LMFDB - Embedded newform 800.4.f.d.49.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,4,Mod(49,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

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

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

chi := DirichletCharacter("800.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	800.f (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$47.2015280046$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 200)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.3
Character	$\chi$	$=$	800.49
Dual form			800.4.f.d.49.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.69300 q^{3} -15.6248i q^{7} +32.1823 q^{9} +O(q^{10})$	$q-7.69300 q^{3} -15.6248i q^{7} +32.1823 q^{9} -46.7640i q^{11} -50.4964 q^{13} -111.080i q^{17} +84.9682i q^{19} +120.201i q^{21} -53.2555i q^{23} -39.8675 q^{27} -229.789i q^{29} +338.615 q^{31} +359.755i q^{33} +71.1508 q^{37} +388.469 q^{39} -3.06105 q^{41} -115.970 q^{43} -574.015i q^{47} +98.8671 q^{49} +854.536i q^{51} +39.0500 q^{53} -653.660i q^{57} -109.940i q^{59} +180.738i q^{61} -502.841i q^{63} -755.845 q^{67} +409.695i q^{69} -127.821 q^{71} +347.937i q^{73} -730.676 q^{77} +456.703 q^{79} -562.221 q^{81} -499.423 q^{83} +1767.77i q^{87} -1301.32 q^{89} +788.995i q^{91} -2604.97 q^{93} +778.230i q^{97} -1504.97i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 216 q^{9}+O(q^{10}) $ 24 * q + 216 * q^9	$ 24 q + 216 q^{9} + 528 q^{31} - 600 q^{39} - 40 q^{41} - 456 q^{49} - 1256 q^{71} + 2232 q^{79} + 2256 q^{81} + 848 q^{89}+O(q^{100}) $ 24 * q + 216 * q^9 + 528 * q^31 - 600 * q^39 - 40 * q^41 - 456 * q^49 - 1256 * q^71 + 2232 * q^79 + 2256 * q^81 + 848 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−7.69300	−1.48052	−0.740260	−	0.672321i	$-0.765297\pi$
$3$	−7.69300	−1.48052	−0.740260	+	0.672321i	$0.765297\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 15.6248i	− 0.843657i	−0.906676	−	0.421829i	$-0.861388\pi$
$7$	− 15.6248i	− 0.843657i	0.906676	−	0.421829i	$-0.138612\pi$
$8$	0	0
$9$	32.1823	1.19194
$10$	0	0
$11$	− 46.7640i	− 1.28181i	−0.767622	−	0.640903i	$-0.778560\pi$
$11$	− 46.7640i	− 1.28181i	0.767622	−	0.640903i	$-0.221440\pi$
$12$	0	0
$13$	−50.4964	−1.07732	−0.538661	−	0.842522i	$-0.681070\pi$
$13$	−50.4964	−1.07732	−0.538661	+	0.842522i	$0.681070\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 111.080i	− 1.58475i	−0.610034	−	0.792375i	$-0.708844\pi$
$17$	− 111.080i	− 1.58475i	0.610034	−	0.792375i	$-0.291156\pi$
$18$	0	0
$19$	84.9682i	1.02595i	0.858404	+	0.512975i	$0.171457\pi$
$19$	84.9682i	1.02595i	−0.858404	+	0.512975i	$0.828543\pi$
$20$	0	0
$21$	120.201i	1.24905i
$22$	0	0
$23$	− 53.2555i	− 0.482806i	−0.970425	−	0.241403i	$-0.922392\pi$
$23$	− 53.2555i	− 0.482806i	0.970425	−	0.241403i	$-0.0776076\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−39.8675	−0.284167
$28$	0	0
$29$	− 229.789i	− 1.47141i	−0.677304	−	0.735703i	$-0.736852\pi$
$29$	− 229.789i	− 1.47141i	0.677304	−	0.735703i	$-0.263148\pi$
$30$	0	0
$31$	338.615	1.96184	0.980920	−	0.194410i	$-0.0622791\pi$
$31$	338.615	1.96184	0.980920	+	0.194410i	$0.0622791\pi$
$32$	0	0
$33$	359.755i	1.89774i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	71.1508	0.316138	0.158069	−	0.987428i	$-0.449473\pi$
$37$	71.1508	0.316138	0.158069	+	0.987428i	$0.449473\pi$
$38$	0	0
$39$	388.469	1.59500
$40$	0	0
$41$	−3.06105	−0.0116599	−0.00582995	−	0.999983i	$-0.501856\pi$
$41$	−3.06105	−0.0116599	−0.00582995	+	0.999983i	$0.501856\pi$
$42$	0	0
$43$	−115.970	−0.411285	−0.205643	−	0.978627i	$-0.565928\pi$
$43$	−115.970	−0.411285	−0.205643	+	0.978627i	$0.565928\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 574.015i	− 1.78146i	−0.454532	−	0.890730i	$-0.650194\pi$
$47$	− 574.015i	− 1.78146i	0.454532	−	0.890730i	$-0.349806\pi$
$48$	0	0
$49$	98.8671	0.288242
$50$	0	0
$51$	854.536i	2.34625i
$52$	0	0
$53$	39.0500	0.101206	0.0506031	−	0.998719i	$-0.483886\pi$
$53$	39.0500	0.101206	0.0506031	+	0.998719i	$0.483886\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 653.660i	− 1.51894i
$58$	0	0
$59$	− 109.940i	− 0.242593i	−0.992616	−	0.121297i	$-0.961295\pi$
$59$	− 109.940i	− 0.242593i	0.992616	−	0.121297i	$-0.0387053\pi$
$60$	0	0
$61$	180.738i	0.379363i	0.981846	+	0.189682i	$0.0607456\pi$
$61$	180.738i	0.379363i	−0.981846	+	0.189682i	$0.939254\pi$
$62$	0	0
$63$	− 502.841i	− 1.00559i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−755.845	−1.37823	−0.689113	−	0.724654i	$-0.742000\pi$
$67$	−755.845	−1.37823	−0.689113	+	0.724654i	$0.742000\pi$
$68$	0	0
$69$	409.695i	0.714804i
$70$	0	0
$71$	−127.821	−0.213656	−0.106828	−	0.994278i	$-0.534069\pi$
$71$	−127.821	−0.213656	−0.106828	+	0.994278i	$0.534069\pi$
$72$	0	0
$73$	347.937i	0.557849i	0.960313	+	0.278924i	$0.0899779\pi$
