LMFDB - Embedded newform 800.4.d.b.401.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,4,Mod(401,800)]

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, 0]))

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

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

chi := DirichletCharacter("800.401");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	800.d (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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 10x^{9} - 32x^{8} + 56x^{7} + 160x^{6} + 448x^{5} - 2048x^{4} - 5120x^{3} - 32768x + 262144 $ x^12 - x^11 - 10x^9 - 32x^8 + 56x^7 + 160x^6 + 448x^5 - 2048x^4 - 5120x^3 - 32768x + 262144
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{35} $
Twist minimal:	no (minimal twist has level 200)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			401.11
Root			$-0.412714 - 2.79815i$ of defining polynomial
Character	$\chi$	$=$	800.401
Dual form			800.4.d.b.401.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.69300i q^{3} +15.6248 q^{7} -32.1823 q^{9} +O(q^{10})$	$q+7.69300i q^{3} +15.6248 q^{7} -32.1823 q^{9} -46.7640i q^{11} +50.4964i q^{13} +111.080 q^{17} -84.9682i q^{19} +120.201i q^{21} -53.2555 q^{23} -39.8675i q^{27} +229.789i q^{29} +338.615 q^{31} +359.755 q^{33} +71.1508i q^{37} -388.469 q^{39} -3.06105 q^{41} +115.970i q^{43} +574.015 q^{47} -98.8671 q^{49} +854.536i q^{51} -39.0500i q^{53} +653.660 q^{57} +109.940i q^{59} +180.738i q^{61} -502.841 q^{63} -755.845i q^{67} -409.695i q^{69} -127.821 q^{71} +347.937 q^{73} -730.676i q^{77} -456.703 q^{79} -562.221 q^{81} +499.423i q^{83} -1767.77 q^{87} +1301.32 q^{89} +788.995i q^{91} +2604.97i q^{93} -778.230 q^{97} +1504.97i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 28 q^{7} - 108 q^{9}+O(q^{10}) $ 12 * q - 28 * q^7 - 108 * q^9	$ 12 q - 28 q^{7} - 108 q^{9} - 112 q^{23} + 264 q^{31} + 116 q^{33} + 300 q^{39} - 20 q^{41} + 940 q^{47} + 228 q^{49} - 176 q^{57} + 1240 q^{63} - 628 q^{71} + 432 q^{73} - 1116 q^{79} + 1128 q^{81} - 4116 q^{87} - 424 q^{89} + 792 q^{97}+O(q^{100}) $ 12 * q - 28 * q^7 - 108 * q^9 - 112 * q^23 + 264 * q^31 + 116 * q^33 + 300 * q^39 - 20 * q^41 + 940 * q^47 + 228 * q^49 - 176 * q^57 + 1240 * q^63 - 628 * q^71 + 432 * q^73 - 1116 * q^79 + 1128 * q^81 - 4116 * q^87 - 424 * q^89 + 792 * q^97

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
$3$	7.69300i	1.48052i	0.672321	+	0.740260i	$0.265297\pi$
$3$	7.69300i	1.48052i	−0.672321	+	0.740260i	$0.734703\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	15.6248	0.843657	0.421829	−	0.906676i	$-0.361388\pi$
$7$	15.6248	0.843657	0.421829	+	0.906676i	$0.361388\pi$
$8$	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
$13$	50.4964i	1.07732i	0.842522	+	0.538661i	$0.181070\pi$
$13$	50.4964i	1.07732i	−0.842522	+	0.538661i	$0.818930\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	111.080	1.58475	0.792375	−	0.610034i	$-0.208844\pi$
$17$	111.080	1.58475	0.792375	+	0.610034i	$0.208844\pi$
$18$	0	0
$19$	− 84.9682i	− 1.02595i	−0.858404	−	0.512975i	$-0.828543\pi$
$19$	− 84.9682i	− 1.02595i	0.858404	−	0.512975i	$-0.171457\pi$
$20$	0	0
$21$	120.201i	1.24905i
$22$	0	0
$23$	−53.2555	−0.482806	−0.241403	−	0.970425i	$-0.577608\pi$
$23$	−53.2555	−0.482806	−0.241403	+	0.970425i	$0.577608\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 39.8675i	− 0.284167i
$28$	0	0
$29$	229.789i	1.47141i	0.677304	+	0.735703i	$0.263148\pi$
$29$	229.789i	1.47141i	−0.677304	+	0.735703i	$0.736852\pi$
$30$	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
$33$	359.755	1.89774
$34$	0	0
$35$	0	0
$36$	0	0
$37$	71.1508i	0.316138i	0.987428	+	0.158069i	$0.0505269\pi$
$37$	71.1508i	0.316138i	−0.987428	+	0.158069i	$0.949473\pi$
$38$	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
$43$	115.970i	0.411285i	0.978627	+	0.205643i	$0.0659285\pi$
$43$	115.970i	0.411285i	−0.978627	+	0.205643i	$0.934072\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	574.015	1.78146	0.890730	−	0.454532i	$-0.150194\pi$
$47$	574.015	1.78146	0.890730	+	0.454532i	$0.150194\pi$
$48$	0	0
$49$	−98.8671	−0.288242
$50$	0	0
$51$	854.536i	2.34625i
$52$	0	0
$53$	− 39.0500i	− 0.101206i	−0.998719	−	0.0506031i	$-0.983886\pi$
