LMFDB - Embedded newform 3888.2.a.bk.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3888,2,Mod(1,3888)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3888, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3888.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3888 = 2^{4} \cdot 3^{5} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3888.a (trivial)

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:	yes
Analytic conductor:	$31.0458363059$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 243)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.347296$ of defining polynomial
Character	$\chi$	$=$	3888.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.65270 q^{5} -2.41147 q^{7} +O(q^{10})$	$q+1.65270 q^{5} -2.41147 q^{7} -5.94356 q^{11} -3.22668 q^{13} +3.00000 q^{17} +6.63816 q^{19} +2.94356 q^{23} -2.26857 q^{25} -1.29086 q^{29} +0.588526 q^{31} -3.98545 q^{35} +0.0418891 q^{37} -4.90167 q^{41} +5.18479 q^{43} +3.73648 q^{47} -1.18479 q^{49} +11.6382 q^{53} -9.82295 q^{55} +7.34730 q^{59} +11.0496 q^{61} -5.33275 q^{65} -1.85710 q^{67} +5.51249 q^{71} +5.55438 q^{73} +14.3327 q^{77} +3.78106 q^{79} -3.98545 q^{83} +4.95811 q^{85} +8.15064 q^{89} +7.78106 q^{91} +10.9709 q^{95} +0.260830 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{5} + 3 q^{7}+O(q^{10}) $ 3 * q + 6 * q^5 + 3 * q^7	$ 3 q + 6 q^{5} + 3 q^{7} - 3 q^{11} - 3 q^{13} + 9 q^{17} + 3 q^{19} - 6 q^{23} + 3 q^{25} + 12 q^{29} + 12 q^{31} + 6 q^{35} - 3 q^{37} - 3 q^{41} + 12 q^{43} + 6 q^{47} + 18 q^{53} - 9 q^{55} + 21 q^{59} + 6 q^{61} + 3 q^{65} - 6 q^{67} + 9 q^{71} + 6 q^{73} + 24 q^{77} - 6 q^{79} + 6 q^{83} + 18 q^{85} + 6 q^{91} - 3 q^{95} + 15 q^{97}+O(q^{100}) $ 3 * q + 6 * q^5 + 3 * q^7 - 3 * q^11 - 3 * q^13 + 9 * q^17 + 3 * q^19 - 6 * q^23 + 3 * q^25 + 12 * q^29 + 12 * q^31 + 6 * q^35 - 3 * q^37 - 3 * q^41 + 12 * q^43 + 6 * q^47 + 18 * q^53 - 9 * q^55 + 21 * q^59 + 6 * q^61 + 3 * q^65 - 6 * q^67 + 9 * q^71 + 6 * q^73 + 24 * q^77 - 6 * q^79 + 6 * q^83 + 18 * q^85 + 6 * q^91 - 3 * q^95 + 15 * q^97

Display 100 coefficients 10 coefficients

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
$5$	1.65270	0.739112	0.369556	−	0.929209i	$-0.379510\pi$
$5$	1.65270	0.739112	0.369556	+	0.929209i	$0.379510\pi$
$6$	0	0
$7$	−2.41147	−0.911452	−0.455726	−	0.890120i	$-0.650620\pi$
$7$	−2.41147	−0.911452	−0.455726	+	0.890120i	$0.650620\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.94356	−1.79205	−0.896026	−	0.444002i	$-0.853558\pi$
$11$	−5.94356	−1.79205	−0.896026	+	0.444002i	$0.853558\pi$
$12$	0	0
$13$	−3.22668	−0.894920	−0.447460	−	0.894304i	$-0.647671\pi$
$13$	−3.22668	−0.894920	−0.447460	+	0.894304i	$0.647671\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	6.63816	1.52290	0.761449	−	0.648225i	$-0.224488\pi$
$19$	6.63816	1.52290	0.761449	+	0.648225i	$0.224488\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.94356	0.613775	0.306888	−	0.951746i	$-0.400712\pi$
$23$	2.94356	0.613775	0.306888	+	0.951746i	$0.400712\pi$
$24$	0	0
$25$	−2.26857	−0.453714
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.29086	−0.239707	−0.119853	−	0.992792i	$-0.538242\pi$
$29$	−1.29086	−0.239707	−0.119853	+	0.992792i	$0.538242\pi$
$30$	0	0
$31$	0.588526	0.105702	0.0528512	−	0.998602i	$-0.483169\pi$
$31$	0.588526	0.105702	0.0528512	+	0.998602i	$0.483169\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.98545	−0.673664
$36$	0	0
$37$	0.0418891	0.00688652	0.00344326	−	0.999994i	$-0.498904\pi$
$37$	0.0418891	0.00688652	0.00344326	+	0.999994i	$0.498904\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.90167	−0.765513	−0.382756	−	0.923849i	$-0.625025\pi$
$41$	−4.90167	−0.765513	−0.382756	+	0.923849i	$0.625025\pi$
$42$	0	0
$43$	5.18479	0.790673	0.395337	−	0.918536i	$-0.370628\pi$
$43$	5.18479	0.790673	0.395337	+	0.918536i	$0.370628\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.73648	0.545022	0.272511	−	0.962153i	$-0.412146\pi$
$47$	3.73648	0.545022	0.272511	+	0.962153i	$0.412146\pi$
$48$	0	0
$49$	−1.18479	−0.169256
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.6382	1.59862	0.799312	−	0.600916i	$-0.205198\pi$
$53$	11.6382	1.59862	0.799312	+	0.600916i	$0.205198\pi$
$54$	0	0
$55$	−9.82295	−1.32453
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.34730	0.956537	0.478268	−	0.878214i	$-0.341265\pi$
$59$	7.34730	0.956537	0.478268	+	0.878214i	$0.341265\pi$
$60$	0	0
$61$	11.0496	1.41476	0.707380	−	0.706833i	$-0.249877\pi$
$61$	11.0496	1.41476	0.707380	+	0.706833i	$0.249877\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.33275	−0.661446
$66$	0	0
$67$	−1.85710	−0.226880	−0.113440	−	0.993545i	$-0.536187\pi$
