LMFDB - Embedded newform 1920.2.a.y.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1920,2,Mod(1,1920)] 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("1920.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1920, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1920 = 2^{7} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1920.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,-2,0,2,0,2,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$15.3312771881$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	1920.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +5.12311 q^{7} +1.00000 q^{9} -2.00000 q^{11} -5.12311 q^{13} +1.00000 q^{15} -1.12311 q^{17} -5.12311 q^{19} -5.12311 q^{21} -5.12311 q^{23} +1.00000 q^{25} -1.00000 q^{27} +8.24621 q^{29} -7.12311 q^{31} +2.00000 q^{33} -5.12311 q^{35} +5.12311 q^{37} +5.12311 q^{39} -2.00000 q^{41} +6.24621 q^{43} -1.00000 q^{45} -13.1231 q^{47} +19.2462 q^{49} +1.12311 q^{51} -10.0000 q^{53} +2.00000 q^{55} +5.12311 q^{57} -6.00000 q^{59} +2.00000 q^{61} +5.12311 q^{63} +5.12311 q^{65} +6.24621 q^{67} +5.12311 q^{69} -8.00000 q^{71} -4.24621 q^{73} -1.00000 q^{75} -10.2462 q^{77} -4.87689 q^{79} +1.00000 q^{81} +4.00000 q^{83} +1.12311 q^{85} -8.24621 q^{87} -10.0000 q^{89} -26.2462 q^{91} +7.12311 q^{93} +5.12311 q^{95} +10.0000 q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 4 q^{11} - 2 q^{13} + 2 q^{15} + 6 q^{17} - 2 q^{19} - 2 q^{21} - 2 q^{23} + 2 q^{25} - 2 q^{27} - 6 q^{31} + 4 q^{33} - 2 q^{35} + 2 q^{37} + 2 q^{39} - 4 q^{41}+ \cdots - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 - 4 * q^11 - 2 * q^13 + 2 * q^15 + 6 * q^17 - 2 * q^19 - 2 * q^21 - 2 * q^23 + 2 * q^25 - 2 * q^27 - 6 * q^31 + 4 * q^33 - 2 * q^35 + 2 * q^37 + 2 * q^39 - 4 * q^41 - 4 * q^43 - 2 * q^45 - 18 * q^47 + 22 * q^49 - 6 * q^51 - 20 * q^53 + 4 * q^55 + 2 * q^57 - 12 * q^59 + 4 * q^61 + 2 * q^63 + 2 * q^65 - 4 * q^67 + 2 * q^69 - 16 * q^71 + 8 * q^73 - 2 * q^75 - 4 * q^77 - 18 * q^79 + 2 * q^81 + 8 * q^83 - 6 * q^85 - 20 * q^89 - 36 * q^91 + 6 * q^93 + 2 * q^95 + 20 * q^97 - 4 * q^99

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	5.12311	1.93635	0.968176	−	0.250270i	$-0.0805195\pi$
$7$	5.12311	1.93635	0.968176	+	0.250270i	$0.0805195\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−5.12311	−1.42089	−0.710447	−	0.703751i	$-0.751507\pi$
$13$	−5.12311	−1.42089	−0.710447	+	0.703751i	$0.751507\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−1.12311	−0.272393	−0.136197	−	0.990682i	$-0.543488\pi$
$17$	−1.12311	−0.272393	−0.136197	+	0.990682i	$0.543488\pi$
$18$	0	0
$19$	−5.12311	−1.17532	−0.587661	−	0.809108i	$-0.699951\pi$
$19$	−5.12311	−1.17532	−0.587661	+	0.809108i	$0.699951\pi$
$20$	0	0
$21$	−5.12311	−1.11795
$22$	0	0
$23$	−5.12311	−1.06824	−0.534121	−	0.845408i	$-0.679357\pi$
$23$	−5.12311	−1.06824	−0.534121	+	0.845408i	$0.679357\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	8.24621	1.53128	0.765641	−	0.643268i	$-0.222422\pi$
$29$	8.24621	1.53128	0.765641	+	0.643268i	$0.222422\pi$
$30$	0	0
$31$	−7.12311	−1.27935	−0.639674	−	0.768647i	$-0.720931\pi$
$31$	−7.12311	−1.27935	−0.639674	+	0.768647i	$0.720931\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	−5.12311	−0.865963
$36$	0	0
$37$	5.12311	0.842233	0.421117	−	0.907006i	$-0.361638\pi$
$37$	5.12311	0.842233	0.421117	+	0.907006i	$0.361638\pi$
$38$	0	0
$39$	5.12311	0.820353
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	6.24621	0.952538	0.476269	−	0.879300i	$-0.341989\pi$
$43$	6.24621	0.952538	0.476269	+	0.879300i	$0.341989\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−13.1231	−1.91420	−0.957101	−	0.289755i	$-0.906426\pi$
$47$	−13.1231	−1.91420	−0.957101	+	0.289755i	$0.906426\pi$
$48$	0	0
$49$	19.2462	2.74946
$50$	0	0
$51$	1.12311	0.157266
$52$	0	0
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	5.12311	0.678572
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	5.12311	0.645451
$64$	0	0
$65$	5.12311	0.635443
$66$	0	0
$67$	6.24621	0.763096	0.381548	−	0.924349i	$-0.375391\pi$
$67$	6.24621	0.763096	0.381548	+	0.924349i	$0.375391\pi$
$68$	0	0
$69$	5.12311	0.616749
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	−4.24621	−0.496981	−0.248491	−	0.968634i	$-0.579935\pi$
$73$	−4.24621	−0.496981	−0.248491	+	0.968634i	$0.579935\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−10.2462	−1.16766
$78$	0	0
$79$	−4.87689	−0.548693	−0.274347	−	0.961631i	$-0.588462\pi$
