LMFDB - Embedded newform 2808.2.cw.b.1585.11

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			1585.11
Character	$\chi$	$=$	2808.1585
Dual form			2808.2.cw.b.2521.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.55919 + 0.900198i) q^{5} +(2.30144 + 1.32873i) q^{7} +(1.01492 + 0.585964i) q^{11} +(-1.18440 - 3.40547i) q^{13} -1.33805 q^{17} -7.81675i q^{19} +(-2.83598 - 4.91206i) q^{23} +(-0.879286 + 1.52297i) q^{25} +(-3.00413 + 5.20331i) q^{29} +(2.23496 - 1.29036i) q^{31} -4.78450 q^{35} -5.90191i q^{37} +(-6.23926 + 3.60224i) q^{41} +(-5.04619 + 8.74025i) q^{43} +(-8.20794 - 4.73885i) q^{47} +(0.0310704 + 0.0538156i) q^{49} +12.3419 q^{53} -2.10994 q^{55} +(10.6080 - 6.12456i) q^{59} +(5.22284 - 9.04622i) q^{61} +(4.91230 + 4.24357i) q^{65} +(5.17129 - 2.98565i) q^{67} -1.35532i q^{71} -9.98135i q^{73} +(1.55718 + 2.69712i) q^{77} +(-2.14171 + 3.70956i) q^{79} +(-1.70208 - 0.982698i) q^{83} +(2.08627 - 1.20451i) q^{85} -8.41529i q^{89} +(1.79914 - 9.41121i) q^{91} +(7.03663 + 12.1878i) q^{95} +(0.765238 + 0.441810i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.55919	+	0.900198i	−0.697291	+	0.402581i	−0.806338	−	0.591456i	$-0.798553\pi$
$5$	−1.55919	+	0.900198i	−0.697291	+	0.402581i	0.109047	+	0.994037i	$0.465220\pi$
$6$		0			0
$7$	2.30144	+	1.32873i	0.869861	+	0.502214i	0.867302	−	0.497782i	$-0.165852\pi$
$7$	2.30144	+	1.32873i	0.869861	+	0.502214i	0.00255888	+	0.999997i	$0.499185\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.01492	+	0.585964i	0.306010	+	0.176675i	0.645140	−	0.764065i	$-0.276799\pi$
$11$	1.01492	+	0.585964i	0.306010	+	0.176675i	−0.339130	+	0.940740i	$0.610133\pi$
$12$		0			0
$13$	−1.18440	−	3.40547i	−0.328493	−	0.944506i
$13$	−1.18440	−	3.40547i	−0.328493	−	0.944506i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.33805			−0.324524			−0.162262	−	0.986748i	$-0.551879\pi$
$17$	−1.33805			−0.324524			−0.162262	+	0.986748i	$0.551879\pi$
$18$		0			0
$19$		−	7.81675i		−	1.79329i	−0.442754	−	0.896643i	$-0.645999\pi$
$19$		−	7.81675i		−	1.79329i	0.442754	−	0.896643i	$-0.354001\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.83598	−	4.91206i	−0.591342	−	1.02423i	−0.994052	−	0.108907i	$-0.965265\pi$
$23$	−2.83598	−	4.91206i	−0.591342	−	1.02423i	0.402710	−	0.915328i	$-0.368068\pi$
$24$		0			0
$25$	−0.879286	+	1.52297i	−0.175857	+	0.304594i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.00413	+	5.20331i	−0.557853	+	0.966230i	0.439822	+	0.898085i	$0.355042\pi$
$29$	−3.00413	+	5.20331i	−0.557853	+	0.966230i	−0.997675	+	0.0681455i	$0.978292\pi$
$30$		0			0
$31$	2.23496	−	1.29036i	0.401411	−	0.231755i	−0.285682	−	0.958325i	$-0.592220\pi$
$31$	2.23496	−	1.29036i	0.401411	−	0.231755i	0.687093	+	0.726570i	$0.258887\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.78450			−0.808728
$36$		0			0
$37$		−	5.90191i		−	0.970268i	−0.874440	−	0.485134i	$-0.838771\pi$
$37$		−	5.90191i		−	0.970268i	0.874440	−	0.485134i	$-0.161229\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−6.23926	+	3.60224i	−0.974408	+	0.562575i	−0.900577	−	0.434696i	$-0.856856\pi$
$41$	−6.23926	+	3.60224i	−0.974408	+	0.562575i	−0.0738310	+	0.997271i	$0.523523\pi$
$42$		0			0
$43$	−5.04619	+	8.74025i	−0.769536	+	1.33288i	0.168279	+	0.985739i	$0.446179\pi$
$43$	−5.04619	+	8.74025i	−0.769536	+	1.33288i	−0.937815	+	0.347136i	$0.887154\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−8.20794	−	4.73885i	−1.19725	−	0.691233i	−0.237309	−	0.971434i	$-0.576265\pi$
$47$	−8.20794	−	4.73885i	−1.19725	−	0.691233i	−0.959941	+	0.280201i	$0.909599\pi$
$48$		0			0
$49$	0.0310704	+	0.0538156i	0.00443864	+	0.00768794i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	12.3419			1.69529			0.847645	−	0.530564i	$-0.178020\pi$
$53$	12.3419			1.69529			0.847645	+	0.530564i	$0.178020\pi$
$54$		0			0
$55$	−2.10994			−0.284504
$56$		0			0
$57$		0			0
$58$		0			0
$59$	10.6080	−	6.12456i	1.38105	−	0.797349i	0.388766	−	0.921337i	$-0.372901\pi$
$59$	10.6080	−	6.12456i	1.38105	−	0.797349i	0.992284	+	0.123987i	$0.0395682\pi$
$60$		0			0
$61$	5.22284	−	9.04622i	0.668716	−	1.15825i	−0.309547	−	0.950884i	$-0.600178\pi$
$61$	5.22284	−	9.04622i	0.668716	−	1.15825i	0.978263	−	0.207366i	$-0.0664891\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.91230	+	4.24357i	0.609295	+	0.526350i
$66$		0			0
$67$	5.17129	−	2.98565i	0.631774	−	0.364755i	−0.149665	−	0.988737i	$-0.547819\pi$
$67$	5.17129	−	2.98565i	0.631774	−	0.364755i	0.781439	+	0.623982i	$0.214486\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	1.35532i		−	0.160847i	−0.996761	−	0.0804236i	$-0.974373\pi$
$71$		−	1.35532i		−	0.160847i	0.996761	−	0.0804236i	$-0.0256273\pi$
$72$		0			0
$73$		−	9.98135i		−	1.16823i	−0.811672	−	0.584114i	$-0.801442\pi$
$73$		−	9.98135i		−	1.16823i	0.811672	−	0.584114i	$-0.198558\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.55718	+	2.69712i	0.177457	+	0.307365i
$78$		0			0
$79$	−2.14171	+	3.70956i	−0.240962	+	0.417358i	−0.960988	−	0.276588i	$-0.910796\pi$
$79$	−2.14171	+	3.70956i	−0.240962	+	0.417358i	0.720027	+	0.693946i	$0.244129\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.70208	−	0.982698i	−0.186828	−	0.107865i	0.403669	−	0.914905i	$-0.367735\pi$
$83$	−1.70208	−	0.982698i	−0.186828	−	0.107865i	−0.590497	+	0.807040i	$0.701068\pi$
$84$		0			0
$85$	2.08627	−	1.20451i	0.226287	−	0.130647i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	8.41529i		−	0.892019i	−0.895028	−	0.446009i	$-0.852845\pi$
$89$		−	8.41529i		−	0.892019i	0.895028	−	0.446009i	$-0.147155\pi$
$90$		0			0
$91$	1.79914	−	9.41121i	0.188602	−	0.986563i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	7.03663	+	12.1878i	0.721943	+	1.25044i
$96$		0			0
