LMFDB - Embedded newform 528.6.a.i.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [528,6,Mod(1,528)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("528.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(528, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 528 = 2^{4} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	528.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,9,0,46,0,-148] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$84.6826568613$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	528.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+9.00000 q^{3} +46.0000 q^{5} -148.000 q^{7} +81.0000 q^{9} -121.000 q^{11} +574.000 q^{13} +414.000 q^{15} -722.000 q^{17} -2160.00 q^{19} -1332.00 q^{21} +2536.00 q^{23} -1009.00 q^{25} +729.000 q^{27} +4650.00 q^{29} -5032.00 q^{31} -1089.00 q^{33} -6808.00 q^{35} +8118.00 q^{37} +5166.00 q^{39} -5138.00 q^{41} -8304.00 q^{43} +3726.00 q^{45} -24728.0 q^{47} +5097.00 q^{49} -6498.00 q^{51} -28746.0 q^{53} -5566.00 q^{55} -19440.0 q^{57} +5860.00 q^{59} -53658.0 q^{61} -11988.0 q^{63} +26404.0 q^{65} -30908.0 q^{67} +22824.0 q^{69} +69648.0 q^{71} -18446.0 q^{73} -9081.00 q^{75} +17908.0 q^{77} +25300.0 q^{79} +6561.00 q^{81} +17556.0 q^{83} -33212.0 q^{85} +41850.0 q^{87} +132570. q^{89} -84952.0 q^{91} -45288.0 q^{93} -99360.0 q^{95} +70658.0 q^{97} -9801.00 q^{99} +O(q^{100})$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	0	0
$5$	46.0000	0.822873	0.411437	−	0.911438i	$-0.365027\pi$
$5$	46.0000	0.822873	0.411437	+	0.911438i	$0.365027\pi$
$6$	0	0
$7$	−148.000	−1.14161	−0.570803	−	0.821087i	$-0.693368\pi$
$7$	−148.000	−1.14161	−0.570803	+	0.821087i	$0.693368\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	574.000	0.942006	0.471003	−	0.882132i	$-0.343892\pi$
$13$	574.000	0.942006	0.471003	+	0.882132i	$0.343892\pi$
$14$	0	0
$15$	414.000	0.475086
$16$	0	0
$17$	−722.000	−0.605919	−0.302960	−	0.953003i	$-0.597975\pi$
$17$	−722.000	−0.605919	−0.302960	+	0.953003i	$0.597975\pi$
$18$	0	0
$19$	−2160.00	−1.37268	−0.686341	−	0.727280i	$-0.740784\pi$
$19$	−2160.00	−1.37268	−0.686341	+	0.727280i	$0.740784\pi$
$20$	0	0
$21$	−1332.00	−0.659107
$22$	0	0
$23$	2536.00	0.999608	0.499804	−	0.866139i	$-0.333405\pi$
$23$	2536.00	0.999608	0.499804	+	0.866139i	$0.333405\pi$
$24$	0	0
$25$	−1009.00	−0.322880
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	4650.00	1.02673	0.513367	−	0.858169i	$-0.328398\pi$
$29$	4650.00	1.02673	0.513367	+	0.858169i	$0.328398\pi$
$30$	0	0
$31$	−5032.00	−0.940451	−0.470226	−	0.882546i	$-0.655828\pi$
$31$	−5032.00	−0.940451	−0.470226	+	0.882546i	$0.655828\pi$
$32$	0	0
$33$	−1089.00	−0.174078
$34$	0	0
$35$	−6808.00	−0.939398
$36$	0	0
$37$	8118.00	0.974866	0.487433	−	0.873161i	$-0.337933\pi$
$37$	8118.00	0.974866	0.487433	+	0.873161i	$0.337933\pi$
$38$	0	0
$39$	5166.00	0.543867
$40$	0	0
$41$	−5138.00	−0.477347	−0.238674	−	0.971100i	$-0.576713\pi$
$41$	−5138.00	−0.477347	−0.238674	+	0.971100i	$0.576713\pi$
$42$	0	0
$43$	−8304.00	−0.684883	−0.342441	−	0.939539i	$-0.611254\pi$
$43$	−8304.00	−0.684883	−0.342441	+	0.939539i	$0.611254\pi$
$44$	0	0
$45$	3726.00	0.274291
$46$	0	0
$47$	−24728.0	−1.63284	−0.816421	−	0.577457i	$-0.804045\pi$
$47$	−24728.0	−1.63284	−0.816421	+	0.577457i	$0.804045\pi$
$48$	0	0
$49$	5097.00	0.303266
$50$	0	0
$51$	−6498.00	−0.349828
$52$	0	0
$53$	−28746.0	−1.40568	−0.702842	−	0.711346i	$-0.748086\pi$
$53$	−28746.0	−1.40568	−0.702842	+	0.711346i	$0.748086\pi$
$54$	0	0
$55$	−5566.00	−0.248106
$56$	0	0
$57$	−19440.0	−0.792518
$58$	0	0
$59$	5860.00	0.219163	0.109582	−	0.993978i	$-0.465049\pi$
$59$	5860.00	0.219163	0.109582	+	0.993978i	$0.465049\pi$
$60$	0	0
$61$	−53658.0	−1.84633	−0.923166	−	0.384401i	$-0.874408\pi$
$61$	−53658.0	−1.84633	−0.923166	+	0.384401i	$0.874408\pi$
$62$	0	0
$63$	−11988.0	−0.380536
$64$	0	0
$65$	26404.0	0.775151
$66$	0	0
$67$	−30908.0	−0.841170	−0.420585	−	0.907253i	$-0.638175\pi$
$67$	−30908.0	−0.841170	−0.420585	+	0.907253i	$0.638175\pi$
$68$	0	0
$69$	22824.0	0.577124
$70$	0	0
$71$	69648.0	1.63969	0.819847	−	0.572583i	$-0.194058\pi$
$71$	69648.0	1.63969	0.819847	+	0.572583i	$0.194058\pi$
