LMFDB - Embedded newform 200.6.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [200,6,Mod(1,200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(200, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("200.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$200 = 2^{3} \cdot 5^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	200.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:	$32.0767639626$
Analytic rank:	$1$
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	$\chi$	$=$	200.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q+18.0000 q^{3} -242.000 q^{7} +81.0000 q^{9} +656.000 q^{11} +206.000 q^{13} -1690.00 q^{17} -1364.00 q^{19} -4356.00 q^{21} -2198.00 q^{23} -2916.00 q^{27} -2218.00 q^{29} -1700.00 q^{31} +11808.0 q^{33} +846.000 q^{37} +3708.00 q^{39} -1818.00 q^{41} -10534.0 q^{43} -12074.0 q^{47} +41757.0 q^{49} -30420.0 q^{51} -32586.0 q^{53} -24552.0 q^{57} +8668.00 q^{59} -34670.0 q^{61} -19602.0 q^{63} +47566.0 q^{67} -39564.0 q^{69} +948.000 q^{71} +63102.0 q^{73} -158752. q^{77} +46536.0 q^{79} -72171.0 q^{81} +88778.0 q^{83} -39924.0 q^{87} -104934. q^{89} -49852.0 q^{91} -30600.0 q^{93} +36254.0 q^{97} +53136.0 q^{99} +O(q^{100})

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
$3$	18.0000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	18.0000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−242.000	−1.86668	−0.933341	−	0.358991i	$-0.883121\pi$
$7$	−242.000	−1.86668	−0.933341	+	0.358991i	$0.883121\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	656.000	1.63464	0.817320	−	0.576184i	$-0.195459\pi$
$11$	656.000	1.63464	0.817320	+	0.576184i	$0.195459\pi$
$12$	0	0
$13$	206.000	0.338072	0.169036	−	0.985610i	$-0.445935\pi$
$13$	206.000	0.338072	0.169036	+	0.985610i	$0.445935\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1690.00	−1.41829	−0.709144	−	0.705064i	$-0.750918\pi$
$17$	−1690.00	−1.41829	−0.709144	+	0.705064i	$0.750918\pi$
$18$	0	0
$19$	−1364.00	−0.866823	−0.433411	−	0.901196i	$-0.642690\pi$
$19$	−1364.00	−0.866823	−0.433411	+	0.901196i	$0.642690\pi$
$20$	0	0
$21$	−4356.00	−2.15546
$22$	0	0
$23$	−2198.00	−0.866379	−0.433190	−	0.901303i	$-0.642612\pi$
$23$	−2198.00	−0.866379	−0.433190	+	0.901303i	$0.642612\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2916.00	−0.769800
$28$	0	0
$29$	−2218.00	−0.489741	−0.244871	−	0.969556i	$-0.578745\pi$
$29$	−2218.00	−0.489741	−0.244871	+	0.969556i	$0.578745\pi$
$30$	0	0
$31$	−1700.00	−0.317720	−0.158860	−	0.987301i	$-0.550782\pi$
$31$	−1700.00	−0.317720	−0.158860	+	0.987301i	$0.550782\pi$
$32$	0	0
$33$	11808.0	1.88752
$34$	0	0
$35$	0	0
$36$	0	0
$37$	846.000	0.101594	0.0507968	−	0.998709i	$-0.483824\pi$
$37$	846.000	0.101594	0.0507968	+	0.998709i	$0.483824\pi$
$38$	0	0
$39$	3708.00	0.390372
$40$	0	0
$41$	−1818.00	−0.168902	−0.0844509	−	0.996428i	$-0.526914\pi$
$41$	−1818.00	−0.168902	−0.0844509	+	0.996428i	$0.526914\pi$
$42$	0	0
$43$	−10534.0	−0.868805	−0.434402	−	0.900719i	$-0.643040\pi$
$43$	−10534.0	−0.868805	−0.434402	+	0.900719i	$0.643040\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12074.0	−0.797272	−0.398636	−	0.917109i	$-0.630516\pi$
$47$	−12074.0	−0.797272	−0.398636	+	0.917109i	$0.630516\pi$
$48$	0	0
$49$	41757.0	2.48450
$50$	0	0
$51$	−30420.0	−1.63770
$52$	0	0
$53$	−32586.0	−1.59346	−0.796730	−	0.604335i	$-0.793439\pi$
$53$	−32586.0	−1.59346	−0.796730	+	0.604335i	$0.793439\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−24552.0	−1.00092
$58$	0	0
$59$	8668.00	0.324182	0.162091	−	0.986776i	$-0.448176\pi$
$59$	8668.00	0.324182	0.162091	+	0.986776i	$0.448176\pi$
$60$	0	0
$61$	−34670.0	−1.19297	−0.596485	−	0.802624i	$-0.703436\pi$
$61$	−34670.0	−1.19297	−0.596485	+	0.802624i	$0.703436\pi$
$62$	0	0
$63$	−19602.0	−0.622227
$64$	0	0
$65$	0	0
$66$	0	0
$67$	47566.0	1.29452	0.647261	−	0.762268i	$-0.275914\pi$
$67$	47566.0	1.29452	0.647261	+	0.762268i	$0.275914\pi$
$68$	0	0
$69$	−39564.0	−1.00041
$70$	0	0
$71$	948.000	0.0223184	0.0111592	−	0.999938i	$-0.496448\pi$
$71$	948.000	0.0223184	0.0111592	+	0.999938i	$0.496448\pi$
$72$	0	0
$73$	63102.0	1.38591	0.692957	−	0.720979i	$-0.256308\pi$
$73$	63102.0	1.38591	0.692957	+	0.720979i	$0.256308\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−158752.	−3.05135
$78$	0	0
$79$	46536.0	0.838921	0.419461	−	0.907773i	$-0.362219\pi$
$79$	46536.0	0.838921	0.419461	+	0.907773i	$0.362219\pi$
$80$	0	0
$81$	−72171.0	−1.22222
$82$	0	0
$83$	88778.0	1.41452	0.707262	−	0.706952i	$-0.249930\pi$
