LMFDB - Embedded newform 1728.4.c.l.1727.17

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1728,4,Mod(1727,1728)] 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([1, 0, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1728.1727"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1728.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,0,0,0,0,0,0,0,0,0,72] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$101.955300490$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 864)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1727.17
Character	$\chi$	$=$	1728.1727
Dual form			1728.4.c.l.1727.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+8.05443i q^{5} -20.0504i q^{7} -14.7358 q^{11} -12.7687 q^{13} -58.0032i q^{17} -10.9489i q^{19} +74.3941 q^{23} +60.1262 q^{25} +144.541i q^{29} -181.662i q^{31} +161.495 q^{35} -99.7875 q^{37} +249.867i q^{41} -201.712i q^{43} -416.814 q^{47} -59.0193 q^{49} +420.474i q^{53} -118.689i q^{55} +266.159 q^{59} -199.785 q^{61} -102.845i q^{65} -55.6890i q^{67} -704.878 q^{71} -1109.82 q^{73} +295.460i q^{77} +751.695i q^{79} +746.773 q^{83} +467.182 q^{85} -1251.58i q^{89} +256.018i q^{91} +88.1871 q^{95} +575.314 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 72 q^{13} - 264 q^{25} - 24 q^{37} + 456 q^{61} + 2184 q^{73} - 3072 q^{85} - 3672 q^{97}+O(q^{100}) $ 24 * q + 72 * q^13 - 264 * q^25 - 24 * q^37 + 456 * q^61 + 2184 * q^73 - 3072 * q^85 - 3672 * 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
$5$	8.05443i	0.720410i	0.932873	+	0.360205i	$0.117293\pi$
$5$	8.05443i	0.720410i	−0.932873	+	0.360205i	$0.882707\pi$
$6$	0	0
$7$	− 20.0504i	− 1.08262i	−0.840823	−	0.541310i	$-0.817928\pi$
$7$	− 20.0504i	− 1.08262i	0.840823	−	0.541310i	$-0.182072\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−14.7358	−0.403911	−0.201956	−	0.979395i	$-0.564730\pi$
$11$	−14.7358	−0.403911	−0.201956	+	0.979395i	$0.564730\pi$
$12$	0	0
$13$	−12.7687	−0.272416	−0.136208	−	0.990680i	$-0.543492\pi$
$13$	−12.7687	−0.272416	−0.136208	+	0.990680i	$0.543492\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 58.0032i	− 0.827520i	−0.910386	−	0.413760i	$-0.864215\pi$
$17$	− 58.0032i	− 0.827520i	0.910386	−	0.413760i	$-0.135785\pi$
$18$	0	0
$19$	− 10.9489i	− 0.132203i	−0.997813	−	0.0661013i	$-0.978944\pi$
$19$	− 10.9489i	− 0.132203i	0.997813	−	0.0661013i	$-0.0210561\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	74.3941	0.674445	0.337223	−	0.941425i	$-0.390512\pi$
$23$	74.3941	0.674445	0.337223	+	0.941425i	$0.390512\pi$
$24$	0	0
$25$	60.1262	0.481010
$26$	0	0
$27$	0	0
$28$	0	0
$29$	144.541i	0.925537i	0.886479	+	0.462769i	$0.153144\pi$
$29$	144.541i	0.925537i	−0.886479	+	0.462769i	$0.846856\pi$
$30$	0	0
$31$	− 181.662i	− 1.05250i	−0.850331	−	0.526248i	$-0.823598\pi$
$31$	− 181.662i	− 1.05250i	0.850331	−	0.526248i	$-0.176402\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	161.495	0.779931
$36$	0	0
$37$	−99.7875	−0.443378	−0.221689	−	0.975117i	$-0.571157\pi$
$37$	−99.7875	−0.443378	−0.221689	+	0.975117i	$0.571157\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	249.867i	0.951774i	0.879506	+	0.475887i	$0.157873\pi$
$41$	249.867i	0.951774i	−0.879506	+	0.475887i	$0.842127\pi$
$42$	0	0
$43$	− 201.712i	− 0.715368i	−0.933843	−	0.357684i	$-0.883567\pi$
$43$	− 201.712i	− 0.715368i	0.933843	−	0.357684i	$-0.116433\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−416.814	−1.29359	−0.646793	−	0.762666i	$-0.723890\pi$
$47$	−416.814	−1.29359	−0.646793	+	0.762666i	$0.723890\pi$
$48$	0	0
$49$	−59.0193	−0.172068
$50$	0	0
$51$	0	0
$52$	0	0
$53$	420.474i	1.08975i	0.838518	+	0.544873i	$0.183422\pi$
$53$	420.474i	1.08975i	−0.838518	+	0.544873i	$0.816578\pi$
$54$	0	0
$55$	− 118.689i	− 0.290982i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	266.159	0.587304	0.293652	−	0.955912i	$-0.405129\pi$
$59$	266.159	0.587304	0.293652	+	0.955912i	$0.405129\pi$
$60$	0	0
$61$	−199.785	−0.419342	−0.209671	−	0.977772i	$-0.567239\pi$
$61$	−199.785	−0.419342	−0.209671	+	0.977772i	$0.567239\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 102.845i	− 0.196251i
$66$	0	0
$67$	− 55.6890i	− 0.101545i	−0.998710	−	0.0507724i	$-0.983832\pi$
$67$	− 55.6890i	− 0.101545i	0.998710	−	0.0507724i	$-0.0161683\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−704.878	−1.17822	−0.589110	−	0.808053i	$-0.700522\pi$
$71$	−704.878	−1.17822	−0.589110	+	0.808053i	$0.700522\pi$
$72$	0	0
$73$	−1109.82	−1.77937	−0.889687	−	0.456572i	$-0.849077\pi$
$73$	−1109.82	−1.77937	−0.889687	+	0.456572i	$0.849077\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	295.460i	0.437283i
