LMFDB - Embedded newform 800.4.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,4,Mod(1,800)]

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, 0, 0]))

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

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

chi := DirichletCharacter("800.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	800.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$47.2015280046$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.5685.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 11x + 6 $ x^3 - x^2 - 11*x + 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.59535$ 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)

$f(q)$	$=$	$q-8.19070 q^{3} +21.7061 q^{7} +40.0875 q^{9} +37.8968 q^{11} +52.7369 q^{13} -99.9636 q^{17} +116.747 q^{19} -177.788 q^{21} +37.7320 q^{23} -107.196 q^{27} +218.664 q^{29} -67.3808 q^{31} -310.401 q^{33} -362.591 q^{37} -431.952 q^{39} -291.803 q^{41} +183.160 q^{43} -443.479 q^{47} +128.155 q^{49} +818.772 q^{51} -416.349 q^{53} -956.240 q^{57} +828.632 q^{59} +442.983 q^{61} +870.143 q^{63} +570.007 q^{67} -309.051 q^{69} +341.295 q^{71} +506.312 q^{73} +822.591 q^{77} +426.757 q^{79} -204.356 q^{81} -982.438 q^{83} -1791.01 q^{87} +926.879 q^{89} +1144.71 q^{91} +551.895 q^{93} +1889.63 q^{97} +1519.19 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 5 q^{3} + 2 q^{7} + 18 q^{9} + 31 q^{11} + 8 q^{13} - 89 q^{17} + 87 q^{19} - 70 q^{21} + 122 q^{23} - 143 q^{27} + 84 q^{29} + 294 q^{31} - 209 q^{33} - 94 q^{37} - 32 q^{39} - 345 q^{41} + 412 q^{43}+ \cdots + 1954 q^{99}+O(q^{100}) $ 3 * q - 5 * q^3 + 2 * q^7 + 18 * q^9 + 31 * q^11 + 8 * q^13 - 89 * q^17 + 87 * q^19 - 70 * q^21 + 122 * q^23 - 143 * q^27 + 84 * q^29 + 294 * q^31 - 209 * q^33 - 94 * q^37 - 32 * q^39 - 345 * q^41 + 412 * q^43 - 824 * q^47 + 359 * q^49 + 1351 * q^51 + 74 * q^53 - 913 * q^57 + 1040 * q^59 + 482 * q^61 + 1532 * q^63 - 735 * q^67 - 614 * q^69 + 2048 * q^71 - 15 * q^73 + 1474 * q^77 + 998 * q^79 - 549 * q^81 + 1221 * q^83 - 2564 * q^87 + 1189 * q^89 + 4032 * q^91 + 454 * q^93 - 1010 * q^97 + 1954 * 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
$3$	−8.19070	−1.57630	−0.788150	−	0.615483i	$-0.788961\pi$
$3$	−8.19070	−1.57630	−0.788150	+	0.615483i	$0.788961\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	21.7061	1.17202	0.586010	−	0.810304i	$-0.300698\pi$
$7$	21.7061	1.17202	0.586010	+	0.810304i	$0.300698\pi$
$8$	0	0
$9$	40.0875	1.48472
$10$	0	0
$11$	37.8968	1.03876	0.519378	−	0.854545i	$-0.326164\pi$
$11$	37.8968	1.03876	0.519378	+	0.854545i	$0.326164\pi$
$12$	0	0
$13$	52.7369	1.12512	0.562561	−	0.826756i	$-0.309816\pi$
$13$	52.7369	1.12512	0.562561	+	0.826756i	$0.309816\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−99.9636	−1.42616	−0.713081	−	0.701082i	$-0.752701\pi$
$17$	−99.9636	−1.42616	−0.713081	+	0.701082i	$0.752701\pi$
$18$	0	0
$19$	116.747	1.40966	0.704832	−	0.709374i	$-0.251022\pi$
$19$	116.747	1.40966	0.704832	+	0.709374i	$0.251022\pi$
$20$	0	0
$21$	−177.788	−1.84745
$22$	0	0
$23$	37.7320	0.342072	0.171036	−	0.985265i	$-0.445289\pi$
$23$	37.7320	0.342072	0.171036	+	0.985265i	$0.445289\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−107.196	−0.764067
$28$	0	0
$29$	218.664	1.40017	0.700085	−	0.714060i	$-0.253146\pi$
$29$	218.664	1.40017	0.700085	+	0.714060i	$0.253146\pi$
$30$	0	0
$31$	−67.3808	−0.390385	−0.195193	−	0.980765i	$-0.562533\pi$
$31$	−67.3808	−0.390385	−0.195193	+	0.980765i	$0.562533\pi$
$32$	0	0
$33$	−310.401	−1.63739
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−362.591	−1.61107	−0.805536	−	0.592547i	$-0.798123\pi$
$37$	−362.591	−1.61107	−0.805536	+	0.592547i	$0.798123\pi$
$38$	0	0
$39$	−431.952	−1.77353
$40$	0	0
$41$	−291.803	−1.11151	−0.555757	−	0.831345i	$-0.687571\pi$
$41$	−291.803	−1.11151	−0.555757	+	0.831345i	$0.687571\pi$
$42$	0	0
$43$	183.160	0.649571	0.324786	−	0.945788i	$-0.394708\pi$
$43$	183.160	0.649571	0.324786	+	0.945788i	$0.394708\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−443.479	−1.37634	−0.688170	−	0.725549i	$-0.741586\pi$
$47$	−443.479	−1.37634	−0.688170	+	0.725549i	$0.741586\pi$
$48$	0	0
$49$	128.155	0.373629
$50$	0	0
$51$	818.772	2.24806
$52$	0	0
$53$	−416.349	−1.07905	−0.539527	−	0.841968i	$-0.681397\pi$
$53$	−416.349	−1.07905	−0.539527	+	0.841968i	$0.681397\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−956.240	−2.22205
$58$	0	0
$59$	828.632	1.82845	0.914226	−	0.405204i	$-0.132799\pi$
$59$	828.632	1.82845	0.914226	+	0.405204i	$0.132799\pi$
$60$	0	0
$61$	442.983	0.929807	0.464903	−	0.885361i	$-0.346089\pi$
$61$	442.983	0.929807	0.464903	+	0.885361i	$0.346089\pi$
$62$	0	0
