LMFDB - Embedded newform 1584.3.i.d.881.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1584,3,Mod(881,1584)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1584, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1584.881"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1584 = 2^{4} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1584.i (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,16,0,0,0,0,0,8] 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:	$43.1608738747$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.765751005184.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 7x^{6} - 38x^{5} - 3x^{4} + 212x^{3} - 140x^{2} - 352x + 396 $ x^8 - 2x^7 + 7x^6 - 38x^5 - 3x^4 + 212x^3 - 140x^2 - 352*x + 396
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 792)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.4
Root			$-0.817659 + 3.72320i$ of defining polynomial
Character	$\chi$	$=$	1584.881
Dual form			1584.3.i.d.881.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.898476i q^{5} +5.84039 q^{7} +3.31662i q^{11} -10.9410 q^{13} +5.82749i q^{17} -27.0227 q^{19} -15.9738i q^{23} +24.1927 q^{25} +47.6224i q^{29} -14.6305 q^{31} -5.24745i q^{35} -37.4148 q^{37} +9.20811i q^{41} -73.0471 q^{43} +8.20831i q^{47} -14.8898 q^{49} +39.1670i q^{53} +2.97991 q^{55} -60.4173i q^{59} +94.3077 q^{61} +9.83019i q^{65} +4.58065 q^{67} -94.4174i q^{71} +2.60239 q^{73} +19.3704i q^{77} -21.1802 q^{79} +95.0013i q^{83} +5.23586 q^{85} +58.0557i q^{89} -63.8995 q^{91} +24.2792i q^{95} -11.5304 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{7} + 8 q^{13} - 8 q^{19} + 16 q^{25} - 8 q^{31} + 8 q^{37} + 40 q^{43} - 160 q^{49} + 72 q^{61} + 16 q^{67} - 64 q^{79} + 240 q^{85} - 416 q^{91} - 24 q^{97}+O(q^{100}) $ 8 * q + 16 * q^7 + 8 * q^13 - 8 * q^19 + 16 * q^25 - 8 * q^31 + 8 * q^37 + 40 * q^43 - 160 * q^49 + 72 * q^61 + 16 * q^67 - 64 * q^79 + 240 * q^85 - 416 * q^91 - 24 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1584\mathbb{Z}\right)^\times$.

$n$	$145$	$353$	$991$	$1189$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$5$	− 0.898476i	− 0.179695i	−0.995956	−	0.0898476i	$-0.971362\pi$
$5$	− 0.898476i	− 0.179695i	0.995956	−	0.0898476i	$-0.0286380\pi$
$6$	0	0
$7$	5.84039	0.834342	0.417171	−	0.908828i	$-0.363022\pi$
$7$	5.84039	0.834342	0.417171	+	0.908828i	$0.363022\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.31662i	0.301511i
$11$	3.31662i	0.301511i
$12$	0	0
$13$	−10.9410	−0.841612	−0.420806	−	0.907151i	$-0.638253\pi$
$13$	−10.9410	−0.841612	−0.420806	+	0.907151i	$0.638253\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.82749i	0.342794i	0.985202	+	0.171397i	$0.0548280\pi$
$17$	5.82749i	0.342794i	−0.985202	+	0.171397i	$0.945172\pi$
$18$	0	0
$19$	−27.0227	−1.42225	−0.711123	−	0.703068i	$-0.751813\pi$
$19$	−27.0227	−1.42225	−0.711123	+	0.703068i	$0.751813\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 15.9738i	− 0.694513i	−0.937770	−	0.347256i	$-0.887113\pi$
$23$	− 15.9738i	− 0.694513i	0.937770	−	0.347256i	$-0.112887\pi$
$24$	0	0
$25$	24.1927	0.967710
$26$	0	0
$27$	0	0
$28$	0	0
$29$	47.6224i	1.64215i	0.570818	+	0.821076i	$0.306626\pi$
$29$	47.6224i	1.64215i	−0.570818	+	0.821076i	$0.693374\pi$
$30$	0	0
$31$	−14.6305	−0.471953	−0.235976	−	0.971759i	$-0.575829\pi$
$31$	−14.6305	−0.471953	−0.235976	+	0.971759i	$0.575829\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 5.24745i	− 0.149927i
$36$	0	0
$37$	−37.4148	−1.01121	−0.505605	−	0.862765i	$-0.668731\pi$
$37$	−37.4148	−1.01121	−0.505605	+	0.862765i	$0.668731\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.20811i	0.224588i	0.993675	+	0.112294i	$0.0358199\pi$
$41$	9.20811i	0.224588i	−0.993675	+	0.112294i	$0.964180\pi$
$42$	0	0
$43$	−73.0471	−1.69877	−0.849385	−	0.527774i	$-0.823027\pi$
$43$	−73.0471	−1.69877	−0.849385	+	0.527774i	$0.823027\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.20831i	0.174645i	0.996180	+	0.0873224i	$0.0278310\pi$
$47$	8.20831i	0.174645i	−0.996180	+	0.0873224i	$0.972169\pi$
$48$	0	0
$49$	−14.8898	−0.303874
$50$	0	0
$51$	0	0
$52$	0	0
$53$	39.1670i	0.739000i	0.929231	+	0.369500i	$0.120471\pi$
$53$	39.1670i	0.739000i	−0.929231	+	0.369500i	$0.879529\pi$
$54$	0	0
$55$	2.97991	0.0541801
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 60.4173i	− 1.02402i	−0.858979	−	0.512011i	$-0.828901\pi$
$59$	− 60.4173i	− 1.02402i	0.858979	−	0.512011i	$-0.171099\pi$
$60$	0	0
$61$	94.3077	1.54603	0.773014	−	0.634389i	$-0.218748\pi$
$61$	94.3077	1.54603	0.773014	+	0.634389i	$0.218748\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	9.83019i	0.151234i
