LMFDB - Embedded newform 800.2.a.k.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [800,2,Mod(1,800)] 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("800.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	800.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,4,0,0,0,-8] 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:	$6.38803216170$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	800.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607 q^{3} +4.47214 q^{7} +2.00000 q^{9} -2.23607 q^{11} -4.00000 q^{13} +7.00000 q^{17} +6.70820 q^{19} +10.0000 q^{21} -4.47214 q^{23} -2.23607 q^{27} -4.47214 q^{31} -5.00000 q^{33} -2.00000 q^{37} -8.94427 q^{39} +5.00000 q^{41} -8.94427 q^{47} +13.0000 q^{49} +15.6525 q^{51} -6.00000 q^{53} +15.0000 q^{57} -8.94427 q^{59} +10.0000 q^{61} +8.94427 q^{63} -2.23607 q^{67} -10.0000 q^{69} +8.94427 q^{71} +9.00000 q^{73} -10.0000 q^{77} +4.47214 q^{79} -11.0000 q^{81} -11.1803 q^{83} -5.00000 q^{89} -17.8885 q^{91} -10.0000 q^{93} -2.00000 q^{97} -4.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{9} - 8 q^{13} + 14 q^{17} + 20 q^{21} - 10 q^{33} - 4 q^{37} + 10 q^{41} + 26 q^{49} - 12 q^{53} + 30 q^{57} + 20 q^{61} - 20 q^{69} + 18 q^{73} - 20 q^{77} - 22 q^{81} - 10 q^{89} - 20 q^{93}+ \cdots - 4 q^{97}+O(q^{100}) $ 2 * q + 4 * q^9 - 8 * q^13 + 14 * q^17 + 20 * q^21 - 10 * q^33 - 4 * q^37 + 10 * q^41 + 26 * q^49 - 12 * q^53 + 30 * q^57 + 20 * q^61 - 20 * q^69 + 18 * q^73 - 20 * q^77 - 22 * q^81 - 10 * q^89 - 20 * q^93 - 4 * q^97

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
$3$	2.23607	1.29099	0.645497	−	0.763763i	$-0.276650\pi$
$3$	2.23607	1.29099	0.645497	+	0.763763i	$0.276650\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.47214	1.69031	0.845154	−	0.534522i	$-0.179509\pi$
$7$	4.47214	1.69031	0.845154	+	0.534522i	$0.179509\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	−2.23607	−0.674200	−0.337100	−	0.941469i	$-0.609446\pi$
$11$	−2.23607	−0.674200	−0.337100	+	0.941469i	$0.609446\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.00000	1.69775	0.848875	−	0.528594i	$-0.177281\pi$
$17$	7.00000	1.69775	0.848875	+	0.528594i	$0.177281\pi$
$18$	0	0
$19$	6.70820	1.53897	0.769484	−	0.638666i	$-0.220514\pi$
$19$	6.70820	1.53897	0.769484	+	0.638666i	$0.220514\pi$
$20$	0	0
$21$	10.0000	2.18218
$22$	0	0
$23$	−4.47214	−0.932505	−0.466252	−	0.884652i	$-0.654396\pi$
$23$	−4.47214	−0.932505	−0.466252	+	0.884652i	$0.654396\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2.23607	−0.430331
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−4.47214	−0.803219	−0.401610	−	0.915811i	$-0.631549\pi$
$31$	−4.47214	−0.803219	−0.401610	+	0.915811i	$0.631549\pi$
$32$	0	0
$33$	−5.00000	−0.870388
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	−8.94427	−1.43223
$40$	0	0
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.94427	−1.30466	−0.652328	−	0.757937i	$-0.726208\pi$
$47$	−8.94427	−1.30466	−0.652328	+	0.757937i	$0.726208\pi$
$48$	0	0
$49$	13.0000	1.85714
$50$	0	0
$51$	15.6525	2.19179
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	15.0000	1.98680
$58$	0	0
$59$	−8.94427	−1.16445	−0.582223	−	0.813029i	$-0.697817\pi$
$59$	−8.94427	−1.16445	−0.582223	+	0.813029i	$0.697817\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	8.94427	1.12687
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−2.23607	−0.273179	−0.136590	−	0.990628i	$-0.543614\pi$
$67$	−2.23607	−0.273179	−0.136590	+	0.990628i	$0.543614\pi$
$68$	0	0
$69$	−10.0000	−1.20386
$70$	0	0
$71$	8.94427	1.06149	0.530745	−	0.847532i	$-0.321912\pi$
$71$	8.94427	1.06149	0.530745	+	0.847532i	$0.321912\pi$
$72$	0	0
$73$	9.00000	1.05337	0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000	1.05337	0.526685	+	0.850060i	$0.323435\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−10.0000	−1.13961
$78$	0	0
$79$	4.47214	0.503155	0.251577	−	0.967837i	$-0.419051\pi$
$79$	4.47214	0.503155	0.251577	+	0.967837i	$0.419051\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	−11.1803	−1.22720	−0.613601	−	0.789616i	$-0.710280\pi$
