LMFDB - Embedded newform 1728.3.h.i.161.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1728,3,Mod(161,1728)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1728.161"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1728, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,0,0,-12,0,0,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0, 0,0,0,0,0,0,-252,0,0,0,0,0,0,0,0,0,0,0,0,0,168] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(49)] == traces)

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

Self dual:	no
Analytic conductor:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 13x^{10} + 129x^{8} - 512x^{6} + 1548x^{4} - 160x^{2} + 16 $ x^12 - 13x^10 + 129x^8 - 512x^6 + 1548x^4 - 160*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{26}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.9
Root			$1.88569 + 1.08870i$ of defining polynomial
Character	$\chi$	$=$	1728.161
Dual form			1728.3.h.i.161.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.32298 q^{5} +4.21979 q^{7} -16.0645 q^{11} -17.6528i q^{13} -4.44668i q^{17} +26.5756i q^{19} -5.59004i q^{23} -13.9578 q^{25} -50.0589 q^{29} -11.4803 q^{31} +14.0223 q^{35} +24.5810i q^{37} +18.5756i q^{41} -37.2356i q^{43} +73.5339i q^{47} -31.1934 q^{49} -2.32602 q^{53} -53.3819 q^{55} -26.8934 q^{59} +12.3091i q^{61} -58.6600i q^{65} +45.2356i q^{67} +83.3476i q^{71} -3.04219 q^{73} -67.7886 q^{77} +17.2115 q^{79} -92.8734 q^{83} -14.7762i q^{85} -160.431i q^{89} -74.4912i q^{91} +88.3101i q^{95} +122.538 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{11} + 72 q^{25} - 252 q^{35} + 168 q^{49} - 264 q^{59} - 276 q^{73} - 396 q^{83} - 396 q^{97}+O(q^{100}) $ 12 * q - 12 * q^11 + 72 * q^25 - 252 * q^35 + 168 * q^49 - 264 * q^59 - 276 * q^73 - 396 * q^83 - 396 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	3.32298	0.664596	0.332298	−	0.943174i	$-0.392176\pi$
$5$	3.32298	0.664596	0.332298	+	0.943174i	$0.392176\pi$
$6$	0	0
$7$	4.21979	0.602827	0.301414	−	0.953494i	$-0.402542\pi$
$7$	4.21979	0.602827	0.301414	+	0.953494i	$0.402542\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−16.0645	−1.46041	−0.730203	−	0.683231i	$-0.760574\pi$
$11$	−16.0645	−1.46041	−0.730203	+	0.683231i	$0.760574\pi$
$12$	0	0
$13$	− 17.6528i	− 1.35791i	−0.734180	−	0.678955i	$-0.762433\pi$
$13$	− 17.6528i	− 1.35791i	0.734180	−	0.678955i	$-0.237567\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 4.44668i	− 0.261569i	−0.991411	−	0.130785i	$-0.958250\pi$
$17$	− 4.44668i	− 0.261569i	0.991411	−	0.130785i	$-0.0417496\pi$
$18$	0	0
$19$	26.5756i	1.39872i	0.714772	+	0.699358i	$0.246531\pi$
$19$	26.5756i	1.39872i	−0.714772	+	0.699358i	$0.753469\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 5.59004i	− 0.243045i	−0.992589	−	0.121523i	$-0.961222\pi$
$23$	− 5.59004i	− 0.243045i	0.992589	−	0.121523i	$-0.0387777\pi$
$24$	0	0
$25$	−13.9578	−0.558313
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−50.0589	−1.72617	−0.863084	−	0.505060i	$-0.831470\pi$
$29$	−50.0589	−1.72617	−0.863084	+	0.505060i	$0.831470\pi$
$30$	0	0
$31$	−11.4803	−0.370333	−0.185166	−	0.982707i	$-0.559282\pi$
$31$	−11.4803	−0.370333	−0.185166	+	0.982707i	$0.559282\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	14.0223	0.400636
$36$	0	0
$37$	24.5810i	0.664352i	0.943217	+	0.332176i	$0.107783\pi$
$37$	24.5810i	0.664352i	−0.943217	+	0.332176i	$0.892217\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	18.5756i	0.453063i	0.974004	+	0.226532i	$0.0727387\pi$
$41$	18.5756i	0.453063i	−0.974004	+	0.226532i	$0.927261\pi$
$42$	0	0
$43$	− 37.2356i	− 0.865943i	−0.901408	−	0.432972i	$-0.857465\pi$
$43$	− 37.2356i	− 0.865943i	0.901408	−	0.432972i	$-0.142535\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	73.5339i	1.56455i	0.622932	+	0.782276i	$0.285941\pi$
$47$	73.5339i	1.56455i	−0.622932	+	0.782276i	$0.714059\pi$
$48$	0	0
$49$	−31.1934	−0.636599
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.32602	−0.0438872	−0.0219436	−	0.999759i	$-0.506985\pi$
$53$	−2.32602	−0.0438872	−0.0219436	+	0.999759i	$0.506985\pi$
$54$	0	0
$55$	−53.3819	−0.970579
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−26.8934	−0.455820	−0.227910	−	0.973682i	$-0.573189\pi$
$59$	−26.8934	−0.455820	−0.227910	+	0.973682i	$0.573189\pi$
$60$	0	0
$61$	12.3091i	0.201788i	0.994897	+	0.100894i	$0.0321703\pi$
$61$	12.3091i	0.201788i	−0.994897	+	0.100894i	$0.967830\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 58.6600i	− 0.902461i
$66$	0	0
$67$	45.2356i	0.675158i	0.941297	+	0.337579i	$0.109608\pi$