$78$	0	0
$79$	751.695i	1.07053i	0.844683	+	0.535267i	$0.179789\pi$
$79$	751.695i	1.07053i	−0.844683	+	0.535267i	$0.820211\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	746.773	0.987578	0.493789	−	0.869582i	$-0.335612\pi$
$83$	746.773	0.987578	0.493789	+	0.869582i	$0.335612\pi$
$84$	0	0
$85$	467.182	0.596153
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 1251.58i	− 1.49064i	−0.666707	−	0.745320i	$-0.732297\pi$
$89$	− 1251.58i	− 1.49064i	0.666707	−	0.745320i	$-0.267703\pi$
$90$	0	0
$91$	256.018i	0.294923i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	88.1871	0.0952401
$96$	0	0
$97$	575.314	0.602209	0.301104	−	0.953591i	$-0.402645\pi$
$97$	575.314	0.602209	0.301104	+	0.953591i	$0.402645\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	849.450i	0.836866i	0.908248	+	0.418433i	$0.137421\pi$
$101$	849.450i	0.836866i	−0.908248	+	0.418433i	$0.862579\pi$
$102$	0	0
$103$	247.993i	0.237237i	0.992940	+	0.118619i	$0.0378466\pi$
$103$	247.993i	0.237237i	−0.992940	+	0.118619i	$0.962153\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−535.255	−0.483599	−0.241799	−	0.970326i	$-0.577738\pi$
$107$	−535.255	−0.483599	−0.241799	+	0.970326i	$0.577738\pi$
$108$	0	0
$109$	−1864.26	−1.63820	−0.819098	−	0.573654i	$-0.805525\pi$
$109$	−1864.26	−1.63820	−0.819098	+	0.573654i	$0.805525\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1606.33i	− 1.33726i	−0.743594	−	0.668632i	$-0.766880\pi$
$113$	− 1606.33i	− 1.33726i	0.743594	−	0.668632i	$-0.233120\pi$
$114$	0	0
$115$	599.202i	0.485877i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1162.99	−0.895890
$120$	0	0
$121$	−1113.86	−0.836856
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1491.09i	1.06693i
$126$	0	0
$127$	342.133i	0.239050i	0.992831	+	0.119525i	$0.0381372\pi$
$127$	342.133i	0.239050i	−0.992831	+	0.119525i	$0.961863\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2849.78	1.90066	0.950328	−	0.311249i	$-0.100747\pi$
$131$	2849.78	1.90066	0.950328	+	0.311249i	$0.100747\pi$
$132$	0	0
$133$	−219.530	−0.143125
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 609.068i	− 0.379826i	−0.981801	−	0.189913i	$-0.939179\pi$
$137$	− 609.068i	− 0.379826i	0.981801	−	0.189913i	$-0.0608206\pi$
$138$	0	0
$139$	925.200i	0.564565i	0.959331	+	0.282282i	$0.0910915\pi$
$139$	925.200i	0.564565i	−0.959331	+	0.282282i	$0.908909\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	188.158	0.110032
$144$	0	0
$145$	−1164.19	−0.666766
$146$	0	0
$147$	0	0
$148$	0	0
$149$	770.540i	0.423658i	0.977307	+	0.211829i	$0.0679420\pi$
$149$	770.540i	0.423658i	−0.977307	+	0.211829i	$0.932058\pi$
$150$	0	0
$151$	− 298.289i	− 0.160758i	−0.996764	−	0.0803788i	$-0.974387\pi$
$151$	− 298.289i	− 0.160758i	0.996764	−	0.0803788i	$-0.0256130\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1463.18	0.758229
$156$	0	0
$157$	−2757.64	−1.40181	−0.700903	−	0.713257i	$-0.747219\pi$
$157$	−2757.64	−1.40181	−0.700903	+	0.713257i	$0.747219\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 1491.63i	− 0.730168i
$162$	0	0
$163$	− 247.512i	− 0.118936i	−0.998230	−	0.0594682i	$-0.981060\pi$
$163$	− 247.512i	− 0.118936i	0.998230	−	0.0594682i	$-0.0189405\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3501.34	−1.62240	−0.811202	−	0.584766i	$-0.801186\pi$
$167$	−3501.34	−1.62240	−0.811202	+	0.584766i	$0.801186\pi$
$168$	0	0
$169$	−2033.96	−0.925790
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 2591.04i	− 1.13869i	−0.822099	−	0.569345i	$-0.807197\pi$
$173$	− 2591.04i	− 1.13869i	0.822099	−	0.569345i	$-0.192803\pi$
$174$	0	0
$175$	− 1205.56i	− 0.520751i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1183.11	−0.494020	−0.247010	−	0.969013i	$-0.579448\pi$
$179$	−1183.11	−0.494020	−0.247010	+	0.969013i	$0.579448\pi$
$180$	0	0
$181$	−1955.27	−0.802949	−0.401475	−	0.915870i	$-0.631502\pi$
$181$	−1955.27	−0.802949	−0.401475	+	0.915870i	$0.631502\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 803.731i	− 0.319414i
$186$	0	0
$187$	854.725i	0.334244i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1888.71	−0.715510	−0.357755	−	0.933815i	$-0.616458\pi$
$191$	−1888.71	−0.715510	−0.357755	+	0.933815i	$0.616458\pi$
$192$	0	0
$193$	−346.356	−0.129178	−0.0645888	−	0.997912i	$-0.520574\pi$
$193$	−346.356	−0.129178	−0.0645888	+	0.997912i	$0.520574\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	642.954i	0.232531i	0.993218	+	0.116265i	$0.0370923\pi$
$197$	642.954i	0.232531i	−0.993218	+	0.116265i	$0.962908\pi$
$198$	0	0
$199$	− 1992.53i	− 0.709783i	−0.934907	−	0.354892i	$-0.884518\pi$