$67$	−1.85710	−0.226880	−0.113440	+	0.993545i	$0.536187\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.51249	0.654212	0.327106	−	0.944988i	$-0.393927\pi$
$71$	5.51249	0.654212	0.327106	+	0.944988i	$0.393927\pi$
$72$	0	0
$73$	5.55438	0.650091	0.325045	−	0.945698i	$-0.394620\pi$
$73$	5.55438	0.650091	0.325045	+	0.945698i	$0.394620\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	14.3327	1.63337
$78$	0	0
$79$	3.78106	0.425402	0.212701	−	0.977117i	$-0.431774\pi$
$79$	3.78106	0.425402	0.212701	+	0.977117i	$0.431774\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.98545	−0.437460	−0.218730	−	0.975785i	$-0.570191\pi$
$83$	−3.98545	−0.437460	−0.218730	+	0.975785i	$0.570191\pi$
$84$	0	0
$85$	4.95811	0.537783
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8.15064	0.863967	0.431983	−	0.901882i	$-0.357814\pi$
$89$	8.15064	0.863967	0.431983	+	0.901882i	$0.357814\pi$
$90$	0	0
$91$	7.78106	0.815677
$92$	0	0
$93$	0	0
$94$	0	0
$95$	10.9709	1.12559
$96$	0	0
$97$	0.260830	0.0264833	0.0132416	−	0.999912i	$-0.495785\pi$
$97$	0.260830	0.0264833	0.0132416	+	0.999912i	$0.495785\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	11.0273	1.09726	0.548631	−	0.836065i	$-0.315149\pi$
$101$	11.0273	1.09726	0.548631	+	0.836065i	$0.315149\pi$
$102$	0	0
$103$	3.90673	0.384941	0.192471	−	0.981303i	$-0.438350\pi$
$103$	3.90673	0.384941	0.192471	+	0.981303i	$0.438350\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.63816	0.255040	0.127520	−	0.991836i	$-0.459298\pi$
$107$	2.63816	0.255040	0.127520	+	0.991836i	$0.459298\pi$
$108$	0	0
$109$	−8.95811	−0.858031	−0.429016	−	0.903297i	$-0.641140\pi$
$109$	−8.95811	−0.858031	−0.429016	+	0.903297i	$0.641140\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	15.9290	1.49848	0.749238	−	0.662301i	$-0.230420\pi$
$113$	15.9290	1.49848	0.749238	+	0.662301i	$0.230420\pi$
$114$	0	0
$115$	4.86484	0.453648
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−7.23442	−0.663178
$120$	0	0
$121$	24.3259	2.21145
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−12.0128	−1.07446
$126$	0	0
$127$	−3.59627	−0.319117	−0.159559	−	0.987188i	$-0.551007\pi$
$127$	−3.59627	−0.319117	−0.159559	+	0.987188i	$0.551007\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	17.7074	1.54710	0.773551	−	0.633734i	$-0.218479\pi$
$131$	17.7074	1.54710	0.773551	+	0.633734i	$0.218479\pi$
$132$	0	0
$133$	−16.0077	−1.38805
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.92902	−0.335678	−0.167839	−	0.985814i	$-0.553679\pi$
$137$	−3.92902	−0.335678	−0.167839	+	0.985814i	$0.553679\pi$
$138$	0	0
$139$	−11.9659	−1.01493	−0.507465	−	0.861672i	$-0.669417\pi$
$139$	−11.9659	−1.01493	−0.507465	+	0.861672i	$0.669417\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	19.1780	1.60374
$144$	0	0
$145$	−2.13341	−0.177170
$146$	0	0
$147$	0	0
$148$	0	0
$149$	20.5253	1.68150	0.840748	−	0.541426i	$-0.182115\pi$
$149$	20.5253	1.68150	0.840748	+	0.541426i	$0.182115\pi$
$150$	0	0
$151$	−16.0077	−1.30269	−0.651346	−	0.758781i	$-0.725795\pi$
$151$	−16.0077	−1.30269	−0.651346	+	0.758781i	$0.725795\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.972659	0.0781258
$156$	0	0
$157$	−21.9736	−1.75368	−0.876842	−	0.480779i	$-0.840354\pi$
$157$	−21.9736	−1.75368	−0.876842	+	0.480779i	$0.840354\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−7.09833	−0.559426
$162$	0	0
$163$	−20.5107	−1.60652	−0.803262	−	0.595625i	$-0.796904\pi$
$163$	−20.5107	−1.60652	−0.803262	+	0.595625i	$0.796904\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.29086	−0.332037	−0.166018	−	0.986123i	$-0.553091\pi$
$167$	−4.29086	−0.332037	−0.166018	+	0.986123i	$0.553091\pi$
$168$	0	0
$169$	−2.58853	−0.199117
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.79292	0.288370	0.144185	−	0.989551i	$-0.453944\pi$
$173$	3.79292	0.288370	0.144185	+	0.989551i	$0.453944\pi$
$174$	0	0
$175$	5.47060	0.413538
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−8.27631	−0.618601	−0.309300	−	0.950964i	$-0.600095\pi$
$179$	−8.27631	−0.618601	−0.309300	+	0.950964i	$0.600095\pi$
$180$	0	0
$181$	6.72193	0.499637	0.249819	−	0.968293i	$-0.419629\pi$
$181$	6.72193	0.499637	0.249819	+	0.968293i	$0.419629\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0.0692302	0.00508991
$186$	0	0
$187$	−17.8307	−1.30391
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.01455	0.362840	0.181420	−	0.983406i	$-0.441931\pi$
$191$	5.01455	0.362840	0.181420	+	0.983406i	$0.441931\pi$
$192$	0	0
$193$	−17.8648	−1.28594	−0.642970	−	0.765892i	$-0.722298\pi$
$193$	−17.8648	−1.28594	−0.642970	+	0.765892i	$0.722298\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.723689	0.0515607	0.0257803	−	0.999668i	$-0.491793\pi$
$197$	0.723689	0.0515607	0.0257803	+	0.999668i	$0.491793\pi$
$198$	0	0
$199$	10.1925	0.722530	0.361265	−	0.932463i	$-0.382345\pi$
