LMFDB - Embedded newform 1296.3.o.be.271.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,3,Mod(271,1296)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 2]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1296.271");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1296.o (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$35.3134422611$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			271.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1296.271
Dual form			1296.3.o.be.703.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.13397 - 3.69615i) q^{5} +(2.19615 - 1.26795i) q^{7} +O(q^{10})$	$q+(2.13397 - 3.69615i) q^{5} +(2.19615 - 1.26795i) q^{7} +(3.80385 - 2.19615i) q^{11} +(1.69615 - 2.93782i) q^{13} -3.33975 q^{17} -26.5359i q^{19} +(-24.5885 - 14.1962i) q^{23} +(3.39230 + 5.87564i) q^{25} +(-2.13397 - 3.69615i) q^{29} +(7.60770 + 4.39230i) q^{31} -10.8231i q^{35} +36.1769 q^{37} +(20.7846 - 36.0000i) q^{41} +(-21.8038 + 12.5885i) q^{43} +(-20.7846 + 12.0000i) q^{47} +(-21.2846 + 36.8660i) q^{49} +56.7846 q^{53} -18.7461i q^{55} +(-83.1384 - 48.0000i) q^{59} +(-34.4808 - 59.7224i) q^{61} +(-7.23909 - 12.5385i) q^{65} +(66.5885 + 38.4449i) q^{67} -135.531i q^{71} -34.1384 q^{73} +(5.56922 - 9.64617i) q^{77} +(-94.9808 + 54.8372i) q^{79} +(-100.392 + 57.9615i) q^{83} +(-7.12693 + 12.3442i) q^{85} +29.6936 q^{89} -8.60254i q^{91} +(-98.0807 - 56.6269i) q^{95} +(-19.7846 - 34.2679i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{5} - 12 q^{7}+O(q^{10}) $ 4 * q + 12 * q^5 - 12 * q^7	$ 4 q + 12 q^{5} - 12 q^{7} + 36 q^{11} - 14 q^{13} - 48 q^{17} - 36 q^{23} - 28 q^{25} - 12 q^{29} + 72 q^{31} + 20 q^{37} - 108 q^{43} - 2 q^{49} + 144 q^{53} - 34 q^{61} + 120 q^{65} + 204 q^{67} + 196 q^{73} - 144 q^{77} - 276 q^{79} - 360 q^{83} - 174 q^{85} - 96 q^{89} + 252 q^{95} + 4 q^{97}+O(q^{100}) $ 4 * q + 12 * q^5 - 12 * q^7 + 36 * q^11 - 14 * q^13 - 48 * q^17 - 36 * q^23 - 28 * q^25 - 12 * q^29 + 72 * q^31 + 20 * q^37 - 108 * q^43 - 2 * q^49 + 144 * q^53 - 34 * q^61 + 120 * q^65 + 204 * q^67 + 196 * q^73 - 144 * q^77 - 276 * q^79 - 360 * q^83 - 174 * q^85 - 96 * q^89 + 252 * q^95 + 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.13397	−	3.69615i	0.426795	−	0.739230i	−0.569791	−	0.821789i	$-0.692976\pi$
$5$	2.13397	−	3.69615i	0.426795	−	0.739230i	0.996586	+	0.0825590i	$0.0263093\pi$
$6$		0			0
$7$	2.19615	−	1.26795i	0.313736	−	0.181136i	−0.334861	−	0.942268i	$-0.608689\pi$
$7$	2.19615	−	1.26795i	0.313736	−	0.181136i	0.648597	+	0.761132i	$0.275356\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.80385	−	2.19615i	0.345804	−	0.199650i	−0.317031	−	0.948415i	$-0.602686\pi$
$11$	3.80385	−	2.19615i	0.345804	−	0.199650i	0.662836	+	0.748765i	$0.269353\pi$
$12$		0			0
$13$	1.69615	−	2.93782i	0.130473	−	0.225986i	−0.793386	−	0.608719i	$-0.791684\pi$
$13$	1.69615	−	2.93782i	0.130473	−	0.225986i	0.923859	+	0.382733i	$0.125017\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.33975			−0.196456			−0.0982278	−	0.995164i	$-0.531317\pi$
$17$	−3.33975			−0.196456			−0.0982278	+	0.995164i	$0.531317\pi$
$18$		0			0
$19$		−	26.5359i		−	1.39663i	−0.715792	−	0.698313i	$-0.753934\pi$
$19$		−	26.5359i		−	1.39663i	0.715792	−	0.698313i	$-0.246066\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−24.5885	−	14.1962i	−1.06906	−	0.617224i	−0.141138	−	0.989990i	$-0.545076\pi$
$23$	−24.5885	−	14.1962i	−1.06906	−	0.617224i	−0.927925	+	0.372766i	$0.878409\pi$
$24$		0			0
$25$	3.39230	+	5.87564i	0.135692	+	0.235026i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.13397	−	3.69615i	−0.0735853	−	0.127454i	0.826885	−	0.562371i	$-0.190111\pi$
$29$	−2.13397	−	3.69615i	−0.0735853	−	0.127454i	−0.900470	+	0.434918i	$0.856777\pi$
$30$		0			0
$31$	7.60770	+	4.39230i	0.245410	+	0.141687i	0.617660	−	0.786445i	$-0.288081\pi$
$31$	7.60770	+	4.39230i	0.245410	+	0.141687i	−0.372251	+	0.928132i	$0.621414\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	10.8231i		−	0.309231i
$36$		0			0
$37$	36.1769			0.977754			0.488877	−	0.872353i	$-0.337407\pi$
$37$	36.1769			0.977754			0.488877	+	0.872353i	$0.337407\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	20.7846	−	36.0000i	0.506942	−	0.878049i	−0.493026	−	0.870015i	$-0.664109\pi$
$41$	20.7846	−	36.0000i	0.506942	−	0.878049i	0.999968	−	0.00803422i	$-0.00255740\pi$
$42$		0			0
$43$	−21.8038	+	12.5885i	−0.507066	+	0.292755i	−0.731627	−	0.681705i	$-0.761239\pi$
$43$	−21.8038	+	12.5885i	−0.507066	+	0.292755i	0.224561	+	0.974460i	$0.427905\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−20.7846	+	12.0000i	−0.442226	+	0.255319i	−0.704541	−	0.709663i	$-0.748847\pi$
$47$	−20.7846	+	12.0000i	−0.442226	+	0.255319i	0.262316	+	0.964982i	$0.415514\pi$
$48$		0			0
$49$	−21.2846	+	36.8660i	−0.434380	+	0.752368i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	56.7846			1.07141			0.535704	−	0.844406i	$-0.320046\pi$
$53$	56.7846			1.07141			0.535704	+	0.844406i	$0.320046\pi$
$54$		0			0
$55$		−	18.7461i		−	0.340839i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−83.1384	−	48.0000i	−1.40913	−	0.813559i	−0.413822	−	0.910358i	$-0.635806\pi$
$59$	−83.1384	−	48.0000i	−1.40913	−	0.813559i	−0.995304	+	0.0967985i	$0.969140\pi$
$60$		0			0
$61$	−34.4808	−	59.7224i	−0.565258	−	0.979056i	−0.997026	−	0.0770712i	$-0.975443\pi$
$61$	−34.4808	−	59.7224i	−0.565258	−	0.979056i	0.431767	−	0.901985i	$-0.357890\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−7.23909	−	12.5385i	−0.111371	−	0.192900i
$66$		0			0
$67$	66.5885	+	38.4449i	0.993858	+	0.573804i	0.906425	−	0.422367i	$-0.138800\pi$
$67$	66.5885	+	38.4449i	0.993858	+	0.573804i	0.0874324	+	0.996170i	$0.472134\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	135.531i		−	1.90888i	−0.298397	−	0.954442i	$-0.596452\pi$
$71$		−	135.531i		−	1.90888i	0.298397	−	0.954442i	$-0.403548\pi$
$72$		0			0
$73$	−34.1384			−0.467650			−0.233825	−	0.972279i	$-0.575124\pi$
