LMFDB - Embedded newform 459.3.c.g.458.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [459,3,Mod(458,459)] 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("459.458"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(459, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 459 = 3^{3} \cdot 17 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	459.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [20,0,0,-36,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$12.5068441341$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 68 x^{18} + 1808 x^{16} + 24602 x^{14} + 187648 x^{12} + 817824 x^{10} + 1959913 x^{8} + \cdots + 1600 $ x^20 + 68x^18 + 1808x^16 + 24602x^14 + 187648x^12 + 817824x^10 + 1959913x^8 + 2273312x^6 + 888560x^4 + 73664*x^2 + 1600
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			458.6
Root			$-2.71920i$ of defining polynomial
Character	$\chi$	$=$	459.458
Dual form			459.3.c.g.458.16

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.71920i q^{2} +1.04434 q^{4} +8.53267 q^{5} -9.34999i q^{7} -8.67225i q^{8} -14.6694i q^{10} -13.4508 q^{11} -8.93379 q^{13} -16.0745 q^{14} -10.7320 q^{16} +(-14.0862 - 9.51737i) q^{17} +30.4441 q^{19} +8.91099 q^{20} +23.1247i q^{22} -2.38425 q^{23} +47.8064 q^{25} +15.3590i q^{26} -9.76455i q^{28} -11.9481 q^{29} +56.3406i q^{31} -16.2385i q^{32} +(-16.3623 + 24.2170i) q^{34} -79.7803i q^{35} -37.7305i q^{37} -52.3396i q^{38} -73.9974i q^{40} +2.40034 q^{41} +74.9007 q^{43} -14.0472 q^{44} +4.09902i q^{46} +86.4381i q^{47} -38.4222 q^{49} -82.1889i q^{50} -9.32990 q^{52} +31.3667i q^{53} -114.771 q^{55} -81.0854 q^{56} +20.5413i q^{58} -56.8958i q^{59} +21.8369i q^{61} +96.8610 q^{62} -70.8453 q^{64} -76.2290 q^{65} +73.3810 q^{67} +(-14.7107 - 9.93935i) q^{68} -137.159 q^{70} +40.3162 q^{71} -71.4708i q^{73} -64.8665 q^{74} +31.7939 q^{76} +125.765i q^{77} +94.7951i q^{79} -91.5726 q^{80} -4.12668i q^{82} -78.0386i q^{83} +(-120.192 - 81.2085i) q^{85} -128.770i q^{86} +116.649i q^{88} +3.65743i q^{89} +83.5308i q^{91} -2.48997 q^{92} +148.605 q^{94} +259.769 q^{95} -58.7499i q^{97} +66.0556i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 36 q^{4} + 100 q^{13} + 228 q^{16} + 172 q^{19} + 272 q^{25} - 80 q^{34} + 124 q^{43} + 4 q^{49} - 888 q^{52} + 268 q^{55} - 1408 q^{64} + 80 q^{67} - 300 q^{70} - 252 q^{76} - 484 q^{85} + 172 q^{94}+O(q^{100}) $ 20 * q - 36 * q^4 + 100 * q^13 + 228 * q^16 + 172 * q^19 + 272 * q^25 - 80 * q^34 + 124 * q^43 + 4 * q^49 - 888 * q^52 + 268 * q^55 - 1408 * q^64 + 80 * q^67 - 300 * q^70 - 252 * q^76 - 484 * q^85 + 172 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/459\mathbb{Z}\right)^\times$.

$n$	$137$	$190$
$\chi(n)$	$-1$	$-1$

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 $

$2$		−	1.71920i		−	0.859602i	−0.902924	−	0.429801i	$-0.858584\pi$
$2$		−	1.71920i		−	0.859602i	0.902924	−	0.429801i	$-0.141416\pi$
$3$		0			0
$3$		0			0
$4$	1.04434			0.261085
$5$	8.53267			1.70653			0.853267	−	0.521475i	$-0.174618\pi$
$5$	8.53267			1.70653			0.853267	+	0.521475i	$0.174618\pi$
$6$		0			0
$7$		−	9.34999i		−	1.33571i	−0.744290	−	0.667856i	$-0.767212\pi$
$7$		−	9.34999i		−	1.33571i	0.744290	−	0.667856i	$-0.232788\pi$
$8$		−	8.67225i		−	1.08403i
$9$		0			0
$10$		−	14.6694i		−	1.46694i
$11$	−13.4508			−1.22280			−0.611401	−	0.791321i	$-0.709394\pi$
$11$	−13.4508			−1.22280			−0.611401	+	0.791321i	$0.709394\pi$
$12$		0			0
$13$	−8.93379			−0.687214			−0.343607	−	0.939113i	$-0.611649\pi$
$13$	−8.93379			−0.687214			−0.343607	+	0.939113i	$0.611649\pi$
$14$	−16.0745			−1.14818
$15$		0			0
$16$	−10.7320			−0.670750
$17$	−14.0862	−	9.51737i	−0.828597	−	0.559845i
$17$	−14.0862	−	9.51737i	−0.828597	−	0.559845i
$18$		0			0
$19$	30.4441			1.60232			0.801161	−	0.598450i	$-0.204216\pi$
$19$	30.4441			1.60232			0.801161	+	0.598450i	$0.204216\pi$
$20$	8.91099			0.445550
$21$		0			0
$22$			23.1247i			1.05112i
$23$	−2.38425			−0.103663			−0.0518316	−	0.998656i	$-0.516506\pi$
$23$	−2.38425			−0.103663			−0.0518316	+	0.998656i	$0.516506\pi$
$24$		0			0
$25$	47.8064			1.91226
$26$			15.3590i			0.590731i
$27$		0			0
$28$		−	9.76455i		−	0.348734i
$29$	−11.9481			−0.412005			−0.206002	−	0.978552i	$-0.566045\pi$
$29$	−11.9481			−0.412005			−0.206002	+	0.978552i	$0.566045\pi$
$30$		0			0
$31$			56.3406i			1.81744i	0.417407	+	0.908720i	$0.362939\pi$
$31$			56.3406i			1.81744i	−0.417407	+	0.908720i	$0.637061\pi$
$32$		−	16.2385i		−	0.507453i
$33$		0			0
$34$	−16.3623	+	24.2170i	−0.481244	+	0.712264i
$35$		−	79.7803i		−	2.27944i
$36$		0			0
$37$		−	37.7305i		−	1.01974i	−0.860250	−	0.509872i	$-0.829693\pi$
$37$		−	37.7305i		−	1.01974i	0.860250	−	0.509872i	$-0.170307\pi$
$38$		−	52.3396i		−	1.37736i
$39$		0			0
$40$		−	73.9974i		−	1.84993i
$41$	2.40034			0.0585449			0.0292725	−	0.999571i	$-0.490681\pi$
$41$	2.40034			0.0585449			0.0292725	+	0.999571i	$0.490681\pi$
$42$		0			0
$43$	74.9007			1.74188			0.870938	−	0.491393i	$-0.163512\pi$
$43$	74.9007			1.74188			0.870938	+	0.491393i	$0.163512\pi$
$44$	−14.0472			−0.319255
$45$		0			0
$46$			4.09902i			0.0891090i
$47$			86.4381i			1.83911i	0.392963	+	0.919554i	$0.371450\pi$
$47$			86.4381i			1.83911i	−0.392963	+	0.919554i	$0.628550\pi$
$48$		0			0
$49$	−38.4222			−0.784127
$50$		−	82.1889i		−	1.64378i
$51$		0			0
$52$	−9.32990			−0.179421
$53$			31.3667i			0.591824i	0.955215	+	0.295912i	$0.0956235\pi$
$53$			31.3667i			0.591824i	−0.955215	+	0.295912i	$0.904377\pi$
$54$		0			0
$55$	−114.771			−2.08675
$56$	−81.0854			−1.44795
$57$		0			0
$58$			20.5413i			0.354160i
$59$		−	56.8958i		−	0.964336i	−0.876079	−	0.482168i	$-0.839849\pi$
$59$		−	56.8958i		−	0.964336i	0.876079	−	0.482168i	$-0.160151\pi$
$60$		0			0
$61$			21.8369i			0.357982i	0.983851	+	0.178991i	$0.0572833\pi$
