LMFDB - Embedded newform 2808.2.cw.b.1585.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2808,2,Mod(1585,2808)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2808, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4, 3])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2808.1585"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2808 = 2^{3} \cdot 3^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2808.cw (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$22.4219928876$
Analytic rank:	$0$
Dimension:	$80$
Relative dimension:	$40$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 936)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1585.3
Character	$\chi$	$=$	2808.1585
Dual form			2808.2.cw.b.2521.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.40435 + 1.96550i) q^{5} +(-4.16592 - 2.40520i) q^{7} +(2.91097 + 1.68065i) q^{11} +(0.448090 - 3.57760i) q^{13} +1.07207 q^{17} -6.57228i q^{19} +(-0.0427423 - 0.0740317i) q^{23} +(5.22640 - 9.05238i) q^{25} +(0.514326 - 0.890839i) q^{29} +(-6.40180 + 3.69608i) q^{31} +18.9097 q^{35} +4.76927i q^{37} +(-2.17095 + 1.25340i) q^{41} +(1.85904 - 3.21995i) q^{43} +(-8.71820 - 5.03346i) q^{47} +(8.06995 + 13.9776i) q^{49} +5.80367 q^{53} -13.2133 q^{55} +(-10.4025 + 6.00590i) q^{59} +(-2.30828 + 3.99806i) q^{61} +(5.50632 + 13.0601i) q^{65} +(2.45580 - 1.41786i) q^{67} -1.36430i q^{71} +2.67779i q^{73} +(-8.08457 - 14.0029i) q^{77} +(0.872958 - 1.51201i) q^{79} +(12.2034 + 7.04566i) q^{83} +(-3.64971 + 2.10716i) q^{85} +9.35462i q^{89} +(-10.4715 + 13.8263i) q^{91} +(12.9178 + 22.3743i) q^{95} +(6.64133 + 3.83437i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q + 6 q^{13} + 4 q^{17} - 10 q^{23} + 44 q^{25} + 52 q^{35} - 26 q^{43} + 48 q^{49} - 60 q^{53} - 16 q^{55} - 10 q^{61} + 26 q^{65} - 32 q^{77} + 6 q^{79} - 4 q^{91} - 8 q^{95}+O(q^{100}) $ 80 * q + 6 * q^13 + 4 * q^17 - 10 * q^23 + 44 * q^25 + 52 * q^35 - 26 * q^43 + 48 * q^49 - 60 * q^53 - 16 * q^55 - 10 * q^61 + 26 * q^65 - 32 * q^77 + 6 * q^79 - 4 * q^91 - 8 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2808\mathbb{Z}\right)^\times$.

$n$	$703$	$1081$	$1405$	$2081$
$\chi(n)$	$1$	$-1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−3.40435	+	1.96550i	−1.52247	+	0.878999i	−0.522824	+	0.852441i	$0.675122\pi$
$5$	−3.40435	+	1.96550i	−1.52247	+	0.878999i	−0.999647	+	0.0265586i	$0.991545\pi$
$6$		0			0
$7$	−4.16592	−	2.40520i	−1.57457	−	0.909079i	−0.995597	−	0.0937328i	$-0.970120\pi$
$7$	−4.16592	−	2.40520i	−1.57457	−	0.909079i	−0.578974	−	0.815346i	$-0.696547\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.91097	+	1.68065i	0.877689	+	0.506734i	0.869896	−	0.493236i	$-0.164186\pi$
$11$	2.91097	+	1.68065i	0.877689	+	0.506734i	0.00779327	+	0.999970i	$0.497519\pi$
$12$		0			0
$13$	0.448090	−	3.57760i	0.124278	−	0.992247i
$13$	0.448090	−	3.57760i	0.124278	−	0.992247i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.07207			0.260016			0.130008	−	0.991513i	$-0.458500\pi$
$17$	1.07207			0.260016			0.130008	+	0.991513i	$0.458500\pi$
$18$		0			0
$19$		−	6.57228i		−	1.50778i	−0.656998	−	0.753892i	$-0.728174\pi$
$19$		−	6.57228i		−	1.50778i	0.656998	−	0.753892i	$-0.271826\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.0427423	−	0.0740317i	−0.00891238	−	0.0154367i	0.861535	−	0.507698i	$-0.169504\pi$
$23$	−0.0427423	−	0.0740317i	−0.00891238	−	0.0154367i	−0.870447	+	0.492262i	$0.836170\pi$
$24$		0			0
$25$	5.22640	−	9.05238i	1.04528	−	1.81048i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.514326	−	0.890839i	0.0955080	−	0.165425i	−0.814313	−	0.580427i	$-0.802886\pi$
$29$	0.514326	−	0.890839i	0.0955080	−	0.165425i	0.909821	+	0.415002i	$0.136219\pi$
$30$		0			0
$31$	−6.40180	+	3.69608i	−1.14980	+	0.663836i	−0.948838	−	0.315764i	$-0.897739\pi$
$31$	−6.40180	+	3.69608i	−1.14980	+	0.663836i	−0.200960	+	0.979600i	$0.564406\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	18.9097			3.19632
$36$		0			0
$37$			4.76927i			0.784063i	0.919952	+	0.392031i	$0.128228\pi$
$37$			4.76927i			0.784063i	−0.919952	+	0.392031i	$0.871772\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.17095	+	1.25340i	−0.339045	+	0.195748i	−0.659850	−	0.751398i	$-0.729380\pi$
$41$	−2.17095	+	1.25340i	−0.339045	+	0.195748i	0.320804	+	0.947145i	$0.396047\pi$
$42$		0			0
$43$	1.85904	−	3.21995i	0.283500	−	0.491037i	−0.688744	−	0.725005i	$-0.741838\pi$
$43$	1.85904	−	3.21995i	0.283500	−	0.491037i	0.972244	+	0.233967i	$0.0751709\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−8.71820	−	5.03346i	−1.27168	−	0.734205i	−0.296377	−	0.955071i	$-0.595778\pi$
$47$	−8.71820	−	5.03346i	−1.27168	−	0.734205i	−0.975304	+	0.220866i	$0.929112\pi$
$48$		0			0
$49$	8.06995	+	13.9776i	1.15285	+	1.99679i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.80367			0.797195			0.398598	−	0.917126i	$-0.369497\pi$
$53$	5.80367			0.797195			0.398598	+	0.917126i	$0.369497\pi$
$54$		0			0
$55$	−13.2133			−1.78168
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−10.4025	+	6.00590i	−1.35429	+	0.781901i	−0.988847	−	0.148932i	$-0.952416\pi$
$59$	−10.4025	+	6.00590i	−1.35429	+	0.781901i	−0.365445	+	0.930833i	$0.619083\pi$
$60$		0			0
$61$	−2.30828	+	3.99806i	−0.295545	+	0.511899i	−0.975112	−	0.221715i	$-0.928835\pi$
$61$	−2.30828	+	3.99806i	−0.295545	+	0.511899i	0.679566	+	0.733614i	$0.262168\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	5.50632	+	13.0601i	0.682975	+	1.61991i
$66$		0			0
$67$	2.45580	−	1.41786i	0.300024	−	0.173219i	−0.342430	−	0.939543i	$-0.611250\pi$
$67$	2.45580	−	1.41786i	0.300024	−	0.173219i	0.642454	+	0.766325i	$0.277916\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	1.36430i		−	0.161913i	−0.996718	−	0.0809564i	$-0.974203\pi$
$71$		−	1.36430i		−	0.161913i	0.996718	−	0.0809564i	$-0.0257975\pi$
$72$		0			0
$73$			2.67779i			0.313412i	0.987645	+	0.156706i	$0.0500874\pi$
