LMFDB - Embedded newform 40.6.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [40,6,Mod(1,40)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(40, 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("40.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 40 = 2^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	40.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:	$6.41535279252$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	40.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{3} -25.0000 q^{5} -62.0000 q^{7} -239.000 q^{9} +O(q^{10})$

$q-2.00000 q^{3} -25.0000 q^{5} -62.0000 q^{7} -239.000 q^{9} -144.000 q^{11} -654.000 q^{13} +50.0000 q^{15} -1190.00 q^{17} +556.000 q^{19} +124.000 q^{21} +2182.00 q^{23} +625.000 q^{25} +964.000 q^{27} -1578.00 q^{29} +9660.00 q^{31} +288.000 q^{33} +1550.00 q^{35} -3534.00 q^{37} +1308.00 q^{39} +7462.00 q^{41} -7114.00 q^{43} +5975.00 q^{45} -28294.0 q^{47} -12963.0 q^{49} +2380.00 q^{51} -13046.0 q^{53} +3600.00 q^{55} -1112.00 q^{57} -37092.0 q^{59} +39570.0 q^{61} +14818.0 q^{63} +16350.0 q^{65} -56734.0 q^{67} -4364.00 q^{69} +45588.0 q^{71} +11842.0 q^{73} -1250.00 q^{75} +8928.00 q^{77} +94216.0 q^{79} +56149.0 q^{81} -31482.0 q^{83} +29750.0 q^{85} +3156.00 q^{87} -94054.0 q^{89} +40548.0 q^{91} -19320.0 q^{93} -13900.0 q^{95} +23714.0 q^{97} +34416.0 q^{99} +O(q^{100})$

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
$3$	−2.00000	−0.128300	−0.0641500	−	0.997940i	$-0.520434\pi$
$3$	−2.00000	−0.128300	−0.0641500	+	0.997940i	$0.520434\pi$
$4$	0	0
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	−62.0000	−0.478241	−0.239120	−	0.970990i	$-0.576859\pi$
$7$	−62.0000	−0.478241	−0.239120	+	0.970990i	$0.576859\pi$
$8$	0	0
$9$	−239.000	−0.983539
$10$	0	0
$11$	−144.000	−0.358823	−0.179412	−	0.983774i	$-0.557419\pi$
$11$	−144.000	−0.358823	−0.179412	+	0.983774i	$0.557419\pi$
$12$	0	0
$13$	−654.000	−1.07330	−0.536648	−	0.843806i	$-0.680310\pi$
$13$	−654.000	−1.07330	−0.536648	+	0.843806i	$0.680310\pi$
$14$	0	0
$15$	50.0000	0.0573775
$16$	0	0
$17$	−1190.00	−0.998676	−0.499338	−	0.866407i	$-0.666423\pi$
$17$	−1190.00	−0.998676	−0.499338	+	0.866407i	$0.666423\pi$
$18$	0	0
$19$	556.000	0.353338	0.176669	−	0.984270i	$-0.443468\pi$
$19$	556.000	0.353338	0.176669	+	0.984270i	$0.443468\pi$
$20$	0	0
$21$	124.000	0.0613583
$22$	0	0
$23$	2182.00	0.860073	0.430036	−	0.902812i	$-0.358501\pi$
$23$	2182.00	0.860073	0.430036	+	0.902812i	$0.358501\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	964.000	0.254488
$28$	0	0
$29$	−1578.00	−0.348427	−0.174214	−	0.984708i	$-0.555738\pi$
$29$	−1578.00	−0.348427	−0.174214	+	0.984708i	$0.555738\pi$
$30$	0	0
$31$	9660.00	1.80540	0.902699	−	0.430273i	$-0.141583\pi$
$31$	9660.00	1.80540	0.902699	+	0.430273i	$0.141583\pi$
$32$	0	0
$33$	288.000	0.0460371
$34$	0	0
$35$	1550.00	0.213876
$36$	0	0
$37$	−3534.00	−0.424387	−0.212194	−	0.977228i	$-0.568061\pi$
$37$	−3534.00	−0.424387	−0.212194	+	0.977228i	$0.568061\pi$
$38$	0	0
$39$	1308.00	0.137704
$40$	0	0
$41$	7462.00	0.693259	0.346630	−	0.938002i	$-0.387326\pi$
$41$	7462.00	0.693259	0.346630	+	0.938002i	$0.387326\pi$
$42$	0	0
$43$	−7114.00	−0.586736	−0.293368	−	0.956000i	$-0.594776\pi$
$43$	−7114.00	−0.586736	−0.293368	+	0.956000i	$0.594776\pi$
$44$	0	0
$45$	5975.00	0.439852
$46$	0	0
$47$	−28294.0	−1.86831	−0.934157	−	0.356863i	$-0.883846\pi$
$47$	−28294.0	−1.86831	−0.934157	+	0.356863i	$0.883846\pi$
$48$	0	0
$49$	−12963.0	−0.771286
$50$	0	0
$51$	2380.00	0.128130
$52$	0	0
$53$	−13046.0	−0.637952	−0.318976	−	0.947763i	$-0.603339\pi$
$53$	−13046.0	−0.637952	−0.318976	+	0.947763i	$0.603339\pi$
$54$	0	0
$55$	3600.00	0.160471
$56$	0	0
$57$	−1112.00	−0.0453333
$58$	0	0
$59$	−37092.0	−1.38724	−0.693618	−	0.720343i	$-0.743984\pi$
$59$	−37092.0	−1.38724	−0.693618	+	0.720343i	$0.743984\pi$
$60$	0	0
$61$	39570.0	1.36157	0.680787	−	0.732481i	$-0.261638\pi$
$61$	39570.0	1.36157	0.680787	+	0.732481i	$0.261638\pi$
$62$	0	0
$63$	14818.0	0.470368
$64$	0	0
$65$	16350.0	0.479992
$66$	0	0
$67$	−56734.0	−1.54403	−0.772016	−	0.635603i	$-0.780752\pi$
$67$	−56734.0	−1.54403	−0.772016	+	0.635603i	$0.780752\pi$
$68$	0	0
$69$	−4364.00	−0.110347
$70$	0	0
$71$	45588.0	1.07326	0.536630	−	0.843818i	$-0.319697\pi$
$71$	45588.0	1.07326	0.536630	+	0.843818i	$0.319697\pi$
$72$	0	0
$73$	11842.0	0.260087	0.130043	−	0.991508i	$-0.458488\pi$
$73$	11842.0	0.260087	0.130043	+	0.991508i	$0.458488\pi$
$74$	0	0
$75$	−1250.00	−0.0256600
$76$	0	0
$77$	8928.00	0.171604
$78$	0	0
$79$	94216.0	1.69847	0.849233	−	0.528018i	$-0.177065\pi$
