LMFDB - Embedded newform 2808.2.cw.b.1585.12

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)

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);

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")

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			1585.12
Character	$\chi$	$=$	2808.1585
Dual form			2808.2.cw.b.2521.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.43066 + 0.825993i) q^{5} +(-3.63744 - 2.10008i) q^{7} +(2.28296 + 1.31807i) q^{11} +(2.56310 + 2.53585i) q^{13} +2.77519 q^{17} -4.71234i q^{19} +(-0.483145 - 0.836832i) q^{23} +(-1.13547 + 1.96669i) q^{25} +(-3.43884 + 5.95624i) q^{29} +(-4.99896 + 2.88615i) q^{31} +6.93860 q^{35} -5.12978i q^{37} +(2.39938 - 1.38528i) q^{41} +(0.835205 - 1.44662i) q^{43} +(-4.26985 - 2.46520i) q^{47} +(5.32066 + 9.21565i) q^{49} -7.54405 q^{53} -4.35485 q^{55} +(3.31701 - 1.91508i) q^{59} +(4.79798 - 8.31035i) q^{61} +(-5.76152 - 1.51083i) q^{65} +(7.15411 - 4.13043i) q^{67} +0.0532321i q^{71} +0.987000i q^{73} +(-5.53608 - 9.58878i) q^{77} +(7.24877 - 12.5552i) q^{79} +(-6.94144 - 4.00764i) q^{83} +(-3.97036 + 2.29229i) q^{85} -8.80436i q^{89} +(-3.99766 - 14.6067i) q^{91} +(3.89236 + 6.74176i) q^{95} +(1.11958 + 0.646389i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.43066	+	0.825993i	−0.639811	+	0.369395i	−0.784542	−	0.620076i	$-0.787102\pi$
$5$	−1.43066	+	0.825993i	−0.639811	+	0.369395i	0.144731	+	0.989471i	$0.453768\pi$
$6$		0			0
$7$	−3.63744	−	2.10008i	−1.37482	−	0.793755i	−0.383293	−	0.923627i	$-0.625210\pi$
$7$	−3.63744	−	2.10008i	−1.37482	−	0.793755i	−0.991531	+	0.129872i	$0.958543\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.28296	+	1.31807i	0.688337	+	0.397412i	0.802989	−	0.595994i	$-0.203242\pi$
$11$	2.28296	+	1.31807i	0.688337	+	0.397412i	−0.114652	+	0.993406i	$0.536575\pi$
$12$		0			0
$13$	2.56310	+	2.53585i	0.710876	+	0.703317i
$13$	2.56310	+	2.53585i	0.710876	+	0.703317i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.77519			0.673083			0.336542	−	0.941669i	$-0.390743\pi$
$17$	2.77519			0.673083			0.336542	+	0.941669i	$0.390743\pi$
$18$		0			0
$19$		−	4.71234i		−	1.08108i	−0.841317	−	0.540542i	$-0.818219\pi$
$19$		−	4.71234i		−	1.08108i	0.841317	−	0.540542i	$-0.181781\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.483145	−	0.836832i	−0.100743	−	0.174492i	0.811248	−	0.584702i	$-0.198789\pi$
$23$	−0.483145	−	0.836832i	−0.100743	−	0.174492i	−0.911991	+	0.410210i	$0.865455\pi$
$24$		0			0
$25$	−1.13547	+	1.96669i	−0.227094	+	0.393339i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.43884	+	5.95624i	−0.638576	+	1.10605i	0.347170	+	0.937802i	$0.387143\pi$
$29$	−3.43884	+	5.95624i	−0.638576	+	1.10605i	−0.985745	+	0.168244i	$0.946190\pi$
$30$		0			0
$31$	−4.99896	+	2.88615i	−0.897841	+	0.518368i	−0.876499	−	0.481404i	$-0.840127\pi$
$31$	−4.99896	+	2.88615i	−0.897841	+	0.518368i	−0.0213416	+	0.999772i	$0.506794\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.93860			1.17284
$36$		0			0
$37$		−	5.12978i		−	0.843331i	−0.906751	−	0.421665i	$-0.861446\pi$
$37$		−	5.12978i		−	0.843331i	0.906751	−	0.421665i	$-0.138554\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.39938	−	1.38528i	0.374720	−	0.216345i	−0.300798	−	0.953688i	$-0.597253\pi$
$41$	2.39938	−	1.38528i	0.374720	−	0.216345i	0.675519	+	0.737343i	$0.263920\pi$
$42$		0			0
$43$	0.835205	−	1.44662i	0.127368	−	0.220607i	−0.795288	−	0.606231i	$-0.792681\pi$
$43$	0.835205	−	1.44662i	0.127368	−	0.220607i	0.922656	+	0.385624i	$0.126014\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.26985	−	2.46520i	−0.622822	−	0.359586i	0.155145	−	0.987892i	$-0.450416\pi$
$47$	−4.26985	−	2.46520i	−0.622822	−	0.359586i	−0.777967	+	0.628305i	$0.783749\pi$
$48$		0			0
$49$	5.32066	+	9.21565i	0.760094	+	1.31652i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−7.54405			−1.03626			−0.518128	−	0.855303i	$-0.673371\pi$
$53$	−7.54405			−1.03626			−0.518128	+	0.855303i	$0.673371\pi$
$54$		0			0
$55$	−4.35485			−0.587208
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.31701	−	1.91508i	0.431838	−	0.249322i	−0.268291	−	0.963338i	$-0.586459\pi$
$59$	3.31701	−	1.91508i	0.431838	−	0.249322i	0.700129	+	0.714016i	$0.253126\pi$
$60$		0			0
$61$	4.79798	−	8.31035i	0.614319	−	1.06403i	−0.376185	−	0.926545i	$-0.622764\pi$
$61$	4.79798	−	8.31035i	0.614319	−	1.06403i	0.990504	−	0.137487i	$-0.0439025\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−5.76152	−	1.51083i	−0.714629	−	0.187396i
$66$		0			0
$67$	7.15411	−	4.13043i	0.874013	−	0.504612i	0.00533352	−	0.999986i	$-0.498302\pi$
$67$	7.15411	−	4.13043i	0.874013	−	0.504612i	0.868680	+	0.495374i	$0.164969\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			0.0532321i			0.00631748i	0.999995	+	0.00315874i	$0.00100546\pi$
$71$			0.0532321i			0.00631748i	−0.999995	+	0.00315874i	$0.998995\pi$
$72$		0			0
$73$			0.987000i			0.115520i	0.998331	+	0.0577598i	$0.0183957\pi$
$73$			0.987000i			0.115520i	−0.998331	+	0.0577598i	$0.981604\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−5.53608	−	9.58878i	−0.630895	−	1.09274i
$78$		0			0
$79$	7.24877	−	12.5552i	0.815550	−	1.41257i	−0.0933822	−	0.995630i	$-0.529768\pi$
$79$	7.24877	−	12.5552i	0.815550	−	1.41257i	0.908932	−	0.416944i	$-0.136899\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.94144	−	4.00764i	−0.761922	−	0.439896i	0.0680636	−	0.997681i	$-0.478318\pi$
$83$	−6.94144	−	4.00764i	−0.761922	−	0.439896i	−0.829985	+	0.557785i	$0.811651\pi$
$84$		0			0
$85$	−3.97036	+	2.29229i	−0.430646	+	0.248634i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	8.80436i		−	0.933260i	−0.884453	−	0.466630i	$-0.845468\pi$
$89$		−	8.80436i		−	0.933260i	0.884453	−	0.466630i	$-0.154532\pi$
$90$		0			0