$66$	0	0
$67$	4.58065	0.0683679	0.0341840	−	0.999416i	$-0.489117\pi$
$67$	4.58065	0.0683679	0.0341840	+	0.999416i	$0.489117\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 94.4174i	− 1.32982i	−0.746922	−	0.664911i	$-0.768469\pi$
$71$	− 94.4174i	− 1.32982i	0.746922	−	0.664911i	$-0.231531\pi$
$72$	0	0
$73$	2.60239	0.0356491	0.0178246	−	0.999841i	$-0.494326\pi$
$73$	2.60239	0.0356491	0.0178246	+	0.999841i	$0.494326\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	19.3704i	0.251564i
$78$	0	0
$79$	−21.1802	−0.268103	−0.134052	−	0.990974i	$-0.542799\pi$
$79$	−21.1802	−0.268103	−0.134052	+	0.990974i	$0.542799\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	95.0013i	1.14459i	0.820046	+	0.572297i	$0.193948\pi$
$83$	95.0013i	1.14459i	−0.820046	+	0.572297i	$0.806052\pi$
$84$	0	0
$85$	5.23586	0.0615984
$86$	0	0
$87$	0	0
$88$	0	0
$89$	58.0557i	0.652311i	0.945316	+	0.326155i	$0.105753\pi$
$89$	58.0557i	0.652311i	−0.945316	+	0.326155i	$0.894247\pi$
$90$	0	0
$91$	−63.8995	−0.702192
$92$	0	0
$93$	0	0
$94$	0	0
$95$	24.2792i	0.255571i
$96$	0	0
$97$	−11.5304	−0.118870	−0.0594351	−	0.998232i	$-0.518930\pi$
$97$	−11.5304	−0.118870	−0.0594351	+	0.998232i	$0.518930\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	157.413i	1.55854i	0.626686	+	0.779272i	$0.284411\pi$
$101$	157.413i	1.55854i	−0.626686	+	0.779272i	$0.715589\pi$
$102$	0	0
$103$	−81.8179	−0.794348	−0.397174	−	0.917743i	$-0.630009\pi$
$103$	−81.8179	−0.794348	−0.397174	+	0.917743i	$0.630009\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	61.9228i	0.578717i	0.957221	+	0.289359i	$0.0934420\pi$
$107$	61.9228i	0.578717i	−0.957221	+	0.289359i	$0.906558\pi$
$108$	0	0
$109$	−131.444	−1.20591	−0.602953	−	0.797776i	$-0.706010\pi$
$109$	−131.444	−1.20591	−0.602953	+	0.797776i	$0.706010\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	25.0028i	0.221263i	0.993861	+	0.110632i	$0.0352874\pi$
$113$	25.0028i	0.221263i	−0.993861	+	0.110632i	$0.964713\pi$
$114$	0	0
$115$	−14.3521	−0.124801
$116$	0	0
$117$	0	0
$118$	0	0
$119$	34.0348i	0.286007i
$120$	0	0
$121$	−11.0000	−0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 44.1985i	− 0.353588i
$126$	0	0
$127$	−41.5597	−0.327242	−0.163621	−	0.986523i	$-0.552317\pi$
$127$	−41.5597	−0.327242	−0.163621	+	0.986523i	$0.552317\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 135.626i	− 1.03531i	−0.855588	−	0.517657i	$-0.826804\pi$
$131$	− 135.626i	− 1.03531i	0.855588	−	0.517657i	$-0.173196\pi$
$132$	0	0
$133$	−157.823	−1.18664
$134$	0	0
$135$	0	0
$136$	0	0
$137$	132.459i	0.966852i	0.875385	+	0.483426i	$0.160608\pi$
$137$	132.459i	0.966852i	−0.875385	+	0.483426i	$0.839392\pi$
$138$	0	0
$139$	−189.749	−1.36510	−0.682549	−	0.730840i	$-0.739129\pi$
$139$	−189.749	−1.36510	−0.682549	+	0.730840i	$0.739129\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 36.2871i	− 0.253756i
$144$	0	0
$145$	42.7876	0.295087
$146$	0	0
$147$	0	0
$148$	0	0
$149$	161.453i	1.08358i	0.840515	+	0.541789i	$0.182253\pi$
$149$	161.453i	1.08358i	−0.840515	+	0.541789i	$0.817747\pi$
$150$	0	0
$151$	−213.279	−1.41245	−0.706223	−	0.707989i	$-0.749602\pi$
$151$	−213.279	−1.41245	−0.706223	+	0.707989i	$0.749602\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	13.1452i	0.0848076i
$156$	0	0
$157$	49.2024	0.313391	0.156695	−	0.987647i	$-0.449916\pi$
$157$	49.2024	0.313391	0.156695	+	0.987647i	$0.449916\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 93.2932i	− 0.579461i
$162$	0	0
$163$	121.597	0.745994	0.372997	−	0.927832i	$-0.378330\pi$
$163$	121.597	0.745994	0.372997	+	0.927832i	$0.378330\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	297.610i	1.78210i	0.453907	+	0.891049i	$0.350030\pi$
$167$	297.610i	1.78210i	−0.453907	+	0.891049i	$0.649970\pi$
$168$	0	0
$169$	−49.2954	−0.291689
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 90.9506i	− 0.525726i	−0.964833	−	0.262863i	$-0.915333\pi$
$173$	− 90.9506i	− 0.525726i	0.964833	−	0.262863i	$-0.0846667\pi$
$174$	0	0
$175$	141.295	0.807401
$176$	0	0
$177$	0	0
$178$	0	0
$179$	168.760i	0.942791i	0.881922	+	0.471396i	$0.156250\pi$
$179$	168.760i	0.942791i	−0.881922	+	0.471396i	$0.843750\pi$
$180$	0	0
$181$	−264.143	−1.45936	−0.729678	−	0.683791i	$-0.760330\pi$
$181$	−264.143	−1.45936	−0.729678	+	0.683791i	$0.760330\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	33.6163i	0.181710i
$186$	0	0
$187$	−19.3276	−0.103356
$188$	0	0
$189$	0	0
$190$	0	0
$191$	31.0641i	0.162639i	0.996688	+	0.0813197i	$0.0259135\pi$
$191$	31.0641i	0.162639i	−0.996688	+	0.0813197i	$0.974087\pi$
$192$	0	0
$193$	−130.552	−0.676434	−0.338217	−	0.941068i	$-0.609824\pi$