$73$	−34.1384			−0.467650			−0.233825	+	0.972279i	$0.575124\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	5.56922	−	9.64617i	0.0723275	−	0.125275i
$78$		0			0
$79$	−94.9808	+	54.8372i	−1.20229	+	0.694141i	−0.961063	−	0.276328i	$-0.910882\pi$
$79$	−94.9808	+	54.8372i	−1.20229	+	0.694141i	−0.241225	+	0.970469i	$0.577549\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−100.392	+	57.9615i	−1.20955	+	0.698332i	−0.962660	−	0.270713i	$-0.912741\pi$
$83$	−100.392	+	57.9615i	−1.20955	+	0.698332i	−0.246886	+	0.969045i	$0.579407\pi$
$84$		0			0
$85$	−7.12693	+	12.3442i	−0.0838463	+	0.145226i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	29.6936			0.333636			0.166818	−	0.985988i	$-0.446651\pi$
$89$	29.6936			0.333636			0.166818	+	0.985988i	$0.446651\pi$
$90$		0			0
$91$		−	8.60254i		−	0.0945334i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−98.0807	−	56.6269i	−1.03243	−	0.596073i
$96$		0			0
$97$	−19.7846	−	34.2679i	−0.203965	−	0.353278i	0.745837	−	0.666128i	$-0.232050\pi$
$97$	−19.7846	−	34.2679i	−0.203965	−	0.353278i	−0.949802	+	0.312850i	$0.898716\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−22.8231	−	39.5307i	−0.225971	−	0.391394i	0.730639	−	0.682764i	$-0.239222\pi$
$101$	−22.8231	−	39.5307i	−0.225971	−	0.391394i	−0.956610	+	0.291370i	$0.905889\pi$
$102$		0			0
$103$	97.1769	+	56.1051i	0.943465	+	0.544710i	0.891045	−	0.453915i	$-0.149973\pi$
$103$	97.1769	+	56.1051i	0.943465	+	0.544710i	0.0524203	+	0.998625i	$0.483306\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	111.215i		−	1.03940i	−0.854350	−	0.519698i	$-0.826044\pi$
$107$		−	111.215i		−	1.03940i	0.854350	−	0.519698i	$-0.173956\pi$
$108$		0			0
$109$	101.392			0.930205			0.465102	−	0.885257i	$-0.346018\pi$
$109$	101.392			0.930205			0.465102	+	0.885257i	$0.346018\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	107.631	−	186.423i	0.952490	−	1.64976i	0.212481	−	0.977165i	$-0.431846\pi$
$113$	107.631	−	186.423i	0.952490	−	1.64976i	0.740010	−	0.672596i	$-0.234821\pi$
$114$		0			0
$115$	−104.942	+	60.5885i	−0.912542	+	0.526856i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−7.33459	+	4.23463i	−0.0616352	+	0.0355851i
$120$		0			0
$121$	−50.8538	+	88.0814i	−0.420280	+	0.727946i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	135.655			1.08524
$126$		0			0
$127$		−	129.962i		−	1.02332i	−0.859188	−	0.511660i	$-0.829031\pi$
$127$		−	129.962i		−	1.02332i	0.859188	−	0.511660i	$-0.170969\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−134.081	−	77.4115i	−1.02352	−	0.590928i	−0.108396	−	0.994108i	$-0.534572\pi$
$131$	−134.081	−	77.4115i	−1.02352	−	0.590928i	−0.915121	+	0.403180i	$0.867905\pi$
$132$		0			0
$133$	−33.6462	−	58.2769i	−0.252979	−	0.438172i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−79.2391	−	137.246i	−0.578388	−	1.00180i	−0.995664	−	0.0930172i	$-0.970349\pi$
$137$	−79.2391	−	137.246i	−0.578388	−	1.00180i	0.417277	−	0.908779i	$-0.362985\pi$
$138$		0			0
$139$	40.3923	+	23.3205i	0.290592	+	0.167773i	0.638209	−	0.769863i	$-0.279676\pi$
$139$	40.3923	+	23.3205i	0.290592	+	0.167773i	−0.347617	+	0.937637i	$0.613009\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	14.9000i		−	0.104196i
$144$		0			0
$145$	−18.2154			−0.125623
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−9.74167	+	16.8731i	−0.0653803	+	0.113242i	−0.896863	−	0.442309i	$-0.854159\pi$
$149$	−9.74167	+	16.8731i	−0.0653803	+	0.113242i	0.831482	+	0.555551i	$0.187493\pi$
$150$		0			0
$151$	15.5307	−	8.96668i	0.102853	−	0.0593820i	−0.447691	−	0.894188i	$-0.647754\pi$
$151$	15.5307	−	8.96668i	0.102853	−	0.0593820i	0.550544	+	0.834806i	$0.314420\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	32.4693	−	18.7461i	0.209479	−	0.120943i
$156$		0			0
$157$	−6.95002	+	12.0378i	−0.0442676	+	0.0766738i	−0.887310	−	0.461173i	$-0.847429\pi$
$157$	−6.95002	+	12.0378i	−0.0442676	+	0.0766738i	0.843043	+	0.537847i	$0.180762\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−72.0000			−0.447205
$162$		0			0
$163$		−	216.995i		−	1.33126i	−0.746283	−	0.665628i	$-0.768164\pi$
$163$		−	216.995i		−	1.33126i	0.746283	−	0.665628i	$-0.231836\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−261.373	−	150.904i	−1.56511	−	0.903616i	−0.996726	−	0.0808508i	$-0.974236\pi$
$167$	−261.373	−	150.904i	−1.56511	−	0.903616i	−0.568382	−	0.822765i	$-0.692430\pi$
$168$		0			0
$169$	78.7461	+	136.392i	0.465953	+	0.807055i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	119.043	+	206.188i	0.688109	+	1.19184i	0.972449	+	0.233117i	$0.0748925\pi$
$173$	119.043	+	206.188i	0.688109	+	1.19184i	−0.284339	+	0.958724i	$0.591774\pi$
$174$		0			0
$175$	14.9000	+	8.60254i	0.0851431	+	0.0491574i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	142.277i		−	0.794843i	−0.917636	−	0.397421i	$-0.869905\pi$
$179$		−	142.277i		−	0.794843i	0.917636	−	0.397421i	$-0.130095\pi$
$180$		0			0
$181$	201.138			1.11126			0.555631	−	0.831429i	$-0.312477\pi$
$181$	201.138			1.11126			0.555631	+	0.831429i	$0.312477\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	77.2006	−	133.715i	0.417301	−	0.722786i
$186$		0			0
$187$	−12.7039	+	7.33459i	−0.0679352	+	0.0392224i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−91.0192	+	52.5500i	−0.476541	+	0.275131i	−0.718974	−	0.695037i	$-0.755388\pi$
$191$	−91.0192	+	52.5500i	−0.476541	+	0.275131i	0.242433	+	0.970168i	$0.422054\pi$
$192$		0			0
$193$	82.4615	−	142.828i	0.427262	−	0.740039i	−0.569367	−	0.822084i	$-0.692812\pi$
$193$	82.4615	−	142.828i	0.427262	−	0.740039i	0.996629	+	0.0820444i	$0.0261449\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	15.4064			0.0782050			0.0391025	−	0.999235i	$-0.487550\pi$
$197$	15.4064			0.0782050			0.0391025	+	0.999235i	$0.487550\pi$
$198$		0			0
$199$		−	57.1487i		−	0.287180i	−0.989637	−	0.143590i	$-0.954135\pi$
