LMFDB - Embedded newform 432.2.l.a.323.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,2,Mod(107,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("432.107");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	432.l (of order $4$, degree $2$, 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:	$3.44953736732$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			323.4
Character	$\chi$	$=$	432.323
Dual form			432.2.l.a.107.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.06417 + 0.931414i) q^{2} +(0.264934 - 1.98237i) q^{4} +(2.29090 + 2.29090i) q^{5} +3.92541 q^{7} +(1.56448 + 2.35636i) q^{8} +O(q^{10})$	\(q+(-1.06417 + 0.931414i) q^{2} +(0.264934 - 1.98237i) q^{4} +(2.29090 + 2.29090i) q^{5} +3.92541 q^{7} +(1.56448 + 2.35636i) q^{8} +(-4.57170 - 0.304140i) q^{10} +(1.48221 - 1.48221i) q^{11} +(-2.94883 - 2.94883i) q^{13} +(-4.17732 + 3.65618i) q^{14} +(-3.85962 - 1.05040i) q^{16} -4.04232i q^{17} +(3.09377 - 3.09377i) q^{19} +(5.14836 - 3.93449i) q^{20} +(-0.196778 + 2.95789i) q^{22} +6.25991i q^{23} +5.49645i q^{25} +(5.88465 + 0.391486i) q^{26} +(1.03998 - 7.78164i) q^{28} +(-1.44083 + 1.44083i) q^{29} +8.70311i q^{31} +(5.08566 - 2.47710i) q^{32} +(3.76508 + 4.30174i) q^{34} +(8.99273 + 8.99273i) q^{35} +(4.01719 - 4.01719i) q^{37} +(-0.410728 + 6.17389i) q^{38} +(-1.81412 + 8.98224i) q^{40} -7.82334 q^{41} +(-0.354484 - 0.354484i) q^{43} +(-2.54561 - 3.33099i) q^{44} +(-5.83057 - 6.66164i) q^{46} -3.25028 q^{47} +8.40885 q^{49} +(-5.11948 - 5.84919i) q^{50} +(-6.62693 + 5.06444i) q^{52} +(-7.21705 - 7.21705i) q^{53} +6.79120 q^{55} +(6.14121 + 9.24966i) q^{56} +(0.191285 - 2.87531i) q^{58} +(-1.07589 + 1.07589i) q^{59} +(8.81820 + 8.81820i) q^{61} +(-8.10621 - 9.26163i) q^{62} +(-3.10483 + 7.37293i) q^{64} -13.5110i q^{65} +(-4.72284 + 4.72284i) q^{67} +(-8.01340 - 1.07095i) q^{68} +(-17.9458 - 1.19387i) q^{70} +9.37491i q^{71} +7.02186i q^{73} +(-0.533322 + 8.01666i) q^{74} +(-5.31336 - 6.95265i) q^{76} +(5.81829 - 5.81829i) q^{77} -9.40022i q^{79} +(-6.43565 - 11.2484i) q^{80} +(8.32540 - 7.28677i) q^{82} +(-4.41582 - 4.41582i) q^{83} +(9.26056 - 9.26056i) q^{85} +(0.707405 + 0.0470613i) q^{86} +(5.81150 + 1.17373i) q^{88} +4.59791 q^{89} +(-11.5754 - 11.5754i) q^{91} +(12.4095 + 1.65846i) q^{92} +(3.45886 - 3.02735i) q^{94} +14.1750 q^{95} -19.5022 q^{97} +(-8.94848 + 7.83212i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q - 4 q^{10} - 20 q^{16} + 8 q^{19} + 4 q^{22} - 12 q^{28} - 36 q^{34} - 12 q^{40} + 32 q^{43} - 16 q^{46} + 32 q^{49} - 60 q^{52} + 64 q^{55} - 48 q^{58} - 16 q^{61} + 48 q^{64} - 32 q^{67} - 72 q^{70} - 96 q^{76} + 40 q^{82} - 16 q^{85} + 36 q^{88} + 24 q^{91} - 36 q^{94}+O(q^{100}) $ 32 * q - 4 * q^10 - 20 * q^16 + 8 * q^19 + 4 * q^22 - 12 * q^28 - 36 * q^34 - 12 * q^40 + 32 * q^43 - 16 * q^46 + 32 * q^49 - 60 * q^52 + 64 * q^55 - 48 * q^58 - 16 * q^61 + 48 * q^64 - 32 * q^67 - 72 * q^70 - 96 * q^76 + 40 * q^82 - 16 * q^85 + 36 * q^88 + 24 * q^91 - 36 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

$n$	$271$	$325$	$353$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$	$-1$

Coefficient data

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

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

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

$2$	−1.06417	+	0.931414i	−0.752485	+	0.658609i
$2$	−1.06417	+	0.931414i	−0.752485	+	0.658609i
$3$		0			0
$3$		0			0
$4$	0.264934	−	1.98237i	0.132467	−	0.991187i
$5$	2.29090	+	2.29090i	1.02452	+	1.02452i	0.999692	+	0.0248303i	$0.00790456\pi$
$5$	2.29090	+	2.29090i	1.02452	+	1.02452i	0.0248303	+	0.999692i	$0.492095\pi$
$6$		0			0
$7$	3.92541			1.48367			0.741833	−	0.670585i	$-0.233957\pi$
$7$	3.92541			1.48367			0.741833	+	0.670585i	$0.233957\pi$
$8$	1.56448	+	2.35636i	0.553126	+	0.833098i
$9$		0			0
$10$	−4.57170	−	0.304140i	−1.44570	−	0.0961774i
$11$	1.48221	−	1.48221i	0.446904	−	0.446904i	−0.447420	−	0.894324i	$-0.647657\pi$
$11$	1.48221	−	1.48221i	0.446904	−	0.446904i	0.894324	+	0.447420i	$0.147657\pi$
$12$		0			0
$13$	−2.94883	−	2.94883i	−0.817858	−	0.817858i	0.167939	−	0.985797i	$-0.446289\pi$
$13$	−2.94883	−	2.94883i	−0.817858	−	0.817858i	−0.985797	+	0.167939i	$0.946289\pi$
$14$	−4.17732	+	3.65618i	−1.11644	+	0.977156i
$15$		0			0
$16$	−3.85962	−	1.05040i	−0.964905	−	0.262599i
$17$		−	4.04232i		−	0.980408i	−0.871608	−	0.490204i	$-0.836922\pi$
$17$		−	4.04232i		−	0.980408i	0.871608	−	0.490204i	$-0.163078\pi$
$18$		0			0
$19$	3.09377	−	3.09377i	0.709759	−	0.709759i	−0.256725	−	0.966484i	$-0.582644\pi$
$19$	3.09377	−	3.09377i	0.709759	−	0.709759i	0.966484	+	0.256725i	$0.0826436\pi$
$20$	5.14836	−	3.93449i	1.15121	−	0.879778i
$21$		0			0
$22$	−0.196778	+	2.95789i	−0.0419533	+	0.630623i
$23$			6.25991i			1.30528i	0.757667	+	0.652641i	$0.226339\pi$
$23$			6.25991i			1.30528i	−0.757667	+	0.652641i	$0.773661\pi$
$24$		0			0
$25$			5.49645i			1.09929i
$26$	5.88465	+	0.391486i	1.15408	+	0.0767768i
$27$		0			0
$28$	1.03998	−	7.78164i	0.196537	−	1.47059i
$29$	−1.44083	+	1.44083i	−0.267556	+	0.267556i	−0.828115	−	0.560559i	$-0.810586\pi$
$29$	−1.44083	+	1.44083i	−0.267556	+	0.267556i	0.560559	+	0.828115i	$0.310586\pi$
$30$		0			0
$31$			8.70311i			1.56313i	0.623827	+	0.781563i	$0.285577\pi$
$31$			8.70311i			1.56313i	−0.623827	+	0.781563i	$0.714423\pi$
$32$	5.08566	−	2.47710i	0.899027	−	0.437894i
$33$		0			0
$34$	3.76508	+	4.30174i	0.645706	+	0.737742i
$35$	8.99273	+	8.99273i	1.52005	+	1.52005i
$36$		0			0
$37$	4.01719	−	4.01719i	0.660422	−	0.660422i	−0.295057	−	0.955480i	$-0.595339\pi$
$37$	4.01719	−	4.01719i	0.660422	−	0.660422i	0.955480	+	0.295057i	$0.0953387\pi$
$38$	−0.410728	+	6.17389i	−0.0666289	+	1.00154i
$39$		0			0
$40$	−1.81412	+	8.98224i	−0.286837	+	1.42022i
$41$	−7.82334			−1.22180			−0.610900	−	0.791708i	$-0.709192\pi$
$41$	−7.82334			−1.22180			−0.610900	+	0.791708i	$0.709192\pi$
$42$		0			0
$43$	−0.354484	−	0.354484i	−0.0540583	−	0.0540583i	0.679561	−	0.733619i	$-0.262170\pi$
$43$	−0.354484	−	0.354484i	−0.0540583	−	0.0540583i	−0.733619	+	0.679561i	$0.762170\pi$
$44$	−2.54561	−	3.33099i	−0.383765	−	0.502165i
$45$		0			0
$46$	−5.83057	−	6.66164i	−0.859671	−	0.982205i
$47$	−3.25028			−0.474101			−0.237051	−	0.971497i	$-0.576181\pi$
$47$	−3.25028			−0.474101			−0.237051	+	0.971497i	$0.576181\pi$
$48$		0			0
$49$	8.40885			1.20126
$50$	−5.11948	−	5.84919i	−0.724003	−	0.827200i
$51$		0			0
$52$	−6.62693	+	5.06444i	−0.918990	+	0.702312i
$53$	−7.21705	−	7.21705i	−0.991338	−	0.991338i	0.00862435	−	0.999963i	$-0.497255\pi$
$53$	−7.21705	−	7.21705i	−0.991338	−	0.991338i	−0.999963	+	0.00862435i	$0.997255\pi$
$54$		0			0
$55$	6.79120			0.915725
$56$	6.14121	+	9.24966i	0.820654	+	1.23604i
$57$		0			0
$58$	0.191285	−	2.87531i	0.0251169	−	0.377547i
$59$	−1.07589	+	1.07589i	−0.140069	+	0.140069i	−0.773665	−	0.633595i	$-0.781578\pi$
$59$	−1.07589	+	1.07589i	−0.140069	+	0.140069i	0.633595	+	0.773665i	$0.281578\pi$
$60$		0			0
$61$	8.81820	+	8.81820i	1.12905	+	1.12905i	0.990331	+	0.138723i	$0.0442999\pi$
$61$	8.81820	+	8.81820i	1.12905	+	1.12905i	0.138723	+	0.990331i	$0.455700\pi$
$62$	−8.10621	−	9.26163i	−1.02949	−	1.17623i
$63$		0			0
$64$	−3.10483	+	7.37293i	−0.388103	+	0.921616i
$65$		−	13.5110i		−	1.67583i
$66$		0			0