$73$			2.67779i			0.313412i	−0.987645	+	0.156706i	$0.949913\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−8.08457	−	14.0029i	−0.921322	−	1.59578i
$78$		0			0
$79$	0.872958	−	1.51201i	0.0982155	−	0.170114i	−0.812731	−	0.582640i	$-0.802020\pi$
$79$	0.872958	−	1.51201i	0.0982155	−	0.170114i	0.910946	+	0.412525i	$0.135353\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	12.2034	+	7.04566i	1.33950	+	0.773362i	0.986733	−	0.162349i	$-0.0519070\pi$
$83$	12.2034	+	7.04566i	1.33950	+	0.773362i	0.352768	+	0.935711i	$0.385240\pi$
$84$		0			0
$85$	−3.64971	+	2.10716i	−0.395867	+	0.228554i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			9.35462i			0.991587i	0.868440	+	0.495794i	$0.165123\pi$
$89$			9.35462i			0.991587i	−0.868440	+	0.495794i	$0.834877\pi$
$90$		0			0
$91$	−10.4715	+	13.8263i	−1.09772	+	1.44939i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	12.9178	+	22.3743i	1.32534	+	2.29556i
$96$		0			0
$97$	6.64133	+	3.83437i	0.674325	+	0.389322i	0.797713	−	0.603037i	$-0.206043\pi$
$97$	6.64133	+	3.83437i	0.674325	+	0.389322i	−0.123388	+	0.992358i	$0.539376\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−7.44051	+	12.8873i	−0.740358	+	1.28234i	0.211974	+	0.977275i	$0.432011\pi$
$101$	−7.44051	+	12.8873i	−0.740358	+	1.28234i	−0.952332	+	0.305063i	$0.901322\pi$
$102$		0			0
$103$	9.46797	+	16.3990i	0.932907	+	1.61584i	0.778325	+	0.627862i	$0.216070\pi$
$103$	9.46797	+	16.3990i	0.932907	+	1.61584i	0.154582	+	0.987980i	$0.450597\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−14.6995			−1.42105			−0.710526	−	0.703671i	$-0.751543\pi$
$107$	−14.6995			−1.42105			−0.710526	+	0.703671i	$0.751543\pi$
$108$		0			0
$109$		−	4.39621i		−	0.421081i	−0.977585	−	0.210540i	$-0.932478\pi$
$109$		−	4.39621i		−	0.421081i	0.977585	−	0.210540i	$-0.0675224\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.23196	+	2.13381i	0.115893	+	0.200732i	0.918136	−	0.396265i	$-0.129694\pi$
$113$	1.23196	+	2.13381i	0.115893	+	0.200732i	−0.802244	+	0.596997i	$0.796360\pi$
$114$		0			0
$115$	0.291019	+	0.168020i	0.0271377	+	0.0156679i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−4.46618	−	2.57855i	−0.409414	−	0.236375i
$120$		0			0
$121$	0.149145	+	0.258327i	0.0135587	+	0.0234843i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			21.4350i			1.91720i
$126$		0			0
$127$	17.2139			1.52749			0.763743	−	0.645520i	$-0.223359\pi$
$127$	17.2139			1.52749			0.763743	+	0.645520i	$0.223359\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−4.22778	−	7.32272i	−0.369382	−	0.639789i	0.620087	−	0.784533i	$-0.287097\pi$
$131$	−4.22778	−	7.32272i	−0.369382	−	0.639789i	−0.989469	+	0.144744i	$0.953764\pi$
$132$		0			0
$133$	−15.8076	+	27.3796i	−1.37070	+	2.37411i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	15.7874	+	9.11483i	1.34880	+	0.778733i	0.988080	−	0.153940i	$-0.0491962\pi$
$137$	15.7874	+	9.11483i	1.34880	+	0.778733i	0.360724	+	0.932672i	$0.382530\pi$
$138$		0			0
$139$	5.76010	+	9.97679i	0.488565	+	0.846220i	0.999913	−	0.0131535i	$-0.00418701\pi$
$139$	5.76010	+	9.97679i	0.488565	+	0.846220i	−0.511348	+	0.859374i	$0.670854\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	7.31705	−	9.66119i	0.611883	−	0.807909i
$144$		0			0
$145$			4.04364i			0.335806i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.77205	+	2.17780i	−0.309019	+	0.178412i	−0.646487	−	0.762925i	$-0.723763\pi$
$149$	−3.77205	+	2.17780i	−0.309019	+	0.178412i	0.337469	+	0.941337i	$0.390429\pi$
$150$		0			0
$151$	3.70820	+	2.14093i	0.301769	+	0.174226i	0.643237	−	0.765667i	$-0.277591\pi$
$151$	3.70820	+	2.14093i	0.301769	+	0.174226i	−0.341468	+	0.939893i	$0.610924\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	14.5293	−	25.1655i	1.16702	−	2.02134i
$156$		0			0
$157$	−0.0630158	−	0.109147i	−0.00502921	−	0.00871085i	0.863500	−	0.504349i	$-0.168268\pi$
$157$	−0.0630158	−	0.109147i	−0.00502921	−	0.00871085i	−0.868529	+	0.495638i	$0.834934\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			0.411214i			0.0324082i
$162$		0			0
$163$		−	16.6292i		−	1.30250i	−0.758862	−	0.651252i	$-0.774244\pi$
$163$		−	16.6292i		−	1.30250i	0.758862	−	0.651252i	$-0.225756\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.79761	−	1.03785i	0.139103	−	0.0803113i	−0.428833	−	0.903384i	$-0.641075\pi$
$167$	1.79761	−	1.03785i	0.139103	−	0.0803113i	0.567937	+	0.823072i	$0.307742\pi$
$168$		0			0
$169$	−12.5984	−	3.20617i	−0.969110	−	0.246629i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−7.17771	+	12.4322i	−0.545711	+	0.945200i	0.452851	+	0.891586i	$0.350407\pi$
$173$	−7.17771	+	12.4322i	−0.545711	+	0.945200i	−0.998562	+	0.0536131i	$0.982926\pi$
$174$		0			0
$175$	−43.5455	+	25.1410i	−3.29173	+	1.90048i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4.08797			0.305549			0.152775	−	0.988261i	$-0.451179\pi$
$179$	4.08797			0.305549			0.152775	+	0.988261i	$0.451179\pi$
$180$		0			0
$181$	−15.9509			−1.18562			−0.592810	−	0.805342i	$-0.701981\pi$
$181$	−15.9509			−1.18562			−0.592810	+	0.805342i	$0.701981\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−9.37400	−	16.2363i	−0.689191	−	1.19371i
$186$		0			0
$187$	3.12077	+	1.80178i	0.228213	+	0.131759i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4.96091	−	8.59255i	0.358959	−	0.621735i	−0.628828	−	0.777544i	$-0.716465\pi$
$191$	4.96091	−	8.59255i	0.358959	−	0.621735i	0.987787	+	0.155809i	$0.0497985\pi$
$192$		0			0
$193$	−13.2392	+	7.64366i	−0.952979	+	0.550203i	−0.894005	−	0.448057i	$-0.852116\pi$
$193$	−13.2392	+	7.64366i	−0.952979	+	0.550203i	−0.0589743	+	0.998260i	$0.518783\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			1.23481i			0.0879768i	0.999032	+	0.0439884i	$0.0140065\pi$
$197$			1.23481i			0.0879768i	−0.999032	+	0.0439884i	$0.985994\pi$
$198$		0			0
$199$	−5.64801			−0.400377			−0.200188	−	0.979757i	$-0.564155\pi$