$61$			21.8369i			0.357982i	−0.983851	+	0.178991i	$0.942717\pi$
$62$	96.8610			1.56227
$63$		0			0
$64$	−70.8453			−1.10696
$65$	−76.2290			−1.17275
$66$		0			0
$67$	73.3810			1.09524			0.547620	−	0.836727i	$-0.315534\pi$
$67$	73.3810			1.09524			0.547620	+	0.836727i	$0.315534\pi$
$68$	−14.7107	−	9.93935i	−0.216334	−	0.146167i
$69$		0			0
$70$	−137.159			−1.95941
$71$	40.3162			0.567833			0.283917	−	0.958849i	$-0.408366\pi$
$71$	40.3162			0.567833			0.283917	+	0.958849i	$0.408366\pi$
$72$		0			0
$73$		−	71.4708i		−	0.979051i	−0.871989	−	0.489526i	$-0.837170\pi$
$73$		−	71.4708i		−	0.979051i	0.871989	−	0.489526i	$-0.162830\pi$
$74$	−64.8665			−0.876574
$75$		0			0
$76$	31.7939			0.418341
$77$			125.765i			1.63331i
$78$		0			0
$79$			94.7951i			1.19994i	0.800023	+	0.599969i	$0.204820\pi$
$79$			94.7951i			1.19994i	−0.800023	+	0.599969i	$0.795180\pi$
$80$	−91.5726			−1.14466
$81$		0			0
$82$		−	4.12668i		−	0.0503253i
$83$		−	78.0386i		−	0.940224i	−0.882607	−	0.470112i	$-0.844214\pi$
$83$		−	78.0386i		−	0.940224i	0.882607	−	0.470112i	$-0.155786\pi$
$84$		0			0
$85$	−120.192	−	81.2085i	−1.41403	−	0.955394i
$86$		−	128.770i		−	1.49732i
$87$		0			0
$88$			116.649i			1.32555i
$89$			3.65743i			0.0410947i	0.999789	+	0.0205474i	$0.00654089\pi$
$89$			3.65743i			0.0410947i	−0.999789	+	0.0205474i	$0.993459\pi$
$90$		0			0
$91$			83.5308i			0.917921i
$92$	−2.48997			−0.0270649
$93$		0			0
$94$	148.605			1.58090
$95$	259.769			2.73441
$96$		0			0
$97$		−	58.7499i		−	0.605669i	−0.953043	−	0.302835i	$-0.902067\pi$
$97$		−	58.7499i		−	0.605669i	0.953043	−	0.302835i	$-0.0979330\pi$
$98$			66.0556i			0.674037i
$99$		0			0
$100$	49.9260			0.499260
$101$		−	40.3589i		−	0.399593i	−0.979837	−	0.199796i	$-0.935972\pi$
$101$		−	40.3589i		−	0.399593i	0.979837	−	0.199796i	$-0.0640281\pi$
$102$		0			0
$103$	64.2929			0.624203			0.312101	−	0.950049i	$-0.398967\pi$
$103$	64.2929			0.624203			0.312101	+	0.950049i	$0.398967\pi$
$104$			77.4760i			0.744962i
$105$		0			0
$106$	53.9257			0.508733
$107$	−17.3426			−0.162081			−0.0810404	−	0.996711i	$-0.525824\pi$
$107$	−17.3426			−0.162081			−0.0810404	+	0.996711i	$0.525824\pi$
$108$		0			0
$109$		−	25.0665i		−	0.229968i	−0.993367	−	0.114984i	$-0.963318\pi$
$109$		−	25.0665i		−	0.229968i	0.993367	−	0.114984i	$-0.0366817\pi$
$110$			197.315i			1.79378i
$111$		0			0
$112$			100.344i			0.895929i
$113$	−59.5562			−0.527046			−0.263523	−	0.964653i	$-0.584885\pi$
$113$	−59.5562			−0.527046			−0.263523	+	0.964653i	$0.584885\pi$
$114$		0			0
$115$	−20.3440			−0.176905
$116$	−12.4779			−0.107568
$117$		0			0
$118$	−97.8155			−0.828945
$119$	−88.9872	+	131.705i	−0.747792	+	1.10677i
$120$		0			0
$121$	59.9245			0.495243
$122$	37.5421			0.307722
$123$		0			0
$124$			58.8387i			0.474505i
$125$	194.599			1.55679
$126$		0			0
$127$	147.382			1.16049			0.580245	−	0.814442i	$-0.302957\pi$
$127$	147.382			1.16049			0.580245	+	0.814442i	$0.302957\pi$
$128$			56.8435i			0.444090i
$129$		0			0
$130$			131.053i			1.00810i
$131$	196.794			1.50224			0.751120	−	0.660165i	$-0.229514\pi$
$131$	196.794			1.50224			0.751120	+	0.660165i	$0.229514\pi$
$132$		0			0
$133$		−	284.652i		−	2.14024i
$134$		−	126.157i		−	0.941470i
$135$		0			0
$136$	−82.5369	+	122.159i	−0.606889	+	0.898225i
$137$			40.6712i			0.296870i	0.988922	+	0.148435i	$0.0474235\pi$
$137$			40.6712i			0.296870i	−0.988922	+	0.148435i	$0.952576\pi$
$138$		0			0
$139$		−	61.6175i		−	0.443292i	−0.975127	−	0.221646i	$-0.928857\pi$
$139$		−	61.6175i		−	0.443292i	0.975127	−	0.221646i	$-0.0711429\pi$
$140$		−	83.3176i		−	0.595126i
$141$		0			0
$142$		−	69.3117i		−	0.488111i
$143$	120.167			0.840327
$144$		0			0
$145$	−101.949			−0.703100
$146$	−122.873			−0.841594
$147$		0			0
$148$		−	39.4035i		−	0.266240i
$149$			194.025i			1.30218i	0.759000	+	0.651091i	$0.225688\pi$
$149$			194.025i			1.30218i	−0.759000	+	0.651091i	$0.774312\pi$
$150$		0			0
$151$	115.633			0.765781			0.382890	−	0.923794i	$-0.374929\pi$
$151$	115.633			0.765781			0.382890	+	0.923794i	$0.374929\pi$
$152$		−	264.019i		−	1.73697i
$153$		0			0
$154$	216.216			1.40400
$155$			480.736i			3.10152i
$156$		0			0
$157$	−144.433			−0.919956			−0.459978	−	0.887930i	$-0.652143\pi$
$157$	−144.433			−0.919956			−0.459978	+	0.887930i	$0.652143\pi$
$158$	162.972			1.03147
$159$		0			0
$160$		−	138.558i		−	0.865985i
$161$			22.2927i			0.138464i
$162$		0			0
$163$		−	2.30381i		−	0.0141338i	−0.999975	−	0.00706689i	$-0.997751\pi$
$163$		−	2.30381i		−	0.0141338i	0.999975	−	0.00706689i	$-0.00224948\pi$
$164$	2.50677			0.0152852
$165$		0			0
$166$	−134.164			−0.808218
$167$	−220.550			−1.32066			−0.660329	−	0.750977i	$-0.729583\pi$
$167$	−220.550			−1.32066			−0.660329	+	0.750977i	$0.729583\pi$
$168$		0			0
$169$	−89.1874			−0.527736
$170$	−139.614	+	206.635i	−0.821259	+	1.21550i
$171$		0			0
$172$	78.2217			0.454777
$173$	170.238			0.984033			0.492017	−	0.870586i	$-0.336260\pi$
$173$	170.238			0.984033			0.492017	+	0.870586i	$0.336260\pi$
$174$		0			0
$175$		−	446.989i		−	2.55422i
$176$	144.354			0.820194
$177$		0			0
$178$	6.28787			0.0353251
$179$			275.548i			1.53938i	0.638420	+	0.769688i	$0.279588\pi$
$179$			275.548i			1.53938i	−0.638420	+	0.769688i	$0.720412\pi$
$180$		0			0
$181$		−	27.8961i		−	0.154122i	−0.997026	−	0.0770610i	$-0.975446\pi$
$181$		−	27.8961i		−	0.154122i	0.997026	−	0.0770610i	$-0.0245536\pi$
$182$	143.606			0.789046
$183$		0			0
$184$			20.6768i			0.112374i
$185$		−	321.942i		−	1.74023i
$186$		0			0
$187$	189.470	+	128.016i	1.01321	+	0.684579i
$188$			90.2706i			0.480163i
$189$		0			0
$190$		−	446.596i		−	2.35051i
$191$			115.048i			0.602348i	0.953569	+	0.301174i	$0.0973785\pi$
$191$			115.048i			0.602348i	−0.953569	+	0.301174i	$0.902622\pi$