$72$	0	0
$73$	−18446.0	−0.405131	−0.202565	−	0.979269i	$-0.564928\pi$
$73$	−18446.0	−0.405131	−0.202565	+	0.979269i	$0.564928\pi$
$74$	0	0
$75$	−9081.00	−0.186415
$76$	0	0
$77$	17908.0	0.344207
$78$	0	0
$79$	25300.0	0.456092	0.228046	−	0.973650i	$-0.426766\pi$
$79$	25300.0	0.456092	0.228046	+	0.973650i	$0.426766\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	17556.0	0.279724	0.139862	−	0.990171i	$-0.455334\pi$
$83$	17556.0	0.279724	0.139862	+	0.990171i	$0.455334\pi$
$84$	0	0
$85$	−33212.0	−0.498595
$86$	0	0
$87$	41850.0	0.592785
$88$	0	0
$89$	132570.	1.77407	0.887034	−	0.461704i	$-0.152762\pi$
$89$	132570.	1.77407	0.887034	+	0.461704i	$0.152762\pi$
$90$	0	0
$91$	−84952.0	−1.07540
$92$	0	0
$93$	−45288.0	−0.542970
$94$	0	0
$95$	−99360.0	−1.12954
$96$	0	0
$97$	70658.0	0.762486	0.381243	−	0.924475i	$-0.375496\pi$
$97$	70658.0	0.762486	0.381243	+	0.924475i	$0.375496\pi$
$98$	0	0
$99$	−9801.00	−0.100504
$100$	0	0
$101$	−101998.	−0.994920	−0.497460	−	0.867487i	$-0.665734\pi$
$101$	−101998.	−0.994920	−0.497460	+	0.867487i	$0.665734\pi$
$102$	0	0
$103$	−130904.	−1.21579	−0.607897	−	0.794016i	$-0.707987\pi$
$103$	−130904.	−1.21579	−0.607897	+	0.794016i	$0.707987\pi$
$104$	0	0
$105$	−61272.0	−0.542361
$106$	0	0
$107$	141612.	1.19575	0.597875	−	0.801589i	$-0.296012\pi$
$107$	141612.	1.19575	0.597875	+	0.801589i	$0.296012\pi$
$108$	0	0
$109$	−239810.	−1.93331	−0.966654	−	0.256086i	$-0.917567\pi$
$109$	−239810.	−1.93331	−0.966654	+	0.256086i	$0.917567\pi$
$110$	0	0
$111$	73062.0	0.562839
$112$	0	0
$113$	−42726.0	−0.314772	−0.157386	−	0.987537i	$-0.550307\pi$
$113$	−42726.0	−0.314772	−0.157386	+	0.987537i	$0.550307\pi$
$114$	0	0
$115$	116656.	0.822550
$116$	0	0
$117$	46494.0	0.314002
$118$	0	0
$119$	106856.	0.691722
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	−46242.0	−0.275597
$124$	0	0
$125$	−190164.	−1.08856
$126$	0	0
$127$	−51788.0	−0.284918	−0.142459	−	0.989801i	$-0.545501\pi$
$127$	−51788.0	−0.284918	−0.142459	+	0.989801i	$0.545501\pi$
$128$	0	0
$129$	−74736.0	−0.395417
$130$	0	0
$131$	−53652.0	−0.273154	−0.136577	−	0.990629i	$-0.543610\pi$
$131$	−53652.0	−0.273154	−0.136577	+	0.990629i	$0.543610\pi$
$132$	0	0
$133$	319680.	1.56706
$134$	0	0
$135$	33534.0	0.158362
$136$	0	0
$137$	−228862.	−1.04177	−0.520886	−	0.853627i	$-0.674398\pi$
$137$	−228862.	−1.04177	−0.520886	+	0.853627i	$0.674398\pi$
$138$	0	0
$139$	−374920.	−1.64589	−0.822947	−	0.568119i	$-0.807671\pi$
$139$	−374920.	−1.64589	−0.822947	+	0.568119i	$0.807671\pi$
$140$	0	0
$141$	−222552.	−0.942722
$142$	0	0
$143$	−69454.0	−0.284025
$144$	0	0
$145$	213900.	0.844872
$146$	0	0
$147$	45873.0	0.175091
$148$	0	0
$149$	−65830.0	−0.242917	−0.121459	−	0.992597i	$-0.538757\pi$
$149$	−65830.0	−0.242917	−0.121459	+	0.992597i	$0.538757\pi$
$150$	0	0
$151$	−154052.	−0.549826	−0.274913	−	0.961469i	$-0.588649\pi$
$151$	−154052.	−0.549826	−0.274913	+	0.961469i	$0.588649\pi$
$152$	0	0
$153$	−58482.0	−0.201973
$154$	0	0
$155$	−231472.	−0.773872
$156$	0	0
$157$	287678.	0.931446	0.465723	−	0.884931i	$-0.345794\pi$
$157$	287678.	0.931446	0.465723	+	0.884931i	$0.345794\pi$
$158$	0	0
$159$	−258714.	−0.811572
$160$	0	0
$161$	−375328.	−1.14116
$162$	0	0
$163$	−105124.	−0.309908	−0.154954	−	0.987922i	$-0.549523\pi$
$163$	−105124.	−0.309908	−0.154954	+	0.987922i	$0.549523\pi$
$164$	0	0
$165$	−50094.0	−0.143244
$166$	0	0
$167$	−150528.	−0.417663	−0.208832	−	0.977952i	$-0.566966\pi$
$167$	−150528.	−0.417663	−0.208832	+	0.977952i	$0.566966\pi$
$168$	0	0
$169$	−41817.0	−0.112625
$170$	0	0
$171$	−174960.	−0.457560
$172$	0	0
$173$	−2166.00	−0.00550229	−0.00275114	−	0.999996i	$-0.500876\pi$
$173$	−2166.00	−0.00550229	−0.00275114	+	0.999996i	$0.500876\pi$
$174$	0	0
$175$	149332.	0.368602
$176$	0	0
$177$	52740.0	0.126534
$178$	0	0
$179$	−672780.	−1.56942	−0.784712	−	0.619860i	$-0.787189\pi$
$179$	−672780.	−1.56942	−0.784712	+	0.619860i	$0.787189\pi$
$180$	0	0
$181$	−526778.	−1.19517	−0.597587	−	0.801804i	$-0.703874\pi$
$181$	−526778.	−1.19517	−0.597587	+	0.801804i	$0.703874\pi$
$182$	0	0
$183$	−482922.	−1.06598
$184$	0	0
$185$	373428.	0.802191
$186$	0	0