$73$	347.937i	0.557849i	−0.960313	+	0.278924i	$0.910022\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−730.676	−1.08141
$78$	0	0
$79$	456.703	0.650419	0.325210	−	0.945642i	$-0.394565\pi$
$79$	456.703	0.650419	0.325210	+	0.945642i	$0.394565\pi$
$80$	0	0
$81$	−562.221	−0.771223
$82$	0	0
$83$	−499.423	−0.660467	−0.330233	−	0.943899i	$-0.607127\pi$
$83$	−499.423	−0.660467	−0.330233	+	0.943899i	$0.607127\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1767.77i	2.17844i
$88$	0	0
$89$	−1301.32	−1.54989	−0.774944	−	0.632030i	$-0.782222\pi$
$89$	−1301.32	−1.54989	−0.774944	+	0.632030i	$0.782222\pi$
$90$	0	0
$91$	788.995i	0.908891i
$92$	0	0
$93$	−2604.97	−2.90454
$94$	0	0
$95$	0	0
$96$	0	0
$97$	778.230i	0.814611i	0.913292	+	0.407306i	$0.133532\pi$
$97$	778.230i	0.814611i	−0.913292	+	0.407306i	$0.866468\pi$
$98$	0	0
$99$	− 1504.97i	− 1.52783i
$100$	0	0
$101$	1147.12i	1.13012i	0.825048	+	0.565062i	$0.191148\pi$
$101$	1147.12i	1.13012i	−0.825048	+	0.565062i	$0.808852\pi$
$102$	0	0
$103$	− 852.432i	− 0.815462i	−0.913102	−	0.407731i	$-0.866320\pi$
$103$	− 852.432i	− 0.815462i	0.913102	−	0.407731i	$-0.133680\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	42.6157	0.0385030	0.0192515	−	0.999815i	$-0.493872\pi$
$107$	42.6157	0.0385030	0.0192515	+	0.999815i	$0.493872\pi$
$108$	0	0
$109$	− 1490.77i	− 1.31000i	−0.755628	−	0.655001i	$-0.772668\pi$
$109$	− 1490.77i	− 1.31000i	0.755628	−	0.655001i	$-0.227332\pi$
$110$	0	0
$111$	−547.363	−0.468049
$112$	0	0
$113$	− 846.817i	− 0.704972i	−0.935817	−	0.352486i	$-0.885336\pi$
$113$	− 846.817i	− 0.704972i	0.935817	−	0.352486i	$-0.114664\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1625.09	−1.28410
$118$	0	0
$119$	−1735.59	−1.33699
$120$	0	0
$121$	−855.870	−0.643028
$122$	0	0
$123$	23.5487	0.0172627
$124$	0	0
$125$	0	0
$126$	0	0
$127$	65.6584i	0.0458759i	0.999737	+	0.0229380i	$0.00730202\pi$
$127$	65.6584i	0.0458759i	−0.999737	+	0.0229380i	$0.992698\pi$
$128$	0	0
$129$	892.158	0.608916
$130$	0	0
$131$	646.301i	0.431050i	0.976498	+	0.215525i	$0.0691463\pi$
$131$	646.301i	0.431050i	−0.976498	+	0.215525i	$0.930854\pi$
$132$	0	0
$133$	1327.61	0.865550
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1794.01i	1.11878i	0.828906	+	0.559388i	$0.188964\pi$
$137$	1794.01i	1.11878i	−0.828906	+	0.559388i	$0.811036\pi$
$138$	0	0
$139$	− 1.37996i	0 0.000842065i	−1.00000		0.000421032i	$-0.999866\pi$
$139$	− 1.37996i	0 0.000842065i	1.00000		0.000421032i	$-0.000134019\pi$
$140$	0	0
$141$	4415.90i	2.63749i
$142$	0	0
$143$	2361.41i	1.38092i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−760.585	−0.426748
$148$	0	0
$149$	− 743.959i	− 0.409044i	−0.978862	−	0.204522i	$-0.934436\pi$
$149$	− 743.959i	− 0.409044i	0.978862	−	0.204522i	$-0.0655640\pi$
$150$	0	0
$151$	−2411.32	−1.29954	−0.649770	−	0.760131i	$-0.725135\pi$
$151$	−2411.32	−1.29954	−0.649770	+	0.760131i	$0.725135\pi$
$152$	0	0
$153$	− 3574.80i	− 1.88892i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−1497.53	−0.761249	−0.380624	−	0.924730i	$-0.624291\pi$
$157$	−1497.53	−0.761249	−0.380624	+	0.924730i	$0.624291\pi$
$158$	0	0
$159$	−300.412	−0.149838
$160$	0	0
$161$	−832.104	−0.407323
$162$	0	0
$163$	−2895.86	−1.39154	−0.695772	−	0.718263i	$-0.744937\pi$
$163$	−2895.86	−1.39154	−0.695772	+	0.718263i	$0.744937\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 1923.39i	− 0.891237i	−0.895223	−	0.445619i	$-0.852984\pi$
$167$	− 1923.39i	− 0.891237i	0.895223	−	0.445619i	$-0.147016\pi$
$168$	0	0
$169$	352.891	0.160624
$170$	0	0
$171$	2734.47i	1.22287i
$172$	0	0
$173$	351.016	0.154262	0.0771309	−	0.997021i	$-0.475424\pi$
$173$	351.016	0.154262	0.0771309	+	0.997021i	$0.475424\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	845.772i	0.359164i
$178$	0	0
$179$	2001.39i	0.835705i	0.908515	+	0.417853i	$0.137217\pi$
$179$	2001.39i	0.835705i	−0.908515	+	0.417853i	$0.862783\pi$
$180$	0	0
$181$	− 4293.36i	− 1.76311i	−0.472082	−	0.881555i	$-0.656497\pi$
$181$	− 4293.36i	− 1.76311i	0.472082	−	0.881555i	$-0.343503\pi$
$182$	0	0
$183$	− 1390.42i	− 0.561655i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−5194.52	−2.03134
$188$	0	0
$189$	622.920i	0.239739i
$190$	0	0
$191$	−442.947	−0.167804	−0.0839019	−	0.996474i	$-0.526738\pi$
$191$	−442.947	−0.167804	−0.0839019	+	0.996474i	$0.526738\pi$
$192$	0	0
$193$	2844.03i	1.06071i	0.847775	+	0.530356i	$0.177942\pi$
$193$	2844.03i	1.06071i	−0.847775	+	0.530356i	$0.822058\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4472.05	1.61736	0.808681	−	0.588247i	$-0.200182\pi$
$197$	4472.05	1.61736	0.808681	+	0.588247i	$0.200182\pi$
$198$	0	0
$199$	−1382.43	−0.492452	−0.246226	−	0.969212i	$-0.579191\pi$
$199$	−1382.43	−0.492452	−0.246226	+	0.969212i	$0.579191\pi$
$200$	0	0