$192$		0			0
$193$			260.938i			1.35201i	0.736896	+	0.676006i	$0.236291\pi$
$193$			260.938i			1.35201i	−0.736896	+	0.676006i	$0.763709\pi$
$194$	−101.003			−0.520635
$195$		0			0
$196$	−40.1258			−0.204724
$197$	−260.581			−1.32274			−0.661372	−	0.750058i	$-0.730026\pi$
$197$	−260.581			−1.32274			−0.661372	+	0.750058i	$0.730026\pi$
$198$		0			0
$199$			154.073i			0.774234i	0.922031	+	0.387117i	$0.126529\pi$
$199$			154.073i			0.774234i	−0.922031	+	0.387117i	$0.873471\pi$
$200$		−	414.589i		−	2.07294i
$201$		0			0
$202$	−69.3851			−0.343491
$203$			111.715i			0.550320i
$204$		0			0
$205$	20.4813			0.0999089
$206$		−	110.533i		−	0.536566i
$207$		0			0
$208$	95.8774			0.460949
$209$	−409.498			−1.95932
$210$		0			0
$211$			86.2840i			0.408929i	0.978874	+	0.204464i	$0.0655453\pi$
$211$			86.2840i			0.408929i	−0.978874	+	0.204464i	$0.934455\pi$
$212$			32.7574i			0.154516i
$213$		0			0
$214$			29.8155i			0.139325i
$215$	639.103			2.97257
$216$		0			0
$217$	526.784			2.42758
$218$	−43.0944			−0.197681
$219$		0			0
$220$	−119.860			−0.544819
$221$	125.843	+	85.0261i	0.569424	+	0.384734i
$222$		0			0
$223$	−111.627			−0.500569			−0.250285	−	0.968172i	$-0.580524\pi$
$223$	−111.627			−0.500569			−0.250285	+	0.968172i	$0.580524\pi$
$224$	−151.830			−0.677811
$225$		0			0
$226$			102.389i			0.453050i
$227$	42.5230			0.187326			0.0936631	−	0.995604i	$-0.470142\pi$
$227$	42.5230			0.187326			0.0936631	+	0.995604i	$0.470142\pi$
$228$		0			0
$229$	−183.725			−0.802293			−0.401147	−	0.916014i	$-0.631388\pi$
$229$	−183.725			−0.802293			−0.401147	+	0.916014i	$0.631388\pi$
$230$			34.9755i			0.152068i
$231$		0			0
$232$			103.617i			0.446626i
$233$	232.499			0.997849			0.498924	−	0.866646i	$-0.333729\pi$
$233$	232.499			0.997849			0.498924	+	0.866646i	$0.333729\pi$
$234$		0			0
$235$			737.547i			3.13850i
$236$		−	59.4185i		−	0.251773i
$237$		0			0
$238$	226.428	+	152.987i	0.951379	+	0.642803i
$239$			61.8517i			0.258794i	0.991593	+	0.129397i	$0.0413041\pi$
$239$			61.8517i			0.258794i	−0.991593	+	0.129397i	$0.958696\pi$
$240$		0			0
$241$			177.471i			0.736394i	0.929748	+	0.368197i	$0.120025\pi$
$241$			177.471i			0.736394i	−0.929748	+	0.368197i	$0.879975\pi$
$242$		−	103.022i		−	0.425712i
$243$		0			0
$244$			22.8051i			0.0934636i
$245$	−327.844			−1.33814
$246$		0			0
$247$	−271.981			−1.10114
$248$	488.600			1.97016
$249$		0			0
$250$		−	334.556i		−	1.33822i
$251$		−	16.8764i		−	0.0672367i	−0.999435	−	0.0336184i	$-0.989297\pi$
$251$		−	16.8764i		−	0.0672367i	0.999435	−	0.0336184i	$-0.0107031\pi$
$252$		0			0
$253$	32.0701			0.126759
$254$		−	253.380i		−	0.997559i
$255$		0			0
$256$	−185.655			−0.725217
$257$		−	321.160i		−	1.24965i	−0.780764	−	0.624825i	$-0.785170\pi$
$257$		−	321.160i		−	1.24965i	0.780764	−	0.624825i	$-0.214830\pi$
$258$		0			0
$259$	−352.780			−1.36209
$260$	−79.6089			−0.306188
$261$		0			0
$262$		−	338.328i		−	1.29133i
$263$		−	339.214i		−	1.28979i	−0.764273	−	0.644893i	$-0.776902\pi$
$263$		−	339.214i		−	1.28979i	0.764273	−	0.644893i	$-0.223098\pi$
$264$		0			0
$265$			267.641i			1.00997i
$266$	−489.375			−1.83975
$267$		0			0
$268$	76.6346			0.285950
$269$	99.2887			0.369103			0.184551	−	0.982823i	$-0.440917\pi$
$269$	99.2887			0.369103			0.184551	+	0.982823i	$0.440917\pi$
$270$		0			0
$271$	−87.2805			−0.322068			−0.161034	−	0.986949i	$-0.551483\pi$
$271$	−87.2805			−0.322068			−0.161034	+	0.986949i	$0.551483\pi$
$272$	151.173	+	102.140i	0.555782	+	0.375516i
$273$		0			0
$274$	69.9220			0.255190
$275$	−643.035			−2.33831
$276$		0			0
$277$			410.030i			1.48025i	0.672468	+	0.740126i	$0.265234\pi$
$277$			410.030i			1.48025i	−0.672468	+	0.740126i	$0.734766\pi$
$278$	−105.933			−0.381054
$279$		0			0
$280$	−691.874			−2.47098
$281$		−	185.604i		−	0.660512i	−0.943891	−	0.330256i	$-0.892865\pi$
$281$		−	185.604i		−	0.660512i	0.943891	−	0.330256i	$-0.107135\pi$
$282$		0			0
$283$		−	17.2021i		−	0.0607850i	−0.999538	−	0.0303925i	$-0.990324\pi$
$283$		−	17.2021i		−	0.0607850i	0.999538	−	0.0303925i	$-0.00967572\pi$
$284$	42.1037			0.148253
$285$		0			0
$286$		−	206.591i		−	0.722347i
$287$		−	22.4432i		−	0.0781992i
$288$		0			0
$289$	107.840	+	268.126i	0.373147	+	0.927772i
$290$			175.272i			0.604386i
$291$		0			0
$292$		−	74.6397i		−	0.255615i
$293$			266.078i			0.908117i	0.890972	+	0.454058i	$0.150024\pi$
$293$			266.078i			0.908117i	−0.890972	+	0.454058i	$0.849976\pi$
$294$		0			0
$295$		−	485.473i		−	1.64567i
$296$	−327.209			−1.10543
$297$		0			0
$298$	333.569			1.11936
$299$	21.3004			0.0712388
$300$		0			0
$301$		−	700.320i		−	2.32665i
$302$		−	198.797i		−	0.658267i
$303$		0			0
$304$	−326.726			−1.07476
$305$			186.327i			0.610908i
$306$		0			0
$307$	−456.017			−1.48540			−0.742698	−	0.669626i	$-0.766454\pi$
$307$	−456.017			−1.48540			−0.742698	+	0.669626i	$0.766454\pi$
$308$			131.341i			0.426432i
$309$		0			0
$310$	826.483			2.66607
$311$	−203.172			−0.653285			−0.326642	−	0.945148i	$-0.605917\pi$
$311$	−203.172			−0.653285			−0.326642	+	0.945148i	$0.605917\pi$
$312$		0			0
$313$		−	283.907i		−	0.907052i	−0.891243	−	0.453526i	$-0.850166\pi$
$313$		−	283.907i		−	0.907052i	0.891243	−	0.453526i	$-0.149834\pi$
$314$			248.310i			0.790796i
$315$		0			0
$316$			98.9981i			0.313285i
$317$	147.243			0.464487			0.232244	−	0.972658i	$-0.425393\pi$
$317$	147.243			0.464487			0.232244	+	0.972658i	$0.425393\pi$
$318$		0			0
$319$	160.712			0.503800
$320$	−604.499			−1.88906
$321$		0			0
$322$	38.3257			0.119024
$323$	−428.840	−	289.748i	−1.32768	−	0.897051i
$324$		0			0
$325$	−427.092			−1.31413
$326$	−3.96071			−0.0121494
$327$		0			0
$328$		−	20.8164i		−	0.0634645i
$329$	808.195			2.45652
$330$		0			0
$331$	−129.981			−0.392693			−0.196347	−	0.980535i	$-0.562908\pi$
