LMFDB - Embedded newform 4400.2.a.cd.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4400,2,Mod(1,4400)] 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("4400.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4400 = 2^{4} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4400.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,10,0,-4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$35.1341768894$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{19})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 11x^{2} + 16 $ x^4 - 11*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 220)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.31342$ of defining polynomial
Character	$\chi$	$=$	4400.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.31342 q^{3} -3.46410 q^{7} -1.27492 q^{9} -1.00000 q^{11} +6.09095 q^{13} -3.46410 q^{17} -4.00000 q^{19} +4.54983 q^{21} +8.24163 q^{23} +5.61478 q^{27} +6.54983 q^{29} -1.72508 q^{31} +1.31342 q^{33} +8.24163 q^{37} -8.00000 q^{39} -6.54983 q^{41} -3.46410 q^{43} +2.62685 q^{47} +5.00000 q^{49} +4.54983 q^{51} +5.25370 q^{57} -14.2749 q^{59} +6.54983 q^{61} +4.41644 q^{63} +10.8685 q^{67} -10.8248 q^{69} +2.27492 q^{71} -11.3446 q^{73} +3.46410 q^{77} -4.54983 q^{79} -3.54983 q^{81} +1.78959 q^{83} -8.60271 q^{87} +8.27492 q^{89} -21.0997 q^{91} +2.26577 q^{93} -3.94027 q^{97} +1.27492 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 10 q^{9} - 4 q^{11} - 16 q^{19} - 12 q^{21} - 4 q^{29} - 22 q^{31} - 32 q^{39} + 4 q^{41} + 20 q^{49} - 12 q^{51} - 42 q^{59} - 4 q^{61} + 2 q^{69} - 6 q^{71} + 12 q^{79} + 16 q^{81} + 18 q^{89} - 24 q^{91}+ \cdots - 10 q^{99}+O(q^{100}) $ 4 * q + 10 * q^9 - 4 * q^11 - 16 * q^19 - 12 * q^21 - 4 * q^29 - 22 * q^31 - 32 * q^39 + 4 * q^41 + 20 * q^49 - 12 * q^51 - 42 * q^59 - 4 * q^61 + 2 * q^69 - 6 * q^71 + 12 * q^79 + 16 * q^81 + 18 * q^89 - 24 * q^91 - 10 * 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
$3$	−1.31342	−0.758306	−0.379153	−	0.925334i	$-0.623785\pi$
$3$	−1.31342	−0.758306	−0.379153	+	0.925334i	$0.623785\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.46410	−1.30931	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	−3.46410	−1.30931	−0.654654	+	0.755929i	$0.727186\pi$
$8$	0	0
$9$	−1.27492	−0.424972
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	6.09095	1.68933	0.844663	−	0.535299i	$-0.179801\pi$
$13$	6.09095	1.68933	0.844663	+	0.535299i	$0.179801\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	4.54983	0.992855
$22$	0	0
$23$	8.24163	1.71850	0.859249	−	0.511558i	$-0.170931\pi$
$23$	8.24163	1.71850	0.859249	+	0.511558i	$0.170931\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.61478	1.08056
$28$	0	0
$29$	6.54983	1.21627	0.608137	−	0.793832i	$-0.291917\pi$
$29$	6.54983	1.21627	0.608137	+	0.793832i	$0.291917\pi$
$30$	0	0
$31$	−1.72508	−0.309834	−0.154917	−	0.987927i	$-0.549511\pi$
$31$	−1.72508	−0.309834	−0.154917	+	0.987927i	$0.549511\pi$
$32$	0	0
$33$	1.31342	0.228638
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.24163	1.35492	0.677458	−	0.735562i	$-0.263082\pi$
$37$	8.24163	1.35492	0.677458	+	0.735562i	$0.263082\pi$
$38$	0	0
$39$	−8.00000	−1.28103
$40$	0	0
$41$	−6.54983	−1.02291	−0.511456	−	0.859309i	$-0.670894\pi$
$41$	−6.54983	−1.02291	−0.511456	+	0.859309i	$0.670894\pi$
$42$	0	0
$43$	−3.46410	−0.528271	−0.264135	−	0.964486i	$-0.585087\pi$
$43$	−3.46410	−0.528271	−0.264135	+	0.964486i	$0.585087\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.62685	0.383165	0.191583	−	0.981476i	$-0.438638\pi$
$47$	2.62685	0.383165	0.191583	+	0.981476i	$0.438638\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	4.54983	0.637104
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.25370	0.695869
$58$	0	0
$59$	−14.2749	−1.85844	−0.929218	−	0.369532i	$-0.879518\pi$
$59$	−14.2749	−1.85844	−0.929218	+	0.369532i	$0.879518\pi$
$60$	0	0
$61$	6.54983	0.838620	0.419310	−	0.907843i	$-0.362272\pi$
$61$	6.54983	0.838620	0.419310	+	0.907843i	$0.362272\pi$
$62$	0	0
$63$	4.41644	0.556419
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.8685	1.32780	0.663898	−	0.747823i	$-0.268901\pi$
$67$	10.8685	1.32780	0.663898	+	0.747823i	$0.268901\pi$
$68$	0	0
$69$	−10.8248	−1.30315
$70$	0	0
$71$	2.27492	0.269983	0.134992	−	0.990847i	$-0.456899\pi$
$71$	2.27492	0.269983	0.134992	+	0.990847i	$0.456899\pi$
$72$	0	0
$73$	−11.3446	−1.32779	−0.663895	−	0.747826i	$-0.731098\pi$
$73$	−11.3446	−1.32779	−0.663895	+	0.747826i	$0.731098\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.46410	0.394771