$53$	− 39.0500i	− 0.101206i	0.998719	−	0.0506031i	$-0.0161144\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	653.660	1.51894
$58$	0	0
$59$	109.940i	0.242593i	0.992616	+	0.121297i	$0.0387053\pi$
$59$	109.940i	0.242593i	−0.992616	+	0.121297i	$0.961295\pi$
$60$	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
$63$	−502.841	−1.00559
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 755.845i	− 1.37823i	−0.724654	−	0.689113i	$-0.758000\pi$
$67$	− 755.845i	− 1.37823i	0.724654	−	0.689113i	$-0.242000\pi$
$68$	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
$73$	347.937	0.557849	0.278924	−	0.960313i	$-0.410022\pi$
$73$	347.937	0.557849	0.278924	+	0.960313i	$0.410022\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 730.676i	− 1.08141i
$78$	0	0
$79$	−456.703	−0.650419	−0.325210	−	0.945642i	$-0.605435\pi$
$79$	−456.703	−0.650419	−0.325210	+	0.945642i	$0.605435\pi$
$80$	0	0
$81$	−562.221	−0.771223
$82$	0	0
$83$	499.423i	0.660467i	0.943899	+	0.330233i	$0.107127\pi$
$83$	499.423i	0.660467i	−0.943899	+	0.330233i	$0.892873\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1767.77	−2.17844
$88$	0	0
$89$	1301.32	1.54989	0.774944	−	0.632030i	$-0.217778\pi$
$89$	1301.32	1.54989	0.774944	+	0.632030i	$0.217778\pi$
$90$	0	0
$91$	788.995i	0.908891i
$92$	0	0
$93$	2604.97i	2.90454i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−778.230	−0.814611	−0.407306	−	0.913292i	$-0.633532\pi$
$97$	−778.230	−0.814611	−0.407306	+	0.913292i	$0.633532\pi$
$98$	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
$103$	−852.432	−0.815462	−0.407731	−	0.913102i	$-0.633680\pi$
$103$	−852.432	−0.815462	−0.407731	+	0.913102i	$0.633680\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	42.6157i	0.0385030i	0.999815	+	0.0192515i	$0.00612832\pi$
$107$	42.6157i	0.0385030i	−0.999815	+	0.0192515i	$0.993872\pi$
$108$	0	0
$109$	1490.77i	1.31000i	0.755628	+	0.655001i	$0.227332\pi$
$109$	1490.77i	1.31000i	−0.755628	+	0.655001i	$0.772668\pi$
$110$	0	0
$111$	−547.363	−0.468049
$112$	0	0
$113$	−846.817	−0.704972	−0.352486	−	0.935817i	$-0.614664\pi$
$113$	−846.817	−0.704972	−0.352486	+	0.935817i	$0.614664\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 1625.09i	− 1.28410i
$118$	0	0
$119$	1735.59	1.33699
$120$	0	0
$121$	−855.870	−0.643028
$122$	0	0
$123$	− 23.5487i	− 0.0172627i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−65.6584	−0.0458759	−0.0229380	−	0.999737i	$-0.507302\pi$
$127$	−65.6584	−0.0458759	−0.0229380	+	0.999737i	$0.507302\pi$
$128$	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
$133$	− 1327.61i	− 0.865550i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1794.01	−1.11878	−0.559388	−	0.828906i	$-0.688964\pi$
$137$	−1794.01	−1.11878	−0.559388	+	0.828906i	$0.688964\pi$
$138$	0	0
$139$	1.37996i	0 0.000842065i	1.00000		0.000421032i	$0.000134019\pi$
$139$	1.37996i	0 0.000842065i	−1.00000		0.000421032i	$0.999866\pi$
$140$	0	0
$141$	4415.90i	2.63749i
$142$	0	0
$143$	2361.41	1.38092
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 760.585i	− 0.426748i
$148$	0	0
$149$	743.959i	0.409044i	0.978862	+	0.204522i	$0.0655640\pi$
$149$	743.959i	0.409044i	−0.978862	+	0.204522i	$0.934436\pi$
$150$	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
$153$	−3574.80	−1.88892
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 1497.53i	− 0.761249i	−0.924730	−	0.380624i	$-0.875709\pi$
$157$	− 1497.53i	− 0.761249i	0.924730	−	0.380624i	$-0.124291\pi$
$158$	0	0
$159$	300.412	0.149838
$160$	0	0
$161$	−832.104	−0.407323
$162$	0	0
$163$	2895.86i	1.39154i	0.718263	+	0.695772i	$0.244937\pi$
$163$	2895.86i	1.39154i	−0.718263	+	0.695772i	$0.755063\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1923.39	0.891237	0.445619	−	0.895223i	$-0.352984\pi$
$167$	1923.39	0.891237	0.445619	+	0.895223i	$0.352984\pi$
$168$	0	0
$169$	−352.891	−0.160624
$170$	0	0
$171$	2734.47i	1.22287i
$172$	0	0
$173$	− 351.016i	− 0.154262i	−0.997021	−	0.0771309i	$-0.975424\pi$
$173$	− 351.016i	− 0.154262i	0.997021	−	0.0771309i	$-0.0245759\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−845.772	−0.359164
$178$	0	0
$179$	− 2001.39i	− 0.835705i	−0.908515	−	0.417853i	$-0.862783\pi$
$179$	− 2001.39i	− 0.835705i	0.908515	−	0.417853i	$-0.137217\pi$
$180$	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
