LMFDB - Embedded newform 1728.3.h.j.161.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1728,3,Mod(161,1728)] 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("1728.161"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1728, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0, 0,0,0,0,0,252,0,0,0,0,0,0,0,0,0,0,0,0,0,168] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(49)] == traces)

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

Self dual:	no
Analytic conductor:	$47.0845896815$
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} - 13x^{10} + 129x^{8} - 512x^{6} + 1548x^{4} - 160x^{2} + 16 $ x^12 - 13x^10 + 129x^8 - 512x^6 + 1548x^4 - 160*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{26}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.8
Root			$2.47317 + 1.42789i$ of defining polynomial
Character	$\chi$	$=$	1728.161
Dual form			1728.3.h.j.161.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.519033 q^{5} +10.4117 q^{7} -14.1346 q^{11} +5.39052i q^{13} -24.9326i q^{17} +13.3367i q^{19} -41.1084i q^{23} -24.7306 q^{25} +7.85538 q^{29} -26.1945 q^{31} +5.40403 q^{35} +1.53769i q^{37} -21.3367i q^{41} -64.1345i q^{43} +19.8630i q^{47} +59.4039 q^{49} -68.6521 q^{53} -7.33635 q^{55} +67.8652 q^{59} -58.6563i q^{61} +2.79786i q^{65} +56.1345i q^{67} -107.615i q^{71} +7.73060 q^{73} -147.166 q^{77} +64.3579 q^{79} +125.192 q^{83} -12.9409i q^{85} +59.6159i q^{89} +56.1246i q^{91} +6.92217i q^{95} -138.481 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{11} + 72 q^{25} + 252 q^{35} + 168 q^{49} + 264 q^{59} - 276 q^{73} + 396 q^{83} - 396 q^{97}+O(q^{100}) $ 12 * q + 12 * q^11 + 72 * q^25 + 252 * q^35 + 168 * q^49 + 264 * q^59 - 276 * q^73 + 396 * q^83 - 396 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0.519033	0.103807	0.0519033	−	0.998652i	$-0.483471\pi$
$5$	0.519033	0.103807	0.0519033	+	0.998652i	$0.483471\pi$
$6$	0	0
$7$	10.4117	1.48739	0.743694	−	0.668520i	$-0.233072\pi$
$7$	10.4117	1.48739	0.743694	+	0.668520i	$0.233072\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−14.1346	−1.28497	−0.642483	−	0.766300i	$-0.722096\pi$
$11$	−14.1346	−1.28497	−0.642483	+	0.766300i	$0.722096\pi$
$12$	0	0
$13$	5.39052i	0.414655i	0.978272	+	0.207328i	$0.0664766\pi$
$13$	5.39052i	0.414655i	−0.978272	+	0.207328i	$0.933523\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 24.9326i	− 1.46662i	−0.679892	−	0.733312i	$-0.737973\pi$
$17$	− 24.9326i	− 1.46662i	0.679892	−	0.733312i	$-0.262027\pi$
$18$	0	0
$19$	13.3367i	0.701929i	0.936389	+	0.350964i	$0.114146\pi$
$19$	13.3367i	0.701929i	−0.936389	+	0.350964i	$0.885854\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 41.1084i	− 1.78732i	−0.448742	−	0.893662i	$-0.648128\pi$
$23$	− 41.1084i	− 1.78732i	0.448742	−	0.893662i	$-0.351872\pi$
$24$	0	0
$25$	−24.7306	−0.989224
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.85538	0.270875	0.135438	−	0.990786i	$-0.456756\pi$
$29$	7.85538	0.270875	0.135438	+	0.990786i	$0.456756\pi$
$30$	0	0
$31$	−26.1945	−0.844985	−0.422493	−	0.906366i	$-0.638845\pi$
$31$	−26.1945	−0.844985	−0.422493	+	0.906366i	$0.638845\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	5.40403	0.154401
$36$	0	0
$37$	1.53769i	0.0415591i	0.999784	+	0.0207795i	$0.00661481\pi$
$37$	1.53769i	0.0415591i	−0.999784	+	0.0207795i	$0.993385\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 21.3367i	− 0.520406i	−0.965554	−	0.260203i	$-0.916210\pi$
$41$	− 21.3367i	− 0.520406i	0.965554	−	0.260203i	$-0.0837895\pi$
$42$	0	0
$43$	− 64.1345i	− 1.49150i	−0.666226	−	0.745750i	$-0.732091\pi$
$43$	− 64.1345i	− 1.49150i	0.666226	−	0.745750i	$-0.267909\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	19.8630i	0.422618i	0.977419	+	0.211309i	$0.0677726\pi$
$47$	19.8630i	0.422618i	−0.977419	+	0.211309i	$0.932227\pi$
$48$	0	0
$49$	59.4039	1.21232
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−68.6521	−1.29532	−0.647661	−	0.761928i	$-0.724253\pi$
$53$	−68.6521	−1.29532	−0.647661	+	0.761928i	$0.724253\pi$
$54$	0	0
$55$	−7.33635	−0.133388
$56$	0	0
$57$	0	0
$58$	0	0
$59$	67.8652	1.15026	0.575129	−	0.818063i	$-0.304952\pi$
$59$	67.8652	1.15026	0.575129	+	0.818063i	$0.304952\pi$
$60$	0	0
$61$	− 58.6563i	− 0.961579i	−0.876836	−	0.480789i	$-0.840350\pi$
$61$	− 58.6563i	− 0.961579i	0.876836	−	0.480789i	$-0.159650\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.79786i	0.0430440i
$66$	0	0
$67$	56.1345i	0.837828i	0.908026	+	0.418914i	$0.137589\pi$
$67$	56.1345i	0.837828i	−0.908026	+	0.418914i	$0.862411\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 107.615i	− 1.51570i	−0.652430	−	0.757849i	$-0.726250\pi$