$78$	0	0
$79$	−4.54983	−0.511896	−0.255948	−	0.966691i	$-0.582388\pi$
$79$	−4.54983	−0.511896	−0.255948	+	0.966691i	$0.582388\pi$
$80$	0	0
$81$	−3.54983	−0.394426
$82$	0	0
$83$	1.78959	0.196434	0.0982168	−	0.995165i	$-0.468686\pi$
$83$	1.78959	0.196434	0.0982168	+	0.995165i	$0.468686\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−8.60271	−0.922307
$88$	0	0
$89$	8.27492	0.877139	0.438570	−	0.898697i	$-0.355485\pi$
$89$	8.27492	0.877139	0.438570	+	0.898697i	$0.355485\pi$
$90$	0	0
$91$	−21.0997	−2.21185
$92$	0	0
$93$	2.26577	0.234949
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−3.94027	−0.400074	−0.200037	−	0.979788i	$-0.564106\pi$
$97$	−3.94027	−0.400074	−0.200037	+	0.979788i	$0.564106\pi$
$98$	0	0
$99$	1.27492	0.128134
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	−9.55505	−0.941487	−0.470744	−	0.882270i	$-0.656014\pi$
$103$	−9.55505	−0.941487	−0.470744	+	0.882270i	$0.656014\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.46410	−0.334887	−0.167444	−	0.985882i	$-0.553551\pi$
$107$	−3.46410	−0.334887	−0.167444	+	0.985882i	$0.553551\pi$
$108$	0	0
$109$	−19.0997	−1.82942	−0.914708	−	0.404115i	$-0.867580\pi$
$109$	−19.0997	−1.82942	−0.914708	+	0.404115i	$0.867580\pi$
$110$	0	0
$111$	−10.8248	−1.02744
$112$	0	0
$113$	−3.94027	−0.370670	−0.185335	−	0.982675i	$-0.559337\pi$
$113$	−3.94027	−0.370670	−0.185335	+	0.982675i	$0.559337\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−7.76546	−0.717917
$118$	0	0
$119$	12.0000	1.10004
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	8.60271	0.775680
$124$	0	0
$125$	0	0
$126$	0	0
$127$	6.09095	0.540484	0.270242	−	0.962792i	$-0.412896\pi$
$127$	6.09095	0.540484	0.270242	+	0.962792i	$0.412896\pi$
$128$	0	0
$129$	4.54983	0.400591
$130$	0	0
$131$	−8.54983	−0.747003	−0.373501	−	0.927630i	$-0.621843\pi$
$131$	−8.54983	−0.747003	−0.373501	+	0.927630i	$0.621843\pi$
$132$	0	0
$133$	13.8564	1.20150
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.94027	0.336640	0.168320	−	0.985732i	$-0.446166\pi$
$137$	3.94027	0.336640	0.168320	+	0.985732i	$0.446166\pi$
$138$	0	0
$139$	8.54983	0.725187	0.362594	−	0.931947i	$-0.381891\pi$
$139$	8.54983	0.725187	0.362594	+	0.931947i	$0.381891\pi$
$140$	0	0
$141$	−3.45017	−0.290556
$142$	0	0
$143$	−6.09095	−0.509351
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−6.56712	−0.541647
$148$	0	0
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	−3.45017	−0.280770	−0.140385	−	0.990097i	$-0.544834\pi$
$151$	−3.45017	−0.280770	−0.140385	+	0.990097i	$0.544834\pi$
$152$	0	0
$153$	4.41644	0.357048
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−23.0504	−1.83962	−0.919810	−	0.392364i	$-0.871657\pi$
$157$	−23.0504	−1.83962	−0.919810	+	0.392364i	$0.871657\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−28.5498	−2.25004
$162$	0	0
$163$	−0.952341	−0.0745931	−0.0372966	−	0.999304i	$-0.511875\pi$
$163$	−0.952341	−0.0745931	−0.0372966	+	0.999304i	$0.511875\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.09095	0.471332	0.235666	−	0.971834i	$-0.424273\pi$
$167$	6.09095	0.471332	0.235666	+	0.971834i	$0.424273\pi$
$168$	0	0
$169$	24.0997	1.85382
$170$	0	0
$171$	5.09967	0.389981
$172$	0	0
$173$	−13.9715	−1.06223	−0.531117	−	0.847299i	$-0.678227\pi$
$173$	−13.9715	−1.06223	−0.531117	+	0.847299i	$0.678227\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	18.7490	1.40926
$178$	0	0
$179$	10.2749	0.767983	0.383992	−	0.923337i	$-0.374549\pi$
$179$	10.2749	0.767983	0.383992	+	0.923337i	$0.374549\pi$
$180$	0	0
$181$	3.72508	0.276883	0.138442	−	0.990371i	$-0.455791\pi$
$181$	3.72508	0.276883	0.138442	+	0.990371i	$0.455791\pi$
$182$	0	0
$183$	−8.60271	−0.635931
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.46410	0.253320
$188$	0	0
$189$	−19.4502	−1.41479
$190$	0	0
$191$	−6.27492	−0.454037	−0.227019	−	0.973890i	$-0.572898\pi$
$191$	−6.27492	−0.454037	−0.227019	+	0.973890i	$0.572898\pi$
$192$	0	0
$193$	−19.9474	−1.43584	−0.717921	−	0.696125i	$-0.754906\pi$
$193$	−19.9474	−1.43584	−0.717921	+	0.696125i	$0.754906\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.78959	0.127503	0.0637517	−	0.997966i	$-0.479693\pi$
$197$	1.78959	0.127503	0.0637517	+	0.997966i	$0.479693\pi$
$198$	0	0
$199$	5.09967	0.361506	0.180753	−	0.983529i	$-0.442147\pi$
$199$	5.09967	0.361506	0.180753	+	0.983529i	$0.442147\pi$
$200$	0	0
$201$	−14.2749	−1.00688
$202$	0	0
$203$	−22.6893	−1.59248
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−10.5074	−0.730314