$187$	87362.0	0.182692
$188$	0	0
$189$	−107892.	−0.219702
$190$	0	0
$191$	305608.	0.606152	0.303076	−	0.952966i	$-0.401986\pi$
$191$	305608.	0.606152	0.303076	+	0.952966i	$0.401986\pi$
$192$	0	0
$193$	116434.	0.225002	0.112501	−	0.993652i	$-0.464114\pi$
$193$	116434.	0.225002	0.112501	+	0.993652i	$0.464114\pi$
$194$	0	0
$195$	237636.	0.447534
$196$	0	0
$197$	−247742.	−0.454814	−0.227407	−	0.973800i	$-0.573025\pi$
$197$	−247742.	−0.454814	−0.227407	+	0.973800i	$0.573025\pi$
$198$	0	0
$199$	513360.	0.918945	0.459472	−	0.888192i	$-0.348039\pi$
$199$	513360.	0.918945	0.459472	+	0.888192i	$0.348039\pi$
$200$	0	0
$201$	−278172.	−0.485650
$202$	0	0
$203$	−688200.	−1.17213
$204$	0	0
$205$	−236348.	−0.392796
$206$	0	0
$207$	205416.	0.333203
$208$	0	0
$209$	261360.	0.413879
$210$	0	0
$211$	620688.	0.959770	0.479885	−	0.877331i	$-0.340678\pi$
$211$	620688.	0.959770	0.479885	+	0.877331i	$0.340678\pi$
$212$	0	0
$213$	626832.	0.946678
$214$	0	0
$215$	−381984.	−0.563571
$216$	0	0
$217$	744736.	1.07363
$218$	0	0
$219$	−166014.	−0.233902
$220$	0	0
$221$	−414428.	−0.570780
$222$	0	0
$223$	1.31802e6	1.77484	0.887419	−	0.460964i	$-0.152496\pi$
$223$	1.31802e6	1.77484	0.887419	+	0.460964i	$0.152496\pi$
$224$	0	0
$225$	−81729.0	−0.107627
$226$	0	0
$227$	887412.	1.14304	0.571519	−	0.820589i	$-0.306354\pi$
$227$	887412.	1.14304	0.571519	+	0.820589i	$0.306354\pi$
$228$	0	0
$229$	−237450.	−0.299215	−0.149608	−	0.988745i	$-0.547801\pi$
$229$	−237450.	−0.299215	−0.149608	+	0.988745i	$0.547801\pi$
$230$	0	0
$231$	161172.	0.198728
$232$	0	0
$233$	−914706.	−1.10380	−0.551902	−	0.833909i	$-0.686098\pi$
$233$	−914706.	−1.10380	−0.551902	+	0.833909i	$0.686098\pi$
$234$	0	0
$235$	−1.13749e6	−1.34362
$236$	0	0
$237$	227700.	0.263325
$238$	0	0
$239$	−1.40892e6	−1.59548	−0.797740	−	0.603001i	$-0.793971\pi$
$239$	−1.40892e6	−1.59548	−0.797740	+	0.603001i	$0.793971\pi$
$240$	0	0
$241$	−826358.	−0.916486	−0.458243	−	0.888827i	$-0.651521\pi$
$241$	−826358.	−0.916486	−0.458243	+	0.888827i	$0.651521\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	234462.	0.249550
$246$	0	0
$247$	−1.23984e6	−1.29307
$248$	0	0
$249$	158004.	0.161499
$250$	0	0
$251$	1.60387e6	1.60688	0.803442	−	0.595384i	$-0.203000\pi$
$251$	1.60387e6	1.60688	0.803442	+	0.595384i	$0.203000\pi$
$252$	0	0
$253$	−306856.	−0.301393
$254$	0	0
$255$	−298908.	−0.287864
$256$	0	0
$257$	397618.	0.375520	0.187760	−	0.982215i	$-0.439877\pi$
$257$	397618.	0.375520	0.187760	+	0.982215i	$0.439877\pi$
$258$	0	0
$259$	−1.20146e6	−1.11291
$260$	0	0
$261$	376650.	0.342245
$262$	0	0
$263$	−2.13166e6	−1.90033	−0.950166	−	0.311745i	$-0.899087\pi$
$263$	−2.13166e6	−1.90033	−0.950166	+	0.311745i	$0.899087\pi$
$264$	0	0
$265$	−1.32232e6	−1.15670
$266$	0	0
$267$	1.19313e6	1.02426
$268$	0	0
$269$	−725810.	−0.611564	−0.305782	−	0.952101i	$-0.598918\pi$
$269$	−725810.	−0.611564	−0.305782	+	0.952101i	$0.598918\pi$
$270$	0	0
$271$	1.46787e6	1.21413	0.607063	−	0.794654i	$-0.292348\pi$
$271$	1.46787e6	1.21413	0.607063	+	0.794654i	$0.292348\pi$
$272$	0	0
$273$	−764568.	−0.620883
$274$	0	0
$275$	122089.	0.0973520
$276$	0	0
$277$	1.52100e6	1.19105	0.595524	−	0.803338i	$-0.296944\pi$
$277$	1.52100e6	1.19105	0.595524	+	0.803338i	$0.296944\pi$
$278$	0	0
$279$	−407592.	−0.313484
$280$	0	0
$281$	464382.	0.350840	0.175420	−	0.984494i	$-0.443872\pi$
$281$	464382.	0.350840	0.175420	+	0.984494i	$0.443872\pi$
$282$	0	0
$283$	415136.	0.308123	0.154062	−	0.988061i	$-0.450765\pi$
$283$	415136.	0.308123	0.154062	+	0.988061i	$0.450765\pi$
$284$	0	0
$285$	−894240.	−0.652142
$286$	0	0
$287$	760424.	0.544943
$288$	0	0
$289$	−898573.	−0.632862
$290$	0	0
$291$	635922.	0.440222
$292$	0	0
$293$	−2.59321e6	−1.76469	−0.882344	−	0.470605i	$-0.844036\pi$
$293$	−2.59321e6	−1.76469	−0.882344	+	0.470605i	$0.844036\pi$
$294$	0	0
$295$	269560.	0.180343
$296$	0	0
$297$	−88209.0	−0.0580259
$298$	0	0
$299$	1.45566e6	0.941636
$300$	0	0
$301$	1.22899e6	0.781867
$302$	0	0
$303$	−917982.	−0.574417
$304$	0	0
$305$	−2.46827e6	−1.51930
$306$	0	0
$307$	930832.	0.563671	0.281835	−	0.959463i	$-0.409057\pi$
$307$	930832.	0.563671	0.281835	+	0.959463i	$0.409057\pi$
$308$	0	0