$199$	10.1925	0.722530	0.361265	+	0.932463i	$0.382345\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.11287	0.218481
$204$	0	0
$205$	−8.10101	−0.565799
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−39.4543	−2.72911
$210$	0	0
$211$	14.8648	1.02334	0.511669	−	0.859183i	$-0.329027\pi$
$211$	14.8648	1.02334	0.511669	+	0.859183i	$0.329027\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	8.56893	0.584396
$216$	0	0
$217$	−1.41921	−0.0963426
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−9.68004	−0.651150
$222$	0	0
$223$	10.9486	0.733174	0.366587	−	0.930384i	$-0.380526\pi$
$223$	10.9486	0.733174	0.366587	+	0.930384i	$0.380526\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	17.3327	1.15041	0.575207	−	0.818008i	$-0.304921\pi$
$227$	17.3327	1.15041	0.575207	+	0.818008i	$0.304921\pi$
$228$	0	0
$229$	1.56212	0.103228	0.0516138	−	0.998667i	$-0.483563\pi$
$229$	1.56212	0.103228	0.0516138	+	0.998667i	$0.483563\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.7888	1.09987	0.549935	−	0.835207i	$-0.314652\pi$
$233$	16.7888	1.09987	0.549935	+	0.835207i	$0.314652\pi$
$234$	0	0
$235$	6.17530	0.402832
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4.02910	0.260621	0.130310	−	0.991473i	$-0.458403\pi$
$239$	4.02910	0.260621	0.130310	+	0.991473i	$0.458403\pi$
$240$	0	0
$241$	−3.35235	−0.215944	−0.107972	−	0.994154i	$-0.534436\pi$
$241$	−3.35235	−0.215944	−0.107972	+	0.994154i	$0.534436\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.95811	−0.125099
$246$	0	0
$247$	−21.4192	−1.36287
$248$	0	0
$249$	0	0
$250$	0	0
$251$	23.1506	1.46126	0.730628	−	0.682776i	$-0.239227\pi$
$251$	23.1506	1.46126	0.730628	+	0.682776i	$0.239227\pi$
$252$	0	0
$253$	−17.4953	−1.09992
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.0128	0.749337	0.374669	−	0.927159i	$-0.377756\pi$
$257$	12.0128	0.749337	0.374669	+	0.927159i	$0.377756\pi$
$258$	0	0
$259$	−0.101014	−0.00627673
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−16.9017	−1.04220	−0.521101	−	0.853495i	$-0.674478\pi$
$263$	−16.9017	−1.04220	−0.521101	+	0.853495i	$0.674478\pi$
$264$	0	0
$265$	19.2344	1.18156
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7.91447	−0.482554	−0.241277	−	0.970456i	$-0.577566\pi$
$269$	−7.91447	−0.482554	−0.241277	+	0.970456i	$0.577566\pi$
$270$	0	0
$271$	17.2344	1.04692	0.523458	−	0.852051i	$-0.324642\pi$
$271$	17.2344	1.04692	0.523458	+	0.852051i	$0.324642\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	13.4834	0.813079
$276$	0	0
$277$	26.4347	1.58831	0.794153	−	0.607717i	$-0.207915\pi$
$277$	26.4347	1.58831	0.794153	+	0.607717i	$0.207915\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−18.9959	−1.13320	−0.566600	−	0.823993i	$-0.691741\pi$
$281$	−18.9959	−1.13320	−0.566600	+	0.823993i	$0.691741\pi$
$282$	0	0
$283$	16.5868	0.985981	0.492991	−	0.870035i	$-0.335904\pi$
$283$	16.5868	0.985981	0.492991	+	0.870035i	$0.335904\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	11.8203	0.697728
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	0	0
$292$	0	0
$293$	19.3577	1.13089	0.565445	−	0.824786i	$-0.308704\pi$
$293$	19.3577	1.13089	0.565445	+	0.824786i	$0.308704\pi$
$294$	0	0
$295$	12.1429	0.706987
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−9.49794	−0.549280
$300$	0	0
$301$	−12.5030	−0.720661
$302$	0	0
$303$	0	0
$304$	0	0
$305$	18.2618	1.04567
$306$	0	0
$307$	−20.8057	−1.18744	−0.593722	−	0.804670i	$-0.702342\pi$
$307$	−20.8057	−1.18744	−0.593722	+	0.804670i	$0.702342\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	10.6655	0.604785	0.302392	−	0.953184i	$-0.402215\pi$
$311$	10.6655	0.604785	0.302392	+	0.953184i	$0.402215\pi$
$312$	0	0
$313$	3.81521	0.215648	0.107824	−	0.994170i	$-0.465612\pi$
$313$	3.81521	0.215648	0.107824	+	0.994170i	$0.465612\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	26.3892	1.48216	0.741082	−	0.671414i	$-0.234313\pi$
$317$	26.3892	1.48216	0.741082	+	0.671414i	$0.234313\pi$
$318$	0	0
$319$	7.67230	0.429567
$320$	0	0
$321$	0	0
$322$	0	0
$323$	19.9145	1.10807
$324$	0	0
$325$	7.31996	0.406038
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−9.01043	−0.496761
$330$	0	0
$331$	1.57161	0.0863837	0.0431919	−	0.999067i	$-0.486247\pi$
$331$	1.57161	0.0863837	0.0431919	+	0.999067i	$0.486247\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3.06923	−0.167690
$336$	0	0
$337$	−8.01548	−0.436631	−0.218316	−	0.975878i	$-0.570056\pi$
$337$	−8.01548	−0.436631	−0.218316	+	0.975878i	$0.570056\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3.49794	−0.189424
$342$	0	0
$343$	19.7374	1.06572
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−19.8452	−1.06535	−0.532674	−	0.846320i	$-0.678813\pi$