$97$	0.765238	+	0.441810i	0.0776982	+	0.0448591i	0.538346	−	0.842724i	$-0.319049\pi$
$97$	0.765238	+	0.441810i	0.0776982	+	0.0448591i	−0.460648	+	0.887583i	$0.652383\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	0.204568	−	0.354322i	0.0203553	−	0.0352564i	−0.855668	−	0.517525i	$-0.826854\pi$
$101$	0.204568	−	0.354322i	0.0203553	−	0.0352564i	0.876024	+	0.482268i	$0.160187\pi$
$102$		0			0
$103$	5.03049	+	8.71307i	0.495669	+	0.858524i	0.999988	−	0.00499380i	$-0.00158958\pi$
$103$	5.03049	+	8.71307i	0.495669	+	0.858524i	−0.504319	+	0.863518i	$0.668256\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.44751			0.429957			0.214979	−	0.976619i	$-0.431032\pi$
$107$	4.44751			0.429957			0.214979	+	0.976619i	$0.431032\pi$
$108$		0			0
$109$			8.51488i			0.815578i	0.913076	+	0.407789i	$0.133700\pi$
$109$			8.51488i			0.815578i	−0.913076	+	0.407789i	$0.866300\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6.92238	−	11.9899i	−0.651203	−	1.12792i	−0.982831	−	0.184506i	$-0.940931\pi$
$113$	−6.92238	−	11.9899i	−0.651203	−	1.12792i	0.331629	−	0.943410i	$-0.392402\pi$
$114$		0			0
$115$	8.84365	+	5.10588i	0.824675	+	0.476126i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.07943	−	1.77791i	−0.282291	−	0.162980i
$120$		0			0
$121$	−4.81329	−	8.33687i	−0.437572	−	0.757897i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	12.1681i		−	1.08835i
$126$		0			0
$127$	−8.12200			−0.720711			−0.360356	−	0.932815i	$-0.617345\pi$
$127$	−8.12200			−0.720711			−0.360356	+	0.932815i	$0.617345\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.89862	−	10.2167i	−0.515364	−	0.892637i	−0.999841	−	0.0178331i	$-0.994323\pi$
$131$	−5.89862	−	10.2167i	−0.515364	−	0.892637i	0.484477	−	0.874804i	$-0.339010\pi$
$132$		0			0
$133$	10.3864	−	17.9898i	0.900614	−	1.55991i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−17.9997	−	10.3921i	−1.53782	−	0.887861i	−0.998966	−	0.0454612i	$-0.985524\pi$
$137$	−17.9997	−	10.3921i	−1.53782	−	0.887861i	−0.538854	−	0.842399i	$-0.681142\pi$
$138$		0			0
$139$	−2.54165	−	4.40227i	−0.215580	−	0.373396i	0.737872	−	0.674941i	$-0.235831\pi$
$139$	−2.54165	−	4.40227i	−0.215580	−	0.373396i	−0.953452	+	0.301545i	$0.902498\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.793412	−	4.15029i	0.0663484	−	0.347065i
$144$		0			0
$145$		−	10.8173i		−	0.898325i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	11.9131	−	6.87805i	0.975962	−	0.563472i	0.0749137	−	0.997190i	$-0.476132\pi$
$149$	11.9131	−	6.87805i	0.975962	−	0.563472i	0.901049	+	0.433718i	$0.142799\pi$
$150$		0			0
$151$	16.0415	+	9.26157i	1.30544	+	0.753696i	0.981331	−	0.192324i	$-0.0616024\pi$
$151$	16.0415	+	9.26157i	1.30544	+	0.753696i	0.324108	+	0.946020i	$0.394936\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.32315	+	4.02382i	−0.186600	+	0.323201i
$156$		0			0
$157$	−4.00643	−	6.93935i	−0.319748	−	0.553820i	0.660687	−	0.750661i	$-0.270265\pi$
$157$	−4.00643	−	6.93935i	−0.319748	−	0.553820i	−0.980435	+	0.196841i	$0.936932\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	15.0730i		−	1.18792i
$162$		0			0
$163$		−	0.362314i		−	0.0283786i	−0.999899	−	0.0141893i	$-0.995483\pi$
$163$		−	0.362314i		−	0.0283786i	0.999899	−	0.0141893i	$-0.00451675\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−11.3437	+	6.54926i	−0.877798	+	0.506797i	−0.869932	−	0.493172i	$-0.835837\pi$
$167$	−11.3437	+	6.54926i	−0.877798	+	0.506797i	−0.00786643	+	0.999969i	$0.502504\pi$
$168$		0			0
$169$	−10.1944	+	8.06686i	−0.784185	+	0.620527i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−0.104110	+	0.180324i	−0.00791535	+	0.0137098i	−0.869956	−	0.493129i	$-0.835853\pi$
$173$	−0.104110	+	0.180324i	−0.00791535	+	0.0137098i	0.862041	+	0.506839i	$0.169186\pi$
$174$		0			0
$175$	−4.04724	+	2.33668i	−0.305943	+	0.176636i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−15.4820			−1.15718			−0.578589	−	0.815619i	$-0.696397\pi$
$179$	−15.4820			−1.15718			−0.578589	+	0.815619i	$0.696397\pi$
$180$		0			0
$181$	9.66553			0.718433			0.359217	−	0.933254i	$-0.383044\pi$
$181$	9.66553			0.718433			0.359217	+	0.933254i	$0.383044\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	5.31289	+	9.20219i	0.390611	+	0.676559i
$186$		0			0
$187$	−1.35801	−	0.784047i	−0.0993074	−	0.0573352i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2.25148	−	3.89967i	0.162911	−	0.282171i	−0.773000	−	0.634406i	$-0.781245\pi$
$191$	2.25148	−	3.89967i	0.162911	−	0.282171i	0.935912	+	0.352235i	$0.114578\pi$
$192$		0			0
$193$	13.6732	−	7.89423i	0.984219	−	0.568239i	0.0806778	−	0.996740i	$-0.474292\pi$
$193$	13.6732	−	7.89423i	0.984219	−	0.568239i	0.903541	+	0.428501i	$0.140958\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	2.32267i		−	0.165484i	−0.996571	−	0.0827418i	$-0.973632\pi$
$197$		−	2.32267i		−	0.165484i	0.996571	−	0.0827418i	$-0.0263677\pi$
$198$		0			0
$199$	−23.4999			−1.66586			−0.832932	−	0.553375i	$-0.813340\pi$
$199$	−23.4999			−1.66586			−0.832932	+	0.553375i	$0.813340\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−13.8276	+	7.98339i	−0.970510	+	0.560324i
$204$		0			0
$205$	6.48546	−	11.2331i	0.452964	−	0.784556i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.58034	−	7.93337i	0.316828	−	0.548763i
$210$		0			0
$211$	7.49343	+	12.9790i	0.515869	+	0.893511i	0.999830	+	0.0184218i	$0.00586417\pi$
$211$	7.49343	+	12.9790i	0.515869	+	0.893511i	−0.483961	+	0.875089i	$0.660802\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	18.1703i		−	1.23920i
$216$		0			0
$217$	6.85816			0.465562
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.58478	+	4.55667i	0.106604	+	0.306515i
$222$		0			0
$223$	1.22733	+	0.708599i	0.0821880	+	0.0474513i	0.540531	−	0.841324i	$-0.318223\pi$
