LMFDB - Embedded newform 1728.4.a.by.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1728,4,Mod(1,1728)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1728, 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("1728.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1728.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,-3,0,-3,0,0,0,-21] 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:	$101.955300490$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 864)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.11491$ of defining polynomial
Character	$\chi$	$=$	1728.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+9.83700 q^{5} +13.5419 q^{7} +40.0529 q^{11} -46.4627 q^{13} -42.4627 q^{17} -39.9691 q^{19} -38.8236 q^{23} -28.2334 q^{25} -155.062 q^{29} -132.806 q^{31} +133.212 q^{35} -116.929 q^{37} +7.90737 q^{41} -455.172 q^{43} -270.670 q^{47} -159.617 q^{49} -71.3743 q^{53} +394.000 q^{55} -779.199 q^{59} +790.952 q^{61} -457.053 q^{65} -835.630 q^{67} +249.458 q^{71} +619.260 q^{73} +542.392 q^{77} +262.480 q^{79} -465.347 q^{83} -417.705 q^{85} +891.182 q^{89} -629.192 q^{91} -393.176 q^{95} -248.259 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} - 3 q^{7} - 21 q^{11} + 24 q^{13} + 36 q^{17} - 66 q^{19} - 42 q^{23} + 12 q^{25} + 90 q^{29} + 351 q^{31} - 165 q^{35} + 6 q^{37} - 138 q^{41} - 498 q^{43} - 114 q^{47} + 78 q^{49} - 345 q^{53}+ \cdots + 549 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 - 3 * q^7 - 21 * q^11 + 24 * q^13 + 36 * q^17 - 66 * q^19 - 42 * q^23 + 12 * q^25 + 90 * q^29 + 351 * q^31 - 165 * q^35 + 6 * q^37 - 138 * q^41 - 498 * q^43 - 114 * q^47 + 78 * q^49 - 345 * q^53 + 1005 * q^55 - 228 * q^59 - 168 * q^61 + 96 * q^65 - 1506 * q^67 - 1020 * q^71 - 453 * q^73 + 621 * q^77 + 2448 * q^79 - 957 * q^83 + 84 * q^85 + 1038 * q^89 - 3168 * q^91 - 1254 * q^95 + 549 * 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
$5$	9.83700	0.879848	0.439924	−	0.898035i	$-0.355005\pi$
$5$	9.83700	0.879848	0.439924	+	0.898035i	$0.355005\pi$
$6$	0	0
$7$	13.5419	0.731193	0.365597	−	0.930773i	$-0.380865\pi$
$7$	13.5419	0.731193	0.365597	+	0.930773i	$0.380865\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	40.0529	1.09785	0.548927	−	0.835870i	$-0.315036\pi$
$11$	40.0529	1.09785	0.548927	+	0.835870i	$0.315036\pi$
$12$	0	0
$13$	−46.4627	−0.991263	−0.495632	−	0.868533i	$-0.665063\pi$
$13$	−46.4627	−0.991263	−0.495632	+	0.868533i	$0.665063\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−42.4627	−0.605806	−0.302903	−	0.953021i	$-0.597956\pi$
$17$	−42.4627	−0.605806	−0.302903	+	0.953021i	$0.597956\pi$
$18$	0	0
$19$	−39.9691	−0.482608	−0.241304	−	0.970450i	$-0.577575\pi$
$19$	−39.9691	−0.482608	−0.241304	+	0.970450i	$0.577575\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−38.8236	−0.351969	−0.175984	−	0.984393i	$-0.556311\pi$
$23$	−38.8236	−0.351969	−0.175984	+	0.984393i	$0.556311\pi$
$24$	0	0
$25$	−28.2334	−0.225867
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−155.062	−0.992907	−0.496453	−	0.868063i	$-0.665365\pi$
$29$	−155.062	−0.992907	−0.496453	+	0.868063i	$0.665365\pi$
$30$	0	0
$31$	−132.806	−0.769443	−0.384721	−	0.923033i	$-0.625702\pi$
$31$	−132.806	−0.769443	−0.384721	+	0.923033i	$0.625702\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	133.212	0.643339
$36$	0	0
$37$	−116.929	−0.519543	−0.259771	−	0.965670i	$-0.583647\pi$
$37$	−116.929	−0.519543	−0.259771	+	0.965670i	$0.583647\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.90737	0.0301201	0.0150600	−	0.999887i	$-0.495206\pi$
$41$	7.90737	0.0301201	0.0150600	+	0.999887i	$0.495206\pi$
$42$	0	0
$43$	−455.172	−1.61426	−0.807129	−	0.590376i	$-0.798980\pi$
$43$	−455.172	−1.61426	−0.807129	+	0.590376i	$0.798980\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−270.670	−0.840028	−0.420014	−	0.907518i	$-0.637975\pi$
$47$	−270.670	−0.840028	−0.420014	+	0.907518i	$0.637975\pi$
$48$	0	0
$49$	−159.617	−0.465357
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−71.3743	−0.184981	−0.0924907	−	0.995714i	$-0.529483\pi$
$53$	−71.3743	−0.184981	−0.0924907	+	0.995714i	$0.529483\pi$
$54$	0	0
$55$	394.000	0.965946
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−779.199	−1.71937	−0.859687	−	0.510821i	$-0.829342\pi$
$59$	−779.199	−1.71937	−0.859687	+	0.510821i	$0.829342\pi$
$60$	0	0
$61$	790.952	1.66018	0.830091	−	0.557629i	$-0.188289\pi$
$61$	790.952	1.66018	0.830091	+	0.557629i	$0.188289\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−457.053	−0.872161
$66$	0	0
$67$	−835.630	−1.52371	−0.761855	−	0.647748i	$-0.775711\pi$