$83$	88778.0	1.41452	0.707262	+	0.706952i	$0.249930\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−39924.0	−0.565504
$88$	0	0
$89$	−104934.	−1.40424	−0.702120	−	0.712059i	$-0.747763\pi$
$89$	−104934.	−1.40424	−0.702120	+	0.712059i	$0.747763\pi$
$90$	0	0
$91$	−49852.0	−0.631072
$92$	0	0
$93$	−30600.0	−0.366872
$94$	0	0
$95$	0	0
$96$	0	0
$97$	36254.0	0.391225	0.195612	−	0.980681i	$-0.437331\pi$
$97$	36254.0	0.391225	0.195612	+	0.980681i	$0.437331\pi$
$98$	0	0
$99$	53136.0	0.544880
$100$	0	0
$101$	42486.0	0.414422	0.207211	−	0.978296i	$-0.433561\pi$
$101$	42486.0	0.414422	0.207211	+	0.978296i	$0.433561\pi$
$102$	0	0
$103$	−147934.	−1.37396	−0.686981	−	0.726675i	$-0.741065\pi$
$103$	−147934.	−1.37396	−0.686981	+	0.726675i	$0.741065\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18390.0	0.155282	0.0776412	−	0.996981i	$-0.475261\pi$
$107$	18390.0	0.155282	0.0776412	+	0.996981i	$0.475261\pi$
$108$	0	0
$109$	−145006.	−1.16901	−0.584507	−	0.811389i	$-0.698712\pi$
$109$	−145006.	−1.16901	−0.584507	+	0.811389i	$0.698712\pi$
$110$	0	0
$111$	15228.0	0.117310
$112$	0	0
$113$	−82746.0	−0.609608	−0.304804	−	0.952415i	$-0.598591\pi$
$113$	−82746.0	−0.609608	−0.304804	+	0.952415i	$0.598591\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	16686.0	0.112691
$118$	0	0
$119$	408980.	2.64749
$120$	0	0
$121$	269285.	1.67205
$122$	0	0
$123$	−32724.0	−0.195031
$124$	0	0
$125$	0	0
$126$	0	0
$127$	274446.	1.50990	0.754949	−	0.655784i	$-0.227662\pi$
$127$	274446.	1.50990	0.754949	+	0.655784i	$0.227662\pi$
$128$	0	0
$129$	−189612.	−1.00321
$130$	0	0
$131$	202608.	1.03152	0.515761	−	0.856733i	$-0.327509\pi$
$131$	202608.	1.03152	0.515761	+	0.856733i	$0.327509\pi$
$132$	0	0
$133$	330088.	1.61808
$134$	0	0
$135$	0	0
$136$	0	0
$137$	48142.0	0.219141	0.109570	−	0.993979i	$-0.465053\pi$
$137$	48142.0	0.219141	0.109570	+	0.993979i	$0.465053\pi$
$138$	0	0
$139$	−111156.	−0.487973	−0.243987	−	0.969779i	$-0.578455\pi$
$139$	−111156.	−0.487973	−0.243987	+	0.969779i	$0.578455\pi$
$140$	0	0
$141$	−217332.	−0.920610
$142$	0	0
$143$	135136.	0.552626
$144$	0	0
$145$	0	0
$146$	0	0
$147$	751626.	2.86885
$148$	0	0
$149$	243178.	0.897343	0.448672	−	0.893697i	$-0.351897\pi$
$149$	243178.	0.897343	0.448672	+	0.893697i	$0.351897\pi$
$150$	0	0
$151$	368852.	1.31647	0.658233	−	0.752814i	$-0.271304\pi$
$151$	368852.	1.31647	0.658233	+	0.752814i	$0.271304\pi$
$152$	0	0
$153$	−136890.	−0.472763
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−319546.	−1.03463	−0.517314	−	0.855796i	$-0.673068\pi$
$157$	−319546.	−1.03463	−0.517314	+	0.855796i	$0.673068\pi$
$158$	0	0
$159$	−586548.	−1.83997
$160$	0	0
$161$	531916.	1.61725
$162$	0	0
$163$	−69862.0	−0.205955	−0.102977	−	0.994684i	$-0.532837\pi$
$163$	−69862.0	−0.205955	−0.102977	+	0.994684i	$0.532837\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	343422.	0.952877	0.476439	−	0.879208i	$-0.341927\pi$
$167$	343422.	0.952877	0.476439	+	0.879208i	$0.341927\pi$
$168$	0	0
$169$	−328857.	−0.885708
$170$	0	0
$171$	−110484.	−0.288941
$172$	0	0
$173$	1142.00	0.00290102	0.00145051	−	0.999999i	$-0.499538\pi$
$173$	1142.00	0.00290102	0.00145051	+	0.999999i	$0.499538\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	156024.	0.374333
$178$	0	0
$179$	−86684.0	−0.202212	−0.101106	−	0.994876i	$-0.532238\pi$
$179$	−86684.0	−0.202212	−0.101106	+	0.994876i	$0.532238\pi$
$180$	0	0
$181$	−651418.	−1.47796	−0.738981	−	0.673726i	$-0.764693\pi$
$181$	−651418.	−1.47796	−0.738981	+	0.673726i	$0.764693\pi$
$182$	0	0
$183$	−624060.	−1.37752
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.10864e6	−2.31839
$188$	0	0
$189$	705672.	1.43697
$190$	0	0
$191$	29140.0	0.0577971	0.0288986	−	0.999582i	$-0.490800\pi$
$191$	29140.0	0.0577971	0.0288986	+	0.999582i	$0.490800\pi$
$192$	0	0
$193$	646406.	1.24914	0.624571	−	0.780968i	$-0.285274\pi$
$193$	646406.	1.24914	0.624571	+	0.780968i	$0.285274\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−431138.	−0.791500	−0.395750	−	0.918358i	$-0.629515\pi$
$197$	−431138.	−0.791500	−0.395750	+	0.918358i	$0.629515\pi$
$198$	0	0
$199$	−131608.	−0.235586	−0.117793	−	0.993038i	$-0.537582\pi$
$199$	−131608.	−0.235586	−0.117793	+	0.993038i	$0.537582\pi$
$200$	0	0
$201$	856188.	1.49479
$202$	0	0
$203$	536756.	0.914191
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−178038.	−0.288793
$208$	0	0
$209$	−894784.	−1.41694
$210$	0	0
$211$	1.21078e6	1.87224	0.936118	−	0.351686i	$-0.114392\pi$
$211$	1.21078e6	1.87224	0.936118	+	0.351686i	$0.114392\pi$