$223$	1.22733	+	0.708599i	0.0821880	+	0.0474513i	−0.458343	+	0.888776i	$0.651557\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−5.77352	−	3.33334i	−0.383202	−	0.221242i	0.296008	−	0.955185i	$-0.404344\pi$
$227$	−5.77352	−	3.33334i	−0.383202	−	0.221242i	−0.679210	+	0.733944i	$0.737678\pi$
$228$		0			0
$229$	8.33635	−	4.81300i	0.550882	−	0.318052i	−0.198596	−	0.980081i	$-0.563638\pi$
$229$	8.33635	−	4.81300i	0.550882	−	0.318052i	0.749478	+	0.662030i	$0.230305\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4.63723			0.303795			0.151897	−	0.988396i	$-0.451462\pi$
$233$	4.63723			0.303795			0.151897	+	0.988396i	$0.451462\pi$
$234$		0			0
$235$	17.0636			1.11311
$236$		0			0
$237$		0			0
$238$		0			0
$239$	20.5980	−	11.8922i	1.33237	−	0.769245i	0.346708	−	0.937973i	$-0.387299\pi$
$239$	20.5980	−	11.8922i	1.33237	−	0.769245i	0.985663	+	0.168728i	$0.0539660\pi$
$240$		0			0
$241$	19.5297	+	11.2755i	1.25802	+	0.726318i	0.972689	−	0.232111i	$-0.0745634\pi$
$241$	19.5297	+	11.2755i	1.25802	+	0.726318i	0.285330	+	0.958429i	$0.407897\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.0968894	−	0.0559391i	−0.00619004	−	0.00357382i
$246$		0			0
$247$	−26.6197	+	9.25815i	−1.69377	+	0.589082i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−28.2420			−1.78262			−0.891309	−	0.453396i	$-0.850213\pi$
$251$	−28.2420			−1.78262			−0.891309	+	0.453396i	$0.850213\pi$
$252$		0			0
$253$		−	6.64712i		−	0.417901i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3.48493	−	6.03608i	−0.217384	−	0.376520i	0.736623	−	0.676303i	$-0.236419\pi$
$257$	−3.48493	−	6.03608i	−0.217384	−	0.376520i	−0.954007	+	0.299783i	$0.903086\pi$
$258$		0			0
$259$	7.84207	−	13.5829i	0.487282	−	0.843998i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	0.777511	−	1.34669i	0.0479434	−	0.0830404i	−0.841058	−	0.540945i	$-0.818067\pi$
$263$	0.777511	−	1.34669i	0.0479434	−	0.0830404i	0.889001	+	0.457905i	$0.151400\pi$
$264$		0			0
$265$	−19.2434	+	11.1102i	−1.18211	+	0.682491i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	10.3894			0.633455			0.316727	−	0.948517i	$-0.397416\pi$
$269$	10.3894			0.633455			0.316727	+	0.948517i	$0.397416\pi$
$270$		0			0
$271$		−	23.8498i		−	1.44877i	−0.689394	−	0.724387i	$-0.742123\pi$
$271$		−	23.8498i		−	1.44877i	0.689394	−	0.724387i	$-0.257877\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.78481	+	1.03046i	−0.107628	+	0.0621391i
$276$		0			0
$277$	−10.8536	+	18.7990i	−0.652130	+	1.12952i	0.330475	+	0.943815i	$0.392791\pi$
$277$	−10.8536	+	18.7990i	−0.652130	+	1.12952i	−0.982605	+	0.185707i	$0.940542\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.843083	−	0.486754i	−0.0502941	−	0.0290373i	0.474642	−	0.880179i	$-0.342577\pi$
$281$	−0.843083	−	0.486754i	−0.0502941	−	0.0290373i	−0.524936	+	0.851142i	$0.675911\pi$
$282$		0			0
$283$	12.2668	+	21.2468i	0.729188	+	1.26299i	0.957227	+	0.289338i	$0.0934352\pi$
$283$	12.2668	+	21.2468i	0.729188	+	1.26299i	−0.228039	+	0.973652i	$0.573231\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−19.1457			−1.13013
$288$		0			0
$289$	−15.2096			−0.894684
$290$		0			0
$291$		0			0
$292$		0			0
$293$	23.1545	−	13.3682i	1.35270	−	0.780981i	0.364073	−	0.931371i	$-0.381386\pi$
$293$	23.1545	−	13.3682i	1.35270	−	0.780981i	0.988627	+	0.150389i	$0.0480527\pi$
$294$		0			0
$295$	−11.0266	+	19.0987i	−0.641995	+	1.11197i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−13.3689	+	15.4757i	−0.773145	+	0.894980i
$300$		0			0
$301$	−23.2269	+	13.4101i	−1.33878	+	0.772944i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			18.8064i			1.07685i
$306$		0			0
$307$			19.0123i			1.08509i	0.840027	+	0.542544i	$0.182539\pi$
$307$			19.0123i			1.08509i	−0.840027	+	0.542544i	$0.817461\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	4.62413	+	8.00923i	0.262211	+	0.454162i	0.966829	−	0.255424i	$-0.0822152\pi$
$311$	4.62413	+	8.00923i	0.262211	+	0.454162i	−0.704618	+	0.709586i	$0.748882\pi$
$312$		0			0
$313$	14.0904	−	24.4053i	0.796436	−	1.37947i	−0.125488	−	0.992095i	$-0.540050\pi$
$313$	14.0904	−	24.4053i	0.796436	−	1.37947i	0.921923	−	0.387372i	$-0.126617\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	4.02813	+	2.32564i	0.226242	+	0.130621i	0.608837	−	0.793295i	$-0.291636\pi$
$317$	4.02813	+	2.32564i	0.226242	+	0.130621i	−0.382595	+	0.923916i	$0.624970\pi$
$318$		0			0
$319$	−6.09791	+	3.52063i	−0.341417	+	0.197117i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			10.4592i			0.581964i
$324$		0			0
$325$	6.22784	+	1.19058i	0.345458	+	0.0660414i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−12.5934	−	21.8123i	−0.694294	−	1.20255i
$330$		0			0
$331$	11.1406	+	6.43205i	0.612345	+	0.353538i	0.773883	−	0.633329i	$-0.218312\pi$
$331$	11.1406	+	6.43205i	0.612345	+	0.353538i	−0.161538	+	0.986867i	$0.551645\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5.37535	+	9.31038i	−0.293687	+	0.508680i
$336$		0			0
$337$	−4.44018	−	7.69061i	−0.241872	−	0.418934i	0.719376	−	0.694621i	$-0.244428\pi$
$337$	−4.44018	−	7.69061i	−0.241872	−	0.418934i	−0.961247	+	0.275687i	$0.911095\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	3.02441			0.163781
$342$		0			0
$343$		−	18.4371i		−	0.995512i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.97419	+	3.41939i	0.105980	+	0.183563i	0.914138	−	0.405403i	$-0.132869\pi$
$347$	1.97419	+	3.41939i	0.105980	+	0.183563i	−0.808158	+	0.588965i	$0.799535\pi$
$348$		0			0
$349$	26.8533	+	15.5037i	1.43742	+	0.829896i	0.997670	−	0.0682238i	$-0.0217332\pi$
$349$	26.8533	+	15.5037i	1.43742	+	0.829896i	0.439752	+	0.898119i	$0.355067\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4.29737	−	2.48109i	−0.228726	−	0.132055i	0.381258	−	0.924469i	$-0.375491\pi$