$79$	94216.0	1.69847	0.849233	+	0.528018i	$0.177065\pi$
$80$	0	0
$81$	56149.0	0.950888
$82$	0	0
$83$	−31482.0	−0.501611	−0.250806	−	0.968037i	$-0.580695\pi$
$83$	−31482.0	−0.501611	−0.250806	+	0.968037i	$0.580695\pi$
$84$	0	0
$85$	29750.0	0.446622
$86$	0	0
$87$	3156.00	0.0447032
$88$	0	0
$89$	−94054.0	−1.25864	−0.629321	−	0.777145i	$-0.716667\pi$
$89$	−94054.0	−1.25864	−0.629321	+	0.777145i	$0.716667\pi$
$90$	0	0
$91$	40548.0	0.513294
$92$	0	0
$93$	−19320.0	−0.231633
$94$	0	0
$95$	−13900.0	−0.158018
$96$	0	0
$97$	23714.0	0.255903	0.127952	−	0.991780i	$-0.459160\pi$
$97$	23714.0	0.255903	0.127952	+	0.991780i	$0.459160\pi$
$98$	0	0
$99$	34416.0	0.352917
$100$	0	0
$101$	−129674.	−1.26488	−0.632440	−	0.774609i	$-0.717947\pi$
$101$	−129674.	−1.26488	−0.632440	+	0.774609i	$0.717947\pi$
$102$	0	0
$103$	136846.	1.27098	0.635490	−	0.772109i	$-0.280798\pi$
$103$	136846.	1.27098	0.635490	+	0.772109i	$0.280798\pi$
$104$	0	0
$105$	−3100.00	−0.0274403
$106$	0	0
$107$	−193190.	−1.63127	−0.815634	−	0.578569i	$-0.803612\pi$
$107$	−193190.	−1.63127	−0.815634	+	0.578569i	$0.803612\pi$
$108$	0	0
$109$	−120046.	−0.967791	−0.483895	−	0.875126i	$-0.660778\pi$
$109$	−120046.	−0.967791	−0.483895	+	0.875126i	$0.660778\pi$
$110$	0	0
$111$	7068.00	0.0544489
$112$	0	0
$113$	−152646.	−1.12458	−0.562289	−	0.826941i	$-0.690079\pi$
$113$	−152646.	−1.12458	−0.562289	+	0.826941i	$0.690079\pi$
$114$	0	0
$115$	−54550.0	−0.384636
$116$	0	0
$117$	156306.	1.05563
$118$	0	0
$119$	73780.0	0.477608
$120$	0	0
$121$	−140315.	−0.871246
$122$	0	0
$123$	−14924.0	−0.0889452
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	107906.	0.593658	0.296829	−	0.954931i	$-0.404071\pi$
$127$	107906.	0.593658	0.296829	+	0.954931i	$0.404071\pi$
$128$	0	0
$129$	14228.0	0.0752783
$130$	0	0
$131$	−233072.	−1.18662	−0.593310	−	0.804974i	$-0.702179\pi$
$131$	−233072.	−1.18662	−0.593310	+	0.804974i	$0.702179\pi$
$132$	0	0
$133$	−34472.0	−0.168981
$134$	0	0
$135$	−24100.0	−0.113811
$136$	0	0
$137$	356082.	1.62087	0.810436	−	0.585827i	$-0.199230\pi$
$137$	356082.	1.62087	0.810436	+	0.585827i	$0.199230\pi$
$138$	0	0
$139$	312204.	1.37057	0.685285	−	0.728275i	$-0.259677\pi$
$139$	312204.	1.37057	0.685285	+	0.728275i	$0.259677\pi$
$140$	0	0
$141$	56588.0	0.239705
$142$	0	0
$143$	94176.0	0.385124
$144$	0	0
$145$	39450.0	0.155821
$146$	0	0
$147$	25926.0	0.0989560
$148$	0	0
$149$	27498.0	0.101469	0.0507347	−	0.998712i	$-0.483844\pi$
$149$	27498.0	0.101469	0.0507347	+	0.998712i	$0.483844\pi$
$150$	0	0
$151$	−136908.	−0.488637	−0.244319	−	0.969695i	$-0.578564\pi$
$151$	−136908.	−0.488637	−0.244319	+	0.969695i	$0.578564\pi$
$152$	0	0
$153$	284410.	0.982237
$154$	0	0
$155$	−241500.	−0.807398
$156$	0	0
$157$	406714.	1.31686	0.658431	−	0.752641i	$-0.271221\pi$
$157$	406714.	1.31686	0.658431	+	0.752641i	$0.271221\pi$
$158$	0	0
$159$	26092.0	0.0818492
$160$	0	0
$161$	−135284.	−0.411322
$162$	0	0
$163$	−13642.0	−0.0402169	−0.0201085	−	0.999798i	$-0.506401\pi$
$163$	−13642.0	−0.0402169	−0.0201085	+	0.999798i	$0.506401\pi$
$164$	0	0
$165$	−7200.00	−0.0205884
$166$	0	0
$167$	−203438.	−0.564470	−0.282235	−	0.959345i	$-0.591076\pi$
$167$	−203438.	−0.564470	−0.282235	+	0.959345i	$0.591076\pi$
$168$	0	0
$169$	56423.0	0.151964
$170$	0	0
$171$	−132884.	−0.347522
$172$	0	0
$173$	127242.	0.323233	0.161616	−	0.986854i	$-0.448329\pi$
$173$	127242.	0.323233	0.161616	+	0.986854i	$0.448329\pi$
$174$	0	0
$175$	−38750.0	−0.0956482
$176$	0	0
$177$	74184.0	0.177982
$178$	0	0
$179$	−94684.0	−0.220874	−0.110437	−	0.993883i	$-0.535225\pi$
$179$	−94684.0	−0.220874	−0.110437	+	0.993883i	$0.535225\pi$
$180$	0	0
$181$	−517018.	−1.17303	−0.586515	−	0.809938i	$-0.699501\pi$
$181$	−517018.	−1.17303	−0.586515	+	0.809938i	$0.699501\pi$
$182$	0	0
$183$	−79140.0	−0.174690
$184$	0	0
$185$	88350.0	0.189792
$186$	0	0
$187$	171360.	0.358348
$188$	0	0
$189$	−59768.0	−0.121707
$190$	0	0
$191$	−412300.	−0.817768	−0.408884	−	0.912586i	$-0.634082\pi$
$191$	−412300.	−0.817768	−0.408884	+	0.912586i	$0.634082\pi$
$192$	0	0
$193$	−771654.	−1.49118	−0.745589	−	0.666406i	$-0.767832\pi$
$193$	−771654.	−1.49118	−0.745589	+	0.666406i	$0.767832\pi$
$194$	0	0
$195$	−32700.0	−0.0615831
$196$	0	0
$197$	−190238.	−0.349246	−0.174623	−	0.984635i	$-0.555871\pi$
$197$	−190238.	−0.349246	−0.174623	+	0.984635i	$0.555871\pi$
$198$	0	0
$199$	132072.	0.236417	0.118208	−	0.992989i	$-0.462285\pi$
$199$	132072.	0.236417	0.118208	+	0.992989i	$0.462285\pi$
$200$	0	0
$201$	113468.	0.198099
$202$	0	0
$203$	97836.0	0.166632
$204$	0	0
