i	−0.0548103	−	0.0949341i	0.837318	−	0.546715i	$-0.184122\pi$
$787$	−43.1357	−	74.7132i	−0.0548103	−	0.0949341i	−0.892129	+	0.451781i	$0.850789\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			295.025i			0.372978i
$792$		0			0
$793$	227.196			0.286502
$794$		0			0
$795$		0			0
$796$		0			0
$797$	865.089	−	499.459i	1.08543	−	0.626674i	0.153075	−	0.988215i	$-0.451082\pi$
$797$	865.089	−	499.459i	1.08543	−	0.626674i	0.932356	+	0.361540i	$0.117749\pi$
$798$		0			0
$799$	210.765	+	121.685i	0.263786	+	0.152297i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	150.328	−	260.376i	0.187208	−	0.324253i
$804$		0			0
$805$	65.8760	−	38.0335i	0.0818336	−	0.0472466i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−526.876			−0.651268			−0.325634	−	0.945496i	$-0.605578\pi$
$809$	−526.876			−0.651268			−0.325634	+	0.945496i	$0.605578\pi$
$810$		0			0
$811$	−980.131			−1.20855			−0.604273	−	0.796777i	$-0.706536\pi$
$811$	−980.131			−1.20855			−0.604273	+	0.796777i	$0.706536\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	44.4813	−	25.6813i	0.0545783	−	0.0315108i
$816$		0			0
$817$	−631.889	+	1094.46i	−0.773426	+	1.33961i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1151.54	+	664.844i	1.40261	+	0.809798i	0.994660	−	0.103205i	$-0.0329099\pi$
$821$	1151.54	+	664.844i	1.40261	+	0.809798i	0.407951	+	0.913004i	$0.366243\pi$
$822$		0			0
$823$	−354.999	+	204.959i	−0.431347	+	0.249038i	−0.699920	−	0.714221i	$-0.746781\pi$
$823$	−354.999	+	204.959i	−0.431347	+	0.249038i	0.268573	+	0.963259i	$0.413448\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	756.431			0.914668			0.457334	−	0.889295i	$-0.348804\pi$
$827$	756.431			0.914668			0.457334	+	0.889295i	$0.348804\pi$
$828$		0			0
$829$			1028.26i			1.24036i	0.784460	+	0.620179i	$0.212940\pi$
$829$			1028.26i			1.24036i	−0.784460	+	0.620179i	$0.787060\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−122.156	−	211.580i	−0.146646	−	0.253998i
$834$		0			0
$835$	−128.755	+	223.010i	−0.154198	+	0.267078i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−140.269	−	80.9842i	−0.167186	−	0.0965247i	0.414072	−	0.910244i	$-0.364106\pi$
$839$	−140.269	−	80.9842i	−0.167186	−	0.0965247i	−0.581258	+	0.813719i	$0.697439\pi$
$840$		0			0
$841$	687.130	+	1190.14i	0.817040	+	1.41515i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	284.294i		−	0.336443i
$846$		0			0
$847$			272.834i			0.322118i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	15.2773	+	26.4610i	0.0179521	+	0.0310940i
$852$		0			0
$853$	−64.6457	−	37.3232i	−0.0757863	−	0.0437552i	0.461628	−	0.887074i	$-0.347266\pi$
$853$	−64.6457	−	37.3232i	−0.0757863	−	0.0437552i	−0.537414	+	0.843318i	$0.680599\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−619.475	+	1072.96i	−0.722841	+	1.25200i	0.237015	+	0.971506i	$0.423831\pi$
$857$	−619.475	+	1072.96i	−0.722841	+	1.25200i	−0.959856	+	0.280492i	$0.909502\pi$
$858$		0			0
