LMFDB - Embedded newform 504.3.cu.b.305.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [504,3,Mod(233,504)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("504.233"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(504, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 2])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 504 = 2^{3} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	504.cu (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$13.7330053238$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 38 x^{14} - 120 x^{13} + 1059 x^{12} - 3540 x^{11} + 20690 x^{10} - 73200 x^{9} + 269971 x^{8} + \cdots + 352836 $ x^16 + 38x^14 - 120x^13 + 1059x^12 - 3540x^11 + 20690x^10 - 73200x^9 + 269971x^8 - 612180x^7 + 1101306x^6 - 1098360x^5 + 603810x^4 + 832680x^3 - 439020x^2 - 320760x + 352836
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			305.5
Root			$2.24311 - 2.47096i$ of defining polynomial
Character	$\chi$	$=$	504.305
Dual form			504.3.cu.b.233.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.0443209 - 0.0255887i) q^{5} +(-6.98675 - 0.430549i) q^{7} +(-6.88786 - 3.97671i) q^{11} +8.73248 q^{13} +(25.1822 + 14.5390i) q^{17} +(15.5493 + 26.9321i) q^{19} +(25.1822 - 14.5390i) q^{23} +(-12.4987 + 21.6484i) q^{25} -10.1650i q^{29} +(-16.3725 + 28.3580i) q^{31} +(-0.320676 + 0.159699i) q^{35} +(13.7520 + 23.8192i) q^{37} -2.97791i q^{41} -7.18571 q^{43} +(16.3348 - 9.43088i) q^{47} +(48.6293 + 6.01627i) q^{49} +(86.8096 + 50.1196i) q^{53} -0.407035 q^{55} +(-33.9854 - 19.6215i) q^{59} +(-5.31052 - 9.19810i) q^{61} +(0.387031 - 0.223452i) q^{65} +(-49.6552 + 86.0054i) q^{67} -16.0955i q^{71} +(-7.28156 + 12.6120i) q^{73} +(46.4116 + 30.7498i) q^{77} +(55.2149 + 95.6350i) q^{79} -106.655i q^{83} +1.48813 q^{85} +(-13.3293 + 7.69565i) q^{89} +(-61.0116 - 3.75976i) q^{91} +(1.37831 + 0.795770i) q^{95} +116.379 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 40 q^{13} - 20 q^{19} - 4 q^{25} - 56 q^{31} + 76 q^{37} + 72 q^{43} - 48 q^{49} + 648 q^{55} - 72 q^{61} - 156 q^{67} + 124 q^{73} + 184 q^{79} - 864 q^{85} - 116 q^{91} + 1112 q^{97}+O(q^{100}) $ 16 * q + 40 * q^13 - 20 * q^19 - 4 * q^25 - 56 * q^31 + 76 * q^37 + 72 * q^43 - 48 * q^49 + 648 * q^55 - 72 * q^61 - 156 * q^67 + 124 * q^73 + 184 * q^79 - 864 * q^85 - 116 * q^91 + 1112 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/504\mathbb{Z}\right)^\times$.

$n$	$73$	$127$	$253$	$281$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.0443209	−	0.0255887i	0.00886417	−	0.00511773i	−0.495561	−	0.868573i	$-0.665038\pi$
$5$	0.0443209	−	0.0255887i	0.00886417	−	0.00511773i	0.504426	+	0.863455i	$0.331704\pi$
$6$		0			0
$7$	−6.98675	−	0.430549i	−0.998107	−	0.0615069i
$7$	−6.98675	−	0.430549i	−0.998107	−	0.0615069i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−6.88786	−	3.97671i	−0.626169	−	0.361519i	0.153098	−	0.988211i	$-0.451075\pi$
$11$	−6.88786	−	3.97671i	−0.626169	−	0.361519i	−0.779267	+	0.626692i	$0.784408\pi$
$12$		0			0
$13$	8.73248			0.671729			0.335865	−	0.941910i	$-0.390972\pi$
$13$	8.73248			0.671729			0.335865	+	0.941910i	$0.390972\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	25.1822	+	14.5390i	1.48131	+	0.855234i	0.999775	−	0.0211937i	$-0.00674668\pi$
$17$	25.1822	+	14.5390i	1.48131	+	0.855234i	0.481533	+	0.876428i	$0.340080\pi$
$18$		0			0
$19$	15.5493	+	26.9321i	0.818383	+	1.41748i	0.906873	+	0.421404i	$0.138462\pi$
$19$	15.5493	+	26.9321i	0.818383	+	1.41748i	−0.0884901	+	0.996077i	$0.528204\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	25.1822	−	14.5390i	1.09488	−	0.632129i	0.160009	−	0.987116i	$-0.448848\pi$
$23$	25.1822	−	14.5390i	1.09488	−	0.632129i	0.934872	+	0.354986i	$0.115514\pi$
$24$		0			0
$25$	−12.4987	+	21.6484i	−0.499948	+	0.865935i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	10.1650i		−	0.350518i	−0.984522	−	0.175259i	$-0.943924\pi$
$29$		−	10.1650i		−	0.350518i	0.984522	−	0.175259i	$-0.0560763\pi$
$30$		0			0
$31$	−16.3725	+	28.3580i	−0.528146	+	0.914775i	0.471316	+	0.881964i	$0.343779\pi$
$31$	−16.3725	+	28.3580i	−0.528146	+	0.914775i	−0.999462	+	0.0328106i	$0.989554\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.320676	+	0.159699i	−0.00916217	+	0.00456283i
$36$		0			0
$37$	13.7520	+	23.8192i	0.371676	+	0.643761i	0.989823	−	0.142301i	$-0.0454500\pi$
$37$	13.7520	+	23.8192i	0.371676	+	0.643761i	−0.618148	+	0.786062i	$0.712117\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	2.97791i		−	0.0726319i	−0.999340	−	0.0363159i	$-0.988438\pi$
$41$		−	2.97791i		−	0.0726319i	0.999340	−	0.0363159i	$-0.0115623\pi$
$42$		0			0
$43$	−7.18571			−0.167110			−0.0835548	−	0.996503i	$-0.526627\pi$
$43$	−7.18571			−0.167110			−0.0835548	+	0.996503i	$0.526627\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	16.3348	−	9.43088i	0.347548	−	0.200657i	−0.316057	−	0.948740i	$-0.602359\pi$
$47$	16.3348	−	9.43088i	0.347548	−	0.200657i	0.663605	+	0.748083i	$0.269026\pi$
$48$		0			0
$49$	48.6293	+	6.01627i	0.992434	+	0.122781i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	86.8096	+	50.1196i	1.63792	+	0.945652i	0.981549	+	0.191212i	$0.0612418\pi$
$53$	86.8096	+	50.1196i	1.63792	+	0.945652i	0.656369	+	0.754440i	$0.272092\pi$
$54$		0			0
$55$	−0.407035			−0.00740063
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−33.9854	−	19.6215i	−0.576024	−	0.332568i	0.183528	−	0.983015i	$-0.441248\pi$
$59$	−33.9854	−	19.6215i	−0.576024	−	0.332568i	−0.759552	+	0.650447i	$0.774582\pi$
$60$		0			0
$61$	−5.31052	−	9.19810i	−0.0870578	−	0.150788i	0.819208	−	0.573496i	$-0.194413\pi$
$61$	−5.31052	−	9.19810i	−0.0870578	−	0.150788i	−0.906266	+	0.422707i	$0.861080\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.387031	−	0.223452i	0.00595432	−	0.00343773i
$66$		0			0
$67$	−49.6552	+	86.0054i	−0.741123	+	1.28366i	0.210862	+	0.977516i	$0.432373\pi$
$67$	−49.6552	+	86.0054i	−0.741123	+	1.28366i	−0.951984	+	0.306146i	$0.900960\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	16.0955i		−	0.226697i	−0.993555	−	0.113349i	$-0.963842\pi$