$199$	− 1992.53i	− 0.709783i	0.934907	−	0.354892i	$-0.115482\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2898.11	1.00201
$204$	0	0
$205$	−2012.54	−0.685667
$206$	0	0
$207$	0	0
$208$	0	0
$209$	161.341i	0.0533981i
$210$	0	0
$211$	− 4698.88i	− 1.53310i	−0.642185	−	0.766550i	$-0.721972\pi$
$211$	− 4698.88i	− 1.53310i	0.642185	−	0.766550i	$-0.278028\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1624.68	0.515358
$216$	0	0
$217$	−3642.39	−1.13945
$218$	0	0
$219$	0	0
$220$	0	0
$221$	740.626i	0.225430i
$222$	0	0
$223$	322.474i	0.0968362i	0.998827	+	0.0484181i	$0.0154180\pi$
$223$	322.474i	0.0968362i	−0.998827	+	0.0484181i	$0.984582\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1065.03	0.311404	0.155702	−	0.987804i	$-0.450236\pi$
$227$	1065.03	0.311404	0.155702	+	0.987804i	$0.450236\pi$
$228$	0	0
$229$	−2006.11	−0.578898	−0.289449	−	0.957193i	$-0.593472\pi$
$229$	−2006.11	−0.578898	−0.289449	+	0.957193i	$0.593472\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 3146.22i	− 0.884617i	−0.896863	−	0.442308i	$-0.854160\pi$
$233$	− 3146.22i	− 0.884617i	0.896863	−	0.442308i	$-0.145840\pi$
$234$	0	0
$235$	− 3357.20i	− 0.931912i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4032.30	1.09133	0.545665	−	0.838004i	$-0.316277\pi$
$239$	4032.30	1.09133	0.545665	+	0.838004i	$0.316277\pi$
$240$	0	0
$241$	−5876.62	−1.57073	−0.785366	−	0.619031i	$-0.787525\pi$
$241$	−5876.62	−1.57073	−0.785366	+	0.619031i	$0.787525\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 475.366i	− 0.123959i
$246$	0	0
$247$	139.803i	0.0360141i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−7225.11	−1.81691	−0.908457	−	0.417979i	$-0.862738\pi$
$251$	−7225.11	−1.81691	−0.908457	+	0.417979i	$0.862738\pi$
$252$	0	0
$253$	−1096.26	−0.272416
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3390.88i	0.823024i	0.911404	+	0.411512i	$0.134999\pi$
$257$	3390.88i	0.823024i	−0.911404	+	0.411512i	$0.865001\pi$
$258$	0	0
$259$	2000.78i	0.480010i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3851.88	−0.903107	−0.451553	−	0.892244i	$-0.649130\pi$
$263$	−3851.88	−0.903107	−0.451553	+	0.892244i	$0.649130\pi$
$264$	0	0
$265$	−3386.68	−0.785064
$266$	0	0
$267$	0	0
$268$	0	0
$269$	8562.30i	1.94072i	0.241672	+	0.970358i	$0.422304\pi$
$269$	8562.30i	1.94072i	−0.241672	+	0.970358i	$0.577696\pi$
$270$	0	0
$271$	2404.22i	0.538916i	0.963012	+	0.269458i	$0.0868445\pi$
$271$	2404.22i	0.538916i	−0.963012	+	0.269458i	$0.913155\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−886.010	−0.194285
$276$	0	0
$277$	−2936.82	−0.637026	−0.318513	−	0.947918i	$-0.603184\pi$
$277$	−2936.82	−0.637026	−0.318513	+	0.947918i	$0.603184\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5696.37i	1.20931i	0.796486	+	0.604657i	$0.206690\pi$
$281$	5696.37i	1.20931i	−0.796486	+	0.604657i	$0.793310\pi$
$282$	0	0
$283$	1138.94i	0.239233i	0.992820	+	0.119617i	$0.0381665\pi$
$283$	1138.94i	0.239233i	−0.992820	+	0.119617i	$0.961833\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5009.95	1.03041
$288$	0	0
$289$	1548.63	0.315211
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 6467.29i	− 1.28950i	−0.764394	−	0.644750i	$-0.776962\pi$
$293$	− 6467.29i	− 1.28950i	0.764394	−	0.644750i	$-0.223038\pi$
$294$	0	0
$295$	2143.76i	0.423099i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−949.917	−0.183730
$300$	0	0
$301$	−4044.41	−0.774472
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 1609.15i	− 0.302098i
$306$	0	0
$307$	427.736i	0.0795185i	0.999209	+	0.0397592i	$0.0126591\pi$
$307$	427.736i	0.0795185i	−0.999209	+	0.0397592i	$0.987341\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2414.58	0.440252	0.220126	−	0.975471i	$-0.429353\pi$
$311$	2414.58	0.440252	0.220126	+	0.975471i	$0.429353\pi$
$312$	0	0
$313$	−4562.86	−0.823988	−0.411994	−	0.911187i	$-0.635167\pi$
$313$	−4562.86	−0.823988	−0.411994	+	0.911187i	$0.635167\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 4388.37i	− 0.777525i	−0.921338	−	0.388762i	$-0.872903\pi$
$317$	− 4388.37i	− 0.777525i	0.921338	−	0.388762i	$-0.127097\pi$
$318$	0	0
$319$	− 2129.93i	− 0.373835i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−635.071	−0.109400
$324$	0	0
$325$	−767.735	−0.131035
$326$	0	0
$327$	0	0
$328$	0	0
$329$	8357.29i	1.40046i
$330$	0	0
$331$	10042.7i	1.66767i	0.552013	+	0.833836i	$0.313860\pi$
$331$	10042.7i	1.66767i	−0.552013	+	0.833836i	$0.686140\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	448.543	0.0731538
$336$	0	0
$337$	−6324.52	−1.02231	−0.511155	−	0.859488i	$-0.670782\pi$
$337$	−6324.52	−1.02231	−0.511155	+	0.859488i	$0.670782\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2676.94i	0.425115i
