LMFDB - Embedded newform 864.4.a.g.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [864,4,Mod(1,864)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(864, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("864.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	864.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$50.9776502450$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{73}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 18 $ x^2 - x - 18
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.77200$ of defining polynomial
Character	$\chi$	$=$	864.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+8.54400 q^{5} +8.54400 q^{7} +1.00000 q^{11} -8.00000 q^{13} -102.528 q^{17} -119.616 q^{19} -18.0000 q^{23} -52.0000 q^{25} -119.616 q^{29} +93.9840 q^{31} +73.0000 q^{35} -146.000 q^{37} +85.4400 q^{41} +222.144 q^{43} -106.000 q^{47} -270.000 q^{49} +299.040 q^{53} +8.54400 q^{55} +20.0000 q^{59} -408.000 q^{61} -68.3520 q^{65} -598.080 q^{67} +20.0000 q^{71} -591.000 q^{73} +8.54400 q^{77} +683.520 q^{79} +345.000 q^{83} -876.000 q^{85} +222.144 q^{89} -68.3520 q^{91} -1022.00 q^{95} -1241.00 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{11} - 16 q^{13} - 36 q^{23} - 104 q^{25} + 146 q^{35} - 292 q^{37} - 212 q^{47} - 540 q^{49} + 40 q^{59} - 816 q^{61} + 40 q^{71} - 1182 q^{73} + 690 q^{83} - 1752 q^{85} - 2044 q^{95} - 2482 q^{97}+O(q^{100}) $ 2 * q + 2 * q^11 - 16 * q^13 - 36 * q^23 - 104 * q^25 + 146 * q^35 - 292 * q^37 - 212 * q^47 - 540 * q^49 + 40 * q^59 - 816 * q^61 + 40 * q^71 - 1182 * q^73 + 690 * q^83 - 1752 * q^85 - 2044 * q^95 - 2482 * q^97

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
$5$	8.54400	0.764199	0.382099	−	0.924121i	$-0.375201\pi$
$5$	8.54400	0.764199	0.382099	+	0.924121i	$0.375201\pi$
$6$	0	0
$7$	8.54400	0.461333	0.230666	−	0.973033i	$-0.425909\pi$
$7$	8.54400	0.461333	0.230666	+	0.973033i	$0.425909\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000	0.0274101	0.0137051	−	0.999906i	$-0.495637\pi$
$11$	1.00000	0.0274101	0.0137051	+	0.999906i	$0.495637\pi$
$12$	0	0
$13$	−8.00000	−0.170677	−0.0853385	−	0.996352i	$-0.527197\pi$
$13$	−8.00000	−0.170677	−0.0853385	+	0.996352i	$0.527197\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−102.528	−1.46275	−0.731374	−	0.681977i	$-0.761120\pi$
$17$	−102.528	−1.46275	−0.731374	+	0.681977i	$0.761120\pi$
$18$	0	0
$19$	−119.616	−1.44431	−0.722153	−	0.691734i	$-0.756847\pi$
$19$	−119.616	−1.44431	−0.722153	+	0.691734i	$0.756847\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−18.0000	−0.163185	−0.0815926	−	0.996666i	$-0.526001\pi$
$23$	−18.0000	−0.163185	−0.0815926	+	0.996666i	$0.526001\pi$
$24$	0	0
$25$	−52.0000	−0.416000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−119.616	−0.765936	−0.382968	−	0.923762i	$-0.625098\pi$
$29$	−119.616	−0.765936	−0.382968	+	0.923762i	$0.625098\pi$
$30$	0	0
$31$	93.9840	0.544517	0.272259	−	0.962224i	$-0.412229\pi$
$31$	93.9840	0.544517	0.272259	+	0.962224i	$0.412229\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	73.0000	0.352550
$36$	0	0
$37$	−146.000	−0.648710	−0.324355	−	0.945936i	$-0.605147\pi$
$37$	−146.000	−0.648710	−0.324355	+	0.945936i	$0.605147\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	85.4400	0.325451	0.162726	−	0.986671i	$-0.447971\pi$
$41$	85.4400	0.325451	0.162726	+	0.986671i	$0.447971\pi$
$42$	0	0
$43$	222.144	0.787829	0.393915	−	0.919147i	$-0.371121\pi$
$43$	222.144	0.787829	0.393915	+	0.919147i	$0.371121\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−106.000	−0.328972	−0.164486	−	0.986379i	$-0.552597\pi$
$47$	−106.000	−0.328972	−0.164486	+	0.986379i	$0.552597\pi$
$48$	0	0
$49$	−270.000	−0.787172
$50$	0	0
$51$	0	0
$52$	0	0
$53$	299.040	0.775025	0.387512	−	0.921864i	$-0.373334\pi$
$53$	299.040	0.775025	0.387512	+	0.921864i	$0.373334\pi$
$54$	0	0
$55$	8.54400	0.0209468
$56$	0	0
$57$	0	0
$58$	0	0
$59$	20.0000	0.0441318	0.0220659	−	0.999757i	$-0.492976\pi$
$59$	20.0000	0.0441318	0.0220659	+	0.999757i	$0.492976\pi$
$60$	0	0
$61$	−408.000	−0.856378	−0.428189	−	0.903689i	$-0.640848\pi$
$61$	−408.000	−0.856378	−0.428189	+	0.903689i	$0.640848\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−68.3520	−0.130431
$66$	0	0
$67$	−598.080	−1.09055	−0.545277	−	0.838256i	$-0.683576\pi$
$67$	−598.080	−1.09055	−0.545277	+	0.838256i	$0.683576\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	20.0000	0.0334305	0.0167152	−	0.999860i	$-0.494679\pi$
$71$	20.0000	0.0334305	0.0167152	+	0.999860i	$0.494679\pi$
$72$	0	0