$71$	− 107.615i	− 1.51570i	0.652430	−	0.757849i	$-0.273750\pi$
$72$	0	0
$73$	7.73060	0.105899	0.0529493	−	0.998597i	$-0.483138\pi$
$73$	7.73060	0.105899	0.0529493	+	0.998597i	$0.483138\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−147.166	−1.91124
$78$	0	0
$79$	64.3579	0.814657	0.407329	−	0.913282i	$-0.366460\pi$
$79$	64.3579	0.814657	0.407329	+	0.913282i	$0.366460\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	125.192	1.50834	0.754168	−	0.656682i	$-0.228041\pi$
$83$	125.192	1.50834	0.754168	+	0.656682i	$0.228041\pi$
$84$	0	0
$85$	− 12.9409i	− 0.152245i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	59.6159i	0.669841i	0.942246	+	0.334921i	$0.108710\pi$
$89$	59.6159i	0.669841i	−0.942246	+	0.334921i	$0.891290\pi$
$90$	0	0
$91$	56.1246i	0.616753i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	6.92217i	0.0728649i
$96$	0	0
$97$	−138.481	−1.42764	−0.713820	−	0.700329i	$-0.753037\pi$
$97$	−138.481	−1.42764	−0.713820	+	0.700329i	$0.753037\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−137.234	−1.35876	−0.679378	−	0.733789i	$-0.737750\pi$
$101$	−137.234	−1.35876	−0.679378	+	0.733789i	$0.737750\pi$
$102$	0	0
$103$	−69.7312	−0.677002	−0.338501	−	0.940966i	$-0.609920\pi$
$103$	−69.7312	−0.677002	−0.338501	+	0.940966i	$0.609920\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	102.077	0.953994	0.476997	−	0.878905i	$-0.341725\pi$
$107$	102.077	0.953994	0.476997	+	0.878905i	$0.341725\pi$
$108$	0	0
$109$	172.600i	1.58349i	0.610853	+	0.791744i	$0.290827\pi$
$109$	172.600i	1.58349i	−0.610853	+	0.791744i	$0.709173\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 72.7981i	− 0.644231i	−0.946700	−	0.322116i	$-0.895606\pi$
$113$	− 72.7981i	− 0.644231i	0.946700	−	0.322116i	$-0.104394\pi$
$114$	0	0
$115$	− 21.3367i	− 0.185536i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 259.591i	− 2.18144i
$120$	0	0
$121$	78.7879	0.651140
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−25.8118	−0.206495
$126$	0	0
$127$	88.0992	0.693695	0.346847	−	0.937922i	$-0.387252\pi$
$127$	88.0992	0.693695	0.346847	+	0.937922i	$0.387252\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−43.4038	−0.331327	−0.165663	−	0.986182i	$-0.552976\pi$
$131$	−43.4038	−0.331327	−0.165663	+	0.986182i	$0.552976\pi$
$132$	0	0
$133$	138.857i	1.04404i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 152.154i	− 1.11062i	−0.831645	−	0.555308i	$-0.812600\pi$
$137$	− 152.154i	− 1.11062i	0.831645	−	0.555308i	$-0.187400\pi$
$138$	0	0
$139$	− 104.000i	− 0.748201i	−0.927388	−	0.374101i	$-0.877951\pi$
$139$	− 104.000i	− 0.748201i	0.927388	−	0.374101i	$-0.122049\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 76.1930i	− 0.532818i
$144$	0	0
$145$	4.07721	0.0281187
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−19.4437	−0.130495	−0.0652473	−	0.997869i	$-0.520784\pi$
$149$	−19.4437	−0.130495	−0.0652473	+	0.997869i	$0.520784\pi$
$150$	0	0
$151$	21.5254	0.142552	0.0712762	−	0.997457i	$-0.477293\pi$
$151$	21.5254	0.142552	0.0712762	+	0.997457i	$0.477293\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−13.5958	−0.0877151
$156$	0	0
$157$	− 272.808i	− 1.73763i	−0.495134	−	0.868817i	$-0.664881\pi$
$157$	− 272.808i	− 1.73763i	0.495134	−	0.868817i	$-0.335119\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 428.009i	− 2.65844i
$162$	0	0
$163$	− 24.1445i	− 0.148126i	−0.997254	−	0.0740628i	$-0.976403\pi$
$163$	− 24.1445i	− 0.148126i	0.997254	−	0.0740628i	$-0.0235965\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 207.823i	− 1.24445i	−0.782839	−	0.622224i	$-0.786229\pi$
$167$	− 207.823i	− 1.24445i	0.782839	−	0.622224i	$-0.213771\pi$
$168$	0	0
$169$	139.942	0.828061
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−302.257	−1.74715	−0.873575	−	0.486689i	$-0.838204\pi$
$173$	−302.257	−1.74715	−0.873575	+	0.486689i	$0.838204\pi$
$174$	0	0
$175$	−257.488	−1.47136
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−10.9997	−0.0614511	−0.0307256	−	0.999528i	$-0.509782\pi$
$179$	−10.9997	−0.0614511	−0.0307256	+	0.999528i	$0.509782\pi$
$180$	0	0
$181$	− 196.011i	− 1.08293i	−0.840723	−	0.541466i	$-0.817870\pi$
$181$	− 196.011i	− 1.08293i	0.840723	−	0.541466i	$-0.182130\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0.798110i	0.00431411i
$186$	0	0
$187$	352.413i	1.88456i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 267.896i	− 1.40260i	−0.712868	−	0.701298i	$-0.752604\pi$
$191$	− 267.896i	− 1.40260i	0.712868	−	0.701298i	$-0.247396\pi$
$192$	0	0
$193$	−33.9996	−0.176164	−0.0880819	−	0.996113i	$-0.528074\pi$