$212$	0	0
$213$	17064.0	0.0257710
$214$	0	0
$215$	0	0
$216$	0	0
$217$	411400.	0.593082
$218$	0	0
$219$	1.13584e6	1.60031
$220$	0	0
$221$	−348140.	−0.479483
$222$	0	0
$223$	−34886.0	−0.0469774	−0.0234887	−	0.999724i	$-0.507477\pi$
$223$	−34886.0	−0.0469774	−0.0234887	+	0.999724i	$0.507477\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	124182.	0.159954	0.0799768	−	0.996797i	$-0.474515\pi$
$227$	124182.	0.159954	0.0799768	+	0.996797i	$0.474515\pi$
$228$	0	0
$229$	−456386.	−0.575100	−0.287550	−	0.957766i	$-0.592841\pi$
$229$	−456386.	−0.575100	−0.287550	+	0.957766i	$0.592841\pi$
$230$	0	0
$231$	−2.85754e6	−3.52340
$232$	0	0
$233$	−252666.	−0.304900	−0.152450	−	0.988311i	$-0.548716\pi$
$233$	−252666.	−0.304900	−0.152450	+	0.988311i	$0.548716\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	837648.	0.968703
$238$	0	0
$239$	65064.0	0.0736794	0.0368397	−	0.999321i	$-0.488271\pi$
$239$	65064.0	0.0736794	0.0368397	+	0.999321i	$0.488271\pi$
$240$	0	0
$241$	−1.40600e6	−1.55935	−0.779675	−	0.626185i	$-0.784615\pi$
$241$	−1.40600e6	−1.55935	−0.779675	+	0.626185i	$0.784615\pi$
$242$	0	0
$243$	−590490.	−0.641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−280984.	−0.293048
$248$	0	0
$249$	1.59800e6	1.63335
$250$	0	0
$251$	−548400.	−0.549431	−0.274715	−	0.961526i	$-0.588584\pi$
$251$	−548400.	−0.549431	−0.274715	+	0.961526i	$0.588584\pi$
$252$	0	0
$253$	−1.44189e6	−1.41622
$254$	0	0
$255$	0	0
$256$	0	0
$257$	493830.	0.466385	0.233193	−	0.972431i	$-0.425083\pi$
$257$	493830.	0.466385	0.233193	+	0.972431i	$0.425083\pi$
$258$	0	0
$259$	−204732.	−0.189643
$260$	0	0
$261$	−179658.	−0.163247
$262$	0	0
$263$	1.07181e6	0.955495	0.477748	−	0.878497i	$-0.341453\pi$
$263$	1.07181e6	0.955495	0.477748	+	0.878497i	$0.341453\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1.88881e6	−1.62148
$268$	0	0
$269$	999394.	0.842085	0.421043	−	0.907041i	$-0.361664\pi$
$269$	999394.	0.842085	0.421043	+	0.907041i	$0.361664\pi$
$270$	0	0
$271$	1.00760e6	0.833425	0.416713	−	0.909038i	$-0.363182\pi$
$271$	1.00760e6	0.833425	0.416713	+	0.909038i	$0.363182\pi$
$272$	0	0
$273$	−897336.	−0.728700
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−1.02286e6	−0.800969	−0.400485	−	0.916303i	$-0.631158\pi$
$277$	−1.02286e6	−0.800969	−0.400485	+	0.916303i	$0.631158\pi$
$278$	0	0
$279$	−137700.	−0.105907
$280$	0	0
$281$	−1.18172e6	−0.892790	−0.446395	−	0.894836i	$-0.647292\pi$
$281$	−1.18172e6	−0.892790	−0.446395	+	0.894836i	$0.647292\pi$
$282$	0	0
$283$	917506.	0.680993	0.340497	−	0.940246i	$-0.389405\pi$
$283$	917506.	0.680993	0.340497	+	0.940246i	$0.389405\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	439956.	0.315286
$288$	0	0
$289$	1.43624e6	1.01154
$290$	0	0
$291$	652572.	0.451748
$292$	0	0
$293$	512302.	0.348624	0.174312	−	0.984690i	$-0.444230\pi$
$293$	512302.	0.348624	0.174312	+	0.984690i	$0.444230\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.91290e6	−1.25835
$298$	0	0
$299$	−452788.	−0.292898
$300$	0	0
$301$	2.54923e6	1.62178
$302$	0	0
$303$	764748.	0.478533
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1.40946e6	−0.853505	−0.426753	−	0.904368i	$-0.640342\pi$
$307$	−1.40946e6	−0.853505	−0.426753	+	0.904368i	$0.640342\pi$
$308$	0	0
$309$	−2.66281e6	−1.58652
$310$	0	0
$311$	−2.78604e6	−1.63337	−0.816687	−	0.577081i	$-0.804192\pi$
$311$	−2.78604e6	−1.63337	−0.816687	+	0.577081i	$0.804192\pi$
$312$	0	0
$313$	−1.55086e6	−0.894770	−0.447385	−	0.894342i	$-0.647645\pi$
$313$	−1.55086e6	−0.894770	−0.447385	+	0.894342i	$0.647645\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−377322.	−0.210894	−0.105447	−	0.994425i	$-0.533627\pi$
$317$	−377322.	−0.210894	−0.105447	+	0.994425i	$0.533627\pi$
$318$	0	0
$319$	−1.45501e6	−0.800550
$320$	0	0
$321$	331020.	0.179305
$322$	0	0
$323$	2.30516e6	1.22940
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2.61011e6	−1.34986
$328$	0	0
$329$	2.92191e6	1.48825
$330$	0	0
$331$	−1.63063e6	−0.818062	−0.409031	−	0.912521i	$-0.634133\pi$
$331$	−1.63063e6	−0.818062	−0.409031	+	0.912521i	$0.634133\pi$
$332$	0	0
$333$	68526.0	0.0338645
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.36717e6	1.61506	0.807532	−	0.589824i	$-0.200803\pi$
$337$	3.36717e6	1.61506	0.807532	+	0.589824i	$0.200803\pi$
$338$	0	0
$339$	−1.48943e6	−0.703915
$340$	0	0
$341$	−1.11520e6	−0.519358
$342$	0	0
$343$	−6.03790e6	−2.77109
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−837202.	−0.373256	−0.186628	−	0.982431i	$-0.559756\pi$