$331$	−129.981			−0.392693			−0.196347	+	0.980535i	$0.562908\pi$
$332$		−	81.4987i		−	0.245478i
$333$		0			0
$334$			379.170i			1.13524i
$335$	626.136			1.86906
$336$		0			0
$337$			4.12634i			0.0122443i	0.999981	+	0.00612217i	$0.00194876\pi$
$337$			4.12634i			0.0122443i	−0.999981	+	0.00612217i	$0.998051\pi$
$338$			153.331i			0.453643i
$339$		0			0
$340$	−125.522	−	84.8092i	−0.369181	−	0.249439i
$341$		−	757.827i		−	2.22237i
$342$		0			0
$343$		−	98.9020i		−	0.288344i
$344$		−	649.557i		−	1.88825i
$345$		0			0
$346$		−	292.673i		−	0.845877i
$347$	−278.661			−0.803059			−0.401529	−	0.915846i	$-0.631521\pi$
$347$	−278.661			−0.803059			−0.401529	+	0.915846i	$0.631521\pi$
$348$		0			0
$349$	−556.044			−1.59325			−0.796625	−	0.604474i	$-0.793383\pi$
$349$	−556.044			−1.59325			−0.796625	+	0.604474i	$0.793383\pi$
$350$	−768.465			−2.19562
$351$		0			0
$352$			218.421i			0.620514i
$353$			147.427i			0.417639i	0.977954	+	0.208820i	$0.0669622\pi$
$353$			147.427i			0.417639i	−0.977954	+	0.208820i	$0.933038\pi$
$354$		0			0
$355$	344.004			0.969026
$356$			3.81959i			0.0107292i
$357$		0			0
$358$	473.724			1.32325
$359$		−	705.032i		−	1.96388i	−0.189198	−	0.981939i	$-0.560589\pi$
$359$		−	705.032i		−	1.96388i	0.189198	−	0.981939i	$-0.439411\pi$
$360$		0			0
$361$	565.843			1.56743
$362$	−47.9591			−0.132484
$363$		0			0
$364$			87.2344i			0.239655i
$365$		−	609.836i		−	1.67078i
$366$		0			0
$367$			425.527i			1.15947i	0.814804	+	0.579737i	$0.196845\pi$
$367$			425.527i			1.15947i	−0.814804	+	0.579737i	$0.803155\pi$
$368$	25.5878			0.0695321
$369$		0			0
$370$	−553.484			−1.49590
$371$	293.278			0.790506
$372$		0			0
$373$	215.808			0.578573			0.289287	−	0.957243i	$-0.406582\pi$
$373$	215.808			0.578573			0.289287	+	0.957243i	$0.406582\pi$
$374$	220.086	−	325.738i	0.588466	−	0.870957i
$375$		0			0
$376$	749.612			1.99365
$377$	106.742			0.283136
$378$		0			0
$379$		−	620.035i		−	1.63598i	−0.575235	−	0.817988i	$-0.695089\pi$
$379$		−	620.035i		−	1.63598i	0.575235	−	0.817988i	$-0.304911\pi$
$380$	271.287			0.713913
$381$		0			0
$382$	197.792			0.517779
$383$			275.359i			0.718953i	0.933154	+	0.359477i	$0.117045\pi$
$383$			275.359i			0.718953i	−0.933154	+	0.359477i	$0.882955\pi$
$384$		0			0
$385$			1073.11i			2.78730i
$386$	448.606			1.16219
$387$		0			0
$388$		−	61.3548i		−	0.158131i
$389$			40.5825i			0.104325i	0.998639	+	0.0521626i	$0.0166114\pi$
$389$			40.5825i			0.104325i	−0.998639	+	0.0521626i	$0.983389\pi$
$390$		0			0
$391$	33.5850	+	22.6918i	0.0858950	+	0.0580353i
$392$			333.207i			0.850018i
$393$		0			0
$394$			447.991i			1.13703i
$395$			808.855i			2.04773i
$396$		0			0
$397$		−	46.9811i		−	0.118340i	−0.998248	−	0.0591701i	$-0.981155\pi$
$397$		−	46.9811i		−	0.118340i	0.998248	−	0.0591701i	$-0.0188454\pi$
$398$	264.882			0.665533
$399$		0			0
$400$	−513.058			−1.28265
$401$	−52.1419			−0.130030			−0.0650148	−	0.997884i	$-0.520709\pi$
$401$	−52.1419			−0.130030			−0.0650148	+	0.997884i	$0.520709\pi$
$402$		0			0
$403$		−	503.335i		−	1.24897i
$404$		−	42.1483i		−	0.104328i
$405$		0			0
$406$	192.061			0.473056
$407$			507.507i			1.24695i
$408$		0			0
$409$	−440.299			−1.07653			−0.538263	−	0.842777i	$-0.680919\pi$
$409$	−440.299			−1.07653			−0.538263	+	0.842777i	$0.680919\pi$
$410$		−	35.2116i		−	0.0858819i
$411$		0			0
$412$	67.1435			0.162970
$413$	−531.975			−1.28808
$414$		0			0
$415$		−	665.877i		−	1.60452i
$416$			145.071i			0.348729i
$417$		0			0
$418$			704.010i			1.68424i
$419$	−547.542			−1.30678			−0.653392	−	0.757020i	$-0.726654\pi$
$419$	−547.542			−1.30678			−0.653392	+	0.757020i	$0.726654\pi$
$420$		0			0
$421$	481.528			1.14377			0.571886	−	0.820333i	$-0.306212\pi$
$421$	481.528			1.14377			0.571886	+	0.820333i	$0.306212\pi$
$422$	148.340			0.351516
$423$		0			0
$424$	272.019			0.641555
$425$	−673.408	−	454.991i	−1.58449	−	1.07057i
$426$		0			0
$427$	204.175			0.478161
$428$	−18.1116			−0.0423168
$429$		0			0
$430$		−	1098.75i		−	2.55523i
$431$	−202.265			−0.469292			−0.234646	−	0.972081i	$-0.575393\pi$
$431$	−202.265			−0.469292			−0.234646	+	0.972081i	$0.575393\pi$
$432$		0			0
$433$	564.913			1.30465			0.652324	−	0.757940i	$-0.273794\pi$
$433$	564.913			1.30465			0.652324	+	0.757940i	$0.273794\pi$
$434$		−	905.649i		−	2.08675i
$435$		0			0
$436$		−	26.1779i		−	0.0600411i
$437$	−72.5864			−0.166102
$438$		0			0
$439$			345.773i			0.787637i	0.919188	+	0.393818i	$0.128846\pi$
$439$			345.773i			0.787637i	−0.919188	+	0.393818i	$0.871154\pi$
$440$			995.325i			2.26210i
$441$		0			0
$442$	146.177	−	216.349i	0.330718	−	0.489478i
$443$			398.946i			0.900556i	0.892888	+	0.450278i	$0.148675\pi$
$443$			398.946i			0.900556i	−0.892888	+	0.450278i	$0.851325\pi$
$444$		0			0
$445$			31.2076i			0.0701295i
$446$			191.909i			0.430290i
$447$		0			0
$448$			662.402i			1.47858i
$449$	130.517			0.290684			0.145342	−	0.989381i	$-0.453572\pi$
$449$	130.517			0.290684			0.145342	+	0.989381i	$0.453572\pi$
$450$		0			0
$451$	−32.2866			−0.0715888
$452$	−62.1969			−0.137604
$453$		0			0
$454$		−	73.1058i		−	0.161026i
$455$			712.740i			1.56646i
$456$		0			0
$457$	−474.760			−1.03886			−0.519431	−	0.854512i	$-0.673856\pi$
$457$	−474.760			−1.03886			−0.519431	+	0.854512i	$0.673856\pi$
$458$			315.861i			0.689653i
$459$		0			0
$460$	−21.2461			−0.0461871
$461$		−	805.445i		−	1.74717i	−0.486673	−	0.873584i	$-0.661790\pi$
$461$		−	805.445i		−	1.74717i	0.486673	−	0.873584i	$-0.338210\pi$
$462$		0			0
$463$	−449.688			−0.971249			−0.485624	−	0.874168i	$-0.661408\pi$
$463$	−449.688			−0.971249			−0.485624	+	0.874168i	$0.661408\pi$
$464$	128.227			0.276352
$465$		0			0
$466$		−	399.713i		−	0.857752i
$467$			46.7144i			0.100031i	0.998748	+	0.0500154i	$0.0159270\pi$
$467$			46.7144i			0.100031i	−0.998748	+	0.0500154i	$0.984073\pi$