$183$	−1390.42	−0.561655
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 5194.52i	− 2.03134i
$188$	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
$193$	2844.03	1.06071	0.530356	−	0.847775i	$-0.322058\pi$
$193$	2844.03	1.06071	0.530356	+	0.847775i	$0.322058\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4472.05i	1.61736i	0.588247	+	0.808681i	$0.299818\pi$
$197$	4472.05i	1.61736i	−0.588247	+	0.808681i	$0.700182\pi$
$198$	0	0
$199$	1382.43	0.492452	0.246226	−	0.969212i	$-0.420809\pi$
$199$	1382.43	0.492452	0.246226	+	0.969212i	$0.420809\pi$
$200$	0	0
$201$	5814.72	2.04049
$202$	0	0
$203$	3590.40i	1.24136i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1713.88	0.575474
$208$	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
$213$	− 983.326i	− 0.316321i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	5290.78	1.65512
$218$	0	0
$219$	2676.68i	0.825906i
$220$	0	0
$221$	5609.12i	1.70729i
$222$	0	0
$223$	4547.35	1.36553	0.682765	−	0.730638i	$-0.260777\pi$
$223$	4547.35	1.36553	0.682765	+	0.730638i	$0.260777\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3747.56i	1.09575i	0.836561	+	0.547873i	$0.184562\pi$
$227$	3747.56i	1.09575i	−0.836561	+	0.547873i	$0.815438\pi$
$228$	0	0
$229$	− 2061.75i	− 0.594954i	−0.954729	−	0.297477i	$-0.903855\pi$
$229$	− 2061.75i	− 0.594954i	0.954729	−	0.297477i	$-0.0961451\pi$
$230$	0	0
$231$	5621.09	1.60104
$232$	0	0
$233$	809.747	0.227675	0.113837	−	0.993499i	$-0.463686\pi$
$233$	809.747	0.227675	0.113837	+	0.993499i	$0.463686\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 3513.42i	− 0.962958i
$238$	0	0
$239$	−1159.14	−0.313717	−0.156858	−	0.987621i	$-0.550137\pi$
$239$	−1159.14	−0.313717	−0.156858	+	0.987621i	$0.550137\pi$
$240$	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
$243$	− 5401.59i	− 1.42598i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4290.59	1.10528
$248$	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
$253$	2490.44i	0.618864i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2378.19	0.577226	0.288613	−	0.957446i	$-0.406806\pi$
$257$	2378.19	0.577226	0.288613	+	0.957446i	$0.406806\pi$
$258$	0	0
$259$	1111.71i	0.266712i
$260$	0	0
$261$	− 7395.14i	− 1.75382i
$262$	0	0
$263$	3933.72	0.922296	0.461148	−	0.887323i	$-0.347438\pi$
$263$	3933.72	0.922296	0.461148	+	0.887323i	$0.347438\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	10011.1i	2.29464i
$268$	0	0
$269$	3629.38i	0.822628i	0.911494	+	0.411314i	$0.134930\pi$
$269$	3629.38i	0.822628i	−0.911494	+	0.411314i	$0.865070\pi$
$270$	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
$273$	−6069.74	−1.34563
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5566.24i	1.20737i	0.797221	+	0.603687i	$0.206302\pi$
$277$	5566.24i	1.20737i	−0.797221	+	0.603687i	$0.793698\pi$
$278$	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
$283$	− 2632.57i	− 0.552968i	−0.961018	−	0.276484i	$-0.910831\pi$
$283$	− 2632.57i	− 0.552968i	0.961018	−	0.276484i	$-0.0891693\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−47.8282	−0.00983696
$288$	0	0
$289$	7425.67	1.51143
$290$	0	0
$291$	− 5986.93i	− 1.20605i
$292$	0	0
$293$	− 569.395i	− 0.113530i	−0.998388	−	0.0567652i	$-0.981921\pi$
$293$	− 569.395i	− 0.113530i	0.998388	−	0.0567652i	$-0.0180787\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1864.36	−0.364247
$298$	0	0
$299$	− 2689.21i	− 0.520138i
$300$	0	0
$301$	1812.00i	0.346984i
$302$	0	0
$303$	−8824.79	−1.67317
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 527.093i	− 0.0979895i	−0.998799	−	0.0489947i	$-0.984398\pi$
$307$	− 527.093i	− 0.0979895i	0.998799	−	0.0489947i	$-0.0156018\pi$
$308$	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
$313$	9479.08	1.71179	0.855893	−	0.517153i	$-0.173008\pi$
$313$	9479.08	1.71179	0.855893	+	0.517153i	$0.173008\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 9286.72i	− 1.64541i	−0.568471	−	0.822704i	$-0.692465\pi$
$317$	− 9286.72i	− 1.64541i	0.568471	−	0.822704i	$-0.307535\pi$
$318$	0	0
$319$	10745.9	1.88606
$320$	0	0
$321$	−327.843	−0.0570044
$322$	0	0
$323$	− 9438.23i	− 1.62587i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−11468.5	−1.93948
$328$	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