$71$		−	16.0955i		−	0.226697i	0.993555	−	0.113349i	$-0.0361577\pi$
$72$		0			0
$73$	−7.28156	+	12.6120i	−0.0997474	+	0.172768i	−0.911580	−	0.411123i	$-0.865137\pi$
$73$	−7.28156	+	12.6120i	−0.0997474	+	0.172768i	0.811833	+	0.583890i	$0.198470\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	46.4116	+	30.7498i	0.602748	+	0.399348i
$78$		0			0
$79$	55.2149	+	95.6350i	0.698923	+	1.21057i	0.968840	+	0.247686i	$0.0796702\pi$
$79$	55.2149	+	95.6350i	0.698923	+	1.21057i	−0.269918	+	0.962883i	$0.586997\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	106.655i		−	1.28500i	−0.766284	−	0.642502i	$-0.777896\pi$
$83$		−	106.655i		−	1.28500i	0.766284	−	0.642502i	$-0.222104\pi$
$84$		0			0
$85$	1.48813			0.0175074
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−13.3293	+	7.69565i	−0.149767	+	0.0864680i	−0.573011	−	0.819548i	$-0.694225\pi$
$89$	−13.3293	+	7.69565i	−0.149767	+	0.0864680i	0.423244	+	0.906016i	$0.360891\pi$
$90$		0			0
$91$	−61.0116	−	3.75976i	−0.670457	−	0.0413160i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.37831	+	0.795770i	0.0145086	+	0.00837653i
$96$		0			0
$97$	116.379			1.19978			0.599890	−	0.800083i	$-0.295211\pi$
$97$	116.379			1.19978			0.599890	+	0.800083i	$0.295211\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	52.5722	+	30.3526i	0.520517	+	0.300520i	0.737146	−	0.675734i	$-0.236173\pi$
$101$	52.5722	+	30.3526i	0.520517	+	0.300520i	−0.216629	+	0.976254i	$0.569506\pi$
$102$		0			0
$103$	−80.8811	−	140.090i	−0.785253	−	1.36010i	−0.928848	−	0.370462i	$-0.879199\pi$
$103$	−80.8811	−	140.090i	−0.785253	−	1.36010i	0.143595	−	0.989637i	$-0.454134\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−38.5837	+	22.2763i	−0.360595	+	0.208190i	−0.669342	−	0.742955i	$-0.733424\pi$
$107$	−38.5837	+	22.2763i	−0.360595	+	0.208190i	0.308747	+	0.951144i	$0.400090\pi$
$108$		0			0
$109$	50.2502	−	87.0359i	0.461011	−	0.798495i	−0.538001	−	0.842944i	$-0.680820\pi$
$109$	50.2502	−	87.0359i	0.461011	−	0.798495i	0.999012	+	0.0444499i	$0.0141535\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			181.815i			1.60899i	0.593963	+	0.804493i	$0.297563\pi$
$113$			181.815i			1.60899i	−0.593963	+	0.804493i	$0.702437\pi$
$114$		0			0
$115$	0.744066	−	1.28876i	0.00647014	−	0.0112066i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−169.682	−	112.422i	−1.42590	−	0.944726i
$120$		0			0
$121$	−28.8716	−	50.0070i	−0.238608	−	0.413281i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			2.55873i			0.0204699i
$126$		0			0
$127$	−87.6886			−0.690461			−0.345231	−	0.938518i	$-0.612199\pi$
$127$	−87.6886			−0.690461			−0.345231	+	0.938518i	$0.612199\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−65.7880	+	37.9827i	−0.502198	+	0.289944i	−0.729621	−	0.683852i	$-0.760303\pi$
$131$	−65.7880	+	37.9827i	−0.502198	+	0.289944i	0.227423	+	0.973796i	$0.426970\pi$
$132$		0			0
$133$	−97.0433	−	194.863i	−0.729649	−	1.46513i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−129.676	−	74.8682i	−0.946537	−	0.546484i	−0.0545337	−	0.998512i	$-0.517367\pi$
$137$	−129.676	−	74.8682i	−0.946537	−	0.546484i	−0.892004	+	0.452028i	$0.850701\pi$
$138$		0			0
$139$	127.565			0.917734			0.458867	−	0.888505i	$-0.348255\pi$
$139$	127.565			0.917734			0.458867	+	0.888505i	$0.348255\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−60.1481	−	34.7265i	−0.420616	−	0.242843i
$144$		0			0
$145$	−0.260109	−	0.450522i	−0.00179386	−	0.00310705i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	162.279	−	93.6918i	1.08912	−	0.628804i	0.155779	−	0.987792i	$-0.450211\pi$
$149$	162.279	−	93.6918i	1.08912	−	0.628804i	0.933342	+	0.358988i	$0.116878\pi$
$150$		0			0
$151$	−12.4372	+	21.5419i	−0.0823657	+	0.142662i	−0.904266	−	0.426970i	$-0.859581\pi$
$151$	−12.4372	+	21.5419i	−0.0823657	+	0.142662i	0.821900	+	0.569632i	$0.192914\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			1.67580i			0.0108116i
$156$		0			0
$157$	71.7550	−	124.283i	0.457038	−	0.791613i	−0.541765	−	0.840530i	$-0.682244\pi$
$157$	71.7550	−	124.283i	0.457038	−	0.791613i	0.998803	+	0.0489172i	$0.0155770\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−182.202	+	90.7380i	−1.13169	+	0.563590i
$162$		0			0
$163$	−90.4979	−	156.747i	−0.555202	−	0.961637i	−0.997888	−	0.0649608i	$-0.979308\pi$
$163$	−90.4979	−	156.747i	−0.555202	−	0.961637i	0.442686	−	0.896677i	$-0.354026\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			15.7017i			0.0940219i	0.998894	+	0.0470110i	$0.0149696\pi$
$167$			15.7017i			0.0940219i	−0.998894	+	0.0470110i	$0.985030\pi$
$168$		0			0
$169$	−92.7438			−0.548780
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−206.741	+	119.362i	−1.19503	+	0.689952i	−0.959444	−	0.281901i	$-0.909035\pi$
$173$	−206.741	+	119.362i	−1.19503	+	0.689952i	−0.235589	+	0.971853i	$0.575702\pi$
$174$		0			0
$175$	96.6459	−	145.870i	0.552262	−	0.833545i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	156.153	+	90.1549i	0.872362	+	0.503659i	0.868133	−	0.496332i	$-0.165320\pi$
$179$	156.153	+	90.1549i	0.872362	+	0.503659i	0.00422981	+	0.999991i	$0.498654\pi$
$180$		0			0
$181$	161.018			0.889602			0.444801	−	0.895630i	$-0.353274\pi$
$181$	161.018			0.889602			0.444801	+	0.895630i	$0.353274\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.21900	+	0.703791i	0.00658920	+	0.00380427i
$186$		0			0
$187$	−115.635	−	200.285i	−0.618367	−	1.07104i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−98.1384	+	56.6602i	−0.513813	+	0.296650i	−0.734400	−	0.678717i	$-0.762536\pi$
$191$	−98.1384	+	56.6602i	−0.513813	+	0.296650i	0.220586	+	0.975367i	$0.429203\pi$
$192$		0			0
$193$	−176.314	+	305.384i	−0.913542	+	1.58230i	−0.104520	+	0.994523i	$0.533331\pi$
$193$	−176.314	+	305.384i	−0.913542	+	1.58230i	−0.809022	+	0.587779i	$0.800003\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	236.056i		−	1.19825i	−0.800654	−	0.599127i	$-0.795515\pi$