$67$	−4.72284	+	4.72284i	−0.576987	+	0.576987i	−0.934072	−	0.357085i	$-0.883771\pi$
$67$	−4.72284	+	4.72284i	−0.576987	+	0.576987i	0.357085	+	0.934072i	$0.383771\pi$
$68$	−8.01340	−	1.07095i	−0.971768	−	0.129872i
$69$		0			0
$70$	−17.9458	−	1.19387i	−2.14493	−	0.142695i
$71$			9.37491i			1.11260i	0.830982	+	0.556299i	$0.187779\pi$
$71$			9.37491i			1.11260i	−0.830982	+	0.556299i	$0.812221\pi$
$72$		0			0
$73$			7.02186i			0.821846i	0.911670	+	0.410923i	$0.134794\pi$
$73$			7.02186i			0.821846i	−0.911670	+	0.410923i	$0.865206\pi$
$74$	−0.533322	+	8.01666i	−0.0619974	+	0.931918i
$75$		0			0
$76$	−5.31336	−	6.95265i	−0.609484	−	0.797524i
$77$	5.81829	−	5.81829i	0.663056	−	0.663056i
$78$		0			0
$79$		−	9.40022i		−	1.05761i	−0.848744	−	0.528804i	$-0.822641\pi$
$79$		−	9.40022i		−	1.05761i	0.848744	−	0.528804i	$-0.177359\pi$
$80$	−6.43565	−	11.2484i	−0.719528	−	1.25761i
$81$		0			0
$82$	8.32540	−	7.28677i	0.919386	−	0.804689i
$83$	−4.41582	−	4.41582i	−0.484699	−	0.484699i	0.421929	−	0.906629i	$-0.361353\pi$
$83$	−4.41582	−	4.41582i	−0.484699	−	0.484699i	−0.906629	+	0.421929i	$0.861353\pi$
$84$		0			0
$85$	9.26056	−	9.26056i	1.00445	−	1.00445i
$86$	0.707405	+	0.0470613i	0.0762814	+	0.00507475i
$87$		0			0
$88$	5.81150	+	1.17373i	0.619508	+	0.125120i
$89$	4.59791			0.487377			0.243689	−	0.969854i	$-0.421643\pi$
$89$	4.59791			0.487377			0.243689	+	0.969854i	$0.421643\pi$
$90$		0			0
$91$	−11.5754	−	11.5754i	−1.21343	−	1.21343i
$92$	12.4095	+	1.65846i	1.29378	+	0.172907i
$93$		0			0
$94$	3.45886	−	3.02735i	0.356754	−	0.312248i
$95$	14.1750			1.45433
$96$		0			0
$97$	−19.5022			−1.98015			−0.990074	−	0.140550i	$-0.955113\pi$
$97$	−19.5022			−1.98015			−0.990074	+	0.140550i	$0.955113\pi$
$98$	−8.94848	+	7.83212i	−0.903933	+	0.791164i
$99$		0			0
$100$	10.8960	+	1.45620i	1.08960	+	0.145620i
$101$	−9.26033	−	9.26033i	−0.921438	−	0.921438i	0.0756936	−	0.997131i	$-0.475883\pi$
$101$	−9.26033	−	9.26033i	−0.921438	−	0.921438i	−0.997131	+	0.0756936i	$0.975883\pi$
$102$		0			0
$103$	8.33468			0.821240			0.410620	−	0.911807i	$-0.365312\pi$
$103$	8.33468			0.821240			0.410620	+	0.911807i	$0.365312\pi$
$104$	2.33512	−	11.5619i	0.228977	−	1.13373i
$105$		0			0
$106$	14.4023	+	0.958135i	1.39887	+	0.0930623i
$107$	2.94719	−	2.94719i	0.284915	−	0.284915i	−0.550150	−	0.835066i	$-0.685430\pi$
$107$	2.94719	−	2.94719i	0.284915	−	0.284915i	0.835066	+	0.550150i	$0.185430\pi$
$108$		0			0
$109$	−6.23851	−	6.23851i	−0.597541	−	0.597541i	0.342117	−	0.939657i	$-0.388856\pi$
$109$	−6.23851	−	6.23851i	−0.597541	−	0.597541i	−0.939657	+	0.342117i	$0.888856\pi$
$110$	−7.22702	+	6.32542i	−0.689070	+	0.603105i
$111$		0			0
$112$	−15.1506	−	4.12324i	−1.43160	−	0.389610i
$113$			3.00449i			0.282638i	0.989964	+	0.141319i	$0.0451344\pi$
$113$			3.00449i			0.282638i	−0.989964	+	0.141319i	$0.954866\pi$
$114$		0			0
$115$	−14.3408	+	14.3408i	−1.33729	+	1.33729i
$116$	2.47454	+	3.23800i	0.229756	+	0.300640i
$117$		0			0
$118$	0.142835	−	2.14704i	0.0131491	−	0.197651i
$119$		−	15.8678i		−	1.45460i
$120$		0			0
$121$			6.60610i			0.600554i
$122$	−17.5975	−	1.17070i	−1.59320	−	0.105990i
$123$		0			0
$124$	17.2528	+	2.30575i	1.54935	+	0.207063i
$125$	−1.13733	+	1.13733i	−0.101726	+	0.101726i
$126$		0			0
$127$		−	6.65448i		−	0.590489i	−0.955422	−	0.295245i	$-0.904599\pi$
$127$		−	6.65448i		−	0.590489i	0.955422	−	0.295245i	$-0.0954012\pi$
$128$	−3.56317	−	10.7380i	−0.314943	−	0.949111i
$129$		0			0
$130$	12.5843	+	14.3780i	1.10372	+	1.26103i
$131$	2.30064	+	2.30064i	0.201008	+	0.201008i	0.800432	−	0.599424i	$-0.204604\pi$
$131$	2.30064	+	2.30064i	0.201008	+	0.201008i	−0.599424	+	0.800432i	$0.704604\pi$
$132$		0			0
$133$	12.1443	−	12.1443i	1.05305	−	1.05305i
$134$	0.627003	−	9.42484i	0.0541649	−	0.814183i
$135$		0			0
$136$	9.52515	−	6.32412i	0.816775	−	0.542289i
$137$	−12.5130			−1.06906			−0.534530	−	0.845150i	$-0.679511\pi$
$137$	−12.5130			−1.06906			−0.534530	+	0.845150i	$0.679511\pi$
$138$		0			0
$139$	−4.50508	−	4.50508i	−0.382116	−	0.382116i	0.489748	−	0.871864i	$-0.337089\pi$
$139$	−4.50508	−	4.50508i	−0.382116	−	0.382116i	−0.871864	+	0.489748i	$0.837089\pi$
$140$	20.2094	−	15.4445i	1.70801	−	1.30530i
$141$		0			0
$142$	−8.73193	−	9.97654i	−0.732767	−	0.837213i
$143$	−8.74158			−0.731008
$144$		0			0
$145$	−6.60161			−0.548234
$146$	−6.54026	−	7.47248i	−0.541276	−	0.618427i
$147$		0			0
$148$	−6.89929	−	9.02787i	−0.567118	−	0.742087i
$149$	0.250368	+	0.250368i	0.0205110	+	0.0205110i	0.717288	−	0.696777i	$-0.245383\pi$
$149$	0.250368	+	0.250368i	0.0205110	+	0.0205110i	−0.696777	+	0.717288i	$0.745383\pi$
$150$		0			0
$151$	12.9004			1.04982			0.524910	−	0.851158i	$-0.324099\pi$
$151$	12.9004			1.04982			0.524910	+	0.851158i	$0.324099\pi$
$152$	12.1301	+	2.44989i	0.983885	+	0.198712i
$153$		0			0
$154$	−0.772436	+	11.6109i	−0.0622446	+	0.935634i
$155$	−19.9380	+	19.9380i	−1.60146	+	1.60146i
$156$		0			0
$157$	−12.6525	−	12.6525i	−1.00978	−	1.00978i	−0.999952	−	0.00982948i	$-0.996871\pi$
$157$	−12.6525	−	12.6525i	−1.00978	−	1.00978i	−0.00982948	−	0.999952i	$-0.503129\pi$
$158$	8.75550	+	10.0035i	0.696551	+	0.795834i
$159$		0			0
$160$	17.3255	+	5.97596i	1.36970	+	0.472441i
$161$			24.5727i			1.93660i
$162$		0			0
$163$	1.83086	−	1.83086i	0.143404	−	0.143404i	−0.631760	−	0.775164i	$-0.717667\pi$
$163$	1.83086	−	1.83086i	0.143404	−	0.143404i	0.775164	+	0.631760i	$0.217667\pi$
$164$	−2.07267	+	15.5088i	−0.161848	+	1.21103i
$165$		0			0
$166$	8.81216	+	0.586244i	0.683957	+	0.0455014i
$167$			6.95190i			0.537954i	0.963147	+	0.268977i	$0.0866856\pi$
$167$			6.95190i			0.537954i	−0.963147	+	0.268977i	$0.913314\pi$
$168$		0			0
$169$			4.39120i			0.337784i
$170$	−1.22943	+	18.4803i	−0.0942931	+	1.41737i
$171$		0			0
$172$	−0.796635	+	0.608806i	−0.0607429	+	0.0464210i
$173$	3.65405	−	3.65405i	0.277812	−	0.277812i	−0.554423	−	0.832235i	$-0.687061\pi$
$173$	3.65405	−	3.65405i	0.277812	−	0.277812i	0.832235	+	0.554423i	$0.187061\pi$
$174$		0			0
$175$			21.5758i			1.63098i
$176$	−7.27769	+	4.16386i	−0.548576	+	0.313863i
$177$		0			0
$178$	−4.89297	+	4.28256i	−0.366744	+	0.320991i
$179$	−3.58052	−	3.58052i	−0.267621	−	0.267621i	0.560520	−	0.828141i	$-0.310601\pi$
$179$	−3.58052	−	3.58052i	−0.267621	−	0.267621i	−0.828141	+	0.560520i	$0.810601\pi$
$180$		0			0
$181$	−14.5964	+	14.5964i	−1.08494	+	1.08494i	−0.0889003	+	0.996041i	$0.528335\pi$
$181$	−14.5964	+	14.5964i	−1.08494	+	1.08494i	−0.996041	+	0.0889003i	$0.971665\pi$
$182$	23.0997	+	1.53674i	1.71226	+	0.113911i
$183$		0			0
$184$	−14.7506	+	9.79349i	−1.08743	+	0.721985i
$185$	18.4060			1.35323
$186$		0			0
$187$	−5.99158	−	5.99158i	−0.438148	−	0.438148i
$188$	−0.861109	+	6.44327i	−0.0628028	+	0.469923i
$189$		0			0
$190$	−15.0847	+	13.2028i	−1.09436	+	0.957834i
$191$	26.5048			1.91782			0.958909	−	0.283714i	$-0.0915667\pi$
$191$	26.5048			1.91782			0.958909	+	0.283714i	$0.0915667\pi$
$192$		0			0
$193$	−16.8336			−1.21171			−0.605856	−	0.795574i	$-0.707169\pi$
$193$	−16.8336			−1.21171			−0.605856	+	0.795574i	$0.707169\pi$
$194$	20.7537	−	18.1646i	1.49003	−	1.30414i
$195$		0			0
$196$	2.22779	−	16.6695i	0.159128	−	1.19068i
$197$	−7.59408	−	7.59408i	−0.541056	−	0.541056i	0.382782	−	0.923839i	$-0.374966\pi$
