LMFDB - Embedded newform 1008.3.dc.g.305.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1008,3,Mod(305,1008)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 4])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1008.305"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1008.dc (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,24,0,0,0,0,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$27.4660106475$
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} + 90x^{14} + 2793x^{12} + 37090x^{10} + 214104x^{8} + 463326x^{6} + 257641x^{4} + 28374x^{2} + 9 $ x^16 + 90x^14 + 2793x^12 + 37090x^10 + 214104x^8 + 463326x^6 + 257641x^4 + 28374*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			305.5
Root			$1.76745i$ of defining polynomial
Character	$\chi$	$=$	1008.305
Dual form			1008.3.dc.g.737.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.29207 - 1.32333i) q^{5} +(-5.75292 - 3.98797i) q^{7} +(-1.93387 - 1.11652i) q^{11} +2.75937 q^{13} +(-0.189165 - 0.109214i) q^{17} +(-6.60668 - 11.4431i) q^{19} +(-17.4712 + 10.0870i) q^{23} +(-8.99762 + 15.5843i) q^{25} -13.5750i q^{29} +(0.993552 - 1.72088i) q^{31} +(-18.4635 - 1.52772i) q^{35} +(22.5458 + 39.0505i) q^{37} -28.7877i q^{41} -53.1897 q^{43} +(-62.8200 + 36.2691i) q^{47} +(17.1921 + 45.8850i) q^{49} +(-52.0559 - 30.0545i) q^{53} -5.91007 q^{55} +(-56.8045 - 32.7961i) q^{59} +(37.7262 + 65.3436i) q^{61} +(6.32465 - 3.65154i) q^{65} +(-16.1113 + 27.9057i) q^{67} -17.2308i q^{71} +(-7.18849 + 12.4508i) q^{73} +(6.67274 + 14.1355i) q^{77} +(18.0972 + 31.3453i) q^{79} -43.6639i q^{83} -0.578105 q^{85} +(10.2082 - 5.89370i) q^{89} +(-15.8744 - 11.0043i) q^{91} +(-30.2859 - 17.4856i) q^{95} -112.554 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 24 q^{7} - 24 q^{13} + 12 q^{19} + 92 q^{25} - 32 q^{31} - 132 q^{37} - 24 q^{43} + 64 q^{49} + 440 q^{55} + 184 q^{61} - 332 q^{67} + 188 q^{73} + 112 q^{79} + 256 q^{85} + 252 q^{91} - 1032 q^{97}+O(q^{100}) $ 16 * q + 24 * q^7 - 24 * q^13 + 12 * q^19 + 92 * q^25 - 32 * q^31 - 132 * q^37 - 24 * q^43 + 64 * q^49 + 440 * q^55 + 184 * q^61 - 332 * q^67 + 188 * q^73 + 112 * q^79 + 256 * q^85 + 252 * q^91 - 1032 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$-1$

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
$5$	2.29207	−	1.32333i	0.458413	−	0.264665i	−0.252963	−	0.967476i	$-0.581405\pi$
$5$	2.29207	−	1.32333i	0.458413	−	0.264665i	0.711377	+	0.702811i	$0.248072\pi$
$6$		0			0
$7$	−5.75292	−	3.98797i	−0.821846	−	0.569710i
$7$	−5.75292	−	3.98797i	−0.821846	−	0.569710i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.93387	−	1.11652i	−0.175806	−	0.101502i	0.409515	−	0.912304i	$-0.365698\pi$
$11$	−1.93387	−	1.11652i	−0.175806	−	0.101502i	−0.585321	+	0.810802i	$0.699031\pi$
$12$		0			0
$13$	2.75937			0.212259			0.106129	−	0.994352i	$-0.466154\pi$
$13$	2.75937			0.212259			0.106129	+	0.994352i	$0.466154\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.189165	−	0.109214i	−0.0111274	−	0.00642438i	0.494426	−	0.869220i	$-0.335378\pi$
$17$	−0.189165	−	0.109214i	−0.0111274	−	0.00642438i	−0.505553	+	0.862795i	$0.668712\pi$
$18$		0			0
$19$	−6.60668	−	11.4431i	−0.347720	−	0.602269i	0.638124	−	0.769934i	$-0.279711\pi$
$19$	−6.60668	−	11.4431i	−0.347720	−	0.602269i	−0.985844	+	0.167665i	$0.946377\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−17.4712	+	10.0870i	−0.759619	+	0.438566i	−0.829159	−	0.559013i	$-0.811180\pi$
$23$	−17.4712	+	10.0870i	−0.759619	+	0.438566i	0.0695400	+	0.997579i	$0.477847\pi$
$24$		0			0
$25$	−8.99762	+	15.5843i	−0.359905	+	0.623373i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	13.5750i		−	0.468103i	−0.972224	−	0.234052i	$-0.924801\pi$
$29$		−	13.5750i		−	0.468103i	0.972224	−	0.234052i	$-0.0751985\pi$
$30$		0			0
$31$	0.993552	−	1.72088i	0.0320501	−	0.0555123i	−0.849555	−	0.527499i	$-0.823130\pi$
$31$	0.993552	−	1.72088i	0.0320501	−	0.0555123i	0.881606	+	0.471987i	$0.156463\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−18.4635	−	1.52772i	−0.527527	−	0.0436490i
$36$		0			0
$37$	22.5458	+	39.0505i	0.609346	+	1.05542i	0.991348	+	0.131257i	$0.0419014\pi$
$37$	22.5458	+	39.0505i	0.609346	+	1.05542i	−0.382002	+	0.924161i	$0.624765\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	28.7877i		−	0.702139i	−0.936349	−	0.351070i	$-0.885818\pi$
$41$		−	28.7877i		−	0.702139i	0.936349	−	0.351070i	$-0.114182\pi$
$42$		0			0
$43$	−53.1897			−1.23697			−0.618485	−	0.785797i	$-0.712253\pi$
$43$	−53.1897			−1.23697			−0.618485	+	0.785797i	$0.712253\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−62.8200	+	36.2691i	−1.33660	+	0.771684i	−0.986301	−	0.164956i	$-0.947252\pi$
$47$	−62.8200	+	36.2691i	−1.33660	+	0.771684i	−0.350294	+	0.936640i	$0.613918\pi$
$48$		0			0
$49$	17.1921	+	45.8850i	0.350860	+	0.936428i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−52.0559	−	30.0545i	−0.982187	−	0.567066i	−0.0792573	−	0.996854i	$-0.525255\pi$
$53$	−52.0559	−	30.0545i	−0.982187	−	0.567066i	−0.902930	+	0.429788i	$0.858588\pi$
$54$		0			0
$55$	−5.91007			−0.107456
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−56.8045	−	32.7961i	−0.962788	−	0.555866i	−0.0657580	−	0.997836i	$-0.520947\pi$
$59$	−56.8045	−	32.7961i	−0.962788	−	0.555866i	−0.897030	+	0.441970i	$0.854280\pi$
$60$		0			0
$61$	37.7262	+	65.3436i	0.618462	+	1.07121i	0.989767	+	0.142696i	$0.0455771\pi$
$61$	37.7262	+	65.3436i	0.618462	+	1.07121i	−0.371305	+	0.928511i	$0.621090\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.32465	−	3.65154i	0.0973024	−	0.0561775i
$66$		0			0
$67$	−16.1113	+	27.9057i	−0.240468	+	0.416502i	−0.960848	−	0.277077i	$-0.910634\pi$
$67$	−16.1113	+	27.9057i	−0.240468	+	0.416502i	0.720380	+	0.693580i	$0.243968\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	17.2308i		−	0.242688i	−0.992611	−	0.121344i	$-0.961280\pi$
$71$		−	17.2308i		−	0.242688i	0.992611	−	0.121344i	$-0.0387204\pi$
$72$		0			0
