LMFDB - Embedded newform 1728.2.c.g.1727.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1728,2,Mod(1727,1728)] 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("1728.1727"); S:= CuspForms(chi, 2); N := Newforms(S);

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, 2, names="a")

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

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,16] 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:	$13.7981494693$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 864)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1727.3
Root			$-0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1728.1727
Dual form			1728.2.c.g.1727.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.317837i q^{5} -1.44949i q^{7} +1.09638 q^{11} -2.89898 q^{13} -3.46410i q^{17} +4.89898i q^{19} +2.82843 q^{23} +4.89898 q^{25} -9.12096i q^{29} -7.44949i q^{31} -0.460702 q^{35} -4.89898 q^{37} -9.75663i q^{41} +6.89898i q^{43} -9.12096 q^{47} +4.89898 q^{49} -4.41761i q^{53} -0.348469i q^{55} -9.12096 q^{59} -4.00000 q^{61} +0.921404i q^{65} -5.10102i q^{67} +7.56388 q^{71} +1.89898 q^{73} -1.58919i q^{77} +2.00000i q^{79} +15.8742 q^{83} -1.10102 q^{85} -11.9494i q^{89} +4.20204i q^{91} +1.55708 q^{95} -5.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{13} - 32 q^{61} - 24 q^{73} - 48 q^{85} - 40 q^{97}+O(q^{100}) $ 8 * q + 16 * q^13 - 32 * q^61 - 24 * q^73 - 48 * q^85 - 40 * 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
$5$	− 0.317837i	− 0.142141i	−0.997471	−	0.0710706i	$-0.977358\pi$
$5$	− 0.317837i	− 0.142141i	0.997471	−	0.0710706i	$-0.0226416\pi$
$6$	0	0
$7$	− 1.44949i	− 0.547856i	−0.961750	−	0.273928i	$-0.911677\pi$
$7$	− 1.44949i	− 0.547856i	0.961750	−	0.273928i	$-0.0883229\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.09638	0.330570	0.165285	−	0.986246i	$-0.447146\pi$
$11$	1.09638	0.330570	0.165285	+	0.986246i	$0.447146\pi$
$12$	0	0
$13$	−2.89898	−0.804032	−0.402016	−	0.915633i	$-0.631690\pi$
$13$	−2.89898	−0.804032	−0.402016	+	0.915633i	$0.631690\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.46410i	− 0.840168i	−0.907485	−	0.420084i	$-0.862001\pi$
$17$	− 3.46410i	− 0.840168i	0.907485	−	0.420084i	$-0.137999\pi$
$18$	0	0
$19$	4.89898i	1.12390i	0.827170	+	0.561951i	$0.189949\pi$
$19$	4.89898i	1.12390i	−0.827170	+	0.561951i	$0.810051\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.82843	0.589768	0.294884	−	0.955533i	$-0.404719\pi$
$23$	2.82843	0.589768	0.294884	+	0.955533i	$0.404719\pi$
$24$	0	0
$25$	4.89898	0.979796
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 9.12096i	− 1.69372i	−0.531817	−	0.846859i	$-0.678490\pi$
$29$	− 9.12096i	− 1.69372i	0.531817	−	0.846859i	$-0.321510\pi$
$30$	0	0
$31$	− 7.44949i	− 1.33797i	−0.743277	−	0.668984i	$-0.766729\pi$
$31$	− 7.44949i	− 1.33797i	0.743277	−	0.668984i	$-0.233271\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.460702	−0.0778728
$36$	0	0
$37$	−4.89898	−0.805387	−0.402694	−	0.915335i	$-0.631926\pi$
$37$	−4.89898	−0.805387	−0.402694	+	0.915335i	$0.631926\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 9.75663i	− 1.52373i	−0.647736	−	0.761865i	$-0.724284\pi$
$41$	− 9.75663i	− 1.52373i	0.647736	−	0.761865i	$-0.275716\pi$
$42$	0	0
$43$	6.89898i	1.05208i	0.850458	+	0.526042i	$0.176325\pi$
$43$	6.89898i	1.05208i	−0.850458	+	0.526042i	$0.823675\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.12096	−1.33043	−0.665214	−	0.746653i	$-0.731660\pi$
$47$	−9.12096	−1.33043	−0.665214	+	0.746653i	$0.731660\pi$
$48$	0	0
$49$	4.89898	0.699854
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 4.41761i	− 0.606806i	−0.952862	−	0.303403i	$-0.901877\pi$
$53$	− 4.41761i	− 0.606806i	0.952862	−	0.303403i	$-0.0981228\pi$
$54$	0	0
$55$	− 0.348469i	− 0.0469876i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−9.12096	−1.18745	−0.593724	−	0.804669i	$-0.702343\pi$
$59$	−9.12096	−1.18745	−0.593724	+	0.804669i	$0.702343\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.921404i	0.114286i
$66$	0	0
$67$	− 5.10102i	− 0.623189i	−0.950215	−	0.311594i	$-0.899137\pi$
$67$	− 5.10102i	− 0.623189i	0.950215	−	0.311594i	$-0.100863\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	7.56388	0.897667	0.448834	−	0.893615i	$-0.351840\pi$
$71$	7.56388	0.897667	0.448834	+	0.893615i	$0.351840\pi$
$72$	0	0
$73$	1.89898	0.222259	0.111129	−	0.993806i	$-0.464553\pi$
$73$	1.89898	0.222259	0.111129	+	0.993806i	$0.464553\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.58919i	− 0.181105i
$78$	0	0
$79$	2.00000i	0.225018i	0.993651	+	0.112509i	$0.0358886\pi$