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	13.0997	0.901818	0.450909	−	0.892570i	$-0.351100\pi$
$211$	13.0997	0.901818	0.450909	+	0.892570i	$0.351100\pi$
$212$	0	0
$213$	−2.98793	−0.204730
$214$	0	0
$215$	0	0
$216$	0	0
$217$	5.97586	0.405668
$218$	0	0
$219$	14.9003	1.00687
$220$	0	0
$221$	−21.0997	−1.41932
$222$	0	0
$223$	13.4953	0.903714	0.451857	−	0.892090i	$-0.350762\pi$
$223$	13.4953	0.903714	0.451857	+	0.892090i	$0.350762\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−10.3923	−0.689761	−0.344881	−	0.938647i	$-0.612081\pi$
$227$	−10.3923	−0.689761	−0.344881	+	0.938647i	$0.612081\pi$
$228$	0	0
$229$	−20.2749	−1.33980	−0.669902	−	0.742449i	$-0.733664\pi$
$229$	−20.2749	−1.33980	−0.669902	+	0.742449i	$0.733664\pi$
$230$	0	0
$231$	−4.54983	−0.299357
$232$	0	0
$233$	19.9474	1.30679	0.653397	−	0.757015i	$-0.273343\pi$
$233$	19.9474	1.30679	0.653397	+	0.757015i	$0.273343\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	5.97586	0.388174
$238$	0	0
$239$	−4.54983	−0.294304	−0.147152	−	0.989114i	$-0.547011\pi$
$239$	−4.54983	−0.294304	−0.147152	+	0.989114i	$0.547011\pi$
$240$	0	0
$241$	−6.54983	−0.421912	−0.210956	−	0.977496i	$-0.567658\pi$
$241$	−6.54983	−0.421912	−0.210956	+	0.977496i	$0.567658\pi$
$242$	0	0
$243$	−12.1819	−0.781469
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−24.3638	−1.55023
$248$	0	0
$249$	−2.35050	−0.148957
$250$	0	0
$251$	−2.82475	−0.178297	−0.0891484	−	0.996018i	$-0.528415\pi$
$251$	−2.82475	−0.178297	−0.0891484	+	0.996018i	$0.528415\pi$
$252$	0	0
$253$	−8.24163	−0.518147
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−19.1101	−1.19206	−0.596028	−	0.802964i	$-0.703255\pi$
$257$	−19.1101	−1.19206	−0.596028	+	0.802964i	$0.703255\pi$
$258$	0	0
$259$	−28.5498	−1.77400
$260$	0	0
$261$	−8.35050	−0.516883
$262$	0	0
$263$	−3.46410	−0.213606	−0.106803	−	0.994280i	$-0.534061\pi$
$263$	−3.46410	−0.213606	−0.106803	+	0.994280i	$0.534061\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−10.8685	−0.665140
$268$	0	0
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	−25.0997	−1.52470	−0.762348	−	0.647167i	$-0.775954\pi$
$271$	−25.0997	−1.52470	−0.762348	+	0.647167i	$0.775954\pi$
$272$	0	0
$273$	27.7128	1.67726
$274$	0	0
$275$	0	0
$276$	0	0
$277$	13.9715	0.839466	0.419733	−	0.907648i	$-0.362124\pi$
$277$	13.9715	0.839466	0.419733	+	0.907648i	$0.362124\pi$
$278$	0	0
$279$	2.19934	0.131671
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	13.0192	0.773908	0.386954	−	0.922099i	$-0.373527\pi$
$283$	13.0192	0.773908	0.386954	+	0.922099i	$0.373527\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	22.6893	1.33931
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	5.17525	0.303378
$292$	0	0
$293$	−13.0192	−0.760587	−0.380294	−	0.924866i	$-0.624177\pi$
$293$	−13.0192	−0.760587	−0.380294	+	0.924866i	$0.624177\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−5.61478	−0.325803
$298$	0	0
$299$	50.1993	2.90310
$300$	0	0
$301$	12.0000	0.691669
$302$	0	0
$303$	13.1342	0.754542
$304$	0	0
$305$	0	0
$306$	0	0
$307$	26.8756	1.53387	0.766935	−	0.641725i	$-0.221781\pi$
$307$	26.8756	1.53387	0.766935	+	0.641725i	$0.221781\pi$
$308$	0	0
$309$	12.5498	0.713935
$310$	0	0
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	0	0
$313$	8.24163	0.465844	0.232922	−	0.972495i	$-0.425171\pi$
$313$	8.24163	0.465844	0.232922	+	0.972495i	$0.425171\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−14.4477	−0.811462	−0.405731	−	0.913993i	$-0.632983\pi$
$317$	−14.4477	−0.811462	−0.405731	+	0.913993i	$0.632983\pi$
$318$	0	0
$319$	−6.54983	−0.366720
$320$	0	0
$321$	4.54983	0.253947
$322$	0	0
$323$	13.8564	0.770991
$324$	0	0
$325$	0	0
$326$	0	0
$327$	25.0860	1.38726
$328$	0	0
$329$	−9.09967	−0.501681
$330$	0	0
$331$	−21.7251	−1.19412	−0.597059	−	0.802197i	$-0.703664\pi$
$331$	−21.7251	−1.19412	−0.597059	+	0.802197i	$0.703664\pi$
$332$	0	0
$333$	−10.5074	−0.575802
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−13.9715	−0.761076	−0.380538	−	0.924765i	$-0.624261\pi$
$337$	−13.9715	−0.761076	−0.380538	+	0.924765i	$0.624261\pi$
$338$	0	0
$339$	5.17525	0.281081
$340$	0	0
$341$	1.72508	0.0934185
$342$	0	0
$343$	6.92820	0.374088
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−11.3446	−0.609013	−0.304506	−	0.952510i	$-0.598491\pi$
$347$	−11.3446	−0.609013	−0.304506	+	0.952510i	$0.598491\pi$
$348$	0	0
$349$	−31.0997	−1.66473	−0.832364	−	0.554230i	$-0.813013\pi$
$349$	−31.0997	−1.66473	−0.832364	+	0.554230i	$0.813013\pi$