$342$	0	0
$343$	− 5693.93i	− 0.896337i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5525.21	−0.854781	−0.427390	−	0.904067i	$-0.640567\pi$
$347$	−5525.21	−0.854781	−0.427390	+	0.904067i	$0.640567\pi$
$348$	0	0
$349$	−5467.82	−0.838641	−0.419321	−	0.907838i	$-0.637732\pi$
$349$	−5467.82	−0.838641	−0.419321	+	0.907838i	$0.637732\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	11837.4i	1.78482i	0.451230	+	0.892408i	$0.350985\pi$
$353$	11837.4i	1.78482i	−0.451230	+	0.892408i	$0.649015\pi$
$354$	0	0
$355$	− 5677.39i	− 0.848802i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11818.4	1.73748	0.868738	−	0.495272i	$-0.164932\pi$
$359$	11818.4	1.73748	0.868738	+	0.495272i	$0.164932\pi$
$360$	0	0
$361$	6739.12	0.982522
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 8938.94i	− 1.28188i
$366$	0	0
$367$	11931.5i	1.69705i	0.529156	+	0.848525i	$0.322509\pi$
$367$	11931.5i	1.69705i	−0.529156	+	0.848525i	$0.677491\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8430.68	1.17978
$372$	0	0
$373$	−8170.05	−1.13413	−0.567063	−	0.823674i	$-0.691921\pi$
$373$	−8170.05	−1.13413	−0.567063	+	0.823674i	$0.691921\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 1845.60i	− 0.252131i
$378$	0	0
$379$	3653.30i	0.495138i	0.968870	+	0.247569i	$0.0796317\pi$
$379$	3653.30i	0.495138i	−0.968870	+	0.247569i	$0.920368\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−6593.77	−0.879702	−0.439851	−	0.898071i	$-0.644969\pi$
$383$	−6593.77	−0.879702	−0.439851	+	0.898071i	$0.644969\pi$
$384$	0	0
$385$	−2379.76	−0.315023
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 6674.21i	− 0.869912i	−0.900452	−	0.434956i	$-0.856764\pi$
$389$	− 6674.21i	− 0.869912i	0.900452	−	0.434956i	$-0.143236\pi$
$390$	0	0
$391$	− 4315.09i	− 0.558117i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−6054.47	−0.771224
$396$	0	0
$397$	3658.10	0.462455	0.231228	−	0.972900i	$-0.425726\pi$
$397$	3658.10	0.462455	0.231228	+	0.972900i	$0.425726\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5849.71i	0.728481i	0.931305	+	0.364240i	$0.118671\pi$
$401$	5849.71i	0.728481i	−0.931305	+	0.364240i	$0.881329\pi$
$402$	0	0
$403$	2319.59i	0.286717i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1470.45	0.179085
$408$	0	0
$409$	−3027.49	−0.366014	−0.183007	−	0.983112i	$-0.558583\pi$
$409$	−3027.49	−0.366014	−0.183007	+	0.983112i	$0.558583\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 5336.59i	− 0.635827i
$414$	0	0
$415$	6014.83i	0.711461i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−11155.5	−1.30067	−0.650337	−	0.759646i	$-0.725372\pi$
$419$	−11155.5	−1.30067	−0.650337	+	0.759646i	$0.725372\pi$
$420$	0	0
$421$	−10107.7	−1.17011	−0.585057	−	0.810992i	$-0.698928\pi$
$421$	−10107.7	−1.17011	−0.585057	+	0.810992i	$0.698928\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 3487.51i	− 0.398045i
$426$	0	0
$427$	4005.77i	0.453988i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10898.7	1.21804	0.609019	−	0.793156i	$-0.291563\pi$
$431$	10898.7	1.21804	0.609019	+	0.793156i	$0.291563\pi$
$432$	0	0
$433$	−10256.8	−1.13836	−0.569179	−	0.822213i	$-0.692739\pi$
$433$	−10256.8	−1.13836	−0.569179	+	0.822213i	$0.692739\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 814.534i	− 0.0891634i
$438$	0	0
$439$	2765.38i	0.300648i	0.988637	+	0.150324i	$0.0480317\pi$
$439$	2765.38i	0.300648i	−0.988637	+	0.150324i	$0.951968\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2418.62	−0.259395	−0.129698	−	0.991554i	$-0.541401\pi$
$443$	−2418.62	−0.259395	−0.129698	+	0.991554i	$0.541401\pi$
$444$	0	0
$445$	10080.7	1.07387
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 14158.1i	− 1.48811i	−0.668120	−	0.744054i	$-0.732901\pi$
$449$	− 14158.1i	− 1.48811i	0.668120	−	0.744054i	$-0.267099\pi$
$450$	0	0
$451$	− 3682.01i	− 0.384432i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2062.08	−0.212466
$456$	0	0
$457$	11498.5	1.17697	0.588487	−	0.808507i	$-0.299724\pi$
$457$	11498.5	1.17697	0.588487	+	0.808507i	$0.299724\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 10805.2i	− 1.09165i	−0.837900	−	0.545824i	$-0.816217\pi$
$461$	− 10805.2i	− 1.09165i	0.837900	−	0.545824i	$-0.183783\pi$
$462$	0	0
$463$	− 13005.2i	− 1.30541i	−0.757612	−	0.652705i	$-0.773634\pi$
$463$	− 13005.2i	− 1.30541i	0.757612	−	0.652705i	$-0.226366\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−19806.1	−1.96257	−0.981284	−	0.192567i	$-0.938319\pi$
$467$	−19806.1	−1.96257	−0.981284	+	0.192567i	$0.938319\pi$
$468$	0	0
$469$	−1116.59	−0.109934