$309$	−1.17814e6	−0.701939
$310$	0	0
$311$	−2.48527e6	−1.45704	−0.728522	−	0.685022i	$-0.759793\pi$
$311$	−2.48527e6	−1.45704	−0.728522	+	0.685022i	$0.759793\pi$
$312$	0	0
$313$	1.31719e6	0.759957	0.379978	−	0.924995i	$-0.375931\pi$
$313$	1.31719e6	0.759957	0.379978	+	0.924995i	$0.375931\pi$
$314$	0	0
$315$	−551448.	−0.313133
$316$	0	0
$317$	2.25540e6	1.26059	0.630297	−	0.776354i	$-0.282933\pi$
$317$	2.25540e6	1.26059	0.630297	+	0.776354i	$0.282933\pi$
$318$	0	0
$319$	−562650.	−0.309572
$320$	0	0
$321$	1.27451e6	0.690367
$322$	0	0
$323$	1.55952e6	0.831734
$324$	0	0
$325$	−579166.	−0.304155
$326$	0	0
$327$	−2.15829e6	−1.11620
$328$	0	0
$329$	3.65974e6	1.86406
$330$	0	0
$331$	3.17071e6	1.59069	0.795346	−	0.606155i	$-0.207289\pi$
$331$	3.17071e6	1.59069	0.795346	+	0.606155i	$0.207289\pi$
$332$	0	0
$333$	657558.	0.324955
$334$	0	0
$335$	−1.42177e6	−0.692176
$336$	0	0
$337$	1.27630e6	0.612177	0.306089	−	0.952003i	$-0.400980\pi$
$337$	1.27630e6	0.612177	0.306089	+	0.952003i	$0.400980\pi$
$338$	0	0
$339$	−384534.	−0.181734
$340$	0	0
$341$	608872.	0.283557
$342$	0	0
$343$	1.73308e6	0.795396
$344$	0	0
$345$	1.04990e6	0.474900
$346$	0	0
$347$	−3.69303e6	−1.64649	−0.823245	−	0.567687i	$-0.807838\pi$
$347$	−3.69303e6	−1.64649	−0.823245	+	0.567687i	$0.807838\pi$
$348$	0	0
$349$	1.70919e6	0.751150	0.375575	−	0.926792i	$-0.377445\pi$
$349$	1.70919e6	0.751150	0.375575	+	0.926792i	$0.377445\pi$
$350$	0	0
$351$	418446.	0.181289
$352$	0	0
$353$	4.36859e6	1.86597	0.932986	−	0.359914i	$-0.117194\pi$
$353$	4.36859e6	1.86597	0.932986	+	0.359914i	$0.117194\pi$
$354$	0	0
$355$	3.20381e6	1.34926
$356$	0	0
$357$	961704.	0.399366
$358$	0	0
$359$	3.51284e6	1.43854	0.719271	−	0.694730i	$-0.244476\pi$
$359$	3.51284e6	1.43854	0.719271	+	0.694730i	$0.244476\pi$
$360$	0	0
$361$	2.18950e6	0.884254
$362$	0	0
$363$	131769.	0.0524864
$364$	0	0
$365$	−848516.	−0.333371
$366$	0	0
$367$	2.15259e6	0.834251	0.417125	−	0.908849i	$-0.363038\pi$
$367$	2.15259e6	0.834251	0.417125	+	0.908849i	$0.363038\pi$
$368$	0	0
$369$	−416178.	−0.159116
$370$	0	0
$371$	4.25441e6	1.60474
$372$	0	0
$373$	−2.24247e6	−0.834553	−0.417276	−	0.908780i	$-0.637015\pi$
$373$	−2.24247e6	−0.834553	−0.417276	+	0.908780i	$0.637015\pi$
$374$	0	0
$375$	−1.71148e6	−0.628482
$376$	0	0
$377$	2.66910e6	0.967189
$378$	0	0
$379$	2.40986e6	0.861775	0.430887	−	0.902406i	$-0.358201\pi$
$379$	2.40986e6	0.861775	0.430887	+	0.902406i	$0.358201\pi$
$380$	0	0
$381$	−466092.	−0.164497
$382$	0	0
$383$	1.01066e6	0.352052	0.176026	−	0.984386i	$-0.443676\pi$
$383$	1.01066e6	0.352052	0.176026	+	0.984386i	$0.443676\pi$
$384$	0	0
$385$	823768.	0.283239
$386$	0	0
$387$	−672624.	−0.228294
$388$	0	0
$389$	1.27779e6	0.428140	0.214070	−	0.976818i	$-0.431328\pi$
$389$	1.27779e6	0.428140	0.214070	+	0.976818i	$0.431328\pi$
$390$	0	0
$391$	−1.83099e6	−0.605682
$392$	0	0
$393$	−482868.	−0.157706
$394$	0	0
$395$	1.16380e6	0.375306
$396$	0	0
$397$	5.45400e6	1.73676	0.868378	−	0.495903i	$-0.165163\pi$
$397$	5.45400e6	1.73676	0.868378	+	0.495903i	$0.165163\pi$
$398$	0	0
$399$	2.87712e6	0.904744
$400$	0	0
$401$	−1.48980e6	−0.462665	−0.231332	−	0.972875i	$-0.574308\pi$
$401$	−1.48980e6	−0.462665	−0.231332	+	0.972875i	$0.574308\pi$
$402$	0	0
$403$	−2.88837e6	−0.885911
$404$	0	0
$405$	301806.	0.0914303
$406$	0	0
$407$	−982278.	−0.293933
$408$	0	0
$409$	−4.39899e6	−1.30030	−0.650152	−	0.759804i	$-0.725295\pi$
$409$	−4.39899e6	−1.30030	−0.650152	+	0.759804i	$0.725295\pi$
$410$	0	0
$411$	−2.05976e6	−0.601467
$412$	0	0
$413$	−867280.	−0.250198
$414$	0	0
$415$	807576.	0.230178
$416$	0	0
$417$	−3.37428e6	−0.950257
$418$	0	0
$419$	280420.	0.0780322	0.0390161	−	0.999239i	$-0.487578\pi$
$419$	280420.	0.0780322	0.0390161	+	0.999239i	$0.487578\pi$
$420$	0	0
$421$	817462.	0.224782	0.112391	−	0.993664i	$-0.464149\pi$
$421$	817462.	0.224782	0.112391	+	0.993664i	$0.464149\pi$
$422$	0	0
$423$	−2.00297e6	−0.544281
$424$	0	0
$425$	728498.	0.195639
$426$	0	0
$427$	7.94138e6	2.10779
$428$	0	0
$429$	−625086.	−0.163982
$430$	0	0
$431$	−1.88599e6	−0.489043	−0.244521	−	0.969644i	$-0.578631\pi$
$431$	−1.88599e6	−0.489043	−0.244521	+	0.969644i	$0.578631\pi$