$859$	59.7685	+	103.522i	0.0695792	+	0.120515i	0.898716	−	0.438531i	$-0.144501\pi$
$859$	59.7685	+	103.522i	0.0695792	+	0.120515i	−0.829137	+	0.559046i	$0.811168\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1262.84i		−	1.46332i	−0.681672	−	0.731658i	$-0.738747\pi$
$863$		−	1262.84i		−	1.46332i	0.681672	−	0.731658i	$-0.261253\pi$
$864$		0			0
$865$	−361.515			−0.417936
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1026.41	+	592.596i	−1.18114	+	0.681929i
$870$		0			0
$871$	230.904	+	133.313i	0.265103	+	0.153057i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−455.719	+	789.328i	−0.520822	+	0.902090i
$876$		0			0
$877$	−329.652	+	190.325i	−0.375886	+	0.217018i	−0.676027	−	0.736877i	$-0.736300\pi$
$877$	−329.652	+	190.325i	−0.375886	+	0.217018i	0.300141	+	0.953895i	$0.402966\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1191.72			1.35268			0.676342	−	0.736587i	$-0.263564\pi$
$881$	1191.72			1.35268			0.676342	+	0.736587i	$0.263564\pi$
$882$		0			0
$883$	−1390.74			−1.57502			−0.787509	−	0.616303i	$-0.788630\pi$
$883$	−1390.74			−1.57502			−0.787509	+	0.616303i	$0.788630\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1389.31	−	802.119i	1.56630	−	0.904306i	0.569709	−	0.821847i	$-0.307056\pi$
$887$	1389.31	−	802.119i	1.56630	−	0.904306i	0.996594	−	0.0824589i	$-0.0262773\pi$
$888$		0			0
$889$	564.642	−	977.989i	0.635143	−	1.10010i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−917.910	−	529.955i	−1.02789	−	0.593455i
$894$		0			0
$895$	41.6334	−	24.0371i	0.0465178	−	0.0268570i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2713.71			−3.01859
$900$		0			0
$901$		−	240.462i		−	0.266883i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−100.014	−	173.229i	−0.110512	−	0.191413i
$906$		0			0
$907$	321.020	−	556.023i	0.353936	−	0.613035i	−0.632999	−	0.774152i	$-0.718176\pi$
$907$	321.020	−	556.023i	0.353936	−	0.613035i	0.986935	+	0.161117i	$0.0515098\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−23.6696	−	13.6656i	−0.0259820	−	0.0150007i	0.486953	−	0.873428i	$-0.338108\pi$
$911$	−23.6696	−	13.6656i	−0.0259820	−	0.0150007i	−0.512935	+	0.858428i	$0.671442\pi$
$912$		0			0
$913$	−2.28184	−	3.95227i	−0.00249928	−	0.00432888i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	1791.12i		−	1.95323i
$918$		0			0
$919$			678.306i			0.738091i	0.929411	+	0.369046i	$0.120315\pi$
$919$			678.306i			0.738091i	−0.929411	+	0.369046i	$0.879685\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−328.248	−	568.542i	−0.355631	−	0.615971i
$924$		0			0
$925$	−145.221	−	83.8435i	−0.156996	−	0.0906416i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	635.493	−	1100.71i	0.684062	−	1.18483i	−0.289669	−	0.957127i	$-0.593545\pi$
$929$	635.493	−	1100.71i	0.684062	−	1.18483i	0.973731	−	0.227703i	$-0.0731215\pi$
$930$		0			0
$931$	532.006	+	921.461i	0.571435	+	0.989754i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	112.904i		−	0.120753i
$936$		0			0