$197$	−7.59408	−	7.59408i	−0.541056	−	0.541056i	−0.923839	+	0.382782i	$0.874966\pi$
$198$		0			0
$199$	6.50530			0.461148			0.230574	−	0.973055i	$-0.425940\pi$
$199$	6.50530			0.461148			0.230574	+	0.973055i	$0.425940\pi$
$200$	−12.9516	+	8.59907i	−0.915817	+	0.608046i
$201$		0			0
$202$	18.4798	+	1.22940i	1.30024	+	0.0865003i
$203$	−5.65586	+	5.65586i	−0.396964	+	0.396964i
$204$		0			0
$205$	−17.9225	−	17.9225i	−1.25176	−	1.25176i
$206$	−8.86955	+	7.76304i	−0.617971	+	0.540877i
$207$		0			0
$208$	8.28392	+	14.4788i	0.574386	+	1.00392i
$209$		−	9.17124i		−	0.634388i
$210$		0			0
$211$	−4.08987	+	4.08987i	−0.281558	+	0.281558i	−0.833730	−	0.552172i	$-0.813799\pi$
$211$	−4.08987	+	4.08987i	−0.281558	+	0.281558i	0.552172	+	0.833730i	$0.313799\pi$
$212$	−16.2189	+	12.3949i	−1.11392	+	0.851283i
$213$		0			0
$214$	−0.391268	+	5.88137i	−0.0267465	+	0.402042i
$215$		−	1.62418i		−	0.110768i
$216$		0			0
$217$			34.1633i			2.31916i
$218$	12.4495	+	0.828223i	0.843186	+	0.0560944i
$219$		0			0
$220$	1.79922	−	13.4627i	0.121303	−	0.907656i
$221$	−11.9201	+	11.9201i	−0.801835	+	0.801835i
$222$		0			0
$223$			3.52931i			0.236340i	0.992993	+	0.118170i	$0.0377027\pi$
$223$			3.52931i			0.236340i	−0.992993	+	0.118170i	$0.962297\pi$
$224$	19.9633	−	9.72363i	1.33386	−	0.649688i
$225$		0			0
$226$	−2.79842	−	3.19730i	−0.186148	−	0.212681i
$227$	3.43473	+	3.43473i	0.227971	+	0.227971i	0.811845	−	0.583874i	$-0.198464\pi$
$227$	3.43473	+	3.43473i	0.227971	+	0.227971i	−0.583874	+	0.811845i	$0.698464\pi$
$228$		0			0
$229$	9.48142	−	9.48142i	0.626550	−	0.626550i	−0.320649	−	0.947198i	$-0.603901\pi$
$229$	9.48142	−	9.48142i	0.626550	−	0.626550i	0.947198	+	0.320649i	$0.103901\pi$
$230$	1.90389	−	28.6184i	0.125539	−	1.88704i
$231$		0			0
$232$	−5.64926	−	1.14097i	−0.370892	−	0.0749081i
$233$	14.2756			0.935228			0.467614	−	0.883933i	$-0.345114\pi$
$233$	14.2756			0.935228			0.467614	+	0.883933i	$0.345114\pi$
$234$		0			0
$235$	−7.44606	−	7.44606i	−0.485727	−	0.485727i
$236$	1.84778	+	2.41786i	0.120280	+	0.157390i
$237$		0			0
$238$	14.7795	+	16.8861i	0.958012	+	1.09456i
$239$	1.92495			0.124515			0.0622575	−	0.998060i	$-0.480170\pi$
$239$	1.92495			0.124515			0.0622575	+	0.998060i	$0.480170\pi$
$240$		0			0
$241$	11.2699			0.725959			0.362980	−	0.931797i	$-0.381759\pi$
$241$	11.2699			0.725959			0.362980	+	0.931797i	$0.381759\pi$
$242$	−6.15301	−	7.03004i	−0.395531	−	0.451908i
$243$		0			0
$244$	19.8172	−	15.1447i	1.26867	−	0.969542i
$245$	19.2638	+	19.2638i	1.23072	+	1.23072i
$246$		0			0
$247$	−18.2460			−1.16096
$248$	−20.5076	+	13.6158i	−1.30224	+	0.864605i
$249$		0			0
$250$	0.150991	−	2.26964i	0.00954953	−	0.143544i
$251$	19.5469	−	19.5469i	1.23379	−	1.23379i	0.271297	−	0.962496i	$-0.412548\pi$
$251$	19.5469	−	19.5469i	1.23379	−	1.23379i	0.962496	−	0.271297i	$-0.0874524\pi$
$252$		0			0
$253$	9.27852	+	9.27852i	0.583335	+	0.583335i
$254$	6.19808	+	7.08152i	0.388902	+	0.444334i
$255$		0			0
$256$	13.7933	+	8.10827i	0.862083	+	0.506767i
$257$		−	14.7006i		−	0.917000i	−0.888695	−	0.458500i	$-0.848387\pi$
$257$		−	14.7006i		−	0.917000i	0.888695	−	0.458500i	$-0.151613\pi$
$258$		0			0
$259$	15.7691	−	15.7691i	0.979846	−	0.979846i
$260$	−26.7838	−	3.57951i	−1.66106	−	0.221992i
$261$		0			0
$262$	−4.59113	−	0.305432i	−0.283641	−	0.0188697i
$263$			11.0052i			0.678611i	0.940676	+	0.339305i	$0.110192\pi$
$263$			11.0052i			0.678611i	−0.940676	+	0.339305i	$0.889808\pi$
$264$		0			0
$265$		−	33.0671i		−	2.03130i
$266$	−1.61228	+	24.2350i	−0.0988550	+	1.48595i
$267$		0			0
$268$	8.11120	+	10.6137i	0.495470	+	0.648334i
$269$	14.7164	−	14.7164i	0.897277	−	0.897277i	−0.0979172	−	0.995195i	$-0.531218\pi$
$269$	14.7164	−	14.7164i	0.897277	−	0.897277i	0.995195	+	0.0979172i	$0.0312180\pi$
$270$		0			0
$271$			5.89744i			0.358244i	0.983827	+	0.179122i	$0.0573257\pi$
$271$			5.89744i			0.358244i	−0.983827	+	0.179122i	$0.942674\pi$
$272$	−4.24605	+	15.6018i	−0.257454	+	0.946000i
$273$		0			0
$274$	13.3160	−	11.6548i	0.804451	−	0.704093i
$275$	8.14691	+	8.14691i	0.491277	+	0.491277i
$276$		0			0
$277$	−11.0395	+	11.0395i	−0.663300	+	0.663300i	−0.956156	−	0.292856i	$-0.905394\pi$
$277$	−11.0395	+	11.0395i	−0.663300	+	0.663300i	0.292856	+	0.956156i	$0.405394\pi$
$278$	8.99029	+	0.598094i	0.539202	+	0.0358713i
$279$		0			0
$280$	−7.12116	+	35.2590i	−0.425570	+	2.10713i
$281$	−13.9387			−0.831512			−0.415756	−	0.909476i	$-0.636483\pi$
$281$	−13.9387			−0.831512			−0.415756	+	0.909476i	$0.636483\pi$
$282$		0			0
$283$	−7.85301	−	7.85301i	−0.466813	−	0.466813i	0.434067	−	0.900880i	$-0.357078\pi$
$283$	−7.85301	−	7.85301i	−0.466813	−	0.466813i	−0.900880	+	0.434067i	$0.857078\pi$
$284$	18.5846	+	2.48373i	1.10279	+	0.147383i
$285$		0			0
$286$	9.30257	−	8.14204i	0.550072	−	0.481449i
$287$	−30.7098			−1.81274
$288$		0			0
$289$	0.659614			0.0388008
$290$	7.02526	−	6.14884i	0.412538	−	0.361072i
$291$		0			0
$292$	13.9200	+	1.86033i	0.814604	+	0.108868i
$293$	7.03191	+	7.03191i	0.410809	+	0.410809i	0.882020	−	0.471211i	$-0.156183\pi$
$293$	7.03191	+	7.03191i	0.410809	+	0.410809i	−0.471211	+	0.882020i	$0.656183\pi$
$294$		0			0
$295$	−4.92953			−0.287008
$296$	15.7507	+	3.18113i	0.915493	+	0.184900i
$297$		0			0
$298$	−0.499632	−	0.0332389i	−0.0289429	−	0.00192547i
$299$	18.4594	−	18.4594i	1.06754	−	1.06754i
$300$		0			0
$301$	−1.39150	−	1.39150i	−0.0802045	−	0.0802045i
$302$	−13.7283	+	12.0156i	−0.789973	+	0.691421i
$303$		0			0
$304$	−15.1905	+	8.69108i	−0.871232	+	0.498468i
$305$			40.4032i			2.31348i
$306$		0			0
$307$	−2.27226	+	2.27226i	−0.129685	+	0.129685i	−0.768970	−	0.639285i	$-0.779230\pi$
$307$	−2.27226	+	2.27226i	−0.129685	+	0.129685i	0.639285	+	0.768970i	$0.279230\pi$
$308$	−9.99257	−	13.0755i	−0.569380	−	0.745046i
$309$		0			0
$310$	2.64696	−	39.7880i	0.150337	−	2.25981i
$311$		−	25.5297i		−	1.44766i	−0.689980	−	0.723829i	$-0.742381\pi$
$311$		−	25.5297i		−	1.44766i	0.689980	−	0.723829i	$-0.257619\pi$
$312$		0			0
$313$			13.4260i			0.758881i	0.925216	+	0.379441i	$0.123884\pi$
$313$			13.4260i			0.758881i	−0.925216	+	0.379441i	$0.876116\pi$
$314$	25.2492	+	1.67975i	1.42490	+	0.0947936i
$315$		0			0
$316$	−18.6348	−	2.49044i	−1.04829	−	0.140098i
$317$	−6.20914	+	6.20914i	−0.348740	+	0.348740i	−0.859640	−	0.510900i	$-0.829312\pi$
$317$	−6.20914	+	6.20914i	−0.348740	+	0.348740i	0.510900	+	0.859640i	$0.329312\pi$
$318$		0			0
$319$			4.27124i			0.239143i
$320$	−24.0035	+	9.77780i	−1.34184	+	0.546595i
$321$		0			0
$322$	−22.8874	−	26.1497i	−1.27546	−	1.45726i
$323$	−12.5060	−	12.5060i	−0.695853	−	0.695853i
$324$		0			0
$325$	16.2081	−	16.2081i	0.899064	−	0.899064i
$326$	−0.243065	+	3.65364i	−0.0134621	+	0.202357i
$327$		0			0
$328$	−12.2394	−	18.4346i	−0.675810	−	1.01788i
$329$	−12.7587			−0.703408
$330$		0			0
$331$	−21.7555	−	21.7555i	−1.19579	−	1.19579i	−0.975415	−	0.220374i	$-0.929272\pi$
$331$	−21.7555	−	21.7555i	−1.19579	−	1.19579i	−0.220374	−	0.975415i	$-0.570728\pi$
$332$	−9.92372	+	7.58391i	−0.544635	+	0.416221i
$333$		0			0
$334$	−6.47510	−	7.39804i	−0.354302	−	0.404803i
$335$	−21.6391			−1.18227
$336$		0			0
$337$	21.6166			1.17753			0.588764	−	0.808305i	$-0.299615\pi$
$337$	21.6166			1.17753			0.588764	+	0.808305i	$0.299615\pi$
$338$	−4.09002	−	4.67300i	−0.222468	−	0.254178i
$339$		0			0
