LMFDB - Embedded newform 1584.3.j.l.1297.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1584,3,Mod(1297,1584)] 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("1584.1297"); S:= CuspForms(chi, 3); N := Newforms(S);

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

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

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,0,0,-8,0,0,0,0,0,0,0,0,0,0,0,104,0,68] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$43.1608738747$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 92x^{10} + 3321x^{8} + 59016x^{6} + 526568x^{4} + 2105440x^{2} + 2650384 $ x^12 + 92x^10 + 3321x^8 + 59016x^6 + 526568x^4 + 2105440*x^2 + 2650384
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 264)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1297.7
Root			$1.49101i$ of defining polynomial
Character	$\chi$	$=$	1584.1297
Dual form			1584.3.j.l.1297.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.84066 q^{5} -10.7489i q^{7} +(-10.9362 + 1.18274i) q^{11} +9.92254i q^{13} -4.87165i q^{17} -12.8316i q^{19} +25.9455 q^{23} -16.9307 q^{25} -21.2097i q^{29} -36.5216 q^{31} -30.5339i q^{35} -33.1150 q^{37} +59.5078i q^{41} -51.9780i q^{43} +27.3249 q^{47} -66.5388 q^{49} -11.7521 q^{53} +(-31.0661 + 3.35976i) q^{55} -52.6417 q^{59} -10.2767i q^{61} +28.1865i q^{65} -111.566 q^{67} +41.4554 q^{71} +74.5876i q^{73} +(12.7131 + 117.552i) q^{77} +140.887i q^{79} -159.940i q^{83} -13.8387i q^{85} +100.338 q^{89} +106.656 q^{91} -36.4502i q^{95} -76.0017 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8 q^{11} + 104 q^{23} + 68 q^{25} - 48 q^{31} + 48 q^{37} - 24 q^{47} + 20 q^{49} - 48 q^{53} + 256 q^{55} - 304 q^{59} - 208 q^{67} - 216 q^{71} - 144 q^{77} + 552 q^{89} + 48 q^{91} - 400 q^{97}+O(q^{100}) $ 12 * q - 8 * q^11 + 104 * q^23 + 68 * q^25 - 48 * q^31 + 48 * q^37 - 24 * q^47 + 20 * q^49 - 48 * q^53 + 256 * q^55 - 304 * q^59 - 208 * q^67 - 216 * q^71 - 144 * q^77 + 552 * q^89 + 48 * q^91 - 400 * 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
$5$	2.84066			0.568132			0.284066	−	0.958805i	$-0.408317\pi$
$5$	2.84066			0.568132			0.284066	+	0.958805i	$0.408317\pi$
$6$		0			0
$7$		−	10.7489i		−	1.53556i	−0.640715	−	0.767778i	$-0.721362\pi$
$7$		−	10.7489i		−	1.53556i	0.640715	−	0.767778i	$-0.278638\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−10.9362	+	1.18274i	−0.994203	+	0.107522i
$11$	−10.9362	+	1.18274i	−0.994203	+	0.107522i
$12$		0			0
$13$			9.92254i			0.763272i	0.924313	+	0.381636i	$0.124639\pi$
$13$			9.92254i			0.763272i	−0.924313	+	0.381636i	$0.875361\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	4.87165i		−	0.286568i	−0.989682	−	0.143284i	$-0.954234\pi$
$17$		−	4.87165i		−	0.286568i	0.989682	−	0.143284i	$-0.0457662\pi$
$18$		0			0
$19$		−	12.8316i		−	0.675347i	−0.941263	−	0.337674i	$-0.890360\pi$
$19$		−	12.8316i		−	0.675347i	0.941263	−	0.337674i	$-0.109640\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	25.9455			1.12807			0.564033	−	0.825752i	$-0.309249\pi$
$23$	25.9455			1.12807			0.564033	+	0.825752i	$0.309249\pi$
$24$		0			0
$25$	−16.9307			−0.677226
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	21.2097i		−	0.731369i	−0.930739	−	0.365684i	$-0.880835\pi$
$29$		−	21.2097i		−	0.731369i	0.930739	−	0.365684i	$-0.119165\pi$
$30$		0			0
$31$	−36.5216			−1.17812			−0.589058	−	0.808091i	$-0.700501\pi$
$31$	−36.5216			−1.17812			−0.589058	+	0.808091i	$0.700501\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	30.5339i		−	0.872398i
$36$		0			0
$37$	−33.1150			−0.895001			−0.447500	−	0.894284i	$-0.647686\pi$
$37$	−33.1150			−0.895001			−0.447500	+	0.894284i	$0.647686\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			59.5078i			1.45141i	0.688006	+	0.725705i	$0.258486\pi$
$41$			59.5078i			1.45141i	−0.688006	+	0.725705i	$0.741514\pi$
$42$		0			0
$43$		−	51.9780i		−	1.20879i	−0.796685	−	0.604395i	$-0.793415\pi$
$43$		−	51.9780i		−	1.20879i	0.796685	−	0.604395i	$-0.206585\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	27.3249			0.581380			0.290690	−	0.956817i	$-0.406115\pi$
$47$	27.3249			0.581380			0.290690	+	0.956817i	$0.406115\pi$
$48$		0			0
$49$	−66.5388			−1.35793
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−11.7521			−0.221738			−0.110869	−	0.993835i	$-0.535363\pi$
$53$	−11.7521			−0.221738			−0.110869	+	0.993835i	$0.535363\pi$
$54$		0			0
$55$	−31.0661	+	3.35976i	−0.564838	+	0.0610865i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−52.6417			−0.892231			−0.446116	−	0.894975i	$-0.647193\pi$
$59$	−52.6417			−0.892231			−0.446116	+	0.894975i	$0.647193\pi$
$60$		0			0
$61$		−	10.2767i		−	0.168471i	−0.996446	−	0.0842353i	$-0.973155\pi$
$61$		−	10.2767i		−	0.168471i	0.996446	−	0.0842353i	$-0.0268447\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			28.1865i			0.433639i
$66$		0			0
$67$	−111.566			−1.66516			−0.832582	−	0.553902i	$-0.813138\pi$
$67$	−111.566			−1.66516			−0.832582	+	0.553902i	$0.813138\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	41.4554			0.583878			0.291939	−	0.956437i	$-0.405700\pi$
