LMFDB - Embedded newform 1584.3.j.m.1297.1

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 = [12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,68] 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} + 62x^{10} + 1168x^{8} + 7496x^{6} + 15956x^{4} + 10664x^{2} + 64 $ x^12 + 62x^10 + 1168x^8 + 7496x^6 + 15956x^4 + 10664*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	no (minimal twist has level 792)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1297.1
Root			$-1.33639i$ of defining polynomial
Character	$\chi$	$=$	1584.1297
Dual form			1584.3.j.m.1297.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-9.10620 q^{5} -1.25857i q^{7} +(6.90989 + 8.55882i) q^{11} -4.36462i q^{13} -25.9613i q^{17} -22.6502i q^{19} -1.15381 q^{23} +57.9228 q^{25} +37.4221i q^{29} +14.5068 q^{31} +11.4608i q^{35} -10.5068 q^{37} +43.0789i q^{41} -56.7041i q^{43} -56.2103 q^{47} +47.4160 q^{49} -2.62859 q^{53} +(-62.9228 - 77.9383i) q^{55} -70.7646 q^{59} +39.7579i q^{61} +39.7451i q^{65} -107.339 q^{67} +41.3293 q^{71} -72.6658i q^{73} +(10.7719 - 8.69658i) q^{77} +117.885i q^{79} -52.6394i q^{83} +236.409i q^{85} -75.9958 q^{89} -5.49318 q^{91} +206.257i q^{95} +64.4160 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 68 q^{25} + 48 q^{37} + 116 q^{49} - 128 q^{55} - 208 q^{67} - 240 q^{91} + 320 q^{97}+O(q^{100}) $ 12 * q + 68 * q^25 + 48 * q^37 + 116 * q^49 - 128 * q^55 - 208 * q^67 - 240 * q^91 + 320 * 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
$5$	−9.10620			−1.82124			−0.910620	−	0.413245i	$-0.864395\pi$
$5$	−9.10620			−1.82124			−0.910620	+	0.413245i	$0.864395\pi$
$6$		0			0
$7$		−	1.25857i		−	0.179796i	−0.995951	−	0.0898979i	$-0.971346\pi$
$7$		−	1.25857i		−	0.179796i	0.995951	−	0.0898979i	$-0.0286541\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	6.90989	+	8.55882i	0.628172	+	0.778075i
$11$	6.90989	+	8.55882i	0.628172	+	0.778075i
$12$		0			0
$13$		−	4.36462i		−	0.335740i	−0.985809	−	0.167870i	$-0.946311\pi$
$13$		−	4.36462i		−	0.335740i	0.985809	−	0.167870i	$-0.0536888\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	25.9613i		−	1.52714i	−0.645728	−	0.763568i	$-0.723446\pi$
$17$		−	25.9613i		−	1.52714i	0.645728	−	0.763568i	$-0.276554\pi$
$18$		0			0
$19$		−	22.6502i		−	1.19211i	−0.802942	−	0.596057i	$-0.796733\pi$
$19$		−	22.6502i		−	1.19211i	0.802942	−	0.596057i	$-0.203267\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.15381			−0.0501657			−0.0250828	−	0.999685i	$-0.507985\pi$
$23$	−1.15381			−0.0501657			−0.0250828	+	0.999685i	$0.507985\pi$
$24$		0			0
$25$	57.9228			2.31691
$26$		0			0
$27$		0			0
$28$		0			0
$29$			37.4221i			1.29042i	0.764007	+	0.645208i	$0.223229\pi$
$29$			37.4221i			1.29042i	−0.764007	+	0.645208i	$0.776771\pi$
$30$		0			0
$31$	14.5068			0.467962			0.233981	−	0.972241i	$-0.424825\pi$
$31$	14.5068			0.467962			0.233981	+	0.972241i	$0.424825\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			11.4608i			0.327451i
$36$		0			0
$37$	−10.5068			−0.283968			−0.141984	−	0.989869i	$-0.545348\pi$
$37$	−10.5068			−0.283968			−0.141984	+	0.989869i	$0.545348\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			43.0789i			1.05071i	0.850884	+	0.525353i	$0.176067\pi$
$41$			43.0789i			1.05071i	−0.850884	+	0.525353i	$0.823933\pi$
$42$		0			0
$43$		−	56.7041i		−	1.31870i	−0.751836	−	0.659350i	$-0.770831\pi$
$43$		−	56.7041i		−	1.31870i	0.751836	−	0.659350i	$-0.229169\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−56.2103			−1.19596			−0.597982	−	0.801510i	$-0.704031\pi$
$47$	−56.2103			−1.19596			−0.597982	+	0.801510i	$0.704031\pi$
$48$		0			0
$49$	47.4160			0.967673
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.62859			−0.0495961			−0.0247980	−	0.999692i	$-0.507894\pi$
$53$	−2.62859			−0.0495961			−0.0247980	+	0.999692i	$0.507894\pi$
$54$		0			0
$55$	−62.9228	−	77.9383i	−1.14405	−	1.41706i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−70.7646			−1.19940			−0.599700	−	0.800225i	$-0.704713\pi$
$59$	−70.7646			−1.19940			−0.599700	+	0.800225i	$0.704713\pi$
$60$		0			0
$61$			39.7579i			0.651769i	0.945410	+	0.325884i	$0.105662\pi$
$61$			39.7579i			0.651769i	−0.945410	+	0.325884i	$0.894338\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			39.7451i			0.611462i
$66$		0			0
$67$	−107.339			−1.60207			−0.801036	−	0.598616i	$-0.795717\pi$
$67$	−107.339			−1.60207			−0.801036	+	0.598616i	$0.795717\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	41.3293			0.582102			0.291051	−	0.956707i	$-0.405995\pi$
$71$	41.3293			0.582102			0.291051	+	0.956707i	$0.405995\pi$
$72$		0			0
$73$		−	72.6658i		−	0.995423i	−0.867343	−	0.497711i	$-0.834174\pi$
$73$		−	72.6658i		−	0.995423i	0.867343	−	0.497711i	$-0.165826\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	10.7719	−	8.69658i	0.139894	−	0.112943i
$78$		0			0
$79$			117.885i			1.49222i	0.665822	+	0.746110i	$0.268081\pi$
