LMFDB - Embedded newform 1728.3.h.g.161.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(161,1728)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1728.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$47.0845896815$
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:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.3
Root			$0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	1728.161
Dual form			1728.3.h.g.161.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.89898 q^{5} +1.73205 q^{7} +O(q^{10})$	$q-4.89898 q^{5} +1.73205 q^{7} -8.48528 q^{11} -5.19615i q^{13} +25.4558i q^{17} -5.00000i q^{19} +24.4949i q^{23} -1.00000 q^{25} +39.1918 q^{29} +13.8564 q^{31} -8.48528 q^{35} -32.9090i q^{37} -16.9706i q^{41} -34.0000i q^{43} -4.89898i q^{47} -46.0000 q^{49} -9.79796 q^{53} +41.5692 q^{55} -76.3675 q^{59} -22.5167i q^{61} +25.4558i q^{65} -19.0000i q^{67} -68.5857i q^{71} -25.0000 q^{73} -14.6969 q^{77} +140.296 q^{79} +101.823 q^{83} -124.708i q^{85} +8.48528i q^{89} -9.00000i q^{91} +24.4949i q^{95} -71.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 8 q^{25} - 368 q^{49} - 200 q^{73} - 568 q^{97}+O(q^{100}) $ 8 * q - 8 * q^25 - 368 * q^49 - 200 * q^73 - 568 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−4.89898	−0.979796	−0.489898	−	0.871780i	$-0.662966\pi$
$5$	−4.89898	−0.979796	−0.489898	+	0.871780i	$0.662966\pi$
$6$	0	0
$7$	1.73205	0.247436	0.123718	−	0.992317i	$-0.460518\pi$
$7$	1.73205	0.247436	0.123718	+	0.992317i	$0.460518\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−8.48528	−0.771389	−0.385695	−	0.922627i	$-0.626038\pi$
$11$	−8.48528	−0.771389	−0.385695	+	0.922627i	$0.626038\pi$
$12$	0	0
$13$	− 5.19615i	− 0.399704i	−0.979826	−	0.199852i	$-0.935954\pi$
$13$	− 5.19615i	− 0.399704i	0.979826	−	0.199852i	$-0.0640461\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	25.4558i	1.49740i	0.662908	+	0.748701i	$0.269322\pi$
$17$	25.4558i	1.49740i	−0.662908	+	0.748701i	$0.730678\pi$
$18$	0	0
$19$	− 5.00000i	− 0.263158i	−0.991306	−	0.131579i	$-0.957995\pi$
$19$	− 5.00000i	− 0.263158i	0.991306	−	0.131579i	$-0.0420047\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	24.4949i	1.06500i	0.846431	+	0.532498i	$0.178747\pi$
$23$	24.4949i	1.06500i	−0.846431	+	0.532498i	$0.821253\pi$
$24$	0	0
$25$	−1.00000	−0.0400000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	39.1918	1.35144	0.675721	−	0.737157i	$-0.263832\pi$
$29$	39.1918	1.35144	0.675721	+	0.737157i	$0.263832\pi$
$30$	0	0
$31$	13.8564	0.446981	0.223490	−	0.974706i	$-0.428255\pi$
$31$	13.8564	0.446981	0.223490	+	0.974706i	$0.428255\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−8.48528	−0.242437
$36$	0	0
$37$	− 32.9090i	− 0.889431i	−0.895672	−	0.444716i	$-0.853305\pi$
$37$	− 32.9090i	− 0.889431i	0.895672	−	0.444716i	$-0.146695\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 16.9706i	− 0.413916i	−0.978350	−	0.206958i	$-0.933644\pi$
$41$	− 16.9706i	− 0.413916i	0.978350	−	0.206958i	$-0.0663564\pi$
$42$	0	0
$43$	− 34.0000i	− 0.790698i	−0.918531	−	0.395349i	$-0.870624\pi$
$43$	− 34.0000i	− 0.790698i	0.918531	−	0.395349i	$-0.129376\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 4.89898i	− 0.104234i	−0.998641	−	0.0521168i	$-0.983403\pi$
$47$	− 4.89898i	− 0.104234i	0.998641	−	0.0521168i	$-0.0165968\pi$
$48$	0	0
$49$	−46.0000	−0.938776
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.79796	−0.184867	−0.0924336	−	0.995719i	$-0.529465\pi$
$53$	−9.79796	−0.184867	−0.0924336	+	0.995719i	$0.529465\pi$
$54$	0	0
$55$	41.5692	0.755804
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−76.3675	−1.29436	−0.647182	−	0.762335i	$-0.724053\pi$
$59$	−76.3675	−1.29436	−0.647182	+	0.762335i	$0.724053\pi$
$60$	0	0
$61$	− 22.5167i	− 0.369126i	−0.982821	−	0.184563i	$-0.940913\pi$
$61$	− 22.5167i	− 0.369126i	0.982821	−	0.184563i	$-0.0590869\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	25.4558i	0.391628i
$66$	0	0
$67$	− 19.0000i	− 0.283582i	−0.989897	−	0.141791i	$-0.954714\pi$
$67$	− 19.0000i	− 0.283582i	0.989897	−	0.141791i	$-0.0452861\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 68.5857i	− 0.965996i	−0.875621	−	0.482998i	$-0.839548\pi$
$71$	− 68.5857i	− 0.965996i	0.875621	−	0.482998i	$-0.160452\pi$
$72$	0	0
$73$	−25.0000	−0.342466	−0.171233	−	0.985231i	$-0.554775\pi$
$73$	−25.0000	−0.342466	−0.171233	+	0.985231i	$0.554775\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−14.6969	−0.190869
$78$	0	0
$79$	140.296	1.77590	0.887950	−	0.459940i	$-0.152129\pi$