$432$	0	0
$433$	5.84067e6	1.49707	0.748537	−	0.663093i	$-0.230757\pi$
$433$	5.84067e6	1.49707	0.748537	+	0.663093i	$0.230757\pi$
$434$	0	0
$435$	1.92510e6	0.487787
$436$	0	0
$437$	−5.47776e6	−1.37214
$438$	0	0
$439$	509540.	0.126188	0.0630938	−	0.998008i	$-0.479903\pi$
$439$	509540.	0.126188	0.0630938	+	0.998008i	$0.479903\pi$
$440$	0	0
$441$	412857.	0.101089
$442$	0	0
$443$	−4.10268e6	−0.993250	−0.496625	−	0.867965i	$-0.665428\pi$
$443$	−4.10268e6	−0.993250	−0.496625	+	0.867965i	$0.665428\pi$
$444$	0	0
$445$	6.09822e6	1.45983
$446$	0	0
$447$	−592470.	−0.140248
$448$	0	0
$449$	513410.	0.120185	0.0600923	−	0.998193i	$-0.480861\pi$
$449$	513410.	0.120185	0.0600923	+	0.998193i	$0.480861\pi$
$450$	0	0
$451$	621698.	0.143926
$452$	0	0
$453$	−1.38647e6	−0.317442
$454$	0	0
$455$	−3.90779e6	−0.884918
$456$	0	0
$457$	1.22738e6	0.274908	0.137454	−	0.990508i	$-0.456108\pi$
$457$	1.22738e6	0.274908	0.137454	+	0.990508i	$0.456108\pi$
$458$	0	0
$459$	−526338.	−0.116609
$460$	0	0
$461$	−6.41000e6	−1.40477	−0.702386	−	0.711797i	$-0.747882\pi$
$461$	−6.41000e6	−1.40477	−0.702386	+	0.711797i	$0.747882\pi$
$462$	0	0
$463$	−6.63030e6	−1.43741	−0.718705	−	0.695315i	$-0.755265\pi$
$463$	−6.63030e6	−1.43741	−0.718705	+	0.695315i	$0.755265\pi$
$464$	0	0
$465$	−2.08325e6	−0.446795
$466$	0	0
$467$	4.14769e6	0.880064	0.440032	−	0.897982i	$-0.354967\pi$
$467$	4.14769e6	0.880064	0.440032	+	0.897982i	$0.354967\pi$
$468$	0	0
$469$	4.57438e6	0.960286
$470$	0	0
$471$	2.58910e6	0.537770
$472$	0	0
$473$	1.00478e6	0.206500
$474$	0	0
$475$	2.17944e6	0.443211
$476$	0	0
$477$	−2.32843e6	−0.468561
$478$	0	0
$479$	5.05132e6	1.00593	0.502963	−	0.864308i	$-0.332243\pi$
$479$	5.05132e6	1.00593	0.502963	+	0.864308i	$0.332243\pi$
$480$	0	0
$481$	4.65973e6	0.918329
$482$	0	0
$483$	−3.37795e6	−0.658849
$484$	0	0
$485$	3.25027e6	0.627429
$486$	0	0
$487$	−2.66221e6	−0.508651	−0.254325	−	0.967119i	$-0.581853\pi$
$487$	−2.66221e6	−0.508651	−0.254325	+	0.967119i	$0.581853\pi$
$488$	0	0
$489$	−946116.	−0.178925
$490$	0	0
$491$	5.54659e6	1.03830	0.519149	−	0.854684i	$-0.326249\pi$
$491$	5.54659e6	1.03830	0.519149	+	0.854684i	$0.326249\pi$
$492$	0	0
$493$	−3.35730e6	−0.622118
$494$	0	0
$495$	−450846.	−0.0827018
$496$	0	0
$497$	−1.03079e7	−1.87189
$498$	0	0
$499$	6820.00	0.00122612	0.000613060	−	1.00000i	$-0.499805\pi$
$499$	6820.00	0.00122612	0.000613060		1.00000i	$0.499805\pi$
$500$	0	0
$501$	−1.35475e6	−0.241138
$502$	0	0
$503$	451136.	0.0795037	0.0397519	−	0.999210i	$-0.487343\pi$
$503$	451136.	0.0795037	0.0397519	+	0.999210i	$0.487343\pi$
$504$	0	0
$505$	−4.69191e6	−0.818693
$506$	0	0
$507$	−376353.	−0.0650243
$508$	0	0
$509$	393390.	0.0673021	0.0336511	−	0.999434i	$-0.489287\pi$
$509$	393390.	0.0673021	0.0336511	+	0.999434i	$0.489287\pi$
$510$	0	0
$511$	2.73001e6	0.462500
$512$	0	0
$513$	−1.57464e6	−0.264173
$514$	0	0
$515$	−6.02158e6	−1.00044
$516$	0	0
$517$	2.99209e6	0.492321
$518$	0	0
$519$	−19494.0	−0.00317675
$520$	0	0
$521$	3.28432e6	0.530092	0.265046	−	0.964236i	$-0.414613\pi$
$521$	3.28432e6	0.530092	0.265046	+	0.964236i	$0.414613\pi$
$522$	0	0
$523$	1.68266e6	0.268993	0.134497	−	0.990914i	$-0.457058\pi$
$523$	1.68266e6	0.268993	0.134497	+	0.990914i	$0.457058\pi$
$524$	0	0
$525$	1.34399e6	0.212812
$526$	0	0
$527$	3.63310e6	0.569838
$528$	0	0
$529$	−5047.00	−0.000784141 0
$530$	0	0
$531$	474660.	0.0730544
$532$	0	0
$533$	−2.94921e6	−0.449664
$534$	0	0
$535$	6.51415e6	0.983951
$536$	0	0
$537$	−6.05502e6	−0.906108
$538$	0	0
$539$	−616737.	−0.0914383
$540$	0	0
$541$	9.48158e6	1.39280	0.696398	−	0.717656i	$-0.254785\pi$
$541$	9.48158e6	1.39280	0.696398	+	0.717656i	$0.254785\pi$
$542$	0	0
$543$	−4.74100e6	−0.690034
$544$	0	0
$545$	−1.10313e7	−1.59087
$546$	0	0
$547$	6.09239e6	0.870602	0.435301	−	0.900285i	$-0.356642\pi$
$547$	6.09239e6	0.870602	0.435301	+	0.900285i	$0.356642\pi$
$548$	0	0
$549$	−4.34630e6	−0.615444
$550$	0	0
$551$	−1.00440e7	−1.40938
$552$	0	0
$553$	−3.74440e6	−0.520678
$554$	0	0
$555$	3.36085e6	0.463145
$556$	0	0
$557$	8.49594e6	1.16031	0.580154	−	0.814507i	$-0.302992\pi$
$557$	8.49594e6	1.16031	0.580154	+	0.814507i	$0.302992\pi$
$558$	0	0