$71$	41.4554			0.583878			0.291939	+	0.956437i	$0.405700\pi$
$72$		0			0
$73$			74.5876i			1.02175i	0.859656	+	0.510874i	$0.170678\pi$
$73$			74.5876i			1.02175i	−0.859656	+	0.510874i	$0.829322\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	12.7131	+	117.552i	0.165106	+	1.52665i
$78$		0			0
$79$			140.887i			1.78338i	0.452651	+	0.891688i	$0.350478\pi$
$79$			140.887i			1.78338i	−0.452651	+	0.891688i	$0.649522\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	159.940i		−	1.92699i	−0.267735	−	0.963493i	$-0.586275\pi$
$83$		−	159.940i		−	1.92699i	0.267735	−	0.963493i	$-0.413725\pi$
$84$		0			0
$85$		−	13.8387i		−	0.162808i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	100.338			1.12739			0.563695	−	0.825983i	$-0.309379\pi$
$89$	100.338			1.12739			0.563695	+	0.825983i	$0.309379\pi$
$90$		0			0
$91$	106.656			1.17205
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	36.4502i		−	0.383686i
$96$		0			0
$97$	−76.0017			−0.783523			−0.391762	−	0.920067i	$-0.628134\pi$
$97$	−76.0017			−0.783523			−0.391762	+	0.920067i	$0.628134\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			103.725i			1.02698i	0.858096	+	0.513490i	$0.171647\pi$
$101$			103.725i			1.02698i	−0.858096	+	0.513490i	$0.828353\pi$
$102$		0			0
$103$	−166.247			−1.61404			−0.807022	−	0.590521i	$-0.798922\pi$
$103$	−166.247			−1.61404			−0.807022	+	0.590521i	$0.798922\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	0.519084i		−	0.00485125i	−0.999997	−	0.00242563i	$-0.999228\pi$
$107$		−	0.519084i		−	0.00485125i	0.999997	−	0.00242563i	$-0.000772101\pi$
$108$		0			0
$109$			5.10030i			0.0467917i	0.999726	+	0.0233959i	$0.00744782\pi$
$109$			5.10030i			0.0467917i	−0.999726	+	0.0233959i	$0.992552\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−121.417			−1.07448			−0.537242	−	0.843428i	$-0.680534\pi$
$113$	−121.417			−1.07448			−0.537242	+	0.843428i	$0.680534\pi$
$114$		0			0
$115$	73.7023			0.640890
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−52.3649			−0.440041
$120$		0			0
$121$	118.202	−	25.8694i	0.976878	−	0.213797i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−119.111			−0.952885
$126$		0			0
$127$		−	83.6600i		−	0.658740i	−0.944201	−	0.329370i	$-0.893164\pi$
$127$		−	83.6600i		−	0.658740i	0.944201	−	0.329370i	$-0.106836\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			56.7651i			0.433322i	0.976247	+	0.216661i	$0.0695166\pi$
$131$			56.7651i			0.433322i	−0.976247	+	0.216661i	$0.930483\pi$
$132$		0			0
$133$	−137.925			−1.03703
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−144.518			−1.05487			−0.527437	−	0.849594i	$-0.676847\pi$
$137$	−144.518			−1.05487			−0.527437	+	0.849594i	$0.676847\pi$
$138$		0			0
$139$		−	174.164i		−	1.25298i	−0.779430	−	0.626489i	$-0.784491\pi$
$139$		−	174.164i		−	1.25298i	0.779430	−	0.626489i	$-0.215509\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−11.7358	−	108.515i	−0.0820684	−	0.758848i
$144$		0			0
$145$		−	60.2495i		−	0.415514i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	58.6794i		−	0.393822i	−0.980421	−	0.196911i	$-0.936909\pi$
$149$		−	58.6794i		−	0.393822i	0.980421	−	0.196911i	$-0.0630910\pi$
$150$		0			0
$151$		−	205.068i		−	1.35807i	−0.734108	−	0.679033i	$-0.762399\pi$
$151$		−	205.068i		−	1.35807i	0.734108	−	0.679033i	$-0.237601\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−103.745			−0.669325
$156$		0			0
$157$	−134.810			−0.858664			−0.429332	−	0.903147i	$-0.641251\pi$
$157$	−134.810			−0.858664			−0.429332	+	0.903147i	$0.641251\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	278.886i		−	1.73221i
$162$		0			0
$163$	72.6274			0.445567			0.222783	−	0.974868i	$-0.428486\pi$
$163$	72.6274			0.445567			0.222783	+	0.974868i	$0.428486\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			268.435i			1.60739i	0.595039	+	0.803697i	$0.297137\pi$
$167$			268.435i			1.60739i	−0.595039	+	0.803697i	$0.702863\pi$
$168$		0			0
$169$	70.5432			0.417415
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	161.494i		−	0.933490i	−0.884392	−	0.466745i	$-0.845427\pi$
$173$		−	161.494i		−	0.933490i	0.884392	−	0.466745i	$-0.154573\pi$
$174$		0			0
$175$			181.986i			1.03992i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−223.528			−1.24876			−0.624380	−	0.781121i	$-0.714648\pi$
$179$	−223.528			−1.24876			−0.624380	+	0.781121i	$0.714648\pi$
$180$		0			0
$181$	81.3398			0.449391			0.224696	−	0.974429i	$-0.427861\pi$
$181$	81.3398			0.449391			0.224696	+	0.974429i	$0.427861\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−94.0684			−0.508478
$186$		0			0
$187$	5.76190	+	53.2775i	0.0308123	+	0.284907i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−44.4338			−0.232638			−0.116319	−	0.993212i	$-0.537109\pi$
$191$	−44.4338			−0.232638			−0.116319	+	0.993212i	$0.537109\pi$
$192$		0			0
$193$		−	288.171i		−	1.49311i	−0.665323	−	0.746556i	$-0.731706\pi$
$193$		−	288.171i		−	1.49311i	0.665323	−	0.746556i	$-0.268294\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	248.267i		−	1.26024i	−0.776498	−	0.630120i	$-0.783006\pi$
