LMFDB - Embedded newform 3888.2.a.bk.1.3

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			1.3
Root			$1.87939$ 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)

$f(q)$	$=$	$q+3.87939 q^{5} +2.18479 q^{7} +O(q^{10})$	$q+3.87939 q^{5} +2.18479 q^{7} -0.162504 q^{11} +2.41147 q^{13} +3.00000 q^{17} -3.59627 q^{19} -2.83750 q^{23} +10.0496 q^{25} +6.71688 q^{29} +5.18479 q^{31} +8.47565 q^{35} -6.63816 q^{37} -5.80066 q^{41} +6.22668 q^{43} -7.39693 q^{47} -2.22668 q^{49} +1.40373 q^{53} -0.630415 q^{55} +5.12061 q^{59} -3.78106 q^{61} +9.35504 q^{65} +5.86484 q^{67} +15.3182 q^{71} +8.68004 q^{73} -0.355037 q^{77} +1.26857 q^{79} +8.47565 q^{83} +11.6382 q^{85} +7.72193 q^{89} +5.26857 q^{91} -13.9513 q^{95} -3.90673 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	3.87939	1.73491	0.867457	−	0.497512i	$-0.165753\pi$
$5$	3.87939	1.73491	0.867457	+	0.497512i	$0.165753\pi$
$6$	0	0
$7$	2.18479	0.825774	0.412887	−	0.910782i	$-0.364520\pi$
$7$	2.18479	0.825774	0.412887	+	0.910782i	$0.364520\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.162504	−0.0489967	−0.0244984	−	0.999700i	$-0.507799\pi$
$11$	−0.162504	−0.0489967	−0.0244984	+	0.999700i	$0.507799\pi$
$12$	0	0
$13$	2.41147	0.668823	0.334411	−	0.942427i	$-0.391463\pi$
$13$	2.41147	0.668823	0.334411	+	0.942427i	$0.391463\pi$
$14$	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
$19$	−3.59627	−0.825040	−0.412520	−	0.910949i	$-0.635351\pi$
$19$	−3.59627	−0.825040	−0.412520	+	0.910949i	$0.635351\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.83750	−0.591659	−0.295829	−	0.955241i	$-0.595596\pi$
$23$	−2.83750	−0.591659	−0.295829	+	0.955241i	$0.595596\pi$
$24$	0	0
$25$	10.0496	2.00993
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.71688	1.24729	0.623647	−	0.781706i	$-0.285650\pi$
$29$	6.71688	1.24729	0.623647	+	0.781706i	$0.285650\pi$
$30$	0	0
$31$	5.18479	0.931216	0.465608	−	0.884991i	$-0.345836\pi$
$31$	5.18479	0.931216	0.465608	+	0.884991i	$0.345836\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	8.47565	1.43265
$36$	0	0
$37$	−6.63816	−1.09131	−0.545653	−	0.838011i	$-0.683718\pi$
$37$	−6.63816	−1.09131	−0.545653	+	0.838011i	$0.683718\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.80066	−0.905911	−0.452955	−	0.891533i	$-0.649630\pi$
$41$	−5.80066	−0.905911	−0.452955	+	0.891533i	$0.649630\pi$
$42$	0	0
$43$	6.22668	0.949560	0.474780	−	0.880105i	$-0.342528\pi$
$43$	6.22668	0.949560	0.474780	+	0.880105i	$0.342528\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.39693	−1.07895	−0.539476	−	0.842001i	$-0.681378\pi$
$47$	−7.39693	−1.07895	−0.539476	+	0.842001i	$0.681378\pi$
$48$	0	0
$49$	−2.22668	−0.318097
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.40373	0.192818	0.0964088	−	0.995342i	$-0.469264\pi$
$53$	1.40373	0.192818	0.0964088	+	0.995342i	$0.469264\pi$
$54$	0	0
$55$	−0.630415	−0.0850051
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.12061	0.666647	0.333324	−	0.942812i	$-0.391830\pi$
$59$	5.12061	0.666647	0.333324	+	0.942812i	$0.391830\pi$
$60$	0	0
$61$	−3.78106	−0.484115	−0.242058	−	0.970262i	$-0.577822\pi$
$61$	−3.78106	−0.484115	−0.242058	+	0.970262i	$0.577822\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	9.35504	1.16035
$66$	0	0
$67$	5.86484	0.716504	0.358252	−	0.933625i	$-0.383373\pi$
$67$	5.86484	0.716504	0.358252	+	0.933625i	$0.383373\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	15.3182	1.81794	0.908968	−	0.416866i	$-0.136872\pi$
$71$	15.3182	1.81794	0.908968	+	0.416866i	$0.136872\pi$
$72$	0	0
$73$	8.68004	1.01592	0.507961	−	0.861380i	$-0.330399\pi$
$73$	8.68004	1.01592	0.507961	+	0.861380i	$0.330399\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.355037	−0.0404602
$78$	0	0
$79$	1.26857	0.142725	0.0713627	−	0.997450i	$-0.477265\pi$
$79$	1.26857	0.142725	0.0713627	+	0.997450i	$0.477265\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.47565	0.930324	0.465162	−	0.885226i	$-0.345996\pi$
$83$	8.47565	0.930324	0.465162	+	0.885226i	$0.345996\pi$
$84$	0	0
$85$	11.6382	1.26234
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.72193	0.818523	0.409262	−	0.912417i	$-0.365786\pi$
$89$	7.72193	0.818523	0.409262	+	0.912417i	$0.365786\pi$
$90$	0	0
$91$	5.26857	0.552296
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−13.9513	−1.43137
$96$	0	0
$97$	−3.90673	−0.396668	−0.198334	−	0.980134i	$-0.563553\pi$
$97$	−3.90673	−0.396668	−0.198334	+	0.980134i	$0.563553\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−8.11381	−0.807354	−0.403677	−	0.914902i	$-0.632268\pi$