$333$	− 2289.80i	− 0.376817i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	62.1362	0.0100438	0.00502192	−	0.999987i	$-0.498401\pi$
$337$	62.1362	0.0100438	0.00502192	+	0.999987i	$0.498401\pi$
$338$	0	0
$339$	− 6514.57i	− 1.04372i
$340$	0	0
$341$	− 15835.0i	− 2.51470i
$342$	0	0
$343$	−6904.06	−1.08684
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2384.20i	− 0.368849i	−0.982847	−	0.184424i	$-0.940958\pi$
$347$	− 2384.20i	− 0.368849i	0.982847	−	0.184424i	$-0.0590421\pi$
$348$	0	0
$349$	− 7940.30i	− 1.21786i	−0.793223	−	0.608932i	$-0.791598\pi$
$349$	− 7940.30i	− 1.21786i	0.793223	−	0.608932i	$-0.208402\pi$
$350$	0	0
$351$	2013.17	0.306139
$352$	0	0
$353$	8718.60	1.31457	0.657286	−	0.753641i	$-0.271704\pi$
$353$	8718.60	1.31457	0.657286	+	0.753641i	$0.271704\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	13351.9i	1.97943i
$358$	0	0
$359$	−5920.09	−0.870336	−0.435168	−	0.900349i	$-0.643311\pi$
$359$	−5920.09	−0.870336	−0.435168	+	0.900349i	$0.643311\pi$
$360$	0	0
$361$	−360.591	−0.0525719
$362$	0	0
$363$	− 6584.21i	− 0.952015i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−8779.02	−1.24867	−0.624334	−	0.781158i	$-0.714629\pi$
$367$	−8779.02	−1.24867	−0.624334	+	0.781158i	$0.714629\pi$
$368$	0	0
$369$	98.5117	0.0138979
$370$	0	0
$371$	− 610.147i	− 0.0853834i
$372$	0	0
$373$	− 10346.8i	− 1.43629i	−0.695893	−	0.718145i	$-0.744991\pi$
$373$	− 10346.8i	− 1.43629i	0.695893	−	0.718145i	$-0.255009\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−11603.5	−1.58518
$378$	0	0
$379$	− 5650.98i	− 0.765888i	−0.923772	−	0.382944i	$-0.874910\pi$
$379$	− 5650.98i	− 0.765888i	0.923772	−	0.382944i	$-0.125090\pi$
$380$	0	0
$381$	− 505.110i	− 0.0679202i
$382$	0	0
$383$	4349.97	0.580348	0.290174	−	0.956974i	$-0.406287\pi$
$383$	4349.97	0.580348	0.290174	+	0.956974i	$0.406287\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 3732.18i	− 0.490226i
$388$	0	0
$389$	3876.59i	0.505273i	0.967561	+	0.252636i	$0.0812976\pi$
$389$	3876.59i	0.505273i	−0.967561	+	0.252636i	$0.918702\pi$
$390$	0	0
$391$	−5915.60	−0.765127
$392$	0	0
$393$	−4972.00	−0.638178
$394$	0	0
$395$	0	0
$396$	0	0
$397$	10301.1i	1.30226i	0.758966	+	0.651130i	$0.225705\pi$
$397$	10301.1i	1.30226i	−0.758966	+	0.651130i	$0.774295\pi$
$398$	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
$403$	17098.9i	2.11354i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3327.29	0.405228
$408$	0	0
$409$	−4045.02	−0.489031	−0.244515	−	0.969645i	$-0.578629\pi$
$409$	−4045.02	−0.489031	−0.244515	+	0.969645i	$0.578629\pi$
$410$	0	0
$411$	− 13801.3i	− 1.65637i
$412$	0	0
$413$	1717.79i	0.204666i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−10.6161	−0.00124669
$418$	0	0
$419$	10923.6i	1.27363i	0.771016	+	0.636815i	$0.219749\pi$
$419$	10923.6i	1.27363i	−0.771016	+	0.636815i	$0.780251\pi$
$420$	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
$423$	−18473.1	−2.12339
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2823.99i	0.320053i
$428$	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
$433$	−4671.88	−0.518513	−0.259257	−	0.965808i	$-0.583478\pi$
$433$	−4671.88	−0.518513	−0.259257	+	0.965808i	$0.583478\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4525.02i	0.495334i
$438$	0	0
$439$	1304.49	0.141823	0.0709113	−	0.997483i	$-0.477409\pi$
$439$	1304.49	0.141823	0.0709113	+	0.997483i	$0.477409\pi$
$440$	0	0
$441$	3181.77	0.343567
$442$	0	0
$443$	− 8613.52i	− 0.923794i	−0.886934	−	0.461897i	$-0.847169\pi$
$443$	− 8613.52i	− 0.923794i	0.886934	−	0.461897i	$-0.152831\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−5723.28	−0.605597
$448$	0	0
$449$	16155.3	1.69803	0.849016	−	0.528367i	$-0.177195\pi$
$449$	16155.3	1.69803	0.849016	+	0.528367i	$0.177195\pi$
$450$	0	0
$451$	143.147i	0.0149457i
$452$	0	0
$453$	− 18550.3i	− 1.92399i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1325.32	−0.135659	−0.0678293	−	0.997697i	$-0.521607\pi$
$457$	−1325.32	−0.135659	−0.0678293	+	0.997697i	$0.521607\pi$
$458$	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
$463$	12348.5	1.23949	0.619743	−	0.784805i	$-0.287237\pi$
$463$	12348.5	1.23949	0.619743	+	0.784805i	$0.287237\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11587.9i	1.14823i	0.818776	+	0.574113i	$0.194653\pi$
$467$	11587.9i	1.14823i	−0.818776	+	0.574113i	$0.805347\pi$