$67$	45.2356i	0.675158i	−0.941297	+	0.337579i	$0.890392\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	83.3476i	1.17391i	0.809620	+	0.586955i	$0.199673\pi$
$71$	83.3476i	1.17391i	−0.809620	+	0.586955i	$0.800327\pi$
$72$	0	0
$73$	−3.04219	−0.0416738	−0.0208369	−	0.999783i	$-0.506633\pi$
$73$	−3.04219	−0.0416738	−0.0208369	+	0.999783i	$0.506633\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−67.7886	−0.880372
$78$	0	0
$79$	17.2115	0.217867	0.108933	−	0.994049i	$-0.465256\pi$
$79$	17.2115	0.217867	0.108933	+	0.994049i	$0.465256\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−92.8734	−1.11896	−0.559479	−	0.828845i	$-0.688999\pi$
$83$	−92.8734	−1.11896	−0.559479	+	0.828845i	$0.688999\pi$
$84$	0	0
$85$	− 14.7762i	− 0.173838i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 160.431i	− 1.80260i	−0.433197	−	0.901299i	$-0.642614\pi$
$89$	− 160.431i	− 1.80260i	0.433197	−	0.901299i	$-0.357386\pi$
$90$	0	0
$91$	− 74.4912i	− 0.818585i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	88.3101i	0.929580i
$96$	0	0
$97$	122.538	1.26328	0.631639	−	0.775263i	$-0.282383\pi$
$97$	122.538	1.26328	0.631639	+	0.775263i	$0.282383\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−104.556	−1.03520	−0.517602	−	0.855622i	$-0.673175\pi$
$101$	−104.556	−1.03520	−0.517602	+	0.855622i	$0.673175\pi$
$102$	0	0
$103$	172.509	1.67484	0.837420	−	0.546560i	$-0.184063\pi$
$103$	172.509	1.67484	0.837420	+	0.546560i	$0.184063\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.82654	−0.0264163	−0.0132081	−	0.999913i	$-0.504204\pi$
$107$	−2.82654	−0.0264163	−0.0132081	+	0.999913i	$0.504204\pi$
$108$	0	0
$109$	− 59.6686i	− 0.547419i	−0.961812	−	0.273709i	$-0.911749\pi$
$109$	− 59.6686i	− 0.547419i	0.961812	−	0.273709i	$-0.0882506\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 153.682i	− 1.36002i	−0.733203	−	0.680010i	$-0.761975\pi$
$113$	− 153.682i	− 1.36002i	0.733203	−	0.680010i	$-0.238025\pi$
$114$	0	0
$115$	− 18.5756i	− 0.161527i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 18.7640i	− 0.157681i
$120$	0	0
$121$	137.067	1.13278
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−129.456	−1.03565
$126$	0	0
$127$	52.3530	0.412228	0.206114	−	0.978528i	$-0.433918\pi$
$127$	52.3530	0.412228	0.206114	+	0.978528i	$0.433918\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−118.364	−0.903546	−0.451773	−	0.892133i	$-0.649208\pi$
$131$	−118.364	−0.903546	−0.451773	+	0.892133i	$0.649208\pi$
$132$	0	0
$133$	112.143i	0.843184i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	188.689i	1.37729i	0.725097	+	0.688646i	$0.241795\pi$
$137$	188.689i	1.37729i	−0.725097	+	0.688646i	$0.758205\pi$
$138$	0	0
$139$	104.000i	0.748201i	0.927388	+	0.374101i	$0.122049\pi$
$139$	104.000i	0.748201i	−0.927388	+	0.374101i	$0.877951\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	283.583i	1.98310i
$144$	0	0
$145$	−166.345	−1.14720
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−289.387	−1.94220	−0.971098	−	0.238679i	$-0.923286\pi$
$149$	−289.387	−1.94220	−0.971098	+	0.238679i	$0.923286\pi$
$150$	0	0
$151$	−151.374	−1.00247	−0.501237	−	0.865310i	$-0.667122\pi$
$151$	−151.374	−1.00247	−0.501237	+	0.865310i	$0.667122\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−38.1488	−0.246121
$156$	0	0
$157$	− 176.964i	− 1.12716i	−0.826061	−	0.563581i	$-0.809423\pi$
$157$	− 176.964i	− 1.12716i	0.826061	−	0.563581i	$-0.190577\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 23.5888i	− 0.146514i
$162$	0	0
$163$	− 196.962i	− 1.20836i	−0.796849	−	0.604179i	$-0.793501\pi$
$163$	− 196.962i	− 1.20836i	0.796849	−	0.604179i	$-0.206499\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	319.981i	1.91605i	0.286682	+	0.958026i	$0.407448\pi$
$167$	319.981i	1.91605i	−0.286682	+	0.958026i	$0.592552\pi$
$168$	0	0
$169$	−142.622	−0.843919
$170$	0	0
$171$	0	0
$172$	0	0
$173$	14.3852	0.0831512	0.0415756	−	0.999135i	$-0.486762\pi$
$173$	14.3852	0.0831512	0.0415756	+	0.999135i	$0.486762\pi$
$174$	0	0
$175$	−58.8990	−0.336566
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−131.342	−0.733755	−0.366878	−	0.930269i	$-0.619573\pi$
$179$	−131.342	−0.733755	−0.366878	+	0.930269i	$0.619573\pi$
$180$	0	0
$181$	339.492i	1.87565i	0.347110	+	0.937824i	$0.387163\pi$
$181$	339.492i	1.87565i	−0.347110	+	0.937824i	$0.612837\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	81.6822i	0.441526i
$186$	0	0
$187$	71.4335i	0.381997i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	34.4036i	0.180124i	0.995936	+	0.0900618i	$0.0287065\pi$
$191$	34.4036i	0.180124i	−0.995936	+	0.0900618i	$0.971294\pi$
$192$	0	0
$193$	179.513	0.930121	0.465060	−	0.885279i	$-0.346033\pi$