$79$	−4.87689	−0.548693	−0.274347	+	0.961631i	$0.588462\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	1.12311	0.121818
$86$	0	0
$87$	−8.24621	−0.884087
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−26.2462	−2.75135
$92$	0	0
$93$	7.12311	0.738632
$94$	0	0
$95$	5.12311	0.525620
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	−11.3693	−1.12025	−0.560126	−	0.828407i	$-0.689247\pi$
$103$	−11.3693	−1.12025	−0.560126	+	0.828407i	$0.689247\pi$
$104$	0	0
$105$	5.12311	0.499964
$106$	0	0
$107$	−2.24621	−0.217149	−0.108575	−	0.994088i	$-0.534629\pi$
$107$	−2.24621	−0.217149	−0.108575	+	0.994088i	$0.534629\pi$
$108$	0	0
$109$	−4.24621	−0.406713	−0.203357	−	0.979105i	$-0.565185\pi$
$109$	−4.24621	−0.406713	−0.203357	+	0.979105i	$0.565185\pi$
$110$	0	0
$111$	−5.12311	−0.486264
$112$	0	0
$113$	6.87689	0.646924	0.323462	−	0.946241i	$-0.395153\pi$
$113$	6.87689	0.646924	0.323462	+	0.946241i	$0.395153\pi$
$114$	0	0
$115$	5.12311	0.477732
$116$	0	0
$117$	−5.12311	−0.473631
$118$	0	0
$119$	−5.75379	−0.527449
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	2.00000	0.180334
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−6.87689	−0.610226	−0.305113	−	0.952316i	$-0.598694\pi$
$127$	−6.87689	−0.610226	−0.305113	+	0.952316i	$0.598694\pi$
$128$	0	0
$129$	−6.24621	−0.549948
$130$	0	0
$131$	8.24621	0.720475	0.360237	−	0.932861i	$-0.382696\pi$
$131$	8.24621	0.720475	0.360237	+	0.932861i	$0.382696\pi$
$132$	0	0
$133$	−26.2462	−2.27584
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−2.87689	−0.245790	−0.122895	−	0.992420i	$-0.539218\pi$
$137$	−2.87689	−0.245790	−0.122895	+	0.992420i	$0.539218\pi$
$138$	0	0
$139$	−15.3693	−1.30361	−0.651804	−	0.758387i	$-0.725988\pi$
$139$	−15.3693	−1.30361	−0.651804	+	0.758387i	$0.725988\pi$
$140$	0	0
$141$	13.1231	1.10516
$142$	0	0
$143$	10.2462	0.856831
$144$	0	0
$145$	−8.24621	−0.684811
$146$	0	0
$147$	−19.2462	−1.58740
$148$	0	0
$149$	−20.2462	−1.65863	−0.829317	−	0.558778i	$-0.811270\pi$
$149$	−20.2462	−1.65863	−0.829317	+	0.558778i	$0.811270\pi$
$150$	0	0
$151$	17.3693	1.41349	0.706747	−	0.707466i	$-0.250162\pi$
$151$	17.3693	1.41349	0.706747	+	0.707466i	$0.250162\pi$
$152$	0	0
$153$	−1.12311	−0.0907977
$154$	0	0
$155$	7.12311	0.572142
$156$	0	0
$157$	19.3693	1.54584	0.772920	−	0.634504i	$-0.218795\pi$
$157$	19.3693	1.54584	0.772920	+	0.634504i	$0.218795\pi$
$158$	0	0
$159$	10.0000	0.793052
$160$	0	0
$161$	−26.2462	−2.06849
$162$	0	0
$163$	−8.49242	−0.665178	−0.332589	−	0.943072i	$-0.607922\pi$
$163$	−8.49242	−0.665178	−0.332589	+	0.943072i	$0.607922\pi$
$164$	0	0
$165$	−2.00000	−0.155700
$166$	0	0
$167$	−10.8769	−0.841679	−0.420840	−	0.907135i	$-0.638265\pi$
$167$	−10.8769	−0.841679	−0.420840	+	0.907135i	$0.638265\pi$
$168$	0	0
$169$	13.2462	1.01894
$170$	0	0
$171$	−5.12311	−0.391774
$172$	0	0
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	5.12311	0.387270
$176$	0	0
$177$	6.00000	0.450988
$178$	0	0
$179$	12.2462	0.915325	0.457662	−	0.889126i	$-0.348687\pi$
$179$	12.2462	0.915325	0.457662	+	0.889126i	$0.348687\pi$
$180$	0	0
$181$	0.246211	0.0183007	0.00915037	−	0.999958i	$-0.497087\pi$
$181$	0.246211	0.0183007	0.00915037	+	0.999958i	$0.497087\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	−5.12311	−0.376658
$186$	0	0
$187$	2.24621	0.164259
$188$	0	0
$189$	−5.12311	−0.372651
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	24.2462	1.74528	0.872640	−	0.488364i	$-0.162406\pi$
$193$	24.2462	1.74528	0.872640	+	0.488364i	$0.162406\pi$
$194$	0	0
$195$	−5.12311	−0.366873
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	−19.6155	−1.39051	−0.695254	−	0.718764i	$-0.744708\pi$
$199$	−19.6155	−1.39051	−0.695254	+	0.718764i	$0.744708\pi$
$200$	0	0
$201$	−6.24621	−0.440574
$202$	0	0
$203$	42.2462	2.96510
$204$	0	0
$205$	2.00000	0.139686
$206$	0	0
$207$	−5.12311	−0.356080
$208$	0	0
$209$	10.2462	0.708745
$210$	0	0
$211$	−10.8769	−0.748796	−0.374398	−	0.927268i	$-0.622151\pi$
$211$	−10.8769	−0.748796	−0.374398	+	0.927268i	$0.622151\pi$
$212$	0	0
$213$	8.00000	0.548151
$214$	0	0
$215$	−6.24621	−0.425988
$216$	0	0
$217$	−36.4924	−2.47727
$218$	0	0
$219$	4.24621	0.286932
$220$	0	0
$221$	5.75379	0.387042
$222$	0	0
$223$	14.8769	0.996231	0.498115	−	0.867111i	$-0.334026\pi$
$223$	14.8769	0.996231	0.498115	+	0.867111i	$0.334026\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	10.2462	0.674151
$232$	0	0
$233$	25.6155	1.67813	0.839065	−	0.544032i	$-0.183103\pi$
$233$	25.6155	1.67813	0.839065	+	0.544032i	$0.183103\pi$
$234$	0	0