$350$	0	0
$351$	34.1993	1.82543
$352$	0	0
$353$	−27.3517	−1.45579	−0.727893	−	0.685691i	$-0.759500\pi$
$353$	−27.3517	−1.45579	−0.727893	+	0.685691i	$0.759500\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−15.7611	−0.834165
$358$	0	0
$359$	33.0997	1.74693	0.873467	−	0.486884i	$-0.161866\pi$
$359$	33.0997	1.74693	0.873467	+	0.486884i	$0.161866\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	−1.31342	−0.0689369
$364$	0	0
$365$	0	0
$366$	0	0
$367$	5.61478	0.293089	0.146545	−	0.989204i	$-0.453185\pi$
$367$	5.61478	0.293089	0.146545	+	0.989204i	$0.453185\pi$
$368$	0	0
$369$	8.35050	0.434709
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−8.71780	−0.451390	−0.225695	−	0.974198i	$-0.572465\pi$
$373$	−8.71780	−0.451390	−0.225695	+	0.974198i	$0.572465\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	39.8947	2.05468
$378$	0	0
$379$	−35.9244	−1.84531	−0.922657	−	0.385622i	$-0.873987\pi$
$379$	−35.9244	−1.84531	−0.922657	+	0.385622i	$0.873987\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	0	0
$383$	31.6531	1.61740	0.808699	−	0.588223i	$-0.200172\pi$
$383$	31.6531	1.61740	0.808699	+	0.588223i	$0.200172\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.41644	0.224500
$388$	0	0
$389$	20.2749	1.02798	0.513990	−	0.857796i	$-0.328167\pi$
$389$	20.2749	1.02798	0.513990	+	0.857796i	$0.328167\pi$
$390$	0	0
$391$	−28.5498	−1.44383
$392$	0	0
$393$	11.2296	0.566456
$394$	0	0
$395$	0	0
$396$	0	0
$397$	26.0383	1.30683	0.653413	−	0.757002i	$-0.273337\pi$
$397$	26.0383	1.30683	0.653413	+	0.757002i	$0.273337\pi$
$398$	0	0
$399$	−18.1993	−0.911106
$400$	0	0
$401$	24.1993	1.20846	0.604229	−	0.796811i	$-0.293481\pi$
$401$	24.1993	1.20846	0.604229	+	0.796811i	$0.293481\pi$
$402$	0	0
$403$	−10.5074	−0.523411
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−8.24163	−0.408522
$408$	0	0
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	0	0
$411$	−5.17525	−0.255276
$412$	0	0
$413$	49.4498	2.43326
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−11.2296	−0.549914
$418$	0	0
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	−19.0997	−0.930861	−0.465430	−	0.885084i	$-0.654100\pi$
$421$	−19.0997	−0.930861	−0.465430	+	0.885084i	$0.654100\pi$
$422$	0	0
$423$	−3.34901	−0.162835
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−22.6893	−1.09801
$428$	0	0
$429$	8.00000	0.386244
$430$	0	0
$431$	11.4502	0.551535	0.275768	−	0.961224i	$-0.411068\pi$
$431$	11.4502	0.551535	0.275768	+	0.961224i	$0.411068\pi$
$432$	0	0
$433$	15.1698	0.729016	0.364508	−	0.931200i	$-0.381237\pi$
$433$	15.1698	0.729016	0.364508	+	0.931200i	$0.381237\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−32.9665	−1.57700
$438$	0	0
$439$	−25.0997	−1.19794	−0.598971	−	0.800771i	$-0.704424\pi$
$439$	−25.0997	−1.19794	−0.598971	+	0.800771i	$0.704424\pi$
$440$	0	0
$441$	−6.37459	−0.303552
$442$	0	0
$443$	−34.2799	−1.62869	−0.814344	−	0.580382i	$-0.802903\pi$
$443$	−34.2799	−1.62869	−0.814344	+	0.580382i	$0.802903\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.62685	0.124246
$448$	0	0
$449$	−3.72508	−0.175798	−0.0878988	−	0.996129i	$-0.528015\pi$
$449$	−3.72508	−0.175798	−0.0878988	+	0.996129i	$0.528015\pi$
$450$	0	0
$451$	6.54983	0.308420
$452$	0	0
$453$	4.53153	0.212910
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−2.51176	−0.117495	−0.0587476	−	0.998273i	$-0.518711\pi$
$457$	−2.51176	−0.117495	−0.0587476	+	0.998273i	$0.518711\pi$
$458$	0	0
$459$	−19.4502	−0.907856
$460$	0	0
$461$	−14.5498	−0.677653	−0.338827	−	0.940849i	$-0.610030\pi$
$461$	−14.5498	−0.677653	−0.338827	+	0.940849i	$0.610030\pi$
$462$	0	0
$463$	13.4953	0.627181	0.313590	−	0.949558i	$-0.398468\pi$
$463$	13.4953	0.627181	0.313590	+	0.949558i	$0.398468\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−32.6054	−1.50880	−0.754400	−	0.656415i	$-0.772072\pi$
$467$	−32.6054	−1.50880	−0.754400	+	0.656415i	$0.772072\pi$
$468$	0	0
$469$	−37.6495	−1.73849
$470$	0	0
$471$	30.2749	1.39499
$472$	0	0
$473$	3.46410	0.159280
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−41.0997	−1.87789	−0.938946	−	0.344065i	$-0.888196\pi$
$479$	−41.0997	−1.87789	−0.938946	+	0.344065i	$0.888196\pi$
$480$	0	0
$481$	50.1993	2.28889
$482$	0	0
$483$	37.4980	1.70622
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−8.24163	−0.373464	−0.186732	−	0.982411i	$-0.559790\pi$
$487$	−8.24163	−0.373464	−0.186732	+	0.982411i	$0.559790\pi$
$488$	0	0
$489$	1.25083	0.0565644
$490$	0	0
$491$	−21.0997	−0.952215	−0.476107	−	0.879387i	$-0.657953\pi$