$199$		−	57.1487i		−	0.287180i	0.989637	−	0.143590i	$-0.0458646\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−9.37307	−	5.41154i	−0.0461727	−	0.0266578i
$204$		0			0
$205$	−88.7077	−	153.646i	−0.432720	−	0.749494i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−58.2769	−	100.939i	−0.278837	−	0.482959i
$210$		0			0
$211$	−0.157677	−	0.0910347i	−0.000747283	−	0.000431444i	0.499626	−	0.866241i	$-0.333471\pi$
$211$	−0.157677	−	0.0910347i	−0.000747283	−	0.000431444i	−0.500374	+	0.865810i	$0.666804\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			107.454i			0.499785i
$216$		0			0
$217$	22.2769			0.102658
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−5.66472	+	9.81158i	−0.0256322	+	0.0443963i
$222$		0			0
$223$	84.1577	−	48.5885i	0.377389	−	0.217885i	−0.299293	−	0.954161i	$-0.596751\pi$
$223$	84.1577	−	48.5885i	0.377389	−	0.217885i	0.676682	+	0.736276i	$0.263417\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	88.9808	−	51.3731i	0.391986	−	0.226313i	−0.291034	−	0.956713i	$-0.593999\pi$
$227$	88.9808	−	51.3731i	0.391986	−	0.226313i	0.683020	+	0.730400i	$0.260666\pi$
$228$		0			0
$229$	−163.404	+	283.024i	−0.713554	+	1.23591i	0.249961	+	0.968256i	$0.419582\pi$
$229$	−163.404	+	283.024i	−0.713554	+	1.23591i	−0.963515	+	0.267655i	$0.913751\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	79.4167			0.340844			0.170422	−	0.985371i	$-0.445487\pi$
$233$	79.4167			0.340844			0.170422	+	0.985371i	$0.445487\pi$
$234$		0			0
$235$			102.431i			0.435876i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	187.608	+	108.315i	0.784969	+	0.453202i	0.838189	−	0.545381i	$-0.183615\pi$
$239$	187.608	+	108.315i	0.784969	+	0.453202i	−0.0532191	+	0.998583i	$0.516948\pi$
$240$		0			0
$241$	−40.0307	−	69.3353i	−0.166103	−	0.287698i	0.770944	−	0.636903i	$-0.219785\pi$
$241$	−40.0307	−	69.3353i	−0.166103	−	0.287698i	−0.937046	+	0.349205i	$0.886452\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	90.8416	+	157.342i	0.370782	+	0.642214i
$246$		0			0
$247$	−77.9578	−	45.0089i	−0.315618	−	0.182222i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			135.531i			0.539963i	0.962865	+	0.269982i	$0.0870176\pi$
$251$			135.531i			0.539963i	−0.962865	+	0.269982i	$0.912982\pi$
$252$		0			0
$253$	−124.708			−0.492916
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−34.1391	+	59.1307i	−0.132837	+	0.230081i	−0.924769	−	0.380529i	$-0.875742\pi$
$257$	−34.1391	+	59.1307i	−0.132837	+	0.230081i	0.791932	+	0.610609i	$0.209075\pi$
$258$		0			0
$259$	79.4500	−	45.8705i	0.306757	−	0.177106i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−55.2923	+	31.9230i	−0.210237	+	0.121380i	−0.601422	−	0.798932i	$-0.705399\pi$
$263$	−55.2923	+	31.9230i	−0.210237	+	0.121380i	0.391185	+	0.920312i	$0.372065\pi$
$264$		0			0
$265$	121.177	−	209.885i	0.457271	−	0.792017i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−8.34490			−0.0310219			−0.0155110	−	0.999880i	$-0.504937\pi$
$269$	−8.34490			−0.0310219			−0.0155110	+	0.999880i	$0.504937\pi$
$270$		0			0
$271$			112.756i			0.416075i	0.978121	+	0.208038i	$0.0667077\pi$
$271$			112.756i			0.416075i	−0.978121	+	0.208038i	$0.933292\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	25.8076	+	14.9000i	0.0938459	+	0.0541820i
$276$		0			0
$277$	41.7077	+	72.2398i	0.150569	+	0.260793i	0.931437	−	0.363903i	$-0.118556\pi$
$277$	41.7077	+	72.2398i	0.150569	+	0.260793i	−0.780868	+	0.624696i	$0.785223\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−191.507	−	331.700i	−0.681520	−	1.18043i	−0.974517	−	0.224314i	$-0.927986\pi$
$281$	−191.507	−	331.700i	−0.681520	−	1.18043i	0.292997	−	0.956113i	$-0.405347\pi$
$282$		0			0
$283$	51.2154	+	29.5692i	0.180973	+	0.104485i	0.587750	−	0.809043i	$-0.300014\pi$
$283$	51.2154	+	29.5692i	0.180973	+	0.104485i	−0.406777	+	0.913528i	$0.633347\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	105.415i		−	0.367301i
$288$		0			0
$289$	−277.846			−0.961405
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−213.320	+	369.481i	−0.728054	+	1.26103i	0.229651	+	0.973273i	$0.426242\pi$
$293$	−213.320	+	369.481i	−0.728054	+	1.26103i	−0.957705	+	0.287753i	$0.907092\pi$
$294$		0			0
$295$	−354.831	+	204.862i	−1.20282	+	0.694446i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−83.4115	+	48.1577i	−0.278968	+	0.161062i
$300$		0			0
$301$	−31.9230	+	55.2923i	−0.106057	+	0.183695i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−294.324			−0.964998
$306$		0			0
$307$			533.338i			1.73726i	0.495463	+	0.868629i	$0.334999\pi$
$307$			533.338i			1.73726i	−0.495463	+	0.868629i	$0.665001\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	446.669	+	257.885i	1.43624	+	0.829211i	0.997586	−	0.0694473i	$-0.0221236\pi$
$311$	446.669	+	257.885i	1.43624	+	0.829211i	0.438650	+	0.898658i	$0.355457\pi$
$312$		0			0
$313$	220.031	+	381.104i	0.702974	+	1.21759i	0.967418	+	0.253185i	$0.0814782\pi$
$313$	220.031	+	381.104i	0.702974	+	1.21759i	−0.264444	+	0.964401i	$0.585189\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	243.942	+	422.519i	0.769532	+	1.33287i	0.937817	+	0.347130i	$0.112844\pi$
$317$	243.942	+	422.519i	0.769532	+	1.33287i	−0.168285	+	0.985738i	$0.553823\pi$
$318$		0			0
$319$	−16.2346	−	9.37307i	−0.0508923	−	0.0293827i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			88.6232i			0.274375i
$324$		0			0
$325$	23.0155			0.0708168
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−30.4308	+	52.7077i	−0.0924948	+	0.160206i
$330$		0			0
$331$	252.158	−	145.583i	0.761806	−	0.439829i	−0.0681380	−	0.997676i	$-0.521706\pi$
$331$	252.158	−	145.583i	0.761806	−	0.439829i	0.829944	+	0.557847i	$0.188372\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	284.196	−	164.081i	0.848347	−	0.489793i
$336$		0			0
$337$	223.200	−	386.594i	0.662314	−	1.14716i	−0.317692	−	0.948194i	$-0.602908\pi$
$337$	223.200	−	386.594i	0.662314	−	1.14716i	0.980006	−	0.198968i	$-0.0637590\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	38.5847			0.113152