$193$	−33.9996	−0.176164	−0.0880819	+	0.996113i	$0.528074\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−114.706	−0.582265	−0.291133	−	0.956683i	$-0.594032\pi$
$197$	−114.706	−0.582265	−0.291133	+	0.956683i	$0.594032\pi$
$198$	0	0
$199$	189.845	0.953996	0.476998	−	0.878904i	$-0.341725\pi$
$199$	189.845	0.953996	0.476998	+	0.878904i	$0.341725\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	81.7880	0.402897
$204$	0	0
$205$	− 11.0744i	− 0.0540216i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 188.509i	− 0.901955i
$210$	0	0
$211$	− 373.875i	− 1.77192i	−0.463764	−	0.885959i	$-0.653501\pi$
$211$	− 373.875i	− 1.77192i	0.463764	−	0.885959i	$-0.346499\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 33.2880i	− 0.154828i
$216$	0	0
$217$	−272.730	−1.25682
$218$	0	0
$219$	0	0
$220$	0	0
$221$	134.400	0.608144
$222$	0	0
$223$	−97.7546	−0.438362	−0.219181	−	0.975684i	$-0.570338\pi$
$223$	−97.7546	−0.438362	−0.219181	+	0.975684i	$0.570338\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	103.347	0.455271	0.227636	−	0.973746i	$-0.426900\pi$
$227$	103.347	0.455271	0.227636	+	0.973746i	$0.426900\pi$
$228$	0	0
$229$	− 126.850i	− 0.553931i	−0.960880	−	0.276966i	$-0.910671\pi$
$229$	− 126.850i	− 0.553931i	0.960880	−	0.276966i	$-0.0893288\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	380.942i	1.63494i	0.575968	+	0.817472i	$0.304625\pi$
$233$	380.942i	1.63494i	−0.575968	+	0.817472i	$0.695375\pi$
$234$	0	0
$235$	10.3096i	0.0438705i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 266.998i	− 1.11715i	−0.829456	−	0.558573i	$-0.811349\pi$
$239$	− 266.998i	− 1.11715i	0.829456	−	0.558573i	$-0.188651\pi$
$240$	0	0
$241$	341.231	1.41590	0.707949	−	0.706264i	$-0.249621\pi$
$241$	341.231	1.41590	0.707949	+	0.706264i	$0.249621\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	30.8326	0.125847
$246$	0	0
$247$	−71.8914	−0.291058
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−52.9627	−0.211007	−0.105503	−	0.994419i	$-0.533645\pi$
$251$	−52.9627	−0.211007	−0.105503	+	0.994419i	$0.533645\pi$
$252$	0	0
$253$	581.053i	2.29665i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 120.664i	− 0.469508i	−0.972055	−	0.234754i	$-0.924572\pi$
$257$	− 120.664i	− 0.469508i	0.972055	−	0.234754i	$-0.0754285\pi$
$258$	0	0
$259$	16.0100i	0.0618145i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	441.123i	1.67728i	0.544690	+	0.838638i	$0.316647\pi$
$263$	441.123i	1.67728i	−0.544690	+	0.838638i	$0.683353\pi$
$264$	0	0
$265$	−35.6327	−0.134463
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−455.412	−1.69298	−0.846490	−	0.532404i	$-0.821289\pi$
$269$	−455.412	−1.69298	−0.846490	+	0.532404i	$0.821289\pi$
$270$	0	0
$271$	304.415	1.12330	0.561652	−	0.827374i	$-0.310166\pi$
$271$	304.415	1.12330	0.561652	+	0.827374i	$0.310166\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	349.558	1.27112
$276$	0	0
$277$	541.056i	1.95327i	0.214906	+	0.976635i	$0.431055\pi$
$277$	541.056i	1.95327i	−0.214906	+	0.976635i	$0.568945\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	276.414i	0.983680i	0.870686	+	0.491840i	$0.163675\pi$
$281$	276.414i	0.983680i	−0.870686	+	0.491840i	$0.836325\pi$
$282$	0	0
$283$	− 231.990i	− 0.819753i	−0.912141	−	0.409876i	$-0.865572\pi$
$283$	− 231.990i	− 0.819753i	0.912141	−	0.409876i	$-0.134428\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 222.151i	− 0.774046i
$288$	0	0
$289$	−332.636	−1.15099
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−548.289	−1.87129	−0.935646	−	0.352939i	$-0.885182\pi$
$293$	−548.289	−1.87129	−0.935646	+	0.352939i	$0.885182\pi$
$294$	0	0
$295$	35.2243	0.119405
$296$	0	0
$297$	0	0
$298$	0	0
$299$	221.596	0.741123
$300$	0	0
$301$	− 667.751i	− 2.21844i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 30.4446i	− 0.0998183i
$306$	0	0
$307$	437.740i	1.42586i	0.701233	+	0.712932i	$0.252633\pi$
$307$	437.740i	1.42586i	−0.701233	+	0.712932i	$0.747367\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	349.215i	1.12288i	0.827519	+	0.561438i	$0.189752\pi$
$311$	349.215i	1.12288i	−0.827519	+	0.561438i	$0.810248\pi$
$312$	0	0
$313$	−304.558	−0.973030	−0.486515	−	0.873672i	$-0.661732\pi$
$313$	−304.558	−0.973030	−0.486515	+	0.873672i	$0.661732\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	387.488	1.22236	0.611180	−	0.791491i	$-0.290695\pi$
$317$	387.488	1.22236	0.611180	+	0.791491i	$0.290695\pi$
$318$	0	0
$319$	−111.033	−0.348066
$320$	0	0
$321$	0	0
$322$	0	0
$323$	332.518	1.02947
$324$	0	0
$325$	− 133.311i	− 0.410187i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	206.808i	0.628597i
$330$	0	0
$331$	293.690i	0.887282i	0.896205	+	0.443641i	$0.146314\pi$