$73$	−591.000	−0.947552	−0.473776	−	0.880645i	$-0.657109\pi$
$73$	−591.000	−0.947552	−0.473776	+	0.880645i	$0.657109\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.54400	0.0126452
$78$	0	0
$79$	683.520	0.973444	0.486722	−	0.873557i	$-0.338193\pi$
$79$	683.520	0.973444	0.486722	+	0.873557i	$0.338193\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	345.000	0.456249	0.228125	−	0.973632i	$-0.426741\pi$
$83$	345.000	0.456249	0.228125	+	0.973632i	$0.426741\pi$
$84$	0	0
$85$	−876.000	−1.11783
$86$	0	0
$87$	0	0
$88$	0	0
$89$	222.144	0.264576	0.132288	−	0.991211i	$-0.457768\pi$
$89$	222.144	0.264576	0.132288	+	0.991211i	$0.457768\pi$
$90$	0	0
$91$	−68.3520	−0.0787389
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1022.00	−1.10374
$96$	0	0
$97$	−1241.00	−1.29902	−0.649508	−	0.760355i	$-0.725025\pi$
$97$	−1241.00	−1.29902	−0.649508	+	0.760355i	$0.725025\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	743.328	0.732316	0.366158	−	0.930553i	$-0.380673\pi$
$101$	743.328	0.732316	0.366158	+	0.930553i	$0.380673\pi$
$102$	0	0
$103$	102.528	0.0980814	0.0490407	−	0.998797i	$-0.484384\pi$
$103$	102.528	0.0980814	0.0490407	+	0.998797i	$0.484384\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1301.00	1.17544	0.587722	−	0.809063i	$-0.300025\pi$
$107$	1301.00	1.17544	0.587722	+	0.809063i	$0.300025\pi$
$108$	0	0
$109$	−1426.00	−1.25308	−0.626541	−	0.779388i	$-0.715530\pi$
$109$	−1426.00	−1.25308	−0.626541	+	0.779388i	$0.715530\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1315.78	1.09538	0.547690	−	0.836682i	$-0.315507\pi$
$113$	1315.78	1.09538	0.547690	+	0.836682i	$0.315507\pi$
$114$	0	0
$115$	−153.792	−0.124706
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−876.000	−0.674813
$120$	0	0
$121$	−1330.00	−0.999249
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1512.29	−1.08211
$126$	0	0
$127$	−2674.27	−1.86853	−0.934265	−	0.356579i	$-0.883943\pi$
$127$	−2674.27	−1.86853	−0.934265	+	0.356579i	$0.883943\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	277.000	0.184745	0.0923725	−	0.995725i	$-0.470555\pi$
$131$	277.000	0.184745	0.0923725	+	0.995725i	$0.470555\pi$
$132$	0	0
$133$	−1022.00	−0.666306
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1691.71	−1.05498	−0.527492	−	0.849560i	$-0.676867\pi$
$137$	−1691.71	−1.05498	−0.527492	+	0.849560i	$0.676867\pi$
$138$	0	0
$139$	1606.27	0.980160	0.490080	−	0.871677i	$-0.336968\pi$
$139$	1606.27	0.980160	0.490080	+	0.871677i	$0.336968\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−8.00000	−0.00467828
$144$	0	0
$145$	−1022.00	−0.585327
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2998.95	−1.64888	−0.824440	−	0.565949i	$-0.808510\pi$
$149$	−2998.95	−1.64888	−0.824440	+	0.565949i	$0.808510\pi$
$150$	0	0
$151$	128.160	0.0690697	0.0345348	−	0.999403i	$-0.489005\pi$
$151$	128.160	0.0690697	0.0345348	+	0.999403i	$0.489005\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	803.000	0.416119
$156$	0	0
$157$	−1660.00	−0.843837	−0.421919	−	0.906634i	$-0.638643\pi$
$157$	−1660.00	−0.843837	−0.421919	+	0.906634i	$0.638643\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−153.792	−0.0752827
$162$	0	0
$163$	2187.26	1.05104	0.525521	−	0.850781i	$-0.323870\pi$
$163$	2187.26	1.05104	0.525521	+	0.850781i	$0.323870\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2974.00	−1.37805	−0.689027	−	0.724736i	$-0.741962\pi$
$167$	−2974.00	−1.37805	−0.689027	+	0.724736i	$0.741962\pi$
$168$	0	0
$169$	−2133.00	−0.970869
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2435.04	1.07013	0.535066	−	0.844810i	$-0.320287\pi$
$173$	2435.04	1.07013	0.535066	+	0.844810i	$0.320287\pi$
$174$	0	0
$175$	−444.288	−0.191914
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4659.00	1.94542	0.972710	−	0.232026i	$-0.0745355\pi$
$179$	4659.00	1.94542	0.972710	+	0.232026i	$0.0745355\pi$
$180$	0	0
$181$	−2080.00	−0.854172	−0.427086	−	0.904211i	$-0.640460\pi$
$181$	−2080.00	−0.854172	−0.427086	+	0.904211i	$0.640460\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1247.42	−0.495743
$186$	0	0
$187$	−102.528	−0.0400941
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3168.00	−1.20015	−0.600074	−	0.799944i	$-0.704862\pi$
$191$	−3168.00	−1.20015	−0.600074	+	0.799944i	$0.704862\pi$
$192$	0	0
$193$	−675.000	−0.251749	−0.125875	−	0.992046i	$-0.540174\pi$
$193$	−675.000	−0.251749	−0.125875	+	0.992046i	$0.540174\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3648.29	1.31944	0.659720	−	0.751511i	$-0.270675\pi$
$197$	3648.29	1.31944	0.659720	+	0.751511i	$0.270675\pi$
