LMFDB - Embedded newform 800.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,6,Mod(1,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("800.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	800.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$128.307055850$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 32)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-8.00000 q^{3} -208.000 q^{7} -179.000 q^{9} +O(q^{10})$

$q-8.00000 q^{3} -208.000 q^{7} -179.000 q^{9} +536.000 q^{11} -694.000 q^{13} +1278.00 q^{17} -1112.00 q^{19} +1664.00 q^{21} +3216.00 q^{23} +3376.00 q^{27} +2918.00 q^{29} +2624.00 q^{31} -4288.00 q^{33} +9458.00 q^{37} +5552.00 q^{39} +170.000 q^{41} -19928.0 q^{43} +32.0000 q^{47} +26457.0 q^{49} -10224.0 q^{51} +22178.0 q^{53} +8896.00 q^{57} -41480.0 q^{59} +15462.0 q^{61} +37232.0 q^{63} -20744.0 q^{67} -25728.0 q^{69} -28592.0 q^{71} +53670.0 q^{73} -111488. q^{77} +69152.0 q^{79} +16489.0 q^{81} -37800.0 q^{83} -23344.0 q^{87} -126806. q^{89} +144352. q^{91} -20992.0 q^{93} -62290.0 q^{97} -95944.0 q^{99} +O(q^{100})$

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
$3$	−8.00000	−0.513200	−0.256600	−	0.966518i	$-0.582602\pi$
$3$	−8.00000	−0.513200	−0.256600	+	0.966518i	$0.582602\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−208.000	−1.60442	−0.802210	−	0.597042i	$-0.796343\pi$
$7$	−208.000	−1.60442	−0.802210	+	0.597042i	$0.796343\pi$
$8$	0	0
$9$	−179.000	−0.736626
$10$	0	0
$11$	536.000	1.33562	0.667810	−	0.744332i	$-0.267232\pi$
$11$	536.000	1.33562	0.667810	+	0.744332i	$0.267232\pi$
$12$	0	0
$13$	−694.000	−1.13894	−0.569470	−	0.822012i	$-0.692852\pi$
$13$	−694.000	−1.13894	−0.569470	+	0.822012i	$0.692852\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1278.00	1.07253	0.536264	−	0.844050i	$-0.319835\pi$
$17$	1278.00	1.07253	0.536264	+	0.844050i	$0.319835\pi$
$18$	0	0
$19$	−1112.00	−0.706677	−0.353338	−	0.935496i	$-0.614954\pi$
$19$	−1112.00	−0.706677	−0.353338	+	0.935496i	$0.614954\pi$
$20$	0	0
$21$	1664.00	0.823389
$22$	0	0
$23$	3216.00	1.26764	0.633821	−	0.773480i	$-0.281486\pi$
$23$	3216.00	1.26764	0.633821	+	0.773480i	$0.281486\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3376.00	0.891237
$28$	0	0
$29$	2918.00	0.644303	0.322152	−	0.946688i	$-0.395594\pi$
$29$	2918.00	0.644303	0.322152	+	0.946688i	$0.395594\pi$
$30$	0	0
$31$	2624.00	0.490410	0.245205	−	0.969471i	$-0.421145\pi$
$31$	2624.00	0.490410	0.245205	+	0.969471i	$0.421145\pi$
$32$	0	0
$33$	−4288.00	−0.685441
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9458.00	1.13578	0.567891	−	0.823104i	$-0.307759\pi$
$37$	9458.00	1.13578	0.567891	+	0.823104i	$0.307759\pi$
$38$	0	0
$39$	5552.00	0.584505
$40$	0	0
$41$	170.000	0.0157939	0.00789695	−	0.999969i	$-0.497486\pi$
$41$	170.000	0.0157939	0.00789695	+	0.999969i	$0.497486\pi$
$42$	0	0
$43$	−19928.0	−1.64359	−0.821793	−	0.569786i	$-0.807026\pi$
$43$	−19928.0	−1.64359	−0.821793	+	0.569786i	$0.807026\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	32.0000	0.00211303	0.00105651	−	0.999999i	$-0.499664\pi$
$47$	32.0000	0.00211303	0.00105651	+	0.999999i	$0.499664\pi$
$48$	0	0
$49$	26457.0	1.57417
$50$	0	0
$51$	−10224.0	−0.550422
$52$	0	0
$53$	22178.0	1.08451	0.542254	−	0.840215i	$-0.317571\pi$
$53$	22178.0	1.08451	0.542254	+	0.840215i	$0.317571\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	8896.00	0.362667
$58$	0	0
$59$	−41480.0	−1.55135	−0.775673	−	0.631135i	$-0.782589\pi$
$59$	−41480.0	−1.55135	−0.775673	+	0.631135i	$0.782589\pi$
$60$	0	0
$61$	15462.0	0.532036	0.266018	−	0.963968i	$-0.414292\pi$
$61$	15462.0	0.532036	0.266018	+	0.963968i	$0.414292\pi$
$62$	0	0
$63$	37232.0	1.18186
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−20744.0	−0.564554	−0.282277	−	0.959333i	$-0.591090\pi$
$67$	−20744.0	−0.564554	−0.282277	+	0.959333i	$0.591090\pi$
$68$	0	0
$69$	−25728.0	−0.650554
$70$	0	0
$71$	−28592.0	−0.673130	−0.336565	−	0.941660i	$-0.609265\pi$
$71$	−28592.0	−0.673130	−0.336565	+	0.941660i	$0.609265\pi$
$72$	0	0
$73$	53670.0	1.17876	0.589379	−	0.807857i	$-0.299373\pi$
$73$	53670.0	1.17876	0.589379	+	0.807857i	$0.299373\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−111488.	−2.14290
$78$	0	0
$79$	69152.0	1.24663	0.623314	−	0.781971i	$-0.285786\pi$
$79$	69152.0	1.24663	0.623314	+	0.781971i	$0.285786\pi$
$80$	0	0
$81$	16489.0	0.279243
$82$	0	0
$83$	−37800.0	−0.602277	−0.301139	−	0.953580i	$-0.597367\pi$
$83$	−37800.0	−0.602277	−0.301139	+	0.953580i	$0.597367\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−23344.0	−0.330657