$331$	293.690i	0.887282i	−0.896205	+	0.443641i	$0.853686\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	29.1357i	0.0869722i
$336$	0	0
$337$	415.845	1.23396	0.616980	−	0.786978i	$-0.288356\pi$
$337$	415.845	1.23396	0.616980	+	0.786978i	$0.288356\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	370.250	1.08578
$342$	0	0
$343$	108.323	0.315809
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−202.882	−0.584676	−0.292338	−	0.956315i	$-0.594433\pi$
$347$	−202.882	−0.584676	−0.292338	+	0.956315i	$0.594433\pi$
$348$	0	0
$349$	− 12.3532i	− 0.0353960i	−0.999843	−	0.0176980i	$-0.994366\pi$
$349$	− 12.3532i	− 0.0353960i	0.999843	−	0.0176980i	$-0.00563374\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	25.3361i	0.0717738i	0.999356	+	0.0358869i	$0.0114256\pi$
$353$	25.3361i	0.0717738i	−0.999356	+	0.0358869i	$0.988574\pi$
$354$	0	0
$355$	− 55.8555i	− 0.157340i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	264.158i	0.735816i	0.929862	+	0.367908i	$0.119926\pi$
$359$	264.158i	0.735816i	−0.929862	+	0.367908i	$0.880074\pi$
$360$	0	0
$361$	183.134	0.507296
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.01244	0.0109930
$366$	0	0
$367$	417.797	1.13841	0.569206	−	0.822195i	$-0.307251\pi$
$367$	417.797	1.13841	0.569206	+	0.822195i	$0.307251\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−714.787	−1.92665
$372$	0	0
$373$	− 529.014i	− 1.41827i	−0.705074	−	0.709134i	$-0.749086\pi$
$373$	− 529.014i	− 1.41827i	0.705074	−	0.709134i	$-0.250914\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	42.3446i	0.112320i
$378$	0	0
$379$	− 280.329i	− 0.739654i	−0.929101	−	0.369827i	$-0.879417\pi$
$379$	− 280.329i	− 0.739654i	0.929101	−	0.369827i	$-0.120583\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	57.2283i	0.149421i	0.997205	+	0.0747105i	$0.0238033\pi$
$383$	57.2283i	0.149421i	−0.997205	+	0.0747105i	$0.976197\pi$
$384$	0	0
$385$	−76.3840	−0.198400
$386$	0	0
$387$	0	0
$388$	0	0
$389$	279.589	0.718738	0.359369	−	0.933196i	$-0.382992\pi$
$389$	279.589	0.718738	0.359369	+	0.933196i	$0.382992\pi$
$390$	0	0
$391$	−1024.94	−2.62133
$392$	0	0
$393$	0	0
$394$	0	0
$395$	33.4039	0.0845668
$396$	0	0
$397$	458.764i	1.15558i	0.816187	+	0.577788i	$0.196084\pi$
$397$	458.764i	1.15558i	−0.816187	+	0.577788i	$0.803916\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 620.518i	− 1.54743i	−0.633537	−	0.773713i	$-0.718397\pi$
$401$	− 620.518i	− 1.54743i	0.633537	−	0.773713i	$-0.281603\pi$
$402$	0	0
$403$	− 141.202i	− 0.350378i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 21.7346i	− 0.0534020i
$408$	0	0
$409$	645.114	1.57730	0.788648	−	0.614845i	$-0.210782\pi$
$409$	645.114	1.57730	0.788648	+	0.614845i	$0.210782\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	706.594	1.71088
$414$	0	0
$415$	64.9787	0.156575
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−586.059	−1.39871	−0.699355	−	0.714775i	$-0.746529\pi$
$419$	−586.059	−1.39871	−0.699355	+	0.714775i	$0.746529\pi$
$420$	0	0
$421$	− 59.0014i	− 0.140146i	−0.997542	−	0.0700729i	$-0.977677\pi$
$421$	− 59.0014i	− 0.140146i	0.997542	−	0.0700729i	$-0.0223232\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	616.599i	1.45082i
$426$	0	0
$427$	− 610.713i	− 1.43024i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 14.3336i	− 0.0332565i	−0.999862	−	0.0166283i	$-0.994707\pi$
$431$	− 14.3336i	− 0.0332565i	0.999862	−	0.0166283i	$-0.00529318\pi$
$432$	0	0
$433$	−224.980	−0.519585	−0.259792	−	0.965665i	$-0.583654\pi$
$433$	−224.980	−0.519585	−0.259792	+	0.965665i	$0.583654\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	548.249	1.25457
$438$	0	0
$439$	223.743	0.509666	0.254833	−	0.966985i	$-0.417980\pi$
$439$	223.743	0.509666	0.254833	+	0.966985i	$0.417980\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−107.058	−0.241665	−0.120833	−	0.992673i	$-0.538556\pi$
$443$	−107.058	−0.241665	−0.120833	+	0.992673i	$0.538556\pi$
$444$	0	0
$445$	30.9426i	0.0695340i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 381.815i	− 0.850367i	−0.905107	−	0.425184i	$-0.860209\pi$
$449$	− 381.815i	− 0.850367i	0.905107	−	0.425184i	$-0.139791\pi$
$450$	0	0
$451$	301.586i	0.668705i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	29.1305i	0.0640231i
$456$	0	0
$457$	−349.194	−0.764101	−0.382051	−	0.924141i	$-0.624782\pi$
$457$	−349.194	−0.764101	−0.382051	+	0.924141i	$0.624782\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−4.02277	−0.00872619	−0.00436309	−	0.999990i	$-0.501389\pi$
$461$	−4.02277	−0.00872619	−0.00436309	+	0.999990i	$0.501389\pi$
$462$	0	0
$463$	−851.739	−1.83961	−0.919804	−	0.392377i	$-0.871653\pi$
$463$	−851.739	−1.83961	−0.919804	+	0.392377i	$0.871653\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−503.521	−1.07820	−0.539101	−	0.842241i	$-0.681236\pi$