$347$	−837202.	−0.373256	−0.186628	+	0.982431i	$0.559756\pi$
$348$	0	0
$349$	−1.51910e6	−0.667609	−0.333805	−	0.942642i	$-0.608333\pi$
$349$	−1.51910e6	−0.667609	−0.333805	+	0.942642i	$0.608333\pi$
$350$	0	0
$351$	−600696.	−0.260248
$352$	0	0
$353$	3.51851e6	1.50287	0.751436	−	0.659806i	$-0.229362\pi$
$353$	3.51851e6	1.50287	0.751436	+	0.659806i	$0.229362\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	7.36164e6	3.05706
$358$	0	0
$359$	−3.57089e6	−1.46231	−0.731156	−	0.682210i	$-0.761019\pi$
$359$	−3.57089e6	−1.46231	−0.731156	+	0.682210i	$0.761019\pi$
$360$	0	0
$361$	−615603.	−0.248618
$362$	0	0
$363$	4.84713e6	1.93071
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.58231e6	1.38835	0.694173	−	0.719808i	$-0.255770\pi$
$367$	3.58231e6	1.38835	0.694173	+	0.719808i	$0.255770\pi$
$368$	0	0
$369$	−147258.	−0.0563006
$370$	0	0
$371$	7.88581e6	2.97448
$372$	0	0
$373$	−635530.	−0.236518	−0.118259	−	0.992983i	$-0.537731\pi$
$373$	−635530.	−0.236518	−0.118259	+	0.992983i	$0.537731\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−456908.	−0.165568
$378$	0	0
$379$	67060.0	0.0239809	0.0119905	−	0.999928i	$-0.496183\pi$
$379$	67060.0	0.0239809	0.0119905	+	0.999928i	$0.496183\pi$
$380$	0	0
$381$	4.94003e6	1.74348
$382$	0	0
$383$	−4.45129e6	−1.55056	−0.775280	−	0.631618i	$-0.782391\pi$
$383$	−4.45129e6	−1.55056	−0.775280	+	0.631618i	$0.782391\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−853254.	−0.289602
$388$	0	0
$389$	5.79825e6	1.94278	0.971388	−	0.237496i	$-0.0763267\pi$
$389$	5.79825e6	1.94278	0.971388	+	0.237496i	$0.0763267\pi$
$390$	0	0
$391$	3.71462e6	1.22878
$392$	0	0
$393$	3.64694e6	1.19110
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−333874.	−0.106318	−0.0531589	−	0.998586i	$-0.516929\pi$
$397$	−333874.	−0.106318	−0.0531589	+	0.998586i	$0.516929\pi$
$398$	0	0
$399$	5.94158e6	1.86840
$400$	0	0
$401$	−2.55689e6	−0.794057	−0.397029	−	0.917806i	$-0.629959\pi$
$401$	−2.55689e6	−0.794057	−0.397029	+	0.917806i	$0.629959\pi$
$402$	0	0
$403$	−350200.	−0.107412
$404$	0	0
$405$	0	0
$406$	0	0
$407$	554976.	0.166069
$408$	0	0
$409$	−3.05511e6	−0.903063	−0.451531	−	0.892255i	$-0.649122\pi$
$409$	−3.05511e6	−0.903063	−0.451531	+	0.892255i	$0.649122\pi$
$410$	0	0
$411$	866556.	0.253042
$412$	0	0
$413$	−2.09766e6	−0.605145
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−2.00081e6	−0.563463
$418$	0	0
$419$	−3.54347e6	−0.986038	−0.493019	−	0.870019i	$-0.664107\pi$
$419$	−3.54347e6	−0.986038	−0.493019	+	0.870019i	$0.664107\pi$
$420$	0	0
$421$	1.97294e6	0.542511	0.271255	−	0.962507i	$-0.412561\pi$
$421$	1.97294e6	0.542511	0.271255	+	0.962507i	$0.412561\pi$
$422$	0	0
$423$	−977994.	−0.265757
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8.39014e6	2.22689
$428$	0	0
$429$	2.43245e6	0.638117
$430$	0	0
$431$	1.37396e6	0.356270	0.178135	−	0.984006i	$-0.442994\pi$
$431$	1.37396e6	0.356270	0.178135	+	0.984006i	$0.442994\pi$
$432$	0	0
$433$	−5.18813e6	−1.32981	−0.664907	−	0.746926i	$-0.731529\pi$
$433$	−5.18813e6	−1.32981	−0.664907	+	0.746926i	$0.731529\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.99807e6	0.750997
$438$	0	0
$439$	−2.94082e6	−0.728296	−0.364148	−	0.931341i	$-0.618640\pi$
$439$	−2.94082e6	−0.728296	−0.364148	+	0.931341i	$0.618640\pi$
$440$	0	0
$441$	3.38232e6	0.828167
$442$	0	0
$443$	1.28347e6	0.310724	0.155362	−	0.987858i	$-0.450346\pi$
$443$	1.28347e6	0.310724	0.155362	+	0.987858i	$0.450346\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4.37720e6	1.03616
$448$	0	0
$449$	−4.95263e6	−1.15937	−0.579683	−	0.814842i	$-0.696824\pi$
$449$	−4.95263e6	−1.15937	−0.579683	+	0.814842i	$0.696824\pi$
$450$	0	0
$451$	−1.19261e6	−0.276094
$452$	0	0
$453$	6.63934e6	1.52012
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−7.91315e6	−1.77239	−0.886194	−	0.463315i	$-0.846660\pi$
$457$	−7.91315e6	−1.77239	−0.886194	+	0.463315i	$0.846660\pi$
$458$	0	0
$459$	4.92804e6	1.09180
$460$	0	0
$461$	−6.18530e6	−1.35553	−0.677764	−	0.735280i	$-0.737051\pi$
$461$	−6.18530e6	−1.35553	−0.677764	+	0.735280i	$0.737051\pi$
$462$	0	0
$463$	−491934.	−0.106648	−0.0533242	−	0.998577i	$-0.516982\pi$
$463$	−491934.	−0.106648	−0.0533242	+	0.998577i	$0.516982\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−447442.	−0.0949390	−0.0474695	−	0.998873i	$-0.515116\pi$
$467$	−447442.	−0.0949390	−0.0474695	+	0.998873i	$0.515116\pi$
$468$	0	0
$469$	−1.15110e7	−2.41646
$470$	0	0
$471$	−5.75183e6	−1.19469
$472$	0	0
$473$	−6.91030e6	−1.42018
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2.63947e6	−0.531154