$73$	−7.18849	+	12.4508i	−0.0984724	+	0.170559i	−0.911053	−	0.412290i	$-0.864729\pi$
$73$	−7.18849	+	12.4508i	−0.0984724	+	0.170559i	0.812580	+	0.582850i	$0.198062\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	6.67274	+	14.1355i	0.0866589	+	0.183577i
$78$		0			0
$79$	18.0972	+	31.3453i	0.229079	+	0.396776i	0.957535	−	0.288316i	$-0.0930953\pi$
$79$	18.0972	+	31.3453i	0.229079	+	0.396776i	−0.728457	+	0.685092i	$0.759762\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	43.6639i		−	0.526071i	−0.964786	−	0.263035i	$-0.915276\pi$
$83$		−	43.6639i		−	0.526071i	0.964786	−	0.263035i	$-0.0847236\pi$
$84$		0			0
$85$	−0.578105			−0.00680124
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.2082	−	5.89370i	0.114699	−	0.0662213i	−0.441553	−	0.897235i	$-0.645572\pi$
$89$	10.2082	−	5.89370i	0.114699	−	0.0662213i	0.556252	+	0.831014i	$0.312239\pi$
$90$		0			0
$91$	−15.8744	−	11.0043i	−0.174444	−	0.120926i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−30.2859	−	17.4856i	−0.318799	−	0.184059i
$96$		0			0
$97$	−112.554			−1.16035			−0.580175	−	0.814492i	$-0.697016\pi$
$97$	−112.554			−1.16035			−0.580175	+	0.814492i	$0.697016\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−116.838	−	67.4566i	−1.15681	−	0.667887i	−0.206276	−	0.978494i	$-0.566135\pi$
$101$	−116.838	−	67.4566i	−1.15681	−	0.667887i	−0.950539	+	0.310606i	$0.899468\pi$
$102$		0			0
$103$	17.4709	+	30.2604i	0.169620	+	0.293791i	0.938286	−	0.345860i	$-0.112413\pi$
$103$	17.4709	+	30.2604i	0.169620	+	0.293791i	−0.768666	+	0.639650i	$0.779079\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	22.6307	−	13.0658i	0.211501	−	0.122110i	−0.390508	−	0.920600i	$-0.627700\pi$
$107$	22.6307	−	13.0658i	0.211501	−	0.122110i	0.602009	+	0.798489i	$0.294367\pi$
$108$		0			0
$109$	4.03319	−	6.98568i	0.0370017	−	0.0640888i	−0.846931	−	0.531702i	$-0.821553\pi$
$109$	4.03319	−	6.98568i	0.0370017	−	0.0640888i	0.883933	+	0.467613i	$0.154886\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	30.1803i		−	0.267083i	−0.991043	−	0.133541i	$-0.957365\pi$
$113$		−	30.1803i		−	0.267083i	0.991043	−	0.133541i	$-0.0426349\pi$
$114$		0			0
$115$	−26.6968	+	46.2403i	−0.232146	+	0.402089i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.652707	+	1.38269i	0.00548493	+	0.0116192i
$120$		0			0
$121$	−58.0068	−	100.471i	−0.479395	−	0.830336i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			113.793i			0.910347i
$126$		0			0
$127$	−70.8602			−0.557954			−0.278977	−	0.960298i	$-0.589995\pi$
$127$	−70.8602			−0.557954			−0.278977	+	0.960298i	$0.589995\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−80.2222	+	46.3163i	−0.612383	+	0.353559i	−0.773897	−	0.633311i	$-0.781695\pi$
$131$	−80.2222	+	46.3163i	−0.612383	+	0.353559i	0.161515	+	0.986870i	$0.448362\pi$
$132$		0			0
$133$	−7.62710	+	92.1786i	−0.0573466	+	0.693072i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−178.775	−	103.216i	−1.30492	−	0.753398i	−0.323680	−	0.946167i	$-0.604920\pi$
$137$	−178.775	−	103.216i	−1.30492	−	0.753398i	−0.981244	+	0.192769i	$0.938253\pi$
$138$		0			0
$139$	194.003			1.39570			0.697852	−	0.716242i	$-0.254139\pi$
$139$	194.003			1.39570			0.697852	+	0.716242i	$0.254139\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−5.33625	−	3.08089i	−0.0373164	−	0.0215447i
$144$		0			0
$145$	−17.9641	−	31.1148i	−0.123891	−	0.214585i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−140.695	+	81.2302i	−0.944260	+	0.545169i	−0.891293	−	0.453427i	$-0.850201\pi$
$149$	−140.695	+	81.2302i	−0.944260	+	0.545169i	−0.0529671	+	0.998596i	$0.516868\pi$
$150$		0			0
$151$	−115.700	+	200.399i	−0.766227	+	1.32714i	0.173368	+	0.984857i	$0.444535\pi$
$151$	−115.700	+	200.399i	−0.766227	+	1.32714i	−0.939595	+	0.342287i	$0.888798\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	5.25917i		−	0.0339301i
$156$		0			0
$157$	52.2334	−	90.4709i	0.332697	−	0.576248i	−0.650343	−	0.759641i	$-0.725375\pi$
$157$	52.2334	−	90.4709i	0.332697	−	0.576248i	0.983040	+	0.183393i	$0.0587081\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	140.737	+	11.6450i	0.874145	+	0.0723291i
$162$		0			0
$163$	103.060	+	178.505i	0.632269	+	1.09512i	0.987087	+	0.160187i	$0.0512098\pi$
$163$	103.060	+	178.505i	0.632269	+	1.09512i	−0.354817	+	0.934936i	$0.615457\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	239.559i		−	1.43448i	−0.696824	−	0.717242i	$-0.745404\pi$
$167$		−	239.559i		−	1.43448i	0.696824	−	0.717242i	$-0.254596\pi$
$168$		0			0
$169$	−161.386			−0.954946
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−271.139	+	156.542i	−1.56728	+	0.904870i	−0.570796	+	0.821092i	$0.693365\pi$
$173$	−271.139	+	156.542i	−1.56728	+	0.904870i	−0.996485	+	0.0837773i	$0.973302\pi$
$174$		0			0
$175$	113.912	−	53.7731i	0.650928	−	0.307275i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−0.835046	−	0.482114i	−0.00466506	−	0.00269337i	0.497666	−	0.867369i	$-0.334191\pi$
$179$	−0.835046	−	0.482114i	−0.00466506	−	0.00269337i	−0.502331	+	0.864676i	$0.667524\pi$
$180$		0			0
$181$	−182.012			−1.00559			−0.502795	−	0.864406i	$-0.667695\pi$
$181$	−182.012			−1.00559			−0.502795	+	0.864406i	$0.667695\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	103.353	+	59.6709i	0.558665	+	0.322545i
$186$		0			0
$187$	0.243880	+	0.422413i	0.00130417	+	0.00225889i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	225.335	−	130.097i	1.17976	−	0.681136i	0.223802	−	0.974635i	$-0.428153\pi$
$191$	225.335	−	130.097i	1.17976	−	0.681136i	0.955959	+	0.293499i	$0.0948196\pi$
$192$		0			0
$193$	128.058	−	221.803i	0.663512	−	1.14924i	−0.316175	−	0.948701i	$-0.602399\pi$
$193$	128.058	−	221.803i	0.663512	−	1.14924i	0.979687	−	0.200535i	$-0.0642680\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	8.84852i		−	0.0449163i	−0.999748	−	0.0224582i	$-0.992851\pi$
$197$		−	8.84852i		−	0.0449163i	0.999748	−	0.0224582i	$-0.00714926\pi$
$198$		0			0
$199$	23.3812	−	40.4974i	0.117494	−	0.203505i	−0.801280	−	0.598289i	$-0.795847\pi$
