LMFDB - Embedded newform 648.4.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,4,Mod(1,648)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("648.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	648.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$38.2332376837$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.72153.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 3x + 12 $ x^4 - x^3 - 8x^2 + 3x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 72)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.51317$ of defining polynomial
Character	$\chi$	$=$	648.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.01626 q^{5} +0.937231 q^{7} +O(q^{10})$	$q-8.01626 q^{5} +0.937231 q^{7} +5.33669 q^{11} -3.11587 q^{13} +132.704 q^{17} -86.6565 q^{19} +41.2424 q^{23} -60.7396 q^{25} -202.794 q^{29} +318.733 q^{31} -7.51308 q^{35} -363.510 q^{37} +9.83459 q^{41} -362.460 q^{43} -75.1760 q^{47} -342.122 q^{49} +403.981 q^{53} -42.7803 q^{55} -430.702 q^{59} +321.565 q^{61} +24.9776 q^{65} -726.648 q^{67} -829.702 q^{71} -160.106 q^{73} +5.00171 q^{77} -924.628 q^{79} +511.687 q^{83} -1063.79 q^{85} -320.459 q^{89} -2.92029 q^{91} +694.661 q^{95} -601.643 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 5 q^{5} - 3 q^{7}+O(q^{10}) $ 4 * q + 5 * q^5 - 3 * q^7	$ 4 q + 5 q^{5} - 3 q^{7} - 16 q^{11} - 29 q^{13} - 17 q^{17} - 109 q^{19} - 37 q^{23} - 97 q^{25} + 3 q^{29} - 331 q^{31} - 171 q^{35} - 366 q^{37} + 378 q^{41} - 506 q^{43} - 171 q^{47} - 829 q^{49} + 410 q^{53} - 1163 q^{55} - 616 q^{59} - 1331 q^{61} + 815 q^{65} - 1162 q^{67} - 344 q^{71} - 1307 q^{73} + 741 q^{77} - 1853 q^{79} - 1421 q^{83} - 2074 q^{85} + 816 q^{89} - 1995 q^{91} - 1292 q^{95} - 2506 q^{97}+O(q^{100}) $ 4 * q + 5 * q^5 - 3 * q^7 - 16 * q^11 - 29 * q^13 - 17 * q^17 - 109 * q^19 - 37 * q^23 - 97 * q^25 + 3 * q^29 - 331 * q^31 - 171 * q^35 - 366 * q^37 + 378 * q^41 - 506 * q^43 - 171 * q^47 - 829 * q^49 + 410 * q^53 - 1163 * q^55 - 616 * q^59 - 1331 * q^61 + 815 * q^65 - 1162 * q^67 - 344 * q^71 - 1307 * q^73 + 741 * q^77 - 1853 * q^79 - 1421 * q^83 - 2074 * q^85 + 816 * q^89 - 1995 * q^91 - 1292 * q^95 - 2506 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−8.01626	−0.716996	−0.358498	−	0.933531i	$-0.616711\pi$
$5$	−8.01626	−0.716996	−0.358498	+	0.933531i	$0.616711\pi$
$6$	0	0
$7$	0.937231	0.0506057	0.0253028	−	0.999680i	$-0.491945\pi$
$7$	0.937231	0.0506057	0.0253028	+	0.999680i	$0.491945\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.33669	0.146279	0.0731397	−	0.997322i	$-0.476698\pi$
$11$	5.33669	0.146279	0.0731397	+	0.997322i	$0.476698\pi$
$12$	0	0
$13$	−3.11587	−0.0664760	−0.0332380	−	0.999447i	$-0.510582\pi$
$13$	−3.11587	−0.0664760	−0.0332380	+	0.999447i	$0.510582\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	132.704	1.89327	0.946634	−	0.322311i	$-0.104460\pi$
$17$	132.704	1.89327	0.946634	+	0.322311i	$0.104460\pi$
$18$	0	0
$19$	−86.6565	−1.04634	−0.523168	−	0.852230i	$-0.675250\pi$
$19$	−86.6565	−1.04634	−0.523168	+	0.852230i	$0.675250\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	41.2424	0.373897	0.186949	−	0.982370i	$-0.440140\pi$
$23$	41.2424	0.373897	0.186949	+	0.982370i	$0.440140\pi$
$24$	0	0
$25$	−60.7396	−0.485917
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−202.794	−1.29855	−0.649275	−	0.760553i	$-0.724928\pi$
$29$	−202.794	−1.29855	−0.649275	+	0.760553i	$0.724928\pi$
$30$	0	0
$31$	318.733	1.84665	0.923325	−	0.384019i	$-0.125460\pi$
$31$	318.733	1.84665	0.923325	+	0.384019i	$0.125460\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−7.51308	−0.0362841
$36$	0	0
$37$	−363.510	−1.61515	−0.807577	−	0.589762i	$-0.799222\pi$
$37$	−363.510	−1.61515	−0.807577	+	0.589762i	$0.799222\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.83459	0.0374611	0.0187305	−	0.999825i	$-0.494038\pi$
$41$	9.83459	0.0374611	0.0187305	+	0.999825i	$0.494038\pi$
$42$	0	0
$43$	−362.460	−1.28546	−0.642728	−	0.766095i	$-0.722197\pi$
$43$	−362.460	−1.28546	−0.642728	+	0.766095i	$0.722197\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−75.1760	−0.233309	−0.116655	−	0.993173i	$-0.537217\pi$
$47$	−75.1760	−0.233309	−0.116655	+	0.993173i	$0.537217\pi$
$48$	0	0
$49$	−342.122	−0.997439
$50$	0	0
$51$	0	0
$52$	0	0
$53$	403.981	1.04700	0.523501	−	0.852025i	$-0.324626\pi$
$53$	403.981	1.04700	0.523501	+	0.852025i	$0.324626\pi$
$54$	0	0
$55$	−42.7803	−0.104882
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−430.702	−0.950384	−0.475192	−	0.879882i	$-0.657621\pi$
$59$	−430.702	−0.950384	−0.475192	+	0.879882i	$0.657621\pi$
$60$	0	0
$61$	321.565	0.674954	0.337477	−	0.941334i	$-0.390426\pi$