$468$		0			0
$469$		−	686.112i		−	1.46292i
$470$	1267.99			2.69786
$471$		0			0
$472$	−493.415			−1.04537
$473$	−1007.48			−2.12997
$474$		0			0
$475$	1455.42			3.06405
$476$	−92.9328	+	137.545i	−0.195237	+	0.288960i
$477$		0			0
$478$	106.336			0.222460
$479$	−36.8170			−0.0768621			−0.0384311	−	0.999261i	$-0.512236\pi$
$479$	−36.8170			−0.0768621			−0.0384311	+	0.999261i	$0.512236\pi$
$480$		0			0
$481$			337.077i			0.700783i
$482$	305.109			0.633006
$483$		0			0
$484$	62.5814			0.129300
$485$		−	501.294i		−	1.03359i
$486$		0			0
$487$			227.007i			0.466133i	0.972461	+	0.233067i	$0.0748760\pi$
$487$			227.007i			0.466133i	−0.972461	+	0.233067i	$0.925124\pi$
$488$	189.375			0.388064
$489$		0			0
$490$			563.631i			1.15027i
$491$			848.072i			1.72723i	0.504149	+	0.863617i	$0.331806\pi$
$491$			848.072i			1.72723i	−0.504149	+	0.863617i	$0.668194\pi$
$492$		0			0
$493$	168.303	+	113.715i	0.341386	+	0.230659i
$494$			467.591i			0.946540i
$495$		0			0
$496$		−	604.648i		−	1.21905i
$497$		−	376.956i		−	0.758462i
$498$		0			0
$499$			63.7180i			0.127691i	0.997960	+	0.0638457i	$0.0203366\pi$
$499$			63.7180i			0.127691i	−0.997960	+	0.0638457i	$0.979663\pi$
$500$	203.228			0.406455
$501$		0			0
$502$	−29.0140			−0.0577968
$503$	785.062			1.56076			0.780380	−	0.625306i	$-0.215026\pi$
$503$	785.062			1.56076			0.780380	+	0.625306i	$0.215026\pi$
$504$		0			0
$505$		−	344.369i		−	0.681918i
$506$		−	55.1351i		−	0.108963i
$507$		0			0
$508$	153.917			0.302986
$509$			635.582i			1.24869i	0.781149	+	0.624344i	$0.214634\pi$
$509$			635.582i			1.24869i	−0.781149	+	0.624344i	$0.785366\pi$
$510$		0			0
$511$	−668.251			−1.30773
$512$			546.554i			1.06749i
$513$		0			0
$514$	−552.140			−1.07420
$515$	548.590			1.06522
$516$		0			0
$517$		−	1162.66i		−	2.24886i
$518$			606.501i			1.17085i
$519$		0			0
$520$			661.077i			1.27130i
$521$	235.641			0.452286			0.226143	−	0.974094i	$-0.427388\pi$
$521$	235.641			0.452286			0.226143	+	0.974094i	$0.427388\pi$
$522$		0			0
$523$	−825.930			−1.57922			−0.789608	−	0.613612i	$-0.789716\pi$
$523$	−825.930			−1.57922			−0.789608	+	0.613612i	$0.789716\pi$
$524$	205.519			0.392212
$525$		0			0
$526$	−583.178			−1.10870
$527$	536.214	−	793.623i	1.01748	−	1.50593i
$528$		0			0
$529$	−523.315			−0.989254
$530$	460.130			0.868169
$531$		0			0
$532$		−	297.273i		−	0.558784i
$533$	−21.4442			−0.0402329
$534$		0			0
$535$	−147.979			−0.276596
$536$		−	636.378i		−	1.18727i
$537$		0			0
$538$		−	170.697i		−	0.317282i
$539$	516.810			0.958832
$540$		0			0
$541$			0.859781i			0.00158924i	1.00000		0.000794622i	$0.000252936\pi$
$541$			0.859781i			0.00158924i	−1.00000		0.000794622i	$0.999747\pi$
$542$			150.053i			0.276851i
$543$		0			0
$544$	−154.548	+	228.738i	−0.284095	+	0.420474i
$545$		−	213.884i		−	0.392448i
$546$		0			0
$547$		−	613.906i		−	1.12231i	−0.827709	−	0.561157i	$-0.810356\pi$
$547$		−	613.906i		−	1.12231i	0.827709	−	0.561157i	$-0.189644\pi$
$548$			42.4745i			0.0775081i
$549$		0			0
$550$			1105.51i			2.01001i
$551$	−363.750			−0.660164
$552$		0			0
$553$	886.332			1.60277
$554$	704.925			1.27243
$555$		0			0
$556$		−	64.3496i		−	0.115737i
$557$		−	196.628i		−	0.353012i	−0.984300	−	0.176506i	$-0.943521\pi$
$557$		−	196.628i		−	0.353012i	0.984300	−	0.176506i	$-0.0564794\pi$
$558$		0			0
$559$	−669.147			−1.19704
$560$			856.203i			1.52893i
$561$		0			0
$562$	−319.091			−0.567777
$563$			730.838i			1.29811i	0.760740	+	0.649057i	$0.224836\pi$
$563$			730.838i			1.29811i	−0.760740	+	0.649057i	$0.775164\pi$
$564$		0			0
$565$	−508.173			−0.899422
$566$	−29.5740			−0.0522509
$567$		0			0
$568$		−	349.632i		−	0.615549i
$569$		−	26.5371i		−	0.0466381i	−0.999728	−	0.0233190i	$-0.992577\pi$
$569$		−	26.5371i		−	0.0466381i	0.999728	−	0.0233190i	$-0.00742335\pi$
$570$		0			0
$571$		−	69.3373i		−	0.121431i	−0.998155	−	0.0607157i	$-0.980662\pi$
$571$		−	69.3373i		−	0.121431i	0.998155	−	0.0607157i	$-0.0193383\pi$
$572$	125.495			0.219396
$573$		0			0
$574$	−38.5844			−0.0672202
$575$	−113.983			−0.198230
$576$		0			0
$577$	−76.6280			−0.132804			−0.0664021	−	0.997793i	$-0.521152\pi$
$577$	−76.6280			−0.132804			−0.0664021	+	0.997793i	$0.521152\pi$
$578$	460.964	−	185.398i	0.797515	−	0.320758i
$579$		0			0
$580$	−106.470			−0.183568
$581$	−729.659			−1.25587
$582$		0			0
$583$		−	421.907i		−	0.723683i
$584$	−619.812			−1.06132
$585$		0			0
$586$	457.443			0.780619
$587$			170.819i			0.291003i	0.989358	+	0.145501i	$0.0464795\pi$
$587$			170.819i			0.291003i	−0.989358	+	0.145501i	$0.953520\pi$
$588$		0			0
$589$			1715.24i			2.91212i
$590$	−834.627			−1.41462
$591$		0			0
$592$			404.924i			0.683994i
$593$			1102.03i			1.85839i	0.369588	+	0.929196i	$0.379499\pi$
$593$			1102.03i			1.85839i	−0.369588	+	0.929196i	$0.620501\pi$
$594$		0			0
$595$	−759.298	+	1123.80i	−1.27613	+	1.88874i
$596$			202.628i			0.339980i
$597$		0			0
$598$		−	36.6197i		−	0.0612370i
$599$		−	226.804i		−	0.378637i	−0.981916	−	0.189319i	$-0.939372\pi$
$599$		−	226.804i		−	0.378637i	0.981916	−	0.189319i	$-0.0606279\pi$
$600$		0			0
$601$			200.595i			0.333768i	0.985976	+	0.166884i	$0.0533706\pi$
$601$			200.595i			0.333768i	−0.985976	+	0.166884i	$0.946629\pi$
$602$	−1203.99			−1.99999
$603$		0			0
$604$	120.760			0.199934
$605$	511.315			0.845149
$606$		0			0
$607$			533.493i			0.878902i	0.898267	+	0.439451i	$0.144827\pi$
$607$			533.493i			0.878902i	−0.898267	+	0.439451i	$0.855173\pi$
$608$		−	494.366i		−	0.813102i
$609$		0			0
$610$	320.334			0.525138
$611$		−	772.220i		−	1.26386i
$612$		0			0
$613$	599.964			0.978735			0.489367	−	0.872078i	$-0.337228\pi$
$613$	599.964			0.978735			0.489367	+	0.872078i	$0.337228\pi$
$614$			783.985i			1.27685i
$615$		0			0
$616$	1090.66			1.77056