$199$	23.3812	−	40.4974i	0.117494	−	0.203505i	0.918774	+	0.394784i	$0.129181\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−54.1367	+	78.0959i	−0.266683	+	0.384709i
$204$		0			0
$205$	−38.0955	−	65.9833i	−0.185832	−	0.321870i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			29.5060i			0.141177i
$210$		0			0
$211$	−64.8732			−0.307456			−0.153728	−	0.988113i	$-0.549128\pi$
$211$	−64.8732			−0.307456			−0.153728	+	0.988113i	$0.549128\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−121.914	+	70.3873i	−0.567043	+	0.327383i
$216$		0			0
$217$	−12.5787	+	5.93784i	−0.0579662	+	0.0273633i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.521976	−	0.301363i	−0.00236188	−	0.00136363i
$222$		0			0
$223$	224.551			1.00696			0.503478	−	0.864008i	$-0.332054\pi$
$223$	224.551			1.00696			0.503478	+	0.864008i	$0.332054\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	351.802	+	203.113i	1.54979	+	0.894771i	0.998157	+	0.0606824i	$0.0193277\pi$
$227$	351.802	+	203.113i	1.54979	+	0.894771i	0.551631	+	0.834088i	$0.314006\pi$
$228$		0			0
$229$	185.758	+	321.742i	0.811171	+	1.40499i	0.912045	+	0.410090i	$0.134503\pi$
$229$	185.758	+	321.742i	0.811171	+	1.40499i	−0.100875	+	0.994899i	$0.532164\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	187.339	−	108.160i	0.804032	−	0.464208i	−0.0408473	−	0.999165i	$-0.513006\pi$
$233$	187.339	−	108.160i	0.804032	−	0.464208i	0.844879	+	0.534957i	$0.179672\pi$
$234$		0			0
$235$	−95.9917	+	166.263i	−0.408475	+	0.707500i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	0.556048i		−	0.00232656i	−0.999999	−	0.00116328i	$-0.999630\pi$
$239$		−	0.556048i		−	0.00232656i	0.999999	−	0.00116328i	$-0.000370284\pi$
$240$		0			0
$241$	117.405	−	203.352i	0.487160	−	0.843785i	−0.512731	−	0.858549i	$-0.671366\pi$
$241$	117.405	−	203.352i	0.487160	−	0.843785i	0.999891	+	0.0147639i	$0.00469968\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	100.126	+	82.4206i	0.408679	+	0.336411i
$246$		0			0
$247$	−18.2303	−	31.5757i	−0.0738067	−	0.127837i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	169.210i		−	0.674145i	−0.941479	−	0.337073i	$-0.890563\pi$
$251$		−	169.210i		−	0.674145i	0.941479	−	0.337073i	$-0.109437\pi$
$252$		0			0
$253$	45.0494			0.178061
$254$		0			0
$255$		0			0
$256$		0			0
$257$	303.857	−	175.432i	1.18232	−	0.682614i	0.225772	−	0.974180i	$-0.427510\pi$
$257$	303.857	−	175.432i	1.18232	−	0.682614i	0.956551	+	0.291566i	$0.0941763\pi$
$258$		0			0
$259$	26.0281	−	314.566i	0.100494	−	1.21454i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	306.035	+	176.689i	1.16363	+	0.671823i	0.952171	−	0.305564i	$-0.0988451\pi$
$263$	306.035	+	176.689i	1.16363	+	0.671823i	0.211459	+	0.977387i	$0.432178\pi$
$264$		0			0
$265$	−159.088			−0.600330
$266$		0			0
$267$		0			0
$268$		0			0
$269$	191.178	+	110.376i	0.710698	+	0.410321i	0.811319	−	0.584603i	$-0.198750\pi$
$269$	191.178	+	110.376i	0.710698	+	0.410321i	−0.100622	+	0.994925i	$0.532083\pi$
$270$		0			0
$271$	−171.040	−	296.249i	−0.631142	−	1.09317i	−0.987319	−	0.158752i	$-0.949253\pi$
$271$	−171.040	−	296.249i	−0.631142	−	1.09317i	0.356176	−	0.934419i	$-0.384080\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	34.8004	−	20.0920i	0.126547	−	0.0730619i
$276$		0			0
$277$	90.0492	−	155.970i	0.325088	−	0.563068i	−0.656443	−	0.754376i	$-0.727940\pi$
$277$	90.0492	−	155.970i	0.325088	−	0.563068i	0.981530	+	0.191308i	$0.0612729\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	531.477i		−	1.89138i	−0.325074	−	0.945688i	$-0.605389\pi$
$281$		−	531.477i		−	1.89138i	0.325074	−	0.945688i	$-0.394611\pi$
$282$		0			0
$283$	116.483	−	201.755i	0.411601	−	0.712914i	−0.583464	−	0.812139i	$-0.698303\pi$
$283$	116.483	−	201.755i	0.411601	−	0.712914i	0.995065	+	0.0992253i	$0.0316364\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−114.805	+	165.613i	−0.400016	+	0.577050i
$288$		0			0
$289$	−144.476	−	250.240i	−0.499917	−	0.865882i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			421.281i			1.43782i	0.695103	+	0.718910i	$0.255359\pi$
$293$			421.281i			1.43782i	−0.695103	+	0.718910i	$0.744641\pi$
$294$		0			0
$295$	−173.600			−0.588473
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−48.2095	+	27.8338i	−0.161236	+	0.0930896i
$300$		0			0
$301$	305.996	+	212.119i	1.01660	+	0.704714i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	172.942	+	99.8480i	0.567022	+	0.327370i
$306$		0			0
$307$	166.493			0.542323			0.271162	−	0.962534i	$-0.412592\pi$
$307$	166.493			0.542323			0.271162	+	0.962534i	$0.412592\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	139.712	+	80.6630i	0.449236	+	0.259366i	0.707507	−	0.706706i	$-0.249820\pi$
$311$	139.712	+	80.6630i	0.449236	+	0.259366i	−0.258272	+	0.966072i	$0.583153\pi$
$312$		0			0
$313$	109.316	+	189.341i	0.349253	+	0.604924i	0.986117	−	0.166052i	$-0.0531021\pi$
$313$	109.316	+	189.341i	0.349253	+	0.604924i	−0.636864	+	0.770976i	$0.719769\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−139.940	+	80.7944i	−0.441451	+	0.254872i	−0.704213	−	0.709989i	$-0.748700\pi$
$317$	−139.940	+	80.7944i	−0.441451	+	0.254872i	0.262762	+	0.964861i	$0.415367\pi$
$318$		0			0
$319$	−15.1567	+	26.2523i	−0.0475133	+	0.0822955i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			2.88618i			0.00893555i
$324$		0			0
$325$	−24.8277	+	43.0029i	−0.0763930	+	0.132317i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	506.039	+	41.8710i	1.53811	+	0.127267i
$330$		0			0
$331$	−176.578	−	305.841i	−0.533467	−	0.923992i	−0.999236	−	0.0390857i	$-0.987555\pi$
$331$	−176.578	−	305.841i	−0.533467	−	0.923992i	0.465769	−	0.884906i	$-0.345778\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			85.2822i			0.254574i
$336$		0			0
$337$	274.194			0.813632			0.406816	−	0.913510i	$-0.366639\pi$
$337$	274.194			0.813632			0.406816	+	0.913510i	$0.366639\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−3.84280	+	2.21864i	−0.0112692	+	0.00650627i
