LMFDB - Embedded newform 1600.4.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,4,Mod(1,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, 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("1600.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1600.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:	$94.4030560092$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-6.00000 q^{3} +34.0000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q-6.00000 q^{3} +34.0000 q^{7} +9.00000 q^{9} -16.0000 q^{11} +58.0000 q^{13} +70.0000 q^{17} -4.00000 q^{19} -204.000 q^{21} +134.000 q^{23} +108.000 q^{27} +242.000 q^{29} +100.000 q^{31} +96.0000 q^{33} -438.000 q^{37} -348.000 q^{39} -138.000 q^{41} +178.000 q^{43} -22.0000 q^{47} +813.000 q^{49} -420.000 q^{51} +162.000 q^{53} +24.0000 q^{57} +268.000 q^{59} -250.000 q^{61} +306.000 q^{63} +422.000 q^{67} -804.000 q^{69} -852.000 q^{71} -306.000 q^{73} -544.000 q^{77} -456.000 q^{79} -891.000 q^{81} +434.000 q^{83} -1452.00 q^{87} -726.000 q^{89} +1972.00 q^{91} -600.000 q^{93} -1378.00 q^{97} -144.000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

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
$3$	−6.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−6.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	34.0000	1.83583	0.917914	−	0.396780i	$-0.129872\pi$
$7$	34.0000	1.83583	0.917914	+	0.396780i	$0.129872\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−16.0000	−0.438562	−0.219281	−	0.975662i	$-0.570371\pi$
$11$	−16.0000	−0.438562	−0.219281	+	0.975662i	$0.570371\pi$
$12$	0	0
$13$	58.0000	1.23741	0.618704	−	0.785624i	$-0.287658\pi$
$13$	58.0000	1.23741	0.618704	+	0.785624i	$0.287658\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	70.0000	0.998676	0.499338	−	0.866407i	$-0.333577\pi$
$17$	70.0000	0.998676	0.499338	+	0.866407i	$0.333577\pi$
$18$	0	0
$19$	−4.00000	−0.0482980	−0.0241490	−	0.999708i	$-0.507688\pi$
$19$	−4.00000	−0.0482980	−0.0241490	+	0.999708i	$0.507688\pi$
$20$	0	0
$21$	−204.000	−2.11983
$22$	0	0
$23$	134.000	1.21482	0.607412	−	0.794387i	$-0.292208\pi$
$23$	134.000	1.21482	0.607412	+	0.794387i	$0.292208\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	108.000	0.769800
$28$	0	0
$29$	242.000	1.54960	0.774798	−	0.632209i	$-0.217852\pi$
$29$	242.000	1.54960	0.774798	+	0.632209i	$0.217852\pi$
$30$	0	0
$31$	100.000	0.579372	0.289686	−	0.957122i	$-0.406449\pi$
$31$	100.000	0.579372	0.289686	+	0.957122i	$0.406449\pi$
$32$	0	0
$33$	96.0000	0.506408
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−438.000	−1.94613	−0.973064	−	0.230534i	$-0.925953\pi$
$37$	−438.000	−1.94613	−0.973064	+	0.230534i	$0.925953\pi$
$38$	0	0
$39$	−348.000	−1.42884
$40$	0	0
$41$	−138.000	−0.525658	−0.262829	−	0.964842i	$-0.584656\pi$
$41$	−138.000	−0.525658	−0.262829	+	0.964842i	$0.584656\pi$
$42$	0	0
$43$	178.000	0.631273	0.315637	−	0.948880i	$-0.397782\pi$
$43$	178.000	0.631273	0.315637	+	0.948880i	$0.397782\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−22.0000	−0.0682772	−0.0341386	−	0.999417i	$-0.510869\pi$
$47$	−22.0000	−0.0682772	−0.0341386	+	0.999417i	$0.510869\pi$
$48$	0	0
$49$	813.000	2.37026
$50$	0	0
$51$	−420.000	−1.15317
$52$	0	0
$53$	162.000	0.419857	0.209928	−	0.977717i	$-0.432677\pi$
$53$	162.000	0.419857	0.209928	+	0.977717i	$0.432677\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	24.0000	0.0557698
$58$	0	0
$59$	268.000	0.591367	0.295683	−	0.955286i	$-0.404453\pi$
$59$	268.000	0.591367	0.295683	+	0.955286i	$0.404453\pi$
$60$	0	0
$61$	−250.000	−0.524741	−0.262371	−	0.964967i	$-0.584504\pi$
$61$	−250.000	−0.524741	−0.262371	+	0.964967i	$0.584504\pi$
$62$	0	0
$63$	306.000	0.611942
$64$	0	0
$65$	0	0
$66$	0	0
$67$	422.000	0.769485	0.384743	−	0.923024i	$-0.374290\pi$
$67$	422.000	0.769485	0.384743	+	0.923024i	$0.374290\pi$
$68$	0	0
$69$	−804.000	−1.40276
$70$	0	0
$71$	−852.000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−852.000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	−306.000	−0.490611	−0.245305	−	0.969446i	$-0.578888\pi$
$73$	−306.000	−0.490611	−0.245305	+	0.969446i	$0.578888\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−544.000	−0.805124
$78$	0	0
$79$	−456.000	−0.649418	−0.324709	−	0.945814i	$-0.605266\pi$
$79$	−456.000	−0.649418	−0.324709	+	0.945814i	$0.605266\pi$
$80$	0	0
$81$	−891.000	−1.22222
$82$	0	0
$83$	434.000	0.573948	0.286974	−	0.957938i	$-0.407351\pi$