$61$	321.565	0.674954	0.337477	+	0.941334i	$0.390426\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	24.9776	0.0476630
$66$	0	0
$67$	−726.648	−1.32499	−0.662494	−	0.749067i	$-0.730502\pi$
$67$	−726.648	−1.32499	−0.662494	+	0.749067i	$0.730502\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−829.702	−1.38687	−0.693433	−	0.720521i	$-0.743903\pi$
$71$	−829.702	−1.38687	−0.693433	+	0.720521i	$0.743903\pi$
$72$	0	0
$73$	−160.106	−0.256699	−0.128349	−	0.991729i	$-0.540968\pi$
$73$	−160.106	−0.256699	−0.128349	+	0.991729i	$0.540968\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.00171	0.00740257
$78$	0	0
$79$	−924.628	−1.31682	−0.658410	−	0.752659i	$-0.728771\pi$
$79$	−924.628	−1.31682	−0.658410	+	0.752659i	$0.728771\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	511.687	0.676686	0.338343	−	0.941023i	$-0.390134\pi$
$83$	511.687	0.676686	0.338343	+	0.941023i	$0.390134\pi$
$84$	0	0
$85$	−1063.79	−1.35746
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−320.459	−0.381669	−0.190835	−	0.981622i	$-0.561119\pi$
$89$	−320.459	−0.381669	−0.190835	+	0.981622i	$0.561119\pi$
$90$	0	0
$91$	−2.92029	−0.00336406
$92$	0	0
$93$	0	0
$94$	0	0
$95$	694.661	0.750218
$96$	0	0
$97$	−601.643	−0.629769	−0.314884	−	0.949130i	$-0.601966\pi$
$97$	−601.643	−0.629769	−0.314884	+	0.949130i	$0.601966\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	876.325	0.863342	0.431671	−	0.902031i	$-0.357924\pi$
$101$	876.325	0.863342	0.431671	+	0.902031i	$0.357924\pi$
$102$	0	0
$103$	−223.437	−0.213747	−0.106873	−	0.994273i	$-0.534084\pi$
$103$	−223.437	−0.213747	−0.106873	+	0.994273i	$0.534084\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1265.34	−1.14322	−0.571612	−	0.820524i	$-0.693682\pi$
$107$	−1265.34	−1.14322	−0.571612	+	0.820524i	$0.693682\pi$
$108$	0	0
$109$	−213.009	−0.187180	−0.0935898	−	0.995611i	$-0.529834\pi$
$109$	−213.009	−0.187180	−0.0935898	+	0.995611i	$0.529834\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1613.98	1.34363	0.671814	−	0.740720i	$-0.265515\pi$
$113$	1613.98	1.34363	0.671814	+	0.740720i	$0.265515\pi$
$114$	0	0
$115$	−330.610	−0.268083
$116$	0	0
$117$	0	0
$118$	0	0
$119$	124.375	0.0958101
$120$	0	0
$121$	−1302.52	−0.978602
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1488.94	1.06540
$126$	0	0
$127$	−1234.19	−0.862333	−0.431167	−	0.902272i	$-0.641898\pi$
$127$	−1234.19	−0.862333	−0.431167	+	0.902272i	$0.641898\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1447.44	−0.965369	−0.482685	−	0.875794i	$-0.660338\pi$
$131$	−1447.44	−0.965369	−0.482685	+	0.875794i	$0.660338\pi$
$132$	0	0
$133$	−81.2171	−0.0529505
$134$	0	0
$135$	0	0
$136$	0	0
$137$	553.782	0.345349	0.172674	−	0.984979i	$-0.444759\pi$
$137$	553.782	0.345349	0.172674	+	0.984979i	$0.444759\pi$
$138$	0	0
$139$	−1589.06	−0.969654	−0.484827	−	0.874610i	$-0.661117\pi$
$139$	−1589.06	−0.969654	−0.484827	+	0.874610i	$0.661117\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−16.6285	−0.00972406
$144$	0	0
$145$	1625.65	0.931055
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1212.81	−0.666828	−0.333414	−	0.942781i	$-0.608201\pi$
$149$	−1212.81	−0.666828	−0.333414	+	0.942781i	$0.608201\pi$
$150$	0	0
$151$	531.858	0.286636	0.143318	−	0.989677i	$-0.454223\pi$
$151$	531.858	0.286636	0.143318	+	0.989677i	$0.454223\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2555.05	−1.32404
$156$	0	0
$157$	2162.94	1.09950	0.549749	−	0.835330i	$-0.314723\pi$
$157$	2162.94	1.09950	0.549749	+	0.835330i	$0.314723\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	38.6537	0.0189213
$162$	0	0
$163$	−1602.11	−0.769859	−0.384930	−	0.922946i	$-0.625774\pi$
$163$	−1602.11	−0.769859	−0.384930	+	0.922946i	$0.625774\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1733.60	−0.803294	−0.401647	−	0.915795i	$-0.631562\pi$
$167$	−1733.60	−0.803294	−0.401647	+	0.915795i	$0.631562\pi$
$168$	0	0
$169$	−2187.29	−0.995581
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1874.65	0.823854	0.411927	−	0.911217i	$-0.364856\pi$
$173$	1874.65	0.823854	0.411927	+	0.911217i	$0.364856\pi$
$174$	0	0
$175$	−56.9271	−0.0245902
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2058.01	0.859344	0.429672	−	0.902985i	$-0.358629\pi$
$179$	2058.01	0.859344	0.429672	+	0.902985i	$0.358629\pi$
$180$	0	0
$181$	−3938.45	−1.61736	−0.808681	−	0.588248i	$-0.799818\pi$
$181$	−3938.45	−1.61736	−0.808681	+	0.588248i	$0.799818\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2913.99	1.15806
$186$	0	0
$187$	708.203	0.276946
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4465.09	−1.69153	−0.845765	−	0.533555i	$-0.820856\pi$
$191$	−4465.09	−1.69153	−0.845765	+	0.533555i	$0.820856\pi$
$192$	0	0
$193$	4150.03	1.54780	0.773901	−	0.633307i	$-0.218303\pi$