$193$	179.513	0.930121	0.465060	+	0.885279i	$0.346033\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−261.378	−1.32679	−0.663396	−	0.748268i	$-0.730886\pi$
$197$	−261.378	−1.32679	−0.663396	+	0.748268i	$0.730886\pi$
$198$	0	0
$199$	−208.861	−1.04955	−0.524776	−	0.851240i	$-0.675851\pi$
$199$	−208.861	−1.04955	−0.524776	+	0.851240i	$0.675851\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−211.238	−1.04058
$204$	0	0
$205$	61.7263i	0.301104i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 426.923i	− 2.04269i
$210$	0	0
$211$	− 71.5178i	− 0.338947i	−0.985535	−	0.169474i	$-0.945793\pi$
$211$	− 71.5178i	− 0.338947i	0.985535	−	0.169474i	$-0.0542067\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 123.733i	− 0.575502i
$216$	0	0
$217$	−48.4445	−0.223247
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−78.4964	−0.355187
$222$	0	0
$223$	−33.5721	−0.150548	−0.0752739	−	0.997163i	$-0.523983\pi$
$223$	−33.5721	−0.150548	−0.0752739	+	0.997163i	$0.523983\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	56.3024	0.248028	0.124014	−	0.992280i	$-0.460423\pi$
$227$	56.3024	0.248028	0.124014	+	0.992280i	$0.460423\pi$
$228$	0	0
$229$	− 382.162i	− 1.66883i	−0.551136	−	0.834416i	$-0.685805\pi$
$229$	− 382.162i	− 1.66883i	0.551136	−	0.834416i	$-0.314195\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 43.9645i	− 0.188689i	−0.995540	−	0.0943445i	$-0.969924\pi$
$233$	− 43.9645i	− 0.188689i	0.995540	−	0.0943445i	$-0.0300755\pi$
$234$	0	0
$235$	244.352i	1.03979i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 172.049i	− 0.719872i	−0.932977	−	0.359936i	$-0.882798\pi$
$239$	− 172.049i	− 0.719872i	0.932977	−	0.359936i	$-0.117202\pi$
$240$	0	0
$241$	−383.547	−1.59148	−0.795741	−	0.605638i	$-0.792918\pi$
$241$	−383.547	−1.59148	−0.795741	+	0.605638i	$0.792918\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−103.655	−0.423081
$246$	0	0
$247$	469.134	1.89933
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−184.391	−0.734627	−0.367314	−	0.930097i	$-0.619722\pi$
$251$	−184.391	−0.734627	−0.367314	+	0.930097i	$0.619722\pi$
$252$	0	0
$253$	89.8010i	0.354945i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 302.918i	− 1.17867i	−0.807889	−	0.589334i	$-0.799390\pi$
$257$	− 302.918i	− 1.17867i	0.807889	−	0.589334i	$-0.200610\pi$
$258$	0	0
$259$	103.727i	0.400490i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	397.614i	1.51184i	0.654664	+	0.755920i	$0.272810\pi$
$263$	397.614i	1.51184i	−0.654664	+	0.755920i	$0.727190\pi$
$264$	0	0
$265$	−7.72932	−0.0291672
$266$	0	0
$267$	0	0
$268$	0	0
$269$	73.1579	0.271962	0.135981	−	0.990711i	$-0.456581\pi$
$269$	73.1579	0.271962	0.135981	+	0.990711i	$0.456581\pi$
$270$	0	0
$271$	−325.980	−1.20288	−0.601439	−	0.798919i	$-0.705406\pi$
$271$	−325.980	−1.20288	−0.601439	+	0.798919i	$0.705406\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	224.225	0.815363
$276$	0	0
$277$	− 341.933i	− 1.23441i	−0.786800	−	0.617207i	$-0.788264\pi$
$277$	− 341.933i	− 1.23441i	0.786800	−	0.617207i	$-0.211736\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	137.251i	0.488438i	0.969720	+	0.244219i	$0.0785315\pi$
$281$	137.251i	0.488438i	−0.969720	+	0.244219i	$0.921468\pi$
$282$	0	0
$283$	351.727i	1.24285i	0.783473	+	0.621425i	$0.213446\pi$
$283$	351.727i	1.24285i	−0.783473	+	0.621425i	$0.786554\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	78.3851i	0.273119i
$288$	0	0
$289$	269.227	0.931582
$290$	0	0
$291$	0	0
$292$	0	0
$293$	417.493	1.42489	0.712445	−	0.701728i	$-0.247588\pi$
$293$	417.493	1.42489	0.712445	+	0.701728i	$0.247588\pi$
$294$	0	0
$295$	−89.3660	−0.302936
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−98.6801	−0.330034
$300$	0	0
$301$	− 157.126i	− 0.522014i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	40.9028i	0.134107i
$306$	0	0
$307$	− 93.7177i	− 0.305269i	−0.988283	−	0.152635i	$-0.951224\pi$
$307$	− 93.7177i	− 0.305269i	0.988283	−	0.152635i	$-0.0487758\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	183.230i	0.589162i	0.955626	+	0.294581i	$0.0951802\pi$
$311$	183.230i	0.589162i	−0.955626	+	0.294581i	$0.904820\pi$
$312$	0	0
$313$	126.882	0.405375	0.202688	−	0.979243i	$-0.435032\pi$
$313$	126.882	0.405375	0.202688	+	0.979243i	$0.435032\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−371.872	−1.17310	−0.586548	−	0.809914i	$-0.699514\pi$
$317$	−371.872	−1.17310	−0.586548	+	0.809914i	$0.699514\pi$
$318$	0	0
$319$	804.169	2.52090
$320$	0	0
$321$	0	0
$322$	0	0
$323$	118.173	0.365861
$324$	0	0
$325$	246.395i	0.758138i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	310.298i	0.943154i
$330$	0	0
$331$	− 548.352i	− 1.65665i	−0.560247	−	0.828326i	$-0.689294\pi$