$83$	434.000	0.573948	0.286974	+	0.957938i	$0.407351\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1452.00	−1.78932
$88$	0	0
$89$	−726.000	−0.864672	−0.432336	−	0.901712i	$-0.642311\pi$
$89$	−726.000	−0.864672	−0.432336	+	0.901712i	$0.642311\pi$
$90$	0	0
$91$	1972.00	2.27167
$92$	0	0
$93$	−600.000	−0.669001
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1378.00	−1.44242	−0.721210	−	0.692717i	$-0.756414\pi$
$97$	−1378.00	−1.44242	−0.721210	+	0.692717i	$0.756414\pi$
$98$	0	0
$99$	−144.000	−0.146187
$100$	0	0
$101$	−126.000	−0.124133	−0.0620667	−	0.998072i	$-0.519769\pi$
$101$	−126.000	−0.124133	−0.0620667	+	0.998072i	$0.519769\pi$
$102$	0	0
$103$	1262.00	1.20727	0.603634	−	0.797262i	$-0.293719\pi$
$103$	1262.00	1.20727	0.603634	+	0.797262i	$0.293719\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	510.000	0.460781	0.230390	−	0.973098i	$-0.426000\pi$
$107$	510.000	0.460781	0.230390	+	0.973098i	$0.426000\pi$
$108$	0	0
$109$	−26.0000	−0.0228472	−0.0114236	−	0.999935i	$-0.503636\pi$
$109$	−26.0000	−0.0228472	−0.0114236	+	0.999935i	$0.503636\pi$
$110$	0	0
$111$	2628.00	2.24720
$112$	0	0
$113$	−1242.00	−1.03396	−0.516980	−	0.855997i	$-0.672944\pi$
$113$	−1242.00	−1.03396	−0.516980	+	0.855997i	$0.672944\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	522.000	0.412469
$118$	0	0
$119$	2380.00	1.83340
$120$	0	0
$121$	−1075.00	−0.807663
$122$	0	0
$123$	828.000	0.606978
$124$	0	0
$125$	0	0
$126$	0	0
$127$	978.000	0.683334	0.341667	−	0.939821i	$-0.389008\pi$
$127$	978.000	0.683334	0.341667	+	0.939821i	$0.389008\pi$
$128$	0	0
$129$	−1068.00	−0.728931
$130$	0	0
$131$	912.000	0.608258	0.304129	−	0.952631i	$-0.401635\pi$
$131$	912.000	0.608258	0.304129	+	0.952631i	$0.401635\pi$
$132$	0	0
$133$	−136.000	−0.0886669
$134$	0	0
$135$	0	0
$136$	0	0
$137$	926.000	0.577471	0.288735	−	0.957409i	$-0.406765\pi$
$137$	926.000	0.577471	0.288735	+	0.957409i	$0.406765\pi$
$138$	0	0
$139$	−516.000	−0.314867	−0.157434	−	0.987530i	$-0.550322\pi$
$139$	−516.000	−0.314867	−0.157434	+	0.987530i	$0.550322\pi$
$140$	0	0
$141$	132.000	0.0788398
$142$	0	0
$143$	−928.000	−0.542680
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−4878.00	−2.73694
$148$	0	0
$149$	958.000	0.526728	0.263364	−	0.964697i	$-0.415168\pi$
$149$	958.000	0.526728	0.263364	+	0.964697i	$0.415168\pi$
$150$	0	0
$151$	332.000	0.178926	0.0894628	−	0.995990i	$-0.471485\pi$
$151$	332.000	0.178926	0.0894628	+	0.995990i	$0.471485\pi$
$152$	0	0
$153$	630.000	0.332892
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−1022.00	−0.519519	−0.259759	−	0.965673i	$-0.583643\pi$
$157$	−1022.00	−0.519519	−0.259759	+	0.965673i	$0.583643\pi$
$158$	0	0
$159$	−972.000	−0.484809
$160$	0	0
$161$	4556.00	2.23021
$162$	0	0
$163$	−926.000	−0.444969	−0.222484	−	0.974936i	$-0.571417\pi$
$163$	−926.000	−0.444969	−0.222484	+	0.974936i	$0.571417\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−654.000	−0.303042	−0.151521	−	0.988454i	$-0.548417\pi$
$167$	−654.000	−0.303042	−0.151521	+	0.988454i	$0.548417\pi$
$168$	0	0
$169$	1167.00	0.531179
$170$	0	0
$171$	−36.0000	−0.0160993
$172$	0	0
$173$	−1294.00	−0.568676	−0.284338	−	0.958724i	$-0.591774\pi$
$173$	−1294.00	−0.568676	−0.284338	+	0.958724i	$0.591774\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1608.00	−0.682851
$178$	0	0
$179$	2836.00	1.18420	0.592102	−	0.805863i	$-0.298298\pi$
$179$	2836.00	1.18420	0.592102	+	0.805863i	$0.298298\pi$
$180$	0	0
$181$	−1742.00	−0.715369	−0.357685	−	0.933842i	$-0.616434\pi$
$181$	−1742.00	−0.715369	−0.357685	+	0.933842i	$0.616434\pi$
$182$	0	0
$183$	1500.00	0.605919
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1120.00	−0.437981
$188$	0	0
$189$	3672.00	1.41322
$190$	0	0
$191$	4460.00	1.68960	0.844802	−	0.535079i	$-0.179718\pi$
$191$	4460.00	1.68960	0.844802	+	0.535079i	$0.179718\pi$
$192$	0	0
$193$	3782.00	1.41054	0.705270	−	0.708939i	$-0.250826\pi$
$193$	3782.00	1.41054	0.705270	+	0.708939i	$0.250826\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4474.00	1.61807	0.809034	−	0.587762i	$-0.199991\pi$
$197$	4474.00	1.61807	0.809034	+	0.587762i	$0.199991\pi$
$198$	0	0
$199$	3608.00	1.28525	0.642624	−	0.766182i	$-0.277846\pi$
$199$	3608.00	1.28525	0.642624	+	0.766182i	$0.277846\pi$
$200$	0	0
$201$	−2532.00	−0.888525
$202$	0	0
$203$	8228.00	2.84479
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1206.00	0.404941
$208$	0	0
$209$	64.0000	0.0211817
$210$	0	0
$211$	256.000	0.0835250	0.0417625	−	0.999128i	$-0.486703\pi$
