LMFDB - Embedded newform 2808.2.cw.b.1585.10

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			1585.10
Character	$\chi$	$=$	2808.1585
Dual form			2808.2.cw.b.2521.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.91469 + 1.10545i) q^{5} +(-1.00877 - 0.582415i) q^{7} +(0.252566 + 0.145819i) q^{11} +(1.26769 + 3.37535i) q^{13} -2.15546 q^{17} -3.83640i q^{19} +(3.65862 + 6.33691i) q^{23} +(-0.0559674 + 0.0969384i) q^{25} +(-0.370932 + 0.642474i) q^{29} +(5.53094 - 3.19329i) q^{31} +2.57532 q^{35} -5.81635i q^{37} +(-5.12394 + 2.95831i) q^{41} +(-1.75717 + 3.04352i) q^{43} +(-8.10996 - 4.68229i) q^{47} +(-2.82159 - 4.88713i) q^{49} -5.41101 q^{53} -0.644781 q^{55} +(-8.56121 + 4.94282i) q^{59} +(2.34926 - 4.06904i) q^{61} +(-6.15851 - 5.06138i) q^{65} +(7.47721 - 4.31697i) q^{67} -1.57202i q^{71} -1.79440i q^{73} +(-0.169854 - 0.294196i) q^{77} +(-2.80768 + 4.86305i) q^{79} +(-7.17113 - 4.14026i) q^{83} +(4.12704 - 2.38275i) q^{85} +16.0139i q^{89} +(0.687038 - 4.14328i) q^{91} +(4.24094 + 7.34553i) q^{95} +(-14.9442 - 8.62804i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.91469	+	1.10545i	−0.856277	+	0.494372i	−0.862764	−	0.505607i	$-0.831268\pi$
$5$	−1.91469	+	1.10545i	−0.856277	+	0.494372i	0.00648699	+	0.999979i	$0.497935\pi$
$6$		0			0
$7$	−1.00877	−	0.582415i	−0.381280	−	0.220132i	0.297095	−	0.954848i	$-0.403982\pi$
$7$	−1.00877	−	0.582415i	−0.381280	−	0.220132i	−0.678375	+	0.734716i	$0.737316\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.252566	+	0.145819i	0.0761514	+	0.0439661i	0.537592	−	0.843205i	$-0.319334\pi$
$11$	0.252566	+	0.145819i	0.0761514	+	0.0439661i	−0.461441	+	0.887171i	$0.652667\pi$
$12$		0			0
$13$	1.26769	+	3.37535i	0.351595	+	0.936152i
$13$	1.26769	+	3.37535i	0.351595	+	0.936152i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.15546			−0.522776			−0.261388	−	0.965234i	$-0.584180\pi$
$17$	−2.15546			−0.522776			−0.261388	+	0.965234i	$0.584180\pi$
$18$		0			0
$19$		−	3.83640i		−	0.880130i	−0.897966	−	0.440065i	$-0.854955\pi$
$19$		−	3.83640i		−	0.880130i	0.897966	−	0.440065i	$-0.145045\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.65862	+	6.33691i	0.762874	+	1.32134i	0.941363	+	0.337395i	$0.109546\pi$
$23$	3.65862	+	6.33691i	0.762874	+	1.32134i	−0.178489	+	0.983942i	$0.557121\pi$
$24$		0			0
$25$	−0.0559674	+	0.0969384i	−0.0111935	+	0.0193877i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.370932	+	0.642474i	−0.0688804	+	0.119304i	−0.898409	−	0.439160i	$-0.855276\pi$
$29$	−0.370932	+	0.642474i	−0.0688804	+	0.119304i	0.829528	+	0.558465i	$0.188609\pi$
$30$		0			0
$31$	5.53094	−	3.19329i	0.993386	−	0.573532i	0.0871016	−	0.996199i	$-0.472240\pi$
$31$	5.53094	−	3.19329i	0.993386	−	0.573532i	0.906285	+	0.422668i	$0.138906\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.57532			0.435308
$36$		0			0
$37$		−	5.81635i		−	0.956202i	−0.878305	−	0.478101i	$-0.841325\pi$
$37$		−	5.81635i		−	0.956202i	0.878305	−	0.478101i	$-0.158675\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−5.12394	+	2.95831i	−0.800225	+	0.462010i	−0.843550	−	0.537051i	$-0.819538\pi$
$41$	−5.12394	+	2.95831i	−0.800225	+	0.462010i	0.0433247	+	0.999061i	$0.486205\pi$
$42$		0			0
$43$	−1.75717	+	3.04352i	−0.267967	+	0.464132i	−0.968337	−	0.249648i	$-0.919685\pi$
$43$	−1.75717	+	3.04352i	−0.267967	+	0.464132i	0.700370	+	0.713780i	$0.253018\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−8.10996	−	4.68229i	−1.18296	−	0.682982i	−0.226262	−	0.974067i	$-0.572650\pi$
$47$	−8.10996	−	4.68229i	−1.18296	−	0.682982i	−0.956697	+	0.291085i	$0.905984\pi$
$48$		0			0
$49$	−2.82159	−	4.88713i	−0.403084	−	0.698161i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.41101			−0.743259			−0.371630	−	0.928381i	$-0.621201\pi$
$53$	−5.41101			−0.743259			−0.371630	+	0.928381i	$0.621201\pi$
$54$		0			0
$55$	−0.644781			−0.0869423
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−8.56121	+	4.94282i	−1.11458	+	0.643500i	−0.940011	−	0.341145i	$-0.889185\pi$
$59$	−8.56121	+	4.94282i	−1.11458	+	0.643500i	−0.174565	+	0.984646i	$0.555852\pi$
$60$		0			0
$61$	2.34926	−	4.06904i	0.300792	−	0.520988i	−0.675523	−	0.737339i	$-0.736082\pi$
$61$	2.34926	−	4.06904i	0.300792	−	0.520988i	0.976316	+	0.216351i	$0.0694156\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−6.15851	−	5.06138i	−0.763869	−	0.627787i
$66$		0			0
$67$	7.47721	−	4.31697i	0.913487	−	0.527402i	0.0319356	−	0.999490i	$-0.489833\pi$
$67$	7.47721	−	4.31697i	0.913487	−	0.527402i	0.881551	+	0.472088i	$0.156500\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	1.57202i		−	0.186564i	−0.995640	−	0.0932822i	$-0.970264\pi$
$71$		−	1.57202i		−	0.186564i	0.995640	−	0.0932822i	$-0.0297359\pi$
$72$		0			0
$73$		−	1.79440i		−	0.210018i	−0.994471	−	0.105009i	$-0.966513\pi$
$73$		−	1.79440i		−	0.210018i	0.994471	−	0.105009i	$-0.0334872\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.169854	−	0.294196i	−0.0193567	−	0.0335268i
$78$		0			0
$79$	−2.80768	+	4.86305i	−0.315889	+	0.547136i	−0.979626	−	0.200830i	$-0.935636\pi$
$79$	−2.80768	+	4.86305i	−0.315889	+	0.547136i	0.663737	+	0.747966i	$0.268969\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7.17113	−	4.14026i	−0.787134	−	0.454452i	0.0518185	−	0.998657i	$-0.483498\pi$
$83$	−7.17113	−	4.14026i	−0.787134	−	0.454452i	−0.838953	+	0.544204i	$0.816832\pi$
$84$		0			0
$85$	4.12704	−	2.38275i	0.447641	−	0.258446i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			16.0139i			1.69747i	0.528822	+	0.848733i	$0.322634\pi$
$89$			16.0139i			1.69747i	−0.528822	+	0.848733i	$0.677366\pi$
$90$		0			0
$91$	0.687038	−	4.14328i	0.0720212	−	0.434334i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.24094	+	7.34553i	0.435111	+	0.753635i
