LMFDB - Embedded newform 672.3.l.a.433.30

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [672,3,Mod(433,672)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(672, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("672.433"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	672.l (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$18.3106737650$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			433.30
Character	$\chi$	$=$	672.433
Dual form			672.3.l.a.433.19

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} +5.53884 q^{5} +(5.95316 + 3.68238i) q^{7} +3.00000 q^{9} +16.1324i q^{11} +14.1524 q^{13} +9.59355 q^{15} +7.47600i q^{17} -28.5884 q^{19} +(10.3112 + 6.37806i) q^{21} -8.09656 q^{23} +5.67874 q^{25} +5.19615 q^{27} -5.91522i q^{29} +27.0356i q^{31} +27.9422i q^{33} +(32.9736 + 20.3961i) q^{35} -20.5381i q^{37} +24.5127 q^{39} -49.1096i q^{41} -42.4974i q^{43} +16.6165 q^{45} +48.6891i q^{47} +(21.8802 + 43.8435i) q^{49} +12.9488i q^{51} -77.1902i q^{53} +89.3549i q^{55} -49.5165 q^{57} -40.7234 q^{59} -10.9064 q^{61} +(17.8595 + 11.0471i) q^{63} +78.3879 q^{65} -116.132i q^{67} -14.0237 q^{69} +104.767 q^{71} +82.8647i q^{73} +9.83587 q^{75} +(-59.4057 + 96.0389i) q^{77} +44.7667 q^{79} +9.00000 q^{81} +65.8500 q^{83} +41.4083i q^{85} -10.2455i q^{87} +134.189i q^{89} +(84.2515 + 52.1145i) q^{91} +46.8270i q^{93} -158.346 q^{95} -37.4629i q^{97} +48.3973i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 96 q^{9} - 64 q^{23} + 160 q^{25} - 16 q^{49} + 96 q^{57} + 640 q^{71} + 64 q^{79} + 288 q^{81} + 768 q^{95}+O(q^{100}) $ 32 * q + 96 * q^9 - 64 * q^23 + 160 * q^25 - 16 * q^49 + 96 * q^57 + 640 * q^71 + 64 * q^79 + 288 * q^81 + 768 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$	1.73205			0.577350
$3$	1.73205			0.577350
$4$		0			0
$5$	5.53884			1.10777			0.553884	−	0.832594i	$-0.313145\pi$
$5$	5.53884			1.10777			0.553884	+	0.832594i	$0.313145\pi$
$6$		0			0
$7$	5.95316	+	3.68238i	0.850451	+	0.526054i
$7$	5.95316	+	3.68238i	0.850451	+	0.526054i
$8$		0			0
$9$	3.00000			0.333333
$10$		0			0
$11$			16.1324i			1.46658i	0.679914	+	0.733292i	$0.262017\pi$
$11$			16.1324i			1.46658i	−0.679914	+	0.733292i	$0.737983\pi$
$12$		0			0
$13$	14.1524			1.08865			0.544323	−	0.838876i	$-0.316786\pi$
$13$	14.1524			1.08865			0.544323	+	0.838876i	$0.316786\pi$
$14$		0			0
$15$	9.59355			0.639570
$16$		0			0
$17$			7.47600i			0.439764i	0.975526	+	0.219882i	$0.0705673\pi$
$17$			7.47600i			0.439764i	−0.975526	+	0.219882i	$0.929433\pi$
$18$		0			0
$19$	−28.5884			−1.50465			−0.752326	−	0.658791i	$-0.771068\pi$
$19$	−28.5884			−1.50465			−0.752326	+	0.658791i	$0.771068\pi$
$20$		0			0
$21$	10.3112	+	6.37806i	0.491008	+	0.303717i
$22$		0			0
$23$	−8.09656			−0.352024			−0.176012	−	0.984388i	$-0.556320\pi$
$23$	−8.09656			−0.352024			−0.176012	+	0.984388i	$0.556320\pi$
$24$		0			0
$25$	5.67874			0.227150
$26$		0			0
$27$	5.19615			0.192450
$28$		0			0
$29$		−	5.91522i		−	0.203973i	−0.994786	−	0.101987i	$-0.967480\pi$
$29$		−	5.91522i		−	0.203973i	0.994786	−	0.101987i	$-0.0325199\pi$
$30$		0			0
$31$			27.0356i			0.872116i	0.899919	+	0.436058i	$0.143626\pi$
$31$			27.0356i			0.872116i	−0.899919	+	0.436058i	$0.856374\pi$
$32$		0			0
$33$			27.9422i			0.846733i
$34$		0			0
$35$	32.9736	+	20.3961i	0.942103	+	0.582745i
$36$		0			0
$37$		−	20.5381i		−	0.555083i	−0.960714	−	0.277541i	$-0.910480\pi$
$37$		−	20.5381i		−	0.555083i	0.960714	−	0.277541i	$-0.0895196\pi$
$38$		0			0
$39$	24.5127			0.628530
$40$		0			0
$41$		−	49.1096i		−	1.19779i	−0.800826	−	0.598897i	$-0.795606\pi$
$41$		−	49.1096i		−	1.19779i	0.800826	−	0.598897i	$-0.204394\pi$
$42$		0			0
$43$		−	42.4974i		−	0.988311i	−0.869374	−	0.494155i	$-0.835477\pi$
$43$		−	42.4974i		−	0.988311i	0.869374	−	0.494155i	$-0.164523\pi$
$44$		0			0
$45$	16.6165			0.369256
$46$		0			0
$47$			48.6891i			1.03594i	0.855399	+	0.517969i	$0.173312\pi$
$47$			48.6891i			1.03594i	−0.855399	+	0.517969i	$0.826688\pi$
$48$		0			0
$49$	21.8802	+	43.8435i	0.446535	+	0.894766i
$50$		0			0
$51$			12.9488i			0.253898i
$52$		0			0
$53$		−	77.1902i		−	1.45642i	−0.685355	−	0.728209i	$-0.740353\pi$
$53$		−	77.1902i		−	1.45642i	0.685355	−	0.728209i	$-0.259647\pi$
$54$		0			0
$55$			89.3549i			1.62464i
$56$		0			0
$57$	−49.5165			−0.868711
$58$		0			0
$59$	−40.7234			−0.690228			−0.345114	−	0.938561i	$-0.612160\pi$
$59$	−40.7234			−0.690228			−0.345114	+	0.938561i	$0.612160\pi$
$60$		0			0
$61$	−10.9064			−0.178794			−0.0893969	−	0.995996i	$-0.528494\pi$
$61$	−10.9064			−0.178794			−0.0893969	+	0.995996i	$0.528494\pi$
$62$		0			0
$63$	17.8595	+	11.0471i	0.283484	+	0.175351i
$64$		0			0
$65$	78.3879			1.20597
$66$		0			0
$67$		−	116.132i		−	1.73331i	−0.498909	−	0.866655i	$-0.666266\pi$
$67$		−	116.132i		−	1.73331i	0.498909	−	0.866655i	$-0.333734\pi$
$68$		0			0
$69$	−14.0237			−0.203241
$70$		0			0
$71$	104.767			1.47560			0.737799	−	0.675021i	$-0.235865\pi$
$71$	104.767			1.47560			0.737799	+	0.675021i	$0.235865\pi$
$72$		0			0
$73$			82.8647i			1.13513i	0.823328	+	0.567566i	$0.192115\pi$
$73$			82.8647i			1.13513i	−0.823328	+	0.567566i	$0.807885\pi$
$74$		0			0
$75$	9.83587			0.131145
$76$		0			0
$77$	−59.4057	+	96.0389i	−0.771502	+	1.24726i
$78$		0			0
$79$	44.7667			0.566667			0.283333	−	0.959022i	$-0.408560\pi$
$79$	44.7667			0.566667			0.283333	+	0.959022i	$0.408560\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$	65.8500			0.793374			0.396687	−	0.917954i	$-0.370160\pi$