$88$	0	0
$89$	−126806.	−1.69693	−0.848467	−	0.529249i	$-0.822474\pi$
$89$	−126806.	−1.69693	−0.848467	+	0.529249i	$0.822474\pi$
$90$	0	0
$91$	144352.	1.82734
$92$	0	0
$93$	−20992.0	−0.251679
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−62290.0	−0.672185	−0.336093	−	0.941829i	$-0.609106\pi$
$97$	−62290.0	−0.672185	−0.336093	+	0.941829i	$0.609106\pi$
$98$	0	0
$99$	−95944.0	−0.983852
$100$	0	0
$101$	6414.00	0.0625641	0.0312821	−	0.999511i	$-0.490041\pi$
$101$	6414.00	0.0625641	0.0312821	+	0.999511i	$0.490041\pi$
$102$	0	0
$103$	−108432.	−1.00708	−0.503541	−	0.863972i	$-0.667970\pi$
$103$	−108432.	−1.00708	−0.503541	+	0.863972i	$0.667970\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	103976.	0.877958	0.438979	−	0.898497i	$-0.355340\pi$
$107$	103976.	0.877958	0.438979	+	0.898497i	$0.355340\pi$
$108$	0	0
$109$	2486.00	0.0200417	0.0100209	−	0.999950i	$-0.496810\pi$
$109$	2486.00	0.0200417	0.0100209	+	0.999950i	$0.496810\pi$
$110$	0	0
$111$	−75664.0	−0.582884
$112$	0	0
$113$	−15794.0	−0.116358	−0.0581790	−	0.998306i	$-0.518529\pi$
$113$	−15794.0	−0.116358	−0.0581790	+	0.998306i	$0.518529\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	124226.	0.838973
$118$	0	0
$119$	−265824.	−1.72079
$120$	0	0
$121$	126245.	0.783882
$122$	0	0
$123$	−1360.00	−0.00810543
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1024.00	0.00563366	0.00281683	−	0.999996i	$-0.499103\pi$
$127$	1024.00	0.00563366	0.00281683	+	0.999996i	$0.499103\pi$
$128$	0	0
$129$	159424.	0.843489
$130$	0	0
$131$	22664.0	0.115387	0.0576937	−	0.998334i	$-0.481625\pi$
$131$	22664.0	0.115387	0.0576937	+	0.998334i	$0.481625\pi$
$132$	0	0
$133$	231296.	1.13381
$134$	0	0
$135$	0	0
$136$	0	0
$137$	53238.0	0.242337	0.121169	−	0.992632i	$-0.461336\pi$
$137$	53238.0	0.242337	0.121169	+	0.992632i	$0.461336\pi$
$138$	0	0
$139$	−19816.0	−0.0869919	−0.0434960	−	0.999054i	$-0.513850\pi$
$139$	−19816.0	−0.0869919	−0.0434960	+	0.999054i	$0.513850\pi$
$140$	0	0
$141$	−256.000	−0.00108441
$142$	0	0
$143$	−371984.	−1.52119
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−211656.	−0.807862
$148$	0	0
$149$	452190.	1.66861	0.834306	−	0.551302i	$-0.185869\pi$
$149$	452190.	1.66861	0.834306	+	0.551302i	$0.185869\pi$
$150$	0	0
$151$	263280.	0.939670	0.469835	−	0.882754i	$-0.344313\pi$
$151$	263280.	0.939670	0.469835	+	0.882754i	$0.344313\pi$
$152$	0	0
$153$	−228762.	−0.790051
$154$	0	0
$155$	0	0
$156$	0	0
$157$	353530.	1.14466	0.572331	−	0.820023i	$-0.306039\pi$
$157$	353530.	1.14466	0.572331	+	0.820023i	$0.306039\pi$
$158$	0	0
$159$	−177424.	−0.556570
$160$	0	0
$161$	−668928.	−2.03383
$162$	0	0
$163$	−100936.	−0.297562	−0.148781	−	0.988870i	$-0.547535\pi$
$163$	−100936.	−0.297562	−0.148781	+	0.988870i	$0.547535\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−284944.	−0.790621	−0.395310	−	0.918548i	$-0.629363\pi$
$167$	−284944.	−0.790621	−0.395310	+	0.918548i	$0.629363\pi$
$168$	0	0
$169$	110343.	0.297186
$170$	0	0
$171$	199048.	0.520556
$172$	0	0
$173$	−484374.	−1.23045	−0.615227	−	0.788350i	$-0.710936\pi$
$173$	−484374.	−1.23045	−0.615227	+	0.788350i	$0.710936\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	331840.	0.796151
$178$	0	0
$179$	−406680.	−0.948681	−0.474341	−	0.880341i	$-0.657313\pi$
$179$	−406680.	−0.948681	−0.474341	+	0.880341i	$0.657313\pi$
$180$	0	0
$181$	570302.	1.29392	0.646962	−	0.762523i	$-0.276039\pi$
$181$	570302.	1.29392	0.646962	+	0.762523i	$0.276039\pi$
$182$	0	0
$183$	−123696.	−0.273041
$184$	0	0
$185$	0	0
$186$	0	0
$187$	685008.	1.43249
$188$	0	0
$189$	−702208.	−1.42992
$190$	0	0
$191$	−138624.	−0.274951	−0.137475	−	0.990505i	$-0.543899\pi$
$191$	−138624.	−0.274951	−0.137475	+	0.990505i	$0.543899\pi$
$192$	0	0
$193$	−34482.0	−0.0666345	−0.0333173	−	0.999445i	$-0.510607\pi$
$193$	−34482.0	−0.0666345	−0.0333173	+	0.999445i	$0.510607\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−643598.	−1.18154	−0.590771	−	0.806839i	$-0.701176\pi$
$197$	−643598.	−1.18154	−0.590771	+	0.806839i	$0.701176\pi$
$198$	0	0
$199$	−1.10738e6	−1.98227	−0.991134	−	0.132865i	$-0.957582\pi$
$199$	−1.10738e6	−1.98227	−0.991134	+	0.132865i	$0.957582\pi$
$200$	0	0
$201$	165952.	0.289729
$202$	0	0
$203$	−606944.	−1.03373
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−575664.	−0.933777
$208$	0	0
$209$	−596032.	−0.943852
$210$	0	0
$211$	−229976.	−0.355612	−0.177806	−	0.984066i	$-0.556900\pi$
$211$	−229976.	−0.355612	−0.177806	+	0.984066i	$0.556900\pi$
$212$	0	0
$213$	228736.	0.345450