$63$	870.143	1.74012
$64$	0	0
$65$	0	0
$66$	0	0
$67$	570.007	1.03936	0.519682	−	0.854360i	$-0.326050\pi$
$67$	570.007	1.03936	0.519682	+	0.854360i	$0.326050\pi$
$68$	0	0
$69$	−309.051	−0.539208
$70$	0	0
$71$	341.295	0.570483	0.285241	−	0.958456i	$-0.407926\pi$
$71$	341.295	0.570483	0.285241	+	0.958456i	$0.407926\pi$
$72$	0	0
$73$	506.312	0.811772	0.405886	−	0.913924i	$-0.366963\pi$
$73$	506.312	0.811772	0.405886	+	0.913924i	$0.366963\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	822.591	1.21744
$78$	0	0
$79$	426.757	0.607771	0.303886	−	0.952708i	$-0.401716\pi$
$79$	426.757	0.607771	0.303886	+	0.952708i	$0.401716\pi$
$80$	0	0
$81$	−204.356	−0.280323
$82$	0	0
$83$	−982.438	−1.29924	−0.649618	−	0.760261i	$-0.725071\pi$
$83$	−982.438	−1.29924	−0.649618	+	0.760261i	$0.725071\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1791.01	−2.20709
$88$	0	0
$89$	926.879	1.10392	0.551961	−	0.833870i	$-0.313880\pi$
$89$	926.879	1.10392	0.551961	+	0.833870i	$0.313880\pi$
$90$	0	0
$91$	1144.71	1.31866
$92$	0	0
$93$	551.895	0.615364
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1889.63	1.97797	0.988984	−	0.148024i	$-0.0472912\pi$
$97$	1889.63	1.97797	0.988984	+	0.148024i	$0.0472912\pi$
$98$	0	0
$99$	1519.19	1.54226
$100$	0	0
$101$	96.2748	0.0948486	0.0474243	−	0.998875i	$-0.484899\pi$
$101$	96.2748	0.0948486	0.0474243	+	0.998875i	$0.484899\pi$
$102$	0	0
$103$	1701.85	1.62804	0.814019	−	0.580838i	$-0.197275\pi$
$103$	1701.85	1.62804	0.814019	+	0.580838i	$0.197275\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−746.877	−0.674798	−0.337399	−	0.941362i	$-0.609547\pi$
$107$	−746.877	−0.674798	−0.337399	+	0.941362i	$0.609547\pi$
$108$	0	0
$109$	−2150.44	−1.88968	−0.944840	−	0.327532i	$-0.893783\pi$
$109$	−2150.44	−1.88968	−0.944840	+	0.327532i	$0.893783\pi$
$110$	0	0
$111$	2969.88	2.53953
$112$	0	0
$113$	661.058	0.550328	0.275164	−	0.961397i	$-0.411268\pi$
$113$	661.058	0.550328	0.275164	+	0.961397i	$0.411268\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2114.09	1.67049
$118$	0	0
$119$	−2169.82	−1.67149
$120$	0	0
$121$	105.167	0.0790134
$122$	0	0
$123$	2390.07	1.75208
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−989.619	−0.691452	−0.345726	−	0.938335i	$-0.612367\pi$
$127$	−989.619	−0.691452	−0.345726	+	0.938335i	$0.612367\pi$
$128$	0	0
$129$	−1500.20	−1.02392
$130$	0	0
$131$	−6.42246	−0.00428346	−0.00214173	−	0.999998i	$-0.500682\pi$
$131$	−6.42246	−0.00428346	−0.00214173	+	0.999998i	$0.500682\pi$
$132$	0	0
$133$	2534.12	1.65215
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1033.75	−0.644668	−0.322334	−	0.946626i	$-0.604467\pi$
$137$	−1033.75	−0.644668	−0.322334	+	0.946626i	$0.604467\pi$
$138$	0	0
$139$	−56.8412	−0.0346850	−0.0173425	−	0.999850i	$-0.505521\pi$
$139$	−56.8412	−0.0346850	−0.0173425	+	0.999850i	$0.505521\pi$
$140$	0	0
$141$	3632.40	2.16953
$142$	0	0
$143$	1998.56	1.16873
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1049.68	−0.588951
$148$	0	0
$149$	542.238	0.298133	0.149067	−	0.988827i	$-0.452373\pi$
$149$	542.238	0.298133	0.149067	+	0.988827i	$0.452373\pi$
$150$	0	0
$151$	−390.225	−0.210305	−0.105152	−	0.994456i	$-0.533533\pi$
$151$	−390.225	−0.210305	−0.105152	+	0.994456i	$0.533533\pi$
$152$	0	0
$153$	−4007.29	−2.11745
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3674.37	1.86782	0.933908	−	0.357514i	$-0.116376\pi$
$157$	3674.37	1.86782	0.933908	+	0.357514i	$0.116376\pi$
$158$	0	0
$159$	3410.19	1.70091
$160$	0	0
$161$	819.014	0.400915
$162$	0	0
$163$	−1219.38	−0.585944	−0.292972	−	0.956121i	$-0.594644\pi$
$163$	−1219.38	−0.585944	−0.292972	+	0.956121i	$0.594644\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	610.547	0.282907	0.141454	−	0.989945i	$-0.454822\pi$
$167$	610.547	0.282907	0.141454	+	0.989945i	$0.454822\pi$
$168$	0	0
$169$	584.181	0.265899
$170$	0	0
$171$	4680.10	2.09296
$172$	0	0
$173$	−50.4300	−0.0221626	−0.0110813	−	0.999939i	$-0.503527\pi$
$173$	−50.4300	−0.0221626	−0.0110813	+	0.999939i	$0.503527\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−6787.07	−2.88219
$178$	0	0
$179$	373.064	0.155777	0.0778885	−	0.996962i	$-0.475182\pi$
$179$	373.064	0.155777	0.0778885	+	0.996962i	$0.475182\pi$
$180$	0	0
$181$	−232.883	−0.0956358	−0.0478179	−	0.998856i	$-0.515227\pi$
$181$	−232.883	−0.0956358	−0.0478179	+	0.998856i	$0.515227\pi$
$182$	0	0
$183$	−3628.34	−1.46565
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−3788.30	−1.48143
$188$	0	0
$189$	−2326.80	−0.895501
$190$	0	0
$191$	2769.18	1.04906	0.524531	−	0.851391i	$-0.324241\pi$
$191$	2769.18	1.04906	0.524531	+	0.851391i	$0.324241\pi$
$192$	0	0
$193$	−307.412	−0.114653	−0.0573264	−	0.998355i	$-0.518258\pi$