$101$	−8.11381	−0.807354	−0.403677	+	0.914902i	$0.632268\pi$
$102$	0	0
$103$	−18.6459	−1.83723	−0.918617	−	0.395148i	$-0.870693\pi$
$103$	−18.6459	−1.83723	−0.918617	+	0.395148i	$0.870693\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.59627	−0.734359	−0.367179	−	0.930150i	$-0.619676\pi$
$107$	−7.59627	−0.734359	−0.367179	+	0.930150i	$0.619676\pi$
$108$	0	0
$109$	−15.6382	−1.49786	−0.748932	−	0.662647i	$-0.769433\pi$
$109$	−15.6382	−1.49786	−0.748932	+	0.662647i	$0.769433\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.31315	−0.217603	−0.108801	−	0.994064i	$-0.534701\pi$
$113$	−2.31315	−0.217603	−0.108801	+	0.994064i	$0.534701\pi$
$114$	0	0
$115$	−11.0077	−1.02648
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.55438	0.600839
$120$	0	0
$121$	−10.9736	−0.997599
$122$	0	0
$123$	0	0
$124$	0	0
$125$	19.5895	1.75213
$126$	0	0
$127$	−0.0418891	−0.00371705	−0.00185853	−	0.999998i	$-0.500592\pi$
$127$	−0.0418891	−0.00371705	−0.00185853	+	0.999998i	$0.500592\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−18.3482	−1.60309	−0.801546	−	0.597933i	$-0.795989\pi$
$131$	−18.3482	−1.60309	−0.801546	+	0.597933i	$0.795989\pi$
$132$	0	0
$133$	−7.85710	−0.681297
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.3131	1.22285	0.611427	−	0.791301i	$-0.290596\pi$
$137$	14.3131	1.22285	0.611427	+	0.791301i	$0.290596\pi$
$138$	0	0
$139$	−10.4953	−0.890196	−0.445098	−	0.895482i	$-0.646831\pi$
$139$	−10.4953	−0.890196	−0.445098	+	0.895482i	$0.646831\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.391874	−0.0327701
$144$	0	0
$145$	26.0574	2.16395
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.27126	−0.104146	−0.0520728	−	0.998643i	$-0.516583\pi$
$149$	−1.27126	−0.104146	−0.0520728	+	0.998643i	$0.516583\pi$
$150$	0	0
$151$	−7.85710	−0.639401	−0.319701	−	0.947519i	$-0.603582\pi$
$151$	−7.85710	−0.639401	−0.319701	+	0.947519i	$0.603582\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	20.1138	1.61558
$156$	0	0
$157$	−12.3523	−0.985825	−0.492912	−	0.870079i	$-0.664068\pi$
$157$	−12.3523	−0.985825	−0.492912	+	0.870079i	$0.664068\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−6.19934	−0.488576
$162$	0	0
$163$	13.7469	1.07674	0.538371	−	0.842708i	$-0.319040\pi$
$163$	13.7469	1.07674	0.538371	+	0.842708i	$0.319040\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.71688	0.287621	0.143810	−	0.989605i	$-0.454064\pi$
$167$	3.71688	0.287621	0.143810	+	0.989605i	$0.454064\pi$
$168$	0	0
$169$	−7.18479	−0.552676
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1.55943	−0.118561	−0.0592806	−	0.998241i	$-0.518881\pi$
$173$	−1.55943	−0.118561	−0.0592806	+	0.998241i	$0.518881\pi$
$174$	0	0
$175$	21.9564	1.65974
$176$	0	0
$177$	0	0
$178$	0	0
$179$	12.1925	0.911313	0.455656	−	0.890156i	$-0.349405\pi$
$179$	12.1925	0.911313	0.455656	+	0.890156i	$0.349405\pi$
$180$	0	0
$181$	−16.8726	−1.25413	−0.627064	−	0.778967i	$-0.715744\pi$
$181$	−16.8726	−1.25413	−0.627064	+	0.778967i	$0.715744\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−25.7520	−1.89332
$186$	0	0
$187$	−0.487511	−0.0356504
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.4757	1.26449	0.632247	−	0.774767i	$-0.282133\pi$
$191$	17.4757	1.26449	0.632247	+	0.774767i	$0.282133\pi$
$192$	0	0
$193$	−1.99226	−0.143406	−0.0717030	−	0.997426i	$-0.522843\pi$
$193$	−1.99226	−0.143406	−0.0717030	+	0.997426i	$0.522843\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.1925	1.50991	0.754953	−	0.655779i	$-0.227660\pi$
$197$	21.1925	1.50991	0.754953	+	0.655779i	$0.227660\pi$
$198$	0	0
$199$	3.08378	0.218603	0.109302	−	0.994009i	$-0.465139\pi$
$199$	3.08378	0.218603	0.109302	+	0.994009i	$0.465139\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	14.6750	1.02998
$204$	0	0
$205$	−22.5030	−1.57168
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.584407	0.0404243
$210$	0	0
$211$	−1.00774	−0.0693757	−0.0346879	−	0.999398i	$-0.511044\pi$
$211$	−1.00774	−0.0693757	−0.0346879	+	0.999398i	$0.511044\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	24.1557	1.64740
$216$	0	0
$217$	11.3277	0.768974
$218$	0	0
$219$	0	0
$220$	0	0
$221$	7.23442	0.486640
$222$	0	0
$223$	−18.2841	−1.22439	−0.612195	−	0.790707i	$-0.709713\pi$
$223$	−18.2841	−1.22439	−0.612195	+	0.790707i	$0.709713\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.64496	0.175552	0.0877762	−	0.996140i	$-0.472024\pi$
$227$	2.64496	0.175552	0.0877762	+	0.996140i	$0.472024\pi$