$197$		−	248.267i		−	1.26024i	0.776498	−	0.630120i	$-0.216994\pi$
$198$		0			0
$199$	−321.247			−1.61431			−0.807154	−	0.590340i	$-0.798994\pi$
$199$	−321.247			−1.61431			−0.807154	+	0.590340i	$0.798994\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−227.981			−1.12306
$204$		0			0
$205$			169.041i			0.824592i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	15.1764	+	140.329i	0.0726145	+	0.671432i
$210$		0			0
$211$			132.688i			0.628854i	0.949282	+	0.314427i	$0.101812\pi$
$211$			132.688i			0.628854i	−0.949282	+	0.314427i	$0.898188\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	147.652i		−	0.686752i
$216$		0			0
$217$			392.567i			1.80906i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	48.3392			0.218729
$222$		0			0
$223$	−98.0007			−0.439465			−0.219733	−	0.975560i	$-0.570518\pi$
$223$	−98.0007			−0.439465			−0.219733	+	0.975560i	$0.570518\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			405.082i			1.78450i	0.451542	+	0.892250i	$0.350874\pi$
$227$			405.082i			1.78450i	−0.451542	+	0.892250i	$0.649126\pi$
$228$		0			0
$229$	−12.1031			−0.0528519			−0.0264260	−	0.999651i	$-0.508413\pi$
$229$	−12.1031			−0.0528519			−0.0264260	+	0.999651i	$0.508413\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			27.5250i			0.118133i	0.998254	+	0.0590666i	$0.0188124\pi$
$233$			27.5250i			0.118133i	−0.998254	+	0.0590666i	$0.981188\pi$
$234$		0			0
$235$	77.6206			0.330300
$236$		0			0
$237$		0			0
$238$		0			0
$239$			392.732i			1.64323i	0.570043	+	0.821615i	$0.306927\pi$
$239$			392.732i			1.64323i	−0.570043	+	0.821615i	$0.693073\pi$
$240$		0			0
$241$		−	210.803i		−	0.874699i	−0.899292	−	0.437350i	$-0.855917\pi$
$241$		−	210.803i		−	0.874699i	0.899292	−	0.437350i	$-0.144083\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−189.014			−0.771486
$246$		0			0
$247$	127.322			0.515474
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−44.2332			−0.176228			−0.0881139	−	0.996110i	$-0.528084\pi$
$251$	−44.2332			−0.176228			−0.0881139	+	0.996110i	$0.528084\pi$
$252$		0			0
$253$	−283.746	+	30.6868i	−1.12153	+	0.121292i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	212.264			0.825930			0.412965	−	0.910747i	$-0.364493\pi$
$257$	212.264			0.825930			0.412965	+	0.910747i	$0.364493\pi$
$258$		0			0
$259$			355.950i			1.37432i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	304.576i		−	1.15808i	−0.815298	−	0.579042i	$-0.803427\pi$
$263$		−	304.576i		−	1.15808i	0.815298	−	0.579042i	$-0.196573\pi$
$264$		0			0
$265$	−33.3838			−0.125976
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−68.3318			−0.254021			−0.127011	−	0.991901i	$-0.540538\pi$
$269$	−68.3318			−0.254021			−0.127011	+	0.991901i	$0.540538\pi$
$270$		0			0
$271$		−	441.019i		−	1.62738i	−0.581301	−	0.813689i	$-0.697456\pi$
$271$		−	441.019i		−	1.62738i	0.581301	−	0.813689i	$-0.302544\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	185.158	−	20.0246i	0.673300	−	0.0728166i
$276$		0			0
$277$		−	359.153i		−	1.29658i	−0.761393	−	0.648290i	$-0.775484\pi$
$277$		−	359.153i		−	1.29658i	0.761393	−	0.648290i	$-0.224516\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			71.6698i			0.255053i	0.991835	+	0.127526i	$0.0407037\pi$
$281$			71.6698i			0.255053i	−0.991835	+	0.127526i	$0.959296\pi$
$282$		0			0
$283$			437.964i			1.54758i	0.633445	+	0.773788i	$0.281640\pi$
$283$			437.964i			1.54758i	−0.633445	+	0.773788i	$0.718360\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	639.643			2.22872
$288$		0			0
$289$	265.267			0.917879
$290$		0			0
$291$		0			0
$292$		0			0
$293$			14.9182i			0.0509154i	0.999676	+	0.0254577i	$0.00810431\pi$
$293$			14.9182i			0.0509154i	−0.999676	+	0.0254577i	$0.991896\pi$
$294$		0			0
$295$	−149.537			−0.506905
$296$		0			0
$297$		0			0
$298$		0			0
$299$			257.445i			0.861021i
$300$		0			0
$301$	−558.706			−1.85617
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	29.1926i		−	0.0957135i
$306$		0			0
$307$		−	5.23270i		−	0.0170446i	−0.999964	−	0.00852231i	$-0.997287\pi$
$307$		−	5.23270i		−	0.0170446i	0.999964	−	0.00852231i	$-0.00271277\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3.18650			0.0102460			0.00512299	−	0.999987i	$-0.498369\pi$
$311$	3.18650			0.0102460			0.00512299	+	0.999987i	$0.498369\pi$
$312$		0			0
$313$	526.784			1.68302			0.841508	−	0.540244i	$-0.181668\pi$
$313$	526.784			1.68302			0.841508	+	0.540244i	$0.181668\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−152.235			−0.480235			−0.240118	−	0.970744i	$-0.577186\pi$
$317$	−152.235			−0.480235			−0.240118	+	0.970744i	$0.577186\pi$
$318$		0			0
$319$	25.0855	+	231.954i	0.0786381	+	0.727129i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−62.5111			−0.193533
$324$		0			0
$325$		−	167.995i		−	0.516908i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	293.712i		−	0.892742i
$330$		0			0
$331$	37.9948			0.114788			0.0573939	−	0.998352i	$-0.481721\pi$
$331$	37.9948			0.114788			0.0573939	+	0.998352i	$0.481721\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−316.921			−0.946032
$336$		0			0
$337$			178.031i			0.528282i	0.964484	+	0.264141i	$0.0850885\pi$