$211$	256.000	0.0835250	0.0417625	+	0.999128i	$0.486703\pi$
$212$	0	0
$213$	5112.00	1.64445
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3400.00	1.06363
$218$	0	0
$219$	1836.00	0.566509
$220$	0	0
$221$	4060.00	1.23577
$222$	0	0
$223$	5158.00	1.54890	0.774451	−	0.632634i	$-0.218026\pi$
$223$	5158.00	1.54890	0.774451	+	0.632634i	$0.218026\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2226.00	−0.650858	−0.325429	−	0.945566i	$-0.605509\pi$
$227$	−2226.00	−0.650858	−0.325429	+	0.945566i	$0.605509\pi$
$228$	0	0
$229$	−2086.00	−0.601951	−0.300975	−	0.953632i	$-0.597312\pi$
$229$	−2086.00	−0.601951	−0.300975	+	0.953632i	$0.597312\pi$
$230$	0	0
$231$	3264.00	0.929677
$232$	0	0
$233$	5718.00	1.60772	0.803860	−	0.594819i	$-0.202776\pi$
$233$	5718.00	1.60772	0.803860	+	0.594819i	$0.202776\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2736.00	0.749883
$238$	0	0
$239$	−3624.00	−0.980825	−0.490412	−	0.871491i	$-0.663154\pi$
$239$	−3624.00	−0.980825	−0.490412	+	0.871491i	$0.663154\pi$
$240$	0	0
$241$	−82.0000	−0.0219174	−0.0109587	−	0.999940i	$-0.503488\pi$
$241$	−82.0000	−0.0219174	−0.0109587	+	0.999940i	$0.503488\pi$
$242$	0	0
$243$	2430.00	0.641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−232.000	−0.0597644
$248$	0	0
$249$	−2604.00	−0.662738
$250$	0	0
$251$	5040.00	1.26742	0.633709	−	0.773571i	$-0.281532\pi$
$251$	5040.00	1.26742	0.633709	+	0.773571i	$0.281532\pi$
$252$	0	0
$253$	−2144.00	−0.532775
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2310.00	0.560676	0.280338	−	0.959901i	$-0.409553\pi$
$257$	2310.00	0.560676	0.280338	+	0.959901i	$0.409553\pi$
$258$	0	0
$259$	−14892.0	−3.57276
$260$	0	0
$261$	2178.00	0.516532
$262$	0	0
$263$	4110.00	0.963625	0.481813	−	0.876274i	$-0.339979\pi$
$263$	4110.00	0.963625	0.481813	+	0.876274i	$0.339979\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4356.00	0.998438
$268$	0	0
$269$	−746.000	−0.169087	−0.0845435	−	0.996420i	$-0.526943\pi$
$269$	−746.000	−0.169087	−0.0845435	+	0.996420i	$0.526943\pi$
$270$	0	0
$271$	−4596.00	−1.03021	−0.515105	−	0.857127i	$-0.672247\pi$
$271$	−4596.00	−1.03021	−0.515105	+	0.857127i	$0.672247\pi$
$272$	0	0
$273$	−11832.0	−2.62310
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−2206.00	−0.478504	−0.239252	−	0.970957i	$-0.576902\pi$
$277$	−2206.00	−0.478504	−0.239252	+	0.970957i	$0.576902\pi$
$278$	0	0
$279$	900.000	0.193124
$280$	0	0
$281$	8278.00	1.75738	0.878691	−	0.477392i	$-0.158418\pi$
$281$	8278.00	1.75738	0.878691	+	0.477392i	$0.158418\pi$
$282$	0	0
$283$	1178.00	0.247438	0.123719	−	0.992317i	$-0.460518\pi$
$283$	1178.00	0.247438	0.123719	+	0.992317i	$0.460518\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4692.00	−0.965017
$288$	0	0
$289$	−13.0000	−0.00264604
$290$	0	0
$291$	8268.00	1.66556
$292$	0	0
$293$	106.000	0.0211351	0.0105676	−	0.999944i	$-0.496636\pi$
$293$	106.000	0.0211351	0.0105676	+	0.999944i	$0.496636\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1728.00	−0.337605
$298$	0	0
$299$	7772.00	1.50323
$300$	0	0
$301$	6052.00	1.15891
$302$	0	0
$303$	756.000	0.143337
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8134.00	1.51216	0.756078	−	0.654482i	$-0.227113\pi$
$307$	8134.00	1.51216	0.756078	+	0.654482i	$0.227113\pi$
$308$	0	0
$309$	−7572.00	−1.39403
$310$	0	0
$311$	−4396.00	−0.801525	−0.400763	−	0.916182i	$-0.631255\pi$
$311$	−4396.00	−0.801525	−0.400763	+	0.916182i	$0.631255\pi$
$312$	0	0
$313$	−4826.00	−0.871507	−0.435753	−	0.900066i	$-0.643518\pi$
$313$	−4826.00	−0.871507	−0.435753	+	0.900066i	$0.643518\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7026.00	1.24486	0.622428	−	0.782677i	$-0.286146\pi$
$317$	7026.00	1.24486	0.622428	+	0.782677i	$0.286146\pi$
$318$	0	0
$319$	−3872.00	−0.679594
$320$	0	0
$321$	−3060.00	−0.532064
$322$	0	0
$323$	−280.000	−0.0482341
$324$	0	0
$325$	0	0
$326$	0	0
$327$	156.000	0.0263817
$328$	0	0
$329$	−748.000	−0.125345
$330$	0	0
$331$	−8808.00	−1.46263	−0.731316	−	0.682038i	$-0.761094\pi$
$331$	−8808.00	−1.46263	−0.731316	+	0.682038i	$0.761094\pi$
$332$	0	0
$333$	−3942.00	−0.648710
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−5602.00	−0.905520	−0.452760	−	0.891632i	$-0.649561\pi$
$337$	−5602.00	−0.905520	−0.452760	+	0.891632i	$0.649561\pi$
$338$	0	0
$339$	7452.00	1.19391
$340$	0	0
$341$	−1600.00	−0.254090
$342$	0	0
$343$	15980.0	2.51557
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6634.00	−1.02632	−0.513158	−	0.858294i	$-0.671525\pi$