$347$	−19.8452	−1.06535	−0.532674	+	0.846320i	$0.678813\pi$
$348$	0	0
$349$	11.0933	0.593809	0.296905	−	0.954907i	$-0.404046\pi$
$349$	11.0933	0.593809	0.296905	+	0.954907i	$0.404046\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2.86390	−0.152430	−0.0762151	−	0.997091i	$-0.524284\pi$
$353$	−2.86390	−0.152430	−0.0762151	+	0.997091i	$0.524284\pi$
$354$	0	0
$355$	9.11051	0.483536
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−28.7888	−1.51941	−0.759707	−	0.650265i	$-0.774658\pi$
$359$	−28.7888	−1.51941	−0.759707	+	0.650265i	$0.774658\pi$
$360$	0	0
$361$	25.0651	1.31922
$362$	0	0
$363$	0	0
$364$	0	0
$365$	9.17974	0.480490
$366$	0	0
$367$	10.9923	0.573791	0.286896	−	0.957962i	$-0.407377\pi$
$367$	10.9923	0.573791	0.286896	+	0.957962i	$0.407377\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−28.0651	−1.45707
$372$	0	0
$373$	33.4097	1.72989	0.864945	−	0.501867i	$-0.167353\pi$
$373$	33.4097	1.72989	0.864945	+	0.501867i	$0.167353\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.16519	0.214518
$378$	0	0
$379$	−20.9394	−1.07559	−0.537794	−	0.843077i	$-0.680742\pi$
$379$	−20.9394	−1.07559	−0.537794	+	0.843077i	$0.680742\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4.11112	0.210068	0.105034	−	0.994469i	$-0.466505\pi$
$383$	4.11112	0.210068	0.105034	+	0.994469i	$0.466505\pi$
$384$	0	0
$385$	23.6878	1.20724
$386$	0	0
$387$	0	0
$388$	0	0
$389$	17.0942	0.866711	0.433355	−	0.901223i	$-0.357330\pi$
$389$	17.0942	0.866711	0.433355	+	0.901223i	$0.357330\pi$
$390$	0	0
$391$	8.83069	0.446587
$392$	0	0
$393$	0	0
$394$	0	0
$395$	6.24897	0.314420
$396$	0	0
$397$	22.4020	1.12432	0.562162	−	0.827027i	$-0.309970\pi$
$397$	22.4020	1.12432	0.562162	+	0.827027i	$0.309970\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−14.5817	−0.728176	−0.364088	−	0.931364i	$-0.618619\pi$
$401$	−14.5817	−0.728176	−0.364088	+	0.931364i	$0.618619\pi$
$402$	0	0
$403$	−1.89899	−0.0945952
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−0.248970	−0.0123410
$408$	0	0
$409$	−17.5030	−0.865467	−0.432734	−	0.901522i	$-0.642451\pi$
$409$	−17.5030	−0.865467	−0.432734	+	0.901522i	$0.642451\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−17.7178	−0.871837
$414$	0	0
$415$	−6.58677	−0.323332
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−18.8621	−0.921476	−0.460738	−	0.887536i	$-0.652415\pi$
$419$	−18.8621	−0.921476	−0.460738	+	0.887536i	$0.652415\pi$
$420$	0	0
$421$	−32.3337	−1.57585	−0.787924	−	0.615773i	$-0.788844\pi$
$421$	−32.3337	−1.57585	−0.787924	+	0.615773i	$0.788844\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6.80571	−0.330126
$426$	0	0
$427$	−26.6459	−1.28949
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−34.3164	−1.65297	−0.826483	−	0.562962i	$-0.809662\pi$
$431$	−34.3164	−1.65297	−0.826483	+	0.562962i	$0.809662\pi$
$432$	0	0
$433$	−25.0669	−1.20464	−0.602318	−	0.798256i	$-0.705756\pi$
$433$	−25.0669	−1.20464	−0.602318	+	0.798256i	$0.705756\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	19.5398	0.934717
$438$	0	0
$439$	23.2080	1.10766	0.553829	−	0.832630i	$-0.313166\pi$
$439$	23.2080	1.10766	0.553829	+	0.832630i	$0.313166\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4.12155	0.195821	0.0979103	−	0.995195i	$-0.468784\pi$
$443$	4.12155	0.195821	0.0979103	+	0.995195i	$0.468784\pi$
$444$	0	0
$445$	13.4706	0.638568
$446$	0	0
$447$	0	0
$448$	0	0
$449$	18.3414	0.865585	0.432793	−	0.901494i	$-0.357528\pi$
$449$	18.3414	0.865585	0.432793	+	0.901494i	$0.357528\pi$
$450$	0	0
$451$	29.1334	1.37184
$452$	0	0
$453$	0	0
$454$	0	0
$455$	12.8598	0.602876
$456$	0	0
$457$	−19.4611	−0.910352	−0.455176	−	0.890401i	$-0.650424\pi$
$457$	−19.4611	−0.910352	−0.455176	+	0.890401i	$0.650424\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	27.7493	1.29241	0.646206	−	0.763163i	$-0.276355\pi$
$461$	27.7493	1.29241	0.646206	+	0.763163i	$0.276355\pi$
$462$	0	0
$463$	−38.6860	−1.79789	−0.898946	−	0.438059i	$-0.855666\pi$
$463$	−38.6860	−1.79789	−0.898946	+	0.438059i	$0.855666\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−29.7638	−1.37731	−0.688653	−	0.725091i	$-0.741798\pi$
$467$	−29.7638	−1.37731	−0.688653	+	0.725091i	$0.741798\pi$
$468$	0	0
$469$	4.47834	0.206791
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−30.8161	−1.41693
$474$	0	0
$475$	−15.0591	−0.690960
$476$	0	0
$477$	0	0
$478$	0	0
$479$	37.6759	1.72146	0.860728	−	0.509064i	$-0.170008\pi$
$479$	37.6759	1.72146	0.860728	+	0.509064i	$0.170008\pi$
$480$	0	0
$481$	−0.135163	−0.00616289
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.431074	0.0195741
$486$	0	0
$487$	0.763823	0.0346121	0.0173061	−	0.999850i	$-0.494491\pi$
$487$	0.763823	0.0346121	0.0173061	+	0.999850i	$0.494491\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0.497941	0.0224717	0.0112359	−	0.999937i	$-0.496423\pi$