$337$			178.031i			0.528282i	−0.964484	+	0.264141i	$0.914911\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	399.408	−	43.1955i	1.17129	−	0.126673i
$342$		0			0
$343$			188.523i			0.549629i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	654.859i		−	1.88720i	−0.331086	−	0.943601i	$-0.607415\pi$
$347$		−	654.859i		−	1.88720i	0.331086	−	0.943601i	$-0.392585\pi$
$348$		0			0
$349$			58.7665i			0.168385i	0.996449	+	0.0841927i	$0.0268311\pi$
$349$			58.7665i			0.168385i	−0.996449	+	0.0841927i	$0.973169\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	464.110			1.31476			0.657380	−	0.753559i	$-0.271665\pi$
$353$	464.110			1.31476			0.657380	+	0.753559i	$0.271665\pi$
$354$		0			0
$355$	117.760			0.331720
$356$		0			0
$357$		0			0
$358$		0			0
$359$			4.61765i			0.0128625i	0.999979	+	0.00643127i	$0.00204715\pi$
$359$			4.61765i			0.0128625i	−0.999979	+	0.00643127i	$0.997953\pi$
$360$		0			0
$361$	196.350			0.543906
$362$		0			0
$363$		0			0
$364$		0			0
$365$			211.878i			0.580487i
$366$		0			0
$367$	−224.275			−0.611102			−0.305551	−	0.952176i	$-0.598841\pi$
$367$	−224.275			−0.611102			−0.305551	+	0.952176i	$0.598841\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			126.322i			0.340492i
$372$		0			0
$373$		−	700.624i		−	1.87835i	−0.343442	−	0.939174i	$-0.611593\pi$
$373$		−	700.624i		−	1.87835i	0.343442	−	0.939174i	$-0.388407\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	210.454			0.558234
$378$		0			0
$379$	57.5884			0.151948			0.0759742	−	0.997110i	$-0.475793\pi$
$379$	57.5884			0.151948			0.0759742	+	0.997110i	$0.475793\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	130.685			0.341215			0.170607	−	0.985339i	$-0.445427\pi$
$383$	130.685			0.341215			0.170607	+	0.985339i	$0.445427\pi$
$384$		0			0
$385$	36.1137	+	333.926i	0.0938018	+	0.867341i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−337.351			−0.867226			−0.433613	−	0.901099i	$-0.642762\pi$
$389$	−337.351			−0.867226			−0.433613	+	0.901099i	$0.642762\pi$
$390$		0			0
$391$		−	126.398i		−	0.323267i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			400.211i			1.01319i
$396$		0			0
$397$	−680.023			−1.71290			−0.856452	−	0.516226i	$-0.827336\pi$
$397$	−680.023			−1.71290			−0.856452	+	0.516226i	$0.827336\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	217.814			0.543178			0.271589	−	0.962413i	$-0.412451\pi$
$401$	217.814			0.543178			0.271589	+	0.962413i	$0.412451\pi$
$402$		0			0
$403$		−	362.387i		−	0.899223i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	362.153	−	39.1664i	0.889812	−	0.0962320i
$408$		0			0
$409$		−	252.720i		−	0.617898i	−0.951079	−	0.308949i	$-0.900023\pi$
$409$		−	252.720i		−	0.617898i	0.951079	−	0.308949i	$-0.0999772\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			565.840i			1.37007i
$414$		0			0
$415$		−	454.334i		−	1.09478i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	299.848			0.715629			0.357814	−	0.933793i	$-0.383522\pi$
$419$	299.848			0.715629			0.357814	+	0.933793i	$0.383522\pi$
$420$		0			0
$421$	656.124			1.55849			0.779245	−	0.626719i	$-0.215603\pi$
$421$	656.124			1.55849			0.779245	+	0.626719i	$0.215603\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			82.4803i			0.194071i
$426$		0			0
$427$	−110.463			−0.258696
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	80.5448i		−	0.186879i	−0.995625	−	0.0934395i	$-0.970214\pi$
$431$		−	80.5448i		−	0.186879i	0.995625	−	0.0934395i	$-0.0297862\pi$
$432$		0			0
$433$	−10.2738			−0.0237271			−0.0118635	−	0.999930i	$-0.503776\pi$
$433$	−10.2738			−0.0237271			−0.0118635	+	0.999930i	$0.503776\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	332.922i		−	0.761836i
$438$		0			0
$439$			261.343i			0.595314i	0.954673	+	0.297657i	$0.0962052\pi$
$439$			261.343i			0.595314i	−0.954673	+	0.297657i	$0.903795\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−165.013			−0.372489			−0.186245	−	0.982503i	$-0.559632\pi$
$443$	−165.013			−0.372489			−0.186245	+	0.982503i	$0.559632\pi$
$444$		0			0
$445$	285.025			0.640506
$446$		0			0
$447$		0			0
$448$		0			0
$449$	834.916			1.85950			0.929751	−	0.368189i	$-0.120022\pi$
$449$	834.916			1.85950			0.929751	+	0.368189i	$0.120022\pi$
$450$		0			0
$451$	−70.3822	−	650.791i	−0.156058	−	1.44300i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	302.974			0.665878
$456$		0			0
$457$			180.496i			0.394959i	0.980307	+	0.197480i	$0.0632756\pi$
$457$			180.496i			0.394959i	−0.980307	+	0.197480i	$0.936724\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	662.314i		−	1.43669i	−0.695687	−	0.718345i	$-0.744900\pi$
$461$		−	662.314i		−	1.43669i	0.695687	−	0.718345i	$-0.255100\pi$
$462$		0			0
$463$	798.486			1.72459			0.862295	−	0.506405i	$-0.169026\pi$
$463$	798.486			1.72459			0.862295	+	0.506405i	$0.169026\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	630.206			1.34948			0.674738	−	0.738057i	$-0.264257\pi$
$467$	630.206			1.34948			0.674738	+	0.738057i	$0.264257\pi$
$468$		0			0
$469$			1199.21i			2.55695i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	61.4764	+	568.443i	0.129971	+	1.20178i