$491$	−21.0997	−0.952215	−0.476107	+	0.879387i	$0.657953\pi$
$492$	0	0
$493$	−22.6893	−1.02187
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−7.88054	−0.353491
$498$	0	0
$499$	−41.0997	−1.83987	−0.919937	−	0.392066i	$-0.871760\pi$
$499$	−41.0997	−1.83987	−0.919937	+	0.392066i	$0.871760\pi$
$500$	0	0
$501$	−8.00000	−0.357414
$502$	0	0
$503$	−12.2970	−0.548296	−0.274148	−	0.961688i	$-0.588396\pi$
$503$	−12.2970	−0.548296	−0.274148	+	0.961688i	$0.588396\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−31.6531	−1.40576
$508$	0	0
$509$	37.9244	1.68097	0.840485	−	0.541835i	$-0.182270\pi$
$509$	37.9244	1.68097	0.840485	+	0.541835i	$0.182270\pi$
$510$	0	0
$511$	39.2990	1.73849
$512$	0	0
$513$	−22.4591	−0.991594
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−2.62685	−0.115529
$518$	0	0
$519$	18.3505	0.805497
$520$	0	0
$521$	30.4743	1.33510	0.667551	−	0.744564i	$-0.267343\pi$
$521$	30.4743	1.33510	0.667551	+	0.744564i	$0.267343\pi$
$522$	0	0
$523$	−23.5265	−1.02874	−0.514372	−	0.857567i	$-0.671975\pi$
$523$	−23.5265	−1.02874	−0.514372	+	0.857567i	$0.671975\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.97586	0.260313
$528$	0	0
$529$	44.9244	1.95324
$530$	0	0
$531$	18.1993	0.789784
$532$	0	0
$533$	−39.8947	−1.72803
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−13.4953	−0.582366
$538$	0	0
$539$	−5.00000	−0.215365
$540$	0	0
$541$	9.45017	0.406294	0.203147	−	0.979148i	$-0.434883\pi$
$541$	9.45017	0.406294	0.203147	+	0.979148i	$0.434883\pi$
$542$	0	0
$543$	−4.89261	−0.209962
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−23.5265	−1.00592	−0.502961	−	0.864309i	$-0.667756\pi$
$547$	−23.5265	−1.00592	−0.502961	+	0.864309i	$0.667756\pi$
$548$	0	0
$549$	−8.35050	−0.356391
$550$	0	0
$551$	−26.1993	−1.11613
$552$	0	0
$553$	15.7611	0.670230
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20.8997	0.885549	0.442774	−	0.896633i	$-0.353994\pi$
$557$	20.8997	0.885549	0.442774	+	0.896633i	$0.353994\pi$
$558$	0	0
$559$	−21.0997	−0.892421
$560$	0	0
$561$	−4.54983	−0.192094
$562$	0	0
$563$	−40.7320	−1.71665	−0.858324	−	0.513108i	$-0.828494\pi$
$563$	−40.7320	−1.71665	−0.858324	+	0.513108i	$0.828494\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	12.2970	0.516425
$568$	0	0
$569$	−2.54983	−0.106895	−0.0534473	−	0.998571i	$-0.517021\pi$
$569$	−2.54983	−0.106895	−0.0534473	+	0.998571i	$0.517021\pi$
$570$	0	0
$571$	2.90033	0.121375	0.0606875	−	0.998157i	$-0.480671\pi$
$571$	2.90033	0.121375	0.0606875	+	0.998157i	$0.480671\pi$
$572$	0	0
$573$	8.24163	0.344299
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−20.4235	−0.850243	−0.425121	−	0.905136i	$-0.639769\pi$
$577$	−20.4235	−0.850243	−0.425121	+	0.905136i	$0.639769\pi$
$578$	0	0
$579$	26.1993	1.08881
$580$	0	0
$581$	−6.19934	−0.257192
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.3879	0.758951	0.379476	−	0.925202i	$-0.376104\pi$
$587$	18.3879	0.758951	0.379476	+	0.925202i	$0.376104\pi$
$588$	0	0
$589$	6.90033	0.284323
$590$	0	0
$591$	−2.35050	−0.0966865
$592$	0	0
$593$	−9.43996	−0.387653	−0.193826	−	0.981036i	$-0.562090\pi$
$593$	−9.43996	−0.387653	−0.193826	+	0.981036i	$0.562090\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6.69803	−0.274132
$598$	0	0
$599$	26.1993	1.07048	0.535238	−	0.844701i	$-0.320222\pi$
$599$	26.1993	1.07048	0.535238	+	0.844701i	$0.320222\pi$
$600$	0	0
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	0	0
$603$	−13.8564	−0.564276
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1.78959	−0.0726374	−0.0363187	−	0.999340i	$-0.511563\pi$
$607$	−1.78959	−0.0726374	−0.0363187	+	0.999340i	$0.511563\pi$
$608$	0	0
$609$	29.8007	1.20758
$610$	0	0
$611$	16.0000	0.647291
$612$	0	0
$613$	34.0339	1.37462	0.687309	−	0.726365i	$-0.258792\pi$
$613$	34.0339	1.37462	0.687309	+	0.726365i	$0.258792\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−34.6410	−1.39459	−0.697297	−	0.716782i	$-0.745614\pi$
$617$	−34.6410	−1.39459	−0.697297	+	0.716782i	$0.745614\pi$
$618$	0	0
$619$	7.92442	0.318509	0.159255	−	0.987238i	$-0.449091\pi$
$619$	7.92442	0.318509	0.159255	+	0.987238i	$0.449091\pi$
$620$	0	0
$621$	46.2749	1.85695
$622$	0	0
$623$	−28.6652	−1.14845
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−5.25370	−0.209812
$628$	0	0
$629$	−28.5498	−1.13836
$630$	0	0
$631$	−43.3746	−1.72672	−0.863358	−	0.504593i	$-0.831643\pi$
$631$	−43.3746	−1.72672	−0.863358	+	0.504593i	$0.831643\pi$
$632$	0	0
$633$	−17.2054	−0.683854
$634$	0	0
$635$	0	0