$193$	−130.552	−0.676434	−0.338217	+	0.941068i	$0.609824\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 85.6037i	− 0.434537i	−0.976112	−	0.217268i	$-0.930285\pi$
$197$	− 85.6037i	− 0.434537i	0.976112	−	0.217268i	$-0.0697147\pi$
$198$	0	0
$199$	−75.3159	−0.378472	−0.189236	−	0.981932i	$-0.560601\pi$
$199$	−75.3159	−0.378472	−0.189236	+	0.981932i	$0.560601\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	278.134i	1.37012i
$204$	0	0
$205$	8.27327	0.0403574
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 89.6241i	− 0.428823i
$210$	0	0
$211$	22.2069	0.105246	0.0526229	−	0.998614i	$-0.483242\pi$
$211$	22.2069	0.105246	0.0526229	+	0.998614i	$0.483242\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	65.6310i	0.305261i
$216$	0	0
$217$	−85.4481	−0.393770
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 63.7584i	− 0.288499i
$222$	0	0
$223$	−23.8473	−0.106939	−0.0534694	−	0.998569i	$-0.517028\pi$
$223$	−23.8473	−0.106939	−0.0534694	+	0.998569i	$0.517028\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	33.6882i	0.148406i	0.997243	+	0.0742032i	$0.0236413\pi$
$227$	33.6882i	0.148406i	−0.997243	+	0.0742032i	$0.976359\pi$
$228$	0	0
$229$	−334.146	−1.45915	−0.729577	−	0.683899i	$-0.760283\pi$
$229$	−334.146	−1.45915	−0.729577	+	0.683899i	$0.760283\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 300.099i	− 1.28798i	−0.765034	−	0.643990i	$-0.777278\pi$
$233$	− 300.099i	− 1.28798i	0.765034	−	0.643990i	$-0.222722\pi$
$234$	0	0
$235$	7.37497	0.0313828
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 123.358i	− 0.516142i	−0.966126	−	0.258071i	$-0.916913\pi$
$239$	− 123.358i	− 0.516142i	0.966126	−	0.258071i	$-0.0830869\pi$
$240$	0	0
$241$	24.6002	0.102076	0.0510378	−	0.998697i	$-0.483747\pi$
$241$	24.6002	0.102076	0.0510378	+	0.998697i	$0.483747\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	13.3781i	0.0546046i
$246$	0	0
$247$	295.654	1.19698
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 119.450i	− 0.475895i	−0.971278	−	0.237947i	$-0.923525\pi$
$251$	− 119.450i	− 0.475895i	0.971278	−	0.237947i	$-0.0764746\pi$
$252$	0	0
$253$	52.9791	0.209403
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 173.980i	− 0.676967i	−0.940972	−	0.338483i	$-0.890086\pi$
$257$	− 173.980i	− 0.676967i	0.940972	−	0.338483i	$-0.109914\pi$
$258$	0	0
$259$	−218.517	−0.843696
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 51.4972i	− 0.195807i	−0.995196	−	0.0979034i	$-0.968786\pi$
$263$	− 51.4972i	− 0.195807i	0.995196	−	0.0979034i	$-0.0312136\pi$
$264$	0	0
$265$	35.1906	0.132795
$266$	0	0
$267$	0	0
$268$	0	0
$269$	373.895i	1.38995i	0.719036	+	0.694973i	$0.244584\pi$
$269$	373.895i	1.38995i	−0.719036	+	0.694973i	$0.755416\pi$
$270$	0	0
$271$	280.242	1.03410	0.517052	−	0.855954i	$-0.327029\pi$
$271$	280.242	1.03410	0.517052	+	0.855954i	$0.327029\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	80.2382i	0.291775i
$276$	0	0
$277$	242.334	0.874852	0.437426	−	0.899254i	$-0.355890\pi$
$277$	242.334	0.874852	0.437426	+	0.899254i	$0.355890\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 469.811i	− 1.67192i	−0.548787	−	0.835962i	$-0.684910\pi$
$281$	− 469.811i	− 1.67192i	0.548787	−	0.835962i	$-0.315090\pi$
$282$	0	0
$283$	334.888	1.18335	0.591676	−	0.806176i	$-0.298467\pi$
$283$	334.888	1.18335	0.591676	+	0.806176i	$0.298467\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	53.7790i	0.187383i
$288$	0	0
$289$	255.040	0.882493
$290$	0	0
$291$	0	0
$292$	0	0
$293$	197.787i	0.675040i	0.941318	+	0.337520i	$0.109588\pi$
$293$	197.787i	0.675040i	−0.941318	+	0.337520i	$0.890412\pi$
$294$	0	0
$295$	−54.2835	−0.184012
$296$	0	0
$297$	0	0
$298$	0	0
$299$	174.769i	0.584510i
$300$	0	0
$301$	−426.624	−1.41735
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 84.7332i	− 0.277814i
$306$	0	0
$307$	132.940	0.433029	0.216514	−	0.976279i	$-0.430531\pi$
$307$	132.940	0.433029	0.216514	+	0.976279i	$0.430531\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	472.346i	1.51880i	0.650626	+	0.759398i	$0.274507\pi$
$311$	472.346i	1.51880i	−0.650626	+	0.759398i	$0.725493\pi$
$312$	0	0
$313$	423.461	1.35291	0.676455	−	0.736484i	$-0.263515\pi$
$313$	423.461	1.35291	0.676455	+	0.736484i	$0.263515\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 218.805i	− 0.690238i	−0.938559	−	0.345119i	$-0.887839\pi$
$317$	− 218.805i	− 0.690238i	0.938559	−	0.345119i	$-0.112161\pi$
$318$	0	0
$319$	−157.946	−0.495128
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 157.474i	− 0.487537i
$324$	0	0
$325$	−264.692	−0.814436
$326$	0	0
$327$	0	0
$328$	0	0
$329$	47.9398i	0.145714i