$91$	−3.99766	−	14.6067i	−0.419069	−	1.53120i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.89236	+	6.74176i	0.399348	+	0.691690i
$96$		0			0
$97$	1.11958	+	0.646389i	0.113676	+	0.0656308i	0.555760	−	0.831343i	$-0.312427\pi$
$97$	1.11958	+	0.646389i	0.113676	+	0.0656308i	−0.442084	+	0.896974i	$0.645761\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.22972	−	10.7902i	0.619881	−	1.07366i	−0.369627	−	0.929180i	$-0.620514\pi$
$101$	6.22972	−	10.7902i	0.619881	−	1.07366i	0.989507	−	0.144484i	$-0.0461523\pi$
$102$		0			0
$103$	−8.74904	−	15.1538i	−0.862068	−	1.49315i	−0.869929	−	0.493177i	$-0.835836\pi$
$103$	−8.74904	−	15.1538i	−0.862068	−	1.49315i	0.00786103	−	0.999969i	$-0.497498\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.21102			0.310421			0.155211	−	0.987881i	$-0.450394\pi$
$107$	3.21102			0.310421			0.155211	+	0.987881i	$0.450394\pi$
$108$		0			0
$109$		−	17.2490i		−	1.65215i	−0.563560	−	0.826075i	$-0.690569\pi$
$109$		−	17.2490i		−	1.65215i	0.563560	−	0.826075i	$-0.309431\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.847508	−	1.46793i	−0.0797269	−	0.138091i	0.823405	−	0.567454i	$-0.192071\pi$
$113$	−0.847508	−	1.46793i	−0.0797269	−	0.138091i	−0.903132	+	0.429363i	$0.858738\pi$
$114$		0			0
$115$	1.38243	+	0.798149i	0.128913	+	0.0744278i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−10.0946	−	5.82812i	−0.925371	−	0.534263i
$120$		0			0
$121$	−2.02541	−	3.50811i	−0.184128	−	0.318919i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	12.0115i		−	1.07434i
$126$		0			0
$127$	4.71212			0.418133			0.209067	−	0.977901i	$-0.432957\pi$
$127$	4.71212			0.418133			0.209067	+	0.977901i	$0.432957\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.61370	−	11.4553i	−0.577842	−	1.00085i	−0.995727	−	0.0923511i	$-0.970562\pi$
$131$	−6.61370	−	11.4553i	−0.577842	−	1.00085i	0.417885	−	0.908500i	$-0.362772\pi$
$132$		0			0
$133$	−9.89628	+	17.1409i	−0.858116	+	1.48630i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.225795	+	0.130363i	0.0192910	+	0.0111376i	0.509614	−	0.860403i	$-0.329788\pi$
$137$	0.225795	+	0.130363i	0.0192910	+	0.0111376i	−0.490324	+	0.871541i	$0.663121\pi$
$138$		0			0
$139$	−5.87771	−	10.1805i	−0.498541	−	0.863498i	0.501458	−	0.865182i	$-0.332797\pi$
$139$	−5.87771	−	10.1805i	−0.498541	−	0.863498i	−0.999999	+	0.00168410i	$0.999464\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.50904	+	9.16756i	0.209816	+	0.766630i
$144$		0			0
$145$		−	11.3618i		−	0.943548i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−18.1431	+	10.4749i	−1.48634	+	0.858137i	−0.999879	−	0.0155668i	$-0.995045\pi$
$149$	−18.1431	+	10.4749i	−1.48634	+	0.858137i	−0.486458	+	0.873704i	$0.661711\pi$
$150$		0			0
$151$	10.8725	+	6.27726i	0.884794	+	0.510836i	0.872236	−	0.489085i	$-0.162669\pi$
$151$	10.8725	+	6.27726i	0.884794	+	0.510836i	0.0125580	+	0.999921i	$0.496003\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4.76788	−	8.25822i	0.382966	−	0.663316i
$156$		0			0
$157$	5.72790	+	9.92101i	0.457136	+	0.791783i	0.998808	−	0.0488068i	$-0.0155419\pi$
$157$	5.72790	+	9.92101i	0.457136	+	0.791783i	−0.541672	+	0.840590i	$0.682209\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			4.05857i			0.319860i
$162$		0			0
$163$		−	11.3497i		−	0.888976i	−0.895785	−	0.444488i	$-0.853386\pi$
$163$		−	11.3497i		−	0.888976i	0.895785	−	0.444488i	$-0.146614\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−20.5639	+	11.8725i	−1.59128	+	0.918725i	−0.598191	+	0.801354i	$0.704113\pi$
$167$	−20.5639	+	11.8725i	−1.59128	+	0.918725i	−0.993088	+	0.117371i	$0.962553\pi$
$168$		0			0
$169$	0.138980	+	12.9993i	0.0106908	+	0.999943i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−9.28380	+	16.0800i	−0.705834	+	1.22254i	0.260556	+	0.965459i	$0.416094\pi$
$173$	−9.28380	+	16.0800i	−0.705834	+	1.22254i	−0.966390	+	0.257082i	$0.917239\pi$
$174$		0			0
$175$	8.26043	−	4.76916i	0.624430	−	0.360515i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	5.12934			0.383385			0.191692	−	0.981455i	$-0.438602\pi$
$179$	5.12934			0.383385			0.191692	+	0.981455i	$0.438602\pi$
$180$		0			0
$181$	20.5034			1.52401			0.762003	−	0.647573i	$-0.224216\pi$
$181$	20.5034			1.52401			0.762003	+	0.647573i	$0.224216\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	4.23716	+	7.33898i	0.311522	+	0.539573i
$186$		0			0
$187$	6.33564	+	3.65789i	0.463308	+	0.267491i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	9.78159	−	16.9422i	0.707771	−	1.22590i	−0.257911	−	0.966169i	$-0.583034\pi$
$191$	9.78159	−	16.9422i	0.707771	−	1.22590i	0.965682	−	0.259726i	$-0.0836324\pi$
$192$		0			0
$193$	−4.89103	+	2.82384i	−0.352064	+	0.203264i	−0.665594	−	0.746314i	$-0.731822\pi$
$193$	−4.89103	+	2.82384i	−0.352064	+	0.203264i	0.313530	+	0.949578i	$0.398488\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	25.6253i		−	1.82573i	−0.408263	−	0.912865i	$-0.633865\pi$
$197$		−	25.6253i		−	1.82573i	0.408263	−	0.912865i	$-0.366135\pi$
$198$		0			0
$199$	−2.09536			−0.148536			−0.0742682	−	0.997238i	$-0.523662\pi$
$199$	−2.09536			−0.148536			−0.0742682	+	0.997238i	$0.523662\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	25.0171	−	14.4437i	1.75586	−	1.01375i
$204$		0			0
$205$	−2.28847	+	3.96374i	−0.159833	+	0.276840i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	6.21117	−	10.7581i	0.429636	−	0.744151i
$210$		0			0
$211$	11.8477	+	20.5209i	0.815631	+	1.41271i	0.908874	+	0.417071i	$0.136943\pi$
$211$	11.8477	+	20.5209i	0.815631	+	1.41271i	−0.0932432	+	0.995643i	$0.529723\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			2.75949i			0.188196i
$216$		0			0
$217$	24.2446			1.64583
$218$		0			0
$219$		0			0
$220$		0			0
$221$	7.11310	+	7.03746i	0.478479	+	0.473391i