$193$	4150.03	1.54780	0.773901	+	0.633307i	$0.218303\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	536.823	0.194148	0.0970738	−	0.995277i	$-0.469052\pi$
$197$	536.823	0.194148	0.0970738	+	0.995277i	$0.469052\pi$
$198$	0	0
$199$	1953.59	0.695911	0.347956	−	0.937511i	$-0.386876\pi$
$199$	1953.59	0.695911	0.347956	+	0.937511i	$0.386876\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−190.065	−0.0657141
$204$	0	0
$205$	−78.8366	−0.0268594
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−462.459	−0.153057
$210$	0	0
$211$	3227.29	1.05296	0.526482	−	0.850186i	$-0.323511\pi$
$211$	3227.29	1.05296	0.526482	+	0.850186i	$0.323511\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2905.57	0.921666
$216$	0	0
$217$	298.726	0.0934510
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−413.490	−0.125857
$222$	0	0
$223$	4655.98	1.39815	0.699075	−	0.715049i	$-0.253596\pi$
$223$	4655.98	1.39815	0.699075	+	0.715049i	$0.253596\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4080.81	−1.19318	−0.596592	−	0.802545i	$-0.703479\pi$
$227$	−4080.81	−1.19318	−0.596592	+	0.802545i	$0.703479\pi$
$228$	0	0
$229$	1298.44	0.374687	0.187344	−	0.982294i	$-0.440012\pi$
$229$	1298.44	0.374687	0.187344	+	0.982294i	$0.440012\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3073.26	−0.864103	−0.432052	−	0.901849i	$-0.642210\pi$
$233$	−3073.26	−0.864103	−0.432052	+	0.901849i	$0.642210\pi$
$234$	0	0
$235$	602.630	0.167282
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3575.52	−0.967704	−0.483852	−	0.875150i	$-0.660763\pi$
$239$	−3575.52	−0.967704	−0.483852	+	0.875150i	$0.660763\pi$
$240$	0	0
$241$	−2717.59	−0.726370	−0.363185	−	0.931717i	$-0.618311\pi$
$241$	−2717.59	−0.726370	−0.363185	+	0.931717i	$0.618311\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2742.53	0.715160
$246$	0	0
$247$	270.011	0.0695561
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4583.36	−1.15259	−0.576293	−	0.817243i	$-0.695501\pi$
$251$	−4583.36	−1.15259	−0.576293	+	0.817243i	$0.695501\pi$
$252$	0	0
$253$	220.098	0.0546935
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5975.43	1.45034	0.725169	−	0.688571i	$-0.241761\pi$
$257$	5975.43	1.45034	0.725169	+	0.688571i	$0.241761\pi$
$258$	0	0
$259$	−340.693	−0.0817360
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6175.17	1.44782	0.723912	−	0.689893i	$-0.242342\pi$
$263$	6175.17	1.44782	0.723912	+	0.689893i	$0.242342\pi$
$264$	0	0
$265$	−3238.42	−0.750696
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6580.02	1.49142	0.745708	−	0.666273i	$-0.232111\pi$
$269$	6580.02	1.49142	0.745708	+	0.666273i	$0.232111\pi$
$270$	0	0
$271$	5817.31	1.30397	0.651986	−	0.758231i	$-0.273936\pi$
$271$	5817.31	1.30397	0.651986	+	0.758231i	$0.273936\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−324.149	−0.0710797
$276$	0	0
$277$	8651.05	1.87650	0.938252	−	0.345953i	$-0.112444\pi$
$277$	8651.05	1.87650	0.938252	+	0.345953i	$0.112444\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	7296.28	1.54897	0.774483	−	0.632595i	$-0.218010\pi$
$281$	7296.28	1.54897	0.774483	+	0.632595i	$0.218010\pi$
$282$	0	0
$283$	−811.916	−0.170542	−0.0852710	−	0.996358i	$-0.527176\pi$
$283$	−811.916	−0.170542	−0.0852710	+	0.996358i	$0.527176\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.21728	0.00189574
$288$	0	0
$289$	12697.5	2.58446
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−4801.91	−0.957443	−0.478722	−	0.877967i	$-0.658900\pi$
$293$	−4801.91	−0.957443	−0.478722	+	0.877967i	$0.658900\pi$
$294$	0	0
$295$	3452.62	0.681421
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−128.506	−0.0248552
$300$	0	0
$301$	−339.708	−0.0650514
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2577.75	−0.483939
$306$	0	0
$307$	−755.147	−0.140386	−0.0701930	−	0.997533i	$-0.522362\pi$
$307$	−755.147	−0.140386	−0.0701930	+	0.997533i	$0.522362\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5266.07	0.960165	0.480082	−	0.877223i	$-0.340607\pi$
$311$	5266.07	0.960165	0.480082	+	0.877223i	$0.340607\pi$
$312$	0	0
$313$	1394.11	0.251756	0.125878	−	0.992046i	$-0.459825\pi$
$313$	1394.11	0.251756	0.125878	+	0.992046i	$0.459825\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5117.11	−0.906642	−0.453321	−	0.891347i	$-0.649761\pi$
$317$	−5117.11	−0.906642	−0.453321	+	0.891347i	$0.649761\pi$
$318$	0	0
$319$	−1082.25	−0.189951
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−11499.7	−1.98099
$324$	0	0
$325$	189.257	0.0323018
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−70.4572	−0.0118068
$330$	0	0
$331$	−242.800	−0.0403187	−0.0201593	−	0.999797i	$-0.506417\pi$
$331$	−242.800	−0.0403187	−0.0201593	+	0.999797i	$0.506417\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5825.00	0.950011