$67$	−835.630	−1.52371	−0.761855	+	0.647748i	$0.775711\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	249.458	0.416976	0.208488	−	0.978025i	$-0.433146\pi$
$71$	249.458	0.416976	0.208488	+	0.978025i	$0.433146\pi$
$72$	0	0
$73$	619.260	0.992862	0.496431	−	0.868076i	$-0.334644\pi$
$73$	619.260	0.992862	0.496431	+	0.868076i	$0.334644\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	542.392	0.802744
$78$	0	0
$79$	262.480	0.373814	0.186907	−	0.982378i	$-0.440154\pi$
$79$	262.480	0.373814	0.186907	+	0.982378i	$0.440154\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−465.347	−0.615404	−0.307702	−	0.951483i	$-0.599560\pi$
$83$	−465.347	−0.615404	−0.307702	+	0.951483i	$0.599560\pi$
$84$	0	0
$85$	−417.705	−0.533018
$86$	0	0
$87$	0	0
$88$	0	0
$89$	891.182	1.06141	0.530703	−	0.847558i	$-0.321928\pi$
$89$	891.182	1.06141	0.530703	+	0.847558i	$0.321928\pi$
$90$	0	0
$91$	−629.192	−0.724805
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−393.176	−0.424622
$96$	0	0
$97$	−248.259	−0.259865	−0.129932	−	0.991523i	$-0.541476\pi$
$97$	−248.259	−0.259865	−0.129932	+	0.991523i	$0.541476\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	164.416	0.161980	0.0809901	−	0.996715i	$-0.474192\pi$
$101$	164.416	0.161980	0.0809901	+	0.996715i	$0.474192\pi$
$102$	0	0
$103$	1514.44	1.44875	0.724377	−	0.689404i	$-0.242127\pi$
$103$	1514.44	1.44875	0.724377	+	0.689404i	$0.242127\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−406.118	−0.366925	−0.183462	−	0.983027i	$-0.558731\pi$
$107$	−406.118	−0.366925	−0.183462	+	0.983027i	$0.558731\pi$
$108$	0	0
$109$	−761.097	−0.668807	−0.334403	−	0.942430i	$-0.608535\pi$
$109$	−761.097	−0.668807	−0.334403	+	0.942430i	$0.608535\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	558.569	0.465007	0.232503	−	0.972596i	$-0.425308\pi$
$113$	558.569	0.465007	0.232503	+	0.972596i	$0.425308\pi$
$114$	0	0
$115$	−381.908	−0.309679
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−575.025	−0.442962
$120$	0	0
$121$	273.235	0.205285
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1507.36	−1.07858
$126$	0	0
$127$	2309.02	1.61332	0.806661	−	0.591014i	$-0.201272\pi$
$127$	2309.02	1.61332	0.806661	+	0.591014i	$0.201272\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	421.046	0.280816	0.140408	−	0.990094i	$-0.455159\pi$
$131$	421.046	0.280816	0.140408	+	0.990094i	$0.455159\pi$
$132$	0	0
$133$	−541.257	−0.352879
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−302.781	−0.188820	−0.0944100	−	0.995533i	$-0.530096\pi$
$137$	−302.781	−0.188820	−0.0944100	+	0.995533i	$0.530096\pi$
$138$	0	0
$139$	−2593.40	−1.58251	−0.791255	−	0.611486i	$-0.790572\pi$
$139$	−2593.40	−1.58251	−0.791255	+	0.611486i	$0.790572\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1860.96	−1.08826
$144$	0	0
$145$	−1525.35	−0.873607
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3351.49	1.84272	0.921358	−	0.388714i	$-0.127081\pi$
$149$	3351.49	1.84272	0.921358	+	0.388714i	$0.127081\pi$
$150$	0	0
$151$	2886.95	1.55587	0.777937	−	0.628342i	$-0.216266\pi$
$151$	2886.95	1.55587	0.777937	+	0.628342i	$0.216266\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1306.42	−0.676993
$156$	0	0
$157$	784.328	0.398702	0.199351	−	0.979928i	$-0.436117\pi$
$157$	784.328	0.398702	0.199351	+	0.979928i	$0.436117\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−525.745	−0.257357
$162$	0	0
$163$	448.525	0.215529	0.107764	−	0.994176i	$-0.465631\pi$
$163$	448.525	0.215529	0.107764	+	0.994176i	$0.465631\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3665.74	−1.69858	−0.849292	−	0.527924i	$-0.822971\pi$
$167$	−3665.74	−1.69858	−0.849292	+	0.527924i	$0.822971\pi$
$168$	0	0
$169$	−38.2208	−0.0173968
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1476.41	−0.648841	−0.324421	−	0.945913i	$-0.605169\pi$
$173$	−1476.41	−0.648841	−0.324421	+	0.945913i	$0.605169\pi$
$174$	0	0
$175$	−382.333	−0.165152
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2940.99	−1.22804	−0.614021	−	0.789290i	$-0.710449\pi$
$179$	−2940.99	−1.22804	−0.614021	+	0.789290i	$0.710449\pi$
$180$	0	0
$181$	684.504	0.281098	0.140549	−	0.990074i	$-0.455113\pi$
$181$	684.504	0.281098	0.140549	+	0.990074i	$0.455113\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1150.23	−0.457119
$186$	0	0
$187$	−1700.75	−0.665087
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1475.64	−0.559022	−0.279511	−	0.960142i	$-0.590172\pi$
$191$	−1475.64	−0.559022	−0.279511	+	0.960142i	$0.590172\pi$
$192$	0	0
$193$	589.569	0.219887	0.109943	−	0.993938i	$-0.464933\pi$