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−545792.	−0.786824
$218$	0	0
$219$	−429360.	−0.604939
$220$	0	0
$221$	−886932.	−1.22155
$222$	0	0
$223$	−1.08947e6	−1.46708	−0.733540	−	0.679646i	$-0.762133\pi$
$223$	−1.08947e6	−1.46708	−0.733540	+	0.679646i	$0.762133\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−687048.	−0.884958	−0.442479	−	0.896779i	$-0.645901\pi$
$227$	−687048.	−0.884958	−0.442479	+	0.896779i	$0.645901\pi$
$228$	0	0
$229$	−699730.	−0.881743	−0.440871	−	0.897570i	$-0.645330\pi$
$229$	−699730.	−0.881743	−0.440871	+	0.897570i	$0.645330\pi$
$230$	0	0
$231$	891904.	1.09974
$232$	0	0
$233$	−937722.	−1.13158	−0.565789	−	0.824550i	$-0.691428\pi$
$233$	−937722.	−1.13158	−0.565789	+	0.824550i	$0.691428\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−553216.	−0.639770
$238$	0	0
$239$	−643488.	−0.728695	−0.364347	−	0.931263i	$-0.618708\pi$
$239$	−643488.	−0.728695	−0.364347	+	0.931263i	$0.618708\pi$
$240$	0	0
$241$	157282.	0.174436	0.0872181	−	0.996189i	$-0.472202\pi$
$241$	157282.	0.174436	0.0872181	+	0.996189i	$0.472202\pi$
$242$	0	0
$243$	−952280.	−1.03454
$244$	0	0
$245$	0	0
$246$	0	0
$247$	771728.	0.804863
$248$	0	0
$249$	302400.	0.309089
$250$	0	0
$251$	1.58604e6	1.58902	0.794511	−	0.607250i	$-0.207727\pi$
$251$	1.58604e6	1.58902	0.794511	+	0.607250i	$0.207727\pi$
$252$	0	0
$253$	1.72378e6	1.69309
$254$	0	0
$255$	0	0
$256$	0	0
$257$	654334.	0.617969	0.308984	−	0.951067i	$-0.400011\pi$
$257$	654334.	0.617969	0.308984	+	0.951067i	$0.400011\pi$
$258$	0	0
$259$	−1.96726e6	−1.82227
$260$	0	0
$261$	−522322.	−0.474610
$262$	0	0
$263$	−330192.	−0.294359	−0.147179	−	0.989110i	$-0.547019\pi$
$263$	−330192.	−0.294359	−0.147179	+	0.989110i	$0.547019\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.01445e6	0.870867
$268$	0	0
$269$	1.56956e6	1.32250	0.661252	−	0.750164i	$-0.270026\pi$
$269$	1.56956e6	1.32250	0.661252	+	0.750164i	$0.270026\pi$
$270$	0	0
$271$	−957792.	−0.792224	−0.396112	−	0.918202i	$-0.629641\pi$
$271$	−957792.	−0.792224	−0.396112	+	0.918202i	$0.629641\pi$
$272$	0	0
$273$	−1.15482e6	−0.937791
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−565438.	−0.442778	−0.221389	−	0.975186i	$-0.571059\pi$
$277$	−565438.	−0.442778	−0.221389	+	0.975186i	$0.571059\pi$
$278$	0	0
$279$	−469696.	−0.361249
$280$	0	0
$281$	−1.34127e6	−1.01333	−0.506664	−	0.862143i	$-0.669122\pi$
$281$	−1.34127e6	−1.01333	−0.506664	+	0.862143i	$0.669122\pi$
$282$	0	0
$283$	−734264.	−0.544987	−0.272494	−	0.962158i	$-0.587848\pi$
$283$	−734264.	−0.544987	−0.272494	+	0.962158i	$0.587848\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−35360.0	−0.0253401
$288$	0	0
$289$	213427.	0.150316
$290$	0	0
$291$	498320.	0.344966
$292$	0	0
$293$	1.13320e6	0.771149	0.385574	−	0.922677i	$-0.374003\pi$
$293$	1.13320e6	0.771149	0.385574	+	0.922677i	$0.374003\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.80954e6	1.19035
$298$	0	0
$299$	−2.23190e6	−1.44377
$300$	0	0
$301$	4.14502e6	2.63700
$302$	0	0
$303$	−51312.0	−0.0321079
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−2.91377e6	−1.76445	−0.882224	−	0.470829i	$-0.843955\pi$
$307$	−2.91377e6	−1.76445	−0.882224	+	0.470829i	$0.843955\pi$
$308$	0	0
$309$	867456.	0.516834
$310$	0	0
$311$	1.43813e6	0.843134	0.421567	−	0.906797i	$-0.361480\pi$
$311$	1.43813e6	0.843134	0.421567	+	0.906797i	$0.361480\pi$
$312$	0	0
$313$	1.37601e6	0.793888	0.396944	−	0.917843i	$-0.370071\pi$
$313$	1.37601e6	0.793888	0.396944	+	0.917843i	$0.370071\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.23494e6	0.690235	0.345118	−	0.938559i	$-0.387839\pi$
$317$	1.23494e6	0.690235	0.345118	+	0.938559i	$0.387839\pi$
$318$	0	0
$319$	1.56405e6	0.860545
$320$	0	0
$321$	−831808.	−0.450568
$322$	0	0
$323$	−1.42114e6	−0.757930
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−19888.0	−0.0102854
$328$	0	0
$329$	−6656.00	−0.00339019
$330$	0	0
$331$	1.48930e6	0.747160	0.373580	−	0.927598i	$-0.378130\pi$
$331$	1.48930e6	0.747160	0.373580	+	0.927598i	$0.378130\pi$
$332$	0	0
$333$	−1.69298e6	−0.836646
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−838226.	−0.402056	−0.201028	−	0.979586i	$-0.564428\pi$
$337$	−838226.	−0.402056	−0.201028	+	0.979586i	$0.564428\pi$
$338$	0	0
$339$	126352.	0.0597149
$340$	0	0
$341$	1.40646e6	0.655002
$342$	0	0
$343$	−2.00720e6	−0.921203
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−350008.	−0.156047	−0.0780233	−	0.996952i	$-0.524861\pi$
$347$	−350008.	−0.156047	−0.0780233	+	0.996952i	$0.524861\pi$
$348$	0	0
$349$	−383642.	−0.168602	−0.0843010	−	0.996440i	$-0.526866\pi$