$83$	−11.1803	−1.22720	−0.613601	+	0.789616i	$0.710280\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−5.00000	−0.529999	−0.264999	−	0.964249i	$-0.585372\pi$
$89$	−5.00000	−0.529999	−0.264999	+	0.964249i	$0.585372\pi$
$90$	0	0
$91$	−17.8885	−1.87523
$92$	0	0
$93$	−10.0000	−1.03695
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	−4.47214	−0.449467
$100$	0	0
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	−8.94427	−0.881305	−0.440653	−	0.897678i	$-0.645253\pi$
$103$	−8.94427	−0.881305	−0.440653	+	0.897678i	$0.645253\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.23607	0.216169	0.108084	−	0.994142i	$-0.465528\pi$
$107$	2.23607	0.216169	0.108084	+	0.994142i	$0.465528\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	−4.47214	−0.424476
$112$	0	0
$113$	1.00000	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−8.00000	−0.739600
$118$	0	0
$119$	31.3050	2.86972
$120$	0	0
$121$	−6.00000	−0.545455
$122$	0	0
$123$	11.1803	1.00810
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.47214	0.396838	0.198419	−	0.980117i	$-0.436419\pi$
$127$	4.47214	0.396838	0.198419	+	0.980117i	$0.436419\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−17.8885	−1.56293	−0.781465	−	0.623949i	$-0.785527\pi$
$131$	−17.8885	−1.56293	−0.781465	+	0.623949i	$0.785527\pi$
$132$	0	0
$133$	30.0000	2.60133
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.00000	0.256307	0.128154	−	0.991754i	$-0.459095\pi$
$137$	3.00000	0.256307	0.128154	+	0.991754i	$0.459095\pi$
$138$	0	0
$139$	−2.23607	−0.189661	−0.0948304	−	0.995493i	$-0.530231\pi$
$139$	−2.23607	−0.189661	−0.0948304	+	0.995493i	$0.530231\pi$
$140$	0	0
$141$	−20.0000	−1.68430
$142$	0	0
$143$	8.94427	0.747958
$144$	0	0
$145$	0	0
$146$	0	0
$147$	29.0689	2.39756
$148$	0	0
$149$	−16.0000	−1.31077	−0.655386	−	0.755295i	$-0.727494\pi$
$149$	−16.0000	−1.31077	−0.655386	+	0.755295i	$0.727494\pi$
$150$	0	0
$151$	13.4164	1.09181	0.545906	−	0.837846i	$-0.316186\pi$
$151$	13.4164	1.09181	0.545906	+	0.837846i	$0.316186\pi$
$152$	0	0
$153$	14.0000	1.13183
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	0	0
$159$	−13.4164	−1.06399
$160$	0	0
$161$	−20.0000	−1.57622
$162$	0	0
$163$	−2.23607	−0.175142	−0.0875712	−	0.996158i	$-0.527911\pi$
$163$	−2.23607	−0.175142	−0.0875712	+	0.996158i	$0.527911\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.94427	0.692129	0.346064	−	0.938211i	$-0.387518\pi$
$167$	8.94427	0.692129	0.346064	+	0.938211i	$0.387518\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	13.4164	1.02598
$172$	0	0
$173$	−16.0000	−1.21646	−0.608229	−	0.793762i	$-0.708120\pi$
$173$	−16.0000	−1.21646	−0.608229	+	0.793762i	$0.708120\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−20.0000	−1.50329
$178$	0	0
$179$	−2.23607	−0.167132	−0.0835658	−	0.996502i	$-0.526631\pi$
$179$	−2.23607	−0.167132	−0.0835658	+	0.996502i	$0.526631\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	22.3607	1.65295
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−15.6525	−1.14462
$188$	0	0
$189$	−10.0000	−0.727393
$190$	0	0
$191$	−13.4164	−0.970777	−0.485389	−	0.874299i	$-0.661322\pi$
$191$	−13.4164	−0.970777	−0.485389	+	0.874299i	$0.661322\pi$
$192$	0	0
$193$	−9.00000	−0.647834	−0.323917	−	0.946085i	$-0.605000\pi$
$193$	−9.00000	−0.647834	−0.323917	+	0.946085i	$0.605000\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	−8.94427	−0.634043	−0.317021	−	0.948418i	$-0.602683\pi$
$199$	−8.94427	−0.634043	−0.317021	+	0.948418i	$0.602683\pi$
$200$	0	0
$201$	−5.00000	−0.352673
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−8.94427	−0.621670
$208$	0	0
$209$	−15.0000	−1.03757
$210$	0	0
$211$	20.1246	1.38544	0.692718	−	0.721209i	$-0.256413\pi$
$211$	20.1246	1.38544	0.692718	+	0.721209i	$0.256413\pi$
$212$	0	0
$213$	20.0000	1.37038
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−20.0000	−1.35769
$218$	0	0
$219$	20.1246	1.35990
$220$	0	0
$221$	−28.0000	−1.88348
$222$	0	0
$223$	−26.8328	−1.79686	−0.898429	−	0.439119i	$-0.855291\pi$