$331$	− 548.352i	− 1.65665i	0.560247	−	0.828326i	$-0.310706\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	150.317i	0.448707i
$336$	0	0
$337$	329.663	0.978227	0.489114	−	0.872220i	$-0.337320\pi$
$337$	329.663	0.978227	0.489114	+	0.872220i	$0.337320\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	184.425	0.540836
$342$	0	0
$343$	−338.399	−0.986587
$344$	0	0
$345$	0	0
$346$	0	0
$347$	567.567	1.63564	0.817820	−	0.575474i	$-0.195182\pi$
$347$	567.567	1.63564	0.817820	+	0.575474i	$0.195182\pi$
$348$	0	0
$349$	425.470i	1.21911i	0.792743	+	0.609556i	$0.208652\pi$
$349$	425.470i	1.21911i	−0.792743	+	0.609556i	$0.791348\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 299.260i	− 0.847762i	−0.905718	−	0.423881i	$-0.860667\pi$
$353$	− 299.260i	− 0.847762i	0.905718	−	0.423881i	$-0.139333\pi$
$354$	0	0
$355$	276.962i	0.780176i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 279.248i	− 0.777850i	−0.921269	−	0.388925i	$-0.872847\pi$
$359$	− 279.248i	− 0.777850i	0.921269	−	0.388925i	$-0.127153\pi$
$360$	0	0
$361$	−345.262	−0.956405
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−10.1091	−0.0276962
$366$	0	0
$367$	−402.044	−1.09549	−0.547744	−	0.836646i	$-0.684513\pi$
$367$	−402.044	−1.09549	−0.547744	+	0.836646i	$0.684513\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−9.81532	−0.0264564
$372$	0	0
$373$	− 342.832i	− 0.919120i	−0.888147	−	0.459560i	$-0.848007\pi$
$373$	− 342.832i	− 0.919120i	0.888147	−	0.459560i	$-0.151993\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	883.681i	2.34398i
$378$	0	0
$379$	− 640.832i	− 1.69085i	−0.534095	−	0.845425i	$-0.679347\pi$
$379$	− 640.832i	− 1.69085i	0.534095	−	0.845425i	$-0.320653\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 563.570i	− 1.47146i	−0.677274	−	0.735731i	$-0.736839\pi$
$383$	− 563.570i	− 1.47146i	0.677274	−	0.735731i	$-0.263161\pi$
$384$	0	0
$385$	−225.260	−0.585091
$386$	0	0
$387$	0	0
$388$	0	0
$389$	342.245	0.879806	0.439903	−	0.898045i	$-0.355013\pi$
$389$	342.245	0.879806	0.439903	+	0.898045i	$0.355013\pi$
$390$	0	0
$391$	−24.8571	−0.0635732
$392$	0	0
$393$	0	0
$394$	0	0
$395$	57.1934	0.144793
$396$	0	0
$397$	403.500i	1.01637i	0.861247	+	0.508187i	$0.169684\pi$
$397$	403.500i	1.01637i	−0.861247	+	0.508187i	$0.830316\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 169.827i	− 0.423508i	−0.977323	−	0.211754i	$-0.932082\pi$
$401$	− 169.827i	− 0.423508i	0.977323	−	0.211754i	$-0.0679176\pi$
$402$	0	0
$403$	202.660i	0.502878i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 394.881i	− 0.970223i
$408$	0	0
$409$	356.191	0.870884	0.435442	−	0.900217i	$-0.356592\pi$
$409$	356.191	0.870884	0.435442	+	0.900217i	$0.356592\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−113.484	−0.274780
$414$	0	0
$415$	−308.616	−0.743654
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−417.045	−0.995334	−0.497667	−	0.867368i	$-0.665810\pi$
$419$	−417.045	−0.995334	−0.497667	+	0.867368i	$0.665810\pi$
$420$	0	0
$421$	660.848i	1.56971i	0.619679	+	0.784855i	$0.287263\pi$
$421$	660.848i	1.56971i	−0.619679	+	0.784855i	$0.712737\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	62.0659i	0.146037i
$426$	0	0
$427$	51.9417i	0.121643i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 639.445i	− 1.48363i	−0.670604	−	0.741816i	$-0.733965\pi$
$431$	− 639.445i	− 1.48363i	0.670604	−	0.741816i	$-0.266035\pi$
$432$	0	0
$433$	−464.454	−1.07264	−0.536321	−	0.844014i	$-0.680186\pi$
$433$	−464.454	−1.07264	−0.536321	+	0.844014i	$0.680186\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	148.559	0.339951
$438$	0	0
$439$	79.8478	0.181886	0.0909428	−	0.995856i	$-0.471012\pi$
$439$	79.8478	0.181886	0.0909428	+	0.995856i	$0.471012\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	389.622	0.879509	0.439754	−	0.898118i	$-0.355066\pi$
$443$	389.622	0.879509	0.439754	+	0.898118i	$0.355066\pi$
$444$	0	0
$445$	− 533.110i	− 1.19800i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 654.843i	− 1.45845i	−0.684275	−	0.729224i	$-0.739881\pi$
$449$	− 654.843i	− 1.45845i	0.684275	−	0.729224i	$-0.260119\pi$
$450$	0	0
$451$	− 298.407i	− 0.661656i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 247.533i	− 0.544028i
$456$	0	0
$457$	470.596	1.02975	0.514876	−	0.857265i	$-0.327838\pi$
$457$	470.596	1.02975	0.514876	+	0.857265i	$0.327838\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	805.808	1.74796	0.873979	−	0.485965i	$-0.161532\pi$
$461$	805.808	1.74796	0.873979	+	0.485965i	$0.161532\pi$
$462$	0	0
$463$	487.786	1.05353	0.526767	−	0.850010i	$-0.323404\pi$