$205$	−186550.	−0.310035
$206$	0	0
$207$	−521498.	−0.845915
$208$	0	0
$209$	−80064.0	−0.126786
$210$	0	0
$211$	928704.	1.43606	0.718028	−	0.696015i	$-0.245045\pi$
$211$	928704.	1.43606	0.718028	+	0.696015i	$0.245045\pi$
$212$	0	0
$213$	−91176.0	−0.137699
$214$	0	0
$215$	177850.	0.262396
$216$	0	0
$217$	−598920.	−0.863415
$218$	0	0
$219$	−23684.0	−0.0333691
$220$	0	0
$221$	778260.	1.07187
$222$	0	0
$223$	421494.	0.567583	0.283791	−	0.958886i	$-0.408408\pi$
$223$	421494.	0.567583	0.283791	+	0.958886i	$0.408408\pi$
$224$	0	0
$225$	−149375.	−0.196708
$226$	0	0
$227$	991962.	1.27770	0.638852	−	0.769329i	$-0.279410\pi$
$227$	991962.	1.27770	0.638852	+	0.769329i	$0.279410\pi$
$228$	0	0
$229$	−266946.	−0.336384	−0.168192	−	0.985754i	$-0.553793\pi$
$229$	−266946.	−0.336384	−0.168192	+	0.985754i	$0.553793\pi$
$230$	0	0
$231$	−17856.0	−0.0220168
$232$	0	0
$233$	960314.	1.15884	0.579420	−	0.815029i	$-0.303279\pi$
$233$	960314.	1.15884	0.579420	+	0.815029i	$0.303279\pi$
$234$	0	0
$235$	707350.	0.835535
$236$	0	0
$237$	−188432.	−0.217913
$238$	0	0
$239$	−492696.	−0.557936	−0.278968	−	0.960300i	$-0.589992\pi$
$239$	−492696.	−0.557936	−0.278968	+	0.960300i	$0.589992\pi$
$240$	0	0
$241$	56078.0	0.0621942	0.0310971	−	0.999516i	$-0.490100\pi$
$241$	56078.0	0.0621942	0.0310971	+	0.999516i	$0.490100\pi$
$242$	0	0
$243$	−346550.	−0.376487
$244$	0	0
$245$	324075.	0.344929
$246$	0	0
$247$	−363624.	−0.379237
$248$	0	0
$249$	62964.0	0.0643567
$250$	0	0
$251$	1.96792e6	1.97162	0.985810	−	0.167866i	$-0.0536876\pi$
$251$	1.96792e6	1.97162	0.985810	+	0.167866i	$0.0536876\pi$
$252$	0	0
$253$	−314208.	−0.308614
$254$	0	0
$255$	−59500.0	−0.0573016
$256$	0	0
$257$	−971910.	−0.917896	−0.458948	−	0.888463i	$-0.651773\pi$
$257$	−971910.	−0.917896	−0.458948	+	0.888463i	$0.651773\pi$
$258$	0	0
$259$	219108.	0.202959
$260$	0	0
$261$	377142.	0.342692
$262$	0	0
$263$	−154770.	−0.137974	−0.0689870	−	0.997618i	$-0.521977\pi$
$263$	−154770.	−0.137974	−0.0689870	+	0.997618i	$0.521977\pi$
$264$	0	0
$265$	326150.	0.285301
$266$	0	0
$267$	188108.	0.161484
$268$	0	0
$269$	1.02371e6	0.862577	0.431289	−	0.902214i	$-0.358059\pi$
$269$	1.02371e6	0.862577	0.431289	+	0.902214i	$0.358059\pi$
$270$	0	0
$271$	−1.14776e6	−0.949350	−0.474675	−	0.880161i	$-0.657434\pi$
$271$	−1.14776e6	−0.949350	−0.474675	+	0.880161i	$0.657434\pi$
$272$	0	0
$273$	−81096.0	−0.0658556
$274$	0	0
$275$	−90000.0	−0.0717647
$276$	0	0
$277$	−2.49676e6	−1.95514	−0.977568	−	0.210619i	$-0.932452\pi$
$277$	−2.49676e6	−1.95514	−0.977568	+	0.210619i	$0.932452\pi$
$278$	0	0
$279$	−2.30874e6	−1.77568
$280$	0	0
$281$	1.69540e6	1.28087	0.640436	−	0.768011i	$-0.278754\pi$
$281$	1.69540e6	1.28087	0.640436	+	0.768011i	$0.278754\pi$
$282$	0	0
$283$	−2.12395e6	−1.57645	−0.788223	−	0.615390i	$-0.788999\pi$
$283$	−2.12395e6	−1.57645	−0.788223	+	0.615390i	$0.788999\pi$
$284$	0	0
$285$	27800.0	0.0202737
$286$	0	0
$287$	−462644.	−0.331545
$288$	0	0
$289$	−3757.00	−0.00264604
$290$	0	0
$291$	−47428.0	−0.0328324
$292$	0	0
$293$	992722.	0.675552	0.337776	−	0.941227i	$-0.390325\pi$
$293$	992722.	0.675552	0.337776	+	0.941227i	$0.390325\pi$
$294$	0	0
$295$	927300.	0.620391
$296$	0	0
$297$	−138816.	−0.0913163
$298$	0	0
$299$	−1.42703e6	−0.923112
$300$	0	0
$301$	441068.	0.280601
$302$	0	0
$303$	259348.	0.162284
$304$	0	0
$305$	−989250.	−0.608915
$306$	0	0
$307$	487522.	0.295222	0.147611	−	0.989046i	$-0.452842\pi$
$307$	487522.	0.295222	0.147611	+	0.989046i	$0.452842\pi$
$308$	0	0
$309$	−273692.	−0.163067
$310$	0	0
$311$	−444116.	−0.260373	−0.130186	−	0.991490i	$-0.541558\pi$
$311$	−444116.	−0.260373	−0.130186	+	0.991490i	$0.541558\pi$
$312$	0	0
$313$	47242.0	0.0272563	0.0136282	−	0.999907i	$-0.495662\pi$
$313$	47242.0	0.0272563	0.0136282	+	0.999907i	$0.495662\pi$
$314$	0	0
$315$	−370450.	−0.210355
$316$	0	0
$317$	694058.	0.387925	0.193962	−	0.981009i	$-0.437866\pi$
$317$	694058.	0.387925	0.193962	+	0.981009i	$0.437866\pi$
$318$	0	0
$319$	227232.	0.125024
$320$	0	0
$321$	386380.	0.209292
$322$	0	0
$323$	−661640.	−0.352871
$324$	0	0
$325$	−408750.	−0.214659
$326$	0	0
$327$	240092.	0.124168
$328$	0	0
$329$	1.75423e6	0.893504
$330$	0	0
$331$	82168.0	0.0412223	0.0206112	−	0.999788i	$-0.493439\pi$
$331$	82168.0	0.0412223	0.0206112	+	0.999788i	$0.493439\pi$
$332$	0	0
$333$	844626.	0.417401
$334$	0	0
$335$	1.41835e6	0.690512
$336$	0	0
$337$	−727934.	−0.349154	−0.174577	−	0.984644i	$-0.555856\pi$
$337$	−727934.	−0.349154	−0.174577	+	0.984644i	$0.555856\pi$
$338$	0	0
$339$	305292.	0.144283
$340$	0	0
$341$	−1.39104e6	−0.647819
$342$	0	0