$559$	−4.76650e6	−0.645163
$560$	0	0
$561$	786258.	0.105477
$562$	0	0
$563$	7.02216e6	0.933683	0.466842	−	0.884341i	$-0.345392\pi$
$563$	7.02216e6	0.933683	0.466842	+	0.884341i	$0.345392\pi$
$564$	0	0
$565$	−1.96540e6	−0.259017
$566$	0	0
$567$	−971028.	−0.126845
$568$	0	0
$569$	9.41847e6	1.21955	0.609775	−	0.792574i	$-0.291260\pi$
$569$	9.41847e6	1.21955	0.609775	+	0.792574i	$0.291260\pi$
$570$	0	0
$571$	−7.29699e6	−0.936599	−0.468299	−	0.883570i	$-0.655133\pi$
$571$	−7.29699e6	−0.936599	−0.468299	+	0.883570i	$0.655133\pi$
$572$	0	0
$573$	2.75047e6	0.349962
$574$	0	0
$575$	−2.55882e6	−0.322753
$576$	0	0
$577$	−3.29590e6	−0.412131	−0.206065	−	0.978538i	$-0.566066\pi$
$577$	−3.29590e6	−0.412131	−0.206065	+	0.978538i	$0.566066\pi$
$578$	0	0
$579$	1.04791e6	0.129905
$580$	0	0
$581$	−2.59829e6	−0.319335
$582$	0	0
$583$	3.47827e6	0.423830
$584$	0	0
$585$	2.13872e6	0.258384
$586$	0	0
$587$	−4.39827e6	−0.526849	−0.263425	−	0.964680i	$-0.584852\pi$
$587$	−4.39827e6	−0.526849	−0.263425	+	0.964680i	$0.584852\pi$
$588$	0	0
$589$	1.08691e7	1.29094
$590$	0	0
$591$	−2.22968e6	−0.262587
$592$	0	0
$593$	9.21781e6	1.07644	0.538222	−	0.842803i	$-0.319096\pi$
$593$	9.21781e6	1.07644	0.538222	+	0.842803i	$0.319096\pi$
$594$	0	0
$595$	4.91538e6	0.569199
$596$	0	0
$597$	4.62024e6	0.530553
$598$	0	0
$599$	−3.77140e6	−0.429473	−0.214736	−	0.976672i	$-0.568889\pi$
$599$	−3.77140e6	−0.429473	−0.214736	+	0.976672i	$0.568889\pi$
$600$	0	0
$601$	4.19724e6	0.473999	0.237000	−	0.971510i	$-0.423836\pi$
$601$	4.19724e6	0.473999	0.237000	+	0.971510i	$0.423836\pi$
$602$	0	0
$603$	−2.50355e6	−0.280390
$604$	0	0
$605$	673486.	0.0748066
$606$	0	0
$607$	1.00133e6	0.110308	0.0551539	−	0.998478i	$-0.482435\pi$
$607$	1.00133e6	0.110308	0.0551539	+	0.998478i	$0.482435\pi$
$608$	0	0
$609$	−6.19380e6	−0.676728
$610$	0	0
$611$	−1.41939e7	−1.53815
$612$	0	0
$613$	−7.38239e6	−0.793498	−0.396749	−	0.917927i	$-0.629862\pi$
$613$	−7.38239e6	−0.793498	−0.396749	+	0.917927i	$0.629862\pi$
$614$	0	0
$615$	−2.12713e6	−0.226781
$616$	0	0
$617$	−1.54025e7	−1.62884	−0.814418	−	0.580279i	$-0.802944\pi$
$617$	−1.54025e7	−1.62884	−0.814418	+	0.580279i	$0.802944\pi$
$618$	0	0
$619$	7.12402e6	0.747306	0.373653	−	0.927569i	$-0.378105\pi$
$619$	7.12402e6	0.747306	0.373653	+	0.927569i	$0.378105\pi$
$620$	0	0
$621$	1.84874e6	0.192375
$622$	0	0
$623$	−1.96204e7	−2.02529
$624$	0	0
$625$	−5.59442e6	−0.572869
$626$	0	0
$627$	2.35224e6	0.238953
$628$	0	0
$629$	−5.86120e6	−0.590690
$630$	0	0
$631$	−1.16696e7	−1.16677	−0.583383	−	0.812197i	$-0.698271\pi$
$631$	−1.16696e7	−1.16677	−0.583383	+	0.812197i	$0.698271\pi$
$632$	0	0
$633$	5.58619e6	0.554124
$634$	0	0
$635$	−2.38225e6	−0.234451
$636$	0	0
$637$	2.92568e6	0.285679
$638$	0	0
$639$	5.64149e6	0.546565
$640$	0	0
$641$	−1.10271e7	−1.06003	−0.530014	−	0.847989i	$-0.677813\pi$
$641$	−1.10271e7	−1.06003	−0.530014	+	0.847989i	$0.677813\pi$
$642$	0	0
$643$	9.56024e6	0.911887	0.455944	−	0.890009i	$-0.349302\pi$
$643$	9.56024e6	0.911887	0.455944	+	0.890009i	$0.349302\pi$
$644$	0	0
$645$	−3.43786e6	−0.325378
$646$	0	0
$647$	1.09942e7	1.03253	0.516263	−	0.856430i	$-0.327323\pi$
$647$	1.09942e7	1.03253	0.516263	+	0.856430i	$0.327323\pi$
$648$	0	0
$649$	−709060.	−0.0660802
$650$	0	0
$651$	6.70262e6	0.619858
$652$	0	0
$653$	−295346.	−0.0271049	−0.0135525	−	0.999908i	$-0.504314\pi$
$653$	−295346.	−0.0271049	−0.0135525	+	0.999908i	$0.504314\pi$
$654$	0	0
$655$	−2.46799e6	−0.224771
$656$	0	0
$657$	−1.49413e6	−0.135044
$658$	0	0
$659$	1.65613e7	1.48553	0.742766	−	0.669551i	$-0.233514\pi$
$659$	1.65613e7	1.48553	0.742766	+	0.669551i	$0.233514\pi$
$660$	0	0
$661$	1.97042e6	0.175411	0.0877053	−	0.996146i	$-0.472047\pi$
$661$	1.97042e6	0.175411	0.0877053	+	0.996146i	$0.472047\pi$
$662$	0	0
$663$	−3.72985e6	−0.329540
$664$	0	0
$665$	1.47053e7	1.28949
$666$	0	0
$667$	1.17924e7	1.02633
$668$	0	0
$669$	1.18621e7	1.02470
$670$	0	0
$671$	6.49262e6	0.556690
$672$	0	0
$673$	−1.63733e6	−0.139347	−0.0696735	−	0.997570i	$-0.522196\pi$
$673$	−1.63733e6	−0.139347	−0.0696735	+	0.997570i	$0.522196\pi$
$674$	0	0
$675$	−735561.	−0.0621383
$676$	0	0