$193$	−307.412	−0.114653	−0.0573264	+	0.998355i	$0.518258\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	317.010	0.114650	0.0573249	−	0.998356i	$-0.481743\pi$
$197$	317.010	0.114650	0.0573249	+	0.998356i	$0.481743\pi$
$198$	0	0
$199$	3917.06	1.39534	0.697671	−	0.716418i	$-0.254220\pi$
$199$	3917.06	1.39534	0.697671	+	0.716418i	$0.254220\pi$
$200$	0	0
$201$	−4668.75	−1.63835
$202$	0	0
$203$	4746.35	1.64103
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1512.58	0.507882
$208$	0	0
$209$	4424.34	1.46430
$210$	0	0
$211$	−2054.34	−0.670267	−0.335133	−	0.942171i	$-0.608781\pi$
$211$	−2054.34	−0.670267	−0.335133	+	0.942171i	$0.608781\pi$
$212$	0	0
$213$	−2795.44	−0.899252
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1462.57	−0.457539
$218$	0	0
$219$	−4147.05	−1.27960
$220$	0	0
$221$	−5271.77	−1.60461
$222$	0	0
$223$	−3087.92	−0.927275	−0.463638	−	0.886025i	$-0.653456\pi$
$223$	−3087.92	−0.927275	−0.463638	+	0.886025i	$0.653456\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3572.18	1.04447	0.522233	−	0.852803i	$-0.325099\pi$
$227$	3572.18	1.04447	0.522233	+	0.852803i	$0.325099\pi$
$228$	0	0
$229$	1799.26	0.519207	0.259603	−	0.965715i	$-0.416408\pi$
$229$	1799.26	0.519207	0.259603	+	0.965715i	$0.416408\pi$
$230$	0	0
$231$	−6737.60	−1.91905
$232$	0	0
$233$	3900.64	1.09674	0.548368	−	0.836237i	$-0.315249\pi$
$233$	3900.64	1.09674	0.548368	+	0.836237i	$0.315249\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−3495.44	−0.958030
$238$	0	0
$239$	60.6726	0.0164209	0.00821043	−	0.999966i	$-0.497387\pi$
$239$	60.6726	0.0164209	0.00821043	+	0.999966i	$0.497387\pi$
$240$	0	0
$241$	−6160.51	−1.64661	−0.823305	−	0.567599i	$-0.807872\pi$
$241$	−6160.51	−1.64661	−0.823305	+	0.567599i	$0.807872\pi$
$242$	0	0
$243$	4568.10	1.20594
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6156.88	1.58604
$248$	0	0
$249$	8046.85	2.04798
$250$	0	0
$251$	−880.907	−0.221523	−0.110762	−	0.993847i	$-0.535329\pi$
$251$	−880.907	−0.221523	−0.110762	+	0.993847i	$0.535329\pi$
$252$	0	0
$253$	1429.92	0.355329
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5845.84	−1.41889	−0.709443	−	0.704763i	$-0.751053\pi$
$257$	−5845.84	−1.41889	−0.709443	+	0.704763i	$0.751053\pi$
$258$	0	0
$259$	−7870.44	−1.88821
$260$	0	0
$261$	8765.70	2.07886
$262$	0	0
$263$	−1994.10	−0.467535	−0.233767	−	0.972293i	$-0.575105\pi$
$263$	−1994.10	−0.467535	−0.233767	+	0.972293i	$0.575105\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−7591.78	−1.74011
$268$	0	0
$269$	−426.120	−0.0965836	−0.0482918	−	0.998833i	$-0.515378\pi$
$269$	−426.120	−0.0965836	−0.0482918	+	0.998833i	$0.515378\pi$
$270$	0	0
$271$	7527.32	1.68728	0.843638	−	0.536912i	$-0.180409\pi$
$271$	7527.32	1.68728	0.843638	+	0.536912i	$0.180409\pi$
$272$	0	0
$273$	−9375.99	−2.07861
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2409.09	0.522557	0.261279	−	0.965263i	$-0.415856\pi$
$277$	2409.09	0.522557	0.261279	+	0.965263i	$0.415856\pi$
$278$	0	0
$279$	−2701.13	−0.579613
$280$	0	0
$281$	−5487.57	−1.16499	−0.582493	−	0.812836i	$-0.697923\pi$
$281$	−5487.57	−1.16499	−0.582493	+	0.812836i	$0.697923\pi$
$282$	0	0
$283$	−4999.15	−1.05006	−0.525032	−	0.851082i	$-0.675947\pi$
$283$	−4999.15	−1.05006	−0.525032	+	0.851082i	$0.675947\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6333.91	−1.30271
$288$	0	0
$289$	5079.73	1.03394
$290$	0	0
$291$	−15477.4	−3.11787
$292$	0	0
$293$	8992.96	1.79309	0.896543	−	0.442956i	$-0.146070\pi$
$293$	8992.96	1.79309	0.896543	+	0.442956i	$0.146070\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−4062.37	−0.793679
$298$	0	0
$299$	1989.87	0.384873
$300$	0	0
$301$	3975.68	0.761310
$302$	0	0
$303$	−788.558	−0.149510
$304$	0	0
$305$	0	0
$306$	0	0
$307$	5584.49	1.03819	0.519094	−	0.854717i	$-0.326270\pi$
$307$	5584.49	1.03819	0.519094	+	0.854717i	$0.326270\pi$
$308$	0	0
$309$	−13939.3	−2.56628
$310$	0	0
$311$	−6218.04	−1.13374	−0.566869	−	0.823808i	$-0.691845\pi$
$311$	−6218.04	−1.13374	−0.566869	+	0.823808i	$0.691845\pi$
$312$	0	0
$313$	5046.00	0.911236	0.455618	−	0.890175i	$-0.349418\pi$
$313$	5046.00	0.911236	0.455618	+	0.890175i	$0.349418\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5370.16	0.951476	0.475738	−	0.879587i	$-0.342181\pi$
$317$	5370.16	0.951476	0.475738	+	0.879587i	$0.342181\pi$
$318$	0	0
$319$	8286.67	1.45443
$320$	0	0
$321$	6117.44	1.06368
$322$	0	0
$323$	−11670.5	−2.01041
$324$	0	0
$325$	0	0
$326$	0	0
$327$	17613.6	2.97870
$328$	0	0
$329$	−9626.19	−1.61310
$330$	0	0
$331$	9.92458	0.00164805	0.000824024	−	1.00000i	$-0.499738\pi$
$331$	9.92458	0.00164805	0.000824024		1.00000i	$0.499738\pi$
$332$	0	0
$333$	−14535.4	−2.39199