$223$	−26.8328	−1.79686	−0.898429	+	0.439119i	$0.855291\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−17.8885	−1.18730	−0.593652	−	0.804722i	$-0.702314\pi$
$227$	−17.8885	−1.18730	−0.593652	+	0.804722i	$0.702314\pi$
$228$	0	0
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	0	0
$231$	−22.3607	−1.47122
$232$	0	0
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	10.0000	0.649570
$238$	0	0
$239$	−17.8885	−1.15711	−0.578557	−	0.815642i	$-0.696384\pi$
$239$	−17.8885	−1.15711	−0.578557	+	0.815642i	$0.696384\pi$
$240$	0	0
$241$	5.00000	0.322078	0.161039	−	0.986948i	$-0.448515\pi$
$241$	5.00000	0.322078	0.161039	+	0.986948i	$0.448515\pi$
$242$	0	0
$243$	−17.8885	−1.14755
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−26.8328	−1.70733
$248$	0	0
$249$	−25.0000	−1.58431
$250$	0	0
$251$	11.1803	0.705697	0.352848	−	0.935681i	$-0.385213\pi$
$251$	11.1803	0.705697	0.352848	+	0.935681i	$0.385213\pi$
$252$	0	0
$253$	10.0000	0.628695
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	−8.94427	−0.555770
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−22.3607	−1.37882	−0.689409	−	0.724372i	$-0.742130\pi$
$263$	−22.3607	−1.37882	−0.689409	+	0.724372i	$0.742130\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−11.1803	−0.684226
$268$	0	0
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	13.4164	0.814989	0.407494	−	0.913208i	$-0.366403\pi$
$271$	13.4164	0.814989	0.407494	+	0.913208i	$0.366403\pi$
$272$	0	0
$273$	−40.0000	−2.42091
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	0	0
$279$	−8.94427	−0.535480
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	6.70820	0.398761	0.199381	−	0.979922i	$-0.436107\pi$
$283$	6.70820	0.398761	0.199381	+	0.979922i	$0.436107\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	22.3607	1.31991
$288$	0	0
$289$	32.0000	1.88235
$290$	0	0
$291$	−4.47214	−0.262161
$292$	0	0
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.00000	0.290129
$298$	0	0
$299$	17.8885	1.03452
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−4.47214	−0.256917
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−15.6525	−0.893334	−0.446667	−	0.894700i	$-0.647389\pi$
$307$	−15.6525	−0.893334	−0.446667	+	0.894700i	$0.647389\pi$
$308$	0	0
$309$	−20.0000	−1.13776
$310$	0	0
$311$	22.3607	1.26796	0.633979	−	0.773350i	$-0.281421\pi$
$311$	22.3607	1.26796	0.633979	+	0.773350i	$0.281421\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	5.00000	0.279073
$322$	0	0
$323$	46.9574	2.61278
$324$	0	0
$325$	0	0
$326$	0	0
$327$	13.4164	0.741929
$328$	0	0
$329$	−40.0000	−2.20527
$330$	0	0
$331$	−11.1803	−0.614527	−0.307264	−	0.951624i	$-0.599413\pi$
$331$	−11.1803	−0.614527	−0.307264	+	0.951624i	$0.599413\pi$
$332$	0	0
$333$	−4.00000	−0.219199
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−27.0000	−1.47078	−0.735392	−	0.677642i	$-0.763002\pi$
$337$	−27.0000	−1.47078	−0.735392	+	0.677642i	$0.763002\pi$
$338$	0	0
$339$	2.23607	0.121447
$340$	0	0
$341$	10.0000	0.541530
$342$	0	0
$343$	26.8328	1.44884
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.1246	1.08035	0.540173	−	0.841554i	$-0.318359\pi$
$347$	20.1246	1.08035	0.540173	+	0.841554i	$0.318359\pi$
$348$	0	0
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	0	0
$351$	8.94427	0.477410
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	70.0000	3.70479
$358$	0	0
$359$	−31.3050	−1.65221	−0.826106	−	0.563515i	$-0.809449\pi$
$359$	−31.3050	−1.65221	−0.826106	+	0.563515i	$0.809449\pi$
$360$	0	0
$361$	26.0000	1.36842
$362$	0	0
$363$	−13.4164	−0.704179
$364$	0	0
$365$	0	0
$366$	0	0
$367$	26.8328	1.40066	0.700331	−	0.713818i	$-0.253036\pi$
$367$	26.8328	1.40066	0.700331	+	0.713818i	$0.253036\pi$
$368$	0	0
$369$	10.0000	0.520579
$370$	0	0
$371$	−26.8328	−1.39309
$372$	0	0
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−15.6525	−0.804014	−0.402007	−	0.915637i	$-0.631687\pi$
$379$	−15.6525	−0.804014	−0.402007	+	0.915637i	$0.631687\pi$
$380$	0	0
$381$	10.0000	0.512316
$382$	0	0
$383$	−4.47214	−0.228515	−0.114258	−	0.993451i	$-0.536449\pi$
$383$	−4.47214	−0.228515	−0.114258	+	0.993451i	$0.536449\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	−31.3050	−1.58316