$197$		−	236.056i		−	1.19825i	0.800654	−	0.599127i	$-0.204485\pi$
$198$		0			0
$199$	−52.5555	+	91.0288i	−0.264098	+	0.457431i	−0.967327	−	0.253532i	$-0.918408\pi$
$199$	−52.5555	+	91.0288i	−0.264098	+	0.457431i	0.703229	+	0.710963i	$0.251741\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−4.37653	+	71.0204i	−0.0215593	+	0.349854i
$204$		0			0
$205$	−0.0762006	−	0.131983i	−0.000371710	−	0.000643821i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	247.340i		−	1.18344i
$210$		0			0
$211$	112.163			0.531577			0.265788	−	0.964031i	$-0.414368\pi$
$211$	112.163			0.531577			0.265788	+	0.964031i	$0.414368\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.318477	+	0.183873i	−0.00148129	+	0.000855222i
$216$		0			0
$217$	126.600	−	191.081i	0.583411	−	0.880558i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	219.903	+	126.961i	0.995038	+	0.574486i
$222$		0			0
$223$	139.127			0.623887			0.311944	−	0.950101i	$-0.399020\pi$
$223$	139.127			0.623887			0.311944	+	0.950101i	$0.399020\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−104.485	−	60.3245i	−0.460287	−	0.265747i	0.251878	−	0.967759i	$-0.418952\pi$
$227$	−104.485	−	60.3245i	−0.460287	−	0.265747i	−0.712165	+	0.702012i	$0.752285\pi$
$228$		0			0
$229$	−162.831	−	282.031i	−0.711051	−	1.23158i	−0.964463	−	0.264219i	$-0.914886\pi$
$229$	−162.831	−	282.031i	−0.711051	−	1.23158i	0.253411	−	0.967359i	$-0.418447\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−171.696	+	99.1288i	−0.736893	+	0.425446i	−0.820939	−	0.571016i	$-0.806549\pi$
$233$	−171.696	+	99.1288i	−0.736893	+	0.425446i	0.0840454	+	0.996462i	$0.473216\pi$
$234$		0			0
$235$	0.482647	−	0.835969i	0.00205382	−	0.00355732i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			268.284i			1.12253i	0.827636	+	0.561265i	$0.189685\pi$
$239$			268.284i			1.12253i	−0.827636	+	0.561265i	$0.810315\pi$
$240$		0			0
$241$	26.9282	−	46.6409i	0.111735	−	0.193531i	−0.804735	−	0.593634i	$-0.797693\pi$
$241$	26.9282	−	46.6409i	0.111735	−	0.193531i	0.916470	+	0.400104i	$0.131026\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.30924	−	0.977712i	0.00942546	−	0.00399066i
$246$		0			0
$247$	135.784	+	235.184i	0.549732	+	0.952163i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	302.100i		−	1.20358i	−0.798653	−	0.601792i	$-0.794454\pi$
$251$		−	302.100i		−	1.20358i	0.798653	−	0.601792i	$-0.205546\pi$
$252$		0			0
$253$	−231.269			−0.914107
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−387.349	+	223.636i	−1.50719	+	0.870178i	−0.507228	+	0.861812i	$0.669330\pi$
$257$	−387.349	+	223.636i	−1.50719	+	0.870178i	−0.999965	+	0.00836616i	$0.997337\pi$
$258$		0			0
$259$	−85.8265	−	172.339i	−0.331376	−	0.665403i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	86.7847	+	50.1052i	0.329980	+	0.190514i	0.655832	−	0.754907i	$-0.272318\pi$
$263$	86.7847	+	50.1052i	0.329980	+	0.190514i	−0.325852	+	0.945421i	$0.605651\pi$
$264$		0			0
$265$	5.12997			0.0193584
$266$		0			0
$267$		0			0
$268$		0			0
$269$	262.872	+	151.769i	0.977219	+	0.564198i	0.901429	−	0.432926i	$-0.142519\pi$
$269$	262.872	+	151.769i	0.977219	+	0.564198i	0.0757896	+	0.997124i	$0.475852\pi$
$270$		0			0
$271$	−66.0670	−	114.431i	−0.243789	−	0.422256i	0.718001	−	0.696042i	$-0.245057\pi$
$271$	−66.0670	−	114.431i	−0.243789	−	0.422256i	−0.961791	+	0.273786i	$0.911724\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	172.178	−	99.4073i	0.626104	−	0.361481i
$276$		0			0
$277$	−177.128	+	306.794i	−0.639450	+	1.10756i	0.346103	+	0.938196i	$0.387505\pi$
$277$	−177.128	+	306.794i	−0.639450	+	1.10756i	−0.985554	+	0.169364i	$0.945829\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	378.390i		−	1.34658i	−0.739376	−	0.673292i	$-0.764880\pi$
$281$		−	378.390i		−	1.34658i	0.739376	−	0.673292i	$-0.235120\pi$
$282$		0			0
$283$	−118.430	+	205.127i	−0.418481	+	0.724830i	−0.995787	−	0.0916981i	$-0.970771\pi$
$283$	−118.430	+	205.127i	−0.418481	+	0.724830i	0.577306	+	0.816528i	$0.304104\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.28213	+	20.8059i	−0.00446736	+	0.0724943i
$288$		0			0
$289$	278.264	+	481.967i	0.962850	+	1.66771i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	329.073i		−	1.12312i	−0.827437	−	0.561558i	$-0.810202\pi$
$293$		−	329.073i		−	1.12312i	0.827437	−	0.561558i	$-0.189798\pi$
$294$		0			0
$295$	−2.00835			−0.00680797
$296$		0			0
$297$		0			0
$298$		0			0
$299$	219.903	−	126.961i	0.735463	−	0.424620i
$300$		0			0
$301$	50.2047	+	3.09380i	0.166793	+	0.0102784i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.470734	−	0.271778i	−0.00154339	−	0.000891077i
$306$		0			0
$307$	559.322			1.82189			0.910947	−	0.412523i	$-0.135352\pi$
$307$	559.322			1.82189			0.910947	+	0.412523i	$0.135352\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	402.938	+	232.636i	1.29562	+	0.748027i	0.979644	−	0.200740i	$-0.0643348\pi$
$311$	402.938	+	232.636i	1.29562	+	0.748027i	0.315976	+	0.948767i	$0.397668\pi$
$312$		0			0
$313$	225.238	+	390.124i	0.719611	+	1.24640i	0.961154	+	0.276013i	$0.0890134\pi$
$313$	225.238	+	390.124i	0.719611	+	1.24640i	−0.241542	+	0.970390i	$0.577653\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	466.859	−	269.541i	1.47274	−	0.850287i	0.473211	−	0.880949i	$-0.343095\pi$
$317$	466.859	−	269.541i	1.47274	−	0.850287i	0.999530	+	0.0306621i	$0.00976158\pi$
$318$		0			0
$319$	−40.4233	+	70.0152i	−0.126719	+	0.219484i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			904.282i			2.79964i
$324$		0			0
$325$	−109.145	+	189.044i	−0.335829	+	0.581674i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−118.187	+	58.8583i	−0.359232	+	0.178900i
$330$		0			0
$331$	214.709	+	371.887i	0.648668	+	1.12353i	0.983441	+	0.181227i	$0.0580070\pi$
$331$	214.709	+	371.887i	0.648668	+	1.12353i	−0.334773	+	0.942299i	$0.608660\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			5.08244i			0.0151715i
$336$		0			0
$337$	−445.340			−1.32148			−0.660742	−	0.750613i	$-0.729758\pi$