$79$	140.296	1.77590	0.887950	+	0.459940i	$0.152129\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	101.823	1.22679	0.613394	−	0.789777i	$-0.289804\pi$
$83$	101.823	1.22679	0.613394	+	0.789777i	$0.289804\pi$
$84$	0	0
$85$	− 124.708i	− 1.46715i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8.48528i	0.0953402i	0.998863	+	0.0476701i	$0.0151796\pi$
$89$	8.48528i	0.0953402i	−0.998863	+	0.0476701i	$0.984820\pi$
$90$	0	0
$91$	− 9.00000i	− 0.0989011i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	24.4949i	0.257841i
$96$	0	0
$97$	−71.0000	−0.731959	−0.365979	−	0.930623i	$-0.619266\pi$
$97$	−71.0000	−0.731959	−0.365979	+	0.930623i	$0.619266\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−127.373	−1.26112	−0.630562	−	0.776139i	$-0.717175\pi$
$101$	−127.373	−1.26112	−0.630562	+	0.776139i	$0.717175\pi$
$102$	0	0
$103$	53.6936	0.521297	0.260648	−	0.965434i	$-0.416064\pi$
$103$	53.6936	0.521297	0.260648	+	0.965434i	$0.416064\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	178.191	1.66534	0.832668	−	0.553773i	$-0.186812\pi$
$107$	178.191	1.66534	0.832668	+	0.553773i	$0.186812\pi$
$108$	0	0
$109$	− 152.420i	− 1.39835i	−0.714949	−	0.699176i	$-0.753550\pi$
$109$	− 152.420i	− 1.39835i	0.714949	−	0.699176i	$-0.246450\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 212.132i	− 1.87727i	−0.344906	−	0.938637i	$-0.612089\pi$
$113$	− 212.132i	− 1.87727i	0.344906	−	0.938637i	$-0.387911\pi$
$114$	0	0
$115$	− 120.000i	− 1.04348i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	44.0908i	0.370511i
$120$	0	0
$121$	−49.0000	−0.404959
$122$	0	0
$123$	0	0
$124$	0	0
$125$	127.373	1.01899
$126$	0	0
$127$	110.851	0.872845	0.436422	−	0.899742i	$-0.356245\pi$
$127$	110.851	0.872845	0.436422	+	0.899742i	$0.356245\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−33.9411	−0.259093	−0.129546	−	0.991573i	$-0.541352\pi$
$131$	−33.9411	−0.259093	−0.129546	+	0.991573i	$0.541352\pi$
$132$	0	0
$133$	− 8.66025i	− 0.0651147i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.48528i	0.0619364i	0.999520	+	0.0309682i	$0.00985905\pi$
$137$	8.48528i	0.0619364i	−0.999520	+	0.0309682i	$0.990141\pi$
$138$	0	0
$139$	− 77.0000i	− 0.553957i	−0.960876	−	0.276978i	$-0.910667\pi$
$139$	− 77.0000i	− 0.553957i	0.960876	−	0.276978i	$-0.0893331\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	44.0908i	0.308327i
$144$	0	0
$145$	−192.000	−1.32414
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−146.969	−0.986372	−0.493186	−	0.869924i	$-0.664168\pi$
$149$	−146.969	−0.986372	−0.493186	+	0.869924i	$0.664168\pi$
$150$	0	0
$151$	261.540	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$151$	261.540	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−67.8823	−0.437950
$156$	0	0
$157$	− 263.272i	− 1.67689i	−0.544986	−	0.838445i	$-0.683465\pi$
$157$	− 263.272i	− 1.67689i	0.544986	−	0.838445i	$-0.316535\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	42.4264i	0.263518i
$162$	0	0
$163$	− 101.000i	− 0.619632i	−0.950797	−	0.309816i	$-0.899733\pi$
$163$	− 101.000i	− 0.619632i	0.950797	−	0.309816i	$-0.100267\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 112.677i	− 0.674710i	−0.941378	−	0.337355i	$-0.890468\pi$
$167$	− 112.677i	− 0.674710i	0.941378	−	0.337355i	$-0.109532\pi$
$168$	0	0
$169$	142.000	0.840237
$170$	0	0
$171$	0	0
$172$	0	0
$173$	127.373	0.736263	0.368131	−	0.929774i	$-0.379998\pi$
$173$	127.373	0.736263	0.368131	+	0.929774i	$0.379998\pi$
$174$	0	0
$175$	−1.73205	−0.00989743
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−118.794	−0.663653	−0.331827	−	0.943340i	$-0.607665\pi$
$179$	−118.794	−0.663653	−0.331827	+	0.943340i	$0.607665\pi$
$180$	0	0
$181$	− 174.937i	− 0.966503i	−0.875481	−	0.483252i	$-0.839456\pi$
$181$	− 174.937i	− 0.966503i	0.875481	−	0.483252i	$-0.160544\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	161.220i	0.871461i
$186$	0	0
$187$	− 216.000i	− 1.15508i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 318.434i	− 1.66719i	−0.552375	−	0.833596i	$-0.686278\pi$
$191$	− 318.434i	− 1.66719i	0.552375	−	0.833596i	$-0.313722\pi$
$192$	0	0
$193$	−265.000	−1.37306	−0.686528	−	0.727103i	$-0.740866\pi$
$193$	−265.000	−1.37306	−0.686528	+	0.727103i	$0.740866\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−44.0908	−0.223811	−0.111906	−	0.993719i	$-0.535695\pi$
$197$	−44.0908	−0.223811	−0.111906	+	0.993719i	$0.535695\pi$
$198$	0	0
$199$	25.9808	0.130557	0.0652783	−	0.997867i	$-0.479206\pi$
$199$	25.9808	0.130557	0.0652783	+	0.997867i	$0.479206\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	67.8823	0.334395