$201$	5814.72	2.04049
$202$	0	0
$203$	−3590.40	−1.24136
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1713.88i	− 0.575474i
$208$	0	0
$209$	3973.45	1.31507
$210$	0	0
$211$	− 350.255i	− 0.114278i	−0.998366	−	0.0571388i	$-0.981802\pi$
$211$	− 350.255i	− 0.114278i	0.998366	−	0.0571388i	$-0.0181978\pi$
$212$	0	0
$213$	983.326	0.316321
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 5290.78i	− 1.65512i
$218$	0	0
$219$	− 2676.68i	− 0.825906i
$220$	0	0
$221$	5609.12i	1.70729i
$222$	0	0
$223$	4547.35i	1.36553i	0.730638	+	0.682765i	$0.239223\pi$
$223$	4547.35i	1.36553i	−0.730638	+	0.682765i	$0.760777\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3747.56	1.09575	0.547873	−	0.836561i	$-0.315438\pi$
$227$	3747.56	1.09575	0.547873	+	0.836561i	$0.315438\pi$
$228$	0	0
$229$	2061.75i	0.594954i	0.954729	+	0.297477i	$0.0961451\pi$
$229$	2061.75i	0.594954i	−0.954729	+	0.297477i	$0.903855\pi$
$230$	0	0
$231$	5621.09	1.60104
$232$	0	0
$233$	809.747i	0.227675i	0.993499	+	0.113837i	$0.0363143\pi$
$233$	809.747i	0.227675i	−0.993499	+	0.113837i	$0.963686\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−3513.42	−0.962958
$238$	0	0
$239$	1159.14	0.313717	0.156858	−	0.987621i	$-0.449863\pi$
$239$	1159.14	0.313717	0.156858	+	0.987621i	$0.449863\pi$
$240$	0	0
$241$	3732.40	0.997615	0.498807	−	0.866713i	$-0.333772\pi$
$241$	3732.40	0.997615	0.498807	+	0.866713i	$0.333772\pi$
$242$	0	0
$243$	5401.59	1.42598
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 4290.59i	− 1.10528i
$248$	0	0
$249$	3842.06	0.977834
$250$	0	0
$251$	− 3609.63i	− 0.907719i	−0.891073	−	0.453860i	$-0.850047\pi$
$251$	− 3609.63i	− 0.907719i	0.891073	−	0.453860i	$-0.149953\pi$
$252$	0	0
$253$	−2490.44	−0.618864
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 2378.19i	− 0.577226i	−0.957446	−	0.288613i	$-0.906806\pi$
$257$	− 2378.19i	− 0.577226i	0.957446	−	0.288613i	$-0.0931941\pi$
$258$	0	0
$259$	− 1111.71i	− 0.266712i
$260$	0	0
$261$	− 7395.14i	− 1.75382i
$262$	0	0
$263$	3933.72i	0.922296i	0.887323	+	0.461148i	$0.152562\pi$
$263$	3933.72i	0.922296i	−0.887323	+	0.461148i	$0.847438\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	10011.1	2.29464
$268$	0	0
$269$	− 3629.38i	− 0.822628i	−0.911494	−	0.411314i	$-0.865070\pi$
$269$	− 3629.38i	− 0.822628i	0.911494	−	0.411314i	$-0.134930\pi$
$270$	0	0
$271$	−4019.80	−0.901054	−0.450527	−	0.892763i	$-0.648764\pi$
$271$	−4019.80	−0.901054	−0.450527	+	0.892763i	$0.648764\pi$
$272$	0	0
$273$	− 6069.74i	− 1.34563i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5566.24	1.20737	0.603687	−	0.797221i	$-0.293698\pi$
$277$	5566.24	1.20737	0.603687	+	0.797221i	$0.293698\pi$
$278$	0	0
$279$	10897.4	2.33839
$280$	0	0
$281$	−8293.84	−1.76074	−0.880372	−	0.474284i	$-0.842707\pi$
$281$	−8293.84	−1.76074	−0.880372	+	0.474284i	$0.842707\pi$
$282$	0	0
$283$	2632.57	0.552968	0.276484	−	0.961018i	$-0.410831\pi$
$283$	2632.57	0.552968	0.276484	+	0.961018i	$0.410831\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	47.8282i	0.00983696i
$288$	0	0
$289$	−7425.67	−1.51143
$290$	0	0
$291$	− 5986.93i	− 1.20605i
$292$	0	0
$293$	569.395	0.113530	0.0567652	−	0.998388i	$-0.481921\pi$
$293$	569.395	0.113530	0.0567652	+	0.998388i	$0.481921\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1864.36i	0.364247i
$298$	0	0
$299$	2689.21i	0.520138i
$300$	0	0
$301$	1812.00i	0.346984i
$302$	0	0
$303$	− 8824.79i	− 1.67317i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−527.093	−0.0979895	−0.0489947	−	0.998799i	$-0.515602\pi$
$307$	−527.093	−0.0979895	−0.0489947	+	0.998799i	$0.515602\pi$
$308$	0	0
$309$	6557.76i	1.20731i
$310$	0	0
$311$	6798.16	1.23951	0.619756	−	0.784795i	$-0.287232\pi$
$311$	6798.16	1.23951	0.619756	+	0.784795i	$0.287232\pi$
$312$	0	0
$313$	9479.08i	1.71179i	0.517153	+	0.855893i	$0.326992\pi$
$313$	9479.08i	1.71179i	−0.517153	+	0.855893i	$0.673008\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9286.72	−1.64541	−0.822704	−	0.568471i	$-0.807535\pi$
$317$	−9286.72	−1.64541	−0.822704	+	0.568471i	$0.807535\pi$
$318$	0	0
$319$	−10745.9	−1.88606
$320$	0	0
$321$	−327.843	−0.0570044
$322$	0	0
$323$	9438.23	1.62587
$324$	0	0
$325$	0	0
$326$	0	0
$327$	11468.5i	1.93948i
$328$	0	0
$329$	−8968.84	−1.50294
$330$	0	0
$331$	− 559.492i	− 0.0929078i	−0.998920	−	0.0464539i	$-0.985208\pi$
$331$	− 559.492i	− 0.0929078i	0.998920	−	0.0464539i	$-0.0147921\pi$
$332$	0	0
$333$	2289.80	0.376817
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 62.1362i	− 0.0100438i	−0.999987	−	0.00502192i	$-0.998401\pi$
$337$	− 62.1362i	− 0.0100438i	0.999987	−	0.00502192i	$-0.00159853\pi$
$338$	0	0
$339$	6514.57i	1.04372i
$340$	0	0
$341$	− 15835.0i	− 2.51470i
$342$	0	0
$343$	− 6904.06i	− 1.08684i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2384.20	−0.368849	−0.184424	−	0.982847i	$-0.559042\pi$