$193$	589.569	0.219887	0.109943	+	0.993938i	$0.464933\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2832.25	−1.02431	−0.512156	−	0.858892i	$-0.671153\pi$
$197$	−2832.25	−1.02431	−0.512156	+	0.858892i	$0.671153\pi$
$198$	0	0
$199$	2174.53	0.774614	0.387307	−	0.921951i	$-0.373405\pi$
$199$	2174.53	0.774614	0.387307	+	0.921951i	$0.373405\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2099.83	−0.726007
$204$	0	0
$205$	77.7848	0.0265011
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1600.88	−0.529833
$210$	0	0
$211$	3971.58	1.29580	0.647902	−	0.761724i	$-0.275647\pi$
$211$	3971.58	1.29580	0.647902	+	0.761724i	$0.275647\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4477.53	−1.42030
$216$	0	0
$217$	−1798.45	−0.562611
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1972.93	0.600514
$222$	0	0
$223$	−40.1758	−0.0120644	−0.00603222	−	0.999982i	$-0.501920\pi$
$223$	−40.1758	−0.0120644	−0.00603222	+	0.999982i	$0.501920\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4563.84	−1.33442	−0.667209	−	0.744870i	$-0.732511\pi$
$227$	−4563.84	−1.33442	−0.667209	+	0.744870i	$0.732511\pi$
$228$	0	0
$229$	2417.80	0.697698	0.348849	−	0.937179i	$-0.386573\pi$
$229$	2417.80	0.697698	0.348849	+	0.937179i	$0.386573\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6625.61	1.86291	0.931455	−	0.363857i	$-0.118540\pi$
$233$	6625.61	1.86291	0.931455	+	0.363857i	$0.118540\pi$
$234$	0	0
$235$	−2662.58	−0.739097
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3075.46	0.832365	0.416183	−	0.909281i	$-0.363368\pi$
$239$	3075.46	0.832365	0.416183	+	0.909281i	$0.363368\pi$
$240$	0	0
$241$	7340.97	1.96213	0.981066	−	0.193675i	$-0.0620406\pi$
$241$	7340.97	1.96213	0.981066	+	0.193675i	$0.0620406\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1570.16	−0.409443
$246$	0	0
$247$	1857.07	0.478391
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5874.10	−1.47717	−0.738586	−	0.674159i	$-0.764506\pi$
$251$	−5874.10	−1.47717	−0.738586	+	0.674159i	$0.764506\pi$
$252$	0	0
$253$	−1555.00	−0.386410
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5242.60	−1.27247	−0.636234	−	0.771496i	$-0.719509\pi$
$257$	−5242.60	−1.27247	−0.636234	+	0.771496i	$0.719509\pi$
$258$	0	0
$259$	−1583.44	−0.379886
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1619.63	−0.379735	−0.189868	−	0.981810i	$-0.560806\pi$
$263$	−1619.63	−0.379735	−0.189868	+	0.981810i	$0.560806\pi$
$264$	0	0
$265$	−702.110	−0.162756
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5686.54	−1.28890	−0.644451	−	0.764646i	$-0.722914\pi$
$269$	−5686.54	−1.28890	−0.644451	+	0.764646i	$0.722914\pi$
$270$	0	0
$271$	−7136.07	−1.59958	−0.799789	−	0.600282i	$-0.795055\pi$
$271$	−7136.07	−1.59958	−0.799789	+	0.600282i	$0.795055\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1130.83	−0.247969
$276$	0	0
$277$	−786.585	−0.170618	−0.0853092	−	0.996355i	$-0.527188\pi$
$277$	−786.585	−0.170618	−0.0853092	+	0.996355i	$0.527188\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−6586.96	−1.39838	−0.699191	−	0.714935i	$-0.746456\pi$
$281$	−6586.96	−1.39838	−0.699191	+	0.714935i	$0.746456\pi$
$282$	0	0
$283$	−1411.45	−0.296474	−0.148237	−	0.988952i	$-0.547360\pi$
$283$	−1411.45	−0.296474	−0.148237	+	0.988952i	$0.547360\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	107.081	0.0220236
$288$	0	0
$289$	−3109.92	−0.632999
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5326.36	1.06201	0.531006	−	0.847368i	$-0.321814\pi$
$293$	5326.36	1.06201	0.531006	+	0.847368i	$0.321814\pi$
$294$	0	0
$295$	−7664.98	−1.51279
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1803.85	0.348894
$300$	0	0
$301$	−6163.89	−1.18033
$302$	0	0
$303$	0	0
$304$	0	0
$305$	7780.60	1.46071
$306$	0	0
$307$	1692.71	0.314684	0.157342	−	0.987544i	$-0.449707\pi$
$307$	1692.71	0.314684	0.157342	+	0.987544i	$0.449707\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6054.86	−1.10399	−0.551993	−	0.833849i	$-0.686132\pi$
$311$	−6054.86	−1.10399	−0.551993	+	0.833849i	$0.686132\pi$
$312$	0	0
$313$	145.232	0.0262269	0.0131134	−	0.999914i	$-0.495826\pi$
$313$	145.232	0.0262269	0.0131134	+	0.999914i	$0.495826\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9070.78	1.60715	0.803574	−	0.595205i	$-0.202929\pi$
$317$	9070.78	1.60715	0.803574	+	0.595205i	$0.202929\pi$
$318$	0	0
$319$	−6210.68	−1.09007
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1697.20	0.292367
$324$	0	0
$325$	1311.80	0.223894
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3665.39	−0.614223
$330$	0	0
$331$	9668.25	1.60548	0.802742	−	0.596326i	$-0.203374\pi$
