LMFDB - Embedded newform 1728.4.f.h.863.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1728,4,Mod(863,1728)] 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, 1, 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("1728.863"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1728.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-384,0,304,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1248,0,-720] 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:	$101.955300490$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 29x^{14} + 601x^{12} - 5608x^{10} + 37420x^{8} - 128832x^{6} + 318736x^{4} - 389376x^{2} + 331776 $ x^16 - 29x^14 + 601x^12 - 5608x^10 + 37420x^8 - 128832x^6 + 318736x^4 - 389376*x^2 + 331776
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{30}\cdot 3^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			863.5
Root			$-2.33286 + 1.34688i$ of defining polynomial
Character	$\chi$	$=$	1728.863
Dual form			1728.4.f.h.863.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-8.34090 q^{5} -21.0128i q^{7} +9.52051i q^{11} +8.89335i q^{13} +42.5915i q^{17} -24.3503 q^{19} +124.587 q^{23} -55.4293 q^{25} -88.0933 q^{29} -15.1632i q^{31} +175.266i q^{35} +79.3110i q^{37} +164.542i q^{41} -393.223 q^{43} +374.718 q^{47} -98.5375 q^{49} +90.1098 q^{53} -79.4096i q^{55} -25.3654i q^{59} +669.799i q^{61} -74.1786i q^{65} +236.666 q^{67} -128.089 q^{71} +152.044 q^{73} +200.052 q^{77} -525.676i q^{79} +761.686i q^{83} -355.252i q^{85} -1587.03i q^{89} +186.874 q^{91} +203.104 q^{95} -1050.34 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 384 q^{23} + 304 q^{25} + 1248 q^{47} - 720 q^{49} + 5088 q^{71} - 128 q^{73} + 11712 q^{95} + 4592 q^{97}+O(q^{100}) $ 16 * q - 384 * q^23 + 304 * q^25 + 1248 * q^47 - 720 * q^49 + 5088 * q^71 - 128 * q^73 + 11712 * q^95 + 4592 * 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
$5$	−8.34090	−0.746033	−0.373017	−	0.927825i	$-0.621677\pi$
$5$	−8.34090	−0.746033	−0.373017	+	0.927825i	$0.621677\pi$
$6$	0	0
$7$	− 21.0128i	− 1.13458i	−0.823517	−	0.567292i	$-0.807991\pi$
$7$	− 21.0128i	− 1.13458i	0.823517	−	0.567292i	$-0.192009\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	9.52051i	0.260958i	0.991451	+	0.130479i	$0.0416516\pi$
$11$	9.52051i	0.260958i	−0.991451	+	0.130479i	$0.958348\pi$
$12$	0	0
$13$	8.89335i	0.189736i	0.995490	+	0.0948682i	$0.0302430\pi$
$13$	8.89335i	0.189736i	−0.995490	+	0.0948682i	$0.969757\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	42.5915i	0.607645i	0.952729	+	0.303822i	$0.0982629\pi$
$17$	42.5915i	0.607645i	−0.952729	+	0.303822i	$0.901737\pi$
$18$	0	0
$19$	−24.3503	−0.294018	−0.147009	−	0.989135i	$-0.546965\pi$
$19$	−24.3503	−0.294018	−0.147009	+	0.989135i	$0.546965\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	124.587	1.12948	0.564742	−	0.825267i	$-0.308976\pi$
$23$	124.587	1.12948	0.564742	+	0.825267i	$0.308976\pi$
$24$	0	0
$25$	−55.4293	−0.443435
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−88.0933	−0.564087	−0.282043	−	0.959402i	$-0.591012\pi$
$29$	−88.0933	−0.564087	−0.282043	+	0.959402i	$0.591012\pi$
$30$	0	0
$31$	− 15.1632i	− 0.0878511i	−0.999035	−	0.0439256i	$-0.986014\pi$
$31$	− 15.1632i	− 0.0878511i	0.999035	−	0.0439256i	$-0.0139864\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	175.266i	0.846437i
$36$	0	0
$37$	79.3110i	0.352396i	0.984355	+	0.176198i	$0.0563799\pi$
$37$	79.3110i	0.352396i	−0.984355	+	0.176198i	$0.943620\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	164.542i	0.626760i	0.949628	+	0.313380i	$0.101461\pi$
$41$	164.542i	0.626760i	−0.949628	+	0.313380i	$0.898539\pi$
$42$	0	0
$43$	−393.223	−1.39456	−0.697278	−	0.716801i	$-0.745606\pi$
$43$	−393.223	−1.39456	−0.697278	+	0.716801i	$0.745606\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	374.718	1.16294	0.581471	−	0.813567i	$-0.302478\pi$
$47$	374.718	1.16294	0.581471	+	0.813567i	$0.302478\pi$
$48$	0	0
$49$	−98.5375	−0.287281
$50$	0	0
$51$	0	0
$52$	0	0
$53$	90.1098	0.233538	0.116769	−	0.993159i	$-0.462746\pi$
$53$	90.1098	0.233538	0.116769	+	0.993159i	$0.462746\pi$
$54$	0	0
$55$	− 79.4096i	− 0.194683i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 25.3654i	− 0.0559711i	−0.999608	−	0.0279855i	$-0.991091\pi$
$59$	− 25.3654i	− 0.0559711i	0.999608	−	0.0279855i	$-0.00890923\pi$
$60$	0	0
$61$	669.799i	1.40588i	0.711247	+	0.702942i	$0.248131\pi$
$61$	669.799i	1.40588i	−0.711247	+	0.702942i	$0.751869\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 74.1786i	− 0.141550i
$66$	0	0
$67$	236.666	0.431543	0.215771	−	0.976444i	$-0.430773\pi$