$347$	−6634.00	−1.02632	−0.513158	+	0.858294i	$0.671525\pi$
$348$	0	0
$349$	−3198.00	−0.490501	−0.245251	−	0.969460i	$-0.578870\pi$
$349$	−3198.00	−0.490501	−0.245251	+	0.969460i	$0.578870\pi$
$350$	0	0
$351$	6264.00	0.952557
$352$	0	0
$353$	5230.00	0.788569	0.394284	−	0.918988i	$-0.370992\pi$
$353$	5230.00	0.788569	0.394284	+	0.918988i	$0.370992\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−14280.0	−2.11702
$358$	0	0
$359$	−312.000	−0.0458683	−0.0229342	−	0.999737i	$-0.507301\pi$
$359$	−312.000	−0.0458683	−0.0229342	+	0.999737i	$0.507301\pi$
$360$	0	0
$361$	−6843.00	−0.997667
$362$	0	0
$363$	6450.00	0.932609
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−10790.0	−1.53470	−0.767348	−	0.641231i	$-0.778424\pi$
$367$	−10790.0	−1.53470	−0.767348	+	0.641231i	$0.778424\pi$
$368$	0	0
$369$	−1242.00	−0.175219
$370$	0	0
$371$	5508.00	0.770785
$372$	0	0
$373$	−4190.00	−0.581635	−0.290818	−	0.956778i	$-0.593927\pi$
$373$	−4190.00	−0.581635	−0.290818	+	0.956778i	$0.593927\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	14036.0	1.91748
$378$	0	0
$379$	6980.00	0.946012	0.473006	−	0.881059i	$-0.343169\pi$
$379$	6980.00	0.946012	0.473006	+	0.881059i	$0.343169\pi$
$380$	0	0
$381$	−5868.00	−0.789047
$382$	0	0
$383$	−13962.0	−1.86273	−0.931364	−	0.364089i	$-0.881380\pi$
$383$	−13962.0	−1.86273	−0.931364	+	0.364089i	$0.881380\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1602.00	0.210424
$388$	0	0
$389$	−3810.00	−0.496593	−0.248296	−	0.968684i	$-0.579871\pi$
$389$	−3810.00	−0.496593	−0.248296	+	0.968684i	$0.579871\pi$
$390$	0	0
$391$	9380.00	1.21321
$392$	0	0
$393$	−5472.00	−0.702356
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−9158.00	−1.15775	−0.578875	−	0.815416i	$-0.696508\pi$
$397$	−9158.00	−1.15775	−0.578875	+	0.815416i	$0.696508\pi$
$398$	0	0
$399$	816.000	0.102384
$400$	0	0
$401$	4866.00	0.605976	0.302988	−	0.952994i	$-0.402016\pi$
$401$	4866.00	0.605976	0.302988	+	0.952994i	$0.402016\pi$
$402$	0	0
$403$	5800.00	0.716920
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7008.00	0.853498
$408$	0	0
$409$	13486.0	1.63042	0.815208	−	0.579169i	$-0.196623\pi$
$409$	13486.0	1.63042	0.815208	+	0.579169i	$0.196623\pi$
$410$	0	0
$411$	−5556.00	−0.666806
$412$	0	0
$413$	9112.00	1.08565
$414$	0	0
$415$	0	0
$416$	0	0
$417$	3096.00	0.363577
$418$	0	0
$419$	−5628.00	−0.656195	−0.328098	−	0.944644i	$-0.606407\pi$
$419$	−5628.00	−0.656195	−0.328098	+	0.944644i	$0.606407\pi$
$420$	0	0
$421$	−7938.00	−0.918942	−0.459471	−	0.888193i	$-0.651961\pi$
$421$	−7938.00	−0.918942	−0.459471	+	0.888193i	$0.651961\pi$
$422$	0	0
$423$	−198.000	−0.0227591
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−8500.00	−0.963334
$428$	0	0
$429$	5568.00	0.626633
$430$	0	0
$431$	1916.00	0.214131	0.107066	−	0.994252i	$-0.465855\pi$
$431$	1916.00	0.214131	0.107066	+	0.994252i	$0.465855\pi$
$432$	0	0
$433$	16510.0	1.83238	0.916189	−	0.400746i	$-0.131249\pi$
$433$	16510.0	1.83238	0.916189	+	0.400746i	$0.131249\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−536.000	−0.0586736
$438$	0	0
$439$	−1256.00	−0.136550	−0.0682752	−	0.997667i	$-0.521750\pi$
$439$	−1256.00	−0.136550	−0.0682752	+	0.997667i	$0.521750\pi$
$440$	0	0
$441$	7317.00	0.790087
$442$	0	0
$443$	−12222.0	−1.31080	−0.655400	−	0.755282i	$-0.727500\pi$
$443$	−12222.0	−1.31080	−0.655400	+	0.755282i	$0.727500\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−5748.00	−0.608213
$448$	0	0
$449$	−5946.00	−0.624965	−0.312482	−	0.949924i	$-0.601160\pi$
$449$	−5946.00	−0.624965	−0.312482	+	0.949924i	$0.601160\pi$
$450$	0	0
$451$	2208.00	0.230534
$452$	0	0
$453$	−1992.00	−0.206606
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1258.00	−0.128768	−0.0643838	−	0.997925i	$-0.520508\pi$
$457$	−1258.00	−0.128768	−0.0643838	+	0.997925i	$0.520508\pi$
$458$	0	0
$459$	7560.00	0.768781
$460$	0	0
$461$	−16422.0	−1.65911	−0.829554	−	0.558426i	$-0.811405\pi$
$461$	−16422.0	−1.65911	−0.829554	+	0.558426i	$0.811405\pi$
$462$	0	0
$463$	−2658.00	−0.266799	−0.133399	−	0.991062i	$-0.542589\pi$
$463$	−2658.00	−0.266799	−0.133399	+	0.991062i	$0.542589\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3686.00	0.365241	0.182621	−	0.983183i	$-0.441542\pi$
$467$	3686.00	0.365241	0.182621	+	0.983183i	$0.441542\pi$
$468$	0	0
$469$	14348.0	1.41264
$470$	0	0
$471$	6132.00	0.599889
$472$	0	0
$473$	−2848.00	−0.276852
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1458.00	0.139952
$478$	0	0