$474$		0			0
$475$			217.247i			0.457363i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			155.371i			0.324366i	0.986761	+	0.162183i	$0.0518535\pi$
$479$			155.371i			0.324366i	−0.986761	+	0.162183i	$0.948146\pi$
$480$		0			0
$481$		−	328.585i		−	0.683129i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−215.895			−0.445144
$486$		0			0
$487$	290.959			0.597452			0.298726	−	0.954339i	$-0.403438\pi$
$487$	290.959			0.597452			0.298726	+	0.954339i	$0.403438\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	318.229i		−	0.648124i	−0.946036	−	0.324062i	$-0.894951\pi$
$491$		−	318.229i		−	0.648124i	0.946036	−	0.324062i	$-0.105049\pi$
$492$		0			0
$493$	−103.326			−0.209587
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	445.599i		−	0.896578i
$498$		0			0
$499$	−337.814			−0.676982			−0.338491	−	0.940970i	$-0.609917\pi$
$499$	−337.814			−0.676982			−0.338491	+	0.940970i	$0.609917\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	92.1346i		−	0.183170i	−0.995797	−	0.0915851i	$-0.970807\pi$
$503$		−	92.1346i		−	0.183170i	0.995797	−	0.0915851i	$-0.0291933\pi$
$504$		0			0
$505$			294.647i			0.583459i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	293.791			0.577192			0.288596	−	0.957451i	$-0.406812\pi$
$509$	293.791			0.577192			0.288596	+	0.957451i	$0.406812\pi$
$510$		0			0
$511$	801.734			1.56895
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−472.250			−0.916990
$516$		0			0
$517$	−298.831	+	32.3182i	−0.578010	+	0.0625110i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−241.256			−0.463063			−0.231531	−	0.972827i	$-0.574374\pi$
$521$	−241.256			−0.463063			−0.231531	+	0.972827i	$0.574374\pi$
$522$		0			0
$523$		−	663.303i		−	1.26827i	−0.773224	−	0.634133i	$-0.781357\pi$
$523$		−	663.303i		−	1.26827i	0.773224	−	0.634133i	$-0.218643\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			177.921i			0.337610i
$528$		0			0
$529$	144.170			0.272532
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−590.469			−1.10782
$534$		0			0
$535$		−	1.47454i		−	0.00275615i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	727.684	−	78.6980i	1.35006	−	0.146008i
$540$		0			0
$541$			752.674i			1.39126i	0.718398	+	0.695632i	$0.244876\pi$
$541$			752.674i			1.39126i	−0.718398	+	0.695632i	$0.755124\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			14.4882i			0.0265839i
$546$		0			0
$547$		−	183.495i		−	0.335457i	−0.985833	−	0.167728i	$-0.946357\pi$
$547$		−	183.495i		−	0.335457i	0.985833	−	0.167728i	$-0.0536432\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−272.154			−0.493928
$552$		0			0
$553$	1514.38			2.73847
$554$		0			0
$555$		0			0
$556$		0			0
$557$			994.695i			1.78581i	0.450247	+	0.892904i	$0.351336\pi$
$557$			994.695i			1.78581i	−0.450247	+	0.892904i	$0.648664\pi$
$558$		0			0
$559$	515.753			0.922636
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	941.678i		−	1.67261i	−0.548266	−	0.836304i	$-0.684712\pi$
$563$		−	941.678i		−	1.67261i	0.548266	−	0.836304i	$-0.315288\pi$
$564$		0			0
$565$	−344.904			−0.610449
$566$		0			0
$567$		0			0
$568$		0			0
$569$			570.966i			1.00346i	0.865026	+	0.501728i	$0.167302\pi$
$569$			570.966i			1.00346i	−0.865026	+	0.501728i	$0.832698\pi$
$570$		0			0
$571$			819.603i			1.43538i	0.696362	+	0.717691i	$0.254801\pi$
$571$			819.603i			1.43538i	−0.696362	+	0.717691i	$0.745199\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−439.275			−0.763956
$576$		0			0
$577$	611.201			1.05927			0.529637	−	0.848224i	$-0.322328\pi$
$577$	611.201			1.05927			0.529637	+	0.848224i	$0.322328\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1719.18			−2.95900
$582$		0			0
$583$	128.524	−	13.8997i	0.220453	−	0.0238417i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−195.407			−0.332891			−0.166445	−	0.986051i	$-0.553229\pi$
$587$	−195.407			−0.332891			−0.166445	+	0.986051i	$0.553229\pi$
$588$		0			0
$589$			468.630i			0.795637i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			968.958i			1.63399i	0.576643	+	0.816996i	$0.304362\pi$
$593$			968.958i			1.63399i	−0.576643	+	0.816996i	$0.695638\pi$
$594$		0			0
$595$	−148.751			−0.250001
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1106.47			−1.84720			−0.923602	−	0.383353i	$-0.874769\pi$
$599$	−1106.47			−1.84720			−0.923602	+	0.383353i	$0.874769\pi$
$600$		0			0
$601$			502.863i			0.836710i	0.908284	+	0.418355i	$0.137393\pi$
$601$			502.863i			0.836710i	−0.908284	+	0.418355i	$0.862607\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	335.772	−	73.4862i	0.554995	−	0.121465i
$606$		0			0
$607$			516.756i			0.851328i	0.904881	+	0.425664i	$0.139960\pi$
$607$			516.756i			0.851328i	−0.904881	+	0.425664i	$0.860040\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			271.132i			0.443751i
$612$		0			0
$613$		−	560.256i		−	0.913957i	−0.889478	−	0.456979i	$-0.848932\pi$
$613$		−	560.256i		−	0.913957i	0.889478	−	0.456979i	$-0.151068\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	650.674			1.05458			0.527289	−	0.849686i	$-0.323209\pi$