$342$		0			0
$343$	84.0830	−	332.534i	0.245140	−	0.969488i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−157.086	−	90.6939i	−0.452698	−	0.261366i	0.256271	−	0.966605i	$-0.417506\pi$
$347$	−157.086	−	90.6939i	−0.452698	−	0.261366i	−0.708969	+	0.705239i	$0.750839\pi$
$348$		0			0
$349$	−40.3342			−0.115571			−0.0577854	−	0.998329i	$-0.518404\pi$
$349$	−40.3342			−0.115571			−0.0577854	+	0.998329i	$0.518404\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	231.822	+	133.843i	0.656720	+	0.379158i	0.791026	−	0.611782i	$-0.209547\pi$
$353$	231.822	+	133.843i	0.656720	+	0.379158i	−0.134306	+	0.990940i	$0.542881\pi$
$354$		0			0
$355$	−22.8020	−	39.4942i	−0.0642310	−	0.111251i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	33.4009	−	19.2840i	0.0930387	−	0.0537159i	−0.452759	−	0.891633i	$-0.649560\pi$
$359$	33.4009	−	19.2840i	0.0930387	−	0.0537159i	0.545797	+	0.837917i	$0.316227\pi$
$360$		0			0
$361$	93.2034	−	161.433i	0.258181	−	0.447183i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			38.0508i			0.104249i
$366$		0			0
$367$	56.8588	−	98.4824i	0.154929	−	0.268344i	−0.778104	−	0.628135i	$-0.783819\pi$
$367$	56.8588	−	98.4824i	0.154929	−	0.268344i	0.933033	+	0.359791i	$0.117152\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	179.617	+	380.499i	0.484143	+	1.02560i
$372$		0			0
$373$	47.7009	+	82.6203i	0.127884	+	0.221502i	0.922857	−	0.385143i	$-0.125848\pi$
$373$	47.7009	+	82.6203i	0.127884	+	0.221502i	−0.794972	+	0.606646i	$0.792515\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	37.4584i		−	0.0993592i
$378$		0			0
$379$	−622.681			−1.64296			−0.821479	−	0.570239i	$-0.806851\pi$
$379$	−622.681			−1.64296			−0.821479	+	0.570239i	$0.806851\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−636.322	+	367.381i	−1.66142	+	0.959219i	−0.689376	+	0.724403i	$0.742115\pi$
$383$	−636.322	+	367.381i	−1.66142	+	0.959219i	−0.972040	+	0.234816i	$0.924551\pi$
$384$		0			0
$385$	34.0002	+	23.5692i	0.0883121	+	0.0612187i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−563.496	−	325.335i	−1.44858	−	0.836336i	−0.450179	−	0.892938i	$-0.648640\pi$
$389$	−563.496	−	325.335i	−1.44858	−	0.836336i	−0.998397	+	0.0566027i	$0.981973\pi$
$390$		0			0
$391$	4.40660			0.0112701
$392$		0			0
$393$		0			0
$394$		0			0
$395$	82.9601	+	47.8970i	0.210026	+	0.121258i
$396$		0			0
$397$	325.871	+	564.425i	0.820833	+	1.42172i	0.905063	+	0.425278i	$0.139824\pi$
$397$	325.871	+	564.425i	0.820833	+	1.42172i	−0.0842294	+	0.996446i	$0.526843\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	505.959	−	292.116i	1.26174	−	0.728468i	0.288332	−	0.957530i	$-0.406899\pi$
$401$	505.959	−	292.116i	1.26174	−	0.728468i	0.973412	+	0.229062i	$0.0735659\pi$
$402$		0			0
$403$	2.74157	−	4.74855i	0.00680291	−	0.0117830i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	100.691i		−	0.247399i
$408$		0			0
$409$	209.966	−	363.672i	0.513364	−	0.889173i	−0.486516	−	0.873672i	$-0.661732\pi$
$409$	209.966	−	363.672i	0.513364	−	0.889173i	0.999880	−	0.0155010i	$-0.00493431\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	196.002	+	415.208i	0.474580	+	1.00535i
$414$		0			0
$415$	−57.7815	−	100.080i	−0.139233	−	0.241158i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	37.2124i		−	0.0888123i	−0.999014	−	0.0444061i	$-0.985860\pi$
$419$		−	37.2124i		−	0.0888123i	0.999014	−	0.0444061i	$-0.0141396\pi$
$420$		0			0
$421$	189.949			0.451186			0.225593	−	0.974222i	$-0.427568\pi$
$421$	189.949			0.451186			0.225593	+	0.974222i	$0.427568\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	3.40407	−	1.96534i	0.00800958	−	0.00462433i
$426$		0			0
$427$	43.5531	−	526.368i	0.101998	−	1.23271i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	360.924	+	208.379i	0.837410	+	0.483479i	0.856383	−	0.516341i	$-0.172706\pi$
$431$	360.924	+	208.379i	0.837410	+	0.483479i	−0.0189728	+	0.999820i	$0.506040\pi$
$432$		0			0
$433$	−838.980			−1.93760			−0.968799	−	0.247846i	$-0.920277\pi$
$433$	−838.980			−1.93760			−0.968799	+	0.247846i	$0.920277\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	230.854	+	133.284i	0.528270	+	0.304997i
$438$		0			0
$439$	49.3253	+	85.4340i	0.112358	+	0.194610i	0.916721	−	0.399529i	$-0.130826\pi$
$439$	49.3253	+	85.4340i	0.112358	+	0.194610i	−0.804362	+	0.594139i	$0.797493\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	396.826	−	229.107i	0.895769	−	0.517172i	0.0199437	−	0.999801i	$-0.493651\pi$
$443$	396.826	−	229.107i	0.895769	−	0.517172i	0.875825	+	0.482629i	$0.160318\pi$
$444$		0			0
$445$	15.5986	−	27.0175i	0.0350530	−	0.0607135i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	476.631i		−	1.06154i	−0.847517	−	0.530769i	$-0.821903\pi$
$449$		−	476.631i		−	1.06154i	0.847517	−	0.530769i	$-0.178097\pi$
$450$		0			0
$451$	−32.1420	+	55.6716i	−0.0712683	+	0.123440i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−50.9475	−	4.21553i	−0.111972	−	0.00926490i
$456$		0			0
$457$	−208.528	−	361.181i	−0.456298	−	0.790331i	0.542464	−	0.840079i	$-0.317492\pi$
$457$	−208.528	−	361.181i	−0.456298	−	0.790331i	−0.998762	+	0.0497477i	$0.984158\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	13.7887i		−	0.0299105i	−0.999888	−	0.0149552i	$-0.995239\pi$
$461$		−	13.7887i		−	0.0299105i	0.999888	−	0.0149552i	$-0.00476058\pi$
$462$		0			0
$463$	599.660			1.29516			0.647581	−	0.761997i	$-0.275781\pi$
$463$	599.660			1.29516			0.647581	+	0.761997i	$0.275781\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−310.275	+	179.137i	−0.664401	+	0.383592i	−0.793952	−	0.607981i	$-0.791980\pi$
$467$	−310.275	+	179.137i	−0.664401	+	0.383592i	0.129551	+	0.991573i	$0.458646\pi$
$468$		0			0
$469$	203.974	−	96.2874i	0.434913	−	0.205304i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	102.862	+	59.3873i	0.217467	+	0.125555i
$474$		0			0
$475$	237.778			0.500585
$476$		0			0
$477$		0			0
$478$		0			0
$479$	573.217	+	330.947i	1.19669	+	0.690912i	0.959817	−	0.280627i	$-0.0905425\pi$