$67$	236.666	0.431543	0.215771	+	0.976444i	$0.430773\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−128.089	−0.214103	−0.107052	−	0.994253i	$-0.534141\pi$
$71$	−128.089	−0.214103	−0.107052	+	0.994253i	$0.534141\pi$
$72$	0	0
$73$	152.044	0.243773	0.121887	−	0.992544i	$-0.461106\pi$
$73$	152.044	0.243773	0.121887	+	0.992544i	$0.461106\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	200.052	0.296079
$78$	0	0
$79$	− 525.676i	− 0.748647i	−0.927298	−	0.374324i	$-0.877875\pi$
$79$	− 525.676i	− 0.748647i	0.927298	−	0.374324i	$-0.122125\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	761.686i	1.00730i	0.863908	+	0.503650i	$0.168010\pi$
$83$	761.686i	1.00730i	−0.863908	+	0.503650i	$0.831990\pi$
$84$	0	0
$85$	− 355.252i	− 0.453323i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 1587.03i	− 1.89017i	−0.326830	−	0.945083i	$-0.605981\pi$
$89$	− 1587.03i	− 1.89017i	0.326830	−	0.945083i	$-0.394019\pi$
$90$	0	0
$91$	186.874	0.215272
$92$	0	0
$93$	0	0
$94$	0	0
$95$	203.104	0.219347
$96$	0	0
$97$	−1050.34	−1.09944	−0.549720	−	0.835349i	$-0.685265\pi$
$97$	−1050.34	−1.09944	−0.549720	+	0.835349i	$0.685265\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	800.038	0.788186	0.394093	−	0.919071i	$-0.371059\pi$
$101$	800.038	0.788186	0.394093	+	0.919071i	$0.371059\pi$
$102$	0	0
$103$	788.835i	0.754623i	0.926086	+	0.377312i	$0.123151\pi$
$103$	788.835i	0.754623i	−0.926086	+	0.377312i	$0.876849\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1368.22i	− 1.23618i	−0.786108	−	0.618089i	$-0.787907\pi$
$107$	− 1368.22i	− 1.23618i	0.786108	−	0.618089i	$-0.212093\pi$
$108$	0	0
$109$	− 953.026i	− 0.837461i	−0.908110	−	0.418731i	$-0.862475\pi$
$109$	− 953.026i	− 0.837461i	0.908110	−	0.418731i	$-0.137525\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1477.75i	− 1.23022i	−0.788440	−	0.615112i	$-0.789111\pi$
$113$	− 1477.75i	− 1.23022i	0.788440	−	0.615112i	$-0.210889\pi$
$114$	0	0
$115$	−1039.17	−0.842633
$116$	0	0
$117$	0	0
$118$	0	0
$119$	894.967	0.689424
$120$	0	0
$121$	1240.36	0.931901
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1504.94	1.07685
$126$	0	0
$127$	410.595i	0.286885i	0.989659	+	0.143442i	$0.0458172\pi$
$127$	410.595i	0.286885i	−0.989659	+	0.143442i	$0.954183\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1599.45i	1.06675i	0.845879	+	0.533375i	$0.179076\pi$
$131$	1599.45i	1.06675i	−0.845879	+	0.533375i	$0.820924\pi$
$132$	0	0
$133$	511.668i	0.333588i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1064.13i	− 0.663611i	−0.943348	−	0.331806i	$-0.892342\pi$
$137$	− 1064.13i	− 0.663611i	0.943348	−	0.331806i	$-0.107658\pi$
$138$	0	0
$139$	2144.33	1.30848	0.654242	−	0.756285i	$-0.272988\pi$
$139$	2144.33	1.30848	0.654242	+	0.756285i	$0.272988\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−84.6692	−0.0495133
$144$	0	0
$145$	734.777	0.420827
$146$	0	0
$147$	0	0
$148$	0	0
$149$	593.265	0.326189	0.163095	−	0.986610i	$-0.447852\pi$
$149$	593.265	0.326189	0.163095	+	0.986610i	$0.447852\pi$
$150$	0	0
$151$	− 506.715i	− 0.273085i	−0.990634	−	0.136543i	$-0.956401\pi$
$151$	− 506.715i	− 0.273085i	0.990634	−	0.136543i	$-0.0435991\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	126.475i	0.0655399i
$156$	0	0
$157$	− 811.801i	− 0.412667i	−0.978482	−	0.206334i	$-0.933847\pi$
$157$	− 811.801i	− 0.412667i	0.978482	−	0.206334i	$-0.0661533\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 2617.92i	− 1.28150i
$162$	0	0
$163$	714.183	0.343185	0.171593	−	0.985168i	$-0.445109\pi$
$163$	714.183	0.343185	0.171593	+	0.985168i	$0.445109\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3332.48	1.54416	0.772080	−	0.635525i	$-0.219216\pi$
$167$	3332.48	1.54416	0.772080	+	0.635525i	$0.219216\pi$
$168$	0	0
$169$	2117.91	0.964000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−427.084	−0.187691	−0.0938456	−	0.995587i	$-0.529916\pi$
$173$	−427.084	−0.187691	−0.0938456	+	0.995587i	$0.529916\pi$
$174$	0	0
$175$	1164.73i	0.503114i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3217.18i	1.34337i	0.740837	+	0.671685i	$0.234429\pi$
$179$	3217.18i	1.34337i	−0.740837	+	0.671685i	$0.765571\pi$
$180$	0	0
$181$	− 1783.70i	− 0.732496i	−0.930517	−	0.366248i	$-0.880642\pi$
$181$	− 1783.70i	− 0.732496i	0.930517	−	0.366248i	$-0.119358\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 661.525i	− 0.262899i
$186$	0	0
$187$	−405.493	−0.158570
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5100.33	1.93218	0.966091	−	0.258203i	$-0.0831303\pi$
$191$	5100.33	1.93218	0.966091	+	0.258203i	$0.0831303\pi$
$192$	0	0
$193$	−2430.38	−0.906440	−0.453220	−	0.891399i	$-0.649725\pi$
$193$	−2430.38	−0.906440	−0.453220	+	0.891399i	$0.649725\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3291.04	1.19024	0.595119	−	0.803637i	$-0.297105\pi$