$491$	0.497941	0.0224717	0.0112359	+	0.999937i	$0.496423\pi$
$492$	0	0
$493$	−3.87258	−0.174412
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−13.2932	−0.596283
$498$	0	0
$499$	−8.96585	−0.401367	−0.200683	−	0.979656i	$-0.564316\pi$
$499$	−8.96585	−0.401367	−0.200683	+	0.979656i	$0.564316\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−18.3618	−0.818714	−0.409357	−	0.912374i	$-0.634247\pi$
$503$	−18.3618	−0.818714	−0.409357	+	0.912374i	$0.634247\pi$
$504$	0	0
$505$	18.2249	0.810999
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−28.3705	−1.25750	−0.628751	−	0.777607i	$-0.716433\pi$
$509$	−28.3705	−1.25750	−0.628751	+	0.777607i	$0.716433\pi$
$510$	0	0
$511$	−13.3942	−0.592526
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.45666	0.284514
$516$	0	0
$517$	−22.2080	−0.976707
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−32.6382	−1.42990	−0.714952	−	0.699174i	$-0.753551\pi$
$521$	−32.6382	−1.42990	−0.714952	+	0.699174i	$0.753551\pi$
$522$	0	0
$523$	22.0232	0.963008	0.481504	−	0.876444i	$-0.340091\pi$
$523$	22.0232	0.963008	0.481504	+	0.876444i	$0.340091\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.76558	0.0769098
$528$	0	0
$529$	−14.3354	−0.623280
$530$	0	0
$531$	0	0
$532$	0	0
$533$	15.8161	0.685073
$534$	0	0
$535$	4.36009	0.188503
$536$	0	0
$537$	0	0
$538$	0	0
$539$	7.04189	0.303316
$540$	0	0
$541$	−15.7870	−0.678738	−0.339369	−	0.940653i	$-0.610214\pi$
$541$	−15.7870	−0.678738	−0.339369	+	0.940653i	$0.610214\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−14.8051	−0.634181
$546$	0	0
$547$	27.4192	1.17236	0.586180	−	0.810180i	$-0.300631\pi$
$547$	27.4192	1.17236	0.586180	+	0.810180i	$0.300631\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.56893	−0.365048
$552$	0	0
$553$	−9.11793	−0.387734
$554$	0	0
$555$	0	0
$556$	0	0
$557$	29.4020	1.24580	0.622901	−	0.782301i	$-0.285954\pi$
$557$	29.4020	1.24580	0.622901	+	0.782301i	$0.285954\pi$
$558$	0	0
$559$	−16.7297	−0.707590
$560$	0	0
$561$	0	0
$562$	0	0
$563$	10.3705	0.437065	0.218533	−	0.975830i	$-0.429873\pi$
$563$	10.3705	0.437065	0.218533	+	0.975830i	$0.429873\pi$
$564$	0	0
$565$	26.3259	1.10754
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−32.9691	−1.38214	−0.691069	−	0.722788i	$-0.742860\pi$
$569$	−32.9691	−1.38214	−0.691069	+	0.722788i	$0.742860\pi$
$570$	0	0
$571$	0.737415	0.0308599	0.0154299	−	0.999881i	$-0.495088\pi$
$571$	0.737415	0.0308599	0.0154299	+	0.999881i	$0.495088\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6.67768	−0.278479
$576$	0	0
$577$	19.3432	0.805267	0.402634	−	0.915361i	$-0.368095\pi$
$577$	19.3432	0.805267	0.402634	+	0.915361i	$0.368095\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	9.61081	0.398724
$582$	0	0
$583$	−69.1721	−2.86482
$584$	0	0
$585$	0	0
$586$	0	0
$587$	31.9121	1.31715	0.658577	−	0.752514i	$-0.271159\pi$
$587$	31.9121	1.31715	0.658577	+	0.752514i	$0.271159\pi$
$588$	0	0
$589$	3.90673	0.160974
$590$	0	0
$591$	0	0
$592$	0	0
$593$	31.6783	1.30087	0.650436	−	0.759561i	$-0.274586\pi$
$593$	31.6783	1.30087	0.650436	+	0.759561i	$0.274586\pi$
$594$	0	0
$595$	−11.9564	−0.490163
$596$	0	0
$597$	0	0
$598$	0	0
$599$	12.6236	0.515787	0.257893	−	0.966173i	$-0.416972\pi$
$599$	12.6236	0.515787	0.257893	+	0.966173i	$0.416972\pi$
$600$	0	0
$601$	−8.90848	−0.363385	−0.181692	−	0.983355i	$-0.558157\pi$
$601$	−8.90848	−0.363385	−0.181692	+	0.983355i	$0.558157\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	40.2036	1.63451
$606$	0	0
$607$	33.1242	1.34447	0.672236	−	0.740337i	$-0.265334\pi$
$607$	33.1242	1.34447	0.672236	+	0.740337i	$0.265334\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−12.0564	−0.487751
$612$	0	0
$613$	17.6800	0.714090	0.357045	−	0.934087i	$-0.383784\pi$
$613$	17.6800	0.714090	0.357045	+	0.934087i	$0.383784\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−25.7324	−1.03595	−0.517973	−	0.855397i	$-0.673313\pi$
$617$	−25.7324	−1.03595	−0.517973	+	0.855397i	$0.673313\pi$
$618$	0	0
$619$	−27.7948	−1.11717	−0.558583	−	0.829448i	$-0.688655\pi$
$619$	−27.7948	−1.11717	−0.558583	+	0.829448i	$0.688655\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−19.6551	−0.787464
$624$	0	0
$625$	−8.51073	−0.340429
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.125667	0.00501068
$630$	0	0
$631$	−26.8138	−1.06744	−0.533720	−	0.845661i	$-0.679206\pi$
$631$	−26.8138	−1.06744	−0.533720	+	0.845661i	$0.679206\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−5.94356	−0.235863
$636$	0	0
$637$	3.82295	0.151471
$638$	0	0
$639$	0	0
$640$	0	0
$641$	12.6905	0.501244	0.250622	−	0.968085i	$-0.419365\pi$
$641$	12.6905	0.501244	0.250622	+	0.968085i	$0.419365\pi$
$642$	0	0