$392$	0	0
$393$	−40.0000	−2.01773
$394$	0	0
$395$	0	0
$396$	0	0
$397$	28.0000	1.40528	0.702640	−	0.711546i	$-0.252005\pi$
$397$	28.0000	1.40528	0.702640	+	0.711546i	$0.252005\pi$
$398$	0	0
$399$	67.0820	3.35830
$400$	0	0
$401$	17.0000	0.848939	0.424470	−	0.905442i	$-0.360461\pi$
$401$	17.0000	0.848939	0.424470	+	0.905442i	$0.360461\pi$
$402$	0	0
$403$	17.8885	0.891092
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.47214	0.221676
$408$	0	0
$409$	29.0000	1.43396	0.716979	−	0.697095i	$-0.245524\pi$
$409$	29.0000	1.43396	0.716979	+	0.697095i	$0.245524\pi$
$410$	0	0
$411$	6.70820	0.330891
$412$	0	0
$413$	−40.0000	−1.96827
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−5.00000	−0.244851
$418$	0	0
$419$	15.6525	0.764673	0.382337	−	0.924023i	$-0.375119\pi$
$419$	15.6525	0.764673	0.382337	+	0.924023i	$0.375119\pi$
$420$	0	0
$421$	20.0000	0.974740	0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000	0.974740	0.487370	+	0.873195i	$0.337956\pi$
$422$	0	0
$423$	−17.8885	−0.869771
$424$	0	0
$425$	0	0
$426$	0	0
$427$	44.7214	2.16422
$428$	0	0
$429$	20.0000	0.965609
$430$	0	0
$431$	4.47214	0.215415	0.107708	−	0.994183i	$-0.465649\pi$
$431$	4.47214	0.215415	0.107708	+	0.994183i	$0.465649\pi$
$432$	0	0
$433$	−1.00000	−0.0480569	−0.0240285	−	0.999711i	$-0.507649\pi$
$433$	−1.00000	−0.0480569	−0.0240285	+	0.999711i	$0.507649\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−30.0000	−1.43509
$438$	0	0
$439$	35.7771	1.70755	0.853774	−	0.520644i	$-0.174308\pi$
$439$	35.7771	1.70755	0.853774	+	0.520644i	$0.174308\pi$
$440$	0	0
$441$	26.0000	1.23810
$442$	0	0
$443$	−38.0132	−1.80606	−0.903030	−	0.429578i	$-0.858662\pi$
$443$	−38.0132	−1.80606	−0.903030	+	0.429578i	$0.858662\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−35.7771	−1.69220
$448$	0	0
$449$	−19.0000	−0.896665	−0.448333	−	0.893867i	$-0.647982\pi$
$449$	−19.0000	−0.896665	−0.448333	+	0.893867i	$0.647982\pi$
$450$	0	0
$451$	−11.1803	−0.526462
$452$	0	0
$453$	30.0000	1.40952
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7.00000	0.327446	0.163723	−	0.986506i	$-0.447650\pi$
$457$	7.00000	0.327446	0.163723	+	0.986506i	$0.447650\pi$
$458$	0	0
$459$	−15.6525	−0.730595
$460$	0	0
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	0	0
$463$	26.8328	1.24703	0.623513	−	0.781813i	$-0.285705\pi$
$463$	26.8328	1.24703	0.623513	+	0.781813i	$0.285705\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−17.8885	−0.827783	−0.413892	−	0.910326i	$-0.635831\pi$
$467$	−17.8885	−0.827783	−0.413892	+	0.910326i	$0.635831\pi$
$468$	0	0
$469$	−10.0000	−0.461757
$470$	0	0
$471$	−40.2492	−1.85459
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−12.0000	−0.549442
$478$	0	0
$479$	−13.4164	−0.613011	−0.306506	−	0.951869i	$-0.599160\pi$
$479$	−13.4164	−0.613011	−0.306506	+	0.951869i	$0.599160\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	0	0
$483$	−44.7214	−2.03489
$484$	0	0
$485$	0	0
$486$	0	0
$487$	4.47214	0.202652	0.101326	−	0.994853i	$-0.467692\pi$
$487$	4.47214	0.202652	0.101326	+	0.994853i	$0.467692\pi$
$488$	0	0
$489$	−5.00000	−0.226108
$490$	0	0
$491$	35.7771	1.61460	0.807299	−	0.590143i	$-0.200929\pi$
$491$	35.7771	1.61460	0.807299	+	0.590143i	$0.200929\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	40.0000	1.79425
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	0	0
$501$	20.0000	0.893534
$502$	0	0
$503$	26.8328	1.19642	0.598208	−	0.801341i	$-0.295880\pi$
$503$	26.8328	1.19642	0.598208	+	0.801341i	$0.295880\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6.70820	0.297922
$508$	0	0
$509$	10.0000	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$510$	0	0
$511$	40.2492	1.78052
$512$	0	0
$513$	−15.0000	−0.662266
$514$	0	0
$515$	0	0
$516$	0	0
$517$	20.0000	0.879599
$518$	0	0
$519$	−35.7771	−1.57044
$520$	0	0
$521$	13.0000	0.569540	0.284770	−	0.958596i	$-0.408083\pi$
$521$	13.0000	0.569540	0.284770	+	0.958596i	$0.408083\pi$
$522$	0	0
$523$	−6.70820	−0.293329	−0.146665	−	0.989186i	$-0.546854\pi$