$330$	0	0
$331$	534.088	1.61356	0.806779	−	0.590853i	$-0.201209\pi$
$331$	534.088	1.61356	0.806779	+	0.590853i	$0.201209\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 4.11561i	− 0.0122854i
$336$	0	0
$337$	−273.875	−0.812685	−0.406343	−	0.913721i	$-0.633196\pi$
$337$	−273.875	−0.812685	−0.406343	+	0.913721i	$0.633196\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 48.5240i	− 0.142299i
$342$	0	0
$343$	−373.142	−1.08788
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 312.657i	− 0.901028i	−0.892769	−	0.450514i	$-0.851241\pi$
$347$	− 312.657i	− 0.901028i	0.892769	−	0.450514i	$-0.148759\pi$
$348$	0	0
$349$	−150.069	−0.429997	−0.214998	−	0.976614i	$-0.568975\pi$
$349$	−150.069	−0.429997	−0.214998	+	0.976614i	$0.568975\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	440.145i	1.24687i	0.781875	+	0.623435i	$0.214263\pi$
$353$	440.145i	1.24687i	−0.781875	+	0.623435i	$0.785737\pi$
$354$	0	0
$355$	−84.8318	−0.238963
$356$	0	0
$357$	0	0
$358$	0	0
$359$	27.7541i	0.0773094i	0.999253	+	0.0386547i	$0.0123072\pi$
$359$	27.7541i	0.0773094i	−0.999253	+	0.0386547i	$0.987693\pi$
$360$	0	0
$361$	369.225	1.02278
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 2.33818i	− 0.00640598i
$366$	0	0
$367$	−327.242	−0.891668	−0.445834	−	0.895116i	$-0.647093\pi$
$367$	−327.242	−0.891668	−0.445834	+	0.895116i	$0.647093\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	228.751i	0.616579i
$372$	0	0
$373$	−284.490	−0.762709	−0.381354	−	0.924429i	$-0.624542\pi$
$373$	−284.490	−0.762709	−0.381354	+	0.924429i	$0.624542\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 521.035i	− 1.38206i
$378$	0	0
$379$	42.2823	0.111563	0.0557814	−	0.998443i	$-0.482235\pi$
$379$	42.2823	0.111563	0.0557814	+	0.998443i	$0.482235\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 1.87506i	− 0.00489572i	−0.999997	−	0.00244786i	$-0.999221\pi$
$383$	− 1.87506i	− 0.00489572i	0.999997	−	0.00244786i	$-0.000779179\pi$
$384$	0	0
$385$	17.4038	0.0452048
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 73.3524i	− 0.188566i	−0.995545	−	0.0942832i	$-0.969944\pi$
$389$	− 73.3524i	− 0.188566i	0.995545	−	0.0942832i	$-0.0300559\pi$
$390$	0	0
$391$	93.0871	0.238075
$392$	0	0
$393$	0	0
$394$	0	0
$395$	19.0299i	0.0481769i
$396$	0	0
$397$	−569.032	−1.43333	−0.716665	−	0.697417i	$-0.754332\pi$
$397$	−569.032	−1.43333	−0.716665	+	0.697417i	$0.754332\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 507.615i	− 1.26587i	−0.774204	−	0.632936i	$-0.781850\pi$
$401$	− 507.615i	− 1.26587i	0.774204	−	0.632936i	$-0.218150\pi$
$402$	0	0
$403$	160.072	0.397201
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 124.091i	− 0.304892i
$408$	0	0
$409$	704.194	1.72175	0.860873	−	0.508820i	$-0.169918\pi$
$409$	704.194	1.72175	0.860873	+	0.508820i	$0.169918\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 352.861i	− 0.854384i
$414$	0	0
$415$	85.3564	0.205678
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 429.788i	− 1.02575i	−0.858464	−	0.512874i	$-0.828581\pi$
$419$	− 429.788i	− 1.02575i	0.858464	−	0.512874i	$-0.171419\pi$
$420$	0	0
$421$	581.393	1.38098	0.690490	−	0.723342i	$-0.257395\pi$
$421$	581.393	1.38098	0.690490	+	0.723342i	$0.257395\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	140.983i	0.331725i
$426$	0	0
$427$	550.794	1.28992
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 45.0355i	− 0.104491i	−0.998634	−	0.0522454i	$-0.983362\pi$
$431$	− 45.0355i	− 0.104491i	0.998634	−	0.0522454i	$-0.0166378\pi$
$432$	0	0
$433$	417.303	0.963749	0.481874	−	0.876240i	$-0.339956\pi$
$433$	417.303	0.963749	0.481874	+	0.876240i	$0.339956\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	431.655i	0.987768i
$438$	0	0
$439$	−20.8288	−0.0474461	−0.0237231	−	0.999719i	$-0.507552\pi$
$439$	−20.8288	−0.0474461	−0.0237231	+	0.999719i	$0.507552\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 508.881i	− 1.14872i	−0.818604	−	0.574358i	$-0.805252\pi$
$443$	− 508.881i	− 1.14872i	0.818604	−	0.574358i	$-0.194748\pi$
$444$	0	0
$445$	52.1616	0.117217
$446$	0	0
$447$	0	0
$448$	0	0
$449$	863.715i	1.92364i	0.273678	+	0.961821i	$0.411760\pi$
$449$	863.715i	1.92364i	−0.273678	+	0.961821i	$0.588240\pi$
$450$	0	0
$451$	−30.5399	−0.0677159
$452$	0	0
$453$	0	0
$454$	0	0
$455$	57.4122i	0.126181i
$456$	0	0
$457$	748.309	1.63744	0.818719	−	0.574194i	$-0.194685\pi$
$457$	748.309	1.63744	0.818719	+	0.574194i	$0.194685\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	458.330i	0.994209i	0.867691	+	0.497105i	$0.165603\pi$
$461$	458.330i	0.994209i	−0.867691	+	0.497105i	$0.834397\pi$
$462$	0	0
$463$	809.594	1.74858	0.874292	−	0.485400i	$-0.161326\pi$
