LMFDB - Embedded newform 60.6.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [60,6,Mod(1,60)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(60, 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("60.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 60 = 2^{2} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	60.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:	$9.62302918878$
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	$\chi$	$=$	60.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+9.00000 q^{3} -25.0000 q^{5} -244.000 q^{7} +81.0000 q^{9} +O(q^{10})$

$q+9.00000 q^{3} -25.0000 q^{5} -244.000 q^{7} +81.0000 q^{9} -144.000 q^{11} +50.0000 q^{13} -225.000 q^{15} -1914.00 q^{17} +140.000 q^{19} -2196.00 q^{21} -624.000 q^{23} +625.000 q^{25} +729.000 q^{27} -3126.00 q^{29} -5176.00 q^{31} -1296.00 q^{33} +6100.00 q^{35} +15698.0 q^{37} +450.000 q^{39} +12570.0 q^{41} +11516.0 q^{43} -2025.00 q^{45} -26736.0 q^{47} +42729.0 q^{49} -17226.0 q^{51} -19158.0 q^{53} +3600.00 q^{55} +1260.00 q^{57} +27984.0 q^{59} +22022.0 q^{61} -19764.0 q^{63} -1250.00 q^{65} -12676.0 q^{67} -5616.00 q^{69} -59520.0 q^{71} -67102.0 q^{73} +5625.00 q^{75} +35136.0 q^{77} +11048.0 q^{79} +6561.00 q^{81} -115284. q^{83} +47850.0 q^{85} -28134.0 q^{87} +73650.0 q^{89} -12200.0 q^{91} -46584.0 q^{93} -3500.00 q^{95} +35522.0 q^{97} -11664.0 q^{99} +O(q^{100})$

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
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	0	0
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	−244.000	−1.88211	−0.941054	−	0.338255i	$-0.890163\pi$
$7$	−244.000	−1.88211	−0.941054	+	0.338255i	$0.890163\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	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
$13$	50.0000	0.0820562	0.0410281	−	0.999158i	$-0.486937\pi$
$13$	50.0000	0.0820562	0.0410281	+	0.999158i	$0.486937\pi$
$14$	0	0
$15$	−225.000	−0.258199
$16$	0	0
$17$	−1914.00	−1.60627	−0.803137	−	0.595794i	$-0.796837\pi$
$17$	−1914.00	−1.60627	−0.803137	+	0.595794i	$0.796837\pi$
$18$	0	0
$19$	140.000	0.0889701	0.0444850	−	0.999010i	$-0.485835\pi$
$19$	140.000	0.0889701	0.0444850	+	0.999010i	$0.485835\pi$
$20$	0	0
$21$	−2196.00	−1.08664
$22$	0	0
$23$	−624.000	−0.245960	−0.122980	−	0.992409i	$-0.539245\pi$
$23$	−624.000	−0.245960	−0.122980	+	0.992409i	$0.539245\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	−3126.00	−0.690230	−0.345115	−	0.938560i	$-0.612160\pi$
$29$	−3126.00	−0.690230	−0.345115	+	0.938560i	$0.612160\pi$
$30$	0	0
$31$	−5176.00	−0.967364	−0.483682	−	0.875244i	$-0.660701\pi$
$31$	−5176.00	−0.967364	−0.483682	+	0.875244i	$0.660701\pi$
$32$	0	0
$33$	−1296.00	−0.207167
$34$	0	0
$35$	6100.00	0.841705
$36$	0	0
$37$	15698.0	1.88512	0.942562	−	0.334031i	$-0.108409\pi$
$37$	15698.0	1.88512	0.942562	+	0.334031i	$0.108409\pi$
$38$	0	0
$39$	450.000	0.0473752
$40$	0	0
$41$	12570.0	1.16782	0.583910	−	0.811819i	$-0.301522\pi$
$41$	12570.0	1.16782	0.583910	+	0.811819i	$0.301522\pi$
$42$	0	0
$43$	11516.0	0.949796	0.474898	−	0.880041i	$-0.342485\pi$
$43$	11516.0	0.949796	0.474898	+	0.880041i	$0.342485\pi$
$44$	0	0
$45$	−2025.00	−0.149071
$46$	0	0
$47$	−26736.0	−1.76544	−0.882718	−	0.469904i	$-0.844289\pi$
$47$	−26736.0	−1.76544	−0.882718	+	0.469904i	$0.844289\pi$
$48$	0	0
$49$	42729.0	2.54233
$50$	0	0
$51$	−17226.0	−0.927383
$52$	0	0
$53$	−19158.0	−0.936829	−0.468415	−	0.883509i	$-0.655175\pi$
$53$	−19158.0	−0.936829	−0.468415	+	0.883509i	$0.655175\pi$
$54$	0	0
$55$	3600.00	0.160471
$56$	0	0
$57$	1260.00	0.0513669
$58$	0	0
$59$	27984.0	1.04660	0.523299	−	0.852149i	$-0.324701\pi$
$59$	27984.0	1.04660	0.523299	+	0.852149i	$0.324701\pi$
$60$	0	0
$61$	22022.0	0.757761	0.378880	−	0.925446i	$-0.376309\pi$
$61$	22022.0	0.757761	0.378880	+	0.925446i	$0.376309\pi$
$62$	0	0
$63$	−19764.0	−0.627370
$64$	0	0
$65$	−1250.00	−0.0366967
$66$	0	0
$67$	−12676.0	−0.344981	−0.172491	−	0.985011i	$-0.555181\pi$
$67$	−12676.0	−0.344981	−0.172491	+	0.985011i	$0.555181\pi$
$68$	0	0
$69$	−5616.00	−0.142005
$70$	0	0
$71$	−59520.0	−1.40125	−0.700627	−	0.713527i	$-0.747096\pi$
$71$	−59520.0	−1.40125	−0.700627	+	0.713527i	$0.747096\pi$
$72$	0	0
$73$	−67102.0	−1.47377	−0.736883	−	0.676021i	$-0.763703\pi$
$73$	−67102.0	−1.47377	−0.736883	+	0.676021i	$0.763703\pi$
$74$	0	0
$75$	5625.00	0.115470
$76$	0	0
$77$	35136.0	0.675345
$78$	0	0
$79$	11048.0	0.199166	0.0995832	−	0.995029i	$-0.468249\pi$
$79$	11048.0	0.199166	0.0995832	+	0.995029i	$0.468249\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−115284.	−1.83685	−0.918425	−	0.395595i	$-0.870539\pi$