$235$	13.1231	0.856057
$236$	0	0
$237$	4.87689	0.316788
$238$	0	0
$239$	−6.24621	−0.404034	−0.202017	−	0.979382i	$-0.564750\pi$
$239$	−6.24621	−0.404034	−0.202017	+	0.979382i	$0.564750\pi$
$240$	0	0
$241$	−7.75379	−0.499465	−0.249733	−	0.968315i	$-0.580343\pi$
$241$	−7.75379	−0.499465	−0.249733	+	0.968315i	$0.580343\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−19.2462	−1.22960
$246$	0	0
$247$	26.2462	1.67001
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	4.24621	0.268018	0.134009	−	0.990980i	$-0.457215\pi$
$251$	4.24621	0.268018	0.134009	+	0.990980i	$0.457215\pi$
$252$	0	0
$253$	10.2462	0.644174
$254$	0	0
$255$	−1.12311	−0.0703316
$256$	0	0
$257$	31.8617	1.98748	0.993740	−	0.111714i	$-0.0356342\pi$
$257$	31.8617	1.98748	0.993740	+	0.111714i	$0.0356342\pi$
$258$	0	0
$259$	26.2462	1.63086
$260$	0	0
$261$	8.24621	0.510428
$262$	0	0
$263$	15.3693	0.947713	0.473856	−	0.880602i	$-0.342862\pi$
$263$	15.3693	0.947713	0.473856	+	0.880602i	$0.342862\pi$
$264$	0	0
$265$	10.0000	0.614295
$266$	0	0
$267$	10.0000	0.611990
$268$	0	0
$269$	8.24621	0.502780	0.251390	−	0.967886i	$-0.419112\pi$
$269$	8.24621	0.502780	0.251390	+	0.967886i	$0.419112\pi$
$270$	0	0
$271$	−7.12311	−0.432698	−0.216349	−	0.976316i	$-0.569415\pi$
$271$	−7.12311	−0.432698	−0.216349	+	0.976316i	$0.569415\pi$
$272$	0	0
$273$	26.2462	1.58849
$274$	0	0
$275$	−2.00000	−0.120605
$276$	0	0
$277$	14.8769	0.893866	0.446933	−	0.894567i	$-0.352516\pi$
$277$	14.8769	0.893866	0.446933	+	0.894567i	$0.352516\pi$
$278$	0	0
$279$	−7.12311	−0.426449
$280$	0	0
$281$	0.246211	0.0146877	0.00734387	−	0.999973i	$-0.497662\pi$
$281$	0.246211	0.0146877	0.00734387	+	0.999973i	$0.497662\pi$
$282$	0	0
$283$	−30.2462	−1.79795	−0.898975	−	0.437999i	$-0.855687\pi$
$283$	−30.2462	−1.79795	−0.898975	+	0.437999i	$0.855687\pi$
$284$	0	0
$285$	−5.12311	−0.303467
$286$	0	0
$287$	−10.2462	−0.604815
$288$	0	0
$289$	−15.7386	−0.925802
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	28.7386	1.67893	0.839464	−	0.543415i	$-0.182869\pi$
$293$	28.7386	1.67893	0.839464	+	0.543415i	$0.182869\pi$
$294$	0	0
$295$	6.00000	0.349334
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	26.2462	1.51786
$300$	0	0
$301$	32.0000	1.84445
$302$	0	0
$303$	14.0000	0.804279
$304$	0	0
$305$	−2.00000	−0.114520
$306$	0	0
$307$	−14.2462	−0.813074	−0.406537	−	0.913634i	$-0.633264\pi$
$307$	−14.2462	−0.813074	−0.406537	+	0.913634i	$0.633264\pi$
$308$	0	0
$309$	11.3693	0.646778
$310$	0	0
$311$	−30.2462	−1.71511	−0.857553	−	0.514396i	$-0.828016\pi$
$311$	−30.2462	−1.71511	−0.857553	+	0.514396i	$0.828016\pi$
$312$	0	0
$313$	20.7386	1.17222	0.586108	−	0.810233i	$-0.300659\pi$
$313$	20.7386	1.17222	0.586108	+	0.810233i	$0.300659\pi$
$314$	0	0
$315$	−5.12311	−0.288654
$316$	0	0
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	0	0
$319$	−16.4924	−0.923398
$320$	0	0
$321$	2.24621	0.125371
$322$	0	0
$323$	5.75379	0.320149
$324$	0	0
$325$	−5.12311	−0.284179
$326$	0	0
$327$	4.24621	0.234816
$328$	0	0
$329$	−67.2311	−3.70657
$330$	0	0
$331$	1.12311	0.0617315	0.0308657	−	0.999524i	$-0.490174\pi$
$331$	1.12311	0.0617315	0.0308657	+	0.999524i	$0.490174\pi$
$332$	0	0
$333$	5.12311	0.280744
$334$	0	0
$335$	−6.24621	−0.341267
$336$	0	0
$337$	−24.7386	−1.34760	−0.673800	−	0.738914i	$-0.735339\pi$
$337$	−24.7386	−1.34760	−0.673800	+	0.738914i	$0.735339\pi$
$338$	0	0
$339$	−6.87689	−0.373502
$340$	0	0
$341$	14.2462	0.771476
$342$	0	0
$343$	62.7386	3.38757
$344$	0	0
$345$	−5.12311	−0.275819
$346$	0	0
$347$	−10.2462	−0.550045	−0.275023	−	0.961438i	$-0.588685\pi$
$347$	−10.2462	−0.550045	−0.275023	+	0.961438i	$0.588685\pi$
$348$	0	0
$349$	−10.4924	−0.561646	−0.280823	−	0.959760i	$-0.590607\pi$
$349$	−10.4924	−0.561646	−0.280823	+	0.959760i	$0.590607\pi$
$350$	0	0
$351$	5.12311	0.273451
$352$	0	0
$353$	−27.8617	−1.48293	−0.741465	−	0.670991i	$-0.765869\pi$
$353$	−27.8617	−1.48293	−0.741465	+	0.670991i	$0.765869\pi$
$354$	0	0
$355$	8.00000	0.424596
$356$	0	0
$357$	5.75379	0.304523
$358$	0	0
$359$	4.00000	0.211112	0.105556	−	0.994413i	$-0.466338\pi$
$359$	4.00000	0.211112	0.105556	+	0.994413i	$0.466338\pi$
$360$	0	0
$361$	7.24621	0.381380
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	4.24621	0.222257
$366$	0	0
$367$	7.36932	0.384675	0.192338	−	0.981329i	$-0.438393\pi$
$367$	7.36932	0.384675	0.192338	+	0.981329i	$0.438393\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	−51.2311	−2.65978
$372$	0	0
$373$	−17.1231	−0.886601	−0.443300	−	0.896373i	$-0.646193\pi$