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2972.40i	0.288945i
$474$	0	0
$475$	− 658.316i	− 0.0635907i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	4540.81	0.433142	0.216571	−	0.976267i	$-0.430513\pi$
$479$	4540.81	0.433142	0.216571	+	0.976267i	$0.430513\pi$
$480$	0	0
$481$	1274.16	0.120783
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4633.82i	0.433837i
$486$	0	0
$487$	− 17236.2i	− 1.60379i	−0.597464	−	0.801896i	$-0.703825\pi$
$487$	− 17236.2i	− 1.60379i	0.597464	−	0.801896i	$-0.296175\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	9264.37	0.851517	0.425759	−	0.904837i	$-0.360007\pi$
$491$	9264.37	0.851517	0.425759	+	0.904837i	$0.360007\pi$
$492$	0	0
$493$	8383.83	0.765900
$494$	0	0
$495$	0	0
$496$	0	0
$497$	14133.1i	1.27557i
$498$	0	0
$499$	− 21195.2i	− 1.90146i	−0.310027	−	0.950728i	$-0.600338\pi$
$499$	− 21195.2i	− 1.90146i	0.310027	−	0.950728i	$-0.399662\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	10426.8	0.924267	0.462134	−	0.886810i	$-0.347084\pi$
$503$	10426.8	0.924267	0.462134	+	0.886810i	$0.347084\pi$
$504$	0	0
$505$	−6841.84	−0.602886
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8235.12i	0.717122i	0.933506	+	0.358561i	$0.116733\pi$
$509$	8235.12i	0.717122i	−0.933506	+	0.358561i	$0.883267\pi$
$510$	0	0
$511$	22252.3i	1.92639i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1997.44	−0.170908
$516$	0	0
$517$	6142.10	0.522494
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5265.62i	0.442785i	0.975185	+	0.221393i	$0.0710602\pi$
$521$	5265.62i	0.442785i	−0.975185	+	0.221393i	$0.928940\pi$
$522$	0	0
$523$	− 12675.7i	− 1.05979i	−0.848064	−	0.529894i	$-0.822232\pi$
$523$	− 12675.7i	− 1.05979i	0.848064	−	0.529894i	$-0.177768\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−10537.0	−0.870962
$528$	0	0
$529$	−6632.52	−0.545124
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 3190.49i	− 0.259278i
$534$	0	0
$535$	− 4311.17i	− 0.348389i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	869.698	0.0695001
$540$	0	0
$541$	4698.09	0.373358	0.186679	−	0.982421i	$-0.440228\pi$
$541$	4698.09	0.373358	0.186679	+	0.982421i	$0.440228\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 15015.5i	− 1.18017i
$546$	0	0
$547$	6668.52i	0.521253i	0.965440	+	0.260627i	$0.0839291\pi$
$547$	6668.52i	0.521253i	−0.965440	+	0.260627i	$0.916071\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1582.56	0.122358
$552$	0	0
$553$	15071.8	1.15898
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2283.58i	0.173714i	0.996221	+	0.0868569i	$0.0276823\pi$
$557$	2283.58i	0.173714i	−0.996221	+	0.0868569i	$0.972318\pi$
$558$	0	0
$559$	2575.61i	0.194878i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	6323.88	0.473392	0.236696	−	0.971584i	$-0.423935\pi$
$563$	6323.88	0.473392	0.236696	+	0.971584i	$0.423935\pi$
$564$	0	0
$565$	12938.1	0.963378
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 4953.43i	− 0.364954i	−0.983210	−	0.182477i	$-0.941589\pi$
$569$	− 4953.43i	− 0.364954i	0.983210	−	0.182477i	$-0.0584114\pi$
$570$	0	0
$571$	11295.9i	0.827877i	0.910305	+	0.413938i	$0.135847\pi$
$571$	11295.9i	0.827877i	−0.910305	+	0.413938i	$0.864153\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4473.03	0.324415
$576$	0	0
$577$	16341.3	1.17903	0.589514	−	0.807758i	$-0.299319\pi$
$577$	16341.3	1.17903	0.589514	+	0.807758i	$0.299319\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 14973.1i	− 1.06917i
$582$	0	0
$583$	− 6196.04i	− 0.440161i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	24609.9	1.73042	0.865212	−	0.501407i	$-0.167184\pi$
$587$	24609.9	1.73042	0.865212	+	0.501407i	$0.167184\pi$
$588$	0	0
$589$	−1989.00	−0.139143
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 6930.38i	− 0.479927i	−0.970782	−	0.239963i	$-0.922865\pi$
$593$	− 6930.38i	− 0.479927i	0.970782	−	0.239963i	$-0.0771354\pi$
$594$	0	0
$595$	− 9367.20i	− 0.645408i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−10182.6	−0.694572	−0.347286	−	0.937759i	$-0.612897\pi$
$599$	−10182.6	−0.694572	−0.347286	+	0.937759i	$0.612897\pi$
$600$	0	0
$601$	9536.14	0.647234	0.323617	−	0.946188i	$-0.395101\pi$
$601$	9536.14	0.647234	0.323617	+	0.946188i	$0.395101\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 8971.46i	− 0.602879i
$606$	0	0
$607$	− 6873.14i	− 0.459592i	−0.973239	−	0.229796i	$-0.926194\pi$
$607$	− 6873.14i	− 0.459592i	0.973239	−	0.229796i	$-0.0738058\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5322.18	0.352393
$612$	0	0
$613$	−11378.4	−0.749704	−0.374852	−	0.927085i	$-0.622306\pi$