$83$	−115284.	−1.83685	−0.918425	+	0.395595i	$0.870539\pi$
$84$	0	0
$85$	47850.0	0.718348
$86$	0	0
$87$	−28134.0	−0.398505
$88$	0	0
$89$	73650.0	0.985593	0.492797	−	0.870145i	$-0.335975\pi$
$89$	73650.0	0.985593	0.492797	+	0.870145i	$0.335975\pi$
$90$	0	0
$91$	−12200.0	−0.154439
$92$	0	0
$93$	−46584.0	−0.558508
$94$	0	0
$95$	−3500.00	−0.0397886
$96$	0	0
$97$	35522.0	0.383326	0.191663	−	0.981461i	$-0.438612\pi$
$97$	35522.0	0.383326	0.191663	+	0.981461i	$0.438612\pi$
$98$	0	0
$99$	−11664.0	−0.119608
$100$	0	0
$101$	−112062.	−1.09309	−0.546544	−	0.837431i	$-0.684057\pi$
$101$	−112062.	−1.09309	−0.546544	+	0.837431i	$0.684057\pi$
$102$	0	0
$103$	−106516.	−0.989286	−0.494643	−	0.869096i	$-0.664701\pi$
$103$	−106516.	−0.989286	−0.494643	+	0.869096i	$0.664701\pi$
$104$	0	0
$105$	54900.0	0.485958
$106$	0	0
$107$	−29292.0	−0.247337	−0.123669	−	0.992324i	$-0.539466\pi$
$107$	−29292.0	−0.247337	−0.123669	+	0.992324i	$0.539466\pi$
$108$	0	0
$109$	125318.	1.01029	0.505146	−	0.863034i	$-0.331439\pi$
$109$	125318.	1.01029	0.505146	+	0.863034i	$0.331439\pi$
$110$	0	0
$111$	141282.	1.08838
$112$	0	0
$113$	166638.	1.22766	0.613830	−	0.789438i	$-0.289628\pi$
$113$	166638.	1.22766	0.613830	+	0.789438i	$0.289628\pi$
$114$	0	0
$115$	15600.0	0.109997
$116$	0	0
$117$	4050.00	0.0273521
$118$	0	0
$119$	467016.	3.02318
$120$	0	0
$121$	−140315.	−0.871246
$122$	0	0
$123$	113130.	0.674241
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	−201724.	−1.10981	−0.554905	−	0.831914i	$-0.687245\pi$
$127$	−201724.	−1.10981	−0.554905	+	0.831914i	$0.687245\pi$
$128$	0	0
$129$	103644.	0.548365
$130$	0	0
$131$	131472.	0.669353	0.334676	−	0.942333i	$-0.391373\pi$
$131$	131472.	0.669353	0.334676	+	0.942333i	$0.391373\pi$
$132$	0	0
$133$	−34160.0	−0.167451
$134$	0	0
$135$	−18225.0	−0.0860663
$136$	0	0
$137$	−102042.	−0.464491	−0.232246	−	0.972657i	$-0.574607\pi$
$137$	−102042.	−0.464491	−0.232246	+	0.972657i	$0.574607\pi$
$138$	0	0
$139$	−243604.	−1.06942	−0.534709	−	0.845036i	$-0.679579\pi$
$139$	−243604.	−1.06942	−0.534709	+	0.845036i	$0.679579\pi$
$140$	0	0
$141$	−240624.	−1.01927
$142$	0	0
$143$	−7200.00	−0.0294437
$144$	0	0
$145$	78150.0	0.308680
$146$	0	0
$147$	384561.	1.46782
$148$	0	0
$149$	−309462.	−1.14194	−0.570968	−	0.820972i	$-0.693432\pi$
$149$	−309462.	−1.14194	−0.570968	+	0.820972i	$0.693432\pi$
$150$	0	0
$151$	−147160.	−0.525227	−0.262614	−	0.964901i	$-0.584584\pi$
$151$	−147160.	−0.525227	−0.262614	+	0.964901i	$0.584584\pi$
$152$	0	0
$153$	−155034.	−0.535425
$154$	0	0
$155$	129400.	0.432618
$156$	0	0
$157$	268082.	0.867998	0.433999	−	0.900913i	$-0.357102\pi$
$157$	268082.	0.867998	0.433999	+	0.900913i	$0.357102\pi$
$158$	0	0
$159$	−172422.	−0.540879
$160$	0	0
$161$	152256.	0.462924
$162$	0	0
$163$	−276580.	−0.815364	−0.407682	−	0.913124i	$-0.633663\pi$
$163$	−276580.	−0.815364	−0.407682	+	0.913124i	$0.633663\pi$
$164$	0	0
$165$	32400.0	0.0926478
$166$	0	0
$167$	−592272.	−1.64335	−0.821675	−	0.569956i	$-0.806960\pi$
$167$	−592272.	−1.64335	−0.821675	+	0.569956i	$0.806960\pi$
$168$	0	0
$169$	−368793.	−0.993267
$170$	0	0
$171$	11340.0	0.0296567
$172$	0	0
$173$	29154.0	0.0740599	0.0370299	−	0.999314i	$-0.488210\pi$
$173$	29154.0	0.0740599	0.0370299	+	0.999314i	$0.488210\pi$
$174$	0	0
$175$	−152500.	−0.376422
$176$	0	0
$177$	251856.	0.604253
$178$	0	0
$179$	131784.	0.307419	0.153709	−	0.988116i	$-0.450878\pi$
$179$	131784.	0.307419	0.153709	+	0.988116i	$0.450878\pi$
$180$	0	0
$181$	−297730.	−0.675501	−0.337751	−	0.941236i	$-0.609666\pi$
$181$	−297730.	−0.675501	−0.337751	+	0.941236i	$0.609666\pi$
$182$	0	0
$183$	198198.	0.437493
$184$	0	0
$185$	−392450.	−0.843053
$186$	0	0
$187$	275616.	0.576369
$188$	0	0
$189$	−177876.	−0.362212
$190$	0	0
$191$	851784.	1.68945	0.844726	−	0.535198i	$-0.179763\pi$
$191$	851784.	1.68945	0.844726	+	0.535198i	$0.179763\pi$
$192$	0	0
$193$	292394.	0.565035	0.282517	−	0.959262i	$-0.408831\pi$
$193$	292394.	0.565035	0.282517	+	0.959262i	$0.408831\pi$
$194$	0	0
$195$	−11250.0	−0.0211868
$196$	0	0
$197$	−869526.	−1.59631	−0.798155	−	0.602453i	$-0.794190\pi$
$197$	−869526.	−1.59631	−0.798155	+	0.602453i	$0.794190\pi$
$198$	0	0
$199$	−143992.	−0.257754	−0.128877	−	0.991661i	$-0.541137\pi$
$199$	−143992.	−0.257754	−0.128877	+	0.991661i	$0.541137\pi$
$200$	0	0
$201$	−114084.	−0.199175
$202$	0	0
$203$	762744.	1.29909
$204$	0	0
$205$	−314250.	−0.522265
$206$	0	0
$207$	−50544.0	−0.0819868
$208$	0	0
$209$	−20160.0	−0.0319246
$210$	0	0