$79$	2.00000i	0.225018i	−0.993651	+	0.112509i	$0.964111\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	15.8742	1.74242	0.871209	−	0.490912i	$-0.163336\pi$
$83$	15.8742	1.74242	0.871209	+	0.490912i	$0.163336\pi$
$84$	0	0
$85$	−1.10102	−0.119422
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 11.9494i	− 1.26663i	−0.773893	−	0.633316i	$-0.781693\pi$
$89$	− 11.9494i	− 1.26663i	0.773893	−	0.633316i	$-0.218307\pi$
$90$	0	0
$91$	4.20204i	0.440494i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.55708	0.159753
$96$	0	0
$97$	−5.00000	−0.507673	−0.253837	−	0.967247i	$-0.581693\pi$
$97$	−5.00000	−0.507673	−0.253837	+	0.967247i	$0.581693\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 15.7313i	− 1.56533i	−0.622446	−	0.782663i	$-0.713861\pi$
$101$	− 15.7313i	− 1.56533i	0.622446	−	0.782663i	$-0.286139\pi$
$102$	0	0
$103$	− 10.0000i	− 0.985329i	−0.870219	−	0.492665i	$-0.836023\pi$
$103$	− 10.0000i	− 0.985329i	0.870219	−	0.492665i	$-0.163977\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.66025	−0.837218	−0.418609	−	0.908166i	$-0.637482\pi$
$107$	−8.66025	−0.837218	−0.418609	+	0.908166i	$0.637482\pi$
$108$	0	0
$109$	4.89898	0.469237	0.234619	−	0.972088i	$-0.424616\pi$
$109$	4.89898	0.469237	0.234619	+	0.972088i	$0.424616\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 14.1421i	− 1.33038i	−0.746674	−	0.665190i	$-0.768350\pi$
$113$	− 14.1421i	− 1.33038i	0.746674	−	0.665190i	$-0.231650\pi$
$114$	0	0
$115$	− 0.898979i	− 0.0838303i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.02118	−0.460291
$120$	0	0
$121$	−9.79796	−0.890724
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 3.14626i	− 0.281410i
$126$	0	0
$127$	15.2474i	1.35299i	0.736446	+	0.676496i	$0.236502\pi$
$127$	15.2474i	1.35299i	−0.736446	+	0.676496i	$0.763498\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3.00340	0.262408	0.131204	−	0.991355i	$-0.458116\pi$
$131$	3.00340	0.262408	0.131204	+	0.991355i	$0.458116\pi$
$132$	0	0
$133$	7.10102	0.615737
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.6780i	0.912286i	0.889906	+	0.456143i	$0.150769\pi$
$137$	10.6780i	0.912286i	−0.889906	+	0.456143i	$0.849231\pi$
$138$	0	0
$139$	− 20.0000i	− 1.69638i	−0.529694	−	0.848189i	$-0.677693\pi$
$139$	− 20.0000i	− 1.69638i	0.529694	−	0.848189i	$-0.322307\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.17837	−0.265789
$144$	0	0
$145$	−2.89898	−0.240747
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.78194i	0.309829i	0.987928	+	0.154914i	$0.0495101\pi$
$149$	3.78194i	0.309829i	−0.987928	+	0.154914i	$0.950490\pi$
$150$	0	0
$151$	17.2474i	1.40358i	0.712385	+	0.701789i	$0.247615\pi$
$151$	17.2474i	1.40358i	−0.712385	+	0.701789i	$0.752385\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.36773	−0.190180
$156$	0	0
$157$	13.7980	1.10120	0.550599	−	0.834770i	$-0.314400\pi$
$157$	13.7980	1.10120	0.550599	+	0.834770i	$0.314400\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 4.09978i	− 0.323108i
$162$	0	0
$163$	6.00000i	0.469956i	0.972001	+	0.234978i	$0.0755019\pi$
$163$	6.00000i	0.469956i	−0.972001	+	0.234978i	$0.924498\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	22.9774	1.77804	0.889021	−	0.457867i	$-0.151386\pi$
$167$	22.9774	1.77804	0.889021	+	0.457867i	$0.151386\pi$
$168$	0	0
$169$	−4.59592	−0.353532
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 5.33902i	− 0.405918i	−0.979187	−	0.202959i	$-0.934944\pi$
$173$	− 5.33902i	− 0.405918i	0.979187	−	0.202959i	$-0.0650558\pi$
$174$	0	0
$175$	− 7.10102i	− 0.536787i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	8.66025	0.647298	0.323649	−	0.946177i	$-0.395090\pi$
$179$	8.66025	0.647298	0.323649	+	0.946177i	$0.395090\pi$
$180$	0	0
$181$	9.79796	0.728277	0.364138	−	0.931345i	$-0.381364\pi$
$181$	9.79796	0.728277	0.364138	+	0.931345i	$0.381364\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.55708i	0.114479i
$186$	0	0
$187$	− 3.79796i	− 0.277734i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−17.6062	−1.27394	−0.636971	−	0.770888i	$-0.719813\pi$
$191$	−17.6062	−1.27394	−0.636971	+	0.770888i	$0.719813\pi$
$192$	0	0
$193$	2.10102	0.151235	0.0756174	−	0.997137i	$-0.475907\pi$
$193$	2.10102	0.151235	0.0756174	+	0.997137i	$0.475907\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.2672i	0.874003i	0.899461	+	0.437002i	$0.143960\pi$
$197$	12.2672i	0.874003i	−0.899461	+	0.437002i	$0.856040\pi$
$198$	0	0
$199$	20.3485i	1.44246i	0.692693	+	0.721232i	$0.256424\pi$
$199$	20.3485i	1.44246i	−0.692693	+	0.721232i	$0.743576\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−13.2207	−0.927913
$204$	0	0
$205$	−3.10102	−0.216585