$353$	−4.29737	−	2.48109i	−0.228726	−	0.132055i	−0.609984	+	0.792414i	$0.708824\pi$
$354$		0			0
$355$	1.22006	+	2.11320i	0.0647540	+	0.112157i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			23.5827i			1.24465i	0.782760	+	0.622324i	$0.213811\pi$
$359$			23.5827i			1.24465i	−0.782760	+	0.622324i	$0.786189\pi$
$360$		0			0
$361$	−42.1016			−2.21587
$362$		0			0
$363$		0			0
$364$		0			0
$365$	8.98519	+	15.5628i	0.470306	+	0.814595i
$366$		0			0
$367$	−1.98759	+	3.44261i	−0.103752	+	0.179703i	−0.913227	−	0.407450i	$-0.866418\pi$
$367$	−1.98759	+	3.44261i	−0.103752	+	0.179703i	0.809476	+	0.587153i	$0.199751\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	28.4041	+	16.3991i	1.47467	+	0.851399i
$372$		0			0
$373$	11.4714	+	19.8691i	0.593969	+	1.02878i	0.993692	+	0.112148i	$0.0357730\pi$
$373$	11.4714	+	19.8691i	0.593969	+	1.02878i	−0.399723	+	0.916636i	$0.630894\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	21.2778	+	4.06768i	1.09586	+	0.209496i
$378$		0			0
$379$		−	14.0412i		−	0.721246i	−0.932712	−	0.360623i	$-0.882564\pi$
$379$		−	14.0412i		−	0.721246i	0.932712	−	0.360623i	$-0.117436\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.22496	−	0.707232i	0.0625926	−	0.0361379i	−0.468377	−	0.883529i	$-0.655161\pi$
$383$	1.22496	−	0.707232i	0.0625926	−	0.0361379i	0.530970	+	0.847391i	$0.321828\pi$
$384$		0			0
$385$	−4.85588	−	2.80354i	−0.247479	−	0.142882i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	12.7767	−	22.1299i	0.647805	−	1.12203i	−0.335841	−	0.941919i	$-0.609020\pi$
$389$	12.7767	−	22.1299i	0.647805	−	1.12203i	0.983646	−	0.180113i	$-0.0576463\pi$
$390$		0			0
$391$	3.79467	+	6.57256i	0.191905	+	0.332389i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	7.71187i		−	0.388026i
$396$		0			0
$397$			0.254531i			0.0127746i	0.999980	+	0.00638728i	$0.00203315\pi$
$397$			0.254531i			0.0127746i	−0.999980	+	0.00638728i	$0.997967\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−0.372180	+	0.214878i	−0.0185858	+	0.0107305i	−0.509264	−	0.860610i	$-0.670082\pi$
$401$	−0.372180	+	0.214878i	−0.0185858	+	0.0107305i	0.490678	+	0.871341i	$0.336749\pi$
$402$		0			0
$403$	−7.04135	−	6.08279i	−0.350755	−	0.303005i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3.45831	−	5.98996i	0.171422	−	0.296911i
$408$		0			0
$409$	−4.51984	+	2.60953i	−0.223492	+	0.129033i	−0.607566	−	0.794269i	$-0.707854\pi$
$409$	−4.51984	+	2.60953i	−0.223492	+	0.129033i	0.384074	+	0.923302i	$0.374521\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	32.5516			1.60176
$414$		0			0
$415$	3.53849			0.173698
$416$		0			0
$417$		0			0
$418$		0			0
$419$	11.5418	+	19.9910i	0.563853	+	0.976622i	0.997155	+	0.0753735i	$0.0240149\pi$
$419$	11.5418	+	19.9910i	0.563853	+	0.976622i	−0.433302	+	0.901249i	$0.642652\pi$
$420$		0			0
$421$	−13.6048	−	7.85475i	−0.663059	−	0.382817i	0.130383	−	0.991464i	$-0.458379\pi$
$421$	−13.6048	−	7.85475i	−0.663059	−	0.382817i	−0.793441	+	0.608647i	$0.791713\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.17652	−	2.03780i	0.0570698	−	0.0988479i
$426$		0			0
$427$	24.0401	−	13.8795i	1.16338	−	0.671678i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			28.2548i			1.36098i	0.732756	+	0.680492i	$0.238234\pi$
$431$			28.2548i			1.36098i	−0.732756	+	0.680492i	$0.761766\pi$
$432$		0			0
$433$	1.00582			0.0483366			0.0241683	−	0.999708i	$-0.492306\pi$
$433$	1.00582			0.0483366			0.0241683	+	0.999708i	$0.492306\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−38.3963	+	22.1681i	−1.83675	+	1.06045i
$438$		0			0
$439$	1.05111	−	1.82057i	0.0501667	−	0.0868912i	−0.839852	−	0.542816i	$-0.817358\pi$
$439$	1.05111	−	1.82057i	0.0501667	−	0.0868912i	0.890018	+	0.455925i	$0.150691\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	11.5606	−	20.0236i	0.549262	−	0.951350i	−0.449063	−	0.893500i	$-0.648242\pi$
$443$	11.5606	−	20.0236i	0.549262	−	0.951350i	0.998325	−	0.0578496i	$-0.0184244\pi$
$444$		0			0
$445$	7.57543	+	13.1210i	0.359110	+	0.621996i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			0.677149i			0.0319566i	0.999872	+	0.0159783i	$0.00508627\pi$
$449$			0.677149i			0.0319566i	−0.999872	+	0.0159783i	$0.994914\pi$
$450$		0			0
$451$	−8.44313			−0.397571
$452$		0			0
$453$		0			0
$454$		0			0
$455$	5.66675	+	16.2934i	0.265661	+	0.763849i
$456$		0			0
$457$	−10.1204	−	5.84300i	−0.473411	−	0.273324i	0.244255	−	0.969711i	$-0.421457\pi$
$457$	−10.1204	−	5.84300i	−0.473411	−	0.273324i	−0.717667	+	0.696387i	$0.754790\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	7.11644	+	4.10868i	0.331446	+	0.191360i	0.656483	−	0.754341i	$-0.272043\pi$
$461$	7.11644	+	4.10868i	0.331446	+	0.191360i	−0.325037	+	0.945701i	$0.605377\pi$
$462$		0			0
$463$	−3.75857	+	2.17001i	−0.174676	+	0.100849i	−0.584789	−	0.811186i	$-0.698823\pi$
$463$	−3.75857	+	2.17001i	−0.174676	+	0.100849i	0.410113	+	0.912035i	$0.365489\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−31.2648			−1.44676			−0.723382	−	0.690448i	$-0.757413\pi$
$467$	−31.2648			−1.44676			−0.723382	+	0.690448i	$0.757413\pi$
$468$		0			0
$469$	15.8685			0.732740
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−10.2429	+	5.91377i	−0.470971	+	0.271915i
$474$		0			0
$475$	11.9047	+	6.87316i	0.546223	+	0.315362i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4.58937	−	2.64967i	−0.209694	−	0.121067i	0.391475	−	0.920189i	$-0.371965\pi$
$479$	−4.58937	−	2.64967i	−0.209694	−	0.121067i	−0.601169	+	0.799122i	$0.705298\pi$
$480$		0			0
$481$	−20.0987	+	6.99021i	−0.916424	+	0.318726i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.59087			−0.0722376
$486$		0			0