$342$		0			0
$343$			232.210i			0.676998i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	440.827	+	254.512i	1.27039	+	0.733463i	0.975062	−	0.221932i	$-0.0712363\pi$
$347$	440.827	+	254.512i	1.27039	+	0.733463i	0.295332	+	0.955395i	$0.404570\pi$
$348$		0			0
$349$	−129.785	−	224.794i	−0.371876	−	0.644108i	0.617978	−	0.786195i	$-0.287952\pi$
$349$	−129.785	−	224.794i	−0.371876	−	0.644108i	−0.989854	+	0.142087i	$0.954619\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−268.708	−	465.415i	−0.761211	−	1.31846i	−0.942226	−	0.334977i	$-0.891272\pi$
$353$	−268.708	−	465.415i	−0.761211	−	1.31846i	0.181015	−	0.983480i	$-0.442062\pi$
$354$		0			0
$355$	−500.942	−	289.219i	−1.41111	−	0.814702i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			302.900i			0.843733i	0.906658	+	0.421866i	$0.138625\pi$
$359$			302.900i			0.843733i	−0.906658	+	0.421866i	$0.861375\pi$
$360$		0			0
$361$	−343.154			−0.950565
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−72.8506	+	126.181i	−0.199591	+	0.345701i
$366$		0			0
$367$	423.531	−	244.526i	1.15403	−	0.666282i	0.204168	−	0.978936i	$-0.434551\pi$
$367$	423.531	−	244.526i	1.15403	−	0.666282i	0.949867	+	0.312654i	$0.101218\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	124.708	−	72.0000i	0.336139	−	0.194070i
$372$		0			0
$373$	−227.277	+	393.655i	−0.609321	+	1.05538i	0.382031	+	0.924149i	$0.375225\pi$
$373$	−227.277	+	393.655i	−0.609321	+	1.05538i	−0.991352	+	0.131226i	$0.958109\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−14.4782			−0.0384037
$378$		0			0
$379$		−	241.359i		−	0.636831i	−0.947951	−	0.318416i	$-0.896849\pi$
$379$		−	241.359i		−	0.636831i	0.947951	−	0.318416i	$-0.103151\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	313.535	+	181.019i	0.818628	+	0.472635i	0.849943	−	0.526874i	$-0.176636\pi$
$383$	313.535	+	181.019i	0.818628	+	0.472635i	−0.0313151	+	0.999510i	$0.509970\pi$
$384$		0			0
$385$	−23.7691	−	41.1694i	−0.0617380	−	0.106933i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	225.100	+	389.885i	0.578663	+	1.00227i	0.995633	+	0.0933537i	$0.0297587\pi$
$389$	225.100	+	389.885i	0.578663	+	1.00227i	−0.416970	+	0.908920i	$0.636908\pi$
$390$		0			0
$391$	82.1192	+	47.4115i	0.210024	+	0.121257i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			468.084i			1.18502i
$396$		0			0
$397$	−55.7461			−0.140418			−0.0702092	−	0.997532i	$-0.522367\pi$
$397$	−55.7461			−0.140418			−0.0702092	+	0.997532i	$0.522367\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	53.6225	−	92.8768i	0.133722	−	0.231613i	−0.791387	−	0.611316i	$-0.790640\pi$
$401$	53.6225	−	92.8768i	0.133722	−	0.231613i	0.925108	+	0.379703i	$0.123974\pi$
$402$		0			0
$403$	25.8076	−	14.9000i	0.0640388	−	0.0369728i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	137.611	−	79.4500i	0.338112	−	0.195209i
$408$		0			0
$409$	298.323	−	516.711i	0.729396	−	1.26335i	−0.227742	−	0.973721i	$-0.573134\pi$
$409$	298.323	−	516.711i	0.729396	−	1.26335i	0.957139	−	0.289630i	$-0.0935323\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−243.446			−0.589458
$414$		0			0
$415$			494.754i			1.19218i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	290.438	+	167.685i	0.693170	+	0.400202i	0.804799	−	0.593548i	$-0.202273\pi$
$419$	290.438	+	167.685i	0.693170	+	0.400202i	−0.111628	+	0.993750i	$0.535607\pi$
$420$		0			0
$421$	143.973	+	249.369i	0.341979	+	0.592324i	0.984800	−	0.173692i	$-0.0555696\pi$
$421$	143.973	+	249.369i	0.341979	+	0.592324i	−0.642821	+	0.766016i	$0.722236\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−11.3294	−	19.6232i	−0.0266575	−	0.0461721i
$426$		0			0
$427$	−151.450	−	87.4397i	−0.354684	−	0.204777i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			408.631i			0.948099i	0.880498	+	0.474050i	$0.157208\pi$
$431$			408.631i			0.948099i	−0.880498	+	0.474050i	$0.842792\pi$
$432$		0			0
$433$	161.200			0.372286			0.186143	−	0.982523i	$-0.440401\pi$
$433$	161.200			0.372286			0.186143	+	0.982523i	$0.440401\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−376.708	+	652.477i	−0.862031	+	1.49308i
$438$		0			0
$439$	554.123	−	319.923i	1.26224	−	0.728754i	0.288732	−	0.957410i	$-0.406766\pi$
$439$	554.123	−	319.923i	1.26224	−	0.728754i	0.973507	+	0.228656i	$0.0734331\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	115.061	−	66.4308i	0.259732	−	0.149957i	−0.364480	−	0.931211i	$-0.618753\pi$
$443$	115.061	−	66.4308i	0.259732	−	0.149957i	0.624213	+	0.781255i	$0.285420\pi$
$444$		0			0
$445$	63.3653	−	109.752i	0.142394	−	0.246634i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−684.400			−1.52428			−0.762138	−	0.647415i	$-0.775850\pi$
$449$	−684.400			−1.52428			−0.762138	+	0.647415i	$0.775850\pi$
$450$		0			0
$451$		−	182.585i		−	0.404844i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−31.7963	−	18.3576i	−0.0698820	−	0.0403464i
$456$		0			0
$457$	248.600	+	430.588i	0.543982	+	0.942205i	0.998670	+	0.0515543i	$0.0164175\pi$
$457$	248.600	+	430.588i	0.543982	+	0.942205i	−0.454688	+	0.890651i	$0.650249\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	452.238	+	783.300i	0.980994	+	1.69913i	0.658537	+	0.752548i	$0.271175\pi$
$461$	452.238	+	783.300i	0.980994	+	1.69913i	0.322457	+	0.946584i	$0.395491\pi$
$462$		0			0
$463$	−701.338	−	404.918i	−1.51477	−	0.874553i	−0.999850	−	0.0173142i	$-0.994488\pi$
$463$	−701.338	−	404.918i	−1.51477	−	0.874553i	−0.514920	−	0.857238i	$-0.672178\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			210.200i			0.450107i	0.974346	+	0.225053i	$0.0722557\pi$
$467$			210.200i			0.450107i	−0.974346	+	0.225053i	$0.927744\pi$
$468$		0			0
$469$	194.985			0.415745
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−55.2923	+	95.7691i	−0.116897	+	0.202472i
$474$		0			0
$475$	155.916	−	90.0179i	0.328243	−	0.189511i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	68.1962	−	39.3731i	0.142372	−	0.0821985i	−0.427122	−	0.904194i	$-0.640473\pi$