$199$	−5.64801			−0.400377			−0.200188	+	0.979757i	$0.564155\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−4.28529	+	2.47411i	−0.300768	+	0.173649i
$204$		0			0
$205$	4.92711	−	8.53401i	0.344125	−	0.596041i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	11.0457	−	19.1317i	0.764046	−	1.32337i
$210$		0			0
$211$	−7.62853	−	13.2130i	−0.525170	−	0.909621i	−0.999570	−	0.0293117i	$-0.990668\pi$
$211$	−7.62853	−	13.2130i	−0.525170	−	0.909621i	0.474401	−	0.880309i	$-0.342665\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			14.6158i			0.996787i
$216$		0			0
$217$	35.5592			2.41392
$218$		0			0
$219$		0			0
$220$		0			0
$221$	0.480385	−	3.83545i	0.0323142	−	0.258000i
$222$		0			0
$223$	13.9167	+	8.03480i	0.931930	+	0.538050i	0.887422	−	0.460959i	$-0.152494\pi$
$223$	13.9167	+	8.03480i	0.931930	+	0.538050i	0.0445087	+	0.999009i	$0.485828\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	4.18451	+	2.41593i	0.277735	+	0.160351i	0.632398	−	0.774644i	$-0.282071\pi$
$227$	4.18451	+	2.41593i	0.277735	+	0.160351i	−0.354662	+	0.934994i	$0.615404\pi$
$228$		0			0
$229$	15.6874	−	9.05711i	1.03665	−	0.598510i	0.117768	−	0.993041i	$-0.462426\pi$
$229$	15.6874	−	9.05711i	1.03665	−	0.598510i	0.918883	+	0.394531i	$0.129093\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.37594			0.0901411			0.0450705	−	0.998984i	$-0.485649\pi$
$233$	1.37594			0.0901411			0.0450705	+	0.998984i	$0.485649\pi$
$234$		0			0
$235$	39.5731			2.58146
$236$		0			0
$237$		0			0
$238$		0			0
$239$	4.84277	−	2.79597i	0.313253	−	0.180856i	−0.335128	−	0.942172i	$-0.608780\pi$
$239$	4.84277	−	2.79597i	0.313253	−	0.180856i	0.648381	+	0.761316i	$0.275446\pi$
$240$		0			0
$241$	−10.4450	−	6.03041i	−0.672820	−	0.388453i	0.124324	−	0.992242i	$-0.460324\pi$
$241$	−10.4450	−	6.03041i	−0.672820	−	0.388453i	−0.797144	+	0.603789i	$0.793657\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−54.9458	−	31.7230i	−3.51036	−	2.02671i
$246$		0			0
$247$	−23.5130	−	2.94497i	−1.49610	−	0.187384i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−9.90201			−0.625009			−0.312505	−	0.949916i	$-0.601168\pi$
$251$	−9.90201			−0.625009			−0.312505	+	0.949916i	$0.601168\pi$
$252$		0			0
$253$		−	0.287338i		−	0.0180648i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	0.0748821	+	0.129700i	0.00467102	+	0.00809044i	0.868351	−	0.495949i	$-0.165180\pi$
$257$	0.0748821	+	0.129700i	0.00467102	+	0.00809044i	−0.863680	+	0.504040i	$0.831846\pi$
$258$		0			0
$259$	11.4710	−	19.8684i	0.712775	−	1.23456i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3.18077	−	5.50926i	0.196135	−	0.339716i	−0.751137	−	0.660146i	$-0.770494\pi$
$263$	3.18077	−	5.50926i	0.196135	−	0.339716i	0.947272	+	0.320431i	$0.103828\pi$
$264$		0			0
$265$	−19.7577	+	11.4071i	−1.21371	+	0.700734i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−8.43403			−0.514232			−0.257116	−	0.966381i	$-0.582772\pi$
$269$	−8.43403			−0.514232			−0.257116	+	0.966381i	$0.582772\pi$
$270$		0			0
$271$		−	3.97794i		−	0.241643i	−0.992674	−	0.120821i	$-0.961447\pi$
$271$		−	3.97794i		−	0.241643i	0.992674	−	0.120821i	$-0.0385528\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	30.4277	−	17.5674i	1.83486	−	1.05936i
$276$		0			0
$277$	−0.463857	+	0.803425i	−0.0278705	+	0.0482731i	−0.879624	−	0.475669i	$-0.842206\pi$
$277$	−0.463857	+	0.803425i	−0.0278705	+	0.0482731i	0.851754	+	0.523942i	$0.175539\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	21.1864	+	12.2320i	1.26387	+	0.729698i	0.973822	−	0.227312i	$-0.0729938\pi$
$281$	21.1864	+	12.2320i	1.26387	+	0.729698i	0.290053	+	0.957011i	$0.406327\pi$
$282$		0			0
$283$	4.48336	+	7.76541i	0.266508	+	0.461606i	0.967958	−	0.251114i	$-0.0807968\pi$
$283$	4.48336	+	7.76541i	0.266508	+	0.461606i	−0.701450	+	0.712719i	$0.747463\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	12.0587			0.711801
$288$		0			0
$289$	−15.8507			−0.932392
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.734208	+	0.423895i	−0.0428929	+	0.0247642i	−0.521293	−	0.853378i	$-0.674550\pi$
$293$	−0.734208	+	0.423895i	−0.0428929	+	0.0247642i	0.478400	+	0.878142i	$0.341217\pi$
$294$		0			0
$295$	23.6092	−	40.8923i	1.37458	−	2.38084i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.284008	+	0.119742i	−0.0164246	+	0.00692485i
$300$		0			0
$301$	−15.4892	+	8.94270i	−0.892783	+	0.515449i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	18.1477i		−	1.03914i
$306$		0			0
$307$		−	22.2228i		−	1.26832i	−0.773201	−	0.634162i	$-0.781345\pi$
$307$		−	22.2228i		−	1.26832i	0.773201	−	0.634162i	$-0.218655\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	13.5320	+	23.4382i	0.767331	+	1.32906i	0.939005	+	0.343903i	$0.111749\pi$
$311$	13.5320	+	23.4382i	0.767331	+	1.32906i	−0.171674	+	0.985154i	$0.554918\pi$
$312$		0			0
$313$	−10.6275	+	18.4074i	−0.600703	+	1.04045i	0.392012	+	0.919960i	$0.371779\pi$
$313$	−10.6275	+	18.4074i	−0.600703	+	1.04045i	−0.992715	+	0.120488i	$0.961554\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	9.41447	+	5.43545i	0.528770	+	0.305285i	0.740515	−	0.672040i	$-0.234582\pi$
$317$	9.41447	+	5.43545i	0.528770	+	0.305285i	−0.211746	+	0.977325i	$0.567915\pi$
$318$		0			0
$319$	2.99437	−	1.72880i	0.167653	−	0.0967943i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	7.04597i		−	0.392048i
$324$		0			0
$325$	−30.0439	−	22.7542i	−1.66654	−	1.26218i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	24.2129	+	41.9380i	1.33490	+	2.31212i
$330$		0			0
$331$	6.42626	+	3.71020i	0.353219	+	0.203931i	0.666102	−	0.745860i	$-0.267961\pi$
$331$	6.42626	+	3.71020i	0.353219	+	0.203931i	−0.312883	+	0.949792i	$0.601295\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5.57361	+	9.65377i	−0.304519	+	0.527442i
$336$		0			0
$337$	10.1784	+	17.6296i	0.554455	+	0.960345i	0.997946	+	0.0640656i	$0.0204067\pi$