$197$	3291.04	1.19024	0.595119	+	0.803637i	$0.297105\pi$
$198$	0	0
$199$	− 66.7409i	− 0.0237746i	−0.999929	−	0.0118873i	$-0.996216\pi$
$199$	− 66.7409i	− 0.0237746i	0.999929	−	0.0118873i	$-0.00378393\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1851.09i	0.640004i
$204$	0	0
$205$	− 1372.43i	− 0.467583i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 231.827i	− 0.0767264i
$210$	0	0
$211$	1777.87	0.580066	0.290033	−	0.957017i	$-0.406334\pi$
$211$	1777.87	0.580066	0.290033	+	0.957017i	$0.406334\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	3279.83	1.04038
$216$	0	0
$217$	−318.621	−0.0996745
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−378.781	−0.115292
$222$	0	0
$223$	139.145i	0.0417841i	0.999782	+	0.0208921i	$0.00665063\pi$
$223$	139.145i	0.0417841i	−0.999782	+	0.0208921i	$0.993349\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4440.80i	1.29844i	0.760600	+	0.649221i	$0.224905\pi$
$227$	4440.80i	1.29844i	−0.760600	+	0.649221i	$0.775095\pi$
$228$	0	0
$229$	4173.79i	1.20442i	0.798338	+	0.602209i	$0.205713\pi$
$229$	4173.79i	1.20442i	−0.798338	+	0.602209i	$0.794287\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 3213.92i	− 0.903653i	−0.892106	−	0.451826i	$-0.850773\pi$
$233$	− 3213.92i	− 0.903653i	0.892106	−	0.451826i	$-0.149227\pi$
$234$	0	0
$235$	−3125.49	−0.867593
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5234.01	1.41657	0.708284	−	0.705928i	$-0.249470\pi$
$239$	5234.01	1.41657	0.708284	+	0.705928i	$0.249470\pi$
$240$	0	0
$241$	−2537.35	−0.678195	−0.339097	−	0.940751i	$-0.610122\pi$
$241$	−2537.35	−0.678195	−0.339097	+	0.940751i	$0.610122\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	821.892	0.214321
$246$	0	0
$247$	− 216.556i	− 0.0557859i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1435.00i	0.360863i	0.983588	+	0.180431i	$0.0577493\pi$
$251$	1435.00i	0.360863i	−0.983588	+	0.180431i	$0.942251\pi$
$252$	0	0
$253$	1186.13i	0.294748i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 2950.34i	− 0.716099i	−0.933703	−	0.358049i	$-0.883442\pi$
$257$	− 2950.34i	− 0.716099i	0.933703	−	0.358049i	$-0.116558\pi$
$258$	0	0
$259$	1666.54	0.399823
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3240.53	0.759771	0.379885	−	0.925034i	$-0.375963\pi$
$263$	3240.53	0.759771	0.379885	+	0.925034i	$0.375963\pi$
$264$	0	0
$265$	−751.597	−0.174227
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−456.419	−0.103451	−0.0517255	−	0.998661i	$-0.516472\pi$
$269$	−456.419	−0.103451	−0.0517255	+	0.998661i	$0.516472\pi$
$270$	0	0
$271$	− 2459.31i	− 0.551263i	−0.961263	−	0.275632i	$-0.911113\pi$
$271$	− 2459.31i	− 0.551263i	0.961263	−	0.275632i	$-0.0888870\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 527.715i	− 0.115718i
$276$	0	0
$277$	− 2754.09i	− 0.597391i	−0.954348	−	0.298695i	$-0.903449\pi$
$277$	− 2754.09i	− 0.597391i	0.954348	−	0.298695i	$-0.0965515\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 3166.15i	− 0.672159i	−0.941834	−	0.336080i	$-0.890899\pi$
$281$	− 3166.15i	− 0.672159i	0.941834	−	0.336080i	$-0.109101\pi$
$282$	0	0
$283$	−1576.28	−0.331095	−0.165548	−	0.986202i	$-0.552939\pi$
$283$	−1576.28	−0.331095	−0.165548	+	0.986202i	$0.552939\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3457.49	0.711112
$288$	0	0
$289$	3098.96	0.630768
$290$	0	0
$291$	0	0
$292$	0	0
$293$	8729.65	1.74058	0.870292	−	0.492535i	$-0.163930\pi$
$293$	8729.65	1.74058	0.870292	+	0.492535i	$0.163930\pi$
$294$	0	0
$295$	211.570i	0.0417563i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1107.99i	0.214304i
$300$	0	0
$301$	8262.70i	1.58224i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 5586.73i	− 1.04884i
$306$	0	0
$307$	−6993.96	−1.30022	−0.650108	−	0.759842i	$-0.725276\pi$
$307$	−6993.96	−1.30022	−0.650108	+	0.759842i	$0.725276\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1684.03	0.307050	0.153525	−	0.988145i	$-0.450937\pi$
$311$	1684.03	0.307050	0.153525	+	0.988145i	$0.450937\pi$
$312$	0	0
$313$	7673.11	1.38565	0.692827	−	0.721104i	$-0.256365\pi$
$313$	7673.11	1.38565	0.692827	+	0.721104i	$0.256365\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2739.03	0.485297	0.242648	−	0.970114i	$-0.421984\pi$
$317$	2739.03	0.485297	0.242648	+	0.970114i	$0.421984\pi$
$318$	0	0
$319$	− 838.693i	− 0.147203i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 1037.12i	− 0.178658i
$324$	0	0
$325$	− 492.952i	− 0.0841356i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 7873.87i	− 1.31945i
$330$	0	0
$331$	351.062	0.0582964	0.0291482	−	0.999575i	$-0.490721\pi$
$331$	351.062	0.0582964	0.0291482	+	0.999575i	$0.490721\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1974.01	−0.321945