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.37113i	0.371528i
$210$	0	0
$211$	− 28.4949i	− 1.96167i	−0.194841	−	0.980835i	$-0.562419\pi$
$211$	− 28.4949i	− 1.96167i	0.194841	−	0.980835i	$-0.437581\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.19275	0.149544
$216$	0	0
$217$	−10.7980	−0.733013
$218$	0	0
$219$	0	0
$220$	0	0
$221$	10.0424i	0.675522i
$222$	0	0
$223$	0.202041i	0.0135297i	0.999977	+	0.00676483i	$0.00215333\pi$
$223$	0.202041i	0.0135297i	−0.999977	+	0.00676483i	$0.997847\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.6637	0.774144	0.387072	−	0.922050i	$-0.373487\pi$
$227$	11.6637	0.774144	0.387072	+	0.922050i	$0.373487\pi$
$228$	0	0
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	13.2207i	0.866119i	0.901365	+	0.433059i	$0.142566\pi$
$233$	13.2207i	0.866119i	−0.901365	+	0.433059i	$0.857434\pi$
$234$	0	0
$235$	2.89898i	0.189109i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−21.0703	−1.36293	−0.681463	−	0.731852i	$-0.738656\pi$
$239$	−21.0703	−1.36293	−0.681463	+	0.731852i	$0.738656\pi$
$240$	0	0
$241$	−19.7980	−1.27530	−0.637649	−	0.770327i	$-0.720093\pi$
$241$	−19.7980	−1.27530	−0.637649	+	0.770327i	$0.720093\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1.55708i	− 0.0994781i
$246$	0	0
$247$	− 14.2020i	− 0.903654i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−17.3205	−1.09326	−0.546630	−	0.837374i	$-0.684090\pi$
$251$	−17.3205	−1.09326	−0.546630	+	0.837374i	$0.684090\pi$
$252$	0	0
$253$	3.10102	0.194959
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.6637i	0.727559i	0.931485	+	0.363779i	$0.118514\pi$
$257$	11.6637i	0.727559i	−0.931485	+	0.363779i	$0.881486\pi$
$258$	0	0
$259$	7.10102i	0.441236i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	24.8844	1.53444	0.767218	−	0.641386i	$-0.221640\pi$
$263$	24.8844	1.53444	0.767218	+	0.641386i	$0.221640\pi$
$264$	0	0
$265$	−1.40408	−0.0862521
$266$	0	0
$267$	0	0
$268$	0	0
$269$	10.3923i	0.633630i	0.948487	+	0.316815i	$0.102613\pi$
$269$	10.3923i	0.633630i	−0.948487	+	0.316815i	$0.897387\pi$
$270$	0	0
$271$	12.5505i	0.762389i	0.924495	+	0.381195i	$0.124487\pi$
$271$	12.5505i	0.762389i	−0.924495	+	0.381195i	$0.875513\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.37113	0.323891
$276$	0	0
$277$	13.7980	0.829039	0.414520	−	0.910040i	$-0.363950\pi$
$277$	13.7980	0.829039	0.414520	+	0.910040i	$0.363950\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 0.921404i	− 0.0549663i	−0.999622	−	0.0274832i	$-0.991251\pi$
$281$	− 0.921404i	− 0.0549663i	0.999622	−	0.0274832i	$-0.00874927\pi$
$282$	0	0
$283$	− 3.10102i	− 0.184337i	−0.995743	−	0.0921683i	$-0.970620\pi$
$283$	− 3.10102i	− 0.184337i	0.995743	−	0.0921683i	$-0.0293798\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−14.1421	−0.834784
$288$	0	0
$289$	5.00000	0.294118
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6.00680i	0.350921i	0.984486	+	0.175460i	$0.0561414\pi$
$293$	6.00680i	0.350921i	−0.984486	+	0.175460i	$0.943859\pi$
$294$	0	0
$295$	2.89898i	0.168785i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−8.19955	−0.474192
$300$	0	0
$301$	10.0000	0.576390
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.27135i	0.0727972i
$306$	0	0
$307$	18.0000i	1.02731i	0.857996	+	0.513657i	$0.171710\pi$
$307$	18.0000i	1.02731i	−0.857996	+	0.513657i	$0.828290\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4.09978	−0.232477	−0.116238	−	0.993221i	$-0.537084\pi$
$311$	−4.09978	−0.232477	−0.116238	+	0.993221i	$0.537084\pi$
$312$	0	0
$313$	29.4949	1.66715	0.833575	−	0.552406i	$-0.186290\pi$
$313$	29.4949	1.66715	0.833575	+	0.552406i	$0.186290\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	29.5877i	1.66181i	0.556413	+	0.830906i	$0.312177\pi$
$317$	29.5877i	1.66181i	−0.556413	+	0.830906i	$0.687823\pi$
$318$	0	0
$319$	− 10.0000i	− 0.559893i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	16.9706	0.944267
$324$	0	0
$325$	−14.2020	−0.787787
$326$	0	0
$327$	0	0
$328$	0	0
$329$	13.2207i	0.728883i
$330$	0	0
$331$	0.696938i	0.0383072i	0.999817	+	0.0191536i	$0.00609715\pi$
$331$	0.696938i	0.0383072i	−0.999817	+	0.0191536i	$0.993903\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.62129	−0.0885808
$336$	0	0
$337$	4.20204	0.228900	0.114450	−	0.993429i	$-0.463489\pi$
$337$	4.20204	0.228900	0.114450	+	0.993429i	$0.463489\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 8.16744i	− 0.442292i
$342$	0	0
$343$	− 17.2474i	− 0.931275i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.48188	0.294283	0.147141	−	0.989115i	$-0.452993\pi$
$347$	5.48188	0.294283	0.147141	+	0.989115i	$0.452993\pi$
$348$	0	0