$211$	836420.	1.29336	0.646678	−	0.762763i	$-0.276158\pi$
$211$	836420.	1.29336	0.646678	+	0.762763i	$0.276158\pi$
$212$	0	0
$213$	−535680.	−0.809015
$214$	0	0
$215$	−287900.	−0.424762
$216$	0	0
$217$	1.26294e6	1.82068
$218$	0	0
$219$	−603918.	−0.850879
$220$	0	0
$221$	−95700.0	−0.131805
$222$	0	0
$223$	1.22090e6	1.64406	0.822031	−	0.569443i	$-0.192841\pi$
$223$	1.22090e6	1.64406	0.822031	+	0.569443i	$0.192841\pi$
$224$	0	0
$225$	50625.0	0.0666667
$226$	0	0
$227$	−173580.	−0.223581	−0.111791	−	0.993732i	$-0.535659\pi$
$227$	−173580.	−0.223581	−0.111791	+	0.993732i	$0.535659\pi$
$228$	0	0
$229$	120806.	0.152230	0.0761149	−	0.997099i	$-0.475748\pi$
$229$	120806.	0.152230	0.0761149	+	0.997099i	$0.475748\pi$
$230$	0	0
$231$	316224.	0.389910
$232$	0	0
$233$	592590.	0.715096	0.357548	−	0.933895i	$-0.383613\pi$
$233$	592590.	0.715096	0.357548	+	0.933895i	$0.383613\pi$
$234$	0	0
$235$	668400.	0.789527
$236$	0	0
$237$	99432.0	0.114989
$238$	0	0
$239$	706008.	0.799493	0.399747	−	0.916626i	$-0.369098\pi$
$239$	706008.	0.799493	0.399747	+	0.916626i	$0.369098\pi$
$240$	0	0
$241$	1.56154e6	1.73185	0.865924	−	0.500175i	$-0.166731\pi$
$241$	1.56154e6	1.73185	0.865924	+	0.500175i	$0.166731\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	−1.06822e6	−1.13697
$246$	0	0
$247$	7000.00	0.00730055
$248$	0	0
$249$	−1.03756e6	−1.06051
$250$	0	0
$251$	−1.48471e6	−1.48750	−0.743752	−	0.668456i	$-0.766955\pi$
$251$	−1.48471e6	−1.48750	−0.743752	+	0.668456i	$0.766955\pi$
$252$	0	0
$253$	89856.0	0.0882563
$254$	0	0
$255$	430650.	0.414738
$256$	0	0
$257$	624558.	0.589848	0.294924	−	0.955521i	$-0.404706\pi$
$257$	624558.	0.589848	0.294924	+	0.955521i	$0.404706\pi$
$258$	0	0
$259$	−3.83031e6	−3.54801
$260$	0	0
$261$	−253206.	−0.230077
$262$	0	0
$263$	778896.	0.694369	0.347184	−	0.937797i	$-0.387138\pi$
$263$	778896.	0.694369	0.347184	+	0.937797i	$0.387138\pi$
$264$	0	0
$265$	478950.	0.418963
$266$	0	0
$267$	662850.	0.569033
$268$	0	0
$269$	−11574.0	−0.00975220	−0.00487610	−	0.999988i	$-0.501552\pi$
$269$	−11574.0	−0.00975220	−0.00487610	+	0.999988i	$0.501552\pi$
$270$	0	0
$271$	118064.	0.0976550	0.0488275	−	0.998807i	$-0.484452\pi$
$271$	118064.	0.0976550	0.0488275	+	0.998807i	$0.484452\pi$
$272$	0	0
$273$	−109800.	−0.0891653
$274$	0	0
$275$	−90000.0	−0.0717647
$276$	0	0
$277$	659282.	0.516264	0.258132	−	0.966110i	$-0.416893\pi$
$277$	659282.	0.516264	0.258132	+	0.966110i	$0.416893\pi$
$278$	0	0
$279$	−419256.	−0.322455
$280$	0	0
$281$	−1.49057e6	−1.12613	−0.563064	−	0.826413i	$-0.690378\pi$
$281$	−1.49057e6	−1.12613	−0.563064	+	0.826413i	$0.690378\pi$
$282$	0	0
$283$	−1.13129e6	−0.839670	−0.419835	−	0.907600i	$-0.637912\pi$
$283$	−1.13129e6	−0.839670	−0.419835	+	0.907600i	$0.637912\pi$
$284$	0	0
$285$	−31500.0	−0.0229720
$286$	0	0
$287$	−3.06708e6	−2.19796
$288$	0	0
$289$	2.24354e6	1.58012
$290$	0	0
$291$	319698.	0.221313
$292$	0	0
$293$	896298.	0.609935	0.304967	−	0.952363i	$-0.401354\pi$
$293$	896298.	0.609935	0.304967	+	0.952363i	$0.401354\pi$
$294$	0	0
$295$	−699600.	−0.468053
$296$	0	0
$297$	−104976.	−0.0690556
$298$	0	0
$299$	−31200.0	−0.0201826
$300$	0	0
$301$	−2.80990e6	−1.78762
$302$	0	0
$303$	−1.00856e6	−0.631094
$304$	0	0
$305$	−550550.	−0.338881
$306$	0	0
$307$	−1.38641e6	−0.839550	−0.419775	−	0.907628i	$-0.637891\pi$
$307$	−1.38641e6	−0.839550	−0.419775	+	0.907628i	$0.637891\pi$
$308$	0	0
$309$	−958644.	−0.571164
$310$	0	0
$311$	3.19238e6	1.87160	0.935802	−	0.352525i	$-0.114677\pi$
$311$	3.19238e6	1.87160	0.935802	+	0.352525i	$0.114677\pi$
$312$	0	0
$313$	2.51683e6	1.45209	0.726045	−	0.687647i	$-0.241356\pi$
$313$	2.51683e6	1.45209	0.726045	+	0.687647i	$0.241356\pi$
$314$	0	0
$315$	494100.	0.280568
$316$	0	0
$317$	−2.21897e6	−1.24024	−0.620118	−	0.784509i	$-0.712915\pi$
$317$	−2.21897e6	−1.24024	−0.620118	+	0.784509i	$0.712915\pi$
$318$	0	0
$319$	450144.	0.247671
$320$	0	0
$321$	−263628.	−0.142800
$322$	0	0
$323$	−267960.	−0.142910
$324$	0	0
$325$	31250.0	0.0164112
$326$	0	0
$327$	1.12786e6	0.583293
$328$	0	0
$329$	6.52358e6	3.32274
$330$	0	0
$331$	−27148.0	−0.0136197	−0.00680985	−	0.999977i	$-0.502168\pi$
$331$	−27148.0	−0.0136197	−0.00680985	+	0.999977i	$0.502168\pi$
$332$	0	0
$333$	1.27154e6	0.628375
$334$	0	0
$335$	316900.	0.154280
$336$	0	0
$337$	2.19670e6	1.05365	0.526824	−	0.849974i	$-0.323383\pi$
$337$	2.19670e6	1.05365	0.526824	+	0.849974i	$0.323383\pi$
$338$	0	0
$339$	1.49974e6	0.708790
$340$	0	0
$341$	745344.	0.347113
$342$	0	0
$343$	−6.32497e6	−2.90284
$344$	0	0
$345$	140400.	0.0635067
$346$	0	0