$79$			117.885i			1.49222i	−0.665822	+	0.746110i	$0.731919\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	52.6394i		−	0.634209i	−0.948391	−	0.317105i	$-0.897289\pi$
$83$		−	52.6394i		−	0.634209i	0.948391	−	0.317105i	$-0.102711\pi$
$84$		0			0
$85$			236.409i			2.78128i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−75.9958			−0.853886			−0.426943	−	0.904279i	$-0.640409\pi$
$89$	−75.9958			−0.853886			−0.426943	+	0.904279i	$0.640409\pi$
$90$		0			0
$91$	−5.49318			−0.0603646
$92$		0			0
$93$		0			0
$94$		0			0
$95$			206.257i			2.17113i
$96$		0			0
$97$	64.4160			0.664082			0.332041	−	0.943265i	$-0.392263\pi$
$97$	64.4160			0.664082			0.332041	+	0.943265i	$0.392263\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			87.9112i			0.870408i	0.900332	+	0.435204i	$0.143324\pi$
$101$			87.9112i			0.870408i	−0.900332	+	0.435204i	$0.856676\pi$
$102$		0			0
$103$	−95.3388			−0.925620			−0.462810	−	0.886458i	$-0.653159\pi$
$103$	−95.3388			−0.925620			−0.462810	+	0.886458i	$0.653159\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			178.120i			1.66467i	0.554273	+	0.832335i	$0.312996\pi$
$107$			178.120i			1.66467i	−0.554273	+	0.832335i	$0.687004\pi$
$108$		0			0
$109$			139.951i			1.28395i	0.766725	+	0.641976i	$0.221885\pi$
$109$			139.951i			1.28395i	−0.766725	+	0.641976i	$0.778115\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−124.982			−1.10604			−0.553020	−	0.833168i	$-0.686525\pi$
$113$	−124.982			−1.10604			−0.553020	+	0.833168i	$0.686525\pi$
$114$		0			0
$115$	10.5068			0.0913637
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−32.6741			−0.274572
$120$		0			0
$121$	−25.5068	+	118.281i	−0.210800	+	0.977529i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−299.802			−2.39841
$126$		0			0
$127$			70.3910i			0.554260i	0.960832	+	0.277130i	$0.0893832\pi$
$127$			70.3910i			0.554260i	−0.960832	+	0.277130i	$0.910617\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			17.8530i			0.136283i	0.997676	+	0.0681414i	$0.0217069\pi$
$131$			17.8530i			0.136283i	−0.997676	+	0.0681414i	$0.978293\pi$
$132$		0			0
$133$	−28.5068			−0.214337
$134$		0			0
$135$		0			0
$136$		0			0
$137$	184.628			1.34765			0.673826	−	0.738890i	$-0.264650\pi$
$137$	184.628			1.34765			0.673826	+	0.738890i	$0.264650\pi$
$138$		0			0
$139$			134.404i			0.966937i	0.875362	+	0.483469i	$0.160623\pi$
$139$			134.404i			0.966937i	−0.875362	+	0.483469i	$0.839377\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	37.3560	−	30.1590i	0.261231	−	0.210902i
$144$		0			0
$145$		−	340.773i		−	2.35016i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			77.1858i			0.518025i	0.965874	+	0.259013i	$0.0833972\pi$
$149$			77.1858i			0.518025i	−0.965874	+	0.259013i	$0.916603\pi$
$150$		0			0
$151$		−	232.318i		−	1.53853i	−0.638929	−	0.769265i	$-0.720622\pi$
$151$		−	232.318i		−	1.53853i	0.638929	−	0.769265i	$-0.279378\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−132.102			−0.852271
$156$		0			0
$157$	−184.859			−1.17745			−0.588724	−	0.808334i	$-0.700369\pi$
$157$	−184.859			−1.17745			−0.588724	+	0.808334i	$0.700369\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			1.45215i			0.00901957i
$162$		0			0
$163$	−119.873			−0.735417			−0.367708	−	0.929941i	$-0.619858\pi$
$163$	−119.873			−0.735417			−0.367708	+	0.929941i	$0.619858\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	91.6490i		−	0.548797i	−0.961616	−	0.274398i	$-0.911521\pi$
$167$		−	91.6490i		−	0.548797i	0.961616	−	0.274398i	$-0.0884787\pi$
$168$		0			0
$169$	149.950			0.887279
$170$		0			0
$171$		0			0
$172$		0			0
$173$			174.069i			1.00618i	0.864234	+	0.503090i	$0.167803\pi$
$173$			174.069i			1.00618i	−0.864234	+	0.503090i	$0.832197\pi$
$174$		0			0
$175$		−	72.8999i		−	0.416571i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−208.740			−1.16614			−0.583072	−	0.812421i	$-0.698149\pi$
$179$	−208.740			−1.16614			−0.583072	+	0.812421i	$0.698149\pi$
$180$		0			0
$181$	−57.8184			−0.319438			−0.159719	−	0.987162i	$-0.551059\pi$
$181$	−57.8184			−0.319438			−0.159719	+	0.987162i	$0.551059\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	95.6772			0.517174
$186$		0			0
$187$	222.198	−	179.390i	1.18823	−	0.959303i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−322.277			−1.68731			−0.843656	−	0.536884i	$-0.819601\pi$
$191$	−322.277			−1.68731			−0.843656	+	0.536884i	$0.819601\pi$
$192$		0			0
$193$		−	111.907i		−	0.579831i	−0.957052	−	0.289916i	$-0.906373\pi$
$193$		−	111.907i		−	0.579831i	0.957052	−	0.289916i	$-0.0936272\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			366.990i			1.86290i	0.363875	+	0.931448i	$0.381454\pi$
$197$			366.990i			1.86290i	−0.363875	+	0.931448i	$0.618546\pi$
$198$		0			0
$199$	387.862			1.94906			0.974528	−	0.224266i	$-0.0719984\pi$
$199$	387.862			1.94906			0.974528	+	0.224266i	$0.0719984\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	47.0983			0.232011
$204$		0			0
$205$		−	392.285i		−	1.91359i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	193.859	−	156.510i	0.927554	−	0.748853i