$636$	0	0
$637$	30.4547	1.20666
$638$	0	0
$639$	−2.90033	−0.114735
$640$	0	0
$641$	−5.37459	−0.212283	−0.106142	−	0.994351i	$-0.533850\pi$
$641$	−5.37459	−0.212283	−0.106142	+	0.994351i	$0.533850\pi$
$642$	0	0
$643$	0.591258	0.0233170	0.0116585	−	0.999932i	$-0.496289\pi$
$643$	0.591258	0.0233170	0.0116585	+	0.999932i	$0.496289\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	23.7725	0.934595	0.467298	−	0.884100i	$-0.345228\pi$
$647$	23.7725	0.934595	0.467298	+	0.884100i	$0.345228\pi$
$648$	0	0
$649$	14.2749	0.560340
$650$	0	0
$651$	−7.84884	−0.307620
$652$	0	0
$653$	16.8443	0.659170	0.329585	−	0.944126i	$-0.393091\pi$
$653$	16.8443	0.659170	0.329585	+	0.944126i	$0.393091\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	14.4635	0.564274
$658$	0	0
$659$	−8.54983	−0.333054	−0.166527	−	0.986037i	$-0.553255\pi$
$659$	−8.54983	−0.333054	−0.166527	+	0.986037i	$0.553255\pi$
$660$	0	0
$661$	−11.1752	−0.434667	−0.217333	−	0.976097i	$-0.569736\pi$
$661$	−11.1752	−0.434667	−0.217333	+	0.976097i	$0.569736\pi$
$662$	0	0
$663$	27.7128	1.07628
$664$	0	0
$665$	0	0
$666$	0	0
$667$	53.9813	2.09016
$668$	0	0
$669$	−17.7251	−0.685291
$670$	0	0
$671$	−6.54983	−0.252854
$672$	0	0
$673$	−43.5890	−1.68023	−0.840116	−	0.542407i	$-0.817513\pi$
$673$	−43.5890	−1.68023	−0.840116	+	0.542407i	$0.817513\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	38.3353	1.47335	0.736673	−	0.676250i	$-0.236396\pi$
$677$	38.3353	1.47335	0.736673	+	0.676250i	$0.236396\pi$
$678$	0	0
$679$	13.6495	0.523820
$680$	0	0
$681$	13.6495	0.523050
$682$	0	0
$683$	−2.62685	−0.100514	−0.0502568	−	0.998736i	$-0.516004\pi$
$683$	−2.62685	−0.100514	−0.0502568	+	0.998736i	$0.516004\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	26.6296	1.01598
$688$	0	0
$689$	0	0
$690$	0	0
$691$	7.37459	0.280542	0.140271	−	0.990113i	$-0.455203\pi$
$691$	7.37459	0.280542	0.140271	+	0.990113i	$0.455203\pi$
$692$	0	0
$693$	−4.41644	−0.167767
$694$	0	0
$695$	0	0
$696$	0	0
$697$	22.6893	0.859418
$698$	0	0
$699$	−26.1993	−0.990950
$700$	0	0
$701$	−48.1993	−1.82046	−0.910232	−	0.414099i	$-0.864097\pi$
$701$	−48.1993	−1.82046	−0.910232	+	0.414099i	$0.864097\pi$
$702$	0	0
$703$	−32.9665	−1.24336
$704$	0	0
$705$	0	0
$706$	0	0
$707$	34.6410	1.30281
$708$	0	0
$709$	30.4743	1.14448	0.572242	−	0.820085i	$-0.306074\pi$
$709$	30.4743	1.14448	0.572242	+	0.820085i	$0.306074\pi$
$710$	0	0
$711$	5.80066	0.217542
$712$	0	0
$713$	−14.2175	−0.532449
$714$	0	0
$715$	0	0
$716$	0	0
$717$	5.97586	0.223173
$718$	0	0
$719$	9.72508	0.362684	0.181342	−	0.983420i	$-0.441956\pi$
$719$	9.72508	0.362684	0.181342	+	0.983420i	$0.441956\pi$
$720$	0	0
$721$	33.0997	1.23270
$722$	0	0
$723$	8.60271	0.319938
$724$	0	0
$725$	0	0
$726$	0	0
$727$	25.6772	0.952315	0.476158	−	0.879360i	$-0.342029\pi$
$727$	25.6772	0.952315	0.476158	+	0.879360i	$0.342029\pi$
$728$	0	0
$729$	26.6495	0.987019
$730$	0	0
$731$	12.0000	0.443836
$732$	0	0
$733$	37.3830	1.38077	0.690385	−	0.723442i	$-0.257441\pi$
$733$	37.3830	1.38077	0.690385	+	0.723442i	$0.257441\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−10.8685	−0.400345
$738$	0	0
$739$	33.6495	1.23782	0.618908	−	0.785463i	$-0.287575\pi$
$739$	33.6495	1.23782	0.618908	+	0.785463i	$0.287575\pi$
$740$	0	0
$741$	32.0000	1.17555
$742$	0	0
$743$	40.7320	1.49431	0.747155	−	0.664649i	$-0.231419\pi$
$743$	40.7320	1.49431	0.747155	+	0.664649i	$0.231419\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−2.28159	−0.0834788
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	−6.82475	−0.249039	−0.124519	−	0.992217i	$-0.539739\pi$
$751$	−6.82475	−0.249039	−0.124519	+	0.992217i	$0.539739\pi$
$752$	0	0
$753$	3.71010	0.135203
$754$	0	0
$755$	0	0
$756$	0	0
$757$	34.6410	1.25905	0.629525	−	0.776981i	$-0.283250\pi$
$757$	34.6410	1.25905	0.629525	+	0.776981i	$0.283250\pi$
$758$	0	0
$759$	10.8248	0.392914
$760$	0	0
$761$	22.5498	0.817431	0.408715	−	0.912662i	$-0.365977\pi$
$761$	22.5498	0.817431	0.408715	+	0.912662i	$0.365977\pi$
$762$	0	0
$763$	66.1632	2.39527
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−86.9478	−3.13950
$768$	0	0
$769$	−4.35050	−0.156883	−0.0784415	−	0.996919i	$-0.524994\pi$
$769$	−4.35050	−0.156883	−0.0784415	+	0.996919i	$0.524994\pi$
$770$	0	0
$771$	25.0997	0.903942
$772$	0	0
$773$	−21.0148	−0.755849	−0.377925	−	0.925836i	$-0.623362\pi$
$773$	−21.0148	−0.755849	−0.377925	+	0.925836i	$0.623362\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	37.4980	1.34523
$778$	0	0
$779$	26.1993	0.938689
$780$	0	0