$617$	650.674			1.05458			0.527289	+	0.849686i	$0.323209\pi$
$618$		0			0
$619$	1214.42			1.96191			0.980956	−	0.194231i	$-0.0622212\pi$
$619$	1214.42			1.96191			0.980956	+	0.194231i	$0.0622212\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	1078.52i		−	1.73117i
$624$		0			0
$625$	84.9139			0.135862
$626$		0			0
$627$		0			0
$628$		0			0
$629$			161.325i			0.256478i
$630$		0			0
$631$	−56.5629			−0.0896400			−0.0448200	−	0.998995i	$-0.514271\pi$
$631$	−56.5629			−0.0896400			−0.0448200	+	0.998995i	$0.514271\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	237.649i		−	0.374251i
$636$		0			0
$637$		−	660.234i		−	1.03647i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−444.698			−0.693757			−0.346878	−	0.937910i	$-0.612758\pi$
$641$	−444.698			−0.693757			−0.346878	+	0.937910i	$0.612758\pi$
$642$		0			0
$643$	−324.009			−0.503901			−0.251951	−	0.967740i	$-0.581072\pi$
$643$	−324.009			−0.503901			−0.251951	+	0.967740i	$0.581072\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1239.64			−1.91598			−0.957992	−	0.286794i	$-0.907410\pi$
$647$	−1239.64			−1.91598			−0.957992	+	0.286794i	$0.907410\pi$
$648$		0			0
$649$	575.701	−	62.2613i	0.887059	−	0.0959343i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	936.961			1.43486			0.717428	−	0.696633i	$-0.245319\pi$
$653$	936.961			1.43486			0.717428	+	0.696633i	$0.245319\pi$
$654$		0			0
$655$			161.250i			0.246184i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			443.451i			0.672915i	0.941699	+	0.336457i	$0.109229\pi$
$659$			443.451i			0.672915i	−0.941699	+	0.336457i	$0.890771\pi$
$660$		0			0
$661$	663.601			1.00393			0.501967	−	0.864887i	$-0.332610\pi$
$661$	663.601			1.00393			0.501967	+	0.864887i	$0.332610\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−391.799			−0.589172
$666$		0			0
$667$		−	550.296i		−	0.825032i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	12.1547	+	112.388i	0.0181143	+	0.167494i
$672$		0			0
$673$		−	1263.95i		−	1.87809i	−0.343797	−	0.939044i	$-0.611713\pi$
$673$		−	1263.95i		−	1.87809i	0.343797	−	0.939044i	$-0.388287\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			91.1059i			0.134573i	0.997734	+	0.0672865i	$0.0214341\pi$
$677$			91.1059i			0.134573i	−0.997734	+	0.0672865i	$0.978566\pi$
$678$		0			0
$679$			816.935i			1.20314i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	883.305			1.29327			0.646636	−	0.762798i	$-0.276175\pi$
$683$	883.305			1.29327			0.646636	+	0.762798i	$0.276175\pi$
$684$		0			0
$685$	−410.525			−0.599307
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	116.611i		−	0.169247i
$690$		0			0
$691$	−77.0201			−0.111462			−0.0557309	−	0.998446i	$-0.517749\pi$
$691$	−77.0201			−0.111462			−0.0557309	+	0.998446i	$0.517749\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	494.740i		−	0.711856i
$696$		0			0
$697$	289.901			0.415927
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	542.108i		−	0.773335i	−0.922219	−	0.386667i	$-0.873626\pi$
$701$		−	542.108i		−	0.773335i	0.922219	−	0.386667i	$-0.126374\pi$
$702$		0			0
$703$			424.919i			0.604436i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1114.93			1.57698
$708$		0			0
$709$	−851.219			−1.20059			−0.600295	−	0.799778i	$-0.704950\pi$
$709$	−851.219			−1.20059			−0.600295	+	0.799778i	$0.704950\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−947.571			−1.32899
$714$		0			0
$715$	−33.3373	−	308.255i	−0.0466256	−	0.431125i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	455.344			0.633302			0.316651	−	0.948542i	$-0.397442\pi$
$719$	455.344			0.633302			0.316651	+	0.948542i	$0.397442\pi$
$720$		0			0
$721$			1786.97i			2.47846i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			359.094i			0.495302i
$726$		0			0
$727$	−806.154			−1.10888			−0.554439	−	0.832225i	$-0.687067\pi$
$727$	−806.154			−1.10888			−0.554439	+	0.832225i	$0.687067\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−253.219			−0.346400
$732$		0			0
$733$		−	928.139i		−	1.26622i	−0.774062	−	0.633110i	$-0.781778\pi$
$733$		−	928.139i		−	1.26622i	0.774062	−	0.633110i	$-0.218222\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1220.11	−	131.953i	1.65551	−	0.179041i
$738$		0			0
$739$			264.669i			0.358145i	0.983836	+	0.179073i	$0.0573097\pi$
$739$			264.669i			0.358145i	−0.983836	+	0.179073i	$0.942690\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			807.427i			1.08671i	0.839502	+	0.543356i	$0.182847\pi$
$743$			807.427i			1.08671i	−0.839502	+	0.543356i	$0.817153\pi$
$744$		0			0
$745$		−	166.688i		−	0.223743i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−5.57958			−0.00744937
$750$		0			0
$751$	813.412			1.08310			0.541552	−	0.840667i	$-0.317837\pi$
$751$	813.412			1.08310			0.541552	+	0.840667i	$0.317837\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	582.528i		−	0.771560i
$756$		0			0
$757$	515.367			0.680801			0.340401	−	0.940280i	$-0.389437\pi$
$757$	515.367			0.680801			0.340401	+	0.940280i	$0.389437\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	1368.82i		−	1.79871i	−0.437214	−	0.899357i	$-0.644035\pi$