$478$	0	0
$479$	8.18487e6	1.62995	0.814973	−	0.579499i	$-0.196752\pi$
$479$	8.18487e6	1.62995	0.814973	+	0.579499i	$0.196752\pi$
$480$	0	0
$481$	174276.	0.0343459
$482$	0	0
$483$	9.57449e6	1.86744
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−6.21524e6	−1.18751	−0.593753	−	0.804648i	$-0.702354\pi$
$487$	−6.21524e6	−1.18751	−0.593753	+	0.804648i	$0.702354\pi$
$488$	0	0
$489$	−1.25752e6	−0.237816
$490$	0	0
$491$	827856.	0.154971	0.0774856	−	0.996993i	$-0.475311\pi$
$491$	827856.	0.154971	0.0774856	+	0.996993i	$0.475311\pi$
$492$	0	0
$493$	3.74842e6	0.694594
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−229416.	−0.0416613
$498$	0	0
$499$	−1.04004e7	−1.86982	−0.934908	−	0.354890i	$-0.884518\pi$
$499$	−1.04004e7	−1.86982	−0.934908	+	0.354890i	$0.884518\pi$
$500$	0	0
$501$	6.18160e6	1.10029
$502$	0	0
$503$	−2.03821e6	−0.359193	−0.179597	−	0.983740i	$-0.557479\pi$
$503$	−2.03821e6	−0.359193	−0.179597	+	0.983740i	$0.557479\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−5.91943e6	−1.02273
$508$	0	0
$509$	3.66133e6	0.626390	0.313195	−	0.949689i	$-0.398601\pi$
$509$	3.66133e6	0.626390	0.313195	+	0.949689i	$0.398601\pi$
$510$	0	0
$511$	−1.52707e7	−2.58706
$512$	0	0
$513$	3.97742e6	0.667281
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−7.92054e6	−1.30325
$518$	0	0
$519$	20556.0	0.00334981
$520$	0	0
$521$	3.24713e6	0.524089	0.262045	−	0.965056i	$-0.415603\pi$
$521$	3.24713e6	0.524089	0.262045	+	0.965056i	$0.415603\pi$
$522$	0	0
$523$	4.97357e6	0.795086	0.397543	−	0.917584i	$-0.369863\pi$
$523$	4.97357e6	0.795086	0.397543	+	0.917584i	$0.369863\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.87300e6	0.450619
$528$	0	0
$529$	−1.60514e6	−0.249387
$530$	0	0
$531$	702108.	0.108061
$532$	0	0
$533$	−374508.	−0.0571009
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−1.56031e6	−0.233494
$538$	0	0
$539$	2.73926e7	4.06126
$540$	0	0
$541$	2.42544e6	0.356285	0.178142	−	0.984005i	$-0.442991\pi$
$541$	2.42544e6	0.356285	0.178142	+	0.984005i	$0.442991\pi$
$542$	0	0
$543$	−1.17255e7	−1.70660
$544$	0	0
$545$	0	0
$546$	0	0
$547$	731254.	0.104496	0.0522480	−	0.998634i	$-0.483361\pi$
$547$	731254.	0.104496	0.0522480	+	0.998634i	$0.483361\pi$
$548$	0	0
$549$	−2.80827e6	−0.397656
$550$	0	0
$551$	3.02535e6	0.424519
$552$	0	0
$553$	−1.12617e7	−1.56600
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−7.71992e6	−1.05433	−0.527163	−	0.849764i	$-0.676744\pi$
$557$	−7.71992e6	−1.05433	−0.527163	+	0.849764i	$0.676744\pi$
$558$	0	0
$559$	−2.17000e6	−0.293718
$560$	0	0
$561$	−1.99555e7	−2.67705
$562$	0	0
$563$	−3.10576e6	−0.412949	−0.206475	−	0.978452i	$-0.566199\pi$
$563$	−3.10576e6	−0.412949	−0.206475	+	0.978452i	$0.566199\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.74654e7	2.28150
$568$	0	0
$569$	−482498.	−0.0624762	−0.0312381	−	0.999512i	$-0.509945\pi$
$569$	−482498.	−0.0624762	−0.0312381	+	0.999512i	$0.509945\pi$
$570$	0	0
$571$	1.38502e7	1.77773	0.888865	−	0.458169i	$-0.151495\pi$
$571$	1.38502e7	1.77773	0.888865	+	0.458169i	$0.151495\pi$
$572$	0	0
$573$	524520.	0.0667384
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−7.09764e6	−0.887513	−0.443756	−	0.896147i	$-0.646354\pi$
$577$	−7.09764e6	−0.887513	−0.443756	+	0.896147i	$0.646354\pi$
$578$	0	0
$579$	1.16353e7	1.44239
$580$	0	0
$581$	−2.14843e7	−2.64046
$582$	0	0
$583$	−2.13764e7	−2.60473
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.56926e6	−0.187975	−0.0939873	−	0.995573i	$-0.529961\pi$
$587$	−1.56926e6	−0.187975	−0.0939873	+	0.995573i	$0.529961\pi$
$588$	0	0
$589$	2.31880e6	0.275407
$590$	0	0
$591$	−7.76048e6	−0.913945
$592$	0	0
$593$	1.30477e7	1.52370	0.761848	−	0.647756i	$-0.224293\pi$
$593$	1.30477e7	1.52370	0.761848	+	0.647756i	$0.224293\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2.36894e6	−0.272031
$598$	0	0
$599$	−5.24688e6	−0.597495	−0.298747	−	0.954332i	$-0.596569\pi$
$599$	−5.24688e6	−0.597495	−0.298747	+	0.954332i	$0.596569\pi$
$600$	0	0
$601$	−5.57316e6	−0.629384	−0.314692	−	0.949194i	$-0.601901\pi$
$601$	−5.57316e6	−0.629384	−0.314692	+	0.949194i	$0.601901\pi$
$602$	0	0
$603$	3.85285e6	0.431508
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1.98249e6	−0.218393	−0.109197	−	0.994020i	$-0.534828\pi$
$607$	−1.98249e6	−0.218393	−0.109197	+	0.994020i	$0.534828\pi$
$608$	0	0
$609$	9.66161e6	1.05562
$610$	0	0
$611$	−2.48724e6	−0.269535
$612$	0	0
$613$	−969810.	−0.104240	−0.0521201	−	0.998641i	$-0.516598\pi$