$349$	−24.8990	−1.33281	−0.666406	−	0.745589i	$-0.732168\pi$
$349$	−24.8990	−1.33281	−0.666406	+	0.745589i	$0.732168\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.1421i	0.752710i	0.926476	+	0.376355i	$0.122823\pi$
$353$	14.1421i	0.752710i	−0.926476	+	0.376355i	$0.877177\pi$
$354$	0	0
$355$	− 2.40408i	− 0.127595i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	34.2911	1.80981	0.904907	−	0.425610i	$-0.139940\pi$
$359$	34.2911	1.80981	0.904907	+	0.425610i	$0.139940\pi$
$360$	0	0
$361$	−5.00000	−0.263158
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 0.603566i	− 0.0315921i
$366$	0	0
$367$	8.55051i	0.446333i	0.974780	+	0.223167i	$0.0716394\pi$
$367$	8.55051i	0.446333i	−0.974780	+	0.223167i	$0.928361\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−6.40329	−0.332442
$372$	0	0
$373$	−34.4949	−1.78608	−0.893039	−	0.449979i	$-0.851431\pi$
$373$	−34.4949	−1.78608	−0.893039	+	0.449979i	$0.851431\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	26.4415i	1.36180i
$378$	0	0
$379$	8.00000i	0.410932i	0.978664	+	0.205466i	$0.0658711\pi$
$379$	8.00000i	0.410932i	−0.978664	+	0.205466i	$0.934129\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.74983	0.191607	0.0958037	−	0.995400i	$-0.469458\pi$
$383$	3.74983	0.191607	0.0958037	+	0.995400i	$0.469458\pi$
$384$	0	0
$385$	−0.505103	−0.0257424
$386$	0	0
$387$	0	0
$388$	0	0
$389$	32.0662i	1.62582i	0.582388	+	0.812911i	$0.302118\pi$
$389$	32.0662i	1.62582i	−0.582388	+	0.812911i	$0.697882\pi$
$390$	0	0
$391$	− 9.79796i	− 0.495504i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0.635674	0.0319843
$396$	0	0
$397$	−8.69694	−0.436487	−0.218243	−	0.975894i	$-0.570033\pi$
$397$	−8.69694	−0.436487	−0.218243	+	0.975894i	$0.570033\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	22.6274i	1.12996i	0.825105	+	0.564980i	$0.191116\pi$
$401$	22.6274i	1.12996i	−0.825105	+	0.564980i	$0.808884\pi$
$402$	0	0
$403$	21.5959i	1.07577i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.37113	−0.266237
$408$	0	0
$409$	−7.00000	−0.346128	−0.173064	−	0.984911i	$-0.555367\pi$
$409$	−7.00000	−0.346128	−0.173064	+	0.984911i	$0.555367\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	13.2207i	0.650550i
$414$	0	0
$415$	− 5.04541i	− 0.247669i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	26.0915	1.27465	0.637327	−	0.770593i	$-0.280040\pi$
$419$	26.0915	1.27465	0.637327	+	0.770593i	$0.280040\pi$
$420$	0	0
$421$	29.7980	1.45226	0.726132	−	0.687556i	$-0.241316\pi$
$421$	29.7980	1.45226	0.726132	+	0.687556i	$0.241316\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 16.9706i	− 0.823193i
$426$	0	0
$427$	5.79796i	0.280583i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−27.3629	−1.31802	−0.659011	−	0.752133i	$-0.729025\pi$
$431$	−27.3629	−1.31802	−0.659011	+	0.752133i	$0.729025\pi$
$432$	0	0
$433$	32.3939	1.55675	0.778375	−	0.627799i	$-0.216044\pi$
$433$	32.3939	1.55675	0.778375	+	0.627799i	$0.216044\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	13.8564i	0.662842i
$438$	0	0
$439$	− 19.2474i	− 0.918631i	−0.888273	−	0.459315i	$-0.848095\pi$
$439$	− 19.2474i	− 0.918631i	0.888273	−	0.459315i	$-0.151905\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−28.6342	−1.36045	−0.680226	−	0.733002i	$-0.738119\pi$
$443$	−28.6342	−1.36045	−0.680226	+	0.733002i	$0.738119\pi$
$444$	0	0
$445$	−3.79796	−0.180041
$446$	0	0
$447$	0	0
$448$	0	0
$449$	22.3417i	1.05437i	0.849751	+	0.527185i	$0.176752\pi$
$449$	22.3417i	1.05437i	−0.849751	+	0.527185i	$0.823248\pi$
$450$	0	0
$451$	− 10.6969i	− 0.503699i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.33557	0.0626123
$456$	0	0
$457$	25.0000	1.16945	0.584725	−	0.811231i	$-0.301202\pi$
$457$	25.0000	1.16945	0.584725	+	0.811231i	$0.301202\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.87492i	0.0873235i	0.999046	+	0.0436618i	$0.0139024\pi$
$461$	1.87492i	0.0873235i	−0.999046	+	0.0436618i	$0.986098\pi$
$462$	0	0
$463$	− 27.0454i	− 1.25691i	−0.777847	−	0.628453i	$-0.783688\pi$
$463$	− 27.0454i	− 1.25691i	0.777847	−	0.628453i	$-0.216312\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0.174973	0.00809677	0.00404838	−	0.999992i	$-0.498711\pi$
$467$	0.174973	0.00809677	0.00404838	+	0.999992i	$0.498711\pi$
$468$	0	0
$469$	−7.39388	−0.341418
$470$	0	0
$471$	0	0
$472$	0	0
$473$	7.56388i	0.347787i
$474$	0	0
$475$	24.0000i	1.10120i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	26.0915	1.19215	0.596076	−	0.802928i	$-0.296726\pi$
$479$	26.0915	1.19215	0.596076	+	0.802928i	$0.296726\pi$
$480$	0	0
$481$	14.2020	0.647557
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.58919i	0.0721612i
$486$	0	0