$337$	10.1784	+	17.6296i	0.554455	+	0.960345i	−0.443490	+	0.896279i	$0.646260\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−24.8472			−1.34555
$342$		0			0
$343$		−	43.9665i		−	2.37397i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	4.83922	+	8.38177i	0.259783	+	0.449957i	0.966184	−	0.257855i	$-0.0830157\pi$
$347$	4.83922	+	8.38177i	0.259783	+	0.449957i	−0.706401	+	0.707812i	$0.749682\pi$
$348$		0			0
$349$	20.3188	+	11.7311i	1.08764	+	0.627949i	0.932947	−	0.360014i	$-0.117228\pi$
$349$	20.3188	+	11.7311i	1.08764	+	0.627949i	0.154693	+	0.987963i	$0.450561\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	9.12721	+	5.26960i	0.485792	+	0.280472i	0.722827	−	0.691029i	$-0.242842\pi$
$353$	9.12721	+	5.26960i	0.485792	+	0.280472i	−0.237035	+	0.971501i	$0.576176\pi$
$354$		0			0
$355$	2.68154	+	4.64456i	0.142321	+	0.246508i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	33.8494i		−	1.78650i	−0.449557	−	0.893252i	$-0.648418\pi$
$359$		−	33.8494i		−	1.78650i	0.449557	−	0.893252i	$-0.351582\pi$
$360$		0			0
$361$	−24.1949			−1.27341
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−5.26320	−	9.11613i	−0.275489	−	0.477160i
$366$		0			0
$367$	−0.752021	+	1.30254i	−0.0392552	+	0.0679920i	−0.884985	−	0.465619i	$-0.845832\pi$
$367$	−0.752021	+	1.30254i	−0.0392552	+	0.0679920i	0.845730	+	0.533611i	$0.179165\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−24.1776	−	13.9590i	−1.25524	−	0.724713i
$372$		0			0
$373$	10.6689	+	18.4790i	0.552414	+	0.956808i	0.998100	+	0.0616191i	$0.0196264\pi$
$373$	10.6689	+	18.4790i	0.552414	+	0.956808i	−0.445686	+	0.895189i	$0.647040\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2.95660	−	2.23923i	−0.152273	−	0.115326i
$378$		0			0
$379$			13.1930i			0.677681i	0.940844	+	0.338840i	$0.110035\pi$
$379$			13.1930i			0.677681i	−0.940844	+	0.338840i	$0.889965\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	6.64542	−	3.83673i	0.339565	−	0.196048i	−0.320515	−	0.947244i	$-0.603856\pi$
$383$	6.64542	−	3.83673i	0.339565	−	0.196048i	0.660080	+	0.751196i	$0.270523\pi$
$384$		0			0
$385$	55.0454	+	31.7805i	2.80537	+	1.61968i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.23245	−	2.13467i	0.0624877	−	0.108232i	−0.833089	−	0.553139i	$-0.813430\pi$
$389$	1.23245	−	2.13467i	0.0624877	−	0.108232i	0.895577	+	0.444907i	$0.146763\pi$
$390$		0			0
$391$	−0.0458228	−	0.0793675i	−0.00231736	−	0.00401379i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			6.86321i			0.345325i
$396$		0			0
$397$			30.6817i			1.53987i	0.638121	+	0.769936i	$0.279712\pi$
$397$			30.6817i			1.53987i	−0.638121	+	0.769936i	$0.720288\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−28.8517	+	16.6575i	−1.44078	+	0.831837i	−0.997902	−	0.0647435i	$-0.979377\pi$
$401$	−28.8517	+	16.6575i	−1.44078	+	0.831837i	−0.442881	+	0.896580i	$0.646044\pi$
$402$		0			0
$403$	10.3545	+	24.5593i	0.515795	+	1.22338i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−8.01545	+	13.8832i	−0.397311	+	0.688163i
$408$		0			0
$409$	29.0463	−	16.7699i	1.43625	−	0.829219i	0.438662	−	0.898652i	$-0.355452\pi$
$409$	29.0463	−	16.7699i	1.43625	−	0.829219i	0.997587	+	0.0694331i	$0.0221190\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	57.7814			2.84324
$414$		0			0
$415$	−55.3930			−2.71914
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2.84913	+	4.93483i	0.139189	+	0.241082i	0.927190	−	0.374592i	$-0.122217\pi$
$419$	2.84913	+	4.93483i	0.139189	+	0.241082i	−0.788001	+	0.615674i	$0.788884\pi$
$420$		0			0
$421$	−18.0959	−	10.4477i	−0.881939	−	0.509188i	−0.0106421	−	0.999943i	$-0.503388\pi$
$421$	−18.0959	−	10.4477i	−0.881939	−	0.509188i	−0.871297	+	0.490755i	$0.836721\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	5.60308	−	9.70482i	0.271789	−	0.470753i
$426$		0			0
$427$	19.2322	−	11.1037i	0.930714	−	0.537348i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			26.6529i			1.28382i	0.766778	+	0.641912i	$0.221859\pi$
$431$			26.6529i			1.28382i	−0.766778	+	0.641912i	$0.778141\pi$
$432$		0			0
$433$	29.5411			1.41965			0.709827	−	0.704376i	$-0.248773\pi$
$433$	29.5411			1.41965			0.709827	+	0.704376i	$0.248773\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−0.486557	+	0.280914i	−0.0232752	+	0.0134379i
$438$		0			0
$439$	3.60721	−	6.24786i	0.172163	−	0.298194i	−0.767013	−	0.641631i	$-0.778258\pi$
$439$	3.60721	−	6.24786i	0.172163	−	0.298194i	0.939176	+	0.343437i	$0.111591\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4.26300	−	7.38373i	0.202541	−	0.350812i	−0.746805	−	0.665043i	$-0.768413\pi$
$443$	4.26300	−	7.38373i	0.202541	−	0.350812i	0.949347	+	0.314231i	$0.101747\pi$
$444$		0			0
$445$	−18.3865	−	31.8464i	−0.871604	−	1.50966i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			14.0039i			0.660883i	0.943826	+	0.330441i	$0.107198\pi$
$449$			14.0039i			0.660883i	−0.943826	+	0.330441i	$0.892802\pi$
$450$		0			0
$451$	−8.42608			−0.396769
$452$		0			0
$453$		0			0
$454$		0			0
$455$	8.47323	−	67.6512i	0.397231	−	3.17154i
$456$		0			0
$457$	29.6357	+	17.1102i	1.38630	+	0.800380i	0.992896	−	0.118987i	$-0.0379646\pi$
$457$	29.6357	+	17.1102i	1.38630	+	0.800380i	0.393402	+	0.919366i	$0.371298\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	7.56297	+	4.36648i	0.352242	+	0.203367i	0.665672	−	0.746244i	$-0.268145\pi$
$461$	7.56297	+	4.36648i	0.352242	+	0.203367i	−0.313430	+	0.949611i	$0.601478\pi$
$462$		0			0
$463$	0.0993345	−	0.0573508i	0.00461646	−	0.00266532i	−0.497690	−	0.867355i	$-0.665818\pi$
$463$	0.0993345	−	0.0573508i	0.00461646	−	0.00266532i	0.502306	+	0.864690i	$0.332485\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−42.6123			−1.97186			−0.985932	−	0.167147i	$-0.946544\pi$
$467$	−42.6123			−1.97186			−0.985932	+	0.167147i	$0.946544\pi$
$468$		0			0
$469$	−13.6409			−0.629879
$470$		0			0
$471$		0			0
$472$		0			0
$473$	10.8232	−	6.24877i	0.497650	−	0.287319i