$83$	65.8500			0.793374			0.396687	+	0.917954i	$0.370160\pi$
$84$		0			0
$85$			41.4083i			0.487157i
$86$		0			0
$87$		−	10.2455i		−	0.117764i
$88$		0			0
$89$			134.189i			1.50774i	0.657025	+	0.753869i	$0.271815\pi$
$89$			134.189i			1.50774i	−0.657025	+	0.753869i	$0.728185\pi$
$90$		0			0
$91$	84.2515	+	52.1145i	0.925841	+	0.572687i
$92$		0			0
$93$			46.8270i			0.503516i
$94$		0			0
$95$	−158.346			−1.66680
$96$		0			0
$97$		−	37.4629i		−	0.386215i	−0.981178	−	0.193107i	$-0.938143\pi$
$97$		−	37.4629i		−	0.386215i	0.981178	−	0.193107i	$-0.0618566\pi$
$98$		0			0
$99$			48.3973i			0.488861i
$100$		0			0
$101$	140.538			1.39147			0.695734	−	0.718299i	$-0.255079\pi$
$101$	140.538			1.39147			0.695734	+	0.718299i	$0.255079\pi$
$102$		0			0
$103$		−	198.347i		−	1.92570i	−0.270033	−	0.962851i	$-0.587035\pi$
$103$		−	198.347i		−	1.92570i	0.270033	−	0.962851i	$-0.412965\pi$
$104$		0			0
$105$	57.1119	+	35.3271i	0.543923	+	0.336448i
$106$		0			0
$107$			21.1655i			0.197809i	0.995097	+	0.0989044i	$0.0315338\pi$
$107$			21.1655i			0.197809i	−0.995097	+	0.0989044i	$0.968466\pi$
$108$		0			0
$109$		−	107.388i		−	0.985210i	−0.870253	−	0.492605i	$-0.836045\pi$
$109$		−	107.388i		−	0.985210i	0.870253	−	0.492605i	$-0.163955\pi$
$110$		0			0
$111$		−	35.5730i		−	0.320477i
$112$		0			0
$113$	112.913			0.999234			0.499617	−	0.866247i	$-0.333474\pi$
$113$	112.913			0.999234			0.499617	+	0.866247i	$0.333474\pi$
$114$		0			0
$115$	−44.8455			−0.389961
$116$		0			0
$117$	42.4572			0.362882
$118$		0			0
$119$	−27.5294	+	44.5058i	−0.231340	+	0.373998i
$120$		0			0
$121$	−139.255			−1.15087
$122$		0			0
$123$		−	85.0603i		−	0.691547i
$124$		0			0
$125$	−107.017			−0.856139
$126$		0			0
$127$	−166.247			−1.30903			−0.654515	−	0.756049i	$-0.727127\pi$
$127$	−166.247			−1.30903			−0.654515	+	0.756049i	$0.727127\pi$
$128$		0			0
$129$		−	73.6076i		−	0.570601i
$130$		0			0
$131$	156.297			1.19311			0.596553	−	0.802574i	$-0.296536\pi$
$131$	156.297			1.19311			0.596553	+	0.802574i	$0.296536\pi$
$132$		0			0
$133$	−170.191	−	105.273i	−1.27963	−	0.791528i
$134$		0			0
$135$	28.7807			0.213190
$136$		0			0
$137$	68.1911			0.497746			0.248873	−	0.968536i	$-0.419940\pi$
$137$	68.1911			0.497746			0.248873	+	0.968536i	$0.419940\pi$
$138$		0			0
$139$	−190.458			−1.37020			−0.685101	−	0.728448i	$-0.740242\pi$
$139$	−190.458			−1.37020			−0.685101	+	0.728448i	$0.740242\pi$
$140$		0			0
$141$			84.3320i			0.598099i
$142$		0			0
$143$			228.313i			1.59659i
$144$		0			0
$145$		−	32.7635i		−	0.225955i
$146$		0			0
$147$	37.8976	+	75.9393i	0.257807	+	0.516594i
$148$		0			0
$149$			41.6094i			0.279257i	0.990204	+	0.139629i	$0.0445909\pi$
$149$			41.6094i			0.279257i	−0.990204	+	0.139629i	$0.955409\pi$
$150$		0			0
$151$	149.930			0.992914			0.496457	−	0.868061i	$-0.334634\pi$
$151$	149.930			0.992914			0.496457	+	0.868061i	$0.334634\pi$
$152$		0			0
$153$			22.4280i			0.146588i
$154$		0			0
$155$			149.746i			0.966102i
$156$		0			0
$157$	81.7752			0.520861			0.260431	−	0.965493i	$-0.416135\pi$
$157$	81.7752			0.520861			0.260431	+	0.965493i	$0.416135\pi$
$158$		0			0
$159$		−	133.697i		−	0.840864i
$160$		0			0
$161$	−48.2001	−	29.8146i	−0.299380	−	0.185184i
$162$		0			0
$163$		−	275.383i		−	1.68946i	−0.535190	−	0.844732i	$-0.679760\pi$
$163$		−	275.383i		−	1.68946i	0.535190	−	0.844732i	$-0.320240\pi$
$164$		0			0
$165$			154.767i			0.937983i
$166$		0			0
$167$		−	263.827i		−	1.57980i	−0.613233	−	0.789902i	$-0.710132\pi$
$167$		−	263.827i		−	1.57980i	0.613233	−	0.789902i	$-0.289868\pi$
$168$		0			0
$169$	31.2906			0.185152
$170$		0			0
$171$	−85.7652			−0.501551
$172$		0			0
$173$	54.6502			0.315897			0.157949	−	0.987447i	$-0.449512\pi$
$173$	54.6502			0.315897			0.157949	+	0.987447i	$0.449512\pi$
$174$		0			0
$175$	33.8064	+	20.9113i	0.193180	+	0.119493i
$176$		0			0
$177$	−70.5351			−0.398503
$178$		0			0
$179$			105.555i			0.589691i	0.955545	+	0.294845i	$0.0952681\pi$
$179$			105.555i			0.589691i	−0.955545	+	0.294845i	$0.904732\pi$
$180$		0			0
$181$	−135.753			−0.750018			−0.375009	−	0.927021i	$-0.622360\pi$
$181$	−135.753			−0.750018			−0.375009	+	0.927021i	$0.622360\pi$
$182$		0			0
$183$	−18.8905			−0.103227
$184$		0			0
$185$		−	113.757i		−	0.614903i
$186$		0			0
$187$	−120.606			−0.644952
$188$		0			0
$189$	30.9335	+	19.1342i	0.163669	+	0.101239i
$190$		0			0
$191$	−21.5720			−0.112943			−0.0564713	−	0.998404i	$-0.517985\pi$
$191$	−21.5720			−0.112943			−0.0564713	+	0.998404i	$0.517985\pi$
$192$		0			0
$193$	−318.072			−1.64804			−0.824021	−	0.566560i	$-0.808274\pi$
$193$	−318.072			−1.64804			−0.824021	+	0.566560i	$0.808274\pi$
$194$		0			0
$195$	135.772			0.696266
$196$		0			0
$197$		−	0.244515i		−	0.00124119i	−1.00000		0.000620597i	$-0.999802\pi$
$197$		−	0.244515i		−	0.00124119i	1.00000		0.000620597i	$-0.000197542\pi$
$198$		0			0
$199$			236.282i			1.18735i	0.804706	+	0.593674i	$0.202323\pi$
$199$			236.282i			1.18735i	−0.804706	+	0.593674i	$0.797677\pi$
$200$		0			0
$201$		−	201.146i		−	1.00073i
$202$		0			0
$203$	21.7821	−	35.2143i	0.107301	−	0.173469i
$204$		0			0
$205$		−	272.010i		−	1.32688i
$206$		0			0
$207$	−24.2897			−0.117341
$208$		0			0
$209$		−	461.200i		−	2.20670i
$210$		0			0
$211$		−	100.465i		−	0.476140i	−0.971248	−	0.238070i	$-0.923485\pi$
$211$		−	100.465i		−	0.476140i	0.971248	−	0.238070i	$-0.0765147\pi$
$212$		0			0