$337$	−445.340			−1.32148			−0.660742	+	0.750613i	$0.729758\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	225.543	−	130.217i	0.661417	−	0.381869i
$342$		0			0
$343$	−337.170	−	62.9714i	−0.983003	−	0.183590i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	512.667	+	295.988i	1.47743	+	0.852992i	0.999675	−	0.0255029i	$-0.00811872\pi$
$347$	512.667	+	295.988i	1.47743	+	0.852992i	0.477751	+	0.878495i	$0.341452\pi$
$348$		0			0
$349$	276.865			0.793311			0.396655	−	0.917968i	$-0.370171\pi$
$349$	276.865			0.793311			0.396655	+	0.917968i	$0.370171\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−514.365	−	296.969i	−1.45712	−	0.841271i	−0.458255	−	0.888821i	$-0.651525\pi$
$353$	−514.365	−	296.969i	−1.45712	−	0.841271i	−0.998869	+	0.0475496i	$0.984859\pi$
$354$		0			0
$355$	−0.411862	−	0.713367i	−0.00116018	−	0.00200948i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	305.428	−	176.339i	0.850775	−	0.491195i	−0.0101374	−	0.999949i	$-0.503227\pi$
$359$	305.428	−	176.339i	0.850775	−	0.491195i	0.860912	+	0.508754i	$0.169894\pi$
$360$		0			0
$361$	−303.060	+	524.915i	−0.839501	+	1.45406i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			0.745302i			0.00204192i
$366$		0			0
$367$	27.6654	−	47.9179i	0.0753826	−	0.130567i	−0.825870	−	0.563861i	$-0.809316\pi$
$367$	27.6654	−	47.9179i	0.0753826	−	0.130567i	0.901253	+	0.433294i	$0.142649\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−584.938	−	387.548i	−1.57665	−	1.04460i
$372$		0			0
$373$	−195.809	−	339.151i	−0.524957	−	0.909252i	−0.999578	−	0.0290617i	$-0.990748\pi$
$373$	−195.809	−	339.151i	−0.524957	−	0.909252i	0.474621	−	0.880190i	$-0.342585\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	88.7658i		−	0.235453i
$378$		0			0
$379$	96.5047			0.254630			0.127315	−	0.991862i	$-0.459364\pi$
$379$	96.5047			0.254630			0.127315	+	0.991862i	$0.459364\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	281.443	−	162.491i	0.734838	−	0.424259i	−0.0853514	−	0.996351i	$-0.527201\pi$
$383$	281.443	−	162.491i	0.734838	−	0.424259i	0.820190	+	0.572092i	$0.193868\pi$
$384$		0			0
$385$	2.84385	+	0.175248i	0.00738662	+	0.000455190i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−275.937	−	159.312i	−0.709350	−	0.409543i	0.101470	−	0.994839i	$-0.467645\pi$
$389$	−275.937	−	159.312i	−0.709350	−	0.409543i	−0.810820	+	0.585295i	$0.800979\pi$
$390$		0			0
$391$	845.528			2.16247
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.89434	+	2.82575i	0.0123907	+	0.00715380i
$396$		0			0
$397$	−67.5631	−	117.023i	−0.170184	−	0.294768i	0.768300	−	0.640090i	$-0.221103\pi$
$397$	−67.5631	−	117.023i	−0.170184	−	0.294768i	−0.938484	+	0.345322i	$0.887770\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	70.0122	−	40.4216i	0.174594	−	0.100802i	−0.410156	−	0.912015i	$-0.634526\pi$
$401$	70.0122	−	40.4216i	0.174594	−	0.100802i	0.584750	+	0.811213i	$0.301192\pi$
$402$		0			0
$403$	−142.973	+	247.636i	−0.354771	+	0.614481i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	218.751i		−	0.537471i
$408$		0			0
$409$	−260.419	+	451.059i	−0.636722	+	1.10283i	0.349426	+	0.936964i	$0.386377\pi$
$409$	−260.419	+	451.059i	−0.636722	+	1.10283i	−0.986148	+	0.165870i	$0.946957\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	228.999	+	151.723i	0.554478	+	0.367367i
$414$		0			0
$415$	−2.72917	−	4.72706i	−0.00657631	−	0.0113905i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			452.306i			1.07949i	0.841829	+	0.539744i	$0.181479\pi$
$419$			452.306i			1.07949i	−0.841829	+	0.539744i	$0.818521\pi$
$420$		0			0
$421$	8.02945			0.0190723			0.00953616	−	0.999955i	$-0.496964\pi$
$421$	8.02945			0.0190723			0.00953616	+	0.999955i	$0.496964\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−629.490	+	363.436i	−1.48115	+	0.855144i
$426$		0			0
$427$	33.1431	+	66.5512i	0.0776184	+	0.155858i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	582.041	+	336.042i	1.35044	+	0.779679i	0.988311	−	0.152448i	$-0.0487157\pi$
$431$	582.041	+	336.042i	1.35044	+	0.779679i	0.362132	+	0.932127i	$0.382049\pi$
$432$		0			0
$433$	161.009			0.371846			0.185923	−	0.982564i	$-0.440472\pi$
$433$	161.009			0.371846			0.185923	+	0.982564i	$0.440472\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	783.132	+	452.141i	1.79206	+	1.03465i
$438$		0			0
$439$	−342.661	−	593.506i	−0.780548	−	1.35195i	−0.931623	−	0.363427i	$-0.881607\pi$
$439$	−342.661	−	593.506i	−0.780548	−	1.35195i	0.151075	−	0.988522i	$-0.451727\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−68.1622	+	39.3535i	−0.153865	+	0.0888340i	−0.574956	−	0.818185i	$-0.694981\pi$
$443$	−68.1622	+	39.3535i	−0.153865	+	0.0888340i	0.421091	+	0.907019i	$0.361647\pi$
$444$		0			0
$445$	−0.393843	+	0.682156i	−0.000885040	+	0.00153293i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			7.47467i			0.0166474i	0.999965	+	0.00832369i	$0.00264954\pi$
$449$			7.47467i			0.0166474i	−0.999965	+	0.00832369i	$0.997350\pi$
$450$		0			0
$451$	−11.8423	+	20.5114i	−0.0262578	+	0.0454798i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2.80029	+	1.39457i	−0.00615449	+	0.00306499i
$456$		0			0
$457$	101.732	+	176.206i	0.222609	+	0.385571i	0.955600	−	0.294669i	$-0.0952092\pi$
$457$	101.732	+	176.206i	0.222609	+	0.385571i	−0.732990	+	0.680239i	$0.761876\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	884.717i		−	1.91913i	−0.281494	−	0.959563i	$-0.590830\pi$
$461$		−	884.717i		−	1.91913i	0.281494	−	0.959563i	$-0.409170\pi$
$462$		0			0
$463$	388.379			0.838831			0.419415	−	0.907794i	$-0.362235\pi$
$463$	388.379			0.838831			0.419415	+	0.907794i	$0.362235\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−55.5028	+	32.0445i	−0.118850	+	0.0686178i	−0.558246	−	0.829675i	$-0.688526\pi$
$467$	−55.5028	+	32.0445i	−0.118850	+	0.0686178i	0.439397	+	0.898293i	$0.355192\pi$
$468$		0			0
$469$	383.958	−	579.519i	0.818674	−	1.23565i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	49.4942	+	28.5755i	0.104639	+	0.0604133i
$474$		0			0