$198$	0	0
$199$	4502.69	1.60396	0.801978	−	0.597354i	$-0.203781\pi$
$199$	4502.69	1.60396	0.801978	+	0.597354i	$0.203781\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1022.00	−0.353351
$204$	0	0
$205$	730.000	0.248709
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−119.616	−0.0395886
$210$	0	0
$211$	−2990.40	−0.975676	−0.487838	−	0.872934i	$-0.662214\pi$
$211$	−2990.40	−0.975676	−0.487838	+	0.872934i	$0.662214\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1898.00	0.602058
$216$	0	0
$217$	803.000	0.251204
$218$	0	0
$219$	0	0
$220$	0	0
$221$	820.224	0.249657
$222$	0	0
$223$	−4989.70	−1.49836	−0.749181	−	0.662365i	$-0.769553\pi$
$223$	−4989.70	−1.49836	−0.749181	+	0.662365i	$0.769553\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2892.00	0.845589	0.422795	−	0.906226i	$-0.361049\pi$
$227$	2892.00	0.845589	0.422795	+	0.906226i	$0.361049\pi$
$228$	0	0
$229$	86.0000	0.0248168	0.0124084	−	0.999923i	$-0.496050\pi$
$229$	86.0000	0.0248168	0.0124084	+	0.999923i	$0.496050\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	51.2640	0.0144138	0.00720691	−	0.999974i	$-0.497706\pi$
$233$	51.2640	0.0144138	0.00720691	+	0.999974i	$0.497706\pi$
$234$	0	0
$235$	−905.664	−0.251400
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6894.00	−1.86584	−0.932920	−	0.360084i	$-0.882748\pi$
$239$	−6894.00	−1.86584	−0.932920	+	0.360084i	$0.882748\pi$
$240$	0	0
$241$	−1538.00	−0.411084	−0.205542	−	0.978648i	$-0.565896\pi$
$241$	−1538.00	−0.411084	−0.205542	+	0.978648i	$0.565896\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2306.88	−0.601556
$246$	0	0
$247$	956.928	0.246510
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6432.00	1.61747	0.808734	−	0.588175i	$-0.200153\pi$
$251$	6432.00	1.61747	0.808734	+	0.588175i	$0.200153\pi$
$252$	0	0
$253$	−18.0000	−0.00447293
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3007.49	0.729969	0.364984	−	0.931014i	$-0.381074\pi$
$257$	3007.49	0.729969	0.364984	+	0.931014i	$0.381074\pi$
$258$	0	0
$259$	−1247.42	−0.299271
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2850.00	−0.668207	−0.334104	−	0.942536i	$-0.608434\pi$
$263$	−2850.00	−0.668207	−0.334104	+	0.942536i	$0.608434\pi$
$264$	0	0
$265$	2555.00	0.592273
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2306.88	−0.522874	−0.261437	−	0.965221i	$-0.584196\pi$
$269$	−2306.88	−0.522874	−0.261437	+	0.965221i	$0.584196\pi$
$270$	0	0
$271$	3357.79	0.752662	0.376331	−	0.926485i	$-0.377186\pi$
$271$	3357.79	0.752662	0.376331	+	0.926485i	$0.377186\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−52.0000	−0.0114026
$276$	0	0
$277$	4816.00	1.04464	0.522320	−	0.852749i	$-0.325067\pi$
$277$	4816.00	1.04464	0.522320	+	0.852749i	$0.325067\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4818.82	1.02301	0.511506	−	0.859279i	$-0.329088\pi$
$281$	4818.82	1.02301	0.511506	+	0.859279i	$0.329088\pi$
$282$	0	0
$283$	3810.63	0.800418	0.400209	−	0.916424i	$-0.368938\pi$
$283$	3810.63	0.800418	0.400209	+	0.916424i	$0.368938\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	730.000	0.150141
$288$	0	0
$289$	5599.00	1.13963
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4972.61	0.991478	0.495739	−	0.868472i	$-0.334897\pi$
$293$	4972.61	0.991478	0.495739	+	0.868472i	$0.334897\pi$
$294$	0	0
$295$	170.880	0.0337255
$296$	0	0
$297$	0	0
$298$	0	0
$299$	144.000	0.0278520
$300$	0	0
$301$	1898.00	0.363451
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3485.95	−0.654443
$306$	0	0
$307$	−4818.82	−0.895845	−0.447923	−	0.894072i	$-0.647836\pi$
$307$	−4818.82	−0.895845	−0.447923	+	0.894072i	$0.647836\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−7098.00	−1.29418	−0.647091	−	0.762413i	$-0.724015\pi$
$311$	−7098.00	−1.29418	−0.647091	+	0.762413i	$0.724015\pi$
$312$	0	0
$313$	997.000	0.180044	0.0900220	−	0.995940i	$-0.471306\pi$
$313$	997.000	0.180044	0.0900220	+	0.995940i	$0.471306\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1922.40	−0.340608	−0.170304	−	0.985392i	$-0.554475\pi$
$317$	−1922.40	−0.340608	−0.170304	+	0.985392i	$0.554475\pi$
$318$	0	0
$319$	−119.616	−0.0209944
$320$	0	0
$321$	0	0
$322$	0	0
$323$	12264.0	2.11265
$324$	0	0
$325$	416.000	0.0710016
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−905.664	−0.151766
$330$	0	0
$331$	−1213.25	−0.201469	−0.100734	−	0.994913i	$-0.532119\pi$
$331$	−1213.25	−0.201469	−0.100734	+	0.994913i	$0.532119\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5110.00	−0.833400
$336$	0	0