$213$	181.463			0.851937
$214$		0			0
$215$		−	235.386i		−	1.09482i
$216$		0			0
$217$	−99.5552	+	160.947i	−0.458780	+	0.741692i
$218$		0			0
$219$			143.526i			0.655369i
$220$		0			0
$221$			105.803i			0.478748i
$222$		0			0
$223$		−	378.232i		−	1.69611i	−0.529912	−	0.848053i	$-0.677775\pi$
$223$		−	378.232i		−	1.69611i	0.529912	−	0.848053i	$-0.322225\pi$
$224$		0			0
$225$	17.0362			0.0757165
$226$		0			0
$227$	−56.6476			−0.249549			−0.124774	−	0.992185i	$-0.539821\pi$
$227$	−56.6476			−0.249549			−0.124774	+	0.992185i	$0.539821\pi$
$228$		0			0
$229$	−296.565			−1.29504			−0.647522	−	0.762047i	$-0.724195\pi$
$229$	−296.565			−1.29504			−0.647522	+	0.762047i	$0.724195\pi$
$230$		0			0
$231$	−102.894	+	166.344i	−0.445427	+	0.720105i
$232$		0			0
$233$	−112.373			−0.482289			−0.241145	−	0.970489i	$-0.577523\pi$
$233$	−112.373			−0.482289			−0.241145	+	0.970489i	$0.577523\pi$
$234$		0			0
$235$			269.681i			1.14758i
$236$		0			0
$237$	77.5381			0.327165
$238$		0			0
$239$	94.4921			0.395364			0.197682	−	0.980266i	$-0.436659\pi$
$239$	94.4921			0.395364			0.197682	+	0.980266i	$0.436659\pi$
$240$		0			0
$241$		−	87.7294i		−	0.364022i	−0.983296	−	0.182011i	$-0.941739\pi$
$241$		−	87.7294i		−	0.364022i	0.983296	−	0.182011i	$-0.0582607\pi$
$242$		0			0
$243$	15.5885			0.0641500
$244$		0			0
$245$	121.191	+	242.842i	0.494657	+	0.991193i
$246$		0			0
$247$	−404.594			−1.63803
$248$		0			0
$249$	114.056			0.458055
$250$		0			0
$251$	−344.939			−1.37426			−0.687130	−	0.726534i	$-0.741130\pi$
$251$	−344.939			−1.37426			−0.687130	+	0.726534i	$0.741130\pi$
$252$		0			0
$253$		−	130.617i		−	0.516273i
$254$		0			0
$255$			71.7213i			0.281260i
$256$		0			0
$257$			75.7961i			0.294927i	0.989068	+	0.147463i	$0.0471108\pi$
$257$			75.7961i			0.294927i	−0.989068	+	0.147463i	$0.952889\pi$
$258$		0			0
$259$	75.6289	−	122.266i	0.292003	−	0.472071i
$260$		0			0
$261$		−	17.7457i		−	0.0679911i
$262$		0			0
$263$	19.8005			0.0752872			0.0376436	−	0.999291i	$-0.488015\pi$
$263$	19.8005			0.0752872			0.0376436	+	0.999291i	$0.488015\pi$
$264$		0			0
$265$		−	427.544i		−	1.61337i
$266$		0			0
$267$			232.422i			0.870493i
$268$		0			0
$269$	185.601			0.689966			0.344983	−	0.938609i	$-0.387885\pi$
$269$	185.601			0.689966			0.344983	+	0.938609i	$0.387885\pi$
$270$		0			0
$271$			138.260i			0.510185i	0.966917	+	0.255093i	$0.0821059\pi$
$271$			138.260i			0.510185i	−0.966917	+	0.255093i	$0.917894\pi$
$272$		0			0
$273$	145.928	+	90.2649i	0.534535	+	0.330641i
$274$		0			0
$275$			91.6119i			0.333134i
$276$		0			0
$277$			110.650i			0.399457i	0.979851	+	0.199728i	$0.0640060\pi$
$277$			110.650i			0.399457i	−0.979851	+	0.199728i	$0.935994\pi$
$278$		0			0
$279$			81.1068i			0.290705i
$280$		0			0
$281$	−531.686			−1.89212			−0.946061	−	0.323989i	$-0.894976\pi$
$281$	−531.686			−1.89212			−0.946061	+	0.323989i	$0.894976\pi$
$282$		0			0
$283$	−439.139			−1.55173			−0.775864	−	0.630900i	$-0.782686\pi$
$283$	−439.139			−1.55173			−0.775864	+	0.630900i	$0.782686\pi$
$284$		0			0
$285$	−274.264			−0.962330
$286$		0			0
$287$	180.840	−	292.357i	0.630104	−	1.01867i
$288$		0			0
$289$	233.109			0.806607
$290$		0			0
$291$		−	64.8876i		−	0.222981i
$292$		0			0
$293$	122.938			0.419585			0.209793	−	0.977746i	$-0.432721\pi$
$293$	122.938			0.419585			0.209793	+	0.977746i	$0.432721\pi$
$294$		0			0
$295$	−225.561			−0.764612
$296$		0			0
$297$			83.8266i			0.282244i
$298$		0			0
$299$	−114.586			−0.383230
$300$		0			0
$301$	156.491	−	252.994i	0.519905	−	0.840510i
$302$		0			0
$303$	243.420			0.803365
$304$		0			0
$305$	−60.4089			−0.198062
$306$		0			0
$307$	5.82104			0.0189611			0.00948053	−	0.999955i	$-0.496982\pi$
$307$	5.82104			0.0189611			0.00948053	+	0.999955i	$0.496982\pi$
$308$		0			0
$309$		−	343.548i		−	1.11180i
$310$		0			0
$311$			208.552i			0.670585i	0.942114	+	0.335293i	$0.108835\pi$
$311$			208.552i			0.670585i	−0.942114	+	0.335293i	$0.891165\pi$
$312$		0			0
$313$		−	25.0482i		−	0.0800261i	−0.999199	−	0.0400130i	$-0.987260\pi$
$313$		−	25.0482i		−	0.0800261i	0.999199	−	0.0400130i	$-0.0127399\pi$
$314$		0			0
$315$	98.9208	+	61.1883i	0.314034	+	0.194248i
$316$		0			0
$317$		−	172.733i		−	0.544899i	−0.962170	−	0.272449i	$-0.912166\pi$
$317$		−	172.733i		−	0.544899i	0.962170	−	0.272449i	$-0.0878338\pi$
$318$		0			0
$319$	95.4269			0.299144
$320$		0			0
$321$			36.6598i			0.114205i
$322$		0			0
$323$		−	213.727i		−	0.661692i
$324$		0			0
$325$	80.3678			0.247286
$326$		0			0
$327$		−	186.001i		−	0.568811i
$328$		0			0
$329$	−179.292	+	289.854i	−0.544959	+	0.881015i
$330$		0			0
$331$		−	285.098i		−	0.861323i	−0.902513	−	0.430662i	$-0.858280\pi$
$331$		−	285.098i		−	0.861323i	0.902513	−	0.430662i	$-0.141720\pi$
$332$		0			0
$333$		−	61.6142i		−	0.185028i
$334$		0			0
$335$		−	643.235i		−	1.92010i
$336$		0			0
$337$	507.150			1.50490			0.752448	−	0.658652i	$-0.228873\pi$
$337$	507.150			1.50490			0.752448	+	0.658652i	$0.228873\pi$
$338$		0			0
$339$	195.572			0.576908
$340$		0			0
$341$	−436.150			−1.27903
$342$		0			0
$343$	−31.1921	+	341.579i	−0.0909391	+	0.995856i
$344$		0			0
$345$	−77.6748			−0.225144
$346$		0			0
$347$		−	4.14103i		−	0.0119338i	−0.999982	−	0.00596691i	$-0.998101\pi$
$347$		−	4.14103i		−	0.0119338i	0.999982	−	0.00596691i	$-0.00189934\pi$
$348$		0			0
$349$	231.206			0.662481			0.331240	−	0.943546i	$-0.392533\pi$
$349$	231.206			0.662481			0.331240	+	0.943546i	$0.392533\pi$
$350$		0			0