$96$		0			0
$97$	−14.9442	−	8.62804i	−1.51735	−	0.876044i	−0.999792	−	0.0203983i	$-0.993507\pi$
$97$	−14.9442	−	8.62804i	−1.51735	−	0.876044i	−0.517561	−	0.855646i	$-0.673160\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.91990	−	6.78947i	0.390045	−	0.675578i	−0.602410	−	0.798187i	$-0.705793\pi$
$101$	3.91990	−	6.78947i	0.390045	−	0.675578i	0.992455	+	0.122609i	$0.0391261\pi$
$102$		0			0
$103$	−3.51672	−	6.09113i	−0.346512	−	0.600177i	0.639115	−	0.769111i	$-0.279301\pi$
$103$	−3.51672	−	6.09113i	−0.346512	−	0.600177i	−0.985627	+	0.168934i	$0.945967\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.67922			−0.742378			−0.371189	−	0.928557i	$-0.621050\pi$
$107$	−7.67922			−0.742378			−0.371189	+	0.928557i	$0.621050\pi$
$108$		0			0
$109$			2.72395i			0.260907i	0.991454	+	0.130454i	$0.0416433\pi$
$109$			2.72395i			0.260907i	−0.991454	+	0.130454i	$0.958357\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.00926	−	5.21219i	−0.283087	−	0.490321i	0.689056	−	0.724708i	$-0.258025\pi$
$113$	−3.00926	−	5.21219i	−0.283087	−	0.490321i	−0.972144	+	0.234386i	$0.924692\pi$
$114$		0			0
$115$	−14.0102	−	8.08882i	−1.30646	−	0.754286i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.17437	+	1.25537i	0.199324	+	0.115080i
$120$		0			0
$121$	−5.45747	−	9.45262i	−0.496134	−	0.859329i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	11.3020i		−	1.01088i
$126$		0			0
$127$	−4.33262			−0.384458			−0.192229	−	0.981350i	$-0.561572\pi$
$127$	−4.33262			−0.384458			−0.192229	+	0.981350i	$0.561572\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.97323	−	6.88184i	−0.347143	−	0.601269i	0.638598	−	0.769541i	$-0.279515\pi$
$131$	−3.97323	−	6.88184i	−0.347143	−	0.601269i	−0.985741	+	0.168272i	$0.946181\pi$
$132$		0			0
$133$	−2.23438	+	3.87005i	−0.193745	+	0.335576i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−8.89322	−	5.13450i	−0.759799	−	0.438670i	0.0694247	−	0.997587i	$-0.477884\pi$
$137$	−8.89322	−	5.13450i	−0.759799	−	0.438670i	−0.829224	+	0.558917i	$0.811217\pi$
$138$		0			0
$139$	4.35935	+	7.55061i	0.369755	+	0.640434i	0.989527	−	0.144348i	$-0.0461083\pi$
$139$	4.35935	+	7.55061i	0.369755	+	0.640434i	−0.619772	+	0.784782i	$0.712775\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.172013	+	1.03735i	−0.0143845	+	0.0867476i
$144$		0			0
$145$		−	1.64019i		−	0.136210i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.53118	+	3.77078i	−0.535055	+	0.308914i	−0.743072	−	0.669211i	$-0.766632\pi$
$149$	−6.53118	+	3.77078i	−0.535055	+	0.308914i	0.208018	+	0.978125i	$0.433299\pi$
$150$		0			0
$151$	−12.9339	−	7.46741i	−1.05255	−	0.607689i	−0.129187	−	0.991620i	$-0.541237\pi$
$151$	−12.9339	−	7.46741i	−1.05255	−	0.607689i	−0.923362	+	0.383931i	$0.874570\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−7.06004	+	12.2283i	−0.567076	+	0.982204i
$156$		0			0
$157$	−4.54448	−	7.87128i	−0.362689	−	0.628196i	0.625713	−	0.780053i	$-0.284808\pi$
$157$	−4.54448	−	7.87128i	−0.362689	−	0.628196i	−0.988402	+	0.151857i	$0.951475\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	8.52333i		−	0.671732i
$162$		0			0
$163$			17.8973i			1.40182i	0.713248	+	0.700912i	$0.247223\pi$
$163$			17.8973i			1.40182i	−0.713248	+	0.700912i	$0.752777\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	10.6547	−	6.15151i	0.824488	−	0.476018i	−0.0274735	−	0.999623i	$-0.508746\pi$
$167$	10.6547	−	6.15151i	0.824488	−	0.476018i	0.851962	+	0.523604i	$0.175413\pi$
$168$		0			0
$169$	−9.78591	+	8.55780i	−0.752762	+	0.658292i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	10.7657	−	18.6467i	0.818499	−	1.41768i	−0.0882885	−	0.996095i	$-0.528140\pi$
$173$	10.7657	−	18.6467i	0.818499	−	1.41768i	0.906788	−	0.421587i	$-0.138527\pi$
$174$		0			0
$175$	0.112917	−	0.0651925i	0.00853570	−	0.00492809i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	14.8784			1.11206			0.556031	−	0.831161i	$-0.312323\pi$
$179$	14.8784			1.11206			0.556031	+	0.831161i	$0.312323\pi$
$180$		0			0
$181$	−11.7742			−0.875171			−0.437585	−	0.899177i	$-0.644166\pi$
$181$	−11.7742			−0.875171			−0.437585	+	0.899177i	$0.644166\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	6.42968	+	11.1365i	0.472719	+	0.818774i
$186$		0			0
$187$	−0.544395	−	0.314307i	−0.0398101	−	0.0229844i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−0.741298	+	1.28397i	−0.0536384	+	0.0929045i	−0.891598	−	0.452828i	$-0.850415\pi$
$191$	−0.741298	+	1.28397i	−0.0536384	+	0.0929045i	0.837959	+	0.545732i	$0.183749\pi$
$192$		0			0
$193$	−12.2920	+	7.09680i	−0.884799	+	0.510839i	−0.872238	−	0.489082i	$-0.837332\pi$
$193$	−12.2920	+	7.09680i	−0.884799	+	0.510839i	−0.0125615	+	0.999921i	$0.503999\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	15.2490i		−	1.08645i	−0.839589	−	0.543223i	$-0.817204\pi$
$197$		−	15.2490i		−	1.08645i	0.839589	−	0.543223i	$-0.182796\pi$
$198$		0			0
$199$	12.2757			0.870202			0.435101	−	0.900382i	$-0.356713\pi$
$199$	12.2757			0.870202			0.435101	+	0.900382i	$0.356713\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.748373	−	0.432073i	0.0525255	−	0.0303256i
$204$		0			0
$205$	6.54052	−	11.3285i	0.456809	−	0.791217i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.559419	−	0.968943i	0.0386958	−	0.0670232i
$210$		0			0
$211$	12.3557	+	21.4007i	0.850600	+	1.47328i	0.880667	+	0.473735i	$0.157095\pi$
$211$	12.3557	+	21.4007i	0.850600	+	1.47328i	−0.0300669	+	0.999548i	$0.509572\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	7.76986i		−	0.529900i
$216$		0			0
$217$	−7.43928			−0.505011
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.73246	−	7.27542i	−0.183805	−	0.489398i
$222$		0			0