$523$	−6.70820	−0.293329	−0.146665	+	0.989186i	$0.546854\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−31.3050	−1.36367
$528$	0	0
$529$	−3.00000	−0.130435
$530$	0	0
$531$	−17.8885	−0.776297
$532$	0	0
$533$	−20.0000	−0.866296
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−5.00000	−0.215766
$538$	0	0
$539$	−29.0689	−1.25209
$540$	0	0
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	0	0
$543$	4.47214	0.191918
$544$	0	0
$545$	0	0
$546$	0	0
$547$	20.1246	0.860466	0.430233	−	0.902718i	$-0.358431\pi$
$547$	20.1246	0.860466	0.430233	+	0.902718i	$0.358431\pi$
$548$	0	0
$549$	20.0000	0.853579
$550$	0	0
$551$	0	0
$552$	0	0
$553$	20.0000	0.850487
$554$	0	0
$555$	0	0
$556$	0	0
$557$	28.0000	1.18640	0.593199	−	0.805056i	$-0.297865\pi$
$557$	28.0000	1.18640	0.593199	+	0.805056i	$0.297865\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−35.0000	−1.47770
$562$	0	0
$563$	44.7214	1.88478	0.942390	−	0.334515i	$-0.108573\pi$
$563$	44.7214	1.88478	0.942390	+	0.334515i	$0.108573\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−49.1935	−2.06593
$568$	0	0
$569$	1.00000	0.0419222	0.0209611	−	0.999780i	$-0.493327\pi$
$569$	1.00000	0.0419222	0.0209611	+	0.999780i	$0.493327\pi$
$570$	0	0
$571$	17.8885	0.748612	0.374306	−	0.927305i	$-0.377881\pi$
$571$	17.8885	0.748612	0.374306	+	0.927305i	$0.377881\pi$
$572$	0	0
$573$	−30.0000	−1.25327
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−43.0000	−1.79011	−0.895057	−	0.445952i	$-0.852865\pi$
$577$	−43.0000	−1.79011	−0.895057	+	0.445952i	$0.852865\pi$
$578$	0	0
$579$	−20.1246	−0.836350
$580$	0	0
$581$	−50.0000	−2.07435
$582$	0	0
$583$	13.4164	0.555651
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20.1246	−0.830632	−0.415316	−	0.909677i	$-0.636329\pi$
$587$	−20.1246	−0.830632	−0.415316	+	0.909677i	$0.636329\pi$
$588$	0	0
$589$	−30.0000	−1.23613
$590$	0	0
$591$	40.2492	1.65563
$592$	0	0
$593$	9.00000	0.369586	0.184793	−	0.982777i	$-0.440839\pi$
$593$	9.00000	0.369586	0.184793	+	0.982777i	$0.440839\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−20.0000	−0.818546
$598$	0	0
$599$	4.47214	0.182727	0.0913633	−	0.995818i	$-0.470878\pi$
$599$	4.47214	0.182727	0.0913633	+	0.995818i	$0.470878\pi$
$600$	0	0
$601$	−25.0000	−1.01977	−0.509886	−	0.860242i	$-0.670312\pi$
$601$	−25.0000	−1.01977	−0.509886	+	0.860242i	$0.670312\pi$
$602$	0	0
$603$	−4.47214	−0.182119
$604$	0	0
$605$	0	0
$606$	0	0
$607$	17.8885	0.726074	0.363037	−	0.931775i	$-0.381740\pi$
$607$	17.8885	0.726074	0.363037	+	0.931775i	$0.381740\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	35.7771	1.44739
$612$	0	0
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	0	0
$619$	−17.8885	−0.719001	−0.359501	−	0.933145i	$-0.617053\pi$
$619$	−17.8885	−0.719001	−0.359501	+	0.933145i	$0.617053\pi$
$620$	0	0
$621$	10.0000	0.401286
$622$	0	0
$623$	−22.3607	−0.895862
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−33.5410	−1.33950
$628$	0	0
$629$	−14.0000	−0.558217
$630$	0	0
$631$	13.4164	0.534099	0.267049	−	0.963683i	$-0.413951\pi$
$631$	13.4164	0.534099	0.267049	+	0.963683i	$0.413951\pi$
$632$	0	0
$633$	45.0000	1.78859
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−52.0000	−2.06032
$638$	0	0
$639$	17.8885	0.707660
$640$	0	0
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	0	0
$643$	−17.8885	−0.705455	−0.352728	−	0.935726i	$-0.614746\pi$
$643$	−17.8885	−0.705455	−0.352728	+	0.935726i	$0.614746\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8.94427	0.351636	0.175818	−	0.984423i	$-0.443743\pi$
$647$	8.94427	0.351636	0.175818	+	0.984423i	$0.443743\pi$
$648$	0	0
$649$	20.0000	0.785069
$650$	0	0
$651$	−44.7214	−1.75277
$652$	0	0
$653$	34.0000	1.33052	0.665261	−	0.746611i	$-0.268320\pi$
$653$	34.0000	1.33052	0.665261	+	0.746611i	$0.268320\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	18.0000	0.702247
$658$	0	0
$659$	−6.70820	−0.261315	−0.130657	−	0.991428i	$-0.541709\pi$
$659$	−6.70820	−0.261315	−0.130657	+	0.991428i	$0.541709\pi$
$660$	0	0
$661$	40.0000	1.55582	0.777910	−	0.628376i	$-0.216280\pi$