$487$	3.79796i	0.172102i	0.996291	+	0.0860510i	$0.0274248\pi$
$487$	3.79796i	0.172102i	−0.996291	+	0.0860510i	$0.972575\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−14.3171	−0.646122	−0.323061	−	0.946378i	$-0.604712\pi$
$491$	−14.3171	−0.646122	−0.323061	+	0.946378i	$0.604712\pi$
$492$	0	0
$493$	−31.5959	−1.42301
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 10.9638i	− 0.491792i
$498$	0	0
$499$	14.8990i	0.666970i	0.942755	+	0.333485i	$0.108225\pi$
$499$	14.8990i	0.666970i	−0.942755	+	0.333485i	$0.891775\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−31.1127	−1.38725	−0.693623	−	0.720338i	$-0.743987\pi$
$503$	−31.1127	−1.38725	−0.693623	+	0.720338i	$0.743987\pi$
$504$	0	0
$505$	−5.00000	−0.222497
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 26.6951i	− 1.18324i	−0.806217	−	0.591619i	$-0.798489\pi$
$509$	− 26.6951i	− 1.18324i	0.806217	−	0.591619i	$-0.201511\pi$
$510$	0	0
$511$	− 2.75255i	− 0.121766i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3.17837	−0.140056
$516$	0	0
$517$	−10.0000	−0.439799
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 21.7060i	− 0.950958i	−0.879727	−	0.475479i	$-0.842275\pi$
$521$	− 21.7060i	− 0.950958i	0.879727	−	0.475479i	$-0.157725\pi$
$522$	0	0
$523$	27.3939i	1.19785i	0.800805	+	0.598925i	$0.204405\pi$
$523$	27.3939i	1.19785i	−0.800805	+	0.598925i	$0.795595\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−25.8058	−1.12412
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	28.2843i	1.22513i
$534$	0	0
$535$	2.75255i	0.119003i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.37113	0.231351
$540$	0	0
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 1.55708i	− 0.0666979i
$546$	0	0
$547$	− 23.7980i	− 1.01753i	−0.860906	−	0.508764i	$-0.830103\pi$
$547$	− 23.7980i	− 1.01753i	0.860906	−	0.508764i	$-0.169897\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	44.6834	1.90358
$552$	0	0
$553$	2.89898	0.123277
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 12.5529i	− 0.531886i	−0.963989	−	0.265943i	$-0.914317\pi$
$557$	− 12.5529i	− 0.531886i	0.963989	−	0.265943i	$-0.0856832\pi$
$558$	0	0
$559$	− 20.0000i	− 0.845910i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	22.8024	0.961006	0.480503	−	0.876993i	$-0.340454\pi$
$563$	22.8024	0.961006	0.480503	+	0.876993i	$0.340454\pi$
$564$	0	0
$565$	−4.49490	−0.189102
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.19955i	0.343743i	0.985119	+	0.171872i	$0.0549814\pi$
$569$	8.19955i	0.343743i	−0.985119	+	0.171872i	$0.945019\pi$
$570$	0	0
$571$	− 19.7980i	− 0.828519i	−0.910159	−	0.414259i	$-0.864041\pi$
$571$	− 19.7980i	− 0.828519i	0.910159	−	0.414259i	$-0.135959\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	13.8564	0.577852
$576$	0	0
$577$	−3.79796	−0.158111	−0.0790556	−	0.996870i	$-0.525190\pi$
$577$	−3.79796	−0.158111	−0.0790556	+	0.996870i	$0.525190\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 23.0095i	− 0.954594i
$582$	0	0
$583$	− 4.84337i	− 0.200592i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−23.0881	−0.952949	−0.476474	−	0.879188i	$-0.658085\pi$
$587$	−23.0881	−0.952949	−0.476474	+	0.879188i	$0.658085\pi$
$588$	0	0
$589$	36.4949	1.50375
$590$	0	0
$591$	0	0
$592$	0	0
$593$	8.13534i	0.334078i	0.985950	+	0.167039i	$0.0534206\pi$
$593$	8.13534i	0.334078i	−0.985950	+	0.167039i	$0.946579\pi$
$594$	0	0
$595$	1.59592i	0.0654263i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−25.5201	−1.04272	−0.521361	−	0.853336i	$-0.674575\pi$
$599$	−25.5201	−1.04272	−0.521361	+	0.853336i	$0.674575\pi$
$600$	0	0
$601$	10.3031	0.420271	0.210135	−	0.977672i	$-0.432610\pi$
$601$	10.3031	0.420271	0.210135	+	0.977672i	$0.432610\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.11416i	0.126608i
$606$	0	0
$607$	23.7980i	0.965929i	0.875640	+	0.482965i	$0.160440\pi$
$607$	23.7980i	0.965929i	−0.875640	+	0.482965i	$0.839560\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	26.4415	1.06971
$612$	0	0
$613$	−30.0000	−1.21169	−0.605844	−	0.795583i	$-0.707165\pi$
$613$	−30.0000	−1.21169	−0.605844	+	0.795583i	$0.707165\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	27.9985i	1.12718i	0.826055	+	0.563589i	$0.190580\pi$
$617$	27.9985i	1.12718i	−0.826055	+	0.563589i	$0.809420\pi$
$618$	0	0
$619$	− 21.5959i	− 0.868013i	−0.900910	−	0.434007i	$-0.857099\pi$
$619$	− 21.5959i	− 0.868013i	0.900910	−	0.434007i	$-0.142901\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−17.3205	−0.693932
$624$	0	0
$625$	23.4949	0.939796
$626$	0	0
$627$	0	0
$628$	0	0
$629$	16.9706i	0.676661i
$630$	0	0
$631$	45.7423i	1.82097i	0.413538	+	0.910487i	$0.364293\pi$