$468$	0	0
$469$	− 11809.9i	− 1.16275i
$470$	0	0
$471$	11520.5	1.12704
$472$	0	0
$473$	5423.22	0.527188
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1256.72i	0.120631i
$478$	0	0
$479$	−7554.03	−0.720568	−0.360284	−	0.932843i	$-0.617320\pi$
$479$	−7554.03	−0.720568	−0.360284	+	0.932843i	$0.617320\pi$
$480$	0	0
$481$	−3592.86	−0.340583
$482$	0	0
$483$	− 6401.38i	− 0.603049i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1536.22	0.142942	0.0714710	−	0.997443i	$-0.477231\pi$
$487$	1536.22	0.142942	0.0714710	+	0.997443i	$0.477231\pi$
$488$	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
$493$	25524.9i	2.33181i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1997.17	−0.180252
$498$	0	0
$499$	− 17543.4i	− 1.57385i	−0.617049	−	0.786925i	$-0.711672\pi$
$499$	− 17543.4i	− 1.57385i	0.617049	−	0.786925i	$-0.288328\pi$
$500$	0	0
$501$	14796.7i	1.31949i
$502$	0	0
$503$	3452.08	0.306005	0.153002	−	0.988226i	$-0.451106\pi$
$503$	3452.08	0.306005	0.153002	+	0.988226i	$0.451106\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 2714.79i	− 0.237807i
$508$	0	0
$509$	2687.40i	0.234021i	0.993131	+	0.117011i	$0.0373311\pi$
$509$	2687.40i	0.234021i	−0.993131	+	0.117011i	$0.962669\pi$
$510$	0	0
$511$	5436.43	0.470633
$512$	0	0
$513$	−3387.47	−0.291541
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 26843.2i	− 2.28349i
$518$	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
$523$	− 1002.24i	− 0.0837952i	−0.999122	−	0.0418976i	$-0.986660\pi$
$523$	− 1002.24i	− 0.0837952i	0.999122	−	0.0418976i	$-0.0133403\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	37613.2	3.10903
$528$	0	0
$529$	−9330.85	−0.766898
$530$	0	0
$531$	− 3538.13i	− 0.289156i
$532$	0	0
$533$	− 154.572i	− 0.0125615i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	15396.7	1.23728
$538$	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
$543$	33028.8	2.61032
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1675.49i	0.130967i	0.997854	+	0.0654833i	$0.0208589\pi$
$547$	1675.49i	0.130967i	−0.997854	+	0.0654833i	$0.979141\pi$
$548$	0	0
$549$	− 5816.58i	− 0.452177i
$550$	0	0
$551$	19524.8	1.50959
$552$	0	0
$553$	−7135.87	−0.548731
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 6407.34i	− 0.487411i	−0.969849	−	0.243705i	$-0.921637\pi$
$557$	− 6407.34i	− 0.487411i	0.969849	−	0.243705i	$-0.0783630\pi$
$558$	0	0
$559$	−5856.08	−0.443087
$560$	0	0
$561$	39961.5	3.00744
$562$	0	0
$563$	5627.44i	0.421258i	0.977566	+	0.210629i	$0.0675512\pi$
$563$	5627.44i	0.421258i	−0.977566	+	0.210629i	$0.932449\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−8784.57	−0.650648
$568$	0	0
$569$	8395.02	0.618520	0.309260	−	0.950978i	$-0.399919\pi$
$569$	8395.02	0.618520	0.309260	+	0.950978i	$0.399919\pi$
$570$	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
$573$	− 3407.59i	− 0.248437i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	19751.5	1.42507	0.712537	−	0.701635i	$-0.247546\pi$
$577$	19751.5	1.42507	0.712537	+	0.701635i	$0.247546\pi$
$578$	0	0
$579$	21879.1i	1.57041i
$580$	0	0
$581$	7803.35i	0.557208i
$582$	0	0
$583$	−1826.13	−0.129727
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8029.42i	0.564582i	0.959329	+	0.282291i	$0.0910944\pi$
$587$	8029.42i	0.564582i	−0.959329	+	0.282291i	$0.908906\pi$
$588$	0	0
$589$	− 28771.5i	− 2.01275i
$590$	0	0
$591$	−34403.5	−2.39454
$592$	0	0
$593$	−18749.1	−1.29837	−0.649186	−	0.760629i	$-0.724890\pi$
$593$	−18749.1	−1.29837	−0.649186	+	0.760629i	$0.724890\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10635.1i	0.729085i
$598$	0	0
$599$	2130.81	0.145346	0.0726732	−	0.997356i	$-0.476847\pi$
$599$	2130.81	0.145346	0.0726732	+	0.997356i	$0.476847\pi$
$600$	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
$603$	24324.8i	1.64276i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	5643.07	0.377340	0.188670	−	0.982041i	$-0.439582\pi$
$607$	5643.07	0.377340	0.188670	+	0.982041i	$0.439582\pi$
$608$	0	0
$609$	−27620.9	−1.83786
$610$	0	0
$611$	28985.7i	1.91921i
$612$	0	0
$613$	11519.3i	0.758988i	0.925194	+	0.379494i	$0.123902\pi$
$613$	11519.3i	0.758988i	−0.925194	+	0.379494i	$0.876098\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1610.86	−0.105107	−0.0525534	−	0.998618i	$-0.516736\pi$