$479$	573.217	+	330.947i	1.19669	+	0.690912i	0.236878	+	0.971539i	$0.423876\pi$
$480$		0			0
$481$	62.2122	+	107.755i	0.129339	+	0.224022i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−257.981	+	148.945i	−0.531920	+	0.307104i
$486$		0			0
$487$	−289.632	+	501.658i	−0.594727	+	1.03010i	0.398858	+	0.917013i	$0.369407\pi$
$487$	−289.632	+	501.658i	−0.594727	+	1.03010i	−0.993585	+	0.113085i	$0.963927\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	717.779i		−	1.46187i	−0.682446	−	0.730936i	$-0.739084\pi$
$491$		−	717.779i		−	1.46187i	0.682446	−	0.730936i	$-0.260916\pi$
$492$		0			0
$493$	−1.48259	+	2.56792i	−0.00300728	+	0.00520875i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−68.7161	+	99.1276i	−0.138262	+	0.199452i
$498$		0			0
$499$	214.779	+	372.008i	0.430419	+	0.745508i	0.996909	−	0.0785608i	$-0.0250325\pi$
$499$	214.779	+	372.008i	0.430419	+	0.745508i	−0.566490	+	0.824068i	$0.691699\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	720.070i		−	1.43155i	−0.698330	−	0.715776i	$-0.746073\pi$
$503$		−	720.070i		−	1.43155i	0.698330	−	0.715776i	$-0.253927\pi$
$504$		0			0
$505$	−357.068			−0.707066
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−286.350	+	165.324i	−0.562573	+	0.324802i	−0.754177	−	0.656671i	$-0.771964\pi$
$509$	−286.350	+	165.324i	−0.562573	+	0.324802i	0.191605	+	0.981472i	$0.438631\pi$
$510$		0			0
$511$	91.0083	−	42.9611i	0.178098	−	0.0840726i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	80.0888	+	46.2393i	0.155512	+	0.0897850i
$516$		0			0
$517$	161.981			0.313309
$518$		0			0
$519$		0			0
$520$		0			0
$521$	261.221	+	150.816i	0.501385	+	0.289475i	0.729285	−	0.684210i	$-0.239853\pi$
$521$	261.221	+	150.816i	0.501385	+	0.289475i	−0.227901	+	0.973684i	$0.573186\pi$
$522$		0			0
$523$	265.175	+	459.297i	0.507028	+	0.878197i	0.999967	+	0.00813378i	$0.00258909\pi$
$523$	265.175	+	459.297i	0.507028	+	0.878197i	−0.492939	+	0.870064i	$0.664078\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−0.375891	+	0.217021i	−0.000713265	+	0.000411804i
$528$		0			0
$529$	−61.0039	+	105.662i	−0.115319	+	0.199739i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	79.4358i		−	0.149035i
$534$		0			0
$535$	34.5806	−	59.8954i	0.0646367	−	0.111954i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	17.9841	−	107.931i	0.0333657	−	0.200243i
$540$		0			0
$541$	−186.318	−	322.713i	−0.344396	−	0.596512i	0.640848	−	0.767668i	$-0.278583\pi$
$541$	−186.318	−	322.713i	−0.344396	−	0.596512i	−0.985244	+	0.171156i	$0.945250\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	21.3489i		−	0.0391722i
$546$		0			0
$547$	−146.893			−0.268543			−0.134272	−	0.990945i	$-0.542869\pi$
$547$	−146.893			−0.268543			−0.134272	+	0.990945i	$0.542869\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−155.340	+	89.6857i	−0.281924	+	0.162769i
$552$		0			0
$553$	20.8924	−	252.498i	0.0377801	−	0.456597i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−542.991	−	313.496i	−0.974850	−	0.562830i	−0.0741383	−	0.997248i	$-0.523621\pi$
$557$	−542.991	−	313.496i	−0.974850	−	0.562830i	−0.900711	+	0.434418i	$0.856954\pi$
$558$		0			0
$559$	−146.770			−0.262558
$560$		0			0
$561$		0			0
$562$		0			0
$563$	13.1507	+	7.59257i	0.0233583	+	0.0134859i	0.511634	−	0.859204i	$-0.329041\pi$
$563$	13.1507	+	7.59257i	0.0233583	+	0.0134859i	−0.488275	+	0.872690i	$0.662374\pi$
$564$		0			0
$565$	−39.9384	−	69.1754i	−0.0706875	−	0.122434i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−181.017	+	104.510i	−0.318132	+	0.183673i	−0.650560	−	0.759455i	$-0.725466\pi$
$569$	−181.017	+	104.510i	−0.318132	+	0.183673i	0.332428	+	0.943129i	$0.392132\pi$
$570$		0			0
$571$	−185.956	+	322.085i	−0.325667	+	0.564071i	−0.981647	−	0.190707i	$-0.938922\pi$
$571$	−185.956	+	322.085i	−0.325667	+	0.564071i	0.655980	+	0.754778i	$0.272255\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	363.037i		−	0.631368i
$576$		0			0
$577$	−22.9361	+	39.7264i	−0.0397506	+	0.0688500i	−0.885216	−	0.465180i	$-0.845990\pi$
$577$	−22.9361	+	39.7264i	−0.0397506	+	0.0688500i	0.845466	+	0.534030i	$0.179323\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−174.130	+	251.195i	−0.299708	+	0.432349i
$582$		0			0
$583$	67.1128	+	116.243i	0.115116	+	0.199387i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			71.7427i			0.122219i	0.998131	+	0.0611096i	$0.0194639\pi$
$587$			71.7427i			0.122219i	−0.998131	+	0.0611096i	$0.980536\pi$
$588$		0			0
$589$	−26.2563			−0.0445778
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−683.978	+	394.895i	−1.15342	+	0.665927i	−0.949718	−	0.313106i	$-0.898631\pi$
$593$	−683.978	+	394.895i	−1.15342	+	0.665927i	−0.203702	+	0.979033i	$0.565297\pi$
$594$		0			0
$595$	3.32579	+	2.30547i	0.00558957	+	0.00387474i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−327.619	−	189.151i	−0.546943	−	0.315777i	0.200945	−	0.979602i	$-0.435599\pi$
$599$	−327.619	−	189.151i	−0.546943	−	0.315777i	−0.747888	+	0.663825i	$0.768932\pi$
$600$		0			0
$601$	−228.767			−0.380644			−0.190322	−	0.981722i	$-0.560953\pi$
$601$	−228.767			−0.380644			−0.190322	+	0.981722i	$0.560953\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−265.911	−	153.524i	−0.439522	−	0.253758i
$606$		0			0
$607$	469.133	+	812.561i	0.772871	+	1.33865i	0.935983	+	0.352044i	$0.114513\pi$
$607$	469.133	+	812.561i	0.772871	+	1.33865i	−0.163113	+	0.986607i	$0.552153\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−173.343	+	100.080i	−0.283704	+	0.163797i
$612$		0			0
$613$	−362.770	+	628.336i	−0.591794	+	1.02502i	0.402197	+	0.915553i	$0.368247\pi$
$613$	−362.770	+	628.336i	−0.591794	+	1.02502i	−0.993991	+	0.109464i	$0.965087\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	203.830i		−	0.330356i	−0.986264	−	0.165178i	$-0.947180\pi$
$617$		−	203.830i		−	0.330356i	0.986264	−	0.165178i	$-0.0528199\pi$
$618$		0			0
$619$	−316.977	+	549.019i	−0.512078	+	0.886946i	0.487824	+	0.872942i	$0.337791\pi$