$631$	45.7423i	1.82097i	−0.413538	+	0.910487i	$0.635707\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.84621	0.192316
$636$	0	0
$637$	−14.2020	−0.562705
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 48.7832i	− 1.92682i	−0.268034	−	0.963409i	$-0.586374\pi$
$641$	− 48.7832i	− 1.92682i	0.268034	−	0.963409i	$-0.413626\pi$
$642$	0	0
$643$	37.3939i	1.47467i	0.675527	+	0.737335i	$0.263916\pi$
$643$	37.3939i	1.47467i	−0.675527	+	0.737335i	$0.736084\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	20.7846	0.817127	0.408564	−	0.912730i	$-0.366030\pi$
$647$	20.7846	0.817127	0.408564	+	0.912730i	$0.366030\pi$
$648$	0	0
$649$	−10.0000	−0.392534
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18.5597i	0.726299i	0.931731	+	0.363150i	$0.118299\pi$
$653$	18.5597i	0.726299i	−0.931731	+	0.363150i	$0.881701\pi$
$654$	0	0
$655$	− 0.954592i	− 0.0372990i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−27.8236	−1.08385	−0.541926	−	0.840426i	$-0.682305\pi$
$659$	−27.8236	−1.08385	−0.541926	+	0.840426i	$0.682305\pi$
$660$	0	0
$661$	4.49490	0.174831	0.0874156	−	0.996172i	$-0.472139\pi$
$661$	4.49490	0.174831	0.0874156	+	0.996172i	$0.472139\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 2.25697i	− 0.0875215i
$666$	0	0
$667$	− 25.7980i	− 0.998901i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.38551	−0.169301
$672$	0	0
$673$	−36.5959	−1.41067	−0.705334	−	0.708875i	$-0.749203\pi$
$673$	−36.5959	−1.41067	−0.705334	+	0.708875i	$0.749203\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 10.9638i	− 0.421372i	−0.977554	−	0.210686i	$-0.932430\pi$
$677$	− 10.9638i	− 0.421372i	0.977554	−	0.210686i	$-0.0675697\pi$
$678$	0	0
$679$	7.24745i	0.278132i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−10.3923	−0.397650	−0.198825	−	0.980035i	$-0.563713\pi$
$683$	−10.3923	−0.397650	−0.198825	+	0.980035i	$0.563713\pi$
$684$	0	0
$685$	3.39388	0.129673
$686$	0	0
$687$	0	0
$688$	0	0
$689$	12.8066i	0.487891i
$690$	0	0
$691$	− 4.00000i	− 0.152167i	−0.997101	−	0.0760836i	$-0.975758\pi$
$691$	− 4.00000i	− 0.152167i	0.997101	−	0.0760836i	$-0.0242416\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6.35674	−0.241125
$696$	0	0
$697$	−33.7980	−1.28019
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 38.7087i	− 1.46201i	−0.682374	−	0.731003i	$-0.739052\pi$
$701$	− 38.7087i	− 1.46201i	0.682374	−	0.731003i	$-0.260948\pi$
$702$	0	0
$703$	− 24.0000i	− 0.905177i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−22.8024	−0.857572
$708$	0	0
$709$	40.0000	1.50223	0.751116	−	0.660171i	$-0.229516\pi$
$709$	40.0000	1.50223	0.751116	+	0.660171i	$0.229516\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 21.0703i	− 0.789090i
$714$	0	0
$715$	1.01021i	0.0377795i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−6.92820	−0.258378	−0.129189	−	0.991620i	$-0.541237\pi$
$719$	−6.92820	−0.258378	−0.129189	+	0.991620i	$0.541237\pi$
$720$	0	0
$721$	−14.4949	−0.539818
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 44.6834i	− 1.65950i
$726$	0	0
$727$	− 7.24745i	− 0.268793i	−0.990928	−	0.134396i	$-0.957090\pi$
$727$	− 7.24745i	− 0.268793i	0.990928	−	0.134396i	$-0.0429096\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	23.8988	0.883928
$732$	0	0
$733$	−4.49490	−0.166023	−0.0830114	−	0.996549i	$-0.526454\pi$
$733$	−4.49490	−0.166023	−0.0830114	+	0.996549i	$0.526454\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 5.59264i	− 0.206007i
$738$	0	0
$739$	− 19.3939i	− 0.713415i	−0.934216	−	0.356708i	$-0.883899\pi$
$739$	− 19.3939i	− 0.713415i	0.934216	−	0.356708i	$-0.116101\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	31.4626	1.15425	0.577126	−	0.816655i	$-0.304174\pi$
$743$	31.4626	1.15425	0.577126	+	0.816655i	$0.304174\pi$
$744$	0	0
$745$	1.20204	0.0440394
$746$	0	0
$747$	0	0
$748$	0	0
$749$	12.5529i	0.458675i
$750$	0	0
$751$	47.7423i	1.74214i	0.491156	+	0.871071i	$0.336574\pi$
$751$	47.7423i	1.74214i	−0.491156	+	0.871071i	$0.663426\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	5.48188	0.199506
$756$	0	0
$757$	39.3939	1.43179	0.715897	−	0.698205i	$-0.246018\pi$
$757$	39.3939	1.43179	0.715897	+	0.698205i	$0.246018\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 52.8829i	− 1.91700i	−0.285085	−	0.958502i	$-0.592022\pi$
$761$	− 52.8829i	− 1.91700i	0.285085	−	0.958502i	$-0.407978\pi$
$762$	0	0
$763$	− 7.10102i	− 0.257074i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	26.4415	0.954746
$768$	0	0
$769$	11.4949	0.414517	0.207258	−	0.978286i	$-0.433546\pi$
$769$	11.4949	0.414517	0.207258	+	0.978286i	$0.433546\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 34.2911i	− 1.23336i	−0.787212	−	0.616682i	$-0.788476\pi$