$204$	0	0
$205$	83.1384i	0.405553i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	42.4264i	0.202997i
$210$	0	0
$211$	53.0000i	0.251185i	0.992082	+	0.125592i	$0.0400832\pi$
$211$	53.0000i	0.251185i	−0.992082	+	0.125592i	$0.959917\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	166.565i	0.774722i
$216$	0	0
$217$	24.0000	0.110599
$218$	0	0
$219$	0	0
$220$	0	0
$221$	132.272	0.598518
$222$	0	0
$223$	401.836	1.80195	0.900977	−	0.433867i	$-0.142851\pi$
$223$	401.836	1.80195	0.900977	+	0.433867i	$0.142851\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−186.676	−0.822362	−0.411181	−	0.911554i	$-0.634884\pi$
$227$	−186.676	−0.822362	−0.411181	+	0.911554i	$0.634884\pi$
$228$	0	0
$229$	− 304.841i	− 1.33118i	−0.746316	−	0.665592i	$-0.768179\pi$
$229$	− 304.841i	− 1.33118i	0.746316	−	0.665592i	$-0.231821\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 118.794i	− 0.509845i	−0.966961	−	0.254923i	$-0.917950\pi$
$233$	− 118.794i	− 0.509845i	0.966961	−	0.254923i	$-0.0820500\pi$
$234$	0	0
$235$	24.0000i	0.102128i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	372.322i	1.55783i	0.627127	+	0.778917i	$0.284231\pi$
$239$	372.322i	1.55783i	−0.627127	+	0.778917i	$0.715769\pi$
$240$	0	0
$241$	−25.0000	−0.103734	−0.0518672	−	0.998654i	$-0.516517\pi$
$241$	−25.0000	−0.103734	−0.0518672	+	0.998654i	$0.516517\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	225.353	0.919808
$246$	0	0
$247$	−25.9808	−0.105185
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−186.676	−0.743730	−0.371865	−	0.928287i	$-0.621282\pi$
$251$	−186.676	−0.743730	−0.371865	+	0.928287i	$0.621282\pi$
$252$	0	0
$253$	− 207.846i	− 0.821526i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	390.323i	1.51877i	0.650644	+	0.759383i	$0.274499\pi$
$257$	390.323i	1.51877i	−0.650644	+	0.759383i	$0.725501\pi$
$258$	0	0
$259$	− 57.0000i	− 0.220077i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	411.514i	1.56469i	0.622843	+	0.782347i	$0.285977\pi$
$263$	411.514i	1.56469i	−0.622843	+	0.782347i	$0.714023\pi$
$264$	0	0
$265$	48.0000	0.181132
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−161.666	−0.600990	−0.300495	−	0.953783i	$-0.597152\pi$
$269$	−161.666	−0.600990	−0.300495	+	0.953783i	$0.597152\pi$
$270$	0	0
$271$	−154.153	−0.568828	−0.284414	−	0.958701i	$-0.591799\pi$
$271$	−154.153	−0.568828	−0.284414	+	0.958701i	$0.591799\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8.48528	0.0308556
$276$	0	0
$277$	318.697i	1.15053i	0.817966	+	0.575266i	$0.195101\pi$
$277$	318.697i	1.15053i	−0.817966	+	0.575266i	$0.804899\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 135.765i	− 0.483148i	−0.970382	−	0.241574i	$-0.922336\pi$
$281$	− 135.765i	− 0.483148i	0.970382	−	0.241574i	$-0.0776636\pi$
$282$	0	0
$283$	226.000i	0.798587i	0.916823	+	0.399293i	$0.130744\pi$
$283$	226.000i	0.798587i	−0.916823	+	0.399293i	$0.869256\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 29.3939i	− 0.102418i
$288$	0	0
$289$	−359.000	−1.24221
$290$	0	0
$291$	0	0
$292$	0	0
$293$	240.050	0.819283	0.409642	−	0.912247i	$-0.365654\pi$
$293$	240.050	0.819283	0.409642	+	0.912247i	$0.365654\pi$
$294$	0	0
$295$	374.123	1.26821
$296$	0	0
$297$	0	0
$298$	0	0
$299$	127.279	0.425683
$300$	0	0
$301$	− 58.8897i	− 0.195647i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	110.309i	0.361668i
$306$	0	0
$307$	− 130.000i	− 0.423453i	−0.977329	−	0.211726i	$-0.932091\pi$
$307$	− 130.000i	− 0.423453i	0.977329	−	0.211726i	$-0.0679086\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 465.403i	− 1.49647i	−0.663432	−	0.748236i	$-0.730901\pi$
$311$	− 465.403i	− 1.49647i	0.663432	−	0.748236i	$-0.269099\pi$
$312$	0	0
$313$	481.000	1.53674	0.768371	−	0.640005i	$-0.221068\pi$
$313$	481.000	1.53674	0.768371	+	0.640005i	$0.221068\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−529.090	−1.66905	−0.834526	−	0.550968i	$-0.814259\pi$
$317$	−529.090	−1.66905	−0.834526	+	0.550968i	$0.814259\pi$
$318$	0	0
$319$	−332.554	−1.04249
$320$	0	0
$321$	0	0
$322$	0	0
$323$	127.279	0.394053
$324$	0	0
$325$	5.19615i	0.0159882i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 8.48528i	− 0.0257911i
$330$	0	0
$331$	− 413.000i	− 1.24773i	−0.781530	−	0.623867i	$-0.785561\pi$
$331$	− 413.000i	− 1.24773i	0.781530	−	0.623867i	$-0.214439\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	93.0806i	0.277853i
$336$	0	0
$337$	−503.000	−1.49258	−0.746291	−	0.665620i	$-0.768167\pi$
$337$	−503.000	−1.49258	−0.746291	+	0.665620i	$0.768167\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−117.576	−0.344796