$463$	487.786	1.05353	0.526767	+	0.850010i	$0.323404\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−449.958	−0.963508	−0.481754	−	0.876306i	$-0.660000\pi$
$467$	−449.958	−0.963508	−0.481754	+	0.876306i	$0.660000\pi$
$468$	0	0
$469$	190.885i	0.407003i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	598.169i	1.26463i
$474$	0	0
$475$	− 370.937i	− 0.780920i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 152.266i	− 0.317883i	−0.987288	−	0.158942i	$-0.949192\pi$
$479$	− 152.266i	− 0.317883i	0.987288	−	0.158942i	$-0.0508082\pi$
$480$	0	0
$481$	433.925	0.902130
$482$	0	0
$483$	0	0
$484$	0	0
$485$	407.191	0.839569
$486$	0	0
$487$	527.351	1.08286	0.541429	−	0.840747i	$-0.317884\pi$
$487$	527.351	1.08286	0.541429	+	0.840747i	$0.317884\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−285.609	−0.581689	−0.290844	−	0.956770i	$-0.593936\pi$
$491$	−285.609	−0.581689	−0.290844	+	0.956770i	$0.593936\pi$
$492$	0	0
$493$	222.596i	0.451512i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	351.709i	0.707665i
$498$	0	0
$499$	420.062i	0.841808i	0.907105	+	0.420904i	$0.138287\pi$
$499$	420.062i	0.841808i	−0.907105	+	0.420904i	$0.861713\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	80.8820i	0.160799i	0.996763	+	0.0803996i	$0.0256197\pi$
$503$	80.8820i	0.160799i	−0.996763	+	0.0803996i	$0.974380\pi$
$504$	0	0
$505$	−347.436	−0.687992
$506$	0	0
$507$	0	0
$508$	0	0
$509$	549.502	1.07957	0.539786	−	0.841802i	$-0.318505\pi$
$509$	549.502	1.07957	0.539786	+	0.841802i	$0.318505\pi$
$510$	0	0
$511$	−12.8374	−0.0251221
$512$	0	0
$513$	0	0
$514$	0	0
$515$	573.242	1.11309
$516$	0	0
$517$	− 1181.28i	− 2.28488i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	990.371i	1.90090i	0.310870	+	0.950452i	$0.399380\pi$
$521$	990.371i	1.90090i	−0.310870	+	0.950452i	$0.600620\pi$
$522$	0	0
$523$	103.016i	0.196971i	0.995139	+	0.0984853i	$0.0313997\pi$
$523$	103.016i	0.196971i	−0.995139	+	0.0984853i	$0.968600\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	51.0492i	0.0968676i
$528$	0	0
$529$	497.751	0.940929
$530$	0	0
$531$	0	0
$532$	0	0
$533$	327.912	0.615219
$534$	0	0
$535$	−9.39254	−0.0175561
$536$	0	0
$537$	0	0
$538$	0	0
$539$	501.105	0.929693
$540$	0	0
$541$	306.170i	0.565934i	0.959130	+	0.282967i	$0.0913187\pi$
$541$	306.170i	0.565934i	−0.959130	+	0.282967i	$0.908681\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 198.278i	− 0.363812i
$546$	0	0
$547$	421.171i	0.769966i	0.922924	+	0.384983i	$0.125793\pi$
$547$	421.171i	0.769966i	−0.922924	+	0.384983i	$0.874207\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 1330.34i	− 2.41442i
$552$	0	0
$553$	72.6288	0.131336
$554$	0	0
$555$	0	0
$556$	0	0
$557$	387.083	0.694942	0.347471	−	0.937691i	$-0.387040\pi$
$557$	387.083	0.694942	0.347471	+	0.937691i	$0.387040\pi$
$558$	0	0
$559$	−657.313	−1.17587
$560$	0	0
$561$	0	0
$562$	0	0
$563$	915.887	1.62680	0.813399	−	0.581706i	$-0.197615\pi$
$563$	915.887	1.62680	0.813399	+	0.581706i	$0.197615\pi$
$564$	0	0
$565$	− 510.683i	− 0.903863i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1080.19i	1.89840i	0.314679	+	0.949198i	$0.398103\pi$
$569$	1080.19i	1.89840i	−0.314679	+	0.949198i	$0.601897\pi$
$570$	0	0
$571$	1025.40i	1.79579i	0.440206	+	0.897897i	$0.354906\pi$
$571$	1025.40i	1.79579i	−0.440206	+	0.897897i	$0.645094\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	78.0248i	0.135695i
$576$	0	0
$577$	−686.792	−1.19028	−0.595140	−	0.803622i	$-0.702903\pi$
$577$	−686.792	−1.19028	−0.595140	+	0.803622i	$0.702903\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−391.906	−0.674538
$582$	0	0
$583$	37.3663	0.0640931
$584$	0	0
$585$	0	0
$586$	0	0
$587$	68.2446	0.116260	0.0581300	−	0.998309i	$-0.481486\pi$
$587$	68.2446	0.116260	0.0581300	+	0.998309i	$0.481486\pi$
$588$	0	0
$589$	− 305.096i	− 0.517990i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 1113.53i	− 1.87779i	−0.344203	−	0.938895i	$-0.611851\pi$
$593$	− 1113.53i	− 1.87779i	0.344203	−	0.938895i	$-0.388149\pi$
$594$	0	0
$595$	− 62.3525i	− 0.104794i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	127.721i	0.213223i	0.994301	+	0.106612i	$0.0340002\pi$
$599$	127.721i	0.213223i	−0.994301	+	0.106612i	$0.966000\pi$
$600$	0	0
$601$	−776.688	−1.29233	−0.646163	−	0.763200i	$-0.723627\pi$
$601$	−776.688	−1.29233	−0.646163	+	0.763200i	$0.723627\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	455.470	0.752843
$606$	0	0
$607$	863.872	1.42318	0.711592	−	0.702593i	$-0.247975\pi$
$607$	863.872	1.42318	0.711592	+	0.702593i	$0.247975\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1298.08	2.12452
$612$	0	0
$613$	− 178.086i	− 0.290516i	−0.989394	−	0.145258i	$-0.953599\pi$