$463$	809.594	1.74858	0.874292	+	0.485400i	$0.161326\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 389.616i	− 0.834295i	−0.908839	−	0.417148i	$-0.863030\pi$
$467$	− 389.616i	− 0.834295i	0.908839	−	0.417148i	$-0.136970\pi$
$468$	0	0
$469$	26.7528	0.0570422
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 242.270i	− 0.512198i
$474$	0	0
$475$	−653.752	−1.37632
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 156.159i	− 0.326010i	−0.986625	−	0.163005i	$-0.947881\pi$
$479$	− 156.159i	− 0.326010i	0.986625	−	0.163005i	$-0.0521187\pi$
$480$	0	0
$481$	409.354	0.851047
$482$	0	0
$483$	0	0
$484$	0	0
$485$	10.3598i	0.0213604i
$486$	0	0
$487$	5.57999	0.0114579	0.00572895	−	0.999984i	$-0.498176\pi$
$487$	5.57999	0.0114579	0.00572895	+	0.999984i	$0.498176\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 152.437i	− 0.310462i	−0.987878	−	0.155231i	$-0.950388\pi$
$491$	− 152.437i	− 0.310462i	0.987878	−	0.155231i	$-0.0496121\pi$
$492$	0	0
$493$	−277.519	−0.562919
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 551.435i	− 1.10953i
$498$	0	0
$499$	−404.485	−0.810591	−0.405295	−	0.914186i	$-0.632831\pi$
$499$	−404.485	−0.810591	−0.405295	+	0.914186i	$0.632831\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 466.860i	− 0.928150i	−0.885796	−	0.464075i	$-0.846387\pi$
$503$	− 466.860i	− 0.928150i	0.885796	−	0.464075i	$-0.153613\pi$
$504$	0	0
$505$	141.432	0.280063
$506$	0	0
$507$	0	0
$508$	0	0
$509$	483.517i	0.949935i	0.880003	+	0.474968i	$0.157540\pi$
$509$	483.517i	0.949935i	−0.880003	+	0.474968i	$0.842460\pi$
$510$	0	0
$511$	15.1990	0.0297436
$512$	0	0
$513$	0	0
$514$	0	0
$515$	73.5114i	0.142741i
$516$	0	0
$517$	−27.2239	−0.0526574
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 473.958i	− 0.909707i	−0.890566	−	0.454854i	$-0.849692\pi$
$521$	− 473.958i	− 0.909707i	0.890566	−	0.454854i	$-0.150308\pi$
$522$	0	0
$523$	−472.250	−0.902964	−0.451482	−	0.892280i	$-0.649105\pi$
$523$	−472.250	−0.902964	−0.451482	+	0.892280i	$0.649105\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 85.2594i	− 0.161782i
$528$	0	0
$529$	273.838	0.517652
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 100.746i	− 0.189016i
$534$	0	0
$535$	55.6361	0.103993
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 49.3839i	− 0.0916213i
$540$	0	0
$541$	630.810	1.16601	0.583004	−	0.812469i	$-0.301877\pi$
$541$	630.810	1.16601	0.583004	+	0.812469i	$0.301877\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	118.099i	0.216696i
$546$	0	0
$547$	−495.637	−0.906101	−0.453051	−	0.891485i	$-0.649664\pi$
$547$	−495.637	−0.906101	−0.453051	+	0.891485i	$0.649664\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 1286.89i	− 2.33554i
$552$	0	0
$553$	−123.701	−0.223690
$554$	0	0
$555$	0	0
$556$	0	0
$557$	430.245i	0.772433i	0.922408	+	0.386217i	$0.126218\pi$
$557$	430.245i	0.772433i	−0.922408	+	0.386217i	$0.873782\pi$
$558$	0	0
$559$	799.205	1.42971
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1087.63i	1.93185i	0.258827	+	0.965924i	$0.416664\pi$
$563$	1087.63i	1.93185i	−0.258827	+	0.965924i	$0.583336\pi$
$564$	0	0
$565$	22.4644	0.0397600
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 363.864i	− 0.639480i	−0.947505	−	0.319740i	$-0.896404\pi$
$569$	− 363.864i	− 0.639480i	0.947505	−	0.319740i	$-0.103596\pi$
$570$	0	0
$571$	1061.89	1.85970	0.929850	−	0.367940i	$-0.119937\pi$
$571$	1061.89	1.85970	0.929850	+	0.367940i	$0.119937\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 386.450i	− 0.672087i
$576$	0	0
$577$	−694.738	−1.20405	−0.602026	−	0.798476i	$-0.705640\pi$
$577$	−694.738	−1.20405	−0.602026	+	0.798476i	$0.705640\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	554.845i	0.954983i
$582$	0	0
$583$	−129.902	−0.222817
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1116.55i	− 1.90213i	−0.308994	−	0.951064i	$-0.599992\pi$
$587$	− 1116.55i	− 1.90213i	0.308994	−	0.951064i	$-0.400008\pi$
$588$	0	0
$589$	395.356	0.671233
$590$	0	0
$591$	0	0
$592$	0	0
$593$	420.459i	0.709037i	0.935049	+	0.354519i	$0.115355\pi$
$593$	420.459i	0.709037i	−0.935049	+	0.354519i	$0.884645\pi$
$594$	0	0
$595$	30.5795	0.0513941
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 537.195i	− 0.896820i	−0.893828	−	0.448410i	$-0.851991\pi$
$599$	− 537.195i	− 0.896820i	0.893828	−	0.448410i	$-0.148009\pi$
$600$	0	0
$601$	838.213	1.39470	0.697349	−	0.716732i	$-0.254363\pi$
$601$	838.213	1.39470	0.697349	+	0.716732i	$0.254363\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	9.88323i	0.0163359i
$606$	0	0
$607$	−410.665	−0.676548	−0.338274	−	0.941048i	$-0.609843\pi$