$479$	88.0000	0.00839420	0.00419710	−	0.999991i	$-0.498664\pi$
$479$	88.0000	0.00839420	0.00419710	+	0.999991i	$0.498664\pi$
$480$	0	0
$481$	−25404.0	−2.40816
$482$	0	0
$483$	−27336.0	−2.57522
$484$	0	0
$485$	0	0
$486$	0	0
$487$	14714.0	1.36911	0.684553	−	0.728963i	$-0.259997\pi$
$487$	14714.0	1.36911	0.684553	+	0.728963i	$0.259997\pi$
$488$	0	0
$489$	5556.00	0.513806
$490$	0	0
$491$	7344.00	0.675010	0.337505	−	0.941324i	$-0.390417\pi$
$491$	7344.00	0.675010	0.337505	+	0.941324i	$0.390417\pi$
$492$	0	0
$493$	16940.0	1.54754
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−28968.0	−2.61447
$498$	0	0
$499$	−1604.00	−0.143898	−0.0719488	−	0.997408i	$-0.522922\pi$
$499$	−1604.00	−0.143898	−0.0719488	+	0.997408i	$0.522922\pi$
$500$	0	0
$501$	3924.00	0.349923
$502$	0	0
$503$	−14802.0	−1.31210	−0.656052	−	0.754715i	$-0.727775\pi$
$503$	−14802.0	−1.31210	−0.656052	+	0.754715i	$0.727775\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−7002.00	−0.613353
$508$	0	0
$509$	22514.0	1.96054	0.980271	−	0.197660i	$-0.0633342\pi$
$509$	22514.0	1.96054	0.980271	+	0.197660i	$0.0633342\pi$
$510$	0	0
$511$	−10404.0	−0.900677
$512$	0	0
$513$	−432.000	−0.0371799
$514$	0	0
$515$	0	0
$516$	0	0
$517$	352.000	0.0299438
$518$	0	0
$519$	7764.00	0.656651
$520$	0	0
$521$	−6710.00	−0.564243	−0.282121	−	0.959379i	$-0.591038\pi$
$521$	−6710.00	−0.564243	−0.282121	+	0.959379i	$0.591038\pi$
$522$	0	0
$523$	7930.00	0.663011	0.331505	−	0.943453i	$-0.392443\pi$
$523$	7930.00	0.663011	0.331505	+	0.943453i	$0.392443\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7000.00	0.578605
$528$	0	0
$529$	5789.00	0.475795
$530$	0	0
$531$	2412.00	0.197122
$532$	0	0
$533$	−8004.00	−0.650454
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−17016.0	−1.36740
$538$	0	0
$539$	−13008.0	−1.03951
$540$	0	0
$541$	−4918.00	−0.390834	−0.195417	−	0.980720i	$-0.562606\pi$
$541$	−4918.00	−0.390834	−0.195417	+	0.980720i	$0.562606\pi$
$542$	0	0
$543$	10452.0	0.826037
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−3922.00	−0.306568	−0.153284	−	0.988182i	$-0.548985\pi$
$547$	−3922.00	−0.306568	−0.153284	+	0.988182i	$0.548985\pi$
$548$	0	0
$549$	−2250.00	−0.174914
$550$	0	0
$551$	−968.000	−0.0748424
$552$	0	0
$553$	−15504.0	−1.19222
$554$	0	0
$555$	0	0
$556$	0	0
$557$	17786.0	1.35299	0.676496	−	0.736446i	$-0.263497\pi$
$557$	17786.0	1.35299	0.676496	+	0.736446i	$0.263497\pi$
$558$	0	0
$559$	10324.0	0.781143
$560$	0	0
$561$	6720.00	0.505737
$562$	0	0
$563$	20266.0	1.51707	0.758535	−	0.651633i	$-0.225916\pi$
$563$	20266.0	1.51707	0.758535	+	0.651633i	$0.225916\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−30294.0	−2.24379
$568$	0	0
$569$	13358.0	0.984177	0.492088	−	0.870545i	$-0.336234\pi$
$569$	13358.0	0.984177	0.492088	+	0.870545i	$0.336234\pi$
$570$	0	0
$571$	−16360.0	−1.19903	−0.599514	−	0.800364i	$-0.704639\pi$
$571$	−16360.0	−1.19903	−0.599514	+	0.800364i	$0.704639\pi$
$572$	0	0
$573$	−26760.0	−1.95099
$574$	0	0
$575$	0	0
$576$	0	0
$577$	15574.0	1.12366	0.561832	−	0.827251i	$-0.310097\pi$
$577$	15574.0	1.12366	0.561832	+	0.827251i	$0.310097\pi$
$578$	0	0
$579$	−22692.0	−1.62875
$580$	0	0
$581$	14756.0	1.05367
$582$	0	0
$583$	−2592.00	−0.184133
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6654.00	0.467870	0.233935	−	0.972252i	$-0.424840\pi$
$587$	6654.00	0.467870	0.233935	+	0.972252i	$0.424840\pi$
$588$	0	0
$589$	−400.000	−0.0279825
$590$	0	0
$591$	−26844.0	−1.86838
$592$	0	0
$593$	17742.0	1.22863	0.614314	−	0.789062i	$-0.289433\pi$
$593$	17742.0	1.22863	0.614314	+	0.789062i	$0.289433\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−21648.0	−1.48408
$598$	0	0
$599$	15840.0	1.08048	0.540238	−	0.841512i	$-0.318334\pi$
$599$	15840.0	1.08048	0.540238	+	0.841512i	$0.318334\pi$
$600$	0	0
$601$	−3002.00	−0.203751	−0.101875	−	0.994797i	$-0.532484\pi$
$601$	−3002.00	−0.203751	−0.101875	+	0.994797i	$0.532484\pi$
$602$	0	0
$603$	3798.00	0.256495
$604$	0	0
$605$	0	0
$606$	0	0
$607$	23610.0	1.57875	0.789374	−	0.613912i	$-0.210405\pi$
$607$	23610.0	1.57875	0.789374	+	0.613912i	$0.210405\pi$
$608$	0	0
$609$	−49368.0	−3.28488
$610$	0	0
$611$	−1276.00	−0.0844868
$612$	0	0
$613$	23850.0	1.57144	0.785720	−	0.618583i	$-0.212293\pi$
$613$	23850.0	1.57144	0.785720	+	0.618583i	$0.212293\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5334.00	0.348037	0.174018	−	0.984742i	$-0.444325\pi$
$617$	5334.00	0.348037	0.174018	+	0.984742i	$0.444325\pi$
$618$	0	0