$342$	0	0
$343$	−164.545	−0.479723
$344$	0	0
$345$	0	0
$346$	0	0
$347$	305.470	0.880317	0.440159	−	0.897920i	$-0.354922\pi$
$347$	305.470	0.880317	0.440159	+	0.897920i	$0.354922\pi$
$348$	0	0
$349$	507.491i	1.45413i	0.686569	+	0.727064i	$0.259116\pi$
$349$	507.491i	1.45413i	−0.686569	+	0.727064i	$0.740884\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 101.823i	− 0.288451i	−0.989545	−	0.144226i	$-0.953931\pi$
$353$	− 101.823i	− 0.288451i	0.989545	−	0.144226i	$-0.0460691\pi$
$354$	0	0
$355$	336.000i	0.946479i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 494.797i	− 1.37826i	−0.724636	−	0.689132i	$-0.757992\pi$
$359$	− 494.797i	− 1.37826i	0.724636	−	0.689132i	$-0.242008\pi$
$360$	0	0
$361$	336.000	0.930748
$362$	0	0
$363$	0	0
$364$	0	0
$365$	122.474	0.335547
$366$	0	0
$367$	219.970	0.599375	0.299687	−	0.954037i	$-0.403118\pi$
$367$	219.970	0.599375	0.299687	+	0.954037i	$0.403118\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−16.9706	−0.0457428
$372$	0	0
$373$	− 254.611i	− 0.682604i	−0.939954	−	0.341302i	$-0.889132\pi$
$373$	− 254.611i	− 0.682604i	0.939954	−	0.341302i	$-0.110868\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 203.647i	− 0.540177i
$378$	0	0
$379$	533.000i	1.40633i	0.711025	+	0.703166i	$0.248231\pi$
$379$	533.000i	1.40633i	−0.711025	+	0.703166i	$0.751769\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 362.524i	− 0.946539i	−0.880918	−	0.473270i	$-0.843074\pi$
$383$	− 362.524i	− 0.946539i	0.880918	−	0.473270i	$-0.156926\pi$
$384$	0	0
$385$	72.0000	0.187013
$386$	0	0
$387$	0	0
$388$	0	0
$389$	93.0806	0.239282	0.119641	−	0.992817i	$-0.461826\pi$
$389$	93.0806	0.239282	0.119641	+	0.992817i	$0.461826\pi$
$390$	0	0
$391$	−623.538	−1.59473
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−687.308	−1.74002
$396$	0	0
$397$	69.2820i	0.174514i	0.996186	+	0.0872570i	$0.0278101\pi$
$397$	69.2820i	0.174514i	−0.996186	+	0.0872570i	$0.972190\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 407.294i	− 1.01569i	−0.861447	−	0.507847i	$-0.830441\pi$
$401$	− 407.294i	− 1.01569i	0.861447	−	0.507847i	$-0.169559\pi$
$402$	0	0
$403$	− 72.0000i	− 0.178660i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	279.242i	0.686098i
$408$	0	0
$409$	431.000	1.05379	0.526895	−	0.849930i	$-0.323356\pi$
$409$	431.000	1.05379	0.526895	+	0.849930i	$0.323356\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−132.272	−0.320272
$414$	0	0
$415$	−498.831	−1.20200
$416$	0	0
$417$	0	0
$418$	0	0
$419$	551.543	1.31633	0.658166	−	0.752873i	$-0.271332\pi$
$419$	551.543	1.31633	0.658166	+	0.752873i	$0.271332\pi$
$420$	0	0
$421$	715.337i	1.69914i	0.527478	+	0.849569i	$0.323138\pi$
$421$	715.337i	1.69914i	−0.527478	+	0.849569i	$0.676862\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 25.4558i	− 0.0598961i
$426$	0	0
$427$	− 39.0000i	− 0.0913349i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	475.201i	1.10255i	0.834322	+	0.551277i	$0.185859\pi$
$431$	475.201i	1.10255i	−0.834322	+	0.551277i	$0.814141\pi$
$432$	0	0
$433$	−218.000	−0.503464	−0.251732	−	0.967797i	$-0.581000\pi$
$433$	−218.000	−0.503464	−0.251732	+	0.967797i	$0.581000\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	122.474	0.280262
$438$	0	0
$439$	290.985	0.662835	0.331417	−	0.943484i	$-0.392473\pi$
$439$	290.985	0.662835	0.331417	+	0.943484i	$0.392473\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	322.441	0.727857	0.363929	−	0.931427i	$-0.381435\pi$
$443$	322.441	0.727857	0.363929	+	0.931427i	$0.381435\pi$
$444$	0	0
$445$	− 41.5692i	− 0.0934140i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 721.249i	− 1.60635i	−0.595747	−	0.803173i	$-0.703144\pi$
$449$	− 721.249i	− 1.60635i	0.595747	−	0.803173i	$-0.296856\pi$
$450$	0	0
$451$	144.000i	0.319290i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	44.0908i	0.0969029i
$456$	0	0
$457$	794.000	1.73742	0.868709	−	0.495323i	$-0.164950\pi$
$457$	794.000	1.73742	0.868709	+	0.495323i	$0.164950\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	132.272	0.286925	0.143463	−	0.989656i	$-0.454176\pi$
$461$	132.272	0.286925	0.143463	+	0.989656i	$0.454176\pi$
$462$	0	0
$463$	−524.811	−1.13350	−0.566751	−	0.823889i	$-0.691800\pi$
$463$	−524.811	−1.13350	−0.566751	+	0.823889i	$0.691800\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	500.632	1.07202	0.536008	−	0.844213i	$-0.319932\pi$
$467$	500.632	1.07202	0.536008	+	0.844213i	$0.319932\pi$
$468$	0	0
$469$	− 32.9090i	− 0.0701684i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	288.500i	0.609936i