$351$	73.5381			0.209510
$352$		0			0
$353$			543.259i			1.53898i	0.638660	+	0.769489i	$0.279489\pi$
$353$			543.259i			1.53898i	−0.638660	+	0.769489i	$0.720511\pi$
$354$		0			0
$355$	580.290			1.63462
$356$		0			0
$357$	−47.6824	+	77.0863i	−0.133564	+	0.215928i
$358$		0			0
$359$	−270.411			−0.753235			−0.376618	−	0.926369i	$-0.622913\pi$
$359$	−270.411			−0.753235			−0.376618	+	0.926369i	$0.622913\pi$
$360$		0			0
$361$	456.296			1.26398
$362$		0			0
$363$	−241.197			−0.664455
$364$		0			0
$365$			458.974i			1.25746i
$366$		0			0
$367$			192.315i			0.524019i	0.965065	+	0.262010i	$0.0843852\pi$
$367$			192.315i			0.524019i	−0.965065	+	0.262010i	$0.915615\pi$
$368$		0			0
$369$		−	147.329i		−	0.399265i
$370$		0			0
$371$	284.243	−	459.525i	0.766154	−	1.23861i
$372$		0			0
$373$			73.8655i			0.198031i	0.995086	+	0.0990155i	$0.0315693\pi$
$373$			73.8655i			0.198031i	−0.995086	+	0.0990155i	$0.968431\pi$
$374$		0			0
$375$	−185.359			−0.494292
$376$		0			0
$377$		−	83.7147i		−	0.222055i
$378$		0			0
$379$		−	10.9952i		−	0.0290111i	−0.999895	−	0.0145056i	$-0.995383\pi$
$379$		−	10.9952i		−	0.0290111i	0.999895	−	0.0145056i	$-0.00461742\pi$
$380$		0			0
$381$	−287.948			−0.755769
$382$		0			0
$383$			557.586i			1.45584i	0.685663	+	0.727919i	$0.259512\pi$
$383$			557.586i			1.45584i	−0.685663	+	0.727919i	$0.740488\pi$
$384$		0			0
$385$	−329.038	+	531.944i	−0.854645	+	1.38167i
$386$		0			0
$387$		−	127.492i		−	0.329437i
$388$		0			0
$389$			41.0393i			0.105499i	0.998608	+	0.0527497i	$0.0167986\pi$
$389$			41.0393i			0.105499i	−0.998608	+	0.0527497i	$0.983201\pi$
$390$		0			0
$391$		−	60.5299i		−	0.154808i
$392$		0			0
$393$	270.714			0.688840
$394$		0			0
$395$	247.955			0.627735
$396$		0			0
$397$	−228.169			−0.574733			−0.287366	−	0.957821i	$-0.592780\pi$
$397$	−228.169			−0.574733			−0.287366	+	0.957821i	$0.592780\pi$
$398$		0			0
$399$	−294.780	−	182.339i	−0.738797	−	0.456989i
$400$		0			0
$401$	582.508			1.45264			0.726319	−	0.687357i	$-0.241229\pi$
$401$	582.508			1.45264			0.726319	+	0.687357i	$0.241229\pi$
$402$		0			0
$403$			382.619i			0.949426i
$404$		0			0
$405$	49.8496			0.123085
$406$		0			0
$407$	331.329			0.814076
$408$		0			0
$409$		−	4.68899i		−	0.0114645i	−0.999984	−	0.00573226i	$-0.998175\pi$
$409$		−	4.68899i		−	0.0114645i	0.999984	−	0.00573226i	$-0.00182465\pi$
$410$		0			0
$411$	118.111			0.287373
$412$		0			0
$413$	−242.433	−	149.959i	−0.587005	−	0.363097i
$414$		0			0
$415$	364.733			0.878874
$416$		0			0
$417$	−329.883			−0.791086
$418$		0			0
$419$	−105.256			−0.251208			−0.125604	−	0.992080i	$-0.540087\pi$
$419$	−105.256			−0.251208			−0.125604	+	0.992080i	$0.540087\pi$
$420$		0			0
$421$			619.528i			1.47156i	0.677220	+	0.735781i	$0.263185\pi$
$421$			619.528i			1.47156i	−0.677220	+	0.735781i	$0.736815\pi$
$422$		0			0
$423$			146.067i			0.345313i
$424$		0			0
$425$			42.4542i			0.0998923i
$426$		0			0
$427$	−64.9277	−	40.1616i	−0.152055	−	0.0940552i
$428$		0			0
$429$			395.449i			0.921793i
$430$		0			0
$431$	438.654			1.01776			0.508879	−	0.860838i	$-0.330060\pi$
$431$	438.654			1.01776			0.508879	+	0.860838i	$0.330060\pi$
$432$		0			0
$433$			43.0393i			0.0993979i	0.998764	+	0.0496990i	$0.0158262\pi$
$433$			43.0393i			0.0993979i	−0.998764	+	0.0496990i	$0.984174\pi$
$434$		0			0
$435$		−	56.7480i		−	0.130455i
$436$		0			0
$437$	231.468			0.529674
$438$		0			0
$439$		−	206.214i		−	0.469736i	−0.972027	−	0.234868i	$-0.924534\pi$
$439$		−	206.214i		−	0.469736i	0.972027	−	0.234868i	$-0.0754658\pi$
$440$		0			0
$441$	65.6406	+	131.531i	0.148845	+	0.298255i
$442$		0			0
$443$			611.291i			1.37989i	0.723862	+	0.689945i	$0.242365\pi$
$443$			611.291i			1.37989i	−0.723862	+	0.689945i	$0.757635\pi$
$444$		0			0
$445$			743.249i			1.67022i
$446$		0			0
$447$			72.0695i			0.161229i
$448$		0			0
$449$	−597.695			−1.33117			−0.665585	−	0.746322i	$-0.731818\pi$
$449$	−597.695			−1.33117			−0.665585	+	0.746322i	$0.731818\pi$
$450$		0			0
$451$	792.257			1.75667
$452$		0			0
$453$	259.686			0.573259
$454$		0			0
$455$	466.656	+	288.654i	1.02562	+	0.634404i
$456$		0			0
$457$	−350.459			−0.766868			−0.383434	−	0.923568i	$-0.625259\pi$
$457$	−350.459			−0.766868			−0.383434	+	0.923568i	$0.625259\pi$
$458$		0			0
$459$			38.8464i			0.0846327i
$460$		0			0
$461$	−182.865			−0.396671			−0.198335	−	0.980134i	$-0.563553\pi$
$461$	−182.865			−0.396671			−0.198335	+	0.980134i	$0.563553\pi$
$462$		0			0
$463$	−62.8699			−0.135788			−0.0678941	−	0.997693i	$-0.521628\pi$
$463$	−62.8699			−0.135788			−0.0678941	+	0.997693i	$0.521628\pi$
$464$		0			0
$465$			259.367i			0.557779i
$466$		0			0
$467$	645.253			1.38170			0.690849	−	0.722999i	$-0.257237\pi$
$467$	645.253			1.38170			0.690849	+	0.722999i	$0.257237\pi$
$468$		0			0
$469$	427.641	−	691.350i	0.911814	−	1.47409i
$470$		0			0
$471$	141.639			0.300719
$472$		0			0
$473$	685.586			1.44944
$474$		0			0
$475$	−162.346			−0.341781
$476$		0			0
$477$		−	231.571i		−	0.485473i
$478$		0			0
$479$			437.654i			0.913684i	0.889548	+	0.456842i	$0.151019\pi$
$479$			437.654i			0.913684i	−0.889548	+	0.456842i	$0.848981\pi$
$480$		0			0
$481$		−	290.663i		−	0.604289i
$482$		0			0
$483$	−83.4850	−	51.6404i	−0.172847	−	0.106916i
$484$		0			0
$485$		−	207.501i		−	0.427837i
$486$		0			0
$487$	14.7391			0.0302650			0.0151325	−	0.999885i	$-0.495183\pi$
$487$	14.7391			0.0302650			0.0151325	+	0.999885i	$0.495183\pi$
$488$		0			0