$617$	−1610.86	−0.105107	−0.0525534	+	0.998618i	$0.516736\pi$
$618$	0	0
$619$	17227.9i	1.11866i	0.828946	+	0.559328i	$0.188941\pi$
$619$	17227.9i	1.11866i	−0.828946	+	0.559328i	$0.811059\pi$
$620$	0	0
$621$	2123.16i	0.137197i
$622$	0	0
$623$	20332.9	1.30757
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 30567.8i	− 1.94698i
$628$	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
$633$	2694.52	0.169190
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 4992.44i	− 0.310530i
$638$	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
$643$	− 17840.2i	− 1.09417i	−0.837078	−	0.547083i	$-0.815738\pi$
$643$	− 17840.2i	− 1.09417i	0.837078	−	0.547083i	$-0.184262\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3785.71	−0.230033	−0.115017	−	0.993364i	$-0.536692\pi$
$647$	−3785.71	−0.230033	−0.115017	+	0.993364i	$0.536692\pi$
$648$	0	0
$649$	5141.25	0.310958
$650$	0	0
$651$	40702.0i	2.45044i
$652$	0	0
$653$	− 17813.4i	− 1.06752i	−0.845636	−	0.533761i	$-0.820778\pi$
$653$	− 17813.4i	− 1.06752i	0.845636	−	0.533761i	$-0.179222\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−11197.4	−0.664921
$658$	0	0
$659$	5929.18i	0.350483i	0.984525	+	0.175241i	$0.0560706\pi$
$659$	5929.18i	0.350483i	−0.984525	+	0.175241i	$0.943929\pi$
$660$	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
$663$	−43151.0	−2.52767
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 12237.5i	− 0.710404i
$668$	0	0
$669$	34982.8i	2.02169i
$670$	0	0
$671$	8452.04	0.486270
$672$	0	0
$673$	12240.5	0.701094	0.350547	−	0.936545i	$-0.385996\pi$
$673$	12240.5	0.701094	0.350547	+	0.936545i	$0.385996\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3035.77i	0.172340i	0.996280	+	0.0861701i	$0.0274628\pi$
$677$	3035.77i	0.172340i	−0.996280	+	0.0861701i	$0.972537\pi$
$678$	0	0
$679$	−12159.7	−0.687253
$680$	0	0
$681$	−28830.0	−1.62227
$682$	0	0
$683$	18824.7i	1.05462i	0.849673	+	0.527311i	$0.176800\pi$
$683$	18824.7i	1.05462i	−0.849673	+	0.527311i	$0.823200\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	15861.1	0.880841
$688$	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
$693$	23514.8i	1.28897i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−340.020	−0.0184780
$698$	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
$703$	6045.55	0.324342
$704$	0	0
$705$	0	0
$706$	0	0
$707$	17923.4i	0.953438i
$708$	0	0
$709$	10236.0i	0.542200i	0.962551	+	0.271100i	$0.0873874\pi$
$709$	10236.0i	0.542200i	−0.962551	+	0.271100i	$0.912613\pi$
$710$	0	0
$711$	14697.8	0.775259
$712$	0	0
$713$	−18033.1	−0.947189
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 8917.24i	− 0.464464i
$718$	0	0
$719$	17107.9	0.887367	0.443684	−	0.896184i	$-0.353671\pi$
$719$	17107.9	0.887367	0.443684	+	0.896184i	$0.353671\pi$
$720$	0	0
$721$	−13319.0	−0.687971
$722$	0	0
$723$	28713.4i	1.47699i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−29399.2	−1.49980	−0.749902	−	0.661549i	$-0.769899\pi$
$727$	−29399.2	−1.49980	−0.749902	+	0.661549i	$0.769899\pi$
$728$	0	0
$729$	26374.5	1.33996
$730$	0	0
$731$	12881.9i	0.651785i
$732$	0	0
$733$	17312.7i	0.872386i	0.899853	+	0.436193i	$0.143673\pi$
$733$	17312.7i	0.872386i	−0.899853	+	0.436193i	$0.856327\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−35346.3	−1.76662
$738$	0	0
$739$	− 20459.6i	− 1.01843i	−0.860639	−	0.509215i	$-0.829936\pi$
$739$	− 20459.6i	− 1.01843i	0.860639	−	0.509215i	$-0.170064\pi$
$740$	0	0
$741$	33007.5i	1.63639i
$742$	0	0
$743$	−15382.4	−0.759524	−0.379762	−	0.925084i	$-0.623994\pi$
$743$	−15382.4	−0.759524	−0.379762	+	0.925084i	$0.623994\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 16072.6i	− 0.787235i
$748$	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
$753$	27768.9	1.34390
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 30322.2i	− 1.45585i	−0.685657	−	0.727924i	$-0.740485\pi$
$757$	− 30322.2i	− 1.45585i	0.685657	−	0.727924i	$-0.259515\pi$
$758$	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
$763$	23293.0i	1.10519i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5551.60	−0.261351
$768$	0	0
$769$	18385.0	0.862131	0.431066	−	0.902321i	$-0.358138\pi$
$769$	18385.0	0.862131	0.431066	+	0.902321i	$0.358138\pi$
$770$	0	0
$771$	18295.4i	0.854594i
$772$	0	0