$479$	68.1962	−	39.3731i	0.142372	−	0.0821985i	0.569494	+	0.821996i	$0.307139\pi$
$480$		0			0
$481$	61.3616	−	106.281i	0.127571	−	0.220959i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−168.879			−0.348205
$486$		0			0
$487$		−	382.641i		−	0.785711i	−0.919600	−	0.392855i	$-0.871487\pi$
$487$		−	382.641i		−	0.785711i	0.919600	−	0.392855i	$-0.128513\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	408.904	+	236.081i	0.832798	+	0.480816i	0.854810	−	0.518942i	$-0.173674\pi$
$491$	408.904	+	236.081i	0.832798	+	0.480816i	−0.0220117	+	0.999758i	$0.507007\pi$
$492$		0			0
$493$	7.12693	+	12.3442i	0.0144563	+	0.0250390i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−171.846	−	297.646i	−0.345767	−	0.598886i
$498$		0			0
$499$	360.473	+	208.119i	0.722391	+	0.417073i	0.815632	−	0.578571i	$-0.196389\pi$
$499$	360.473	+	208.119i	0.722391	+	0.417073i	−0.0932412	+	0.995644i	$0.529723\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			327.215i			0.650528i	0.945623	+	0.325264i	$0.105453\pi$
$503$			327.215i			0.650528i	−0.945623	+	0.325264i	$0.894547\pi$
$504$		0			0
$505$	−194.816			−0.385773
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−153.100	+	265.177i	−0.300786	+	0.520976i	−0.976314	−	0.216358i	$-0.930582\pi$
$509$	−153.100	+	265.177i	−0.300786	+	0.520976i	0.675528	+	0.737334i	$0.263916\pi$
$510$		0			0
$511$	−74.9732	+	43.2858i	−0.146719	+	0.0847080i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	414.746	−	239.454i	0.805332	−	0.464959i
$516$		0			0
$517$	−52.7077	+	91.2923i	−0.101949	+	0.176581i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	429.015			0.823446			0.411723	−	0.911309i	$-0.364927\pi$
$521$	429.015			0.823446			0.411723	+	0.911309i	$0.364927\pi$
$522$		0			0
$523$			561.864i			1.07431i	0.843484	+	0.537155i	$0.180501\pi$
$523$			561.864i			1.07431i	−0.843484	+	0.537155i	$0.819499\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−25.4078	−	14.6692i	−0.0482121	−	0.0278353i
$528$		0			0
$529$	138.561	+	239.996i	0.261931	+	0.453678i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−70.5077	−	122.123i	−0.132285	−	0.229124i
$534$		0			0
$535$	−411.069	−	237.331i	−0.768353	−	0.443609i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			186.977i			0.346896i
$540$		0			0
$541$	225.469			0.416764			0.208382	−	0.978048i	$-0.433180\pi$
$541$	225.469			0.416764			0.208382	+	0.978048i	$0.433180\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	216.369	−	374.761i	0.397007	−	0.687636i
$546$		0			0
$547$	314.585	−	181.626i	0.575109	−	0.332039i	−0.184078	−	0.982912i	$-0.558930\pi$
$547$	314.585	−	181.626i	0.575109	−	0.332039i	0.759187	+	0.650872i	$0.225597\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−98.0807	+	56.6269i	−0.178005	+	0.102771i
$552$		0			0
$553$	−139.061	+	240.862i	−0.251467	+	0.435554i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	571.022			1.02517			0.512587	−	0.858635i	$-0.328687\pi$
$557$	571.022			1.02517			0.512587	+	0.858635i	$0.328687\pi$
$558$		0			0
$559$			85.4078i			0.152787i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	206.354	+	119.138i	0.366525	+	0.211614i	0.671939	−	0.740606i	$-0.265462\pi$
$563$	206.354	+	119.138i	0.366525	+	0.211614i	−0.305414	+	0.952220i	$0.598795\pi$
$564$		0			0
$565$	−459.365	−	795.644i	−0.813036	−	1.40822i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−153.469	−	265.815i	−0.269716	−	0.467162i	0.699072	−	0.715051i	$-0.253597\pi$
$569$	−153.469	−	265.815i	−0.269716	−	0.467162i	−0.968789	+	0.247889i	$0.920263\pi$
$570$		0			0
$571$	−449.569	−	259.559i	−0.787337	−	0.454569i	0.0516874	−	0.998663i	$-0.483540\pi$
$571$	−449.569	−	259.559i	−0.787337	−	0.454569i	−0.839024	+	0.544094i	$0.816873\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	192.631i		−	0.335010i
$576$		0			0
$577$	−891.523			−1.54510			−0.772550	−	0.634954i	$-0.781019\pi$
$577$	−891.523			−1.54510			−0.772550	+	0.634954i	$0.781019\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−146.985	+	254.585i	−0.252985	+	0.438184i
$582$		0			0
$583$	216.000	−	124.708i	0.370497	−	0.213907i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−34.7808	+	20.0807i	−0.0592519	+	0.0342091i	−0.529333	−	0.848414i	$-0.677558\pi$
$587$	−34.7808	+	20.0807i	−0.0592519	+	0.0342091i	0.470081	+	0.882623i	$0.344225\pi$
$588$		0			0
$589$	116.554	−	201.877i	0.197884	−	0.342745i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−715.568			−1.20669			−0.603346	−	0.797480i	$-0.706166\pi$
$593$	−715.568			−1.20669			−0.603346	+	0.797480i	$0.706166\pi$
$594$		0			0
$595$			36.1464i			0.0607502i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	510.115	+	294.515i	0.851612	+	0.491678i	0.861194	−	0.508276i	$-0.169717\pi$
$599$	510.115	+	294.515i	0.851612	+	0.491678i	−0.00958261	+	0.999954i	$0.503050\pi$
$600$		0			0
$601$	347.638	+	602.127i	0.578433	+	1.00188i	0.995659	+	0.0930731i	$0.0296690\pi$
$601$	347.638	+	602.127i	0.578433	+	1.00188i	−0.417226	+	0.908803i	$0.636998\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	217.042	+	375.927i	0.358746	+	0.621367i
$606$		0			0
$607$	700.404	+	404.378i	1.15388	+	0.666191i	0.949829	−	0.312769i	$-0.101257\pi$
$607$	700.404	+	404.378i	1.15388	+	0.666191i	0.204048	+	0.978961i	$0.434590\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			81.4153i			0.133249i
$612$		0			0
$613$	707.415			1.15402			0.577011	−	0.816736i	$-0.304219\pi$
$613$	707.415			1.15402			0.577011	+	0.816736i	$0.304219\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	224.922	−	389.577i	0.364542	−	0.631405i	−0.624161	−	0.781296i	$-0.714559\pi$
$617$	224.922	−	389.577i	0.364542	−	0.631405i	0.988703	+	0.149891i	$0.0478923\pi$
$618$		0			0
$619$	−829.177	+	478.726i	−1.33954	+	0.773385i	−0.986739	−	0.162312i	$-0.948105\pi$
$619$	−829.177	+	478.726i	−1.33954	+	0.773385i	−0.352803	+	0.935698i	$0.614771\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	65.2116	−	37.6499i	0.104674	−	0.0604333i