$223$	−22.2970	−	12.8732i	−1.49312	−	0.862053i	−0.493150	−	0.869944i	$-0.664155\pi$
$223$	−22.2970	−	12.8732i	−1.49312	−	0.862053i	−0.999969	+	0.00789126i	$0.997488\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.90850	+	5.72068i	0.657650	+	0.379695i	0.791381	−	0.611323i	$-0.209362\pi$
$227$	9.90850	+	5.72068i	0.657650	+	0.379695i	−0.133731	+	0.991018i	$0.542696\pi$
$228$		0			0
$229$	11.8918	−	6.86575i	0.785834	−	0.453702i	−0.0526598	−	0.998613i	$-0.516770\pi$
$229$	11.8918	−	6.86575i	0.785834	−	0.453702i	0.838494	+	0.544911i	$0.183437\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−17.8969			−1.17247			−0.586233	−	0.810143i	$-0.699390\pi$
$233$	−17.8969			−1.17247			−0.586233	+	0.810143i	$0.699390\pi$
$234$		0			0
$235$	20.7041			1.35059
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−7.94867	+	4.58917i	−0.514157	+	0.296849i	−0.734541	−	0.678565i	$-0.762602\pi$
$239$	−7.94867	+	4.58917i	−0.514157	+	0.296849i	0.220384	+	0.975413i	$0.429269\pi$
$240$		0			0
$241$	3.50775	+	2.02520i	0.225954	+	0.130455i	0.608704	−	0.793397i	$-0.291690\pi$
$241$	3.50775	+	2.02520i	0.225954	+	0.130455i	−0.382750	+	0.923852i	$0.625023\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	10.8049	+	6.23824i	0.690302	+	0.398546i
$246$		0			0
$247$	12.9492	−	4.86338i	0.823936	−	0.309449i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	7.19260			0.453993			0.226996	−	0.973896i	$-0.427109\pi$
$251$	7.19260			0.453993			0.226996	+	0.973896i	$0.427109\pi$
$252$		0			0
$253$			2.13398i			0.134162i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−5.98687	−	10.3696i	−0.373451	−	0.646836i	0.616643	−	0.787243i	$-0.288492\pi$
$257$	−5.98687	−	10.3696i	−0.373451	−	0.646836i	−0.990094	+	0.140407i	$0.955159\pi$
$258$		0			0
$259$	−3.38753	+	5.86737i	−0.210491	+	0.364581i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−9.94118	+	17.2186i	−0.613000	+	1.06175i	0.377732	+	0.925915i	$0.376704\pi$
$263$	−9.94118	+	17.2186i	−0.613000	+	1.06175i	−0.990732	+	0.135832i	$0.956629\pi$
$264$		0			0
$265$	10.3604	−	5.98159i	0.636435	−	0.367446i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	5.61461			0.342329			0.171165	−	0.985242i	$-0.445247\pi$
$269$	5.61461			0.342329			0.171165	+	0.985242i	$0.445247\pi$
$270$		0			0
$271$			8.77363i			0.532960i	0.963840	+	0.266480i	$0.0858606\pi$
$271$			8.77363i			0.532960i	−0.963840	+	0.266480i	$0.914139\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.0282709	+	0.0163222i	−0.00170480	+	0.000984266i
$276$		0			0
$277$	6.62496	−	11.4748i	0.398055	−	0.689452i	−0.595431	−	0.803407i	$-0.703019\pi$
$277$	6.62496	−	11.4748i	0.398055	−	0.689452i	0.993486	+	0.113955i	$0.0363519\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	3.55941	+	2.05502i	0.212336	+	0.122592i	0.602397	−	0.798197i	$-0.294213\pi$
$281$	3.55941	+	2.05502i	0.212336	+	0.122592i	−0.390060	+	0.920789i	$0.627546\pi$
$282$		0			0
$283$	2.51643	+	4.35858i	0.149586	+	0.259091i	0.931075	−	0.364829i	$-0.118873\pi$
$283$	2.51643	+	4.35858i	0.149586	+	0.259091i	−0.781488	+	0.623920i	$0.785539\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	6.89185			0.406813
$288$		0			0
$289$	−12.3540			−0.726705
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.273708	+	0.158025i	−0.0159902	+	0.00923194i	−0.507974	−	0.861373i	$-0.669605\pi$
$293$	−0.273708	+	0.158025i	−0.0159902	+	0.00923194i	0.491984	+	0.870604i	$0.336272\pi$
$294$		0			0
$295$	10.9281	−	18.9280i	0.636256	−	1.10203i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−16.7512	+	20.3823i	−0.968750	+	1.17874i
$300$		0			0
$301$	3.54518	−	2.04681i	0.204341	−	0.117976i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			10.3880i			0.594813i
$306$		0			0
$307$			5.60861i			0.320100i	0.987109	+	0.160050i	$0.0511656\pi$
$307$			5.60861i			0.320100i	−0.987109	+	0.160050i	$0.948834\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	6.39800	+	11.0817i	0.362797	+	0.628384i	0.988420	−	0.151742i	$-0.0484884\pi$
$311$	6.39800	+	11.0817i	0.362797	+	0.628384i	−0.625623	+	0.780126i	$0.715155\pi$
$312$		0			0
$313$	−5.35958	+	9.28307i	−0.302942	+	0.524710i	−0.976801	−	0.214149i	$-0.931302\pi$
$313$	−5.35958	+	9.28307i	−0.302942	+	0.524710i	0.673859	+	0.738860i	$0.264635\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−26.1325	−	15.0876i	−1.46775	−	0.847405i	−0.468400	−	0.883516i	$-0.655169\pi$
$317$	−26.1325	−	15.0876i	−1.46775	−	0.847405i	−0.999348	+	0.0361118i	$0.988503\pi$
$318$		0			0
$319$	−0.187370	+	0.108178i	−0.0104907	+	0.00605680i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			8.26921i			0.460111i
$324$		0			0
$325$	−0.398150	−	0.0660212i	−0.0220854	−	0.00366220i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	5.45407	+	9.44672i	0.300692	+	0.520815i
$330$		0			0
$331$	12.6168	+	7.28431i	0.693482	+	0.400382i	0.804915	−	0.593390i	$-0.202211\pi$
$331$	12.6168	+	7.28431i	0.693482	+	0.400382i	−0.111433	+	0.993772i	$0.535544\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−9.54438	+	16.5313i	−0.521465	+	0.903204i
$336$		0			0
$337$	14.8456	+	25.7133i	0.808691	+	1.40069i	0.913771	+	0.406230i	$0.133157\pi$
$337$	14.8456	+	25.7133i	0.808691	+	1.40069i	−0.105080	+	0.994464i	$0.533510\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.86257			0.100864
$342$		0			0
$343$			14.7271i			0.795191i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−4.21438	−	7.29952i	−0.226240	−	0.391859i	0.730451	−	0.682965i	$-0.239310\pi$
$347$	−4.21438	−	7.29952i	−0.226240	−	0.391859i	−0.956691	+	0.291106i	$0.905977\pi$
$348$		0			0
$349$	−27.2940	−	15.7582i	−1.46102	−	0.843517i	−0.461957	−	0.886903i	$-0.652852\pi$