$487$		−	13.6948i		−	0.620571i	−0.950643	−	0.310285i	$-0.899575\pi$
$487$		−	13.6948i		−	0.620571i	0.950643	−	0.310285i	$-0.100425\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	0.206712	+	0.358035i	0.00932876	+	0.0161579i	0.870652	−	0.491899i	$-0.163697\pi$
$491$	0.206712	+	0.358035i	0.00932876	+	0.0161579i	−0.861323	+	0.508057i	$0.830364\pi$
$492$		0			0
$493$	4.01967	−	6.96227i	0.181037	−	0.313565i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.80086	−	3.11919i	0.0807798	−	0.139915i
$498$		0			0
$499$	−34.2594	+	19.7796i	−1.53366	+	0.885459i	−0.534471	+	0.845187i	$0.679489\pi$
$499$	−34.2594	+	19.7796i	−1.53366	+	0.885459i	−0.999189	+	0.0402716i	$0.987178\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	5.35395			0.238721			0.119360	−	0.992851i	$-0.461916\pi$
$503$	5.35395			0.238721			0.119360	+	0.992851i	$0.461916\pi$
$504$		0			0
$505$			0.736607i			0.0327786i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−29.8564	+	17.2376i	−1.32336	+	0.764044i	−0.984264	−	0.176706i	$-0.943456\pi$
$509$	−29.8564	+	17.2376i	−1.32336	+	0.764044i	−0.339100	+	0.940750i	$0.610122\pi$
$510$		0			0
$511$	13.2626	−	22.9714i	0.586701	−	1.01620i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−15.6870	−	9.05688i	−0.691251	−	0.399094i
$516$		0			0
$517$	−5.55360	−	9.61911i	−0.244247	−	0.423048i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.293147			−0.0128430			−0.00642149	−	0.999979i	$-0.502044\pi$
$521$	−0.293147			−0.0128430			−0.00642149	+	0.999979i	$0.502044\pi$
$522$		0			0
$523$	−8.15975			−0.356801			−0.178401	−	0.983958i	$-0.557092\pi$
$523$	−8.15975			−0.356801			−0.178401	+	0.983958i	$0.557092\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.99048	+	1.72655i	−0.130267	+	0.0752099i
$528$		0			0
$529$	−4.58554	+	7.94239i	−0.199371	+	0.345321i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	19.6571	+	16.9811i	0.851442	+	0.735533i
$534$		0			0
$535$	−6.93451	+	4.00364i	−0.299805	+	0.173093i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			0.0728247i			0.00313678i
$540$		0			0
$541$		−	24.8891i		−	1.07007i	−0.844831	−	0.535034i	$-0.820299\pi$
$541$		−	24.8891i		−	1.07007i	0.844831	−	0.535034i	$-0.179701\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−7.66508	−	13.2763i	−0.328336	−	0.568695i
$546$		0			0
$547$	−4.99309	+	8.64829i	−0.213489	+	0.369774i	−0.952804	−	0.303586i	$-0.901816\pi$
$547$	−4.99309	+	8.64829i	−0.213489	+	0.369774i	0.739315	+	0.673360i	$0.235150\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	40.6730	+	23.4826i	1.73273	+	1.00039i
$552$		0			0
$553$	−9.85803	+	5.69154i	−0.419206	+	0.242029i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			10.3622i			0.439059i	0.975606	+	0.219530i	$0.0704522\pi$
$557$			10.3622i			0.439059i	−0.975606	+	0.219530i	$0.929548\pi$
$558$		0			0
$559$	35.7413	+	6.83268i	1.51170	+	0.288992i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−0.0915592	−	0.158585i	−0.00385876	−	0.00668357i	0.864090	−	0.503338i	$-0.167895\pi$
$563$	−0.0915592	−	0.158585i	−0.00385876	−	0.00668357i	−0.867948	+	0.496654i	$0.834562\pi$
$564$		0			0
$565$	21.5866	+	12.4630i	0.908155	+	0.524324i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−5.73133	+	9.92696i	−0.240270	+	0.416160i	−0.960791	−	0.277273i	$-0.910569\pi$
$569$	−5.73133	+	9.92696i	−0.240270	+	0.416160i	0.720521	+	0.693433i	$0.243903\pi$
$570$		0			0
$571$	−17.1416	−	29.6902i	−0.717356	−	1.24250i	−0.962044	−	0.272894i	$-0.912019\pi$
$571$	−17.1416	−	29.6902i	−0.717356	−	1.24250i	0.244688	−	0.969602i	$-0.421314\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	9.97454			0.415967
$576$		0			0
$577$		−	0.249728i		−	0.0103963i	−0.999986	−	0.00519815i	$-0.998345\pi$
$577$		−	0.249728i		−	0.0103963i	0.999986	−	0.00519815i	$-0.00165463\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2.61149	−	4.52323i	−0.108343	−	0.187655i
$582$		0			0
$583$	12.5260	+	7.23191i	0.518775	+	0.299515i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	8.57593	+	4.95132i	0.353967	+	0.204363i	0.666431	−	0.745567i	$-0.267821\pi$
$587$	8.57593	+	4.95132i	0.353967	+	0.204363i	−0.312464	+	0.949929i	$0.601154\pi$
$588$		0			0
$589$	−10.0864	−	17.4701i	−0.415603	−	0.719845i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			40.4008i			1.65906i	0.558461	+	0.829531i	$0.311392\pi$
$593$			40.4008i			1.65906i	−0.558461	+	0.829531i	$0.688608\pi$
$594$		0			0
$595$	6.40188			0.262451
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−2.83916	−	4.91757i	−0.116005	−	0.200926i	0.802176	−	0.597087i	$-0.203675\pi$
$599$	−2.83916	−	4.91757i	−0.116005	−	0.200926i	−0.918181	+	0.396161i	$0.870342\pi$
$600$		0			0
$601$	3.12330	−	5.40971i	0.127402	−	0.220667i	−0.795267	−	0.606259i	$-0.792670\pi$
$601$	3.12330	−	5.40971i	0.127402	−	0.220667i	0.922669	+	0.385592i	$0.126003\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	15.0097	+	8.66584i	0.610230	+	0.352316i
$606$		0			0
$607$	−20.8317	−	36.0816i	−0.845532	−	1.46451i	−0.885158	−	0.465291i	$-0.845950\pi$
$607$	−20.8317	−	36.0816i	−0.845532	−	1.46451i	0.0396256	−	0.999215i	$-0.487383\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−6.41654	+	33.5645i	−0.259586	+	1.35788i
$612$		0			0
$613$		−	5.54519i		−	0.223968i	−0.993710	−	0.111984i	$-0.964279\pi$
$613$		−	5.54519i		−	0.223968i	0.993710	−	0.111984i	$-0.0357205\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	28.6171	−	16.5221i	1.15208	−	0.665154i	0.202688	−	0.979243i	$-0.435032\pi$
$617$	28.6171	−	16.5221i	1.15208	−	0.665154i	0.949393	+	0.314089i	$0.101699\pi$
$618$		0			0
$619$	−41.7437	−	24.1007i	−1.67782	−	0.968689i	−0.963047	−	0.269332i	$-0.913197\pi$