$661$	40.0000	1.55582	0.777910	+	0.628376i	$0.216280\pi$
$662$	0	0
$663$	−62.6099	−2.43157
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−60.0000	−2.31973
$670$	0	0
$671$	−22.3607	−0.863224
$672$	0	0
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−32.0000	−1.22986	−0.614930	−	0.788582i	$-0.710816\pi$
$677$	−32.0000	−1.22986	−0.614930	+	0.788582i	$0.710816\pi$
$678$	0	0
$679$	−8.94427	−0.343250
$680$	0	0
$681$	−40.0000	−1.53280
$682$	0	0
$683$	11.1803	0.427804	0.213902	−	0.976855i	$-0.431383\pi$
$683$	11.1803	0.427804	0.213902	+	0.976855i	$0.431383\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	44.7214	1.70623
$688$	0	0
$689$	24.0000	0.914327
$690$	0	0
$691$	−38.0132	−1.44609	−0.723044	−	0.690802i	$-0.757258\pi$
$691$	−38.0132	−1.44609	−0.723044	+	0.690802i	$0.757258\pi$
$692$	0	0
$693$	−20.0000	−0.759737
$694$	0	0
$695$	0	0
$696$	0	0
$697$	35.0000	1.32572
$698$	0	0
$699$	58.1378	2.19897
$700$	0	0
$701$	−20.0000	−0.755390	−0.377695	−	0.925930i	$-0.623283\pi$
$701$	−20.0000	−0.755390	−0.377695	+	0.925930i	$0.623283\pi$
$702$	0	0
$703$	−13.4164	−0.506009
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−8.94427	−0.336384
$708$	0	0
$709$	20.0000	0.751116	0.375558	−	0.926799i	$-0.377451\pi$
$709$	20.0000	0.751116	0.375558	+	0.926799i	$0.377451\pi$
$710$	0	0
$711$	8.94427	0.335436
$712$	0	0
$713$	20.0000	0.749006
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−40.0000	−1.49383
$718$	0	0
$719$	−4.47214	−0.166783	−0.0833913	−	0.996517i	$-0.526575\pi$
$719$	−4.47214	−0.166783	−0.0833913	+	0.996517i	$0.526575\pi$
$720$	0	0
$721$	−40.0000	−1.48968
$722$	0	0
$723$	11.1803	0.415801
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	−7.00000	−0.259259
$730$	0	0
$731$	0	0
$732$	0	0
$733$	44.0000	1.62518	0.812589	−	0.582838i	$-0.198058\pi$
$733$	44.0000	1.62518	0.812589	+	0.582838i	$0.198058\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.00000	0.184177
$738$	0	0
$739$	35.7771	1.31608	0.658041	−	0.752982i	$-0.271385\pi$
$739$	35.7771	1.31608	0.658041	+	0.752982i	$0.271385\pi$
$740$	0	0
$741$	−60.0000	−2.20416
$742$	0	0
$743$	31.3050	1.14847	0.574234	−	0.818691i	$-0.305300\pi$
$743$	31.3050	1.14847	0.574234	+	0.818691i	$0.305300\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−22.3607	−0.818134
$748$	0	0
$749$	10.0000	0.365392
$750$	0	0
$751$	−53.6656	−1.95829	−0.979143	−	0.203171i	$-0.934875\pi$
$751$	−53.6656	−1.95829	−0.979143	+	0.203171i	$0.934875\pi$
$752$	0	0
$753$	25.0000	0.911051
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−8.00000	−0.290765	−0.145382	−	0.989376i	$-0.546441\pi$
$757$	−8.00000	−0.290765	−0.145382	+	0.989376i	$0.546441\pi$
$758$	0	0
$759$	22.3607	0.811641
$760$	0	0
$761$	−43.0000	−1.55875	−0.779374	−	0.626559i	$-0.784463\pi$
$761$	−43.0000	−1.55875	−0.779374	+	0.626559i	$0.784463\pi$
$762$	0	0
$763$	26.8328	0.971413
$764$	0	0
$765$	0	0
$766$	0	0
$767$	35.7771	1.29184
$768$	0	0
$769$	−15.0000	−0.540914	−0.270457	−	0.962732i	$-0.587175\pi$
$769$	−15.0000	−0.540914	−0.270457	+	0.962732i	$0.587175\pi$
$770$	0	0
$771$	4.47214	0.161060
$772$	0	0
$773$	−24.0000	−0.863220	−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000	−0.863220	−0.431610	+	0.902060i	$0.642054\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−20.0000	−0.717496
$778$	0	0
$779$	33.5410	1.20173
$780$	0	0
$781$	−20.0000	−0.715656
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	0	0
$789$	−50.0000	−1.78005
$790$	0	0
$791$	4.47214	0.159011
$792$	0	0
$793$	−40.0000	−1.42044
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	−62.6099	−2.21498
$800$	0	0
$801$	−10.0000	−0.353333
$802$	0	0
$803$	−20.1246	−0.710182
$804$	0	0
$805$	0	0
$806$	0	0
$807$	53.6656	1.88912
$808$	0	0
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	17.8885	0.628152	0.314076	−	0.949398i	$-0.398305\pi$
$811$	17.8885	0.628152	0.314076	+	0.949398i	$0.398305\pi$
$812$	0	0
$813$	30.0000	1.05215
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−35.7771	−1.25015
$820$	0	0
$821$	50.0000	1.74501	0.872506	−	0.488603i	$-0.162493\pi$