$228$	0	0
$229$	−3.46286	−0.228832	−0.114416	−	0.993433i	$-0.536500\pi$
$229$	−3.46286	−0.228832	−0.114416	+	0.993433i	$0.536500\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.12567	0.401306	0.200653	−	0.979662i	$-0.435694\pi$
$233$	6.12567	0.401306	0.200653	+	0.979662i	$0.435694\pi$
$234$	0	0
$235$	−28.6955	−1.87189
$236$	0	0
$237$	0	0
$238$	0	0
$239$	28.9513	1.87270	0.936352	−	0.351062i	$-0.114179\pi$
$239$	28.9513	1.87270	0.936352	+	0.351062i	$0.114179\pi$
$240$	0	0
$241$	22.3259	1.43814	0.719070	−	0.694937i	$-0.244568\pi$
$241$	22.3259	1.43814	0.719070	+	0.694937i	$0.244568\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−8.63816	−0.551872
$246$	0	0
$247$	−8.67230	−0.551805
$248$	0	0
$249$	0	0
$250$	0	0
$251$	22.7219	1.43420	0.717098	−	0.696972i	$-0.245470\pi$
$251$	22.7219	1.43420	0.717098	+	0.696972i	$0.245470\pi$
$252$	0	0
$253$	0.461104	0.0289894
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−19.5895	−1.22196	−0.610978	−	0.791647i	$-0.709224\pi$
$257$	−19.5895	−1.22196	−0.610978	+	0.791647i	$0.709224\pi$
$258$	0	0
$259$	−14.5030	−0.901172
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−17.8007	−1.09764	−0.548818	−	0.835942i	$-0.684922\pi$
$263$	−17.8007	−1.09764	−0.548818	+	0.835942i	$0.684922\pi$
$264$	0	0
$265$	5.44562	0.334522
$266$	0	0
$267$	0	0
$268$	0	0
$269$	22.7888	1.38946	0.694729	−	0.719272i	$-0.255524\pi$
$269$	22.7888	1.38946	0.694729	+	0.719272i	$0.255524\pi$
$270$	0	0
$271$	3.44562	0.209307	0.104653	−	0.994509i	$-0.466627\pi$
$271$	3.44562	0.209307	0.104653	+	0.994509i	$0.466627\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.63310	−0.0984798
$276$	0	0
$277$	−2.61350	−0.157030	−0.0785151	−	0.996913i	$-0.525018\pi$
$277$	−2.61350	−0.157030	−0.0785151	+	0.996913i	$0.525018\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−13.6851	−0.816384	−0.408192	−	0.912896i	$-0.633841\pi$
$281$	−13.6851	−0.816384	−0.408192	+	0.912896i	$0.633841\pi$
$282$	0	0
$283$	−22.8803	−1.36009	−0.680047	−	0.733169i	$-0.738041\pi$
$283$	−22.8803	−1.36009	−0.680047	+	0.733169i	$0.738041\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−12.6732	−0.748078
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	0	0
$292$	0	0
$293$	24.2814	1.41853	0.709266	−	0.704941i	$-0.249026\pi$
$293$	24.2814	1.41853	0.709266	+	0.704941i	$0.249026\pi$
$294$	0	0
$295$	19.8648	1.15658
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−6.84255	−0.395715
$300$	0	0
$301$	13.6040	0.784122
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−14.6682	−0.839898
$306$	0	0
$307$	16.1489	0.921666	0.460833	−	0.887487i	$-0.347551\pi$
$307$	16.1489	0.921666	0.460833	+	0.887487i	$0.347551\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−18.7101	−1.06095	−0.530475	−	0.847700i	$-0.677987\pi$
$311$	−18.7101	−1.06095	−0.530475	+	0.847700i	$0.677987\pi$
$312$	0	0
$313$	2.77332	0.156757	0.0783786	−	0.996924i	$-0.475026\pi$
$313$	2.77332	0.156757	0.0783786	+	0.996924i	$0.475026\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.4825	0.981913	0.490956	−	0.871184i	$-0.336647\pi$
$317$	17.4825	0.981913	0.490956	+	0.871184i	$0.336647\pi$
$318$	0	0
$319$	−1.09152	−0.0611133
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−10.7888	−0.600305
$324$	0	0
$325$	24.2344	1.34428
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−16.1607	−0.890971
$330$	0	0
$331$	32.4593	1.78413	0.892064	−	0.451910i	$-0.149257\pi$
$331$	32.4593	1.78413	0.892064	+	0.451910i	$0.149257\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	22.7520	1.24307
$336$	0	0
$337$	8.28581	0.451357	0.225678	−	0.974202i	$-0.427540\pi$
$337$	8.28581	0.451357	0.225678	+	0.974202i	$0.427540\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−0.842549	−0.0456266
$342$	0	0
$343$	−20.1584	−1.08845
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−14.9632	−0.803265	−0.401632	−	0.915801i	$-0.631557\pi$
$347$	−14.9632	−0.803265	−0.401632	+	0.915801i	$0.631557\pi$
$348$	0	0
$349$	33.6459	1.80102	0.900512	−	0.434832i	$-0.143192\pi$
$349$	33.6459	1.80102	0.900512	+	0.434832i	$0.143192\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−15.7537	−0.838486	−0.419243	−	0.907874i	$-0.637704\pi$
$353$	−15.7537	−0.838486	−0.419243	+	0.907874i	$0.637704\pi$
$354$	0	0
$355$	59.4252	3.15396
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−18.1257	−0.956636	−0.478318	−	0.878187i	$-0.658753\pi$
$359$	−18.1257	−0.956636	−0.478318	+	0.878187i	$0.658753\pi$
$360$	0	0