$937$	1049.34			1.11990			0.559948	−	0.828528i	$-0.310821\pi$
$937$	1049.34			1.11990			0.559948	+	0.828528i	$0.310821\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1206.53	−	696.589i	1.28218	−	0.740265i	0.304931	−	0.952374i	$-0.401367\pi$
$941$	1206.53	−	696.589i	1.28218	−	0.740265i	0.977246	+	0.212109i	$0.0680333\pi$
$942$		0			0
$943$	75.6325	+	43.6665i	0.0802042	+	0.0463059i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	751.624	−	1301.85i	0.793690	−	1.37471i	−0.129978	−	0.991517i	$-0.541491\pi$
$947$	751.624	−	1301.85i	0.793690	−	1.37471i	0.923668	−	0.383194i	$-0.125176\pi$
$948$		0			0
$949$	−105.925	+	61.1561i	−0.111618	+	0.0644427i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−788.992			−0.827903			−0.413951	−	0.910299i	$-0.635852\pi$
$953$	−788.992			−0.827903			−0.413951	+	0.910299i	$0.635852\pi$
$954$		0			0
$955$	−56.8301			−0.0595080
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1114.38	−	643.386i	1.16202	−	0.670893i
$960$		0			0
$961$	1181.66	−	2046.69i	1.22961	−	2.12975i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	486.224	+	280.721i	0.503859	+	0.290903i
$966$		0			0
$967$	396.835	−	229.113i	0.410377	−	0.236931i	−0.280575	−	0.959832i	$-0.590525\pi$
$967$	396.835	−	229.113i	0.410377	−	0.236931i	0.690952	+	0.722901i	$0.257192\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−25.2266			−0.0259800			−0.0129900	−	0.999916i	$-0.504135\pi$
$971$	−25.2266			−0.0259800			−0.0129900	+	0.999916i	$0.504135\pi$
$972$		0			0
$973$		−	1934.32i		−	1.98800i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−736.113	−	1274.98i	−0.753442	−	1.30500i	−0.946145	−	0.323743i	$-0.895059\pi$
$977$	−736.113	−	1274.98i	−0.753442	−	1.30500i	0.192703	−	0.981257i	$-0.438274\pi$
$978$		0			0
$979$	820.559	−	1421.25i	0.838160	−	1.45174i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−909.975	−	525.374i	−0.925712	−	0.534460i	−0.0402592	−	0.999189i	$-0.512818\pi$
$983$	−909.975	−	525.374i	−0.925712	−	0.534460i	−0.885453	+	0.464729i	$0.846152\pi$
$984$		0			0
$985$	188.861	+	327.117i	0.191737	+	0.332099i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			236.996i			0.239632i
$990$		0			0
$991$		−	1021.63i		−	1.03091i	−0.856917	−	0.515454i	$-0.827623\pi$
$991$		−	1021.63i		−	1.03091i	0.856917	−	0.515454i	$-0.172377\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−33.2484	−	57.5880i	−0.0334155	−	0.0578774i
$996$		0			0
$997$	331.758	+	191.540i	0.332756	+	0.192117i	0.657064	−	0.753835i	$-0.271798\pi$
$997$	331.758	+	191.540i	0.332756	+	0.192117i	−0.324308	+	0.945952i	$0.605131\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.3.t.b.559.8		40
3.2	odd	2		288.3.t.b.79.6		40
4.3	odd	2		216.3.p.b.19.13		40
8.3	odd	2	inner	864.3.t.b.559.13		40
8.5	even	2		216.3.p.b.19.1		40
9.2	odd	6		2592.3.b.e.1135.13		20
9.4	even	3	inner	864.3.t.b.847.13		40
9.5	odd	6		288.3.t.b.175.5		40
9.7	even	3		2592.3.b.f.1135.8		20
12.11	even	2		72.3.p.b.43.8	✓	40