$773$	16712.1i	0.777609i	0.921320	+	0.388804i	$0.127112\pi$
$773$	16712.1i	0.777609i	−0.921320	+	0.388804i	$0.872888\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−8552.41	−0.394873
$778$	0	0
$779$	260.092i	0.0119625i
$780$	0	0
$781$	5977.41i	0.273865i
$782$	0	0
$783$	9161.12	0.418125
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 33805.6i	− 1.53118i	−0.643328	−	0.765591i	$-0.722447\pi$
$787$	− 33805.6i	− 1.53118i	0.643328	−	0.765591i	$-0.277553\pi$
$788$	0	0
$789$	30262.1i	1.36548i
$790$	0	0
$791$	−13231.3	−0.594755
$792$	0	0
$793$	−9126.64	−0.408697
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8399.83i	0.373321i	0.982424	+	0.186661i	$0.0597665\pi$
$797$	8399.83i	0.373321i	−0.982424	+	0.186661i	$0.940234\pi$
$798$	0	0
$799$	63761.3	2.82317
$800$	0	0
$801$	−41879.6	−1.84737
$802$	0	0
$803$	− 16270.9i	− 0.715054i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−27920.8	−1.21792
$808$	0	0
$809$	−12503.9	−0.543402	−0.271701	−	0.962382i	$-0.587586\pi$
$809$	−12503.9	−0.543402	−0.271701	+	0.962382i	$0.587586\pi$
$810$	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
$813$	− 30924.3i	− 1.33403i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	9853.76	0.421958
$818$	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
$823$	−39096.4	−1.65591	−0.827955	−	0.560794i	$-0.810496\pi$
$823$	−39096.4	−1.65591	−0.827955	+	0.560794i	$0.810496\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 24525.7i	− 1.03125i	−0.856815	−	0.515625i	$-0.827560\pi$
$827$	− 24525.7i	− 1.03125i	0.856815	−	0.515625i	$-0.172440\pi$
$828$	0	0
$829$	− 22818.8i	− 0.956007i	−0.878358	−	0.478004i	$-0.841361\pi$
$829$	− 22818.8i	− 0.956007i	0.878358	−	0.478004i	$-0.158639\pi$
$830$	0	0
$831$	−42821.1	−1.78754
$832$	0	0
$833$	−10982.1	−0.456792
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 13499.7i	− 0.557490i
$838$	0	0
$839$	−43262.8	−1.78021	−0.890107	−	0.455752i	$-0.849370\pi$
$839$	−43262.8	−1.78021	−0.890107	+	0.455752i	$0.849370\pi$
$840$	0	0
$841$	−28414.0	−1.16503
$842$	0	0
$843$	− 63804.6i	− 2.60682i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−13372.8	−0.542495
$848$	0	0
$849$	20252.4	0.818680
$850$	0	0
$851$	− 3789.17i	− 0.152633i
$852$	0	0
$853$	36605.3i	1.46933i	0.678429	+	0.734666i	$0.262661\pi$
$853$	36605.3i	1.46933i	−0.678429	+	0.734666i	$0.737339\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	145.647	0.00580539	0.00290269	−	0.999996i	$-0.499076\pi$
$857$	145.647	0.00580539	0.00290269	+	0.999996i	$0.499076\pi$
$858$	0	0
$859$	− 48187.2i	− 1.91400i	−0.290088	−	0.957000i	$-0.593685\pi$
$859$	− 48187.2i	− 1.91400i	0.290088	−	0.957000i	$-0.406315\pi$
$860$	0	0
$861$	− 367.942i	− 0.0145638i
$862$	0	0
$863$	−13624.8	−0.537419	−0.268710	−	0.963221i	$-0.586597\pi$
$863$	−13624.8	−0.537419	−0.268710	+	0.963221i	$0.586597\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	57125.7i	2.23771i
$868$	0	0
$869$	21357.3i	0.833711i
$870$	0	0
$871$	38167.5	1.48479
$872$	0	0
$873$	25045.2	0.970965
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 7517.05i	− 0.289433i	−0.989473	−	0.144716i	$-0.953773\pi$
$877$	− 7517.05i	− 0.289433i	0.989473	−	0.144716i	$-0.0462270\pi$
$878$	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
$883$	18051.5i	0.687974i	0.938974	+	0.343987i	$0.111778\pi$
$883$	18051.5i	0.687974i	−0.938974	+	0.343987i	$0.888222\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−11236.4	−0.425345	−0.212673	−	0.977123i	$-0.568217\pi$
$887$	−11236.4	−0.425345	−0.212673	+	0.977123i	$0.568217\pi$
$888$	0	0
$889$	−1025.90	−0.0387036
$890$	0	0
$891$	26291.7i	0.988558i
$892$	0	0
$893$	− 48773.0i	− 1.82769i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	20688.1	0.770074
$898$	0	0
$899$	77810.1i	2.88666i
$900$	0	0
$901$	− 4337.66i	− 0.160387i
$902$	0	0
$903$	−13939.7	−0.513716
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 12021.6i	− 0.440100i	−0.975489	−	0.220050i	$-0.929378\pi$
$907$	− 12021.6i	− 0.440100i	0.975489	−	0.220050i	$-0.0706220\pi$
$908$	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
$913$	23355.0	0.846591
$914$	0	0
$915$	0	0
$916$	0	0
$917$	10098.3i	0.363659i
$918$	0	0
$919$	−29706.9	−1.06631	−0.533156	−	0.846017i	$-0.678994\pi$
$919$	−29706.9	−1.06631	−0.533156	+	0.846017i	$0.678994\pi$