$489$		−	476.977i		−	0.975412i
$490$		0			0
$491$		−	120.149i		−	0.244702i	−0.992487	−	0.122351i	$-0.960957\pi$
$491$		−	120.149i		−	0.244702i	0.992487	−	0.122351i	$-0.0390434\pi$
$492$		0			0
$493$	44.2222			0.0897002
$494$		0			0
$495$			268.065i			0.541545i
$496$		0			0
$497$	623.697	+	385.793i	1.25492	+	0.776244i
$498$		0			0
$499$			4.70185i			0.00942254i	0.999989	+	0.00471127i	$0.00149965\pi$
$499$			4.70185i			0.00942254i	−0.999989	+	0.00471127i	$0.998500\pi$
$500$		0			0
$501$		−	456.962i		−	0.912100i
$502$		0			0
$503$			112.427i			0.223512i	0.993736	+	0.111756i	$0.0356475\pi$
$503$			112.427i			0.223512i	−0.993736	+	0.111756i	$0.964352\pi$
$504$		0			0
$505$	778.419			1.54142
$506$		0			0
$507$	54.1969			0.106897
$508$		0			0
$509$	764.966			1.50288			0.751440	−	0.659801i	$-0.229360\pi$
$509$	764.966			1.50288			0.751440	+	0.659801i	$0.229360\pi$
$510$		0			0
$511$	−305.139	+	493.307i	−0.597141	+	0.965375i
$512$		0			0
$513$	−148.550			−0.289570
$514$		0			0
$515$		−	1098.61i		−	2.13323i
$516$		0			0
$517$	−785.474			−1.51929
$518$		0			0
$519$	94.6569			0.182383
$520$		0			0
$521$			993.978i			1.90783i	0.300083	+	0.953913i	$0.402986\pi$
$521$			993.978i			1.90783i	−0.300083	+	0.953913i	$0.597014\pi$
$522$		0			0
$523$	−460.338			−0.880188			−0.440094	−	0.897952i	$-0.645055\pi$
$523$	−460.338			−0.880188			−0.440094	+	0.897952i	$0.645055\pi$
$524$		0			0
$525$	58.5545	+	36.2194i	0.111532	+	0.0689893i
$526$		0			0
$527$	−202.118			−0.383525
$528$		0			0
$529$	−463.446			−0.876079
$530$		0			0
$531$	−122.170			−0.230076
$532$		0			0
$533$		−	695.019i		−	1.30397i
$534$		0			0
$535$			117.232i			0.219126i
$536$		0			0
$537$			182.826i			0.340458i
$538$		0			0
$539$	−707.303	+	352.981i	−1.31225	+	0.654881i
$540$		0			0
$541$			411.334i			0.760321i	0.924921	+	0.380160i	$0.124131\pi$
$541$			411.334i			0.760321i	−0.924921	+	0.380160i	$0.875869\pi$
$542$		0			0
$543$	−235.131			−0.433023
$544$		0			0
$545$		−	594.804i		−	1.09138i
$546$		0			0
$547$		−	30.2891i		−	0.0553732i	−0.999617	−	0.0276866i	$-0.991186\pi$
$547$		−	30.2891i		−	0.0553732i	0.999617	−	0.0276866i	$-0.00881404\pi$
$548$		0			0
$549$	−32.7193			−0.0595980
$550$		0			0
$551$			169.107i			0.306909i
$552$		0			0
$553$	266.503	+	164.848i	0.481922	+	0.298097i
$554$		0			0
$555$		−	197.033i		−	0.355014i
$556$		0			0
$557$		−	763.819i		−	1.37131i	−0.727927	−	0.685655i	$-0.759516\pi$
$557$		−	763.819i		−	1.37131i	0.727927	−	0.685655i	$-0.240484\pi$
$558$		0			0
$559$		−	601.440i		−	1.07592i
$560$		0			0
$561$	−208.896			−0.372363
$562$		0			0
$563$	40.3472			0.0716646			0.0358323	−	0.999358i	$-0.488592\pi$
$563$	40.3472			0.0716646			0.0358323	+	0.999358i	$0.488592\pi$
$564$		0			0
$565$	625.409			1.10692
$566$		0			0
$567$	53.5784	+	33.1414i	0.0944946	+	0.0584504i
$568$		0			0
$569$	186.637			0.328008			0.164004	−	0.986460i	$-0.447559\pi$
$569$	186.637			0.328008			0.164004	+	0.986460i	$0.447559\pi$
$570$		0			0
$571$			901.865i			1.57945i	0.613462	+	0.789724i	$0.289776\pi$
$571$			901.865i			1.57945i	−0.613462	+	0.789724i	$0.710224\pi$
$572$		0			0
$573$	−37.3639			−0.0652075
$574$		0			0
$575$	−45.9783			−0.0799622
$576$		0			0
$577$			742.756i			1.28727i	0.765332	+	0.643636i	$0.222575\pi$
$577$			742.756i			1.28727i	−0.765332	+	0.643636i	$0.777425\pi$
$578$		0			0
$579$	−550.917			−0.951497
$580$		0			0
$581$	392.016	+	242.485i	0.674726	+	0.417357i
$582$		0			0
$583$	1245.27			2.13596
$584$		0			0
$585$	235.164			0.401989
$586$		0			0
$587$	−514.288			−0.876130			−0.438065	−	0.898943i	$-0.644336\pi$
$587$	−514.288			−0.876130			−0.438065	+	0.898943i	$0.644336\pi$
$588$		0			0
$589$		−	772.904i		−	1.31223i
$590$		0			0
$591$		−	0.423513i		−	0.000716604i
$592$		0			0
$593$		−	466.087i		−	0.785982i	−0.919542	−	0.392991i	$-0.871440\pi$
$593$		−	466.087i		−	0.785982i	0.919542	−	0.392991i	$-0.128560\pi$
$594$		0			0
$595$	−152.481	+	246.510i	−0.256271	+	0.414303i
$596$		0			0
$597$			409.253i			0.685515i
$598$		0			0
$599$	−947.796			−1.58230			−0.791149	−	0.611624i	$-0.790517\pi$
$599$	−947.796			−1.58230			−0.791149	+	0.611624i	$0.790517\pi$
$600$		0			0
$601$			153.107i			0.254754i	0.991854	+	0.127377i	$0.0406559\pi$
$601$			153.107i			0.254754i	−0.991854	+	0.127377i	$0.959344\pi$
$602$		0			0
$603$		−	348.395i		−	0.577770i
$604$		0			0
$605$	−771.312			−1.27490
$606$		0			0
$607$		−	298.930i		−	0.492471i	−0.969210	−	0.246236i	$-0.920806\pi$
$607$		−	298.930i		−	0.492471i	0.969210	−	0.246236i	$-0.0791937\pi$
$608$		0			0
$609$	37.7277	−	60.9929i	0.0619502	−	0.100153i
$610$		0			0
$611$			689.068i			1.12777i
$612$		0			0
$613$		−	291.497i		−	0.475525i	−0.971323	−	0.237762i	$-0.923586\pi$
$613$		−	291.497i		−	0.475525i	0.971323	−	0.237762i	$-0.0764140\pi$
$614$		0			0
$615$		−	471.135i		−	0.766073i
$616$		0			0
$617$	−359.433			−0.582549			−0.291275	−	0.956639i	$-0.594079\pi$
$617$	−359.433			−0.582549			−0.291275	+	0.956639i	$0.594079\pi$
$618$		0			0
$619$	−57.2368			−0.0924666			−0.0462333	−	0.998931i	$-0.514722\pi$
$619$	−57.2368			−0.0924666			−0.0462333	+	0.998931i	$0.514722\pi$
$620$		0			0
$621$	−42.0710			−0.0677471
$622$		0			0
$623$	−494.133	+	798.846i	−0.793151	+	1.28226i
$624$		0			0
$625$	−734.720			−1.17555
$626$		0			0
$627$		−	798.822i		−	1.27404i
$628$		0			0
$629$	153.542			0.244106
$630$		0			0
$631$	55.9357			0.0886461			0.0443230	−	0.999017i	$-0.485887\pi$
$631$	55.9357			0.0886461			0.0443230	+	0.999017i	$0.485887\pi$