$373$	−17.1231	−0.886601	−0.443300	+	0.896373i	$0.646193\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−42.2462	−2.17579
$378$	0	0
$379$	3.36932	0.173070	0.0865351	−	0.996249i	$-0.472421\pi$
$379$	3.36932	0.173070	0.0865351	+	0.996249i	$0.472421\pi$
$380$	0	0
$381$	6.87689	0.352314
$382$	0	0
$383$	5.12311	0.261778	0.130889	−	0.991397i	$-0.458217\pi$
$383$	5.12311	0.261778	0.130889	+	0.991397i	$0.458217\pi$
$384$	0	0
$385$	10.2462	0.522195
$386$	0	0
$387$	6.24621	0.317513
$388$	0	0
$389$	−10.4924	−0.531987	−0.265993	−	0.963975i	$-0.585700\pi$
$389$	−10.4924	−0.531987	−0.265993	+	0.963975i	$0.585700\pi$
$390$	0	0
$391$	5.75379	0.290982
$392$	0	0
$393$	−8.24621	−0.415966
$394$	0	0
$395$	4.87689	0.245383
$396$	0	0
$397$	15.3693	0.771364	0.385682	−	0.922632i	$-0.373966\pi$
$397$	15.3693	0.771364	0.385682	+	0.922632i	$0.373966\pi$
$398$	0	0
$399$	26.2462	1.31395
$400$	0	0
$401$	−11.7538	−0.586956	−0.293478	−	0.955966i	$-0.594813\pi$
$401$	−11.7538	−0.586956	−0.293478	+	0.955966i	$0.594813\pi$
$402$	0	0
$403$	36.4924	1.81782
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−10.2462	−0.507886
$408$	0	0
$409$	18.4924	0.914391	0.457196	−	0.889366i	$-0.348854\pi$
$409$	18.4924	0.914391	0.457196	+	0.889366i	$0.348854\pi$
$410$	0	0
$411$	2.87689	0.141907
$412$	0	0
$413$	−30.7386	−1.51255
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	15.3693	0.752639
$418$	0	0
$419$	−0.246211	−0.0120282	−0.00601410	−	0.999982i	$-0.501914\pi$
$419$	−0.246211	−0.0120282	−0.00601410	+	0.999982i	$0.501914\pi$
$420$	0	0
$421$	−30.9848	−1.51011	−0.755054	−	0.655662i	$-0.772390\pi$
$421$	−30.9848	−1.51011	−0.755054	+	0.655662i	$0.772390\pi$
$422$	0	0
$423$	−13.1231	−0.638067
$424$	0	0
$425$	−1.12311	−0.0544786
$426$	0	0
$427$	10.2462	0.495849
$428$	0	0
$429$	−10.2462	−0.494692
$430$	0	0
$431$	16.4924	0.794412	0.397206	−	0.917729i	$-0.369980\pi$
$431$	16.4924	0.794412	0.397206	+	0.917729i	$0.369980\pi$
$432$	0	0
$433$	0.246211	0.0118322	0.00591608	−	0.999982i	$-0.498117\pi$
$433$	0.246211	0.0118322	0.00591608	+	0.999982i	$0.498117\pi$
$434$	0	0
$435$	8.24621	0.395376
$436$	0	0
$437$	26.2462	1.25553
$438$	0	0
$439$	−2.63068	−0.125556	−0.0627778	−	0.998028i	$-0.519996\pi$
$439$	−2.63068	−0.125556	−0.0627778	+	0.998028i	$0.519996\pi$
$440$	0	0
$441$	19.2462	0.916486
$442$	0	0
$443$	3.50758	0.166650	0.0833250	−	0.996522i	$-0.473446\pi$
$443$	3.50758	0.166650	0.0833250	+	0.996522i	$0.473446\pi$
$444$	0	0
$445$	10.0000	0.474045
$446$	0	0
$447$	20.2462	0.957613
$448$	0	0
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	0	0
$451$	4.00000	0.188353
$452$	0	0
$453$	−17.3693	−0.816082
$454$	0	0
$455$	26.2462	1.23044
$456$	0	0
$457$	6.49242	0.303703	0.151851	−	0.988403i	$-0.451476\pi$
$457$	6.49242	0.303703	0.151851	+	0.988403i	$0.451476\pi$
$458$	0	0
$459$	1.12311	0.0524221
$460$	0	0
$461$	40.2462	1.87445	0.937226	−	0.348721i	$-0.113384\pi$
$461$	40.2462	1.87445	0.937226	+	0.348721i	$0.113384\pi$
$462$	0	0
$463$	37.1231	1.72526	0.862629	−	0.505838i	$-0.168817\pi$
$463$	37.1231	1.72526	0.862629	+	0.505838i	$0.168817\pi$
$464$	0	0
$465$	−7.12311	−0.330326
$466$	0	0
$467$	18.7386	0.867121	0.433560	−	0.901125i	$-0.357257\pi$
$467$	18.7386	0.867121	0.433560	+	0.901125i	$0.357257\pi$
$468$	0	0
$469$	32.0000	1.47762
$470$	0	0
$471$	−19.3693	−0.892491
$472$	0	0
$473$	−12.4924	−0.574402
$474$	0	0
$475$	−5.12311	−0.235064
$476$	0	0
$477$	−10.0000	−0.457869
$478$	0	0
$479$	36.4924	1.66738	0.833691	−	0.552232i	$-0.186224\pi$
$479$	36.4924	1.66738	0.833691	+	0.552232i	$0.186224\pi$
$480$	0	0
$481$	−26.2462	−1.19672
$482$	0	0
$483$	26.2462	1.19424
$484$	0	0
$485$	−10.0000	−0.454077
$486$	0	0
$487$	−13.6155	−0.616978	−0.308489	−	0.951228i	$-0.599823\pi$
$487$	−13.6155	−0.616978	−0.308489	+	0.951228i	$0.599823\pi$
$488$	0	0
$489$	8.49242	0.384041
$490$	0	0
$491$	1.50758	0.0680360	0.0340180	−	0.999421i	$-0.489170\pi$
$491$	1.50758	0.0680360	0.0340180	+	0.999421i	$0.489170\pi$
$492$	0	0
$493$	−9.26137	−0.417111
$494$	0	0
$495$	2.00000	0.0898933
$496$	0	0
$497$	−40.9848	−1.83842
$498$	0	0
$499$	4.63068	0.207298	0.103649	−	0.994614i	$-0.466948\pi$
$499$	4.63068	0.207298	0.103649	+	0.994614i	$0.466948\pi$
$500$	0	0
$501$	10.8769	0.485944
$502$	0	0
$503$	−1.61553	−0.0720328	−0.0360164	−	0.999351i	$-0.511467\pi$
$503$	−1.61553	−0.0720328	−0.0360164	+	0.999351i	$0.511467\pi$
$504$	0	0
$505$	14.0000	0.622992
$506$	0	0
$507$	−13.2462	−0.588285
$508$	0	0
$509$	−14.0000	−0.620539	−0.310270	−	0.950649i	$-0.600419\pi$