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−1684.61	−0.272304	−0.136152	−	0.990688i	$-0.543474\pi$
$337$	−1684.61	−0.272304	−0.136152	+	0.990688i	$0.543474\pi$
$338$	0	0
$339$	−5414.52	−0.867482
$340$	0	0
$341$	−2553.51	−0.405515
$342$	0	0
$343$	−4663.45	−0.734119
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8690.93	1.34453	0.672267	−	0.740308i	$-0.265320\pi$
$347$	8690.93	1.34453	0.672267	+	0.740308i	$0.265320\pi$
$348$	0	0
$349$	6861.60	1.05242	0.526208	−	0.850356i	$-0.323613\pi$
$349$	6861.60	1.05242	0.526208	+	0.850356i	$0.323613\pi$
$350$	0	0
$351$	−5653.16	−0.859668
$352$	0	0
$353$	−5010.05	−0.755405	−0.377703	−	0.925927i	$-0.623286\pi$
$353$	−5010.05	−0.755405	−0.377703	+	0.925927i	$0.623286\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	17772.3	2.63477
$358$	0	0
$359$	−2871.64	−0.422171	−0.211085	−	0.977468i	$-0.567700\pi$
$359$	−2871.64	−0.422171	−0.211085	+	0.977468i	$0.567700\pi$
$360$	0	0
$361$	6770.88	0.987152
$362$	0	0
$363$	−861.390	−0.124549
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−2096.96	−0.298257	−0.149129	−	0.988818i	$-0.547647\pi$
$367$	−2096.96	−0.298257	−0.149129	+	0.988818i	$0.547647\pi$
$368$	0	0
$369$	−11697.7	−1.65029
$370$	0	0
$371$	−9037.30	−1.26467
$372$	0	0
$373$	9530.96	1.32304	0.661521	−	0.749927i	$-0.269911\pi$
$373$	9530.96	1.32304	0.661521	+	0.749927i	$0.269911\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	11531.7	1.57536
$378$	0	0
$379$	9603.41	1.30157	0.650784	−	0.759263i	$-0.274440\pi$
$379$	9603.41	1.30157	0.650784	+	0.759263i	$0.274440\pi$
$380$	0	0
$381$	8105.66	1.08994
$382$	0	0
$383$	7688.07	1.02570	0.512849	−	0.858479i	$-0.328590\pi$
$383$	7688.07	1.02570	0.512849	+	0.858479i	$0.328590\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7342.41	0.964433
$388$	0	0
$389$	3579.00	0.466485	0.233242	−	0.972419i	$-0.425066\pi$
$389$	3579.00	0.466485	0.233242	+	0.972419i	$0.425066\pi$
$390$	0	0
$391$	−3771.83	−0.487850
$392$	0	0
$393$	52.6044	0.00675202
$394$	0	0
$395$	0	0
$396$	0	0
$397$	1262.66	0.159624	0.0798122	−	0.996810i	$-0.474568\pi$
$397$	1262.66	0.159624	0.0798122	+	0.996810i	$0.474568\pi$
$398$	0	0
$399$	−20756.2	−2.60429
$400$	0	0
$401$	1738.14	0.216455	0.108228	−	0.994126i	$-0.465482\pi$
$401$	1738.14	0.216455	0.108228	+	0.994126i	$0.465482\pi$
$402$	0	0
$403$	−3553.45	−0.439231
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−13741.1	−1.67351
$408$	0	0
$409$	2168.41	0.262154	0.131077	−	0.991372i	$-0.458156\pi$
$409$	2168.41	0.262154	0.131077	+	0.991372i	$0.458156\pi$
$410$	0	0
$411$	8467.16	1.01619
$412$	0	0
$413$	17986.4	2.14298
$414$	0	0
$415$	0	0
$416$	0	0
$417$	465.569	0.0546739
$418$	0	0
$419$	−7505.84	−0.875141	−0.437571	−	0.899184i	$-0.644161\pi$
$419$	−7505.84	−0.875141	−0.437571	+	0.899184i	$0.644161\pi$
$420$	0	0
$421$	−11097.3	−1.28468	−0.642341	−	0.766419i	$-0.722037\pi$
$421$	−11097.3	−1.28468	−0.642341	+	0.766419i	$0.722037\pi$
$422$	0	0
$423$	−17777.9	−2.04348
$424$	0	0
$425$	0	0
$426$	0	0
$427$	9615.44	1.08975
$428$	0	0
$429$	−16369.6	−1.84226
$430$	0	0
$431$	−8935.68	−0.998646	−0.499323	−	0.866416i	$-0.666418\pi$
$431$	−8935.68	−0.998646	−0.499323	+	0.866416i	$0.666418\pi$
$432$	0	0
$433$	6453.45	0.716243	0.358122	−	0.933675i	$-0.383417\pi$
$433$	6453.45	0.716243	0.358122	+	0.933675i	$0.383417\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4405.10	0.482207
$438$	0	0
$439$	−5254.88	−0.571303	−0.285651	−	0.958334i	$-0.592210\pi$
$439$	−5254.88	−0.571303	−0.285651	+	0.958334i	$0.592210\pi$
$440$	0	0
$441$	5137.40	0.554735
$442$	0	0
$443$	−242.163	−0.0259718	−0.0129859	−	0.999916i	$-0.504134\pi$
$443$	−242.163	−0.0259718	−0.0129859	+	0.999916i	$0.504134\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−4441.30	−0.469947
$448$	0	0
$449$	10154.7	1.06733	0.533666	−	0.845696i	$-0.320814\pi$
$449$	10154.7	1.06733	0.533666	+	0.845696i	$0.320814\pi$
$450$	0	0
$451$	−11058.4	−1.15459
$452$	0	0
$453$	3196.21	0.331504
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−967.379	−0.0990198	−0.0495099	−	0.998774i	$-0.515766\pi$
$457$	−967.379	−0.0990198	−0.0495099	+	0.998774i	$0.515766\pi$
$458$	0	0
$459$	10715.7	1.08968
$460$	0	0
$461$	17022.8	1.71980	0.859902	−	0.510460i	$-0.170525\pi$
$461$	17022.8	1.71980	0.859902	+	0.510460i	$0.170525\pi$
$462$	0	0
$463$	13976.3	1.40288	0.701440	−	0.712728i	$-0.252541\pi$
$463$	13976.3	1.40288	0.701440	+	0.712728i	$0.252541\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4454.27	−0.441369	−0.220684	−	0.975345i	$-0.570829\pi$
$467$	−4454.27	−0.441369	−0.220684	+	0.975345i	$0.570829\pi$
$468$	0	0
$469$	12372.6	1.21816
$470$	0	0
$471$	−30095.7	−2.94424
$472$	0	0
$473$	6941.16	0.674746