$761$		−	1368.82i		−	1.79871i	0.437214	−	0.899357i	$-0.355965\pi$
$762$		0			0
$763$	54.8226			0.0718514
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	522.339i		−	0.681016i
$768$		0			0
$769$		−	763.597i		−	0.992974i	−0.868044	−	0.496487i	$-0.834623\pi$
$769$		−	763.597i		−	0.992974i	0.868044	−	0.496487i	$-0.165377\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−363.115			−0.469748			−0.234874	−	0.972026i	$-0.575468\pi$
$773$	−363.115			−0.469748			−0.234874	+	0.972026i	$0.575468\pi$
$774$		0			0
$775$	618.335			0.797851
$776$		0			0
$777$		0			0
$778$		0			0
$779$	763.580			0.980205
$780$		0			0
$781$	−453.365	+	49.0309i	−0.580493	+	0.0627796i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−382.950			−0.487834
$786$		0			0
$787$		−	654.403i		−	0.831516i	−0.909475	−	0.415758i	$-0.863516\pi$
$787$		−	654.403i		−	0.831516i	0.909475	−	0.415758i	$-0.136484\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			1305.10i			1.64993i
$792$		0			0
$793$	101.971			0.128589
$794$		0			0
$795$		0			0
$796$		0			0
$797$	714.153			0.896052			0.448026	−	0.894021i	$-0.352127\pi$
$797$	714.153			0.896052			0.448026	+	0.894021i	$0.352127\pi$
$798$		0			0
$799$		−	133.117i		−	0.166605i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−88.2176	−	815.707i	−0.109860	−	1.01582i
$804$		0			0
$805$		−	792.219i		−	0.984123i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	382.349i		−	0.472620i	−0.971678	−	0.236310i	$-0.924062\pi$
$809$		−	382.349i		−	0.472620i	0.971678	−	0.236310i	$-0.0759380\pi$
$810$		0			0
$811$		−	501.644i		−	0.618550i	−0.950973	−	0.309275i	$-0.899914\pi$
$811$		−	501.644i		−	0.618550i	0.950973	−	0.309275i	$-0.100086\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	206.309			0.253140
$816$		0			0
$817$	−666.960			−0.816353
$818$		0			0
$819$		0			0
$820$		0			0
$821$			909.589i			1.10790i	0.832549	+	0.553952i	$0.186881\pi$
$821$			909.589i			1.10790i	−0.832549	+	0.553952i	$0.813119\pi$
$822$		0			0
$823$	−140.947			−0.171260			−0.0856300	−	0.996327i	$-0.527290\pi$
$823$	−140.947			−0.171260			−0.0856300	+	0.996327i	$0.527290\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			911.884i			1.10264i	0.834293	+	0.551321i	$0.185876\pi$
$827$			911.884i			1.10264i	−0.834293	+	0.551321i	$0.814124\pi$
$828$		0			0
$829$	−828.651			−0.999579			−0.499789	−	0.866147i	$-0.666589\pi$
$829$	−828.651			−0.999579			−0.499789	+	0.866147i	$0.666589\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			324.154i			0.389140i
$834$		0			0
$835$			762.532i			0.913211i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−335.872			−0.400324			−0.200162	−	0.979763i	$-0.564147\pi$
$839$	−335.872			−0.400324			−0.200162	+	0.979763i	$0.564147\pi$
$840$		0			0
$841$	391.149			0.465100
$842$		0			0
$843$		0			0
$844$		0			0
$845$	200.389			0.237147
$846$		0			0
$847$	−278.068	−	1270.54i	−0.328297	−	1.50005i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−859.186			−1.00962
$852$		0			0
$853$			931.470i			1.09199i	0.837787	+	0.545997i	$0.183849\pi$
$853$			931.470i			1.09199i	−0.837787	+	0.545997i	$0.816151\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	844.370i		−	0.985262i	−0.870238	−	0.492631i	$-0.836035\pi$
$857$		−	844.370i		−	0.985262i	0.870238	−	0.492631i	$-0.163965\pi$
$858$		0			0
$859$	−1611.22			−1.87569			−0.937847	−	0.347049i	$-0.887184\pi$
$859$	−1611.22			−1.87569			−0.937847	+	0.347049i	$0.887184\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1538.99			1.78330			0.891649	−	0.452728i	$-0.149549\pi$
$863$	1538.99			1.78330			0.891649	+	0.452728i	$0.149549\pi$
$864$		0			0
$865$		−	458.749i		−	0.530345i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−166.632	−	1540.77i	−0.191752	−	1.77304i
$870$		0			0
$871$		−	1107.02i		−	1.27097i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			1280.31i			1.46321i
$876$		0			0
$877$		−	706.007i		−	0.805025i	−0.915415	−	0.402512i	$-0.868137\pi$
$877$		−	706.007i		−	0.805025i	0.915415	−	0.402512i	$-0.131863\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1216.22			1.38050			0.690249	−	0.723572i	$-0.257501\pi$
$881$	1216.22			1.38050			0.690249	+	0.723572i	$0.257501\pi$
$882$		0			0
$883$	268.067			0.303586			0.151793	−	0.988412i	$-0.451495\pi$
$883$	268.067			0.303586			0.151793	+	0.988412i	$0.451495\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			215.223i			0.242642i	0.992613	+	0.121321i	$0.0387130\pi$
$887$			215.223i			0.242642i	−0.992613	+	0.121321i	$0.961287\pi$
$888$		0			0
$889$	−899.253			−1.01153
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	350.621i		−	0.392633i
$894$		0			0
$895$	−634.967			−0.709460
$896$		0			0
$897$		0			0
$898$		0			0
$899$			774.612i			0.861637i
$900$		0			0
$901$			57.2523i			0.0635430i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	231.059			0.255313
$906$		0			0
$907$	−1017.47			−1.12180			−0.560899	−	0.827884i	$-0.689545\pi$
$907$	−1017.47			−1.12180			−0.560899	+	0.827884i	$0.689545\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1430.88			−1.57067			−0.785334	−	0.619072i	$-0.787509\pi$