$613$	− 178.086i	− 0.290516i	0.989394	−	0.145258i	$-0.0464012\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	79.7883i	0.129317i	0.997907	+	0.0646583i	$0.0205957\pi$
$617$	79.7883i	0.129317i	−0.997907	+	0.0646583i	$0.979404\pi$
$618$	0	0
$619$	− 578.660i	− 0.934830i	−0.884038	−	0.467415i	$-0.845185\pi$
$619$	− 578.660i	− 0.934830i	0.884038	−	0.467415i	$-0.154815\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 676.986i	− 1.08666i
$624$	0	0
$625$	−81.2341	−0.129975
$626$	0	0
$627$	0	0
$628$	0	0
$629$	109.304	0.173774
$630$	0	0
$631$	−999.072	−1.58332	−0.791658	−	0.610965i	$-0.790782\pi$
$631$	−999.072	−1.58332	−0.791658	+	0.610965i	$0.790782\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	173.968	0.273965
$636$	0	0
$637$	550.651i	0.864445i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	66.9225i	0.104403i	0.998637	+	0.0522016i	$0.0166239\pi$
$641$	66.9225i	0.104403i	−0.998637	+	0.0522016i	$0.983376\pi$
$642$	0	0
$643$	− 521.676i	− 0.811315i	−0.914025	−	0.405658i	$-0.867043\pi$
$643$	− 521.676i	− 0.811315i	0.914025	−	0.405658i	$-0.132957\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 485.924i	− 0.751041i	−0.926814	−	0.375521i	$-0.877464\pi$
$647$	− 485.924i	− 0.751041i	0.926814	−	0.375521i	$-0.122536\pi$
$648$	0	0
$649$	432.027	0.665681
$650$	0	0
$651$	0	0
$652$	0	0
$653$	370.358	0.567165	0.283582	−	0.958948i	$-0.408477\pi$
$653$	370.358	0.567165	0.283582	+	0.958948i	$0.408477\pi$
$654$	0	0
$655$	−393.323	−0.600493
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−291.218	−0.441910	−0.220955	−	0.975284i	$-0.570917\pi$
$659$	−291.218	−0.441910	−0.220955	+	0.975284i	$0.570917\pi$
$660$	0	0
$661$	441.331i	0.667672i	0.942631	+	0.333836i	$0.108343\pi$
$661$	441.331i	0.667672i	−0.942631	+	0.333836i	$0.891657\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	372.650i	0.560376i
$666$	0	0
$667$	279.831i	0.419537i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 197.739i	− 0.294692i
$672$	0	0
$673$	−432.036	−0.641955	−0.320977	−	0.947087i	$-0.604011\pi$
$673$	−432.036	−0.641955	−0.320977	+	0.947087i	$0.604011\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	106.934	0.157953	0.0789764	−	0.996876i	$-0.474835\pi$
$677$	106.934	0.157953	0.0789764	+	0.996876i	$0.474835\pi$
$678$	0	0
$679$	517.084	0.761538
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−102.329	−0.149823	−0.0749113	−	0.997190i	$-0.523867\pi$
$683$	−102.329	−0.149823	−0.0749113	+	0.997190i	$0.523867\pi$
$684$	0	0
$685$	627.010i	0.915343i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	41.0608i	0.0595948i
$690$	0	0
$691$	− 387.134i	− 0.560252i	−0.959963	−	0.280126i	$-0.909624\pi$
$691$	− 387.134i	− 0.560252i	0.959963	−	0.280126i	$-0.0903763\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	345.590i	0.497251i
$696$	0	0
$697$	82.5997	0.118507
$698$	0	0
$699$	0	0
$700$	0	0
$701$	942.899	1.34508	0.672538	−	0.740062i	$-0.265204\pi$
$701$	942.899	1.34508	0.672538	+	0.740062i	$0.265204\pi$
$702$	0	0
$703$	−653.255	−0.929240
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−441.203	−0.624049
$708$	0	0
$709$	486.394i	0.686028i	0.939330	+	0.343014i	$0.111448\pi$
$709$	486.394i	0.686028i	−0.939330	+	0.343014i	$0.888552\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	64.1754i	0.0900076i
$714$	0	0
$715$	942.341i	1.31796i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	370.683i	0.515553i	0.966205	+	0.257777i	$0.0829899\pi$
$719$	370.683i	0.515553i	−0.966205	+	0.257777i	$0.917010\pi$
$720$	0	0
$721$	727.950	1.00964
$722$	0	0
$723$	0	0
$724$	0	0
$725$	698.712	0.963741
$726$	0	0
$727$	−799.119	−1.09920	−0.549600	−	0.835428i	$-0.685220\pi$
$727$	−799.119	−1.09920	−0.549600	+	0.835428i	$0.685220\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−165.575	−0.226504
$732$	0	0
$733$	− 601.978i	− 0.821252i	−0.911804	−	0.410626i	$-0.865310\pi$
$733$	− 601.978i	− 0.821252i	0.911804	−	0.410626i	$-0.134690\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 726.685i	− 0.986004i
$738$	0	0
$739$	− 320.189i	− 0.433273i	−0.976252	−	0.216637i	$-0.930491\pi$
$739$	− 320.189i	− 0.433273i	0.976252	−	0.216637i	$-0.0695087\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 1138.71i	− 1.53259i	−0.642491	−	0.766293i	$-0.722099\pi$
$743$	− 1138.71i	− 1.53259i	0.642491	−	0.766293i	$-0.277901\pi$
$744$	0	0
$745$	−961.628	−1.29078
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−11.9274	−0.0159244
$750$	0	0
$751$	258.212	0.343824	0.171912	−	0.985112i	$-0.445006\pi$
$751$	258.212	0.343824	0.171912	+	0.985112i	$0.445006\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−503.011	−0.666240
$756$	0	0
$757$	728.371i	0.962181i	0.876671	+	0.481090i	$0.159759\pi$