$773$	− 34.2911i	− 1.23336i	0.787212	−	0.616682i	$-0.211524\pi$
$774$	0	0
$775$	− 36.4949i	− 1.31094i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	47.7975	1.71252
$780$	0	0
$781$	8.29286	0.296742
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 4.38551i	− 0.156525i
$786$	0	0
$787$	− 35.1918i	− 1.25445i	−0.778837	−	0.627227i	$-0.784190\pi$
$787$	− 35.1918i	− 1.25445i	0.778837	−	0.627227i	$-0.215810\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−20.4989	−0.728856
$792$	0	0
$793$	11.5959	0.411783
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7.24604i	0.256668i	0.991731	+	0.128334i	$0.0409629\pi$
$797$	7.24604i	0.256668i	−0.991731	+	0.128334i	$0.959037\pi$
$798$	0	0
$799$	31.5959i	1.11778i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.08200	0.0734720
$804$	0	0
$805$	−1.30306	−0.0459269
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 11.9494i	− 0.420118i	−0.977689	−	0.210059i	$-0.932634\pi$
$809$	− 11.9494i	− 0.420118i	0.977689	−	0.210059i	$-0.0673656\pi$
$810$	0	0
$811$	− 24.4949i	− 0.860132i	−0.902797	−	0.430066i	$-0.858490\pi$
$811$	− 24.4949i	− 0.860132i	0.902797	−	0.430066i	$-0.141510\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.90702	0.0668001
$816$	0	0
$817$	−33.7980	−1.18244
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 22.9774i	− 0.801915i	−0.916097	−	0.400958i	$-0.868677\pi$
$821$	− 22.9774i	− 0.801915i	0.916097	−	0.400958i	$-0.131323\pi$
$822$	0	0
$823$	29.6515i	1.03359i	0.856110	+	0.516794i	$0.172875\pi$
$823$	29.6515i	1.03359i	−0.856110	+	0.516794i	$0.827125\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	16.6206	0.577955	0.288978	−	0.957336i	$-0.406685\pi$
$827$	16.6206	0.577955	0.288978	+	0.957336i	$0.406685\pi$
$828$	0	0
$829$	37.5959	1.30576	0.652880	−	0.757461i	$-0.273561\pi$
$829$	37.5959	1.30576	0.652880	+	0.757461i	$0.273561\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 16.9706i	− 0.587995i
$834$	0	0
$835$	− 7.30306i	− 0.252733i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	3.17837	0.109730	0.0548648	−	0.998494i	$-0.482527\pi$
$839$	3.17837	0.109730	0.0548648	+	0.998494i	$0.482527\pi$
$840$	0	0
$841$	−54.1918	−1.86868
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.46075i	0.0502515i
$846$	0	0
$847$	14.2020i	0.487988i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−13.8564	−0.474991
$852$	0	0
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 11.6637i	− 0.398423i	−0.979957	−	0.199211i	$-0.936162\pi$
$857$	− 11.6637i	− 0.398423i	0.979957	−	0.199211i	$-0.0638380\pi$
$858$	0	0
$859$	39.5959i	1.35100i	0.737362	+	0.675498i	$0.236071\pi$
$859$	39.5959i	1.35100i	−0.737362	+	0.675498i	$0.763929\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	6.92820	0.235839	0.117919	−	0.993023i	$-0.462378\pi$
$863$	6.92820	0.235839	0.117919	+	0.993023i	$0.462378\pi$
$864$	0	0
$865$	−1.69694	−0.0576976
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2.19275i	0.0743840i
$870$	0	0
$871$	14.7878i	0.501064i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−4.56048	−0.154172
$876$	0	0
$877$	16.2020	0.547104	0.273552	−	0.961857i	$-0.411801\pi$
$877$	16.2020	0.547104	0.273552	+	0.961857i	$0.411801\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 7.27815i	− 0.245207i	−0.992456	−	0.122604i	$-0.960876\pi$
$881$	− 7.27815i	− 0.245207i	0.992456	−	0.122604i	$-0.0391244\pi$
$882$	0	0
$883$	− 52.8990i	− 1.78019i	−0.455773	−	0.890096i	$-0.650637\pi$
$883$	− 52.8990i	− 1.78019i	0.455773	−	0.890096i	$-0.349363\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−45.3191	−1.52166	−0.760832	−	0.648948i	$-0.775209\pi$
$887$	−45.3191	−1.52166	−0.760832	+	0.648948i	$0.775209\pi$
$888$	0	0
$889$	22.1010	0.741244
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 44.6834i	− 1.49527i
$894$	0	0
$895$	− 2.75255i	− 0.0920076i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−67.9465	−2.26614
$900$	0	0
$901$	−15.3031	−0.509819
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 3.11416i	− 0.103518i
$906$	0	0
$907$	− 4.49490i	− 0.149251i	−0.997212	−	0.0746253i	$-0.976224\pi$
$907$	− 4.49490i	− 0.149251i	0.997212	−	0.0746253i	$-0.0237761\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−0.921404	−0.0305275	−0.0152637	−	0.999884i	$-0.504859\pi$
$911$	−0.921404	−0.0305275	−0.0152637	+	0.999884i	$0.504859\pi$
$912$	0	0
$913$	17.4041	0.575991
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 4.35340i	− 0.143762i
$918$	0	0
$919$	− 2.75255i	− 0.0907983i	−0.998969	−	0.0453991i	$-0.985544\pi$