$222$		0			0
$223$	8.71542	+	5.03185i	0.583628	+	0.336958i	0.762574	−	0.646901i	$-0.223935\pi$
$223$	8.71542	+	5.03185i	0.583628	+	0.336958i	−0.178946	+	0.983859i	$0.557269\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−9.90016	−	5.71586i	−0.657096	−	0.379375i	0.134073	−	0.990971i	$-0.457194\pi$
$227$	−9.90016	−	5.71586i	−0.657096	−	0.379375i	−0.791170	+	0.611597i	$0.790528\pi$
$228$		0			0
$229$	−22.0810	+	12.7485i	−1.45916	+	0.842445i	−0.998970	−	0.0453774i	$-0.985551\pi$
$229$	−22.0810	+	12.7485i	−1.45916	+	0.842445i	−0.460187	+	0.887822i	$0.652218\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3.82435			−0.250542			−0.125271	−	0.992123i	$-0.539980\pi$
$233$	−3.82435			−0.250542			−0.125271	+	0.992123i	$0.539980\pi$
$234$		0			0
$235$	8.14495			0.531318
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.128051	−	0.0739304i	0.00828294	−	0.00478216i	−0.495853	−	0.868407i	$-0.665144\pi$
$239$	0.128051	−	0.0739304i	0.00828294	−	0.00478216i	0.504136	+	0.863624i	$0.331811\pi$
$240$		0			0
$241$	−2.03972	−	1.17763i	−0.131390	−	0.0758581i	0.432864	−	0.901459i	$-0.357503\pi$
$241$	−2.03972	−	1.17763i	−0.131390	−	0.0758581i	−0.564254	+	0.825601i	$0.690836\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−15.2241	−	8.78965i	−0.972634	−	0.561550i
$246$		0			0
$247$	11.9498	−	12.0782i	0.760345	−	0.768518i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	29.9852			1.89265			0.946325	−	0.323218i	$-0.104765\pi$
$251$	29.9852			1.89265			0.946325	+	0.323218i	$0.104765\pi$
$252$		0			0
$253$		−	2.54727i		−	0.160145i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	12.6703	+	21.9457i	0.790355	+	1.36893i	0.925747	+	0.378142i	$0.123437\pi$
$257$	12.6703	+	21.9457i	0.790355	+	1.36893i	−0.135393	+	0.990792i	$0.543230\pi$
$258$		0			0
$259$	−10.7729	+	18.6593i	−0.669398	+	1.15943i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.08227	+	10.5348i	−0.375049	+	0.649604i	−0.990334	−	0.138700i	$-0.955707\pi$
$263$	−6.08227	+	10.5348i	−0.375049	+	0.649604i	0.615285	+	0.788304i	$0.289041\pi$
$264$		0			0
$265$	10.7930	−	6.23133i	0.663008	−	0.382788i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−17.4224			−1.06226			−0.531132	−	0.847289i	$-0.678233\pi$
$269$	−17.4224			−1.06226			−0.531132	+	0.847289i	$0.678233\pi$
$270$		0			0
$271$			20.5427i			1.24788i	0.781473	+	0.623939i	$0.214469\pi$
$271$			20.5427i			1.24788i	−0.781473	+	0.623939i	$0.785531\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−5.18446	+	2.99325i	−0.312635	+	0.180500i
$276$		0			0
$277$	−10.1748	+	17.6233i	−0.611344	+	1.05888i	0.379670	+	0.925122i	$0.376037\pi$
$277$	−10.1748	+	17.6233i	−0.611344	+	1.05888i	−0.991014	+	0.133757i	$0.957296\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−22.3673	−	12.9138i	−1.33432	−	0.770372i	−0.348364	−	0.937359i	$-0.613263\pi$
$281$	−22.3673	−	12.9138i	−1.33432	−	0.770372i	−0.985959	+	0.166988i	$0.946596\pi$
$282$		0			0
$283$	−12.7895	−	22.1521i	−0.760258	−	1.31681i	−0.942717	−	0.333593i	$-0.891739\pi$
$283$	−12.7895	−	22.1521i	−0.760258	−	1.31681i	0.182459	−	0.983213i	$-0.441594\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−11.6368			−0.686899
$288$		0			0
$289$	−9.29830			−0.546959
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−21.9855	+	12.6933i	−1.28440	+	0.741551i	−0.977650	−	0.210237i	$-0.932576\pi$
$293$	−21.9855	+	12.6933i	−1.28440	+	0.741551i	−0.306754	+	0.951789i	$0.599243\pi$
$294$		0			0
$295$	−3.16368	+	5.47965i	−0.184197	+	0.319038i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.883726	−	3.37007i	0.0511072	−	0.194896i
$300$		0			0
$301$	−6.07602	+	3.50799i	−0.350216	+	0.202197i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			15.8524i			0.907706i
$306$		0			0
$307$		−	10.0799i		−	0.575288i	−0.957737	−	0.287644i	$-0.907128\pi$
$307$		−	10.0799i		−	0.575288i	0.957737	−	0.287644i	$-0.0928720\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−8.02457	−	13.8990i	−0.455032	−	0.788138i	0.543658	−	0.839307i	$-0.317039\pi$
$311$	−8.02457	−	13.8990i	−0.455032	−	0.788138i	−0.998690	+	0.0511686i	$0.983705\pi$
$312$		0			0
$313$	11.7016	−	20.2678i	0.661414	−	1.14560i	−0.318830	−	0.947812i	$-0.603290\pi$
$313$	11.7016	−	20.2678i	0.661414	−	1.14560i	0.980244	−	0.197791i	$-0.0633769\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	20.9411	+	12.0903i	1.17617	+	0.679061i	0.955125	−	0.296203i	$-0.0957206\pi$
$317$	20.9411	+	12.0903i	1.17617	+	0.679061i	0.221043	+	0.975264i	$0.429054\pi$
$318$		0			0
$319$	−15.7014	+	9.06522i	−0.879111	+	0.507555i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	13.0776i		−	0.727660i
$324$		0			0
$325$	−7.89756	+	2.16146i	−0.438078	+	0.119896i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	10.3542	+	17.9340i	0.570847	+	0.988736i
$330$		0			0
$331$	−2.77109	−	1.59989i	−0.152313	−	0.0879379i	0.421906	−	0.906639i	$-0.361361\pi$
$331$	−2.77109	−	1.59989i	−0.152313	−	0.0879379i	−0.574219	+	0.818701i	$0.694694\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−6.82340	+	11.8185i	−0.372802	+	0.645713i
$336$		0			0
$337$	−5.88411	−	10.1916i	−0.320528	−	0.555171i	0.660069	−	0.751205i	$-0.270527\pi$
$337$	−5.88411	−	10.1916i	−0.320528	−	0.555171i	−0.980597	+	0.196034i	$0.937194\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−15.2166			−0.824023
$342$		0			0
$343$		−	15.2941i		−	0.825804i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−18.1366	−	31.4135i	−0.973623	−	1.68636i	−0.684406	−	0.729101i	$-0.739938\pi$
$347$	−18.1366	−	31.4135i	−0.973623	−	1.68636i	−0.289217	−	0.957263i	$-0.593395\pi$
$348$		0			0
$349$	18.1384	+	10.4722i	0.970925	+	0.560564i	0.899518	−	0.436884i	$-0.143918\pi$