$347$	−989652.	−0.441224	−0.220612	−	0.975362i	$-0.570805\pi$
$347$	−989652.	−0.441224	−0.220612	+	0.975362i	$0.570805\pi$
$348$	0	0
$349$	−3.05055e6	−1.34065	−0.670325	−	0.742068i	$-0.733845\pi$
$349$	−3.05055e6	−1.34065	−0.670325	+	0.742068i	$0.733845\pi$
$350$	0	0
$351$	36450.0	0.0157917
$352$	0	0
$353$	−339642.	−0.145072	−0.0725362	−	0.997366i	$-0.523109\pi$
$353$	−339642.	−0.145072	−0.0725362	+	0.997366i	$0.523109\pi$
$354$	0	0
$355$	1.48800e6	0.626660
$356$	0	0
$357$	4.20314e6	1.74544
$358$	0	0
$359$	−2.20678e6	−0.903696	−0.451848	−	0.892095i	$-0.649235\pi$
$359$	−2.20678e6	−0.903696	−0.451848	+	0.892095i	$0.649235\pi$
$360$	0	0
$361$	−2.45650e6	−0.992084
$362$	0	0
$363$	−1.26284e6	−0.503014
$364$	0	0
$365$	1.67755e6	0.659088
$366$	0	0
$367$	2.07132e6	0.802752	0.401376	−	0.915913i	$-0.368532\pi$
$367$	2.07132e6	0.802752	0.401376	+	0.915913i	$0.368532\pi$
$368$	0	0
$369$	1.01817e6	0.389273
$370$	0	0
$371$	4.67455e6	1.76321
$372$	0	0
$373$	−1.94997e6	−0.725699	−0.362850	−	0.931848i	$-0.618196\pi$
$373$	−1.94997e6	−0.725699	−0.362850	+	0.931848i	$0.618196\pi$
$374$	0	0
$375$	−140625.	−0.0516398
$376$	0	0
$377$	−156300.	−0.0566377
$378$	0	0
$379$	3.85003e6	1.37678	0.688392	−	0.725339i	$-0.258317\pi$
$379$	3.85003e6	1.37678	0.688392	+	0.725339i	$0.258317\pi$
$380$	0	0
$381$	−1.81552e6	−0.640749
$382$	0	0
$383$	−67152.0	−0.0233917	−0.0116959	−	0.999932i	$-0.503723\pi$
$383$	−67152.0	−0.0233917	−0.0116959	+	0.999932i	$0.503723\pi$
$384$	0	0
$385$	−878400.	−0.302023
$386$	0	0
$387$	932796.	0.316599
$388$	0	0
$389$	4.32331e6	1.44858	0.724289	−	0.689496i	$-0.242168\pi$
$389$	4.32331e6	1.44858	0.724289	+	0.689496i	$0.242168\pi$
$390$	0	0
$391$	1.19434e6	0.395080
$392$	0	0
$393$	1.18325e6	0.386451
$394$	0	0
$395$	−276200.	−0.0890699
$396$	0	0
$397$	118298.	0.0376705	0.0188352	−	0.999823i	$-0.494004\pi$
$397$	118298.	0.0376705	0.0188352	+	0.999823i	$0.494004\pi$
$398$	0	0
$399$	−307440.	−0.0966781
$400$	0	0
$401$	−3.13901e6	−0.974838	−0.487419	−	0.873168i	$-0.662062\pi$
$401$	−3.13901e6	−0.974838	−0.487419	+	0.873168i	$0.662062\pi$
$402$	0	0
$403$	−258800.	−0.0793783
$404$	0	0
$405$	−164025.	−0.0496904
$406$	0	0
$407$	−2.26051e6	−0.676427
$408$	0	0
$409$	−452614.	−0.133789	−0.0668944	−	0.997760i	$-0.521309\pi$
$409$	−452614.	−0.133789	−0.0668944	+	0.997760i	$0.521309\pi$
$410$	0	0
$411$	−918378.	−0.268174
$412$	0	0
$413$	−6.82810e6	−1.96981
$414$	0	0
$415$	2.88210e6	0.821465
$416$	0	0
$417$	−2.19244e6	−0.617429
$418$	0	0
$419$	3.73579e6	1.03956	0.519778	−	0.854302i	$-0.326015\pi$
$419$	3.73579e6	1.03956	0.519778	+	0.854302i	$0.326015\pi$
$420$	0	0
$421$	−1.87465e6	−0.515484	−0.257742	−	0.966214i	$-0.582978\pi$
$421$	−1.87465e6	−0.515484	−0.257742	+	0.966214i	$0.582978\pi$
$422$	0	0
$423$	−2.16562e6	−0.588478
$424$	0	0
$425$	−1.19625e6	−0.321255
$426$	0	0
$427$	−5.37337e6	−1.42619
$428$	0	0
$429$	−64800.0	−0.0169993
$430$	0	0
$431$	−4.74019e6	−1.22914	−0.614572	−	0.788861i	$-0.710671\pi$
$431$	−4.74019e6	−1.22914	−0.614572	+	0.788861i	$0.710671\pi$
$432$	0	0
$433$	−4.35377e6	−1.11595	−0.557976	−	0.829857i	$-0.688422\pi$
$433$	−4.35377e6	−1.11595	−0.557976	+	0.829857i	$0.688422\pi$
$434$	0	0
$435$	703350.	0.178217
$436$	0	0
$437$	−87360.0	−0.0218831
$438$	0	0
$439$	1.04929e6	0.259856	0.129928	−	0.991523i	$-0.458525\pi$
$439$	1.04929e6	0.259856	0.129928	+	0.991523i	$0.458525\pi$
$440$	0	0
$441$	3.46105e6	0.847445
$442$	0	0
$443$	6.00788e6	1.45449	0.727247	−	0.686375i	$-0.240799\pi$
$443$	6.00788e6	1.45449	0.727247	+	0.686375i	$0.240799\pi$
$444$	0	0
$445$	−1.84125e6	−0.440771
$446$	0	0
$447$	−2.78516e6	−0.659297
$448$	0	0
$449$	1.37222e6	0.321223	0.160612	−	0.987018i	$-0.448653\pi$
$449$	1.37222e6	0.321223	0.160612	+	0.987018i	$0.448653\pi$
$450$	0	0
$451$	−1.81008e6	−0.419041
$452$	0	0
$453$	−1.32444e6	−0.303240
$454$	0	0
$455$	305000.	0.0690671
$456$	0	0
$457$	−389830.	−0.0873142	−0.0436571	−	0.999047i	$-0.513901\pi$
$457$	−389830.	−0.0873142	−0.0436571	+	0.999047i	$0.513901\pi$
$458$	0	0
$459$	−1.39531e6	−0.309128
$460$	0	0
$461$	−3.15137e6	−0.690633	−0.345317	−	0.938486i	$-0.612229\pi$
$461$	−3.15137e6	−0.690633	−0.345317	+	0.938486i	$0.612229\pi$
$462$	0	0
$463$	4.66939e6	1.01230	0.506148	−	0.862447i	$-0.331069\pi$
$463$	4.66939e6	1.01230	0.506148	+	0.862447i	$0.331069\pi$
$464$	0	0
$465$	1.16460e6	0.249772
$466$	0	0
$467$	−4.92445e6	−1.04488	−0.522439	−	0.852677i	$-0.674978\pi$
$467$	−4.92445e6	−1.04488	−0.522439	+	0.852677i	$0.674978\pi$
$468$	0	0
$469$	3.09294e6	0.649292
$470$	0	0
$471$	2.41274e6	0.501139