$920$	0	0
$921$	4054.93	0.145075
$922$	0	0
$923$	− 6454.50i	− 0.230176i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	27433.2	0.971980
$928$	0	0
$929$	−32926.5	−1.16285	−0.581423	−	0.813602i	$-0.697504\pi$
$929$	−32926.5	−1.16285	−0.581423	+	0.813602i	$0.697504\pi$
$930$	0	0
$931$	8400.56i	0.295722i
$932$	0	0
$933$	52298.3i	1.83512i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	31490.5	1.09792	0.548960	−	0.835849i	$-0.315024\pi$
$937$	31490.5	1.09792	0.548960	+	0.835849i	$0.315024\pi$
$938$	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
$943$	163.018	0.00562947
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 17708.2i	− 0.607644i	−0.952729	−	0.303822i	$-0.901737\pi$
$947$	− 17708.2i	− 0.607644i	0.952729	−	0.303822i	$-0.0982629\pi$
$948$	0	0
$949$	17569.6i	0.600983i
$950$	0	0
$951$	71442.8	2.43606
$952$	0	0
$953$	26224.3	0.891382	0.445691	−	0.895187i	$-0.352958\pi$
$953$	26224.3	0.891382	0.445691	+	0.895187i	$0.352958\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	82667.9i	2.79234i
$958$	0	0
$959$	−28030.9	−0.943864
$960$	0	0
$961$	84869.2	2.84882
$962$	0	0
$963$	− 1371.47i	− 0.0458931i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	28936.2	0.962282	0.481141	−	0.876643i	$-0.340223\pi$
$967$	28936.2	0.962282	0.481141	+	0.876643i	$0.340223\pi$
$968$	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
$973$	21.5616i	0 0.000710414i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	10231.8	0.335050	0.167525	−	0.985868i	$-0.446422\pi$
$977$	10231.8	0.335050	0.167525	+	0.985868i	$0.446422\pi$
$978$	0	0
$979$	− 60855.1i	− 1.98666i
$980$	0	0
$981$	− 47976.6i	− 1.56144i
$982$	0	0
$983$	−913.463	−0.0296388	−0.0148194	−	0.999890i	$-0.504717\pi$
$983$	−913.463	−0.0296388	−0.0148194	+	0.999890i	$0.504717\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	68997.3i	2.22514i
$988$	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
$993$	4304.18	0.137552
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 34081.1i	− 1.08261i	−0.840827	−	0.541304i	$-0.817931\pi$
$997$	− 34081.1i	− 1.08261i	0.840827	−	0.541304i	$-0.182069\pi$
$998$	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.d.b.401.11		12
4.3	odd	2		200.4.d.c.101.8	yes	12
5.2	odd	4		800.4.f.d.49.22		24
5.3	odd	4		800.4.f.d.49.3		24
5.4	even	2		800.4.d.c.401.2		12
8.3	odd	2		200.4.d.c.101.7	✓	12
8.5	even	2	inner	800.4.d.b.401.2		12
20.3	even	4		200.4.f.d.149.23		24
20.7	even	4		200.4.f.d.149.2		24
20.19	odd	2		200.4.d.d.101.5	yes	12
40.3	even	4		200.4.f.d.149.1		24
40.13	odd	4		800.4.f.d.49.21		24
40.19	odd	2		200.4.d.d.101.6	yes	12
40.27	even	4		200.4.f.d.149.24		24
40.29	even	2		800.4.d.c.401.11		12
40.37	odd	4		800.4.f.d.49.4		24

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

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

\(f(q)\)	\(=\)	\(q+7.69300i q^{3} +15.6248 q^{7} -32.1823 q^{9} +O(q^{10})\)	\(q+7.69300i q^{3} +15.6248 q^{7} -32.1823 q^{9} -46.7640i q^{11} +50.4964i q^{13} +111.080 q^{17} -84.9682i q^{19} +120.201i q^{21} -53.2555 q^{23} -39.8675i q^{27} +229.789i q^{29} +338.615 q^{31} +359.755 q^{33} +71.1508i q^{37} -388.469 q^{39} -3.06105 q^{41} +115.970i q^{43} +574.015 q^{47} -98.8671 q^{49} +854.536i q^{51} -39.0500i q^{53} +653.660 q^{57} +109.940i q^{59} +180.738i q^{61} -502.841 q^{63} -755.845i q^{67} -409.695i q^{69} -127.821 q^{71} +347.937 q^{73} -730.676i q^{77} -456.703 q^{79} -562.221 q^{81} +499.423i q^{83} -1767.77 q^{87} +1301.32 q^{89} +788.995i q^{91} +2604.97i q^{93} -778.230 q^{97} +1504.97i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 28 q^{7} - 108 q^{9}+O(q^{10}) \) 12 * q - 28 * q^7 - 108 * q^9	\( 12 q - 28 q^{7} - 108 q^{9} - 112 q^{23} + 264 q^{31} + 116 q^{33} + 300 q^{39} - 20 q^{41} + 940 q^{47} + 228 q^{49} - 176 q^{57} + 1240 q^{63} - 628 q^{71} + 432 q^{73} - 1116 q^{79} + 1128 q^{81} - 4116 q^{87} - 424 q^{89} + 792 q^{97}+O(q^{100}) \) 12 * q - 28 * q^7 - 108 * q^9 - 112 * q^23 + 264 * q^31 + 116 * q^33 + 300 * q^39 - 20 * q^41 + 940 * q^47 + 228 * q^49 - 176 * q^57 + 1240 * q^63 - 628 * q^71 + 432 * q^73 - 1116 * q^79 + 1128 * q^81 - 4116 * q^87 - 424 * q^89 + 792 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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