$343$	1.84574e6	0.847101
$344$	0	0
$345$	109100.	0.0493488
$346$	0	0
$347$	2.02298e6	0.901919	0.450959	−	0.892544i	$-0.351082\pi$
$347$	2.02298e6	0.901919	0.450959	+	0.892544i	$0.351082\pi$
$348$	0	0
$349$	4.40858e6	1.93747	0.968736	−	0.248095i	$-0.0798044\pi$
$349$	4.40858e6	1.93747	0.968736	+	0.248095i	$0.0798044\pi$
$350$	0	0
$351$	−630456.	−0.273141
$352$	0	0
$353$	1.06965e6	0.456883	0.228441	−	0.973558i	$-0.426637\pi$
$353$	1.06965e6	0.456883	0.228441	+	0.973558i	$0.426637\pi$
$354$	0	0
$355$	−1.13970e6	−0.479976
$356$	0	0
$357$	−147560.	−0.0612771
$358$	0	0
$359$	−32968.0	−0.0135007	−0.00675035	−	0.999977i	$-0.502149\pi$
$359$	−32968.0	−0.0135007	−0.00675035	+	0.999977i	$0.502149\pi$
$360$	0	0
$361$	−2.16696e6	−0.875152
$362$	0	0
$363$	280630.	0.111781
$364$	0	0
$365$	−296050.	−0.116314
$366$	0	0
$367$	3.64081e6	1.41102	0.705509	−	0.708700i	$-0.250718\pi$
$367$	3.64081e6	1.41102	0.705509	+	0.708700i	$0.250718\pi$
$368$	0	0
$369$	−1.78342e6	−0.681847
$370$	0	0
$371$	808852.	0.305094
$372$	0	0
$373$	−3.17311e6	−1.18090	−0.590450	−	0.807074i	$-0.701050\pi$
$373$	−3.17311e6	−1.18090	−0.590450	+	0.807074i	$0.701050\pi$
$374$	0	0
$375$	31250.0	0.0114755
$376$	0	0
$377$	1.03201e6	0.373965
$378$	0	0
$379$	1.60498e6	0.573947	0.286973	−	0.957939i	$-0.407351\pi$
$379$	1.60498e6	0.573947	0.286973	+	0.957939i	$0.407351\pi$
$380$	0	0
$381$	−215812.	−0.0761664
$382$	0	0
$383$	1.98925e6	0.692936	0.346468	−	0.938062i	$-0.387381\pi$
$383$	1.98925e6	0.692936	0.346468	+	0.938062i	$0.387381\pi$
$384$	0	0
$385$	−223200.	−0.0767436
$386$	0	0
$387$	1.70025e6	0.577078
$388$	0	0
$389$	−5.16495e6	−1.73058	−0.865291	−	0.501270i	$-0.832866\pi$
$389$	−5.16495e6	−1.73058	−0.865291	+	0.501270i	$0.832866\pi$
$390$	0	0
$391$	−2.59658e6	−0.858934
$392$	0	0
$393$	466144.	0.152243
$394$	0	0
$395$	−2.35540e6	−0.759577
$396$	0	0
$397$	937586.	0.298562	0.149281	−	0.988795i	$-0.452304\pi$
$397$	937586.	0.298562	0.149281	+	0.988795i	$0.452304\pi$
$398$	0	0
$399$	68944.0	0.0216802
$400$	0	0
$401$	−5.63657e6	−1.75047	−0.875234	−	0.483699i	$-0.839293\pi$
$401$	−5.63657e6	−1.75047	−0.875234	+	0.483699i	$0.839293\pi$
$402$	0	0
$403$	−6.31764e6	−1.93773
$404$	0	0
$405$	−1.40373e6	−0.425250
$406$	0	0
$407$	508896.	0.152280
$408$	0	0
$409$	4.06137e6	1.20051	0.600254	−	0.799810i	$-0.295066\pi$
$409$	4.06137e6	1.20051	0.600254	+	0.799810i	$0.295066\pi$
$410$	0	0
$411$	−712164.	−0.207958
$412$	0	0
$413$	2.29970e6	0.663433
$414$	0	0
$415$	787050.	0.224327
$416$	0	0
$417$	−624408.	−0.175844
$418$	0	0
$419$	−976108.	−0.271621	−0.135810	−	0.990735i	$-0.543364\pi$
$419$	−976108.	−0.271621	−0.135810	+	0.990735i	$0.543364\pi$
$420$	0	0
$421$	−1.62706e6	−0.447403	−0.223701	−	0.974658i	$-0.571814\pi$
$421$	−1.62706e6	−0.447403	−0.223701	+	0.974658i	$0.571814\pi$
$422$	0	0
$423$	6.76227e6	1.83756
$424$	0	0
$425$	−743750.	−0.199735
$426$	0	0
$427$	−2.45334e6	−0.651161
$428$	0	0
$429$	−188352.	−0.0494114
$430$	0	0
$431$	−4.27900e6	−1.10956	−0.554778	−	0.831998i	$-0.687197\pi$
$431$	−4.27900e6	−1.10956	−0.554778	+	0.831998i	$0.687197\pi$
$432$	0	0
$433$	−3.20195e6	−0.820720	−0.410360	−	0.911924i	$-0.634597\pi$
$433$	−3.20195e6	−0.820720	−0.410360	+	0.911924i	$0.634597\pi$
$434$	0	0
$435$	−78900.0	−0.0199919
$436$	0	0
$437$	1.21319e6	0.303897
$438$	0	0
$439$	5.09246e6	1.26115	0.630574	−	0.776129i	$-0.282820\pi$
$439$	5.09246e6	1.26115	0.630574	+	0.776129i	$0.282820\pi$
$440$	0	0
$441$	3.09816e6	0.758590
$442$	0	0
$443$	−5.43551e6	−1.31593	−0.657963	−	0.753050i	$-0.728582\pi$
$443$	−5.43551e6	−1.31593	−0.657963	+	0.753050i	$0.728582\pi$
$444$	0	0
$445$	2.35135e6	0.562882
$446$	0	0
$447$	−54996.0	−0.0130185
$448$	0	0
$449$	−2.99007e6	−0.699948	−0.349974	−	0.936759i	$-0.613810\pi$
$449$	−2.99007e6	−0.699948	−0.349974	+	0.936759i	$0.613810\pi$
$450$	0	0
$451$	−1.07453e6	−0.248758
$452$	0	0
$453$	273816.	0.0626922
$454$	0	0
$455$	−1.01370e6	−0.229552
$456$	0	0
$457$	8.01759e6	1.79578	0.897891	−	0.440218i	$-0.145099\pi$
$457$	8.01759e6	1.79578	0.897891	+	0.440218i	$0.145099\pi$
$458$	0	0
$459$	−1.14716e6	−0.254151
$460$	0	0
$461$	2.58462e6	0.566428	0.283214	−	0.959057i	$-0.408599\pi$
$461$	2.58462e6	0.566428	0.283214	+	0.959057i	$0.408599\pi$
$462$	0	0
$463$	6.14261e6	1.33168	0.665840	−	0.746094i	$-0.268073\pi$
$463$	6.14261e6	1.33168	0.665840	+	0.746094i	$0.268073\pi$
$464$	0	0
$465$	483000.	0.103589
$466$	0	0
$467$	−1.59270e6	−0.337942	−0.168971	−	0.985621i	$-0.554044\pi$
$467$	−1.59270e6	−0.337942	−0.168971	+	0.985621i	$0.554044\pi$
$468$	0	0
$469$	3.51751e6	0.738419
$470$	0	0