$472$	0	0
$473$	−1.65830e6	−0.340809
$474$	0	0
$475$	87500.0	0.0177940
$476$	0	0
$477$	−1.55180e6	−0.312276
$478$	0	0
$479$	−4.03030e6	−0.802598	−0.401299	−	0.915947i	$-0.631441\pi$
$479$	−4.03030e6	−0.802598	−0.401299	+	0.915947i	$0.631441\pi$
$480$	0	0
$481$	784900.	0.154686
$482$	0	0
$483$	1.37030e6	0.267269
$484$	0	0
$485$	−888050.	−0.171429
$486$	0	0
$487$	4.14213e6	0.791410	0.395705	−	0.918378i	$-0.370500\pi$
$487$	4.14213e6	0.791410	0.395705	+	0.918378i	$0.370500\pi$
$488$	0	0
$489$	−2.48922e6	−0.470751
$490$	0	0
$491$	−1.22854e6	−0.229977	−0.114988	−	0.993367i	$-0.536683\pi$
$491$	−1.22854e6	−0.229977	−0.114988	+	0.993367i	$0.536683\pi$
$492$	0	0
$493$	5.98316e6	1.10870
$494$	0	0
$495$	291600.	0.0534902
$496$	0	0
$497$	1.45229e7	2.63731
$498$	0	0
$499$	3.64756e6	0.655770	0.327885	−	0.944718i	$-0.393664\pi$
$499$	3.64756e6	0.655770	0.327885	+	0.944718i	$0.393664\pi$
$500$	0	0
$501$	−5.33045e6	−0.948788
$502$	0	0
$503$	−1.98454e6	−0.349735	−0.174867	−	0.984592i	$-0.555950\pi$
$503$	−1.98454e6	−0.349735	−0.174867	+	0.984592i	$0.555950\pi$
$504$	0	0
$505$	2.80155e6	0.488844
$506$	0	0
$507$	−3.31914e6	−0.573463
$508$	0	0
$509$	−5.75852e6	−0.985182	−0.492591	−	0.870261i	$-0.663950\pi$
$509$	−5.75852e6	−0.985182	−0.492591	+	0.870261i	$0.663950\pi$
$510$	0	0
$511$	1.63729e7	2.77379
$512$	0	0
$513$	102060.	0.0171223
$514$	0	0
$515$	2.66290e6	0.442422
$516$	0	0
$517$	3.84998e6	0.633480
$518$	0	0
$519$	262386.	0.0427585
$520$	0	0
$521$	−6.84008e6	−1.10399	−0.551997	−	0.833846i	$-0.686134\pi$
$521$	−6.84008e6	−1.10399	−0.551997	+	0.833846i	$0.686134\pi$
$522$	0	0
$523$	7.26991e6	1.16218	0.581092	−	0.813838i	$-0.302626\pi$
$523$	7.26991e6	1.16218	0.581092	+	0.813838i	$0.302626\pi$
$524$	0	0
$525$	−1.37250e6	−0.217327
$526$	0	0
$527$	9.90686e6	1.55385
$528$	0	0
$529$	−6.04697e6	−0.939504
$530$	0	0
$531$	2.26670e6	0.348866
$532$	0	0
$533$	628500.	0.0958269
$534$	0	0
$535$	732300.	0.110613
$536$	0	0
$537$	1.18606e6	0.177488
$538$	0	0
$539$	−6.15298e6	−0.912249
$540$	0	0
$541$	−1.88876e6	−0.277450	−0.138725	−	0.990331i	$-0.544300\pi$
$541$	−1.88876e6	−0.277450	−0.138725	+	0.990331i	$0.544300\pi$
$542$	0	0
$543$	−2.67957e6	−0.390001
$544$	0	0
$545$	−3.13295e6	−0.451817
$546$	0	0
$547$	−2.64867e6	−0.378494	−0.189247	−	0.981929i	$-0.560605\pi$
$547$	−2.64867e6	−0.378494	−0.189247	+	0.981929i	$0.560605\pi$
$548$	0	0
$549$	1.78378e6	0.252587
$550$	0	0
$551$	−437640.	−0.0614098
$552$	0	0
$553$	−2.69571e6	−0.374853
$554$	0	0
$555$	−3.53205e6	−0.486737
$556$	0	0
$557$	−2.82704e6	−0.386095	−0.193047	−	0.981189i	$-0.561837\pi$
$557$	−2.82704e6	−0.386095	−0.193047	+	0.981189i	$0.561837\pi$
$558$	0	0
$559$	575800.	0.0779367
$560$	0	0
$561$	2.48054e6	0.332767
$562$	0	0
$563$	9.22050e6	1.22598	0.612990	−	0.790091i	$-0.289967\pi$
$563$	9.22050e6	1.22598	0.612990	+	0.790091i	$0.289967\pi$
$564$	0	0
$565$	−4.16595e6	−0.549026
$566$	0	0
$567$	−1.60088e6	−0.209123
$568$	0	0
$569$	−2.96705e6	−0.384189	−0.192094	−	0.981376i	$-0.561528\pi$
$569$	−2.96705e6	−0.384189	−0.192094	+	0.981376i	$0.561528\pi$
$570$	0	0
$571$	1.56746e6	0.201190	0.100595	−	0.994927i	$-0.467925\pi$
$571$	1.56746e6	0.201190	0.100595	+	0.994927i	$0.467925\pi$
$572$	0	0
$573$	7.66606e6	0.975406
$574$	0	0
$575$	−390000.	−0.0491921
$576$	0	0
$577$	−1.07923e7	−1.34951	−0.674754	−	0.738043i	$-0.735750\pi$
$577$	−1.07923e7	−1.34951	−0.674754	+	0.738043i	$0.735750\pi$
$578$	0	0
$579$	2.63155e6	0.326223
$580$	0	0
$581$	2.81293e7	3.45715
$582$	0	0
$583$	2.75875e6	0.336156
$584$	0	0
$585$	−101250.	−0.0122322
$586$	0	0
$587$	−1.34368e7	−1.60954	−0.804770	−	0.593586i	$-0.797712\pi$
$587$	−1.34368e7	−1.60954	−0.804770	+	0.593586i	$0.797712\pi$
$588$	0	0
$589$	−724640.	−0.0860665
$590$	0	0
$591$	−7.82573e6	−0.921630
$592$	0	0
$593$	−3.53507e6	−0.412821	−0.206410	−	0.978465i	$-0.566178\pi$
$593$	−3.53507e6	−0.412821	−0.206410	+	0.978465i	$0.566178\pi$
$594$	0	0
$595$	−1.16754e7	−1.35201
$596$	0	0
$597$	−1.29593e6	−0.148814
$598$	0	0
$599$	−8.56850e6	−0.975749	−0.487874	−	0.872914i	$-0.662228\pi$
$599$	−8.56850e6	−0.975749	−0.487874	+	0.872914i	$0.662228\pi$
$600$	0	0
$601$	−1.15733e7	−1.30699	−0.653493	−	0.756932i	$-0.726697\pi$
$601$	−1.15733e7	−1.30699	−0.653493	+	0.756932i	$0.726697\pi$
$602$	0	0
$603$	−1.02676e6	−0.114994
$604$	0	0
$605$	3.50787e6	0.389633
$606$	0	0
$607$	3.76651e6	0.414923	0.207461	−	0.978243i	$-0.433480\pi$
$607$	3.76651e6	0.414923	0.207461	+	0.978243i	$0.433480\pi$
$608$	0	0
$609$	6.86470e6	0.750029
$610$	0	0
$611$	−1.33680e6	−0.144865
$612$	0	0