$757$	728.371i	0.962181i	−0.876671	+	0.481090i	$0.840241\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	613.280i	0.805887i	0.915225	+	0.402943i	$0.132013\pi$
$761$	613.280i	0.805887i	−0.915225	+	0.402943i	$0.867987\pi$
$762$	0	0
$763$	− 251.789i	− 0.329999i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	474.744i	0.618962i
$768$	0	0
$769$	−869.758	−1.13103	−0.565513	−	0.824740i	$-0.691322\pi$
$769$	−869.758	−1.13103	−0.565513	+	0.824740i	$0.691322\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	255.034	0.329928	0.164964	−	0.986300i	$-0.447249\pi$
$773$	255.034	0.329928	0.164964	+	0.986300i	$0.447249\pi$
$774$	0	0
$775$	160.240	0.206761
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−493.657	−0.633707
$780$	0	0
$781$	− 1338.93i	− 1.71438i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 588.049i	− 0.749107i
$786$	0	0
$787$	− 984.076i	− 1.25041i	−0.780459	−	0.625207i	$-0.785014\pi$
$787$	− 984.076i	− 1.25041i	0.780459	−	0.625207i	$-0.214986\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 648.507i	− 0.819857i
$792$	0	0
$793$	217.290	0.274010
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−163.608	−0.205280	−0.102640	−	0.994719i	$-0.532729\pi$
$797$	−163.608	−0.205280	−0.102640	+	0.994719i	$0.532729\pi$
$798$	0	0
$799$	326.982	0.409239
$800$	0	0
$801$	0	0
$802$	0	0
$803$	48.8711	0.0608606
$804$	0	0
$805$	− 78.3851i	− 0.0973728i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1116.97i	1.38067i	0.723488	+	0.690337i	$0.242538\pi$
$809$	1116.97i	1.38067i	−0.723488	+	0.690337i	$0.757462\pi$
$810$	0	0
$811$	596.540i	0.735562i	0.929913	+	0.367781i	$0.119882\pi$
$811$	596.540i	0.735562i	−0.929913	+	0.367781i	$0.880118\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 654.502i	− 0.803070i
$816$	0	0
$817$	989.557	1.21121
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−752.641	−0.916737	−0.458369	−	0.888762i	$-0.651566\pi$
$821$	−752.641	−0.916737	−0.458369	+	0.888762i	$0.651566\pi$
$822$	0	0
$823$	946.576	1.15015	0.575076	−	0.818100i	$-0.304972\pi$
$823$	946.576	1.15015	0.575076	+	0.818100i	$0.304972\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	613.125	0.741385	0.370693	−	0.928756i	$-0.379120\pi$
$827$	613.125	0.741385	0.370693	+	0.928756i	$0.379120\pi$
$828$	0	0
$829$	− 346.852i	− 0.418398i	−0.977873	−	0.209199i	$-0.932914\pi$
$829$	− 346.852i	− 0.418398i	0.977873	−	0.209199i	$-0.0670857\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	138.707i	0.166515i
$834$	0	0
$835$	1063.29i	1.27340i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	717.830i	0.855578i	0.903879	+	0.427789i	$0.140707\pi$
$839$	717.830i	0.855578i	−0.903879	+	0.427789i	$0.859293\pi$
$840$	0	0
$841$	1664.89	1.97966
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−473.931	−0.560865
$846$	0	0
$847$	578.393	0.682873
$848$	0	0
$849$	0	0
$850$	0	0
$851$	137.409	0.161468
$852$	0	0
$853$	510.640i	0.598640i	0.954153	+	0.299320i	$0.0967598\pi$
$853$	510.640i	0.598640i	−0.954153	+	0.299320i	$0.903240\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 210.292i	− 0.245381i	−0.992445	−	0.122691i	$-0.960848\pi$
$857$	− 210.292i	− 0.245381i	0.992445	−	0.122691i	$-0.0391523\pi$
$858$	0	0
$859$	1.80806i	0.00210484i	0.999999	+	0.00105242i	$0.000334996\pi$
$859$	1.80806i	0.00210484i	−0.999999	+	0.00105242i	$0.999665\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	598.210i	0.693174i	0.938018	+	0.346587i	$0.112660\pi$
$863$	598.210i	0.693174i	−0.938018	+	0.346587i	$0.887340\pi$
$864$	0	0
$865$	47.8016	0.0552619
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−276.493	−0.318174
$870$	0	0
$871$	798.536	0.916803
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−546.277	−0.624317
$876$	0	0
$877$	− 177.942i	− 0.202899i	−0.994841	−	0.101449i	$-0.967652\pi$
$877$	− 177.942i	− 0.202899i	0.994841	−	0.101449i	$-0.0323480\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 291.465i	− 0.330834i	−0.986224	−	0.165417i	$-0.947103\pi$
$881$	− 291.465i	− 0.330834i	0.986224	−	0.165417i	$-0.0528970\pi$
$882$	0	0
$883$	− 1218.02i	− 1.37941i	−0.724091	−	0.689704i	$-0.757741\pi$
$883$	− 1218.02i	− 1.37941i	0.724091	−	0.689704i	$-0.242259\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	481.322i	0.542640i	0.962489	+	0.271320i	$0.0874602\pi$
$887$	481.322i	0.542640i	−0.962489	+	0.271320i	$0.912540\pi$
$888$	0	0
$889$	220.919	0.248502
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1954.21	−2.18836
$894$	0	0
$895$	−436.447	−0.487651
$896$	0	0
$897$	0	0
$898$	0	0
$899$	574.691	0.639256
$900$	0	0
$901$	10.3431i	0.0114795i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1128.13i	1.24655i