$361$	−6.06687	−0.319309
$362$	0	0
$363$	0	0
$364$	0	0
$365$	33.6732	1.76254
$366$	0	0
$367$	19.1429	0.999251	0.499626	−	0.866241i	$-0.333471\pi$
$367$	19.1429	0.999251	0.499626	+	0.866241i	$0.333471\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.06687	0.159224
$372$	0	0
$373$	−15.2499	−0.789610	−0.394805	−	0.918765i	$-0.629188\pi$
$373$	−15.2499	−0.789610	−0.394805	+	0.918765i	$0.629188\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	16.1976	0.834218
$378$	0	0
$379$	−9.84760	−0.505837	−0.252919	−	0.967488i	$-0.581390\pi$
$379$	−9.84760	−0.505837	−0.252919	+	0.967488i	$0.581390\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−28.3901	−1.45067	−0.725334	−	0.688397i	$-0.758315\pi$
$383$	−28.3901	−1.45067	−0.725334	+	0.688397i	$0.758315\pi$
$384$	0	0
$385$	−1.37733	−0.0701950
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10.8844	0.551863	0.275931	−	0.961177i	$-0.411014\pi$
$389$	10.8844	0.551863	0.275931	+	0.961177i	$0.411014\pi$
$390$	0	0
$391$	−8.51249	−0.430495
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.92127	0.247616
$396$	0	0
$397$	−18.1070	−0.908764	−0.454382	−	0.890807i	$-0.650140\pi$
$397$	−18.1070	−0.908764	−0.454382	+	0.890807i	$0.650140\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.43376	0.0715987	0.0357993	−	0.999359i	$-0.488602\pi$
$401$	1.43376	0.0715987	0.0357993	+	0.999359i	$0.488602\pi$
$402$	0	0
$403$	12.5030	0.622818
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.07873	0.0534704
$408$	0	0
$409$	8.60401	0.425441	0.212720	−	0.977113i	$-0.431768\pi$
$409$	8.60401	0.425441	0.212720	+	0.977113i	$0.431768\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	11.1875	0.550500
$414$	0	0
$415$	32.8803	1.61403
$416$	0	0
$417$	0	0
$418$	0	0
$419$	12.3114	0.601451	0.300725	−	0.953711i	$-0.402771\pi$
$419$	12.3114	0.601451	0.300725	+	0.953711i	$0.402771\pi$
$420$	0	0
$421$	11.1165	0.541785	0.270892	−	0.962610i	$-0.412681\pi$
$421$	11.1165	0.541785	0.270892	+	0.962610i	$0.412681\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	30.1489	1.46244
$426$	0	0
$427$	−8.26083	−0.399770
$428$	0	0
$429$	0	0
$430$	0	0
$431$	36.8958	1.77721	0.888604	−	0.458675i	$-0.151676\pi$
$431$	36.8958	1.77721	0.888604	+	0.458675i	$0.151676\pi$
$432$	0	0
$433$	−37.9982	−1.82608	−0.913040	−	0.407871i	$-0.866271\pi$
$433$	−37.9982	−1.82608	−0.913040	+	0.407871i	$0.866271\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	10.2044	0.488142
$438$	0	0
$439$	−0.202029	−0.00964231	−0.00482115	−	0.999988i	$-0.501535\pi$
$439$	−0.202029	−0.00964231	−0.00482115	+	0.999988i	$0.501535\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−21.2294	−1.00864	−0.504319	−	0.863517i	$-0.668256\pi$
$443$	−21.2294	−1.00864	−0.504319	+	0.863517i	$0.668256\pi$
$444$	0	0
$445$	29.9564	1.42007
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−33.2594	−1.56961	−0.784804	−	0.619744i	$-0.787236\pi$
$449$	−33.2594	−1.56961	−0.784804	+	0.619744i	$0.787236\pi$
$450$	0	0
$451$	0.942629	0.0443867
$452$	0	0
$453$	0	0
$454$	0	0
$455$	20.4388	0.958186
$456$	0	0
$457$	−0.0341483	−0.00159739	−0.000798695	−	1.00000i	$-0.500254\pi$
$457$	−0.0341483	−0.00159739	−0.000798695		1.00000i	$0.500254\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−14.9864	−0.697986	−0.348993	−	0.937125i	$-0.613476\pi$
$461$	−14.9864	−0.697986	−0.348993	+	0.937125i	$0.613476\pi$
$462$	0	0
$463$	30.4424	1.41478	0.707390	−	0.706823i	$-0.249872\pi$
$463$	30.4424	1.41478	0.707390	+	0.706823i	$0.249872\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0.510734	0.0236339	0.0118170	−	0.999930i	$-0.496238\pi$
$467$	0.510734	0.0236339	0.0118170	+	0.999930i	$0.496238\pi$
$468$	0	0
$469$	12.8135	0.591670
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.01186	−0.0465254
$474$	0	0
$475$	−36.1411	−1.65827
$476$	0	0
$477$	0	0
$478$	0	0
$479$	15.4507	0.705959	0.352980	−	0.935631i	$-0.385168\pi$
$479$	15.4507	0.705959	0.352980	+	0.935631i	$0.385168\pi$
$480$	0	0
$481$	−16.0077	−0.729890
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−15.1557	−0.688185
$486$	0	0
$487$	−29.5107	−1.33726	−0.668629	−	0.743596i	$-0.733119\pi$
$487$	−29.5107	−1.33726	−0.668629	+	0.743596i	$0.733119\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2.15745	−0.0973644	−0.0486822	−	0.998814i	$-0.515502\pi$
$491$	−2.15745	−0.0973644	−0.0486822	+	0.998814i	$0.515502\pi$
$492$	0	0
$493$	20.1506	0.907539
$494$	0	0
$495$	0	0
$496$	0	0
$497$	33.4671	1.50120
$498$	0	0