$349$	−27.2940	−	15.7582i	−1.46102	−	0.843517i	−0.999058	+	0.0433851i	$0.986186\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2.40734	+	1.38988i	0.128130	+	0.0739758i	0.562695	−	0.826665i	$-0.309765\pi$
$353$	2.40734	+	1.38988i	0.128130	+	0.0739758i	−0.434565	+	0.900640i	$0.643098\pi$
$354$		0			0
$355$	1.73779	+	3.00994i	0.0922322	+	0.159751i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			28.1761i			1.48708i	0.668692	+	0.743539i	$0.266854\pi$
$359$			28.1761i			1.48708i	−0.668692	+	0.743539i	$0.733146\pi$
$360$		0			0
$361$	4.28205			0.225371
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.98361	+	3.43572i	0.103827	+	0.179834i
$366$		0			0
$367$	−17.7440	+	30.7335i	−0.926230	+	1.60428i	−0.136659	+	0.990618i	$0.543637\pi$
$367$	−17.7440	+	30.7335i	−0.926230	+	1.60428i	−0.789571	+	0.613660i	$0.789697\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	5.45847	+	3.15145i	0.283390	+	0.163615i
$372$		0			0
$373$	−15.7013	−	27.1954i	−0.812982	−	1.40813i	−0.910768	−	0.412919i	$-0.864509\pi$
$373$	−15.7013	−	27.1954i	−0.812982	−	1.40813i	0.0977853	−	0.995208i	$-0.468824\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2.63880	−	0.437565i	−0.135905	−	0.0225358i
$378$		0			0
$379$		−	20.9265i		−	1.07492i	−0.843288	−	0.537462i	$-0.819383\pi$
$379$		−	20.9265i		−	1.07492i	0.843288	−	0.537462i	$-0.180617\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	0.210574	−	0.121575i	0.0107598	−	0.00621219i	−0.494610	−	0.869115i	$-0.664689\pi$
$383$	0.210574	−	0.121575i	0.0107598	−	0.00621219i	0.505370	+	0.862903i	$0.331356\pi$
$384$		0			0
$385$	0.650437	+	0.375530i	0.0331494	+	0.0191388i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	18.0937	−	31.3391i	0.917385	−	1.58896i	0.114013	−	0.993479i	$-0.463629\pi$
$389$	18.0937	−	31.3391i	0.917385	−	1.58896i	0.803372	−	0.595478i	$-0.203037\pi$
$390$		0			0
$391$	−7.88600	−	13.6590i	−0.398812	−	0.690763i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	12.4150i		−	0.624666i
$396$		0			0
$397$			33.0787i			1.66017i	0.557635	+	0.830086i	$0.311709\pi$
$397$			33.0787i			1.66017i	−0.557635	+	0.830086i	$0.688291\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	4.59095	−	2.65058i	0.229261	−	0.132364i	−0.380970	−	0.924587i	$-0.624410\pi$
$401$	4.59095	−	2.65058i	0.229261	−	0.132364i	0.610231	+	0.792223i	$0.291077\pi$
$402$		0			0
$403$	17.7900	+	14.6207i	0.886183	+	0.728310i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	0.848134	−	1.46901i	0.0420404	−	0.0728162i
$408$		0			0
$409$	−0.738843	+	0.426571i	−0.0365334	+	0.0210926i	−0.518155	−	0.855286i	$-0.673381\pi$
$409$	−0.738843	+	0.426571i	−0.0365334	+	0.0210926i	0.481622	+	0.876379i	$0.340048\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	11.5151			0.566620
$414$		0			0
$415$	18.3074			0.898673
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−19.3543	−	33.5226i	−0.945520	−	1.63769i	−0.754707	−	0.656062i	$-0.772221\pi$
$419$	−19.3543	−	33.5226i	−0.945520	−	1.63769i	−0.190813	−	0.981626i	$-0.561112\pi$
$420$		0			0
$421$	14.2901	+	8.25041i	0.696459	+	0.402101i	0.806027	−	0.591879i	$-0.201614\pi$
$421$	14.2901	+	8.25041i	0.696459	+	0.402101i	−0.109568	+	0.993979i	$0.534947\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.120636	−	0.208947i	0.00585168	−	0.0101354i
$426$		0			0
$427$	−4.73974	+	2.73649i	−0.229372	+	0.132428i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	21.9603i		−	1.05779i	−0.848687	−	0.528895i	$-0.822607\pi$
$431$		−	21.9603i		−	1.05779i	0.848687	−	0.528895i	$-0.177393\pi$
$432$		0			0
$433$	−7.96989			−0.383009			−0.191504	−	0.981492i	$-0.561337\pi$
$433$	−7.96989			−0.383009			−0.191504	+	0.981492i	$0.561337\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	24.3109	−	14.0359i	1.16295	−	0.671428i
$438$		0			0
$439$	12.7371	−	22.0613i	0.607908	−	1.05293i	−0.383677	−	0.923468i	$-0.625342\pi$
$439$	12.7371	−	22.0613i	0.607908	−	1.05293i	0.991585	−	0.129460i	$-0.0413244\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−3.93415	+	6.81415i	−0.186917	+	0.323750i	−0.944221	−	0.329313i	$-0.893183\pi$
$443$	−3.93415	+	6.81415i	−0.186917	+	0.323750i	0.757304	+	0.653063i	$0.226516\pi$
$444$		0			0
$445$	−17.7025	−	30.6616i	−0.839179	−	1.45350i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			20.2221i			0.954342i	0.878811	+	0.477171i	$0.158338\pi$
$449$			20.2221i			0.954342i	−0.878811	+	0.477171i	$0.841662\pi$
$450$		0			0
$451$	−1.72551			−0.0812511
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3.26471	+	8.69259i	0.153052	+	0.407515i
$456$		0			0
$457$	−1.02690	−	0.592879i	−0.0480362	−	0.0277337i	0.475790	−	0.879559i	$-0.342162\pi$
$457$	−1.02690	−	0.592879i	−0.0480362	−	0.0277337i	−0.523826	+	0.851825i	$0.675496\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	17.9921	+	10.3878i	0.837976	+	0.483806i	0.856576	−	0.516021i	$-0.172587\pi$
$461$	17.9921	+	10.3878i	0.837976	+	0.483806i	−0.0185995	+	0.999827i	$0.505921\pi$
$462$		0			0
$463$	21.0641	−	12.1613i	0.978930	−	0.565186i	0.0769833	−	0.997032i	$-0.475471\pi$
$463$	21.0641	−	12.1613i	0.978930	−	0.565186i	0.901947	+	0.431847i	$0.142138\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−10.0899			−0.466905			−0.233453	−	0.972368i	$-0.575002\pi$
$467$	−10.0899			−0.466905			−0.233453	+	0.972368i	$0.575002\pi$
$468$		0			0
$469$	−10.0571			−0.464393
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.887604	+	0.512458i	−0.0408121	+	0.0235629i
$474$		0			0
$475$	0.371894	+	0.214713i	0.0170637	+	0.00985172i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	36.4756	+	21.0592i	1.66661	+	0.962220i	0.969443	+	0.245316i	$0.0788918\pi$
$479$	36.4756	+	21.0592i	1.66661	+	0.962220i	0.697172	+	0.716904i	$0.254442\pi$
$480$		0			0
$481$	19.6322	−	7.37335i	0.895151	−	0.336196i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	38.1514			1.73237