$210$		0			0
$211$			336.912i			1.59674i	0.602167	+	0.798371i	$0.294304\pi$
$211$			336.912i			1.59674i	−0.602167	+	0.798371i	$0.705696\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			516.359i			2.40167i
$216$		0			0
$217$		−	18.2579i		−	0.0841376i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−113.311			−0.512720
$222$		0			0
$223$	−83.5205			−0.374531			−0.187266	−	0.982309i	$-0.559963\pi$
$223$	−83.5205			−0.374531			−0.187266	+	0.982309i	$0.559963\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	21.1613i		−	0.0932216i	−0.998913	−	0.0466108i	$-0.985158\pi$
$227$		−	21.1613i		−	0.0932216i	0.998913	−	0.0466108i	$-0.0148421\pi$
$228$		0			0
$229$	9.67482			0.0422481			0.0211241	−	0.999777i	$-0.493276\pi$
$229$	9.67482			0.0422481			0.0211241	+	0.999777i	$0.493276\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			58.1561i			0.249597i	0.992182	+	0.124799i	$0.0398285\pi$
$233$			58.1561i			0.249597i	−0.992182	+	0.124799i	$0.960172\pi$
$234$		0			0
$235$	511.862			2.17814
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	357.085i		−	1.49408i	−0.664780	−	0.747039i	$-0.731475\pi$
$239$		−	357.085i		−	1.49408i	0.664780	−	0.747039i	$-0.268525\pi$
$240$		0			0
$241$		−	21.6298i		−	0.0897501i	−0.998993	−	0.0448751i	$-0.985711\pi$
$241$		−	21.6298i		−	0.0897501i	0.998993	−	0.0448751i	$-0.0142890\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−431.779			−1.76237
$246$		0			0
$247$	−98.8593			−0.400240
$248$		0			0
$249$		0			0
$250$		0			0
$251$	338.274			1.34771			0.673853	−	0.738866i	$-0.264638\pi$
$251$	338.274			1.34771			0.673853	+	0.738866i	$0.264638\pi$
$252$		0			0
$253$	−7.97270	−	9.87526i	−0.0315127	−	0.0390326i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−260.167			−1.01232			−0.506162	−	0.862438i	$-0.668936\pi$
$257$	−260.167			−1.01232			−0.506162	+	0.862438i	$0.668936\pi$
$258$		0			0
$259$			13.2236i			0.0510563i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	167.684i		−	0.637581i	−0.947825	−	0.318790i	$-0.896723\pi$
$263$		−	167.684i		−	0.637581i	0.947825	−	0.318790i	$-0.103277\pi$
$264$		0			0
$265$	23.9365			0.0903263
$266$		0			0
$267$		0			0
$268$		0			0
$269$	215.463			0.800979			0.400489	−	0.916301i	$-0.368840\pi$
$269$	215.463			0.800979			0.400489	+	0.916301i	$0.368840\pi$
$270$		0			0
$271$			293.827i			1.08423i	0.840303	+	0.542117i	$0.182377\pi$
$271$			293.827i			1.08423i	−0.840303	+	0.542117i	$0.817623\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	400.240	+	495.751i	1.45542	+	1.80273i
$276$		0			0
$277$		−	130.052i		−	0.469502i	−0.972056	−	0.234751i	$-0.924573\pi$
$277$		−	130.052i		−	0.469502i	0.972056	−	0.234751i	$-0.0754274\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	148.133i		−	0.527165i	−0.964637	−	0.263583i	$-0.915096\pi$
$281$		−	148.133i		−	0.527165i	0.964637	−	0.263583i	$-0.0849041\pi$
$282$		0			0
$283$		−	201.334i		−	0.711429i	−0.934595	−	0.355714i	$-0.884238\pi$
$283$		−	201.334i		−	0.711429i	0.934595	−	0.355714i	$-0.115762\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	54.2179			0.188912
$288$		0			0
$289$	−384.989			−1.33214
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	304.856i		−	1.04046i	−0.854025	−	0.520232i	$-0.825845\pi$
$293$		−	304.856i		−	1.04046i	0.854025	−	0.520232i	$-0.174155\pi$
$294$		0			0
$295$	644.396			2.18439
$296$		0			0
$297$		0			0
$298$		0			0
$299$			5.03594i			0.0168426i
$300$		0			0
$301$	−71.3661			−0.237097
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	362.043i		−	1.18703i
$306$		0			0
$307$			286.046i			0.931746i	0.884852	+	0.465873i	$0.154260\pi$
$307$			286.046i			0.931746i	−0.884852	+	0.465873i	$0.845740\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−480.118			−1.54379			−0.771894	−	0.635751i	$-0.780691\pi$
$311$	−480.118			−1.54379			−0.771894	+	0.635751i	$0.780691\pi$
$312$		0			0
$313$	−377.057			−1.20466			−0.602328	−	0.798249i	$-0.705760\pi$
$313$	−377.057			−1.20466			−0.602328	+	0.798249i	$0.705760\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	35.2652			0.111247			0.0556233	−	0.998452i	$-0.482285\pi$
$317$	35.2652			0.111247			0.0556233	+	0.998452i	$0.482285\pi$
$318$		0			0
$319$	−320.289	+	258.583i	−1.00404	+	0.810604i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−588.028			−1.82052
$324$		0			0
$325$		−	252.811i		−	0.777880i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			70.7446i			0.215029i
$330$		0			0
$331$	−398.551			−1.20408			−0.602040	−	0.798466i	$-0.705645\pi$
$331$	−398.551			−1.20408			−0.602040	+	0.798466i	$0.705645\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	977.449			2.91776
$336$		0			0
$337$			620.674i			1.84176i	0.389841	+	0.920882i	$0.372530\pi$
$337$			620.674i			1.84176i	−0.389841	+	0.920882i	$0.627470\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	100.241	+	124.161i	0.293961	+	0.364109i
$342$		0			0