$349$	−383642.	−0.168602	−0.0843010	+	0.996440i	$0.526866\pi$
$350$	0	0
$351$	−2.34294e6	−1.01507
$352$	0	0
$353$	−4.09309e6	−1.74829	−0.874147	−	0.485661i	$-0.838579\pi$
$353$	−4.09309e6	−1.74829	−0.874147	+	0.485661i	$0.838579\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.12659e6	0.883108
$358$	0	0
$359$	−3.14430e6	−1.28762	−0.643811	−	0.765185i	$-0.722648\pi$
$359$	−3.14430e6	−1.28762	−0.643811	+	0.765185i	$0.722648\pi$
$360$	0	0
$361$	−1.23955e6	−0.500608
$362$	0	0
$363$	−1.00996e6	−0.402288
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−1.47619e6	−0.572108	−0.286054	−	0.958214i	$-0.592344\pi$
$367$	−1.47619e6	−0.572108	−0.286054	+	0.958214i	$0.592344\pi$
$368$	0	0
$369$	−30430.0	−0.0116342
$370$	0	0
$371$	−4.61302e6	−1.74001
$372$	0	0
$373$	3.73981e6	1.39180	0.695901	−	0.718138i	$-0.255005\pi$
$373$	3.73981e6	1.39180	0.695901	+	0.718138i	$0.255005\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.02509e6	−0.733823
$378$	0	0
$379$	−1.89966e6	−0.679324	−0.339662	−	0.940548i	$-0.610313\pi$
$379$	−1.89966e6	−0.679324	−0.339662	+	0.940548i	$0.610313\pi$
$380$	0	0
$381$	−8192.00	−0.00289120
$382$	0	0
$383$	1.74310e6	0.607192	0.303596	−	0.952801i	$-0.401813\pi$
$383$	1.74310e6	0.607192	0.303596	+	0.952801i	$0.401813\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.56711e6	1.21071
$388$	0	0
$389$	−2.69147e6	−0.901812	−0.450906	−	0.892571i	$-0.648899\pi$
$389$	−2.69147e6	−0.901812	−0.450906	+	0.892571i	$0.648899\pi$
$390$	0	0
$391$	4.11005e6	1.35958
$392$	0	0
$393$	−181312.	−0.0592168
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−5.37353e6	−1.71113	−0.855565	−	0.517695i	$-0.826790\pi$
$397$	−5.37353e6	−1.71113	−0.855565	+	0.517695i	$0.826790\pi$
$398$	0	0
$399$	−1.85037e6	−0.581870
$400$	0	0
$401$	156418.	0.0485765	0.0242882	−	0.999705i	$-0.492268\pi$
$401$	156418.	0.0485765	0.0242882	+	0.999705i	$0.492268\pi$
$402$	0	0
$403$	−1.82106e6	−0.558548
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.06949e6	1.51697
$408$	0	0
$409$	−306086.	−0.0904764	−0.0452382	−	0.998976i	$-0.514405\pi$
$409$	−306086.	−0.0904764	−0.0452382	+	0.998976i	$0.514405\pi$
$410$	0	0
$411$	−425904.	−0.124368
$412$	0	0
$413$	8.62784e6	2.48901
$414$	0	0
$415$	0	0
$416$	0	0
$417$	158528.	0.0446443
$418$	0	0
$419$	6.70868e6	1.86682	0.933409	−	0.358814i	$-0.116819\pi$
$419$	6.70868e6	1.86682	0.933409	+	0.358814i	$0.116819\pi$
$420$	0	0
$421$	4.02347e6	1.10636	0.553179	−	0.833063i	$-0.313415\pi$
$421$	4.02347e6	1.10636	0.553179	+	0.833063i	$0.313415\pi$
$422$	0	0
$423$	−5728.00	−0.00155651
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−3.21610e6	−0.853610
$428$	0	0
$429$	2.97587e6	0.780676
$430$	0	0
$431$	7.04304e6	1.82628	0.913139	−	0.407648i	$-0.133651\pi$
$431$	7.04304e6	1.82628	0.913139	+	0.407648i	$0.133651\pi$
$432$	0	0
$433$	1.25142e6	0.320763	0.160381	−	0.987055i	$-0.448728\pi$
$433$	1.25142e6	0.320763	0.160381	+	0.987055i	$0.448728\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.57619e6	−0.895813
$438$	0	0
$439$	1.25406e6	0.310569	0.155285	−	0.987870i	$-0.450371\pi$
$439$	1.25406e6	0.310569	0.155285	+	0.987870i	$0.450371\pi$
$440$	0	0
$441$	−4.73580e6	−1.15957
$442$	0	0
$443$	5.18081e6	1.25426	0.627131	−	0.778914i	$-0.284229\pi$
$443$	5.18081e6	1.25426	0.627131	+	0.778914i	$0.284229\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−3.61752e6	−0.856332
$448$	0	0
$449$	−5.73064e6	−1.34149	−0.670745	−	0.741688i	$-0.734025\pi$
$449$	−5.73064e6	−1.34149	−0.670745	+	0.741688i	$0.734025\pi$
$450$	0	0
$451$	91120.0	0.0210947
$452$	0	0
$453$	−2.10624e6	−0.482239
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.24153e6	1.17400	0.586999	−	0.809588i	$-0.300309\pi$
$457$	5.24153e6	1.17400	0.586999	+	0.809588i	$0.300309\pi$
$458$	0	0
$459$	4.31453e6	0.955876
$460$	0	0
$461$	−173994.	−0.0381313	−0.0190657	−	0.999818i	$-0.506069\pi$
$461$	−173994.	−0.0381313	−0.0190657	+	0.999818i	$0.506069\pi$
$462$	0	0
$463$	4.01277e6	0.869945	0.434972	−	0.900444i	$-0.356758\pi$
$463$	4.01277e6	0.869945	0.434972	+	0.900444i	$0.356758\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	774616.	0.164359	0.0821796	−	0.996618i	$-0.473812\pi$
$467$	774616.	0.164359	0.0821796	+	0.996618i	$0.473812\pi$
$468$	0	0
$469$	4.31475e6	0.905782
$470$	0	0
$471$	−2.82824e6	−0.587441
$472$	0	0
$473$	−1.06814e7	−2.19521
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−3.96986e6	−0.798876
$478$	0	0
$479$	−2.33530e6	−0.465054	−0.232527	−	0.972590i	$-0.574699\pi$
$479$	−2.33530e6	−0.465054	−0.232527	+	0.972590i	$0.574699\pi$