$677$	−6.35878e6	−0.533215	−0.266607	−	0.963805i	$-0.585903\pi$
$677$	−6.35878e6	−0.533215	−0.266607	+	0.963805i	$0.585903\pi$
$678$	0	0
$679$	−1.04574e7	−0.870460
$680$	0	0
$681$	7.98671e6	0.659933
$682$	0	0
$683$	−1.11033e7	−0.910751	−0.455376	−	0.890299i	$-0.650495\pi$
$683$	−1.11033e7	−0.910751	−0.455376	+	0.890299i	$0.650495\pi$
$684$	0	0
$685$	−1.05277e7	−0.857245
$686$	0	0
$687$	−2.13705e6	−0.172752
$688$	0	0
$689$	−1.65002e7	−1.32416
$690$	0	0
$691$	−1.70189e7	−1.35592	−0.677962	−	0.735097i	$-0.737136\pi$
$691$	−1.70189e7	−1.35592	−0.677962	+	0.735097i	$0.737136\pi$
$692$	0	0
$693$	1.45055e6	0.114736
$694$	0	0
$695$	−1.72463e7	−1.35436
$696$	0	0
$697$	3.70964e6	0.289234
$698$	0	0
$699$	−8.23235e6	−0.637281
$700$	0	0
$701$	1.58021e7	1.21456	0.607280	−	0.794488i	$-0.292260\pi$
$701$	1.58021e7	1.21456	0.607280	+	0.794488i	$0.292260\pi$
$702$	0	0
$703$	−1.75349e7	−1.33818
$704$	0	0
$705$	−1.02374e7	−0.775741
$706$	0	0
$707$	1.50957e7	1.13581
$708$	0	0
$709$	1.24834e7	0.932643	0.466322	−	0.884615i	$-0.345579\pi$
$709$	1.24834e7	0.932643	0.466322	+	0.884615i	$0.345579\pi$
$710$	0	0
$711$	2.04930e6	0.152031
$712$	0	0
$713$	−1.27612e7	−0.940083
$714$	0	0
$715$	−3.19488e6	−0.233717
$716$	0	0
$717$	−1.26803e7	−0.921151
$718$	0	0
$719$	−2.00724e6	−0.144803	−0.0724014	−	0.997376i	$-0.523066\pi$
$719$	−2.00724e6	−0.144803	−0.0724014	+	0.997376i	$0.523066\pi$
$720$	0	0
$721$	1.93738e7	1.38796
$722$	0	0
$723$	−7.43722e6	−0.529133
$724$	0	0
$725$	−4.69185e6	−0.331512
$726$	0	0
$727$	−6.97301e6	−0.489310	−0.244655	−	0.969610i	$-0.578675\pi$
$727$	−6.97301e6	−0.489310	−0.244655	+	0.969610i	$0.578675\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	5.99549e6	0.414984
$732$	0	0
$733$	−2.34965e7	−1.61527	−0.807633	−	0.589685i	$-0.799252\pi$
$733$	−2.34965e7	−1.61527	−0.807633	+	0.589685i	$0.799252\pi$
$734$	0	0
$735$	2.11016e6	0.144078
$736$	0	0
$737$	3.73987e6	0.253622
$738$	0	0
$739$	1.39901e7	0.942346	0.471173	−	0.882041i	$-0.343831\pi$
$739$	1.39901e7	0.942346	0.471173	+	0.882041i	$0.343831\pi$
$740$	0	0
$741$	−1.11586e7	−0.746556
$742$	0	0
$743$	−2.42745e7	−1.61316	−0.806582	−	0.591123i	$-0.798685\pi$
$743$	−2.42745e7	−1.61316	−0.806582	+	0.591123i	$0.798685\pi$
$744$	0	0
$745$	−3.02818e6	−0.199890
$746$	0	0
$747$	1.42204e6	0.0932415
$748$	0	0
$749$	−2.09586e7	−1.36508
$750$	0	0
$751$	−1.53660e7	−0.994170	−0.497085	−	0.867702i	$-0.665596\pi$
$751$	−1.53660e7	−0.994170	−0.497085	+	0.867702i	$0.665596\pi$
$752$	0	0
$753$	1.44348e7	0.927734
$754$	0	0
$755$	−7.08639e6	−0.452437
$756$	0	0
$757$	2.07605e7	1.31674	0.658368	−	0.752697i	$-0.271247\pi$
$757$	2.07605e7	1.31674	0.658368	+	0.752697i	$0.271247\pi$
$758$	0	0
$759$	−2.76170e6	−0.174009
$760$	0	0
$761$	5.83810e6	0.365435	0.182717	−	0.983165i	$-0.441511\pi$
$761$	5.83810e6	0.365435	0.182717	+	0.983165i	$0.441511\pi$
$762$	0	0
$763$	3.54919e7	2.20708
$764$	0	0
$765$	−2.69017e6	−0.166198
$766$	0	0
$767$	3.36364e6	0.206453
$768$	0	0
$769$	−1.39197e7	−0.848818	−0.424409	−	0.905471i	$-0.639518\pi$
$769$	−1.39197e7	−0.848818	−0.424409	+	0.905471i	$0.639518\pi$
$770$	0	0
$771$	3.57856e6	0.216807
$772$	0	0
$773$	−4.17883e6	−0.251539	−0.125770	−	0.992059i	$-0.540140\pi$
$773$	−4.17883e6	−0.251539	−0.125770	+	0.992059i	$0.540140\pi$
$774$	0	0
$775$	5.07729e6	0.303653
$776$	0	0
$777$	−1.08132e7	−0.642541
$778$	0	0
$779$	1.10981e7	0.655246
$780$	0	0
$781$	−8.42741e6	−0.494386
$782$	0	0
$783$	3.38985e6	0.197595
$784$	0	0
$785$	1.32332e7	0.766461
$786$	0	0
$787$	−9.66705e6	−0.556361	−0.278181	−	0.960529i	$-0.589731\pi$
$787$	−9.66705e6	−0.556361	−0.278181	+	0.960529i	$0.589731\pi$
$788$	0	0
$789$	−1.91850e7	−1.09716
$790$	0	0
$791$	6.32345e6	0.359346
$792$	0	0
$793$	−3.07997e7	−1.73926
$794$	0	0
$795$	−1.19008e7	−0.667821
$796$	0	0
$797$	−5.79884e6	−0.323367	−0.161683	−	0.986843i	$-0.551692\pi$
$797$	−5.79884e6	−0.323367	−0.161683	+	0.986843i	$0.551692\pi$
$798$	0	0
$799$	1.78536e7	0.989371
$800$	0	0
$801$	1.07382e7	0.591356
$802$	0	0
$803$	2.23197e6	0.122151
$804$	0	0
$805$	−1.72651e7	−0.939029
$806$	0	0
$807$	−6.53229e6	−0.353087
$808$	0	0
$809$	−1.92543e7	−1.03433	−0.517163	−	0.855887i	$-0.673012\pi$