$919$	− 2.75255i	− 0.0907983i	0.998969	−	0.0453991i	$-0.0144560\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−21.9275	−0.721753
$924$	0	0
$925$	−24.0000	−0.789115
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 15.1278i	− 0.496326i	−0.968718	−	0.248163i	$-0.920173\pi$
$929$	− 15.1278i	− 0.496326i	0.968718	−	0.248163i	$-0.0798268\pi$
$930$	0	0
$931$	24.0000i	0.786568i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1.20713	−0.0394775
$936$	0	0
$937$	−15.0000	−0.490029	−0.245014	−	0.969519i	$-0.578793\pi$
$937$	−15.0000	−0.490029	−0.245014	+	0.969519i	$0.578793\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 41.5371i	− 1.35407i	−0.735950	−	0.677036i	$-0.763264\pi$
$941$	− 41.5371i	− 1.35407i	0.735950	−	0.677036i	$-0.236736\pi$
$942$	0	0
$943$	− 27.5959i	− 0.898647i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	23.0881	0.750263	0.375132	−	0.926972i	$-0.377598\pi$
$947$	23.0881	0.750263	0.375132	+	0.926972i	$0.377598\pi$
$948$	0	0
$949$	−5.50510	−0.178703
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 10.7423i	− 0.347976i	−0.984748	−	0.173988i	$-0.944335\pi$
$953$	− 10.7423i	− 0.347976i	0.984748	−	0.173988i	$-0.0556653\pi$
$954$	0	0
$955$	5.59592i	0.181080i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	15.4777	0.499801
$960$	0	0
$961$	−24.4949	−0.790158
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 0.667783i	− 0.0214967i
$966$	0	0
$967$	49.2474i	1.58369i	0.610721	+	0.791846i	$0.290880\pi$
$967$	49.2474i	1.58369i	−0.610721	+	0.791846i	$0.709120\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	19.3383	0.620595	0.310298	−	0.950639i	$-0.399571\pi$
$971$	19.3383	0.620595	0.310298	+	0.950639i	$0.399571\pi$
$972$	0	0
$973$	−28.9898	−0.929370
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 31.4626i	− 1.00658i	−0.864118	−	0.503290i	$-0.832123\pi$
$977$	− 31.4626i	− 1.00658i	0.864118	−	0.503290i	$-0.167877\pi$
$978$	0	0
$979$	− 13.1010i	− 0.418710i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−52.5330	−1.67554	−0.837771	−	0.546022i	$-0.816142\pi$
$983$	−52.5330	−1.67554	−0.837771	+	0.546022i	$0.816142\pi$
$984$	0	0
$985$	3.89898	0.124232
$986$	0	0
$987$	0	0
$988$	0	0
$989$	19.5133i	0.620486i
$990$	0	0
$991$	− 8.75255i	− 0.278034i	−0.990290	−	0.139017i	$-0.955606\pi$
$991$	− 8.75255i	− 0.278034i	0.990290	−	0.139017i	$-0.0443943\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	6.46750	0.205034
$996$	0	0
$997$	60.6969	1.92229	0.961146	−	0.276042i	$-0.0890228\pi$
$997$	60.6969	1.92229	0.961146	+	0.276042i	$0.0890228\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.2.c.g.1727.3		8
3.2	odd	2	inner	1728.2.c.g.1727.5		8
4.3	odd	2	inner	1728.2.c.g.1727.4		8
8.3	odd	2		864.2.c.a.863.6	yes	8
8.5	even	2		864.2.c.a.863.5	yes	8
12.11	even	2	inner	1728.2.c.g.1727.6		8
24.5	odd	2		864.2.c.a.863.3	✓	8
24.11	even	2		864.2.c.a.863.4	yes	8
72.5	odd	6		2592.2.s.a.1727.4		8
72.11	even	6		2592.2.s.h.863.2		8
72.13	even	6		2592.2.s.h.1727.2		8
72.29	odd	6		2592.2.s.h.863.1		8
72.43	odd	6		2592.2.s.a.863.4		8
72.59	even	6		2592.2.s.a.1727.3		8
72.61	even	6		2592.2.s.a.863.3		8
72.67	odd	6		2592.2.s.h.1727.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.2.c.a.863.3	✓	8	24.5	odd	2
864.2.c.a.863.4	yes	8	24.11	even	2
864.2.c.a.863.5	yes	8	8.5	even	2
864.2.c.a.863.6	yes	8	8.3	odd	2
1728.2.c.g.1727.3		8	1.1	even	1	trivial
1728.2.c.g.1727.4		8	4.3	odd	2	inner
1728.2.c.g.1727.5		8	3.2	odd	2	inner
1728.2.c.g.1727.6		8	12.11	even	2	inner
2592.2.s.a.863.3		8	72.61	even	6
2592.2.s.a.863.4		8	72.43	odd	6
2592.2.s.a.1727.3		8	72.59	even	6
2592.2.s.a.1727.4		8	72.5	odd	6
2592.2.s.h.863.1		8	72.29	odd	6
2592.2.s.h.863.2		8	72.11	even	6
2592.2.s.h.1727.1		8	72.67	odd	6
2592.2.s.h.1727.2		8	72.13	even	6

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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1728.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.317837i q^{5} -1.44949i q^{7} +1.09638 q^{11} -2.89898 q^{13} -3.46410i q^{17} +4.89898i q^{19} +2.82843 q^{23} +4.89898 q^{25} -9.12096i q^{29} -7.44949i q^{31} -0.460702 q^{35} -4.89898 q^{37} -9.75663i q^{41} +6.89898i q^{43} -9.12096 q^{47} +4.89898 q^{49} -4.41761i q^{53} -0.348469i q^{55} -9.12096 q^{59} -4.00000 q^{61} +0.921404i q^{65} -5.10102i q^{67} +7.56388 q^{71} +1.89898 q^{73} -1.58919i q^{77} +2.00000i q^{79} +15.8742 q^{83} -1.10102 q^{85} -11.9494i q^{89} +4.20204i q^{91} +1.55708 q^{95} -5.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{13} - 32 q^{61} - 24 q^{73} - 48 q^{85} - 40 q^{97}+O(q^{100}) \) 8 * q + 16 * q^13 - 32 * q^61 - 24 * q^73 - 48 * q^85 - 40 * q^97

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