$619$	−316.977	+	549.019i	−0.512078	+	0.886946i	−0.999902	+	0.0140035i	$0.995542\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−82.2308	−	6.80400i	−0.131992	−	0.0109213i
$624$		0			0
$625$	−74.3548	−	128.786i	−0.118968	−	0.206058i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	9.84932i		−	0.0156587i
$630$		0			0
$631$	−916.941			−1.45316			−0.726578	−	0.687084i	$-0.758890\pi$
$631$	−916.941			−1.45316			−0.726578	+	0.687084i	$0.758890\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−162.416	+	93.7711i	−0.255774	+	0.147671i
$636$		0			0
$637$	47.4394	+	126.613i	0.0744732	+	0.198765i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−750.341	−	433.210i	−1.17058	−	0.675834i	−0.216763	−	0.976224i	$-0.569550\pi$
$641$	−750.341	−	433.210i	−1.17058	−	0.675834i	−0.953816	+	0.300390i	$0.902883\pi$
$642$		0			0
$643$	1111.18			1.72811			0.864056	−	0.503395i	$-0.167916\pi$
$643$	1111.18			1.72811			0.864056	+	0.503395i	$0.167916\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−341.420	−	197.119i	−0.527697	−	0.304666i	0.212381	−	0.977187i	$-0.431878\pi$
$647$	−341.420	−	197.119i	−0.527697	−	0.304666i	−0.740078	+	0.672521i	$0.765211\pi$
$648$		0			0
$649$	73.2349	+	126.847i	0.112843	+	0.195449i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−357.841	+	206.600i	−0.547995	+	0.316385i	−0.748313	−	0.663346i	$-0.769136\pi$
$653$	−357.841	+	206.600i	−0.547995	+	0.316385i	0.200318	+	0.979731i	$0.435803\pi$
$654$		0			0
$655$	−122.583	+	212.320i	−0.187150	+	0.324153i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	5.11039i		−	0.00775477i	−0.999992	−	0.00387739i	$-0.998766\pi$
$659$		−	5.11039i		−	0.00775477i	0.999992	−	0.00387739i	$-0.00123421\pi$
$660$		0			0
$661$	242.032	−	419.211i	0.366160	−	0.634208i	−0.622802	−	0.782380i	$-0.714006\pi$
$661$	242.032	−	419.211i	0.366160	−	0.634208i	0.988962	+	0.148172i	$0.0473390\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	104.500	+	221.373i	0.157144	+	0.332891i
$666$		0			0
$667$	136.931	+	237.172i	0.205294	+	0.355580i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	168.488i		−	0.251100i
$672$		0			0
$673$	−1267.99			−1.88409			−0.942043	−	0.335492i	$-0.891098\pi$
$673$	−1267.99			−1.88409			−0.942043	+	0.335492i	$0.891098\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1044.20	+	602.871i	−1.54240	+	0.890503i	−0.543710	+	0.839273i	$0.682981\pi$
$677$	−1044.20	+	602.871i	−1.54240	+	0.890503i	−0.998687	+	0.0512296i	$0.983686\pi$
$678$		0			0
$679$	647.514	+	448.862i	0.953628	+	0.661063i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1111.53	−	641.742i	−1.62742	−	0.939593i	−0.984858	−	0.173363i	$-0.944537\pi$
$683$	−1111.53	−	641.742i	−1.62742	−	0.939593i	−0.642566	−	0.766231i	$-0.722130\pi$
$684$		0			0
$685$	−546.351			−0.797593
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−143.641	−	82.9314i	−0.208478	−	0.120365i
$690$		0			0
$691$	127.035	+	220.032i	0.183843	+	0.318425i	0.943186	−	0.332265i	$-0.107813\pi$
$691$	127.035	+	220.032i	0.183843	+	0.318425i	−0.759343	+	0.650690i	$0.774480\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	444.668	−	256.729i	0.639810	−	0.369394i
$696$		0			0
$697$	−3.14403	+	5.44563i	−0.00451081	+	0.00781295i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	908.558i		−	1.29609i	−0.761603	−	0.648044i	$-0.775587\pi$
$701$		−	908.558i		−	1.29609i	0.761603	−	0.648044i	$-0.224413\pi$
$702$		0			0
$703$	297.906	−	515.989i	0.423764	−	0.733981i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	403.146	+	854.021i	0.570221	+	1.20795i
$708$		0			0
$709$	99.1393	+	171.714i	0.139830	+	0.242192i	0.927432	−	0.373992i	$-0.122011\pi$
$709$	99.1393	+	171.714i	0.139830	+	0.242192i	−0.787602	+	0.616184i	$0.788678\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			40.0879i			0.0562243i
$714$		0			0
$715$	−16.3081			−0.0228085
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−649.862	+	375.198i	−0.903841	+	0.521833i	−0.878444	−	0.477845i	$-0.841418\pi$
$719$	−649.862	+	375.198i	−0.903841	+	0.521833i	−0.0253967	+	0.999677i	$0.508085\pi$
$720$		0			0
$721$	20.1693	−	243.759i	0.0279740	−	0.338085i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	211.557	+	122.143i	0.291803	+	0.168473i
$726$		0			0
$727$	−1338.46			−1.84108			−0.920539	−	0.390651i	$-0.872250\pi$
$727$	−1338.46			−1.84108			−0.920539	+	0.390651i	$0.872250\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	10.0616	+	5.80908i	0.0137642	+	0.00794676i
$732$		0			0
$733$	214.224	+	371.047i	0.292256	+	0.506203i	0.974343	−	0.225069i	$-0.0722606\pi$
$733$	214.224	+	371.047i	0.292256	+	0.506203i	−0.682087	+	0.731271i	$0.738927\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	62.3144	−	35.9773i	0.0845515	−	0.0488158i
$738$		0			0
$739$	−82.4864	+	142.871i	−0.111619	+	0.193330i	−0.916423	−	0.400211i	$-0.868937\pi$
$739$	−82.4864	+	142.871i	−0.111619	+	0.193330i	0.804804	+	0.593540i	$0.202270\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	1326.94i		−	1.78593i	−0.450131	−	0.892963i	$-0.648623\pi$
$743$		−	1326.94i		−	1.78593i	0.450131	−	0.892963i	$-0.351377\pi$
$744$		0			0
$745$	−214.988	+	372.370i	−0.288574	+	0.499826i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−182.298	−	15.0839i	−0.243389	−	0.0201387i
$750$		0			0
$751$	−266.641	−	461.835i	−0.355048	−	0.614960i	0.632079	−	0.774904i	$-0.282202\pi$
$751$	−266.641	−	461.835i	−0.355048	−	0.614960i	−0.987126	+	0.159944i	$0.948869\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			612.437i			0.811175i
$756$		0			0
$757$	−321.071			−0.424136			−0.212068	−	0.977255i	$-0.568020\pi$
$757$	−321.071			−0.424136			−0.212068	+	0.977255i	$0.568020\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	100.748	−	58.1666i	0.132388	−	0.0764344i	−0.432343	−	0.901709i	$-0.642313\pi$
$761$	100.748	−	58.1666i	0.132388	−	0.0764344i	0.564731	+	0.825275i	$0.308980\pi$
$762$		0			0