$467$	−503.521	−1.07820	−0.539101	+	0.842241i	$0.681236\pi$
$468$	0	0
$469$	584.457i	1.24618i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	906.518i	1.91653i
$474$	0	0
$475$	− 329.823i	− 0.694365i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	286.856i	0.598863i	0.954118	+	0.299432i	$0.0967971\pi$
$479$	286.856i	0.598863i	−0.954118	+	0.299432i	$0.903203\pi$
$480$	0	0
$481$	−8.28892	−0.0172327
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−71.8763	−0.148199
$486$	0	0
$487$	873.195	1.79301	0.896504	−	0.443036i	$-0.146099\pi$
$487$	873.195	1.79301	0.896504	+	0.443036i	$0.146099\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−441.288	−0.898754	−0.449377	−	0.893342i	$-0.648354\pi$
$491$	−441.288	−0.898754	−0.449377	+	0.893342i	$0.648354\pi$
$492$	0	0
$493$	− 195.855i	− 0.397272i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1120.45i	− 2.25443i
$498$	0	0
$499$	897.749i	1.79910i	0.436823	+	0.899548i	$0.356104\pi$
$499$	897.749i	1.79910i	−0.436823	+	0.899548i	$0.643896\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	638.356i	1.26910i	0.772883	+	0.634549i	$0.218814\pi$
$503$	638.356i	1.26910i	−0.772883	+	0.634549i	$0.781186\pi$
$504$	0	0
$505$	−71.2292	−0.141048
$506$	0	0
$507$	0	0
$508$	0	0
$509$	684.754	1.34529	0.672647	−	0.739964i	$-0.265157\pi$
$509$	684.754	1.34529	0.672647	+	0.739964i	$0.265157\pi$
$510$	0	0
$511$	80.4889	0.157512
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−36.1928	−0.0702773
$516$	0	0
$517$	− 280.757i	− 0.543050i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	568.644i	1.09145i	0.837965	+	0.545723i	$0.183745\pi$
$521$	568.644i	1.09145i	−0.837965	+	0.545723i	$0.816255\pi$
$522$	0	0
$523$	972.143i	1.85878i	0.369097	+	0.929391i	$0.379667\pi$
$523$	972.143i	1.85878i	−0.369097	+	0.929391i	$0.620333\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	653.099i	1.23928i
$528$	0	0
$529$	−1160.90	−2.19452
$530$	0	0
$531$	0	0
$532$	0	0
$533$	115.016	0.215789
$534$	0	0
$535$	52.9815	0.0990309
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−839.653	−1.55780
$540$	0	0
$541$	213.997i	0.395558i	0.980247	+	0.197779i	$0.0633728\pi$
$541$	213.997i	0.395558i	−0.980247	+	0.197779i	$0.936627\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	89.5853i	0.164377i
$546$	0	0
$547$	− 525.720i	− 0.961097i	−0.876968	−	0.480549i	$-0.840438\pi$
$547$	− 525.720i	− 0.961097i	0.876968	−	0.480549i	$-0.159562\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	104.764i	0.190135i
$552$	0	0
$553$	670.076	1.21171
$554$	0	0
$555$	0	0
$556$	0	0
$557$	171.281	0.307506	0.153753	−	0.988109i	$-0.450864\pi$
$557$	171.281	0.307506	0.153753	+	0.988109i	$0.450864\pi$
$558$	0	0
$559$	345.718	0.618458
$560$	0	0
$561$	0	0
$562$	0	0
$563$	887.405	1.57621	0.788104	−	0.615543i	$-0.211063\pi$
$563$	887.405	1.57621	0.788104	+	0.615543i	$0.211063\pi$
$564$	0	0
$565$	− 37.7847i	− 0.0668755i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	812.458i	1.42787i	0.700212	+	0.713935i	$0.253089\pi$
$569$	812.458i	1.42787i	−0.700212	+	0.713935i	$0.746911\pi$
$570$	0	0
$571$	− 528.085i	− 0.924842i	−0.886660	−	0.462421i	$-0.846981\pi$
$571$	− 528.085i	− 0.924842i	0.886660	−	0.462421i	$-0.153019\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1016.64i	1.76806i
$576$	0	0
$577$	603.116	1.04526	0.522631	−	0.852559i	$-0.324951\pi$
$577$	603.116	1.04526	0.522631	+	0.852559i	$0.324951\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1303.46	2.24348
$582$	0	0
$583$	970.373	1.66445
$584$	0	0
$585$	0	0
$586$	0	0
$587$	496.885	0.846481	0.423241	−	0.906017i	$-0.360892\pi$
$587$	496.885	0.846481	0.423241	+	0.906017i	$0.360892\pi$
$588$	0	0
$589$	− 349.347i	− 0.593120i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	963.674i	1.62508i	0.582904	+	0.812541i	$0.301916\pi$
$593$	963.674i	1.62508i	−0.582904	+	0.812541i	$0.698084\pi$
$594$	0	0
$595$	− 134.737i	− 0.226448i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	305.820i	0.510551i	0.966868	+	0.255276i	$0.0821662\pi$
$599$	305.820i	0.510551i	−0.966868	+	0.255276i	$0.917834\pi$
$600$	0	0
$601$	148.888	0.247733	0.123867	−	0.992299i	$-0.460470\pi$
$601$	148.888	0.247733	0.123867	+	0.992299i	$0.460470\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	40.8936	0.0675927
$606$	0	0
$607$	−257.812	−0.424731	−0.212366	−	0.977190i	$-0.568117\pi$
$607$	−257.812	−0.424731	−0.212366	+	0.977190i	$0.568117\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−107.072	−0.175241
$612$	0	0
$613$	516.885i	0.843206i	0.906781	+	0.421603i	$0.138532\pi$