$781$	−2.27492	−0.0814029
$782$	0	0
$783$	36.7759	1.31426
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−0.837253	−0.0298449	−0.0149224	−	0.999889i	$-0.504750\pi$
$787$	−0.837253	−0.0298449	−0.0149224	+	0.999889i	$0.504750\pi$
$788$	0	0
$789$	4.54983	0.161978
$790$	0	0
$791$	13.6495	0.485320
$792$	0	0
$793$	39.8947	1.41670
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−17.7967	−0.630391	−0.315195	−	0.949027i	$-0.602070\pi$
$797$	−17.7967	−0.630391	−0.315195	+	0.949027i	$0.602070\pi$
$798$	0	0
$799$	−9.09967	−0.321923
$800$	0	0
$801$	−10.5498	−0.372760
$802$	0	0
$803$	11.3446	0.400344
$804$	0	0
$805$	0	0
$806$	0	0
$807$	23.6416	0.832225
$808$	0	0
$809$	35.6495	1.25337	0.626685	−	0.779273i	$-0.284411\pi$
$809$	35.6495	1.25337	0.626685	+	0.779273i	$0.284411\pi$
$810$	0	0
$811$	−41.6495	−1.46251	−0.731256	−	0.682103i	$-0.761065\pi$
$811$	−41.6495	−1.46251	−0.731256	+	0.682103i	$0.761065\pi$
$812$	0	0
$813$	32.9665	1.15619
$814$	0	0
$815$	0	0
$816$	0	0
$817$	13.8564	0.484774
$818$	0	0
$819$	26.9003	0.939974
$820$	0	0
$821$	−47.0997	−1.64379	−0.821895	−	0.569639i	$-0.807083\pi$
$821$	−47.0997	−1.64379	−0.821895	+	0.569639i	$0.807083\pi$
$822$	0	0
$823$	−48.1363	−1.67793	−0.838964	−	0.544187i	$-0.816838\pi$
$823$	−48.1363	−1.67793	−0.838964	+	0.544187i	$0.816838\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	14.9238	0.518953	0.259476	−	0.965749i	$-0.416450\pi$
$827$	14.9238	0.518953	0.259476	+	0.965749i	$0.416450\pi$
$828$	0	0
$829$	19.7251	0.685080	0.342540	−	0.939503i	$-0.388713\pi$
$829$	19.7251	0.685080	0.342540	+	0.939503i	$0.388713\pi$
$830$	0	0
$831$	−18.3505	−0.636572
$832$	0	0
$833$	−17.3205	−0.600120
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−9.68596	−0.334796
$838$	0	0
$839$	2.27492	0.0785389	0.0392694	−	0.999229i	$-0.487497\pi$
$839$	2.27492	0.0785389	0.0392694	+	0.999229i	$0.487497\pi$
$840$	0	0
$841$	13.9003	0.479322
$842$	0	0
$843$	−7.88054	−0.271420
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−3.46410	−0.119028
$848$	0	0
$849$	−17.0997	−0.586859
$850$	0	0
$851$	67.9244	2.32842
$852$	0	0
$853$	−12.0668	−0.413160	−0.206580	−	0.978430i	$-0.566233\pi$
$853$	−12.0668	−0.413160	−0.206580	+	0.978430i	$0.566233\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	56.4931	1.92977	0.964883	−	0.262680i	$-0.0846064\pi$
$857$	56.4931	1.92977	0.964883	+	0.262680i	$0.0846064\pi$
$858$	0	0
$859$	−5.72508	−0.195337	−0.0976687	−	0.995219i	$-0.531139\pi$
$859$	−5.72508	−0.195337	−0.0976687	+	0.995219i	$0.531139\pi$
$860$	0	0
$861$	−29.8007	−1.01560
$862$	0	0
$863$	42.5216	1.44745	0.723725	−	0.690088i	$-0.242428\pi$
$863$	42.5216	1.44745	0.723725	+	0.690088i	$0.242428\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	6.56712	0.223031
$868$	0	0
$869$	4.54983	0.154343
$870$	0	0
$871$	66.1993	2.24308
$872$	0	0
$873$	5.02352	0.170020
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−58.1676	−1.96418	−0.982089	−	0.188415i	$-0.939665\pi$
$877$	−58.1676	−1.96418	−0.982089	+	0.188415i	$0.939665\pi$
$878$	0	0
$879$	17.0997	0.576758
$880$	0	0
$881$	24.8248	0.836367	0.418184	−	0.908363i	$-0.362667\pi$
$881$	24.8248	0.836367	0.418184	+	0.908363i	$0.362667\pi$
$882$	0	0
$883$	−0.952341	−0.0320488	−0.0160244	−	0.999872i	$-0.505101\pi$
$883$	−0.952341	−0.0320488	−0.0160244	+	0.999872i	$0.505101\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−7.04329	−0.236491	−0.118245	−	0.992984i	$-0.537727\pi$
$887$	−7.04329	−0.236491	−0.118245	+	0.992984i	$0.537727\pi$
$888$	0	0
$889$	−21.0997	−0.707660
$890$	0	0
$891$	3.54983	0.118924
$892$	0	0
$893$	−10.5074	−0.351616
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−65.9330	−2.20144
$898$	0	0
$899$	−11.2990	−0.376843
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−15.7611	−0.524496
$904$	0	0
$905$	0	0
$906$	0	0
$907$	14.5786	0.484074	0.242037	−	0.970267i	$-0.422185\pi$
$907$	14.5786	0.484074	0.242037	+	0.970267i	$0.422185\pi$
$908$	0	0
$909$	12.7492	0.422863
$910$	0	0
$911$	17.0997	0.566537	0.283269	−	0.959041i	$-0.408581\pi$
$911$	17.0997	0.566537	0.283269	+	0.959041i	$0.408581\pi$
$912$	0	0
$913$	−1.78959	−0.0592269
$914$	0	0
$915$	0	0
$916$	0	0
$917$	29.6175	0.978056
$918$	0	0
$919$	38.7492	1.27822	0.639109	−	0.769116i	$-0.279303\pi$
$919$	38.7492	1.27822	0.639109	+	0.769116i	$0.279303\pi$
$920$	0	0
$921$	−35.2990	−1.16314
$922$	0	0
$923$	13.8564	0.456089
$924$	0	0
$925$	0	0
$926$	0	0
$927$	12.1819	0.400106
$928$	0	0
$929$	27.0997	0.889111	0.444556	−	0.895751i	$-0.353362\pi$