$499$	−7.49525	−0.335534	−0.167767	−	0.985827i	$-0.553656\pi$
$499$	−7.49525	−0.335534	−0.167767	+	0.985827i	$0.553656\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−28.5963	−1.27504	−0.637522	−	0.770432i	$-0.720041\pi$
$503$	−28.5963	−1.27504	−0.637522	+	0.770432i	$0.720041\pi$
$504$	0	0
$505$	−31.4766	−1.40069
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.69190	−0.0749923	−0.0374962	−	0.999297i	$-0.511938\pi$
$509$	−1.69190	−0.0749923	−0.0374962	+	0.999297i	$0.511938\pi$
$510$	0	0
$511$	18.9641	0.838922
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−72.3346	−3.18744
$516$	0	0
$517$	1.20203	0.0528652
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−22.4037	−0.981525	−0.490763	−	0.871293i	$-0.663282\pi$
$521$	−22.4037	−0.981525	−0.490763	+	0.871293i	$0.663282\pi$
$522$	0	0
$523$	−2.42871	−0.106200	−0.0531000	−	0.998589i	$-0.516910\pi$
$523$	−2.42871	−0.106200	−0.0531000	+	0.998589i	$0.516910\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	15.5544	0.677559
$528$	0	0
$529$	−14.9486	−0.649940
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−13.9881	−0.605894
$534$	0	0
$535$	−29.4688	−1.27405
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.361844	0.0155857
$540$	0	0
$541$	38.9394	1.67414	0.837069	−	0.547098i	$-0.184267\pi$
$541$	38.9394	1.67414	0.837069	+	0.547098i	$0.184267\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−60.6664	−2.59866
$546$	0	0
$547$	14.6723	0.627342	0.313671	−	0.949532i	$-0.398441\pi$
$547$	14.6723	0.627342	0.313671	+	0.949532i	$0.398441\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−24.1557	−1.02907
$552$	0	0
$553$	2.77156	0.117859
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−11.1070	−0.470619	−0.235309	−	0.971921i	$-0.575610\pi$
$557$	−11.1070	−0.470619	−0.235309	+	0.971921i	$0.575610\pi$
$558$	0	0
$559$	15.0155	0.635087
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−16.3081	−0.687304	−0.343652	−	0.939097i	$-0.611664\pi$
$563$	−16.3081	−0.687304	−0.343652	+	0.939097i	$0.611664\pi$
$564$	0	0
$565$	−8.97359	−0.377522
$566$	0	0
$567$	0	0
$568$	0	0
$569$	36.0164	1.50989	0.754943	−	0.655790i	$-0.227664\pi$
$569$	36.0164	1.50989	0.754943	+	0.655790i	$0.227664\pi$
$570$	0	0
$571$	−39.1584	−1.63873	−0.819364	−	0.573274i	$-0.805673\pi$
$571$	−39.1584	−1.63873	−0.819364	+	0.573274i	$0.805673\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−28.5158	−1.18919
$576$	0	0
$577$	11.8057	0.491478	0.245739	−	0.969336i	$-0.420969\pi$
$577$	11.8057	0.491478	0.245739	+	0.969336i	$0.420969\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	18.5175	0.768237
$582$	0	0
$583$	−0.228112	−0.00944744
$584$	0	0
$585$	0	0
$586$	0	0
$587$	39.9614	1.64938	0.824692	−	0.565582i	$-0.191349\pi$
$587$	39.9614	1.64938	0.824692	+	0.565582i	$0.191349\pi$
$588$	0	0
$589$	−18.6459	−0.768291
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−29.2995	−1.20319	−0.601594	−	0.798802i	$-0.705467\pi$
$593$	−29.2995	−1.20319	−0.601594	+	0.798802i	$0.705467\pi$
$594$	0	0
$595$	25.4270	1.04240
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−10.0719	−0.411527	−0.205764	−	0.978602i	$-0.565968\pi$
$599$	−10.0719	−0.411527	−0.205764	+	0.978602i	$0.565968\pi$
$600$	0	0
$601$	−30.4192	−1.24083	−0.620413	−	0.784275i	$-0.713035\pi$
$601$	−30.4192	−1.24083	−0.620413	+	0.784275i	$0.713035\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−42.5708	−1.73075
$606$	0	0
$607$	23.0743	0.936556	0.468278	−	0.883581i	$-0.344875\pi$
$607$	23.0743	0.936556	0.468278	+	0.883581i	$0.344875\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−17.8375	−0.721628
$612$	0	0
$613$	0.765578	0.0309214	0.0154607	−	0.999880i	$-0.495079\pi$
$613$	0.765578	0.0309214	0.0154607	+	0.999880i	$0.495079\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−9.28817	−0.373928	−0.186964	−	0.982367i	$-0.559865\pi$
$617$	−9.28817	−0.373928	−0.186964	+	0.982367i	$0.559865\pi$
$618$	0	0
$619$	35.0823	1.41008	0.705039	−	0.709168i	$-0.250929\pi$
$619$	35.0823	1.41008	0.705039	+	0.709168i	$0.250929\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	16.8708	0.675915
$624$	0	0
$625$	25.7469	1.02988
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−19.9145	−0.794042
$630$	0	0
$631$	−35.7621	−1.42367	−0.711833	−	0.702349i	$-0.752135\pi$
$631$	−35.7621	−1.42367	−0.711833	+	0.702349i	$0.752135\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−0.162504	−0.00644877
$636$	0	0
$637$	−5.36959	−0.212751