$336$	0	0
$337$	−9518.32	−1.53856	−0.769282	−	0.638909i	$-0.779386\pi$
$337$	−9518.32	−1.53856	−0.769282	+	0.638909i	$0.779386\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	144.361	0.0229255
$342$	0	0
$343$	− 5136.84i	− 0.808639i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7173.42i	1.10977i	0.831928	+	0.554884i	$0.187237\pi$
$347$	7173.42i	1.10977i	−0.831928	+	0.554884i	$0.812763\pi$
$348$	0	0
$349$	− 4567.04i	− 0.700482i	−0.936660	−	0.350241i	$-0.886100\pi$
$349$	− 4567.04i	− 0.700482i	0.936660	−	0.350241i	$-0.113900\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 11197.0i	− 1.68826i	−0.536139	−	0.844130i	$-0.680118\pi$
$353$	− 11197.0i	− 1.68826i	0.536139	−	0.844130i	$-0.319882\pi$
$354$	0	0
$355$	1068.37	0.159728
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2666.74	0.392048	0.196024	−	0.980599i	$-0.437197\pi$
$359$	2666.74	0.392048	0.196024	+	0.980599i	$0.437197\pi$
$360$	0	0
$361$	−6266.06	−0.913553
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1268.19	−0.181863
$366$	0	0
$367$	5057.41i	0.719332i	0.933081	+	0.359666i	$0.117109\pi$
$367$	5057.41i	0.719332i	−0.933081	+	0.359666i	$0.882891\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 1893.46i	− 0.264969i
$372$	0	0
$373$	7410.03i	1.02862i	0.857603	+	0.514312i	$0.171953\pi$
$373$	7410.03i	1.02862i	−0.857603	+	0.514312i	$0.828047\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 783.444i	− 0.107028i
$378$	0	0
$379$	−3745.08	−0.507578	−0.253789	−	0.967260i	$-0.581677\pi$
$379$	−3745.08	−0.507578	−0.253789	+	0.967260i	$0.581677\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3335.14	−0.444955	−0.222477	−	0.974938i	$-0.571414\pi$
$383$	−3335.14	−0.444955	−0.222477	+	0.974938i	$0.571414\pi$
$384$	0	0
$385$	−1668.62	−0.220885
$386$	0	0
$387$	0	0
$388$	0	0
$389$	13683.2	1.78346	0.891729	−	0.452570i	$-0.149493\pi$
$389$	13683.2	1.78346	0.891729	+	0.452570i	$0.149493\pi$
$390$	0	0
$391$	5306.34i	0.686325i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4384.61i	0.558516i
$396$	0	0
$397$	− 6688.01i	− 0.845495i	−0.906248	−	0.422747i	$-0.861066\pi$
$397$	− 6688.01i	− 0.845495i	0.906248	−	0.422747i	$-0.138934\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	936.356i	0.116607i	0.998299	+	0.0583034i	$0.0185691\pi$
$401$	936.356i	0.116607i	−0.998299	+	0.0583034i	$0.981431\pi$
$402$	0	0
$403$	134.851	0.0166685
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−755.080	−0.0919606
$408$	0	0
$409$	−2176.86	−0.263176	−0.131588	−	0.991305i	$-0.542008\pi$
$409$	−2176.86	−0.263176	−0.131588	+	0.991305i	$0.542008\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−532.998	−0.0635039
$414$	0	0
$415$	− 6353.15i	− 0.751479i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 11402.9i	− 1.32952i	−0.747057	−	0.664760i	$-0.768534\pi$
$419$	− 11402.9i	− 1.32952i	0.747057	−	0.664760i	$-0.231466\pi$
$420$	0	0
$421$	− 11065.0i	− 1.28094i	−0.767982	−	0.640471i	$-0.778739\pi$
$421$	− 11065.0i	− 1.28094i	0.767982	−	0.640471i	$-0.221261\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 2360.82i	− 0.269451i
$426$	0	0
$427$	14074.3	1.59509
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9230.40	1.03158	0.515792	−	0.856714i	$-0.327498\pi$
$431$	9230.40	1.03158	0.515792	+	0.856714i	$0.327498\pi$
$432$	0	0
$433$	16809.1	1.86558	0.932788	−	0.360425i	$-0.117368\pi$
$433$	16809.1	1.86558	0.932788	+	0.360425i	$0.117368\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3033.73	−0.332089
$438$	0	0
$439$	− 7087.15i	− 0.770504i	−0.922811	−	0.385252i	$-0.874115\pi$
$439$	− 7087.15i	− 0.770504i	0.922811	−	0.385252i	$-0.125885\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1035.88i	0.111098i	0.998456	+	0.0555488i	$0.0176908\pi$
$443$	1035.88i	0.111098i	−0.998456	+	0.0555488i	$0.982309\pi$
$444$	0	0
$445$	13237.3i	1.41013i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	10505.0i	1.10414i	0.833797	+	0.552071i	$0.186162\pi$
$449$	10505.0i	1.10414i	−0.833797	+	0.552071i	$0.813838\pi$
$450$	0	0
$451$	−1566.52	−0.163558
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1558.70	−0.160600
$456$	0	0
$457$	5252.21	0.537610	0.268805	−	0.963195i	$-0.413371\pi$
$457$	5252.21	0.537610	0.268805	+	0.963195i	$0.413371\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−12272.1	−1.23985	−0.619924	−	0.784662i	$-0.712837\pi$
$461$	−12272.1	−1.23985	−0.619924	+	0.784662i	$0.712837\pi$
$462$	0	0
$463$	10710.1i	1.07503i	0.843254	+	0.537516i	$0.180637\pi$
$463$	10710.1i	1.07503i	−0.843254	+	0.537516i	$0.819363\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6462.30i	0.640342i	0.947360	+	0.320171i	$0.103740\pi$