$343$		−	121.346i		−	0.353779i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	296.036i		−	0.853129i	−0.904457	−	0.426565i	$-0.859724\pi$
$347$		−	296.036i		−	0.853129i	0.904457	−	0.426565i	$-0.140276\pi$
$348$		0			0
$349$		−	452.011i		−	1.29516i	−0.761998	−	0.647580i	$-0.775781\pi$
$349$		−	452.011i		−	1.29516i	0.761998	−	0.647580i	$-0.224219\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	329.627			0.933788			0.466894	−	0.884313i	$-0.345373\pi$
$353$	329.627			0.933788			0.466894	+	0.884313i	$0.345373\pi$
$354$		0			0
$355$	−376.352			−1.06015
$356$		0			0
$357$		0			0
$358$		0			0
$359$			258.488i			0.720021i	0.932948	+	0.360011i	$0.117227\pi$
$359$			258.488i			0.720021i	−0.932948	+	0.360011i	$0.882773\pi$
$360$		0			0
$361$	−152.030			−0.421136
$362$		0			0
$363$		0			0
$364$		0			0
$365$			661.710i			1.81290i
$366$		0			0
$367$	−240.044			−0.654070			−0.327035	−	0.945012i	$-0.606050\pi$
$367$	−240.044			−0.654070			−0.327035	+	0.945012i	$0.606050\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			3.30827i			0.00891716i
$372$		0			0
$373$			211.690i			0.567534i	0.958893	+	0.283767i	$0.0915842\pi$
$373$			211.690i			0.567534i	−0.958893	+	0.283767i	$0.908416\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	163.333			0.433244
$378$		0			0
$379$	376.342			0.992986			0.496493	−	0.868041i	$-0.334621\pi$
$379$	376.342			0.992986			0.496493	+	0.868041i	$0.334621\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−79.5789			−0.207778			−0.103889	−	0.994589i	$-0.533129\pi$
$383$	−79.5789			−0.207778			−0.103889	+	0.994589i	$0.533129\pi$
$384$		0			0
$385$	−98.0908	+	79.1928i	−0.254781	+	0.205696i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−288.376			−0.741327			−0.370664	−	0.928767i	$-0.620870\pi$
$389$	−288.376			−0.741327			−0.370664	+	0.928767i	$0.620870\pi$
$390$		0			0
$391$			29.9544i			0.0766098i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	1073.49i		−	2.71769i
$396$		0			0
$397$	−145.312			−0.366024			−0.183012	−	0.983111i	$-0.558585\pi$
$397$	−145.312			−0.366024			−0.183012	+	0.983111i	$0.558585\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	541.242			1.34973			0.674865	−	0.737941i	$-0.264202\pi$
$401$	541.242			1.34973			0.674865	+	0.737941i	$0.264202\pi$
$402$		0			0
$403$		−	63.3167i		−	0.157113i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−72.6010	−	89.9260i	−0.178381	−	0.220948i
$408$		0			0
$409$			456.392i			1.11587i	0.829884	+	0.557936i	$0.188407\pi$
$409$			456.392i			1.11587i	−0.829884	+	0.557936i	$0.811593\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			89.0622i			0.215647i
$414$		0			0
$415$			479.344i			1.15505i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−276.726			−0.660443			−0.330221	−	0.943904i	$-0.607123\pi$
$419$	−276.726			−0.660443			−0.330221	+	0.943904i	$0.607123\pi$
$420$		0			0
$421$	485.074			1.15219			0.576097	−	0.817381i	$-0.304575\pi$
$421$	485.074			1.15219			0.576097	+	0.817381i	$0.304575\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	1503.75i		−	3.53824i
$426$		0			0
$427$	50.0381			0.117185
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	729.003i		−	1.69142i	−0.533640	−	0.845712i	$-0.679176\pi$
$431$		−	729.003i		−	1.69142i	0.533640	−	0.845712i	$-0.320824\pi$
$432$		0			0
$433$	−238.071			−0.549818			−0.274909	−	0.961470i	$-0.588648\pi$
$433$	−238.071			−0.549818			−0.274909	+	0.961470i	$0.588648\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			26.1340i			0.0598032i
$438$		0			0
$439$			128.737i			0.293251i	0.989192	+	0.146625i	$0.0468412\pi$
$439$			128.737i			0.293251i	−0.989192	+	0.146625i	$0.953159\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	437.013			0.986486			0.493243	−	0.869892i	$-0.335811\pi$
$443$	437.013			0.986486			0.493243	+	0.869892i	$0.335811\pi$
$444$		0			0
$445$	692.033			1.55513
$446$		0			0
$447$		0			0
$448$		0			0
$449$	698.820			1.55639			0.778196	−	0.628021i	$-0.216135\pi$
$449$	698.820			1.55639			0.778196	+	0.628021i	$0.216135\pi$
$450$		0			0
$451$	−368.705	+	297.671i	−0.817528	+	0.660024i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	50.0219			0.109938
$456$		0			0
$457$			836.559i			1.83055i	0.402834	+	0.915273i	$0.368025\pi$
$457$			836.559i			1.83055i	−0.402834	+	0.915273i	$0.631975\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			343.200i			0.744469i	0.928139	+	0.372235i	$0.121408\pi$
$461$			343.200i			0.744469i	−0.928139	+	0.372235i	$0.878592\pi$
$462$		0			0
$463$	−567.808			−1.22637			−0.613183	−	0.789941i	$-0.710111\pi$
$463$	−567.808			−1.22637			−0.613183	+	0.789941i	$0.710111\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−157.448			−0.337148			−0.168574	−	0.985689i	$-0.553916\pi$
$467$	−157.448			−0.337148			−0.168574	+	0.985689i	$0.553916\pi$
$468$		0			0
$469$			135.093i			0.288046i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	485.321	−	391.819i	1.02605	−	0.828371i
$474$		0			0