$929$	27.0997	0.889111	0.444556	+	0.895751i	$0.353362\pi$
$930$	0	0
$931$	−20.0000	−0.655474
$932$	0	0
$933$	−5.25370	−0.171998
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−30.2246	−0.987394	−0.493697	−	0.869634i	$-0.664355\pi$
$937$	−30.2246	−0.987394	−0.493697	+	0.869634i	$0.664355\pi$
$938$	0	0
$939$	−10.8248	−0.353252
$940$	0	0
$941$	−0.900331	−0.0293500	−0.0146750	−	0.999892i	$-0.504671\pi$
$941$	−0.900331	−0.0293500	−0.0146750	+	0.999892i	$0.504671\pi$
$942$	0	0
$943$	−53.9813	−1.75787
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−8.24163	−0.267817	−0.133908	−	0.990994i	$-0.542753\pi$
$947$	−8.24163	−0.267817	−0.133908	+	0.990994i	$0.542753\pi$
$948$	0	0
$949$	−69.0997	−2.24307
$950$	0	0
$951$	18.9759	0.615336
$952$	0	0
$953$	6.81312	0.220698	0.110349	−	0.993893i	$-0.464803\pi$
$953$	6.81312	0.220698	0.110349	+	0.993893i	$0.464803\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	8.60271	0.278086
$958$	0	0
$959$	−13.6495	−0.440765
$960$	0	0
$961$	−28.0241	−0.904003
$962$	0	0
$963$	4.41644	0.142318
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−40.7320	−1.30985	−0.654926	−	0.755693i	$-0.727300\pi$
$967$	−40.7320	−1.30985	−0.654926	+	0.755693i	$0.727300\pi$
$968$	0	0
$969$	−18.1993	−0.584647
$970$	0	0
$971$	−29.7251	−0.953923	−0.476962	−	0.878924i	$-0.658262\pi$
$971$	−29.7251	−0.953923	−0.476962	+	0.878924i	$0.658262\pi$
$972$	0	0
$973$	−29.6175	−0.949493
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.31342	−0.0420202	−0.0210101	−	0.999779i	$-0.506688\pi$
$977$	−1.31342	−0.0420202	−0.0210101	+	0.999779i	$0.506688\pi$
$978$	0	0
$979$	−8.27492	−0.264468
$980$	0	0
$981$	24.3505	0.777452
$982$	0	0
$983$	−30.9309	−0.986543	−0.493272	−	0.869875i	$-0.664199\pi$
$983$	−30.9309	−0.986543	−0.493272	+	0.869875i	$0.664199\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	11.9517	0.380428
$988$	0	0
$989$	−28.5498	−0.907832
$990$	0	0
$991$	−34.1993	−1.08638	−0.543189	−	0.839611i	$-0.682783\pi$
$991$	−34.1993	−1.08638	−0.543189	+	0.839611i	$0.682783\pi$
$992$	0	0
$993$	28.5342	0.905507
$994$	0	0
$995$	0	0
$996$	0	0
$997$	5.13861	0.162741	0.0813707	−	0.996684i	$-0.474070\pi$
$997$	5.13861	0.162741	0.0813707	+	0.996684i	$0.474070\pi$
$998$	0	0
$999$	46.2749	1.46407

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.a.cd.1.2		4
4.3	odd	2		1100.2.a.j.1.3		4
5.2	odd	4		880.2.b.i.529.3		4
5.3	odd	4		880.2.b.i.529.2		4
5.4	even	2	inner	4400.2.a.cd.1.3		4
12.11	even	2		9900.2.a.cb.1.4		4
20.3	even	4		220.2.b.b.89.3	yes	4
20.7	even	4		220.2.b.b.89.2	✓	4
20.19	odd	2		1100.2.a.j.1.2		4
60.23	odd	4		1980.2.c.g.1189.2		4
60.47	odd	4		1980.2.c.g.1189.1		4
60.59	even	2		9900.2.a.cb.1.1		4
220.43	odd	4		2420.2.b.e.969.3		4
220.87	odd	4		2420.2.b.e.969.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.2.b.b.89.2	✓	4	20.7	even	4
220.2.b.b.89.3	yes	4	20.3	even	4
880.2.b.i.529.2		4	5.3	odd	4
880.2.b.i.529.3		4	5.2	odd	4
1100.2.a.j.1.2		4	20.19	odd	2
1100.2.a.j.1.3		4	4.3	odd	2
1980.2.c.g.1189.1		4	60.47	odd	4
1980.2.c.g.1189.2		4	60.23	odd	4
2420.2.b.e.969.2		4	220.87	odd	4
2420.2.b.e.969.3		4	220.43	odd	4
4400.2.a.cd.1.2		4	1.1	even	1	trivial
4400.2.a.cd.1.3		4	5.4	even	2	inner
9900.2.a.cb.1.1		4	60.59	even	2
9900.2.a.cb.1.4		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.a (trivial)

\(f(q)\)	\(=\)	\(q-1.31342 q^{3} -3.46410 q^{7} -1.27492 q^{9} -1.00000 q^{11} +6.09095 q^{13} -3.46410 q^{17} -4.00000 q^{19} +4.54983 q^{21} +8.24163 q^{23} +5.61478 q^{27} +6.54983 q^{29} -1.72508 q^{31} +1.31342 q^{33} +8.24163 q^{37} -8.00000 q^{39} -6.54983 q^{41} -3.46410 q^{43} +2.62685 q^{47} +5.00000 q^{49} +4.54983 q^{51} +5.25370 q^{57} -14.2749 q^{59} +6.54983 q^{61} +4.41644 q^{63} +10.8685 q^{67} -10.8248 q^{69} +2.27492 q^{71} -11.3446 q^{73} +3.46410 q^{77} -4.54983 q^{79} -3.54983 q^{81} +1.78959 q^{83} -8.60271 q^{87} +8.27492 q^{89} -21.0997 q^{91} +2.26577 q^{93} -3.94027 q^{97} +1.27492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{9} - 4 q^{11} - 16 q^{19} - 12 q^{21} - 4 q^{29} - 22 q^{31} - 32 q^{39} + 4 q^{41} + 20 q^{49} - 12 q^{51} - 42 q^{59} - 4 q^{61} + 2 q^{69} - 6 q^{71} + 12 q^{79} + 16 q^{81} + 18 q^{89} - 24 q^{91}+ \cdots - 10 q^{99}+O(q^{100}) \) 4 * q + 10 * q^9 - 4 * q^11 - 16 * q^19 - 12 * q^21 - 4 * q^29 - 22 * q^31 - 32 * q^39 + 4 * q^41 + 20 * q^49 - 12 * q^51 - 42 * q^59 - 4 * q^61 + 2 * q^69 - 6 * q^71 + 12 * q^79 + 16 * q^81 + 18 * q^89 - 24 * q^91 - 10 * q^99

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