$474$	0	0
$475$	5.00000i	0.0105263i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 734.847i	− 1.53413i	−0.641571	−	0.767064i	$-0.721717\pi$
$479$	− 734.847i	− 1.53413i	0.641571	−	0.767064i	$-0.278283\pi$
$480$	0	0
$481$	−171.000	−0.355509
$482$	0	0
$483$	0	0
$484$	0	0
$485$	347.828	0.717170
$486$	0	0
$487$	−400.104	−0.821568	−0.410784	−	0.911733i	$-0.634745\pi$
$487$	−400.104	−0.821568	−0.410784	+	0.911733i	$0.634745\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−42.4264	−0.0864082	−0.0432041	−	0.999066i	$-0.513757\pi$
$491$	−42.4264	−0.0864082	−0.0432041	+	0.999066i	$0.513757\pi$
$492$	0	0
$493$	997.661i	2.02365i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 118.794i	− 0.239022i
$498$	0	0
$499$	326.000i	0.653307i	0.945144	+	0.326653i	$0.105921\pi$
$499$	326.000i	0.653307i	−0.945144	+	0.326653i	$0.894079\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	318.434i	0.633069i	0.948581	+	0.316534i	$0.102519\pi$
$503$	318.434i	0.633069i	−0.948581	+	0.316534i	$0.897481\pi$
$504$	0	0
$505$	624.000	1.23564
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−700.554	−1.37633	−0.688167	−	0.725552i	$-0.741584\pi$
$509$	−700.554	−1.37633	−0.688167	+	0.725552i	$0.741584\pi$
$510$	0	0
$511$	−43.3013	−0.0847383
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−263.044	−0.510765
$516$	0	0
$517$	41.5692i	0.0804047i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	602.455i	1.15634i	0.815915	+	0.578172i	$0.196234\pi$
$521$	602.455i	1.15634i	−0.815915	+	0.578172i	$0.803766\pi$
$522$	0	0
$523$	605.000i	1.15679i	0.815758	+	0.578394i	$0.196320\pi$
$523$	605.000i	1.15679i	−0.815758	+	0.578394i	$0.803680\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	352.727i	0.669310i
$528$	0	0
$529$	−71.0000	−0.134216
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−88.1816	−0.165444
$534$	0	0
$535$	−872.954	−1.63169
$536$	0	0
$537$	0	0
$538$	0	0
$539$	390.323	0.724161
$540$	0	0
$541$	174.937i	0.323359i	0.986843	+	0.161679i	$0.0516911\pi$
$541$	174.937i	0.323359i	−0.986843	+	0.161679i	$0.948309\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	746.705i	1.37010i
$546$	0	0
$547$	67.0000i	0.122486i	0.998123	+	0.0612431i	$0.0195065\pi$
$547$	67.0000i	0.122486i	−0.998123	+	0.0612431i	$0.980493\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 195.959i	− 0.355643i
$552$	0	0
$553$	243.000	0.439421
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−455.605	−0.817962	−0.408981	−	0.912543i	$-0.634116\pi$
$557$	−455.605	−0.817962	−0.408981	+	0.912543i	$0.634116\pi$
$558$	0	0
$559$	−176.669	−0.316045
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−967.322	−1.71816	−0.859078	−	0.511844i	$-0.828962\pi$
$563$	−967.322	−1.71816	−0.859078	+	0.511844i	$0.828962\pi$
$564$	0	0
$565$	1039.23i	1.83935i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	432.749i	0.760544i	0.924875	+	0.380272i	$0.124170\pi$
$569$	432.749i	0.760544i	−0.924875	+	0.380272i	$0.875830\pi$
$570$	0	0
$571$	451.000i	0.789842i	0.918715	+	0.394921i	$0.129228\pi$
$571$	451.000i	0.789842i	−0.918715	+	0.394921i	$0.870772\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 24.4949i	− 0.0425998i
$576$	0	0
$577$	−623.000	−1.07972	−0.539861	−	0.841754i	$-0.681523\pi$
$577$	−623.000	−1.07972	−0.539861	+	0.841754i	$0.681523\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	176.363	0.303551
$582$	0	0
$583$	83.1384	0.142605
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−857.013	−1.45999	−0.729994	−	0.683453i	$-0.760477\pi$
$587$	−857.013	−1.45999	−0.729994	+	0.683453i	$0.760477\pi$
$588$	0	0
$589$	− 69.2820i	− 0.117627i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	576.999i	0.973017i	0.873676	+	0.486509i	$0.161730\pi$
$593$	576.999i	0.973017i	−0.873676	+	0.486509i	$0.838270\pi$
$594$	0	0
$595$	− 216.000i	− 0.363025i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	685.857i	1.14500i	0.819903	+	0.572502i	$0.194027\pi$
$599$	685.857i	1.14500i	−0.819903	+	0.572502i	$0.805973\pi$
$600$	0	0
$601$	986.000	1.64060	0.820300	−	0.571934i	$-0.193807\pi$
$601$	986.000	1.64060	0.820300	+	0.571934i	$0.193807\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	240.050	0.396777
$606$	0	0
$607$	−538.668	−0.887426	−0.443713	−	0.896169i	$-0.646339\pi$
$607$	−538.668	−0.887426	−0.443713	+	0.896169i	$0.646339\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−25.4558	−0.0416626
$612$	0	0
$613$	476.314i	0.777021i	0.921444	+	0.388511i	$0.127010\pi$
$613$	476.314i	0.777021i	−0.921444	+	0.388511i	$0.872990\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 958.837i	− 1.55403i	−0.629482	−	0.777015i	$-0.716733\pi$