$475$		−	1311.96i		−	2.76202i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			241.557i			0.504294i	0.967689	+	0.252147i	$0.0811367\pi$
$479$			241.557i			0.504294i	−0.967689	+	0.252147i	$0.918863\pi$
$480$		0			0
$481$			45.8583i			0.0953394i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−586.585			−1.20945
$486$		0			0
$487$	263.085			0.540215			0.270108	−	0.962830i	$-0.412941\pi$
$487$	263.085			0.540215			0.270108	+	0.962830i	$0.412941\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			413.060i			0.841263i	0.907232	+	0.420632i	$0.138191\pi$
$491$			413.060i			0.841263i	−0.907232	+	0.420632i	$0.861809\pi$
$492$		0			0
$493$	971.526			1.97064
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	52.0158i		−	0.104660i
$498$		0			0
$499$	−123.328			−0.247150			−0.123575	−	0.992335i	$-0.539436\pi$
$499$	−123.328			−0.247150			−0.123575	+	0.992335i	$0.539436\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	90.8038i		−	0.180525i	−0.995918	−	0.0902623i	$-0.971229\pi$
$503$		−	90.8038i		−	0.180525i	0.995918	−	0.0902623i	$-0.0287705\pi$
$504$		0			0
$505$		−	800.537i		−	1.58522i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−402.289			−0.790352			−0.395176	−	0.918605i	$-0.629316\pi$
$509$	−402.289			−0.790352			−0.395176	+	0.918605i	$0.629316\pi$
$510$		0			0
$511$	−91.4551			−0.178973
$512$		0			0
$513$		0			0
$514$		0			0
$515$	868.174			1.68578
$516$		0			0
$517$	−388.407	−	481.094i	−0.751271	−	0.930549i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−174.299			−0.334547			−0.167274	−	0.985911i	$-0.553496\pi$
$521$	−174.299			−0.334547			−0.167274	+	0.985911i	$0.553496\pi$
$522$		0			0
$523$		−	345.020i		−	0.659694i	−0.944034	−	0.329847i	$-0.893003\pi$
$523$		−	345.020i		−	0.659694i	0.944034	−	0.329847i	$-0.106997\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	376.616i		−	0.714641i
$528$		0			0
$529$	−527.669			−0.997483
$530$		0			0
$531$		0			0
$532$		0			0
$533$	188.023			0.352764
$534$		0			0
$535$		−	1621.99i		−	3.03176i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	327.639	+	405.825i	0.607865	+	0.752922i
$540$		0			0
$541$			801.666i			1.48182i	0.671603	+	0.740911i	$0.265606\pi$
$541$			801.666i			1.48182i	−0.671603	+	0.740911i	$0.734394\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	1274.42i		−	2.33838i
$546$		0			0
$547$			921.079i			1.68387i	0.539576	+	0.841937i	$0.318585\pi$
$547$			921.079i			1.68387i	−0.539576	+	0.841937i	$0.681415\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	847.617			1.53832
$552$		0			0
$553$	148.367			0.268295
$554$		0			0
$555$		0			0
$556$		0			0
$557$			218.157i			0.391664i	0.980638	+	0.195832i	$0.0627407\pi$
$557$			218.157i			0.391664i	−0.980638	+	0.195832i	$0.937259\pi$
$558$		0			0
$559$	−247.492			−0.442740
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	364.093i		−	0.646702i	−0.946279	−	0.323351i	$-0.895191\pi$
$563$		−	364.093i		−	0.646702i	0.946279	−	0.323351i	$-0.104809\pi$
$564$		0			0
$565$	1138.11			2.01436
$566$		0			0
$567$		0			0
$568$		0			0
$569$			156.954i			0.275842i	0.990443	+	0.137921i	$0.0440420\pi$
$569$			156.954i			0.275842i	−0.990443	+	0.137921i	$0.955958\pi$
$570$		0			0
$571$		−	580.979i		−	1.01748i	−0.860921	−	0.508738i	$-0.830112\pi$
$571$		−	580.979i		−	1.01748i	0.860921	−	0.508738i	$-0.169888\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−66.8320			−0.116229
$576$		0			0
$577$	941.065			1.63096			0.815481	−	0.578784i	$-0.196473\pi$
$577$	941.065			1.63096			0.815481	+	0.578784i	$0.196473\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−66.2503			−0.114028
$582$		0			0
$583$	−18.1633	−	22.4976i	−0.0311549	−	0.0385894i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	345.113			0.587926			0.293963	−	0.955817i	$-0.405026\pi$
$587$	345.113			0.587926			0.293963	+	0.955817i	$0.405026\pi$
$588$		0			0
$589$		−	328.582i		−	0.557864i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			351.117i			0.592103i	0.955172	+	0.296052i	$0.0956701\pi$
$593$			351.117i			0.592103i	−0.955172	+	0.296052i	$0.904330\pi$
$594$		0			0
$595$	297.537			0.500062
$596$		0			0
$597$		0			0
$598$		0			0
$599$	569.700			0.951085			0.475542	−	0.879693i	$-0.342252\pi$
$599$	569.700			0.951085			0.475542	+	0.879693i	$0.342252\pi$
$600$		0			0
$601$			329.704i			0.548593i	0.961645	+	0.274297i	$0.0884450\pi$
$601$			329.704i			0.548593i	−0.961645	+	0.274297i	$0.911555\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	232.270	−	1077.09i	0.383918	−	1.78031i
$606$		0			0
$607$		−	472.818i		−	0.778943i	−0.921039	−	0.389471i	$-0.872658\pi$
$607$		−	472.818i		−	0.778943i	0.921039	−	0.389471i	$-0.127342\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			245.336i			0.401533i
$612$		0			0
$613$		−	396.464i		−	0.646760i	−0.946269	−	0.323380i	$-0.895181\pi$
$613$		−	396.464i		−	0.646760i	0.946269	−	0.323380i	$-0.104819\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	867.347			1.40575			0.702874	−	0.711314i	$-0.251900\pi$
$617$	867.347			1.40575			0.702874	+	0.711314i	$0.251900\pi$