$613$	5.55106e6	0.596657	0.298328	−	0.954463i	$-0.403571\pi$
$613$	5.55106e6	0.596657	0.298328	+	0.954463i	$0.403571\pi$
$614$	0	0
$615$	−2.82825e6	−0.301530
$616$	0	0
$617$	8.66393e6	0.916225	0.458113	−	0.888894i	$-0.348526\pi$
$617$	8.66393e6	0.916225	0.458113	+	0.888894i	$0.348526\pi$
$618$	0	0
$619$	−6.11780e6	−0.641754	−0.320877	−	0.947121i	$-0.603978\pi$
$619$	−6.11780e6	−0.641754	−0.320877	+	0.947121i	$0.603978\pi$
$620$	0	0
$621$	−454896.	−0.0473351
$622$	0	0
$623$	−1.79706e7	−1.85499
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−181440.	−0.0184316
$628$	0	0
$629$	−3.00460e7	−3.02803
$630$	0	0
$631$	4.11315e6	0.411246	0.205623	−	0.978631i	$-0.434078\pi$
$631$	4.11315e6	0.411246	0.205623	+	0.978631i	$0.434078\pi$
$632$	0	0
$633$	7.52778e6	0.746720
$634$	0	0
$635$	5.04310e6	0.496322
$636$	0	0
$637$	2.13645e6	0.208614
$638$	0	0
$639$	−4.82112e6	−0.467085
$640$	0	0
$641$	64674.0	0.00621705	0.00310853	−	0.999995i	$-0.499011\pi$
$641$	64674.0	0.00621705	0.00310853	+	0.999995i	$0.499011\pi$
$642$	0	0
$643$	−2.42852e6	−0.231640	−0.115820	−	0.993270i	$-0.536950\pi$
$643$	−2.42852e6	−0.231640	−0.115820	+	0.993270i	$0.536950\pi$
$644$	0	0
$645$	−2.59110e6	−0.245236
$646$	0	0
$647$	7.43777e6	0.698525	0.349262	−	0.937025i	$-0.386432\pi$
$647$	7.43777e6	0.698525	0.349262	+	0.937025i	$0.386432\pi$
$648$	0	0
$649$	−4.02970e6	−0.375544
$650$	0	0
$651$	1.13665e7	1.05117
$652$	0	0
$653$	1.99762e7	1.83329	0.916644	−	0.399706i	$-0.130888\pi$
$653$	1.99762e7	1.83329	0.916644	+	0.399706i	$0.130888\pi$
$654$	0	0
$655$	−3.28680e6	−0.299344
$656$	0	0
$657$	−5.43526e6	−0.491255
$658$	0	0
$659$	−742752.	−0.0666239	−0.0333120	−	0.999445i	$-0.510605\pi$
$659$	−742752.	−0.0666239	−0.0333120	+	0.999445i	$0.510605\pi$
$660$	0	0
$661$	−1.20838e7	−1.07573	−0.537863	−	0.843032i	$-0.680768\pi$
$661$	−1.20838e7	−1.07573	−0.537863	+	0.843032i	$0.680768\pi$
$662$	0	0
$663$	−861300.	−0.0760975
$664$	0	0
$665$	854000.	0.0748865
$666$	0	0
$667$	1.95062e6	0.169769
$668$	0	0
$669$	1.09881e7	0.949199
$670$	0	0
$671$	−3.17117e6	−0.271902
$672$	0	0
$673$	−7.14487e6	−0.608074	−0.304037	−	0.952660i	$-0.598335\pi$
$673$	−7.14487e6	−0.608074	−0.304037	+	0.952660i	$0.598335\pi$
$674$	0	0
$675$	455625.	0.0384900
$676$	0	0
$677$	2.03255e7	1.70440	0.852198	−	0.523219i	$-0.175269\pi$
$677$	2.03255e7	1.70440	0.852198	+	0.523219i	$0.175269\pi$
$678$	0	0
$679$	−8.66737e6	−0.721461
$680$	0	0
$681$	−1.56222e6	−0.129085
$682$	0	0
$683$	−6.65416e6	−0.545810	−0.272905	−	0.962041i	$-0.587984\pi$
$683$	−6.65416e6	−0.545810	−0.272905	+	0.962041i	$0.587984\pi$
$684$	0	0
$685$	2.55105e6	0.207727
$686$	0	0
$687$	1.08725e6	0.0878899
$688$	0	0
$689$	−957900.	−0.0768727
$690$	0	0
$691$	−1.11430e7	−0.887784	−0.443892	−	0.896080i	$-0.646403\pi$
$691$	−1.11430e7	−0.887784	−0.443892	+	0.896080i	$0.646403\pi$
$692$	0	0
$693$	2.84602e6	0.225115
$694$	0	0
$695$	6.09010e6	0.478258
$696$	0	0
$697$	−2.40590e7	−1.87584
$698$	0	0
$699$	5.33331e6	0.412861
$700$	0	0
$701$	1.47407e7	1.13299	0.566493	−	0.824067i	$-0.308300\pi$
$701$	1.47407e7	1.13299	0.566493	+	0.824067i	$0.308300\pi$
$702$	0	0
$703$	2.19772e6	0.167720
$704$	0	0
$705$	6.01560e6	0.455833
$706$	0	0
$707$	2.73431e7	2.05731
$708$	0	0
$709$	2.01119e7	1.50258	0.751290	−	0.659973i	$-0.229432\pi$
$709$	2.01119e7	1.50258	0.751290	+	0.659973i	$0.229432\pi$
$710$	0	0
$711$	894888.	0.0663888
$712$	0	0
$713$	3.22982e6	0.237933
$714$	0	0
$715$	180000.	0.0131676
$716$	0	0
$717$	6.35407e6	0.461588
$718$	0	0
$719$	9.17933e6	0.662199	0.331100	−	0.943596i	$-0.392580\pi$
$719$	9.17933e6	0.662199	0.331100	+	0.943596i	$0.392580\pi$
$720$	0	0
$721$	2.59899e7	1.86194
$722$	0	0
$723$	1.40538e7	0.999883
$724$	0	0
$725$	−1.95375e6	−0.138046
$726$	0	0
$727$	−1.72524e7	−1.21064	−0.605319	−	0.795983i	$-0.706954\pi$
$727$	−1.72524e7	−1.21064	−0.605319	+	0.795983i	$0.706954\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−2.20416e7	−1.52563
$732$	0	0
$733$	2.79936e7	1.92442	0.962209	−	0.272314i	$-0.0877888\pi$
$733$	2.79936e7	1.92442	0.962209	+	0.272314i	$0.0877888\pi$
$734$	0	0
$735$	−9.61402e6	−0.656428
$736$	0	0
$737$	1.82534e6	0.123787
$738$	0	0
$739$	−6.96778e6	−0.469335	−0.234668	−	0.972076i	$-0.575400\pi$
$739$	−6.96778e6	−0.469335	−0.234668	+	0.972076i	$0.575400\pi$
$740$	0	0
$741$	63000.0	0.00421498
$742$	0	0
$743$	1.35720e7	0.901929	0.450965	−	0.892542i	$-0.351080\pi$
$743$	1.35720e7	0.901929	0.450965	+	0.892542i	$0.351080\pi$
$744$	0	0
$745$	7.73655e6	0.510689
$746$	0	0
$747$	−9.33800e6	−0.612283
$748$	0	0