$607$	−410.665	−0.676548	−0.338274	+	0.941048i	$0.609843\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 89.8068i	− 0.146983i
$612$	0	0
$613$	316.930	0.517014	0.258507	−	0.966009i	$-0.416769\pi$
$613$	316.930	0.517014	0.258507	+	0.966009i	$0.416769\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	213.310i	0.345721i	0.984946	+	0.172861i	$0.0553010\pi$
$617$	213.310i	0.345721i	−0.984946	+	0.172861i	$0.944699\pi$
$618$	0	0
$619$	−767.257	−1.23951	−0.619755	−	0.784795i	$-0.712768\pi$
$619$	−767.257	−1.23951	−0.619755	+	0.784795i	$0.712768\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	339.068i	0.544250i
$624$	0	0
$625$	565.107	0.904172
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 218.034i	− 0.346637i
$630$	0	0
$631$	661.533	1.04839	0.524194	−	0.851599i	$-0.324367\pi$
$631$	661.533	1.04839	0.524194	+	0.851599i	$0.324367\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	37.3404i	0.0588037i
$636$	0	0
$637$	162.909	0.255744
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 1124.47i	− 1.75424i	−0.480275	−	0.877118i	$-0.659463\pi$
$641$	− 1124.47i	− 1.75424i	0.480275	−	0.877118i	$-0.340537\pi$
$642$	0	0
$643$	324.300	0.504354	0.252177	−	0.967681i	$-0.418853\pi$
$643$	324.300	0.504354	0.252177	+	0.967681i	$0.418853\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 282.516i	− 0.436655i	−0.975876	−	0.218327i	$-0.929940\pi$
$647$	− 282.516i	− 0.436655i	0.975876	−	0.218327i	$-0.0700601\pi$
$648$	0	0
$649$	200.381	0.308754
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 405.245i	− 0.620590i	−0.950640	−	0.310295i	$-0.899572\pi$
$653$	− 405.245i	− 0.620590i	0.950640	−	0.310295i	$-0.100428\pi$
$654$	0	0
$655$	−121.857	−0.186041
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 120.502i	− 0.182856i	−0.995812	−	0.0914280i	$-0.970857\pi$
$659$	− 120.502i	− 0.182856i	0.995812	−	0.0914280i	$-0.0291431\pi$
$660$	0	0
$661$	−1036.48	−1.56805	−0.784025	−	0.620729i	$-0.786837\pi$
$661$	−1036.48	−1.56805	−0.784025	+	0.620729i	$0.786837\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	141.800i	0.213233i
$666$	0	0
$667$	760.711	1.14050
$668$	0	0
$669$	0	0
$670$	0	0
$671$	312.783i	0.466145i
$672$	0	0
$673$	1132.60	1.68291	0.841457	−	0.540325i	$-0.181699\pi$
$673$	1132.60	1.68291	0.841457	+	0.540325i	$0.181699\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 400.478i	− 0.591548i	−0.955258	−	0.295774i	$-0.904423\pi$
$677$	− 400.478i	− 0.591548i	0.955258	−	0.295774i	$-0.0955775\pi$
$678$	0	0
$679$	−67.3421	−0.0991784
$680$	0	0
$681$	0	0
$682$	0	0
$683$	67.1523i	0.0983196i	0.998791	+	0.0491598i	$0.0156544\pi$
$683$	67.1523i	0.0983196i	−0.998791	+	0.0491598i	$0.984346\pi$
$684$	0	0
$685$	119.011	0.173739
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 428.525i	− 0.621952i
$690$	0	0
$691$	−350.058	−0.506596	−0.253298	−	0.967388i	$-0.581515\pi$
$691$	−350.058	−0.506596	−0.253298	+	0.967388i	$0.581515\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	170.485i	0.245302i
$696$	0	0
$697$	−53.6602	−0.0769874
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 405.947i	− 0.579097i	−0.957163	−	0.289549i	$-0.906495\pi$
$701$	− 405.947i	− 0.579097i	0.957163	−	0.289549i	$-0.0935052\pi$
$702$	0	0
$703$	1011.05	1.43819
$704$	0	0
$705$	0	0
$706$	0	0
$707$	919.354i	1.30036i
$708$	0	0
$709$	−387.273	−0.546224	−0.273112	−	0.961982i	$-0.588053\pi$
$709$	−387.273	−0.546224	−0.273112	+	0.961982i	$0.588053\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	233.705i	0.327777i
$714$	0	0
$715$	−32.6030	−0.0455987
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 803.931i	− 1.11812i	−0.829126	−	0.559062i	$-0.811161\pi$
$719$	− 803.931i	− 1.11812i	0.829126	−	0.559062i	$-0.188839\pi$
$720$	0	0
$721$	−477.849	−0.662758
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1152.12i	1.58913i
$726$	0	0
$727$	−831.125	−1.14323	−0.571613	−	0.820524i	$-0.693682\pi$
$727$	−831.125	−1.14323	−0.571613	+	0.820524i	$0.693682\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 425.681i	− 0.582327i
$732$	0	0
$733$	−141.098	−0.192494	−0.0962471	−	0.995357i	$-0.530684\pi$
$733$	−141.098	−0.192494	−0.0962471	+	0.995357i	$0.530684\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	15.1923i	0.0206137i
$738$	0	0
$739$	458.947	0.621038	0.310519	−	0.950567i	$-0.399497\pi$
$739$	458.947	0.621038	0.310519	+	0.950567i	$0.399497\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	220.803i	0.297177i	0.988899	+	0.148589i	$0.0474730\pi$
$743$	220.803i	0.297177i	−0.988899	+	0.148589i	$0.952527\pi$
$744$	0	0
$745$	145.062	0.194714
$746$	0	0
$747$	0	0
$748$	0	0
$749$	361.653i	0.482848i
$750$	0	0