$618$		0			0
$619$	−439.648			−0.710254			−0.355127	−	0.934818i	$-0.615562\pi$
$619$	−439.648			−0.710254			−0.355127	+	0.934818i	$0.615562\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			95.6461i			0.153525i
$624$		0			0
$625$	1281.98			2.05117
$626$		0			0
$627$		0			0
$628$		0			0
$629$			272.771i			0.433658i
$630$		0			0
$631$	590.016			0.935050			0.467525	−	0.883980i	$-0.345146\pi$
$631$	590.016			0.935050			0.467525	+	0.883980i	$0.345146\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	640.994i		−	1.00944i
$636$		0			0
$637$		−	206.953i		−	0.324886i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	298.460			0.465616			0.232808	−	0.972523i	$-0.425209\pi$
$641$	298.460			0.465616			0.232808	+	0.972523i	$0.425209\pi$
$642$		0			0
$643$	−452.640			−0.703950			−0.351975	−	0.936009i	$-0.614490\pi$
$643$	−452.640			−0.703950			−0.351975	+	0.936009i	$0.614490\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1121.42			−1.73325			−0.866627	−	0.498957i	$-0.833717\pi$
$647$	−1121.42			−1.73325			−0.866627	+	0.498957i	$0.833717\pi$
$648$		0			0
$649$	−488.976	−	605.661i	−0.753429	−	0.933222i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−33.8597			−0.0518525			−0.0259262	−	0.999664i	$-0.508254\pi$
$653$	−33.8597			−0.0518525			−0.0259262	+	0.999664i	$0.508254\pi$
$654$		0			0
$655$		−	162.573i		−	0.248203i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	1279.30i		−	1.94127i	−0.240551	−	0.970637i	$-0.577328\pi$
$659$		−	1279.30i		−	1.94127i	0.240551	−	0.970637i	$-0.422672\pi$
$660$		0			0
$661$	152.133			0.230155			0.115078	−	0.993356i	$-0.463288\pi$
$661$	152.133			0.230155			0.115078	+	0.993356i	$0.463288\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	259.589			0.390359
$666$		0			0
$667$		−	43.1780i		−	0.0647346i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−340.281	+	274.723i	−0.507125	+	0.409423i
$672$		0			0
$673$			989.620i			1.47046i	0.677817	+	0.735231i	$0.262926\pi$
$673$			989.620i			1.47046i	−0.677817	+	0.735231i	$0.737074\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	584.524i		−	0.863403i	−0.902017	−	0.431701i	$-0.857913\pi$
$677$		−	584.524i		−	0.863403i	0.902017	−	0.431701i	$-0.142087\pi$
$678$		0			0
$679$		−	81.0721i		−	0.119399i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−807.158			−1.18178			−0.590892	−	0.806751i	$-0.701224\pi$
$683$	−807.158			−1.18178			−0.590892	+	0.806751i	$0.701224\pi$
$684$		0			0
$685$	−1681.26			−2.45440
$686$		0			0
$687$		0			0
$688$		0			0
$689$			11.4728i			0.0166514i
$690$		0			0
$691$	807.030			1.16792			0.583958	−	0.811784i	$-0.301503\pi$
$691$	807.030			1.16792			0.583958	+	0.811784i	$0.301503\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	1223.91i		−	1.76102i
$696$		0			0
$697$	1118.39			1.60457
$698$		0			0
$699$		0			0
$700$		0			0
$701$			809.425i			1.15467i	0.816507	+	0.577336i	$0.195908\pi$
$701$			809.425i			1.15467i	−0.816507	+	0.577336i	$0.804092\pi$
$702$		0			0
$703$			237.981i			0.338523i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	110.642			0.156496
$708$		0			0
$709$	−1300.84			−1.83475			−0.917374	−	0.398026i	$-0.869695\pi$
$709$	−1300.84			−1.83475			−0.917374	+	0.398026i	$0.869695\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−16.7381			−0.0234756
$714$		0			0
$715$	−340.171	+	274.634i	−0.475763	+	0.384104i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	286.494			0.398462			0.199231	−	0.979953i	$-0.436156\pi$
$719$	286.494			0.398462			0.199231	+	0.979953i	$0.436156\pi$
$720$		0			0
$721$			119.991i			0.166422i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			2167.59i			2.98978i
$726$		0			0
$727$	−1049.94			−1.44421			−0.722107	−	0.691781i	$-0.756826\pi$
$727$	−1049.94			−1.44421			−0.722107	+	0.691781i	$0.756826\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1472.11			−2.01383
$732$		0			0
$733$			1156.03i			1.57713i	0.614954	+	0.788563i	$0.289174\pi$
$733$			1156.03i			1.57713i	−0.614954	+	0.788563i	$0.710826\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−741.700	−	918.694i	−1.00638	−	1.24653i
$738$		0			0
$739$			382.102i			0.517053i	0.966004	+	0.258526i	$0.0832369\pi$
$739$			382.102i			0.517053i	−0.966004	+	0.258526i	$0.916763\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	32.1110i		−	0.0432180i	−0.999766	−	0.0216090i	$-0.993121\pi$
$743$		−	32.1110i		−	0.0432180i	0.999766	−	0.0216090i	$-0.00687890\pi$
$744$		0			0
$745$		−	702.869i		−	0.943448i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	224.176			0.299301
$750$		0			0
$751$	−1149.40			−1.53049			−0.765246	−	0.643738i	$-0.777383\pi$
$751$	−1149.40			−1.53049			−0.765246	+	0.643738i	$0.777383\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			2115.54i			2.80203i
$756$		0			0
$757$	−701.289			−0.926405			−0.463202	−	0.886252i	$-0.653300\pi$
$757$	−701.289			−0.926405			−0.463202	+	0.886252i	$0.653300\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	243.504i		−	0.319979i	−0.987119	−	0.159989i	$-0.948854\pi$