$749$	7.14725e6	0.465516
$750$	0	0
$751$	8.17647e6	0.529013	0.264506	−	0.964384i	$-0.414791\pi$
$751$	8.17647e6	0.529013	0.264506	+	0.964384i	$0.414791\pi$
$752$	0	0
$753$	−1.33624e7	−0.858810
$754$	0	0
$755$	3.67900e6	0.234889
$756$	0	0
$757$	−8.60741e6	−0.545925	−0.272962	−	0.962025i	$-0.588003\pi$
$757$	−8.60741e6	−0.545925	−0.272962	+	0.962025i	$0.588003\pi$
$758$	0	0
$759$	808704.	0.0509548
$760$	0	0
$761$	−2.28596e7	−1.43089	−0.715446	−	0.698668i	$-0.753777\pi$
$761$	−2.28596e7	−1.43089	−0.715446	+	0.698668i	$0.753777\pi$
$762$	0	0
$763$	−3.05776e7	−1.90148
$764$	0	0
$765$	3.87585e6	0.239449
$766$	0	0
$767$	1.39920e6	0.0858799
$768$	0	0
$769$	1.54790e7	0.943905	0.471952	−	0.881624i	$-0.343549\pi$
$769$	1.54790e7	0.943905	0.471952	+	0.881624i	$0.343549\pi$
$770$	0	0
$771$	5.62102e6	0.340549
$772$	0	0
$773$	−2.99671e7	−1.80383	−0.901915	−	0.431913i	$-0.857839\pi$
$773$	−2.99671e7	−1.80383	−0.901915	+	0.431913i	$0.857839\pi$
$774$	0	0
$775$	−3.23500e6	−0.193473
$776$	0	0
$777$	−3.44728e7	−2.04844
$778$	0	0
$779$	1.75980e6	0.103901
$780$	0	0
$781$	8.57088e6	0.502803
$782$	0	0
$783$	−2.27885e6	−0.132835
$784$	0	0
$785$	−6.70205e6	−0.388180
$786$	0	0
$787$	−1.06695e6	−0.0614054	−0.0307027	−	0.999529i	$-0.509775\pi$
$787$	−1.06695e6	−0.0614054	−0.0307027	+	0.999529i	$0.509775\pi$
$788$	0	0
$789$	7.01006e6	0.400894
$790$	0	0
$791$	−4.06597e7	−2.31059
$792$	0	0
$793$	1.10110e6	0.0621790
$794$	0	0
$795$	4.31055e6	0.241888
$796$	0	0
$797$	−5.28008e6	−0.294438	−0.147219	−	0.989104i	$-0.547032\pi$
$797$	−5.28008e6	−0.294438	−0.147219	+	0.989104i	$0.547032\pi$
$798$	0	0
$799$	5.11727e7	2.83577
$800$	0	0
$801$	5.96565e6	0.328531
$802$	0	0
$803$	9.66269e6	0.528821
$804$	0	0
$805$	−3.80640e6	−0.207026
$806$	0	0
$807$	−104166.	−0.00563044
$808$	0	0
$809$	5.68555e6	0.305422	0.152711	−	0.988271i	$-0.451200\pi$
$809$	5.68555e6	0.305422	0.152711	+	0.988271i	$0.451200\pi$
$810$	0	0
$811$	−2.71746e7	−1.45081	−0.725405	−	0.688322i	$-0.758348\pi$
$811$	−2.71746e7	−1.45081	−0.725405	+	0.688322i	$0.758348\pi$
$812$	0	0
$813$	1.06258e6	0.0563811
$814$	0	0
$815$	6.91450e6	0.364642
$816$	0	0
$817$	1.61224e6	0.0845035
$818$	0	0
$819$	−988200.	−0.0514796
$820$	0	0
$821$	3.30951e7	1.71359	0.856793	−	0.515661i	$-0.172453\pi$
$821$	3.30951e7	1.71359	0.856793	+	0.515661i	$0.172453\pi$
$822$	0	0
$823$	6.90631e6	0.355424	0.177712	−	0.984083i	$-0.443131\pi$
$823$	6.90631e6	0.355424	0.177712	+	0.984083i	$0.443131\pi$
$824$	0	0
$825$	−810000.	−0.0414334
$826$	0	0
$827$	2.37490e7	1.20748	0.603742	−	0.797180i	$-0.293676\pi$
$827$	2.37490e7	1.20748	0.603742	+	0.797180i	$0.293676\pi$
$828$	0	0
$829$	−3.22011e7	−1.62736	−0.813681	−	0.581311i	$-0.802540\pi$
$829$	−3.22011e7	−1.62736	−0.813681	+	0.581311i	$0.802540\pi$
$830$	0	0
$831$	5.93354e6	0.298065
$832$	0	0
$833$	−8.17833e7	−4.08368
$834$	0	0
$835$	1.48068e7	0.734928
$836$	0	0
$837$	−3.77330e6	−0.186169
$838$	0	0
$839$	−1.62821e7	−0.798554	−0.399277	−	0.916830i	$-0.630739\pi$
$839$	−1.62821e7	−0.798554	−0.399277	+	0.916830i	$0.630739\pi$
$840$	0	0
$841$	−1.07393e7	−0.523582
$842$	0	0
$843$	−1.34152e7	−0.650170
$844$	0	0
$845$	9.21982e6	0.444202
$846$	0	0
$847$	3.42369e7	1.63978
$848$	0	0
$849$	−1.01816e7	−0.484784
$850$	0	0
$851$	−9.79555e6	−0.463666
$852$	0	0
$853$	−4.06728e7	−1.91395	−0.956977	−	0.290164i	$-0.906290\pi$
$853$	−4.06728e7	−1.91395	−0.956977	+	0.290164i	$0.906290\pi$
$854$	0	0
$855$	−283500.	−0.0132629
$856$	0	0
$857$	−1.99154e6	−0.0926268	−0.0463134	−	0.998927i	$-0.514747\pi$
$857$	−1.99154e6	−0.0926268	−0.0463134	+	0.998927i	$0.514747\pi$
$858$	0	0
$859$	−3.62242e6	−0.167500	−0.0837502	−	0.996487i	$-0.526690\pi$
$859$	−3.62242e6	−0.167500	−0.0837502	+	0.996487i	$0.526690\pi$
$860$	0	0
$861$	−2.76037e7	−1.26899
$862$	0	0
$863$	2.38014e7	1.08786	0.543932	−	0.839129i	$-0.316935\pi$
$863$	2.38014e7	1.08786	0.543932	+	0.839129i	$0.316935\pi$
$864$	0	0
$865$	−728850.	−0.0331206
$866$	0	0
$867$	2.01919e7	0.912280
$868$	0	0
$869$	−1.59091e6	−0.0714655
$870$	0	0
$871$	−633800.	−0.0283078
$872$	0	0
$873$	2.87728e6	0.127775
$874$	0	0
$875$	3.81250e6	0.168341
$876$	0	0
$877$	1.08955e7	0.478352	0.239176	−	0.970976i	$-0.423123\pi$
$877$	1.08955e7	0.478352	0.239176	+	0.970976i	$0.423123\pi$
$878$	0	0
$879$	8.06668e6	0.352146
$880$	0	0
$881$	1.79389e7	0.778674	0.389337	−	0.921095i	$-0.372704\pi$
$881$	1.79389e7	0.778674	0.389337	+	0.921095i	$0.372704\pi$
$882$	0	0
$883$	7.70601e6	0.332604	0.166302	−	0.986075i	$-0.446817\pi$
$883$	7.70601e6	0.332604	0.166302	+	0.986075i	$0.446817\pi$