$471$	−813428.	−0.168953
$472$	0	0
$473$	1.02442e6	0.210535
$474$	0	0
$475$	347500.	0.0706677
$476$	0	0
$477$	3.11799e6	0.627450
$478$	0	0
$479$	863592.	0.171977	0.0859884	−	0.996296i	$-0.472595\pi$
$479$	863592.	0.171977	0.0859884	+	0.996296i	$0.472595\pi$
$480$	0	0
$481$	2.31124e6	0.455493
$482$	0	0
$483$	270568.	0.0527726
$484$	0	0
$485$	−592850.	−0.114443
$486$	0	0
$487$	−8.20714e6	−1.56808	−0.784042	−	0.620707i	$-0.786846\pi$
$487$	−8.20714e6	−1.56808	−0.784042	+	0.620707i	$0.786846\pi$
$488$	0	0
$489$	27284.0	0.00515984
$490$	0	0
$491$	8.93394e6	1.67240	0.836198	−	0.548428i	$-0.184773\pi$
$491$	8.93394e6	1.67240	0.836198	+	0.548428i	$0.184773\pi$
$492$	0	0
$493$	1.87782e6	0.347966
$494$	0	0
$495$	−860400.	−0.157829
$496$	0	0
$497$	−2.82646e6	−0.513276
$498$	0	0
$499$	1.11960e6	0.201284	0.100642	−	0.994923i	$-0.467910\pi$
$499$	1.11960e6	0.201284	0.100642	+	0.994923i	$0.467910\pi$
$500$	0	0
$501$	406876.	0.0724215
$502$	0	0
$503$	3.68177e6	0.648839	0.324420	−	0.945913i	$-0.394831\pi$
$503$	3.68177e6	0.648839	0.324420	+	0.945913i	$0.394831\pi$
$504$	0	0
$505$	3.24185e6	0.565672
$506$	0	0
$507$	−112846.	−0.0194969
$508$	0	0
$509$	−6.73483e6	−1.15221	−0.576105	−	0.817375i	$-0.695428\pi$
$509$	−6.73483e6	−1.15221	−0.576105	+	0.817375i	$0.695428\pi$
$510$	0	0
$511$	−734204.	−0.124384
$512$	0	0
$513$	535984.	0.0899204
$514$	0	0
$515$	−3.42115e6	−0.568400
$516$	0	0
$517$	4.07434e6	0.670395
$518$	0	0
$519$	−254484.	−0.0414708
$520$	0	0
$521$	441370.	0.0712375	0.0356187	−	0.999365i	$-0.488660\pi$
$521$	441370.	0.0712375	0.0356187	+	0.999365i	$0.488660\pi$
$522$	0	0
$523$	−1.17300e7	−1.87518	−0.937589	−	0.347744i	$-0.886948\pi$
$523$	−1.17300e7	−1.87518	−0.937589	+	0.347744i	$0.886948\pi$
$524$	0	0
$525$	77500.0	0.0122717
$526$	0	0
$527$	−1.14954e7	−1.80301
$528$	0	0
$529$	−1.67522e6	−0.260275
$530$	0	0
$531$	8.86499e6	1.36440
$532$	0	0
$533$	−4.88015e6	−0.744072
$534$	0	0
$535$	4.82975e6	0.729525
$536$	0	0
$537$	189368.	0.0283381
$538$	0	0
$539$	1.86667e6	0.276755
$540$	0	0
$541$	744158.	0.109313	0.0546565	−	0.998505i	$-0.482594\pi$
$541$	744158.	0.109313	0.0546565	+	0.998505i	$0.482594\pi$
$542$	0	0
$543$	1.03404e6	0.150500
$544$	0	0
$545$	3.00115e6	0.432809
$546$	0	0
$547$	−3.24801e6	−0.464139	−0.232070	−	0.972699i	$-0.574550\pi$
$547$	−3.24801e6	−0.464139	−0.232070	+	0.972699i	$0.574550\pi$
$548$	0	0
$549$	−9.45723e6	−1.33916
$550$	0	0
$551$	−877368.	−0.123113
$552$	0	0
$553$	−5.84139e6	−0.812276
$554$	0	0
$555$	−176700.	−0.0243503
$556$	0	0
$557$	−9.94446e6	−1.35814	−0.679068	−	0.734075i	$-0.737616\pi$
$557$	−9.94446e6	−1.35814	−0.679068	+	0.734075i	$0.737616\pi$
$558$	0	0
$559$	4.65256e6	0.629741
$560$	0	0
$561$	−342720.	−0.0459761
$562$	0	0
$563$	3.89374e6	0.517721	0.258861	−	0.965915i	$-0.416653\pi$
$563$	3.89374e6	0.517721	0.258861	+	0.965915i	$0.416653\pi$
$564$	0	0
$565$	3.81615e6	0.502926
$566$	0	0
$567$	−3.48124e6	−0.454754
$568$	0	0
$569$	−1.11951e7	−1.44960	−0.724801	−	0.688958i	$-0.758068\pi$
$569$	−1.11951e7	−1.44960	−0.724801	+	0.688958i	$0.758068\pi$
$570$	0	0
$571$	−844040.	−0.108336	−0.0541680	−	0.998532i	$-0.517251\pi$
$571$	−844040.	−0.108336	−0.0541680	+	0.998532i	$0.517251\pi$
$572$	0	0
$573$	824600.	0.104920
$574$	0	0
$575$	1.36375e6	0.172015
$576$	0	0
$577$	−5.13378e6	−0.641945	−0.320973	−	0.947088i	$-0.604010\pi$
$577$	−5.13378e6	−0.641945	−0.320973	+	0.947088i	$0.604010\pi$
$578$	0	0
$579$	1.54331e6	0.191318
$580$	0	0
$581$	1.95188e6	0.239891
$582$	0	0
$583$	1.87862e6	0.228912
$584$	0	0
$585$	−3.90765e6	−0.472091
$586$	0	0
$587$	9.76156e6	1.16929	0.584647	−	0.811287i	$-0.301233\pi$
$587$	9.76156e6	1.16929	0.584647	+	0.811287i	$0.301233\pi$
$588$	0	0
$589$	5.37096e6	0.637916
$590$	0	0
$591$	380476.	0.0448083
$592$	0	0
$593$	966226.	0.112835	0.0564173	−	0.998407i	$-0.482032\pi$
$593$	966226.	0.112835	0.0564173	+	0.998407i	$0.482032\pi$
$594$	0	0
$595$	−1.84450e6	−0.213593
$596$	0	0
$597$	−264144.	−0.0303323
$598$	0	0
$599$	−7.90000e6	−0.899622	−0.449811	−	0.893124i	$-0.648509\pi$
$599$	−7.90000e6	−0.899622	−0.449811	+	0.893124i	$0.648509\pi$
$600$	0	0
$601$	1.03126e7	1.16461	0.582307	−	0.812969i	$-0.302150\pi$
$601$	1.03126e7	1.16461	0.582307	+	0.812969i	$0.302150\pi$
$602$	0	0
$603$	1.35594e7	1.51862
$604$	0	0
$605$	3.50787e6	0.389633
$606$	0	0
$607$	−9.70767e6	−1.06941	−0.534704	−	0.845040i	$-0.679577\pi$
$607$	−9.70767e6	−1.06941	−0.534704	+	0.845040i	$0.679577\pi$
$608$	0	0
$609$	−195672.	−0.0213789
$610$	0	0
$611$	1.85043e7	2.00525
$612$	0	0
$613$	−1.10568e7	−1.18844	−0.594219	−	0.804304i	$-0.702539\pi$