$474$		0			0
$475$	−59.4948	−	34.3493i	−2.72981	−	1.57606i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	26.1199	+	15.0803i	1.19345	+	0.689037i	0.959086	−	0.283113i	$-0.0913672\pi$
$479$	26.1199	+	15.0803i	1.19345	+	0.689037i	0.234360	+	0.972150i	$0.424701\pi$
$480$		0			0
$481$	17.0625	+	2.13706i	0.777984	+	0.0974415i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−30.1459			−1.36885
$486$		0			0
$487$			1.98683i			0.0900319i	0.998986	+	0.0450160i	$0.0143339\pi$
$487$			1.98683i			0.0900319i	−0.998986	+	0.0450160i	$0.985666\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	9.16415	+	15.8728i	0.413572	+	0.716328i	0.995277	−	0.0970714i	$-0.0309475\pi$
$491$	9.16415	+	15.8728i	0.413572	+	0.716328i	−0.581705	+	0.813400i	$0.697614\pi$
$492$		0			0
$493$	0.551396	−	0.955045i	0.0248336	−	0.0430131i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−3.28141	+	5.68357i	−0.147192	+	0.254943i
$498$		0			0
$499$	1.45294	−	0.838857i	0.0650427	−	0.0375524i	−0.467126	−	0.884191i	$-0.654711\pi$
$499$	1.45294	−	0.838857i	0.0650427	−	0.0375524i	0.532169	+	0.846638i	$0.321377\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−3.47253			−0.154833			−0.0774163	−	0.996999i	$-0.524667\pi$
$503$	−3.47253			−0.154833			−0.0774163	+	0.996999i	$0.524667\pi$
$504$		0			0
$505$		−	58.4973i		−	2.60310i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−16.0585	+	9.27140i	−0.711782	+	0.410948i	−0.811720	−	0.584046i	$-0.801469\pi$
$509$	−16.0585	+	9.27140i	−0.711782	+	0.410948i	0.0999385	+	0.994994i	$0.468135\pi$
$510$		0			0
$511$	6.44061	−	11.1555i	0.284916	−	0.493489i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−64.4645	−	37.2186i	−2.84065	−	1.64005i
$516$		0			0
$517$	−16.9189	−	29.3044i	−0.744093	−	1.28881i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−25.7368			−1.12755			−0.563775	−	0.825929i	$-0.690651\pi$
$521$	−25.7368			−1.12755			−0.563775	+	0.825929i	$0.690651\pi$
$522$		0			0
$523$	−6.85993			−0.299964			−0.149982	−	0.988689i	$-0.547922\pi$
$523$	−6.85993			−0.299964			−0.149982	+	0.988689i	$0.547922\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6.86320	+	3.96247i	−0.298966	+	0.172608i
$528$		0			0
$529$	11.4963	−	19.9123i	0.499841	−	0.865750i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3.51138	+	8.32842i	0.152095	+	0.360744i
$534$		0			0
$535$	50.0422	−	28.8919i	2.16351	−	1.24910i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			54.2509i			2.33675i
$540$		0			0
$541$			19.9869i			0.859306i	0.902994	+	0.429653i	$0.141364\pi$
$541$			19.9869i			0.859306i	−0.902994	+	0.429653i	$0.858636\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	8.64077	+	14.9662i	0.370130	+	0.641084i
$546$		0			0
$547$	−7.24562	+	12.5498i	−0.309800	+	0.536590i	−0.978319	−	0.207106i	$-0.933596\pi$
$547$	−7.24562	+	12.5498i	−0.309800	+	0.536590i	0.668518	+	0.743696i	$0.266929\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−5.85485	−	3.38030i	−0.249425	−	0.144005i
$552$		0			0
$553$	−7.27336	+	4.19927i	−0.309295	+	0.178571i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	12.8651i		−	0.545110i	−0.962140	−	0.272555i	$-0.912131\pi$
$557$		−	12.8651i		−	0.545110i	0.962140	−	0.272555i	$-0.0878687\pi$
$558$		0			0
$559$	−10.6867	−	8.09371i	−0.451998	−	0.342328i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2.23678	+	3.87421i	0.0942689	+	0.163278i	0.909303	−	0.416134i	$-0.136615\pi$
$563$	2.23678	+	3.87421i	0.0942689	+	0.163278i	−0.815034	+	0.579413i	$0.803282\pi$
$564$		0			0
$565$	−8.38802	−	4.84283i	−0.352887	−	0.203739i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−21.7045	+	37.5933i	−0.909900	+	1.57599i	−0.0956977	+	0.995410i	$0.530508\pi$
$569$	−21.7045	+	37.5933i	−0.909900	+	1.57599i	−0.814202	+	0.580582i	$0.802825\pi$
$570$		0			0
$571$	−9.96107	−	17.2531i	−0.416858	−	0.722019i	0.578764	−	0.815495i	$-0.303535\pi$
$571$	−9.96107	−	17.2531i	−0.416858	−	0.722019i	−0.995621	+	0.0934764i	$0.970202\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.893552			−0.0372637
$576$		0			0
$577$			40.2426i			1.67532i	0.546190	+	0.837661i	$0.316078\pi$
$577$			40.2426i			1.67532i	−0.546190	+	0.837661i	$0.683922\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−33.8924	−	58.7034i	−1.40609	−	2.43543i
$582$		0			0
$583$	16.8943	+	9.75392i	0.699690	+	0.403966i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−5.59848	−	3.23229i	−0.231074	−	0.133411i	0.379993	−	0.924989i	$-0.375926\pi$
$587$	−5.59848	−	3.23229i	−0.231074	−	0.133411i	−0.611068	+	0.791578i	$0.709260\pi$
$588$		0			0
$589$	24.2917	+	42.0744i	1.00092	+	1.73365i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	20.8574i		−	0.856510i	−0.903658	−	0.428255i	$-0.859129\pi$
$593$		−	20.8574i		−	0.856510i	0.903658	−	0.428255i	$-0.140871\pi$
$594$		0			0
$595$	20.2726			0.831094
$596$		0			0
$597$		0			0
$598$		0			0
$599$	3.67196	+	6.36003i	0.150032	+	0.259864i	0.931239	−	0.364409i	$-0.118729\pi$
$599$	3.67196	+	6.36003i	0.150032	+	0.259864i	−0.781207	+	0.624272i	$0.785396\pi$
$600$		0			0
$601$	5.37782	−	9.31465i	0.219366	−	0.379953i	−0.735248	−	0.677798i	$-0.762935\pi$
$601$	5.37782	−	9.31465i	0.219366	−	0.379953i	0.954614	+	0.297845i	$0.0962679\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.01549	−	0.586291i	−0.0412854	−	0.0238361i
$606$		0			0
$607$	8.29221	+	14.3625i	0.336570	+	0.582957i	0.983785	−	0.179351i	$-0.0573997\pi$
$607$	8.29221	+	14.3625i	0.336570	+	0.582957i	−0.647215	+	0.762308i	$0.724066\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−21.9142	+	28.9348i	−0.886555	+	1.17058i
$612$		0			0
$613$		−	4.54150i		−	0.183429i	−0.995785	−	0.0917147i	$-0.970765\pi$
$613$		−	4.54150i		−	0.183429i	0.995785	−	0.0917147i	$-0.0292348\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.149964	−	0.0865815i	0.00603731	−	0.00348564i	−0.496978	−	0.867763i	$-0.665557\pi$