$619$	−41.7437	−	24.1007i	−1.67782	−	0.968689i	−0.714772	−	0.699358i	$-0.753470\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	11.1817	−	19.3672i	0.447985	−	0.775932i
$624$		0			0
$625$	6.55728	+	11.3575i	0.262291	+	0.454302i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			7.89702i			0.314875i
$630$		0			0
$631$			47.4191i			1.88773i	0.330337	+	0.943863i	$0.392838\pi$
$631$			47.4191i			1.88773i	−0.330337	+	0.943863i	$0.607162\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	12.6637	−	7.31141i	0.502545	−	0.290145i
$636$		0			0
$637$	0.146467	−	0.169548i	0.00580325	−	0.00671775i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−7.76535	+	13.4500i	−0.306713	+	0.531242i	−0.977641	−	0.210280i	$-0.932563\pi$
$641$	−7.76535	+	13.4500i	−0.306713	+	0.531242i	0.670928	+	0.741522i	$0.265896\pi$
$642$		0			0
$643$	−25.4196	+	14.6760i	−1.00245	+	0.578766i	−0.908973	−	0.416855i	$-0.863132\pi$
$643$	−25.4196	+	14.6760i	−1.00245	+	0.578766i	−0.0934792	+	0.995621i	$0.529799\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	2.49961			0.0982699			0.0491350	−	0.998792i	$-0.484354\pi$
$647$	2.49961			0.0982699			0.0491350	+	0.998792i	$0.484354\pi$
$648$		0			0
$649$	14.3551			0.563486
$650$		0			0
$651$		0			0
$652$		0			0
$653$	0.184801	+	0.320085i	0.00723182	+	0.0125259i	0.869619	−	0.493724i	$-0.164365\pi$
$653$	0.184801	+	0.320085i	0.00723182	+	0.0125259i	−0.862387	+	0.506250i	$0.831031\pi$
$654$		0			0
$655$	18.3941	+	10.6198i	0.718718	+	0.414952i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	7.51113	−	13.0097i	0.292592	−	0.506784i	−0.681830	−	0.731511i	$-0.738816\pi$
$659$	7.51113	−	13.0097i	0.292592	−	0.506784i	0.974422	+	0.224727i	$0.0721489\pi$
$660$		0			0
$661$	−29.4593	+	17.0084i	−1.14584	+	0.661548i	−0.947869	−	0.318660i	$-0.896767\pi$
$661$	−29.4593	+	17.0084i	−1.14584	+	0.661548i	−0.197967	+	0.980209i	$0.563434\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			37.3992i			1.45028i
$666$		0			0
$667$	34.0786			1.31953
$668$		0			0
$669$		0			0
$670$		0			0
$671$	10.6015	−	6.12079i	0.409267	−	0.236291i
$672$		0			0
$673$	−14.0001	+	24.2489i	−0.539664	+	0.934726i	0.459258	+	0.888303i	$0.348116\pi$
$673$	−14.0001	+	24.2489i	−0.539664	+	0.934726i	−0.998922	+	0.0464229i	$0.985218\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−12.4662	+	21.5921i	−0.479115	+	0.829852i	−0.999713	−	0.0239498i	$-0.992376\pi$
$677$	−12.4662	+	21.5921i	−0.479115	+	0.829852i	0.520598	+	0.853802i	$0.325709\pi$
$678$		0			0
$679$	1.17410	+	2.03360i	0.0450577	+	0.0780423i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	33.7051i		−	1.28969i	−0.764313	−	0.644845i	$-0.776922\pi$
$683$		−	33.7051i		−	1.28969i	0.764313	−	0.644845i	$-0.223078\pi$
$684$		0			0
$685$	37.4200			1.42974
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−14.6177	−	42.0299i	−0.556891	−	1.60121i
$690$		0			0
$691$	−4.02127	−	2.32168i	−0.152976	−	0.0883209i	0.421558	−	0.906802i	$-0.361483\pi$
$691$	−4.02127	−	2.32168i	−0.152976	−	0.0883209i	−0.574534	+	0.818481i	$0.694817\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	7.92583	+	4.57598i	0.300644	+	0.173577i
$696$		0			0
$697$	8.34841	−	4.81996i	0.316219	−	0.182569i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	27.8138			1.05051			0.525257	−	0.850944i	$-0.323969\pi$
$701$	27.8138			1.05051			0.525257	+	0.850944i	$0.323969\pi$
$702$		0			0
$703$	−46.1338			−1.73997
$704$		0			0
$705$		0			0
$706$		0			0
$707$	0.941600	−	0.543633i	0.0354125	−	0.0204454i
$708$		0			0
$709$	−34.1616	−	19.7232i	−1.28297	−	0.740721i	−0.305576	−	0.952168i	$-0.598849\pi$
$709$	−34.1616	−	19.7232i	−1.28297	−	0.740721i	−0.977390	+	0.211447i	$0.932182\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−12.6766	−	7.31884i	−0.474743	−	0.274093i
$714$		0			0
$715$	2.49900	+	7.18531i	0.0934574	+	0.268715i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	31.1306			1.16097			0.580487	−	0.814269i	$-0.302862\pi$
$719$	31.1306			1.16097			0.580487	+	0.814269i	$0.302862\pi$
$720$		0			0
$721$			26.7367i			0.995728i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−5.28298	−	9.15040i	−0.196205	−	0.339837i
$726$		0			0
$727$	−13.2138	+	22.8869i	−0.490071	+	0.848829i	−0.999935	−	0.0114268i	$-0.996363\pi$
$727$	−13.2138	+	22.8869i	−0.490071	+	0.848829i	0.509863	+	0.860255i	$0.329696\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	6.75203	−	11.6949i	0.249733	−	0.432550i
$732$		0			0
$733$	−9.72200	+	5.61300i	−0.359090	+	0.207321i	−0.668682	−	0.743549i	$-0.733141\pi$
$733$	−9.72200	+	5.61300i	−0.359090	+	0.207321i	0.309591	+	0.950870i	$0.399808\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	6.99793			0.257772
$738$		0			0
$739$			13.7910i			0.507310i	0.967295	+	0.253655i	$0.0816327\pi$
$739$			13.7910i			0.507310i	−0.967295	+	0.253655i	$0.918367\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−30.7527	+	17.7551i	−1.12821	+	0.651371i	−0.943483	−	0.331420i	$-0.892472\pi$
$743$	−30.7527	+	17.7551i	−1.12821	+	0.651371i	−0.184724	+	0.982791i	$0.559139\pi$
$744$		0			0
$745$	−12.3832	+	21.4484i	−0.453686	+	0.785808i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	10.2357	+	5.90956i	0.374003	+	0.215931i
$750$		0			0
$751$	6.23222	+	10.7945i	0.227417	+	0.393898i	0.957042	−	0.289950i	$-0.0936387\pi$
$751$	6.23222	+	10.7945i	0.227417	+	0.393898i	−0.729625	+	0.683848i	$0.760305\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−33.3490			−1.21369
$756$		0			0
$757$	27.3303			0.993336			0.496668	−	0.867941i	$-0.334557\pi$
$757$	27.3303			0.993336			0.496668	+	0.867941i	$0.334557\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	36.2143	−	20.9083i	1.31277	−	0.757926i	0.330213	−	0.943906i	$-0.392879\pi$