$336$	0	0
$337$	−51.2944	−0.00829135	−0.00414568	−	0.999991i	$-0.501320\pi$
$337$	−51.2944	−0.00829135	−0.00414568	+	0.999991i	$0.501320\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1700.98	0.270127
$342$	0	0
$343$	−642.117	−0.101082
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−861.444	−0.133270	−0.0666350	−	0.997777i	$-0.521226\pi$
$347$	−861.444	−0.133270	−0.0666350	+	0.997777i	$0.521226\pi$
$348$	0	0
$349$	3125.36	0.479361	0.239680	−	0.970852i	$-0.422957\pi$
$349$	3125.36	0.479361	0.239680	+	0.970852i	$0.422957\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−8324.73	−1.25519	−0.627593	−	0.778542i	$-0.715960\pi$
$353$	−8324.73	−1.25519	−0.627593	+	0.778542i	$0.715960\pi$
$354$	0	0
$355$	6651.10	0.994377
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5819.46	−0.855542	−0.427771	−	0.903887i	$-0.640701\pi$
$359$	−5819.46	−0.855542	−0.427771	+	0.903887i	$0.640701\pi$
$360$	0	0
$361$	650.350	0.0948170
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1283.45	0.184052
$366$	0	0
$367$	−6259.27	−0.890277	−0.445138	−	0.895462i	$-0.646845\pi$
$367$	−6259.27	−0.890277	−0.445138	+	0.895462i	$0.646845\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	378.624	0.0529842
$372$	0	0
$373$	480.671	0.0667243	0.0333622	−	0.999443i	$-0.489379\pi$
$373$	480.671	0.0667243	0.0333622	+	0.999443i	$0.489379\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	631.881	0.0863224
$378$	0	0
$379$	7321.57	0.992305	0.496152	−	0.868235i	$-0.334746\pi$
$379$	7321.57	0.992305	0.496152	+	0.868235i	$0.334746\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2646.90	0.353134	0.176567	−	0.984289i	$-0.443501\pi$
$383$	2646.90	0.353134	0.176567	+	0.984289i	$0.443501\pi$
$384$	0	0
$385$	−40.0950	−0.00530761
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4861.43	0.633636	0.316818	−	0.948486i	$-0.397386\pi$
$389$	4861.43	0.633636	0.316818	+	0.948486i	$0.397386\pi$
$390$	0	0
$391$	5473.05	0.707888
$392$	0	0
$393$	0	0
$394$	0	0
$395$	7412.06	0.944155
$396$	0	0
$397$	−7253.21	−0.916948	−0.458474	−	0.888708i	$-0.651604\pi$
$397$	−7253.21	−0.916948	−0.458474	+	0.888708i	$0.651604\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2301.38	−0.286597	−0.143298	−	0.989680i	$-0.545771\pi$
$401$	−2301.38	−0.286597	−0.143298	+	0.989680i	$0.545771\pi$
$402$	0	0
$403$	−993.132	−0.122758
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1939.94	−0.236264
$408$	0	0
$409$	−1804.43	−0.218150	−0.109075	−	0.994034i	$-0.534789\pi$
$409$	−1804.43	−0.218150	−0.109075	+	0.994034i	$0.534789\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−403.667	−0.0480948
$414$	0	0
$415$	−4101.81	−0.485181
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−7476.35	−0.871703	−0.435852	−	0.900019i	$-0.643553\pi$
$419$	−7476.35	−0.871703	−0.435852	+	0.900019i	$0.643553\pi$
$420$	0	0
$421$	12886.0	1.49175	0.745875	−	0.666086i	$-0.232032\pi$
$421$	12886.0	1.49175	0.745875	+	0.666086i	$0.232032\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−8060.42	−0.919971
$426$	0	0
$427$	301.381	0.0341565
$428$	0	0
$429$	0	0
$430$	0	0
$431$	13065.8	1.46022	0.730110	−	0.683329i	$-0.239469\pi$
$431$	13065.8	1.46022	0.730110	+	0.683329i	$0.239469\pi$
$432$	0	0
$433$	−6190.38	−0.687046	−0.343523	−	0.939144i	$-0.611620\pi$
$433$	−6190.38	−0.687046	−0.343523	+	0.939144i	$0.611620\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3573.92	−0.391222
$438$	0	0
$439$	−14543.6	−1.58115	−0.790576	−	0.612363i	$-0.790219\pi$
$439$	−14543.6	−1.58115	−0.790576	+	0.612363i	$0.790219\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−853.715	−0.0915603	−0.0457802	−	0.998952i	$-0.514577\pi$
$443$	−853.715	−0.0915603	−0.0457802	+	0.998952i	$0.514577\pi$
$444$	0	0
$445$	2568.88	0.273655
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6187.67	0.650366	0.325183	−	0.945651i	$-0.394574\pi$
$449$	6187.67	0.650366	0.325183	+	0.945651i	$0.394574\pi$
$450$	0	0
$451$	52.4842	0.00547979
$452$	0	0
$453$	0	0
$454$	0	0
$455$	23.4098	0.00241202
$456$	0	0
$457$	4778.13	0.489084	0.244542	−	0.969639i	$-0.421362\pi$
$457$	4778.13	0.489084	0.244542	+	0.969639i	$0.421362\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−9089.20	−0.918278	−0.459139	−	0.888365i	$-0.651842\pi$
$461$	−9089.20	−0.918278	−0.459139	+	0.888365i	$0.651842\pi$
$462$	0	0
$463$	−7348.21	−0.737582	−0.368791	−	0.929512i	$-0.620228\pi$
$463$	−7348.21	−0.737582	−0.368791	+	0.929512i	$0.620228\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	948.436	0.0939794	0.0469897	−	0.998895i	$-0.485037\pi$
$467$	948.436	0.0939794	0.0469897	+	0.998895i	$0.485037\pi$
$468$	0	0
$469$	−681.037	−0.0670519
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1934.34	−0.188036