$475$	−777.382			−1.63659
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−716.263	−	413.535i	−1.49533	−	0.863329i	−0.495345	−	0.868696i	$-0.664958\pi$
$479$	−716.263	−	413.535i	−1.49533	−	0.863329i	−0.999986	+	0.00536689i	$0.998292\pi$
$480$		0			0
$481$	120.089	+	208.000i	0.249665	+	0.432433i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5.15800	−	2.97797i	0.0106351	−	0.00614015i
$486$		0			0
$487$	−288.976	+	500.522i	−0.593380	+	1.02777i	0.400393	+	0.916344i	$0.368874\pi$
$487$	−288.976	+	500.522i	−0.593380	+	1.02777i	−0.993773	+	0.111421i	$0.964460\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	390.760i		−	0.795845i	−0.917419	−	0.397923i	$-0.869731\pi$
$491$		−	390.760i		−	0.795845i	0.917419	−	0.397923i	$-0.130269\pi$
$492$		0			0
$493$	147.789	−	255.978i	0.299775	−	0.519225i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.92990	+	112.455i	−0.0139435	+	0.226268i
$498$		0			0
$499$	273.835	+	474.296i	0.548767	+	0.950492i	0.998359	+	0.0572584i	$0.0182359\pi$
$499$	273.835	+	474.296i	0.548767	+	0.950492i	−0.449592	+	0.893234i	$0.648431\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	45.8671i		−	0.0911870i	−0.998960	−	0.0455935i	$-0.985482\pi$
$503$		−	45.8671i		−	0.0911870i	0.998960	−	0.0455935i	$-0.0145179\pi$
$504$		0			0
$505$	3.10673			0.00615193
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−370.961	+	214.174i	−0.728804	+	0.420775i	−0.817984	−	0.575241i	$-0.804908\pi$
$509$	−370.961	+	214.174i	−0.728804	+	0.420775i	0.0891808	+	0.996015i	$0.471575\pi$
$510$		0			0
$511$	56.3045	−	84.9820i	0.110185	−	0.166305i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−7.16944	−	4.13928i	−0.0139212	−	0.00803743i
$516$		0			0
$517$	−150.015			−0.290165
$518$		0			0
$519$		0			0
$520$		0			0
$521$	246.599	+	142.374i	0.473318	+	0.273270i	0.717628	−	0.696427i	$-0.245228\pi$
$521$	246.599	+	142.374i	0.473318	+	0.273270i	−0.244310	+	0.969697i	$0.578561\pi$
$522$		0			0
$523$	−491.642	−	851.548i	−0.940041	−	1.62820i	−0.765388	−	0.643569i	$-0.777453\pi$
$523$	−491.642	−	851.548i	−0.940041	−	1.62820i	−0.174653	−	0.984630i	$-0.555880\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−824.593	+	476.079i	−1.56469	+	0.903376i
$528$		0			0
$529$	158.264	−	274.121i	0.299175	−	0.518187i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	26.0045i		−	0.0487889i
$534$		0			0
$535$	−1.14004	+	1.97461i	−0.00213092	+	0.00369086i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−311.027	−	234.824i	−0.577044	−	0.435665i
$540$		0			0
$541$	−70.0577	−	121.343i	−0.129497	−	0.224295i	0.793985	−	0.607937i	$-0.208003\pi$
$541$	−70.0577	−	121.343i	−0.129497	−	0.224295i	−0.923482	+	0.383643i	$0.874669\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	5.14334i		−	0.00943732i
$546$		0			0
$547$	154.927			0.283231			0.141615	−	0.989922i	$-0.454770\pi$
$547$	154.927			0.283231			0.141615	+	0.989922i	$0.454770\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	273.766	−	158.059i	0.496852	−	0.286858i
$552$		0			0
$553$	−344.597	−	691.950i	−0.623141	−	1.25127i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	37.6509	+	21.7378i	0.0675960	+	0.0390265i	0.533417	−	0.845852i	$-0.320908\pi$
$557$	37.6509	+	21.7378i	0.0675960	+	0.0390265i	−0.465821	+	0.884879i	$0.654241\pi$
$558$		0			0
$559$	−62.7491			−0.112252
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−4.46739	−	2.57925i	−0.00793498	−	0.00458126i	0.496027	−	0.868307i	$-0.334792\pi$
$563$	−4.46739	−	2.57925i	−0.00793498	−	0.00458126i	−0.503962	+	0.863726i	$0.668125\pi$
$564$		0			0
$565$	4.65241	+	8.05821i	0.00823436	+	0.0142623i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−262.711	+	151.676i	−0.461706	+	0.266566i	−0.712761	−	0.701407i	$-0.752556\pi$
$569$	−262.711	+	151.676i	−0.461706	+	0.266566i	0.251055	+	0.967973i	$0.419222\pi$
$570$		0			0
$571$	546.563	−	946.674i	0.957202	−	1.65792i	0.227957	−	0.973671i	$-0.426796\pi$
$571$	546.563	−	946.674i	0.957202	−	1.65792i	0.729246	−	0.684252i	$-0.239871\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			726.873i			1.26413i
$576$		0			0
$577$	420.854	−	728.940i	0.729383	−	1.26333i	−0.227762	−	0.973717i	$-0.573141\pi$
$577$	420.854	−	728.940i	0.729383	−	1.26333i	0.957144	−	0.289611i	$-0.0935259\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−45.9203	+	745.174i	−0.0790367	+	1.28257i
$582$		0			0
$583$	−398.622	−	690.433i	−0.683742	−	1.18428i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	803.940i		−	1.36957i	−0.728743	−	0.684787i	$-0.759895\pi$
$587$		−	803.940i		−	1.36957i	0.728743	−	0.684787i	$-0.240105\pi$
$588$		0			0
$589$	−1018.32			−1.72890
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−748.506	+	432.150i	−1.26224	+	0.728752i	−0.973507	−	0.228658i	$-0.926566\pi$
$593$	−748.506	+	432.150i	−1.26224	+	0.728752i	−0.288729	+	0.957411i	$0.593233\pi$
$594$		0			0
$595$	−10.3972	−	0.640713i	−0.0174743	−	0.00107683i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−595.095	−	343.578i	−0.993480	−	0.573586i	−0.0871675	−	0.996194i	$-0.527782\pi$
$599$	−595.095	−	343.578i	−0.993480	−	0.573586i	−0.906313	+	0.422608i	$0.861115\pi$
$600$		0			0
$601$	733.136			1.21986			0.609930	−	0.792455i	$-0.291198\pi$
$601$	733.136			1.21986			0.609930	+	0.792455i	$0.291198\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.55923	−	1.47757i	−0.00423013	−	0.00244226i
$606$		0			0
$607$	216.418	+	374.848i	0.356538	+	0.617542i	0.987380	−	0.158370i	$-0.0506237\pi$
$607$	216.418	+	374.848i	0.356538	+	0.617542i	−0.630842	+	0.775911i	$0.717290\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	142.643	−	82.3550i	0.233458	−	0.134787i
$612$		0			0
$613$	−1.59035	+	2.75456i	−0.00259437	+	0.00449358i	−0.867320	−	0.497752i	$-0.834159\pi$
$613$	−1.59035	+	2.75456i	−0.00259437	+	0.00449358i	0.864725	+	0.502245i	$0.167492\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	692.934i		−	1.12307i	−0.827453	−	0.561535i	$-0.810211\pi$