$613$	−1.10568e7	−1.18844	−0.594219	+	0.804304i	$0.702539\pi$
$614$	0	0
$615$	373100.	0.0397775
$616$	0	0
$617$	8.31174e6	0.878980	0.439490	−	0.898248i	$-0.355159\pi$
$617$	8.31174e6	0.878980	0.439490	+	0.898248i	$0.355159\pi$
$618$	0	0
$619$	1.15451e7	1.21108	0.605539	−	0.795816i	$-0.292958\pi$
$619$	1.15451e7	1.21108	0.605539	+	0.795816i	$0.292958\pi$
$620$	0	0
$621$	2.10345e6	0.218878
$622$	0	0
$623$	5.83135e6	0.601934
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	160128.	0.0162667
$628$	0	0
$629$	4.20546e6	0.423825
$630$	0	0
$631$	−8.20262e6	−0.820123	−0.410062	−	0.912058i	$-0.634493\pi$
$631$	−8.20262e6	−0.820123	−0.410062	+	0.912058i	$0.634493\pi$
$632$	0	0
$633$	−1.85741e6	−0.184246
$634$	0	0
$635$	−2.69765e6	−0.265492
$636$	0	0
$637$	8.47780e6	0.827818
$638$	0	0
$639$	−1.08955e7	−1.05559
$640$	0	0
$641$	−5.39695e6	−0.518804	−0.259402	−	0.965769i	$-0.583525\pi$
$641$	−5.39695e6	−0.518804	−0.259402	+	0.965769i	$0.583525\pi$
$642$	0	0
$643$	−1.33896e7	−1.27715	−0.638573	−	0.769561i	$-0.720475\pi$
$643$	−1.33896e7	−1.27715	−0.638573	+	0.769561i	$0.720475\pi$
$644$	0	0
$645$	−355700.	−0.0336655
$646$	0	0
$647$	6.48254e6	0.608814	0.304407	−	0.952542i	$-0.401542\pi$
$647$	6.48254e6	0.608814	0.304407	+	0.952542i	$0.401542\pi$
$648$	0	0
$649$	5.34125e6	0.497773
$650$	0	0
$651$	1.19784e6	0.110776
$652$	0	0
$653$	1.44907e7	1.32986	0.664931	−	0.746904i	$-0.268461\pi$
$653$	1.44907e7	1.32986	0.664931	+	0.746904i	$0.268461\pi$
$654$	0	0
$655$	5.82680e6	0.530673
$656$	0	0
$657$	−2.83024e6	−0.255805
$658$	0	0
$659$	6.59080e6	0.591187	0.295593	−	0.955314i	$-0.404483\pi$
$659$	6.59080e6	0.591187	0.295593	+	0.955314i	$0.404483\pi$
$660$	0	0
$661$	−3.25233e6	−0.289528	−0.144764	−	0.989466i	$-0.546242\pi$
$661$	−3.25233e6	−0.289528	−0.144764	+	0.989466i	$0.546242\pi$
$662$	0	0
$663$	−1.55652e6	−0.137522
$664$	0	0
$665$	861800.	0.0755705
$666$	0	0
$667$	−3.44320e6	−0.299673
$668$	0	0
$669$	−842988.	−0.0728209
$670$	0	0
$671$	−5.69808e6	−0.488565
$672$	0	0
$673$	3.86655e6	0.329068	0.164534	−	0.986371i	$-0.447388\pi$
$673$	3.86655e6	0.329068	0.164534	+	0.986371i	$0.447388\pi$
$674$	0	0
$675$	602500.	0.0508976
$676$	0	0
$677$	1.23856e6	0.103859	0.0519297	−	0.998651i	$-0.483463\pi$
$677$	1.23856e6	0.103859	0.0519297	+	0.998651i	$0.483463\pi$
$678$	0	0
$679$	−1.47027e6	−0.122383
$680$	0	0
$681$	−1.98392e6	−0.163930
$682$	0	0
$683$	−1.31376e7	−1.07762	−0.538810	−	0.842427i	$-0.681126\pi$
$683$	−1.31376e7	−1.07762	−0.538810	+	0.842427i	$0.681126\pi$
$684$	0	0
$685$	−8.90205e6	−0.724876
$686$	0	0
$687$	533892.	0.0431580
$688$	0	0
$689$	8.53208e6	0.684711
$690$	0	0
$691$	−1.23841e7	−0.986664	−0.493332	−	0.869841i	$-0.664221\pi$
$691$	−1.23841e7	−0.986664	−0.493332	+	0.869841i	$0.664221\pi$
$692$	0	0
$693$	−2.13379e6	−0.168779
$694$	0	0
$695$	−7.80510e6	−0.612938
$696$	0	0
$697$	−8.87978e6	−0.692341
$698$	0	0
$699$	−1.92063e6	−0.148679
$700$	0	0
$701$	−9.78952e6	−0.752430	−0.376215	−	0.926532i	$-0.622775\pi$
$701$	−9.78952e6	−0.752430	−0.376215	+	0.926532i	$0.622775\pi$
$702$	0	0
$703$	−1.96490e6	−0.149952
$704$	0	0
$705$	−1.41470e6	−0.107199
$706$	0	0
$707$	8.03979e6	0.604917
$708$	0	0
$709$	1.22257e7	0.913397	0.456699	−	0.889622i	$-0.349032\pi$
$709$	1.22257e7	0.913397	0.456699	+	0.889622i	$0.349032\pi$
$710$	0	0
$711$	−2.25176e7	−1.67051
$712$	0	0
$713$	2.10781e7	1.55277
$714$	0	0
$715$	−2.35440e6	−0.172233
$716$	0	0
$717$	985392.	0.0715832
$718$	0	0
$719$	−1.35053e7	−0.974276	−0.487138	−	0.873325i	$-0.661959\pi$
$719$	−1.35053e7	−0.974276	−0.487138	+	0.873325i	$0.661959\pi$
$720$	0	0
$721$	−8.48445e6	−0.607835
$722$	0	0
$723$	−112156.	−0.00797952
$724$	0	0
$725$	−986250.	−0.0696854
$726$	0	0
$727$	1.17271e7	0.822916	0.411458	−	0.911429i	$-0.365020\pi$
$727$	1.17271e7	0.822916	0.411458	+	0.911429i	$0.365020\pi$
$728$	0	0
$729$	−1.29511e7	−0.902585
$730$	0	0
$731$	8.46566e6	0.585959
$732$	0	0
$733$	−1.16512e7	−0.800960	−0.400480	−	0.916305i	$-0.631157\pi$
$733$	−1.16512e7	−0.800960	−0.400480	+	0.916305i	$0.631157\pi$
$734$	0	0
$735$	−648150.	−0.0442545
$736$	0	0
$737$	8.16970e6	0.554035
$738$	0	0
$739$	−1.26808e7	−0.854155	−0.427077	−	0.904215i	$-0.640457\pi$
$739$	−1.26808e7	−0.854155	−0.427077	+	0.904215i	$0.640457\pi$
$740$	0	0
$741$	727248.	0.0486561
$742$	0	0
$743$	−197370.	−0.0131162	−0.00655812	−	0.999978i	$-0.502088\pi$
$743$	−197370.	−0.0131162	−0.00655812	+	0.999978i	$0.502088\pi$
$744$	0	0
$745$	−687450.	−0.0453785
$746$	0	0
$747$	7.52420e6	0.493354
$748$	0	0
$749$	1.19778e7	0.780139
$750$	0	0