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−16690.4	−1.60210
$478$	0	0
$479$	−9617.44	−0.917394	−0.458697	−	0.888593i	$-0.651684\pi$
$479$	−9617.44	−0.917394	−0.458697	+	0.888593i	$0.651684\pi$
$480$	0	0
$481$	−19121.9	−1.81265
$482$	0	0
$483$	−6708.29	−0.631963
$484$	0	0
$485$	0	0
$486$	0	0
$487$	8346.22	0.776599	0.388299	−	0.921533i	$-0.373063\pi$
$487$	8346.22	0.776599	0.388299	+	0.921533i	$0.373063\pi$
$488$	0	0
$489$	9987.53	0.923623
$490$	0	0
$491$	11867.9	1.09081	0.545406	−	0.838172i	$-0.316375\pi$
$491$	11867.9	1.09081	0.545406	+	0.838172i	$0.316375\pi$
$492$	0	0
$493$	−21858.5	−1.99687
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7408.19	0.668617
$498$	0	0
$499$	17696.1	1.58755	0.793773	−	0.608214i	$-0.208114\pi$
$499$	17696.1	1.58755	0.793773	+	0.608214i	$0.208114\pi$
$500$	0	0
$501$	−5000.80	−0.445947
$502$	0	0
$503$	9911.58	0.878599	0.439300	−	0.898341i	$-0.355227\pi$
$503$	9911.58	0.878599	0.439300	+	0.898341i	$0.355227\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−4784.85	−0.419137
$508$	0	0
$509$	−3999.53	−0.348283	−0.174141	−	0.984721i	$-0.555715\pi$
$509$	−3999.53	−0.348283	−0.174141	+	0.984721i	$0.555715\pi$
$510$	0	0
$511$	10990.1	0.951413
$512$	0	0
$513$	−12514.8	−1.07708
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−16806.4	−1.42968
$518$	0	0
$519$	413.057	0.0349348
$520$	0	0
$521$	−11301.1	−0.950306	−0.475153	−	0.879903i	$-0.657607\pi$
$521$	−11301.1	−0.950306	−0.475153	+	0.879903i	$0.657607\pi$
$522$	0	0
$523$	−3377.88	−0.282417	−0.141209	−	0.989980i	$-0.545099\pi$
$523$	−3377.88	−0.282417	−0.141209	+	0.989980i	$0.545099\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6735.62	0.556752
$528$	0	0
$529$	−10743.3	−0.882987
$530$	0	0
$531$	33217.8	2.71474
$532$	0	0
$533$	−15388.8	−1.25059
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−3055.65	−0.245551
$538$	0	0
$539$	4856.65	0.388109
$540$	0	0
$541$	−19694.0	−1.56508	−0.782542	−	0.622598i	$-0.786077\pi$
$541$	−19694.0	−1.56508	−0.782542	+	0.622598i	$0.786077\pi$
$542$	0	0
$543$	1907.48	0.150751
$544$	0	0
$545$	0	0
$546$	0	0
$547$	11353.7	0.887474	0.443737	−	0.896157i	$-0.353652\pi$
$547$	11353.7	0.887474	0.443737	+	0.896157i	$0.353652\pi$
$548$	0	0
$549$	17758.1	1.38050
$550$	0	0
$551$	25528.4	1.97377
$552$	0	0
$553$	9263.23	0.712320
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−13675.1	−1.04027	−0.520137	−	0.854083i	$-0.674119\pi$
$557$	−13675.1	−1.04027	−0.520137	+	0.854083i	$0.674119\pi$
$558$	0	0
$559$	9659.27	0.730847
$560$	0	0
$561$	31028.8	2.33518
$562$	0	0
$563$	−8690.01	−0.650515	−0.325258	−	0.945625i	$-0.605451\pi$
$563$	−8690.01	−0.650515	−0.325258	+	0.945625i	$0.605451\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−4435.76	−0.328544
$568$	0	0
$569$	10774.4	0.793827	0.396914	−	0.917856i	$-0.370081\pi$
$569$	10774.4	0.793827	0.396914	+	0.917856i	$0.370081\pi$
$570$	0	0
$571$	−11320.2	−0.829657	−0.414828	−	0.909900i	$-0.636158\pi$
$571$	−11320.2	−0.829657	−0.414828	+	0.909900i	$0.636158\pi$
$572$	0	0
$573$	−22681.5	−1.65364
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−13525.2	−0.975842	−0.487921	−	0.872888i	$-0.662245\pi$
$577$	−13525.2	−0.975842	−0.487921	+	0.872888i	$0.662245\pi$
$578$	0	0
$579$	2517.92	0.180727
$580$	0	0
$581$	−21324.9	−1.52273
$582$	0	0
$583$	−15778.3	−1.12087
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3492.61	−0.245580	−0.122790	−	0.992433i	$-0.539184\pi$
$587$	−3492.61	−0.245580	−0.122790	+	0.992433i	$0.539184\pi$
$588$	0	0
$589$	−7866.50	−0.550312
$590$	0	0
$591$	−2596.53	−0.180722
$592$	0	0
$593$	−8431.49	−0.583878	−0.291939	−	0.956437i	$-0.594300\pi$
$593$	−8431.49	−0.583878	−0.291939	+	0.956437i	$0.594300\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−32083.5	−2.19948
$598$	0	0
$599$	−16783.6	−1.14484	−0.572420	−	0.819960i	$-0.693995\pi$
$599$	−16783.6	−1.14484	−0.572420	+	0.819960i	$0.693995\pi$
$600$	0	0
$601$	−10152.2	−0.689044	−0.344522	−	0.938778i	$-0.611959\pi$
$601$	−10152.2	−0.689044	−0.344522	+	0.938778i	$0.611959\pi$
$602$	0	0
$603$	22850.1	1.54317
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−14130.3	−0.944859	−0.472430	−	0.881368i	$-0.656623\pi$
$607$	−14130.3	−0.944859	−0.472430	+	0.881368i	$0.656623\pi$
$608$	0	0
$609$	−38875.9	−2.58675
$610$	0	0
$611$	−23387.7	−1.54855
$612$	0	0
$613$	5458.31	0.359639	0.179820	−	0.983700i	$-0.442449\pi$
$613$	5458.31	0.359639	0.179820	+	0.983700i	$0.442449\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−4490.56	−0.293004	−0.146502	−	0.989210i	$-0.546801\pi$
$617$	−4490.56	−0.293004	−0.146502	+	0.989210i	$0.546801\pi$
$618$	0	0
$619$	7386.94	0.479655	0.239827	−	0.970816i	$-0.422909\pi$