$761$		−	243.504i		−	0.319979i	0.987119	−	0.159989i	$-0.0511460\pi$
$762$		0			0
$763$	176.138			0.230849
$764$		0			0
$765$		0			0
$766$		0			0
$767$			308.860i			0.402686i
$768$		0			0
$769$		−	198.766i		−	0.258473i	−0.991614	−	0.129237i	$-0.958747\pi$
$769$		−	198.766i		−	0.258473i	0.991614	−	0.129237i	$-0.0412527\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	487.782			0.631024			0.315512	−	0.948922i	$-0.397824\pi$
$773$	487.782			0.631024			0.315512	+	0.948922i	$0.397824\pi$
$774$		0			0
$775$	840.276			1.08423
$776$		0			0
$777$		0			0
$778$		0			0
$779$	975.745			1.25256
$780$		0			0
$781$	285.581	+	353.730i	0.365660	+	0.452919i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1683.37			2.14441
$786$		0			0
$787$			466.423i			0.592660i	0.955086	+	0.296330i	$0.0957628\pi$
$787$			466.423i			0.592660i	−0.955086	+	0.296330i	$0.904237\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			157.299i			0.198861i
$792$		0			0
$793$	173.528			0.218825
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1352.38			−1.69683			−0.848416	−	0.529330i	$-0.822443\pi$
$797$	−1352.38			−1.69683			−0.848416	+	0.529330i	$0.822443\pi$
$798$		0			0
$799$			1459.29i			1.82640i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	621.934	−	502.113i	0.774513	−	0.625296i
$804$		0			0
$805$		−	13.2236i		−	0.0164268i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			180.043i			0.222551i	0.993790	+	0.111275i	$0.0354935\pi$
$809$			180.043i			0.222551i	−0.993790	+	0.111275i	$0.964506\pi$
$810$		0			0
$811$			798.355i			0.984408i	0.870480	+	0.492204i	$0.163809\pi$
$811$			798.355i			0.984408i	−0.870480	+	0.492204i	$0.836191\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1091.59			1.33937
$816$		0			0
$817$	−1284.36			−1.57204
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	1043.41i		−	1.27090i	−0.772142	−	0.635450i	$-0.780815\pi$
$821$		−	1043.41i		−	1.27090i	0.772142	−	0.635450i	$-0.219185\pi$
$822$		0			0
$823$	399.360			0.485250			0.242625	−	0.970120i	$-0.421992\pi$
$823$	399.360			0.485250			0.242625	+	0.970120i	$0.421992\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	663.248i		−	0.801993i	−0.916080	−	0.400996i	$-0.868664\pi$
$827$		−	663.248i		−	0.801993i	0.916080	−	0.400996i	$-0.131336\pi$
$828$		0			0
$829$	−771.532			−0.930678			−0.465339	−	0.885133i	$-0.654068\pi$
$829$	−771.532			−0.930678			−0.465339	+	0.885133i	$0.654068\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	1230.98i		−	1.47777i
$834$		0			0
$835$			834.574i			0.999490i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−342.898			−0.408698			−0.204349	−	0.978898i	$-0.565508\pi$
$839$	−342.898			−0.408698			−0.204349	+	0.978898i	$0.565508\pi$
$840$		0			0
$841$	−559.413			−0.665176
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1365.48			−1.61595
$846$		0			0
$847$	148.865	+	32.1021i	0.175756	+	0.0379010i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	12.1229			0.0142455
$852$		0			0
$853$			1046.87i			1.22728i	0.789587	+	0.613639i	$0.210295\pi$
$853$			1046.87i			1.22728i	−0.789587	+	0.613639i	$0.789705\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			317.760i			0.370782i	0.982665	+	0.185391i	$0.0593551\pi$
$857$			317.760i			0.370782i	−0.982665	+	0.185391i	$0.940645\pi$
$858$		0			0
$859$	1456.77			1.69589			0.847946	−	0.530083i	$-0.177839\pi$
$859$	1456.77			1.69589			0.847946	+	0.530083i	$0.177839\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1165.67			1.35072			0.675360	−	0.737488i	$-0.263988\pi$
$863$	1165.67			1.35072			0.675360	+	0.737488i	$0.263988\pi$
$864$		0			0
$865$		−	1585.11i		−	1.83249i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1008.96	+	814.575i	−1.16106	+	0.937371i
$870$		0			0
$871$			468.493i			0.537879i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			377.321i			0.431225i
$876$		0			0
$877$			1338.84i			1.52661i	0.646037	+	0.763306i	$0.276425\pi$
$877$			1338.84i			1.52661i	−0.646037	+	0.763306i	$0.723575\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1195.51			−1.35699			−0.678493	−	0.734606i	$-0.737367\pi$
$881$	−1195.51			−1.35699			−0.678493	+	0.734606i	$0.737367\pi$
$882$		0			0
$883$	−1333.48			−1.51016			−0.755082	−	0.655630i	$-0.772403\pi$
$883$	−1333.48			−1.51016			−0.755082	+	0.655630i	$0.772403\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			1594.14i			1.79723i	0.438741	+	0.898613i	$0.355424\pi$
$887$			1594.14i			1.79723i	−0.438741	+	0.898613i	$0.644576\pi$
$888$		0			0
$889$	88.5920			0.0996535
$890$		0			0
$891$		0			0
$892$		0			0
$893$			1273.17i			1.42573i
$894$		0			0
$895$	1900.83			2.12383
$896$		0			0
$897$		0			0
$898$		0			0
$899$			542.876i			0.603866i
$900$		0			0
$901$			68.2417i			0.0757399i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	526.505			0.581774
$906$		0			0
$907$	−1203.54			−1.32694			−0.663471	−	0.748202i	$-0.730917\pi$