$337$	4418.00	0.714136	0.357068	−	0.934078i	$-0.383776\pi$
$337$	4418.00	0.714136	0.357068	+	0.934078i	$0.383776\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	93.9840	0.0149253
$342$	0	0
$343$	−5237.47	−0.824481
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5329.00	−0.824426	−0.412213	−	0.911088i	$-0.635244\pi$
$347$	−5329.00	−0.824426	−0.412213	+	0.911088i	$0.635244\pi$
$348$	0	0
$349$	9466.00	1.45187	0.725936	−	0.687762i	$-0.241407\pi$
$349$	9466.00	1.45187	0.725936	+	0.687762i	$0.241407\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6852.29	−1.03317	−0.516587	−	0.856235i	$-0.672798\pi$
$353$	−6852.29	−1.03317	−0.516587	+	0.856235i	$0.672798\pi$
$354$	0	0
$355$	170.880	0.0255475
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7954.00	1.16935	0.584674	−	0.811268i	$-0.301222\pi$
$359$	7954.00	1.16935	0.584674	+	0.811268i	$0.301222\pi$
$360$	0	0
$361$	7449.00	1.08602
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−5049.51	−0.724118
$366$	0	0
$367$	12021.4	1.70984	0.854922	−	0.518757i	$-0.173605\pi$
$367$	12021.4	1.70984	0.854922	+	0.518757i	$0.173605\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2555.00	0.357544
$372$	0	0
$373$	6164.00	0.855656	0.427828	−	0.903860i	$-0.359279\pi$
$373$	6164.00	0.855656	0.427828	+	0.903860i	$0.359279\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	956.928	0.130728
$378$	0	0
$379$	2699.91	0.365923	0.182961	−	0.983120i	$-0.441432\pi$
$379$	2699.91	0.365923	0.182961	+	0.983120i	$0.441432\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−11788.0	−1.57269	−0.786343	−	0.617790i	$-0.788028\pi$
$383$	−11788.0	−1.57269	−0.786343	+	0.617790i	$0.788028\pi$
$384$	0	0
$385$	73.0000	0.00966344
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−6895.01	−0.898691	−0.449346	−	0.893358i	$-0.648343\pi$
$389$	−6895.01	−0.898691	−0.449346	+	0.893358i	$0.648343\pi$
$390$	0	0
$391$	1845.50	0.238699
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5840.00	0.743905
$396$	0	0
$397$	1336.00	0.168897	0.0844483	−	0.996428i	$-0.473087\pi$
$397$	1336.00	0.168897	0.0844483	+	0.996428i	$0.473087\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5570.69	−0.693733	−0.346867	−	0.937914i	$-0.612754\pi$
$401$	−5570.69	−0.693733	−0.346867	+	0.937914i	$0.612754\pi$
$402$	0	0
$403$	−751.872	−0.0929365
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−146.000	−0.0177812
$408$	0	0
$409$	6943.00	0.839387	0.419693	−	0.907666i	$-0.362138\pi$
$409$	6943.00	0.839387	0.419693	+	0.907666i	$0.362138\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	170.880	0.0203595
$414$	0	0
$415$	2947.68	0.348665
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−3428.00	−0.399687	−0.199843	−	0.979828i	$-0.564043\pi$
$419$	−3428.00	−0.399687	−0.199843	+	0.979828i	$0.564043\pi$
$420$	0	0
$421$	10256.0	1.18728	0.593642	−	0.804729i	$-0.297689\pi$
$421$	10256.0	1.18728	0.593642	+	0.804729i	$0.297689\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5331.46	0.608503
$426$	0	0
$427$	−3485.95	−0.395075
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6810.00	0.761082	0.380541	−	0.924764i	$-0.375738\pi$
$431$	6810.00	0.761082	0.380541	+	0.924764i	$0.375738\pi$
$432$	0	0
$433$	13909.0	1.54370	0.771852	−	0.635802i	$-0.219331\pi$
$433$	13909.0	1.54370	0.771852	+	0.635802i	$0.219331\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2153.09	0.235689
$438$	0	0
$439$	−3938.79	−0.428219	−0.214109	−	0.976810i	$-0.568685\pi$
$439$	−3938.79	−0.428219	−0.214109	+	0.976810i	$0.568685\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3900.00	−0.418272	−0.209136	−	0.977887i	$-0.567065\pi$
$443$	−3900.00	−0.418272	−0.209136	+	0.977887i	$0.567065\pi$
$444$	0	0
$445$	1898.00	0.202188
$446$	0	0
$447$	0	0
$448$	0	0
$449$	17139.3	1.80145	0.900726	−	0.434387i	$-0.143035\pi$
$449$	17139.3	1.80145	0.900726	+	0.434387i	$0.143035\pi$
$450$	0	0
$451$	85.4400	0.00892065
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−584.000	−0.0601722
$456$	0	0
$457$	−4705.00	−0.481599	−0.240799	−	0.970575i	$-0.577410\pi$
$457$	−4705.00	−0.481599	−0.240799	+	0.970575i	$0.577410\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	12585.3	1.27149	0.635745	−	0.771899i	$-0.280693\pi$
$461$	12585.3	1.27149	0.635745	+	0.771899i	$0.280693\pi$
$462$	0	0
$463$	−17387.0	−1.74524	−0.872618	−	0.488403i	$-0.837580\pi$
$463$	−17387.0	−1.74524	−0.872618	+	0.488403i	$0.837580\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11881.0	1.17727	0.588637	−	0.808397i	$-0.299665\pi$
$467$	11881.0	1.17727	0.588637	+	0.808397i	$0.299665\pi$