$349$	18.1384	+	10.4722i	0.970925	+	0.560564i	0.0714066	+	0.997447i	$0.477251\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	14.4946	+	8.36846i	0.771469	+	0.445408i	0.833399	−	0.552673i	$-0.186392\pi$
$353$	14.4946	+	8.36846i	0.771469	+	0.445408i	−0.0619292	+	0.998081i	$0.519725\pi$
$354$		0			0
$355$	−0.0439693	−	0.0761571i	−0.00233365	−	0.00404200i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	29.9958i		−	1.58312i	−0.611092	−	0.791560i	$-0.709269\pi$
$359$		−	29.9958i		−	1.58312i	0.611092	−	0.791560i	$-0.290731\pi$
$360$		0			0
$361$	−3.20614			−0.168744
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−0.815255	−	1.41206i	−0.0426724	−	0.0739107i
$366$		0			0
$367$	4.73460	−	8.20057i	0.247144	−	0.428067i	−0.715588	−	0.698523i	$-0.753841\pi$
$367$	4.73460	−	8.20057i	0.247144	−	0.428067i	0.962732	+	0.270456i	$0.0871745\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	27.4411	+	15.8431i	1.42467	+	0.822533i
$372$		0			0
$373$	−7.37007	−	12.7653i	−0.381608	−	0.660964i	0.609685	−	0.792644i	$-0.291296\pi$
$373$	−7.37007	−	12.7653i	−0.381608	−	0.660964i	−0.991292	+	0.131680i	$0.957963\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−23.9182	+	6.54609i	−1.23185	+	0.337141i
$378$		0			0
$379$			14.7901i			0.759717i	0.925045	+	0.379859i	$0.124027\pi$
$379$			14.7901i			0.759717i	−0.925045	+	0.379859i	$0.875973\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	15.3401	−	8.85660i	0.783842	−	0.452551i	−0.0539484	−	0.998544i	$-0.517181\pi$
$383$	15.3401	−	8.85660i	0.783842	−	0.452551i	0.837790	+	0.545993i	$0.183847\pi$
$384$		0			0
$385$	15.8405	+	9.14553i	0.807308	+	0.466099i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−6.20014	+	10.7390i	−0.314360	+	0.544487i	−0.979301	−	0.202408i	$-0.935123\pi$
$389$	−6.20014	+	10.7390i	−0.314360	+	0.544487i	0.664941	+	0.746896i	$0.268457\pi$
$390$		0			0
$391$	−1.34082	−	2.32237i	−0.0678082	−	0.117447i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			23.9497i			1.20504i
$396$		0			0
$397$		−	14.0928i		−	0.707297i	−0.935378	−	0.353649i	$-0.884941\pi$
$397$		−	14.0928i		−	0.707297i	0.935378	−	0.353649i	$-0.115059\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	26.1289	−	15.0856i	1.30482	−	0.753336i	0.323591	−	0.946197i	$-0.395110\pi$
$401$	26.1289	−	15.0856i	1.30482	−	0.753336i	0.981226	+	0.192861i	$0.0617765\pi$
$402$		0			0
$403$	−20.1317	−	5.27909i	−1.00283	−	0.262970i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	6.76139	−	11.7111i	0.335150	−	0.580496i
$408$		0			0
$409$	−13.8705	+	8.00815i	−0.685853	+	0.395977i	−0.802057	−	0.597248i	$-0.796261\pi$
$409$	−13.8705	+	8.00815i	−0.685853	+	0.395977i	0.116204	+	0.993225i	$0.462927\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−16.0872			−0.791602
$414$		0			0
$415$	13.2411			0.649981
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−13.8421	−	23.9752i	−0.676229	−	1.17126i	−0.976108	−	0.217286i	$-0.930280\pi$
$419$	−13.8421	−	23.9752i	−0.676229	−	1.17126i	0.299879	−	0.953977i	$-0.403054\pi$
$420$		0			0
$421$	−14.3628	−	8.29235i	−0.699999	−	0.404145i	0.107348	−	0.994222i	$-0.465764\pi$
$421$	−14.3628	−	8.29235i	−0.699999	−	0.404145i	−0.807347	+	0.590077i	$0.799097\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.15115	+	5.45796i	−0.152853	+	0.264750i
$426$		0			0
$427$	−34.9048	+	20.1523i	−1.68916	+	0.975238i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	0.719688i		−	0.0346661i	−0.999850	−	0.0173331i	$-0.994482\pi$
$431$		−	0.719688i		−	0.0346661i	0.999850	−	0.0173331i	$-0.00551756\pi$
$432$		0			0
$433$	7.69378			0.369739			0.184870	−	0.982763i	$-0.440814\pi$
$433$	7.69378			0.369739			0.184870	+	0.982763i	$0.440814\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3.94344	+	2.27674i	−0.188640	+	0.108911i
$438$		0			0
$439$	13.0575	−	22.6163i	0.623202	−	1.07942i	−0.365684	−	0.930739i	$-0.619165\pi$
$439$	13.0575	−	22.6163i	0.623202	−	1.07942i	0.988886	−	0.148678i	$-0.0475018\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−3.01854	+	5.22826i	−0.143415	+	0.248402i	−0.928781	−	0.370630i	$-0.879142\pi$
$443$	−3.01854	+	5.22826i	−0.143415	+	0.248402i	0.785365	+	0.619032i	$0.212475\pi$
$444$		0			0
$445$	7.27234	+	12.5961i	0.344742	+	0.597110i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			40.5862i			1.91538i	0.287799	+	0.957691i	$0.407077\pi$
$449$			40.5862i			1.91538i	−0.287799	+	0.957691i	$0.592923\pi$
$450$		0			0
$451$	7.30357			0.343912
$452$		0			0
$453$		0			0
$454$		0			0
$455$	17.7843	+	17.5952i	0.833742	+	0.824876i
$456$		0			0
$457$	20.8173	+	12.0188i	0.973790	+	0.562218i	0.900390	−	0.435085i	$-0.143282\pi$
$457$	20.8173	+	12.0188i	0.973790	+	0.562218i	0.0734004	+	0.997303i	$0.476615\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	16.8379	+	9.72135i	0.784218	+	0.452768i	0.837923	−	0.545788i	$-0.183770\pi$
$461$	16.8379	+	9.72135i	0.784218	+	0.452768i	−0.0537052	+	0.998557i	$0.517103\pi$
$462$		0			0
$463$	−0.897269	+	0.518038i	−0.0416996	+	0.0240753i	−0.520705	−	0.853737i	$-0.674331\pi$
$463$	−0.897269	+	0.518038i	−0.0416996	+	0.0240753i	0.479005	+	0.877812i	$0.340997\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	19.0967			0.883688			0.441844	−	0.897092i	$-0.354325\pi$
$467$	19.0967			0.883688			0.441844	+	0.897092i	$0.354325\pi$
$468$		0			0
$469$	−34.6969			−1.60215
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3.81347	−	2.20171i	0.175344	−	0.101235i
$474$		0			0
$475$	9.26773	+	5.35073i	0.425233	+	0.245508i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	17.2977	+	9.98682i	0.790352	+	0.456310i	0.840086	−	0.542453i	$-0.182504\pi$
$479$	17.2977	+	9.98682i	0.790352	+	0.456310i	−0.0497348	+	0.998762i	$0.515838\pi$
$480$		0			0