$617$		−	692.934i		−	1.12307i	0.827453	−	0.561535i	$-0.189789\pi$
$618$		0			0
$619$	−147.819	+	256.030i	−0.238803	+	0.413619i	−0.960371	−	0.278725i	$-0.910088\pi$
$619$	−147.819	+	256.030i	−0.238803	+	0.413619i	0.721568	+	0.692344i	$0.243422\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	96.4415	−	48.0287i	0.154802	−	0.0770925i
$624$		0			0
$625$	−312.402	−	541.096i	−0.499843	−	0.865753i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			799.760i			1.27148i
$630$		0			0
$631$	−355.577			−0.563514			−0.281757	−	0.959486i	$-0.590917\pi$
$631$	−355.577			−0.563514			−0.281757	+	0.959486i	$0.590917\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3.88643	+	2.24383i	−0.00612037	+	0.00353360i
$636$		0			0
$637$	424.654	+	52.5369i	0.666647	+	0.0824755i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−432.469	−	249.686i	−0.674679	−	0.389526i	0.123168	−	0.992386i	$-0.460695\pi$
$641$	−432.469	−	249.686i	−0.674679	−	0.389526i	−0.797847	+	0.602860i	$0.794028\pi$
$642$		0			0
$643$	20.0253			0.0311436			0.0155718	−	0.999879i	$-0.495043\pi$
$643$	20.0253			0.0311436			0.0155718	+	0.999879i	$0.495043\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1045.02	−	603.341i	−1.61517	−	0.932521i	−0.988145	−	0.153526i	$-0.950937\pi$
$647$	−1045.02	−	603.341i	−1.61517	−	0.932521i	−0.627030	−	0.778995i	$-0.715730\pi$
$648$		0			0
$649$	156.058	+	270.300i	0.240459	+	0.416487i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	477.039	−	275.419i	0.730534	−	0.421774i	−0.0880832	−	0.996113i	$-0.528074\pi$
$653$	477.039	−	275.419i	0.730534	−	0.421774i	0.818618	+	0.574339i	$0.194741\pi$
$654$		0			0
$655$	−1.94385	+	3.36685i	−0.00296771	+	0.00514023i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			521.731i			0.791702i	0.918315	+	0.395851i	$0.129550\pi$
$659$			521.731i			0.791702i	−0.918315	+	0.395851i	$0.870450\pi$
$660$		0			0
$661$	506.319	−	876.971i	0.765990	−	1.32673i	−0.173732	−	0.984793i	$-0.555583\pi$
$661$	506.319	−	876.971i	0.765990	−	1.32673i	0.939722	−	0.341940i	$-0.111084\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−9.28732	−	6.15328i	−0.0139659	−	0.00925305i
$666$		0			0
$667$	−147.789	−	255.978i	−0.221573	−	0.383775i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			84.4736i			0.125892i
$672$		0			0
$673$	−229.880			−0.341576			−0.170788	−	0.985308i	$-0.554631\pi$
$673$	−229.880			−0.341576			−0.170788	+	0.985308i	$0.554631\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	258.781	−	149.407i	0.382247	−	0.220690i	−0.296549	−	0.955018i	$-0.595836\pi$
$677$	258.781	−	149.407i	0.382247	−	0.220690i	0.678795	+	0.734328i	$0.262502\pi$
$678$		0			0
$679$	−813.108	−	50.1066i	−1.19751	−	0.0737948i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−275.387	−	158.995i	−0.403202	−	0.232789i	0.284663	−	0.958628i	$-0.408118\pi$
$683$	−275.387	−	158.995i	−0.403202	−	0.232789i	−0.687865	+	0.725839i	$0.741452\pi$
$684$		0			0
$685$	−7.66311			−0.0111870
$686$		0			0
$687$		0			0
$688$		0			0
$689$	758.063	+	437.668i	1.10024	+	0.635222i
$690$		0			0
$691$	30.7568	+	53.2724i	0.0445106	+	0.0770946i	0.887422	−	0.460957i	$-0.152494\pi$
$691$	30.7568	+	53.2724i	0.0445106	+	0.0770946i	−0.842912	+	0.538052i	$0.819160\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	5.65379	−	3.26422i	0.00813495	−	0.00469672i
$696$		0			0
$697$	43.2957	−	74.9904i	0.0621172	−	0.107590i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	58.4919i		−	0.0834407i	−0.999129	−	0.0417204i	$-0.986716\pi$
$701$		−	58.4919i		−	0.0834407i	0.999129	−	0.0417204i	$-0.0132839\pi$
$702$		0			0
$703$	−427.667	+	740.742i	−0.608346	+	1.05369i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−354.240	−	234.701i	−0.501047	−	0.331967i
$708$		0			0
$709$	−99.4793	−	172.303i	−0.140309	−	0.243023i	0.787304	−	0.616565i	$-0.211476\pi$
$709$	−99.4793	−	172.303i	−0.140309	−	0.243023i	−0.927613	+	0.373542i	$0.878143\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			952.158i			1.33543i
$714$		0			0
$715$	−3.55442			−0.00497122
$716$		0			0
$717$		0			0
$718$		0			0
$719$	632.241	−	365.024i	0.879333	−	0.507683i	0.00889476	−	0.999960i	$-0.497169\pi$
$719$	632.241	−	365.024i	0.879333	−	0.507683i	0.870439	+	0.492277i	$0.163835\pi$
$720$		0			0
$721$	504.780	+	1013.60i	0.700111	+	1.40582i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	220.056	+	127.049i	0.303526	+	0.175241i
$726$		0			0
$727$	427.442			0.587953			0.293976	−	0.955813i	$-0.405021\pi$
$727$	427.442			0.587953			0.293976	+	0.955813i	$0.405021\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−180.952	−	104.473i	−0.247541	−	0.142918i
$732$		0			0
$733$	−51.3845	−	89.0005i	−0.0701016	−	0.121420i	0.828844	−	0.559480i	$-0.188999\pi$
$733$	−51.3845	−	89.0005i	−0.0701016	−	0.121420i	−0.898946	+	0.438060i	$0.855666\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	684.037	−	394.929i	0.928136	−	0.535860i
$738$		0			0
$739$	163.820	−	283.744i	0.221678	−	0.383957i	−0.733640	−	0.679539i	$-0.762180\pi$
$739$	163.820	−	283.744i	0.221678	−	0.383957i	0.955318	+	0.295581i	$0.0955134\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			267.882i			0.360541i	0.983617	+	0.180271i	$0.0576973\pi$
$743$			267.882i			0.360541i	−0.983617	+	0.180271i	$0.942303\pi$
$744$		0			0
$745$	4.79490	−	8.30500i	0.00643610	−	0.0111477i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	279.165	−	139.027i	0.372717	−	0.185616i
$750$		0			0
$751$	488.726	+	846.498i	0.650766	+	1.12716i	0.982937	+	0.183941i	$0.0588856\pi$
$751$	488.726	+	846.498i	0.650766	+	1.12716i	−0.332171	+	0.943219i	$0.607781\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			1.27301i			0.00168610i
$756$		0			0
$757$	1371.04			1.81115			0.905575	−	0.424187i	$-0.139440\pi$
$757$	1371.04			1.81115			0.905575	+	0.424187i	$0.139440\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−484.380	+	279.657i	−0.636504	+	0.367486i	−0.783267	−	0.621686i	$-0.786448\pi$
$761$	−484.380	+	279.657i	−0.636504	+	0.367486i	0.146762	+	0.989172i	$0.453115\pi$