$331$	9668.25	1.60548	0.802742	+	0.596326i	$0.203374\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−8220.10	−1.34063
$336$	0	0
$337$	−5730.77	−0.926335	−0.463168	−	0.886271i	$-0.653287\pi$
$337$	−5730.77	−0.926335	−0.463168	+	0.886271i	$0.653287\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−5319.28	−0.844737
$342$	0	0
$343$	−6806.39	−1.07146
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−11934.3	−1.84631	−0.923155	−	0.384429i	$-0.874398\pi$
$347$	−11934.3	−1.84631	−0.923155	+	0.384429i	$0.874398\pi$
$348$	0	0
$349$	10225.6	1.56839	0.784193	−	0.620518i	$-0.213077\pi$
$349$	10225.6	1.56839	0.784193	+	0.620518i	$0.213077\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9972.04	1.50356	0.751782	−	0.659411i	$-0.229194\pi$
$353$	9972.04	1.50356	0.751782	+	0.659411i	$0.229194\pi$
$354$	0	0
$355$	2453.92	0.366875
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6309.46	−0.927578	−0.463789	−	0.885946i	$-0.653510\pi$
$359$	−6309.46	−0.927578	−0.463789	+	0.885946i	$0.653510\pi$
$360$	0	0
$361$	−5261.47	−0.767090
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6091.66	0.873568
$366$	0	0
$367$	−1061.15	−0.150931	−0.0754654	−	0.997148i	$-0.524044\pi$
$367$	−1061.15	−0.150931	−0.0754654	+	0.997148i	$0.524044\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−966.543	−0.135257
$372$	0	0
$373$	−4296.88	−0.596472	−0.298236	−	0.954492i	$-0.596398\pi$
$373$	−4296.88	−0.596472	−0.298236	+	0.954492i	$0.596398\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7204.59	0.984232
$378$	0	0
$379$	3965.26	0.537418	0.268709	−	0.963221i	$-0.413403\pi$
$379$	3965.26	0.537418	0.268709	+	0.963221i	$0.413403\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1981.21	−0.264321	−0.132161	−	0.991228i	$-0.542191\pi$
$383$	−1981.21	−0.264321	−0.132161	+	0.991228i	$0.542191\pi$
$384$	0	0
$385$	5335.51	0.706293
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−3621.21	−0.471986	−0.235993	−	0.971755i	$-0.575834\pi$
$389$	−3621.21	−0.471986	−0.235993	+	0.971755i	$0.575834\pi$
$390$	0	0
$391$	1648.55	0.213225
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2582.02	0.328899
$396$	0	0
$397$	−14052.2	−1.77648	−0.888239	−	0.459382i	$-0.848071\pi$
$397$	−14052.2	−1.77648	−0.888239	+	0.459382i	$0.848071\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−10743.2	−1.33788	−0.668942	−	0.743315i	$-0.733252\pi$
$401$	−10743.2	−1.33788	−0.668942	+	0.743315i	$0.733252\pi$
$402$	0	0
$403$	6170.54	0.762721
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4683.36	−0.570382
$408$	0	0
$409$	−423.339	−0.0511804	−0.0255902	−	0.999673i	$-0.508147\pi$
$409$	−423.339	−0.0511804	−0.0255902	+	0.999673i	$0.508147\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−10551.8	−1.25719
$414$	0	0
$415$	−4577.62	−0.541462
$416$	0	0
$417$	0	0
$418$	0	0
$419$	5959.04	0.694792	0.347396	−	0.937718i	$-0.387066\pi$
$419$	5959.04	0.694792	0.347396	+	0.937718i	$0.387066\pi$
$420$	0	0
$421$	15199.7	1.75959	0.879795	−	0.475352i	$-0.157679\pi$
$421$	15199.7	1.75959	0.879795	+	0.475352i	$0.157679\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1198.86	0.136832
$426$	0	0
$427$	10711.0	1.21391
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4428.98	0.494980	0.247490	−	0.968890i	$-0.420394\pi$
$431$	4428.98	0.494980	0.247490	+	0.968890i	$0.420394\pi$
$432$	0	0
$433$	2497.20	0.277154	0.138577	−	0.990352i	$-0.455747\pi$
$433$	2497.20	0.277154	0.138577	+	0.990352i	$0.455747\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1551.75	0.169863
$438$	0	0
$439$	−4332.64	−0.471038	−0.235519	−	0.971870i	$-0.575679\pi$
$439$	−4332.64	−0.471038	−0.235519	+	0.971870i	$0.575679\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−11865.4	−1.27256	−0.636279	−	0.771459i	$-0.719527\pi$
$443$	−11865.4	−1.27256	−0.636279	+	0.771459i	$0.719527\pi$
$444$	0	0
$445$	8766.56	0.933876
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1724.93	0.181302	0.0906511	−	0.995883i	$-0.471105\pi$
$449$	1724.93	0.181302	0.0906511	+	0.995883i	$0.471105\pi$
$450$	0	0
$451$	316.713	0.0330675
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6189.37	−0.637719
$456$	0	0
$457$	10714.6	1.09673	0.548365	−	0.836239i	$-0.315250\pi$
$457$	10714.6	1.09673	0.548365	+	0.836239i	$0.315250\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	12258.8	1.23850	0.619249	−	0.785195i	$-0.287437\pi$
$461$	12258.8	1.23850	0.619249	+	0.785195i	$0.287437\pi$
$462$	0	0
$463$	−5968.59	−0.599101	−0.299551	−	0.954080i	$-0.596837\pi$
$463$	−5968.59	−0.599101	−0.299551	+	0.954080i	$0.596837\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2367.56	0.234598	0.117299	−	0.993097i	$-0.462576\pi$