$613$	516.885i	0.843206i	−0.906781	+	0.421603i	$0.861468\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	922.184i	1.49463i	0.664472	+	0.747313i	$0.268656\pi$
$617$	922.184i	1.49463i	−0.664472	+	0.747313i	$0.731344\pi$
$618$	0	0
$619$	517.202i	0.835545i	0.908552	+	0.417772i	$0.137189\pi$
$619$	517.202i	0.835545i	−0.908552	+	0.417772i	$0.862811\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	620.704i	0.996314i
$624$	0	0
$625$	604.868	0.967789
$626$	0	0
$627$	0	0
$628$	0	0
$629$	38.3385	0.0609516
$630$	0	0
$631$	−500.632	−0.793395	−0.396698	−	0.917949i	$-0.629844\pi$
$631$	−500.632	−0.793395	−0.396698	+	0.917949i	$0.629844\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	45.7264	0.0720101
$636$	0	0
$637$	320.218i	0.502697i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 171.491i	− 0.267537i	−0.991013	−	0.133769i	$-0.957292\pi$
$641$	− 171.491i	− 0.267537i	0.991013	−	0.133769i	$-0.0427079\pi$
$642$	0	0
$643$	− 614.941i	− 0.956362i	−0.878261	−	0.478181i	$-0.841296\pi$
$643$	− 614.941i	− 0.956362i	0.878261	−	0.478181i	$-0.158704\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 448.046i	− 0.692497i	−0.938143	−	0.346249i	$-0.887455\pi$
$647$	− 448.046i	− 0.692497i	0.938143	−	0.346249i	$-0.112545\pi$
$648$	0	0
$649$	−959.250	−1.47804
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−749.334	−1.14753	−0.573763	−	0.819021i	$-0.694517\pi$
$653$	−749.334	−1.14753	−0.573763	+	0.819021i	$0.694517\pi$
$654$	0	0
$655$	−22.5280	−0.0343939
$656$	0	0
$657$	0	0
$658$	0	0
$659$	504.555	0.765637	0.382819	−	0.923824i	$-0.374953\pi$
$659$	504.555	0.765637	0.382819	+	0.923824i	$0.374953\pi$
$660$	0	0
$661$	− 670.255i	− 1.01400i	−0.861945	−	0.507001i	$-0.830754\pi$
$661$	− 670.255i	− 1.01400i	0.861945	−	0.507001i	$-0.169246\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	72.0717i	0.108378i
$666$	0	0
$667$	− 322.922i	− 0.484142i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	829.086i	1.23560i
$672$	0	0
$673$	458.749	0.681648	0.340824	−	0.940127i	$-0.389294\pi$
$673$	458.749	0.681648	0.340824	+	0.940127i	$0.389294\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−260.944	−0.385442	−0.192721	−	0.981254i	$-0.561731\pi$
$677$	−260.944	−0.385442	−0.192721	+	0.981254i	$0.561731\pi$
$678$	0	0
$679$	−1441.83	−2.12346
$680$	0	0
$681$	0	0
$682$	0	0
$683$	831.346	1.21720	0.608599	−	0.793478i	$-0.291732\pi$
$683$	831.346	1.21720	0.608599	+	0.793478i	$0.291732\pi$
$684$	0	0
$685$	− 78.9732i	− 0.115289i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 370.070i	− 0.537112i
$690$	0	0
$691$	− 693.675i	− 1.00387i	−0.864905	−	0.501936i	$-0.832621\pi$
$691$	− 693.675i	− 1.00387i	0.864905	−	0.501936i	$-0.167379\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 53.9795i	− 0.0776683i
$696$	0	0
$697$	−531.979	−0.763240
$698$	0	0
$699$	0	0
$700$	0	0
$701$	652.155	0.930321	0.465161	−	0.885226i	$-0.345997\pi$
$701$	652.155	0.930321	0.465161	+	0.885226i	$0.345997\pi$
$702$	0	0
$703$	−20.5076	−0.0291715
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1428.85	−2.02100
$708$	0	0
$709$	− 38.0968i	− 0.0537332i	−0.999639	−	0.0268666i	$-0.991447\pi$
$709$	− 38.0968i	− 0.0537332i	0.999639	−	0.0268666i	$-0.00855293\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1076.82i	1.51026i
$714$	0	0
$715$	− 39.5467i	− 0.0553101i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	529.843i	0.736917i	0.929644	+	0.368458i	$0.120114\pi$
$719$	529.843i	0.736917i	−0.929644	+	0.368458i	$0.879886\pi$
$720$	0	0
$721$	−726.022	−1.00696
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−194.268	−0.267956
$726$	0	0
$727$	−918.579	−1.26352	−0.631760	−	0.775164i	$-0.717667\pi$
$727$	−918.579	−1.26352	−0.631760	+	0.775164i	$0.717667\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1599.04	−2.18747
$732$	0	0
$733$	− 1004.73i	− 1.37070i	−0.728212	−	0.685352i	$-0.759648\pi$
$733$	− 1004.73i	− 1.37070i	0.728212	−	0.685352i	$-0.240352\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 793.441i	− 1.07658i
$738$	0	0
$739$	461.471i	0.624454i	0.950008	+	0.312227i	$0.101075\pi$
$739$	461.471i	0.624454i	−0.950008	+	0.312227i	$0.898925\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 335.450i	− 0.451481i	−0.974187	−	0.225740i	$-0.927520\pi$
$743$	− 335.450i	− 0.451481i	0.974187	−	0.225740i	$-0.0724801\pi$
$744$	0	0
$745$	−10.0919	−0.0135462
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1062.80	1.41896
$750$	0	0
$751$	230.848	0.307387	0.153694	−	0.988119i	$-0.450883\pi$
$751$	230.848	0.307387	0.153694	+	0.988119i	$0.450883\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	11.1724	0.0147979
$756$	0	0