$340$	−15.9045	−	20.8113i	−0.862541	−	1.12865i
$341$	12.8999	+	12.8999i	0.698567	+	0.698567i
$342$		0			0
$343$	5.53032			0.298609
$344$	0.280709	−	1.38987i	0.0151348	−	0.0749369i
$345$		0			0
$346$	−0.485112	+	7.29199i	−0.0260798	+	0.392020i
$347$	−6.44820	+	6.44820i	−0.346157	+	0.346157i	−0.858676	−	0.512519i	$-0.828713\pi$
$347$	−6.44820	+	6.44820i	−0.346157	+	0.346157i	0.512519	+	0.858676i	$0.328713\pi$
$348$		0			0
$349$	6.88298	+	6.88298i	0.368438	+	0.368438i	0.866907	−	0.498470i	$-0.166104\pi$
$349$	6.88298	+	6.88298i	0.368438	+	0.368438i	−0.498470	+	0.866907i	$0.666104\pi$
$350$	−20.0960	−	22.9605i	−1.07418	−	1.22729i
$351$		0			0
$352$	3.86644	−	11.2096i	0.206082	−	0.597475i
$353$		−	13.6358i		−	0.725758i	−0.931836	−	0.362879i	$-0.881794\pi$
$353$		−	13.6358i		−	0.725758i	0.931836	−	0.362879i	$-0.118206\pi$
$354$		0			0
$355$	−21.4770	+	21.4770i	−1.13988	+	1.13988i
$356$	1.21814	−	9.11478i	0.0645614	−	0.483082i
$357$		0			0
$358$	7.14524	+	0.475349i	0.377638	+	0.0251230i
$359$			6.92427i			0.365449i	0.983164	+	0.182725i	$0.0584917\pi$
$359$			6.92427i			0.365449i	−0.983164	+	0.182725i	$0.941508\pi$
$360$		0			0
$361$		−	0.142798i		−	0.00751568i
$362$	1.93781	−	29.1284i	0.101849	−	1.53095i
$363$		0			0
$364$	−26.0134	+	19.8800i	−1.36347	+	1.04200i
$365$	−16.0864	+	16.0864i	−0.842000	+	0.842000i
$366$		0			0
$367$		−	7.17865i		−	0.374723i	−0.982291	−	0.187361i	$-0.940006\pi$
$367$		−	7.17865i		−	0.374723i	0.982291	−	0.187361i	$-0.0599935\pi$
$368$	6.57540	−	24.1609i	0.342766	−	1.25947i
$369$		0			0
$370$	−19.5872	+	17.1436i	−1.01829	+	0.891253i
$371$	−28.3299	−	28.3299i	−1.47081	−	1.47081i
$372$		0			0
$373$	18.4024	−	18.4024i	0.952841	−	0.952841i	−0.0460960	−	0.998937i	$-0.514678\pi$
$373$	18.4024	−	18.4024i	0.952841	−	0.952841i	0.998937	+	0.0460960i	$0.0146780\pi$
$374$	11.9567	+	0.795442i	0.618268	+	0.0411313i
$375$		0			0
$376$	−5.08498	−	7.65881i	−0.262238	−	0.394973i
$377$	8.49754			0.437646
$378$		0			0
$379$	23.9539	+	23.9539i	1.23043	+	1.23043i	0.963798	+	0.266633i	$0.0859112\pi$
$379$	23.9539	+	23.9539i	1.23043	+	1.23043i	0.266633	+	0.963798i	$0.414089\pi$
$380$	3.75545	−	28.1002i	0.192650	−	1.44151i
$381$		0			0
$382$	−28.2057	+	24.6869i	−1.44313	+	1.26309i
$383$	−38.6212			−1.97345			−0.986725	−	0.162401i	$-0.948076\pi$
$383$	−38.6212			−1.97345			−0.986725	+	0.162401i	$0.948076\pi$
$384$		0			0
$385$	26.6583			1.35863
$386$	17.9139	−	15.6791i	0.911795	−	0.798045i
$387$		0			0
$388$	−5.16679	+	38.6606i	−0.262304	+	1.96270i
$389$	18.9545	+	18.9545i	0.961030	+	0.961030i	0.999269	−	0.0382386i	$-0.0121747\pi$
$389$	18.9545	+	18.9545i	0.961030	+	0.961030i	−0.0382386	+	0.999269i	$0.512175\pi$
$390$		0			0
$391$	25.3046			1.27971
$392$	13.1554	+	19.8142i	0.664450	+	1.00077i
$393$		0			0
$394$	15.1547	+	1.00819i	0.763481	+	0.0507919i
$395$	21.5350	−	21.5350i	1.08354	−	1.08354i
$396$		0			0
$397$	6.59843	+	6.59843i	0.331166	+	0.331166i	0.853029	−	0.521863i	$-0.174763\pi$
$397$	6.59843	+	6.59843i	0.331166	+	0.331166i	−0.521863	+	0.853029i	$0.674763\pi$
$398$	−6.92277	+	6.05913i	−0.347007	+	0.303717i
$399$		0			0
$400$	5.77346	−	21.2142i	0.288673	−	1.06071i
$401$		−	1.10271i		−	0.0550668i	−0.999621	−	0.0275334i	$-0.991235\pi$
$401$		−	1.10271i		−	0.0550668i	0.999621	−	0.0275334i	$-0.00876526\pi$
$402$		0			0
$403$	25.6640	−	25.6640i	1.27842	−	1.27842i
$404$	−20.8108	+	15.9041i	−1.03538	+	0.791257i
$405$		0			0
$406$	0.750871	−	11.2868i	0.0372651	−	0.560153i
$407$		−	11.9087i		−	0.590290i
$408$		0			0
$409$			11.2837i			0.557943i	0.960299	+	0.278972i	$0.0899936\pi$
$409$			11.2837i			0.557943i	−0.960299	+	0.278972i	$0.910006\pi$
$410$	35.7659	+	2.37939i	1.76635	+	0.117510i
$411$		0			0
$412$	2.20814	−	16.5225i	0.108787	−	0.814003i
$413$	−4.22332	+	4.22332i	−0.207816	+	0.207816i
$414$		0			0
$415$		−	20.2324i		−	0.993170i
$416$	−22.3013	−	7.69221i	−1.09341	−	0.377142i
$417$		0			0
$418$	8.54223	+	9.75980i	0.417814	+	0.477367i
$419$	−15.9684	−	15.9684i	−0.780106	−	0.780106i	0.199742	−	0.979848i	$-0.435989\pi$
$419$	−15.9684	−	15.9684i	−0.780106	−	0.780106i	−0.979848	+	0.199742i	$0.935989\pi$
$420$		0			0
$421$	−8.86432	+	8.86432i	−0.432021	+	0.432021i	−0.889315	−	0.457295i	$-0.848819\pi$
$421$	−8.86432	+	8.86432i	−0.432021	+	0.432021i	0.457295	+	0.889315i	$0.348819\pi$
$422$	0.542970	−	8.16169i	0.0264314	−	0.397305i
$423$		0			0
$424$	5.71504	−	28.2969i	0.277547	−	1.37422i
$425$	22.2184			1.07775
$426$		0			0
$427$	34.6151	+	34.6151i	1.67514	+	1.67514i
$428$	−5.06162	−	6.62324i	−0.244663	−	0.320146i
$429$		0			0
$430$	1.51278	+	1.72841i	0.0729528	+	0.0833512i
$431$	−4.03151			−0.194191			−0.0970955	−	0.995275i	$-0.530955\pi$
$431$	−4.03151			−0.194191			−0.0970955	+	0.995275i	$0.530955\pi$
$432$		0			0
$433$	−16.2331			−0.780114			−0.390057	−	0.920791i	$-0.627545\pi$
$433$	−16.2331			−0.780114			−0.390057	+	0.920791i	$0.627545\pi$
$434$	−31.8202	−	36.3557i	−1.52742	−	1.74513i
$435$		0			0
$436$	−14.0199	+	10.7143i	−0.671429	+	0.513120i
$437$	19.3667	+	19.3667i	0.926436	+	0.926436i
$438$		0			0
$439$	2.58870			0.123552			0.0617759	−	0.998090i	$-0.480324\pi$
$439$	2.58870			0.123552			0.0617759	+	0.998090i	$0.480324\pi$
$440$	10.6247	+	16.0025i	0.506512	+	0.762889i
$441$		0			0
$442$	1.58251	−	23.7877i	0.0752725	−	1.13146i
$443$	−16.9121	+	16.9121i	−0.803516	+	0.803516i	−0.983643	−	0.180127i	$-0.942349\pi$
$443$	−16.9121	+	16.9121i	−0.803516	+	0.803516i	0.180127	+	0.983643i	$0.442349\pi$
$444$		0			0
$445$	10.5333	+	10.5333i	0.499329	+	0.499329i
$446$	−3.28725	−	3.75580i	−0.155656	−	0.177842i
$447$		0			0
$448$	−12.1877	+	28.9418i	−0.575816	+	1.36737i
$449$		−	28.7158i		−	1.35518i	−0.735439	−	0.677591i	$-0.763024\pi$
$449$		−	28.7158i		−	1.35518i	0.735439	−	0.677591i	$-0.236976\pi$
$450$		0			0
$451$	−11.5958	+	11.5958i	−0.546027	+	0.546027i
$452$	5.95602	+	0.795991i	0.280148	+	0.0374403i
$453$		0			0
$454$	−6.85431	−	0.455994i	−0.321689	−	0.0214009i
$455$		−	53.0360i		−	2.48637i
$456$		0			0
$457$			9.36648i			0.438146i	0.975709	+	0.219073i	$0.0703032\pi$
$457$			9.36648i			0.438146i	−0.975709	+	0.219073i	$0.929697\pi$
$458$	−1.25875	+	18.9210i	−0.0588176	+	0.884121i
$459$		0			0
$460$	24.6295	+	32.2283i	1.14836	+	1.50265i
$461$	−5.78729	+	5.78729i	−0.269541	+	0.269541i	−0.828915	−	0.559374i	$-0.811041\pi$
$461$	−5.78729	+	5.78729i	−0.269541	+	0.269541i	0.559374	+	0.828915i	$0.311041\pi$
$462$		0			0
$463$		−	23.8978i		−	1.11063i	−0.831641	−	0.555314i	$-0.812598\pi$
$463$		−	23.8978i		−	1.11063i	0.831641	−	0.555314i	$-0.187402\pi$
$464$	7.07451	−	4.04762i	0.328426	−	0.187906i
$465$		0			0
$466$	−15.1918	+	13.2965i	−0.703745	+	0.615950i
$467$	9.30497	+	9.30497i	0.430583	+	0.430583i	0.888827	−	0.458244i	$-0.151521\pi$
$467$	9.30497	+	9.30497i	0.430583	+	0.430583i	−0.458244	+	0.888827i	$0.651521\pi$
$468$		0			0
$469$	−18.5391	+	18.5391i	−0.856055	+	0.856055i
$470$	14.8593	+	0.988538i	0.685407	+	0.0455979i
$471$		0			0
$472$	−4.21840	−	0.851978i	−0.194167	−	0.0392154i
$473$	−1.05084			−0.0483177
$474$		0			0
$475$	17.0048	+	17.0048i	0.780232	+	0.780232i
$476$	−31.4559	−	4.20392i	−1.44178	−	0.192686i
$477$		0			0
$478$	−2.04849	+	1.79293i	−0.0936956	+	0.0820067i
$479$	39.2595			1.79381			0.896907	−	0.442220i	$-0.145809\pi$