$480$	0	0
$481$	−6.56385e6	−1.29359
$482$	0	0
$483$	5.35142e6	1.04376
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1.03947e6	0.198605	0.0993025	−	0.995057i	$-0.468339\pi$
$487$	1.03947e6	0.198605	0.0993025	+	0.995057i	$0.468339\pi$
$488$	0	0
$489$	807488.	0.152709
$490$	0	0
$491$	−7.85092e6	−1.46966	−0.734830	−	0.678251i	$-0.762738\pi$
$491$	−7.85092e6	−1.46966	−0.734830	+	0.678251i	$0.762738\pi$
$492$	0	0
$493$	3.72920e6	0.691033
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.94714e6	1.07998
$498$	0	0
$499$	−2.71644e6	−0.488370	−0.244185	−	0.969729i	$-0.578520\pi$
$499$	−2.71644e6	−0.488370	−0.244185	+	0.969729i	$0.578520\pi$
$500$	0	0
$501$	2.27955e6	0.405747
$502$	0	0
$503$	−4.62034e6	−0.814242	−0.407121	−	0.913374i	$-0.633467\pi$
$503$	−4.62034e6	−0.814242	−0.407121	+	0.913374i	$0.633467\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−882744.	−0.152516
$508$	0	0
$509$	−4.46198e6	−0.763366	−0.381683	−	0.924293i	$-0.624655\pi$
$509$	−4.46198e6	−0.763366	−0.381683	+	0.924293i	$0.624655\pi$
$510$	0	0
$511$	−1.11634e7	−1.89122
$512$	0	0
$513$	−3.75411e6	−0.629816
$514$	0	0
$515$	0	0
$516$	0	0
$517$	17152.0	0.00282220
$518$	0	0
$519$	3.87499e6	0.631470
$520$	0	0
$521$	3.74375e6	0.604245	0.302122	−	0.953269i	$-0.402305\pi$
$521$	3.74375e6	0.604245	0.302122	+	0.953269i	$0.402305\pi$
$522$	0	0
$523$	9.28433e6	1.48421	0.742107	−	0.670282i	$-0.233827\pi$
$523$	9.28433e6	1.48421	0.742107	+	0.670282i	$0.233827\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.35347e6	0.525979
$528$	0	0
$529$	3.90631e6	0.606915
$530$	0	0
$531$	7.42492e6	1.14276
$532$	0	0
$533$	−117980.	−0.0179883
$534$	0	0
$535$	0	0
$536$	0	0
$537$	3.25344e6	0.486863
$538$	0	0
$539$	1.41810e7	2.10249
$540$	0	0
$541$	862150.	0.126645	0.0633227	−	0.997993i	$-0.479830\pi$
$541$	862150.	0.126645	0.0633227	+	0.997993i	$0.479830\pi$
$542$	0	0
$543$	−4.56242e6	−0.664042
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.75442e6	0.250707	0.125353	−	0.992112i	$-0.459994\pi$
$547$	1.75442e6	0.250707	0.125353	+	0.992112i	$0.459994\pi$
$548$	0	0
$549$	−2.76770e6	−0.391911
$550$	0	0
$551$	−3.24482e6	−0.455314
$552$	0	0
$553$	−1.43836e7	−2.00012
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.00292e7	1.36971	0.684856	−	0.728678i	$-0.259865\pi$
$557$	1.00292e7	1.36971	0.684856	+	0.728678i	$0.259865\pi$
$558$	0	0
$559$	1.38300e7	1.87195
$560$	0	0
$561$	−5.48006e6	−0.735154
$562$	0	0
$563$	−5.27460e6	−0.701324	−0.350662	−	0.936502i	$-0.614043\pi$
$563$	−5.27460e6	−0.701324	−0.350662	+	0.936502i	$0.614043\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−3.42971e6	−0.448023
$568$	0	0
$569$	8.36940e6	1.08371	0.541856	−	0.840471i	$-0.317722\pi$
$569$	8.36940e6	1.08371	0.541856	+	0.840471i	$0.317722\pi$
$570$	0	0
$571$	−4.02702e6	−0.516884	−0.258442	−	0.966027i	$-0.583209\pi$
$571$	−4.02702e6	−0.516884	−0.258442	+	0.966027i	$0.583209\pi$
$572$	0	0
$573$	1.10899e6	0.141105
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−2.37568e6	−0.297063	−0.148532	−	0.988908i	$-0.547455\pi$
$577$	−2.37568e6	−0.297063	−0.148532	+	0.988908i	$0.547455\pi$
$578$	0	0
$579$	275856.	0.0341968
$580$	0	0
$581$	7.86240e6	0.966306
$582$	0	0
$583$	1.18874e7	1.44849
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.44028e6	−0.412096	−0.206048	−	0.978542i	$-0.566060\pi$
$587$	−3.44028e6	−0.412096	−0.206048	+	0.978542i	$0.566060\pi$
$588$	0	0
$589$	−2.91789e6	−0.346562
$590$	0	0
$591$	5.14878e6	0.606368
$592$	0	0
$593$	3.22942e6	0.377127	0.188564	−	0.982061i	$-0.439617\pi$
$593$	3.22942e6	0.377127	0.188564	+	0.982061i	$0.439617\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8.85901e6	1.01730
$598$	0	0
$599$	9.29714e6	1.05872	0.529361	−	0.848397i	$-0.322432\pi$
$599$	9.29714e6	1.05872	0.529361	+	0.848397i	$0.322432\pi$
$600$	0	0
$601$	−1.12782e7	−1.27366	−0.636828	−	0.771006i	$-0.719754\pi$
$601$	−1.12782e7	−1.27366	−0.636828	+	0.771006i	$0.719754\pi$
$602$	0	0
$603$	3.71318e6	0.415865
$604$	0	0
$605$	0	0
$606$	0	0
$607$	7.00115e6	0.771255	0.385627	−	0.922655i	$-0.373985\pi$
$607$	7.00115e6	0.771255	0.385627	+	0.922655i	$0.373985\pi$
$608$	0	0
$609$	4.85555e6	0.530512
$610$	0	0
$611$	−22208.0	−0.00240661
$612$	0	0
$613$	6.19432e6	0.665798	0.332899	−	0.942962i	$-0.391973\pi$
$613$	6.19432e6	0.665798	0.332899	+	0.942962i	$0.391973\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.62407e6	0.277500	0.138750	−	0.990327i	$-0.455692\pi$
$617$	2.62407e6	0.277500	0.138750	+	0.990327i	$0.455692\pi$
$618$	0	0