$509$	−14.0000	−0.620539	−0.310270	+	0.950649i	$0.600419\pi$
$510$	0	0
$511$	−21.7538	−0.962331
$512$	0	0
$513$	5.12311	0.226191
$514$	0	0
$515$	11.3693	0.500992
$516$	0	0
$517$	26.2462	1.15431
$518$	0	0
$519$	14.0000	0.614532
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	0	0
$523$	−18.7386	−0.819383	−0.409692	−	0.912224i	$-0.634364\pi$
$523$	−18.7386	−0.819383	−0.409692	+	0.912224i	$0.634364\pi$
$524$	0	0
$525$	−5.12311	−0.223591
$526$	0	0
$527$	8.00000	0.348485
$528$	0	0
$529$	3.24621	0.141140
$530$	0	0
$531$	−6.00000	−0.260378
$532$	0	0
$533$	10.2462	0.443813
$534$	0	0
$535$	2.24621	0.0971122
$536$	0	0
$537$	−12.2462	−0.528463
$538$	0	0
$539$	−38.4924	−1.65799
$540$	0	0
$541$	8.24621	0.354532	0.177266	−	0.984163i	$-0.443275\pi$
$541$	8.24621	0.354532	0.177266	+	0.984163i	$0.443275\pi$
$542$	0	0
$543$	−0.246211	−0.0105659
$544$	0	0
$545$	4.24621	0.181888
$546$	0	0
$547$	9.75379	0.417042	0.208521	−	0.978018i	$-0.433135\pi$
$547$	9.75379	0.417042	0.208521	+	0.978018i	$0.433135\pi$
$548$	0	0
$549$	2.00000	0.0853579
$550$	0	0
$551$	−42.2462	−1.79975
$552$	0	0
$553$	−24.9848	−1.06246
$554$	0	0
$555$	5.12311	0.217464
$556$	0	0
$557$	−1.50758	−0.0638781	−0.0319391	−	0.999490i	$-0.510168\pi$
$557$	−1.50758	−0.0638781	−0.0319391	+	0.999490i	$0.510168\pi$
$558$	0	0
$559$	−32.0000	−1.35346
$560$	0	0
$561$	−2.24621	−0.0948351
$562$	0	0
$563$	22.7386	0.958319	0.479160	−	0.877728i	$-0.340942\pi$
$563$	22.7386	0.958319	0.479160	+	0.877728i	$0.340942\pi$
$564$	0	0
$565$	−6.87689	−0.289313
$566$	0	0
$567$	5.12311	0.215150
$568$	0	0
$569$	24.7386	1.03710	0.518549	−	0.855048i	$-0.326472\pi$
$569$	24.7386	1.03710	0.518549	+	0.855048i	$0.326472\pi$
$570$	0	0
$571$	−2.38447	−0.0997870	−0.0498935	−	0.998755i	$-0.515888\pi$
$571$	−2.38447	−0.0997870	−0.0498935	+	0.998755i	$0.515888\pi$
$572$	0	0
$573$	4.00000	0.167102
$574$	0	0
$575$	−5.12311	−0.213648
$576$	0	0
$577$	42.9848	1.78948	0.894741	−	0.446585i	$-0.147360\pi$
$577$	42.9848	1.78948	0.894741	+	0.446585i	$0.147360\pi$
$578$	0	0
$579$	−24.2462	−1.00764
$580$	0	0
$581$	20.4924	0.850169
$582$	0	0
$583$	20.0000	0.828315
$584$	0	0
$585$	5.12311	0.211814
$586$	0	0
$587$	46.7386	1.92911	0.964555	−	0.263882i	$-0.0850030\pi$
$587$	46.7386	1.92911	0.964555	+	0.263882i	$0.0850030\pi$
$588$	0	0
$589$	36.4924	1.50364
$590$	0	0
$591$	2.00000	0.0822690
$592$	0	0
$593$	39.8617	1.63693	0.818463	−	0.574560i	$-0.194827\pi$
$593$	39.8617	1.63693	0.818463	+	0.574560i	$0.194827\pi$
$594$	0	0
$595$	5.75379	0.235882
$596$	0	0
$597$	19.6155	0.802810
$598$	0	0
$599$	−24.4924	−1.00073	−0.500367	−	0.865814i	$-0.666801\pi$
$599$	−24.4924	−1.00073	−0.500367	+	0.865814i	$0.666801\pi$
$600$	0	0
$601$	−30.9848	−1.26390	−0.631949	−	0.775010i	$-0.717745\pi$
$601$	−30.9848	−1.26390	−0.631949	+	0.775010i	$0.717745\pi$
$602$	0	0
$603$	6.24621	0.254365
$604$	0	0
$605$	7.00000	0.284590
$606$	0	0
$607$	−13.1231	−0.532650	−0.266325	−	0.963883i	$-0.585810\pi$
$607$	−13.1231	−0.532650	−0.266325	+	0.963883i	$0.585810\pi$
$608$	0	0
$609$	−42.2462	−1.71190
$610$	0	0
$611$	67.2311	2.71988
$612$	0	0
$613$	7.36932	0.297644	0.148822	−	0.988864i	$-0.452452\pi$
$613$	7.36932	0.297644	0.148822	+	0.988864i	$0.452452\pi$
$614$	0	0
$615$	−2.00000	−0.0806478
$616$	0	0
$617$	9.61553	0.387107	0.193553	−	0.981090i	$-0.437999\pi$
$617$	9.61553	0.387107	0.193553	+	0.981090i	$0.437999\pi$
$618$	0	0
$619$	21.1231	0.849009	0.424505	−	0.905426i	$-0.360448\pi$
$619$	21.1231	0.849009	0.424505	+	0.905426i	$0.360448\pi$
$620$	0	0
$621$	5.12311	0.205583
$622$	0	0
$623$	−51.2311	−2.05253
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−10.2462	−0.409194
$628$	0	0
$629$	−5.75379	−0.229419
$630$	0	0
$631$	−13.8617	−0.551827	−0.275914	−	0.961182i	$-0.588980\pi$
$631$	−13.8617	−0.551827	−0.275914	+	0.961182i	$0.588980\pi$
$632$	0	0
$633$	10.8769	0.432318
$634$	0	0
$635$	6.87689	0.272901
$636$	0	0
$637$	−98.6004	−3.90669
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	−36.2462	−1.43164	−0.715820	−	0.698285i	$-0.753947\pi$
$641$	−36.2462	−1.43164	−0.715820	+	0.698285i	$0.753947\pi$
$642$	0	0
$643$	32.4924	1.28138	0.640688	−	0.767801i	$-0.278649\pi$
$643$	32.4924	1.28138	0.640688	+	0.767801i	$0.278649\pi$
$644$	0	0
$645$	6.24621	0.245944
$646$	0	0
$647$	13.1231	0.515923	0.257961	−	0.966155i	$-0.416949\pi$
$647$	13.1231	0.515923	0.257961	+	0.966155i	$0.416949\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	36.4924	1.43025
$652$	0	0
$653$	−44.7386	−1.75076	−0.875379	−	0.483437i	$-0.839388\pi$