$467$	6462.30i	0.640342i	−0.947360	+	0.320171i	$0.896260\pi$
$468$	0	0
$469$	− 4973.02i	− 0.489622i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 3743.68i	− 0.363921i
$474$	0	0
$475$	1349.72	0.130378
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3634.99	0.346737	0.173368	−	0.984857i	$-0.444535\pi$
$479$	3634.99	0.346737	0.173368	+	0.984857i	$0.444535\pi$
$480$	0	0
$481$	−705.340	−0.0668623
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8760.76	0.820218
$486$	0	0
$487$	7438.33i	0.692121i	0.938212	+	0.346061i	$0.112481\pi$
$487$	7438.33i	0.692121i	−0.938212	+	0.346061i	$0.887519\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	13466.3i	1.23773i	0.785498	+	0.618865i	$0.212407\pi$
$491$	13466.3i	1.23773i	−0.785498	+	0.618865i	$0.787593\pi$
$492$	0	0
$493$	− 3752.03i	− 0.342764i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2691.50i	0.242918i
$498$	0	0
$499$	14751.7	1.32340	0.661702	−	0.749767i	$-0.269835\pi$
$499$	14751.7	1.32340	0.661702	+	0.749767i	$0.269835\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	3906.71	0.346306	0.173153	−	0.984895i	$-0.444605\pi$
$503$	3906.71	0.346306	0.173153	+	0.984895i	$0.444605\pi$
$504$	0	0
$505$	−6673.04	−0.588013
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−8809.20	−0.767114	−0.383557	−	0.923517i	$-0.625301\pi$
$509$	−8809.20	−0.767114	−0.383557	+	0.923517i	$0.625301\pi$
$510$	0	0
$511$	− 3194.88i	− 0.276581i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 6579.60i	− 0.562974i
$516$	0	0
$517$	3567.50i	0.303479i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	18473.2i	1.55340i	0.629868	+	0.776702i	$0.283109\pi$
$521$	18473.2i	1.55340i	−0.629868	+	0.776702i	$0.716891\pi$
$522$	0	0
$523$	−8529.50	−0.713133	−0.356567	−	0.934270i	$-0.616053\pi$
$523$	−8529.50	−0.713133	−0.356567	+	0.934270i	$0.616053\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	645.822	0.0533823
$528$	0	0
$529$	3354.86	0.275735
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1463.33	−0.118919
$534$	0	0
$535$	11412.2i	0.922229i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 938.127i	− 0.0749685i
$540$	0	0
$541$	− 18895.8i	− 1.50165i	−0.660501	−	0.750825i	$-0.729656\pi$
$541$	− 18895.8i	− 1.50165i	0.660501	−	0.750825i	$-0.270344\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7949.09i	0.624774i
$546$	0	0
$547$	−13183.5	−1.03050	−0.515251	−	0.857039i	$-0.672301\pi$
$547$	−13183.5	−1.03050	−0.515251	+	0.857039i	$0.672301\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2145.10	0.165852
$552$	0	0
$553$	−11045.9	−0.849403
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−24772.1	−1.88443	−0.942213	−	0.335014i	$-0.891259\pi$
$557$	−24772.1	−1.88443	−0.942213	+	0.335014i	$0.891259\pi$
$558$	0	0
$559$	− 3497.07i	− 0.264598i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 12773.3i	− 0.956184i	−0.878310	−	0.478092i	$-0.841329\pi$
$563$	− 12773.3i	− 0.956184i	0.878310	−	0.478092i	$-0.158671\pi$
$564$	0	0
$565$	12325.8i	0.917788i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	6082.50i	0.448140i	0.974573	+	0.224070i	$0.0719344\pi$
$569$	6082.50i	0.448140i	−0.974573	+	0.224070i	$0.928066\pi$
$570$	0	0
$571$	20646.2	1.51316	0.756581	−	0.653899i	$-0.226868\pi$
$571$	20646.2	1.51316	0.756581	+	0.653899i	$0.226868\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6905.76	−0.500852
$576$	0	0
$577$	16942.5	1.22240	0.611202	−	0.791475i	$-0.290686\pi$
$577$	16942.5	1.22240	0.611202	+	0.791475i	$0.290686\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	16005.2	1.14287
$582$	0	0
$583$	857.891i	0.0609438i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 2780.60i	− 0.195516i	−0.995210	−	0.0977578i	$-0.968833\pi$
$587$	− 2780.60i	− 0.195516i	0.995210	−	0.0977578i	$-0.0311670\pi$
$588$	0	0
$589$	369.228i	0.0258298i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	86.9257i	0.00601958i	0.999995	+	0.00300979i	$0.000958047\pi$
$593$	86.9257i	0.00601958i	−0.999995	+	0.00300979i	$0.999042\pi$
$594$	0	0
$595$	−7464.83	−0.514333
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−5769.53	−0.393550	−0.196775	−	0.980449i	$-0.563047\pi$
$599$	−5769.53	−0.393550	−0.196775	+	0.980449i	$0.563047\pi$
$600$	0	0
$601$	−2807.07	−0.190520	−0.0952601	−	0.995452i	$-0.530368\pi$
$601$	−2807.07	−0.190520	−0.0952601	+	0.995452i	$0.530368\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−10345.7	−0.695229
$606$	0	0
$607$	− 4199.36i	− 0.280802i	−0.990095	−	0.140401i	$-0.955161\pi$
$607$	− 4199.36i	− 0.280802i	0.990095	−	0.140401i	$-0.0448392\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3332.50i	0.220652i
$612$	0	0
$613$	− 20700.8i	− 1.36394i	−0.731378	−	0.681972i	$-0.761123\pi$