$347$	−2384.20	−0.368849	−0.184424	+	0.982847i	$0.559042\pi$
$348$	0	0
$349$	7940.30i	1.21786i	0.793223	+	0.608932i	$0.208402\pi$
$349$	7940.30i	1.21786i	−0.793223	+	0.608932i	$0.791598\pi$
$350$	0	0
$351$	2013.17	0.306139
$352$	0	0
$353$	8718.60i	1.31457i	0.753641	+	0.657286i	$0.228296\pi$
$353$	8718.60i	1.31457i	−0.753641	+	0.657286i	$0.771704\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	13351.9	1.97943
$358$	0	0
$359$	5920.09	0.870336	0.435168	−	0.900349i	$-0.356689\pi$
$359$	5920.09	0.870336	0.435168	+	0.900349i	$0.356689\pi$
$360$	0	0
$361$	−360.591	−0.0525719
$362$	0	0
$363$	6584.21	0.952015
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8779.02i	1.24867i	0.781158	+	0.624334i	$0.214629\pi$
$367$	8779.02i	1.24867i	−0.781158	+	0.624334i	$0.785371\pi$
$368$	0	0
$369$	−98.5117	−0.0138979
$370$	0	0
$371$	− 610.147i	− 0.0853834i
$372$	0	0
$373$	10346.8	1.43629	0.718145	−	0.695893i	$-0.244991\pi$
$373$	10346.8	1.43629	0.718145	+	0.695893i	$0.244991\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	11603.5i	1.58518i
$378$	0	0
$379$	5650.98i	0.765888i	0.923772	+	0.382944i	$0.125090\pi$
$379$	5650.98i	0.765888i	−0.923772	+	0.382944i	$0.874910\pi$
$380$	0	0
$381$	− 505.110i	− 0.0679202i
$382$	0	0
$383$	4349.97i	0.580348i	0.956974	+	0.290174i	$0.0937132\pi$
$383$	4349.97i	0.580348i	−0.956974	+	0.290174i	$0.906287\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3732.18	−0.490226
$388$	0	0
$389$	− 3876.59i	− 0.505273i	−0.967561	−	0.252636i	$-0.918702\pi$
$389$	− 3876.59i	− 0.505273i	0.967561	−	0.252636i	$-0.0812976\pi$
$390$	0	0
$391$	−5915.60	−0.765127
$392$	0	0
$393$	− 4972.00i	− 0.638178i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	10301.1	1.30226	0.651130	−	0.758966i	$-0.274295\pi$
$397$	10301.1	1.30226	0.651130	+	0.758966i	$0.274295\pi$
$398$	0	0
$399$	−10213.3	−1.28146
$400$	0	0
$401$	−7296.55	−0.908660	−0.454330	−	0.890834i	$-0.650121\pi$
$401$	−7296.55	−0.908660	−0.454330	+	0.890834i	$0.650121\pi$
$402$	0	0
$403$	−17098.9	−2.11354
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 3327.29i	− 0.405228i
$408$	0	0
$409$	4045.02	0.489031	0.244515	−	0.969645i	$-0.421371\pi$
$409$	4045.02	0.489031	0.244515	+	0.969645i	$0.421371\pi$
$410$	0	0
$411$	− 13801.3i	− 1.65637i
$412$	0	0
$413$	−1717.79	−0.204666
$414$	0	0
$415$	0	0
$416$	0	0
$417$	10.6161i	0.00124669i
$418$	0	0
$419$	− 10923.6i	− 1.27363i	−0.771016	−	0.636815i	$-0.780251\pi$
$419$	− 10923.6i	− 1.27363i	0.771016	−	0.636815i	$-0.219749\pi$
$420$	0	0
$421$	− 14877.4i	− 1.72228i	−0.508368	−	0.861140i	$-0.669751\pi$
$421$	− 14877.4i	− 1.72228i	0.508368	−	0.861140i	$-0.330249\pi$
$422$	0	0
$423$	− 18473.1i	− 2.12339i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2823.99	0.320053
$428$	0	0
$429$	− 18166.4i	− 2.04448i
$430$	0	0
$431$	−9899.13	−1.10632	−0.553161	−	0.833075i	$-0.686578\pi$
$431$	−9899.13	−1.10632	−0.553161	+	0.833075i	$0.686578\pi$
$432$	0	0
$433$	− 4671.88i	− 0.518513i	−0.965808	−	0.259257i	$-0.916522\pi$
$433$	− 4671.88i	− 0.518513i	0.965808	−	0.259257i	$-0.0834775\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4525.02	0.495334
$438$	0	0
$439$	−1304.49	−0.141823	−0.0709113	−	0.997483i	$-0.522591\pi$
$439$	−1304.49	−0.141823	−0.0709113	+	0.997483i	$0.522591\pi$
$440$	0	0
$441$	3181.77	0.343567
$442$	0	0
$443$	8613.52	0.923794	0.461897	−	0.886934i	$-0.347169\pi$
$443$	8613.52	0.923794	0.461897	+	0.886934i	$0.347169\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5723.28i	0.605597i
$448$	0	0
$449$	−16155.3	−1.69803	−0.849016	−	0.528367i	$-0.822805\pi$
$449$	−16155.3	−1.69803	−0.849016	+	0.528367i	$0.822805\pi$
$450$	0	0
$451$	143.147i	0.0149457i
$452$	0	0
$453$	18550.3	1.92399
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1325.32i	0.135659i	0.997697	+	0.0678293i	$0.0216073\pi$
$457$	1325.32i	0.135659i	−0.997697	+	0.0678293i	$0.978393\pi$
$458$	0	0
$459$	4428.47i	0.450334i
$460$	0	0
$461$	4357.31i	0.440217i	0.975475	+	0.220109i	$0.0706412\pi$
$461$	4357.31i	0.440217i	−0.975475	+	0.220109i	$0.929359\pi$
$462$	0	0
$463$	12348.5i	1.23949i	0.784805	+	0.619743i	$0.212763\pi$
$463$	12348.5i	1.23949i	−0.784805	+	0.619743i	$0.787237\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11587.9	1.14823	0.574113	−	0.818776i	$-0.305347\pi$
$467$	11587.9	1.14823	0.574113	+	0.818776i	$0.305347\pi$
$468$	0	0
$469$	11809.9i	1.16275i
$470$	0	0
$471$	11520.5	1.12704
$472$	0	0
$473$	5423.22i	0.527188i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1256.72	0.120631
$478$	0	0
$479$	7554.03	0.720568	0.360284	−	0.932843i	$-0.382680\pi$
$479$	7554.03	0.720568	0.360284	+	0.932843i	$0.382680\pi$
$480$	0	0
$481$	−3592.86	−0.340583
$482$	0	0
$483$	6401.38	0.603049