$613$	−11378.4	−0.749704	−0.374852	+	0.927085i	$0.622306\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1884.90i	0.122987i	0.998107	+	0.0614937i	$0.0195864\pi$
$617$	1884.90i	0.122987i	−0.998107	+	0.0614937i	$0.980414\pi$
$618$	0	0
$619$	25715.0i	1.66975i	0.550441	+	0.834874i	$0.314460\pi$
$619$	25715.0i	1.66975i	−0.550441	+	0.834874i	$0.685540\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−25094.7	−1.61380
$624$	0	0
$625$	−4494.07	−0.287620
$626$	0	0
$627$	0	0
$628$	0	0
$629$	5787.99i	0.366904i
$630$	0	0
$631$	− 8509.78i	− 0.536876i	−0.963297	−	0.268438i	$-0.913493\pi$
$631$	− 8509.78i	− 0.536876i	0.963297	−	0.268438i	$-0.0865075\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2755.68	−0.172214
$636$	0	0
$637$	753.601	0.0468740
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2720.43i	0.167630i	0.996481	+	0.0838148i	$0.0267104\pi$
$641$	2720.43i	0.167630i	−0.996481	+	0.0838148i	$0.973290\pi$
$642$	0	0
$643$	− 24207.3i	− 1.48467i	−0.670029	−	0.742335i	$-0.733718\pi$
$643$	− 24207.3i	− 1.48467i	0.670029	−	0.742335i	$-0.266282\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−832.754	−0.0506012	−0.0253006	−	0.999680i	$-0.508054\pi$
$647$	−832.754	−0.0506012	−0.0253006	+	0.999680i	$0.508054\pi$
$648$	0	0
$649$	−3922.07	−0.237219
$650$	0	0
$651$	0	0
$652$	0	0
$653$	14518.1i	0.870043i	0.900420	+	0.435021i	$0.143259\pi$
$653$	14518.1i	0.870043i	−0.900420	+	0.435021i	$0.856741\pi$
$654$	0	0
$655$	22953.3i	1.36925i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−10396.2	−0.614532	−0.307266	−	0.951624i	$-0.599414\pi$
$659$	−10396.2	−0.614532	−0.307266	+	0.951624i	$0.599414\pi$
$660$	0	0
$661$	15419.6	0.907343	0.453672	−	0.891169i	$-0.350114\pi$
$661$	15419.6	0.907343	0.453672	+	0.891169i	$0.350114\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1768.19i	− 0.103109i
$666$	0	0
$667$	10753.0i	0.624224i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2944.00	0.169377
$672$	0	0
$673$	32417.9	1.85679	0.928394	−	0.371597i	$-0.121190\pi$
$673$	32417.9	1.85679	0.928394	+	0.371597i	$0.121190\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	27165.3i	1.54217i	0.636735	+	0.771083i	$0.280285\pi$
$677$	27165.3i	1.54217i	−0.636735	+	0.771083i	$0.719715\pi$
$678$	0	0
$679$	− 11535.3i	− 0.651964i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	19995.4	1.12021	0.560104	−	0.828422i	$-0.310761\pi$
$683$	19995.4	1.12021	0.560104	+	0.828422i	$0.310761\pi$
$684$	0	0
$685$	4905.69	0.273630
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 5368.92i	− 0.296864i
$690$	0	0
$691$	28011.7i	1.54213i	0.636755	+	0.771066i	$0.280276\pi$
$691$	28011.7i	1.54213i	−0.636755	+	0.771066i	$0.719724\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7451.96	−0.406718
$696$	0	0
$697$	14493.1	0.787612
$698$	0	0
$699$	0	0
$700$	0	0
$701$	23915.4i	1.28855i	0.764795	+	0.644274i	$0.222840\pi$
$701$	23915.4i	1.28855i	−0.764795	+	0.644274i	$0.777160\pi$
$702$	0	0
$703$	1092.56i	0.0586157i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	17031.8	0.906008
$708$	0	0
$709$	33718.3	1.78606	0.893030	−	0.449997i	$-0.148575\pi$
$709$	33718.3	1.78606	0.893030	+	0.449997i	$0.148575\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 13514.6i	− 0.709851i
$714$	0	0
$715$	1515.50i	0.0792680i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−13347.3	−0.692312	−0.346156	−	0.938177i	$-0.612513\pi$
$719$	−13347.3	−0.692312	−0.346156	+	0.938177i	$0.612513\pi$
$720$	0	0
$721$	4972.36	0.256838
$722$	0	0
$723$	0	0
$724$	0	0
$725$	8690.70i	0.445192i
$726$	0	0
$727$	12331.5i	0.629093i	0.949242	+	0.314547i	$0.101852\pi$
$727$	12331.5i	0.629093i	−0.949242	+	0.314547i	$0.898148\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−11699.9	−0.591981
$732$	0	0
$733$	4599.02	0.231744	0.115872	−	0.993264i	$-0.463034\pi$
$733$	4599.02	0.231744	0.115872	+	0.993264i	$0.463034\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	820.624i	0.0410150i
$738$	0	0
$739$	− 11846.4i	− 0.589687i	−0.955546	−	0.294843i	$-0.904733\pi$
$739$	− 11846.4i	− 0.589687i	0.955546	−	0.294843i	$-0.0952675\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	12096.8	0.597294	0.298647	−	0.954364i	$-0.403465\pi$
$743$	12096.8	0.597294	0.298647	+	0.954364i	$0.403465\pi$
$744$	0	0
$745$	−6206.26	−0.305208
$746$	0	0
$747$	0	0
$748$	0	0
$749$	10732.1i	0.523554i
$750$	0	0
$751$	19161.3i	0.931035i	0.885039	+	0.465518i	$0.154132\pi$
$751$	19161.3i	0.931035i	−0.885039	+	0.465518i	$0.845868\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2402.55	0.115811
$756$	0	0
$757$	3351.80	0.160929	0.0804645	−	0.996757i	$-0.474360\pi$