$643$	−15.4766	−0.610337	−0.305168	−	0.952298i	$-0.598713\pi$
$643$	−15.4766	−0.610337	−0.305168	+	0.952298i	$0.598713\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−11.1506	−0.438377	−0.219189	−	0.975683i	$-0.570341\pi$
$647$	−11.1506	−0.438377	−0.219189	+	0.975683i	$0.570341\pi$
$648$	0	0
$649$	−43.6691	−1.71416
$650$	0	0
$651$	0	0
$652$	0	0
$653$	44.5921	1.74503	0.872513	−	0.488591i	$-0.162489\pi$
$653$	44.5921	1.74503	0.872513	+	0.488591i	$0.162489\pi$
$654$	0	0
$655$	29.2651	1.14348
$656$	0	0
$657$	0	0
$658$	0	0
$659$	14.0966	0.549124	0.274562	−	0.961569i	$-0.411467\pi$
$659$	14.0966	0.549124	0.274562	+	0.961569i	$0.411467\pi$
$660$	0	0
$661$	36.1147	1.40470	0.702350	−	0.711831i	$-0.252134\pi$
$661$	36.1147	1.40470	0.702350	+	0.711831i	$0.252134\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−26.4561	−1.02592
$666$	0	0
$667$	−3.79973	−0.147126
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−65.6742	−2.53532
$672$	0	0
$673$	2.24216	0.0864290	0.0432145	−	0.999066i	$-0.486240\pi$
$673$	2.24216	0.0864290	0.0432145	+	0.999066i	$0.486240\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	35.1762	1.35193	0.675966	−	0.736933i	$-0.263727\pi$
$677$	35.1762	1.35193	0.675966	+	0.736933i	$0.263727\pi$
$678$	0	0
$679$	−0.628984	−0.0241382
$680$	0	0
$681$	0	0
$682$	0	0
$683$	17.7638	0.679714	0.339857	−	0.940477i	$-0.389621\pi$
$683$	17.7638	0.679714	0.339857	+	0.940477i	$0.389621\pi$
$684$	0	0
$685$	−6.49350	−0.248104
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−37.5526	−1.43064
$690$	0	0
$691$	44.0306	1.67500	0.837502	−	0.546434i	$-0.184015\pi$
$691$	44.0306	1.67500	0.837502	+	0.546434i	$0.184015\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−19.7760	−0.750147
$696$	0	0
$697$	−14.7050	−0.556992
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−30.1052	−1.13706	−0.568530	−	0.822663i	$-0.692488\pi$
$701$	−30.1052	−1.13706	−0.568530	+	0.822663i	$0.692488\pi$
$702$	0	0
$703$	0.278066	0.0104875
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−26.5921	−1.00010
$708$	0	0
$709$	−24.7374	−0.929033	−0.464517	−	0.885564i	$-0.653772\pi$
$709$	−24.7374	−0.929033	−0.464517	+	0.885564i	$0.653772\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.73236	0.0648775
$714$	0	0
$715$	31.6955	1.18535
$716$	0	0
$717$	0	0
$718$	0	0
$719$	43.5526	1.62424	0.812119	−	0.583491i	$-0.198314\pi$
$719$	43.5526	1.62424	0.812119	+	0.583491i	$0.198314\pi$
$720$	0	0
$721$	−9.42097	−0.350855
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.92841	0.108758
$726$	0	0
$727$	−20.5371	−0.761680	−0.380840	−	0.924641i	$-0.624365\pi$
$727$	−20.5371	−0.761680	−0.380840	+	0.924641i	$0.624365\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	15.5544	0.575299
$732$	0	0
$733$	14.0060	0.517323	0.258661	−	0.965968i	$-0.416719\pi$
$733$	14.0060	0.517323	0.258661	+	0.965968i	$0.416719\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.0378	0.406581
$738$	0	0
$739$	41.9813	1.54431	0.772154	−	0.635435i	$-0.219179\pi$
$739$	41.9813	1.54431	0.772154	+	0.635435i	$0.219179\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−27.8621	−1.02216	−0.511082	−	0.859532i	$-0.670755\pi$
$743$	−27.8621	−1.02216	−0.511082	+	0.859532i	$0.670755\pi$
$744$	0	0
$745$	33.9222	1.24281
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−6.36184	−0.232457
$750$	0	0
$751$	52.9050	1.93053	0.965265	−	0.261273i	$-0.0841423\pi$
$751$	52.9050	1.93053	0.965265	+	0.261273i	$0.0841423\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−26.4561	−0.962834
$756$	0	0
$757$	−41.4858	−1.50783	−0.753913	−	0.656975i	$-0.771836\pi$
$757$	−41.4858	−1.50783	−0.753913	+	0.656975i	$0.771836\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−45.3874	−1.64529	−0.822647	−	0.568553i	$-0.807503\pi$
$761$	−45.3874	−1.64529	−0.822647	+	0.568553i	$0.807503\pi$
$762$	0	0
$763$	21.6023	0.782054
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−23.7074	−0.856024
$768$	0	0
$769$	5.11650	0.184506	0.0922528	−	0.995736i	$-0.470593\pi$
$769$	5.11650	0.184506	0.0922528	+	0.995736i	$0.470593\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−52.6427	−1.89343	−0.946713	−	0.322077i	$-0.895619\pi$
$773$	−52.6427	−1.89343	−0.946713	+	0.322077i	$0.895619\pi$
$774$	0	0
$775$	−1.33511	−0.0479587
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−32.5381	−1.16580
$780$	0	0
$781$	−32.7638	−1.17238
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−36.3158	−1.29617
$786$	0	0
$787$	20.7624	0.740099	0.370050	−	0.929012i	$-0.379341\pi$
$787$	20.7624	0.740099	0.370050	+	0.929012i	$0.379341\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−38.4124	−1.36579
$792$	0	0
$793$	−35.6536	−1.26610
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−45.5800	−1.61453	−0.807263	−	0.590192i	$-0.799052\pi$