$613$	− 20700.8i	− 1.36394i	0.731378	−	0.681972i	$-0.238877\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11915.5i	0.777472i	0.921349	+	0.388736i	$0.127088\pi$
$617$	11915.5i	0.777472i	−0.921349	+	0.388736i	$0.872912\pi$
$618$	0	0
$619$	−23887.2	−1.55106	−0.775531	−	0.631310i	$-0.782518\pi$
$619$	−23887.2	−1.55106	−0.775531	+	0.631310i	$0.782518\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−33347.9	−2.14455
$624$	0	0
$625$	−5623.92	−0.359931
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3377.97	−0.214131
$630$	0	0
$631$	− 16604.4i	− 1.04756i	−0.851853	−	0.523781i	$-0.824521\pi$
$631$	− 16604.4i	− 1.04756i	0.851853	−	0.523781i	$-0.175479\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 3424.73i	− 0.214026i
$636$	0	0
$637$	− 876.329i	− 0.0545077i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	19290.4i	1.18865i	0.804225	+	0.594325i	$0.202581\pi$
$641$	19290.4i	1.18865i	−0.804225	+	0.594325i	$0.797419\pi$
$642$	0	0
$643$	31649.9	1.94114	0.970569	−	0.240825i	$-0.0774178\pi$
$643$	31649.9	1.94114	0.970569	+	0.240825i	$0.0774178\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−11446.1	−0.695508	−0.347754	−	0.937586i	$-0.613056\pi$
$647$	−11446.1	−0.695508	−0.347754	+	0.937586i	$0.613056\pi$
$648$	0	0
$649$	241.491	0.0146061
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−13206.8	−0.791455	−0.395728	−	0.918368i	$-0.629508\pi$
$653$	−13206.8	−0.791455	−0.395728	+	0.918368i	$0.629508\pi$
$654$	0	0
$655$	− 13340.8i	− 0.795830i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1855.68i	0.109692i	0.998495	+	0.0548461i	$0.0174668\pi$
$659$	1855.68i	0.109692i	−0.998495	+	0.0548461i	$0.982533\pi$
$660$	0	0
$661$	26168.4i	1.53983i	0.638144	+	0.769917i	$0.279703\pi$
$661$	26168.4i	1.53983i	−0.638144	+	0.769917i	$0.720297\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 4267.77i	− 0.248868i
$666$	0	0
$667$	−10975.3	−0.637127
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6376.82	−0.366877
$672$	0	0
$673$	−7781.63	−0.445706	−0.222853	−	0.974852i	$-0.571537\pi$
$673$	−7781.63	−0.445706	−0.222853	+	0.974852i	$0.571537\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	25165.8	1.42866	0.714328	−	0.699811i	$-0.246733\pi$
$677$	25165.8	1.42866	0.714328	+	0.699811i	$0.246733\pi$
$678$	0	0
$679$	22070.5i	1.24741i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 33273.8i	− 1.86411i	−0.362321	−	0.932054i	$-0.618015\pi$
$683$	− 33273.8i	− 1.86411i	0.362321	−	0.932054i	$-0.381985\pi$
$684$	0	0
$685$	8875.80i	0.495076i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	801.378i	0.0443107i
$690$	0	0
$691$	−6250.73	−0.344123	−0.172062	−	0.985086i	$-0.555043\pi$
$691$	−6250.73	−0.344123	−0.172062	+	0.985086i	$0.555043\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−17885.6	−0.976173
$696$	0	0
$697$	−7008.09	−0.380847
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−13370.0	−0.720366	−0.360183	−	0.932882i	$-0.617286\pi$
$701$	−13370.0	−0.720366	−0.360183	+	0.932882i	$0.617286\pi$
$702$	0	0
$703$	− 1931.25i	− 0.103611i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 16811.0i	− 0.894264i
$708$	0	0
$709$	11341.8i	0.600777i	0.953817	+	0.300389i	$0.0971164\pi$
$709$	11341.8i	0.600777i	−0.953817	+	0.300389i	$0.902884\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 1889.13i	− 0.0992265i
$714$	0	0
$715$	706.218	0.0369385
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−12962.9	−0.672370	−0.336185	−	0.941796i	$-0.609137\pi$
$719$	−12962.9	−0.672370	−0.336185	+	0.941796i	$0.609137\pi$
$720$	0	0
$721$	16575.6	0.856184
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4882.95	0.250135
$726$	0	0
$727$	− 33452.0i	− 1.70656i	−0.521457	−	0.853278i	$-0.674611\pi$
$727$	− 33452.0i	− 1.70656i	0.521457	−	0.853278i	$-0.325389\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 16747.9i	− 0.847394i
$732$	0	0
$733$	21088.6i	1.06265i	0.847167	+	0.531326i	$0.178306\pi$
$733$	21088.6i	1.06265i	−0.847167	+	0.531326i	$0.821694\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2253.18i	0.112615i
$738$	0	0
$739$	−27140.6	−1.35099	−0.675497	−	0.737363i	$-0.736071\pi$
$739$	−27140.6	−1.35099	−0.675497	+	0.737363i	$0.736071\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−35192.4	−1.73766	−0.868832	−	0.495106i	$-0.835129\pi$
$743$	−35192.4	−1.73766	−0.868832	+	0.495106i	$0.835129\pi$
$744$	0	0
$745$	−4948.37	−0.243348
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−28750.2	−1.40255
$750$	0	0
$751$	29014.9i	1.40981i	0.709300	+	0.704907i	$0.249011\pi$
$751$	29014.9i	1.40981i	−0.709300	+	0.704907i	$0.750989\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	4226.46i	0.203731i