$617$	0.149964	−	0.0865815i	0.00603731	−	0.00348564i	0.503016	+	0.864277i	$0.332224\pi$
$618$		0			0
$619$	−23.6778	−	13.6704i	−0.951692	−	0.549460i	−0.0580858	−	0.998312i	$-0.518500\pi$
$619$	−23.6778	−	13.6704i	−0.951692	−	0.549460i	−0.893606	+	0.448852i	$0.851833\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	22.4997	−	38.9706i	0.901431	−	1.56132i
$624$		0			0
$625$	−15.9985	−	27.7101i	−0.639938	−	1.10841i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			5.11301i			0.203869i
$630$		0			0
$631$		−	37.8137i		−	1.50534i	−0.658397	−	0.752670i	$-0.728765\pi$
$631$		−	37.8137i		−	1.50534i	0.658397	−	0.752670i	$-0.271235\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−58.6021	+	33.8339i	−2.32555	+	1.34266i
$636$		0			0
$637$	53.6221	−	22.6078i	2.12459	−	0.895755i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	3.77702	−	6.54200i	0.149184	−	0.258393i	−0.781742	−	0.623601i	$-0.785669\pi$
$641$	3.77702	−	6.54200i	0.149184	−	0.258393i	0.930926	+	0.365208i	$0.119002\pi$
$642$		0			0
$643$	24.2155	−	13.9808i	0.954964	−	0.551349i	0.0603447	−	0.998178i	$-0.480780\pi$
$643$	24.2155	−	13.9808i	0.954964	−	0.551349i	0.894620	+	0.446829i	$0.147447\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	5.98842			0.235429			0.117715	−	0.993047i	$-0.462443\pi$
$647$	5.98842			0.235429			0.117715	+	0.993047i	$0.462443\pi$
$648$		0			0
$649$	−40.3751			−1.58486
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−8.90801	−	15.4291i	−0.348597	−	0.603789i	0.637403	−	0.770531i	$-0.280009\pi$
$653$	−8.90801	−	15.4291i	−0.348597	−	0.603789i	−0.986001	+	0.166742i	$0.946675\pi$
$654$		0			0
$655$	28.7857	+	16.6194i	1.12475	+	0.649374i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−23.8222	+	41.2613i	−0.927982	+	1.60731i	−0.141288	+	0.989969i	$0.545124\pi$
$659$	−23.8222	+	41.2613i	−0.927982	+	1.60731i	−0.786694	+	0.617343i	$0.788209\pi$
$660$		0			0
$661$	−4.59633	+	2.65370i	−0.178777	+	0.103217i	−0.586718	−	0.809791i	$-0.699580\pi$
$661$	−4.59633	+	2.65370i	−0.178777	+	0.103217i	0.407941	+	0.913008i	$0.366247\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	124.280i		−	4.81936i
$666$		0			0
$667$	−0.0879339			−0.00340481
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−13.4387	+	7.75881i	−0.518794	+	0.299526i
$672$		0			0
$673$	13.9628	−	24.1842i	0.538225	−	0.932233i	−0.460775	−	0.887517i	$-0.652428\pi$
$673$	13.9628	−	24.1842i	0.538225	−	0.932233i	0.999000	−	0.0447160i	$-0.0142383\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	9.19203	−	15.9211i	0.353278	−	0.611896i	−0.633544	−	0.773707i	$-0.718400\pi$
$677$	9.19203	−	15.9211i	0.353278	−	0.611896i	0.986822	+	0.161811i	$0.0517336\pi$
$678$		0			0
$679$	−18.4449	−	31.9474i	−0.707848	−	1.22603i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	27.5533i		−	1.05430i	−0.849773	−	0.527149i	$-0.823261\pi$
$683$		−	27.5533i		−	1.05430i	0.849773	−	0.527149i	$-0.176739\pi$
$684$		0			0
$685$	−71.6609			−2.73802
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.60056	−	20.7632i	0.0990736	−	0.791015i
$690$		0			0
$691$	18.0882	+	10.4432i	0.688109	+	0.397280i	0.802903	−	0.596110i	$-0.203288\pi$
$691$	18.0882	+	10.4432i	0.688109	+	0.397280i	−0.114795	+	0.993389i	$0.536621\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−39.2188	−	22.6430i	−1.48765	−	0.858897i
$696$		0			0
$697$	−2.32742	+	1.34374i	−0.0881573	+	0.0508976i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−39.0518			−1.47496			−0.737482	−	0.675366i	$-0.763986\pi$
$701$	−39.0518			−1.47496			−0.737482	+	0.675366i	$0.763986\pi$
$702$		0			0
$703$	31.3450			1.18220
$704$		0			0
$705$		0			0
$706$		0			0
$707$	61.9932	−	35.7918i	2.33149	−	1.34609i
$708$		0			0
$709$	2.00983	+	1.16037i	0.0754807	+	0.0435788i	0.537265	−	0.843413i	$-0.319457\pi$
$709$	2.00983	+	1.16037i	0.0754807	+	0.0435788i	−0.461785	+	0.886992i	$0.652791\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.547255	+	0.315958i	0.0204949	+	0.0118327i
$714$		0			0
$715$	−5.92072	+	47.2717i	−0.221423	+	1.76786i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	7.04587			0.262767			0.131383	−	0.991332i	$-0.458058\pi$
$719$	7.04587			0.262767			0.131383	+	0.991332i	$0.458058\pi$
$720$		0			0
$721$		−	91.0893i		−	3.39234i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−5.37615	−	9.31176i	−0.199665	−	0.345830i
$726$		0			0
$727$	−3.57535	+	6.19269i	−0.132602	+	0.229674i	−0.924679	−	0.380748i	$-0.875667\pi$
$727$	−3.57535	+	6.19269i	−0.132602	+	0.229674i	0.792077	+	0.610422i	$0.209000\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.99302	−	3.45202i	0.0737147	−	0.127678i
$732$		0			0
$733$	7.90471	−	4.56378i	0.291967	−	0.168567i	−0.346862	−	0.937916i	$-0.612753\pi$
$733$	7.90471	−	4.56378i	0.291967	−	0.168567i	0.638829	+	0.769349i	$0.279419\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	9.53168			0.351104
$738$		0			0
$739$			44.0839i			1.62165i	0.585286	+	0.810827i	$0.300982\pi$
$739$			44.0839i			1.62165i	−0.585286	+	0.810827i	$0.699018\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	22.0405	−	12.7251i	0.808586	−	0.466838i	−0.0378784	−	0.999282i	$-0.512060\pi$
$743$	22.0405	−	12.7251i	0.808586	−	0.466838i	0.846465	+	0.532445i	$0.178727\pi$
$744$		0			0
$745$	8.56093	−	14.8280i	0.313648	−	0.543255i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	61.2369	+	35.3551i	2.23755	+	1.29185i
$750$		0			0
$751$	1.96715	+	3.40721i	0.0717824	+	0.124331i	0.899683	−	0.436545i	$-0.143798\pi$
$751$	1.96715	+	3.40721i	0.0717824	+	0.124331i	−0.827900	+	0.560876i	$0.810465\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−16.8320			−0.612579
$756$		0			0
$757$	33.9336			1.23334			0.616669	−	0.787223i	$-0.288482\pi$
$757$	33.9336			1.23334			0.616669	+	0.787223i	$0.288482\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−9.27065	+	5.35241i	−0.336061	+	0.194025i	−0.658529	−	0.752556i	$-0.728821\pi$