$474$	0	0
$475$	5263.49	0.508432
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2987.88	0.285010	0.142505	−	0.989794i	$-0.454484\pi$
$479$	2987.88	0.285010	0.142505	+	0.989794i	$0.454484\pi$
$480$	0	0
$481$	1132.65	0.107369
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4822.92	0.451542
$486$	0	0
$487$	−19555.8	−1.81963	−0.909814	−	0.415017i	$-0.863776\pi$
$487$	−19555.8	−1.81963	−0.909814	+	0.415017i	$0.863776\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	858.127	0.0788732	0.0394366	−	0.999222i	$-0.487444\pi$
$491$	858.127	0.0788732	0.0394366	+	0.999222i	$0.487444\pi$
$492$	0	0
$493$	−26911.7	−2.45850
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−777.622	−0.0701833
$498$	0	0
$499$	3689.41	0.330983	0.165491	−	0.986211i	$-0.447079\pi$
$499$	3689.41	0.330983	0.165491	+	0.986211i	$0.447079\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	9034.08	0.800814	0.400407	−	0.916337i	$-0.368869\pi$
$503$	9034.08	0.800814	0.400407	+	0.916337i	$0.368869\pi$
$504$	0	0
$505$	−7024.84	−0.619013
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−6290.41	−0.547775	−0.273888	−	0.961762i	$-0.588310\pi$
$509$	−6290.41	−0.547775	−0.273888	+	0.961762i	$0.588310\pi$
$510$	0	0
$511$	−150.056	−0.0129904
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1791.13	0.153256
$516$	0	0
$517$	−401.191	−0.0341284
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−13010.4	−1.09404	−0.547019	−	0.837120i	$-0.684238\pi$
$521$	−13010.4	−1.09404	−0.547019	+	0.837120i	$0.684238\pi$
$522$	0	0
$523$	1108.69	0.0926956	0.0463478	−	0.998925i	$-0.485242\pi$
$523$	1108.69	0.0926956	0.0463478	+	0.998925i	$0.485242\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	42297.3	3.49620
$528$	0	0
$529$	−10466.1	−0.860201
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−30.6433	−0.00249026
$534$	0	0
$535$	10143.3	0.819686
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1825.80	−0.145905
$540$	0	0
$541$	−38.8964	−0.00309110	−0.00154555	−	0.999999i	$-0.500492\pi$
$541$	−38.8964	−0.00309110	−0.00154555	+	0.999999i	$0.500492\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1707.54	0.134207
$546$	0	0
$547$	10029.8	0.783993	0.391997	−	0.919967i	$-0.371784\pi$
$547$	10029.8	0.783993	0.391997	+	0.919967i	$0.371784\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	17573.5	1.35872
$552$	0	0
$553$	−866.590	−0.0666386
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−7355.74	−0.559556	−0.279778	−	0.960065i	$-0.590261\pi$
$557$	−7355.74	−0.559556	−0.279778	+	0.960065i	$0.590261\pi$
$558$	0	0
$559$	1129.38	0.0854519
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−21459.9	−1.60644	−0.803220	−	0.595683i	$-0.796881\pi$
$563$	−21459.9	−1.60644	−0.803220	+	0.595683i	$0.796881\pi$
$564$	0	0
$565$	−12938.0	−0.963376
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1457.92	−0.107415	−0.0537077	−	0.998557i	$-0.517104\pi$
$569$	−1457.92	−0.107415	−0.0537077	+	0.998557i	$0.517104\pi$
$570$	0	0
$571$	−22400.1	−1.64171	−0.820853	−	0.571140i	$-0.806501\pi$
$571$	−22400.1	−1.64171	−0.820853	+	0.571140i	$0.806501\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2505.05	−0.181683
$576$	0	0
$577$	−12833.5	−0.925936	−0.462968	−	0.886375i	$-0.653215\pi$
$577$	−12833.5	−0.925936	−0.462968	+	0.886375i	$0.653215\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	479.569	0.0342442
$582$	0	0
$583$	2155.92	0.153155
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20754.8	−1.45935	−0.729677	−	0.683792i	$-0.760330\pi$
$587$	−20754.8	−1.45935	−0.729677	+	0.683792i	$0.760330\pi$
$588$	0	0
$589$	−27620.3	−1.93222
$590$	0	0
$591$	0	0
$592$	0	0
$593$	113.272	0.00784406	0.00392203	−	0.999992i	$-0.498752\pi$
$593$	113.272	0.00784406	0.00392203	+	0.999992i	$0.498752\pi$
$594$	0	0
$595$	−997.019	−0.0686955
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−11162.0	−0.761382	−0.380691	−	0.924702i	$-0.624314\pi$
$599$	−11162.0	−0.761382	−0.380691	+	0.924702i	$0.624314\pi$
$600$	0	0
$601$	−1879.82	−0.127586	−0.0637932	−	0.997963i	$-0.520320\pi$
$601$	−1879.82	−0.127586	−0.0637932	+	0.997963i	$0.520320\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	10441.3	0.701654
$606$	0	0
$607$	−21324.3	−1.42591	−0.712956	−	0.701209i	$-0.752644\pi$
$607$	−21324.3	−1.42591	−0.712956	+	0.701209i	$0.752644\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	234.239	0.0155095
$612$	0	0
$613$	26096.2	1.71944	0.859720	−	0.510765i	$-0.170638\pi$
$613$	26096.2	1.71944	0.859720	+	0.510765i	$0.170638\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−19181.3	−1.25155	−0.625777	−	0.780002i	$-0.715218\pi$
$617$	−19181.3	−1.25155	−0.625777	+	0.780002i	$0.715218\pi$
$618$	0	0
$619$	−3363.16	−0.218379	−0.109190	−	0.994021i	$-0.534826\pi$