$757$	3351.80	0.160929	0.0804645	+	0.996757i	$0.474360\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	6196.36i	0.295162i	0.989050	+	0.147581i	$0.0471487\pi$
$761$	6196.36i	0.295162i	−0.989050	+	0.147581i	$0.952851\pi$
$762$	0	0
$763$	37379.1i	1.77354i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3398.51	−0.159991
$768$	0	0
$769$	36368.9	1.70546	0.852728	−	0.522355i	$-0.174946\pi$
$769$	36368.9	1.70546	0.852728	+	0.522355i	$0.174946\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	25925.2i	1.20629i	0.797631	+	0.603146i	$0.206086\pi$
$773$	25925.2i	1.20629i	−0.797631	+	0.603146i	$0.793914\pi$
$774$	0	0
$775$	− 10922.6i	− 0.506261i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2735.77	0.125827
$780$	0	0
$781$	10387.0	0.475896
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 22211.2i	− 1.00987i
$786$	0	0
$787$	− 31849.9i	− 1.44260i	−0.692622	−	0.721301i	$-0.743544\pi$
$787$	− 31849.9i	− 1.44260i	0.692622	−	0.721301i	$-0.256456\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−32207.6	−1.44775
$792$	0	0
$793$	2551.00	0.114235
$794$	0	0
$795$	0	0
$796$	0	0
$797$	10138.1i	0.450576i	0.974292	+	0.225288i	$0.0723323\pi$
$797$	10138.1i	0.450576i	−0.974292	+	0.225288i	$0.927668\pi$
$798$	0	0
$799$	24176.5i	1.07047i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	16354.1	0.718709
$804$	0	0
$805$	12014.2	0.526021
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 14786.4i	− 0.642597i	−0.946978	−	0.321299i	$-0.895881\pi$
$809$	− 14786.4i	− 0.642597i	0.946978	−	0.321299i	$-0.104119\pi$
$810$	0	0
$811$	16494.2i	0.714168i	0.934072	+	0.357084i	$0.116229\pi$
$811$	16494.2i	0.714168i	−0.934072	+	0.357084i	$0.883771\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1993.57	0.0856829
$816$	0	0
$817$	−2208.53	−0.0945735
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16305.6i	0.693142i	0.938024	+	0.346571i	$0.112654\pi$
$821$	16305.6i	0.693142i	−0.938024	+	0.346571i	$0.887346\pi$
$822$	0	0
$823$	− 34129.3i	− 1.44553i	−0.691093	−	0.722766i	$-0.742870\pi$
$823$	− 34129.3i	− 1.44553i	0.691093	−	0.722766i	$-0.257130\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1299.90	0.0546578	0.0273289	−	0.999626i	$-0.491300\pi$
$827$	1299.90	0.0546578	0.0273289	+	0.999626i	$0.491300\pi$
$828$	0	0
$829$	14143.6	0.592555	0.296277	−	0.955102i	$-0.404255\pi$
$829$	14143.6	0.592555	0.296277	+	0.955102i	$0.404255\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3423.30i	0.142390i
$834$	0	0
$835$	− 28201.2i	− 1.16880i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	4548.95	0.187184	0.0935919	−	0.995611i	$-0.470165\pi$
$839$	4548.95	0.187184	0.0935919	+	0.995611i	$0.470165\pi$
$840$	0	0
$841$	3496.92	0.143381
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 16382.4i	− 0.666948i
$846$	0	0
$847$	22333.3i	0.905997i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−7423.60	−0.299034
$852$	0	0
$853$	12354.3	0.495901	0.247951	−	0.968773i	$-0.420243\pi$
$853$	12354.3	0.495901	0.247951	+	0.968773i	$0.420243\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 17157.4i	− 0.683882i	−0.939721	−	0.341941i	$-0.888916\pi$
$857$	− 17157.4i	− 0.683882i	0.939721	−	0.341941i	$-0.111084\pi$
$858$	0	0
$859$	− 13098.1i	− 0.520256i	−0.965574	−	0.260128i	$-0.916235\pi$
$859$	− 13098.1i	− 0.520256i	0.965574	−	0.260128i	$-0.0837648\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−43684.8	−1.72311	−0.861557	−	0.507661i	$-0.830510\pi$
$863$	−43684.8	−1.72311	−0.861557	+	0.507661i	$0.830510\pi$
$864$	0	0
$865$	20869.4	0.820323
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 11076.9i	− 0.432401i
$870$	0	0
$871$	711.078i	0.0276624i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	29896.9	1.15508
$876$	0	0
$877$	11013.2	0.424046	0.212023	−	0.977265i	$-0.431995\pi$
$877$	11013.2	0.424046	0.212023	+	0.977265i	$0.431995\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	8301.14i	0.317449i	0.987323	+	0.158725i	$0.0507382\pi$
$881$	8301.14i	0.317449i	−0.987323	+	0.158725i	$0.949262\pi$
$882$	0	0
$883$	− 41641.5i	− 1.58703i	−0.608550	−	0.793516i	$-0.708248\pi$
$883$	− 41641.5i	− 1.58703i	0.608550	−	0.793516i	$-0.291752\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11662.3	0.441466	0.220733	−	0.975334i	$-0.429155\pi$
$887$	11662.3	0.441466	0.220733	+	0.975334i	$0.429155\pi$
$888$	0	0
$889$	6859.90	0.258801
$890$	0	0
$891$	0	0
$892$	0	0
$893$	4563.65i	0.171015i
$894$	0	0
$895$	− 9529.24i	− 0.355897i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	26257.5	0.974125
$900$	0	0
$901$	24388.8	0.901787
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 15748.5i	− 0.578452i
$906$	0	0