$613$	−969810.	−0.104240	−0.0521201	+	0.998641i	$0.516598\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.12946e7	1.19442	0.597211	−	0.802084i	$-0.296275\pi$
$617$	1.12946e7	1.19442	0.597211	+	0.802084i	$0.296275\pi$
$618$	0	0
$619$	−1.80728e7	−1.89583	−0.947914	−	0.318528i	$-0.896812\pi$
$619$	−1.80728e7	−1.89583	−0.947914	+	0.318528i	$0.896812\pi$
$620$	0	0
$621$	6.40937e6	0.666939
$622$	0	0
$623$	2.53940e7	2.62127
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−1.61061e7	−1.63615
$628$	0	0
$629$	−1.42974e6	−0.144089
$630$	0	0
$631$	4.62634e6	0.462556	0.231278	−	0.972888i	$-0.425709\pi$
$631$	4.62634e6	0.462556	0.231278	+	0.972888i	$0.425709\pi$
$632$	0	0
$633$	2.17941e7	2.16187
$634$	0	0
$635$	0	0
$636$	0	0
$637$	8.60194e6	0.839939
$638$	0	0
$639$	76788.0	0.00743946
$640$	0	0
$641$	1.99058e7	1.91352	0.956762	−	0.290871i	$-0.0939452\pi$
$641$	1.99058e7	1.91352	0.956762	+	0.290871i	$0.0939452\pi$
$642$	0	0
$643$	−1.21078e7	−1.15489	−0.577443	−	0.816431i	$-0.695949\pi$
$643$	−1.21078e7	−1.15489	−0.577443	+	0.816431i	$0.695949\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.53124e6	0.237724	0.118862	−	0.992911i	$-0.462075\pi$
$647$	2.53124e6	0.237724	0.118862	+	0.992911i	$0.462075\pi$
$648$	0	0
$649$	5.68621e6	0.529921
$650$	0	0
$651$	7.40520e6	0.684832
$652$	0	0
$653$	−1.37043e7	−1.25770	−0.628848	−	0.777529i	$-0.716473\pi$
$653$	−1.37043e7	−1.25770	−0.628848	+	0.777529i	$0.716473\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	5.11126e6	0.461971
$658$	0	0
$659$	−9.83320e6	−0.882026	−0.441013	−	0.897501i	$-0.645381\pi$
$659$	−9.83320e6	−0.882026	−0.441013	+	0.897501i	$0.645381\pi$
$660$	0	0
$661$	6.68687e6	0.595278	0.297639	−	0.954679i	$-0.403801\pi$
$661$	6.68687e6	0.595278	0.297639	+	0.954679i	$0.403801\pi$
$662$	0	0
$663$	−6.26652e6	−0.553659
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.87516e6	0.424302
$668$	0	0
$669$	−627948.	−0.0542448
$670$	0	0
$671$	−2.27435e7	−1.95008
$672$	0	0
$673$	727566.	0.0619205	0.0309603	−	0.999521i	$-0.490143\pi$
$673$	727566.	0.0619205	0.0309603	+	0.999521i	$0.490143\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.86951e7	−1.56768	−0.783839	−	0.620964i	$-0.786741\pi$
$677$	−1.86951e7	−1.56768	−0.783839	+	0.620964i	$0.786741\pi$
$678$	0	0
$679$	−8.77347e6	−0.730293
$680$	0	0
$681$	2.23528e6	0.184699
$682$	0	0
$683$	1.79850e7	1.47523	0.737614	−	0.675223i	$-0.235952\pi$
$683$	1.79850e7	1.47523	0.737614	+	0.675223i	$0.235952\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−8.21495e6	−0.664069
$688$	0	0
$689$	−6.71272e6	−0.538704
$690$	0	0
$691$	1.20006e6	0.0956107	0.0478053	−	0.998857i	$-0.484777\pi$
$691$	1.20006e6	0.0956107	0.0478053	+	0.998857i	$0.484777\pi$
$692$	0	0
$693$	−1.28589e7	−1.01712
$694$	0	0
$695$	0	0
$696$	0	0
$697$	3.07242e6	0.239551
$698$	0	0
$699$	−4.54799e6	−0.352068
$700$	0	0
$701$	−1.15194e7	−0.885393	−0.442697	−	0.896671i	$-0.645978\pi$
$701$	−1.15194e7	−0.885393	−0.442697	+	0.896671i	$0.645978\pi$
$702$	0	0
$703$	−1.15394e6	−0.0880636
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.02816e7	−0.773593
$708$	0	0
$709$	−610738.	−0.0456288	−0.0228144	−	0.999740i	$-0.507263\pi$
$709$	−610738.	−0.0456288	−0.0228144	+	0.999740i	$0.507263\pi$
$710$	0	0
$711$	3.76942e6	0.279640
$712$	0	0
$713$	3.73660e6	0.275266
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.17115e6	0.0850776
$718$	0	0
$719$	3.97278e6	0.286597	0.143299	−	0.989680i	$-0.454229\pi$
$719$	3.97278e6	0.286597	0.143299	+	0.989680i	$0.454229\pi$
$720$	0	0
$721$	3.58000e7	2.56475
$722$	0	0
$723$	−2.53080e7	−1.80058
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.23220e7	0.864658	0.432329	−	0.901716i	$-0.357692\pi$
$727$	1.23220e7	0.864658	0.432329	+	0.901716i	$0.357692\pi$
$728$	0	0
$729$	6.90873e6	0.481481
$730$	0	0
$731$	1.78025e7	1.23222
$732$	0	0
$733$	−8.14579e6	−0.559981	−0.279990	−	0.960003i	$-0.590331\pi$
$733$	−8.14579e6	−0.559981	−0.279990	+	0.960003i	$0.590331\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.12033e7	2.11608
$738$	0	0
$739$	7.16653e6	0.482723	0.241361	−	0.970435i	$-0.422406\pi$
$739$	7.16653e6	0.482723	0.241361	+	0.970435i	$0.422406\pi$
$740$	0	0
$741$	−5.05771e6	−0.338383
$742$	0	0
$743$	5.65041e6	0.375498	0.187749	−	0.982217i	$-0.439881\pi$
$743$	5.65041e6	0.375498	0.187749	+	0.982217i	$0.439881\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	7.19102e6	0.471508
$748$	0	0
$749$	−4.45038e6	−0.289863
$750$	0	0
$751$	9.09500e6	0.588441	0.294221	−	0.955738i	$-0.404940\pi$