$617$	− 958.837i	− 1.55403i	0.629482	−	0.777015i	$-0.283267\pi$
$618$	0	0
$619$	− 821.000i	− 1.32633i	−0.748472	−	0.663166i	$-0.769212\pi$
$619$	− 821.000i	− 1.32633i	0.748472	−	0.663166i	$-0.230788\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	14.6969i	0.0235906i
$624$	0	0
$625$	−599.000	−0.958400
$626$	0	0
$627$	0	0
$628$	0	0
$629$	837.725	1.33184
$630$	0	0
$631$	−957.824	−1.51795	−0.758973	−	0.651122i	$-0.774299\pi$
$631$	−957.824	−1.51795	−0.758973	+	0.651122i	$0.774299\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−543.058	−0.855209
$636$	0	0
$637$	239.023i	0.375232i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 220.617i	− 0.344177i	−0.985082	−	0.172088i	$-0.944949\pi$
$641$	− 220.617i	− 0.344177i	0.985082	−	0.172088i	$-0.0550515\pi$
$642$	0	0
$643$	898.000i	1.39658i	0.715816	+	0.698289i	$0.246055\pi$
$643$	898.000i	1.39658i	−0.715816	+	0.698289i	$0.753945\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 303.737i	− 0.469454i	−0.972061	−	0.234727i	$-0.924580\pi$
$647$	− 303.737i	− 0.469454i	0.972061	−	0.234727i	$-0.0754196\pi$
$648$	0	0
$649$	648.000	0.998459
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−176.363	−0.270082	−0.135041	−	0.990840i	$-0.543117\pi$
$653$	−176.363	−0.270082	−0.135041	+	0.990840i	$0.543117\pi$
$654$	0	0
$655$	166.277	0.253858
$656$	0	0
$657$	0	0
$658$	0	0
$659$	424.264	0.643800	0.321900	−	0.946774i	$-0.395679\pi$
$659$	424.264	0.643800	0.321900	+	0.946774i	$0.395679\pi$
$660$	0	0
$661$	310.037i	0.469043i	0.972111	+	0.234521i	$0.0753522\pi$
$661$	310.037i	0.469043i	−0.972111	+	0.234521i	$0.924648\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	42.4264i	0.0637991i
$666$	0	0
$667$	960.000i	1.43928i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	191.060i	0.284739i
$672$	0	0
$673$	−215.000	−0.319465	−0.159733	−	0.987160i	$-0.551063\pi$
$673$	−215.000	−0.319465	−0.159733	+	0.987160i	$0.551063\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	171.464	0.253271	0.126635	−	0.991949i	$-0.459582\pi$
$677$	171.464	0.253271	0.126635	+	0.991949i	$0.459582\pi$
$678$	0	0
$679$	−122.976	−0.181113
$680$	0	0
$681$	0	0
$682$	0	0
$683$	695.793	1.01873	0.509365	−	0.860550i	$-0.329880\pi$
$683$	695.793	1.01873	0.509365	+	0.860550i	$0.329880\pi$
$684$	0	0
$685$	− 41.5692i	− 0.0606850i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	50.9117i	0.0738921i
$690$	0	0
$691$	562.000i	0.813314i	0.913581	+	0.406657i	$0.133306\pi$
$691$	562.000i	0.813314i	−0.913581	+	0.406657i	$0.866694\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	377.221i	0.542765i
$696$	0	0
$697$	432.000	0.619799
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1131.66	1.61436	0.807179	−	0.590307i	$-0.200993\pi$
$701$	1131.66	1.61436	0.807179	+	0.590307i	$0.200993\pi$
$702$	0	0
$703$	−164.545	−0.234061
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−220.617	−0.312047
$708$	0	0
$709$	368.927i	0.520348i	0.965562	+	0.260174i	$0.0837799\pi$
$709$	368.927i	0.520348i	−0.965562	+	0.260174i	$0.916220\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	339.411i	0.476033i
$714$	0	0
$715$	− 216.000i	− 0.302098i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	93.0000	0.128988
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−39.1918	−0.0540577
$726$	0	0
$727$	457.261	0.628970	0.314485	−	0.949262i	$-0.398168\pi$
$727$	457.261	0.628970	0.314485	+	0.949262i	$0.398168\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	865.499	1.18399
$732$	0	0
$733$	96.9948i	0.132326i	0.997809	+	0.0661629i	$0.0210757\pi$
$733$	96.9948i	0.132326i	−0.997809	+	0.0661629i	$0.978924\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	161.220i	0.218752i
$738$	0	0
$739$	− 514.000i	− 0.695535i	−0.937581	−	0.347767i	$-0.886940\pi$
$739$	− 514.000i	− 0.695535i	0.937581	−	0.347767i	$-0.113060\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 1053.28i	− 1.41761i	−0.705407	−	0.708803i	$-0.749236\pi$
$743$	− 1053.28i	− 1.41761i	0.705407	−	0.708803i	$-0.250764\pi$
$744$	0	0
$745$	720.000	0.966443
$746$	0	0
$747$	0	0
$748$	0	0
$749$	308.636	0.412064
$750$	0	0
$751$	−289.252	−0.385156	−0.192578	−	0.981282i	$-0.561685\pi$
$751$	−289.252	−0.385156	−0.192578	+	0.981282i	$0.561685\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1281.28	−1.69706
$756$	0	0
$757$	− 434.745i	− 0.574300i	−0.957886	−	0.287150i	$-0.907292\pi$
$757$	− 434.745i	− 0.574300i	0.957886	−	0.287150i	$-0.0927077\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 1094.60i	− 1.43837i	−0.694817	−	0.719186i	$-0.744515\pi$