$619$	2164.00	0.140515	0.0702573	−	0.997529i	$-0.477618\pi$
$619$	2164.00	0.140515	0.0702573	+	0.997529i	$0.477618\pi$
$620$	0	0
$621$	14472.0	0.935171
$622$	0	0
$623$	−24684.0	−1.58739
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−384.000	−0.0244585
$628$	0	0
$629$	−30660.0	−1.94355
$630$	0	0
$631$	−25220.0	−1.59111	−0.795557	−	0.605879i	$-0.792821\pi$
$631$	−25220.0	−1.59111	−0.795557	+	0.605879i	$0.792821\pi$
$632$	0	0
$633$	−1536.00	−0.0964463
$634$	0	0
$635$	0	0
$636$	0	0
$637$	47154.0	2.93298
$638$	0	0
$639$	−7668.00	−0.474713
$640$	0	0
$641$	−12306.0	−0.758280	−0.379140	−	0.925339i	$-0.623780\pi$
$641$	−12306.0	−0.758280	−0.379140	+	0.925339i	$0.623780\pi$
$642$	0	0
$643$	−27414.0	−1.68134	−0.840671	−	0.541547i	$-0.817839\pi$
$643$	−27414.0	−1.68134	−0.840671	+	0.541547i	$0.817839\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	21834.0	1.32671	0.663356	−	0.748304i	$-0.269131\pi$
$647$	21834.0	1.32671	0.663356	+	0.748304i	$0.269131\pi$
$648$	0	0
$649$	−4288.00	−0.259351
$650$	0	0
$651$	−20400.0	−1.22817
$652$	0	0
$653$	−23998.0	−1.43815	−0.719077	−	0.694931i	$-0.755435\pi$
$653$	−23998.0	−1.43815	−0.719077	+	0.694931i	$0.755435\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−2754.00	−0.163537
$658$	0	0
$659$	32004.0	1.89180	0.945902	−	0.324452i	$-0.105180\pi$
$659$	32004.0	1.89180	0.945902	+	0.324452i	$0.105180\pi$
$660$	0	0
$661$	8526.00	0.501699	0.250849	−	0.968026i	$-0.419290\pi$
$661$	8526.00	0.501699	0.250849	+	0.968026i	$0.419290\pi$
$662$	0	0
$663$	−24360.0	−1.42694
$664$	0	0
$665$	0	0
$666$	0	0
$667$	32428.0	1.88248
$668$	0	0
$669$	−30948.0	−1.78852
$670$	0	0
$671$	4000.00	0.230132
$672$	0	0
$673$	−8178.00	−0.468408	−0.234204	−	0.972187i	$-0.575248\pi$
$673$	−8178.00	−0.468408	−0.234204	+	0.972187i	$0.575248\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−16646.0	−0.944989	−0.472495	−	0.881334i	$-0.656646\pi$
$677$	−16646.0	−0.944989	−0.472495	+	0.881334i	$0.656646\pi$
$678$	0	0
$679$	−46852.0	−2.64803
$680$	0	0
$681$	13356.0	0.751546
$682$	0	0
$683$	−22446.0	−1.25750	−0.628750	−	0.777608i	$-0.716433\pi$
$683$	−22446.0	−1.25750	−0.628750	+	0.777608i	$0.716433\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	12516.0	0.695073
$688$	0	0
$689$	9396.00	0.519534
$690$	0	0
$691$	−35336.0	−1.94536	−0.972681	−	0.232147i	$-0.925425\pi$
$691$	−35336.0	−1.94536	−0.972681	+	0.232147i	$0.925425\pi$
$692$	0	0
$693$	−4896.00	−0.268375
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−9660.00	−0.524962
$698$	0	0
$699$	−34308.0	−1.85643
$700$	0	0
$701$	−3482.00	−0.187608	−0.0938041	−	0.995591i	$-0.529903\pi$
$701$	−3482.00	−0.187608	−0.0938041	+	0.995591i	$0.529903\pi$
$702$	0	0
$703$	1752.00	0.0939942
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4284.00	−0.227887
$708$	0	0
$709$	19402.0	1.02773	0.513863	−	0.857872i	$-0.328214\pi$
$709$	19402.0	1.02773	0.513863	+	0.857872i	$0.328214\pi$
$710$	0	0
$711$	−4104.00	−0.216473
$712$	0	0
$713$	13400.0	0.703834
$714$	0	0
$715$	0	0
$716$	0	0
$717$	21744.0	1.13256
$718$	0	0
$719$	−9896.00	−0.513294	−0.256647	−	0.966505i	$-0.582618\pi$
$719$	−9896.00	−0.513294	−0.256647	+	0.966505i	$0.582618\pi$
$720$	0	0
$721$	42908.0	2.21633
$722$	0	0
$723$	492.000	0.0253080
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−494.000	−0.0252014	−0.0126007	−	0.999921i	$-0.504011\pi$
$727$	−494.000	−0.0252014	−0.0126007	+	0.999921i	$0.504011\pi$
$728$	0	0
$729$	9477.00	0.481481
$730$	0	0
$731$	12460.0	0.630437
$732$	0	0
$733$	9282.00	0.467720	0.233860	−	0.972270i	$-0.424864\pi$
$733$	9282.00	0.467720	0.233860	+	0.972270i	$0.424864\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6752.00	−0.337467
$738$	0	0
$739$	3252.00	0.161877	0.0809383	−	0.996719i	$-0.474208\pi$
$739$	3252.00	0.161877	0.0809383	+	0.996719i	$0.474208\pi$
$740$	0	0
$741$	1392.00	0.0690100
$742$	0	0
$743$	4710.00	0.232561	0.116281	−	0.993216i	$-0.462903\pi$
$743$	4710.00	0.232561	0.116281	+	0.993216i	$0.462903\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3906.00	0.191316
$748$	0	0
$749$	17340.0	0.845914
$750$	0	0
$751$	25764.0	1.25185	0.625927	−	0.779882i	$-0.284721\pi$
$751$	25764.0	1.25185	0.625927	+	0.779882i	$0.284721\pi$
$752$	0	0
$753$	−30240.0	−1.46349
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−30094.0	−1.44489	−0.722447	−	0.691426i	$-0.756983\pi$
$757$	−30094.0	−1.44489	−0.722447	+	0.691426i	$0.756983\pi$
$758$	0	0
$759$	12864.0	0.615196
$760$	0	0