$479$	39.2595			1.79381			0.896907	+	0.442220i	$0.145809\pi$
$480$		0			0
$481$	−23.6920			−1.08026
$482$	−11.9932	+	10.4970i	−0.546273	+	0.478124i
$483$		0			0
$484$	13.0958	+	1.75018i	0.595262	+	0.0795536i
$485$	−44.6776	−	44.6776i	−2.02870	−	2.02870i
$486$		0			0
$487$	−8.11673			−0.367804			−0.183902	−	0.982945i	$-0.558873\pi$
$487$	−8.11673			−0.367804			−0.183902	+	0.982945i	$0.558873\pi$
$488$	−6.98295	+	34.5747i	−0.316103	+	1.56512i
$489$		0			0
$490$	−38.4427	−	2.55747i	−1.73666	−	0.115534i
$491$	−5.14359	+	5.14359i	−0.232127	+	0.232127i	−0.813580	−	0.581453i	$-0.802484\pi$
$491$	−5.14359	+	5.14359i	−0.232127	+	0.232127i	0.581453	+	0.813580i	$0.302484\pi$
$492$		0			0
$493$	5.82431	+	5.82431i	0.262314	+	0.262314i
$494$	19.4169	−	16.9946i	0.873608	−	0.764622i
$495$		0			0
$496$	9.14173	−	33.5907i	0.410476	−	1.50827i
$497$			36.8004i			1.65072i
$498$		0			0
$499$	−9.20531	+	9.20531i	−0.412086	+	0.412086i	−0.882465	−	0.470379i	$-0.844117\pi$
$499$	−9.20531	+	9.20531i	−0.412086	+	0.412086i	0.470379	+	0.882465i	$0.344117\pi$
$500$	1.95329	+	2.55592i	0.0873538	+	0.114304i
$501$		0			0
$502$	−2.59505	+	39.0077i	−0.115823	+	1.74100i
$503$			23.6311i			1.05366i	0.849971	+	0.526829i	$0.176619\pi$
$503$			23.6311i			1.05366i	−0.849971	+	0.526829i	$0.823381\pi$
$504$		0			0
$505$		−	42.4290i		−	1.88807i
$506$	−18.5161	−	1.23181i	−0.823141	−	0.0547609i
$507$		0			0
$508$	−13.1917	−	1.76300i	−0.585286	−	0.0782204i
$509$	11.3879	−	11.3879i	0.504759	−	0.504759i	−0.408154	−	0.912913i	$-0.633827\pi$
$509$	11.3879	−	11.3879i	0.504759	−	0.504759i	0.912913	+	0.408154i	$0.133827\pi$
$510$		0			0
$511$			27.5637i			1.21935i
$512$	−22.2307	+	4.21870i	−0.982466	+	0.186442i
$513$		0			0
$514$	13.6924	+	15.6440i	0.603945	+	0.690028i
$515$	19.0939	+	19.0939i	0.841379	+	0.841379i
$516$		0			0
$517$	−4.81760	+	4.81760i	−0.211878	+	0.211878i
$518$	−2.09351	+	31.4687i	−0.0919834	+	1.38266i
$519$		0			0
$520$	31.8366	−	21.1376i	1.39613	−	0.926944i
$521$	−41.2322			−1.80641			−0.903207	−	0.429206i	$-0.858794\pi$
$521$	−41.2322			−1.80641			−0.903207	+	0.429206i	$0.858794\pi$
$522$		0			0
$523$	−8.31489	−	8.31489i	−0.363585	−	0.363585i	0.501546	−	0.865131i	$-0.332765\pi$
$523$	−8.31489	−	8.31489i	−0.363585	−	0.363585i	−0.865131	+	0.501546i	$0.832765\pi$
$524$	5.17024	−	3.95121i	0.225863	−	0.172609i
$525$		0			0
$526$	−10.2504	−	11.7115i	−0.446939	−	0.510644i
$527$	35.1808			1.53250
$528$		0			0
$529$	−16.1865			−0.703762
$530$	30.7992	+	35.1892i	1.33783	+	1.52852i
$531$		0			0
$532$	−20.8571	−	27.2920i	−0.904271	−	1.18326i
$533$	23.0697	+	23.0697i	0.999260	+	0.999260i
$534$		0			0
$535$	13.5034			0.583804
$536$	−18.5175	−	3.73992i	−0.799832	−	0.161540i
$537$		0			0
$538$	−1.95375	+	29.3680i	−0.0842323	+	1.26614i
$539$	12.4637	−	12.4637i	0.536849	−	0.536849i
$540$		0			0
$541$	18.3380	+	18.3380i	0.788411	+	0.788411i	0.981234	−	0.192822i	$-0.0617641\pi$
$541$	18.3380	+	18.3380i	0.788411	+	0.788411i	−0.192822	+	0.981234i	$0.561764\pi$
$542$	−5.49296	−	6.27591i	−0.235943	−	0.269573i
$543$		0			0
$544$	−10.0132	−	20.5579i	−0.429314	−	0.881413i
$545$		−	28.5836i		−	1.22439i
$546$		0			0
$547$	5.60694	−	5.60694i	0.239736	−	0.239736i	−0.577005	−	0.816741i	$-0.695779\pi$
$547$	5.60694	−	5.60694i	0.239736	−	0.239736i	0.816741	+	0.577005i	$0.195779\pi$
$548$	−3.31513	+	24.8055i	−0.141615	+	1.05964i
$549$		0			0
$550$	−16.2579	−	1.08158i	−0.693238	−	0.0461188i
$551$			8.91520i			0.379800i
$552$		0			0
$553$		−	36.8997i		−	1.56914i
$554$	1.46560	−	22.0303i	0.0622676	−	0.935979i
$555$		0			0
$556$	−10.1243	+	7.73721i	−0.429366	+	0.328131i
$557$	23.1342	−	23.1342i	0.980228	−	0.980228i	−0.0195801	−	0.999808i	$-0.506233\pi$
$557$	23.1342	−	23.1342i	0.980228	−	0.980228i	0.999808	+	0.0195801i	$0.00623292\pi$
$558$		0			0
$559$			2.09063i			0.0884241i
$560$	−25.2626	−	44.1544i	−1.06754	−	1.86587i
$561$		0			0
$562$	14.8332	−	12.9827i	0.625700	−	0.547642i
$563$	−8.43813	−	8.43813i	−0.355625	−	0.355625i	0.506573	−	0.862197i	$-0.330912\pi$
$563$	−8.43813	−	8.43813i	−0.355625	−	0.355625i	−0.862197	+	0.506573i	$0.830912\pi$
$564$		0			0
$565$	−6.88298	+	6.88298i	−0.289569	+	0.289569i
$566$	15.6714	+	1.04257i	0.658717	+	0.0438223i
$567$		0			0
$568$	−22.0906	+	14.6668i	−0.926902	+	0.615407i
$569$	30.8525			1.29341			0.646703	−	0.762742i	$-0.276147\pi$
$569$	30.8525			1.29341			0.646703	+	0.762742i	$0.276147\pi$
$570$		0			0
$571$	15.1081	+	15.1081i	0.632255	+	0.632255i	0.948633	−	0.316378i	$-0.102467\pi$
$571$	15.1081	+	15.1081i	0.632255	+	0.632255i	−0.316378	+	0.948633i	$0.602467\pi$
$572$	−2.31594	+	17.3291i	−0.0968345	+	0.724566i
$573$		0			0
$574$	32.6806	−	28.6036i	1.36406	−	1.19389i
$575$	−34.4073			−1.43488
$576$		0			0
$577$	4.79886			0.199779			0.0998895	−	0.994999i	$-0.468151\pi$
$577$	4.79886			0.199779			0.0998895	+	0.994999i	$0.468151\pi$
$578$	−0.701945	+	0.614374i	−0.0291970	+	0.0255546i
$579$		0			0
$580$	−1.74899	+	13.0869i	−0.0726229	+	0.543403i
$581$	−17.3339	−	17.3339i	−0.719132	−	0.719132i
$582$		0			0
$583$	−21.3944			−0.886066
$584$	−16.5460	+	10.9855i	−0.684678	+	0.454585i
$585$		0			0
$586$	−14.0328	−	0.933556i	−0.579690	−	0.0385648i
$587$	−27.3522	+	27.3522i	−1.12894	+	1.12894i	−0.138595	+	0.990349i	$0.544259\pi$
$587$	−27.3522	+	27.3522i	−1.12894	+	1.12894i	−0.990349	+	0.138595i	$0.955741\pi$
$588$		0			0
$589$	26.9254	+	26.9254i	1.10944	+	1.10944i
$590$	5.24588	−	4.59143i	0.215969	−	0.189026i
$591$		0			0
$592$	−19.7245	+	11.2852i	−0.810671	+	0.463818i
$593$			34.6331i			1.42221i	0.703086	+	0.711105i	$0.251805\pi$
$593$			34.6331i			1.42221i	−0.703086	+	0.711105i	$0.748195\pi$
$594$		0			0
$595$	36.3515	−	36.3515i	1.49027	−	1.49027i
$596$	0.562655	−	0.429993i	0.0230472	−	0.0176132i
$597$		0			0
$598$	−2.45067	+	36.8374i	−0.100215	+	1.50639i
$599$			44.5187i			1.81898i	0.415722	+	0.909492i	$0.363529\pi$
$599$			44.5187i			1.81898i	−0.415722	+	0.909492i	$0.636471\pi$
$600$		0			0
$601$		−	37.0714i		−	1.51217i	−0.654471	−	0.756087i	$-0.727109\pi$
$601$		−	37.0714i		−	1.51217i	0.654471	−	0.756087i	$-0.272891\pi$
$602$	2.77685	+	0.184735i	0.113176	+	0.00752923i
$603$		0			0
$604$	3.41775	−	25.5734i	0.139066	−	1.04057i
$605$	−15.1339	+	15.1339i	−0.615281	+	0.615281i
$606$		0			0
$607$		−	17.7409i		−	0.720081i	−0.932937	−	0.360041i	$-0.882763\pi$
$607$		−	17.7409i		−	0.720081i	0.932937	−	0.360041i	$-0.117237\pi$
$608$	8.07029	−	23.3974i	0.327294	−	0.948891i
$609$		0			0
$610$	−37.6322	−	42.9961i	−1.52368	−	1.74086i
$611$	9.58451	+	9.58451i	0.387748	+	0.387748i
$612$		0			0
$613$	−0.100232	+	0.100232i	−0.00404832	+	0.00404832i	−0.709128	−	0.705080i	$-0.750911\pi$
$613$	−0.100232	+	0.100232i	−0.00404832	+	0.00404832i	0.705080	+	0.709128i	$0.250911\pi$
$614$	0.301665	−	4.53450i	0.0121742	−	0.182997i
$615$		0			0
$616$	22.8125	+	4.60738i	0.919144	+	0.185637i
$617$	3.11363			0.125350			0.0626751	−	0.998034i	$-0.480037\pi$
$617$	3.11363			0.125350			0.0626751	+	0.998034i	$0.480037\pi$
$618$		0			0
$619$	31.4834	+	31.4834i	1.26543	+	1.26543i	0.948426	+	0.317000i	$0.102675\pi$
$619$	31.4834	+	31.4834i	1.26543	+	1.26543i	0.317000	+	0.948426i	$0.397325\pi$
$620$	34.2423	+	44.8068i	1.37520	+	1.79948i
$621$		0			0
$622$	23.7787	+	27.1681i	0.953441	+	1.08934i
$623$	18.0487			0.723105
$624$		0			0