$468$	0	0
$469$	−5110.00	−0.503108
$470$	0	0
$471$	0	0
$472$	0	0
$473$	222.144	0.0215945
$474$	0	0
$475$	6220.03	0.600831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−8538.00	−0.814428	−0.407214	−	0.913333i	$-0.633500\pi$
$479$	−8538.00	−0.814428	−0.407214	+	0.913333i	$0.633500\pi$
$480$	0	0
$481$	1168.00	0.110720
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−10603.1	−0.992706
$486$	0	0
$487$	−10731.3	−0.998522	−0.499261	−	0.866452i	$-0.666395\pi$
$487$	−10731.3	−0.998522	−0.499261	+	0.866452i	$0.666395\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−13109.0	−1.20489	−0.602445	−	0.798160i	$-0.705807\pi$
$491$	−13109.0	−1.20489	−0.602445	+	0.798160i	$0.705807\pi$
$492$	0	0
$493$	12264.0	1.12037
$494$	0	0
$495$	0	0
$496$	0	0
$497$	170.880	0.0154226
$498$	0	0
$499$	7330.76	0.657655	0.328827	−	0.944390i	$-0.393347\pi$
$499$	7330.76	0.657655	0.328827	+	0.944390i	$0.393347\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	12834.0	1.13765	0.568827	−	0.822457i	$-0.307397\pi$
$503$	12834.0	1.13765	0.568827	+	0.822457i	$0.307397\pi$
$504$	0	0
$505$	6351.00	0.559635
$506$	0	0
$507$	0	0
$508$	0	0
$509$	15609.9	1.35932	0.679662	−	0.733525i	$-0.262126\pi$
$509$	15609.9	1.35932	0.679662	+	0.733525i	$0.262126\pi$
$510$	0	0
$511$	−5049.51	−0.437137
$512$	0	0
$513$	0	0
$514$	0	0
$515$	876.000	0.0749537
$516$	0	0
$517$	−106.000	−0.00901717
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−18660.1	−1.56912	−0.784562	−	0.620050i	$-0.787112\pi$
$521$	−18660.1	−1.56912	−0.784562	+	0.620050i	$0.787112\pi$
$522$	0	0
$523$	−1127.81	−0.0942937	−0.0471469	−	0.998888i	$-0.515013\pi$
$523$	−1127.81	−0.0942937	−0.0471469	+	0.998888i	$0.515013\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9636.00	−0.796491
$528$	0	0
$529$	−11843.0	−0.973371
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−683.520	−0.0555470
$534$	0	0
$535$	11115.7	0.898272
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−270.000	−0.0215765
$540$	0	0
$541$	−10244.0	−0.814092	−0.407046	−	0.913408i	$-0.633441\pi$
$541$	−10244.0	−0.814092	−0.407046	+	0.913408i	$0.633441\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−12183.7	−0.957605
$546$	0	0
$547$	24504.2	1.91540	0.957700	−	0.287768i	$-0.0929131\pi$
$547$	24504.2	1.91540	0.957700	+	0.287768i	$0.0929131\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	14308.0	1.10625
$552$	0	0
$553$	5840.00	0.449081
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7800.68	0.593403	0.296701	−	0.954970i	$-0.404113\pi$
$557$	7800.68	0.593403	0.296701	+	0.954970i	$0.404113\pi$
$558$	0	0
$559$	−1777.15	−0.134464
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−1113.00	−0.0833168	−0.0416584	−	0.999132i	$-0.513264\pi$
$563$	−1113.00	−0.0833168	−0.0416584	+	0.999132i	$0.513264\pi$
$564$	0	0
$565$	11242.0	0.837088
$566$	0	0
$567$	0	0
$568$	0	0
$569$	12747.7	0.939208	0.469604	−	0.882877i	$-0.344397\pi$
$569$	12747.7	0.939208	0.469604	+	0.882877i	$0.344397\pi$
$570$	0	0
$571$	−10116.1	−0.741411	−0.370705	−	0.928750i	$-0.620884\pi$
$571$	−10116.1	−0.741411	−0.370705	+	0.928750i	$0.620884\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	936.000	0.0678850
$576$	0	0
$577$	6030.00	0.435064	0.217532	−	0.976053i	$-0.430199\pi$
$577$	6030.00	0.435064	0.217532	+	0.976053i	$0.430199\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2947.68	0.210483
$582$	0	0
$583$	299.040	0.0212435
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−15975.0	−1.12327	−0.561634	−	0.827386i	$-0.689827\pi$
$587$	−15975.0	−1.12327	−0.561634	+	0.827386i	$0.689827\pi$
$588$	0	0
$589$	−11242.0	−0.786449
$590$	0	0
$591$	0	0
$592$	0	0
$593$	7023.17	0.486353	0.243176	−	0.969982i	$-0.421811\pi$
$593$	7023.17	0.486353	0.243176	+	0.969982i	$0.421811\pi$
$594$	0	0
$595$	−7484.55	−0.515692
$596$	0	0
$597$	0	0
$598$	0	0
$599$	11078.0	0.755651	0.377825	−	0.925877i	$-0.376672\pi$
$599$	11078.0	0.755651	0.377825	+	0.925877i	$0.376672\pi$
$600$	0	0
$601$	−19397.0	−1.31651	−0.658253	−	0.752797i	$-0.728704\pi$
$601$	−19397.0	−1.31651	−0.658253	+	0.752797i	$0.728704\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−11363.5	−0.763625
$606$	0	0
$607$	19753.7	1.32089	0.660444	−	0.750875i	$-0.270368\pi$
$607$	19753.7	1.32089	0.660444	+	0.750875i	$0.270368\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	848.000	0.0561480
$612$	0	0
$613$	−16866.0	−1.11127	−0.555637	−	0.831425i	$-0.687526\pi$
$613$	−16866.0	−1.11127	−0.555637	+	0.831425i	$0.687526\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	871.488	0.0568635	0.0284318	−	0.999596i	$-0.490949\pi$