$821$	50.0000	1.74501	0.872506	+	0.488603i	$0.162493\pi$
$822$	0	0
$823$	26.8328	0.935333	0.467667	−	0.883905i	$-0.345095\pi$
$823$	26.8328	0.935333	0.467667	+	0.883905i	$0.345095\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	51.4296	1.78838	0.894191	−	0.447687i	$-0.147752\pi$
$827$	51.4296	1.78838	0.894191	+	0.447687i	$0.147752\pi$
$828$	0	0
$829$	4.00000	0.138926	0.0694629	−	0.997585i	$-0.477871\pi$
$829$	4.00000	0.138926	0.0694629	+	0.997585i	$0.477871\pi$
$830$	0	0
$831$	−17.8885	−0.620547
$832$	0	0
$833$	91.0000	3.15296
$834$	0	0
$835$	0	0
$836$	0	0
$837$	10.0000	0.345651
$838$	0	0
$839$	17.8885	0.617581	0.308791	−	0.951130i	$-0.400076\pi$
$839$	17.8885	0.617581	0.308791	+	0.951130i	$0.400076\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	67.0820	2.31043
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−26.8328	−0.921986
$848$	0	0
$849$	15.0000	0.514799
$850$	0	0
$851$	8.94427	0.306606
$852$	0	0
$853$	−34.0000	−1.16414	−0.582069	−	0.813139i	$-0.697757\pi$
$853$	−34.0000	−1.16414	−0.582069	+	0.813139i	$0.697757\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.00000	0.102478	0.0512390	−	0.998686i	$-0.483683\pi$
$857$	3.00000	0.102478	0.0512390	+	0.998686i	$0.483683\pi$
$858$	0	0
$859$	33.5410	1.14440	0.572202	−	0.820112i	$-0.306089\pi$
$859$	33.5410	1.14440	0.572202	+	0.820112i	$0.306089\pi$
$860$	0	0
$861$	50.0000	1.70400
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	71.5542	2.43011
$868$	0	0
$869$	−10.0000	−0.339227
$870$	0	0
$871$	8.94427	0.303065
$872$	0	0
$873$	−4.00000	−0.135379
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−28.0000	−0.945493	−0.472746	−	0.881199i	$-0.656737\pi$
$877$	−28.0000	−0.945493	−0.472746	+	0.881199i	$0.656737\pi$
$878$	0	0
$879$	−31.3050	−1.05589
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	2.23607	0.0752497	0.0376248	−	0.999292i	$-0.488021\pi$
$883$	2.23607	0.0752497	0.0376248	+	0.999292i	$0.488021\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−8.94427	−0.300319	−0.150160	−	0.988662i	$-0.547979\pi$
$887$	−8.94427	−0.300319	−0.150160	+	0.988662i	$0.547979\pi$
$888$	0	0
$889$	20.0000	0.670778
$890$	0	0
$891$	24.5967	0.824022
$892$	0	0
$893$	−60.0000	−2.00782
$894$	0	0
$895$	0	0
$896$	0	0
$897$	40.0000	1.33556
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−42.0000	−1.39922
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−53.6656	−1.78194	−0.890969	−	0.454064i	$-0.849974\pi$
$907$	−53.6656	−1.78194	−0.890969	+	0.454064i	$0.849974\pi$
$908$	0	0
$909$	−4.00000	−0.132672
$910$	0	0
$911$	−44.7214	−1.48168	−0.740842	−	0.671679i	$-0.765573\pi$
$911$	−44.7214	−1.48168	−0.740842	+	0.671679i	$0.765573\pi$
$912$	0	0
$913$	25.0000	0.827379
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−80.0000	−2.64183
$918$	0	0
$919$	−13.4164	−0.442566	−0.221283	−	0.975210i	$-0.571025\pi$
$919$	−13.4164	−0.442566	−0.221283	+	0.975210i	$0.571025\pi$
$920$	0	0
$921$	−35.0000	−1.15329
$922$	0	0
$923$	−35.7771	−1.17762
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−17.8885	−0.587537
$928$	0	0
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	0	0
$931$	87.2067	2.85808
$932$	0	0
$933$	50.0000	1.63693
$934$	0	0
$935$	0	0
$936$	0	0
$937$	17.0000	0.555366	0.277683	−	0.960673i	$-0.410434\pi$
$937$	17.0000	0.555366	0.277683	+	0.960673i	$0.410434\pi$
$938$	0	0
$939$	13.4164	0.437828
$940$	0	0
$941$	−28.0000	−0.912774	−0.456387	−	0.889781i	$-0.650857\pi$
$941$	−28.0000	−0.912774	−0.456387	+	0.889781i	$0.650857\pi$
$942$	0	0
$943$	−22.3607	−0.728164
$944$	0	0
$945$	0	0
$946$	0	0
$947$	26.8328	0.871949	0.435975	−	0.899959i	$-0.356404\pi$
$947$	26.8328	0.871949	0.435975	+	0.899959i	$0.356404\pi$
$948$	0	0
$949$	−36.0000	−1.16861
$950$	0	0
$951$	26.8328	0.870114
$952$	0	0
$953$	−9.00000	−0.291539	−0.145769	−	0.989319i	$-0.546566\pi$
$953$	−9.00000	−0.291539	−0.145769	+	0.989319i	$0.546566\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	13.4164	0.433238
$960$	0	0
$961$	−11.0000	−0.354839
$962$	0	0
$963$	4.47214	0.144113
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−53.6656	−1.72577	−0.862885	−	0.505400i	$-0.831345\pi$