$619$	−3363.16	−0.218379	−0.109190	+	0.994021i	$0.534826\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−300.344	−0.0193146
$624$	0	0
$625$	−4343.24	−0.277967
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−48239.4	−3.05792
$630$	0	0
$631$	1485.04	0.0936900	0.0468450	−	0.998902i	$-0.485083\pi$
$631$	1485.04	0.0936900	0.0468450	+	0.998902i	$0.485083\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	9893.55	0.618289
$636$	0	0
$637$	1066.01	0.0663057
$638$	0	0
$639$	0	0
$640$	0	0
$641$	16416.2	1.01155	0.505773	−	0.862667i	$-0.331207\pi$
$641$	16416.2	1.01155	0.505773	+	0.862667i	$0.331207\pi$
$642$	0	0
$643$	−20400.0	−1.25116	−0.625580	−	0.780160i	$-0.715138\pi$
$643$	−20400.0	−1.25116	−0.625580	+	0.780160i	$0.715138\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	25011.7	1.51980	0.759900	−	0.650040i	$-0.225248\pi$
$647$	25011.7	1.51980	0.759900	+	0.650040i	$0.225248\pi$
$648$	0	0
$649$	−2298.53	−0.139022
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7527.53	0.451110	0.225555	−	0.974230i	$-0.427580\pi$
$653$	7527.53	0.451110	0.225555	+	0.974230i	$0.427580\pi$
$654$	0	0
$655$	11603.0	0.692166
$656$	0	0
$657$	0	0
$658$	0	0
$659$	29968.0	1.77145	0.885725	−	0.464210i	$-0.153662\pi$
$659$	29968.0	1.77145	0.885725	+	0.464210i	$0.153662\pi$
$660$	0	0
$661$	12307.6	0.724222	0.362111	−	0.932135i	$-0.382056\pi$
$661$	12307.6	0.724222	0.362111	+	0.932135i	$0.382056\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	651.057	0.0379653
$666$	0	0
$667$	−8363.73	−0.485525
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1716.09	0.0987319
$672$	0	0
$673$	2770.84	0.158704	0.0793521	−	0.996847i	$-0.474715\pi$
$673$	2770.84	0.158704	0.0793521	+	0.996847i	$0.474715\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−22465.0	−1.27533	−0.637666	−	0.770313i	$-0.720100\pi$
$677$	−22465.0	−1.27533	−0.637666	+	0.770313i	$0.720100\pi$
$678$	0	0
$679$	−563.878	−0.0318699
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−7924.28	−0.443945	−0.221972	−	0.975053i	$-0.571249\pi$
$683$	−7924.28	−0.443945	−0.221972	+	0.975053i	$0.571249\pi$
$684$	0	0
$685$	−4439.25	−0.247613
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1258.75	−0.0696004
$690$	0	0
$691$	−22780.4	−1.25413	−0.627067	−	0.778965i	$-0.715745\pi$
$691$	−22780.4	−1.25413	−0.627067	+	0.778965i	$0.715745\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	12738.3	0.695238
$696$	0	0
$697$	1305.09	0.0709239
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−13999.0	−0.754256	−0.377128	−	0.926161i	$-0.623088\pi$
$701$	−13999.0	−0.754256	−0.377128	+	0.926161i	$0.623088\pi$
$702$	0	0
$703$	31500.5	1.68999
$704$	0	0
$705$	0	0
$706$	0	0
$707$	821.318	0.0436900
$708$	0	0
$709$	−9026.64	−0.478142	−0.239071	−	0.971002i	$-0.576843\pi$
$709$	−9026.64	−0.478142	−0.239071	+	0.971002i	$0.576843\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	13145.3	0.690458
$714$	0	0
$715$	133.298	0.00697211
$716$	0	0
$717$	0	0
$718$	0	0
$719$	18206.9	0.944371	0.472186	−	0.881499i	$-0.343465\pi$
$719$	18206.9	0.944371	0.472186	+	0.881499i	$0.343465\pi$
$720$	0	0
$721$	−209.412	−0.0108168
$722$	0	0
$723$	0	0
$724$	0	0
$725$	12317.7	0.630988
$726$	0	0
$727$	10334.1	0.527195	0.263598	−	0.964633i	$-0.415091\pi$
$727$	10334.1	0.527195	0.263598	+	0.964633i	$0.415091\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−48100.0	−2.43371
$732$	0	0
$733$	22352.7	1.12635	0.563176	−	0.826337i	$-0.309579\pi$
$733$	22352.7	1.12635	0.563176	+	0.826337i	$0.309579\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3877.90	−0.193818
$738$	0	0
$739$	−22423.2	−1.11617	−0.558086	−	0.829783i	$-0.688464\pi$
$739$	−22423.2	−1.11617	−0.558086	+	0.829783i	$0.688464\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	11655.4	0.575498	0.287749	−	0.957706i	$-0.407093\pi$
$743$	11655.4	0.575498	0.287749	+	0.957706i	$0.407093\pi$
$744$	0	0
$745$	9722.21	0.478113
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1185.91	−0.0578536
$750$	0	0
$751$	2638.13	0.128185	0.0640923	−	0.997944i	$-0.479585\pi$
$751$	2638.13	0.128185	0.0640923	+	0.997944i	$0.479585\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−4263.51	−0.205517
$756$	0	0
$757$	−19939.5	−0.957350	−0.478675	−	0.877992i	$-0.658883\pi$
$757$	−19939.5	−0.957350	−0.478675	+	0.877992i	$0.658883\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−26761.4	−1.27477	−0.637385	−	0.770546i	$-0.719984\pi$
$761$	−26761.4	−1.27477	−0.637385	+	0.770546i	$0.719984\pi$
$762$	0	0
$763$	−199.639	−0.00947235
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1342.01	0.0631777
$768$	0	0
$769$	7576.81	0.355301	0.177651	−	0.984094i	$-0.443150\pi$