$617$	−789.387			−1.27940			−0.639698	−	0.768627i	$-0.720940\pi$
$617$	−789.387			−1.27940			−0.639698	+	0.768627i	$0.720940\pi$
$618$		0			0
$619$			263.810i			0.426187i	0.977032	+	0.213093i	$0.0683538\pi$
$619$			263.810i			0.426187i	−0.977032	+	0.213093i	$0.931646\pi$
$620$			502.051i			0.809759i
$621$		0			0
$622$			349.293i			0.561565i
$623$	34.1969			0.0548907
$624$		0			0
$625$	465.291			0.744466
$626$	−488.094			−0.779703
$627$		0			0
$628$	−150.837			−0.240186
$629$	−359.095	+	531.478i	−0.570899	+	0.844958i
$630$		0			0
$631$	847.348			1.34287			0.671433	−	0.741065i	$-0.265679\pi$
$631$	847.348			1.34287			0.671433	+	0.741065i	$0.265679\pi$
$632$	822.086			1.30077
$633$		0			0
$634$		−	253.140i		−	0.399274i
$635$	1257.56			1.98041
$636$		0			0
$637$	343.256			0.538863
$638$		−	276.297i		−	0.433067i
$639$		0			0
$640$			485.027i			0.757855i
$641$	−496.117			−0.773973			−0.386987	−	0.922085i	$-0.626484\pi$
$641$	−496.117			−0.773973			−0.386987	+	0.922085i	$0.626484\pi$
$642$		0			0
$643$		−	675.075i		−	1.04988i	−0.851138	−	0.524942i	$-0.824087\pi$
$643$		−	675.075i		−	1.04988i	0.851138	−	0.524942i	$-0.175913\pi$
$644$			23.2812i			0.0361509i
$645$		0			0
$646$	−498.135	+	737.264i	−0.771107	+	1.14128i
$647$		−	8.13757i		−	0.0125774i	−0.999980	−	0.00628869i	$-0.997998\pi$
$647$		−	8.13757i		−	0.0125774i	0.999980	−	0.00628869i	$-0.00200177\pi$
$648$		0			0
$649$			765.295i			1.17919i
$650$			734.258i			1.12963i
$651$		0			0
$652$		−	2.40595i		−	0.00369011i
$653$	1169.10			1.79035			0.895177	−	0.445710i	$-0.147049\pi$
$653$	1169.10			1.79035			0.895177	+	0.445710i	$0.147049\pi$
$654$		0			0
$655$	1679.17			2.56362
$656$	−25.7605			−0.0392690
$657$		0			0
$658$		−	1389.45i		−	2.11163i
$659$		−	68.6538i		−	0.104179i	−0.998642	−	0.0520893i	$-0.983412\pi$
$659$		−	68.6538i		−	0.104179i	0.998642	−	0.0520893i	$-0.0165881\pi$
$660$		0			0
$661$	−232.907			−0.352356			−0.176178	−	0.984358i	$-0.556373\pi$
$661$	−232.907			−0.352356			−0.176178	+	0.984358i	$0.556373\pi$
$662$			223.465i			0.337560i
$663$		0			0
$664$	−676.770			−1.01923
$665$		−	2428.84i		−	3.65239i
$666$		0			0
$667$	28.4874			0.0427097
$668$	−230.329			−0.344803
$669$		0			0
$670$		−	1076.46i		−	1.60665i
$671$		−	293.724i		−	0.437741i
$672$		0			0
$673$			179.762i			0.267106i	0.991042	+	0.133553i	$0.0426386\pi$
$673$			179.762i			0.267106i	−0.991042	+	0.133553i	$0.957361\pi$
$674$	7.09402			0.0105253
$675$		0			0
$676$	−93.1419			−0.137784
$677$	−322.648			−0.476584			−0.238292	−	0.971194i	$-0.576588\pi$
$677$	−322.648			−0.476584			−0.238292	+	0.971194i	$0.576588\pi$
$678$		0			0
$679$	−549.311			−0.809000
$680$	−704.260	+	1042.34i	−1.03568	+	1.53285i
$681$		0			0
$682$	−1302.86			−1.91035
$683$	1289.59			1.88813			0.944066	−	0.329755i	$-0.106966\pi$
$683$	1289.59			1.88813			0.944066	+	0.329755i	$0.106966\pi$
$684$		0			0
$685$			347.033i			0.506618i
$686$	−170.033			−0.247861
$687$		0			0
$688$	−803.834			−1.16836
$689$		−	280.223i		−	0.406710i
$690$		0			0
$691$		−	723.167i		−	1.04655i	−0.852164	−	0.523276i	$-0.824710\pi$
$691$		−	723.167i		−	1.04655i	0.852164	−	0.523276i	$-0.175290\pi$
$692$	177.786			0.256916
$693$		0			0
$694$			479.076i			0.690311i
$695$		−	525.762i		−	0.756492i
$696$		0			0
$697$	−33.8116	−	22.8449i	−0.0485102	−	0.0327761i
$698$			955.953i			1.36956i
$699$		0			0
$700$		−	466.808i		−	0.666868i
$701$		−	296.143i		−	0.422458i	−0.977437	−	0.211229i	$-0.932253\pi$
$701$		−	296.143i		−	0.422458i	0.977437	−	0.211229i	$-0.0677466\pi$
$702$		0			0
$703$		−	1148.67i		−	1.63396i
$704$	952.927			1.35359
$705$		0			0
$706$	253.457			0.359004
$707$	−377.355			−0.533741
$708$		0			0
$709$		−	1213.51i		−	1.71158i	−0.517324	−	0.855790i	$-0.673072\pi$
$709$		−	1213.51i		−	1.71158i	0.517324	−	0.855790i	$-0.326928\pi$
$710$		−	591.414i		−	0.832977i
$711$		0			0
$712$	31.7181			0.0445479
$713$		−	134.330i		−	0.188402i
$714$		0			0
$715$	1025.34			1.43405
$716$			287.766i			0.401907i
$717$		0			0
$718$	−1212.09			−1.68815
$719$	−769.633			−1.07042			−0.535211	−	0.844719i	$-0.679768\pi$
$719$	−769.633			−1.07042			−0.535211	+	0.844719i	$0.679768\pi$
$720$		0			0
$721$		−	601.138i		−	0.833755i
$722$		−	972.800i		−	1.34737i
$723$		0			0
$724$		−	29.1330i		−	0.0402389i
$725$	−571.197			−0.787858
$726$		0			0
$727$	251.418			0.345829			0.172915	−	0.984937i	$-0.444682\pi$
$727$	251.418			0.345829			0.172915	+	0.984937i	$0.444682\pi$
$728$	724.400			0.995054
$729$		0			0
$730$	−1048.43			−1.43621
$731$	−1055.06	−	712.857i	−1.44331	−	0.975181i
$732$		0			0
$733$	588.784			0.803253			0.401626	−	0.915804i	$-0.368445\pi$
$733$	588.784			0.803253			0.401626	+	0.915804i	$0.368445\pi$
$734$	731.568			0.996686
$735$		0			0
$736$			38.7166i			0.0526041i
$737$	−987.035			−1.33926
$738$		0			0
$739$	522.425			0.706935			0.353467	−	0.935447i	$-0.385003\pi$
$739$	522.425			0.706935			0.353467	+	0.935447i	$0.385003\pi$
$740$		−	336.217i		−	0.454347i
$741$		0			0
$742$		−	504.204i		−	0.679521i
$743$	−359.980			−0.484495			−0.242248	−	0.970214i	$-0.577885\pi$
$743$	−359.980			−0.484495			−0.242248	+	0.970214i	$0.577885\pi$
$744$		0			0
$745$			1655.55i			2.22222i
$746$		−	371.018i		−	0.497343i
$747$		0			0
$748$	197.871	+	133.692i	0.264534	+	0.178733i
$749$			162.153i			0.216493i
$750$		0			0
$751$			337.419i			0.449293i	0.974440	+	0.224647i	$0.0721228\pi$
$751$			337.419i			0.449293i	−0.974440	+	0.224647i	$0.927877\pi$
$752$		−	927.654i		−	1.23358i
$753$		0			0
$754$		−	183.511i		−	0.243384i
$755$	986.657			1.30683
$756$		0			0
$757$	−613.931			−0.811005			−0.405503	−	0.914094i	$-0.632904\pi$
$757$	−613.931			−0.811005			−0.405503	+	0.914094i	$0.632904\pi$
$758$	−1065.97			−1.40629
$759$		0			0
$760$		−	2252.78i		−	2.96419i