$761$	− 1094.60i	− 1.43837i	0.694817	−	0.719186i	$-0.255485\pi$
$762$	0	0
$763$	− 264.000i	− 0.346003i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	396.817i	0.517363i
$768$	0	0
$769$	−529.000	−0.687906	−0.343953	−	0.938987i	$-0.611766\pi$
$769$	−529.000	−0.687906	−0.343953	+	0.938987i	$0.611766\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1332.52	−1.72383	−0.861916	−	0.507051i	$-0.830736\pi$
$773$	−1332.52	−1.72383	−0.861916	+	0.507051i	$0.830736\pi$
$774$	0	0
$775$	−13.8564	−0.0178792
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−84.8528	−0.108925
$780$	0	0
$781$	581.969i	0.745159i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1289.76i	1.64301i
$786$	0	0
$787$	− 197.000i	− 0.250318i	−0.992137	−	0.125159i	$-0.960056\pi$
$787$	− 197.000i	− 0.250318i	0.992137	−	0.125159i	$-0.0399440\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 367.423i	− 0.464505i
$792$	0	0
$793$	−117.000	−0.147541
$794$	0	0
$795$	0	0
$796$	0	0
$797$	421.312	0.528623	0.264311	−	0.964437i	$-0.414855\pi$
$797$	421.312	0.528623	0.264311	+	0.964437i	$0.414855\pi$
$798$	0	0
$799$	124.708	0.156080
$800$	0	0
$801$	0	0
$802$	0	0
$803$	212.132	0.264174
$804$	0	0
$805$	− 207.846i	− 0.258194i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	220.617i	0.272704i	0.990660	+	0.136352i	$0.0435378\pi$
$809$	220.617i	0.272704i	−0.990660	+	0.136352i	$0.956462\pi$
$810$	0	0
$811$	110.000i	0.135635i	0.997698	+	0.0678175i	$0.0216036\pi$
$811$	110.000i	0.135635i	−0.997698	+	0.0678175i	$0.978396\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	494.797i	0.607113i
$816$	0	0
$817$	−170.000	−0.208078
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1268.84	−1.54548	−0.772738	−	0.634725i	$-0.781113\pi$
$821$	−1268.84	−1.54548	−0.772738	+	0.634725i	$0.781113\pi$
$822$	0	0
$823$	−389.711	−0.473525	−0.236763	−	0.971568i	$-0.576086\pi$
$823$	−389.711	−0.473525	−0.236763	+	0.971568i	$0.576086\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	415.779	0.502755	0.251378	−	0.967889i	$-0.419116\pi$
$827$	415.779	0.502755	0.251378	+	0.967889i	$0.419116\pi$
$828$	0	0
$829$	213.042i	0.256987i	0.991710	+	0.128494i	$0.0410142\pi$
$829$	213.042i	0.256987i	−0.991710	+	0.128494i	$0.958986\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 1170.97i	− 1.40572i
$834$	0	0
$835$	552.000i	0.661078i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 107.778i	− 0.128460i	−0.997935	−	0.0642298i	$-0.979541\pi$
$839$	− 107.778i	− 0.128460i	0.997935	−	0.0642298i	$-0.0204591\pi$
$840$	0	0
$841$	695.000	0.826397
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−695.655	−0.823260
$846$	0	0
$847$	−84.8705	−0.100201
$848$	0	0
$849$	0	0
$850$	0	0
$851$	806.102	0.947241
$852$	0	0
$853$	− 684.160i	− 0.802063i	−0.916064	−	0.401032i	$-0.868652\pi$
$853$	− 684.160i	− 0.802063i	0.916064	−	0.401032i	$-0.131348\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1035.20i	− 1.20794i	−0.797007	−	0.603970i	$-0.793585\pi$
$857$	− 1035.20i	− 1.20794i	0.797007	−	0.603970i	$-0.206415\pi$
$858$	0	0
$859$	341.000i	0.396973i	0.980104	+	0.198487i	$0.0636026\pi$
$859$	341.000i	0.396973i	−0.980104	+	0.198487i	$0.936397\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1376.61i	− 1.59515i	−0.603221	−	0.797574i	$-0.706116\pi$
$863$	− 1376.61i	− 1.59515i	0.603221	−	0.797574i	$-0.293884\pi$
$864$	0	0
$865$	−624.000	−0.721387
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1190.45	−1.36991
$870$	0	0
$871$	−98.7269	−0.113349
$872$	0	0
$873$	0	0
$874$	0	0
$875$	220.617	0.252134
$876$	0	0
$877$	− 1030.57i	− 1.17511i	−0.809185	−	0.587554i	$-0.800091\pi$
$877$	− 1030.57i	− 1.17511i	0.809185	−	0.587554i	$-0.199909\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 347.897i	− 0.394888i	−0.980314	−	0.197444i	$-0.936736\pi$
$881$	− 347.897i	− 0.394888i	0.980314	−	0.197444i	$-0.0632641\pi$
$882$	0	0
$883$	− 1613.00i	− 1.82673i	−0.407145	−	0.913364i	$-0.633476\pi$
$883$	− 1613.00i	− 1.82673i	0.407145	−	0.913364i	$-0.366524\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	117.576i	0.132554i	0.997801	+	0.0662771i	$0.0211121\pi$
$887$	117.576i	0.132554i	−0.997801	+	0.0662771i	$0.978888\pi$
$888$	0	0
$889$	192.000	0.215973
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−24.4949	−0.0274299
$894$	0	0
$895$	581.969	0.650245
$896$	0	0
$897$	0	0
$898$	0	0
$899$	543.058	0.604069
$900$	0	0
$901$	− 249.415i	− 0.276821i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	857.013i	0.946976i
$906$	0	0