$486$		0			0
$487$			25.4950i			1.15529i	0.816289	+	0.577644i	$0.196028\pi$
$487$			25.4950i			1.15529i	−0.816289	+	0.577644i	$0.803972\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−11.2578	−	19.4991i	−0.508059	−	0.879984i	−0.999956	−	0.00933096i	$-0.997030\pi$
$491$	−11.2578	−	19.4991i	−0.508059	−	0.879984i	0.491897	−	0.870653i	$-0.336304\pi$
$492$		0			0
$493$	0.799530	−	1.38483i	0.0360090	−	0.0623695i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.915568	+	1.58581i	−0.0410688	+	0.0711333i
$498$		0			0
$499$	15.5398	−	8.97193i	0.695659	−	0.401639i	−0.110070	−	0.993924i	$-0.535107\pi$
$499$	15.5398	−	8.97193i	0.695659	−	0.401639i	0.805729	+	0.592285i	$0.201774\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	9.20678			0.410510			0.205255	−	0.978709i	$-0.434198\pi$
$503$	9.20678			0.410510			0.205255	+	0.978709i	$0.434198\pi$
$504$		0			0
$505$			17.3330i			0.771309i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.565564	−	0.326529i	0.0250682	−	0.0144731i	−0.487414	−	0.873171i	$-0.662060\pi$
$509$	0.565564	−	0.326529i	0.0250682	−	0.0144731i	0.512482	+	0.858698i	$0.328726\pi$
$510$		0			0
$511$	−1.04508	+	1.81014i	−0.0462318	+	0.0800758i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	13.4669	+	7.77510i	0.593421	+	0.342612i
$516$		0			0
$517$	−1.36553	−	2.36517i	−0.0600560	−	0.104020i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−31.8369			−1.39480			−0.697399	−	0.716683i	$-0.745659\pi$
$521$	−31.8369			−1.39480			−0.697399	+	0.716683i	$0.745659\pi$
$522$		0			0
$523$	33.3595			1.45871			0.729355	−	0.684136i	$-0.239820\pi$
$523$	33.3595			1.45871			0.729355	+	0.684136i	$0.239820\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−11.9217	+	6.88301i	−0.519319	+	0.299829i
$528$		0			0
$529$	−15.2709	+	26.4500i	−0.663953	+	1.15000i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−16.4809	−	13.5448i	−0.713867	−	0.586692i
$534$		0			0
$535$	14.7034	−	8.48898i	0.635681	−	0.367011i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	1.64576i		−	0.0708880i
$540$		0			0
$541$			10.1445i			0.436146i	0.975932	+	0.218073i	$0.0699771\pi$
$541$			10.1445i			0.436146i	−0.975932	+	0.218073i	$0.930023\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3.01119	−	5.21553i	−0.128985	−	0.223409i
$546$		0			0
$547$	8.35259	−	14.4671i	0.357131	−	0.618569i	−0.630349	−	0.776312i	$-0.717088\pi$
$547$	8.35259	−	14.4671i	0.357131	−	0.618569i	0.987480	+	0.157742i	$0.0504216\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	2.46479	+	1.42304i	0.105003	+	0.0606237i
$552$		0			0
$553$	5.66463	−	3.27047i	0.240884	−	0.139075i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			31.3082i			1.32657i	0.748367	+	0.663285i	$0.230838\pi$
$557$			31.3082i			1.32657i	−0.748367	+	0.663285i	$0.769162\pi$
$558$		0			0
$559$	−12.5005	−	2.07283i	−0.528714	−	0.0876713i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−7.58162	−	13.1318i	−0.319527	−	0.553438i	0.660862	−	0.750507i	$-0.270191\pi$
$563$	−7.58162	−	13.1318i	−0.319527	−	0.553438i	−0.980389	+	0.197070i	$0.936857\pi$
$564$		0			0
$565$	11.5236	+	6.65316i	0.484802	+	0.279901i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	7.13422	−	12.3568i	0.299082	−	0.518025i	−0.676844	−	0.736126i	$-0.736653\pi$
$569$	7.13422	−	12.3568i	0.299082	−	0.518025i	0.975926	+	0.218101i	$0.0699862\pi$
$570$		0			0
$571$	3.56135	+	6.16844i	0.149038	+	0.258141i	0.930872	−	0.365345i	$-0.119049\pi$
$571$	3.56135	+	6.16844i	0.149038	+	0.258141i	−0.781834	+	0.623486i	$0.785716\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.819053			−0.0341569
$576$		0			0
$577$			44.7223i			1.86181i	0.365255	+	0.930907i	$0.380982\pi$
$577$			44.7223i			1.86181i	−0.365255	+	0.930907i	$0.619018\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	4.82269	+	8.35315i	0.200079	+	0.346547i
$582$		0			0
$583$	−1.36664	−	0.789027i	−0.0566002	−	0.0326782i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−6.86543	−	3.96376i	−0.283367	−	0.163602i	0.351580	−	0.936158i	$-0.385645\pi$
$587$	−6.86543	−	3.96376i	−0.283367	−	0.163602i	−0.634947	+	0.772556i	$0.718978\pi$
$588$		0			0
$589$	−12.2507	−	21.2189i	−0.504783	−	0.874309i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	15.7522i		−	0.646866i	−0.946251	−	0.323433i	$-0.895163\pi$
$593$		−	15.7522i		−	0.646866i	0.946251	−	0.323433i	$-0.104837\pi$
$594$		0			0
$595$	−5.55100			−0.227569
$596$		0			0
$597$		0			0
$598$		0			0
$599$	11.0365	+	19.1157i	0.450938	+	0.781047i	0.998445	−	0.0557544i	$-0.0177564\pi$
$599$	11.0365	+	19.1157i	0.450938	+	0.781047i	−0.547507	+	0.836801i	$0.684423\pi$
$600$		0			0
$601$	8.79460	−	15.2327i	0.358739	−	0.621354i	−0.629011	−	0.777396i	$-0.716540\pi$
$601$	8.79460	−	15.2327i	0.358739	−	0.621354i	0.987750	+	0.156042i	$0.0498734\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	20.8988	+	12.0659i	0.849656	+	0.490549i
$606$		0			0
$607$	0.449016	+	0.777719i	0.0182250	+	0.0315666i	0.874994	−	0.484134i	$-0.160865\pi$
$607$	0.449016	+	0.777719i	0.0182250	+	0.0315666i	−0.856769	+	0.515700i	$0.827532\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	5.52340	−	33.3096i	0.223453	−	1.34756i
$612$		0			0
$613$		−	0.130728i		−	0.00528005i	−0.999997	−	0.00264002i	$-0.999160\pi$
$613$		−	0.130728i		−	0.00528005i	0.999997	−	0.00264002i	$-0.000840346\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−10.4725	+	6.04630i	−0.421607	+	0.243415i	−0.695765	−	0.718270i	$-0.744934\pi$
$617$	−10.4725	+	6.04630i	−0.421607	+	0.243415i	0.274158	+	0.961685i	$0.411601\pi$
$618$		0			0
$619$	−38.1937	−	22.0511i	−1.53513	−	0.886310i	−0.999113	−	0.0421046i	$-0.986594\pi$