$761$	36.2143	−	20.9083i	1.31277	−	0.757926i	0.982553	+	0.185981i	$0.0595461\pi$
$762$		0			0
$763$	−11.3140	+	19.5965i	−0.409595	+	0.709439i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−33.4211	−	28.8714i	−1.20677	−	1.04249i
$768$		0			0
$769$	−38.4184	+	22.1809i	−1.38540	+	0.799864i	−0.992793	−	0.119841i	$-0.961762\pi$
$769$	−38.4184	+	22.1809i	−1.38540	+	0.799864i	−0.392611	+	0.919704i	$0.628428\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	29.3084i		−	1.05415i	−0.849818	−	0.527076i	$-0.823288\pi$
$773$		−	29.3084i		−	1.05415i	0.849818	−	0.527076i	$-0.176712\pi$
$774$		0			0
$775$			4.53837i			0.163023i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	28.1578	+	48.7707i	1.00886	+	1.74739i
$780$		0			0
$781$	0.794170	−	1.37554i	0.0284176	−	0.0492208i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	12.4936	+	7.21317i	0.445915	+	0.257449i
$786$		0			0
$787$	17.9606	−	10.3696i	0.640228	−	0.369636i	−0.144475	−	0.989509i	$-0.546149\pi$
$787$	17.9606	−	10.3696i	0.640228	−	0.369636i	0.784702	+	0.619873i	$0.212816\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	36.7920i		−	1.30817i
$792$		0			0
$793$	−36.9925	−	7.07187i	−1.31364	−	0.251130i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.07189	−	1.85657i	−0.0379683	−	0.0657630i	0.846417	−	0.532521i	$-0.178755\pi$
$797$	−1.07189	−	1.85657i	−0.0379683	−	0.0657630i	−0.884385	+	0.466758i	$0.845422\pi$
$798$		0			0
$799$	10.9826	+	6.34080i	0.388536	+	0.224322i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	5.84871	−	10.1303i	0.206396	−	0.357489i
$804$		0			0
$805$	13.5687	+	23.5017i	0.478235	+	0.828327i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−35.4976			−1.24803			−0.624014	−	0.781413i	$-0.714499\pi$
$809$	−35.4976			−1.24803			−0.624014	+	0.781413i	$0.714499\pi$
$810$		0			0
$811$			47.1960i			1.65728i	0.559785	+	0.828638i	$0.310884\pi$
$811$			47.1960i			1.65728i	−0.559785	+	0.828638i	$0.689116\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	0.326154	+	0.564916i	0.0114247	+	0.0197881i
$816$		0			0
$817$	68.3204	+	39.4448i	2.39023	+	1.38000i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−31.4493	−	18.1573i	−1.09759	−	0.633693i	−0.162002	−	0.986790i	$-0.551795\pi$
$821$	−31.4493	−	18.1573i	−1.09759	−	0.633693i	−0.935586	+	0.353098i	$0.885128\pi$
$822$		0			0
$823$	8.48337	+	14.6936i	0.295712	+	0.512188i	0.975150	−	0.221545i	$-0.0711099\pi$
$823$	8.48337	+	14.6936i	0.295712	+	0.512188i	−0.679438	+	0.733732i	$0.737777\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			24.9508i			0.867626i	0.901003	+	0.433813i	$0.142832\pi$
$827$			24.9508i			0.867626i	−0.901003	+	0.433813i	$0.857168\pi$
$828$		0			0
$829$	8.49384			0.295003			0.147502	−	0.989062i	$-0.452877\pi$
$829$	8.49384			0.295003			0.147502	+	0.989062i	$0.452877\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.0415737	−	0.0720077i	−0.00144044	−	0.00249492i
$834$		0			0
$835$	11.7913	−	20.4231i	0.408054	−	0.706770i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	33.7446	+	19.4825i	1.16499	+	0.672610i	0.952496	−	0.304551i	$-0.0985065\pi$
$839$	33.7446	+	19.4825i	1.16499	+	0.672610i	0.212499	+	0.977161i	$0.431840\pi$
$840$		0			0
$841$	−3.54962	−	6.14813i	−0.122401	−	0.212004i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	8.63323	−	21.7547i	0.296992	−	0.748386i
$846$		0			0
$847$		−	25.5823i		−	0.879020i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−28.9905	+	16.7377i	−0.993782	+	0.573760i
$852$		0			0
$853$	−13.4418	−	7.76062i	−0.460238	−	0.265719i	0.251906	−	0.967752i	$-0.418943\pi$
$853$	−13.4418	−	7.76062i	−0.460238	−	0.265719i	−0.712144	+	0.702033i	$0.752276\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−20.9017	+	36.2027i	−0.713987	+	1.23666i	0.249362	+	0.968410i	$0.419779\pi$
$857$	−20.9017	+	36.2027i	−0.713987	+	1.23666i	−0.963349	+	0.268252i	$0.913554\pi$
$858$		0			0
$859$	23.7115	+	41.0695i	0.809025	+	1.40127i	0.913540	+	0.406749i	$0.133338\pi$
$859$	23.7115	+	41.0695i	0.809025	+	1.40127i	−0.104515	+	0.994523i	$0.533329\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			49.1903i			1.67446i	0.546853	+	0.837229i	$0.315826\pi$
$863$			49.1903i			1.67446i	−0.546853	+	0.837229i	$0.684174\pi$
$864$		0			0
$865$		−	0.374879i		−	0.0127463i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−4.34733	+	2.50993i	−0.147473	+	0.0851437i
$870$		0			0
$871$	−16.2924	−	14.0745i	−0.552046	−	0.476895i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	16.1682	−	28.0041i	0.546584	−	0.946712i
$876$		0			0
$877$	8.93127	−	5.15647i	0.301587	−	0.174122i	−0.341568	−	0.939857i	$-0.610958\pi$
$877$	8.93127	−	5.15647i	0.301587	−	0.174122i	0.643156	+	0.765735i	$0.277625\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	10.1849			0.343139			0.171570	−	0.985172i	$-0.445116\pi$
$881$	10.1849			0.343139			0.171570	+	0.985172i	$0.445116\pi$
$882$		0			0
$883$	−12.2488			−0.412206			−0.206103	−	0.978530i	$-0.566078\pi$
$883$	−12.2488			−0.412206			−0.206103	+	0.978530i	$0.566078\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−5.18061	−	8.97309i	−0.173948	−	0.301287i	0.765849	−	0.643021i	$-0.222319\pi$
$887$	−5.18061	−	8.97309i	−0.173948	−	0.301287i	−0.939797	+	0.341734i	$0.888986\pi$
$888$		0			0
$889$	−18.6923	−	10.7920i	−0.626918	−	0.361952i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−37.0425	+	64.1594i	−1.23958	+	2.14701i
$894$		0			0
$895$	24.1394	−	13.9369i	0.806890	−	0.465858i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			15.5056i			0.517141i
$900$		0			0
$901$	−16.5140			−0.550162
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−15.0704	+	8.70089i	−0.500957	+	0.289227i
$906$		0			0