$624$		0			0
$625$	204.677	−	354.511i	0.327483	−	0.567217i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−120.822			−0.192085
$630$		0			0
$631$			526.179i			0.833882i	0.908934	+	0.416941i	$0.136898\pi$
$631$			526.179i			0.833882i	−0.908934	+	0.416941i	$0.863102\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−480.358	−	277.335i	−0.756469	−	0.436747i
$636$		0			0
$637$	72.2039	+	125.061i	0.113350	+	0.196328i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−266.301	−	461.246i	−0.415446	−	0.719573i	0.580030	−	0.814595i	$-0.303041\pi$
$641$	−266.301	−	461.246i	−0.415446	−	0.719573i	−0.995475	+	0.0950227i	$0.969708\pi$
$642$		0			0
$643$	778.131	+	449.254i	1.21016	+	0.698684i	0.962793	−	0.270239i	$-0.0871027\pi$
$643$	778.131	+	449.254i	1.21016	+	0.698684i	0.247363	+	0.968923i	$0.420436\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	992.008i		−	1.53324i	−0.642100	−	0.766621i	$-0.721937\pi$
$647$		−	992.008i		−	1.53324i	0.642100	−	0.766621i	$-0.278063\pi$
$648$		0			0
$649$	−421.661			−0.649709
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−187.608	+	324.946i	−0.287301	+	0.497620i	−0.973165	−	0.230110i	$-0.926091\pi$
$653$	−187.608	+	324.946i	−0.287301	+	0.497620i	0.685863	+	0.727730i	$0.259425\pi$
$654$		0			0
$655$	−572.250	+	330.389i	−0.873664	+	0.504410i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−991.019	+	572.165i	−1.50382	+	0.868233i	−0.503833	+	0.863801i	$0.668077\pi$
$659$	−991.019	+	572.165i	−1.50382	+	0.868233i	−0.999990	+	0.00443122i	$0.998589\pi$
$660$		0			0
$661$	−218.242	+	378.007i	−0.330170	+	0.571871i	−0.982545	−	0.186025i	$-0.940439\pi$
$661$	−218.242	+	378.007i	−0.330170	+	0.571871i	0.652375	+	0.757896i	$0.273773\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−287.200			−0.431880
$666$		0			0
$667$			121.177i			0.181675i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−262.319	−	151.450i	−0.390938	−	0.225708i
$672$		0			0
$673$	−374.169	−	648.080i	−0.555972	−	0.962972i	−0.997827	−	0.0658854i	$-0.979013\pi$
$673$	−374.169	−	648.080i	−0.555972	−	0.962972i	0.441855	−	0.897086i	$-0.354321\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	28.3923	+	49.1769i	0.0419384	+	0.0726395i	0.886233	−	0.463240i	$-0.153313\pi$
$677$	28.3923	+	49.1769i	0.0419384	+	0.0726395i	−0.844294	+	0.535880i	$0.819980\pi$
$678$		0			0
$679$	−86.9000	−	50.1718i	−0.127982	−	0.0738907i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	711.384i		−	1.04156i	−0.853691	−	0.520779i	$-0.825641\pi$
$683$		−	711.384i		−	1.04156i	0.853691	−	0.520779i	$-0.174359\pi$
$684$		0			0
$685$	−676.377			−0.987411
$686$		0			0
$687$		0			0
$688$		0			0
$689$	96.3154	−	166.823i	0.139790	−	0.242123i
$690$		0			0
$691$	228.158	−	131.727i	0.330185	−	0.190632i	−0.325738	−	0.945460i	$-0.605613\pi$
$691$	228.158	−	131.727i	0.330185	−	0.190632i	0.655923	+	0.754828i	$0.272280\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	172.392	−	99.5307i	0.248046	−	0.143210i
$696$		0			0
$697$	−69.4153	+	120.231i	−0.0995916	+	0.172498i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1177.07			1.67913			0.839565	−	0.543259i	$-0.182810\pi$
$701$	1177.07			1.67913			0.839565	+	0.543259i	$0.182810\pi$
$702$		0			0
$703$		−	959.987i		−	1.36556i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−100.246	−	57.8770i	−0.141791	−	0.0818628i
$708$		0			0
$709$	−215.481	−	373.224i	−0.303922	−	0.526409i	0.673099	−	0.739553i	$-0.264963\pi$
$709$	−215.481	−	373.224i	−0.303922	−	0.526409i	−0.977021	+	0.213144i	$0.931630\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−124.708	−	216.000i	−0.174906	−	0.302945i
$714$		0			0
$715$	−55.0728	−	31.7963i	−0.0770249	−	0.0444704i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			514.361i			0.715384i	0.933840	+	0.357692i	$0.116436\pi$
$719$			514.361i			0.715384i	−0.933840	+	0.357692i	$0.883564\pi$
$720$		0			0
$721$	284.554			0.394665
$722$		0			0
$723$		0			0
$724$		0			0
$725$	14.4782	−	25.0770i	0.0199699	−	0.0345889i
$726$		0			0
$727$	−569.412	+	328.750i	−0.783235	+	0.452201i	−0.837575	−	0.546322i	$-0.816028\pi$
$727$	−569.412	+	328.750i	−0.783235	+	0.452201i	0.0543408	+	0.998522i	$0.482694\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	72.8193	−	42.0422i	0.0996160	−	0.0575133i
$732$		0			0
$733$	157.000	−	271.932i	0.214188	−	0.370985i	−0.738833	−	0.673889i	$-0.764623\pi$
$733$	157.000	−	271.932i	0.214188	−	0.370985i	0.953021	+	0.302904i	$0.0979561\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	337.723			0.458240
$738$		0			0
$739$			150.431i			0.203560i	0.994807	+	0.101780i	$0.0324538\pi$
$739$			150.431i			0.203560i	−0.994807	+	0.101780i	$0.967546\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−443.538	−	256.077i	−0.596956	−	0.344653i	0.170887	−	0.985291i	$-0.445337\pi$
$743$	−443.538	−	256.077i	−0.596956	−	0.344653i	−0.767843	+	0.640638i	$0.778670\pi$
$744$		0			0
$745$	41.5770	+	72.0134i	0.0558080	+	0.0966623i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−141.015	−	244.246i	−0.188272	−	0.326096i
$750$		0			0
$751$	630.904	+	364.252i	0.840085	+	0.485023i	0.857293	−	0.514829i	$-0.172144\pi$
$751$	630.904	+	364.252i	0.840085	+	0.485023i	−0.0172081	+	0.999852i	$0.505478\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	76.5387i		−	0.101376i
$756$		0			0
$757$	1154.25			1.52476			0.762382	−	0.647128i	$-0.224030\pi$
$757$	1154.25			1.52476			0.762382	+	0.647128i	$0.224030\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	438.884	−	760.169i	0.576720	−	0.998908i	−0.419132	−	0.907925i	$-0.637666\pi$
$761$	438.884	−	760.169i	0.576720	−	0.998908i	0.995852	−	0.0909832i	$-0.0290009\pi$
$762$		0			0
$763$	222.673	−	128.560i	0.291839	−	0.168493i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−282.031	+	162.831i	−0.367707	+	0.212295i
$768$		0			0