$797$	−45.5800	−1.61453	−0.807263	+	0.590192i	$0.799052\pi$
$798$	0	0
$799$	11.2094	0.396562
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−33.0128	−1.16500
$804$	0	0
$805$	−11.7314	−0.413479
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−4.21120	−0.148058	−0.0740290	−	0.997256i	$-0.523586\pi$
$809$	−4.21120	−0.148058	−0.0740290	+	0.997256i	$0.523586\pi$
$810$	0	0
$811$	−11.3618	−0.398968	−0.199484	−	0.979901i	$-0.563927\pi$
$811$	−11.3618	−0.398968	−0.199484	+	0.979901i	$0.563927\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−33.8982	−1.18740
$816$	0	0
$817$	34.4175	1.20411
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.64227	0.0573158	0.0286579	−	0.999589i	$-0.490877\pi$
$821$	1.64227	0.0573158	0.0286579	+	0.999589i	$0.490877\pi$
$822$	0	0
$823$	−11.2189	−0.391068	−0.195534	−	0.980697i	$-0.562644\pi$
$823$	−11.2189	−0.391068	−0.195534	+	0.980697i	$0.562644\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9.61318	0.334283	0.167141	−	0.985933i	$-0.446546\pi$
$827$	9.61318	0.334283	0.167141	+	0.985933i	$0.446546\pi$
$828$	0	0
$829$	33.4938	1.16329	0.581644	−	0.813443i	$-0.302410\pi$
$829$	33.4938	1.16329	0.581644	+	0.813443i	$0.302410\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.55438	−0.123152
$834$	0	0
$835$	−7.09152	−0.245412
$836$	0	0
$837$	0	0
$838$	0	0
$839$	31.9941	1.10456	0.552280	−	0.833659i	$-0.313758\pi$
$839$	31.9941	1.10456	0.552280	+	0.833659i	$0.313758\pi$
$840$	0	0
$841$	−27.3337	−0.942541
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−4.27807	−0.147170
$846$	0	0
$847$	−58.6614	−2.01563
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0.123303	0.00422678
$852$	0	0
$853$	35.2422	1.20667	0.603334	−	0.797488i	$-0.293838\pi$
$853$	35.2422	1.20667	0.603334	+	0.797488i	$0.293838\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−21.2490	−0.725851	−0.362925	−	0.931818i	$-0.618222\pi$
$857$	−21.2490	−0.725851	−0.362925	+	0.931818i	$0.618222\pi$
$858$	0	0
$859$	−51.8384	−1.76870	−0.884352	−	0.466820i	$-0.845399\pi$
$859$	−51.8384	−1.76870	−0.884352	+	0.466820i	$0.845399\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	22.6783	0.771978	0.385989	−	0.922503i	$-0.373860\pi$
$863$	22.6783	0.771978	0.385989	+	0.922503i	$0.373860\pi$
$864$	0	0
$865$	6.26857	0.213138
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−22.4730	−0.762343
$870$	0	0
$871$	5.99226	0.203040
$872$	0	0
$873$	0	0
$874$	0	0
$875$	28.9685	0.979315
$876$	0	0
$877$	1.13341	0.0382725	0.0191362	−	0.999817i	$-0.493908\pi$
$877$	1.13341	0.0382725	0.0191362	+	0.999817i	$0.493908\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−30.8289	−1.03865	−0.519327	−	0.854576i	$-0.673817\pi$
$881$	−30.8289	−1.03865	−0.519327	+	0.854576i	$0.673817\pi$
$882$	0	0
$883$	9.33511	0.314152	0.157076	−	0.987587i	$-0.449793\pi$
$883$	9.33511	0.314152	0.157076	+	0.987587i	$0.449793\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	14.0942	0.473237	0.236619	−	0.971603i	$-0.423961\pi$
$887$	14.0942	0.473237	0.236619	+	0.971603i	$0.423961\pi$
$888$	0	0
$889$	8.67230	0.290860
$890$	0	0
$891$	0	0
$892$	0	0
$893$	24.8033	0.830012
$894$	0	0
$895$	−13.6783	−0.457215
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−0.759704	−0.0253376
$900$	0	0
$901$	34.9145	1.16317
$902$	0	0
$903$	0	0
$904$	0	0
$905$	11.1094	0.369288
$906$	0	0
$907$	8.73917	0.290179	0.145090	−	0.989419i	$-0.453653\pi$
$907$	8.73917	0.290179	0.145090	+	0.989419i	$0.453653\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	20.6509	0.684196	0.342098	−	0.939664i	$-0.388862\pi$
$911$	20.6509	0.684196	0.342098	+	0.939664i	$0.388862\pi$
$912$	0	0
$913$	23.6878	0.783951
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−42.7009	−1.41011
$918$	0	0
$919$	2.36009	0.0778522	0.0389261	−	0.999242i	$-0.487606\pi$
$919$	2.36009	0.0778522	0.0389261	+	0.999242i	$0.487606\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−17.7870	−0.585468
$924$	0	0
$925$	−0.0950283	−0.00312451
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−26.8203	−0.879944	−0.439972	−	0.898011i	$-0.645012\pi$
$929$	−26.8203	−0.879944	−0.439972	+	0.898011i	$0.645012\pi$
$930$	0	0
$931$	−7.86484	−0.257760
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−29.4688	−0.963734
$936$	0	0
$937$	1.93313	0.0631527	0.0315764	−	0.999501i	$-0.489947\pi$
$937$	1.93313	0.0631527	0.0315764	+	0.999501i	$0.489947\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	11.9103	0.388266	0.194133	−	0.980975i	$-0.437811\pi$
$941$	11.9103	0.388266	0.194133	+	0.980975i	$0.437811\pi$
$942$	0	0
$943$	−14.4284	−0.469853
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0.315836	0.0102633	0.00513165	−	0.999987i	$-0.498367\pi$