$763$	−51.0613	+	24.1038i	−0.0669218	+	0.0315909i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−156.744	−	90.4964i	−0.204360	−	0.117988i
$768$		0			0
$769$	1219.26			1.58551			0.792755	−	0.609541i	$-0.208646\pi$
$769$	1219.26			1.58551			0.792755	+	0.609541i	$0.208646\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	326.955	+	188.768i	0.422969	+	0.244202i	0.696347	−	0.717705i	$-0.254807\pi$
$773$	326.955	+	188.768i	0.422969	+	0.244202i	−0.273378	+	0.961907i	$0.588141\pi$
$774$		0			0
$775$	17.8792	+	30.9677i	0.0230699	+	0.0399583i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−329.421	+	190.191i	−0.422877	+	0.244148i
$780$		0			0
$781$	−19.2386	+	33.3221i	−0.0246332	+	0.0426660i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	276.487i		−	0.352213i
$786$		0			0
$787$	215.504	−	373.264i	0.273830	−	0.474287i	−0.696010	−	0.718033i	$-0.745043\pi$
$787$	215.504	−	373.264i	0.273830	−	0.474287i	0.969839	+	0.243746i	$0.0783762\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−120.358	+	173.625i	−0.152160	+	0.219501i
$792$		0			0
$793$	104.100	+	180.307i	0.131274	+	0.227373i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	548.830i		−	0.688619i	−0.938856	−	0.344310i	$-0.888113\pi$
$797$		−	548.830i		−	0.688619i	0.938856	−	0.344310i	$-0.111887\pi$
$798$		0			0
$799$	15.8445			0.0198304
$800$		0			0
$801$		0			0
$802$		0			0
$803$	27.8032	−	16.0522i	0.0346241	−	0.0199902i
$804$		0			0
$805$	337.990	−	159.550i	0.419863	−	0.198199i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−58.8905	−	34.0004i	−0.0727942	−	0.0420277i	0.463161	−	0.886274i	$-0.346715\pi$
$809$	−58.8905	−	34.0004i	−0.0727942	−	0.0420277i	−0.535955	+	0.844246i	$0.680048\pi$
$810$		0			0
$811$	−1515.41			−1.86857			−0.934285	−	0.356528i	$-0.883960\pi$
$811$	−1515.41			−1.86857			−0.934285	+	0.356528i	$0.883960\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	472.441	+	272.764i	0.579682	+	0.334679i
$816$		0			0
$817$	351.407	+	608.656i	0.430119	+	0.744988i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1255.76	−	725.013i	1.52955	−	0.883086i	0.530169	−	0.847892i	$-0.322129\pi$
$821$	1255.76	−	725.013i	1.52955	−	0.883086i	0.999381	−	0.0351935i	$-0.0112048\pi$
$822$		0			0
$823$	396.116	−	686.093i	0.481308	−	0.833649i	−0.518462	−	0.855100i	$-0.673495\pi$
$823$	396.116	−	686.093i	0.481308	−	0.833649i	0.999770	+	0.0214514i	$0.00682870\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			58.8654i			0.0711794i	0.999366	+	0.0355897i	$0.0113309\pi$
$827$			58.8654i			0.0711794i	−0.999366	+	0.0355897i	$0.988669\pi$
$828$		0			0
$829$	−168.454	+	291.770i	−0.203201	+	0.351955i	−0.949558	−	0.313591i	$-0.898468\pi$
$829$	−168.454	+	291.770i	−0.203201	+	0.351955i	0.746357	+	0.665546i	$0.231801\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.75915	−	10.5575i	0.00211182	−	0.0126740i
$834$		0			0
$835$	−317.014	−	549.085i	−0.379658	−	0.657587i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	399.017i		−	0.475586i	−0.971316	−	0.237793i	$-0.923576\pi$
$839$		−	399.017i		−	0.475586i	0.971316	−	0.237793i	$-0.0764240\pi$
$840$		0			0
$841$	656.719			0.780879
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−369.907	+	213.566i	−0.437760	+	0.252741i
$846$		0			0
$847$	−66.9661	+	809.329i	−0.0790627	+	0.955524i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−787.806	−	454.840i	−0.925742	−	0.534477i
$852$		0			0
$853$	34.9637			0.0409891			0.0204946	−	0.999790i	$-0.493476\pi$
$853$	34.9637			0.0409891			0.0204946	+	0.999790i	$0.493476\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	196.472	+	113.433i	0.229256	+	0.132361i	0.610229	−	0.792225i	$-0.291078\pi$
$857$	196.472	+	113.433i	0.229256	+	0.132361i	−0.380973	+	0.924586i	$0.624411\pi$
$858$		0			0
$859$	−41.7024	−	72.2307i	−0.0485476	−	0.0840870i	0.840730	−	0.541454i	$-0.182126\pi$
$859$	−41.7024	−	72.2307i	−0.0485476	−	0.0840870i	−0.889278	+	0.457367i	$0.848793\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−221.552	+	127.913i	−0.256723	+	0.148219i	−0.622839	−	0.782350i	$-0.714021\pi$
$863$	−221.552	+	127.913i	−0.256723	+	0.148219i	0.366116	+	0.930569i	$0.380687\pi$
$864$		0			0
$865$	−414.313	+	717.612i	−0.478975	+	0.829609i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	80.8236i		−	0.0930076i
$870$		0			0
$871$	−44.4571	+	77.0020i	−0.0510415	+	0.0884064i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	453.805	−	654.644i	0.518634	−	0.748165i
$876$		0			0
$877$	428.595	+	742.348i	0.488706	+	0.846463i	0.999916	−	0.0129926i	$-0.00413579\pi$
$877$	428.595	+	742.348i	0.488706	+	0.846463i	−0.511210	+	0.859456i	$0.670802\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1047.82i		−	1.18935i	−0.803965	−	0.594677i	$-0.797280\pi$
$881$		−	1047.82i		−	1.18935i	0.803965	−	0.594677i	$-0.202720\pi$
$882$		0			0
$883$	−30.0703			−0.0340547			−0.0170273	−	0.999855i	$-0.505420\pi$
$883$	−30.0703			−0.0340547			−0.0170273	+	0.999855i	$0.505420\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1087.85	+	628.072i	−1.22644	+	0.708086i	−0.966284	−	0.257480i	$-0.917108\pi$
$887$	−1087.85	+	628.072i	−1.22644	+	0.708086i	−0.260157	+	0.965566i	$0.583774\pi$
$888$		0			0
$889$	407.653	+	282.589i	0.458552	+	0.317872i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	830.064	+	479.237i	0.929523	+	0.536660i
$894$		0			0
$895$	−2.55197			−0.00285137
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−23.3610	−	13.4875i	−0.0259855	−	0.0150027i
$900$		0			0
$901$	6.56477	+	11.3705i	0.00728610	+	0.0126199i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−417.183	+	240.861i	−0.460976	+	0.266145i
$906$		0			0
$907$	−108.351	+	187.669i	−0.119461	+	0.206912i	−0.919554	−	0.392963i	$-0.871450\pi$
$907$	−108.351	+	187.669i	−0.119461	+	0.206912i	0.800093	+	0.599876i	$0.204783\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1212.23i			1.33065i	0.746552	+	0.665327i	$0.231708\pi$
$911$			1212.23i			1.33065i	−0.746552	+	0.665327i	$0.768292\pi$
$912$		0			0