$751$	9.09500e6	0.588441	0.294221	+	0.955738i	$0.404940\pi$
$752$	0	0
$753$	−9.87120e6	−0.634428
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.12880e7	0.715944	0.357972	−	0.933732i	$-0.383468\pi$
$757$	1.12880e7	0.715944	0.357972	+	0.933732i	$0.383468\pi$
$758$	0	0
$759$	−2.59540e7	−1.63531
$760$	0	0
$761$	1.52933e7	0.957283	0.478641	−	0.878011i	$-0.341129\pi$
$761$	1.52933e7	0.957283	0.478641	+	0.878011i	$0.341129\pi$
$762$	0	0
$763$	3.50915e7	2.18218
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.78561e6	0.109597
$768$	0	0
$769$	1.77402e6	0.108179	0.0540894	−	0.998536i	$-0.482774\pi$
$769$	1.77402e6	0.108179	0.0540894	+	0.998536i	$0.482774\pi$
$770$	0	0
$771$	8.88894e6	0.538535
$772$	0	0
$773$	1.46441e7	0.881484	0.440742	−	0.897634i	$-0.354715\pi$
$773$	1.46441e7	0.881484	0.440742	+	0.897634i	$0.354715\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−3.68518e6	−0.218981
$778$	0	0
$779$	2.47975e6	0.146408
$780$	0	0
$781$	621888.	0.0364825
$782$	0	0
$783$	6.46769e6	0.377003
$784$	0	0
$785$	0	0
$786$	0	0
$787$	5.97074e6	0.343630	0.171815	−	0.985129i	$-0.445037\pi$
$787$	5.97074e6	0.343630	0.171815	+	0.985129i	$0.445037\pi$
$788$	0	0
$789$	1.92926e7	1.10331
$790$	0	0
$791$	2.00245e7	1.13794
$792$	0	0
$793$	−7.14202e6	−0.403309
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.40500e7	1.89876	0.949382	−	0.314125i	$-0.101711\pi$
$797$	3.40500e7	1.89876	0.949382	+	0.314125i	$0.101711\pi$
$798$	0	0
$799$	2.04051e7	1.13076
$800$	0	0
$801$	−8.49965e6	−0.468080
$802$	0	0
$803$	4.13949e7	2.26547
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.79891e7	0.972356
$808$	0	0
$809$	−2.63540e7	−1.41571	−0.707857	−	0.706356i	$-0.750338\pi$
$809$	−2.63540e7	−1.41571	−0.707857	+	0.706356i	$0.750338\pi$
$810$	0	0
$811$	−9.49658e6	−0.507008	−0.253504	−	0.967334i	$-0.581583\pi$
$811$	−9.49658e6	−0.507008	−0.253504	+	0.967334i	$0.581583\pi$
$812$	0	0
$813$	1.81369e7	0.962357
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.43684e7	0.753100
$818$	0	0
$819$	−4.03801e6	−0.210357
$820$	0	0
$821$	−1.59887e7	−0.827856	−0.413928	−	0.910310i	$-0.635843\pi$
$821$	−1.59887e7	−0.827856	−0.413928	+	0.910310i	$0.635843\pi$
$822$	0	0
$823$	−3.18347e7	−1.63833	−0.819164	−	0.573559i	$-0.805562\pi$
$823$	−3.18347e7	−1.63833	−0.819164	+	0.573559i	$0.805562\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.27575e7	−0.648635	−0.324317	−	0.945948i	$-0.605135\pi$
$827$	−1.27575e7	−0.648635	−0.324317	+	0.945948i	$0.605135\pi$
$828$	0	0
$829$	6.18613e6	0.312631	0.156316	−	0.987707i	$-0.450038\pi$
$829$	6.18613e6	0.312631	0.156316	+	0.987707i	$0.450038\pi$
$830$	0	0
$831$	−1.84114e7	−0.924880
$832$	0	0
$833$	−7.05693e7	−3.52374
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4.95720e6	0.244581
$838$	0	0
$839$	−5.66754e6	−0.277965	−0.138982	−	0.990295i	$-0.544383\pi$
$839$	−5.66754e6	−0.277965	−0.138982	+	0.990295i	$0.544383\pi$
$840$	0	0
$841$	−1.55916e7	−0.760154
$842$	0	0
$843$	−2.12710e7	−1.03091
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−6.51670e7	−3.12118
$848$	0	0
$849$	1.65151e7	0.786343
$850$	0	0
$851$	−1.85951e6	−0.0880185
$852$	0	0
$853$	1.76010e7	0.828257	0.414129	−	0.910218i	$-0.364086\pi$
$853$	1.76010e7	0.828257	0.414129	+	0.910218i	$0.364086\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−162162.	−0.00754218	−0.00377109	−	0.999993i	$-0.501200\pi$
$857$	−162162.	−0.00754218	−0.00377109	+	0.999993i	$0.501200\pi$
$858$	0	0
$859$	−7.10520e6	−0.328544	−0.164272	−	0.986415i	$-0.552527\pi$
$859$	−7.10520e6	−0.328544	−0.164272	+	0.986415i	$0.552527\pi$
$860$	0	0
$861$	7.91921e6	0.364061
$862$	0	0
$863$	−4.08956e6	−0.186917	−0.0934586	−	0.995623i	$-0.529792\pi$
$863$	−4.08956e6	−0.186917	−0.0934586	+	0.995623i	$0.529792\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2.58524e7	1.16803
$868$	0	0
$869$	3.05276e7	1.37133
$870$	0	0
$871$	9.79860e6	0.437641
$872$	0	0
$873$	2.93657e6	0.130408
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−809194.	−0.0355266	−0.0177633	−	0.999842i	$-0.505655\pi$
$877$	−809194.	−0.0355266	−0.0177633	+	0.999842i	$0.505655\pi$
$878$	0	0
$879$	9.22144e6	0.402556
$880$	0	0
$881$	3.90411e6	0.169466	0.0847329	−	0.996404i	$-0.472996\pi$
$881$	3.90411e6	0.169466	0.0847329	+	0.996404i	$0.472996\pi$
$882$	0	0
$883$	−3.58290e7	−1.54644	−0.773220	−	0.634138i	$-0.781355\pi$
$883$	−3.58290e7	−1.54644	−0.773220	+	0.634138i	$0.781355\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.77571e7	−1.18458	−0.592290	−	0.805725i	$-0.701776\pi$