$619$	2.83721e6	0.297622	0.148811	−	0.988866i	$-0.452455\pi$
$619$	2.83721e6	0.297622	0.148811	+	0.988866i	$0.452455\pi$
$620$	0	0
$621$	1.08572e7	1.12977
$622$	0	0
$623$	2.63756e7	2.72259
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.76826e6	0.484385
$628$	0	0
$629$	1.20873e7	1.21816
$630$	0	0
$631$	−1.29656e7	−1.29634	−0.648170	−	0.761496i	$-0.724465\pi$
$631$	−1.29656e7	−1.29634	−0.648170	+	0.761496i	$0.724465\pi$
$632$	0	0
$633$	1.83981e6	0.182500
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−1.83612e7	−1.79288
$638$	0	0
$639$	5.11797e6	0.495844
$640$	0	0
$641$	−1.16798e7	−1.12276	−0.561382	−	0.827557i	$-0.689730\pi$
$641$	−1.16798e7	−1.12276	−0.561382	+	0.827557i	$0.689730\pi$
$642$	0	0
$643$	7.02732e6	0.670289	0.335145	−	0.942167i	$-0.391215\pi$
$643$	7.02732e6	0.670289	0.335145	+	0.942167i	$0.391215\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−9.72821e6	−0.913634	−0.456817	−	0.889561i	$-0.651011\pi$
$647$	−9.72821e6	−0.913634	−0.456817	+	0.889561i	$0.651011\pi$
$648$	0	0
$649$	−2.22333e7	−2.07201
$650$	0	0
$651$	4.36634e6	0.403798
$652$	0	0
$653$	9.81425e6	0.900688	0.450344	−	0.892855i	$-0.351301\pi$
$653$	9.81425e6	0.900688	0.450344	+	0.892855i	$0.351301\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−9.60693e6	−0.868303
$658$	0	0
$659$	−1.46652e7	−1.31545	−0.657724	−	0.753259i	$-0.728481\pi$
$659$	−1.46652e7	−1.31545	−0.657724	+	0.753259i	$0.728481\pi$
$660$	0	0
$661$	−1.41836e7	−1.26265	−0.631327	−	0.775517i	$-0.717489\pi$
$661$	−1.41836e7	−1.26265	−0.631327	+	0.775517i	$0.717489\pi$
$662$	0	0
$663$	7.09546e6	0.626897
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.38429e6	0.816745
$668$	0	0
$669$	8.71578e6	0.752906
$670$	0	0
$671$	8.28763e6	0.710598
$672$	0	0
$673$	−5.49941e6	−0.468035	−0.234018	−	0.972232i	$-0.575187\pi$
$673$	−5.49941e6	−0.468035	−0.234018	+	0.972232i	$0.575187\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.77375e7	−1.48737	−0.743687	−	0.668528i	$-0.766925\pi$
$677$	−1.77375e7	−1.48737	−0.743687	+	0.668528i	$0.766925\pi$
$678$	0	0
$679$	1.29563e7	1.07847
$680$	0	0
$681$	5.49638e6	0.454160
$682$	0	0
$683$	−9.39670e6	−0.770768	−0.385384	−	0.922756i	$-0.625931\pi$
$683$	−9.39670e6	−0.770768	−0.385384	+	0.922756i	$0.625931\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	5.59784e6	0.452510
$688$	0	0
$689$	−1.53915e7	−1.23519
$690$	0	0
$691$	1.34767e7	1.07371	0.536857	−	0.843673i	$-0.319611\pi$
$691$	1.34767e7	1.07371	0.536857	+	0.843673i	$0.319611\pi$
$692$	0	0
$693$	1.99564e7	1.57851
$694$	0	0
$695$	0	0
$696$	0	0
$697$	217260.	0.0169394
$698$	0	0
$699$	7.50178e6	0.580726
$700$	0	0
$701$	2.15594e7	1.65707	0.828536	−	0.559935i	$-0.189174\pi$
$701$	2.15594e7	1.65707	0.828536	+	0.559935i	$0.189174\pi$
$702$	0	0
$703$	−1.05173e7	−0.802631
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.33411e6	−0.100379
$708$	0	0
$709$	−6.38165e6	−0.476779	−0.238390	−	0.971170i	$-0.576620\pi$
$709$	−6.38165e6	−0.476779	−0.238390	+	0.971170i	$0.576620\pi$
$710$	0	0
$711$	−1.23782e7	−0.918298
$712$	0	0
$713$	8.43878e6	0.621664
$714$	0	0
$715$	0	0
$716$	0	0
$717$	5.14790e6	0.373966
$718$	0	0
$719$	1.63566e7	1.17997	0.589986	−	0.807413i	$-0.299133\pi$
$719$	1.63566e7	1.17997	0.589986	+	0.807413i	$0.299133\pi$
$720$	0	0
$721$	2.25539e7	1.61578
$722$	0	0
$723$	−1.25826e6	−0.0895207
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−2.13130e7	−1.49558	−0.747788	−	0.663937i	$-0.768884\pi$
$727$	−2.13130e7	−1.49558	−0.747788	+	0.663937i	$0.768884\pi$
$728$	0	0
$729$	3.61141e6	0.251686
$730$	0	0
$731$	−2.54680e7	−1.76279
$732$	0	0
$733$	−1.21571e7	−0.835737	−0.417869	−	0.908507i	$-0.637223\pi$
$733$	−1.21571e7	−0.835737	−0.417869	+	0.908507i	$0.637223\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.11188e7	−0.754030
$738$	0	0
$739$	1.92337e7	1.29555	0.647773	−	0.761834i	$-0.275701\pi$
$739$	1.92337e7	1.29555	0.647773	+	0.761834i	$0.275701\pi$
$740$	0	0
$741$	−6.17382e6	−0.413056
$742$	0	0
$743$	1.66565e6	0.110691	0.0553454	−	0.998467i	$-0.482374\pi$
$743$	1.66565e6	0.110691	0.0553454	+	0.998467i	$0.482374\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.76620e6	0.443653
$748$	0	0
$749$	−2.16270e7	−1.40861
$750$	0	0
$751$	−9.81290e6	−0.634888	−0.317444	−	0.948277i	$-0.602825\pi$
$751$	−9.81290e6	−0.634888	−0.317444	+	0.948277i	$0.602825\pi$
$752$	0	0
$753$	−1.26883e7	−0.815486
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.92753e7	1.22254	0.611269	−	0.791423i	$-0.290659\pi$
$757$	1.92753e7	1.22254	0.611269	+	0.791423i	$0.290659\pi$
$758$	0	0