$751$	−1.33282e7	−0.862326	−0.431163	−	0.902274i	$-0.641897\pi$
$751$	−1.33282e7	−0.862326	−0.431163	+	0.902274i	$0.641897\pi$
$752$	0	0
$753$	−3.93584e6	−0.252959
$754$	0	0
$755$	3.42270e6	0.218525
$756$	0	0
$757$	−3.86122e6	−0.244898	−0.122449	−	0.992475i	$-0.539075\pi$
$757$	−3.86122e6	−0.244898	−0.122449	+	0.992475i	$0.539075\pi$
$758$	0	0
$759$	628416.	0.0395952
$760$	0	0
$761$	−8.31756e6	−0.520636	−0.260318	−	0.965523i	$-0.583827\pi$
$761$	−8.31756e6	−0.520636	−0.260318	+	0.965523i	$0.583827\pi$
$762$	0	0
$763$	7.44285e6	0.462837
$764$	0	0
$765$	−7.11025e6	−0.439270
$766$	0	0
$767$	2.42582e7	1.48891
$768$	0	0
$769$	2.76358e7	1.68522	0.842609	−	0.538527i	$-0.181019\pi$
$769$	2.76358e7	1.68522	0.842609	+	0.538527i	$0.181019\pi$
$770$	0	0
$771$	1.94382e6	0.117766
$772$	0	0
$773$	1.78842e7	1.07652	0.538259	−	0.842780i	$-0.319082\pi$
$773$	1.78842e7	1.07652	0.538259	+	0.842780i	$0.319082\pi$
$774$	0	0
$775$	6.03750e6	0.361080
$776$	0	0
$777$	−438216.	−0.0260397
$778$	0	0
$779$	4.14887e6	0.244955
$780$	0	0
$781$	−6.56467e6	−0.385111
$782$	0	0
$783$	−1.52119e6	−0.0886706
$784$	0	0
$785$	−1.01678e7	−0.588918
$786$	0	0
$787$	2.15691e7	1.24135	0.620676	−	0.784067i	$-0.286858\pi$
$787$	2.15691e7	1.24135	0.620676	+	0.784067i	$0.286858\pi$
$788$	0	0
$789$	309540.	0.0177021
$790$	0	0
$791$	9.46405e6	0.537819
$792$	0	0
$793$	−2.58788e7	−1.46137
$794$	0	0
$795$	−652300.	−0.0366041
$796$	0	0
$797$	−1.03060e7	−0.574705	−0.287353	−	0.957825i	$-0.592775\pi$
$797$	−1.03060e7	−0.574705	−0.287353	+	0.957825i	$0.592775\pi$
$798$	0	0
$799$	3.36699e7	1.86584
$800$	0	0
$801$	2.24789e7	1.23792
$802$	0	0
$803$	−1.70525e6	−0.0933251
$804$	0	0
$805$	3.38210e6	0.183949
$806$	0	0
$807$	−2.04743e6	−0.110669
$808$	0	0
$809$	372378.	0.0200038	0.0100019	−	0.999950i	$-0.496816\pi$
$809$	372378.	0.0200038	0.0100019	+	0.999950i	$0.496816\pi$
$810$	0	0
$811$	−1.94795e7	−1.03998	−0.519990	−	0.854173i	$-0.674064\pi$
$811$	−1.94795e7	−1.03998	−0.519990	+	0.854173i	$0.674064\pi$
$812$	0	0
$813$	2.29551e6	0.121802
$814$	0	0
$815$	341050.	0.0179856
$816$	0	0
$817$	−3.95538e6	−0.207316
$818$	0	0
$819$	−9.69097e6	−0.504844
$820$	0	0
$821$	−469318.	−0.0243002	−0.0121501	−	0.999926i	$-0.503868\pi$
$821$	−469318.	−0.0243002	−0.0121501	+	0.999926i	$0.503868\pi$
$822$	0	0
$823$	1.78622e7	0.919253	0.459626	−	0.888112i	$-0.347983\pi$
$823$	1.78622e7	0.919253	0.459626	+	0.888112i	$0.347983\pi$
$824$	0	0
$825$	180000.	0.00920741
$826$	0	0
$827$	9.42560e6	0.479231	0.239616	−	0.970868i	$-0.422979\pi$
$827$	9.42560e6	0.479231	0.239616	+	0.970868i	$0.422979\pi$
$828$	0	0
$829$	−1.48622e7	−0.751098	−0.375549	−	0.926803i	$-0.622546\pi$
$829$	−1.48622e7	−0.751098	−0.375549	+	0.926803i	$0.622546\pi$
$830$	0	0
$831$	4.99352e6	0.250844
$832$	0	0
$833$	1.54260e7	0.770265
$834$	0	0
$835$	5.08595e6	0.252439
$836$	0	0
$837$	9.31224e6	0.459452
$838$	0	0
$839$	−4.71170e6	−0.231085	−0.115543	−	0.993303i	$-0.536861\pi$
$839$	−4.71170e6	−0.231085	−0.115543	+	0.993303i	$0.536861\pi$
$840$	0	0
$841$	−1.80211e7	−0.878599
$842$	0	0
$843$	−3.39080e6	−0.164336
$844$	0	0
$845$	−1.41058e6	−0.0679602
$846$	0	0
$847$	8.69953e6	0.416665
$848$	0	0
$849$	4.24791e6	0.202258
$850$	0	0
$851$	−7.71119e6	−0.365004
$852$	0	0
$853$	1.62685e7	0.765552	0.382776	−	0.923841i	$-0.374968\pi$
$853$	1.62685e7	0.765552	0.382776	+	0.923841i	$0.374968\pi$
$854$	0	0
$855$	3.32210e6	0.155417
$856$	0	0
$857$	−2.92667e7	−1.36120	−0.680600	−	0.732656i	$-0.738281\pi$
$857$	−2.92667e7	−1.36120	−0.680600	+	0.732656i	$0.738281\pi$
$858$	0	0
$859$	−3.31062e7	−1.53083	−0.765413	−	0.643539i	$-0.777465\pi$
$859$	−3.31062e7	−1.53083	−0.765413	+	0.643539i	$0.777465\pi$
$860$	0	0
$861$	925288.	0.0425372
$862$	0	0
$863$	−1.58052e7	−0.722391	−0.361196	−	0.932490i	$-0.617631\pi$
$863$	−1.58052e7	−0.722391	−0.361196	+	0.932490i	$0.617631\pi$
$864$	0	0
$865$	−3.18105e6	−0.144554
$866$	0	0
$867$	7514.00	0.000339487 0
$868$	0	0
$869$	−1.35671e7	−0.609449
$870$	0	0
$871$	3.71040e7	1.65720
$872$	0	0
$873$	−5.66765e6	−0.251691
$874$	0	0
$875$	968750.	0.0427752
$876$	0	0
$877$	−4.26834e7	−1.87396	−0.936980	−	0.349384i	$-0.886391\pi$
$877$	−4.26834e7	−1.87396	−0.936980	+	0.349384i	$0.886391\pi$
$878$	0	0
$879$	−1.98544e6	−0.0866733
$880$	0	0
$881$	−3.57397e6	−0.155135	−0.0775677	−	0.996987i	$-0.524715\pi$
$881$	−3.57397e6	−0.155135	−0.0775677	+	0.996987i	$0.524715\pi$
$882$	0	0
$883$	−1.68471e7	−0.727149	−0.363574	−	0.931565i	$-0.618444\pi$
$883$	−1.68471e7	−0.727149	−0.363574	+	0.931565i	$0.618444\pi$
$884$	0	0
$885$	−1.85460e6	−0.0795962