$769$	7576.81	0.355301	0.177651	+	0.984094i	$0.443150\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	27041.3	1.25823	0.629113	−	0.777314i	$-0.283418\pi$
$773$	27041.3	1.25823	0.629113	+	0.777314i	$0.283418\pi$
$774$	0	0
$775$	−19359.7	−0.897319
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−852.231	−0.0391969
$780$	0	0
$781$	−4427.86	−0.202870
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−17338.7	−0.788336
$786$	0	0
$787$	33604.3	1.52206	0.761031	−	0.648715i	$-0.224693\pi$
$787$	33604.3	1.52206	0.761031	+	0.648715i	$0.224693\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1512.67	0.0679953
$792$	0	0
$793$	−1001.96	−0.0448682
$794$	0	0
$795$	0	0
$796$	0	0
$797$	43206.2	1.92025	0.960127	−	0.279563i	$-0.0901894\pi$
$797$	43206.2	1.92025	0.960127	+	0.279563i	$0.0901894\pi$
$798$	0	0
$799$	−9976.18	−0.441717
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−854.437	−0.0375497
$804$	0	0
$805$	−309.858	−0.0135665
$806$	0	0
$807$	0	0
$808$	0	0
$809$	33276.5	1.44616	0.723078	−	0.690766i	$-0.242727\pi$
$809$	33276.5	1.44616	0.723078	+	0.690766i	$0.242727\pi$
$810$	0	0
$811$	21917.4	0.948980	0.474490	−	0.880261i	$-0.342633\pi$
$811$	21917.4	0.948980	0.474490	+	0.880261i	$0.342633\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	12842.9	0.551986
$816$	0	0
$817$	31409.5	1.34502
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1073.46	−0.0456324	−0.0228162	−	0.999740i	$-0.507263\pi$
$821$	−1073.46	−0.0456324	−0.0228162	+	0.999740i	$0.507263\pi$
$822$	0	0
$823$	12458.4	0.527671	0.263835	−	0.964568i	$-0.415012\pi$
$823$	12458.4	0.527671	0.263835	+	0.964568i	$0.415012\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−902.282	−0.0379388	−0.0189694	−	0.999820i	$-0.506039\pi$
$827$	−902.282	−0.0379388	−0.0189694	+	0.999820i	$0.506039\pi$
$828$	0	0
$829$	29605.8	1.24035	0.620176	−	0.784462i	$-0.287061\pi$
$829$	29605.8	1.24035	0.620176	+	0.784462i	$0.287061\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−45401.0	−1.88842
$834$	0	0
$835$	13897.0	0.575958
$836$	0	0
$837$	0	0
$838$	0	0
$839$	45453.5	1.87036	0.935178	−	0.354177i	$-0.115239\pi$
$839$	45453.5	1.87036	0.935178	+	0.354177i	$0.115239\pi$
$840$	0	0
$841$	16736.6	0.686234
$842$	0	0
$843$	0	0
$844$	0	0
$845$	17533.9	0.713827
$846$	0	0
$847$	−1220.76	−0.0495229
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−14992.0	−0.603902
$852$	0	0
$853$	5591.84	0.224456	0.112228	−	0.993682i	$-0.464201\pi$
$853$	5591.84	0.224456	0.112228	+	0.993682i	$0.464201\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−40981.6	−1.63349	−0.816747	−	0.576997i	$-0.804225\pi$
$857$	−40981.6	−1.63349	−0.816747	+	0.576997i	$0.804225\pi$
$858$	0	0
$859$	42298.0	1.68008	0.840040	−	0.542524i	$-0.182531\pi$
$859$	42298.0	1.68008	0.840040	+	0.542524i	$0.182531\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	19236.0	0.758749	0.379374	−	0.925243i	$-0.376139\pi$
$863$	19236.0	0.758749	0.379374	+	0.925243i	$0.376139\pi$
$864$	0	0
$865$	−15027.6	−0.590700
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−4934.46	−0.192624
$870$	0	0
$871$	2264.14	0.0880798
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1395.48	0.0539151
$876$	0	0
$877$	−24120.8	−0.928735	−0.464368	−	0.885643i	$-0.653718\pi$
$877$	−24120.8	−0.928735	−0.464368	+	0.885643i	$0.653718\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	34292.4	1.31140	0.655698	−	0.755023i	$-0.272375\pi$
$881$	34292.4	1.31140	0.655698	+	0.755023i	$0.272375\pi$
$882$	0	0
$883$	−15818.9	−0.602887	−0.301444	−	0.953484i	$-0.597469\pi$
$883$	−15818.9	−0.602887	−0.301444	+	0.953484i	$0.597469\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−20763.6	−0.785991	−0.392995	−	0.919540i	$-0.628561\pi$
$887$	−20763.6	−0.785991	−0.392995	+	0.919540i	$0.628561\pi$
$888$	0	0
$889$	−1156.72	−0.0436390
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6514.49	0.244120
$894$	0	0
$895$	−16497.5	−0.616146
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−64637.3	−2.39797
$900$	0	0
$901$	53610.1	1.98225
$902$	0	0
$903$	0	0
$904$	0	0
$905$	31571.6	1.15964
$906$	0	0
$907$	−12675.5	−0.464040	−0.232020	−	0.972711i	$-0.574534\pi$
$907$	−12675.5	−0.464040	−0.232020	+	0.972711i	$0.574534\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−18325.7	−0.666472	−0.333236	−	0.942843i	$-0.608141\pi$
$911$	−18325.7	−0.666472	−0.333236	+	0.942843i	$0.608141\pi$
$912$	0	0
$913$	2730.72	0.0989852
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1356.58	−0.0488532
$918$	0	0
$919$	2706.30	0.0971409	0.0485705	−	0.998820i	$-0.484533\pi$
$919$	2706.30	0.0971409	0.0485705	+	0.998820i	$0.484533\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2585.25	0.0921933
$924$	0	0
$925$	22079.5	0.784831
$926$	0	0
$927$	0	0
$928$	0	0