$632$		0			0
$633$		−	174.011i		−	0.274899i
$634$		0			0
$635$	−920.815			−1.45010
$636$		0			0
$637$	309.658	+	620.492i	0.486119	+	0.974084i
$638$		0			0
$639$	314.302			0.491866
$640$		0			0
$641$	−459.772			−0.717273			−0.358636	−	0.933477i	$-0.616758\pi$
$641$	−459.772			−0.717273			−0.358636	+	0.933477i	$0.616758\pi$
$642$		0			0
$643$	712.301			1.10778			0.553888	−	0.832591i	$-0.313143\pi$
$643$	712.301			1.10778			0.553888	+	0.832591i	$0.313143\pi$
$644$		0			0
$645$		−	407.701i		−	0.632094i
$646$		0			0
$647$		−	10.9490i		−	0.0169228i	−0.999964	−	0.00846139i	$-0.997307\pi$
$647$		−	10.9490i		−	0.0169228i	0.999964	−	0.00846139i	$-0.00269338\pi$
$648$		0			0
$649$		−	656.968i		−	1.01228i
$650$		0			0
$651$	−172.435	+	278.769i	−0.264877	+	0.428216i
$652$		0			0
$653$			257.268i			0.393979i	0.980406	+	0.196989i	$0.0631164\pi$
$653$			257.268i			0.393979i	−0.980406	+	0.196989i	$0.936884\pi$
$654$		0			0
$655$	865.704			1.32168
$656$		0			0
$657$			248.594i			0.378378i
$658$		0			0
$659$		−	1016.92i		−	1.54312i	−0.636157	−	0.771560i	$-0.719477\pi$
$659$		−	1016.92i		−	1.54312i	0.636157	−	0.771560i	$-0.280523\pi$
$660$		0			0
$661$	740.204			1.11982			0.559912	−	0.828552i	$-0.310835\pi$
$661$	740.204			1.11982			0.559912	+	0.828552i	$0.310835\pi$
$662$		0			0
$663$			183.257i			0.276405i
$664$		0			0
$665$	−942.662	−	583.091i	−1.41754	−	0.876829i
$666$		0			0
$667$			47.8930i			0.0718036i
$668$		0			0
$669$		−	655.116i		−	0.979247i
$670$		0			0
$671$		−	175.947i		−	0.262216i
$672$		0			0
$673$	−1037.02			−1.54089			−0.770445	−	0.637506i	$-0.779966\pi$
$673$	−1037.02			−1.54089			−0.770445	+	0.637506i	$0.779966\pi$
$674$		0			0
$675$	29.5076			0.0437150
$676$		0			0
$677$	−809.104			−1.19513			−0.597566	−	0.801820i	$-0.703865\pi$
$677$	−809.104			−1.19513			−0.597566	+	0.801820i	$0.703865\pi$
$678$		0			0
$679$	137.952	−	223.022i	0.203170	−	0.328457i
$680$		0			0
$681$	−98.1165			−0.144077
$682$		0			0
$683$		−	306.279i		−	0.448432i	−0.974539	−	0.224216i	$-0.928018\pi$
$683$		−	306.279i		−	0.448432i	0.974539	−	0.224216i	$-0.0719822\pi$
$684$		0			0
$685$	377.700			0.551386
$686$		0			0
$687$	−513.666			−0.747694
$688$		0			0
$689$		−	1092.43i		−	1.58553i
$690$		0			0
$691$	−77.2654			−0.111817			−0.0559084	−	0.998436i	$-0.517805\pi$
$691$	−77.2654			−0.111817			−0.0559084	+	0.998436i	$0.517805\pi$
$692$		0			0
$693$	−178.217	+	288.117i	−0.257167	+	0.415753i
$694$		0			0
$695$	−1054.92			−1.51787
$696$		0			0
$697$	367.143			0.526747
$698$		0			0
$699$	−194.636			−0.278450
$700$		0			0
$701$			57.3914i			0.0818708i	0.999162	+	0.0409354i	$0.0130338\pi$
$701$			57.3914i			0.0818708i	−0.999162	+	0.0409354i	$0.986966\pi$
$702$		0			0
$703$			587.150i			0.835206i
$704$		0			0
$705$			467.102i			0.662555i
$706$		0			0
$707$	836.647	+	517.515i	1.18338	+	0.731987i
$708$		0			0
$709$			759.488i			1.07121i	0.844468	+	0.535605i	$0.179916\pi$
$709$			759.488i			1.07121i	−0.844468	+	0.535605i	$0.820084\pi$
$710$		0			0
$711$	134.300			0.188889
$712$		0			0
$713$		−	218.895i		−	0.307006i
$714$		0			0
$715$			1264.59i			1.76865i
$716$		0			0
$717$	163.665			0.228264
$718$		0			0
$719$		−	1249.77i		−	1.73821i	−0.494632	−	0.869103i	$-0.664697\pi$
$719$		−	1249.77i		−	1.73821i	0.494632	−	0.869103i	$-0.335303\pi$
$720$		0			0
$721$	730.389	−	1180.79i	1.01302	−	1.63772i
$722$		0			0
$723$		−	151.952i		−	0.210168i
$724$		0			0
$725$		−	33.5910i		−	0.0463324i
$726$		0			0
$727$		−	980.653i		−	1.34890i	−0.738319	−	0.674452i	$-0.764380\pi$
$727$		−	980.653i		−	1.34890i	0.738319	−	0.674452i	$-0.235620\pi$
$728$		0			0
$729$	27.0000			0.0370370
$730$		0			0
$731$	317.710			0.434624
$732$		0			0
$733$	532.385			0.726309			0.363155	−	0.931729i	$-0.381700\pi$
$733$	532.385			0.726309			0.363155	+	0.931729i	$0.381700\pi$
$734$		0			0
$735$	209.909	+	420.615i	0.285590	+	0.572266i
$736$		0			0
$737$	1873.49			2.54204
$738$		0			0
$739$			1061.19i			1.43599i	0.696049	+	0.717994i	$0.254939\pi$
$739$			1061.19i			1.43599i	−0.696049	+	0.717994i	$0.745061\pi$
$740$		0			0
$741$	−700.778			−0.945720
$742$		0			0
$743$	237.425			0.319549			0.159774	−	0.987154i	$-0.448923\pi$
$743$	237.425			0.319549			0.159774	+	0.987154i	$0.448923\pi$
$744$		0			0
$745$			230.468i			0.309352i
$746$		0			0
$747$	197.550			0.264458
$748$		0			0
$749$	−77.9395	+	126.002i	−0.104058	+	0.168227i
$750$		0			0
$751$	−572.550			−0.762384			−0.381192	−	0.924496i	$-0.624486\pi$
$751$	−572.550			−0.762384			−0.381192	+	0.924496i	$0.624486\pi$
$752$		0			0
$753$	−597.453			−0.793430
$754$		0			0
$755$	830.438			1.09992
$756$		0			0
$757$			1232.38i			1.62798i	0.580876	+	0.813992i	$0.302710\pi$
$757$			1232.38i			1.62798i	−0.580876	+	0.813992i	$0.697290\pi$
$758$		0			0
$759$		−	226.236i		−	0.298071i
$760$		0			0
$761$		−	474.738i		−	0.623834i	−0.950109	−	0.311917i	$-0.899029\pi$
$761$		−	474.738i		−	0.623834i	0.950109	−	0.311917i	$-0.100971\pi$
$762$		0			0
$763$	395.443	−	639.297i	0.518273	−	0.837873i
$764$		0			0
$765$			124.225i			0.162386i
$766$		0			0
$767$	−576.335			−0.751414
$768$		0			0
$769$		−	812.878i		−	1.05706i	−0.848915	−	0.528529i	$-0.822744\pi$
$769$		−	812.878i		−	1.05706i	0.848915	−	0.528529i	$-0.177256\pi$
$770$		0			0
$771$			131.283i			0.170276i
$772$		0			0
$773$	−864.876			−1.11886			−0.559428	−	0.828879i	$-0.688979\pi$
$773$	−864.876			−1.11886			−0.559428	+	0.828879i	$0.688979\pi$
$774$		0			0