$653$	−44.7386	−1.75076	−0.875379	+	0.483437i	$0.839388\pi$
$654$	0	0
$655$	−8.24621	−0.322206
$656$	0	0
$657$	−4.24621	−0.165660
$658$	0	0
$659$	−26.0000	−1.01282	−0.506408	−	0.862294i	$-0.669027\pi$
$659$	−26.0000	−1.01282	−0.506408	+	0.862294i	$0.669027\pi$
$660$	0	0
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	0	0
$663$	−5.75379	−0.223459
$664$	0	0
$665$	26.2462	1.01778
$666$	0	0
$667$	−42.2462	−1.63578
$668$	0	0
$669$	−14.8769	−0.575174
$670$	0	0
$671$	−4.00000	−0.154418
$672$	0	0
$673$	−20.2462	−0.780434	−0.390217	−	0.920723i	$-0.627600\pi$
$673$	−20.2462	−0.780434	−0.390217	+	0.920723i	$0.627600\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−4.73863	−0.182120	−0.0910602	−	0.995845i	$-0.529026\pi$
$677$	−4.73863	−0.182120	−0.0910602	+	0.995845i	$0.529026\pi$
$678$	0	0
$679$	51.2311	1.96607
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9.75379	0.373218	0.186609	−	0.982434i	$-0.440250\pi$
$683$	9.75379	0.373218	0.186609	+	0.982434i	$0.440250\pi$
$684$	0	0
$685$	2.87689	0.109920
$686$	0	0
$687$	−10.0000	−0.381524
$688$	0	0
$689$	51.2311	1.95175
$690$	0	0
$691$	13.1231	0.499226	0.249613	−	0.968346i	$-0.419697\pi$
$691$	13.1231	0.499226	0.249613	+	0.968346i	$0.419697\pi$
$692$	0	0
$693$	−10.2462	−0.389221
$694$	0	0
$695$	15.3693	0.582991
$696$	0	0
$697$	2.24621	0.0850813
$698$	0	0
$699$	−25.6155	−0.968868
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	−26.2462	−0.989895
$704$	0	0
$705$	−13.1231	−0.494245
$706$	0	0
$707$	−71.7235	−2.69744
$708$	0	0
$709$	50.9848	1.91478	0.957388	−	0.288805i	$-0.0932578\pi$
$709$	50.9848	1.91478	0.957388	+	0.288805i	$0.0932578\pi$
$710$	0	0
$711$	−4.87689	−0.182898
$712$	0	0
$713$	36.4924	1.36665
$714$	0	0
$715$	−10.2462	−0.383187
$716$	0	0
$717$	6.24621	0.233269
$718$	0	0
$719$	22.2462	0.829644	0.414822	−	0.909903i	$-0.363844\pi$
$719$	22.2462	0.829644	0.414822	+	0.909903i	$0.363844\pi$
$720$	0	0
$721$	−58.2462	−2.16920
$722$	0	0
$723$	7.75379	0.288367
$724$	0	0
$725$	8.24621	0.306257
$726$	0	0
$727$	−29.6155	−1.09838	−0.549190	−	0.835698i	$-0.685064\pi$
$727$	−29.6155	−1.09838	−0.549190	+	0.835698i	$0.685064\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−7.01515	−0.259465
$732$	0	0
$733$	−37.6155	−1.38936	−0.694681	−	0.719318i	$-0.744454\pi$
$733$	−37.6155	−1.38936	−0.694681	+	0.719318i	$0.744454\pi$
$734$	0	0
$735$	19.2462	0.709907
$736$	0	0
$737$	−12.4924	−0.460164
$738$	0	0
$739$	33.1231	1.21845	0.609227	−	0.792996i	$-0.291480\pi$
$739$	33.1231	1.21845	0.609227	+	0.792996i	$0.291480\pi$
$740$	0	0
$741$	−26.2462	−0.964179
$742$	0	0
$743$	19.8617	0.728657	0.364328	−	0.931271i	$-0.381299\pi$
$743$	19.8617	0.728657	0.364328	+	0.931271i	$0.381299\pi$
$744$	0	0
$745$	20.2462	0.741764
$746$	0	0
$747$	4.00000	0.146352
$748$	0	0
$749$	−11.5076	−0.420478
$750$	0	0
$751$	−2.63068	−0.0959950	−0.0479975	−	0.998847i	$-0.515284\pi$
$751$	−2.63068	−0.0959950	−0.0479975	+	0.998847i	$0.515284\pi$
$752$	0	0
$753$	−4.24621	−0.154741
$754$	0	0
$755$	−17.3693	−0.632134
$756$	0	0
$757$	−13.6155	−0.494865	−0.247432	−	0.968905i	$-0.579587\pi$
$757$	−13.6155	−0.494865	−0.247432	+	0.968905i	$0.579587\pi$
$758$	0	0
$759$	−10.2462	−0.371914
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	0	0
$763$	−21.7538	−0.787540
$764$	0	0
$765$	1.12311	0.0406060
$766$	0	0
$767$	30.7386	1.10991
$768$	0	0
$769$	31.7538	1.14507	0.572535	−	0.819880i	$-0.305960\pi$
$769$	31.7538	1.14507	0.572535	+	0.819880i	$0.305960\pi$
$770$	0	0
$771$	−31.8617	−1.14747
$772$	0	0
$773$	−47.4773	−1.70764	−0.853819	−	0.520569i	$-0.825720\pi$
$773$	−47.4773	−1.70764	−0.853819	+	0.520569i	$0.825720\pi$
$774$	0	0
$775$	−7.12311	−0.255870
$776$	0	0
$777$	−26.2462	−0.941578
$778$	0	0
$779$	10.2462	0.367109
$780$	0	0
$781$	16.0000	0.572525
$782$	0	0
$783$	−8.24621	−0.294696
$784$	0	0
$785$	−19.3693	−0.691321
$786$	0	0
$787$	−38.2462	−1.36333	−0.681665	−	0.731664i	$-0.738744\pi$
$787$	−38.2462	−1.36333	−0.681665	+	0.731664i	$0.738744\pi$
$788$	0	0
$789$	−15.3693	−0.547162
$790$	0	0
$791$	35.2311	1.25267
$792$	0	0
$793$	−10.2462	−0.363854
$794$	0	0
$795$	−10.0000	−0.354663
$796$	0	0
$797$	−11.7538	−0.416341	−0.208170	−	0.978093i	$-0.566751\pi$
$797$	−11.7538	−0.416341	−0.208170	+	0.978093i	$0.566751\pi$
$798$	0	0
$799$	14.7386	0.521415
$800$	0	0
$801$	−10.0000	−0.353333
$802$	0	0
$803$	8.49242	0.299691
$804$	0	0
$805$	26.2462	0.925057
$806$	0	0
$807$	−8.24621	−0.290280
$808$	0	0
$809$	−12.2462	−0.430554	−0.215277	−	0.976553i	$-0.569065\pi$