$617$	871.488	0.0568635	0.0284318	+	0.999596i	$0.490949\pi$
$618$	0	0
$619$	−26862.3	−1.74425	−0.872124	−	0.489286i	$-0.837258\pi$
$619$	−26862.3	−1.74425	−0.872124	+	0.489286i	$0.837258\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1898.00	0.122057
$624$	0	0
$625$	−6421.00	−0.410944
$626$	0	0
$627$	0	0
$628$	0	0
$629$	14969.1	0.948898
$630$	0	0
$631$	4861.54	0.306711	0.153356	−	0.988171i	$-0.450992\pi$
$631$	4861.54	0.306711	0.153356	+	0.988171i	$0.450992\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−22849.0	−1.42793
$636$	0	0
$637$	2160.00	0.134352
$638$	0	0
$639$	0	0
$640$	0	0
$641$	17.0880	0.00105294	0.000526471	−	1.00000i	$-0.499832\pi$
$641$	17.0880	0.00105294	0.000526471		1.00000i	$0.499832\pi$
$642$	0	0
$643$	6049.15	0.371004	0.185502	−	0.982644i	$-0.440609\pi$
$643$	6049.15	0.371004	0.185502	+	0.982644i	$0.440609\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	23016.0	1.39854	0.699268	−	0.714860i	$-0.253510\pi$
$647$	23016.0	1.39854	0.699268	+	0.714860i	$0.253510\pi$
$648$	0	0
$649$	20.0000	0.00120966
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−29485.4	−1.76700	−0.883500	−	0.468431i	$-0.844819\pi$
$653$	−29485.4	−1.76700	−0.883500	+	0.468431i	$0.844819\pi$
$654$	0	0
$655$	2366.69	0.141182
$656$	0	0
$657$	0	0
$658$	0	0
$659$	3951.00	0.233549	0.116775	−	0.993158i	$-0.462744\pi$
$659$	3951.00	0.233549	0.116775	+	0.993158i	$0.462744\pi$
$660$	0	0
$661$	−15262.0	−0.898068	−0.449034	−	0.893515i	$-0.648232\pi$
$661$	−15262.0	−0.898068	−0.449034	+	0.893515i	$0.648232\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−8731.97	−0.509190
$666$	0	0
$667$	2153.09	0.124989
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−408.000	−0.0234734
$672$	0	0
$673$	−23827.0	−1.36473	−0.682365	−	0.731012i	$-0.739049\pi$
$673$	−23827.0	−1.36473	−0.682365	+	0.731012i	$0.739049\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−21616.3	−1.22715	−0.613577	−	0.789635i	$-0.710270\pi$
$677$	−21616.3	−1.22715	−0.613577	+	0.789635i	$0.710270\pi$
$678$	0	0
$679$	−10603.1	−0.599278
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−21828.0	−1.22288	−0.611439	−	0.791292i	$-0.709409\pi$
$683$	−21828.0	−1.22288	−0.611439	+	0.791292i	$0.709409\pi$
$684$	0	0
$685$	−14454.0	−0.806217
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2392.32	−0.132279
$690$	0	0
$691$	−8578.18	−0.472257	−0.236128	−	0.971722i	$-0.575879\pi$
$691$	−8578.18	−0.472257	−0.236128	+	0.971722i	$0.575879\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13724.0	0.749037
$696$	0	0
$697$	−8760.00	−0.476053
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−12072.7	−0.650469	−0.325234	−	0.945633i	$-0.605443\pi$
$701$	−12072.7	−0.650469	−0.325234	+	0.945633i	$0.605443\pi$
$702$	0	0
$703$	17463.9	0.936935
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6351.00	0.337841
$708$	0	0
$709$	3340.00	0.176920	0.0884600	−	0.996080i	$-0.471805\pi$
$709$	3340.00	0.176920	0.0884600	+	0.996080i	$0.471805\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1691.71	−0.0888571
$714$	0	0
$715$	−68.3520	−0.00357513
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−3196.00	−0.165773	−0.0828864	−	0.996559i	$-0.526414\pi$
$719$	−3196.00	−0.165773	−0.0828864	+	0.996559i	$0.526414\pi$
$720$	0	0
$721$	876.000	0.0452482
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6220.03	0.318629
$726$	0	0
$727$	−12636.6	−0.644656	−0.322328	−	0.946628i	$-0.604465\pi$
$727$	−12636.6	−0.644656	−0.322328	+	0.946628i	$0.604465\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−22776.0	−1.15240
$732$	0	0
$733$	−10978.0	−0.553181	−0.276591	−	0.960988i	$-0.589205\pi$
$733$	−10978.0	−0.553181	−0.276591	+	0.960988i	$0.589205\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−598.080	−0.0298922
$738$	0	0
$739$	34893.7	1.73692	0.868461	−	0.495757i	$-0.165109\pi$
$739$	34893.7	1.73692	0.868461	+	0.495757i	$0.165109\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	21408.0	1.05704	0.528522	−	0.848920i	$-0.322746\pi$
$743$	21408.0	1.05704	0.528522	+	0.848920i	$0.322746\pi$
$744$	0	0
$745$	−25623.0	−1.26007
$746$	0	0
$747$	0	0
$748$	0	0
$749$	11115.7	0.542271
$750$	0	0
$751$	−8210.79	−0.398956	−0.199478	−	0.979902i	$-0.563925\pi$
$751$	−8210.79	−0.398956	−0.199478	+	0.979902i	$0.563925\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1095.00	0.0527830
$756$	0	0
$757$	−38550.0	−1.85089	−0.925445	−	0.378882i	$-0.876309\pi$