$775$			153.528i			0.198101i
$776$		0			0
$777$	130.993	−	211.772i	0.168588	−	0.272550i
$778$		0			0
$779$			1403.96i			1.80226i
$780$		0			0
$781$			1690.15i			2.16409i
$782$		0			0
$783$		−	30.7364i		−	0.0392547i
$784$		0			0
$785$	452.940			0.576993
$786$		0			0
$787$	1030.93			1.30995			0.654975	−	0.755651i	$-0.272679\pi$
$787$	1030.93			1.30995			0.654975	+	0.755651i	$0.272679\pi$
$788$		0			0
$789$	34.2955			0.0434671
$790$		0			0
$791$	672.191	+	415.790i	0.849799	+	0.525651i
$792$		0			0
$793$	−154.352			−0.194643
$794$		0			0
$795$		−	740.528i		−	0.931482i
$796$		0			0
$797$	−742.857			−0.932067			−0.466033	−	0.884767i	$-0.654317\pi$
$797$	−742.857			−0.932067			−0.466033	+	0.884767i	$0.654317\pi$
$798$		0			0
$799$	−364.000			−0.455569
$800$		0			0
$801$			402.566i			0.502579i
$802$		0			0
$803$	−1336.81			−1.66477
$804$		0			0
$805$	−266.973	−	165.138i	−0.331643	−	0.205141i
$806$		0			0
$807$	321.470			0.398352
$808$		0			0
$809$	−947.608			−1.17133			−0.585667	−	0.810552i	$-0.699167\pi$
$809$	−947.608			−1.17133			−0.585667	+	0.810552i	$0.699167\pi$
$810$		0			0
$811$	580.571			0.715870			0.357935	−	0.933747i	$-0.383481\pi$
$811$	580.571			0.715870			0.357935	+	0.933747i	$0.383481\pi$
$812$		0			0
$813$			239.474i			0.294555i
$814$		0			0
$815$		−	1525.30i		−	1.87153i
$816$		0			0
$817$			1214.93i			1.48706i
$818$		0			0
$819$	252.755	+	156.343i	0.308614	+	0.190896i
$820$		0			0
$821$		−	1364.03i		−	1.66143i	−0.556700	−	0.830714i	$-0.687933\pi$
$821$		−	1364.03i		−	1.66143i	0.556700	−	0.830714i	$-0.312067\pi$
$822$		0			0
$823$	246.998			0.300119			0.150059	−	0.988677i	$-0.452053\pi$
$823$	246.998			0.300119			0.150059	+	0.988677i	$0.452053\pi$
$824$		0			0
$825$			158.676i			0.192335i
$826$		0			0
$827$		−	737.549i		−	0.891837i	−0.895074	−	0.445918i	$-0.852877\pi$
$827$		−	737.549i		−	0.891837i	0.895074	−	0.445918i	$-0.147123\pi$
$828$		0			0
$829$	−1260.35			−1.52032			−0.760162	−	0.649733i	$-0.774881\pi$
$829$	−1260.35			−1.52032			−0.760162	+	0.649733i	$0.774881\pi$
$830$		0			0
$831$			191.651i			0.230626i
$832$		0			0
$833$	−327.774	+	163.576i	−0.393486	+	0.196370i
$834$		0			0
$835$		−	1461.30i		−	1.75006i
$836$		0			0
$837$			140.481i			0.167839i
$838$		0			0
$839$		−	272.430i		−	0.324708i	−0.986733	−	0.162354i	$-0.948091\pi$
$839$		−	272.430i		−	0.324708i	0.986733	−	0.162354i	$-0.0519087\pi$
$840$		0			0
$841$	806.010			0.958395
$842$		0			0
$843$	−920.908			−1.09242
$844$		0			0
$845$	173.314			0.205105
$846$		0			0
$847$	−829.009	−	512.790i	−0.978759	−	0.605419i
$848$		0			0
$849$	−760.612			−0.895891
$850$		0			0
$851$			166.288i			0.195403i
$852$		0			0
$853$	167.153			0.195959			0.0979794	−	0.995188i	$-0.468762\pi$
$853$	167.153			0.195959			0.0979794	+	0.995188i	$0.468762\pi$
$854$		0			0
$855$	−475.039			−0.555602
$856$		0			0
$857$		−	13.0480i		−	0.0152252i	−0.999971	−	0.00761258i	$-0.997577\pi$
$857$		−	13.0480i		−	0.0152252i	0.999971	−	0.00761258i	$-0.00242318\pi$
$858$		0			0
$859$	−872.946			−1.01624			−0.508118	−	0.861288i	$-0.669658\pi$
$859$	−872.946			−1.01624			−0.508118	+	0.861288i	$0.669658\pi$
$860$		0			0
$861$	313.224	−	506.377i	0.363791	−	0.588127i
$862$		0			0
$863$	−560.236			−0.649173			−0.324587	−	0.945856i	$-0.605225\pi$
$863$	−560.236			−0.649173			−0.324587	+	0.945856i	$0.605225\pi$
$864$		0			0
$865$	302.699			0.349941
$866$		0			0
$867$	403.757			0.465695
$868$		0			0
$869$			722.195i			0.831064i
$870$		0			0
$871$		−	1643.54i		−	1.88696i
$872$		0			0
$873$		−	112.389i		−	0.128738i
$874$		0			0
$875$	−637.091	−	394.078i	−0.728104	−	0.450375i
$876$		0			0
$877$		−	819.789i		−	0.934766i	−0.884055	−	0.467383i	$-0.845197\pi$
$877$		−	819.789i		−	0.934766i	0.884055	−	0.467383i	$-0.154803\pi$
$878$		0			0
$879$	212.936			0.242248
$880$		0			0
$881$			930.214i			1.05586i	0.849287	+	0.527931i	$0.177032\pi$
$881$			930.214i			1.05586i	−0.849287	+	0.527931i	$0.822968\pi$
$882$		0			0
$883$			444.483i			0.503378i	0.967808	+	0.251689i	$0.0809860\pi$
$883$			444.483i			0.503378i	−0.967808	+	0.251689i	$0.919014\pi$
$884$		0			0
$885$	−390.682			−0.441449
$886$		0			0
$887$		−	159.932i		−	0.180307i	−0.995928	−	0.0901536i	$-0.971264\pi$
$887$		−	159.932i		−	0.180307i	0.995928	−	0.0901536i	$-0.0287358\pi$
$888$		0			0
$889$	−989.694	−	612.183i	−1.11327	−	0.688620i
$890$		0			0
$891$			145.192i			0.162954i
$892$		0			0
$893$		−	1391.94i		−	1.55873i
$894$		0			0
$895$			584.650i			0.653240i
$896$		0			0
$897$	−198.468			−0.221258
$898$		0			0
$899$	159.922			0.177888
$900$		0			0
$901$	577.073			0.640481
$902$		0			0
$903$	271.051	−	438.198i	0.300167	−	0.485269i
$904$		0			0
$905$	−751.915			−0.830845
$906$		0			0
$907$		−	66.5490i		−	0.0733727i	−0.999327	−	0.0366863i	$-0.988320\pi$
$907$		−	66.5490i		−	0.0733727i	0.999327	−	0.0366863i	$-0.0116802\pi$
$908$		0			0
$909$	421.615			0.463823
$910$		0			0
$911$	254.406			0.279260			0.139630	−	0.990204i	$-0.455409\pi$
$911$	254.406			0.279260			0.139630	+	0.990204i	$0.455409\pi$
$912$		0			0
$913$			1062.32i			1.16355i
$914$		0			0
$915$	−104.631			−0.114351
$916$		0			0
$917$	930.460	+	575.544i	1.01468	+	0.627638i
$918$		0			0
$919$	−1688.22			−1.83702			−0.918511	−	0.395395i	$-0.870608\pi$
$919$	−1688.22			−1.83702			−0.918511	+	0.395395i	$0.870608\pi$
$920$		0			0
$921$	10.0823			0.0109472
$922$		0			0
$923$	1482.71			1.60640
$924$		0			0
$925$		−	116.630i		−	0.126087i
$926$		0			0