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2.92633	0.115583	0.0577915	−	0.998329i	$-0.481594\pi$
$641$	2.92633	0.115583	0.0577915	+	0.998329i	$0.481594\pi$
$642$	0	0
$643$	20.2517	0.798647	0.399324	−	0.916810i	$-0.369245\pi$
$643$	20.2517	0.798647	0.399324	+	0.916810i	$0.369245\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−10.7219	−0.421523	−0.210761	−	0.977538i	$-0.567594\pi$
$647$	−10.7219	−0.421523	−0.210761	+	0.977538i	$0.567594\pi$
$648$	0	0
$649$	−0.832119	−0.0326635
$650$	0	0
$651$	0	0
$652$	0	0
$653$	35.7270	1.39811	0.699053	−	0.715070i	$-0.253605\pi$
$653$	35.7270	1.39811	0.699053	+	0.715070i	$0.253605\pi$
$654$	0	0
$655$	−71.1799	−2.78123
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−30.8658	−1.20236	−0.601180	−	0.799114i	$-0.705302\pi$
$659$	−30.8658	−1.20236	−0.601180	+	0.799114i	$0.705302\pi$
$660$	0	0
$661$	−9.84793	−0.383040	−0.191520	−	0.981489i	$-0.561342\pi$
$661$	−9.84793	−0.383040	−0.191520	+	0.981489i	$0.561342\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−30.4807	−1.18199
$666$	0	0
$667$	−19.0591	−0.737972
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0.614437	0.0237201
$672$	0	0
$673$	−19.6973	−0.759274	−0.379637	−	0.925135i	$-0.623951\pi$
$673$	−19.6973	−0.759274	−0.379637	+	0.925135i	$0.623951\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−28.4570	−1.09369	−0.546845	−	0.837234i	$-0.684171\pi$
$677$	−28.4570	−1.09369	−0.546845	+	0.837234i	$0.684171\pi$
$678$	0	0
$679$	−8.53539	−0.327558
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−12.5107	−0.478710	−0.239355	−	0.970932i	$-0.576936\pi$
$683$	−12.5107	−0.478710	−0.239355	+	0.970932i	$0.576936\pi$
$684$	0	0
$685$	55.5262	2.12155
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3.38507	0.128961
$690$	0	0
$691$	−42.6255	−1.62155	−0.810775	−	0.585358i	$-0.800954\pi$
$691$	−42.6255	−1.62155	−0.810775	+	0.585358i	$0.800954\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−40.7151	−1.54441
$696$	0	0
$697$	−17.4020	−0.659147
$698$	0	0
$699$	0	0
$700$	0	0
$701$	51.7701	1.95533	0.977665	−	0.210167i	$-0.0674008\pi$
$701$	51.7701	1.95533	0.977665	+	0.210167i	$0.0674008\pi$
$702$	0	0
$703$	23.8726	0.900371
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−17.7270	−0.666692
$708$	0	0
$709$	15.1584	0.569285	0.284643	−	0.958634i	$-0.408125\pi$
$709$	15.1584	0.569285	0.284643	+	0.958634i	$0.408125\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−14.7118	−0.550962
$714$	0	0
$715$	−1.52023	−0.0568534
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.61493	0.0975206	0.0487603	−	0.998811i	$-0.484473\pi$
$719$	2.61493	0.0975206	0.0487603	+	0.998811i	$0.484473\pi$
$720$	0	0
$721$	−40.7374	−1.51714
$722$	0	0
$723$	0	0
$724$	0	0
$725$	67.5022	2.50697
$726$	0	0
$727$	4.09926	0.152033	0.0760166	−	0.997107i	$-0.475780\pi$
$727$	4.09926	0.152033	0.0760166	+	0.997107i	$0.475780\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	18.6800	0.690906
$732$	0	0
$733$	−38.2080	−1.41125	−0.705623	−	0.708588i	$-0.749333\pi$
$733$	−38.2080	−1.41125	−0.705623	+	0.708588i	$0.749333\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.953058	−0.0351064
$738$	0	0
$739$	24.2094	0.890559	0.445279	−	0.895392i	$-0.353104\pi$
$739$	24.2094	0.890559	0.445279	+	0.895392i	$0.353104\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	3.31139	0.121483	0.0607416	−	0.998154i	$-0.480653\pi$
$743$	3.31139	0.121483	0.0607416	+	0.998154i	$0.480653\pi$
$744$	0	0
$745$	−4.93170	−0.180684
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−16.5963	−0.606414
$750$	0	0
$751$	−13.7110	−0.500322	−0.250161	−	0.968204i	$-0.580484\pi$
$751$	−13.7110	−0.500322	−0.250161	+	0.968204i	$0.580484\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−30.4807	−1.10931
$756$	0	0
$757$	12.3833	0.450079	0.225040	−	0.974350i	$-0.427749\pi$
$757$	12.3833	0.450079	0.225040	+	0.974350i	$0.427749\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	7.58265	0.274871	0.137435	−	0.990511i	$-0.456114\pi$
$761$	7.58265	0.274871	0.137435	+	0.990511i	$0.456114\pi$
$762$	0	0
$763$	−34.1661	−1.23690
$764$	0	0
$765$	0	0
$766$	0	0
$767$	12.3482	0.445869
$768$	0	0
$769$	3.21719	0.116015	0.0580073	−	0.998316i	$-0.481525\pi$
$769$	3.21719	0.116015	0.0580073	+	0.998316i	$0.481525\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−0.184468	−0.00663486	−0.00331743	−	0.999994i	$-0.501056\pi$
$773$	−0.184468	−0.00663486	−0.00331743	+	0.999994i	$0.501056\pi$
$774$	0	0
$775$	52.1052	1.87168
$776$	0	0
$777$	0	0
$778$	0	0
$779$	20.8607	0.747413