$757$	−38550.0	−1.85089	−0.925445	+	0.378882i	$0.876309\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	21941.0	1.04515	0.522576	−	0.852593i	$-0.324971\pi$
$761$	21941.0	1.04515	0.522576	+	0.852593i	$0.324971\pi$
$762$	0	0
$763$	−12183.7	−0.578088
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−160.000	−0.00753229
$768$	0	0
$769$	−11783.0	−0.552543	−0.276272	−	0.961080i	$-0.589099\pi$
$769$	−11783.0	−0.552543	−0.276272	+	0.961080i	$0.589099\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	5485.25	0.255227	0.127614	−	0.991824i	$-0.459268\pi$
$773$	5485.25	0.255227	0.127614	+	0.991824i	$0.459268\pi$
$774$	0	0
$775$	−4887.17	−0.226519
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−10220.0	−0.470051
$780$	0	0
$781$	20.0000	0.000916333 0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−14183.0	−0.644859
$786$	0	0
$787$	−36089.9	−1.63464	−0.817322	−	0.576181i	$-0.804542\pi$
$787$	−36089.9	−1.63464	−0.817322	+	0.576181i	$0.804542\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	11242.0	0.505334
$792$	0	0
$793$	3264.00	0.146164
$794$	0	0
$795$	0	0
$796$	0	0
$797$	18429.4	0.819076	0.409538	−	0.912293i	$-0.365690\pi$
$797$	18429.4	0.819076	0.409538	+	0.912293i	$0.365690\pi$
$798$	0	0
$799$	10868.0	0.481203
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−591.000	−0.0259725
$804$	0	0
$805$	−1314.00	−0.0575309
$806$	0	0
$807$	0	0
$808$	0	0
$809$	4425.79	0.192339	0.0961697	−	0.995365i	$-0.469341\pi$
$809$	4425.79	0.192339	0.0961697	+	0.995365i	$0.469341\pi$
$810$	0	0
$811$	2546.11	0.110242	0.0551209	−	0.998480i	$-0.482446\pi$
$811$	2546.11	0.110242	0.0551209	+	0.998480i	$0.482446\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	18688.0	0.803205
$816$	0	0
$817$	−26572.0	−1.13787
$818$	0	0
$819$	0	0
$820$	0	0
$821$	32894.4	1.39832	0.699161	−	0.714964i	$-0.253557\pi$
$821$	32894.4	1.39832	0.699161	+	0.714964i	$0.253557\pi$
$822$	0	0
$823$	3750.82	0.158864	0.0794321	−	0.996840i	$-0.474689\pi$
$823$	3750.82	0.158864	0.0794321	+	0.996840i	$0.474689\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−39728.0	−1.67047	−0.835235	−	0.549894i	$-0.814668\pi$
$827$	−39728.0	−1.67047	−0.835235	+	0.549894i	$0.814668\pi$
$828$	0	0
$829$	−16774.0	−0.702756	−0.351378	−	0.936234i	$-0.614287\pi$
$829$	−16774.0	−0.702756	−0.351378	+	0.936234i	$0.614287\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	27682.6	1.15143
$834$	0	0
$835$	−25409.9	−1.05311
$836$	0	0
$837$	0	0
$838$	0	0
$839$	24236.0	0.997282	0.498641	−	0.866809i	$-0.333833\pi$
$839$	24236.0	0.997282	0.498641	+	0.866809i	$0.333833\pi$
$840$	0	0
$841$	−10081.0	−0.413342
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−18224.4	−0.741937
$846$	0	0
$847$	−11363.5	−0.460986
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2628.00	0.105860
$852$	0	0
$853$	23022.0	0.924101	0.462051	−	0.886854i	$-0.347114\pi$
$853$	23022.0	0.924101	0.462051	+	0.886854i	$0.347114\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−10697.1	−0.426378	−0.213189	−	0.977011i	$-0.568385\pi$
$857$	−10697.1	−0.426378	−0.213189	+	0.977011i	$0.568385\pi$
$858$	0	0
$859$	26930.7	1.06969	0.534845	−	0.844950i	$-0.320370\pi$
$859$	26930.7	1.06969	0.534845	+	0.844950i	$0.320370\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−21748.0	−0.857834	−0.428917	−	0.903344i	$-0.641105\pi$
$863$	−21748.0	−0.857834	−0.428917	+	0.903344i	$0.641105\pi$
$864$	0	0
$865$	20805.0	0.817793
$866$	0	0
$867$	0	0
$868$	0	0
$869$	683.520	0.0266822
$870$	0	0
$871$	4784.64	0.186133
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−12921.0	−0.499211
$876$	0	0
$877$	−21494.0	−0.827595	−0.413797	−	0.910369i	$-0.635798\pi$
$877$	−21494.0	−0.827595	−0.413797	+	0.910369i	$0.635798\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	615.168	0.0235250	0.0117625	−	0.999931i	$-0.496256\pi$
$881$	615.168	0.0235250	0.0117625	+	0.999931i	$0.496256\pi$
$882$	0	0
$883$	−12901.4	−0.491697	−0.245848	−	0.969308i	$-0.579067\pi$
$883$	−12901.4	−0.491697	−0.245848	+	0.969308i	$0.579067\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	37176.0	1.40727	0.703635	−	0.710562i	$-0.251559\pi$
$887$	37176.0	1.40727	0.703635	+	0.710562i	$0.251559\pi$
$888$	0	0
$889$	−22849.0	−0.862014
$890$	0	0
$891$	0	0
$892$	0	0
$893$	12679.3	0.475136
$894$	0	0
$895$	39806.5	1.48669
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−11242.0	−0.417065
$900$	0	0
$901$	−30660.0	−1.13367
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−17771.5	−0.652758
$906$	0	0
$907$	36790.5	1.34687	0.673433	−	0.739248i	$-0.264819\pi$