$769$	−738.877	+	1279.77i	−0.960828	+	1.66420i	−0.240400	+	0.970674i	$0.577278\pi$
$769$	−738.877	+	1279.77i	−0.960828	+	1.66420i	−0.720429	+	0.693529i	$0.756055\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−115.270			−0.149121			−0.0745604	−	0.997217i	$-0.523755\pi$
$773$	−115.270			−0.149121			−0.0745604	+	0.997217i	$0.523755\pi$
$774$		0			0
$775$			59.6001i			0.0769034i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−955.292	−	551.538i	−1.22631	−	0.708008i
$780$		0			0
$781$	−297.646	−	515.538i	−0.381109	−	0.660100i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	29.6623	+	51.3767i	0.0377864	+	0.0654480i
$786$		0			0
$787$	−933.657	−	539.047i	−1.18635	−	0.684940i	−0.228875	−	0.973456i	$-0.573505\pi$
$787$	−933.657	−	539.047i	−1.18635	−	0.684940i	−0.957475	+	0.288516i	$0.906838\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	545.885i		−	0.690120i
$792$		0			0
$793$	−233.939			−0.295004
$794$		0			0
$795$		0			0
$796$		0			0
$797$	472.381	−	818.188i	0.592699	−	1.02659i	−0.401168	−	0.916005i	$-0.631396\pi$
$797$	472.381	−	818.188i	0.592699	−	1.02659i	0.993867	−	0.110581i	$-0.0352711\pi$
$798$		0			0
$799$	69.4153	−	40.0770i	0.0868777	−	0.0501589i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−129.857	+	74.9732i	−0.161715	+	0.0933664i
$804$		0			0
$805$	−153.646	+	266.123i	−0.190865	+	0.330588i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	623.894			0.771191			0.385595	−	0.922668i	$-0.373996\pi$
$809$	623.894			0.771191			0.385595	+	0.922668i	$0.373996\pi$
$810$		0			0
$811$		−	1364.09i		−	1.68199i	−0.541045	−	0.840994i	$-0.681971\pi$
$811$		−	1364.09i		−	1.68199i	0.541045	−	0.840994i	$-0.318029\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−802.046	−	463.061i	−0.984106	−	0.568174i
$816$		0			0
$817$	334.046	+	578.585i	0.408869	+	0.708182i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−158.027	−	273.712i	−0.192482	−	0.333388i	0.753590	−	0.657344i	$-0.228320\pi$
$821$	−158.027	−	273.712i	−0.192482	−	0.333388i	−0.946072	+	0.323956i	$0.894987\pi$
$822$		0			0
$823$	886.508	+	511.825i	1.07717	+	0.621902i	0.930131	−	0.367229i	$-0.119693\pi$
$823$	886.508	+	511.825i	1.07717	+	0.621902i	0.147036	+	0.989131i	$0.453027\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1100.24i		−	1.33040i	−0.746667	−	0.665199i	$-0.768347\pi$
$827$		−	1100.24i		−	1.33040i	0.746667	−	0.665199i	$-0.231653\pi$
$828$		0			0
$829$	927.108			1.11834			0.559172	−	0.829052i	$-0.311119\pi$
$829$	927.108			1.11834			0.559172	+	0.829052i	$0.311119\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	71.0852	−	123.123i	0.0853364	−	0.147807i
$834$		0			0
$835$	−1115.53	+	644.050i	−1.33596	+	0.771317i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−261.773	+	151.135i	−0.312006	+	0.180137i	−0.647824	−	0.761790i	$-0.724321\pi$
$839$	−261.773	+	151.135i	−0.312006	+	0.180137i	0.335818	+	0.941927i	$0.390987\pi$
$840$		0			0
$841$	411.392	−	712.552i	0.489170	−	0.847268i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	672.169			0.795466
$846$		0			0
$847$			257.920i			0.304510i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−889.535	−	513.573i	−1.04528	−	0.603494i
$852$		0			0
$853$	15.7077	+	27.2065i	0.0184146	+	0.0318950i	0.875086	−	0.483968i	$-0.160805\pi$
$853$	15.7077	+	27.2065i	0.0184146	+	0.0318950i	−0.856671	+	0.515863i	$0.827471\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	539.822	+	935.000i	0.629898	+	1.09101i	0.987572	+	0.157168i	$0.0502366\pi$
$857$	539.822	+	935.000i	0.629898	+	1.09101i	−0.357674	+	0.933847i	$0.616430\pi$
$858$		0			0
$859$	878.985	+	507.482i	1.02326	+	0.590782i	0.915048	−	0.403345i	$-0.132153\pi$
$859$	878.985	+	507.482i	1.02326	+	0.590782i	0.108217	+	0.994127i	$0.465486\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	35.9155i		−	0.0416170i	−0.999783	−	0.0208085i	$-0.993376\pi$
$863$		−	35.9155i		−	0.0416170i	0.999783	−	0.0208085i	$-0.00662403\pi$
$864$		0			0
$865$	1016.14			1.17473
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−240.862	+	417.184i	−0.277171	+	0.480074i
$870$		0			0
$871$	225.888	−	130.417i	0.259344	−	0.149732i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	297.919	−	172.004i	0.340479	−	0.196576i
$876$		0			0
$877$	12.0270	−	20.8313i	0.0137138	−	0.0237529i	−0.859087	−	0.511829i	$-0.828968\pi$
$877$	12.0270	−	20.8313i	0.0137138	−	0.0237529i	0.872801	+	0.488077i	$0.162301\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1163.14			−1.32025			−0.660124	−	0.751157i	$-0.729496\pi$
$881$	−1163.14			−1.32025			−0.660124	+	0.751157i	$0.729496\pi$
$882$		0			0
$883$		−	1382.49i		−	1.56567i	−0.622229	−	0.782835i	$-0.713773\pi$
$883$		−	1382.49i		−	1.56567i	0.622229	−	0.782835i	$-0.286227\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1260.53	−	727.765i	−1.42111	−	0.820480i	−0.424718	−	0.905326i	$-0.639627\pi$
$887$	−1260.53	−	727.765i	−1.42111	−	0.820480i	−0.996394	+	0.0848460i	$0.972960\pi$
$888$		0			0
$889$	−164.785	−	285.415i	−0.185360	−	0.321052i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	318.431	+	551.538i	0.356585	+	0.617624i
$894$		0			0
$895$	−525.877	−	303.615i	−0.587572	−	0.339235i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	37.4923i		−	0.0417044i
$900$		0			0
$901$	−189.646			−0.210484
$902$		0			0
$903$		0			0
$904$		0			0
$905$	429.224	−	743.438i	0.474281	−	0.821479i
$906$		0			0
$907$	681.731	−	393.597i	0.751632	−	0.433955i	−0.0746510	−	0.997210i	$-0.523784\pi$
$907$	681.731	−	393.597i	0.751632	−	0.433955i	0.826283	+	0.563255i	$0.190451\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−804.904	+	464.711i	−0.883539	+	0.510111i	−0.871824	−	0.489820i	$-0.837062\pi$
$911$	−804.904	+	464.711i	−0.883539	+	0.510111i	−0.0117152	+	0.999931i	$0.503729\pi$
$912$		0			0
$913$	−254.585	+	440.954i	−0.278844	+	0.482972i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−392.616			−0.428152
$918$		0			0
$919$		−	1652.18i		−	1.79780i	−0.438156	−	0.898899i	$-0.644368\pi$