$780$	0	0
$781$	−2.48927	−0.0890729
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−47.9195	−1.71032
$786$	0	0
$787$	0.478016	0.0170394	0.00851971	−	0.999964i	$-0.497288\pi$
$787$	0.478016	0.0170394	0.00851971	+	0.999964i	$0.497288\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5.05375	−0.179691
$792$	0	0
$793$	−9.11793	−0.323787
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14.4989	0.513576	0.256788	−	0.966468i	$-0.417336\pi$
$797$	14.4989	0.513576	0.256788	+	0.966468i	$0.417336\pi$
$798$	0	0
$799$	−22.1908	−0.785053
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.41054	−0.0497769
$804$	0	0
$805$	−24.0496	−0.847638
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−14.8743	−0.522954	−0.261477	−	0.965210i	$-0.584210\pi$
$809$	−14.8743	−0.522954	−0.261477	+	0.965210i	$0.584210\pi$
$810$	0	0
$811$	−21.5963	−0.758347	−0.379174	−	0.925325i	$-0.623792\pi$
$811$	−21.5963	−0.758347	−0.379174	+	0.925325i	$0.623792\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	53.3296	1.86805
$816$	0	0
$817$	−22.3928	−0.783425
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−3.28136	−0.114520	−0.0572602	−	0.998359i	$-0.518236\pi$
$821$	−3.28136	−0.114520	−0.0572602	+	0.998359i	$0.518236\pi$
$822$	0	0
$823$	−13.7314	−0.478648	−0.239324	−	0.970940i	$-0.576926\pi$
$823$	−13.7314	−0.478648	−0.239324	+	0.970940i	$0.576926\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−20.2327	−0.703559	−0.351779	−	0.936083i	$-0.614423\pi$
$827$	−20.2327	−0.703559	−0.351779	+	0.936083i	$0.614423\pi$
$828$	0	0
$829$	25.5276	0.886612	0.443306	−	0.896370i	$-0.353806\pi$
$829$	25.5276	0.886612	0.443306	+	0.896370i	$0.353806\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6.68004	−0.231450
$834$	0	0
$835$	14.4192	0.498998
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−17.3800	−0.600025	−0.300012	−	0.953935i	$-0.596991\pi$
$839$	−17.3800	−0.600025	−0.300012	+	0.953935i	$0.596991\pi$
$840$	0	0
$841$	16.1165	0.555741
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−27.8726	−0.958846
$846$	0	0
$847$	−23.9750	−0.823792
$848$	0	0
$849$	0	0
$850$	0	0
$851$	18.8357	0.645681
$852$	0	0
$853$	13.3027	0.455476	0.227738	−	0.973722i	$-0.426867\pi$
$853$	13.3027	0.455476	0.227738	+	0.973722i	$0.426867\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−19.9213	−0.680498	−0.340249	−	0.940335i	$-0.610511\pi$
$857$	−19.9213	−0.680498	−0.340249	+	0.940335i	$0.610511\pi$
$858$	0	0
$859$	−26.3446	−0.898866	−0.449433	−	0.893314i	$-0.648374\pi$
$859$	−26.3446	−0.898866	−0.449433	+	0.893314i	$0.648374\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−38.2995	−1.30373	−0.651866	−	0.758334i	$-0.726013\pi$
$863$	−38.2995	−1.30373	−0.651866	+	0.758334i	$0.726013\pi$
$864$	0	0
$865$	−6.04963	−0.205694
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−0.206148	−0.00699308
$870$	0	0
$871$	14.1429	0.479214
$872$	0	0
$873$	0	0
$874$	0	0
$875$	42.7989	1.44687
$876$	0	0
$877$	−27.0574	−0.913662	−0.456831	−	0.889553i	$-0.651016\pi$
$877$	−27.0574	−0.913662	−0.456831	+	0.889553i	$0.651016\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	30.5776	1.03019	0.515093	−	0.857134i	$-0.327757\pi$
$881$	30.5776	1.03019	0.515093	+	0.857134i	$0.327757\pi$
$882$	0	0
$883$	−44.1052	−1.48426	−0.742130	−	0.670256i	$-0.766184\pi$
$883$	−44.1052	−1.48426	−0.742130	+	0.670256i	$0.766184\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	7.88444	0.264734	0.132367	−	0.991201i	$-0.457742\pi$
$887$	7.88444	0.264734	0.132367	+	0.991201i	$0.457742\pi$
$888$	0	0
$889$	−0.0915189	−0.00306945
$890$	0	0
$891$	0	0
$892$	0	0
$893$	26.6013	0.890179
$894$	0	0
$895$	47.2995	1.58105
$896$	0	0
$897$	0	0
$898$	0	0
$899$	34.8256	1.16150
$900$	0	0
$901$	4.21120	0.140295
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−65.4552	−2.17581
$906$	0	0
$907$	12.9067	0.428561	0.214280	−	0.976772i	$-0.431259\pi$
$907$	12.9067	0.428561	0.214280	+	0.976772i	$0.431259\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−21.1857	−0.701914	−0.350957	−	0.936391i	$-0.614144\pi$
$911$	−21.1857	−0.701914	−0.350957	+	0.936391i	$0.614144\pi$
$912$	0	0
$913$	−1.37733	−0.0455828
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−40.0871	−1.32379
$918$	0	0
$919$	−31.4688	−1.03806	−0.519031	−	0.854756i	$-0.673707\pi$
$919$	−31.4688	−1.03806	−0.519031	+	0.854756i	$0.673707\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	36.9394	1.21588
$924$	0	0
$925$	−66.7110	−2.19344