$481$	13.0083	−	13.1482i	0.593129	−	0.599504i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.13565			−0.0969749
$486$		0			0
$487$		−	32.5937i		−	1.47696i	−0.674275	−	0.738480i	$-0.735544\pi$
$487$		−	32.5937i		−	1.47696i	0.674275	−	0.738480i	$-0.264456\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	9.26692	+	16.0508i	0.418210	+	0.724362i	0.995760	−	0.0919943i	$-0.0293241\pi$
$491$	9.26692	+	16.0508i	0.418210	+	0.724362i	−0.577549	+	0.816356i	$0.695991\pi$
$492$		0			0
$493$	−9.54343	+	16.5297i	−0.429815	+	0.744461i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	0.111792	−	0.193629i	0.00501454	−	0.00868543i
$498$		0			0
$499$	−15.6180	+	9.01707i	−0.699159	+	0.403660i	−0.807034	−	0.590505i	$-0.798929\pi$
$499$	−15.6180	+	9.01707i	−0.699159	+	0.403660i	0.107875	+	0.994164i	$0.465595\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−0.317040			−0.0141361			−0.00706806	−	0.999975i	$-0.502250\pi$
$503$	−0.317040			−0.0141361			−0.00706806	+	0.999975i	$0.502250\pi$
$504$		0			0
$505$			20.5828i			0.915924i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	24.9432	−	14.4010i	1.10559	−	0.638312i	0.167905	−	0.985803i	$-0.446300\pi$
$509$	24.9432	−	14.4010i	1.10559	−	0.638312i	0.937683	+	0.347491i	$0.112966\pi$
$510$		0			0
$511$	2.07278	−	3.59015i	0.0916942	−	0.158819i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	25.0338	+	14.4533i	1.10312	+	0.636888i
$516$		0			0
$517$	−6.49859	−	11.2559i	−0.285808	−	0.495033i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	34.4420			1.50893			0.754466	−	0.656339i	$-0.227896\pi$
$521$	34.4420			1.50893			0.754466	+	0.656339i	$0.227896\pi$
$522$		0			0
$523$	−42.3521			−1.85193			−0.925965	−	0.377610i	$-0.876746\pi$
$523$	−42.3521			−1.85193			−0.925965	+	0.377610i	$0.876746\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−13.8731	+	8.00963i	−0.604321	+	0.348905i
$528$		0			0
$529$	11.0331	−	19.1100i	0.479702	−	0.830868i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	9.66271	+	2.53383i	0.418539	+	0.109753i
$534$		0			0
$535$	−4.59389	+	2.65228i	−0.198611	+	0.114668i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			28.0519i			1.20828i
$540$		0			0
$541$			19.8244i			0.852318i	0.904648	+	0.426159i	$0.140134\pi$
$541$			19.8244i			0.852318i	−0.904648	+	0.426159i	$0.859866\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	14.2475	+	24.6774i	0.610297	+	1.05706i
$546$		0			0
$547$	1.37157	−	2.37563i	0.0586442	−	0.101575i	−0.835213	−	0.549927i	$-0.814656\pi$
$547$	1.37157	−	2.37563i	0.0586442	−	0.101575i	0.893857	+	0.448352i	$0.147989\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	28.0678	+	16.2050i	1.19573	+	0.690355i
$552$		0			0
$553$	−52.7340	+	30.4460i	−2.24248	+	1.29469i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			36.5778i			1.54985i	0.632052	+	0.774926i	$0.282213\pi$
$557$			36.5778i			1.54985i	−0.632052	+	0.774926i	$0.717787\pi$
$558$		0			0
$559$	5.80911	−	1.58988i	0.245699	−	0.0672446i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6.64821	−	11.5150i	−0.280188	−	0.485301i	0.691243	−	0.722623i	$-0.257064\pi$
$563$	−6.64821	−	11.5150i	−0.280188	−	0.485301i	−0.971431	+	0.237322i	$0.923730\pi$
$564$		0			0
$565$	2.42500	+	1.40007i	0.102020	+	0.0589015i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	14.5215	−	25.1519i	0.608772	−	1.05442i	−0.382671	−	0.923885i	$-0.624996\pi$
$569$	14.5215	−	25.1519i	0.608772	−	1.05442i	0.991443	−	0.130540i	$-0.0416710\pi$
$570$		0			0
$571$	−0.708741	−	1.22757i	−0.0296599	−	0.0513724i	0.850814	−	0.525466i	$-0.176109\pi$
$571$	−0.708741	−	1.22757i	−0.0296599	−	0.0513724i	−0.880474	+	0.474094i	$0.842776\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.19439			0.0915124
$576$		0			0
$577$		−	17.5665i		−	0.731302i	−0.930752	−	0.365651i	$-0.880846\pi$
$577$		−	17.5665i		−	0.731302i	0.930752	−	0.365651i	$-0.119154\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	16.8327	+	29.1551i	0.698339	+	1.20956i
$582$		0			0
$583$	−17.2227	−	9.94356i	−0.713293	−	0.411820i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−25.1528	−	14.5220i	−1.03817	−	0.599387i	−0.118855	−	0.992912i	$-0.537922\pi$
$587$	−25.1528	−	14.5220i	−1.03817	−	0.599387i	−0.919314	+	0.393524i	$0.871256\pi$
$588$		0			0
$589$	13.6005	+	23.5568i	0.560400	+	0.970642i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			4.30919i			0.176957i	0.996078	+	0.0884786i	$0.0282005\pi$
$593$			4.30919i			0.176957i	−0.996078	+	0.0884786i	$0.971800\pi$
$594$		0			0
$595$	19.2560			0.789417
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−4.61748	−	7.99771i	−0.188665	−	0.326777i	0.756140	−	0.654409i	$-0.227083\pi$
$599$	−4.61748	−	7.99771i	−0.188665	−	0.326777i	−0.944805	+	0.327632i	$0.893749\pi$
$600$		0			0
$601$	9.05716	−	15.6875i	0.369449	−	0.639905i	−0.620030	−	0.784578i	$-0.712880\pi$
$601$	9.05716	−	15.6875i	0.369449	−	0.639905i	0.989480	+	0.144673i	$0.0462131\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	5.79534	+	3.34594i	0.235614	+	0.136032i
$606$		0			0
$607$	5.22492	+	9.04982i	0.212073	+	0.367321i	0.952363	−	0.304966i	$-0.0986452\pi$
$607$	5.22492	+	9.04982i	0.212073	+	0.367321i	−0.740290	+	0.672287i	$0.765312\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−4.69270	−	17.1462i	−0.189846	−	0.693663i
$612$		0			0
$613$		−	24.1176i		−	0.974102i	−0.873373	−	0.487051i	$-0.838073\pi$
$613$		−	24.1176i		−	0.974102i	0.873373	−	0.487051i	$-0.161927\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	14.9897	−	8.65431i	0.603463	−	0.348410i	−0.166940	−	0.985967i	$-0.553389\pi$
$617$	14.9897	−	8.65431i	0.603463	−	0.348410i	0.770403	+	0.637558i	$0.220055\pi$
$618$		0			0
$619$	41.6042	+	24.0202i	1.67221	+	0.965453i	0.966395	+	0.257060i	$0.0827539\pi$