$761$	22362.0	1.06521	0.532603	−	0.846365i	$-0.321214\pi$
$761$	22362.0	1.06521	0.532603	+	0.846365i	$0.321214\pi$
$762$	0	0
$763$	−884.000	−0.0419436
$764$	0	0
$765$	0	0
$766$	0	0
$767$	15544.0	0.731762
$768$	0	0
$769$	−30398.0	−1.42546	−0.712731	−	0.701438i	$-0.752542\pi$
$769$	−30398.0	−1.42546	−0.712731	+	0.701438i	$0.752542\pi$
$770$	0	0
$771$	−13860.0	−0.647413
$772$	0	0
$773$	1290.00	0.0600234	0.0300117	−	0.999550i	$-0.490446\pi$
$773$	1290.00	0.0600234	0.0300117	+	0.999550i	$0.490446\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	89352.0	4.12546
$778$	0	0
$779$	552.000	0.0253883
$780$	0	0
$781$	13632.0	0.624573
$782$	0	0
$783$	26136.0	1.19288
$784$	0	0
$785$	0	0
$786$	0	0
$787$	14.0000	0.000634112 0	0.000317056	−	1.00000i	$-0.499899\pi$
$787$	14.0000	0.000634112 0	0.000317056		1.00000i	$0.499899\pi$
$788$	0	0
$789$	−24660.0	−1.11270
$790$	0	0
$791$	−42228.0	−1.89817
$792$	0	0
$793$	−14500.0	−0.649319
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−38814.0	−1.72505	−0.862523	−	0.506017i	$-0.831117\pi$
$797$	−38814.0	−1.72505	−0.862523	+	0.506017i	$0.831117\pi$
$798$	0	0
$799$	−1540.00	−0.0681868
$800$	0	0
$801$	−6534.00	−0.288224
$802$	0	0
$803$	4896.00	0.215163
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4476.00	0.195245
$808$	0	0
$809$	27402.0	1.19086	0.595428	−	0.803408i	$-0.296982\pi$
$809$	27402.0	1.19086	0.595428	+	0.803408i	$0.296982\pi$
$810$	0	0
$811$	28576.0	1.23729	0.618643	−	0.785672i	$-0.287683\pi$
$811$	28576.0	1.23729	0.618643	+	0.785672i	$0.287683\pi$
$812$	0	0
$813$	27576.0	1.18958
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−712.000	−0.0304893
$818$	0	0
$819$	17748.0	0.757223
$820$	0	0
$821$	−31762.0	−1.35018	−0.675092	−	0.737733i	$-0.735896\pi$
$821$	−31762.0	−1.35018	−0.675092	+	0.737733i	$0.735896\pi$
$822$	0	0
$823$	−20506.0	−0.868523	−0.434261	−	0.900787i	$-0.642991\pi$
$823$	−20506.0	−0.868523	−0.434261	+	0.900787i	$0.642991\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13014.0	0.547208	0.273604	−	0.961842i	$-0.411784\pi$
$827$	13014.0	0.547208	0.273604	+	0.961842i	$0.411784\pi$
$828$	0	0
$829$	22790.0	0.954800	0.477400	−	0.878686i	$-0.341579\pi$
$829$	22790.0	0.954800	0.477400	+	0.878686i	$0.341579\pi$
$830$	0	0
$831$	13236.0	0.552529
$832$	0	0
$833$	56910.0	2.36712
$834$	0	0
$835$	0	0
$836$	0	0
$837$	10800.0	0.446001
$838$	0	0
$839$	23696.0	0.975062	0.487531	−	0.873106i	$-0.337898\pi$
$839$	23696.0	0.975062	0.487531	+	0.873106i	$0.337898\pi$
$840$	0	0
$841$	34175.0	1.40125
$842$	0	0
$843$	−49668.0	−2.02925
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−36550.0	−1.48273
$848$	0	0
$849$	−7068.00	−0.285716
$850$	0	0
$851$	−58692.0	−2.36420
$852$	0	0
$853$	5306.00	0.212982	0.106491	−	0.994314i	$-0.466038\pi$
$853$	5306.00	0.212982	0.106491	+	0.994314i	$0.466038\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	21054.0	0.839196	0.419598	−	0.907710i	$-0.362171\pi$
$857$	21054.0	0.839196	0.419598	+	0.907710i	$0.362171\pi$
$858$	0	0
$859$	−7364.00	−0.292499	−0.146249	−	0.989248i	$-0.546720\pi$
$859$	−7364.00	−0.292499	−0.146249	+	0.989248i	$0.546720\pi$
$860$	0	0
$861$	28152.0	1.11431
$862$	0	0
$863$	−17226.0	−0.679467	−0.339733	−	0.940522i	$-0.610337\pi$
$863$	−17226.0	−0.679467	−0.339733	+	0.940522i	$0.610337\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	78.0000	0.00305539
$868$	0	0
$869$	7296.00	0.284810
$870$	0	0
$871$	24476.0	0.952167
$872$	0	0
$873$	−12402.0	−0.480807
$874$	0	0
$875$	0	0
$876$	0	0
$877$	21202.0	0.816352	0.408176	−	0.912903i	$-0.366165\pi$
$877$	21202.0	0.816352	0.408176	+	0.912903i	$0.366165\pi$
$878$	0	0
$879$	−636.000	−0.0244047
$880$	0	0
$881$	−29490.0	−1.12774	−0.563872	−	0.825862i	$-0.690689\pi$
$881$	−29490.0	−1.12774	−0.563872	+	0.825862i	$0.690689\pi$
$882$	0	0
$883$	2570.00	0.0979472	0.0489736	−	0.998800i	$-0.484405\pi$
$883$	2570.00	0.0979472	0.0489736	+	0.998800i	$0.484405\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−36334.0	−1.37540	−0.687698	−	0.725997i	$-0.741379\pi$
$887$	−36334.0	−1.37540	−0.687698	+	0.725997i	$0.741379\pi$
$888$	0	0
$889$	33252.0	1.25448
$890$	0	0
$891$	14256.0	0.536020
$892$	0	0
$893$	88.0000	0.00329766
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−46632.0	−1.73578
$898$	0	0
$899$	24200.0	0.897792
$900$	0	0
$901$	11340.0	0.419301
$902$	0	0
$903$	−36312.0	−1.33819
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−12474.0	−0.456662	−0.228331	−	0.973584i	$-0.573327\pi$