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−2.32676	−0.0763386	−0.0381693	−	0.999271i	$-0.512153\pi$
$929$	−2.32676	−0.0763386	−0.0381693	+	0.999271i	$0.512153\pi$
$930$	0	0
$931$	8.00774	0.262443
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1.89124	−0.0618503
$936$	0	0
$937$	−10.9982	−0.359297	−0.179649	−	0.983731i	$-0.557496\pi$
$937$	−10.9982	−0.359297	−0.179649	+	0.983731i	$0.557496\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−24.1037	−0.785758	−0.392879	−	0.919590i	$-0.628521\pi$
$941$	−24.1037	−0.785758	−0.392879	+	0.919590i	$0.628521\pi$
$942$	0	0
$943$	16.4593	0.535990
$944$	0	0
$945$	0	0
$946$	0	0
$947$	11.9195	0.387332	0.193666	−	0.981067i	$-0.437962\pi$
$947$	11.9195	0.387332	0.193666	+	0.981067i	$0.437962\pi$
$948$	0	0
$949$	20.9317	0.679472
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−36.8289	−1.19301	−0.596503	−	0.802611i	$-0.703444\pi$
$953$	−36.8289	−1.19301	−0.596503	+	0.802611i	$0.703444\pi$
$954$	0	0
$955$	67.7948	2.19379
$956$	0	0
$957$	0	0
$958$	0	0
$959$	31.2713	1.00980
$960$	0	0
$961$	−4.11793	−0.132836
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−7.72874	−0.248797
$966$	0	0
$967$	−53.5604	−1.72239	−0.861193	−	0.508279i	$-0.830282\pi$
$967$	−53.5604	−1.72239	−0.861193	+	0.508279i	$0.830282\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−53.2327	−1.70832	−0.854159	−	0.520012i	$-0.825927\pi$
$971$	−53.2327	−1.70832	−0.854159	+	0.520012i	$0.825927\pi$
$972$	0	0
$973$	−22.9299	−0.735100
$974$	0	0
$975$	0	0
$976$	0	0
$977$	13.4570	0.430527	0.215264	−	0.976556i	$-0.430939\pi$
$977$	13.4570	0.430527	0.215264	+	0.976556i	$0.430939\pi$
$978$	0	0
$979$	−1.25484	−0.0401050
$980$	0	0
$981$	0	0
$982$	0	0
$983$	10.2412	0.326644	0.163322	−	0.986573i	$-0.447779\pi$
$983$	10.2412	0.326644	0.163322	+	0.986573i	$0.447779\pi$
$984$	0	0
$985$	82.2140	2.61956
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−17.6682	−0.561816
$990$	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
$995$	11.9632	0.379258
$996$	0	0
$997$	38.5016	1.21936	0.609678	−	0.792649i	$-0.291299\pi$
$997$	38.5016	1.21936	0.609678	+	0.792649i	$0.291299\pi$
$998$	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.1.3		3
3.2	odd	2		3888.2.a.bd.1.1		3
4.3	odd	2		243.2.a.f.1.1	yes	3
12.11	even	2		243.2.a.e.1.3	✓	3
20.19	odd	2		6075.2.a.bq.1.3		3
36.7	odd	6		243.2.c.e.82.3		6
36.11	even	6		243.2.c.f.82.1		6
36.23	even	6		243.2.c.f.163.1		6
36.31	odd	6		243.2.c.e.163.3		6
60.59	even	2		6075.2.a.bv.1.1		3
108.7	odd	18		729.2.e.b.325.1		6
108.11	even	18		729.2.e.h.82.1		6
108.23	even	18		729.2.e.g.406.1		6
108.31	odd	18		729.2.e.b.406.1		6
108.43	odd	18		729.2.e.c.82.1		6
108.47	even	18		729.2.e.g.325.1		6
108.59	even	18		729.2.e.h.649.1		6
108.67	odd	18		729.2.e.i.163.1		6
108.79	odd	18		729.2.e.i.568.1		6
108.83	even	18		729.2.e.a.568.1		6
108.95	even	18		729.2.e.a.163.1		6
108.103	odd	18		729.2.e.c.649.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.a.e.1.3	✓	3	12.11	even	2
243.2.a.f.1.1	yes	3	4.3	odd	2
243.2.c.e.82.3		6	36.7	odd	6
243.2.c.e.163.3		6	36.31	odd	6
243.2.c.f.82.1		6	36.11	even	6
243.2.c.f.163.1		6	36.23	even	6
729.2.e.a.163.1		6	108.95	even	18
729.2.e.a.568.1		6	108.83	even	18
729.2.e.b.325.1		6	108.7	odd	18
729.2.e.b.406.1		6	108.31	odd	18
729.2.e.c.82.1		6	108.43	odd	18
729.2.e.c.649.1		6	108.103	odd	18
729.2.e.g.325.1		6	108.47	even	18
729.2.e.g.406.1		6	108.23	even	18
729.2.e.h.82.1		6	108.11	even	18
729.2.e.h.649.1		6	108.59	even	18
729.2.e.i.163.1		6	108.67	odd	18
729.2.e.i.568.1		6	108.79	odd	18
3888.2.a.bd.1.1		3	3.2	odd	2
3888.2.a.bk.1.3		3	1.1	even	1	trivial
6075.2.a.bq.1.3		3	20.19	odd	2
6075.2.a.bv.1.1		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)

\(f(q)\)	\(=\)	\(q+3.87939 q^{5} +2.18479 q^{7} +O(q^{10})\)	\(q+3.87939 q^{5} +2.18479 q^{7} -0.162504 q^{11} +2.41147 q^{13} +3.00000 q^{17} -3.59627 q^{19} -2.83750 q^{23} +10.0496 q^{25} +6.71688 q^{29} +5.18479 q^{31} +8.47565 q^{35} -6.63816 q^{37} -5.80066 q^{41} +6.22668 q^{43} -7.39693 q^{47} -2.22668 q^{49} +1.40373 q^{53} -0.630415 q^{55} +5.12061 q^{59} -3.78106 q^{61} +9.35504 q^{65} +5.86484 q^{67} +15.3182 q^{71} +8.68004 q^{73} -0.355037 q^{77} +1.26857 q^{79} +8.47565 q^{83} +11.6382 q^{85} +7.72193 q^{89} +5.26857 q^{91} -13.9513 q^{95} -3.90673 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{5} + 3 q^{7}+O(q^{10}) \) 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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