$619$	−38.1937	−	22.0511i	−1.53513	−	0.886310i	−0.536020	−	0.844205i	$-0.680073\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	9.32671	−	16.1543i	0.373667	−	0.647210i
$624$		0			0
$625$	12.2139	+	21.1551i	0.488556	+	0.846204i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			12.5369i			0.499880i
$630$		0			0
$631$			20.5084i			0.816427i	0.912886	+	0.408214i	$0.133848\pi$
$631$			20.5084i			0.816427i	−0.912886	+	0.408214i	$0.866152\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	8.29564	−	4.78949i	0.329202	−	0.190065i
$636$		0			0
$637$	12.9188	−	15.7192i	0.511863	−	0.622818i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.24570	−	2.15761i	0.0492022	−	0.0852206i	−0.840375	−	0.542005i	$-0.817665\pi$
$641$	1.24570	−	2.15761i	0.0492022	−	0.0852206i	0.889578	+	0.456784i	$0.150999\pi$
$642$		0			0
$643$	−4.31818	+	2.49310i	−0.170292	+	0.0983184i	−0.582724	−	0.812670i	$-0.698013\pi$
$643$	−4.31818	+	2.49310i	−0.170292	+	0.0983184i	0.412431	+	0.910989i	$0.364680\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	9.16284			0.360229			0.180114	−	0.983646i	$-0.442353\pi$
$647$	9.16284			0.360229			0.180114	+	0.983646i	$0.442353\pi$
$648$		0			0
$649$	−2.88303			−0.113169
$650$		0			0
$651$		0			0
$652$		0			0
$653$	12.1359	+	21.0201i	0.474916	+	0.822579i	0.999587	−	0.0287260i	$-0.00914504\pi$
$653$	12.1359	+	21.0201i	0.474916	+	0.822579i	−0.524671	+	0.851305i	$0.675812\pi$
$654$		0			0
$655$	15.2150	+	8.78440i	0.594501	+	0.343235i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−5.82249	+	10.0848i	−0.226812	+	0.392850i	−0.956862	−	0.290544i	$-0.906164\pi$
$659$	−5.82249	+	10.0848i	−0.226812	+	0.392850i	0.730050	+	0.683394i	$0.239497\pi$
$660$		0			0
$661$	−30.6874	+	17.7174i	−1.19360	+	0.689125i	−0.959121	−	0.282996i	$-0.908672\pi$
$661$	−30.6874	+	17.7174i	−1.19360	+	0.689125i	−0.234479	+	0.972121i	$0.575338\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	9.87995i		−	0.383128i
$666$		0			0
$667$	−5.42840			−0.210188
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.18669	−	0.685134i	0.0458115	−	0.0264493i
$672$		0			0
$673$	−23.3151	+	40.3829i	−0.898729	+	1.55664i	−0.0696090	+	0.997574i	$0.522175\pi$
$673$	−23.3151	+	40.3829i	−0.898729	+	1.55664i	−0.829120	+	0.559070i	$0.811158\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	11.0245	−	19.0950i	0.423707	−	0.733882i	−0.572592	−	0.819841i	$-0.694062\pi$
$677$	11.0245	−	19.0950i	0.423707	−	0.733882i	0.996299	+	0.0859589i	$0.0273954\pi$
$678$		0			0
$679$	10.0502	+	17.4074i	0.385691	+	0.668036i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			44.9210i			1.71885i	0.511259	+	0.859427i	$0.329179\pi$
$683$			44.9210i			1.71885i	−0.511259	+	0.859427i	$0.670821\pi$
$684$		0			0
$685$	22.7037			0.867464
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−6.85950	−	18.2640i	−0.261326	−	0.695804i
$690$		0			0
$691$	34.7246	+	20.0482i	1.32099	+	0.762671i	0.983886	−	0.178798i	$-0.0572209\pi$
$691$	34.7246	+	20.0482i	1.32099	+	0.762671i	0.337099	+	0.941469i	$0.390554\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−16.6936	−	9.63806i	−0.633225	−	0.365593i
$696$		0			0
$697$	11.0445	−	6.37652i	0.418339	−	0.241528i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−23.7205			−0.895913			−0.447956	−	0.894055i	$-0.647848\pi$
$701$	−23.7205			−0.895913			−0.447956	+	0.894055i	$0.647848\pi$
$702$		0			0
$703$	−22.3138			−0.841583
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−7.90858	+	4.56602i	−0.297433	+	0.171723i
$708$		0			0
$709$	−7.38775	−	4.26532i	−0.277453	−	0.160187i	0.354817	−	0.934936i	$-0.384543\pi$
$709$	−7.38775	−	4.26532i	−0.277453	−	0.160187i	−0.632270	+	0.774748i	$0.717877\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	40.4712	+	23.3660i	1.51566	+	0.875065i
$714$		0			0
$715$	−0.817384	−	2.17636i	−0.0305684	−	0.0813912i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−24.4342			−0.911240			−0.455620	−	0.890174i	$-0.650582\pi$
$719$	−24.4342			−0.911240			−0.455620	+	0.890174i	$0.650582\pi$
$720$		0			0
$721$			8.19275i			0.305114i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−0.0415202	−	0.0719152i	−0.00154202	−	0.00267086i
$726$		0			0
$727$	6.87491	−	11.9077i	0.254976	−	0.441632i	−0.709913	−	0.704290i	$-0.751266\pi$
$727$	6.87491	−	11.9077i	0.254976	−	0.441632i	0.964889	+	0.262658i	$0.0845990\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	3.78752	−	6.56018i	0.140086	−	0.242637i
$732$		0			0
$733$	30.1277	−	17.3942i	1.11279	−	0.642470i	0.173240	−	0.984880i	$-0.444576\pi$
$733$	30.1277	−	17.3942i	1.11279	−	0.642470i	0.939551	+	0.342410i	$0.111243\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2.51798			0.0927511
$738$		0			0
$739$			16.3259i			0.600560i	0.953851	+	0.300280i	$0.0970800\pi$
$739$			16.3259i			0.600560i	−0.953851	+	0.300280i	$0.902920\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	45.0029	−	25.9824i	1.65100	−	0.953203i	0.674333	−	0.738427i	$-0.264431\pi$
$743$	45.0029	−	25.9824i	1.65100	−	0.953203i	0.976663	−	0.214776i	$-0.0689022\pi$
$744$		0			0
$745$	8.33680	−	14.4398i	0.305437	−	0.529032i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	7.74659	+	4.47249i	0.283054	+	0.163421i
$750$		0			0
$751$	−12.2956	−	21.2966i	−0.448674	−	0.777126i	0.549626	−	0.835411i	$-0.314770\pi$
$751$	−12.2956	−	21.2966i	−0.448674	−	0.777126i	−0.998300	+	0.0582850i	$0.981437\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	33.0193			1.20170
$756$		0			0
$757$	13.3525			0.485305			0.242652	−	0.970113i	$-0.421983\pi$
$757$	13.3525			0.485305			0.242652	+	0.970113i	$0.421983\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	32.6916	−	18.8745i	1.18507	−	0.684201i	0.227888	−	0.973687i	$-0.426818\pi$