$761$		−	1355.73i		−	1.78151i	−0.454487	−	0.890754i	$-0.650177\pi$
$761$		−	1355.73i		−	1.78151i	0.454487	−	0.890754i	$-0.349823\pi$
$762$		0			0
$763$	−234.372			−0.307171
$764$			120.149i			0.157264i
$765$		0			0
$766$	473.398			0.618013
$767$			508.295i			0.662706i
$768$		0			0
$769$	−3.26282			−0.00424295			−0.00212147	−	0.999998i	$-0.500675\pi$
$769$	−3.26282			−0.00424295			−0.00212147	+	0.999998i	$0.500675\pi$
$770$	1844.90			2.39597
$771$		0			0
$772$			272.508i			0.352990i
$773$			1253.84i			1.62204i	0.585018	+	0.811020i	$0.301087\pi$
$773$			1253.84i			1.62204i	−0.585018	+	0.811020i	$0.698913\pi$
$774$		0			0
$775$			2693.44i			3.47541i
$776$	−509.494			−0.656564
$777$		0			0
$778$	69.7697			0.0896782
$779$	73.0763			0.0938078
$780$		0			0
$781$	−542.285			−0.694347
$782$	39.0118	−	57.7394i	0.0498873	−	0.0738355i
$783$		0			0
$784$	412.347			0.525953
$785$	−1232.40			−1.56994
$786$		0			0
$787$		−	989.567i		−	1.25739i	−0.777651	−	0.628696i	$-0.783589\pi$
$787$		−	989.567i		−	1.25739i	0.777651	−	0.628696i	$-0.216411\pi$
$788$	−272.134			−0.345348
$789$		0			0
$790$	1390.59			1.76024
$791$			556.850i			0.703982i
$792$		0			0
$793$		−	195.086i		−	0.246010i
$794$	−80.7700			−0.101725
$795$		0			0
$796$			160.904i			0.202141i
$797$		−	977.351i		−	1.22629i	−0.789971	−	0.613144i	$-0.789905\pi$
$797$		−	977.351i		−	1.22629i	0.789971	−	0.613144i	$-0.210095\pi$
$798$		0			0
$799$	822.663	−	1217.58i	1.02962	−	1.52388i
$800$		−	776.303i		−	0.970379i
$801$		0			0
$802$			89.6425i			0.111774i
$803$			961.340i			1.19719i
$804$		0			0
$805$			190.216i			0.236294i
$806$	−865.336			−1.07362
$807$		0			0
$808$	−350.002			−0.433171
$809$	1.39391			0.00172301			0.000861503	−	1.00000i	$-0.499726\pi$
$809$	1.39391			0.00172301			0.000861503		1.00000i	$0.499726\pi$
$810$		0			0
$811$		−	750.546i		−	0.925457i	−0.886500	−	0.462729i	$-0.846870\pi$
$811$		−	750.546i		−	0.925457i	0.886500	−	0.462729i	$-0.153130\pi$
$812$			116.668i			0.143680i
$813$		0			0
$814$	872.507			1.07188
$815$		−	19.6576i		−	0.0241198i
$816$		0			0
$817$	2280.28			2.79105
$818$			756.964i			0.925384i
$819$		0			0
$820$	21.3894			0.0260847
$821$	218.234			0.265815			0.132907	−	0.991128i	$-0.457569\pi$
$821$	218.234			0.265815			0.132907	+	0.991128i	$0.457569\pi$
$822$		0			0
$823$		−	602.891i		−	0.732553i	−0.930506	−	0.366276i	$-0.880633\pi$
$823$		−	602.891i		−	0.732553i	0.930506	−	0.366276i	$-0.119367\pi$
$824$		−	557.564i		−	0.676655i
$825$		0			0
$826$			914.574i			1.10723i
$827$	−290.111			−0.350799			−0.175399	−	0.984497i	$-0.556122\pi$
$827$	−290.111			−0.350799			−0.175399	+	0.984497i	$0.556122\pi$
$828$		0			0
$829$	−103.693			−0.125082			−0.0625409	−	0.998042i	$-0.519920\pi$
$829$	−103.693			−0.125082			−0.0625409	+	0.998042i	$0.519920\pi$
$830$	−1144.78			−1.37925
$831$		0			0
$832$	632.917			0.760717
$833$	541.221	+	365.678i	0.649726	+	0.438990i
$834$		0			0
$835$	−1881.88			−2.25375
$836$	−427.654			−0.511548
$837$		0			0
$838$			941.337i			1.12331i
$839$	−692.406			−0.825275			−0.412638	−	0.910895i	$-0.635392\pi$
$839$	−692.406			−0.825275			−0.412638	+	0.910895i	$0.635392\pi$
$840$		0			0
$841$	−698.242			−0.830252
$842$		−	827.845i		−	0.983189i
$843$		0			0
$844$			90.1097i			0.106765i
$845$	−761.007			−0.900599
$846$		0			0
$847$		−	560.293i		−	0.661503i
$848$		−	336.627i		−	0.396966i
$849$		0			0
$850$	−782.222	+	1157.73i	−0.920261	+	1.36203i
$851$			89.9592i			0.105710i
$852$		0			0
$853$		−	1424.87i		−	1.67042i	−0.549929	−	0.835211i	$-0.685345\pi$
$853$		−	1424.87i		−	1.67042i	0.549929	−	0.835211i	$-0.314655\pi$
$854$		−	351.018i		−	0.411028i
$855$		0			0
$856$			150.400i			0.175700i
$857$	−618.934			−0.722211			−0.361105	−	0.932525i	$-0.617601\pi$
$857$	−618.934			−0.722211			−0.361105	+	0.932525i	$0.617601\pi$
$858$		0			0
$859$	−1111.61			−1.29407			−0.647035	−	0.762460i	$-0.723991\pi$
$859$	−1111.61			−1.29407			−0.647035	+	0.762460i	$0.723991\pi$
$860$	667.439			0.776092
$861$		0			0
$862$			347.735i			0.403405i
$863$		−	35.3248i		−	0.0409326i	−0.999791	−	0.0204663i	$-0.993485\pi$
$863$		−	35.3248i		−	0.0409326i	0.999791	−	0.0204663i	$-0.00651508\pi$
$864$		0			0
$865$	1452.58			1.67929
$866$		−	971.200i		−	1.12148i
$867$		0			0
$868$	550.141			0.633803
$869$		−	1275.07i		−	1.46729i
$870$		0			0
$871$	−655.571			−0.752664
$872$	−217.383			−0.249292
$873$		0			0
$874$			124.791i			0.142781i
$875$		−	1819.50i		−	2.07943i
$876$		0			0
$877$		−	467.872i		−	0.533492i	−0.963767	−	0.266746i	$-0.914052\pi$
$877$		−	467.872i		−	0.533492i	0.963767	−	0.266746i	$-0.0859485\pi$
$878$	594.454			0.677054
$879$		0			0
$880$	1231.73			1.39969
$881$	644.199			0.731213			0.365607	−	0.930770i	$-0.380862\pi$
$881$	644.199			0.731213			0.365607	+	0.930770i	$0.380862\pi$
$882$		0			0
$883$	24.2582			0.0274725			0.0137363	−	0.999906i	$-0.495627\pi$
$883$	24.2582			0.0274725			0.0137363	+	0.999906i	$0.495627\pi$
$884$	131.422	+	88.7960i	0.148668	+	0.100448i
$885$		0			0
$886$	685.870			0.774120
$887$	−371.912			−0.419292			−0.209646	−	0.977777i	$-0.567231\pi$
$887$	−371.912			−0.419292			−0.209646	+	0.977777i	$0.567231\pi$
$888$		0			0
$889$		−	1378.02i		−	1.55008i
$890$	53.6523			0.0602834
$891$		0			0
$892$	−116.576			−0.130691
$893$			2631.53i			2.94684i
$894$		0			0
$895$			2351.16i			2.62700i
$896$	531.486			0.593177
$897$		0			0
$898$		−	224.386i		−	0.249873i
$899$		−	673.165i		−	0.748793i
$900$		0			0
$901$	298.528	−	441.836i	0.331330	−	0.490384i
$902$			55.5072i			0.0615379i
$903$		0			0
$904$			516.486i			0.571334i
$905$		−	238.028i		−	0.263014i
$906$		0			0
$907$			469.749i			0.517915i	0.965889	+	0.258958i	$0.0833790\pi$
$907$			469.749i			0.517915i	−0.965889	+	0.258958i	$0.916621\pi$
$908$	44.4084			0.0489080
$909$		0			0
$910$	1225.35			1.34653