$906$	0	0
$907$	− 808.482i	− 0.891381i	−0.895187	−	0.445690i	$-0.852958\pi$
$907$	− 808.482i	− 0.891381i	0.895187	−	0.445690i	$-0.147042\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 7.69529i	− 0.00844708i	−0.999991	−	0.00422354i	$-0.998656\pi$
$911$	− 7.69529i	− 0.00844708i	0.999991	−	0.00422354i	$-0.00134440\pi$
$912$	0	0
$913$	1491.96	1.63413
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−499.473	−0.544682
$918$	0	0
$919$	460.757	0.501368	0.250684	−	0.968069i	$-0.419345\pi$
$919$	460.757	0.501368	0.250684	+	0.968069i	$0.419345\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1471.32	1.59406
$924$	0	0
$925$	− 343.097i	− 0.370916i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 353.817i	− 0.380858i	−0.981701	−	0.190429i	$-0.939012\pi$
$929$	− 353.817i	− 0.380858i	0.981701	−	0.190429i	$-0.0609880\pi$
$930$	0	0
$931$	− 828.982i	− 0.890422i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	237.372i	0.253874i
$936$	0	0
$937$	−850.015	−0.907166	−0.453583	−	0.891214i	$-0.649854\pi$
$937$	−850.015	−0.907166	−0.453583	+	0.891214i	$0.649854\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−826.634	−0.878464	−0.439232	−	0.898374i	$-0.644749\pi$
$941$	−826.634	−0.878464	−0.439232	+	0.898374i	$0.644749\pi$
$942$	0	0
$943$	103.838	0.110115
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1722.44	−1.81884	−0.909418	−	0.415884i	$-0.863472\pi$
$947$	−1722.44	−1.81884	−0.909418	+	0.415884i	$0.863472\pi$
$948$	0	0
$949$	53.7032i	0.0565893i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 404.315i	− 0.424255i	−0.977242	−	0.212128i	$-0.931961\pi$
$953$	− 404.315i	− 0.424255i	0.977242	−	0.212128i	$-0.0680393\pi$
$954$	0	0
$955$	114.322i	0.119709i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	796.228i	0.830269i
$960$	0	0
$961$	−829.202	−0.862854
$962$	0	0
$963$	0	0
$964$	0	0
$965$	596.519	0.618154
$966$	0	0
$967$	−1239.73	−1.28203	−0.641016	−	0.767527i	$-0.721487\pi$
$967$	−1239.73	−1.28203	−0.641016	+	0.767527i	$0.721487\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−905.483	−0.932526	−0.466263	−	0.884646i	$-0.654400\pi$
$971$	−905.483	−0.932526	−0.466263	+	0.884646i	$0.654400\pi$
$972$	0	0
$973$	438.858i	0.451036i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1736.17i	− 1.77704i	−0.458836	−	0.888521i	$-0.651733\pi$
$977$	− 1736.17i	− 1.77704i	0.458836	−	0.888521i	$-0.348267\pi$
$978$	0	0
$979$	2577.24i	2.63252i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1014.57i	1.03212i	0.856552	+	0.516060i	$0.172602\pi$
$983$	1014.57i	1.03212i	−0.856552	+	0.516060i	$0.827398\pi$
$984$	0	0
$985$	−868.554	−0.881781
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−208.148	−0.210463
$990$	0	0
$991$	1470.74	1.48410	0.742048	−	0.670347i	$-0.233855\pi$
$991$	1470.74	1.48410	0.742048	+	0.670347i	$0.233855\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−694.041	−0.697528
$996$	0	0
$997$	974.749i	0.977683i	0.872373	+	0.488841i	$0.162580\pi$
$997$	974.749i	0.977683i	−0.872373	+	0.488841i	$0.837420\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.h.i.161.9	yes	12
3.2	odd	2		1728.3.h.j.161.3	yes	12
4.3	odd	2		1728.3.h.j.161.9	yes	12
8.3	odd	2	inner	1728.3.h.i.161.4	yes	12
8.5	even	2		1728.3.h.j.161.4	yes	12
12.11	even	2	inner	1728.3.h.i.161.3	✓	12
24.5	odd	2	inner	1728.3.h.i.161.10	yes	12
24.11	even	2		1728.3.h.j.161.10	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.3.h.i.161.3	✓	12	12.11	even	2	inner
1728.3.h.i.161.4	yes	12	8.3	odd	2	inner
1728.3.h.i.161.9	yes	12	1.1	even	1	trivial
1728.3.h.i.161.10	yes	12	24.5	odd	2	inner
1728.3.h.j.161.3	yes	12	3.2	odd	2
1728.3.h.j.161.4	yes	12	8.5	even	2
1728.3.h.j.161.9	yes	12	4.3	odd	2
1728.3.h.j.161.10	yes	12	24.11	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.32298 q^{5} +4.21979 q^{7} -16.0645 q^{11} -17.6528i q^{13} -4.44668i q^{17} +26.5756i q^{19} -5.59004i q^{23} -13.9578 q^{25} -50.0589 q^{29} -11.4803 q^{31} +14.0223 q^{35} +24.5810i q^{37} +18.5756i q^{41} -37.2356i q^{43} +73.5339i q^{47} -31.1934 q^{49} -2.32602 q^{53} -53.3819 q^{55} -26.8934 q^{59} +12.3091i q^{61} -58.6600i q^{65} +45.2356i q^{67} +83.3476i q^{71} -3.04219 q^{73} -67.7886 q^{77} +17.2115 q^{79} -92.8734 q^{83} -14.7762i q^{85} -160.431i q^{89} -74.4912i q^{91} +88.3101i q^{95} +122.538 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{11} + 72 q^{25} - 252 q^{35} + 168 q^{49} - 264 q^{59} - 276 q^{73} - 396 q^{83} - 396 q^{97}+O(q^{100}) \) 12 * q - 12 * q^11 + 72 * q^25 - 252 * q^35 + 168 * q^49 - 264 * q^59 - 276 * q^73 - 396 * q^83 - 396 * q^97

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