$757$	115.377i	0.152414i	0.997092	+	0.0762069i	$0.0242810\pi$
$757$	115.377i	0.152414i	−0.997092	+	0.0762069i	$0.975719\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	121.617i	0.159812i	0.996802	+	0.0799061i	$0.0254621\pi$
$761$	121.617i	0.159812i	−0.996802	+	0.0799061i	$0.974538\pi$
$762$	0	0
$763$	1797.06i	2.35526i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	365.829i	0.476961i
$768$	0	0
$769$	873.135	1.13542	0.567708	−	0.823230i	$-0.307830\pi$
$769$	873.135	1.13542	0.567708	+	0.823230i	$0.307830\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	559.288	0.723529	0.361765	−	0.932269i	$-0.382174\pi$
$773$	559.288	0.723529	0.361765	+	0.932269i	$0.382174\pi$
$774$	0	0
$775$	647.807	0.835880
$776$	0	0
$777$	0	0
$778$	0	0
$779$	284.559	0.365288
$780$	0	0
$781$	1521.09i	1.94762i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 141.597i	− 0.180378i
$786$	0	0
$787$	− 502.213i	− 0.638136i	−0.947732	−	0.319068i	$-0.896630\pi$
$787$	− 502.213i	− 0.638136i	0.947732	−	0.319068i	$-0.103370\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 757.954i	− 0.958222i
$792$	0	0
$793$	316.188	0.398724
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−655.828	−0.822871	−0.411436	−	0.911439i	$-0.634972\pi$
$797$	−655.828	−0.822871	−0.411436	+	0.911439i	$0.634972\pi$
$798$	0	0
$799$	495.237	0.619822
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−109.269	−0.136076
$804$	0	0
$805$	− 222.151i	− 0.275964i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 272.193i	− 0.336456i	−0.985748	−	0.168228i	$-0.946195\pi$
$809$	− 272.193i	− 0.336456i	0.985748	−	0.168228i	$-0.0538045\pi$
$810$	0	0
$811$	− 483.162i	− 0.595760i	−0.954603	−	0.297880i	$-0.903720\pi$
$811$	− 483.162i	− 0.595760i	0.954603	−	0.297880i	$-0.0962796\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 12.5318i	− 0.0153764i
$816$	0	0
$817$	855.340	1.04693
$818$	0	0
$819$	0	0
$820$	0	0
$821$	695.215	0.846791	0.423395	−	0.905945i	$-0.360838\pi$
$821$	695.215	0.846791	0.423395	+	0.905945i	$0.360838\pi$
$822$	0	0
$823$	1496.48	1.81832	0.909161	−	0.416446i	$-0.136724\pi$
$823$	1496.48	1.81832	0.909161	+	0.416446i	$0.136724\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−347.515	−0.420212	−0.210106	−	0.977679i	$-0.567381\pi$
$827$	−347.515	−0.420212	−0.210106	+	0.977679i	$0.567381\pi$
$828$	0	0
$829$	− 1014.09i	− 1.22327i	−0.791140	−	0.611635i	$-0.790512\pi$
$829$	− 1014.09i	− 1.22327i	0.791140	−	0.611635i	$-0.209488\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 1481.09i	− 1.77803i
$834$	0	0
$835$	− 107.867i	− 0.129182i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 207.818i	− 0.247697i	−0.992301	−	0.123848i	$-0.960476\pi$
$839$	− 207.818i	− 0.247697i	0.992301	−	0.123848i	$-0.0395237\pi$
$840$	0	0
$841$	−779.293	−0.926627
$842$	0	0
$843$	0	0
$844$	0	0
$845$	72.6347	0.0859583
$846$	0	0
$847$	820.318	0.968498
$848$	0	0
$849$	0	0
$850$	0	0
$851$	63.2118	0.0742795
$852$	0	0
$853$	− 184.331i	− 0.216098i	−0.994146	−	0.108049i	$-0.965540\pi$
$853$	− 184.331i	− 0.216098i	0.994146	−	0.108049i	$-0.0344603\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	246.050i	0.287106i	0.989643	+	0.143553i	$0.0458528\pi$
$857$	246.050i	0.287106i	−0.989643	+	0.143553i	$0.954147\pi$
$858$	0	0
$859$	− 677.137i	− 0.788286i	−0.919049	−	0.394143i	$-0.871042\pi$
$859$	− 677.137i	− 0.788286i	0.919049	−	0.394143i	$-0.128958\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 766.243i	− 0.887882i	−0.896056	−	0.443941i	$-0.853580\pi$
$863$	− 766.243i	− 0.887882i	0.896056	−	0.443941i	$-0.146420\pi$
$864$	0	0
$865$	−156.881	−0.181366
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−909.675	−1.04681
$870$	0	0
$871$	−302.594	−0.347410
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−268.746	−0.307138
$876$	0	0
$877$	917.942i	1.04668i	0.852123	+	0.523342i	$0.175315\pi$
$877$	917.942i	1.04668i	−0.852123	+	0.523342i	$0.824685\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 1051.56i	− 1.19360i	−0.802389	−	0.596801i	$-0.796438\pi$
$881$	− 1051.56i	− 1.19360i	0.802389	−	0.596801i	$-0.203562\pi$
$882$	0	0
$883$	232.927i	0.263790i	0.991264	+	0.131895i	$0.0421062\pi$
$883$	232.927i	0.263790i	−0.991264	+	0.131895i	$0.957894\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 905.279i	− 1.02061i	−0.859994	−	0.510304i	$-0.829533\pi$
$887$	− 905.279i	− 1.02061i	0.859994	−	0.510304i	$-0.170467\pi$
$888$	0	0
$889$	917.264	1.03179
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−264.906	−0.296648
$894$	0	0
$895$	−5.70924	−0.00637904
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−205.768	−0.228886
$900$	0	0
$901$	1711.68i	1.89975i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 101.736i	− 0.112416i