$756$	0	0
$757$	− 20465.9i	− 0.982621i	−0.870984	−	0.491311i	$-0.836518\pi$
$757$	− 20465.9i	− 0.982621i	0.870984	−	0.491311i	$-0.163482\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	28142.8i	1.34057i	0.742102	+	0.670287i	$0.233829\pi$
$761$	28142.8i	1.34057i	−0.742102	+	0.670287i	$0.766171\pi$
$762$	0	0
$763$	−20025.7	−0.950170
$764$	0	0
$765$	0	0
$766$	0	0
$767$	225.583	0.0106197
$768$	0	0
$769$	−14678.3	−0.688313	−0.344156	−	0.938912i	$-0.611835\pi$
$769$	−14678.3	−0.688313	−0.344156	+	0.938912i	$0.611835\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	28245.5	1.31426	0.657129	−	0.753778i	$-0.271771\pi$
$773$	28245.5	1.31426	0.657129	+	0.753778i	$0.271771\pi$
$774$	0	0
$775$	840.484i	0.0389562i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 4006.65i	− 0.184279i
$780$	0	0
$781$	− 1219.47i	− 0.0558720i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	6771.16i	0.307864i
$786$	0	0
$787$	20062.3	0.908698	0.454349	−	0.890824i	$-0.349872\pi$
$787$	20062.3	0.908698	0.454349	+	0.890824i	$0.349872\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−31051.7	−1.39579
$792$	0	0
$793$	−5956.76	−0.266747
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−27596.3	−1.22649	−0.613244	−	0.789893i	$-0.710136\pi$
$797$	−27596.3	−1.22649	−0.613244	+	0.789893i	$0.710136\pi$
$798$	0	0
$799$	15959.8i	0.706655i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1447.54i	0.0636146i
$804$	0	0
$805$	21835.8i	0.956038i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2338.39i	0.101623i	0.998708	+	0.0508117i	$0.0161808\pi$
$809$	2338.39i	0.101623i	−0.998708	+	0.0508117i	$0.983819\pi$
$810$	0	0
$811$	4326.82	0.187343	0.0936715	−	0.995603i	$-0.470140\pi$
$811$	4326.82	0.187343	0.0936715	+	0.995603i	$0.470140\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−5956.94	−0.256027
$816$	0	0
$817$	9575.09	0.410024
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−29931.7	−1.27238	−0.636190	−	0.771532i	$-0.719491\pi$
$821$	−29931.7	−1.27238	−0.636190	+	0.771532i	$0.719491\pi$
$822$	0	0
$823$	9285.33i	0.393276i	0.980476	+	0.196638i	$0.0630024\pi$
$823$	9285.33i	0.393276i	−0.980476	+	0.196638i	$0.936998\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 17292.6i	− 0.727113i	−0.931572	−	0.363557i	$-0.881562\pi$
$827$	− 17292.6i	− 0.727113i	0.931572	−	0.363557i	$-0.118438\pi$
$828$	0	0
$829$	− 18165.0i	− 0.761033i	−0.924774	−	0.380516i	$-0.875746\pi$
$829$	− 18165.0i	− 0.761033i	0.924774	−	0.380516i	$-0.124254\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 4196.86i	− 0.174565i
$834$	0	0
$835$	−27795.9	−1.15200
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−3851.73	−0.158494	−0.0792469	−	0.996855i	$-0.525252\pi$
$839$	−3851.73	−0.158494	−0.0792469	+	0.996855i	$0.525252\pi$
$840$	0	0
$841$	−16628.6	−0.681806
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−17665.3	−0.719176
$846$	0	0
$847$	− 26063.4i	− 1.05732i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	9881.10i	0.398025i
$852$	0	0
$853$	− 38776.8i	− 1.55650i	−0.627956	−	0.778249i	$-0.716108\pi$
$853$	− 38776.8i	− 1.55650i	0.627956	−	0.778249i	$-0.283892\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	43562.3i	1.73636i	0.496251	+	0.868179i	$0.334709\pi$
$857$	43562.3i	1.73636i	−0.496251	+	0.868179i	$0.665291\pi$
$858$	0	0
$859$	44787.5	1.77896	0.889481	−	0.456972i	$-0.151066\pi$
$859$	44787.5	1.77896	0.889481	+	0.456972i	$0.151066\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−35822.5	−1.41299	−0.706496	−	0.707717i	$-0.749725\pi$
$863$	−35822.5	−1.41299	−0.706496	+	0.707717i	$0.749725\pi$
$864$	0	0
$865$	3562.26	0.140024
$866$	0	0
$867$	0	0
$868$	0	0
$869$	5004.70	0.195366
$870$	0	0
$871$	2104.76i	0.0818793i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 31623.1i	− 1.22178i
$876$	0	0
$877$	41341.0i	1.59178i	0.605444	+	0.795888i	$0.292995\pi$
$877$	41341.0i	1.59178i	−0.605444	+	0.795888i	$0.707005\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	39795.2i	1.52183i	0.648849	+	0.760917i	$0.275250\pi$
$881$	39795.2i	1.52183i	−0.648849	+	0.760917i	$0.724750\pi$
$882$	0	0
$883$	−8467.92	−0.322727	−0.161364	−	0.986895i	$-0.551589\pi$
$883$	−8467.92	−0.322727	−0.161364	+	0.986895i	$0.551589\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	24074.8	0.911333	0.455667	−	0.890151i	$-0.349401\pi$
$887$	24074.8	0.911333	0.455667	+	0.890151i	$0.349401\pi$
$888$	0	0
$889$	8627.74	0.325495
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−9124.50	−0.341926
$894$	0	0
$895$	− 26834.2i	− 1.00220i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1335.77i	0.0495556i
$900$	0	0
$901$	3837.91i	0.141908i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	14877.7i	0.546466i