$762$		0			0
$763$	−388.559	+	586.463i	−0.509251	+	0.768627i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−296.777	−	171.344i	−0.386932	−	0.223395i
$768$		0			0
$769$	−861.068			−1.11972			−0.559862	−	0.828586i	$-0.689146\pi$
$769$	−861.068			−1.11972			−0.559862	+	0.828586i	$0.689146\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−105.336	−	60.8156i	−0.136269	−	0.0786748i	0.430316	−	0.902678i	$-0.358402\pi$
$773$	−105.336	−	60.8156i	−0.136269	−	0.0786748i	−0.566585	+	0.824004i	$0.691736\pi$
$774$		0			0
$775$	−409.270	−	708.876i	−0.528090	−	0.914679i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	80.2014	−	46.3043i	0.102954	−	0.0594407i
$780$		0			0
$781$	−64.0071	+	110.864i	−0.0819554	+	0.141951i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	7.34445i		−	0.00935599i
$786$		0			0
$787$	−52.1249	+	90.2830i	−0.0662324	+	0.114718i	−0.897240	−	0.441543i	$-0.854431\pi$
$787$	−52.1249	+	90.2830i	−0.0662324	+	0.114718i	0.831008	+	0.556261i	$0.187765\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	78.2803	−	1270.30i	0.0989637	−	1.60594i
$792$		0			0
$793$	−46.3740	−	80.3222i	−0.0584792	−	0.101289i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			529.515i			0.664386i	0.943212	+	0.332193i	$0.107788\pi$
$797$			529.515i			0.664386i	−0.943212	+	0.332193i	$0.892212\pi$
$798$		0			0
$799$	548.461			0.686435
$800$		0			0
$801$		0			0
$802$		0			0
$803$	100.309	−	57.9133i	0.124918	−	0.0721212i
$804$		0			0
$805$	−5.75347	+	8.68388i	−0.00714717	+	0.0107874i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	173.384	+	100.104i	0.214319	+	0.123737i	0.603317	−	0.797501i	$-0.293845\pi$
$809$	173.384	+	100.104i	0.214319	+	0.123737i	−0.388998	+	0.921239i	$0.627179\pi$
$810$		0			0
$811$	−1042.03			−1.28487			−0.642435	−	0.766340i	$-0.722076\pi$
$811$	−1042.03			−1.28487			−0.642435	+	0.766340i	$0.722076\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−8.02189	−	4.63144i	−0.00984281	−	0.00568275i
$816$		0			0
$817$	−111.733	−	193.527i	−0.136760	−	0.236875i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−597.404	+	344.911i	−0.727654	+	0.420111i	−0.817563	−	0.575839i	$-0.804676\pi$
$821$	−597.404	+	344.911i	−0.727654	+	0.420111i	0.0899092	+	0.995950i	$0.471342\pi$
$822$		0			0
$823$	−142.393	+	246.632i	−0.173017	+	0.299674i	−0.939473	−	0.342622i	$-0.888685\pi$
$823$	−142.393	+	246.632i	−0.173017	+	0.299674i	0.766456	+	0.642296i	$0.222018\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	420.531i		−	0.508502i	−0.967138	−	0.254251i	$-0.918171\pi$
$827$		−	420.531i		−	0.508502i	0.967138	−	0.254251i	$-0.0818289\pi$
$828$		0			0
$829$	322.601	−	558.761i	0.389145	−	0.674019i	−0.603190	−	0.797598i	$-0.706104\pi$
$829$	322.601	−	558.761i	0.389145	−	0.674019i	0.992335	+	0.123579i	$0.0394372\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1137.12	+	858.523i	1.36509	+	1.03064i
$834$		0			0
$835$	0.401784	+	0.695911i	0.000481179	+	0.000833426i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			156.230i			0.186209i	0.995656	+	0.0931046i	$0.0296791\pi$
$839$			156.230i			0.186209i	−0.995656	+	0.0931046i	$0.970321\pi$
$840$		0			0
$841$	737.672			0.877137
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−4.11049	+	2.37319i	−0.00486448	+	0.00280851i
$846$		0			0
$847$	180.188	+	361.817i	0.212737	+	0.427175i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	692.613	+	399.880i	0.813881	+	0.469894i
$852$		0			0
$853$	−1185.28			−1.38955			−0.694773	−	0.719229i	$-0.744495\pi$
$853$	−1185.28			−1.38955			−0.694773	+	0.719229i	$0.744495\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	677.280	+	391.028i	0.790292	+	0.456275i	0.840065	−	0.542485i	$-0.182517\pi$
$857$	677.280	+	391.028i	0.790292	+	0.456275i	−0.0497736	+	0.998761i	$0.515850\pi$
$858$		0			0
$859$	590.775	+	1023.25i	0.687747	+	1.19121i	0.972565	+	0.232631i	$0.0747336\pi$
$859$	590.775	+	1023.25i	0.687747	+	1.19121i	−0.284818	+	0.958582i	$0.591933\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1055.01	+	609.108i	−1.22249	+	0.705803i	−0.965448	−	0.260597i	$-0.916080\pi$
$863$	−1055.01	+	609.108i	−1.22249	+	0.705803i	−0.257040	+	0.966401i	$0.582747\pi$
$864$		0			0
$865$	−6.10861	+	10.5804i	−0.00706198	+	0.0122317i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	878.294i		−	1.01070i
$870$		0			0
$871$	−433.613	+	751.040i	−0.497834	+	0.862273i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.10166	−	17.8772i	0.00125904	−	0.0204311i
$876$		0			0
$877$	342.141	+	592.605i	0.390126	+	0.675718i	0.992466	−	0.122522i	$-0.0390980\pi$
$877$	342.141	+	592.605i	0.390126	+	0.675718i	−0.602340	+	0.798240i	$0.705765\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	932.418i		−	1.05836i	−0.848509	−	0.529181i	$-0.822499\pi$
$881$		−	932.418i		−	1.05836i	0.848509	−	0.529181i	$-0.177501\pi$
$882$		0			0
$883$	−155.535			−0.176143			−0.0880717	−	0.996114i	$-0.528070\pi$
$883$	−155.535			−0.176143			−0.0880717	+	0.996114i	$0.528070\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1207.39	−	697.087i	1.36121	−	0.785893i	0.371422	−	0.928464i	$-0.378870\pi$
$887$	1207.39	−	697.087i	1.36121	−	0.785893i	0.989785	+	0.142571i	$0.0455369\pi$
$888$		0			0
$889$	612.658	+	37.7542i	0.689154	+	0.0424681i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	507.987	+	293.287i	0.568855	+	0.328429i
$894$		0			0
$895$	9.22777			0.0103104
$896$		0			0
$897$		0			0
$898$		0			0
$899$	288.260	+	166.427i	0.320645	+	0.185124i
$900$		0			0
$901$	1457.37	+	2524.25i	1.61751	+	2.80161i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	7.13645	−	4.12023i	0.00788558	−	0.00455274i
$906$		0			0
$907$	−16.3106	+	28.2507i	−0.0179830	+	0.0311474i	−0.874877	−	0.484345i	$-0.839058\pi$
$907$	−16.3106	+	28.2507i	−0.0179830	+	0.0311474i	0.856894	+	0.515493i	$0.172391\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			4.10899i			0.00451042i	0.999997	+	0.00225521i	$0.000717856\pi$