$919$		−	1652.18i		−	1.79780i	0.438156	−	0.898899i	$-0.355632\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−398.165	−	229.881i	−0.431382	−	0.249058i
$924$		0			0
$925$	122.723	+	212.563i	0.132674	+	0.229797i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−736.530	−	1275.71i	−0.792820	−	1.37321i	−0.924214	−	0.381875i	$-0.875279\pi$
$929$	−736.530	−	1275.71i	−0.792820	−	1.37321i	0.131394	−	0.991330i	$-0.458055\pi$
$930$		0			0
$931$	978.273	+	564.806i	1.05078	+	0.606666i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			62.6073i			0.0669597i
$936$		0			0
$937$	−941.246			−1.00453			−0.502266	−	0.864713i	$-0.667500\pi$
$937$	−941.246			−1.00453			−0.502266	+	0.864713i	$0.667500\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	281.243	−	487.127i	0.298877	−	0.517669i	−0.677003	−	0.735981i	$-0.736721\pi$
$941$	281.243	−	487.127i	0.298877	−	0.517669i	0.975879	+	0.218311i	$0.0700548\pi$
$942$		0			0
$943$	−1022.12	+	590.123i	−1.08391	+	0.625793i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	876.358	−	505.965i	0.925404	−	0.534282i	0.0400489	−	0.999198i	$-0.487249\pi$
$947$	876.358	−	505.965i	0.925404	−	0.534282i	0.885355	+	0.464915i	$0.153915\pi$
$948$		0			0
$949$	−57.9040	+	100.293i	−0.0610158	+	0.105682i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1274.88			1.33775			0.668876	−	0.743374i	$-0.266776\pi$
$953$	1274.88			1.33775			0.668876	+	0.743374i	$0.266776\pi$
$954$		0			0
$955$			448.561i			0.469698i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−348.042	−	200.942i	−0.362922	−	0.209533i
$960$		0			0
$961$	−441.915	−	765.420i	−0.459849	−	0.796483i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−351.942	−	609.581i	−0.364706	−	0.631690i
$966$		0			0
$967$	−573.657	−	331.201i	−0.593234	−	0.342504i	0.173141	−	0.984897i	$-0.444608\pi$
$967$	−573.657	−	331.201i	−0.593234	−	0.342504i	−0.766375	+	0.642393i	$0.777942\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	321.731i		−	0.331340i	−0.986181	−	0.165670i	$-0.947021\pi$
$971$		−	321.731i		−	0.331340i	0.986181	−	0.165670i	$-0.0529786\pi$
$972$		0			0
$973$	118.277			0.121559
$974$		0			0
$975$		0			0
$976$		0			0
$977$	172.939	−	299.538i	0.177010	−	0.306590i	−0.763845	−	0.645399i	$-0.776691\pi$
$977$	172.939	−	299.538i	0.177010	−	0.306590i	0.940855	+	0.338810i	$0.110024\pi$
$978$		0			0
$979$	112.950	−	65.2116i	0.115373	−	0.0666104i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−621.100	+	358.592i	−0.631841	+	0.364794i	−0.781465	−	0.623949i	$-0.785527\pi$
$983$	−621.100	+	358.592i	−0.631841	+	0.364794i	0.149624	+	0.988743i	$0.452194\pi$
$984$		0			0
$985$	32.8768	−	56.9444i	0.0333775	−	0.0578115i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	714.831			0.722781
$990$		0			0
$991$		−	646.095i		−	0.651962i	−0.945376	−	0.325981i	$-0.894305\pi$
$991$		−	646.095i		−	0.651962i	0.945376	−	0.325981i	$-0.105695\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−211.230	−	121.954i	−0.212292	−	0.122567i
$996$		0			0
$997$	−490.481	−	849.538i	−0.491957	−	0.852094i	0.508000	−	0.861357i	$-0.330385\pi$
$997$	−490.481	−	849.538i	−0.491957	−	0.852094i	−0.999957	+	0.00926291i	$0.997051\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.3.o.be.271.1		4
3.2	odd	2		1296.3.o.q.271.2		4
4.3	odd	2		1296.3.o.bf.271.1		4
9.2	odd	6		1296.3.o.r.703.2		4
9.4	even	3		1296.3.g.c.1135.4	yes	4
9.5	odd	6		1296.3.g.i.1135.2	yes	4
9.7	even	3		1296.3.o.bf.703.1		4
12.11	even	2		1296.3.o.r.271.2		4
36.7	odd	6	inner	1296.3.o.be.703.1		4
36.11	even	6		1296.3.o.q.703.2		4
36.23	even	6		1296.3.g.i.1135.1	yes	4
36.31	odd	6		1296.3.g.c.1135.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1296.3.g.c.1135.3	✓	4	36.31	odd	6
1296.3.g.c.1135.4	yes	4	9.4	even	3
1296.3.g.i.1135.1	yes	4	36.23	even	6
1296.3.g.i.1135.2	yes	4	9.5	odd	6
1296.3.o.q.271.2		4	3.2	odd	2
1296.3.o.q.703.2		4	36.11	even	6
1296.3.o.r.271.2		4	12.11	even	2
1296.3.o.r.703.2		4	9.2	odd	6
1296.3.o.be.271.1		4	1.1	even	1	trivial
1296.3.o.be.703.1		4	36.7	odd	6	inner
1296.3.o.bf.271.1		4	4.3	odd	2
1296.3.o.bf.703.1		4	9.7	even	3

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1296.o (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.13397 - 3.69615i) q^{5} +(2.19615 - 1.26795i) q^{7} +O(q^{10})\)	\(q+(2.13397 - 3.69615i) q^{5} +(2.19615 - 1.26795i) q^{7} +(3.80385 - 2.19615i) q^{11} +(1.69615 - 2.93782i) q^{13} -3.33975 q^{17} -26.5359i q^{19} +(-24.5885 - 14.1962i) q^{23} +(3.39230 + 5.87564i) q^{25} +(-2.13397 - 3.69615i) q^{29} +(7.60770 + 4.39230i) q^{31} -10.8231i q^{35} +36.1769 q^{37} +(20.7846 - 36.0000i) q^{41} +(-21.8038 + 12.5885i) q^{43} +(-20.7846 + 12.0000i) q^{47} +(-21.2846 + 36.8660i) q^{49} +56.7846 q^{53} -18.7461i q^{55} +(-83.1384 - 48.0000i) q^{59} +(-34.4808 - 59.7224i) q^{61} +(-7.23909 - 12.5385i) q^{65} +(66.5885 + 38.4449i) q^{67} -135.531i q^{71} -34.1384 q^{73} +(5.56922 - 9.64617i) q^{77} +(-94.9808 + 54.8372i) q^{79} +(-100.392 + 57.9615i) q^{83} +(-7.12693 + 12.3442i) q^{85} +29.6936 q^{89} -8.60254i q^{91} +(-98.0807 - 56.6269i) q^{95} +(-19.7846 - 34.2679i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{5} - 12 q^{7}+O(q^{10}) \) 4 * q + 12 * q^5 - 12 * q^7	\( 4 q + 12 q^{5} - 12 q^{7} + 36 q^{11} - 14 q^{13} - 48 q^{17} - 36 q^{23} - 28 q^{25} - 12 q^{29} + 72 q^{31} + 20 q^{37} - 108 q^{43} - 2 q^{49} + 144 q^{53} - 34 q^{61} + 120 q^{65} + 204 q^{67} + 196 q^{73} - 144 q^{77} - 276 q^{79} - 360 q^{83} - 174 q^{85} - 96 q^{89} + 252 q^{95} + 4 q^{97}+O(q^{100}) \) 4 * q + 12 * q^5 - 12 * q^7 + 36 * q^11 - 14 * q^13 - 48 * q^17 - 36 * q^23 - 28 * q^25 - 12 * q^29 + 72 * q^31 + 20 * q^37 - 108 * q^43 - 2 * q^49 + 144 * q^53 - 34 * q^61 + 120 * q^65 + 204 * q^67 + 196 * q^73 - 144 * q^77 - 276 * q^79 - 360 * q^83 - 174 * q^85 - 96 * q^89 + 252 * q^95 + 4 * q^97

Introduction

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