$887$	−2.77571e7	−1.18458	−0.592290	+	0.805725i	$0.701776\pi$
$888$	0	0
$889$	−6.64159e7	−2.81850
$890$	0	0
$891$	−4.73442e7	−1.99789
$892$	0	0
$893$	1.64689e7	0.691094
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−8.15018e6	−0.338210
$898$	0	0
$899$	3.77060e6	0.155601
$900$	0	0
$901$	5.50703e7	2.25999
$902$	0	0
$903$	4.58861e7	1.87267
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−2.01914e7	−0.814981	−0.407490	−	0.913209i	$-0.633596\pi$
$907$	−2.01914e7	−0.814981	−0.407490	+	0.913209i	$0.633596\pi$
$908$	0	0
$909$	3.44137e6	0.138141
$910$	0	0
$911$	2.75179e7	1.09855	0.549274	−	0.835642i	$-0.314904\pi$
$911$	2.75179e7	1.09855	0.549274	+	0.835642i	$0.314904\pi$
$912$	0	0
$913$	5.82384e7	2.31224
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4.90311e7	−1.92552
$918$	0	0
$919$	1.31786e7	0.514730	0.257365	−	0.966314i	$-0.417146\pi$
$919$	1.31786e7	0.514730	0.257365	+	0.966314i	$0.417146\pi$
$920$	0	0
$921$	−2.53702e7	−0.985543
$922$	0	0
$923$	195288.	0.00754521
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.19827e7	−0.457988
$928$	0	0
$929$	4.00688e7	1.52323	0.761617	−	0.648027i	$-0.224406\pi$
$929$	4.00688e7	1.52323	0.761617	+	0.648027i	$0.224406\pi$
$930$	0	0
$931$	−5.69565e7	−2.15362
$932$	0	0
$933$	−5.01486e7	−1.88606
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−3.04258e7	−1.13212	−0.566060	−	0.824364i	$-0.691533\pi$
$937$	−3.04258e7	−1.13212	−0.566060	+	0.824364i	$0.691533\pi$
$938$	0	0
$939$	−2.79154e7	−1.03319
$940$	0	0
$941$	−3.26349e7	−1.20146	−0.600729	−	0.799452i	$-0.705123\pi$
$941$	−3.26349e7	−1.20146	−0.600729	+	0.799452i	$0.705123\pi$
$942$	0	0
$943$	3.99596e6	0.146333
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.01534e7	−1.09260	−0.546300	−	0.837589i	$-0.683964\pi$
$947$	−3.01534e7	−1.09260	−0.546300	+	0.837589i	$0.683964\pi$
$948$	0	0
$949$	1.29990e7	0.468538
$950$	0	0
$951$	−6.79180e6	−0.243519
$952$	0	0
$953$	−303066.	−0.0108095	−0.00540474	−	0.999985i	$-0.501720\pi$
$953$	−303066.	−0.0108095	−0.00540474	+	0.999985i	$0.501720\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−2.61901e7	−0.924396
$958$	0	0
$959$	−1.16504e7	−0.409066
$960$	0	0
$961$	−2.57392e7	−0.899054
$962$	0	0
$963$	1.48959e6	0.0517608
$964$	0	0
$965$	0	0
$966$	0	0
$967$	7.39863e6	0.254440	0.127220	−	0.991875i	$-0.459395\pi$
$967$	7.39863e6	0.254440	0.127220	+	0.991875i	$0.459395\pi$
$968$	0	0
$969$	4.14929e7	1.41959
$970$	0	0
$971$	−6.18414e6	−0.210490	−0.105245	−	0.994446i	$-0.533563\pi$
$971$	−6.18414e6	−0.210490	−0.105245	+	0.994446i	$0.533563\pi$
$972$	0	0
$973$	2.68998e7	0.910891
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.63928e6	−0.0549436	−0.0274718	−	0.999623i	$-0.508746\pi$
$977$	−1.63928e6	−0.0549436	−0.0274718	+	0.999623i	$0.508746\pi$
$978$	0	0
$979$	−6.88367e7	−2.29543
$980$	0	0
$981$	−1.17455e7	−0.389671
$982$	0	0
$983$	1.13020e7	0.373052	0.186526	−	0.982450i	$-0.440277\pi$
$983$	1.13020e7	0.373052	0.186526	+	0.982450i	$0.440277\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	5.25943e7	1.71849
$988$	0	0
$989$	2.31537e7	0.752714
$990$	0	0
$991$	3.12643e6	0.101126	0.0505632	−	0.998721i	$-0.483898\pi$
$991$	3.12643e6	0.101126	0.0505632	+	0.998721i	$0.483898\pi$
$992$	0	0
$993$	−2.93514e7	−0.944616
$994$	0	0
$995$	0	0
$996$	0	0
$997$	3.55827e7	1.13371	0.566854	−	0.823818i	$-0.308160\pi$
$997$	3.55827e7	1.13371	0.566854	+	0.823818i	$0.308160\pi$
$998$	0	0
$999$	−2.46694e6	−0.0782067

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.6.a.d.1.1		1
4.3	odd	2		400.6.a.b.1.1		1
5.2	odd	4		200.6.c.b.49.1		2
5.3	odd	4		200.6.c.b.49.2		2
5.4	even	2		40.6.a.a.1.1	✓	1
15.14	odd	2		360.6.a.i.1.1		1
20.3	even	4		400.6.c.e.49.1		2
20.7	even	4		400.6.c.e.49.2		2
20.19	odd	2		80.6.a.g.1.1		1
40.19	odd	2		320.6.a.d.1.1		1
40.29	even	2		320.6.a.m.1.1		1
60.59	even	2		720.6.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.a.a.1.1	✓	1	5.4	even	2
80.6.a.g.1.1		1	20.19	odd	2
200.6.a.d.1.1		1	1.1	even	1	trivial
200.6.c.b.49.1		2	5.2	odd	4
200.6.c.b.49.2		2	5.3	odd	4
320.6.a.d.1.1		1	40.19	odd	2
320.6.a.m.1.1		1	40.29	even	2
360.6.a.i.1.1		1	15.14	odd	2
400.6.a.b.1.1		1	4.3	odd	2
400.6.c.e.49.1		2	20.3	even	4
400.6.c.e.49.2		2	20.7	even	4
720.6.a.k.1.1		1	60.59	even	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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	200.6.a.d.1.1

Level	$200$
Weight	$6$
Character	200.1
Self dual	yes
Analytic conductor	$32.077$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$