$967$	−53.6656	−1.72577	−0.862885	+	0.505400i	$0.831345\pi$
$968$	0	0
$969$	105.000	3.37309
$970$	0	0
$971$	−55.9017	−1.79397	−0.896985	−	0.442060i	$-0.854248\pi$
$971$	−55.9017	−1.79397	−0.896985	+	0.442060i	$0.854248\pi$
$972$	0	0
$973$	−10.0000	−0.320585
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.00000	0.0959785	0.0479893	−	0.998848i	$-0.484719\pi$
$977$	3.00000	0.0959785	0.0479893	+	0.998848i	$0.484719\pi$
$978$	0	0
$979$	11.1803	0.357325
$980$	0	0
$981$	12.0000	0.383131
$982$	0	0
$983$	−4.47214	−0.142639	−0.0713195	−	0.997454i	$-0.522721\pi$
$983$	−4.47214	−0.142639	−0.0713195	+	0.997454i	$0.522721\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−89.4427	−2.84699
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−13.4164	−0.426186	−0.213093	−	0.977032i	$-0.568354\pi$
$991$	−13.4164	−0.426186	−0.213093	+	0.977032i	$0.568354\pi$
$992$	0	0
$993$	−25.0000	−0.793351
$994$	0	0
$995$	0	0
$996$	0	0
$997$	52.0000	1.64686	0.823428	−	0.567420i	$-0.192059\pi$
$997$	52.0000	1.64686	0.823428	+	0.567420i	$0.192059\pi$
$998$	0	0
$999$	4.47214	0.141492

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.2.a.k.1.2	yes	2
3.2	odd	2		7200.2.a.cf.1.2		2
4.3	odd	2	inner	800.2.a.k.1.1	✓	2
5.2	odd	4		800.2.c.g.449.1		4
5.3	odd	4		800.2.c.g.449.3		4
5.4	even	2		800.2.a.l.1.1	yes	2
8.3	odd	2		1600.2.a.bb.1.2		2
8.5	even	2		1600.2.a.bb.1.1		2
12.11	even	2		7200.2.a.cf.1.1		2
15.2	even	4		7200.2.f.bg.6049.4		4
15.8	even	4		7200.2.f.bg.6049.2		4
15.14	odd	2		7200.2.a.cn.1.1		2
20.3	even	4		800.2.c.g.449.2		4
20.7	even	4		800.2.c.g.449.4		4
20.19	odd	2		800.2.a.l.1.2	yes	2
40.3	even	4		1600.2.c.o.449.3		4
40.13	odd	4		1600.2.c.o.449.2		4
40.19	odd	2		1600.2.a.ba.1.1		2
40.27	even	4		1600.2.c.o.449.1		4
40.29	even	2		1600.2.a.ba.1.2		2
40.37	odd	4		1600.2.c.o.449.4		4
60.23	odd	4		7200.2.f.bg.6049.3		4
60.47	odd	4		7200.2.f.bg.6049.1		4
60.59	even	2		7200.2.a.cn.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
800.2.a.k.1.1	✓	2	4.3	odd	2	inner
800.2.a.k.1.2	yes	2	1.1	even	1	trivial
800.2.a.l.1.1	yes	2	5.4	even	2
800.2.a.l.1.2	yes	2	20.19	odd	2
800.2.c.g.449.1		4	5.2	odd	4
800.2.c.g.449.2		4	20.3	even	4
800.2.c.g.449.3		4	5.3	odd	4
800.2.c.g.449.4		4	20.7	even	4
1600.2.a.ba.1.1		2	40.19	odd	2
1600.2.a.ba.1.2		2	40.29	even	2
1600.2.a.bb.1.1		2	8.5	even	2
1600.2.a.bb.1.2		2	8.3	odd	2
1600.2.c.o.449.1		4	40.27	even	4
1600.2.c.o.449.2		4	40.13	odd	4
1600.2.c.o.449.3		4	40.3	even	4
1600.2.c.o.449.4		4	40.37	odd	4
7200.2.a.cf.1.1		2	12.11	even	2
7200.2.a.cf.1.2		2	3.2	odd	2
7200.2.a.cn.1.1		2	15.14	odd	2
7200.2.a.cn.1.2		2	60.59	even	2
7200.2.f.bg.6049.1		4	60.47	odd	4
7200.2.f.bg.6049.2		4	15.8	even	4
7200.2.f.bg.6049.3		4	60.23	odd	4
7200.2.f.bg.6049.4		4	15.2	even	4

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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	800.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23607 q^{3} +4.47214 q^{7} +2.00000 q^{9} -2.23607 q^{11} -4.00000 q^{13} +7.00000 q^{17} +6.70820 q^{19} +10.0000 q^{21} -4.47214 q^{23} -2.23607 q^{27} -4.47214 q^{31} -5.00000 q^{33} -2.00000 q^{37} -8.94427 q^{39} +5.00000 q^{41} -8.94427 q^{47} +13.0000 q^{49} +15.6525 q^{51} -6.00000 q^{53} +15.0000 q^{57} -8.94427 q^{59} +10.0000 q^{61} +8.94427 q^{63} -2.23607 q^{67} -10.0000 q^{69} +8.94427 q^{71} +9.00000 q^{73} -10.0000 q^{77} +4.47214 q^{79} -11.0000 q^{81} -11.1803 q^{83} -5.00000 q^{89} -17.8885 q^{91} -10.0000 q^{93} -2.00000 q^{97} -4.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{9} - 8 q^{13} + 14 q^{17} + 20 q^{21} - 10 q^{33} - 4 q^{37} + 10 q^{41} + 26 q^{49} - 12 q^{53} + 30 q^{57} + 20 q^{61} - 20 q^{69} + 18 q^{73} - 20 q^{77} - 22 q^{81} - 10 q^{89} - 20 q^{93}+ \cdots - 4 q^{97}+O(q^{100}) \) 2 * q + 4 * q^9 - 8 * q^13 + 14 * q^17 + 20 * q^21 - 10 * q^33 - 4 * q^37 + 10 * q^41 + 26 * q^49 - 12 * q^53 + 30 * q^57 + 20 * q^61 - 20 * q^69 + 18 * q^73 - 20 * q^77 - 22 * q^81 - 10 * q^89 - 20 * q^93 - 4 * q^97

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