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 1536.22i	− 0.142942i	−0.997443	−	0.0714710i	$-0.977231\pi$
$487$	− 1536.22i	− 0.142942i	0.997443	−	0.0714710i	$-0.0227693\pi$
$488$	0	0
$489$	22277.9	2.06021
$490$	0	0
$491$	− 1242.72i	− 0.114223i	−0.998368	−	0.0571113i	$-0.981811\pi$
$491$	− 1242.72i	− 0.114223i	0.998368	−	0.0571113i	$-0.0181890\pi$
$492$	0	0
$493$	−25524.9	−2.33181
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1997.17i	0.180252i
$498$	0	0
$499$	17543.4i	1.57385i	0.617049	+	0.786925i	$0.288328\pi$
$499$	17543.4i	1.57385i	−0.617049	+	0.786925i	$0.711672\pi$
$500$	0	0
$501$	14796.7i	1.31949i
$502$	0	0
$503$	3452.08i	0.306005i	0.988226	+	0.153002i	$0.0488942\pi$
$503$	3452.08i	0.306005i	−0.988226	+	0.153002i	$0.951106\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−2714.79	−0.237807
$508$	0	0
$509$	− 2687.40i	− 0.234021i	−0.993131	−	0.117011i	$-0.962669\pi$
$509$	− 2687.40i	− 0.234021i	0.993131	−	0.117011i	$-0.0373311\pi$
$510$	0	0
$511$	5436.43	0.470633
$512$	0	0
$513$	− 3387.47i	− 0.291541i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−26843.2	−2.28349
$518$	0	0
$519$	−2700.37	−0.228387
$520$	0	0
$521$	−1131.78	−0.0951711	−0.0475855	−	0.998867i	$-0.515153\pi$
$521$	−1131.78	−0.0951711	−0.0475855	+	0.998867i	$0.515153\pi$
$522$	0	0
$523$	1002.24	0.0837952	0.0418976	−	0.999122i	$-0.486660\pi$
$523$	1002.24	0.0837952	0.0418976	+	0.999122i	$0.486660\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 37613.2i	− 3.10903i
$528$	0	0
$529$	9330.85	0.766898
$530$	0	0
$531$	− 3538.13i	− 0.289156i
$532$	0	0
$533$	154.572	0.0125615
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 15396.7i	− 1.23728i
$538$	0	0
$539$	− 4623.42i	− 0.369471i
$540$	0	0
$541$	6785.77i	0.539266i	0.962963	+	0.269633i	$0.0869024\pi$
$541$	6785.77i	0.539266i	−0.962963	+	0.269633i	$0.913098\pi$
$542$	0	0
$543$	33028.8i	2.61032i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1675.49	0.130967	0.0654833	−	0.997854i	$-0.479141\pi$
$547$	1675.49	0.130967	0.0654833	+	0.997854i	$0.479141\pi$
$548$	0	0
$549$	5816.58i	0.452177i
$550$	0	0
$551$	19524.8	1.50959
$552$	0	0
$553$	− 7135.87i	− 0.548731i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−6407.34	−0.487411	−0.243705	−	0.969849i	$-0.578363\pi$
$557$	−6407.34	−0.487411	−0.243705	+	0.969849i	$0.578363\pi$
$558$	0	0
$559$	5856.08	0.443087
$560$	0	0
$561$	39961.5	3.00744
$562$	0	0
$563$	−5627.44	−0.421258	−0.210629	−	0.977566i	$-0.567551\pi$
$563$	−5627.44	−0.421258	−0.210629	+	0.977566i	$0.567551\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	8784.57i	0.650648i
$568$	0	0
$569$	−8395.02	−0.618520	−0.309260	−	0.950978i	$-0.600081\pi$
$569$	−8395.02	−0.618520	−0.309260	+	0.950978i	$0.600081\pi$
$570$	0	0
$571$	2784.73i	0.204093i	0.994780	+	0.102047i	$0.0325391\pi$
$571$	2784.73i	0.204093i	−0.994780	+	0.102047i	$0.967461\pi$
$572$	0	0
$573$	3407.59	0.248437
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 19751.5i	− 1.42507i	−0.701635	−	0.712537i	$-0.747546\pi$
$577$	− 19751.5i	− 1.42507i	0.701635	−	0.712537i	$-0.252454\pi$
$578$	0	0
$579$	− 21879.1i	− 1.57041i
$580$	0	0
$581$	7803.35i	0.557208i
$582$	0	0
$583$	− 1826.13i	− 0.129727i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8029.42	0.564582	0.282291	−	0.959329i	$-0.408906\pi$
$587$	8029.42	0.564582	0.282291	+	0.959329i	$0.408906\pi$
$588$	0	0
$589$	28771.5i	2.01275i
$590$	0	0
$591$	−34403.5	−2.39454
$592$	0	0
$593$	− 18749.1i	− 1.29837i	−0.760629	−	0.649186i	$-0.775110\pi$
$593$	− 18749.1i	− 1.29837i	0.760629	−	0.649186i	$-0.224890\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10635.1	0.729085
$598$	0	0
$599$	−2130.81	−0.145346	−0.0726732	−	0.997356i	$-0.523153\pi$
$599$	−2130.81	−0.145346	−0.0726732	+	0.997356i	$0.523153\pi$
$600$	0	0
$601$	14697.1	0.997517	0.498759	−	0.866741i	$-0.333789\pi$
$601$	14697.1	0.997517	0.498759	+	0.866741i	$0.333789\pi$
$602$	0	0
$603$	−24324.8	−1.64276
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 5643.07i	− 0.377340i	−0.982041	−	0.188670i	$-0.939582\pi$
$607$	− 5643.07i	− 0.377340i	0.982041	−	0.188670i	$-0.0604176\pi$
$608$	0	0
$609$	27620.9	1.83786
$610$	0	0
$611$	28985.7i	1.91921i
$612$	0	0
$613$	−11519.3	−0.758988	−0.379494	−	0.925194i	$-0.623902\pi$
$613$	−11519.3	−0.758988	−0.379494	+	0.925194i	$0.623902\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1610.86i	0.105107i	0.998618	+	0.0525534i	$0.0167360\pi$
$617$	1610.86i	0.105107i	−0.998618	+	0.0525534i	$0.983264\pi$
$618$	0	0
$619$	− 17227.9i	− 1.11866i	−0.828946	−	0.559328i	$-0.811059\pi$
$619$	− 17227.9i	− 1.11866i	0.828946	−	0.559328i	$-0.188941\pi$
$620$	0	0
$621$	2123.16i	0.137197i
$622$	0	0
$623$	20332.9i	1.30757i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−30567.8	−1.94698
$628$	0	0
$629$	− 7903.40i	− 0.501000i