$619$	7386.94	0.479655	0.239827	+	0.970816i	$0.422909\pi$
$620$	0	0
$621$	−4044.70	−0.261366
$622$	0	0
$623$	20118.9	1.29382
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−36238.4	−2.30817
$628$	0	0
$629$	36246.0	2.29765
$630$	0	0
$631$	26268.5	1.65726	0.828631	−	0.559796i	$-0.189120\pi$
$631$	26268.5	1.65726	0.828631	+	0.559796i	$0.189120\pi$
$632$	0	0
$633$	16826.4	1.05654
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6758.48	0.420378
$638$	0	0
$639$	13681.7	0.847008
$640$	0	0
$641$	21533.8	1.32688	0.663442	−	0.748228i	$-0.269095\pi$
$641$	21533.8	1.32688	0.663442	+	0.748228i	$0.269095\pi$
$642$	0	0
$643$	8517.72	0.522405	0.261202	−	0.965284i	$-0.415881\pi$
$643$	8517.72	0.522405	0.261202	+	0.965284i	$0.415881\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−17887.7	−1.08692	−0.543459	−	0.839435i	$-0.682886\pi$
$647$	−17887.7	−1.08692	−0.543459	+	0.839435i	$0.682886\pi$
$648$	0	0
$649$	31402.5	1.89932
$650$	0	0
$651$	11979.5	0.721219
$652$	0	0
$653$	11322.5	0.678537	0.339269	−	0.940690i	$-0.389820\pi$
$653$	11322.5	0.678537	0.339269	+	0.940690i	$0.389820\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	20296.8	1.20526
$658$	0	0
$659$	−23067.9	−1.36358	−0.681788	−	0.731550i	$-0.738797\pi$
$659$	−23067.9	−1.36358	−0.681788	+	0.731550i	$0.738797\pi$
$660$	0	0
$661$	16492.0	0.970445	0.485222	−	0.874391i	$-0.338739\pi$
$661$	16492.0	0.970445	0.485222	+	0.874391i	$0.338739\pi$
$662$	0	0
$663$	43179.5	2.52934
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8250.63	0.478959
$668$	0	0
$669$	25292.2	1.46166
$670$	0	0
$671$	16787.6	0.965842
$672$	0	0
$673$	−6314.44	−0.361670	−0.180835	−	0.983513i	$-0.557880\pi$
$673$	−6314.44	−0.361670	−0.180835	+	0.983513i	$0.557880\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	12486.8	0.708872	0.354436	−	0.935080i	$-0.384673\pi$
$677$	12486.8	0.708872	0.354436	+	0.935080i	$0.384673\pi$
$678$	0	0
$679$	41016.5	2.31822
$680$	0	0
$681$	−29258.6	−1.64639
$682$	0	0
$683$	9593.65	0.537468	0.268734	−	0.963214i	$-0.413395\pi$
$683$	9593.65	0.537468	0.268734	+	0.963214i	$0.413395\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−14737.2	−0.818425
$688$	0	0
$689$	−21956.9	−1.21407
$690$	0	0
$691$	31333.7	1.72502	0.862510	−	0.506040i	$-0.168891\pi$
$691$	31333.7	1.72502	0.862510	+	0.506040i	$0.168891\pi$
$692$	0	0
$693$	32975.6	1.80756
$694$	0	0
$695$	0	0
$696$	0	0
$697$	29169.7	1.58520
$698$	0	0
$699$	−31949.0	−1.72879
$700$	0	0
$701$	4337.52	0.233703	0.116851	−	0.993149i	$-0.462720\pi$
$701$	4337.52	0.233703	0.116851	+	0.993149i	$0.462720\pi$
$702$	0	0
$703$	−42331.5	−2.27107
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2089.75	0.111164
$708$	0	0
$709$	−26437.5	−1.40040	−0.700198	−	0.713948i	$-0.746905\pi$
$709$	−26437.5	−1.40040	−0.700198	+	0.713948i	$0.746905\pi$
$710$	0	0
$711$	17107.6	0.902371
$712$	0	0
$713$	−2542.41	−0.133540
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−496.951	−0.0258842
$718$	0	0
$719$	26143.2	1.35602	0.678009	−	0.735053i	$-0.262843\pi$
$719$	26143.2	1.35602	0.678009	+	0.735053i	$0.262843\pi$
$720$	0	0
$721$	36940.5	1.90809
$722$	0	0
$723$	50458.8	2.59555
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1355.22	−0.0691366	−0.0345683	−	0.999402i	$-0.511006\pi$
$727$	−1355.22	−0.0691366	−0.0345683	+	0.999402i	$0.511006\pi$
$728$	0	0
$729$	−31898.3	−1.62060
$730$	0	0
$731$	−18309.3	−0.926394
$732$	0	0
$733$	632.549	0.0318741	0.0159371	−	0.999873i	$-0.494927\pi$
$733$	632.549	0.0318741	0.0159371	+	0.999873i	$0.494927\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	21601.4	1.07965
$738$	0	0
$739$	10611.4	0.528209	0.264104	−	0.964494i	$-0.414924\pi$
$739$	10611.4	0.528209	0.264104	+	0.964494i	$0.414924\pi$
$740$	0	0
$741$	−50429.1	−2.50008
$742$	0	0
$743$	5293.51	0.261373	0.130686	−	0.991424i	$-0.458282\pi$
$743$	5293.51	0.261373	0.130686	+	0.991424i	$0.458282\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−39383.5	−1.92900
$748$	0	0
$749$	−16211.8	−0.790876
$750$	0	0
$751$	19063.6	0.926287	0.463143	−	0.886283i	$-0.346721\pi$
$751$	19063.6	0.926287	0.463143	+	0.886283i	$0.346721\pi$
$752$	0	0
$753$	7215.24	0.349187
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−12698.5	−0.609687	−0.304844	−	0.952402i	$-0.598604\pi$
$757$	−12698.5	−0.609687	−0.304844	+	0.952402i	$0.598604\pi$
$758$	0	0
$759$	−11712.0	−0.560106
$760$	0	0
$761$	19828.4	0.944517	0.472259	−	0.881460i	$-0.343439\pi$
$761$	19828.4	0.944517	0.472259	+	0.881460i	$0.343439\pi$
$762$	0	0
$763$	−46677.7	−2.21474
$764$	0	0
$765$	0	0
$766$	0	0
$767$	43699.5	2.05723
$768$	0	0
$769$	−2141.74	−0.100433	−0.0502166	−	0.998738i	$-0.515991\pi$
$769$	−2141.74	−0.100433	−0.0502166	+	0.998738i	$0.515991\pi$