$907$	31427.5i	1.15053i	0.817967	+	0.575266i	$0.195101\pi$
$907$	31427.5i	1.15053i	−0.817967	+	0.575266i	$0.804899\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	11268.3	0.409810	0.204905	−	0.978782i	$-0.434311\pi$
$911$	11268.3	0.409810	0.204905	+	0.978782i	$0.434311\pi$
$912$	0	0
$913$	−11004.3	−0.398894
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 57139.2i	− 2.05769i
$918$	0	0
$919$	− 24439.4i	− 0.877239i	−0.898673	−	0.438619i	$-0.855468\pi$
$919$	− 24439.4i	− 0.877239i	0.898673	−	0.438619i	$-0.144532\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	9000.39	0.320966
$924$	0	0
$925$	−5999.85	−0.213269
$926$	0	0
$927$	0	0
$928$	0	0
$929$	13333.3i	0.470884i	0.971888	+	0.235442i	$0.0756538\pi$
$929$	13333.3i	0.470884i	−0.971888	+	0.235442i	$0.924346\pi$
$930$	0	0
$931$	646.196i	0.0227478i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−6884.32	−0.240793
$936$	0	0
$937$	7456.96	0.259988	0.129994	−	0.991515i	$-0.458504\pi$
$937$	7456.96	0.259988	0.129994	+	0.991515i	$0.458504\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	7265.74i	0.251707i	0.992049	+	0.125853i	$0.0401669\pi$
$941$	7265.74i	0.251707i	−0.992049	+	0.125853i	$0.959833\pi$
$942$	0	0
$943$	18588.7i	0.641919i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−24711.6	−0.847962	−0.423981	−	0.905671i	$-0.639368\pi$
$947$	−24711.6	−0.847962	−0.423981	+	0.905671i	$0.639368\pi$
$948$	0	0
$949$	14170.9	0.484730
$950$	0	0
$951$	0	0
$952$	0	0
$953$	13880.3i	0.471802i	0.971777	+	0.235901i	$0.0758041\pi$
$953$	13880.3i	0.471802i	−0.971777	+	0.235901i	$0.924196\pi$
$954$	0	0
$955$	− 15212.5i	− 0.515461i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−12212.1	−0.411208
$960$	0	0
$961$	−3209.94	−0.107749
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 2789.70i	− 0.0930608i
$966$	0	0
$967$	− 51087.4i	− 1.69892i	−0.527650	−	0.849462i	$-0.676927\pi$
$967$	− 51087.4i	− 1.69892i	0.527650	−	0.849462i	$-0.323073\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−7695.91	−0.254350	−0.127175	−	0.991880i	$-0.540591\pi$
$971$	−7695.91	−0.254350	−0.127175	+	0.991880i	$0.540591\pi$
$972$	0	0
$973$	18550.7	0.611209
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 19421.0i	− 0.635959i	−0.948098	−	0.317980i	$-0.896996\pi$
$977$	− 19421.0i	− 0.635959i	0.948098	−	0.317980i	$-0.103004\pi$
$978$	0	0
$979$	18443.1i	0.602086i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	8871.76	0.287859	0.143929	−	0.989588i	$-0.454026\pi$
$983$	8871.76	0.287859	0.143929	+	0.989588i	$0.454026\pi$
$984$	0	0
$985$	−5178.62	−0.167517
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 15006.2i	− 0.482476i
$990$	0	0
$991$	− 7872.58i	− 0.252352i	−0.992008	−	0.126176i	$-0.959730\pi$
$991$	− 7872.58i	− 0.252352i	0.992008	−	0.126176i	$-0.0402704\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	16048.7	0.511335
$996$	0	0
$997$	1073.89	0.0341127	0.0170564	−	0.999855i	$-0.494571\pi$
$997$	1073.89	0.0341127	0.0170564	+	0.999855i	$0.494571\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.4.c.l.1727.17		24
3.2	odd	2	inner	1728.4.c.l.1727.7		24
4.3	odd	2	inner	1728.4.c.l.1727.18		24
8.3	odd	2		864.4.c.a.863.8	yes	24
8.5	even	2		864.4.c.a.863.7	✓	24
12.11	even	2	inner	1728.4.c.l.1727.8		24
24.5	odd	2		864.4.c.a.863.17	yes	24
24.11	even	2		864.4.c.a.863.18	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.4.c.a.863.7	✓	24	8.5	even	2
864.4.c.a.863.8	yes	24	8.3	odd	2
864.4.c.a.863.17	yes	24	24.5	odd	2
864.4.c.a.863.18	yes	24	24.11	even	2
1728.4.c.l.1727.7		24	3.2	odd	2	inner
1728.4.c.l.1727.8		24	12.11	even	2	inner
1728.4.c.l.1727.17		24	1.1	even	1	trivial
1728.4.c.l.1727.18		24	4.3	odd	2	inner

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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+8.05443i q^{5} -20.0504i q^{7} -14.7358 q^{11} -12.7687 q^{13} -58.0032i q^{17} -10.9489i q^{19} +74.3941 q^{23} +60.1262 q^{25} +144.541i q^{29} -181.662i q^{31} +161.495 q^{35} -99.7875 q^{37} +249.867i q^{41} -201.712i q^{43} -416.814 q^{47} -59.0193 q^{49} +420.474i q^{53} -118.689i q^{55} +266.159 q^{59} -199.785 q^{61} -102.845i q^{65} -55.6890i q^{67} -704.878 q^{71} -1109.82 q^{73} +295.460i q^{77} +751.695i q^{79} +746.773 q^{83} +467.182 q^{85} -1251.58i q^{89} +256.018i q^{91} +88.1871 q^{95} +575.314 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 72 q^{13} - 264 q^{25} - 24 q^{37} + 456 q^{61} + 2184 q^{73} - 3072 q^{85} - 3672 q^{97}+O(q^{100}) \) 24 * q + 72 * q^13 - 264 * q^25 - 24 * q^37 + 456 * q^61 + 2184 * q^73 - 3072 * q^85 - 3672 * q^97

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