$619$	41.6042	+	24.0202i	1.67221	+	0.965453i	0.705818	+	0.708393i	$0.250579\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−18.4898	+	32.0254i	−0.740780	+	1.28307i
$624$		0			0
$625$	4.24405	+	7.35091i	0.169762	+	0.294036i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	14.2361i		−	0.567632i
$630$		0			0
$631$			23.5460i			0.937351i	0.883370	+	0.468676i	$0.155269\pi$
$631$			23.5460i			0.937351i	−0.883370	+	0.468676i	$0.844731\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−6.74145	+	3.89218i	−0.267526	+	0.154456i
$636$		0			0
$637$	−9.73208	+	37.1130i	−0.385599	+	1.47047i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2.63766	−	4.56856i	0.104181	−	0.180447i	−0.809222	−	0.587503i	$-0.800111\pi$
$641$	2.63766	−	4.56856i	0.104181	−	0.180447i	0.913403	+	0.407056i	$0.133444\pi$
$642$		0			0
$643$	−37.6481	+	21.7361i	−1.48469	+	0.857189i	−0.999848	−	0.0174104i	$-0.994458\pi$
$643$	−37.6481	+	21.7361i	−1.48469	+	0.857189i	−0.484846	+	0.874599i	$0.661124\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−4.53802			−0.178408			−0.0892040	−	0.996013i	$-0.528432\pi$
$647$	−4.53802			−0.178408			−0.0892040	+	0.996013i	$0.528432\pi$
$648$		0			0
$649$	10.0968			0.396333
$650$		0			0
$651$		0			0
$652$		0			0
$653$	5.38366	+	9.32478i	0.210679	+	0.364907i	0.951927	−	0.306324i	$-0.0990992\pi$
$653$	5.38366	+	9.32478i	0.210679	+	0.364907i	−0.741248	+	0.671231i	$0.765766\pi$
$654$		0			0
$655$	18.9239	+	10.9257i	0.739419	+	0.426904i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−7.79180	+	13.4958i	−0.303526	+	0.525722i	−0.976932	−	0.213551i	$-0.931497\pi$
$659$	−7.79180	+	13.4958i	−0.303526	+	0.525722i	0.673406	+	0.739273i	$0.264830\pi$
$660$		0			0
$661$	−22.1283	+	12.7758i	−0.860692	+	0.496921i	−0.864244	−	0.503073i	$-0.832203\pi$
$661$	−22.1283	+	12.7758i	−0.860692	+	0.496921i	0.00355165	+	0.999994i	$0.498869\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	32.6970i		−	1.26794i
$666$		0			0
$667$	6.64583			0.257328
$668$		0			0
$669$		0			0
$670$		0			0
$671$	21.9072	−	12.6481i	0.845717	−	0.488275i
$672$		0			0
$673$	1.05029	−	1.81916i	0.0404857	−	0.0701233i	−0.845072	−	0.534652i	$-0.820443\pi$
$673$	1.05029	−	1.81916i	0.0404857	−	0.0701233i	0.885558	+	0.464528i	$0.153776\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	11.2822	−	19.5414i	0.433611	−	0.751036i	−0.563570	−	0.826068i	$-0.690573\pi$
$677$	11.2822	−	19.5414i	0.433611	−	0.751036i	0.997181	+	0.0750320i	$0.0239059\pi$
$678$		0			0
$679$	−2.71493	−	4.70240i	−0.104190	−	0.180462i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			50.4385i			1.92998i	0.262293	+	0.964988i	$0.415521\pi$
$683$			50.4385i			1.92998i	−0.262293	+	0.964988i	$0.584479\pi$
$684$		0			0
$685$	−0.430715			−0.0164568
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−19.3362	−	19.1305i	−0.736650	−	0.728816i
$690$		0			0
$691$	39.0601	+	22.5513i	1.48592	+	0.857894i	0.999871	−	0.0160406i	$-0.00510609\pi$
$691$	39.0601	+	22.5513i	1.48592	+	0.857894i	0.486044	+	0.873934i	$0.338439\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	16.8180	+	9.70989i	0.637944	+	0.368317i
$696$		0			0
$697$	6.65874	−	3.84443i	0.252218	−	0.145618i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−27.9606			−1.05606			−0.528028	−	0.849227i	$-0.677068\pi$
$701$	−27.9606			−1.05606			−0.528028	+	0.849227i	$0.677068\pi$
$702$		0			0
$703$	−24.1733			−0.911712
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−45.3205	+	26.1658i	−1.70445	+	0.984067i
$708$		0			0
$709$	−26.7050	−	15.4182i	−1.00293	−	0.579041i	−0.0938149	−	0.995590i	$-0.529906\pi$
$709$	−26.7050	−	15.4182i	−1.00293	−	0.579041i	−0.909113	+	0.416549i	$0.863240\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	4.83045	+	2.78886i	0.180902	+	0.104444i
$714$		0			0
$715$	−11.1619	−	11.0432i	−0.417432	−	0.412993i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	24.0344			0.896330			0.448165	−	0.893951i	$-0.352078\pi$
$719$	24.0344			0.896330			0.448165	+	0.893951i	$0.352078\pi$
$720$		0			0
$721$			73.4946i			2.73708i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−7.80940	−	13.5263i	−0.290034	−	0.502354i
$726$		0			0
$727$	−4.84666	+	8.39466i	−0.179753	+	0.311341i	−0.941796	−	0.336186i	$-0.890863\pi$
$727$	−4.84666	+	8.39466i	−0.179753	+	0.311341i	0.762043	+	0.647526i	$0.224196\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	2.31785	−	4.01464i	0.0857289	−	0.148487i
$732$		0			0
$733$	13.5294	−	7.81123i	0.499721	−	0.288514i	−0.228877	−	0.973455i	$-0.573505\pi$
$733$	13.5294	−	7.81123i	0.499721	−	0.288514i	0.728598	+	0.684941i	$0.240172\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	21.7767			0.802155
$738$		0			0
$739$			40.9286i			1.50558i	0.658258	+	0.752792i	$0.271293\pi$
$739$			40.9286i			1.50558i	−0.658258	+	0.752792i	$0.728707\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	27.1375	−	15.6678i	0.995578	−	0.574797i	0.0886409	−	0.996064i	$-0.471748\pi$
$743$	27.1375	−	15.6678i	0.995578	−	0.574797i	0.906937	+	0.421267i	$0.138414\pi$
$744$		0			0
$745$	17.3044	−	29.9721i	0.633983	−	1.09809i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−11.6799	−	6.74340i	−0.426775	−	0.246398i
$750$		0			0
$751$	9.96795	+	17.2650i	0.363736	+	0.630009i	0.988572	−	0.150747i	$-0.0481678\pi$
$751$	9.96795	+	17.2650i	0.363736	+	0.630009i	−0.624837	+	0.780755i	$0.714834\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−20.7399			−0.754802
$756$		0			0
$757$	−17.0157			−0.618447			−0.309224	−	0.950989i	$-0.600069\pi$
$757$	−17.0157			−0.618447			−0.309224	+	0.950989i	$0.600069\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−11.6270	+	6.71285i	−0.421478	+	0.243341i	−0.695710	−	0.718323i	$-0.744910\pi$