$630$	0	0
$631$	−22325.0	−1.40847	−0.704236	−	0.709966i	$-0.748710\pi$
$631$	−22325.0	−1.40847	−0.704236	+	0.709966i	$0.748710\pi$
$632$	0	0
$633$	2694.52i	0.169190i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−4992.44	−0.310530
$638$	0	0
$639$	−4113.57	−0.254664
$640$	0	0
$641$	−17545.4	−1.08113	−0.540564	−	0.841303i	$-0.681789\pi$
$641$	−17545.4	−1.08113	−0.540564	+	0.841303i	$0.681789\pi$
$642$	0	0
$643$	17840.2	1.09417	0.547083	−	0.837078i	$-0.315738\pi$
$643$	17840.2	1.09417	0.547083	+	0.837078i	$0.315738\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3785.71i	0.230033i	0.993364	+	0.115017i	$0.0366921\pi$
$647$	3785.71i	0.230033i	−0.993364	+	0.115017i	$0.963308\pi$
$648$	0	0
$649$	−5141.25	−0.310958
$650$	0	0
$651$	40702.0i	2.45044i
$652$	0	0
$653$	17813.4	1.06752	0.533761	−	0.845636i	$-0.320778\pi$
$653$	17813.4	1.06752	0.533761	+	0.845636i	$0.320778\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	11197.4i	0.664921i
$658$	0	0
$659$	− 5929.18i	− 0.350483i	−0.984525	−	0.175241i	$-0.943929\pi$
$659$	− 5929.18i	− 0.350483i	0.984525	−	0.175241i	$-0.0560706\pi$
$660$	0	0
$661$	13203.3i	0.776927i	0.921464	+	0.388464i	$0.126994\pi$
$661$	13203.3i	0.776927i	−0.921464	+	0.388464i	$0.873006\pi$
$662$	0	0
$663$	− 43151.0i	− 2.52767i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−12237.5	−0.710404
$668$	0	0
$669$	− 34982.8i	− 2.02169i
$670$	0	0
$671$	8452.04	0.486270
$672$	0	0
$673$	12240.5i	0.701094i	0.936545	+	0.350547i	$0.114004\pi$
$673$	12240.5i	0.701094i	−0.936545	+	0.350547i	$0.885996\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3035.77	0.172340	0.0861701	−	0.996280i	$-0.472537\pi$
$677$	3035.77	0.172340	0.0861701	+	0.996280i	$0.472537\pi$
$678$	0	0
$679$	12159.7	0.687253
$680$	0	0
$681$	−28830.0	−1.62227
$682$	0	0
$683$	−18824.7	−1.05462	−0.527311	−	0.849673i	$-0.676800\pi$
$683$	−18824.7	−1.05462	−0.527311	+	0.849673i	$0.676800\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 15861.1i	− 0.880841i
$688$	0	0
$689$	−1971.89	−0.109032
$690$	0	0
$691$	− 22287.6i	− 1.22700i	−0.789694	−	0.613502i	$-0.789760\pi$
$691$	− 22287.6i	− 1.22700i	0.789694	−	0.613502i	$-0.210240\pi$
$692$	0	0
$693$	−23514.8	−1.28897
$694$	0	0
$695$	0	0
$696$	0	0
$697$	340.020i	0.0184780i
$698$	0	0
$699$	− 6229.38i	− 0.337077i
$700$	0	0
$701$	− 12975.8i	− 0.699129i	−0.936912	−	0.349565i	$-0.886330\pi$
$701$	− 12975.8i	− 0.699129i	0.936912	−	0.349565i	$-0.113670\pi$
$702$	0	0
$703$	6045.55i	0.324342i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	17923.4	0.953438
$708$	0	0
$709$	− 10236.0i	− 0.542200i	−0.962551	−	0.271100i	$-0.912613\pi$
$709$	− 10236.0i	− 0.542200i	0.962551	−	0.271100i	$-0.0873874\pi$
$710$	0	0
$711$	14697.8	0.775259
$712$	0	0
$713$	− 18033.1i	− 0.947189i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−8917.24	−0.464464
$718$	0	0
$719$	−17107.9	−0.887367	−0.443684	−	0.896184i	$-0.646329\pi$
$719$	−17107.9	−0.887367	−0.443684	+	0.896184i	$0.646329\pi$
$720$	0	0
$721$	−13319.0	−0.687971
$722$	0	0
$723$	−28713.4	−1.47699
$724$	0	0
$725$	0	0
$726$	0	0
$727$	29399.2i	1.49980i	0.661549	+	0.749902i	$0.269899\pi$
$727$	29399.2i	1.49980i	−0.661549	+	0.749902i	$0.730101\pi$
$728$	0	0
$729$	−26374.5	−1.33996
$730$	0	0
$731$	12881.9i	0.651785i
$732$	0	0
$733$	−17312.7	−0.872386	−0.436193	−	0.899853i	$-0.643673\pi$
$733$	−17312.7	−0.872386	−0.436193	+	0.899853i	$0.643673\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	35346.3i	1.76662i
$738$	0	0
$739$	20459.6i	1.01843i	0.860639	+	0.509215i	$0.170064\pi$
$739$	20459.6i	1.01843i	−0.860639	+	0.509215i	$0.829936\pi$
$740$	0	0
$741$	33007.5i	1.63639i
$742$	0	0
$743$	− 15382.4i	− 0.759524i	−0.925084	−	0.379762i	$-0.876006\pi$
$743$	− 15382.4i	− 0.759524i	0.925084	−	0.379762i	$-0.123994\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−16072.6	−0.787235
$748$	0	0
$749$	− 665.860i	− 0.0324833i
$750$	0	0
$751$	16396.7	0.796705	0.398352	−	0.917232i	$-0.369582\pi$
$751$	16396.7	0.796705	0.398352	+	0.917232i	$0.369582\pi$
$752$	0	0
$753$	27768.9i	1.34390i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−30322.2	−1.45585	−0.727924	−	0.685657i	$-0.759515\pi$
$757$	−30322.2	−1.45585	−0.727924	+	0.685657i	$0.759515\pi$
$758$	0	0
$759$	19159.0	0.916240
$760$	0	0
$761$	29611.4	1.41053	0.705264	−	0.708944i	$-0.250828\pi$
$761$	29611.4	1.41053	0.705264	+	0.708944i	$0.250828\pi$
$762$	0	0
$763$	−23293.0	−1.10519
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5551.60i	0.261351i
$768$	0	0
$769$	−18385.0	−0.862131	−0.431066	−	0.902321i	$-0.641862\pi$
$769$	−18385.0	−0.862131	−0.431066	+	0.902321i	$0.641862\pi$
$770$	0	0
$771$	18295.4i	0.854594i
$772$	0	0
$773$	−16712.1	−0.777609	−0.388804	−	0.921320i	$-0.627112\pi$