$911$			4.10899i			0.00451042i	−0.999997	+	0.00225521i	$0.999282\pi$
$912$		0			0
$913$	−424.137	+	734.627i	−0.464553	+	0.804630i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	475.997	−	237.051i	0.519081	−	0.258507i
$918$		0			0
$919$	120.625	+	208.928i	0.131256	+	0.227343i	0.924161	−	0.382003i	$-0.124766\pi$
$919$	120.625	+	208.928i	0.131256	+	0.227343i	−0.792905	+	0.609346i	$0.791432\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	140.554i		−	0.152279i
$924$		0			0
$925$	−687.528			−0.743274
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1485.20	−	857.482i	1.59871	−	0.923016i	0.606974	−	0.794722i	$-0.292383\pi$
$929$	1485.20	−	857.482i	1.59871	−	0.923016i	0.991736	−	0.128294i	$-0.0409502\pi$
$930$		0			0
$931$	594.119	+	1403.24i	0.638151	+	1.50724i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−10.2500	−	5.91787i	−0.0109626	−	0.00632927i
$936$		0			0
$937$	−671.057			−0.716176			−0.358088	−	0.933688i	$-0.616571\pi$
$937$	−671.057			−0.716176			−0.358088	+	0.933688i	$0.616571\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−805.187	−	464.875i	−0.855672	−	0.494022i	0.00688889	−	0.999976i	$-0.497807\pi$
$941$	−805.187	−	464.875i	−0.855672	−	0.494022i	−0.862560	+	0.505954i	$0.831141\pi$
$942$		0			0
$943$	−43.2957	−	74.9904i	−0.0459127	−	0.0795232i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−462.810	+	267.203i	−0.488711	+	0.282158i	−0.724040	−	0.689758i	$-0.757717\pi$
$947$	−462.810	+	267.203i	−0.488711	+	0.282158i	0.235328	+	0.971916i	$0.424383\pi$
$948$		0			0
$949$	−63.5861	+	110.134i	−0.0670033	+	0.116053i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			381.768i			0.400596i	0.979735	+	0.200298i	$0.0641911\pi$
$953$			381.768i			0.400596i	−0.979735	+	0.200298i	$0.935809\pi$
$954$		0			0
$955$	−2.89972	+	5.02246i	−0.00303635	+	0.00525912i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	873.776	+	578.917i	0.911133	+	0.603667i
$960$		0			0
$961$	−55.6183	−	96.3338i	−0.0578755	−	0.100243i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			18.0465i			0.0187011i
$966$		0			0
$967$	1035.54			1.07087			0.535437	−	0.844575i	$-0.320147\pi$
$967$	1035.54			1.07087			0.535437	+	0.844575i	$0.320147\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1611.66	−	930.493i	1.65979	−	0.958283i	0.686987	−	0.726670i	$-0.258933\pi$
$971$	1611.66	−	930.493i	1.65979	−	0.958283i	0.972808	−	0.231613i	$-0.0744003\pi$
$972$		0			0
$973$	−891.265	−	54.9229i	−0.915996	−	0.0564470i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1196.81	−	690.978i	−1.22498	−	0.707245i	−0.259008	−	0.965875i	$-0.583396\pi$
$977$	−1196.81	−	690.978i	−1.22498	−	0.707245i	−0.965976	+	0.258630i	$0.916729\pi$
$978$		0			0
$979$	122.413			0.125039
$980$		0			0
$981$		0			0
$982$		0			0
$983$	443.915	+	256.294i	0.451592	+	0.260727i	0.708502	−	0.705708i	$-0.249371\pi$
$983$	443.915	+	256.294i	0.451592	+	0.260727i	−0.256910	+	0.966435i	$0.582704\pi$
$984$		0			0
$985$	−6.04036	−	10.4622i	−0.00613234	−	0.0106215i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−180.952	+	104.473i	−0.182965	+	0.105635i
$990$		0			0
$991$	760.070	−	1316.48i	0.766973	−	1.32844i	−0.172225	−	0.985058i	$-0.555096\pi$
$991$	760.070	−	1316.48i	0.766973	−	1.32844i	0.939197	−	0.343378i	$-0.111571\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			5.37930i			0.00540633i
$996$		0			0
$997$	−666.258	+	1153.99i	−0.668263	+	1.15747i	0.310126	+	0.950695i	$0.399629\pi$
$997$	−666.258	+	1153.99i	−0.668263	+	1.15747i	−0.978390	+	0.206770i	$0.933705\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.3.cu.b.305.5	yes	16
3.2	odd	2	inner	504.3.cu.b.305.4	yes	16
4.3	odd	2		1008.3.dc.f.305.5		16
7.2	even	3	inner	504.3.cu.b.233.4	✓	16
7.3	odd	6		3528.3.d.f.1961.4		8
7.4	even	3		3528.3.d.k.1961.5		8
12.11	even	2		1008.3.dc.f.305.4		16
21.2	odd	6	inner	504.3.cu.b.233.5	yes	16
21.11	odd	6		3528.3.d.k.1961.4		8
21.17	even	6		3528.3.d.f.1961.5		8
28.23	odd	6		1008.3.dc.f.737.4		16
84.23	even	6		1008.3.dc.f.737.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.3.cu.b.233.4	✓	16	7.2	even	3	inner
504.3.cu.b.233.5	yes	16	21.2	odd	6	inner
504.3.cu.b.305.4	yes	16	3.2	odd	2	inner
504.3.cu.b.305.5	yes	16	1.1	even	1	trivial
1008.3.dc.f.305.4		16	12.11	even	2
1008.3.dc.f.305.5		16	4.3	odd	2
1008.3.dc.f.737.4		16	28.23	odd	6
1008.3.dc.f.737.5		16	84.23	even	6
3528.3.d.f.1961.4		8	7.3	odd	6
3528.3.d.f.1961.5		8	21.17	even	6
3528.3.d.k.1961.4		8	21.11	odd	6
3528.3.d.k.1961.5		8	7.4	even	3

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:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	504.cu (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.0443209 - 0.0255887i) q^{5} +(-6.98675 - 0.430549i) q^{7} +(-6.88786 - 3.97671i) q^{11} +8.73248 q^{13} +(25.1822 + 14.5390i) q^{17} +(15.5493 + 26.9321i) q^{19} +(25.1822 - 14.5390i) q^{23} +(-12.4987 + 21.6484i) q^{25} -10.1650i q^{29} +(-16.3725 + 28.3580i) q^{31} +(-0.320676 + 0.159699i) q^{35} +(13.7520 + 23.8192i) q^{37} -2.97791i q^{41} -7.18571 q^{43} +(16.3348 - 9.43088i) q^{47} +(48.6293 + 6.01627i) q^{49} +(86.8096 + 50.1196i) q^{53} -0.407035 q^{55} +(-33.9854 - 19.6215i) q^{59} +(-5.31052 - 9.19810i) q^{61} +(0.387031 - 0.223452i) q^{65} +(-49.6552 + 86.0054i) q^{67} -16.0955i q^{71} +(-7.28156 + 12.6120i) q^{73} +(46.4116 + 30.7498i) q^{77} +(55.2149 + 95.6350i) q^{79} -106.655i q^{83} +1.48813 q^{85} +(-13.3293 + 7.69565i) q^{89} +(-61.0116 - 3.75976i) q^{91} +(1.37831 + 0.795770i) q^{95} +116.379 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 40 q^{13} - 20 q^{19} - 4 q^{25} - 56 q^{31} + 76 q^{37} + 72 q^{43} - 48 q^{49} + 648 q^{55} - 72 q^{61} - 156 q^{67} + 124 q^{73} + 184 q^{79} - 864 q^{85} - 116 q^{91} + 1112 q^{97}+O(q^{100}) \) 16 * q + 40 * q^13 - 20 * q^19 - 4 * q^25 - 56 * q^31 + 76 * q^37 + 72 * q^43 - 48 * q^49 + 648 * q^55 - 72 * q^61 - 156 * q^67 + 124 * q^73 + 184 * q^79 - 864 * q^85 - 116 * q^91 + 1112 * q^97

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