$759$	−1.37902e7	−0.868893
$760$	0	0
$761$	−1.17863e7	−0.737762	−0.368881	−	0.929477i	$-0.620259\pi$
$761$	−1.17863e7	−0.737762	−0.368881	+	0.929477i	$0.620259\pi$
$762$	0	0
$763$	−517088.	−0.0321553
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.87871e7	1.76689
$768$	0	0
$769$	1.22941e7	0.749690	0.374845	−	0.927087i	$-0.377696\pi$
$769$	1.22941e7	0.749690	0.374845	+	0.927087i	$0.377696\pi$
$770$	0	0
$771$	−5.23467e6	−0.317142
$772$	0	0
$773$	−2.57086e6	−0.154750	−0.0773749	−	0.997002i	$-0.524654\pi$
$773$	−2.57086e6	−0.154750	−0.0773749	+	0.997002i	$0.524654\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.57381e7	0.935190
$778$	0	0
$779$	−189040.	−0.0111612
$780$	0	0
$781$	−1.53253e7	−0.899046
$782$	0	0
$783$	9.85117e6	0.574227
$784$	0	0
$785$	0	0
$786$	0	0
$787$	5.54594e6	0.319182	0.159591	−	0.987183i	$-0.448982\pi$
$787$	5.54594e6	0.319182	0.159591	+	0.987183i	$0.448982\pi$
$788$	0	0
$789$	2.64154e6	0.151065
$790$	0	0
$791$	3.28515e6	0.186687
$792$	0	0
$793$	−1.07306e7	−0.605958
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.09610e7	−1.16887	−0.584436	−	0.811440i	$-0.698684\pi$
$797$	−2.09610e7	−1.16887	−0.584436	+	0.811440i	$0.698684\pi$
$798$	0	0
$799$	40896.0	0.00226628
$800$	0	0
$801$	2.26983e7	1.25000
$802$	0	0
$803$	2.87671e7	1.57437
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.25565e7	−0.678709
$808$	0	0
$809$	2.70297e7	1.45201	0.726005	−	0.687690i	$-0.241375\pi$
$809$	2.70297e7	1.45201	0.726005	+	0.687690i	$0.241375\pi$
$810$	0	0
$811$	2.13052e6	0.113745	0.0568727	−	0.998381i	$-0.481887\pi$
$811$	2.13052e6	0.113745	0.0568727	+	0.998381i	$0.481887\pi$
$812$	0	0
$813$	7.66234e6	0.406570
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.21599e7	1.16148
$818$	0	0
$819$	−2.58390e7	−1.34607
$820$	0	0
$821$	1.58060e7	0.818396	0.409198	−	0.912446i	$-0.365809\pi$
$821$	1.58060e7	0.818396	0.409198	+	0.912446i	$0.365809\pi$
$822$	0	0
$823$	−2.28848e7	−1.17773	−0.588867	−	0.808230i	$-0.700426\pi$
$823$	−2.28848e7	−1.17773	−0.588867	+	0.808230i	$0.700426\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.55336e7	−1.29822	−0.649109	−	0.760695i	$-0.724858\pi$
$827$	−2.55336e7	−1.29822	−0.649109	+	0.760695i	$0.724858\pi$
$828$	0	0
$829$	8.31786e6	0.420364	0.210182	−	0.977662i	$-0.432594\pi$
$829$	8.31786e6	0.420364	0.210182	+	0.977662i	$0.432594\pi$
$830$	0	0
$831$	4.52350e6	0.227234
$832$	0	0
$833$	3.38120e7	1.68834
$834$	0	0
$835$	0	0
$836$	0	0
$837$	8.85862e6	0.437072
$838$	0	0
$839$	−3.66261e7	−1.79633	−0.898164	−	0.439660i	$-0.855099\pi$
$839$	−3.66261e7	−1.79633	−0.898164	+	0.439660i	$0.855099\pi$
$840$	0	0
$841$	−1.19964e7	−0.584873
$842$	0	0
$843$	1.07302e7	0.520041
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−2.62590e7	−1.25768
$848$	0	0
$849$	5.87411e6	0.279687
$850$	0	0
$851$	3.04169e7	1.43976
$852$	0	0
$853$	−1.74802e7	−0.822571	−0.411286	−	0.911507i	$-0.634920\pi$
$853$	−1.74802e7	−0.822571	−0.411286	+	0.911507i	$0.634920\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−9.31062e6	−0.433038	−0.216519	−	0.976278i	$-0.569470\pi$
$857$	−9.31062e6	−0.433038	−0.216519	+	0.976278i	$0.569470\pi$
$858$	0	0
$859$	3.49525e7	1.61620	0.808101	−	0.589045i	$-0.200496\pi$
$859$	3.49525e7	1.61620	0.808101	+	0.589045i	$0.200496\pi$
$860$	0	0
$861$	282880.	0.0130045
$862$	0	0
$863$	−2.02349e7	−0.924858	−0.462429	−	0.886656i	$-0.653022\pi$
$863$	−2.02349e7	−0.924858	−0.462429	+	0.886656i	$0.653022\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−1.70742e6	−0.0771421
$868$	0	0
$869$	3.70655e7	1.66502
$870$	0	0
$871$	1.43963e7	0.642994
$872$	0	0
$873$	1.11499e7	0.495149
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.98342e7	0.870797	0.435398	−	0.900238i	$-0.356608\pi$
$877$	1.98342e7	0.870797	0.435398	+	0.900238i	$0.356608\pi$
$878$	0	0
$879$	−9.06562e6	−0.395754
$880$	0	0
$881$	−3.81023e7	−1.65391	−0.826954	−	0.562270i	$-0.809928\pi$
$881$	−3.81023e7	−1.65391	−0.826954	+	0.562270i	$0.809928\pi$
$882$	0	0
$883$	−2.41560e7	−1.04261	−0.521307	−	0.853369i	$-0.674555\pi$
$883$	−2.41560e7	−1.04261	−0.521307	+	0.853369i	$0.674555\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.02368e7	0.863638	0.431819	−	0.901960i	$-0.357872\pi$
$887$	2.02368e7	0.863638	0.431819	+	0.901960i	$0.357872\pi$
$888$	0	0
$889$	−212992.	−0.00903876
$890$	0	0
$891$	8.83810e6	0.372962
$892$	0	0
$893$	−35584.0	−0.00149323
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.78552e7	0.740942
$898$	0	0
$899$	7.65683e6	0.315973
$900$	0	0
$901$	2.83435e7	1.16316
$902$	0	0
$903$	−3.31602e7	−1.35331