$773$	−16712.1	−0.777609	−0.388804	+	0.921320i	$0.627112\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	8552.41i	0.394873i
$778$	0	0
$779$	− 260.092i	− 0.0119625i
$780$	0	0
$781$	5977.41i	0.273865i
$782$	0	0
$783$	9161.12i	0.418125i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−33805.6	−1.53118	−0.765591	−	0.643328i	$-0.777553\pi$
$787$	−33805.6	−1.53118	−0.765591	+	0.643328i	$0.777553\pi$
$788$	0	0
$789$	− 30262.1i	− 1.36548i
$790$	0	0
$791$	−13231.3	−0.594755
$792$	0	0
$793$	− 9126.64i	− 0.408697i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8399.83	0.373321	0.186661	−	0.982424i	$-0.440234\pi$
$797$	8399.83	0.373321	0.186661	+	0.982424i	$0.440234\pi$
$798$	0	0
$799$	−63761.3	−2.82317
$800$	0	0
$801$	−41879.6	−1.84737
$802$	0	0
$803$	16270.9	0.715054
$804$	0	0
$805$	0	0
$806$	0	0
$807$	27920.8i	1.21792i
$808$	0	0
$809$	12503.9	0.543402	0.271701	−	0.962382i	$-0.412414\pi$
$809$	12503.9	0.543402	0.271701	+	0.962382i	$0.412414\pi$
$810$	0	0
$811$	− 31097.9i	− 1.34648i	−0.739425	−	0.673239i	$-0.764902\pi$
$811$	− 31097.9i	− 1.34648i	0.739425	−	0.673239i	$-0.235098\pi$
$812$	0	0
$813$	30924.3	1.33403
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 9853.76i	− 0.421958i
$818$	0	0
$819$	25391.7i	1.08334i
$820$	0	0
$821$	38043.1i	1.61719i	0.588367	+	0.808594i	$0.299771\pi$
$821$	38043.1i	1.61719i	−0.588367	+	0.808594i	$0.700229\pi$
$822$	0	0
$823$	− 39096.4i	− 1.65591i	−0.560794	−	0.827955i	$-0.689504\pi$
$823$	− 39096.4i	− 1.65591i	0.560794	−	0.827955i	$-0.310496\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−24525.7	−1.03125	−0.515625	−	0.856815i	$-0.672440\pi$
$827$	−24525.7	−1.03125	−0.515625	+	0.856815i	$0.672440\pi$
$828$	0	0
$829$	22818.8i	0.956007i	0.878358	+	0.478004i	$0.158639\pi$
$829$	22818.8i	0.956007i	−0.878358	+	0.478004i	$0.841361\pi$
$830$	0	0
$831$	−42821.1	−1.78754
$832$	0	0
$833$	− 10982.1i	− 0.456792i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−13499.7	−0.557490
$838$	0	0
$839$	43262.8	1.78021	0.890107	−	0.455752i	$-0.150630\pi$
$839$	43262.8	1.78021	0.890107	+	0.455752i	$0.150630\pi$
$840$	0	0
$841$	−28414.0	−1.16503
$842$	0	0
$843$	63804.6	2.60682
$844$	0	0
$845$	0	0
$846$	0	0
$847$	13372.8i	0.542495i
$848$	0	0
$849$	−20252.4	−0.818680
$850$	0	0
$851$	− 3789.17i	− 0.152633i
$852$	0	0
$853$	−36605.3	−1.46933	−0.734666	−	0.678429i	$-0.762661\pi$
$853$	−36605.3	−1.46933	−0.734666	+	0.678429i	$0.762661\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 145.647i	− 0.00580539i	−0.999996	−	0.00290269i	$-0.999076\pi$
$857$	− 145.647i	− 0.00580539i	0.999996	−	0.00290269i	$-0.000923958\pi$
$858$	0	0
$859$	48187.2i	1.91400i	0.290088	+	0.957000i	$0.406315\pi$
$859$	48187.2i	1.91400i	−0.290088	+	0.957000i	$0.593685\pi$
$860$	0	0
$861$	− 367.942i	− 0.0145638i
$862$	0	0
$863$	− 13624.8i	− 0.537419i	−0.963221	−	0.268710i	$-0.913403\pi$
$863$	− 13624.8i	− 0.537419i	0.963221	−	0.268710i	$-0.0865972\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	57125.7	2.23771
$868$	0	0
$869$	− 21357.3i	− 0.833711i
$870$	0	0
$871$	38167.5	1.48479
$872$	0	0
$873$	25045.2i	0.970965i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−7517.05	−0.289433	−0.144716	−	0.989473i	$-0.546227\pi$
$877$	−7517.05	−0.289433	−0.144716	+	0.989473i	$0.546227\pi$
$878$	0	0
$879$	−4380.36	−0.168084
$880$	0	0
$881$	−28282.2	−1.08156	−0.540778	−	0.841165i	$-0.681870\pi$
$881$	−28282.2	−1.08156	−0.540778	+	0.841165i	$0.681870\pi$
$882$	0	0
$883$	−18051.5	−0.687974	−0.343987	−	0.938974i	$-0.611778\pi$
$883$	−18051.5	−0.687974	−0.343987	+	0.938974i	$0.611778\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11236.4i	0.425345i	0.977123	+	0.212673i	$0.0682168\pi$
$887$	11236.4i	0.425345i	−0.977123	+	0.212673i	$0.931783\pi$
$888$	0	0
$889$	1025.90	0.0387036
$890$	0	0
$891$	26291.7i	0.988558i
$892$	0	0
$893$	48773.0	1.82769
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 20688.1i	− 0.770074i
$898$	0	0
$899$	− 77810.1i	− 2.88666i
$900$	0	0
$901$	− 4337.66i	− 0.160387i
$902$	0	0
$903$	− 13939.7i	− 0.513716i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−12021.6	−0.440100	−0.220050	−	0.975489i	$-0.570622\pi$
$907$	−12021.6	−0.440100	−0.220050	+	0.975489i	$0.570622\pi$
$908$	0	0
$909$	36916.9i	1.34704i
$910$	0	0
$911$	−15718.1	−0.571642	−0.285821	−	0.958283i	$-0.592266\pi$
$911$	−15718.1	−0.571642	−0.285821	+	0.958283i	$0.592266\pi$
$912$	0	0
$913$	23355.0i	0.846591i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	10098.3	0.363659
$918$	0	0
$919$	29706.9	1.06631	0.533156	−	0.846017i	$-0.321006\pi$
$919$	29706.9	1.06631	0.533156	+	0.846017i	$0.321006\pi$
$920$	0	0
$921$	4054.93	0.145075
$922$	0	0
$923$	6454.50	0.230176
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 27433.2i	− 0.971980i
$928$	0	0
$929$	32926.5	1.16285	0.581423	−	0.813602i	$-0.302496\pi$