$809$	−12.2462	−0.430554	−0.215277	+	0.976553i	$0.569065\pi$
$810$	0	0
$811$	17.1231	0.601274	0.300637	−	0.953739i	$-0.402801\pi$
$811$	17.1231	0.601274	0.300637	+	0.953739i	$0.402801\pi$
$812$	0	0
$813$	7.12311	0.249818
$814$	0	0
$815$	8.49242	0.297477
$816$	0	0
$817$	−32.0000	−1.11954
$818$	0	0
$819$	−26.2462	−0.917117
$820$	0	0
$821$	−24.7386	−0.863384	−0.431692	−	0.902021i	$-0.642083\pi$
$821$	−24.7386	−0.863384	−0.431692	+	0.902021i	$0.642083\pi$
$822$	0	0
$823$	20.6307	0.719140	0.359570	−	0.933118i	$-0.382923\pi$
$823$	20.6307	0.719140	0.359570	+	0.933118i	$0.382923\pi$
$824$	0	0
$825$	2.00000	0.0696311
$826$	0	0
$827$	−23.5076	−0.817439	−0.408719	−	0.912660i	$-0.634024\pi$
$827$	−23.5076	−0.817439	−0.408719	+	0.912660i	$0.634024\pi$
$828$	0	0
$829$	20.7386	0.720283	0.360141	−	0.932898i	$-0.382728\pi$
$829$	20.7386	0.720283	0.360141	+	0.932898i	$0.382728\pi$
$830$	0	0
$831$	−14.8769	−0.516074
$832$	0	0
$833$	−21.6155	−0.748934
$834$	0	0
$835$	10.8769	0.376410
$836$	0	0
$837$	7.12311	0.246211
$838$	0	0
$839$	44.0000	1.51905	0.759524	−	0.650479i	$-0.225432\pi$
$839$	44.0000	1.51905	0.759524	+	0.650479i	$0.225432\pi$
$840$	0	0
$841$	39.0000	1.34483
$842$	0	0
$843$	−0.246211	−0.00847997
$844$	0	0
$845$	−13.2462	−0.455684
$846$	0	0
$847$	−35.8617	−1.23222
$848$	0	0
$849$	30.2462	1.03805
$850$	0	0
$851$	−26.2462	−0.899709
$852$	0	0
$853$	−21.1231	−0.723241	−0.361621	−	0.932325i	$-0.617776\pi$
$853$	−21.1231	−0.723241	−0.361621	+	0.932325i	$0.617776\pi$
$854$	0	0
$855$	5.12311	0.175207
$856$	0	0
$857$	30.8769	1.05473	0.527367	−	0.849637i	$-0.323179\pi$
$857$	30.8769	1.05473	0.527367	+	0.849637i	$0.323179\pi$
$858$	0	0
$859$	−25.1231	−0.857189	−0.428595	−	0.903497i	$-0.640991\pi$
$859$	−25.1231	−0.857189	−0.428595	+	0.903497i	$0.640991\pi$
$860$	0	0
$861$	10.2462	0.349190
$862$	0	0
$863$	−47.3693	−1.61247	−0.806235	−	0.591595i	$-0.798498\pi$
$863$	−47.3693	−1.61247	−0.806235	+	0.591595i	$0.798498\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	0	0
$867$	15.7386	0.534512
$868$	0	0
$869$	9.75379	0.330875
$870$	0	0
$871$	−32.0000	−1.08428
$872$	0	0
$873$	10.0000	0.338449
$874$	0	0
$875$	−5.12311	−0.173193
$876$	0	0
$877$	−53.6155	−1.81047	−0.905234	−	0.424914i	$-0.860304\pi$
$877$	−53.6155	−1.81047	−0.905234	+	0.424914i	$0.860304\pi$
$878$	0	0
$879$	−28.7386	−0.969330
$880$	0	0
$881$	−14.4924	−0.488262	−0.244131	−	0.969742i	$-0.578503\pi$
$881$	−14.4924	−0.488262	−0.244131	+	0.969742i	$0.578503\pi$
$882$	0	0
$883$	0.492423	0.0165713	0.00828567	−	0.999966i	$-0.497363\pi$
$883$	0.492423	0.0165713	0.00828567	+	0.999966i	$0.497363\pi$
$884$	0	0
$885$	−6.00000	−0.201688
$886$	0	0
$887$	−6.38447	−0.214370	−0.107185	−	0.994239i	$-0.534184\pi$
$887$	−6.38447	−0.214370	−0.107185	+	0.994239i	$0.534184\pi$
$888$	0	0
$889$	−35.2311	−1.18161
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	67.2311	2.24980
$894$	0	0
$895$	−12.2462	−0.409346
$896$	0	0
$897$	−26.2462	−0.876335
$898$	0	0
$899$	−58.7386	−1.95904
$900$	0	0
$901$	11.2311	0.374161
$902$	0	0
$903$	−32.0000	−1.06489
$904$	0	0
$905$	−0.246211	−0.00818434
$906$	0	0
$907$	−28.9848	−0.962426	−0.481213	−	0.876604i	$-0.659804\pi$
$907$	−28.9848	−0.962426	−0.481213	+	0.876604i	$0.659804\pi$
$908$	0	0
$909$	−14.0000	−0.464351
$910$	0	0
$911$	9.26137	0.306843	0.153421	−	0.988161i	$-0.450971\pi$
$911$	9.26137	0.306843	0.153421	+	0.988161i	$0.450971\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	0	0
$915$	2.00000	0.0661180
$916$	0	0
$917$	42.2462	1.39509
$918$	0	0
$919$	−39.1231	−1.29055	−0.645276	−	0.763949i	$-0.723258\pi$
$919$	−39.1231	−1.29055	−0.645276	+	0.763949i	$0.723258\pi$
$920$	0	0
$921$	14.2462	0.469429
$922$	0	0
$923$	40.9848	1.34903
$924$	0	0
$925$	5.12311	0.168447
$926$	0	0
$927$	−11.3693	−0.373417
$928$	0	0
$929$	14.4924	0.475481	0.237740	−	0.971329i	$-0.423593\pi$
$929$	14.4924	0.475481	0.237740	+	0.971329i	$0.423593\pi$
$930$	0	0
$931$	−98.6004	−3.23150
$932$	0	0
$933$	30.2462	0.990217
$934$	0	0
$935$	−2.24621	−0.0734590
$936$	0	0
$937$	−14.0000	−0.457360	−0.228680	−	0.973502i	$-0.573441\pi$
$937$	−14.0000	−0.457360	−0.228680	+	0.973502i	$0.573441\pi$
$938$	0	0
$939$	−20.7386	−0.676780
$940$	0	0
$941$	−43.4773	−1.41732	−0.708659	−	0.705551i	$-0.750700\pi$
$941$	−43.4773	−1.41732	−0.708659	+	0.705551i	$0.750700\pi$
$942$	0	0
$943$	10.2462	0.333663
$944$	0	0
$945$	5.12311	0.166655
$946$	0	0
$947$	26.7386	0.868889	0.434444	−	0.900699i	$-0.356945\pi$
$947$	26.7386	0.868889	0.434444	+	0.900699i	$0.356945\pi$