$761$	−11.6270	+	6.71285i	−0.421478	+	0.243341i	0.274231	+	0.961664i	$0.411577\pi$
$762$		0			0
$763$	−36.2242	+	62.7421i	−1.31140	+	2.27142i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	13.3582	+	3.50289i	0.482336	+	0.126482i
$768$		0			0
$769$	39.9043	−	23.0388i	1.43899	−	0.830799i	0.441206	−	0.897406i	$-0.354551\pi$
$769$	39.9043	−	23.0388i	1.43899	−	0.830799i	0.997779	+	0.0666070i	$0.0212174\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	38.1681i		−	1.37281i	−0.727219	−	0.686405i	$-0.759188\pi$
$773$		−	38.1681i		−	1.37281i	0.727219	−	0.686405i	$-0.240812\pi$
$774$		0			0
$775$		−	13.1086i		−	0.470874i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−6.52792	−	11.3067i	−0.233887	−	0.405104i
$780$		0			0
$781$	−0.0701634	+	0.121526i	−0.00251064	+	0.00434856i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−16.3894	−	9.46241i	−0.584962	−	0.337728i
$786$		0			0
$787$	−9.73274	+	5.61920i	−0.346935	+	0.200303i	−0.663334	−	0.748323i	$-0.730859\pi$
$787$	−9.73274	+	5.61920i	−0.346935	+	0.200303i	0.316400	+	0.948626i	$0.397526\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			7.11934i			0.253134i
$792$		0			0
$793$	33.3715	−	9.13333i	1.18506	−	0.324334i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−25.9716	−	44.9841i	−0.919961	−	1.59342i	−0.799471	−	0.600704i	$-0.794887\pi$
$797$	−25.9716	−	44.9841i	−0.919961	−	1.59342i	−0.120489	−	0.992715i	$-0.538446\pi$
$798$		0			0
$799$	−11.8497	−	6.84140i	−0.419211	−	0.242032i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.30093	+	2.25328i	−0.0459088	+	0.0795164i
$804$		0			0
$805$	−3.35235	−	5.80644i	−0.118155	−	0.204650i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−13.2120			−0.464508			−0.232254	−	0.972655i	$-0.574610\pi$
$809$	−13.2120			−0.464508			−0.232254	+	0.972655i	$0.574610\pi$
$810$		0			0
$811$		−	26.9033i		−	0.944704i	−0.881410	−	0.472352i	$-0.843405\pi$
$811$		−	26.9033i		−	0.944704i	0.881410	−	0.472352i	$-0.156595\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	9.37476	+	16.2376i	0.328383	+	0.568777i
$816$		0			0
$817$	−6.81695	−	3.93577i	−0.238495	−	0.137695i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	13.3222	+	7.69156i	0.464947	+	0.268437i	0.714122	−	0.700021i	$-0.246826\pi$
$821$	13.3222	+	7.69156i	0.464947	+	0.268437i	−0.249175	+	0.968458i	$0.580159\pi$
$822$		0			0
$823$	−10.0259	−	17.3653i	−0.349480	−	0.605318i	0.636677	−	0.771131i	$-0.280309\pi$
$823$	−10.0259	−	17.3653i	−0.349480	−	0.605318i	−0.986157	+	0.165813i	$0.946975\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	11.6593i		−	0.405435i	−0.979237	−	0.202718i	$-0.935023\pi$
$827$		−	11.6593i		−	0.405435i	0.979237	−	0.202718i	$-0.0649774\pi$
$828$		0			0
$829$	−32.7895			−1.13883			−0.569413	−	0.822052i	$-0.692829\pi$
$829$	−32.7895			−1.13883			−0.569413	+	0.822052i	$0.692829\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	14.7659	+	25.5752i	0.511607	+	0.886129i
$834$		0			0
$835$	19.6133	−	33.9712i	0.678745	−	1.17562i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−9.39251	−	5.42277i	−0.324266	−	0.187215i	0.329027	−	0.944321i	$-0.393279\pi$
$839$	−9.39251	−	5.42277i	−0.324266	−	0.187215i	−0.653292	+	0.757106i	$0.726613\pi$
$840$		0			0
$841$	−9.15119	−	15.8503i	−0.315558	−	0.546563i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−10.9361	−	18.4827i	−0.376214	−	0.635826i
$846$		0			0
$847$			17.0141i			0.584610i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−4.29277	+	2.47843i	−0.147154	+	0.0849595i
$852$		0			0
$853$	29.3981	+	16.9730i	1.00657	+	0.581144i	0.910186	−	0.414200i	$-0.135938\pi$
$853$	29.3981	+	16.9730i	1.00657	+	0.581144i	0.0963857	+	0.995344i	$0.469272\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.273868	−	0.474353i	0.00935515	−	0.0162036i	−0.861310	−	0.508080i	$-0.830355\pi$
$857$	0.273868	−	0.474353i	0.00935515	−	0.0162036i	0.870665	+	0.491876i	$0.163689\pi$
$858$		0			0
$859$	13.3494	+	23.1218i	0.455474	+	0.788904i	0.998715	−	0.0506723i	$-0.0161364\pi$
$859$	13.3494	+	23.1218i	0.455474	+	0.788904i	−0.543241	+	0.839577i	$0.682803\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			8.84803i			0.301190i	0.988596	+	0.150595i	$0.0481190\pi$
$863$			8.84803i			0.301190i	−0.988596	+	0.150595i	$0.951881\pi$
$864$		0			0
$865$		−	30.6734i		−	1.04293i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	33.0972	−	19.1087i	1.12275	−	0.648218i
$870$		0			0
$871$	28.8108	+	7.55501i	0.976218	+	0.255992i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−25.2251	+	43.6911i	−0.852763	+	1.47703i
$876$		0			0
$877$	11.4060	−	6.58525i	0.385153	−	0.222368i	−0.294905	−	0.955527i	$-0.595288\pi$
$877$	11.4060	−	6.58525i	0.385153	−	0.222368i	0.680058	+	0.733159i	$0.261955\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	3.52038			0.118605			0.0593024	−	0.998240i	$-0.481112\pi$
$881$	3.52038			0.118605			0.0593024	+	0.998240i	$0.481112\pi$
$882$		0			0
$883$	−23.7425			−0.799000			−0.399500	−	0.916733i	$-0.630816\pi$
$883$	−23.7425			−0.799000			−0.399500	+	0.916733i	$0.630816\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	2.39354	+	4.14574i	0.0803673	+	0.139200i	0.903408	−	0.428783i	$-0.141057\pi$
$887$	2.39354	+	4.14574i	0.0803673	+	0.139200i	−0.823040	+	0.567983i	$0.807724\pi$
$888$		0			0
$889$	−17.1401	−	9.89583i	−0.574860	−	0.331895i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−11.6169	+	20.1210i	−0.388743	+	0.673323i
$894$		0			0
$895$	−7.33834	+	4.23679i	−0.245294	+	0.141620i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	39.7000i		−	1.32407i
$900$		0			0
$901$	−20.9362			−0.697486
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−29.3334	+	16.9357i	−0.975076	+	0.562961i
$906$		0			0