$467$	2367.56	0.234598	0.117299	+	0.993097i	$0.462576\pi$
$468$	0	0
$469$	−11316.0	−1.11413
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−18231.0	−1.77222
$474$	0	0
$475$	1128.46	0.109005
$476$	0	0
$477$	0	0
$478$	0	0
$479$	13579.2	1.29530	0.647649	−	0.761939i	$-0.275752\pi$
$479$	13579.2	1.29530	0.647649	+	0.761939i	$0.275752\pi$
$480$	0	0
$481$	5432.85	0.515004
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2442.12	−0.228642
$486$	0	0
$487$	−3108.88	−0.289274	−0.144637	−	0.989485i	$-0.546201\pi$
$487$	−3108.88	−0.289274	−0.144637	+	0.989485i	$0.546201\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−7443.00	−0.684110	−0.342055	−	0.939680i	$-0.611123\pi$
$491$	−7443.00	−0.684110	−0.342055	+	0.939680i	$0.611123\pi$
$492$	0	0
$493$	6584.35	0.601509
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3378.14	0.304890
$498$	0	0
$499$	16436.9	1.47458	0.737289	−	0.675577i	$-0.236106\pi$
$499$	16436.9	1.47458	0.737289	+	0.675577i	$0.236106\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	14606.3	1.29476	0.647379	−	0.762168i	$-0.275865\pi$
$503$	14606.3	1.29476	0.647379	+	0.762168i	$0.275865\pi$
$504$	0	0
$505$	1617.36	0.142518
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−2530.62	−0.220369	−0.110184	−	0.993911i	$-0.535144\pi$
$509$	−2530.62	−0.220369	−0.110184	+	0.993911i	$0.535144\pi$
$510$	0	0
$511$	8385.95	0.725974
$512$	0	0
$513$	0	0
$514$	0	0
$515$	14897.5	1.27468
$516$	0	0
$517$	−10841.1	−0.922229
$518$	0	0
$519$	0	0
$520$	0	0
$521$	12477.8	1.04925	0.524627	−	0.851332i	$-0.324205\pi$
$521$	12477.8	1.04925	0.524627	+	0.851332i	$0.324205\pi$
$522$	0	0
$523$	13147.6	1.09924	0.549621	−	0.835414i	$-0.314772\pi$
$523$	13147.6	1.09924	0.549621	+	0.835414i	$0.314772\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5639.31	0.466133
$528$	0	0
$529$	−10659.7	−0.876118
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−367.397	−0.0298569
$534$	0	0
$535$	−3994.99	−0.322838
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−6393.14	−0.510894
$540$	0	0
$541$	−418.706	−0.0332746	−0.0166373	−	0.999862i	$-0.505296\pi$
$541$	−418.706	−0.0332746	−0.0166373	+	0.999862i	$0.505296\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−7486.92	−0.588448
$546$	0	0
$547$	−9793.86	−0.765549	−0.382774	−	0.923842i	$-0.625031\pi$
$547$	−9793.86	−0.765549	−0.382774	+	0.923842i	$0.625031\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6197.69	0.479184
$552$	0	0
$553$	3554.47	0.273330
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−15790.7	−1.20121	−0.600604	−	0.799546i	$-0.705073\pi$
$557$	−15790.7	−1.20121	−0.600604	+	0.799546i	$0.705073\pi$
$558$	0	0
$559$	21148.5	1.60015
$560$	0	0
$561$	0	0
$562$	0	0
$563$	16327.4	1.22223	0.611117	−	0.791540i	$-0.290721\pi$
$563$	16327.4	1.22223	0.611117	+	0.791540i	$0.290721\pi$
$564$	0	0
$565$	5494.64	0.409135
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−13708.7	−1.01002	−0.505009	−	0.863114i	$-0.668511\pi$
$569$	−13708.7	−1.01002	−0.505009	+	0.863114i	$0.668511\pi$
$570$	0	0
$571$	−17060.9	−1.25039	−0.625197	−	0.780467i	$-0.714981\pi$
$571$	−17060.9	−1.25039	−0.625197	+	0.780467i	$0.714981\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1096.12	0.0794981
$576$	0	0
$577$	−24581.0	−1.77352	−0.886761	−	0.462229i	$-0.847050\pi$
$577$	−24581.0	−1.77352	−0.886761	+	0.462229i	$0.847050\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−6301.68	−0.449979
$582$	0	0
$583$	−2858.75	−0.203083
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8728.10	−0.613709	−0.306854	−	0.951756i	$-0.599276\pi$
$587$	−8728.10	−0.613709	−0.306854	+	0.951756i	$0.599276\pi$
$588$	0	0
$589$	5308.15	0.371339
$590$	0	0
$591$	0	0
$592$	0	0
$593$	18309.9	1.26796	0.633979	−	0.773350i	$-0.281421\pi$
$593$	18309.9	1.26796	0.633979	+	0.773350i	$0.281421\pi$
$594$	0	0
$595$	−5656.52	−0.389739
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−2905.85	−0.198213	−0.0991067	−	0.995077i	$-0.531599\pi$
$599$	−2905.85	−0.198213	−0.0991067	+	0.995077i	$0.531599\pi$
$600$	0	0
$601$	15925.0	1.08085	0.540427	−	0.841391i	$-0.318263\pi$
$601$	15925.0	1.08085	0.540427	+	0.841391i	$0.318263\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2687.81	0.180620
$606$	0	0
$607$	7953.05	0.531803	0.265901	−	0.964000i	$-0.414330\pi$
$607$	7953.05	0.531803	0.265901	+	0.964000i	$0.414330\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12576.1	0.832689
$612$	0	0
$613$	−18452.8	−1.21582	−0.607912	−	0.794005i	$-0.707993\pi$