$751$	−1027.04	−1.36757	−0.683783	−	0.729685i	$-0.739667\pi$
$751$	−1027.04	−1.36757	−0.683783	+	0.729685i	$0.739667\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	191.626i	0.253810i
$756$	0	0
$757$	−869.131	−1.14813	−0.574063	−	0.818811i	$-0.694633\pi$
$757$	−869.131	−1.14813	−0.574063	+	0.818811i	$0.694633\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1270.01i	1.66887i	0.551107	+	0.834435i	$0.314206\pi$
$761$	1270.01i	1.66887i	−0.551107	+	0.834435i	$0.685794\pi$
$762$	0	0
$763$	−767.684	−1.00614
$764$	0	0
$765$	0	0
$766$	0	0
$767$	661.023i	0.861829i
$768$	0	0
$769$	134.675	0.175130	0.0875649	−	0.996159i	$-0.472091\pi$
$769$	134.675	0.175130	0.0875649	+	0.996159i	$0.472091\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 1123.01i	− 1.45280i	−0.687272	−	0.726400i	$-0.741192\pi$
$773$	− 1123.01i	− 1.45280i	0.687272	−	0.726400i	$-0.258808\pi$
$774$	0	0
$775$	−353.953	−0.456713
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 248.828i	− 0.319420i
$780$	0	0
$781$	313.147	0.400957
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 44.2071i	− 0.0563148i
$786$	0	0
$787$	−607.426	−0.771824	−0.385912	−	0.922536i	$-0.626113\pi$
$787$	−607.426	−0.771824	−0.385912	+	0.922536i	$0.626113\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	146.026i	0.184609i
$792$	0	0
$793$	−1031.82	−1.30116
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1216.12i	1.52588i	0.646472	+	0.762938i	$0.276244\pi$
$797$	1216.12i	1.52588i	−0.646472	+	0.762938i	$0.723756\pi$
$798$	0	0
$799$	−47.8338	−0.0598671
$800$	0	0
$801$	0	0
$802$	0	0
$803$	8.63114i	0.0107486i
$804$	0	0
$805$	−83.8217	−0.104126
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 1238.74i	− 1.53120i	−0.643320	−	0.765598i	$-0.722443\pi$
$809$	− 1238.74i	− 1.53120i	0.643320	−	0.765598i	$-0.277557\pi$
$810$	0	0
$811$	319.770	0.394292	0.197146	−	0.980374i	$-0.436833\pi$
$811$	319.770	0.394292	0.197146	+	0.980374i	$0.436833\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 109.252i	− 0.134052i
$816$	0	0
$817$	1973.93	2.41607
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1300.21i	1.58369i	0.610725	+	0.791843i	$0.290878\pi$
$821$	1300.21i	1.58369i	−0.610725	+	0.791843i	$0.709122\pi$
$822$	0	0
$823$	1465.67	1.78088	0.890441	−	0.455099i	$-0.150396\pi$
$823$	1465.67	1.78088	0.890441	+	0.455099i	$0.150396\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1485.76i	1.79656i	0.439424	+	0.898280i	$0.355183\pi$
$827$	1485.76i	1.79656i	−0.439424	+	0.898280i	$0.644817\pi$
$828$	0	0
$829$	−714.538	−0.861928	−0.430964	−	0.902369i	$-0.641826\pi$
$829$	−714.538	−0.861928	−0.430964	+	0.902369i	$0.641826\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 86.7702i	− 0.104166i
$834$	0	0
$835$	267.396	0.320234
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1399.41i	1.66794i	0.551807	+	0.833972i	$0.313939\pi$
$839$	1399.41i	1.66794i	−0.551807	+	0.833972i	$0.686061\pi$
$840$	0	0
$841$	−1426.90	−1.69666
$842$	0	0
$843$	0	0
$844$	0	0
$845$	44.2907i	0.0524151i
$846$	0	0
$847$	−64.2443	−0.0758493
$848$	0	0
$849$	0	0
$850$	0	0
$851$	597.656i	0.702299i
$852$	0	0
$853$	−937.783	−1.09939	−0.549697	−	0.835364i	$-0.685257\pi$
$853$	−937.783	−1.09939	−0.549697	+	0.835364i	$0.685257\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 764.800i	− 0.892415i	−0.894929	−	0.446208i	$-0.852774\pi$
$857$	− 764.800i	− 0.892415i	0.894929	−	0.446208i	$-0.147226\pi$
$858$	0	0
$859$	565.571	0.658407	0.329203	−	0.944259i	$-0.393220\pi$
$859$	565.571	0.658407	0.329203	+	0.944259i	$0.393220\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 828.368i	− 0.959870i	−0.877304	−	0.479935i	$-0.840660\pi$
$863$	− 828.368i	− 0.959870i	0.877304	−	0.479935i	$-0.159340\pi$
$864$	0	0
$865$	−81.7169	−0.0944704
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 70.2467i	− 0.0808362i
$870$	0	0
$871$	−50.1167	−0.0575393
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 258.137i	− 0.295013i
$876$	0	0
$877$	−1168.62	−1.33252	−0.666259	−	0.745721i	$-0.732105\pi$
$877$	−1168.62	−1.33252	−0.666259	+	0.745721i	$0.732105\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 1134.19i	− 1.28738i	−0.765284	−	0.643692i	$-0.777402\pi$
$881$	− 1134.19i	− 1.28738i	0.765284	−	0.643692i	$-0.222598\pi$
$882$	0	0
$883$	−891.474	−1.00960	−0.504798	−	0.863237i	$-0.668433\pi$
$883$	−891.474	−1.00960	−0.504798	+	0.863237i	$0.668433\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1720.89i	1.94012i	0.242863	+	0.970061i	$0.421913\pi$
$887$	1720.89i	1.94012i	−0.242863	+	0.970061i	$0.578087\pi$
$888$	0	0
$889$	−242.725	−0.273031