$770$	0	0
$771$	47881.5	2.23659
$772$	0	0
$773$	9705.16	0.451579	0.225789	−	0.974176i	$-0.427504\pi$
$773$	9705.16	0.451579	0.225789	+	0.974176i	$0.427504\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	64464.4	2.97638
$778$	0	0
$779$	−34067.2	−1.56686
$780$	0	0
$781$	12934.0	0.592592
$782$	0	0
$783$	−23439.8	−1.06982
$784$	0	0
$785$	0	0
$786$	0	0
$787$	24334.7	1.10221	0.551104	−	0.834437i	$-0.314207\pi$
$787$	24334.7	1.10221	0.551104	+	0.834437i	$0.314207\pi$
$788$	0	0
$789$	16333.1	0.736975
$790$	0	0
$791$	14349.0	0.644995
$792$	0	0
$793$	23361.6	1.04615
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−5913.01	−0.262797	−0.131399	−	0.991330i	$-0.541947\pi$
$797$	−5913.01	−0.262797	−0.131399	+	0.991330i	$0.541947\pi$
$798$	0	0
$799$	44331.7	1.96288
$800$	0	0
$801$	37156.3	1.63902
$802$	0	0
$803$	19187.6	0.843233
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3490.22	0.152245
$808$	0	0
$809$	−38547.6	−1.67523	−0.837614	−	0.546262i	$-0.816050\pi$
$809$	−38547.6	−1.67523	−0.837614	+	0.546262i	$0.816050\pi$
$810$	0	0
$811$	−11741.9	−0.508400	−0.254200	−	0.967152i	$-0.581812\pi$
$811$	−11741.9	−0.508400	−0.254200	+	0.967152i	$0.581812\pi$
$812$	0	0
$813$	−61654.0	−2.65965
$814$	0	0
$815$	0	0
$816$	0	0
$817$	21383.3	0.915677
$818$	0	0
$819$	45888.6	1.95785
$820$	0	0
$821$	−8066.49	−0.342902	−0.171451	−	0.985193i	$-0.554845\pi$
$821$	−8066.49	−0.342902	−0.171451	+	0.985193i	$0.554845\pi$
$822$	0	0
$823$	34620.2	1.46633	0.733163	−	0.680053i	$-0.238043\pi$
$823$	34620.2	1.46633	0.733163	+	0.680053i	$0.238043\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−37537.2	−1.57835	−0.789176	−	0.614167i	$-0.789492\pi$
$827$	−37537.2	−1.57835	−0.789176	+	0.614167i	$0.789492\pi$
$828$	0	0
$829$	−32392.5	−1.35710	−0.678551	−	0.734554i	$-0.737392\pi$
$829$	−32392.5	−1.35710	−0.678551	+	0.734554i	$0.737392\pi$
$830$	0	0
$831$	−19732.1	−0.823707
$832$	0	0
$833$	−12810.8	−0.532855
$834$	0	0
$835$	0	0
$836$	0	0
$837$	7222.92	0.298280
$838$	0	0
$839$	46962.7	1.93246	0.966229	−	0.257684i	$-0.0829594\pi$
$839$	46962.7	1.93246	0.966229	+	0.257684i	$0.0829594\pi$
$840$	0	0
$841$	23425.0	0.960475
$842$	0	0
$843$	44947.0	1.83637
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2282.76	0.0926052
$848$	0	0
$849$	40946.5	1.65522
$850$	0	0
$851$	−13681.3	−0.551103
$852$	0	0
$853$	−3879.81	−0.155735	−0.0778676	−	0.996964i	$-0.524811\pi$
$853$	−3879.81	−0.155735	−0.0778676	+	0.996964i	$0.524811\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−37194.7	−1.48255	−0.741276	−	0.671201i	$-0.765779\pi$
$857$	−37194.7	−1.48255	−0.741276	+	0.671201i	$0.765779\pi$
$858$	0	0
$859$	7966.82	0.316443	0.158221	−	0.987404i	$-0.449424\pi$
$859$	7966.82	0.316443	0.158221	+	0.987404i	$0.449424\pi$
$860$	0	0
$861$	51879.2	2.05347
$862$	0	0
$863$	−21821.7	−0.860742	−0.430371	−	0.902652i	$-0.641617\pi$
$863$	−21821.7	−0.860742	−0.430371	+	0.902652i	$0.641617\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−41606.5	−1.62979
$868$	0	0
$869$	16172.7	0.631326
$870$	0	0
$871$	30060.4	1.16941
$872$	0	0
$873$	75750.5	2.93673
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−9394.86	−0.361735	−0.180868	−	0.983507i	$-0.557891\pi$
$877$	−9394.86	−0.361735	−0.180868	+	0.983507i	$0.557891\pi$
$878$	0	0
$879$	−73658.6	−2.82644
$880$	0	0
$881$	−27007.4	−1.03280	−0.516402	−	0.856346i	$-0.672729\pi$
$881$	−27007.4	−1.03280	−0.516402	+	0.856346i	$0.672729\pi$
$882$	0	0
$883$	1376.56	0.0524633	0.0262317	−	0.999656i	$-0.491649\pi$
$883$	1376.56	0.0524633	0.0262317	+	0.999656i	$0.491649\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−49523.6	−1.87468	−0.937339	−	0.348420i	$-0.886718\pi$
$887$	−49523.6	−1.87468	−0.937339	+	0.348420i	$0.886718\pi$
$888$	0	0
$889$	−21480.8	−0.810395
$890$	0	0
$891$	−7744.42	−0.291187
$892$	0	0
$893$	−51774.8	−1.94018
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−16298.4	−0.606675
$898$	0	0
$899$	−14733.8	−0.546605
$900$	0	0
$901$	41619.7	1.53891
$902$	0	0
$903$	−32563.6	−1.20005
$904$	0	0
$905$	0	0
$906$	0	0
$907$	27592.7	1.01015	0.505073	−	0.863077i	$-0.331466\pi$
$907$	27592.7	1.01015	0.505073	+	0.863077i	$0.331466\pi$
$908$	0	0
$909$	3859.42	0.140824
$910$	0	0
$911$	−34374.2	−1.25013	−0.625065	−	0.780572i	$-0.714928\pi$
$911$	−34374.2	−1.25013	−0.625065	+	0.780572i	$0.714928\pi$
$912$	0	0
$913$	−37231.2	−1.34959
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−139.407	−0.00502030
$918$	0	0
$919$	−49860.3	−1.78970	−0.894852	−	0.446362i	$-0.852719\pi$
$919$	−49860.3	−1.78970	−0.894852	+	0.446362i	$0.852719\pi$
$920$	0	0
$921$	−45740.8	−1.63649
$922$	0	0
$923$	17998.9	0.641863
$924$	0	0
$925$	0	0
$926$	0	0
$927$	68222.8	2.41718