$613$	−18452.8	−1.21582	−0.607912	+	0.794005i	$0.707993\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1253.46	0.0817868	0.0408934	−	0.999164i	$-0.486980\pi$
$617$	1253.46	0.0817868	0.0408934	+	0.999164i	$0.486980\pi$
$618$	0	0
$619$	1697.15	0.110200	0.0551002	−	0.998481i	$-0.482452\pi$
$619$	1697.15	0.110200	0.0551002	+	0.998481i	$0.482452\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	12068.3	0.776093
$624$	0	0
$625$	−11298.7	−0.723117
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4965.13	0.314742
$630$	0	0
$631$	6132.61	0.386902	0.193451	−	0.981110i	$-0.438032\pi$
$631$	6132.61	0.386902	0.193451	+	0.981110i	$0.438032\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	22713.8	1.41948
$636$	0	0
$637$	7416.24	0.461291
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2298.20	0.141612	0.0708062	−	0.997490i	$-0.477443\pi$
$641$	2298.20	0.141612	0.0708062	+	0.997490i	$0.477443\pi$
$642$	0	0
$643$	−28441.5	−1.74436	−0.872179	−	0.489186i	$-0.837294\pi$
$643$	−28441.5	−1.74436	−0.872179	+	0.489186i	$0.837294\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−30736.3	−1.86765	−0.933823	−	0.357734i	$-0.883549\pi$
$647$	−30736.3	−1.86765	−0.933823	+	0.357734i	$0.883549\pi$
$648$	0	0
$649$	−31209.2	−1.88762
$650$	0	0
$651$	0	0
$652$	0	0
$653$	21858.1	1.30991	0.654957	−	0.755666i	$-0.272687\pi$
$653$	21858.1	1.30991	0.654957	+	0.755666i	$0.272687\pi$
$654$	0	0
$655$	4141.83	0.247076
$656$	0	0
$657$	0	0
$658$	0	0
$659$	16535.4	0.977435	0.488717	−	0.872442i	$-0.337465\pi$
$659$	16535.4	0.977435	0.488717	+	0.872442i	$0.337465\pi$
$660$	0	0
$661$	5431.14	0.319587	0.159793	−	0.987150i	$-0.448917\pi$
$661$	5431.14	0.319587	0.159793	+	0.987150i	$0.448917\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5324.35	−0.310480
$666$	0	0
$667$	6020.06	0.349472
$668$	0	0
$669$	0	0
$670$	0	0
$671$	31679.9	1.82264
$672$	0	0
$673$	30532.5	1.74880	0.874398	−	0.485209i	$-0.161256\pi$
$673$	30532.5	1.74880	0.874398	+	0.485209i	$0.161256\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−135.741	−0.00770598	−0.00385299	−	0.999993i	$-0.501226\pi$
$677$	−135.741	−0.00770598	−0.00385299	+	0.999993i	$0.501226\pi$
$678$	0	0
$679$	−3361.89	−0.190011
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−9086.82	−0.509074	−0.254537	−	0.967063i	$-0.581923\pi$
$683$	−9086.82	−0.509074	−0.254537	+	0.967063i	$0.581923\pi$
$684$	0	0
$685$	−2978.46	−0.166133
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3316.24	0.183365
$690$	0	0
$691$	−10738.1	−0.591166	−0.295583	−	0.955317i	$-0.595514\pi$
$691$	−10738.1	−0.591166	−0.295583	+	0.955317i	$0.595514\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−25511.2	−1.39237
$696$	0	0
$697$	−335.768	−0.0182469
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−2349.68	−0.126599	−0.0632996	−	0.997995i	$-0.520162\pi$
$701$	−2349.68	−0.126599	−0.0632996	+	0.997995i	$0.520162\pi$
$702$	0	0
$703$	4673.57	0.250735
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2226.50	0.118439
$708$	0	0
$709$	9778.28	0.517956	0.258978	−	0.965883i	$-0.416614\pi$
$709$	9778.28	0.517956	0.258978	+	0.965883i	$0.416614\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5156.02	0.270820
$714$	0	0
$715$	−18306.3	−0.957507
$716$	0	0
$717$	0	0
$718$	0	0
$719$	30812.8	1.59822	0.799112	−	0.601182i	$-0.205303\pi$
$719$	30812.8	1.59822	0.799112	+	0.601182i	$0.205303\pi$
$720$	0	0
$721$	20508.3	1.05932
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4377.92	0.224265
$726$	0	0
$727$	−3541.20	−0.180654	−0.0903271	−	0.995912i	$-0.528791\pi$
$727$	−3541.20	−0.180654	−0.0903271	+	0.995912i	$0.528791\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	19327.8	0.977927
$732$	0	0
$733$	−13935.0	−0.702182	−0.351091	−	0.936341i	$-0.614189\pi$
$733$	−13935.0	−0.702182	−0.351091	+	0.936341i	$0.614189\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−33469.4	−1.67281
$738$	0	0
$739$	−34285.5	−1.70665	−0.853323	−	0.521382i	$-0.825417\pi$
$739$	−34285.5	−1.70665	−0.853323	+	0.521382i	$0.825417\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	18362.9	0.906690	0.453345	−	0.891335i	$-0.350231\pi$
$743$	18362.9	0.906690	0.453345	+	0.891335i	$0.350231\pi$
$744$	0	0
$745$	32968.6	1.62131
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−5499.61	−0.268293
$750$	0	0
$751$	27069.3	1.31528	0.657638	−	0.753334i	$-0.271556\pi$
$751$	27069.3	1.31528	0.657638	+	0.753334i	$0.271556\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	28399.0	1.36893
$756$	0	0
$757$	2165.38	0.103966	0.0519830	−	0.998648i	$-0.483446\pi$
$757$	2165.38	0.103966	0.0519830	+	0.998648i	$0.483446\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	28055.8	1.33643	0.668214	−	0.743969i	$-0.267059\pi$