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 221.810i	− 0.248388i
$894$	0	0
$895$	151.626	0.169415
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 696.742i	− 0.775019i
$900$	0	0
$901$	−228.245	−0.253325
$902$	0	0
$903$	0	0
$904$	0	0
$905$	237.326i	0.262239i
$906$	0	0
$907$	1704.81	1.87961	0.939806	−	0.341709i	$-0.111006\pi$
$907$	1704.81	1.87961	0.939806	+	0.341709i	$0.111006\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	457.753i	0.502473i	0.967926	+	0.251237i	$0.0808372\pi$
$911$	457.753i	0.502473i	−0.967926	+	0.251237i	$0.919163\pi$
$912$	0	0
$913$	−315.084	−0.345108
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 792.110i	− 0.863805i
$918$	0	0
$919$	−1689.93	−1.83888	−0.919440	−	0.393231i	$-0.871357\pi$
$919$	−1689.93	−1.83888	−0.919440	+	0.393231i	$0.871357\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1033.02i	1.11920i
$924$	0	0
$925$	−905.167	−0.978558
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1093.30i	− 1.17686i	−0.808548	−	0.588430i	$-0.799746\pi$
$929$	− 1093.30i	− 1.17686i	0.808548	−	0.588430i	$-0.200254\pi$
$930$	0	0
$931$	402.362	0.432183
$932$	0	0
$933$	0	0
$934$	0	0
$935$	17.3654i	0.0185726i
$936$	0	0
$937$	1615.41	1.72402	0.862010	−	0.506891i	$-0.169205\pi$
$937$	1615.41	1.72402	0.862010	+	0.506891i	$0.169205\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 646.356i	− 0.686883i	−0.939174	−	0.343441i	$-0.888407\pi$
$941$	− 646.356i	− 0.686883i	0.939174	−	0.343441i	$-0.111593\pi$
$942$	0	0
$943$	147.088	0.155979
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1036.68i	− 1.09470i	−0.836904	−	0.547350i	$-0.815637\pi$
$947$	− 1036.68i	− 1.09470i	0.836904	−	0.547350i	$-0.184363\pi$
$948$	0	0
$949$	−28.4726	−0.0300028
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 868.771i	− 0.911617i	−0.890078	−	0.455808i	$-0.849350\pi$
$953$	− 868.771i	− 0.911617i	0.890078	−	0.455808i	$-0.150650\pi$
$954$	0	0
$955$	27.9104	0.0292255
$956$	0	0
$957$	0	0
$958$	0	0
$959$	773.611i	0.806685i
$960$	0	0
$961$	−746.947	−0.777260
$962$	0	0
$963$	0	0
$964$	0	0
$965$	117.298i	0.121552i
$966$	0	0
$967$	−230.718	−0.238591	−0.119296	−	0.992859i	$-0.538064\pi$
$967$	−230.718	−0.238591	−0.119296	+	0.992859i	$0.538064\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	233.399i	0.240369i	0.992752	+	0.120185i	$0.0383487\pi$
$971$	233.399i	0.240369i	−0.992752	+	0.120185i	$0.961651\pi$
$972$	0	0
$973$	−1108.21	−1.13896
$974$	0	0
$975$	0	0
$976$	0	0
$977$	533.116i	0.545666i	0.962061	+	0.272833i	$0.0879606\pi$
$977$	533.116i	0.545666i	−0.962061	+	0.272833i	$0.912039\pi$
$978$	0	0
$979$	−192.549	−0.196679
$980$	0	0
$981$	0	0
$982$	0	0
$983$	879.218i	0.894423i	0.894428	+	0.447211i	$0.147583\pi$
$983$	879.218i	0.894423i	−0.894428	+	0.447211i	$0.852417\pi$
$984$	0	0
$985$	−76.9129	−0.0780841
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1166.84i	1.17982i
$990$	0	0
$991$	771.452	0.778458	0.389229	−	0.921141i	$-0.372741\pi$
$991$	771.452	0.778458	0.389229	+	0.921141i	$0.372741\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	67.6695i	0.0680096i
$996$	0	0
$997$	864.654	0.867256	0.433628	−	0.901092i	$-0.357233\pi$
$997$	864.654	0.867256	0.433628	+	0.901092i	$0.357233\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1584.3.i.d.881.4		8
3.2	odd	2	inner	1584.3.i.d.881.5		8
4.3	odd	2		792.3.i.a.89.4	✓	8
12.11	even	2		792.3.i.a.89.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
792.3.i.a.89.4	✓	8	4.3	odd	2
792.3.i.a.89.5	yes	8	12.11	even	2
1584.3.i.d.881.4		8	1.1	even	1	trivial
1584.3.i.d.881.5		8	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1584.i (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.898476i q^{5} +5.84039 q^{7} +3.31662i q^{11} -10.9410 q^{13} +5.82749i q^{17} -27.0227 q^{19} -15.9738i q^{23} +24.1927 q^{25} +47.6224i q^{29} -14.6305 q^{31} -5.24745i q^{35} -37.4148 q^{37} +9.20811i q^{41} -73.0471 q^{43} +8.20831i q^{47} -14.8898 q^{49} +39.1670i q^{53} +2.97991 q^{55} -60.4173i q^{59} +94.3077 q^{61} +9.83019i q^{65} +4.58065 q^{67} -94.4174i q^{71} +2.60239 q^{73} +19.3704i q^{77} -21.1802 q^{79} +95.0013i q^{83} +5.23586 q^{85} +58.0557i q^{89} -63.8995 q^{91} +24.2792i q^{95} -11.5304 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{7} + 8 q^{13} - 8 q^{19} + 16 q^{25} - 8 q^{31} + 8 q^{37} + 40 q^{43} - 160 q^{49} + 72 q^{61} + 16 q^{67} - 64 q^{79} + 240 q^{85} - 416 q^{91} - 24 q^{97}+O(q^{100}) \) 8 * q + 16 * q^7 + 8 * q^13 - 8 * q^19 + 16 * q^25 - 8 * q^31 + 8 * q^37 + 40 * q^43 - 160 * q^49 + 72 * q^61 + 16 * q^67 - 64 * q^79 + 240 * q^85 - 416 * q^91 - 24 * q^97

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