$761$	−9.27065	+	5.35241i	−0.336061	+	0.194025i	0.322468	+	0.946580i	$0.395488\pi$
$762$		0			0
$763$	−10.5738	+	18.3143i	−0.382796	+	0.663022i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	16.8254	+	39.9072i	0.607531	+	1.44097i
$768$		0			0
$769$	11.8513	−	6.84235i	0.427369	−	0.246741i	−0.270856	−	0.962620i	$-0.587307\pi$
$769$	11.8513	−	6.84235i	0.427369	−	0.246741i	0.698225	+	0.715878i	$0.253974\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	11.4346i		−	0.411273i	−0.978628	−	0.205637i	$-0.934074\pi$
$773$		−	11.4346i		−	0.411273i	0.978628	−	0.205637i	$-0.0659265\pi$
$774$		0			0
$775$			77.2688i			2.77558i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	8.23769	+	14.2681i	0.295146	+	0.511207i
$780$		0			0
$781$	2.29291	−	3.97143i	0.0820467	−	0.142109i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	0.429056	+	0.247715i	0.0153137	+	0.00884134i
$786$		0			0
$787$	−38.5078	+	22.2325i	−1.37265	+	0.792503i	−0.991262	−	0.131910i	$-0.957889\pi$
$787$	−38.5078	+	22.2325i	−1.37265	+	0.792503i	−0.381393	+	0.924413i	$0.624556\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	11.8524i		−	0.421423i
$792$		0			0
$793$	13.2691	+	10.0496i	0.471201	+	0.356872i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−4.39354	−	7.60984i	−0.155627	−	0.269554i	0.777660	−	0.628685i	$-0.216407\pi$
$797$	−4.39354	−	7.60984i	−0.155627	−	0.269554i	−0.933287	+	0.359131i	$0.883073\pi$
$798$		0			0
$799$	−9.34656	−	5.39624i	−0.330657	−	0.190905i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−4.50042	+	7.79495i	−0.158816	+	0.275078i
$804$		0			0
$805$	−0.808242	−	1.39992i	−0.0284868	−	0.0493406i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	51.4550			1.80906			0.904530	−	0.426410i	$-0.140222\pi$
$809$	51.4550			1.80906			0.904530	+	0.426410i	$0.140222\pi$
$810$		0			0
$811$			0.149585i			0.00525263i	0.999997	+	0.00262632i	$0.000835984\pi$
$811$			0.149585i			0.00525263i	−0.999997	+	0.00262632i	$0.999164\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	32.6848	+	56.6118i	1.14490	+	1.98302i
$816$		0			0
$817$	−21.1624	−	12.2181i	−0.740378	−	0.427458i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−3.57618	−	2.06471i	−0.124809	−	0.0720588i	0.436295	−	0.899803i	$-0.356290\pi$
$821$	−3.57618	−	2.06471i	−0.124809	−	0.0720588i	−0.561105	+	0.827745i	$0.689624\pi$
$822$		0			0
$823$	−12.2798	−	21.2693i	−0.428047	−	0.741400i	0.568652	−	0.822578i	$-0.307465\pi$
$823$	−12.2798	−	21.2693i	−0.428047	−	0.741400i	−0.996700	+	0.0811784i	$0.974132\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	41.7582i		−	1.45207i	−0.687656	−	0.726037i	$-0.741360\pi$
$827$		−	41.7582i		−	1.45207i	0.687656	−	0.726037i	$-0.258640\pi$
$828$		0			0
$829$	30.9427			1.07468			0.537342	−	0.843365i	$-0.319429\pi$
$829$	30.9427			1.07468			0.537342	+	0.843365i	$0.319429\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	8.65158	+	14.9850i	0.299759	+	0.519198i
$834$		0			0
$835$	−4.07979	+	7.06641i	−0.141187	+	0.244543i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−11.2119	−	6.47318i	−0.387077	−	0.223479i	0.293816	−	0.955862i	$-0.405075\pi$
$839$	−11.2119	−	6.47318i	−0.387077	−	0.223479i	−0.680893	+	0.732383i	$0.738408\pi$
$840$		0			0
$841$	13.9709	+	24.1984i	0.481756	+	0.834427i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	49.1912	−	13.8473i	1.69223	−	0.476362i
$846$		0			0
$847$		−	1.43490i		−	0.0493036i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	0.353077	−	0.203849i	0.0121033	−	0.00698786i
$852$		0			0
$853$	20.6627	+	11.9296i	0.707476	+	0.408462i	0.810126	−	0.586256i	$-0.199399\pi$
$853$	20.6627	+	11.9296i	0.707476	+	0.408462i	−0.102650	+	0.994718i	$0.532732\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−8.96321	+	15.5247i	−0.306177	+	0.530315i	−0.977523	−	0.210830i	$-0.932383\pi$
$857$	−8.96321	+	15.5247i	−0.306177	+	0.530315i	0.671345	+	0.741145i	$0.265717\pi$
$858$		0			0
$859$	−7.61224	−	13.1848i	−0.259726	−	0.449859i	0.706442	−	0.707771i	$-0.250299\pi$
$859$	−7.61224	−	13.1848i	−0.259726	−	0.449859i	−0.966169	+	0.257911i	$0.916966\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1.21450i		−	0.0413422i	−0.999786	−	0.0206711i	$-0.993420\pi$
$863$		−	1.21450i		−	0.0413422i	0.999786	−	0.0206711i	$-0.00658029\pi$
$864$		0			0
$865$		−	56.4312i		−	1.91872i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	5.08230	−	2.93427i	0.172405	−	0.0995382i
$870$		0			0
$871$	−3.97211	−	9.42121i	−0.134590	−	0.319225i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	51.5553	−	89.2964i	1.74289	−	3.01877i
$876$		0			0
$877$	−35.9366	+	20.7480i	−1.21349	+	0.700611i	−0.963519	−	0.267641i	$-0.913756\pi$
$877$	−35.9366	+	20.7480i	−1.21349	+	0.700611i	−0.249975	+	0.968252i	$0.580422\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	20.3311			0.684972			0.342486	−	0.939523i	$-0.388731\pi$
$881$	20.3311			0.684972			0.342486	+	0.939523i	$0.388731\pi$
$882$		0			0
$883$	28.4196			0.956395			0.478197	−	0.878252i	$-0.341290\pi$
$883$	28.4196			0.956395			0.478197	+	0.878252i	$0.341290\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−5.94448	−	10.2961i	−0.199596	−	0.345710i	0.748802	−	0.662794i	$-0.230630\pi$
$887$	−5.94448	−	10.2961i	−0.199596	−	0.345710i	−0.948397	+	0.317084i	$0.897296\pi$
$888$		0			0
$889$	−71.7118	−	41.4028i	−2.40514	−	1.38861i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−33.0813	+	57.2985i	−1.10702	+	1.91742i
$894$		0			0
$895$	−13.9169	+	8.03491i	−0.465190	+	0.268578i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			7.60397i			0.253607i
$900$		0			0
$901$	6.22196			0.207284
$902$		0			0
$903$		0			0
$904$		0			0
$905$	54.3024	−	31.3515i	1.80507	−	1.04216i
$906$		0			0
$907$	−7.14235	+	12.3709i	−0.237158	+	0.410770i	−0.959898	−	0.280351i	$-0.909549\pi$