$761$	32.6916	−	18.8745i	1.18507	−	0.684201i	0.957182	+	0.289487i	$0.0934846\pi$
$762$		0			0
$763$	1.58647	−	2.74784i	0.0574340	−	0.0994787i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−27.5367	−	22.6311i	−0.994293	−	0.817161i
$768$		0			0
$769$	24.9841	−	14.4246i	0.900950	−	0.520164i	0.0234417	−	0.999725i	$-0.492538\pi$
$769$	24.9841	−	14.4246i	0.900950	−	0.520164i	0.877508	+	0.479561i	$0.159204\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			7.64931i			0.275127i	0.990493	+	0.137563i	$0.0439270\pi$
$773$			7.64931i			0.275127i	−0.990493	+	0.137563i	$0.956073\pi$
$774$		0			0
$775$			0.714881i			0.0256793i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	11.3493	+	19.6575i	0.406629	+	0.704302i
$780$		0			0
$781$	0.229230	−	0.397038i	0.00820250	−	0.0142072i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	17.4026	+	10.0474i	0.621125	+	0.358607i
$786$		0			0
$787$	−37.0064	+	21.3656i	−1.31913	+	0.761602i	−0.983589	−	0.180422i	$-0.942254\pi$
$787$	−37.0064	+	21.3656i	−1.31913	+	0.761602i	−0.335545	+	0.942024i	$0.608920\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			7.01055i			0.249266i
$792$		0			0
$793$	16.7126	+	2.77128i	0.593481	+	0.0984109i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	22.8947	+	39.6549i	0.810974	+	1.40465i	0.912183	+	0.409782i	$0.134395\pi$
$797$	22.8947	+	39.6549i	0.810974	+	1.40465i	−0.101210	+	0.994865i	$0.532271\pi$
$798$		0			0
$799$	17.4807	+	10.0925i	0.618422	+	0.357046i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	0.261657	−	0.453203i	0.00923368	−	0.0159932i
$804$		0			0
$805$	9.42210	+	16.3196i	0.332085	+	0.575189i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−28.9221			−1.01685			−0.508424	−	0.861107i	$-0.669772\pi$
$809$	−28.9221			−1.01685			−0.508424	+	0.861107i	$0.669772\pi$
$810$		0			0
$811$		−	23.5093i		−	0.825521i	−0.910839	−	0.412761i	$-0.864565\pi$
$811$		−	23.5093i		−	0.825521i	0.910839	−	0.412761i	$-0.135435\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−19.7845	−	34.2678i	−0.693022	−	1.20035i
$816$		0			0
$817$	11.6761	+	6.74122i	0.408496	+	0.235845i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.471666	−	0.272317i	−0.0164613	−	0.00950392i	0.491747	−	0.870738i	$-0.336359\pi$
$821$	−0.471666	−	0.272317i	−0.0164613	−	0.00950392i	−0.508208	+	0.861234i	$0.669692\pi$
$822$		0			0
$823$	5.13512	+	8.89429i	0.178999	+	0.310035i	0.941538	−	0.336907i	$-0.109381\pi$
$823$	5.13512	+	8.89429i	0.178999	+	0.310035i	−0.762539	+	0.646942i	$0.776047\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	27.5250i		−	0.957138i	−0.878050	−	0.478569i	$-0.841156\pi$
$827$		−	27.5250i		−	0.957138i	0.878050	−	0.478569i	$-0.158844\pi$
$828$		0			0
$829$	35.9132			1.24732			0.623658	−	0.781697i	$-0.285646\pi$
$829$	35.9132			1.24732			0.623658	+	0.781697i	$0.285646\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	6.08182	+	10.5340i	0.210722	+	0.364982i
$834$		0			0
$835$	−13.6004	+	23.5565i	−0.470660	+	0.815207i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	36.6064	+	21.1347i	1.26379	+	0.729652i	0.973807	−	0.227378i	$-0.0730153\pi$
$839$	36.6064	+	21.1347i	1.26379	+	0.729652i	0.289988	+	0.957030i	$0.406349\pi$
$840$		0			0
$841$	14.2248	+	24.6381i	0.490511	+	0.849590i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	9.27680	−	27.2034i	0.319132	−	0.935825i
$846$		0			0
$847$			12.7141i			0.436860i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	36.8577	−	21.2798i	1.26347	−	0.729462i
$852$		0			0
$853$	−41.9467	−	24.2179i	−1.43623	−	0.829206i	−0.438642	−	0.898662i	$-0.644540\pi$
$853$	−41.9467	−	24.2179i	−1.43623	−	0.829206i	−0.997585	+	0.0694563i	$0.977874\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−21.4039	+	37.0727i	−0.731145	+	1.26638i	0.225250	+	0.974301i	$0.427680\pi$
$857$	−21.4039	+	37.0727i	−0.731145	+	1.26638i	−0.956394	+	0.292078i	$0.905653\pi$
$858$		0			0
$859$	−19.8720	−	34.4194i	−0.678025	−	1.17437i	−0.975575	−	0.219668i	$-0.929503\pi$
$859$	−19.8720	−	34.4194i	−0.678025	−	1.17437i	0.297549	−	0.954706i	$-0.403831\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	12.1271i		−	0.412810i	−0.978467	−	0.206405i	$-0.933824\pi$
$863$		−	12.1271i		−	0.412810i	0.978467	−	0.206405i	$-0.0661765\pi$
$864$		0			0
$865$			47.6036i			1.61857i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.41825	+	0.818827i	−0.0481108	+	0.0277768i
$870$		0			0
$871$	24.0501	+	19.7656i	0.814906	+	0.669731i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−6.58243	+	11.4011i	−0.222527	+	0.385428i
$876$		0			0
$877$	−46.7146	+	26.9707i	−1.57744	+	0.910735i	−0.582224	+	0.813029i	$0.697817\pi$
$877$	−46.7146	+	26.9707i	−1.57744	+	0.910735i	−0.995215	+	0.0977062i	$0.968849\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−33.2571			−1.12046			−0.560230	−	0.828337i	$-0.689287\pi$
$881$	−33.2571			−1.12046			−0.560230	+	0.828337i	$0.689287\pi$
$882$		0			0
$883$	−41.6999			−1.40331			−0.701656	−	0.712516i	$-0.747556\pi$
$883$	−41.6999			−1.40331			−0.701656	+	0.712516i	$0.747556\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−5.88946	−	10.2008i	−0.197749	−	0.342511i	0.750049	−	0.661382i	$-0.230030\pi$
$887$	−5.88946	−	10.2008i	−0.197749	−	0.342511i	−0.947798	+	0.318871i	$0.896696\pi$
$888$		0			0
$889$	4.37063	+	2.52338i	0.146586	+	0.0846315i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−17.9631	+	31.1130i	−0.601113	+	1.04116i
$894$		0			0
$895$	−28.4875	+	16.4473i	−0.952234	+	0.549772i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			4.73798i			0.158020i
$900$		0			0
$901$	11.6632			0.388558
$902$		0			0
$903$		0			0
$904$		0			0
$905$	22.5440	−	13.0158i	0.749388	−	0.432659i
$906$		0			0