24.5	odd	2		72.3.p.b.43.20	yes	40
24.11	even	2		288.3.t.b.79.5		40
36.7	odd	6		648.3.b.e.163.15		20
36.11	even	6		648.3.b.f.163.6		20
36.23	even	6		72.3.p.b.67.20	yes	40
36.31	odd	6		216.3.p.b.91.1		40
72.5	odd	6		72.3.p.b.67.8	yes	40
72.11	even	6		2592.3.b.e.1135.8		20
72.13	even	6		216.3.p.b.91.13		40
72.29	odd	6		648.3.b.f.163.5		20
72.43	odd	6		2592.3.b.f.1135.13		20
72.59	even	6		288.3.t.b.175.6		40
72.61	even	6		648.3.b.e.163.16		20
72.67	odd	6	inner	864.3.t.b.847.8		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.3.p.b.43.8	✓	40	12.11	even	2
72.3.p.b.43.20	yes	40	24.5	odd	2
72.3.p.b.67.8	yes	40	72.5	odd	6
72.3.p.b.67.20	yes	40	36.23	even	6
216.3.p.b.19.1		40	8.5	even	2
216.3.p.b.19.13		40	4.3	odd	2
216.3.p.b.91.1		40	36.31	odd	6
216.3.p.b.91.13		40	72.13	even	6
288.3.t.b.79.5		40	24.11	even	2
288.3.t.b.79.6		40	3.2	odd	2
288.3.t.b.175.5		40	9.5	odd	6
288.3.t.b.175.6		40	72.59	even	6
648.3.b.e.163.15		20	36.7	odd	6
648.3.b.e.163.16		20	72.61	even	6
648.3.b.f.163.5		20	72.29	odd	6
648.3.b.f.163.6		20	36.11	even	6
864.3.t.b.559.8		40	1.1	even	1	trivial
864.3.t.b.559.13		40	8.3	odd	2	inner
864.3.t.b.847.8		40	72.67	odd	6	inner
864.3.t.b.847.13		40	9.4	even	3	inner
2592.3.b.e.1135.8		20	72.11	even	6
2592.3.b.e.1135.13		20	9.2	odd	6
2592.3.b.f.1135.8		20	9.7	even	3
2592.3.b.f.1135.13		20	72.43	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:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	864.t (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.70411 + 0.983869i) q^{5} +(-8.69613 - 5.02071i) q^{7} +O(q^{10})\)	\(q+(-1.70411 + 0.983869i) q^{5} +(-8.69613 - 5.02071i) q^{7} +(6.08627 - 10.5417i) q^{11} +(-4.28857 + 2.47601i) q^{13} -4.71370 q^{17} +20.5288 q^{19} +(3.33400 - 1.92488i) q^{23} +(-10.5640 + 18.2974i) q^{25} +(40.7608 + 23.5333i) q^{29} +(-49.9323 + 28.8285i) q^{31} +19.7589 q^{35} +7.93672i q^{37} +(11.3426 + 19.6460i) q^{41} +(-30.7806 + 53.3136i) q^{43} +(-44.7133 - 25.8152i) q^{47} +(25.9151 + 44.8862i) q^{49} +51.0133i q^{53} +23.9524i q^{55} +(16.7094 + 28.9415i) q^{59} +(-39.7329 - 22.9398i) q^{61} +(4.87213 - 8.43878i) q^{65} +(-26.9209 - 46.6284i) q^{67} +132.571i q^{71} +24.6995 q^{73} +(-105.854 + 61.1148i) q^{77} +(-84.3214 - 48.6830i) q^{79} +(0.187458 - 0.324687i) q^{83} +(8.03267 - 4.63767i) q^{85} +134.821 q^{89} +49.7252 q^{91} +(-34.9834 + 20.1977i) q^{95} +(10.8826 - 18.8492i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q+O(q^{10}) \) 40 * q	\( 40 q - 16 q^{11} + 4 q^{17} + 76 q^{19} + 118 q^{25} - 108 q^{35} - 20 q^{41} + 16 q^{43} + 166 q^{49} - 64 q^{59} + 102 q^{65} + 64 q^{67} - 292 q^{73} + 554 q^{83} + 688 q^{89} + 204 q^{91} + 92 q^{97}+O(q^{100}) \) 40 * q - 16 * q^11 + 4 * q^17 + 76 * q^19 + 118 * q^25 - 108 * q^35 - 20 * q^41 + 16 * q^43 + 166 * q^49 - 64 * q^59 + 102 * q^65 + 64 * q^67 - 292 * q^73 + 554 * q^83 + 688 * q^89 + 204 * q^91 + 92 * q^97

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