$907$	−1203.54			−1.32694			−0.663471	+	0.748202i	$0.730917\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1221.10			−1.34039			−0.670197	−	0.742184i	$-0.733790\pi$
$911$	−1221.10			−1.34039			−0.670197	+	0.742184i	$0.733790\pi$
$912$		0			0
$913$	450.531	−	363.732i	0.493462	−	0.398392i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	22.4693			0.0245030
$918$		0			0
$919$			559.507i			0.608822i	0.952541	+	0.304411i	$0.0984596\pi$
$919$			559.507i			0.608822i	−0.952541	+	0.304411i	$0.901540\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	180.386i		−	0.195435i
$924$		0			0
$925$	−608.585			−0.657930
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−552.578			−0.594809			−0.297405	−	0.954752i	$-0.596121\pi$
$929$	−552.578			−0.594809			−0.297405	+	0.954752i	$0.596121\pi$
$930$		0			0
$931$		−	1073.98i		−	1.15358i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−2023.38	+	1633.56i	−2.16404	+	1.74712i
$936$		0			0
$937$		−	1326.72i		−	1.41593i	−0.706249	−	0.707963i	$-0.749614\pi$
$937$		−	1326.72i		−	1.41593i	0.706249	−	0.707963i	$-0.250386\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			254.810i			0.270787i	0.990792	+	0.135393i	$0.0432298\pi$
$941$			254.810i			0.270787i	−0.990792	+	0.135393i	$0.956770\pi$
$942$		0			0
$943$		−	49.7049i		−	0.0527094i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1118.78			−1.18139			−0.590697	−	0.806893i	$-0.701147\pi$
$947$	−1118.78			−1.18139			−0.590697	+	0.806893i	$0.701147\pi$
$948$		0			0
$949$	−317.159			−0.334203
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1253.24i		−	1.31504i	−0.753436	−	0.657521i	$-0.771605\pi$
$953$		−	1253.24i		−	1.31504i	0.753436	−	0.657521i	$-0.228395\pi$
$954$		0			0
$955$	2934.71			3.07300
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	232.368i		−	0.242302i
$960$		0			0
$961$	−750.552			−0.781011
$962$		0			0
$963$		0			0
$964$		0			0
$965$			1019.05i			1.05601i
$966$		0			0
$967$		−	416.117i		−	0.430317i	−0.976579	−	0.215159i	$-0.930973\pi$
$967$		−	416.117i		−	0.430317i	0.976579	−	0.215159i	$-0.0690268\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1842.23			1.89725			0.948623	−	0.316409i	$-0.102477\pi$
$971$	1842.23			1.89725			0.948623	+	0.316409i	$0.102477\pi$
$972$		0			0
$973$	169.157			0.173851
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−282.069			−0.288710			−0.144355	−	0.989526i	$-0.546111\pi$
$977$	−282.069			−0.288710			−0.144355	+	0.989526i	$0.546111\pi$
$978$		0			0
$979$	−525.123	−	650.435i	−0.536387	−	0.664387i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1132.30			1.15188			0.575941	−	0.817491i	$-0.304636\pi$
$983$	1132.30			1.15188			0.575941	+	0.817491i	$0.304636\pi$
$984$		0			0
$985$		−	3341.89i		−	3.39278i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			65.4258i			0.0661535i
$990$		0			0
$991$	−715.520			−0.722019			−0.361009	−	0.932562i	$-0.617568\pi$
$991$	−715.520			−0.722019			−0.361009	+	0.932562i	$0.617568\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−3531.95			−3.54970
$996$		0			0
$997$		−	1583.16i		−	1.58793i	−0.607966	−	0.793963i	$-0.708014\pi$
$997$		−	1583.16i		−	1.58793i	0.607966	−	0.793963i	$-0.291986\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1584.3.j.m.1297.1		12
3.2	odd	2	inner	1584.3.j.m.1297.11		12
4.3	odd	2		792.3.j.b.505.2	yes	12
11.10	odd	2	inner	1584.3.j.m.1297.2		12
12.11	even	2		792.3.j.b.505.12	yes	12
33.32	even	2	inner	1584.3.j.m.1297.12		12
44.43	even	2		792.3.j.b.505.1	✓	12
132.131	odd	2		792.3.j.b.505.11	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
792.3.j.b.505.1	✓	12	44.43	even	2
792.3.j.b.505.2	yes	12	4.3	odd	2
792.3.j.b.505.11	yes	12	132.131	odd	2
792.3.j.b.505.12	yes	12	12.11	even	2
1584.3.j.m.1297.1		12	1.1	even	1	trivial
1584.3.j.m.1297.2		12	11.10	odd	2	inner
1584.3.j.m.1297.11		12	3.2	odd	2	inner
1584.3.j.m.1297.12		12	33.32	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 $\F_{q}$
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)

\(f(q)\)	\(=\)	\(q-9.10620 q^{5} -1.25857i q^{7} +(6.90989 + 8.55882i) q^{11} -4.36462i q^{13} -25.9613i q^{17} -22.6502i q^{19} -1.15381 q^{23} +57.9228 q^{25} +37.4221i q^{29} +14.5068 q^{31} +11.4608i q^{35} -10.5068 q^{37} +43.0789i q^{41} -56.7041i q^{43} -56.2103 q^{47} +47.4160 q^{49} -2.62859 q^{53} +(-62.9228 - 77.9383i) q^{55} -70.7646 q^{59} +39.7579i q^{61} +39.7451i q^{65} -107.339 q^{67} +41.3293 q^{71} -72.6658i q^{73} +(10.7719 - 8.69658i) q^{77} +117.885i q^{79} -52.6394i q^{83} +236.409i q^{85} -75.9958 q^{89} -5.49318 q^{91} +206.257i q^{95} +64.4160 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 68 q^{25} + 48 q^{37} + 116 q^{49} - 128 q^{55} - 208 q^{67} - 240 q^{91} + 320 q^{97}+O(q^{100}) \) 12 * q + 68 * q^25 + 48 * q^37 + 116 * q^49 - 128 * q^55 - 208 * q^67 - 240 * q^91 + 320 * q^97

Introduction

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Database

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