$761$	28055.8	1.33643	0.668214	+	0.743969i	$0.267059\pi$
$762$	0	0
$763$	−10306.7	−0.489027
$764$	0	0
$765$	0	0
$766$	0	0
$767$	36203.7	1.70435
$768$	0	0
$769$	−8848.59	−0.414939	−0.207470	−	0.978241i	$-0.566523\pi$
$769$	−8848.59	−0.414939	−0.207470	+	0.978241i	$0.566523\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	27932.1	1.29968	0.649838	−	0.760073i	$-0.274837\pi$
$773$	27932.1	1.29968	0.649838	+	0.760073i	$0.274837\pi$
$774$	0	0
$775$	3749.57	0.173792
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−316.051	−0.0145362
$780$	0	0
$781$	9991.53	0.457779
$782$	0	0
$783$	0	0
$784$	0	0
$785$	7715.44	0.350797
$786$	0	0
$787$	−10334.4	−0.468083	−0.234042	−	0.972227i	$-0.575195\pi$
$787$	−10334.4	−0.468083	−0.234042	+	0.972227i	$0.575195\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	7564.08	0.340010
$792$	0	0
$793$	−36749.7	−1.64568
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−13243.4	−0.588590	−0.294295	−	0.955715i	$-0.595085\pi$
$797$	−13243.4	−0.588590	−0.294295	+	0.955715i	$0.595085\pi$
$798$	0	0
$799$	11493.4	0.508894
$800$	0	0
$801$	0	0
$802$	0	0
$803$	24803.2	1.09002
$804$	0	0
$805$	−5171.75	−0.226435
$806$	0	0
$807$	0	0
$808$	0	0
$809$	29201.0	1.26904	0.634518	−	0.772908i	$-0.281198\pi$
$809$	29201.0	1.26904	0.634518	+	0.772908i	$0.281198\pi$
$810$	0	0
$811$	−31462.8	−1.36228	−0.681140	−	0.732153i	$-0.738516\pi$
$811$	−31462.8	−1.36228	−0.681140	+	0.732153i	$0.738516\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	4412.14	0.189633
$816$	0	0
$817$	18192.8	0.779053
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−30830.8	−1.31060	−0.655299	−	0.755370i	$-0.727457\pi$
$821$	−30830.8	−1.31060	−0.655299	+	0.755370i	$0.727457\pi$
$822$	0	0
$823$	4109.58	0.174059	0.0870297	−	0.996206i	$-0.472263\pi$
$823$	4109.58	0.174059	0.0870297	+	0.996206i	$0.472263\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2476.04	−0.104112	−0.0520559	−	0.998644i	$-0.516577\pi$
$827$	−2476.04	−0.104112	−0.0520559	+	0.998644i	$0.516577\pi$
$828$	0	0
$829$	−22798.8	−0.955169	−0.477584	−	0.878586i	$-0.658487\pi$
$829$	−22798.8	−0.955169	−0.477584	+	0.878586i	$0.658487\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6777.78	0.281916
$834$	0	0
$835$	−36059.9	−1.49450
$836$	0	0
$837$	0	0
$838$	0	0
$839$	7937.63	0.326624	0.163312	−	0.986574i	$-0.447782\pi$
$839$	7937.63	0.326624	0.163312	+	0.986574i	$0.447782\pi$
$840$	0	0
$841$	−344.776	−0.0141365
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−375.978	−0.0153066
$846$	0	0
$847$	3700.11	0.150103
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4539.62	0.182863
$852$	0	0
$853$	40012.6	1.60610	0.803051	−	0.595911i	$-0.203209\pi$
$853$	40012.6	1.60610	0.803051	+	0.595911i	$0.203209\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−9326.30	−0.371739	−0.185870	−	0.982574i	$-0.559510\pi$
$857$	−9326.30	−0.371739	−0.185870	+	0.982574i	$0.559510\pi$
$858$	0	0
$859$	14787.1	0.587344	0.293672	−	0.955906i	$-0.405123\pi$
$859$	14787.1	0.587344	0.293672	+	0.955906i	$0.405123\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	40306.3	1.58985	0.794927	−	0.606706i	$-0.207509\pi$
$863$	40306.3	1.58985	0.794927	+	0.606706i	$0.207509\pi$
$864$	0	0
$865$	−14523.5	−0.570882
$866$	0	0
$867$	0	0
$868$	0	0
$869$	10513.1	0.410393
$870$	0	0
$871$	38825.6	1.51040
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−20412.5	−0.788648
$876$	0	0
$877$	4433.72	0.170714	0.0853569	−	0.996350i	$-0.472797\pi$
$877$	4433.72	0.170714	0.0853569	+	0.996350i	$0.472797\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−40297.7	−1.54105	−0.770524	−	0.637411i	$-0.780005\pi$
$881$	−40297.7	−1.54105	−0.770524	+	0.637411i	$0.780005\pi$
$882$	0	0
$883$	43495.8	1.65770	0.828851	−	0.559469i	$-0.188995\pi$
$883$	43495.8	1.65770	0.828851	+	0.559469i	$0.188995\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	15548.5	0.588575	0.294287	−	0.955717i	$-0.404918\pi$
$887$	15548.5	0.588575	0.294287	+	0.955717i	$0.404918\pi$
$888$	0	0
$889$	31268.4	1.17965
$890$	0	0
$891$	0	0
$892$	0	0
$893$	10818.5	0.405404
$894$	0	0
$895$	−28930.5	−1.08049
$896$	0	0
$897$	0	0
$898$	0	0
$899$	20593.2	0.763985
$900$	0	0
$901$	3030.74	0.112063
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6733.47	0.247324
$906$	0	0
$907$	13144.9	0.481222	0.240611	−	0.970622i	$-0.422652\pi$
$907$	13144.9	0.481222	0.240611	+	0.970622i	$0.422652\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	28750.1	1.04559	0.522795	−	0.852459i	$-0.324889\pi$
$911$	28750.1	1.04559	0.522795	+	0.852459i	$0.324889\pi$