$907$	−6.54410	+	11.3347i	−0.217293	+	0.376363i	−0.953980	−	0.299872i	$-0.903056\pi$
$907$	−6.54410	+	11.3347i	−0.217293	+	0.376363i	0.736686	+	0.676235i	$0.236389\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	13.2230	−	22.9029i	0.438097	−	0.758806i	−0.559446	−	0.828867i	$-0.688986\pi$
$911$	13.2230	−	22.9029i	0.438097	−	0.758806i	0.997543	+	0.0700608i	$0.0223193\pi$
$912$		0			0
$913$	−1.20746	−	2.09137i	−0.0399609	−	0.0692144i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			9.25628i			0.305669i
$918$		0			0
$919$	21.1580			0.697939			0.348970	−	0.937134i	$-0.386532\pi$
$919$	21.1580			0.697939			0.348970	+	0.937134i	$0.386532\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	5.30611	−	1.99284i	0.174653	−	0.0655951i
$924$		0			0
$925$	0.563828	+	0.325526i	0.0185385	+	0.0107032i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−33.6129	−	19.4064i	−1.10281	−	0.636705i	−0.165849	−	0.986151i	$-0.553036\pi$
$929$	−33.6129	−	19.4064i	−1.10281	−	0.636705i	−0.936956	+	0.349446i	$0.886370\pi$
$930$		0			0
$931$	−18.7490	+	10.8247i	−0.614473	+	0.354766i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1.38980			0.0454513
$936$		0			0
$937$	30.4501			0.994762			0.497381	−	0.867532i	$-0.334295\pi$
$937$	30.4501			0.994762			0.497381	+	0.867532i	$0.334295\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	5.89665	−	3.40443i	0.192225	−	0.110981i	−0.400799	−	0.916166i	$-0.631267\pi$
$941$	5.89665	−	3.40443i	0.192225	−	0.110981i	0.593024	+	0.805185i	$0.297934\pi$
$942$		0			0
$943$	−37.4931	−	21.6466i	−1.22094	−	0.704911i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	44.8064	+	25.8690i	1.45601	+	0.840629i	0.998812	−	0.0487345i	$-0.0155188\pi$
$947$	44.8064	+	25.8690i	1.45601	+	0.840629i	0.457201	+	0.889364i	$0.348852\pi$
$948$		0			0
$949$	6.05671	−	2.27474i	0.196609	−	0.0738413i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−57.7337			−1.87018			−0.935090	−	0.354412i	$-0.884681\pi$
$953$	−57.7337			−1.87018			−0.935090	+	0.354412i	$0.884681\pi$
$954$		0			0
$955$		−	3.27787i		−	0.106069i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	5.98082	+	10.3591i	0.193131	+	0.334512i
$960$		0			0
$961$	4.89421	−	8.47701i	0.157878	−	0.273452i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	15.6903	−	27.1764i	0.505089	−	0.874839i
$966$		0			0
$967$	−38.4590	+	22.2043i	−1.23676	+	0.714042i	−0.968430	−	0.249286i	$-0.919804\pi$
$967$	−38.4590	+	22.2043i	−1.23676	+	0.714042i	−0.268327	+	0.963328i	$0.586471\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	41.2827			1.32482			0.662412	−	0.749140i	$-0.269533\pi$
$971$	41.2827			1.32482			0.662412	+	0.749140i	$0.269533\pi$
$972$		0			0
$973$		−	10.1558i		−	0.325580i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−2.44748	+	1.41306i	−0.0783019	+	0.0452076i	−0.538640	−	0.842536i	$-0.681062\pi$
$977$	−2.44748	+	1.41306i	−0.0783019	+	0.0452076i	0.460338	+	0.887744i	$0.347728\pi$
$978$		0			0
$979$	−2.33512	+	4.04455i	−0.0746309	+	0.129264i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	9.36302	+	5.40574i	0.298634	+	0.172416i	0.641829	−	0.766848i	$-0.278176\pi$
$983$	9.36302	+	5.40574i	0.298634	+	0.172416i	−0.343195	+	0.939264i	$0.611509\pi$
$984$		0			0
$985$	16.8570	+	29.1971i	0.537108	+	0.930298i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−25.7153			−0.817699
$990$		0			0
$991$	−21.3052			−0.676782			−0.338391	−	0.941006i	$-0.609883\pi$
$991$	−21.3052			−0.676782			−0.338391	+	0.941006i	$0.609883\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−23.5042	+	13.5702i	−0.745134	+	0.430203i
$996$		0			0
$997$	4.63121	−	8.02150i	0.146672	−	0.254043i	−0.783323	−	0.621614i	$-0.786477\pi$
$997$	4.63121	−	8.02150i	0.146672	−	0.254043i	0.929996	+	0.367571i	$0.119810\pi$
$998$		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.1585.10		80
3.2	odd	2		936.2.cw.b.25.4	yes	80
9.4	even	3	inner	2808.2.cw.b.2521.31		80
9.5	odd	6		936.2.cw.b.337.3	yes	80
13.12	even	2	inner	2808.2.cw.b.1585.31		80
39.38	odd	2		936.2.cw.b.25.3	✓	80
117.77	odd	6		936.2.cw.b.337.4	yes	80
117.103	even	6	inner	2808.2.cw.b.2521.10		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
936.2.cw.b.25.3	✓	80	39.38	odd	2
936.2.cw.b.25.4	yes	80	3.2	odd	2
936.2.cw.b.337.3	yes	80	9.5	odd	6
936.2.cw.b.337.4	yes	80	117.77	odd	6
2808.2.cw.b.1585.10		80	1.1	even	1	trivial
2808.2.cw.b.1585.31		80	13.12	even	2	inner
2808.2.cw.b.2521.10		80	117.103	even	6	inner
2808.2.cw.b.2521.31		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)

\(f(q)\)	\(=\)	\(q+(-1.91469 + 1.10545i) q^{5} +(-1.00877 - 0.582415i) q^{7} +(0.252566 + 0.145819i) q^{11} +(1.26769 + 3.37535i) q^{13} -2.15546 q^{17} -3.83640i q^{19} +(3.65862 + 6.33691i) q^{23} +(-0.0559674 + 0.0969384i) q^{25} +(-0.370932 + 0.642474i) q^{29} +(5.53094 - 3.19329i) q^{31} +2.57532 q^{35} -5.81635i q^{37} +(-5.12394 + 2.95831i) q^{41} +(-1.75717 + 3.04352i) q^{43} +(-8.10996 - 4.68229i) q^{47} +(-2.82159 - 4.88713i) q^{49} -5.41101 q^{53} -0.644781 q^{55} +(-8.56121 + 4.94282i) q^{59} +(2.34926 - 4.06904i) q^{61} +(-6.15851 - 5.06138i) q^{65} +(7.47721 - 4.31697i) q^{67} -1.57202i q^{71} -1.79440i q^{73} +(-0.169854 - 0.294196i) q^{77} +(-2.80768 + 4.86305i) q^{79} +(-7.17113 - 4.14026i) q^{83} +(4.12704 - 2.38275i) q^{85} +16.0139i q^{89} +(0.687038 - 4.14328i) q^{91} +(4.24094 + 7.34553i) q^{95} +(-14.9442 - 8.62804i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 6 q^{13} + 4 q^{17} - 10 q^{23} + 44 q^{25} + 52 q^{35} - 26 q^{43} + 48 q^{49} - 60 q^{53} - 16 q^{55} - 10 q^{61} + 26 q^{65} - 32 q^{77} + 6 q^{79} - 4 q^{91} - 8 q^{95}+O(q^{100}) \) 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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