$625$	22.2713			0.890851
$626$	−12.5052	−	14.2876i	−0.499807	−	0.571047i
$627$		0			0
$628$	−28.4341	+	21.7300i	−1.13465	+	0.867120i
$629$	−16.2388	−	16.2388i	−0.647483	−	0.647483i
$630$		0			0
$631$	−0.355819			−0.0141649			−0.00708247	−	0.999975i	$-0.502254\pi$
$631$	−0.355819			−0.0141649			−0.00708247	+	0.999975i	$0.502254\pi$
$632$	22.1503	−	14.7064i	0.881091	−	0.584990i
$633$		0			0
$634$	0.824324	−	12.3909i	0.0327381	−	0.492105i
$635$	15.2447	−	15.2447i	0.604969	−	0.604969i
$636$		0			0
$637$	−24.7963	−	24.7963i	−0.982464	−	0.982464i
$638$	−3.97829	−	4.54534i	−0.157502	−	0.179952i
$639$		0			0
$640$	16.4367	−	32.7625i	0.649719	−	1.29505i
$641$			14.7273i			0.581693i	0.956770	+	0.290847i	$0.0939369\pi$
$641$			14.7273i			0.581693i	−0.956770	+	0.290847i	$0.906063\pi$
$642$		0			0
$643$	29.4236	−	29.4236i	1.16035	−	1.16035i	0.175956	−	0.984398i	$-0.443698\pi$
$643$	29.4236	−	29.4236i	1.16035	−	1.16035i	0.984398	−	0.175956i	$-0.0563017\pi$
$644$	48.7124	+	6.51015i	1.91954	+	0.256536i
$645$		0			0
$646$	24.9569	+	1.66030i	0.981914	+	0.0653235i
$647$		−	13.2502i		−	0.520918i	−0.965485	−	0.260459i	$-0.916126\pi$
$647$		−	13.2502i		−	0.520918i	0.965485	−	0.260459i	$-0.0838738\pi$
$648$		0			0
$649$			3.18940i			0.125195i
$650$	−2.15179	+	32.3447i	−0.0844000	+	1.26866i
$651$		0			0
$652$	−3.14439	−	4.11451i	−0.123144	−	0.161137i
$653$	12.7276	−	12.7276i	0.498068	−	0.498068i	−0.412768	−	0.910836i	$-0.635438\pi$
$653$	12.7276	−	12.7276i	0.498068	−	0.498068i	0.910836	+	0.412768i	$0.135438\pi$
$654$		0			0
$655$			10.5411i			0.411874i
$656$	30.1951	+	8.21762i	1.17892	+	0.320844i
$657$		0			0
$658$	13.5774	−	11.8836i	0.529304	−	0.463271i
$659$	13.2415	+	13.2415i	0.515816	+	0.515816i	0.916302	−	0.400487i	$-0.131159\pi$
$659$	13.2415	+	13.2415i	0.515816	+	0.515816i	−0.400487	+	0.916302i	$0.631159\pi$
$660$		0			0
$661$	−29.7820	+	29.7820i	−1.15839	+	1.15839i	−0.173562	+	0.984823i	$0.555528\pi$
$661$	−29.7820	+	29.7820i	−1.15839	+	1.15839i	−0.984823	+	0.173562i	$0.944472\pi$
$662$	43.4150	+	2.88826i	1.68737	+	0.112255i
$663$		0			0
$664$	3.49680	−	17.3137i	0.135702	−	0.671902i
$665$	55.6428			2.15774
$666$		0			0
$667$	−9.01949	−	9.01949i	−0.349236	−	0.349236i
$668$	13.7813	+	1.84180i	0.533214	+	0.0712612i
$669$		0			0
$670$	23.0278	−	20.1550i	0.889641	−	0.778655i
$671$	26.1409			1.00916
$672$		0			0
$673$	36.8616			1.42091			0.710455	−	0.703743i	$-0.248489\pi$
$673$	36.8616			1.42091			0.710455	+	0.703743i	$0.248489\pi$
$674$	−23.0038	+	20.1340i	−0.886073	+	0.775532i
$675$		0			0
$676$	8.70500	+	1.16338i	0.334808	+	0.0447453i
$677$	12.4269	+	12.4269i	0.477604	+	0.477604i	0.904365	−	0.426760i	$-0.140345\pi$
$677$	12.4269	+	12.4269i	0.477604	+	0.477604i	−0.426760	+	0.904365i	$0.640345\pi$
$678$		0			0
$679$	−76.5541			−2.93788
$680$	36.3091	+	7.33325i	1.39239	+	0.281217i
$681$		0			0
$682$	−25.7428	−	1.71258i	−0.985743	−	0.0655782i
$683$	−28.7706	+	28.7706i	−1.10088	+	1.10088i	−0.106572	+	0.994305i	$0.533987\pi$
$683$	−28.7706	+	28.7706i	−1.10088	+	1.10088i	−0.994305	+	0.106572i	$0.966013\pi$
$684$		0			0
$685$	−28.6661	−	28.6661i	−1.09528	−	1.09528i
$686$	−5.88522	+	5.15102i	−0.224699	+	0.196667i
$687$		0			0
$688$	0.995825	+	1.74052i	0.0379655	+	0.0663568i
$689$			42.5637i			1.62155i
$690$		0			0
$691$	−28.4129	+	28.4129i	−1.08088	+	1.08088i	−0.0844497	+	0.996428i	$0.526913\pi$
$691$	−28.4129	+	28.4129i	−1.08088	+	1.08088i	−0.996428	+	0.0844497i	$0.973087\pi$
$692$	−6.27562	−	8.21178i	−0.238563	−	0.312165i
$693$		0			0
$694$	0.856062	−	12.8680i	0.0324957	−	0.488461i
$695$		−	20.6414i		−	0.782972i
$696$		0			0
$697$			31.6245i			1.19786i
$698$	−13.7356	−	0.913784i	−0.519900	−	0.0345872i
$699$		0			0
$700$	42.7714	+	5.71618i	1.61661	+	0.216051i
$701$	−19.8190	+	19.8190i	−0.748553	+	0.748553i	−0.974207	−	0.225654i	$-0.927548\pi$
$701$	−19.8190	+	19.8190i	−0.748553	+	0.748553i	0.225654	+	0.974207i	$0.427548\pi$
$702$		0			0
$703$		−	24.8565i		−	0.937481i
$704$	6.32623	+	15.5303i	0.238429	+	0.585318i
$705$		0			0
$706$	12.7005	+	14.5108i	0.477991	+	0.546122i
$707$	−36.3506	−	36.3506i	−1.36711	−	1.36711i
$708$		0			0
$709$	−8.72547	+	8.72547i	−0.327692	+	0.327692i	−0.851708	−	0.524016i	$-0.824433\pi$
$709$	−8.72547	+	8.72547i	−0.327692	+	0.327692i	0.524016	+	0.851708i	$0.324433\pi$
$710$	2.85128	−	42.8593i	0.107007	−	1.60848i
$711$		0			0
$712$	7.19332	+	10.8343i	0.269581	+	0.406033i
$713$	−54.4807			−2.04032
$714$		0			0
$715$	−20.0261	−	20.0261i	−0.748934	−	0.748934i
$716$	−8.04653	+	6.14933i	−0.300713	+	0.229811i
$717$		0			0
$718$	−6.44937	−	7.36863i	−0.240688	−	0.274995i
$719$	−0.962462			−0.0358938			−0.0179469	−	0.999839i	$-0.505713\pi$
$719$	−0.962462			−0.0358938			−0.0179469	+	0.999839i	$0.505713\pi$
$720$		0			0
$721$	32.7170			1.21845
$722$	0.133004	+	0.151962i	0.00494989	+	0.00565543i
$723$		0			0
$724$	25.0684	+	32.8026i	0.931661	+	1.21910i
$725$	−7.91947	−	7.91947i	−0.294122	−	0.294122i
$726$		0			0
$727$	−47.3108			−1.75466			−0.877330	−	0.479888i	$-0.840677\pi$
$727$	−47.3108			−1.75466			−0.877330	+	0.479888i	$0.840677\pi$
$728$	9.16630	−	45.3851i	0.339726	−	1.68208i
$729$		0			0
$730$	2.13563	−	32.1018i	0.0790431	−	1.18814i
$731$	−1.43294	+	1.43294i	−0.0529992	+	0.0529992i
$732$		0			0
$733$	−15.9046	−	15.9046i	−0.587449	−	0.587449i	0.349491	−	0.936940i	$-0.386355\pi$
$733$	−15.9046	−	15.9046i	−0.587449	−	0.587449i	−0.936940	+	0.349491i	$0.886355\pi$
$734$	6.68630	+	7.63934i	0.246796	+	0.281973i
$735$		0			0
$736$	15.5064	+	31.8358i	0.571575	+	1.17348i
$737$			14.0005i			0.515715i
$738$		0			0
$739$	−5.75402	+	5.75402i	−0.211665	+	0.211665i	−0.804974	−	0.593309i	$-0.797821\pi$
$739$	−5.75402	+	5.75402i	−0.211665	+	0.211665i	0.593309	+	0.804974i	$0.297821\pi$
$740$	4.87637	−	36.4876i	0.179259	−	1.34131i
$741$		0			0
$742$	56.5348	+	3.76107i	2.07546	+	0.138073i
$743$			0.582783i			0.0213802i	0.999943	+	0.0106901i	$0.00340283\pi$
$743$			0.582783i			0.0213802i	−0.999943	+	0.0106901i	$0.996597\pi$
$744$		0			0
$745$			1.14714i			0.0420279i
$746$	−2.44310	+	36.7237i	−0.0894483	+	1.34455i
$747$		0			0
$748$	−13.4649	+	10.2902i	−0.492327	+	0.376246i
$749$	11.5689	−	11.5689i	0.422719	−	0.422719i
$750$		0			0
$751$			3.49645i			0.127587i	0.997963	+	0.0637936i	$0.0203199\pi$
$751$			3.49645i			0.127587i	−0.997963	+	0.0637936i	$0.979680\pi$
$752$	12.5448	+	3.41408i	0.457463	+	0.124499i
$753$		0			0
$754$	−9.04287	+	7.91473i	−0.329322	+	0.288238i
$755$	29.5535	+	29.5535i	1.07556	+	1.07556i
$756$		0			0
$757$	13.4628	−	13.4628i	0.489313	−	0.489313i	−0.418776	−	0.908089i	$-0.637541\pi$
$757$	13.4628	−	13.4628i	0.489313	−	0.489313i	0.908089	+	0.418776i	$0.137541\pi$
$758$	−47.8022	−	3.18012i	−1.73625	−	0.115507i
$759$		0			0
$760$	22.1765	+	33.4014i	0.804426	+	1.21160i
$761$	−14.1627			−0.513399			−0.256699	−	0.966491i	$-0.582635\pi$
$761$	−14.1627			−0.513399			−0.256699	+	0.966491i	$0.582635\pi$
$762$		0			0
$763$	−24.4887	−	24.4887i	−0.886551	−	0.886551i
$764$	7.02202	−	52.5424i	0.254048	−	1.90092i
$765$		0			0
$766$	41.0997	−	35.9723i	1.48499	−	1.29973i
$767$	6.34525			0.229114
$768$		0			0
$769$	−23.2136			−0.837105			−0.418552	−	0.908193i	$-0.637462\pi$
$769$	−23.2136			−0.837105			−0.418552	+	0.908193i	$0.637462\pi$