$928$	0	0
$929$	−33348.6	−1.17775	−0.588876	−	0.808223i	$-0.700429\pi$
$929$	−33348.6	−1.17775	−0.588876	+	0.808223i	$0.700429\pi$
$930$	0	0
$931$	14961.7	0.526691
$932$	0	0
$933$	50930.0	1.78711
$934$	0	0
$935$	0	0
$936$	0	0
$937$	20404.8	0.711415	0.355707	−	0.934597i	$-0.384240\pi$
$937$	20404.8	0.711415	0.355707	+	0.934597i	$0.384240\pi$
$938$	0	0
$939$	−41330.3	−1.43638
$940$	0	0
$941$	−47495.4	−1.64538	−0.822692	−	0.568488i	$-0.807529\pi$
$941$	−47495.4	−1.64538	−0.822692	+	0.568488i	$0.807529\pi$
$942$	0	0
$943$	−11010.3	−0.380218
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−38728.7	−1.32895	−0.664473	−	0.747312i	$-0.731344\pi$
$947$	−38728.7	−1.32895	−0.664473	+	0.747312i	$0.731344\pi$
$948$	0	0
$949$	26701.3	0.913343
$950$	0	0
$951$	−43985.3	−1.49981
$952$	0	0
$953$	−14200.9	−0.482699	−0.241349	−	0.970438i	$-0.577590\pi$
$953$	−14200.9	−0.482699	−0.241349	+	0.970438i	$0.577590\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−67873.6	−2.29262
$958$	0	0
$959$	−22438.7	−0.755563
$960$	0	0
$961$	−25250.8	−0.847599
$962$	0	0
$963$	−29940.4	−1.00189
$964$	0	0
$965$	0	0
$966$	0	0
$967$	38637.2	1.28489	0.642445	−	0.766332i	$-0.277920\pi$
$967$	38637.2	1.28489	0.642445	+	0.766332i	$0.277920\pi$
$968$	0	0
$969$	95589.2	3.16901
$970$	0	0
$971$	−18828.7	−0.622288	−0.311144	−	0.950363i	$-0.600712\pi$
$971$	−18828.7	−0.622288	−0.311144	+	0.950363i	$0.600712\pi$
$972$	0	0
$973$	−1233.80	−0.0406514
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−59016.1	−1.93254	−0.966271	−	0.257527i	$-0.917092\pi$
$977$	−59016.1	−1.93254	−0.966271	+	0.257527i	$0.917092\pi$
$978$	0	0
$979$	35125.7	1.14670
$980$	0	0
$981$	−86205.9	−2.80565
$982$	0	0
$983$	42329.5	1.37345	0.686724	−	0.726918i	$-0.259048\pi$
$983$	42329.5	1.37345	0.686724	+	0.726918i	$0.259048\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	78845.2	2.54273
$988$	0	0
$989$	6910.97	0.222200
$990$	0	0
$991$	−6895.28	−0.221025	−0.110513	−	0.993875i	$-0.535249\pi$
$991$	−6895.28	−0.221025	−0.110513	+	0.993875i	$0.535249\pi$
$992$	0	0
$993$	−81.2892	−0.00259782
$994$	0	0
$995$	0	0
$996$	0	0
$997$	34160.9	1.08514	0.542570	−	0.840010i	$-0.317451\pi$
$997$	34160.9	1.08514	0.542570	+	0.840010i	$0.317451\pi$
$998$	0	0
$999$	38868.2	1.23097

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.4.a.v.1.1	yes	3
4.3	odd	2		800.4.a.w.1.3	yes	3
5.2	odd	4		800.4.c.n.449.6		6
5.3	odd	4		800.4.c.n.449.1		6
5.4	even	2		800.4.a.x.1.3	yes	3
8.3	odd	2		1600.4.a.cr.1.1		3
8.5	even	2		1600.4.a.cs.1.3		3
20.3	even	4		800.4.c.m.449.6		6
20.7	even	4		800.4.c.m.449.1		6
20.19	odd	2		800.4.a.u.1.1	✓	3
40.19	odd	2		1600.4.a.ct.1.3		3
40.29	even	2		1600.4.a.cq.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
800.4.a.u.1.1	✓	3	20.19	odd	2
800.4.a.v.1.1	yes	3	1.1	even	1	trivial
800.4.a.w.1.3	yes	3	4.3	odd	2
800.4.a.x.1.3	yes	3	5.4	even	2
800.4.c.m.449.1		6	20.7	even	4
800.4.c.m.449.6		6	20.3	even	4
800.4.c.n.449.1		6	5.3	odd	4
800.4.c.n.449.6		6	5.2	odd	4
1600.4.a.cq.1.1		3	40.29	even	2
1600.4.a.cr.1.1		3	8.3	odd	2
1600.4.a.cs.1.3		3	8.5	even	2
1600.4.a.ct.1.3		3	40.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	800.a (trivial)

\(f(q)\)	\(=\)	\(q-8.19070 q^{3} +21.7061 q^{7} +40.0875 q^{9} +37.8968 q^{11} +52.7369 q^{13} -99.9636 q^{17} +116.747 q^{19} -177.788 q^{21} +37.7320 q^{23} -107.196 q^{27} +218.664 q^{29} -67.3808 q^{31} -310.401 q^{33} -362.591 q^{37} -431.952 q^{39} -291.803 q^{41} +183.160 q^{43} -443.479 q^{47} +128.155 q^{49} +818.772 q^{51} -416.349 q^{53} -956.240 q^{57} +828.632 q^{59} +442.983 q^{61} +870.143 q^{63} +570.007 q^{67} -309.051 q^{69} +341.295 q^{71} +506.312 q^{73} +822.591 q^{77} +426.757 q^{79} -204.356 q^{81} -982.438 q^{83} -1791.01 q^{87} +926.879 q^{89} +1144.71 q^{91} +551.895 q^{93} +1889.63 q^{97} +1519.19 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 5 q^{3} + 2 q^{7} + 18 q^{9} + 31 q^{11} + 8 q^{13} - 89 q^{17} + 87 q^{19} - 70 q^{21} + 122 q^{23} - 143 q^{27} + 84 q^{29} + 294 q^{31} - 209 q^{33} - 94 q^{37} - 32 q^{39} - 345 q^{41} + 412 q^{43}+ \cdots + 1954 q^{99}+O(q^{100}) \) 3 * q - 5 * q^3 + 2 * q^7 + 18 * q^9 + 31 * q^11 + 8 * q^13 - 89 * q^17 + 87 * q^19 - 70 * q^21 + 122 * q^23 - 143 * q^27 + 84 * q^29 + 294 * q^31 - 209 * q^33 - 94 * q^37 - 32 * q^39 - 345 * q^41 + 412 * q^43 - 824 * q^47 + 359 * q^49 + 1351 * q^51 + 74 * q^53 - 913 * q^57 + 1040 * q^59 + 482 * q^61 + 1532 * q^63 - 735 * q^67 - 614 * q^69 + 2048 * q^71 - 15 * q^73 + 1474 * q^77 + 998 * q^79 - 549 * q^81 + 1221 * q^83 - 2564 * q^87 + 1189 * q^89 + 4032 * q^91 + 454 * q^93 - 1010 * q^97 + 1954 * q^99

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