$907$	11.2973	−	19.5674i	0.375119	−	0.649726i	−0.615226	−	0.788351i	$-0.710935\pi$
$907$	11.2973	−	19.5674i	0.375119	−	0.649726i	0.990345	+	0.138625i	$0.0442684\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−10.9088	+	18.8945i	−0.361423	+	0.626003i	−0.988195	−	0.153199i	$-0.951042\pi$
$911$	−10.9088	+	18.8945i	−0.361423	+	0.626003i	0.626772	+	0.779203i	$0.284376\pi$
$912$		0			0
$913$	−1.15165	−	1.99472i	−0.0381141	−	0.0660156i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	31.3508i		−	1.03529i
$918$		0			0
$919$	−13.4021			−0.442094			−0.221047	−	0.975263i	$-0.570947\pi$
$919$	−13.4021			−0.442094			−0.221047	+	0.975263i	$0.570947\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−4.61551	+	1.60524i	−0.151921	+	0.0528372i
$924$		0			0
$925$	8.98842	+	5.18947i	0.295537	+	0.170629i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	8.30493	+	4.79485i	0.272476	+	0.157314i	0.630012	−	0.776585i	$-0.283050\pi$
$929$	8.30493	+	4.79485i	0.272476	+	0.157314i	−0.357536	+	0.933899i	$0.616383\pi$
$930$		0			0
$931$	0.420663	−	0.242870i	0.0137867	−	0.00795974i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	2.82319			0.0923282
$936$		0			0
$937$	−48.2050			−1.57479			−0.787394	−	0.616450i	$-0.788570\pi$
$937$	−48.2050			−1.57479			−0.787394	+	0.616450i	$0.788570\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	35.4422	−	20.4625i	1.15538	−	0.667060i	0.205189	−	0.978722i	$-0.434219\pi$
$941$	35.4422	−	20.4625i	1.15538	−	0.667060i	0.950193	+	0.311663i	$0.100886\pi$
$942$		0			0
$943$	35.3888	+	20.4317i	1.15242	+	0.665349i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−16.5907	−	9.57865i	−0.539125	−	0.311264i	0.205599	−	0.978636i	$-0.434086\pi$
$947$	−16.5907	−	9.57865i	−0.539125	−	0.311264i	−0.744724	+	0.667372i	$0.767419\pi$
$948$		0			0
$949$	−33.9911	+	11.8219i	−1.10340	+	0.383755i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−58.5490			−1.89659			−0.948294	−	0.317393i	$-0.897193\pi$
$953$	−58.5490			−1.89659			−0.948294	+	0.317393i	$0.897193\pi$
$954$		0			0
$955$			8.10711i			0.262340i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−27.6168	−	47.8337i	−0.891793	−	1.54463i
$960$		0			0
$961$	−12.1700	+	21.0790i	−0.392579	+	0.679968i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−14.2127	+	24.6172i	−0.457524	+	0.792456i
$966$		0			0
$967$	−12.5783	+	7.26210i	−0.404491	+	0.233533i	−0.688420	−	0.725312i	$-0.741695\pi$
$967$	−12.5783	+	7.26210i	−0.404491	+	0.233533i	0.283929	+	0.958845i	$0.408362\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	4.10205			0.131641			0.0658206	−	0.997831i	$-0.479033\pi$
$971$	4.10205			0.131641			0.0658206	+	0.997831i	$0.479033\pi$
$972$		0			0
$973$		−	13.5087i		−	0.433070i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−33.7487	+	19.4848i	−1.07972	+	0.623374i	−0.930820	−	0.365479i	$-0.880905\pi$
$977$	−33.7487	+	19.4848i	−1.07972	+	0.623374i	−0.148896	+	0.988853i	$0.547572\pi$
$978$		0			0
$979$	4.93106	−	8.54084i	0.157597	−	0.272966i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	17.1004	+	9.87292i	0.545418	+	0.314897i	0.747272	−	0.664518i	$-0.231363\pi$
$983$	17.1004	+	9.87292i	0.545418	+	0.314897i	−0.201854	+	0.979416i	$0.564697\pi$
$984$		0			0
$985$	2.09087	+	3.62149i	0.0666205	+	0.115390i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	57.2435			1.82024
$990$		0			0
$991$	−15.2644			−0.484890			−0.242445	−	0.970165i	$-0.577949\pi$
$991$	−15.2644			−0.484890			−0.242445	+	0.970165i	$0.577949\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	36.6408	−	21.1546i	1.16159	−	0.670645i
$996$		0			0
$997$	26.5012	−	45.9014i	0.839301	−	1.45371i	−0.0511787	−	0.998690i	$-0.516298\pi$
$997$	26.5012	−	45.9014i	0.839301	−	1.45371i	0.890480	−	0.455023i	$-0.150369\pi$
$998$		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.1585.11		80
3.2	odd	2		936.2.cw.b.25.24	yes	80
9.4	even	3	inner	2808.2.cw.b.2521.30		80
9.5	odd	6		936.2.cw.b.337.23	yes	80
13.12	even	2	inner	2808.2.cw.b.1585.30		80
39.38	odd	2		936.2.cw.b.25.23	✓	80
117.77	odd	6		936.2.cw.b.337.24	yes	80
117.103	even	6	inner	2808.2.cw.b.2521.11		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
936.2.cw.b.25.23	✓	80	39.38	odd	2
936.2.cw.b.25.24	yes	80	3.2	odd	2
936.2.cw.b.337.23	yes	80	9.5	odd	6
936.2.cw.b.337.24	yes	80	117.77	odd	6
2808.2.cw.b.1585.11		80	1.1	even	1	trivial
2808.2.cw.b.1585.30		80	13.12	even	2	inner
2808.2.cw.b.2521.11		80	117.103	even	6	inner
2808.2.cw.b.2521.30		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)

\(f(q)\)	\(=\)	\(q+(-1.55919 + 0.900198i) q^{5} +(2.30144 + 1.32873i) q^{7} +(1.01492 + 0.585964i) q^{11} +(-1.18440 - 3.40547i) q^{13} -1.33805 q^{17} -7.81675i q^{19} +(-2.83598 - 4.91206i) q^{23} +(-0.879286 + 1.52297i) q^{25} +(-3.00413 + 5.20331i) q^{29} +(2.23496 - 1.29036i) q^{31} -4.78450 q^{35} -5.90191i q^{37} +(-6.23926 + 3.60224i) q^{41} +(-5.04619 + 8.74025i) q^{43} +(-8.20794 - 4.73885i) q^{47} +(0.0310704 + 0.0538156i) q^{49} +12.3419 q^{53} -2.10994 q^{55} +(10.6080 - 6.12456i) q^{59} +(5.22284 - 9.04622i) q^{61} +(4.91230 + 4.24357i) q^{65} +(5.17129 - 2.98565i) q^{67} -1.35532i q^{71} -9.98135i q^{73} +(1.55718 + 2.69712i) q^{77} +(-2.14171 + 3.70956i) q^{79} +(-1.70208 - 0.982698i) q^{83} +(2.08627 - 1.20451i) q^{85} -8.41529i q^{89} +(1.79914 - 9.41121i) q^{91} +(7.03663 + 12.1878i) q^{95} +(0.765238 + 0.441810i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 6 q^{13} + 4 q^{17} - 10 q^{23} + 44 q^{25} + 52 q^{35} - 26 q^{43} + 48 q^{49} - 60 q^{53} - 16 q^{55} - 10 q^{61} + 26 q^{65} - 32 q^{77} + 6 q^{79} - 4 q^{91} - 8 q^{95}+O(q^{100}) \) 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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