$907$	− 1171.00i	− 1.29107i	−0.763731	−	0.645535i	$-0.776635\pi$
$907$	− 1171.00i	− 1.29107i	0.763731	−	0.645535i	$-0.223365\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	186.161i	0.204348i	0.994767	+	0.102174i	$0.0325799\pi$
$911$	186.161i	0.204348i	−0.994767	+	0.102174i	$0.967420\pi$
$912$	0	0
$913$	−864.000	−0.946331
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−58.7878	−0.0641088
$918$	0	0
$919$	706.677	0.768963	0.384481	−	0.923133i	$-0.374380\pi$
$919$	706.677	0.768963	0.384481	+	0.923133i	$0.374380\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−356.382	−0.386112
$924$	0	0
$925$	32.9090i	0.0355773i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1086.12i	− 1.16912i	−0.811349	−	0.584562i	$-0.801266\pi$
$929$	− 1086.12i	− 1.16912i	0.811349	−	0.584562i	$-0.198734\pi$
$930$	0	0
$931$	230.000i	0.247046i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1058.18i	1.13174i
$936$	0	0
$937$	1.00000	0.00106724	0.000533618	−	1.00000i	$-0.499830\pi$
$937$	1.00000	0.00106724	0.000533618		1.00000i	$0.499830\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	759.342	0.806952	0.403476	−	0.914990i	$-0.367802\pi$
$941$	759.342	0.806952	0.403476	+	0.914990i	$0.367802\pi$
$942$	0	0
$943$	415.692	0.440819
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1077.63	1.13794	0.568971	−	0.822358i	$-0.307342\pi$
$947$	1077.63	1.13794	0.568971	+	0.822358i	$0.307342\pi$
$948$	0	0
$949$	129.904i	0.136885i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	93.3381i	0.0979413i	0.998800	+	0.0489707i	$0.0155941\pi$
$953$	93.3381i	0.0979413i	−0.998800	+	0.0489707i	$0.984406\pi$
$954$	0	0
$955$	1560.00i	1.63351i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	14.6969i	0.0153253i
$960$	0	0
$961$	−769.000	−0.800208
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1298.23	1.34532
$966$	0	0
$967$	1151.81	1.19112	0.595560	−	0.803311i	$-0.296930\pi$
$967$	1151.81	1.19112	0.595560	+	0.803311i	$0.296930\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1077.63	1.10982	0.554908	−	0.831912i	$-0.312754\pi$
$971$	1077.63	1.10982	0.554908	+	0.831912i	$0.312754\pi$
$972$	0	0
$973$	− 133.368i	− 0.137069i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	373.352i	0.382142i	0.981576	+	0.191071i	$0.0611960\pi$
$977$	373.352i	0.382142i	−0.981576	+	0.191071i	$0.938804\pi$
$978$	0	0
$979$	− 72.0000i	− 0.0735444i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 631.968i	− 0.642898i	−0.946927	−	0.321449i	$-0.895830\pi$
$983$	− 631.968i	− 0.642898i	0.946927	−	0.321449i	$-0.104170\pi$
$984$	0	0
$985$	216.000	0.219289
$986$	0	0
$987$	0	0
$988$	0	0
$989$	832.827	0.842089
$990$	0	0
$991$	219.970	0.221968	0.110984	−	0.993822i	$-0.464600\pi$
$991$	219.970	0.221968	0.110984	+	0.993822i	$0.464600\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−127.279	−0.127919
$996$	0	0
$997$	− 1039.23i	− 1.04236i	−0.853448	−	0.521179i	$-0.825492\pi$
$997$	− 1039.23i	− 1.04236i	0.853448	−	0.521179i	$-0.174508\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.h.g.161.3	yes	8
3.2	odd	2	inner	1728.3.h.g.161.7	yes	8
4.3	odd	2	inner	1728.3.h.g.161.1	✓	8
8.3	odd	2	inner	1728.3.h.g.161.6	yes	8
8.5	even	2	inner	1728.3.h.g.161.8	yes	8
12.11	even	2	inner	1728.3.h.g.161.5	yes	8
24.5	odd	2	inner	1728.3.h.g.161.4	yes	8
24.11	even	2	inner	1728.3.h.g.161.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.3.h.g.161.1	✓	8	4.3	odd	2	inner
1728.3.h.g.161.2	yes	8	24.11	even	2	inner
1728.3.h.g.161.3	yes	8	1.1	even	1	trivial
1728.3.h.g.161.4	yes	8	24.5	odd	2	inner
1728.3.h.g.161.5	yes	8	12.11	even	2	inner
1728.3.h.g.161.6	yes	8	8.3	odd	2	inner
1728.3.h.g.161.7	yes	8	3.2	odd	2	inner
1728.3.h.g.161.8	yes	8	8.5	even	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-4.89898 q^{5} +1.73205 q^{7} +O(q^{10})\)	\(q-4.89898 q^{5} +1.73205 q^{7} -8.48528 q^{11} -5.19615i q^{13} +25.4558i q^{17} -5.00000i q^{19} +24.4949i q^{23} -1.00000 q^{25} +39.1918 q^{29} +13.8564 q^{31} -8.48528 q^{35} -32.9090i q^{37} -16.9706i q^{41} -34.0000i q^{43} -4.89898i q^{47} -46.0000 q^{49} -9.79796 q^{53} +41.5692 q^{55} -76.3675 q^{59} -22.5167i q^{61} +25.4558i q^{65} -19.0000i q^{67} -68.5857i q^{71} -25.0000 q^{73} -14.6969 q^{77} +140.296 q^{79} +101.823 q^{83} -124.708i q^{85} +8.48528i q^{89} -9.00000i q^{91} +24.4949i q^{95} -71.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 8 q^{25} - 368 q^{49} - 200 q^{73} - 568 q^{97}+O(q^{100}) \) 8 * q - 8 * q^25 - 368 * q^49 - 200 * q^73 - 568 * q^97

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