$770$	−28.3690	+	24.8299i	−1.02235	+	0.894807i
$771$		0			0
$772$	−4.45981	+	33.3706i	−0.160512	+	1.20103i
$773$	−27.6986	−	27.6986i	−0.996251	−	0.996251i	0.00374182	−	0.999993i	$-0.498809\pi$
$773$	−27.6986	−	27.6986i	−0.996251	−	0.996251i	−0.999993	+	0.00374182i	$0.998809\pi$
$774$		0			0
$775$	−47.8363			−1.71833
$776$	−30.5107	−	45.9541i	−1.09527	−	1.64966i
$777$		0			0
$778$	−37.8253	−	2.51639i	−1.35610	−	0.0902171i
$779$	−24.2036	+	24.2036i	−0.867184	+	0.867184i
$780$		0			0
$781$	13.8956	+	13.8956i	0.497224	+	0.497224i
$782$	−26.9285	+	23.5691i	−0.962961	+	0.842828i
$783$		0			0
$784$	−32.4550	−	8.83263i	−1.15911	−	0.315451i
$785$		−	57.9713i		−	2.06909i
$786$		0			0
$787$	14.9190	−	14.9190i	0.531806	−	0.531806i	−0.389304	−	0.921109i	$-0.627284\pi$
$787$	14.9190	−	14.9190i	0.531806	−	0.531806i	0.921109	+	0.389304i	$0.127284\pi$
$788$	−17.0663	+	13.0424i	−0.607960	+	0.464616i
$789$		0			0
$790$	−2.85898	+	42.9750i	−0.101718	+	1.52898i
$791$			11.7938i			0.419341i
$792$		0			0
$793$		−	52.0067i		−	1.84681i
$794$	−13.1678	−	0.876007i	−0.467306	−	0.0310883i
$795$		0			0
$796$	1.72348	−	12.8959i	0.0610870	−	0.457084i
$797$	12.0255	−	12.0255i	0.425964	−	0.425964i	−0.461287	−	0.887251i	$-0.652612\pi$
$797$	12.0255	−	12.0255i	0.425964	−	0.425964i	0.887251	+	0.461287i	$0.152612\pi$
$798$		0			0
$799$			13.1387i			0.464813i
$800$	13.6153	+	27.9531i	0.481372	+	0.988292i
$801$		0			0
$802$	1.02708	+	1.17348i	0.0362675	+	0.0414369i
$803$	10.4079	+	10.4079i	0.367286	+	0.367286i
$804$		0			0
$805$	−56.2937	+	56.2937i	−1.98409	+	1.98409i
$806$	−3.40715	+	51.2148i	−0.120012	+	1.80396i
$807$		0			0
$808$	7.33307	−	36.3082i	0.257976	−	1.27732i
$809$	40.4180			1.42102			0.710511	−	0.703686i	$-0.248464\pi$
$809$	40.4180			1.42102			0.710511	+	0.703686i	$0.248464\pi$
$810$		0			0
$811$	16.0921	+	16.0921i	0.565069	+	0.565069i	0.930743	−	0.365674i	$-0.119162\pi$
$811$	16.0921	+	16.0921i	0.565069	+	0.565069i	−0.365674	+	0.930743i	$0.619162\pi$
$812$	9.71361	+	12.7105i	0.340881	+	0.446050i
$813$		0			0
$814$	11.0919	+	12.6729i	0.388771	+	0.444185i
$815$	8.38864			0.293841
$816$		0			0
$817$	−2.19338			−0.0767368
$818$	−10.5098	−	12.0078i	−0.367467	−	0.419844i
$819$		0			0
$820$	−40.2774	+	30.7808i	−1.40655	+	1.07491i
$821$	−7.19331	−	7.19331i	−0.251048	−	0.251048i	0.570352	−	0.821400i	$-0.306807\pi$
$821$	−7.19331	−	7.19331i	−0.251048	−	0.251048i	−0.821400	+	0.570352i	$0.806807\pi$
$822$		0			0
$823$	4.73086			0.164907			0.0824536	−	0.996595i	$-0.473724\pi$
$823$	4.73086			0.164907			0.0824536	+	0.996595i	$0.473724\pi$
$824$	13.0394	+	19.6395i	0.454249	+	0.684173i
$825$		0			0
$826$	0.560688	−	8.42802i	0.0195088	−	0.293248i
$827$	24.6255	−	24.6255i	0.856311	−	0.856311i	−0.134590	−	0.990901i	$-0.542972\pi$
$827$	24.6255	−	24.6255i	0.856311	−	0.856311i	0.990901	+	0.134590i	$0.0429718\pi$
$828$		0			0
$829$	6.33048	+	6.33048i	0.219867	+	0.219867i	0.808442	−	0.588576i	$-0.200311\pi$
$829$	6.33048	+	6.33048i	0.219867	+	0.219867i	−0.588576	+	0.808442i	$0.700311\pi$
$830$	18.8448	+	21.5308i	0.654111	+	0.747346i
$831$		0			0
$832$	30.8971	−	12.5859i	1.07116	−	0.436338i
$833$		−	33.9913i		−	1.17773i
$834$		0			0
$835$	−15.9261	+	15.9261i	−0.551146	+	0.551146i
$836$	−18.1808	−	2.42977i	−0.628797	−	0.0840355i
$837$		0			0
$838$	31.8663	+	2.11996i	1.10080	+	0.0732328i
$839$			7.39471i			0.255294i	0.991820	+	0.127647i	$0.0407424\pi$
$839$			7.39471i			0.255294i	−0.991820	+	0.127647i	$0.959258\pi$
$840$		0			0
$841$			24.8480i			0.856828i
$842$	1.17683	−	17.6895i	0.0405561	−	0.609622i
$843$		0			0
$844$	7.02411	+	9.19120i	0.241780	+	0.316374i
$845$	−10.0598	+	10.0598i	−0.346067	+	0.346067i
$846$		0			0
$847$			25.9316i			0.891022i
$848$	20.2743	+	35.4359i	0.696223	+	1.21687i
$849$		0			0
$850$	−23.6443	+	20.6946i	−0.810993	+	0.709818i
$851$	25.1473	+	25.1473i	0.862038	+	0.862038i
$852$		0			0
$853$	4.27715	−	4.27715i	0.146447	−	0.146447i	−0.630082	−	0.776529i	$-0.716979\pi$
$853$	4.27715	−	4.27715i	0.146447	−	0.146447i	0.776529	+	0.630082i	$0.216979\pi$
$854$	−69.0774	−	4.59549i	−2.36378	−	0.157254i
$855$		0			0
$856$	11.5554	+	2.33382i	0.394956	+	0.0797682i
$857$	9.88470			0.337655			0.168827	−	0.985646i	$-0.446002\pi$
$857$	9.88470			0.337655			0.168827	+	0.985646i	$0.446002\pi$
$858$		0			0
$859$	−29.0135	−	29.0135i	−0.989927	−	0.989927i	0.0100226	−	0.999950i	$-0.496810\pi$
$859$	−29.0135	−	29.0135i	−0.989927	−	0.989927i	−0.999950	+	0.0100226i	$0.996810\pi$
$860$	−3.21973	−	0.430300i	−0.109792	−	0.0146731i
$861$		0			0
$862$	4.29023	−	3.75501i	0.146126	−	0.127896i
$863$	21.8097			0.742410			0.371205	−	0.928551i	$-0.378945\pi$
$863$	21.8097			0.742410			0.371205	+	0.928551i	$0.378945\pi$
$864$		0			0
$865$	16.7421			0.569250
$866$	17.2749	−	15.1198i	0.587024	−	0.513790i
$867$		0			0
$868$	67.7245	+	9.05102i	2.29872	+	0.307212i
$869$	−13.9331	−	13.9331i	−0.472649	−	0.472649i
$870$		0			0
$871$	27.8537			0.943787
$872$	4.94015	−	24.4601i	0.167294	−	0.828325i
$873$		0			0
$874$	−38.6480	−	2.57112i	−1.30729	−	0.0869695i
$875$	−4.46447	+	4.46447i	−0.150927	+	0.150927i
$876$		0			0
$877$	6.55195	+	6.55195i	0.221244	+	0.221244i	0.809022	−	0.587778i	$-0.199997\pi$
$877$	6.55195	+	6.55195i	0.221244	+	0.221244i	−0.587778	+	0.809022i	$0.699997\pi$
$878$	−2.75483	+	2.41115i	−0.0929708	+	0.0813724i
$879$		0			0
$880$	−26.2115	−	7.13346i	−0.883588	−	0.240469i
$881$		−	16.2849i		−	0.548652i	−0.961637	−	0.274326i	$-0.911545\pi$
$881$		−	16.2849i		−	0.548652i	0.961637	−	0.274326i	$-0.0884548\pi$
$882$		0			0
$883$	−10.8315	+	10.8315i	−0.364509	+	0.364509i	−0.865470	−	0.500961i	$-0.832980\pi$
$883$	−10.8315	+	10.8315i	−0.364509	+	0.364509i	0.500961	+	0.865470i	$0.332980\pi$
$884$	20.4721	+	26.7882i	0.688552	+	0.900985i
$885$		0			0
$886$	2.24524	−	33.7495i	0.0754304	−	1.13384i
$887$			47.6155i			1.59877i	0.600818	+	0.799386i	$0.294841\pi$
$887$			47.6155i			1.59877i	−0.600818	+	0.799386i	$0.705159\pi$
$888$		0			0
$889$		−	26.1216i		−	0.876089i
$890$	−21.0202	−	1.39841i	−0.704600	−	0.0468747i
$891$		0			0
$892$	6.99641	+	0.935033i	0.234257	+	0.0313072i
$893$	−10.0556	+	10.0556i	−0.336498	+	0.336498i
$894$		0			0
$895$		−	16.4052i		−	0.548366i
$896$	−13.9869	−	42.1509i	−0.467270	−	1.40816i
$897$		0			0
$898$	26.7463	+	30.5586i	0.892536	+	1.01975i
$899$	−12.5397	−	12.5397i	−0.418223	−	0.418223i
$900$		0			0
$901$	−29.1737	+	29.1737i	−0.971916	+	0.971916i
$902$	1.53946	−	23.1405i	0.0512585	−	0.770496i
$903$		0			0
$904$	−7.07964	+	4.70045i	−0.235465	+	0.156335i
$905$	−66.8777			−2.22309
$906$		0			0
$907$	30.6909	+	30.6909i	1.01907	+	1.01907i	0.999815	+	0.0192587i	$0.00613062\pi$
$907$	30.6909	+	30.6909i	1.01907	+	1.01907i	0.0192587	+	0.999815i	$0.493869\pi$
$908$	7.71890	−	5.89894i	0.256161	−	0.195763i
$909$		0			0
$910$	49.3985	+	56.4396i	1.63755	+	1.87095i
$911$	14.8927			0.493416			0.246708	−	0.969090i	$-0.420651\pi$
$911$	14.8927			0.493416			0.246708	+	0.969090i	$0.420651\pi$
$912$		0			0
$913$	−13.0904			−0.433228
$914$	−8.72408	−	9.96757i	−0.288567	−	0.329698i
$915$		0			0
$916$	−16.2838	−	21.3077i	−0.538031	−	0.704025i
$917$	9.03095	+	9.03095i	0.298228	+	0.298228i
$918$		0			0
$919$	−15.5644			−0.513422			−0.256711	−	0.966488i	$-0.582639\pi$
$919$	−15.5644			−0.513422			−0.256711	+	0.966488i	$0.582639\pi$