$907$	36790.5	1.34687	0.673433	+	0.739248i	$0.264819\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1278.00	0.0464786	0.0232393	−	0.999730i	$-0.492602\pi$
$911$	1278.00	0.0464786	0.0232393	+	0.999730i	$0.492602\pi$
$912$	0	0
$913$	345.000	0.0125058
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2366.69	0.0852290
$918$	0	0
$919$	13149.2	0.471983	0.235992	−	0.971755i	$-0.424166\pi$
$919$	13149.2	0.471983	0.235992	+	0.971755i	$0.424166\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−160.000	−0.00570581
$924$	0	0
$925$	7592.00	0.269863
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−29630.6	−1.04645	−0.523223	−	0.852196i	$-0.675271\pi$
$929$	−29630.6	−1.04645	−0.523223	+	0.852196i	$0.675271\pi$
$930$	0	0
$931$	32296.3	1.13692
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−876.000	−0.0306399
$936$	0	0
$937$	1643.00	0.0572833	0.0286417	−	0.999590i	$-0.490882\pi$
$937$	1643.00	0.0572833	0.0286417	+	0.999590i	$0.490882\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−28665.1	−0.993046	−0.496523	−	0.868023i	$-0.665390\pi$
$941$	−28665.1	−0.993046	−0.496523	+	0.868023i	$0.665390\pi$
$942$	0	0
$943$	−1537.92	−0.0531088
$944$	0	0
$945$	0	0
$946$	0	0
$947$	17367.0	0.595936	0.297968	−	0.954576i	$-0.403691\pi$
$947$	17367.0	0.595936	0.297968	+	0.954576i	$0.403691\pi$
$948$	0	0
$949$	4728.00	0.161725
$950$	0	0
$951$	0	0
$952$	0	0
$953$	26759.8	0.909586	0.454793	−	0.890597i	$-0.349713\pi$
$953$	26759.8	0.909586	0.454793	+	0.890597i	$0.349713\pi$
$954$	0	0
$955$	−27067.4	−0.917152
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−14454.0	−0.486699
$960$	0	0
$961$	−20958.0	−0.703501
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−5767.20	−0.192386
$966$	0	0
$967$	−38268.6	−1.27263	−0.636316	−	0.771429i	$-0.719543\pi$
$967$	−38268.6	−1.27263	−0.636316	+	0.771429i	$0.719543\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−38855.0	−1.28416	−0.642078	−	0.766639i	$-0.721928\pi$
$971$	−38855.0	−1.28416	−0.642078	+	0.766639i	$0.721928\pi$
$972$	0	0
$973$	13724.0	0.452180
$974$	0	0
$975$	0	0
$976$	0	0
$977$	32279.2	1.05702	0.528508	−	0.848928i	$-0.322752\pi$
$977$	32279.2	1.05702	0.528508	+	0.848928i	$0.322752\pi$
$978$	0	0
$979$	222.144	0.00725205
$980$	0	0
$981$	0	0
$982$	0	0
$983$	34134.0	1.10753	0.553767	−	0.832672i	$-0.313190\pi$
$983$	34134.0	1.10753	0.553767	+	0.832672i	$0.313190\pi$
$984$	0	0
$985$	31171.0	1.00832
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3998.59	−0.128562
$990$	0	0
$991$	−36969.9	−1.18505	−0.592527	−	0.805551i	$-0.701869\pi$
$991$	−36969.9	−1.18505	−0.592527	+	0.805551i	$0.701869\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	38471.0	1.22574
$996$	0	0
$997$	20448.0	0.649543	0.324772	−	0.945792i	$-0.394713\pi$
$997$	20448.0	0.649543	0.324772	+	0.945792i	$0.394713\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.4.a.g.1.2	yes	2
3.2	odd	2		864.4.a.f.1.1	✓	2
4.3	odd	2		864.4.a.f.1.2	yes	2
8.3	odd	2		1728.4.a.bn.1.1		2
8.5	even	2		1728.4.a.bm.1.1		2
12.11	even	2	inner	864.4.a.g.1.1	yes	2
24.5	odd	2		1728.4.a.bn.1.2		2
24.11	even	2		1728.4.a.bm.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.4.a.f.1.1	✓	2	3.2	odd	2
864.4.a.f.1.2	yes	2	4.3	odd	2
864.4.a.g.1.1	yes	2	12.11	even	2	inner
864.4.a.g.1.2	yes	2	1.1	even	1	trivial
1728.4.a.bm.1.1		2	8.5	even	2
1728.4.a.bm.1.2		2	24.11	even	2
1728.4.a.bn.1.1		2	8.3	odd	2
1728.4.a.bn.1.2		2	24.5	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	864.a (trivial)

\(f(q)\)	\(=\)	\(q+8.54400 q^{5} +8.54400 q^{7} +1.00000 q^{11} -8.00000 q^{13} -102.528 q^{17} -119.616 q^{19} -18.0000 q^{23} -52.0000 q^{25} -119.616 q^{29} +93.9840 q^{31} +73.0000 q^{35} -146.000 q^{37} +85.4400 q^{41} +222.144 q^{43} -106.000 q^{47} -270.000 q^{49} +299.040 q^{53} +8.54400 q^{55} +20.0000 q^{59} -408.000 q^{61} -68.3520 q^{65} -598.080 q^{67} +20.0000 q^{71} -591.000 q^{73} +8.54400 q^{77} +683.520 q^{79} +345.000 q^{83} -876.000 q^{85} +222.144 q^{89} -68.3520 q^{91} -1022.00 q^{95} -1241.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{11} - 16 q^{13} - 36 q^{23} - 104 q^{25} + 146 q^{35} - 292 q^{37} - 212 q^{47} - 540 q^{49} + 40 q^{59} - 816 q^{61} + 40 q^{71} - 1182 q^{73} + 690 q^{83} - 1752 q^{85} - 2044 q^{95} - 2482 q^{97}+O(q^{100}) \) 2 * q + 2 * q^11 - 16 * q^13 - 36 * q^23 - 104 * q^25 + 146 * q^35 - 292 * q^37 - 212 * q^47 - 540 * q^49 + 40 * q^59 - 816 * q^61 + 40 * q^71 - 1182 * q^73 + 690 * q^83 - 1752 * q^85 - 2044 * q^95 - 2482 * q^97

Introduction

L-functions

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