$886$	0	0
$887$	−8.36792e6	−0.357115	−0.178558	−	0.983929i	$-0.557143\pi$
$887$	−8.36792e6	−0.357115	−0.178558	+	0.983929i	$0.557143\pi$
$888$	0	0
$889$	−6.69017e6	−0.283911
$890$	0	0
$891$	−8.08546e6	−0.341201
$892$	0	0
$893$	−1.57315e7	−0.660147
$894$	0	0
$895$	2.36710e6	0.0987777
$896$	0	0
$897$	2.85406e6	0.118435
$898$	0	0
$899$	−1.52435e7	−0.629050
$900$	0	0
$901$	1.55247e7	0.637107
$902$	0	0
$903$	−882136.	−0.0360011
$904$	0	0
$905$	1.29255e7	0.524595
$906$	0	0
$907$	−2.57230e7	−1.03825	−0.519127	−	0.854697i	$-0.673743\pi$
$907$	−2.57230e7	−1.03825	−0.519127	+	0.854697i	$0.673743\pi$
$908$	0	0
$909$	3.09921e7	1.24406
$910$	0	0
$911$	3.42108e7	1.36574	0.682869	−	0.730540i	$-0.260732\pi$
$911$	3.42108e7	1.36574	0.682869	+	0.730540i	$0.260732\pi$
$912$	0	0
$913$	4.53341e6	0.179990
$914$	0	0
$915$	1.97850e6	0.0781238
$916$	0	0
$917$	1.44505e7	0.567490
$918$	0	0
$919$	2.44034e6	0.0953149	0.0476575	−	0.998864i	$-0.484824\pi$
$919$	2.44034e6	0.0953149	0.0476575	+	0.998864i	$0.484824\pi$
$920$	0	0
$921$	−975044.	−0.0378770
$922$	0	0
$923$	−2.98146e7	−1.15192
$924$	0	0
$925$	−2.20875e6	−0.0848774
$926$	0	0
$927$	−3.27062e7	−1.25006
$928$	0	0
$929$	1.34361e7	0.510781	0.255390	−	0.966838i	$-0.417796\pi$
$929$	1.34361e7	0.510781	0.255390	+	0.966838i	$0.417796\pi$
$930$	0	0
$931$	−7.20743e6	−0.272525
$932$	0	0
$933$	888232.	0.0334058
$934$	0	0
$935$	−4.28400e6	−0.160258
$936$	0	0
$937$	7.96529e6	0.296383	0.148191	−	0.988959i	$-0.452655\pi$
$937$	7.96529e6	0.296383	0.148191	+	0.988959i	$0.452655\pi$
$938$	0	0
$939$	−94484.0	−0.00349699
$940$	0	0
$941$	9.08025e6	0.334290	0.167145	−	0.985932i	$-0.446545\pi$
$941$	9.08025e6	0.334290	0.167145	+	0.985932i	$0.446545\pi$
$942$	0	0
$943$	1.62821e7	0.596253
$944$	0	0
$945$	1.49420e6	0.0544289
$946$	0	0
$947$	−3.21769e7	−1.16592	−0.582961	−	0.812500i	$-0.698106\pi$
$947$	−3.21769e7	−1.16592	−0.582961	+	0.812500i	$0.698106\pi$
$948$	0	0
$949$	−7.74467e6	−0.279150
$950$	0	0
$951$	−1.38812e6	−0.0497708
$952$	0	0
$953$	5.33807e6	0.190394	0.0951968	−	0.995458i	$-0.469652\pi$
$953$	5.33807e6	0.190394	0.0951968	+	0.995458i	$0.469652\pi$
$954$	0	0
$955$	1.03075e7	0.365717
$956$	0	0
$957$	−454464.	−0.0160406
$958$	0	0
$959$	−2.20771e7	−0.775167
$960$	0	0
$961$	6.46864e7	2.25946
$962$	0	0
$963$	4.61724e7	1.60442
$964$	0	0
$965$	1.92914e7	0.666875
$966$	0	0
$967$	3.71522e7	1.27767	0.638834	−	0.769345i	$-0.279417\pi$
$967$	3.71522e7	1.27767	0.638834	+	0.769345i	$0.279417\pi$
$968$	0	0
$969$	1.32328e6	0.0452733
$970$	0	0
$971$	−1.09865e7	−0.373949	−0.186975	−	0.982365i	$-0.559868\pi$
$971$	−1.09865e7	−0.373949	−0.186975	+	0.982365i	$0.559868\pi$
$972$	0	0
$973$	−1.93566e7	−0.655463
$974$	0	0
$975$	817500.	0.0275408
$976$	0	0
$977$	−2.65054e7	−0.888379	−0.444190	−	0.895933i	$-0.646508\pi$
$977$	−2.65054e7	−0.888379	−0.444190	+	0.895933i	$0.646508\pi$
$978$	0	0
$979$	1.35438e7	0.451630
$980$	0	0
$981$	2.86910e7	0.951860
$982$	0	0
$983$	−4.75726e7	−1.57027	−0.785133	−	0.619327i	$-0.787406\pi$
$983$	−4.75726e7	−1.57027	−0.785133	+	0.619327i	$0.787406\pi$
$984$	0	0
$985$	4.75595e6	0.156188
$986$	0	0
$987$	−3.50846e6	−0.114637
$988$	0	0
$989$	−1.55227e7	−0.504636
$990$	0	0
$991$	3.22149e7	1.04201	0.521006	−	0.853553i	$-0.325557\pi$
$991$	3.22149e7	1.04201	0.521006	+	0.853553i	$0.325557\pi$
$992$	0	0
$993$	−164336.	−0.00528883
$994$	0	0
$995$	−3.30180e6	−0.105729
$996$	0	0
$997$	3.87072e7	1.23326	0.616630	−	0.787253i	$-0.288498\pi$
$997$	3.87072e7	1.23326	0.616630	+	0.787253i	$0.288498\pi$
$998$	0	0
$999$	−3.40678e6	−0.108002

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	40.6.a.c.1.1	✓	1
3.2	odd	2		360.6.a.f.1.1		1
4.3	odd	2		80.6.a.d.1.1		1
5.2	odd	4		200.6.c.d.49.2		2
5.3	odd	4		200.6.c.d.49.1		2
5.4	even	2		200.6.a.b.1.1		1
8.3	odd	2		320.6.a.h.1.1		1
8.5	even	2		320.6.a.i.1.1		1
12.11	even	2		720.6.a.t.1.1		1
20.3	even	4		400.6.c.k.49.2		2
20.7	even	4		400.6.c.k.49.1		2
20.19	odd	2		400.6.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.a.c.1.1	✓	1	1.1	even	1	trivial
80.6.a.d.1.1		1	4.3	odd	2
200.6.a.b.1.1		1	5.4	even	2
200.6.c.d.49.1		2	5.3	odd	4
200.6.c.d.49.2		2	5.2	odd	4
320.6.a.h.1.1		1	8.3	odd	2
320.6.a.i.1.1		1	8.5	even	2
360.6.a.f.1.1		1	3.2	odd	2
400.6.a.h.1.1		1	20.19	odd	2
400.6.c.k.49.1		2	20.7	even	4
400.6.c.k.49.2		2	20.3	even	4
720.6.a.t.1.1		1	12.11	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 40 = 2^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	40.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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