$884$	0	0
$885$	−6.29640e6	−0.270230
$886$	0	0
$887$	−3.56776e7	−1.52260	−0.761301	−	0.648399i	$-0.775439\pi$
$887$	−3.56776e7	−1.52260	−0.761301	+	0.648399i	$0.775439\pi$
$888$	0	0
$889$	4.92207e7	2.08878
$890$	0	0
$891$	−944784.	−0.0398693
$892$	0	0
$893$	−3.74304e6	−0.157071
$894$	0	0
$895$	−3.29460e6	−0.137482
$896$	0	0
$897$	−280800.	−0.0116524
$898$	0	0
$899$	1.61802e7	0.667704
$900$	0	0
$901$	3.66684e7	1.50480
$902$	0	0
$903$	−2.52891e7	−1.03208
$904$	0	0
$905$	7.44325e6	0.302093
$906$	0	0
$907$	1.90905e7	0.770547	0.385274	−	0.922802i	$-0.374107\pi$
$907$	1.90905e7	0.770547	0.385274	+	0.922802i	$0.374107\pi$
$908$	0	0
$909$	−9.07702e6	−0.364363
$910$	0	0
$911$	7.40453e6	0.295598	0.147799	−	0.989017i	$-0.452781\pi$
$911$	7.40453e6	0.295598	0.147799	+	0.989017i	$0.452781\pi$
$912$	0	0
$913$	1.66009e7	0.659105
$914$	0	0
$915$	−4.95495e6	−0.195653
$916$	0	0
$917$	−3.20792e7	−1.25979
$918$	0	0
$919$	2.47415e7	0.966356	0.483178	−	0.875522i	$-0.339482\pi$
$919$	2.47415e7	0.966356	0.483178	+	0.875522i	$0.339482\pi$
$920$	0	0
$921$	−1.24777e7	−0.484714
$922$	0	0
$923$	−2.97600e6	−0.114982
$924$	0	0
$925$	9.81125e6	0.377025
$926$	0	0
$927$	−8.62780e6	−0.329762
$928$	0	0
$929$	8.85830e6	0.336753	0.168376	−	0.985723i	$-0.446148\pi$
$929$	8.85830e6	0.336753	0.168376	+	0.985723i	$0.446148\pi$
$930$	0	0
$931$	5.98206e6	0.226192
$932$	0	0
$933$	2.87315e7	1.08057
$934$	0	0
$935$	−6.89040e6	−0.257760
$936$	0	0
$937$	−3.06569e7	−1.14072	−0.570360	−	0.821395i	$-0.693196\pi$
$937$	−3.06569e7	−1.14072	−0.570360	+	0.821395i	$0.693196\pi$
$938$	0	0
$939$	2.26515e7	0.838365
$940$	0	0
$941$	1.29302e7	0.476027	0.238013	−	0.971262i	$-0.423504\pi$
$941$	1.29302e7	0.476027	0.238013	+	0.971262i	$0.423504\pi$
$942$	0	0
$943$	−7.84368e6	−0.287237
$944$	0	0
$945$	4.44690e6	0.161986
$946$	0	0
$947$	2.05701e7	0.745354	0.372677	−	0.927961i	$-0.378440\pi$
$947$	2.05701e7	0.745354	0.372677	+	0.927961i	$0.378440\pi$
$948$	0	0
$949$	−3.35510e6	−0.120932
$950$	0	0
$951$	−1.99708e7	−0.716050
$952$	0	0
$953$	2.00489e7	0.715087	0.357544	−	0.933896i	$-0.383614\pi$
$953$	2.00489e7	0.715087	0.357544	+	0.933896i	$0.383614\pi$
$954$	0	0
$955$	−2.12946e7	−0.755546
$956$	0	0
$957$	4.05130e6	0.142993
$958$	0	0
$959$	2.48982e7	0.874223
$960$	0	0
$961$	−1.83818e6	−0.0642064
$962$	0	0
$963$	−2.37265e6	−0.0824458
$964$	0	0
$965$	−7.30985e6	−0.252691
$966$	0	0
$967$	−4.37266e6	−0.150376	−0.0751882	−	0.997169i	$-0.523956\pi$
$967$	−4.37266e6	−0.150376	−0.0751882	+	0.997169i	$0.523956\pi$
$968$	0	0
$969$	−2.41164e6	−0.0825093
$970$	0	0
$971$	2.61184e7	0.888994	0.444497	−	0.895780i	$-0.353383\pi$
$971$	2.61184e7	0.888994	0.444497	+	0.895780i	$0.353383\pi$
$972$	0	0
$973$	5.94394e7	2.01276
$974$	0	0
$975$	281250.	0.00947504
$976$	0	0
$977$	−2.61443e7	−0.876275	−0.438138	−	0.898908i	$-0.644362\pi$
$977$	−2.61443e7	−0.876275	−0.438138	+	0.898908i	$0.644362\pi$
$978$	0	0
$979$	−1.06056e7	−0.353654
$980$	0	0
$981$	1.01508e7	0.336764
$982$	0	0
$983$	1.52466e7	0.503258	0.251629	−	0.967824i	$-0.419034\pi$
$983$	1.52466e7	0.503258	0.251629	+	0.967824i	$0.419034\pi$
$984$	0	0
$985$	2.17382e7	0.713891
$986$	0	0
$987$	5.87123e7	1.91839
$988$	0	0
$989$	−7.18598e6	−0.233612
$990$	0	0
$991$	−2.90665e7	−0.940175	−0.470088	−	0.882620i	$-0.655778\pi$
$991$	−2.90665e7	−0.940175	−0.470088	+	0.882620i	$0.655778\pi$
$992$	0	0
$993$	−244332.	−0.00786334
$994$	0	0
$995$	3.59980e6	0.115271
$996$	0	0
$997$	2.92663e6	0.0932461	0.0466230	−	0.998913i	$-0.485154\pi$
$997$	2.92663e6	0.0932461	0.0466230	+	0.998913i	$0.485154\pi$
$998$	0	0
$999$	1.14438e7	0.362792

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.6.a.c.1.1	✓	1
3.2	odd	2		180.6.a.c.1.1		1
4.3	odd	2		240.6.a.d.1.1		1
5.2	odd	4		300.6.d.c.49.1		2
5.3	odd	4		300.6.d.c.49.2		2
5.4	even	2		300.6.a.c.1.1		1
8.3	odd	2		960.6.a.bb.1.1		1
8.5	even	2		960.6.a.g.1.1		1
12.11	even	2		720.6.a.x.1.1		1
15.2	even	4		900.6.d.f.649.1		2
15.8	even	4		900.6.d.f.649.2		2
15.14	odd	2		900.6.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.6.a.c.1.1	✓	1	1.1	even	1	trivial
180.6.a.c.1.1		1	3.2	odd	2
240.6.a.d.1.1		1	4.3	odd	2
300.6.a.c.1.1		1	5.4	even	2
300.6.d.c.49.1		2	5.2	odd	4
300.6.d.c.49.2		2	5.3	odd	4
720.6.a.x.1.1		1	12.11	even	2
900.6.a.k.1.1		1	15.14	odd	2
900.6.d.f.649.1		2	15.2	even	4
900.6.d.f.649.2		2	15.8	even	4
960.6.a.g.1.1		1	8.5	even	2
960.6.a.bb.1.1		1	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	60.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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