$927$		−	595.042i		−	0.641901i
$928$		0			0
$929$		−	905.524i		−	0.974730i	−0.873198	−	0.487365i	$-0.837958\pi$
$929$		−	905.524i		−	0.974730i	0.873198	−	0.487365i	$-0.162042\pi$
$930$		0			0
$931$	−625.520	−	1253.42i	−0.671879	−	1.34631i
$932$		0			0
$933$			361.223i			0.387162i
$934$		0			0
$935$	−668.017			−0.714457
$936$		0			0
$937$		−	1465.43i		−	1.56396i	−0.623301	−	0.781982i	$-0.714209\pi$
$937$		−	1465.43i		−	1.56396i	0.623301	−	0.781982i	$-0.285791\pi$
$938$		0			0
$939$		−	43.3847i		−	0.0462031i
$940$		0			0
$941$	250.000			0.265675			0.132837	−	0.991138i	$-0.457591\pi$
$941$	250.000			0.265675			0.132837	+	0.991138i	$0.457591\pi$
$942$		0			0
$943$			397.619i			0.421653i
$944$		0			0
$945$	171.336	+	105.981i	0.181308	+	0.112149i
$946$		0			0
$947$		−	638.465i		−	0.674197i	−0.941469	−	0.337099i	$-0.890554\pi$
$947$		−	638.465i		−	0.674197i	0.941469	−	0.337099i	$-0.109446\pi$
$948$		0			0
$949$			1172.73i			1.23576i
$950$		0			0
$951$		−	299.182i		−	0.314598i
$952$		0			0
$953$	436.356			0.457876			0.228938	−	0.973441i	$-0.426475\pi$
$953$	436.356			0.457876			0.228938	+	0.973441i	$0.426475\pi$
$954$		0			0
$955$	−119.484			−0.125114
$956$		0			0
$957$	165.284			0.172711
$958$		0			0
$959$	405.953	+	251.105i	0.423308	+	0.261841i
$960$		0			0
$961$	230.077			0.239414
$962$		0			0
$963$			63.4966i			0.0659362i
$964$		0			0
$965$	−1761.75			−1.82565
$966$		0			0
$967$	891.342			0.921760			0.460880	−	0.887463i	$-0.347534\pi$
$967$	891.342			0.921760			0.460880	+	0.887463i	$0.347534\pi$
$968$		0			0
$969$		−	370.185i		−	0.382028i
$970$		0			0
$971$	−162.475			−0.167327			−0.0836635	−	0.996494i	$-0.526662\pi$
$971$	−162.475			−0.167327			−0.0836635	+	0.996494i	$0.526662\pi$
$972$		0			0
$973$	−1133.83	−	701.338i	−1.16529	−	0.720800i
$974$		0			0
$975$	139.201			0.142770
$976$		0			0
$977$	499.904			0.511673			0.255836	−	0.966720i	$-0.417649\pi$
$977$	499.904			0.511673			0.255836	+	0.966720i	$0.417649\pi$
$978$		0			0
$979$	−2164.79			−2.21122
$980$		0			0
$981$		−	322.164i		−	0.328403i
$982$		0			0
$983$			544.456i			0.553872i	0.960888	+	0.276936i	$0.0893190\pi$
$983$			544.456i			0.553872i	−0.960888	+	0.276936i	$0.910681\pi$
$984$		0			0
$985$		−	1.35433i		−	0.00137496i
$986$		0			0
$987$	−310.542	+	502.042i	−0.314632	+	0.508654i
$988$		0			0
$989$			344.082i			0.347909i
$990$		0			0
$991$	−778.083			−0.785149			−0.392575	−	0.919720i	$-0.628415\pi$
$991$	−778.083			−0.785149			−0.392575	+	0.919720i	$0.628415\pi$
$992$		0			0
$993$		−	493.804i		−	0.497285i
$994$		0			0
$995$			1308.73i			1.31530i
$996$		0			0
$997$	−366.247			−0.367349			−0.183674	−	0.982987i	$-0.558799\pi$
$997$	−366.247			−0.367349			−0.183674	+	0.982987i	$0.558799\pi$
$998$		0			0
$999$		−	106.719i		−	0.106826i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.3.l.a.433.30		32
3.2	odd	2		2016.3.l.h.433.8		32
4.3	odd	2		168.3.l.a.13.15	yes	32
7.6	odd	2	inner	672.3.l.a.433.14		32
8.3	odd	2		168.3.l.a.13.14	yes	32
8.5	even	2	inner	672.3.l.a.433.3		32
12.11	even	2		504.3.l.h.181.17		32
21.20	even	2		2016.3.l.h.433.26		32
24.5	odd	2		2016.3.l.h.433.25		32
24.11	even	2		504.3.l.h.181.20		32
28.27	even	2		168.3.l.a.13.16	yes	32
56.13	odd	2	inner	672.3.l.a.433.19		32
56.27	even	2		168.3.l.a.13.13	✓	32
84.83	odd	2		504.3.l.h.181.18		32
168.83	odd	2		504.3.l.h.181.19		32
168.125	even	2		2016.3.l.h.433.7		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.3.l.a.13.13	✓	32	56.27	even	2
168.3.l.a.13.14	yes	32	8.3	odd	2
168.3.l.a.13.15	yes	32	4.3	odd	2
168.3.l.a.13.16	yes	32	28.27	even	2
504.3.l.h.181.17		32	12.11	even	2
504.3.l.h.181.18		32	84.83	odd	2
504.3.l.h.181.19		32	168.83	odd	2
504.3.l.h.181.20		32	24.11	even	2
672.3.l.a.433.3		32	8.5	even	2	inner
672.3.l.a.433.14		32	7.6	odd	2	inner
672.3.l.a.433.19		32	56.13	odd	2	inner
672.3.l.a.433.30		32	1.1	even	1	trivial
2016.3.l.h.433.7		32	168.125	even	2
2016.3.l.h.433.8		32	3.2	odd	2
2016.3.l.h.433.25		32	24.5	odd	2
2016.3.l.h.433.26		32	21.20	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:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	672.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} +5.53884 q^{5} +(5.95316 + 3.68238i) q^{7} +3.00000 q^{9} +16.1324i q^{11} +14.1524 q^{13} +9.59355 q^{15} +7.47600i q^{17} -28.5884 q^{19} +(10.3112 + 6.37806i) q^{21} -8.09656 q^{23} +5.67874 q^{25} +5.19615 q^{27} -5.91522i q^{29} +27.0356i q^{31} +27.9422i q^{33} +(32.9736 + 20.3961i) q^{35} -20.5381i q^{37} +24.5127 q^{39} -49.1096i q^{41} -42.4974i q^{43} +16.6165 q^{45} +48.6891i q^{47} +(21.8802 + 43.8435i) q^{49} +12.9488i q^{51} -77.1902i q^{53} +89.3549i q^{55} -49.5165 q^{57} -40.7234 q^{59} -10.9064 q^{61} +(17.8595 + 11.0471i) q^{63} +78.3879 q^{65} -116.132i q^{67} -14.0237 q^{69} +104.767 q^{71} +82.8647i q^{73} +9.83587 q^{75} +(-59.4057 + 96.0389i) q^{77} +44.7667 q^{79} +9.00000 q^{81} +65.8500 q^{83} +41.4083i q^{85} -10.2455i q^{87} +134.189i q^{89} +(84.2515 + 52.1145i) q^{91} +46.8270i q^{93} -158.346 q^{95} -37.4629i q^{97} +48.3973i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 96 q^{9} - 64 q^{23} + 160 q^{25} - 16 q^{49} + 96 q^{57} + 640 q^{71} + 64 q^{79} + 288 q^{81} + 768 q^{95}+O(q^{100}) \) 32 * q + 96 * q^9 - 64 * q^23 + 160 * q^25 - 16 * q^49 + 96 * q^57 + 640 * q^71 + 64 * q^79 + 288 * q^81 + 768 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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