$907$	−12474.0	−0.456662	−0.228331	+	0.973584i	$0.573327\pi$
$908$	0	0
$909$	−1134.00	−0.0413778
$910$	0	0
$911$	−41132.0	−1.49590	−0.747949	−	0.663756i	$-0.768961\pi$
$911$	−41132.0	−1.49590	−0.747949	+	0.663756i	$0.768961\pi$
$912$	0	0
$913$	−6944.00	−0.251712
$914$	0	0
$915$	0	0
$916$	0	0
$917$	31008.0	1.11666
$918$	0	0
$919$	−38416.0	−1.37892	−0.689460	−	0.724324i	$-0.742152\pi$
$919$	−38416.0	−1.37892	−0.689460	+	0.724324i	$0.742152\pi$
$920$	0	0
$921$	−48804.0	−1.74609
$922$	0	0
$923$	−49416.0	−1.76224
$924$	0	0
$925$	0	0
$926$	0	0
$927$	11358.0	0.402423
$928$	0	0
$929$	41302.0	1.45864	0.729319	−	0.684174i	$-0.239837\pi$
$929$	41302.0	1.45864	0.729319	+	0.684174i	$0.239837\pi$
$930$	0	0
$931$	−3252.00	−0.114479
$932$	0	0
$933$	26376.0	0.925521
$934$	0	0
$935$	0	0
$936$	0	0
$937$	26150.0	0.911722	0.455861	−	0.890051i	$-0.349331\pi$
$937$	26150.0	0.911722	0.455861	+	0.890051i	$0.349331\pi$
$938$	0	0
$939$	28956.0	1.00633
$940$	0	0
$941$	−35254.0	−1.22130	−0.610652	−	0.791899i	$-0.709093\pi$
$941$	−35254.0	−1.22130	−0.610652	+	0.791899i	$0.709093\pi$
$942$	0	0
$943$	−18492.0	−0.638582
$944$	0	0
$945$	0	0
$946$	0	0
$947$	18550.0	0.636530	0.318265	−	0.948002i	$-0.396900\pi$
$947$	18550.0	0.636530	0.318265	+	0.948002i	$0.396900\pi$
$948$	0	0
$949$	−17748.0	−0.607086
$950$	0	0
$951$	−42156.0	−1.43744
$952$	0	0
$953$	−17322.0	−0.588788	−0.294394	−	0.955684i	$-0.595118\pi$
$953$	−17322.0	−0.588788	−0.294394	+	0.955684i	$0.595118\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	23232.0	0.784727
$958$	0	0
$959$	31484.0	1.06014
$960$	0	0
$961$	−19791.0	−0.664328
$962$	0	0
$963$	4590.00	0.153594
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−35190.0	−1.17025	−0.585126	−	0.810942i	$-0.698955\pi$
$967$	−35190.0	−1.17025	−0.585126	+	0.810942i	$0.698955\pi$
$968$	0	0
$969$	1680.00	0.0556960
$970$	0	0
$971$	40696.0	1.34500	0.672501	−	0.740096i	$-0.265220\pi$
$971$	40696.0	1.34500	0.672501	+	0.740096i	$0.265220\pi$
$972$	0	0
$973$	−17544.0	−0.578042
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−44306.0	−1.45084	−0.725422	−	0.688304i	$-0.758355\pi$
$977$	−44306.0	−1.45084	−0.725422	+	0.688304i	$0.758355\pi$
$978$	0	0
$979$	11616.0	0.379212
$980$	0	0
$981$	−234.000	−0.00761574
$982$	0	0
$983$	18798.0	0.609932	0.304966	−	0.952363i	$-0.401355\pi$
$983$	18798.0	0.609932	0.304966	+	0.952363i	$0.401355\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4488.00	0.144736
$988$	0	0
$989$	23852.0	0.766885
$990$	0	0
$991$	2468.00	0.0791106	0.0395553	−	0.999217i	$-0.487406\pi$
$991$	2468.00	0.0791106	0.0395553	+	0.999217i	$0.487406\pi$
$992$	0	0
$993$	52848.0	1.68890
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−61086.0	−1.94043	−0.970217	−	0.242237i	$-0.922119\pi$
$997$	−61086.0	−1.94043	−0.970217	+	0.242237i	$0.922119\pi$
$998$	0	0
$999$	−47304.0	−1.49813

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.4.a.j.1.1		1
4.3	odd	2		1600.4.a.br.1.1		1
5.4	even	2		320.4.a.l.1.1		1
8.3	odd	2		400.4.a.e.1.1		1
8.5	even	2		200.4.a.i.1.1		1
20.19	odd	2		320.4.a.c.1.1		1
24.5	odd	2		1800.4.a.bi.1.1		1
40.3	even	4		400.4.c.f.49.1		2
40.13	odd	4		200.4.c.c.49.2		2
40.19	odd	2		80.4.a.e.1.1		1
40.27	even	4		400.4.c.f.49.2		2
40.29	even	2		40.4.a.a.1.1	✓	1
40.37	odd	4		200.4.c.c.49.1		2
80.19	odd	4		1280.4.d.a.641.1		2
80.29	even	4		1280.4.d.p.641.2		2
80.59	odd	4		1280.4.d.a.641.2		2
80.69	even	4		1280.4.d.p.641.1		2
120.29	odd	2		360.4.a.h.1.1		1
120.53	even	4		1800.4.f.j.649.1		2
120.59	even	2		720.4.a.bd.1.1		1
120.77	even	4		1800.4.f.j.649.2		2
280.69	odd	2		1960.4.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.4.a.a.1.1	✓	1	40.29	even	2
80.4.a.e.1.1		1	40.19	odd	2
200.4.a.i.1.1		1	8.5	even	2
200.4.c.c.49.1		2	40.37	odd	4
200.4.c.c.49.2		2	40.13	odd	4
320.4.a.c.1.1		1	20.19	odd	2
320.4.a.l.1.1		1	5.4	even	2
360.4.a.h.1.1		1	120.29	odd	2
400.4.a.e.1.1		1	8.3	odd	2
400.4.c.f.49.1		2	40.3	even	4
400.4.c.f.49.2		2	40.27	even	4
720.4.a.bd.1.1		1	120.59	even	2
1280.4.d.a.641.1		2	80.19	odd	4
1280.4.d.a.641.2		2	80.59	odd	4
1280.4.d.p.641.1		2	80.69	even	4
1280.4.d.p.641.2		2	80.29	even	4
1600.4.a.j.1.1		1	1.1	even	1	trivial
1600.4.a.br.1.1		1	4.3	odd	2
1800.4.a.bi.1.1		1	24.5	odd	2
1800.4.f.j.649.1		2	120.53	even	4
1800.4.f.j.649.2		2	120.77	even	4
1960.4.a.h.1.1		1	280.69	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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1600.a (trivial)

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