$912$	0	0
$913$	−18638.5	−0.675624
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5701.75	0.205331
$918$	0	0
$919$	21287.8	0.764112	0.382056	−	0.924139i	$-0.375216\pi$
$919$	21287.8	0.764112	0.382056	+	0.924139i	$0.375216\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−11590.5	−0.413333
$924$	0	0
$925$	3301.31	0.117347
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−16505.5	−0.582914	−0.291457	−	0.956584i	$-0.594140\pi$
$929$	−16505.5	−0.582914	−0.291457	+	0.956584i	$0.594140\pi$
$930$	0	0
$931$	6379.76	0.224585
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−16730.3	−0.585176
$936$	0	0
$937$	−30510.3	−1.06374	−0.531872	−	0.846825i	$-0.678511\pi$
$937$	−30510.3	−1.06374	−0.531872	+	0.846825i	$0.678511\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−46736.0	−1.61908	−0.809538	−	0.587067i	$-0.800283\pi$
$941$	−46736.0	−1.61908	−0.809538	+	0.587067i	$0.800283\pi$
$942$	0	0
$943$	−306.993	−0.0106013
$944$	0	0
$945$	0	0
$946$	0	0
$947$	29288.1	1.00500	0.502500	−	0.864577i	$-0.332414\pi$
$947$	29288.1	1.00500	0.502500	+	0.864577i	$0.332414\pi$
$948$	0	0
$949$	−28772.5	−0.984188
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−45705.6	−1.55357	−0.776783	−	0.629768i	$-0.783150\pi$
$953$	−45705.6	−1.55357	−0.776783	+	0.629768i	$0.783150\pi$
$954$	0	0
$955$	−14515.8	−0.491855
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−4100.23	−0.138064
$960$	0	0
$961$	−12153.5	−0.407958
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5799.60	0.193467
$966$	0	0
$967$	31074.6	1.03339	0.516696	−	0.856169i	$-0.327162\pi$
$967$	31074.6	1.03339	0.516696	+	0.856169i	$0.327162\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−6532.66	−0.215904	−0.107952	−	0.994156i	$-0.534429\pi$
$971$	−6532.66	−0.215904	−0.107952	+	0.994156i	$0.534429\pi$
$972$	0	0
$973$	−35119.5	−1.15712
$974$	0	0
$975$	0	0
$976$	0	0
$977$	32822.6	1.07481	0.537404	−	0.843325i	$-0.319405\pi$
$977$	32822.6	1.07481	0.537404	+	0.843325i	$0.319405\pi$
$978$	0	0
$979$	35694.4	1.16527
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−53969.2	−1.75112	−0.875559	−	0.483110i	$-0.839507\pi$
$983$	−53969.2	−1.75112	−0.875559	+	0.483110i	$0.839507\pi$
$984$	0	0
$985$	−27860.9	−0.901240
$986$	0	0
$987$	0	0
$988$	0	0
$989$	17671.4	0.568168
$990$	0	0
$991$	−1680.09	−0.0538546	−0.0269273	−	0.999637i	$-0.508572\pi$
$991$	−1680.09	−0.0538546	−0.0269273	+	0.999637i	$0.508572\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	21390.8	0.681543
$996$	0	0
$997$	21961.5	0.697619	0.348810	−	0.937194i	$-0.386586\pi$
$997$	21961.5	0.697619	0.348810	+	0.937194i	$0.386586\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.4.a.by.1.3		3
3.2	odd	2		1728.4.a.ca.1.1		3
4.3	odd	2		1728.4.a.bz.1.3		3
8.3	odd	2		864.4.a.p.1.1	yes	3
8.5	even	2		864.4.a.o.1.1	yes	3
12.11	even	2		1728.4.a.cb.1.1		3
24.5	odd	2		864.4.a.m.1.3	✓	3
24.11	even	2		864.4.a.n.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.4.a.m.1.3	✓	3	24.5	odd	2
864.4.a.n.1.3	yes	3	24.11	even	2
864.4.a.o.1.1	yes	3	8.5	even	2
864.4.a.p.1.1	yes	3	8.3	odd	2
1728.4.a.by.1.3		3	1.1	even	1	trivial
1728.4.a.bz.1.3		3	4.3	odd	2
1728.4.a.ca.1.1		3	3.2	odd	2
1728.4.a.cb.1.1		3	12.11	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1728.a (trivial)

\(f(q)\)	\(=\)	\(q+9.83700 q^{5} +13.5419 q^{7} +40.0529 q^{11} -46.4627 q^{13} -42.4627 q^{17} -39.9691 q^{19} -38.8236 q^{23} -28.2334 q^{25} -155.062 q^{29} -132.806 q^{31} +133.212 q^{35} -116.929 q^{37} +7.90737 q^{41} -455.172 q^{43} -270.670 q^{47} -159.617 q^{49} -71.3743 q^{53} +394.000 q^{55} -779.199 q^{59} +790.952 q^{61} -457.053 q^{65} -835.630 q^{67} +249.458 q^{71} +619.260 q^{73} +542.392 q^{77} +262.480 q^{79} -465.347 q^{83} -417.705 q^{85} +891.182 q^{89} -629.192 q^{91} -393.176 q^{95} -248.259 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} - 3 q^{7} - 21 q^{11} + 24 q^{13} + 36 q^{17} - 66 q^{19} - 42 q^{23} + 12 q^{25} + 90 q^{29} + 351 q^{31} - 165 q^{35} + 6 q^{37} - 138 q^{41} - 498 q^{43} - 114 q^{47} + 78 q^{49} - 345 q^{53}+ \cdots + 549 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 - 3 * q^7 - 21 * q^11 + 24 * q^13 + 36 * q^17 - 66 * q^19 - 42 * q^23 + 12 * q^25 + 90 * q^29 + 351 * q^31 - 165 * q^35 + 6 * q^37 - 138 * q^41 - 498 * q^43 - 114 * q^47 + 78 * q^49 - 345 * q^53 + 1005 * q^55 - 228 * q^59 - 168 * q^61 + 96 * q^65 - 1506 * q^67 - 1020 * q^71 - 453 * q^73 + 621 * q^77 + 2448 * q^79 - 957 * q^83 + 84 * q^85 + 1038 * q^89 - 3168 * q^91 - 1254 * q^95 + 549 * q^97

Introduction

L-functions

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