$907$	−7.14235	+	12.3709i	−0.237158	+	0.410770i	0.722740	+	0.691120i	$0.242883\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−23.5622	+	40.8109i	−0.780649	+	1.35212i	0.150915	+	0.988547i	$0.451778\pi$
$911$	−23.5622	+	40.8109i	−0.780649	+	1.35212i	−0.931564	+	0.363578i	$0.881555\pi$
$912$		0			0
$913$	23.6825	+	41.0193i	0.783777	+	1.35754i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			40.6745i			1.34319i
$918$		0			0
$919$	−44.6273			−1.47212			−0.736060	−	0.676916i	$-0.763316\pi$
$919$	−44.6273			−1.47212			−0.736060	+	0.676916i	$0.763316\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−4.88092	−	0.611329i	−0.160658	−	0.0201222i
$924$		0			0
$925$	43.1732	+	24.9261i	1.41953	+	0.819565i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−11.2761	−	6.51027i	−0.369958	−	0.213595i	0.303482	−	0.952837i	$-0.401851\pi$
$929$	−11.2761	−	6.51027i	−0.369958	−	0.213595i	−0.673440	+	0.739242i	$0.735184\pi$
$930$		0			0
$931$	91.8644	−	53.0379i	3.01073	−	1.73825i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−14.1656			−0.463264
$936$		0			0
$937$	−0.395642			−0.0129251			−0.00646253	−	0.999979i	$-0.502057\pi$
$937$	−0.395642			−0.0129251			−0.00646253	+	0.999979i	$0.502057\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	29.0237	−	16.7568i	0.946145	−	0.546257i	0.0542640	−	0.998527i	$-0.482719\pi$
$941$	29.0237	−	16.7568i	0.946145	−	0.546257i	0.891881	+	0.452269i	$0.149385\pi$
$942$		0			0
$943$	0.185583	+	0.107146i	0.00604340	+	0.00348916i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−22.1028	−	12.7611i	−0.718246	−	0.414679i	0.0958609	−	0.995395i	$-0.469440\pi$
$947$	−22.1028	−	12.7611i	−0.718246	−	0.414679i	−0.814107	+	0.580715i	$0.802773\pi$
$948$		0			0
$949$	9.58006	+	1.19989i	0.310982	+	0.0389501i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	13.0504			0.422743			0.211372	−	0.977406i	$-0.432207\pi$
$953$	13.0504			0.422743			0.211372	+	0.977406i	$0.432207\pi$
$954$		0			0
$955$			39.0027i			1.26210i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−43.8459	−	75.9434i	−1.41586	−	2.45234i
$960$		0			0
$961$	11.8220	−	20.4764i	0.381356	−	0.660528i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	30.0473	−	52.0434i	0.967256	−	1.67534i
$966$		0			0
$967$	3.09895	−	1.78918i	0.0996555	−	0.0575361i	−0.449344	−	0.893359i	$-0.648342\pi$
$967$	3.09895	−	1.78918i	0.0996555	−	0.0575361i	0.548999	+	0.835823i	$0.315009\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	35.7804			1.14825			0.574125	−	0.818768i	$-0.305342\pi$
$971$	35.7804			1.14825			0.574125	+	0.818768i	$0.305342\pi$
$972$		0			0
$973$		−	55.4167i		−	1.77658i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−13.0418	+	7.52970i	−0.417245	+	0.240896i	−0.693898	−	0.720074i	$-0.744108\pi$
$977$	−13.0418	+	7.52970i	−0.417245	+	0.240896i	0.276653	+	0.960970i	$0.410775\pi$
$978$		0			0
$979$	−15.7218	+	27.2310i	−0.502471	+	0.870305i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	28.4653	+	16.4345i	0.907903	+	0.524178i	0.879756	−	0.475425i	$-0.157706\pi$
$983$	28.4653	+	16.4345i	0.907903	+	0.524178i	0.0281475	+	0.999604i	$0.491039\pi$
$984$		0			0
$985$	−2.42703	−	4.20374i	−0.0773316	−	0.133942i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−0.317838			−0.0101066
$990$		0			0
$991$	13.4770			0.428112			0.214056	−	0.976821i	$-0.431333\pi$
$991$	13.4770			0.428112			0.214056	+	0.976821i	$0.431333\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	19.2278	−	11.1012i	0.609562	−	0.351931i
$996$		0			0
$997$	−5.73848	+	9.93934i	−0.181740	+	0.314782i	−0.942473	−	0.334282i	$-0.891506\pi$
$997$	−5.73848	+	9.93934i	−0.181740	+	0.314782i	0.760733	+	0.649064i	$0.224839\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2808.2.cw.b.1585.3		80
3.2	odd	2		936.2.cw.b.25.26	yes	80
9.4	even	3	inner	2808.2.cw.b.2521.38		80
9.5	odd	6		936.2.cw.b.337.25	yes	80
13.12	even	2	inner	2808.2.cw.b.1585.38		80
39.38	odd	2		936.2.cw.b.25.25	✓	80
117.77	odd	6		936.2.cw.b.337.26	yes	80
117.103	even	6	inner	2808.2.cw.b.2521.3		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
936.2.cw.b.25.25	✓	80	39.38	odd	2
936.2.cw.b.25.26	yes	80	3.2	odd	2
936.2.cw.b.337.25	yes	80	9.5	odd	6
936.2.cw.b.337.26	yes	80	117.77	odd	6
2808.2.cw.b.1585.3		80	1.1	even	1	trivial
2808.2.cw.b.1585.38		80	13.12	even	2	inner
2808.2.cw.b.2521.3		80	117.103	even	6	inner
2808.2.cw.b.2521.38		80	9.4	even	3	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2808 = 2^{3} \cdot 3^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2808.cw (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-3.40435 + 1.96550i) q^{5} +(-4.16592 - 2.40520i) q^{7} +(2.91097 + 1.68065i) q^{11} +(0.448090 - 3.57760i) q^{13} +1.07207 q^{17} -6.57228i q^{19} +(-0.0427423 - 0.0740317i) q^{23} +(5.22640 - 9.05238i) q^{25} +(0.514326 - 0.890839i) q^{29} +(-6.40180 + 3.69608i) q^{31} +18.9097 q^{35} +4.76927i q^{37} +(-2.17095 + 1.25340i) q^{41} +(1.85904 - 3.21995i) q^{43} +(-8.71820 - 5.03346i) q^{47} +(8.06995 + 13.9776i) q^{49} +5.80367 q^{53} -13.2133 q^{55} +(-10.4025 + 6.00590i) q^{59} +(-2.30828 + 3.99806i) q^{61} +(5.50632 + 13.0601i) q^{65} +(2.45580 - 1.41786i) q^{67} -1.36430i q^{71} +2.67779i q^{73} +(-8.08457 - 14.0029i) q^{77} +(0.872958 - 1.51201i) q^{79} +(12.2034 + 7.04566i) q^{83} +(-3.64971 + 2.10716i) q^{85} +9.35462i q^{89} +(-10.4715 + 13.8263i) q^{91} +(12.9178 + 22.3743i) q^{95} +(6.64133 + 3.83437i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 6 q^{13} + 4 q^{17} - 10 q^{23} + 44 q^{25} + 52 q^{35} - 26 q^{43} + 48 q^{49} - 60 q^{53} - 16 q^{55} - 10 q^{61} + 26 q^{65} - 32 q^{77} + 6 q^{79} - 4 q^{91} - 8 q^{95}+O(q^{100}) \) 80 * q + 6 * q^13 + 4 * q^17 - 10 * q^23 + 44 * q^25 + 52 * q^35 - 26 * q^43 + 48 * q^49 - 60 * q^53 - 16 * q^55 - 10 * q^61 + 26 * q^65 - 32 * q^77 + 6 * q^79 - 4 * q^91 - 8 * q^95

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