$906$	0	0
$907$	1253.87i	1.38244i	0.722644	+	0.691221i	$0.242927\pi$
$907$	1253.87i	1.38244i	−0.722644	+	0.691221i	$0.757073\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 533.926i	− 0.586087i	−0.956099	−	0.293044i	$-0.905332\pi$
$911$	− 533.926i	− 0.586087i	0.956099	−	0.293044i	$-0.0946681\pi$
$912$	0	0
$913$	−1769.54	−1.93816
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−451.908	−0.492811
$918$	0	0
$919$	−145.848	−0.158703	−0.0793516	−	0.996847i	$-0.525285\pi$
$919$	−145.848	−0.158703	−0.0793516	+	0.996847i	$0.525285\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	580.098	0.628492
$924$	0	0
$925$	− 38.0279i	− 0.0411112i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 916.827i	− 0.986897i	−0.869775	−	0.493449i	$-0.835736\pi$
$929$	− 916.827i	− 0.986897i	0.869775	−	0.493449i	$-0.164264\pi$
$930$	0	0
$931$	792.249i	0.850966i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	182.914i	0.195630i
$936$	0	0
$937$	282.716	0.301725	0.150862	−	0.988555i	$-0.451795\pi$
$937$	282.716	0.301725	0.150862	+	0.988555i	$0.451795\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1669.95	1.77465	0.887327	−	0.461140i	$-0.152560\pi$
$941$	1669.95	1.77465	0.887327	+	0.461140i	$0.152560\pi$
$942$	0	0
$943$	−877.116	−0.930134
$944$	0	0
$945$	0	0
$946$	0	0
$947$	833.418	0.880062	0.440031	−	0.897983i	$-0.354968\pi$
$947$	833.418	0.880062	0.440031	+	0.897983i	$0.354968\pi$
$948$	0	0
$949$	41.6720i	0.0439114i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	978.132i	1.02637i	0.858278	+	0.513186i	$0.171535\pi$
$953$	978.132i	1.02637i	−0.858278	+	0.513186i	$0.828465\pi$
$954$	0	0
$955$	− 139.047i	− 0.145599i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1584.19i	− 1.65192i
$960$	0	0
$961$	−274.846	−0.286000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−17.6469	−0.0182870
$966$	0	0
$967$	−743.175	−0.768537	−0.384268	−	0.923221i	$-0.625546\pi$
$967$	−743.175	−0.768537	−0.384268	+	0.923221i	$0.625546\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	562.344	0.579139	0.289569	−	0.957157i	$-0.406488\pi$
$971$	562.344	0.579139	0.289569	+	0.957157i	$0.406488\pi$
$972$	0	0
$973$	− 1082.82i	− 1.11287i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 116.016i	− 0.118747i	−0.998236	−	0.0593734i	$-0.981090\pi$
$977$	− 116.016i	− 0.118747i	0.998236	−	0.0593734i	$-0.0189103\pi$
$978$	0	0
$979$	− 842.649i	− 0.860724i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1088.69i	1.10752i	0.832676	+	0.553760i	$0.186808\pi$
$983$	1088.69i	1.10752i	−0.832676	+	0.553760i	$0.813192\pi$
$984$	0	0
$985$	−59.5364	−0.0604430
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2636.47	−2.66579
$990$	0	0
$991$	−938.958	−0.947486	−0.473743	−	0.880663i	$-0.657097\pi$
$991$	−938.958	−0.947486	−0.473743	+	0.880663i	$0.657097\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	98.5360	0.0990312
$996$	0	0
$997$	− 970.068i	− 0.972987i	−0.873684	−	0.486494i	$-0.838276\pi$
$997$	− 970.068i	− 0.972987i	0.873684	−	0.486494i	$-0.161724\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.h.j.161.8	yes	12
3.2	odd	2		1728.3.h.i.161.6	yes	12
4.3	odd	2		1728.3.h.i.161.8	yes	12
8.3	odd	2	inner	1728.3.h.j.161.5	yes	12
8.5	even	2		1728.3.h.i.161.5	✓	12
12.11	even	2	inner	1728.3.h.j.161.6	yes	12
24.5	odd	2	inner	1728.3.h.j.161.7	yes	12
24.11	even	2		1728.3.h.i.161.7	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.3.h.i.161.5	✓	12	8.5	even	2
1728.3.h.i.161.6	yes	12	3.2	odd	2
1728.3.h.i.161.7	yes	12	24.11	even	2
1728.3.h.i.161.8	yes	12	4.3	odd	2
1728.3.h.j.161.5	yes	12	8.3	odd	2	inner
1728.3.h.j.161.6	yes	12	12.11	even	2	inner
1728.3.h.j.161.7	yes	12	24.5	odd	2	inner
1728.3.h.j.161.8	yes	12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.519033 q^{5} +10.4117 q^{7} -14.1346 q^{11} +5.39052i q^{13} -24.9326i q^{17} +13.3367i q^{19} -41.1084i q^{23} -24.7306 q^{25} +7.85538 q^{29} -26.1945 q^{31} +5.40403 q^{35} +1.53769i q^{37} -21.3367i q^{41} -64.1345i q^{43} +19.8630i q^{47} +59.4039 q^{49} -68.6521 q^{53} -7.33635 q^{55} +67.8652 q^{59} -58.6563i q^{61} +2.79786i q^{65} +56.1345i q^{67} -107.615i q^{71} +7.73060 q^{73} -147.166 q^{77} +64.3579 q^{79} +125.192 q^{83} -12.9409i q^{85} +59.6159i q^{89} +56.1246i q^{91} +6.92217i q^{95} -138.481 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{11} + 72 q^{25} + 252 q^{35} + 168 q^{49} + 264 q^{59} - 276 q^{73} + 396 q^{83} - 396 q^{97}+O(q^{100}) \) 12 * q + 12 * q^11 + 72 * q^25 + 252 * q^35 + 168 * q^49 + 264 * q^59 - 276 * q^73 + 396 * q^83 - 396 * q^97

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