$947$	0.315836	0.0102633	0.00513165	+	0.999987i	$0.498367\pi$
$948$	0	0
$949$	−17.9222	−0.581779
$950$	0	0
$951$	0	0
$952$	0	0
$953$	3.25133	0.105321	0.0526605	−	0.998612i	$-0.483230\pi$
$953$	3.25133	0.105321	0.0526605	+	0.998612i	$0.483230\pi$
$954$	0	0
$955$	8.28756	0.268179
$956$	0	0
$957$	0	0
$958$	0	0
$959$	9.47472	0.305955
$960$	0	0
$961$	−30.6536	−0.988827
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−29.5253	−0.950452
$966$	0	0
$967$	−10.9676	−0.352694	−0.176347	−	0.984328i	$-0.556428\pi$
$967$	−10.9676	−0.352694	−0.176347	+	0.984328i	$0.556428\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−23.3868	−0.750519	−0.375259	−	0.926920i	$-0.622446\pi$
$971$	−23.3868	−0.750519	−0.375259	+	0.926920i	$0.622446\pi$
$972$	0	0
$973$	28.8553	0.925060
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−50.1762	−1.60528	−0.802640	−	0.596464i	$-0.796572\pi$
$977$	−50.1762	−1.60528	−0.802640	+	0.596464i	$0.796572\pi$
$978$	0	0
$979$	−48.4439	−1.54827
$980$	0	0
$981$	0	0
$982$	0	0
$983$	14.6946	0.468685	0.234342	−	0.972154i	$-0.424706\pi$
$983$	14.6946	0.468685	0.234342	+	0.972154i	$0.424706\pi$
$984$	0	0
$985$	1.19604	0.0381091
$986$	0	0
$987$	0	0
$988$	0	0
$989$	15.2618	0.485296
$990$	0	0
$991$	−2.00000	−0.0635321	−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000	−0.0635321	−0.0317660	+	0.999495i	$0.510113\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	16.8452	0.534030
$996$	0	0
$997$	−45.8863	−1.45323	−0.726617	−	0.687043i	$-0.758908\pi$
$997$	−45.8863	−1.45323	−0.726617	+	0.687043i	$0.758908\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3888.2.a.bk.1.2		3
3.2	odd	2		3888.2.a.bd.1.2		3
4.3	odd	2		243.2.a.f.1.2	yes	3
12.11	even	2		243.2.a.e.1.2	✓	3
20.19	odd	2		6075.2.a.bq.1.2		3
36.7	odd	6		243.2.c.e.82.2		6
36.11	even	6		243.2.c.f.82.2		6
36.23	even	6		243.2.c.f.163.2		6
36.31	odd	6		243.2.c.e.163.2		6
60.59	even	2		6075.2.a.bv.1.2		3
108.7	odd	18		729.2.e.c.325.1		6
108.11	even	18		729.2.e.a.82.1		6
108.23	even	18		729.2.e.h.406.1		6
108.31	odd	18		729.2.e.c.406.1		6
108.43	odd	18		729.2.e.i.82.1		6
108.47	even	18		729.2.e.h.325.1		6
108.59	even	18		729.2.e.a.649.1		6
108.67	odd	18		729.2.e.b.163.1		6
108.79	odd	18		729.2.e.b.568.1		6
108.83	even	18		729.2.e.g.568.1		6
108.95	even	18		729.2.e.g.163.1		6
108.103	odd	18		729.2.e.i.649.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.a.e.1.2	✓	3	12.11	even	2
243.2.a.f.1.2	yes	3	4.3	odd	2
243.2.c.e.82.2		6	36.7	odd	6
243.2.c.e.163.2		6	36.31	odd	6
243.2.c.f.82.2		6	36.11	even	6
243.2.c.f.163.2		6	36.23	even	6
729.2.e.a.82.1		6	108.11	even	18
729.2.e.a.649.1		6	108.59	even	18
729.2.e.b.163.1		6	108.67	odd	18
729.2.e.b.568.1		6	108.79	odd	18
729.2.e.c.325.1		6	108.7	odd	18
729.2.e.c.406.1		6	108.31	odd	18
729.2.e.g.163.1		6	108.95	even	18
729.2.e.g.568.1		6	108.83	even	18
729.2.e.h.325.1		6	108.47	even	18
729.2.e.h.406.1		6	108.23	even	18
729.2.e.i.82.1		6	108.43	odd	18
729.2.e.i.649.1		6	108.103	odd	18
3888.2.a.bd.1.2		3	3.2	odd	2
3888.2.a.bk.1.2		3	1.1	even	1	trivial
6075.2.a.bq.1.2		3	20.19	odd	2
6075.2.a.bv.1.2		3	60.59	even	2

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:	\( N \)	\(=\)	\( 3888 = 2^{4} \cdot 3^{5} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3888.a (trivial)

\(f(q)\)	\(=\)	\(q+1.65270 q^{5} -2.41147 q^{7} +O(q^{10})\)	\(q+1.65270 q^{5} -2.41147 q^{7} -5.94356 q^{11} -3.22668 q^{13} +3.00000 q^{17} +6.63816 q^{19} +2.94356 q^{23} -2.26857 q^{25} -1.29086 q^{29} +0.588526 q^{31} -3.98545 q^{35} +0.0418891 q^{37} -4.90167 q^{41} +5.18479 q^{43} +3.73648 q^{47} -1.18479 q^{49} +11.6382 q^{53} -9.82295 q^{55} +7.34730 q^{59} +11.0496 q^{61} -5.33275 q^{65} -1.85710 q^{67} +5.51249 q^{71} +5.55438 q^{73} +14.3327 q^{77} +3.78106 q^{79} -3.98545 q^{83} +4.95811 q^{85} +8.15064 q^{89} +7.78106 q^{91} +10.9709 q^{95} +0.260830 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{5} + 3 q^{7}+O(q^{10}) \) 3 * q + 6 * q^5 + 3 * q^7	\( 3 q + 6 q^{5} + 3 q^{7} - 3 q^{11} - 3 q^{13} + 9 q^{17} + 3 q^{19} - 6 q^{23} + 3 q^{25} + 12 q^{29} + 12 q^{31} + 6 q^{35} - 3 q^{37} - 3 q^{41} + 12 q^{43} + 6 q^{47} + 18 q^{53} - 9 q^{55} + 21 q^{59} + 6 q^{61} + 3 q^{65} - 6 q^{67} + 9 q^{71} + 6 q^{73} + 24 q^{77} - 6 q^{79} + 6 q^{83} + 18 q^{85} + 6 q^{91} - 3 q^{95} + 15 q^{97}+O(q^{100}) \) 3 * q + 6 * q^5 + 3 * q^7 - 3 * q^11 - 3 * q^13 + 9 * q^17 + 3 * q^19 - 6 * q^23 + 3 * q^25 + 12 * q^29 + 12 * q^31 + 6 * q^35 - 3 * q^37 - 3 * q^41 + 12 * q^43 + 6 * q^47 + 18 * q^53 - 9 * q^55 + 21 * q^59 + 6 * q^61 + 3 * q^65 - 6 * q^67 + 9 * q^71 + 6 * q^73 + 24 * q^77 - 6 * q^79 + 6 * q^83 + 18 * q^85 + 6 * q^91 - 3 * q^95 + 15 * q^97

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