$929$	32926.5	1.16285	0.581423	+	0.813602i	$0.302496\pi$
$930$	0	0
$931$	8400.56i	0.295722i
$932$	0	0
$933$	−52298.3	−1.83512
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 31490.5i	− 1.09792i	−0.835849	−	0.548960i	$-0.815024\pi$
$937$	− 31490.5i	− 1.09792i	0.835849	−	0.548960i	$-0.184976\pi$
$938$	0	0
$939$	− 72922.6i	− 2.53433i
$940$	0	0
$941$	− 51748.0i	− 1.79271i	−0.443340	−	0.896353i	$-0.646207\pi$
$941$	− 51748.0i	− 1.79271i	0.443340	−	0.896353i	$-0.353793\pi$
$942$	0	0
$943$	163.018i	0.00562947i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−17708.2	−0.607644	−0.303822	−	0.952729i	$-0.598263\pi$
$947$	−17708.2	−0.607644	−0.303822	+	0.952729i	$0.598263\pi$
$948$	0	0
$949$	− 17569.6i	− 0.600983i
$950$	0	0
$951$	71442.8	2.43606
$952$	0	0
$953$	26224.3i	0.891382i	0.895187	+	0.445691i	$0.147042\pi$
$953$	26224.3i	0.891382i	−0.895187	+	0.445691i	$0.852958\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	82667.9	2.79234
$958$	0	0
$959$	28030.9	0.943864
$960$	0	0
$961$	84869.2	2.84882
$962$	0	0
$963$	1371.47	0.0458931
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 28936.2i	− 0.962282i	−0.876643	−	0.481141i	$-0.840223\pi$
$967$	− 28936.2i	− 0.962282i	0.876643	−	0.481141i	$-0.159777\pi$
$968$	0	0
$969$	−72608.3	−2.40714
$970$	0	0
$971$	3747.04i	0.123840i	0.998081	+	0.0619198i	$0.0197223\pi$
$971$	3747.04i	0.123840i	−0.998081	+	0.0619198i	$0.980278\pi$
$972$	0	0
$973$	−21.5616	−0.000710414 0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 10231.8i	− 0.335050i	−0.985868	−	0.167525i	$-0.946422\pi$
$977$	− 10231.8i	− 0.335050i	0.985868	−	0.167525i	$-0.0535775\pi$
$978$	0	0
$979$	60855.1i	1.98666i
$980$	0	0
$981$	− 47976.6i	− 1.56144i
$982$	0	0
$983$	− 913.463i	− 0.0296388i	−0.999890	−	0.0148194i	$-0.995283\pi$
$983$	− 913.463i	− 0.0296388i	0.999890	−	0.0148194i	$-0.00471733\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	68997.3	2.22514
$988$	0	0
$989$	6176.04i	0.198571i
$990$	0	0
$991$	−20267.9	−0.649677	−0.324838	−	0.945770i	$-0.605310\pi$
$991$	−20267.9	−0.649677	−0.324838	+	0.945770i	$0.605310\pi$
$992$	0	0
$993$	4304.18i	0.137552i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−34081.1	−1.08261	−0.541304	−	0.840827i	$-0.682069\pi$
$997$	−34081.1	−1.08261	−0.541304	+	0.840827i	$0.682069\pi$
$998$	0	0
$999$	−2836.60	−0.0898360

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.4.f.d.49.3		24
4.3	odd	2		200.4.f.d.149.23		24
5.2	odd	4		800.4.d.b.401.11		12
5.3	odd	4		800.4.d.c.401.2		12
5.4	even	2	inner	800.4.f.d.49.22		24
8.3	odd	2		200.4.f.d.149.1		24
8.5	even	2	inner	800.4.f.d.49.21		24
20.3	even	4		200.4.d.d.101.5	yes	12
20.7	even	4		200.4.d.c.101.8	yes	12
20.19	odd	2		200.4.f.d.149.2		24
40.3	even	4		200.4.d.d.101.6	yes	12
40.13	odd	4		800.4.d.c.401.11		12
40.19	odd	2		200.4.f.d.149.24		24
40.27	even	4		200.4.d.c.101.7	✓	12
40.29	even	2	inner	800.4.f.d.49.4		24
40.37	odd	4		800.4.d.b.401.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.4.d.c.101.7	✓	12	40.27	even	4
200.4.d.c.101.8	yes	12	20.7	even	4
200.4.d.d.101.5	yes	12	20.3	even	4
200.4.d.d.101.6	yes	12	40.3	even	4
200.4.f.d.149.1		24	8.3	odd	2
200.4.f.d.149.2		24	20.19	odd	2
200.4.f.d.149.23		24	4.3	odd	2
200.4.f.d.149.24		24	40.19	odd	2
800.4.d.b.401.2		12	40.37	odd	4
800.4.d.b.401.11		12	5.2	odd	4
800.4.d.c.401.2		12	5.3	odd	4
800.4.d.c.401.11		12	40.13	odd	4
800.4.f.d.49.3		24	1.1	even	1	trivial
800.4.f.d.49.4		24	40.29	even	2	inner
800.4.f.d.49.21		24	8.5	even	2	inner
800.4.f.d.49.22		24	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	800.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-7.69300 q^{3} -15.6248i q^{7} +32.1823 q^{9} +O(q^{10})\)	\(q-7.69300 q^{3} -15.6248i q^{7} +32.1823 q^{9} -46.7640i q^{11} -50.4964 q^{13} -111.080i q^{17} +84.9682i q^{19} +120.201i q^{21} -53.2555i q^{23} -39.8675 q^{27} -229.789i q^{29} +338.615 q^{31} +359.755i q^{33} +71.1508 q^{37} +388.469 q^{39} -3.06105 q^{41} -115.970 q^{43} -574.015i q^{47} +98.8671 q^{49} +854.536i q^{51} +39.0500 q^{53} -653.660i q^{57} -109.940i q^{59} +180.738i q^{61} -502.841i q^{63} -755.845 q^{67} +409.695i q^{69} -127.821 q^{71} +347.937i q^{73} -730.676 q^{77} +456.703 q^{79} -562.221 q^{81} -499.423 q^{83} +1767.77i q^{87} -1301.32 q^{89} +788.995i q^{91} -2604.97 q^{93} +778.230i q^{97} -1504.97i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 216 q^{9}+O(q^{10}) \) 24 * q + 216 * q^9	\( 24 q + 216 q^{9} + 528 q^{31} - 600 q^{39} - 40 q^{41} - 456 q^{49} - 1256 q^{71} + 2232 q^{79} + 2256 q^{81} + 848 q^{89}+O(q^{100}) \) 24 * q + 216 * q^9 + 528 * q^31 - 600 * q^39 - 40 * q^41 - 456 * q^49 - 1256 * q^71 + 2232 * q^79 + 2256 * q^81 + 848 * q^89

Introduction

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