$920$	−56.2280	−	11.3562i	−1.85378	−	0.374403i
$921$		0			0
$922$	0.768320	−	11.5490i	0.0253033	−	0.380348i
$923$	27.6450	−	27.6450i	0.909947	−	0.909947i
$924$		0			0
$925$	22.0803	+	22.0803i	0.725996	+	0.725996i
$926$	22.2588	+	25.4315i	0.731470	+	0.835730i
$927$		0			0
$928$	−3.75850	+	10.8967i	−0.123379	+	0.357701i
$929$		−	52.2135i		−	1.71307i	−0.516091	−	0.856534i	$-0.672613\pi$
$929$		−	52.2135i		−	1.71307i	0.516091	−	0.856534i	$-0.327387\pi$
$930$		0			0
$931$	26.0150	−	26.0150i	0.852608	−	0.852608i
$932$	3.78210	−	28.2997i	0.123887	−	0.926987i
$933$		0			0
$934$	−18.5689	−	1.23533i	−0.607593	−	0.0404211i
$935$		−	27.4522i		−	0.897784i
$936$		0			0
$937$		−	2.44912i		−	0.0800094i	−0.999199	−	0.0400047i	$-0.987263\pi$
$937$		−	2.44912i		−	0.0800094i	0.999199	−	0.0400047i	$-0.0127373\pi$
$938$	2.46125	−	36.9964i	0.0803625	−	1.20797i
$939$		0			0
$940$	−16.7336	+	12.7882i	−0.545790	+	0.417104i
$941$	28.8517	−	28.8517i	0.940539	−	0.940539i	−0.0577894	−	0.998329i	$-0.518405\pi$
$941$	28.8517	−	28.8517i	0.940539	−	0.940539i	0.998329	+	0.0577894i	$0.0184052\pi$
$942$		0			0
$943$		−	48.9734i		−	1.59479i
$944$	5.28265	−	3.02242i	0.171936	−	0.0983715i
$945$		0			0
$946$	1.11828	−	0.978769i	0.0363584	−	0.0318225i
$947$	9.62547	+	9.62547i	0.312786	+	0.312786i	0.845988	−	0.533202i	$-0.179011\pi$
$947$	9.62547	+	9.62547i	0.312786	+	0.312786i	−0.533202	+	0.845988i	$0.679011\pi$
$948$		0			0
$949$	20.7063	−	20.7063i	0.672154	−	0.672154i
$950$	−33.9345	−	2.25755i	−1.10098	−	0.0732445i
$951$		0			0
$952$	37.3901	−	24.8248i	1.21182	−	0.804576i
$953$	−35.4241			−1.14750			−0.573749	−	0.819031i	$-0.694511\pi$
$953$	−35.4241			−1.14750			−0.573749	+	0.819031i	$0.694511\pi$
$954$		0			0
$955$	60.7198	+	60.7198i	1.96485	+	1.96485i
$956$	0.509986	−	3.81598i	0.0164941	−	0.123418i
$957$		0			0
$958$	−41.7790	+	36.5669i	−1.34982	+	1.18142i
$959$	−49.1188			−1.58613
$960$		0			0
$961$	−44.7442			−1.44336
$962$	25.2125	−	22.0671i	0.812882	−	0.711472i
$963$		0			0
$964$	2.98578	−	22.3412i	0.0961657	−	0.719561i
$965$	−38.5642	−	38.5642i	−1.24143	−	1.24143i
$966$		0			0
$967$	15.4575			0.497079			0.248539	−	0.968622i	$-0.420049\pi$
$967$	15.4575			0.497079			0.248539	+	0.968622i	$0.420049\pi$
$968$	−15.5663	+	10.3351i	−0.500320	+	0.332182i
$969$		0			0
$970$	89.1581	+	5.93139i	2.86269	+	0.190445i
$971$	19.0542	−	19.0542i	0.611479	−	0.611479i	−0.331852	−	0.943331i	$-0.607674\pi$
$971$	19.0542	−	19.0542i	0.611479	−	0.611479i	0.943331	+	0.331852i	$0.107674\pi$
$972$		0			0
$973$	−17.6843	−	17.6843i	−0.566932	−	0.566932i
$974$	8.63761	−	7.56004i	0.276767	−	0.242239i
$975$		0			0
$976$	−24.7723	−	43.2975i	−0.792941	−	1.38592i
$977$		−	34.7051i		−	1.11031i	−0.831745	−	0.555157i	$-0.812658\pi$
$977$		−	34.7051i		−	1.11031i	0.831745	−	0.555157i	$-0.187342\pi$
$978$		0			0
$979$	6.81507	−	6.81507i	0.217811	−	0.217811i
$980$	43.2918	−	33.0845i	1.38291	−	1.05685i
$981$		0			0
$982$	0.682863	−	10.2645i	0.0217910	−	0.327553i
$983$			20.0262i			0.638735i	0.947631	+	0.319368i	$0.103470\pi$
$983$			20.0262i			0.638735i	−0.947631	+	0.319368i	$0.896530\pi$
$984$		0			0
$985$		−	34.7946i		−	1.10865i
$986$	−11.6229	−	0.773235i	−0.370150	−	0.0246248i
$987$		0			0
$988$	−4.83399	+	36.1704i	−0.153790	+	1.15073i
$989$	2.21904	−	2.21904i	0.0705614	−	0.0705614i
$990$		0			0
$991$			14.3298i			0.455202i	0.973754	+	0.227601i	$0.0730882\pi$
$991$			14.3298i			0.455202i	−0.973754	+	0.227601i	$0.926912\pi$
$992$	21.5585	+	44.2611i	0.684483	+	1.40529i
$993$		0			0
$994$	−34.2764	−	39.1620i	−1.08718	−	1.24214i
$995$	14.9030	+	14.9030i	0.472457	+	0.472457i
$996$		0			0
$997$	−4.86384	+	4.86384i	−0.154039	+	0.154039i	−0.779919	−	0.625880i	$-0.784740\pi$
$997$	−4.86384	+	4.86384i	−0.154039	+	0.154039i	0.625880	+	0.779919i	$0.284740\pi$
$998$	1.22210	−	18.3700i	0.0386847	−	0.581492i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.l.a.323.4	yes	32
3.2	odd	2	inner	432.2.l.a.323.13	yes	32
4.3	odd	2		1728.2.l.a.431.14		32
12.11	even	2		1728.2.l.a.431.3		32
16.5	even	4		1728.2.l.a.1295.3		32
16.11	odd	4	inner	432.2.l.a.107.13	yes	32
48.5	odd	4		1728.2.l.a.1295.14		32
48.11	even	4	inner	432.2.l.a.107.4	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.l.a.107.4	✓	32	48.11	even	4	inner
432.2.l.a.107.13	yes	32	16.11	odd	4	inner
432.2.l.a.323.4	yes	32	1.1	even	1	trivial
432.2.l.a.323.13	yes	32	3.2	odd	2	inner
1728.2.l.a.431.3		32	12.11	even	2
1728.2.l.a.431.14		32	4.3	odd	2
1728.2.l.a.1295.3		32	16.5	even	4
1728.2.l.a.1295.14		32	48.5	odd	4

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:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	432.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.06417 + 0.931414i) q^{2} +(0.264934 - 1.98237i) q^{4} +(2.29090 + 2.29090i) q^{5} +3.92541 q^{7} +(1.56448 + 2.35636i) q^{8} +O(q^{10})\)	\(q+(-1.06417 + 0.931414i) q^{2} +(0.264934 - 1.98237i) q^{4} +(2.29090 + 2.29090i) q^{5} +3.92541 q^{7} +(1.56448 + 2.35636i) q^{8} +(-4.57170 - 0.304140i) q^{10} +(1.48221 - 1.48221i) q^{11} +(-2.94883 - 2.94883i) q^{13} +(-4.17732 + 3.65618i) q^{14} +(-3.85962 - 1.05040i) q^{16} -4.04232i q^{17} +(3.09377 - 3.09377i) q^{19} +(5.14836 - 3.93449i) q^{20} +(-0.196778 + 2.95789i) q^{22} +6.25991i q^{23} +5.49645i q^{25} +(5.88465 + 0.391486i) q^{26} +(1.03998 - 7.78164i) q^{28} +(-1.44083 + 1.44083i) q^{29} +8.70311i q^{31} +(5.08566 - 2.47710i) q^{32} +(3.76508 + 4.30174i) q^{34} +(8.99273 + 8.99273i) q^{35} +(4.01719 - 4.01719i) q^{37} +(-0.410728 + 6.17389i) q^{38} +(-1.81412 + 8.98224i) q^{40} -7.82334 q^{41} +(-0.354484 - 0.354484i) q^{43} +(-2.54561 - 3.33099i) q^{44} +(-5.83057 - 6.66164i) q^{46} -3.25028 q^{47} +8.40885 q^{49} +(-5.11948 - 5.84919i) q^{50} +(-6.62693 + 5.06444i) q^{52} +(-7.21705 - 7.21705i) q^{53} +6.79120 q^{55} +(6.14121 + 9.24966i) q^{56} +(0.191285 - 2.87531i) q^{58} +(-1.07589 + 1.07589i) q^{59} +(8.81820 + 8.81820i) q^{61} +(-8.10621 - 9.26163i) q^{62} +(-3.10483 + 7.37293i) q^{64} -13.5110i q^{65} +(-4.72284 + 4.72284i) q^{67} +(-8.01340 - 1.07095i) q^{68} +(-17.9458 - 1.19387i) q^{70} +9.37491i q^{71} +7.02186i q^{73} +(-0.533322 + 8.01666i) q^{74} +(-5.31336 - 6.95265i) q^{76} +(5.81829 - 5.81829i) q^{77} -9.40022i q^{79} +(-6.43565 - 11.2484i) q^{80} +(8.32540 - 7.28677i) q^{82} +(-4.41582 - 4.41582i) q^{83} +(9.26056 - 9.26056i) q^{85} +(0.707405 + 0.0470613i) q^{86} +(5.81150 + 1.17373i) q^{88} +4.59791 q^{89} +(-11.5754 - 11.5754i) q^{91} +(12.4095 + 1.65846i) q^{92} +(3.45886 - 3.02735i) q^{94} +14.1750 q^{95} -19.5022 q^{97} +(-8.94848 + 7.83212i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q - 4 q^{10} - 20 q^{16} + 8 q^{19} + 4 q^{22} - 12 q^{28} - 36 q^{34} - 12 q^{40} + 32 q^{43} - 16 q^{46} + 32 q^{49} - 60 q^{52} + 64 q^{55} - 48 q^{58} - 16 q^{61} + 48 q^{64} - 32 q^{67} - 72 q^{70} - 96 q^{76} + 40 q^{82} - 16 q^{85} + 36 q^{88} + 24 q^{91} - 36 q^{94}+O(q^{100}) \) 32 * q - 4 * q^10 - 20 * q^16 + 8 * q^19 + 4 * q^22 - 12 * q^28 - 36 * q^34 - 12 * q^40 + 32 * q^43 - 16 * q^46 + 32 * q^49 - 60 * q^52 + 64 * q^55 - 48 * q^58 - 16 * q^61 + 48 * q^64 - 32 * q^67 - 72 * q^70 - 96 * q^76 + 40 * q^82 - 16 * q^85 + 36 * q^88 + 24 * q^91 - 36 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more