$906$	0	0
$907$	39195.4	1.43491	0.717454	−	0.696606i	$-0.245307\pi$
$907$	39195.4	1.43491	0.717454	+	0.696606i	$0.245307\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	31378.8	1.14119	0.570596	−	0.821231i	$-0.306712\pi$
$911$	31378.8	1.14119	0.570596	+	0.821231i	$0.306712\pi$
$912$	0	0
$913$	−7251.64	−0.262863
$914$	0	0
$915$	0	0
$916$	0	0
$917$	33608.8	1.21032
$918$	0	0
$919$	− 19895.4i	− 0.714133i	−0.934079	−	0.357067i	$-0.883777\pi$
$919$	− 19895.4i	− 0.714133i	0.934079	−	0.357067i	$-0.116223\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 1139.14i	− 0.0406231i
$924$	0	0
$925$	− 4396.15i	− 0.156264i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	27091.5i	0.956775i	0.878149	+	0.478388i	$0.158779\pi$
$929$	27091.5i	0.956775i	−0.878149	+	0.478388i	$0.841221\pi$
$930$	0	0
$931$	2399.42	0.0844659
$932$	0	0
$933$	0	0
$934$	0	0
$935$	3382.18	0.118298
$936$	0	0
$937$	41248.6	1.43814	0.719069	−	0.694939i	$-0.244569\pi$
$937$	41248.6	1.43814	0.719069	+	0.694939i	$0.244569\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−53921.1	−1.86799	−0.933994	−	0.357288i	$-0.883701\pi$
$941$	−53921.1	−1.86799	−0.933994	+	0.357288i	$0.883701\pi$
$942$	0	0
$943$	20499.8i	0.707915i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	33948.7i	1.16492i	0.812858	+	0.582462i	$0.197911\pi$
$947$	33948.7i	1.16492i	−0.812858	+	0.582462i	$0.802089\pi$
$948$	0	0
$949$	1352.18i	0.0462526i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 14737.7i	− 0.500946i	−0.968124	−	0.250473i	$-0.919414\pi$
$953$	− 14737.7i	− 0.500946i	0.968124	−	0.250473i	$-0.0805861\pi$
$954$	0	0
$955$	−42541.3	−1.44147
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−22360.3	−0.752923
$960$	0	0
$961$	29561.1	0.992282
$962$	0	0
$963$	0	0
$964$	0	0
$965$	20271.6	0.676234
$966$	0	0
$967$	33349.1i	1.10903i	0.832173	+	0.554516i	$0.187096\pi$
$967$	33349.1i	1.10903i	−0.832173	+	0.554516i	$0.812904\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 10903.8i	− 0.360370i	−0.983633	−	0.180185i	$-0.942330\pi$
$971$	− 10903.8i	− 0.360370i	0.983633	−	0.180185i	$-0.0576697\pi$
$972$	0	0
$973$	− 45058.3i	− 1.48459i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	27911.7i	0.913998i	0.889467	+	0.456999i	$0.151076\pi$
$977$	27911.7i	0.913998i	−0.889467	+	0.456999i	$0.848924\pi$
$978$	0	0
$979$	15109.3	0.493254
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2039.07	0.0661610	0.0330805	−	0.999453i	$-0.489468\pi$
$983$	2039.07	0.0661610	0.0330805	+	0.999453i	$0.489468\pi$
$984$	0	0
$985$	−27450.3	−0.887958
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−48990.3	−1.57513
$990$	0	0
$991$	8879.39i	0.284625i	0.989822	+	0.142312i	$0.0454538\pi$
$991$	8879.39i	0.284625i	−0.989822	+	0.142312i	$0.954546\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	556.680i	0.0177366i
$996$	0	0
$997$	− 32084.2i	− 1.01917i	−0.860419	−	0.509587i	$-0.829798\pi$
$997$	− 32084.2i	− 1.01917i	0.860419	−	0.509587i	$-0.170202\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.4.f.h.863.5	✓	16
3.2	odd	2		1728.4.f.j.863.11	yes	16
4.3	odd	2		1728.4.f.j.863.6	yes	16
8.3	odd	2		1728.4.f.j.863.12	yes	16
8.5	even	2	inner	1728.4.f.h.863.11	yes	16
12.11	even	2	inner	1728.4.f.h.863.12	yes	16
24.5	odd	2		1728.4.f.j.863.5	yes	16
24.11	even	2	inner	1728.4.f.h.863.6	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.4.f.h.863.5	✓	16	1.1	even	1	trivial
1728.4.f.h.863.6	yes	16	24.11	even	2	inner
1728.4.f.h.863.11	yes	16	8.5	even	2	inner
1728.4.f.h.863.12	yes	16	12.11	even	2	inner
1728.4.f.j.863.5	yes	16	24.5	odd	2
1728.4.f.j.863.6	yes	16	4.3	odd	2
1728.4.f.j.863.11	yes	16	3.2	odd	2
1728.4.f.j.863.12	yes	16	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1728.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-8.34090 q^{5} -21.0128i q^{7} +9.52051i q^{11} +8.89335i q^{13} +42.5915i q^{17} -24.3503 q^{19} +124.587 q^{23} -55.4293 q^{25} -88.0933 q^{29} -15.1632i q^{31} +175.266i q^{35} +79.3110i q^{37} +164.542i q^{41} -393.223 q^{43} +374.718 q^{47} -98.5375 q^{49} +90.1098 q^{53} -79.4096i q^{55} -25.3654i q^{59} +669.799i q^{61} -74.1786i q^{65} +236.666 q^{67} -128.089 q^{71} +152.044 q^{73} +200.052 q^{77} -525.676i q^{79} +761.686i q^{83} -355.252i q^{85} -1587.03i q^{89} +186.874 q^{91} +203.104 q^{95} -1050.34 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 384 q^{23} + 304 q^{25} + 1248 q^{47} - 720 q^{49} + 5088 q^{71} - 128 q^{73} + 11712 q^{95} + 4592 q^{97}+O(q^{100}) \) 16 * q - 384 * q^23 + 304 * q^25 + 1248 * q^47 - 720 * q^49 + 5088 * q^71 - 128 * q^73 + 11712 * q^95 + 4592 * q^97

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