$929$	24885.7	0.878874	0.439437	−	0.898273i	$-0.355178\pi$
$929$	24885.7	0.878874	0.439437	+	0.898273i	$0.355178\pi$
$930$	0	0
$931$	29647.1	1.04366
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−5677.13	−0.198569
$936$	0	0
$937$	−33383.1	−1.16390	−0.581952	−	0.813223i	$-0.697711\pi$
$937$	−33383.1	−1.16390	−0.581952	+	0.813223i	$0.697711\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−29179.4	−1.01086	−0.505431	−	0.862867i	$-0.668667\pi$
$941$	−29179.4	−1.01086	−0.505431	+	0.862867i	$0.668667\pi$
$942$	0	0
$943$	405.602	0.0140066
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−38884.1	−1.33428	−0.667140	−	0.744933i	$-0.732482\pi$
$947$	−38884.1	−1.33428	−0.667140	+	0.744933i	$0.732482\pi$
$948$	0	0
$949$	498.870	0.0170643
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−5321.13	−0.180869	−0.0904345	−	0.995902i	$-0.528826\pi$
$953$	−5321.13	−0.180869	−0.0904345	+	0.995902i	$0.528826\pi$
$954$	0	0
$955$	35793.3	1.21282
$956$	0	0
$957$	0	0
$958$	0	0
$959$	519.021	0.0174766
$960$	0	0
$961$	71799.8	2.41012
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−33267.7	−1.10977
$966$	0	0
$967$	49020.4	1.63019	0.815094	−	0.579329i	$-0.196685\pi$
$967$	49020.4	1.63019	0.815094	+	0.579329i	$0.196685\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−11409.6	−0.377087	−0.188544	−	0.982065i	$-0.560377\pi$
$971$	−11409.6	−0.377087	−0.188544	+	0.982065i	$0.560377\pi$
$972$	0	0
$973$	−1489.31	−0.0490700
$974$	0	0
$975$	0	0
$976$	0	0
$977$	18456.7	0.604382	0.302191	−	0.953247i	$-0.402282\pi$
$977$	18456.7	0.604382	0.302191	+	0.953247i	$0.402282\pi$
$978$	0	0
$979$	−1710.19	−0.0558303
$980$	0	0
$981$	0	0
$982$	0	0
$983$	35607.3	1.15534	0.577668	−	0.816272i	$-0.303963\pi$
$983$	35607.3	1.15534	0.577668	+	0.816272i	$0.303963\pi$
$984$	0	0
$985$	−4303.31	−0.139203
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−14948.7	−0.480629
$990$	0	0
$991$	31299.9	1.00330	0.501652	−	0.865070i	$-0.332726\pi$
$991$	31299.9	1.00330	0.501652	+	0.865070i	$0.332726\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−15660.5	−0.498965
$996$	0	0
$997$	−28778.5	−0.914166	−0.457083	−	0.889424i	$-0.651106\pi$
$997$	−28778.5	−0.914166	−0.457083	+	0.889424i	$0.651106\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.4.a.i.1.1		4
3.2	odd	2		648.4.a.h.1.4		4
4.3	odd	2		1296.4.a.ba.1.1		4
9.2	odd	6		216.4.i.a.145.1		8
9.4	even	3		72.4.i.a.25.4	✓	8
9.5	odd	6		216.4.i.a.73.1		8
9.7	even	3		72.4.i.a.49.4	yes	8
12.11	even	2		1296.4.a.y.1.4		4
36.7	odd	6		144.4.i.e.49.1		8
36.11	even	6		432.4.i.e.145.1		8
36.23	even	6		432.4.i.e.289.1		8
36.31	odd	6		144.4.i.e.97.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.4.i.a.25.4	✓	8	9.4	even	3
72.4.i.a.49.4	yes	8	9.7	even	3
144.4.i.e.49.1		8	36.7	odd	6
144.4.i.e.97.1		8	36.31	odd	6
216.4.i.a.73.1		8	9.5	odd	6
216.4.i.a.145.1		8	9.2	odd	6
432.4.i.e.145.1		8	36.11	even	6
432.4.i.e.289.1		8	36.23	even	6
648.4.a.h.1.4		4	3.2	odd	2
648.4.a.i.1.1		4	1.1	even	1	trivial
1296.4.a.y.1.4		4	12.11	even	2
1296.4.a.ba.1.1		4	4.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	648.a (trivial)

\(f(q)\)	\(=\)	\(q-8.01626 q^{5} +0.937231 q^{7} +O(q^{10})\)	\(q-8.01626 q^{5} +0.937231 q^{7} +5.33669 q^{11} -3.11587 q^{13} +132.704 q^{17} -86.6565 q^{19} +41.2424 q^{23} -60.7396 q^{25} -202.794 q^{29} +318.733 q^{31} -7.51308 q^{35} -363.510 q^{37} +9.83459 q^{41} -362.460 q^{43} -75.1760 q^{47} -342.122 q^{49} +403.981 q^{53} -42.7803 q^{55} -430.702 q^{59} +321.565 q^{61} +24.9776 q^{65} -726.648 q^{67} -829.702 q^{71} -160.106 q^{73} +5.00171 q^{77} -924.628 q^{79} +511.687 q^{83} -1063.79 q^{85} -320.459 q^{89} -2.92029 q^{91} +694.661 q^{95} -601.643 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 5 q^{5} - 3 q^{7}+O(q^{10}) \) 4 * q + 5 * q^5 - 3 * q^7	\( 4 q + 5 q^{5} - 3 q^{7} - 16 q^{11} - 29 q^{13} - 17 q^{17} - 109 q^{19} - 37 q^{23} - 97 q^{25} + 3 q^{29} - 331 q^{31} - 171 q^{35} - 366 q^{37} + 378 q^{41} - 506 q^{43} - 171 q^{47} - 829 q^{49} + 410 q^{53} - 1163 q^{55} - 616 q^{59} - 1331 q^{61} + 815 q^{65} - 1162 q^{67} - 344 q^{71} - 1307 q^{73} + 741 q^{77} - 1853 q^{79} - 1421 q^{83} - 2074 q^{85} + 816 q^{89} - 1995 q^{91} - 1292 q^{95} - 2506 q^{97}+O(q^{100}) \) 4 * q + 5 * q^5 - 3 * q^7 - 16 * q^11 - 29 * q^13 - 17 * q^17 - 109 * q^19 - 37 * q^23 - 97 * q^25 + 3 * q^29 - 331 * q^31 - 171 * q^35 - 366 * q^37 + 378 * q^41 - 506 * q^43 - 171 * q^47 - 829 * q^49 + 410 * q^53 - 1163 * q^55 - 616 * q^59 - 1331 * q^61 + 815 * q^65 - 1162 * q^67 - 344 * q^71 - 1307 * q^73 + 741 * q^77 - 1853 * q^79 - 1421 * q^83 - 2074 * q^85 + 816 * q^89 - 1995 * q^91 - 1292 * q^95 - 2506 * q^97

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