$913$	−48.7515	+	84.4401i	−0.0533971	+	0.0924864i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	646.220	+	53.4700i	0.704711	+	0.0583097i
$918$		0			0
$919$	−406.939	−	704.838i	−0.442806	−	0.766962i	0.555091	−	0.831790i	$-0.312684\pi$
$919$	−406.939	−	704.838i	−0.442806	−	0.766962i	−0.997896	+	0.0648277i	$0.979350\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	47.5462i		−	0.0515127i
$924$		0			0
$925$	−811.434			−0.877226
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−751.541	+	433.902i	−0.808978	+	0.467064i	−0.846601	−	0.532228i	$-0.821355\pi$
$929$	−751.541	+	433.902i	−0.808978	+	0.467064i	0.0376226	+	0.999292i	$0.488022\pi$
$930$		0			0
$931$	411.484	−	499.879i	0.441980	−	0.536927i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1.11798	+	0.645466i	0.00119570	+	0.000690338i
$936$		0			0
$937$	1041.39			1.11141			0.555705	−	0.831379i	$-0.312448\pi$
$937$	1041.39			1.11141			0.555705	+	0.831379i	$0.312448\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−293.234	−	169.299i	−0.311619	−	0.179914i	0.336032	−	0.941851i	$-0.390915\pi$
$941$	−293.234	−	169.299i	−0.311619	−	0.179914i	−0.647651	+	0.761937i	$0.724248\pi$
$942$		0			0
$943$	290.382	+	502.957i	0.307934	+	0.533358i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−751.279	+	433.751i	−0.793325	+	0.458027i	−0.841132	−	0.540830i	$-0.818110\pi$
$947$	−751.279	+	433.751i	−0.793325	+	0.458027i	0.0478066	+	0.998857i	$0.484777\pi$
$948$		0			0
$949$	−19.8357	+	34.3564i	−0.0209017	+	0.0362027i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			521.238i			0.546945i	0.961880	+	0.273472i	$0.0881722\pi$
$953$			521.238i			0.546945i	−0.961880	+	0.273472i	$0.911828\pi$
$954$		0			0
$955$	344.321	−	596.382i	0.360546	−	0.624484i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	616.855	+	1306.74i	0.643227	+	1.36261i
$960$		0			0
$961$	478.526	+	828.831i	0.497946	+	0.862467i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	677.848i		−	0.702434i
$966$		0			0
$967$	−4.51741			−0.00467157			−0.00233579	−	0.999997i	$-0.500744\pi$
$967$	−4.51741			−0.00467157			−0.00233579	+	0.999997i	$0.500744\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1117.45	−	645.161i	1.15083	−	0.664429i	0.201737	−	0.979440i	$-0.435341\pi$
$971$	1117.45	−	645.161i	1.15083	−	0.664429i	0.949088	+	0.315010i	$0.102008\pi$
$972$		0			0
$973$	−1116.08	−	773.678i	−1.14705	−	0.795147i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	108.427	+	62.6002i	0.110979	+	0.0640739i	0.554462	−	0.832209i	$-0.312924\pi$
$977$	108.427	+	62.6002i	0.110979	+	0.0640739i	−0.443483	+	0.896283i	$0.646257\pi$
$978$		0			0
$979$	−26.3217			−0.0268863
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−660.866	−	381.551i	−0.672295	−	0.388150i	0.124650	−	0.992201i	$-0.460219\pi$
$983$	−660.866	−	381.551i	−0.672295	−	0.388150i	−0.796946	+	0.604051i	$0.793552\pi$
$984$		0			0
$985$	−11.7095	−	20.2814i	−0.0118878	−	0.0205902i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	929.289	−	536.526i	0.939625	−	0.542493i
$990$		0			0
$991$	416.369	−	721.173i	0.420151	−	0.727722i	−0.575803	−	0.817588i	$-0.695311\pi$
$991$	416.369	−	721.173i	0.420151	−	0.727722i	0.995954	+	0.0898660i	$0.0286439\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	123.764i		−	0.124386i
$996$		0			0
$997$	177.689	−	307.767i	0.178224	−	0.308693i	−0.763048	−	0.646341i	$-0.776298\pi$
$997$	177.689	−	307.767i	0.178224	−	0.308693i	0.941272	+	0.337649i	$0.109632\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.3.dc.g.305.5		16
3.2	odd	2	inner	1008.3.dc.g.305.4		16
4.3	odd	2		504.3.cu.a.305.5	yes	16
7.2	even	3	inner	1008.3.dc.g.737.4		16
12.11	even	2		504.3.cu.a.305.4	yes	16
21.2	odd	6	inner	1008.3.dc.g.737.5		16
28.3	even	6		3528.3.d.j.1961.4		8
28.11	odd	6		3528.3.d.g.1961.5		8
28.23	odd	6		504.3.cu.a.233.4	✓	16
84.11	even	6		3528.3.d.g.1961.4		8
84.23	even	6		504.3.cu.a.233.5	yes	16
84.59	odd	6		3528.3.d.j.1961.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.3.cu.a.233.4	✓	16	28.23	odd	6
504.3.cu.a.233.5	yes	16	84.23	even	6
504.3.cu.a.305.4	yes	16	12.11	even	2
504.3.cu.a.305.5	yes	16	4.3	odd	2
1008.3.dc.g.305.4		16	3.2	odd	2	inner
1008.3.dc.g.305.5		16	1.1	even	1	trivial
1008.3.dc.g.737.4		16	7.2	even	3	inner
1008.3.dc.g.737.5		16	21.2	odd	6	inner
3528.3.d.g.1961.4		8	84.11	even	6
3528.3.d.g.1961.5		8	28.11	odd	6
3528.3.d.j.1961.4		8	28.3	even	6
3528.3.d.j.1961.5		8	84.59	odd	6

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:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1008.dc (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.29207 - 1.32333i) q^{5} +(-5.75292 - 3.98797i) q^{7} +(-1.93387 - 1.11652i) q^{11} +2.75937 q^{13} +(-0.189165 - 0.109214i) q^{17} +(-6.60668 - 11.4431i) q^{19} +(-17.4712 + 10.0870i) q^{23} +(-8.99762 + 15.5843i) q^{25} -13.5750i q^{29} +(0.993552 - 1.72088i) q^{31} +(-18.4635 - 1.52772i) q^{35} +(22.5458 + 39.0505i) q^{37} -28.7877i q^{41} -53.1897 q^{43} +(-62.8200 + 36.2691i) q^{47} +(17.1921 + 45.8850i) q^{49} +(-52.0559 - 30.0545i) q^{53} -5.91007 q^{55} +(-56.8045 - 32.7961i) q^{59} +(37.7262 + 65.3436i) q^{61} +(6.32465 - 3.65154i) q^{65} +(-16.1113 + 27.9057i) q^{67} -17.2308i q^{71} +(-7.18849 + 12.4508i) q^{73} +(6.67274 + 14.1355i) q^{77} +(18.0972 + 31.3453i) q^{79} -43.6639i q^{83} -0.578105 q^{85} +(10.2082 - 5.89370i) q^{89} +(-15.8744 - 11.0043i) q^{91} +(-30.2859 - 17.4856i) q^{95} -112.554 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 24 q^{7} - 24 q^{13} + 12 q^{19} + 92 q^{25} - 32 q^{31} - 132 q^{37} - 24 q^{43} + 64 q^{49} + 440 q^{55} + 184 q^{61} - 332 q^{67} + 188 q^{73} + 112 q^{79} + 256 q^{85} + 252 q^{91} - 1032 q^{97}+O(q^{100}) \) 16 * q + 24 * q^7 - 24 * q^13 + 12 * q^19 + 92 * q^25 - 32 * q^31 - 132 * q^37 - 24 * q^43 + 64 * q^49 + 440 * q^55 + 184 * q^61 - 332 * q^67 + 188 * q^73 + 112 * q^79 + 256 * q^85 + 252 * q^91 - 1032 * q^97

Introduction

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