$948$	0	0
$949$	21.7538	0.706158
$950$	0	0
$951$	2.00000	0.0648544
$952$	0	0
$953$	45.1231	1.46168	0.730840	−	0.682548i	$-0.239128\pi$
$953$	45.1231	1.46168	0.730840	+	0.682548i	$0.239128\pi$
$954$	0	0
$955$	4.00000	0.129437
$956$	0	0
$957$	16.4924	0.533124
$958$	0	0
$959$	−14.7386	−0.475935
$960$	0	0
$961$	19.7386	0.636730
$962$	0	0
$963$	−2.24621	−0.0723831
$964$	0	0
$965$	−24.2462	−0.780513
$966$	0	0
$967$	51.3693	1.65193	0.825963	−	0.563724i	$-0.190632\pi$
$967$	51.3693	1.65193	0.825963	+	0.563724i	$0.190632\pi$
$968$	0	0
$969$	−5.75379	−0.184838
$970$	0	0
$971$	−32.2462	−1.03483	−0.517415	−	0.855735i	$-0.673106\pi$
$971$	−32.2462	−1.03483	−0.517415	+	0.855735i	$0.673106\pi$
$972$	0	0
$973$	−78.7386	−2.52424
$974$	0	0
$975$	5.12311	0.164071
$976$	0	0
$977$	30.8769	0.987839	0.493920	−	0.869508i	$-0.335564\pi$
$977$	30.8769	0.987839	0.493920	+	0.869508i	$0.335564\pi$
$978$	0	0
$979$	20.0000	0.639203
$980$	0	0
$981$	−4.24621	−0.135571
$982$	0	0
$983$	47.3693	1.51085	0.755423	−	0.655237i	$-0.227431\pi$
$983$	47.3693	1.51085	0.755423	+	0.655237i	$0.227431\pi$
$984$	0	0
$985$	2.00000	0.0637253
$986$	0	0
$987$	67.2311	2.13999
$988$	0	0
$989$	−32.0000	−1.01754
$990$	0	0
$991$	15.1231	0.480401	0.240201	−	0.970723i	$-0.422787\pi$
$991$	15.1231	0.480401	0.240201	+	0.970723i	$0.422787\pi$
$992$	0	0
$993$	−1.12311	−0.0356407
$994$	0	0
$995$	19.6155	0.621854
$996$	0	0
$997$	5.61553	0.177846	0.0889228	−	0.996039i	$-0.471658\pi$
$997$	5.61553	0.177846	0.0889228	+	0.996039i	$0.471658\pi$
$998$	0	0
$999$	−5.12311	−0.162088

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1920.2.a.y.1.2	✓	2
3.2	odd	2		5760.2.a.cj.1.2		2
4.3	odd	2		1920.2.a.ba.1.1	yes	2
5.4	even	2		9600.2.a.cw.1.1		2
8.3	odd	2		1920.2.a.z.1.1	yes	2
8.5	even	2		1920.2.a.bb.1.2	yes	2
12.11	even	2		5760.2.a.cg.1.1		2
16.3	odd	4		3840.2.k.bd.1921.4		4
16.5	even	4		3840.2.k.bc.1921.3		4
16.11	odd	4		3840.2.k.bd.1921.2		4
16.13	even	4		3840.2.k.bc.1921.1		4
20.19	odd	2		9600.2.a.ct.1.2		2
24.5	odd	2		5760.2.a.cc.1.2		2
24.11	even	2		5760.2.a.bx.1.1		2
40.19	odd	2		9600.2.a.dg.1.2		2
40.29	even	2		9600.2.a.cl.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.a.y.1.2	✓	2	1.1	even	1	trivial
1920.2.a.z.1.1	yes	2	8.3	odd	2
1920.2.a.ba.1.1	yes	2	4.3	odd	2
1920.2.a.bb.1.2	yes	2	8.5	even	2
3840.2.k.bc.1921.1		4	16.13	even	4
3840.2.k.bc.1921.3		4	16.5	even	4
3840.2.k.bd.1921.2		4	16.11	odd	4
3840.2.k.bd.1921.4		4	16.3	odd	4
5760.2.a.bx.1.1		2	24.11	even	2
5760.2.a.cc.1.2		2	24.5	odd	2
5760.2.a.cg.1.1		2	12.11	even	2
5760.2.a.cj.1.2		2	3.2	odd	2
9600.2.a.cl.1.1		2	40.29	even	2
9600.2.a.ct.1.2		2	20.19	odd	2
9600.2.a.cw.1.1		2	5.4	even	2
9600.2.a.dg.1.2		2	40.19	odd	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:	\( N \)	\(=\)	\( 1920 = 2^{7} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1920.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +5.12311 q^{7} +1.00000 q^{9} -2.00000 q^{11} -5.12311 q^{13} +1.00000 q^{15} -1.12311 q^{17} -5.12311 q^{19} -5.12311 q^{21} -5.12311 q^{23} +1.00000 q^{25} -1.00000 q^{27} +8.24621 q^{29} -7.12311 q^{31} +2.00000 q^{33} -5.12311 q^{35} +5.12311 q^{37} +5.12311 q^{39} -2.00000 q^{41} +6.24621 q^{43} -1.00000 q^{45} -13.1231 q^{47} +19.2462 q^{49} +1.12311 q^{51} -10.0000 q^{53} +2.00000 q^{55} +5.12311 q^{57} -6.00000 q^{59} +2.00000 q^{61} +5.12311 q^{63} +5.12311 q^{65} +6.24621 q^{67} +5.12311 q^{69} -8.00000 q^{71} -4.24621 q^{73} -1.00000 q^{75} -10.2462 q^{77} -4.87689 q^{79} +1.00000 q^{81} +4.00000 q^{83} +1.12311 q^{85} -8.24621 q^{87} -10.0000 q^{89} -26.2462 q^{91} +7.12311 q^{93} +5.12311 q^{95} +10.0000 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 4 q^{11} - 2 q^{13} + 2 q^{15} + 6 q^{17} - 2 q^{19} - 2 q^{21} - 2 q^{23} + 2 q^{25} - 2 q^{27} - 6 q^{31} + 4 q^{33} - 2 q^{35} + 2 q^{37} + 2 q^{39} - 4 q^{41}+ \cdots - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 - 4 * q^11 - 2 * q^13 + 2 * q^15 + 6 * q^17 - 2 * q^19 - 2 * q^21 - 2 * q^23 + 2 * q^25 - 2 * q^27 - 6 * q^31 + 4 * q^33 - 2 * q^35 + 2 * q^37 + 2 * q^39 - 4 * q^41 - 4 * q^43 - 2 * q^45 - 18 * q^47 + 22 * q^49 - 6 * q^51 - 20 * q^53 + 4 * q^55 + 2 * q^57 - 12 * q^59 + 4 * q^61 + 2 * q^63 + 2 * q^65 - 4 * q^67 + 2 * q^69 - 16 * q^71 + 8 * q^73 - 2 * q^75 - 4 * q^77 - 18 * q^79 + 2 * q^81 + 8 * q^83 - 6 * q^85 - 20 * q^89 - 36 * q^91 + 6 * q^93 + 2 * q^95 + 20 * q^97 - 4 * q^99

Introduction

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