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.97976e7	0.799088	0.399544	−	0.916714i	$-0.369169\pi$
$907$	1.97976e7	0.799088	0.399544	+	0.916714i	$0.369169\pi$
$908$	0	0
$909$	−1.14811e6	−0.0460863
$910$	0	0
$911$	2.13242e7	0.851288	0.425644	−	0.904891i	$-0.360048\pi$
$911$	2.13242e7	0.851288	0.425644	+	0.904891i	$0.360048\pi$
$912$	0	0
$913$	−2.02608e7	−0.804414
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4.71411e6	−0.185130
$918$	0	0
$919$	−3.49941e7	−1.36680	−0.683401	−	0.730043i	$-0.739500\pi$
$919$	−3.49941e7	−1.36680	−0.683401	+	0.730043i	$0.739500\pi$
$920$	0	0
$921$	2.33101e7	0.905515
$922$	0	0
$923$	1.98428e7	0.766655
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.94093e7	0.741842
$928$	0	0
$929$	−2.88107e7	−1.09525	−0.547627	−	0.836722i	$-0.684469\pi$
$929$	−2.88107e7	−1.09525	−0.547627	+	0.836722i	$0.684469\pi$
$930$	0	0
$931$	−2.94202e7	−1.11243
$932$	0	0
$933$	−1.15050e7	−0.432697
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−1.28854e7	−0.479457	−0.239729	−	0.970840i	$-0.577059\pi$
$937$	−1.28854e7	−0.479457	−0.239729	+	0.970840i	$0.577059\pi$
$938$	0	0
$939$	−1.10080e7	−0.407424
$940$	0	0
$941$	−5.27615e7	−1.94242	−0.971210	−	0.238227i	$-0.923434\pi$
$941$	−5.27615e7	−1.94242	−0.971210	+	0.238227i	$0.923434\pi$
$942$	0	0
$943$	546720.	0.0200210
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.53961e6	0.200726	0.100363	−	0.994951i	$-0.468000\pi$
$947$	5.53961e6	0.200726	0.100363	+	0.994951i	$0.468000\pi$
$948$	0	0
$949$	−3.72470e7	−1.34253
$950$	0	0
$951$	−9.87950e6	−0.354229
$952$	0	0
$953$	228102.	0.00813574	0.00406787	−	0.999992i	$-0.498705\pi$
$953$	228102.	0.00813574	0.00406787	+	0.999992i	$0.498705\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−1.25124e7	−0.441632
$958$	0	0
$959$	−1.10735e7	−0.388811
$960$	0	0
$961$	−2.17438e7	−0.759498
$962$	0	0
$963$	−1.86117e7	−0.646726
$964$	0	0
$965$	0	0
$966$	0	0
$967$	7.36709e6	0.253355	0.126678	−	0.991944i	$-0.459569\pi$
$967$	7.36709e6	0.253355	0.126678	+	0.991944i	$0.459569\pi$
$968$	0	0
$969$	1.13691e7	0.388970
$970$	0	0
$971$	1.91161e7	0.650654	0.325327	−	0.945602i	$-0.394526\pi$
$971$	1.91161e7	0.650654	0.325327	+	0.945602i	$0.394526\pi$
$972$	0	0
$973$	4.12173e6	0.139572
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−5.57040e7	−1.86702	−0.933512	−	0.358546i	$-0.883273\pi$
$977$	−5.57040e7	−1.86702	−0.933512	+	0.358546i	$0.883273\pi$
$978$	0	0
$979$	−6.79680e7	−2.26646
$980$	0	0
$981$	−444994.	−0.0147632
$982$	0	0
$983$	−1.55469e7	−0.513167	−0.256584	−	0.966522i	$-0.582597\pi$
$983$	−1.55469e7	−0.513167	−0.256584	+	0.966522i	$0.582597\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	53248.0	0.00173984
$988$	0	0
$989$	−6.40884e7	−2.08348
$990$	0	0
$991$	−2.36890e7	−0.766237	−0.383118	−	0.923699i	$-0.625150\pi$
$991$	−2.36890e7	−0.766237	−0.383118	+	0.923699i	$0.625150\pi$
$992$	0	0
$993$	−1.19144e7	−0.383442
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−3.71720e6	−0.118434	−0.0592172	−	0.998245i	$-0.518860\pi$
$997$	−3.71720e6	−0.118434	−0.0592172	+	0.998245i	$0.518860\pi$
$998$	0	0
$999$	3.19302e7	1.01225

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.6.a.a.1.1		1
4.3	odd	2		800.6.a.e.1.1		1
5.2	odd	4		800.6.c.b.449.2		2
5.3	odd	4		800.6.c.b.449.1		2
5.4	even	2		32.6.a.c.1.1	yes	1
15.14	odd	2		288.6.a.e.1.1		1
20.3	even	4		800.6.c.a.449.2		2
20.7	even	4		800.6.c.a.449.1		2
20.19	odd	2		32.6.a.a.1.1	✓	1
40.19	odd	2		64.6.a.e.1.1		1
40.29	even	2		64.6.a.c.1.1		1
60.59	even	2		288.6.a.d.1.1		1
80.19	odd	4		256.6.b.h.129.1		2
80.29	even	4		256.6.b.b.129.2		2
80.59	odd	4		256.6.b.h.129.2		2
80.69	even	4		256.6.b.b.129.1		2
120.29	odd	2		576.6.a.v.1.1		1
120.59	even	2		576.6.a.u.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.6.a.a.1.1	✓	1	20.19	odd	2
32.6.a.c.1.1	yes	1	5.4	even	2
64.6.a.c.1.1		1	40.29	even	2
64.6.a.e.1.1		1	40.19	odd	2
256.6.b.b.129.1		2	80.69	even	4
256.6.b.b.129.2		2	80.29	even	4
256.6.b.h.129.1		2	80.19	odd	4
256.6.b.h.129.2		2	80.59	odd	4
288.6.a.d.1.1		1	60.59	even	2
288.6.a.e.1.1		1	15.14	odd	2
576.6.a.u.1.1		1	120.59	even	2
576.6.a.v.1.1		1	120.29	odd	2
800.6.a.a.1.1		1	1.1	even	1	trivial
800.6.a.e.1.1		1	4.3	odd	2
800.6.c.a.449.1		2	20.7	even	4
800.6.c.a.449.2		2	20.3	even	4
800.6.c.b.449.1		2	5.3	odd	4
800.6.c.b.449.2		2	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	800.a (trivial)

Introduction

L-functions

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Database

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