$907$	2.21433	−	3.83533i	0.0735256	−	0.127350i	−0.826919	−	0.562322i	$-0.809908\pi$
$907$	2.21433	−	3.83533i	0.0735256	−	0.127350i	0.900444	+	0.434972i	$0.143242\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−25.2576	+	43.7474i	−0.836820	+	1.44942i	0.0557194	+	0.998446i	$0.482255\pi$
$911$	−25.2576	+	43.7474i	−0.836820	+	1.44942i	−0.892540	+	0.450969i	$0.851079\pi$
$912$		0			0
$913$	−10.5647	−	18.2985i	−0.349639	−	0.605593i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			55.5572i			1.83466i
$918$		0			0
$919$	−27.0493			−0.892275			−0.446138	−	0.894964i	$-0.647201\pi$
$919$	−27.0493			−0.892275			−0.446138	+	0.894964i	$0.647201\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−0.134988	+	0.136439i	−0.00444319	+	0.00449095i
$924$		0			0
$925$	10.0887	+	5.82472i	0.331715	+	0.191516i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	17.9409	+	10.3582i	0.588621	+	0.339841i	0.764552	−	0.644562i	$-0.222960\pi$
$929$	17.9409	+	10.3582i	0.588621	+	0.339841i	−0.175931	+	0.984403i	$0.556293\pi$
$930$		0			0
$931$	43.4273	−	25.0728i	1.42327	−	0.821726i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−12.0856			−0.395240
$936$		0			0
$937$	12.0164			0.392557			0.196279	−	0.980548i	$-0.437114\pi$
$937$	12.0164			0.392557			0.196279	+	0.980548i	$0.437114\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	16.4914	−	9.52132i	0.537604	−	0.310386i	−0.206503	−	0.978446i	$-0.566208\pi$
$941$	16.4914	−	9.52132i	0.537604	−	0.310386i	0.744108	+	0.668060i	$0.232875\pi$
$942$		0			0
$943$	−2.31850	−	1.33858i	−0.0755006	−	0.0435903i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−36.4342	−	21.0353i	−1.18395	−	0.683554i	−0.227026	−	0.973889i	$-0.572900\pi$
$947$	−36.4342	−	21.0353i	−1.18395	−	0.683554i	−0.956925	+	0.290334i	$0.906233\pi$
$948$		0			0
$949$	−2.50288	+	2.52978i	−0.0812469	+	0.0821201i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−33.9675			−1.10032			−0.550158	−	0.835060i	$-0.685433\pi$
$953$	−33.9675			−1.10032			−0.550158	+	0.835060i	$0.685433\pi$
$954$		0			0
$955$			32.3181i			1.04579i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−0.547544	−	0.948374i	−0.0176811	−	0.0306246i
$960$		0			0
$961$	1.15976	−	2.00877i	0.0374118	−	0.0647991i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	4.66494	−	8.07991i	0.150170	−	0.260102i
$966$		0			0
$967$	0.908277	−	0.524394i	0.0292082	−	0.0168634i	−0.485325	−	0.874334i	$-0.661299\pi$
$967$	0.908277	−	0.524394i	0.0292082	−	0.0168634i	0.514533	+	0.857471i	$0.327965\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1.63762			−0.0525538			−0.0262769	−	0.999655i	$-0.508365\pi$
$971$	−1.63762			−0.0525538			−0.0262769	+	0.999655i	$0.508365\pi$
$972$		0			0
$973$			49.3746i			1.58288i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−33.1964	+	19.1659i	−1.06205	+	0.613173i	−0.925998	−	0.377529i	$-0.876774\pi$
$977$	−33.1964	+	19.1659i	−1.06205	+	0.613173i	−0.136049	+	0.990702i	$0.543440\pi$
$978$		0			0
$979$	11.6047	−	20.1000i	0.370889	−	0.642398i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−48.0181	−	27.7233i	−1.53154	−	0.884236i	−0.999291	−	0.0376482i	$-0.988013\pi$
$983$	−48.0181	−	27.7233i	−1.53154	−	0.884236i	−0.532250	−	0.846587i	$-0.678653\pi$
$984$		0			0
$985$	21.1663	+	36.6612i	0.674416	+	1.16812i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.61410			−0.0513254
$990$		0			0
$991$	39.1442			1.24346			0.621728	−	0.783233i	$-0.286431\pi$
$991$	39.1442			1.24346			0.621728	+	0.783233i	$0.286431\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	2.99776	−	1.73076i	0.0950353	−	0.0548687i
$996$		0			0
$997$	−3.87163	+	6.70585i	−0.122616	+	0.212377i	−0.920798	−	0.390039i	$-0.872462\pi$
$997$	−3.87163	+	6.70585i	−0.122616	+	0.212377i	0.798183	+	0.602415i	$0.205795\pi$
$998$		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.1585.12		80
3.2	odd	2		936.2.cw.b.25.30	yes	80
9.4	even	3	inner	2808.2.cw.b.2521.29		80
9.5	odd	6		936.2.cw.b.337.29	yes	80
13.12	even	2	inner	2808.2.cw.b.1585.29		80
39.38	odd	2		936.2.cw.b.25.29	✓	80
117.77	odd	6		936.2.cw.b.337.30	yes	80
117.103	even	6	inner	2808.2.cw.b.2521.12		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
936.2.cw.b.25.29	✓	80	39.38	odd	2
936.2.cw.b.25.30	yes	80	3.2	odd	2
936.2.cw.b.337.29	yes	80	9.5	odd	6
936.2.cw.b.337.30	yes	80	117.77	odd	6
2808.2.cw.b.1585.12		80	1.1	even	1	trivial
2808.2.cw.b.1585.29		80	13.12	even	2	inner
2808.2.cw.b.2521.12		80	117.103	even	6	inner
2808.2.cw.b.2521.29		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)

\(f(q)\)	\(=\)	\(q+(-1.43066 + 0.825993i) q^{5} +(-3.63744 - 2.10008i) q^{7} +(2.28296 + 1.31807i) q^{11} +(2.56310 + 2.53585i) q^{13} +2.77519 q^{17} -4.71234i q^{19} +(-0.483145 - 0.836832i) q^{23} +(-1.13547 + 1.96669i) q^{25} +(-3.43884 + 5.95624i) q^{29} +(-4.99896 + 2.88615i) q^{31} +6.93860 q^{35} -5.12978i q^{37} +(2.39938 - 1.38528i) q^{41} +(0.835205 - 1.44662i) q^{43} +(-4.26985 - 2.46520i) q^{47} +(5.32066 + 9.21565i) q^{49} -7.54405 q^{53} -4.35485 q^{55} +(3.31701 - 1.91508i) q^{59} +(4.79798 - 8.31035i) q^{61} +(-5.76152 - 1.51083i) q^{65} +(7.15411 - 4.13043i) q^{67} +0.0532321i q^{71} +0.987000i q^{73} +(-5.53608 - 9.58878i) q^{77} +(7.24877 - 12.5552i) q^{79} +(-6.94144 - 4.00764i) q^{83} +(-3.97036 + 2.29229i) q^{85} -8.80436i q^{89} +(-3.99766 - 14.6067i) q^{91} +(3.89236 + 6.74176i) q^{95} +(1.11958 + 0.646389i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 6 q^{13} + 4 q^{17} - 10 q^{23} + 44 q^{25} + 52 q^{35} - 26 q^{43} + 48 q^{49} - 60 q^{53} - 16 q^{55} - 10 q^{61} + 26 q^{65} - 32 q^{77} + 6 q^{79} - 4 q^{91} - 8 q^{95}+O(q^{100}) \) 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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