$911$	485.815			0.533276			0.266638	−	0.963797i	$-0.414087\pi$
$911$	485.815			0.533276			0.266638	+	0.963797i	$0.414087\pi$
$912$		0			0
$913$			1049.68i			1.14971i
$914$			816.209i			0.893008i
$915$		0			0
$916$	−191.871			−0.209466
$917$		−	1840.02i		−	2.00656i
$918$		0			0
$919$	357.345			0.388842			0.194421	−	0.980918i	$-0.437717\pi$
$919$	357.345			0.388842			0.194421	+	0.980918i	$0.437717\pi$
$920$			176.428i			0.191770i
$921$		0			0
$922$	−1384.72			−1.50187
$923$	−360.176			−0.390223
$924$		0			0
$925$		−	1803.76i		−	1.95001i
$926$			773.106i			0.834887i
$927$		0			0
$928$			194.020i			0.209073i
$929$	−683.118			−0.735326			−0.367663	−	0.929959i	$-0.619842\pi$
$929$	−683.118			−0.735326			−0.367663	+	0.929959i	$0.619842\pi$
$930$		0			0
$931$	−1169.73			−1.25642
$932$	242.807			0.260523
$933$		0			0
$934$	80.3115			0.0859866
$935$	1616.69	+	1092.32i	1.72908	+	1.16826i
$936$		0			0
$937$	−360.068			−0.384277			−0.192139	−	0.981368i	$-0.561542\pi$
$937$	−360.068			−0.384277			−0.192139	+	0.981368i	$0.561542\pi$
$938$	−1179.57			−1.25753
$939$		0			0
$940$			770.249i			0.819414i
$941$	−1461.35			−1.55297			−0.776487	−	0.630133i	$-0.783000\pi$
$941$	−1461.35			−1.55297			−0.776487	+	0.630133i	$0.783000\pi$
$942$		0			0
$943$	−5.72302			−0.00606895
$944$			610.606i			0.646829i
$945$		0			0
$946$			1732.06i			1.83093i
$947$	−1700.73			−1.79591			−0.897955	−	0.440088i	$-0.854947\pi$
$947$	−1700.73			−1.79591			−0.897955	+	0.440088i	$0.854947\pi$
$948$		0			0
$949$			638.505i			0.672818i
$950$		−	2502.17i		−	2.63386i
$951$		0			0
$952$	1142.18	+	771.719i	1.19977	+	0.810629i
$953$		−	1768.98i		−	1.85622i	−0.372309	−	0.928109i	$-0.621434\pi$
$953$		−	1768.98i		−	1.85622i	0.372309	−	0.928109i	$-0.378566\pi$
$954$		0			0
$955$			981.670i			1.02793i
$956$			64.5941i			0.0675671i
$957$		0			0
$958$			63.2959i			0.0660708i
$959$	380.275			0.396533
$960$		0			0
$961$	−2213.27			−2.30309
$962$	579.504			0.602395
$963$		0			0
$964$			185.340i			0.192261i
$965$			2226.50i			2.30725i
$966$		0			0
$967$	58.4012			0.0603942			0.0301971	−	0.999544i	$-0.490387\pi$
$967$	58.4012			0.0603942			0.0301971	+	0.999544i	$0.490387\pi$
$968$		−	519.680i		−	0.536859i
$969$		0			0
$970$	−861.826			−0.888480
$971$		−	150.697i		−	0.155197i	−0.996985	−	0.0775987i	$-0.975275\pi$
$971$		−	150.697i		−	0.155197i	0.996985	−	0.0775987i	$-0.0247253\pi$
$972$		0			0
$973$	−576.123			−0.592110
$974$	390.271			0.400689
$975$		0			0
$976$		−	234.354i		−	0.240117i
$977$			2.82108i			0.00288749i	0.999999	+	0.00144375i	$0.000459559\pi$
$977$			2.82108i			0.00288749i	−0.999999	+	0.00144375i	$0.999540\pi$
$978$		0			0
$979$		−	49.1954i		−	0.0502507i
$980$	−342.380			−0.349367
$981$		0			0
$982$	1458.01			1.48473
$983$	1899.35			1.93219			0.966097	−	0.258178i	$-0.0831222\pi$
$983$	1899.35			1.93219			0.966097	+	0.258178i	$0.0831222\pi$
$984$		0			0
$985$	−2223.45			−2.25731
$986$	195.499	−	289.348i	0.198275	−	0.293456i
$987$		0			0
$988$	−284.040			−0.287490
$989$	−178.582			−0.180568
$990$		0			0
$991$		−	1208.58i		−	1.21956i	−0.792571	−	0.609780i	$-0.791258\pi$
$991$		−	1208.58i		−	1.21956i	0.792571	−	0.609780i	$-0.208742\pi$
$992$	914.886			0.922264
$993$		0			0
$994$	−648.063			−0.651975
$995$			1314.65i			1.32126i
$996$		0			0
$997$			1326.19i			1.33018i	0.746765	+	0.665089i	$0.231606\pi$
$997$			1326.19i			1.33018i	−0.746765	+	0.665089i	$0.768394\pi$
$998$	109.544			0.109764
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	459.3.c.g.458.6	yes	20
3.2	odd	2	inner	459.3.c.g.458.15	yes	20
17.16	even	2	inner	459.3.c.g.458.5	✓	20
51.50	odd	2	inner	459.3.c.g.458.16	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
459.3.c.g.458.5	✓	20	17.16	even	2	inner
459.3.c.g.458.6	yes	20	1.1	even	1	trivial
459.3.c.g.458.15	yes	20	3.2	odd	2	inner
459.3.c.g.458.16	yes	20	51.50	odd	2	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:	\( N \)	\(=\)	\( 459 = 3^{3} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	459.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.71920i q^{2} +1.04434 q^{4} +8.53267 q^{5} -9.34999i q^{7} -8.67225i q^{8} -14.6694i q^{10} -13.4508 q^{11} -8.93379 q^{13} -16.0745 q^{14} -10.7320 q^{16} +(-14.0862 - 9.51737i) q^{17} +30.4441 q^{19} +8.91099 q^{20} +23.1247i q^{22} -2.38425 q^{23} +47.8064 q^{25} +15.3590i q^{26} -9.76455i q^{28} -11.9481 q^{29} +56.3406i q^{31} -16.2385i q^{32} +(-16.3623 + 24.2170i) q^{34} -79.7803i q^{35} -37.7305i q^{37} -52.3396i q^{38} -73.9974i q^{40} +2.40034 q^{41} +74.9007 q^{43} -14.0472 q^{44} +4.09902i q^{46} +86.4381i q^{47} -38.4222 q^{49} -82.1889i q^{50} -9.32990 q^{52} +31.3667i q^{53} -114.771 q^{55} -81.0854 q^{56} +20.5413i q^{58} -56.8958i q^{59} +21.8369i q^{61} +96.8610 q^{62} -70.8453 q^{64} -76.2290 q^{65} +73.3810 q^{67} +(-14.7107 - 9.93935i) q^{68} -137.159 q^{70} +40.3162 q^{71} -71.4708i q^{73} -64.8665 q^{74} +31.7939 q^{76} +125.765i q^{77} +94.7951i q^{79} -91.5726 q^{80} -4.12668i q^{82} -78.0386i q^{83} +(-120.192 - 81.2085i) q^{85} -128.770i q^{86} +116.649i q^{88} +3.65743i q^{89} +83.5308i q^{91} -2.48997 q^{92} +148.605 q^{94} +259.769 q^{95} -58.7499i q^{97} +66.0556i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 36 q^{4} + 100 q^{13} + 228 q^{16} + 172 q^{19} + 272 q^{25} - 80 q^{34} + 124 q^{43} + 4 q^{49} - 888 q^{52} + 268 q^{55} - 1408 q^{64} + 80 q^{67} - 300 q^{70} - 252 q^{76} - 484 q^{85} + 172 q^{94}+O(q^{100}) \) 20 * q - 36 * q^4 + 100 * q^13 + 228 * q^16 + 172 * q^19 + 272 * q^25 - 80 * q^34 + 124 * q^43 + 4 * q^49 - 888 * q^52 + 268 * q^55 - 1408 * q^64 + 80 * q^67 - 300 * q^70 - 252 * q^76 - 484 * q^85 + 172 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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