$911$	−1430.88			−1.57067			−0.785334	+	0.619072i	$0.787509\pi$
$912$		0			0
$913$	189.167	+	1749.14i	0.207193	+	1.91581i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	610.163			0.665390
$918$		0			0
$919$		−	470.346i		−	0.511802i	−0.966703	−	0.255901i	$-0.917628\pi$
$919$		−	470.346i		−	0.511802i	0.966703	−	0.255901i	$-0.0823721\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			411.343i			0.445658i
$924$		0			0
$925$	560.659			0.606118
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−343.074			−0.369294			−0.184647	−	0.982805i	$-0.559114\pi$
$929$	−343.074			−0.369294			−0.184647	+	0.982805i	$0.559114\pi$
$930$		0			0
$931$			853.799i			0.917077i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	16.3676	+	151.343i	0.0175054	+	0.161864i
$936$		0			0
$937$			283.393i			0.302447i	0.988500	+	0.151224i	$0.0483214\pi$
$937$			283.393i			0.302447i	−0.988500	+	0.151224i	$0.951679\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	110.847i		−	0.117797i	−0.998264	−	0.0588985i	$-0.981241\pi$
$941$		−	110.847i		−	0.117797i	0.998264	−	0.0588985i	$-0.0187588\pi$
$942$		0			0
$943$			1543.96i			1.63729i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−520.487			−0.549616			−0.274808	−	0.961499i	$-0.588614\pi$
$947$	−520.487			−0.549616			−0.274808	+	0.961499i	$0.588614\pi$
$948$		0			0
$949$	−740.098			−0.779872
$950$		0			0
$951$		0			0
$952$		0			0
$953$			558.050i			0.585572i	0.956178	+	0.292786i	$0.0945824\pi$
$953$			558.050i			0.585572i	−0.956178	+	0.292786i	$0.905418\pi$
$954$		0			0
$955$	−126.221			−0.132169
$956$		0			0
$957$		0			0
$958$		0			0
$959$			1553.41i			1.61982i
$960$		0			0
$961$	372.826			0.387956
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	818.594i		−	0.848284i
$966$		0			0
$967$			1275.82i			1.31936i	0.751547	+	0.659680i	$0.229308\pi$
$967$			1275.82i			1.31936i	−0.751547	+	0.659680i	$0.770692\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	299.701			0.308651			0.154326	−	0.988020i	$-0.450679\pi$
$971$	299.701			0.308651			0.154326	+	0.988020i	$0.450679\pi$
$972$		0			0
$973$	−1872.07			−1.92402
$974$		0			0
$975$		0			0
$976$		0			0
$977$	92.5462			0.0947249			0.0473624	−	0.998878i	$-0.484918\pi$
$977$	92.5462			0.0947249			0.0473624	+	0.998878i	$0.484918\pi$
$978$		0			0
$979$	−1097.32	+	118.673i	−1.12085	+	0.121219i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−325.113			−0.330736			−0.165368	−	0.986232i	$-0.552881\pi$
$983$	−325.113			−0.330736			−0.165368	+	0.986232i	$0.552881\pi$
$984$		0			0
$985$		−	705.242i		−	0.715982i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	1348.59i		−	1.36359i
$990$		0			0
$991$	769.947			0.776940			0.388470	−	0.921461i	$-0.373004\pi$
$991$	769.947			0.776940			0.388470	+	0.921461i	$0.373004\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−912.554			−0.917140
$996$		0			0
$997$			1397.01i			1.40122i	0.713545	+	0.700609i	$0.247088\pi$
$997$			1397.01i			1.40122i	−0.713545	+	0.700609i	$0.752912\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1584.3.j.l.1297.7		12
3.2	odd	2		528.3.j.d.241.3		12
4.3	odd	2		792.3.j.c.505.8		12
11.10	odd	2	inner	1584.3.j.l.1297.8		12
12.11	even	2		264.3.j.a.241.10	yes	12
24.5	odd	2		2112.3.j.g.769.9		12
24.11	even	2		2112.3.j.h.769.4		12
33.32	even	2		528.3.j.d.241.4		12
44.43	even	2		792.3.j.c.505.7		12
132.131	odd	2		264.3.j.a.241.9	✓	12
264.131	odd	2		2112.3.j.h.769.3		12
264.197	even	2		2112.3.j.g.769.10		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
264.3.j.a.241.9	✓	12	132.131	odd	2
264.3.j.a.241.10	yes	12	12.11	even	2
528.3.j.d.241.3		12	3.2	odd	2
528.3.j.d.241.4		12	33.32	even	2
792.3.j.c.505.7		12	44.43	even	2
792.3.j.c.505.8		12	4.3	odd	2
1584.3.j.l.1297.7		12	1.1	even	1	trivial
1584.3.j.l.1297.8		12	11.10	odd	2	inner
2112.3.j.g.769.9		12	24.5	odd	2
2112.3.j.g.769.10		12	264.197	even	2
2112.3.j.h.769.3		12	264.131	odd	2
2112.3.j.h.769.4		12	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.84066 q^{5} -10.7489i q^{7} +(-10.9362 + 1.18274i) q^{11} +9.92254i q^{13} -4.87165i q^{17} -12.8316i q^{19} +25.9455 q^{23} -16.9307 q^{25} -21.2097i q^{29} -36.5216 q^{31} -30.5339i q^{35} -33.1150 q^{37} +59.5078i q^{41} -51.9780i q^{43} +27.3249 q^{47} -66.5388 q^{49} -11.7521 q^{53} +(-31.0661 + 3.35976i) q^{55} -52.6417 q^{59} -10.2767i q^{61} +28.1865i q^{65} -111.566 q^{67} +41.4554 q^{71} +74.5876i q^{73} +(12.7131 + 117.552i) q^{77} +140.887i q^{79} -159.940i q^{83} -13.8387i q^{85} +100.338 q^{89} +106.656 q^{91} -36.4502i q^{95} -76.0017 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{11} + 104 q^{23} + 68 q^{25} - 48 q^{31} + 48 q^{37} - 24 q^{47} + 20 q^{49} - 48 q^{53} + 256 q^{55} - 304 q^{59} - 208 q^{67} - 216 q^{71} - 144 q^{77} + 552 q^{89} + 48 q^{91} - 400 q^{97}+O(q^{100}) \) 12 * q - 8 * q^11 + 104 * q^23 + 68 * q^25 - 48 * q^31 + 48 * q^37 - 24 * q^47 + 20 * q^49 - 48 * q^53 + 256 * q^55 - 304 * q^59 - 208 * q^67 - 216 * q^71 - 144 * q^77 + 552 * q^89 + 48 * q^91 - 400 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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