$809$	−1.92543e7	−1.03433	−0.517163	+	0.855887i	$0.673012\pi$
$810$	0	0
$811$	1.31938e7	0.704396	0.352198	−	0.935926i	$-0.385434\pi$
$811$	1.31938e7	0.704396	0.352198	+	0.935926i	$0.385434\pi$
$812$	0	0
$813$	1.32108e7	0.700976
$814$	0	0
$815$	−4.83570e6	−0.255015
$816$	0	0
$817$	1.79366e7	0.940126
$818$	0	0
$819$	−6.88111e6	−0.358467
$820$	0	0
$821$	1.33779e7	0.692677	0.346338	−	0.938110i	$-0.387425\pi$
$821$	1.33779e7	0.692677	0.346338	+	0.938110i	$0.387425\pi$
$822$	0	0
$823$	1.88613e7	0.970673	0.485336	−	0.874327i	$-0.338697\pi$
$823$	1.88613e7	0.970673	0.485336	+	0.874327i	$0.338697\pi$
$824$	0	0
$825$	1.09880e6	0.0562062
$826$	0	0
$827$	−1.62680e7	−0.827123	−0.413561	−	0.910476i	$-0.635715\pi$
$827$	−1.62680e7	−0.827123	−0.413561	+	0.910476i	$0.635715\pi$
$828$	0	0
$829$	−2.18098e7	−1.10221	−0.551107	−	0.834435i	$-0.685794\pi$
$829$	−2.18098e7	−1.10221	−0.551107	+	0.834435i	$0.685794\pi$
$830$	0	0
$831$	1.36890e7	0.687652
$832$	0	0
$833$	−3.68003e6	−0.183755
$834$	0	0
$835$	−6.92429e6	−0.343684
$836$	0	0
$837$	−3.66833e6	−0.180990
$838$	0	0
$839$	−1.17771e7	−0.577607	−0.288804	−	0.957388i	$-0.593257\pi$
$839$	−1.17771e7	−0.577607	−0.288804	+	0.957388i	$0.593257\pi$
$840$	0	0
$841$	1.11135e6	0.0541828
$842$	0	0
$843$	4.17944e6	0.202558
$844$	0	0
$845$	−1.92358e6	−0.0926764
$846$	0	0
$847$	−2.16687e6	−0.103782
$848$	0	0
$849$	3.73622e6	0.177895
$850$	0	0
$851$	2.05872e7	0.974483
$852$	0	0
$853$	−1.43993e7	−0.677591	−0.338796	−	0.940860i	$-0.610020\pi$
$853$	−1.43993e7	−0.677591	−0.338796	+	0.940860i	$0.610020\pi$
$854$	0	0
$855$	−8.04816e6	−0.376514
$856$	0	0
$857$	6.27604e6	0.291900	0.145950	−	0.989292i	$-0.453376\pi$
$857$	6.27604e6	0.291900	0.145950	+	0.989292i	$0.453376\pi$
$858$	0	0
$859$	4.71738e6	0.218131	0.109066	−	0.994035i	$-0.465214\pi$
$859$	4.71738e6	0.218131	0.109066	+	0.994035i	$0.465214\pi$
$860$	0	0
$861$	6.84382e6	0.314623
$862$	0	0
$863$	−7.53926e6	−0.344589	−0.172295	−	0.985045i	$-0.555118\pi$
$863$	−7.53926e6	−0.344589	−0.172295	+	0.985045i	$0.555118\pi$
$864$	0	0
$865$	−99636.0	−0.00452768
$866$	0	0
$867$	−8.08716e6	−0.365383
$868$	0	0
$869$	−3.06130e6	−0.137517
$870$	0	0
$871$	−1.77412e7	−0.792387
$872$	0	0
$873$	5.72330e6	0.254162
$874$	0	0
$875$	2.81443e7	1.24271
$876$	0	0
$877$	−1.04331e7	−0.458051	−0.229025	−	0.973420i	$-0.573554\pi$
$877$	−1.04331e7	−0.458051	−0.229025	+	0.973420i	$0.573554\pi$
$878$	0	0
$879$	−2.33389e7	−1.01884
$880$	0	0
$881$	3.91076e7	1.69755	0.848774	−	0.528756i	$-0.177342\pi$
$881$	3.91076e7	1.69755	0.848774	+	0.528756i	$0.177342\pi$
$882$	0	0
$883$	−1.29282e7	−0.558003	−0.279001	−	0.960291i	$-0.590003\pi$
$883$	−1.29282e7	−0.558003	−0.279001	+	0.960291i	$0.590003\pi$
$884$	0	0
$885$	2.42604e6	0.104121
$886$	0	0
$887$	−3.36466e7	−1.43592	−0.717962	−	0.696082i	$-0.754925\pi$
$887$	−3.36466e7	−1.43592	−0.717962	+	0.696082i	$0.754925\pi$
$888$	0	0
$889$	7.66462e6	0.325264
$890$	0	0
$891$	−793881.	−0.0335013
$892$	0	0
$893$	5.34125e7	2.24137
$894$	0	0
$895$	−3.09479e7	−1.29144
$896$	0	0
$897$	1.31010e7	0.543654
$898$	0	0
$899$	−2.33988e7	−0.965594
$900$	0	0
$901$	2.07546e7	0.851731
$902$	0	0
$903$	1.10609e7	0.451411
$904$	0	0
$905$	−2.42318e7	−0.983477
$906$	0	0
$907$	4.19629e7	1.69374	0.846872	−	0.531797i	$-0.178483\pi$
$907$	4.19629e7	1.69374	0.846872	+	0.531797i	$0.178483\pi$
$908$	0	0
$909$	−8.26184e6	−0.331640
$910$	0	0
$911$	1.92521e6	0.0768567	0.0384283	−	0.999261i	$-0.487765\pi$
$911$	1.92521e6	0.0768567	0.0384283	+	0.999261i	$0.487765\pi$
$912$	0	0
$913$	−2.12428e6	−0.0843401
$914$	0	0
$915$	−2.22144e7	−0.877167
$916$	0	0
$917$	7.94050e6	0.311835
$918$	0	0
$919$	1.72481e7	0.673678	0.336839	−	0.941562i	$-0.390642\pi$
$919$	1.72481e7	0.673678	0.336839	+	0.941562i	$0.390642\pi$
$920$	0	0
$921$	8.37749e6	0.325435
$922$	0	0
$923$	3.99780e7	1.54460
$924$	0	0
$925$	−8.19106e6	−0.314765
$926$	0	0
$927$	−1.06032e7	−0.405265
$928$	0	0
$929$	2.51145e6	0.0954740	0.0477370	−	0.998860i	$-0.484799\pi$
$929$	2.51145e6	0.0954740	0.0477370	+	0.998860i	$0.484799\pi$
$930$	0	0
$931$	−1.10095e7	−0.416288
$932$	0	0
$933$	−2.23674e7	−0.841225
$934$	0	0
$935$	4.01865e6	0.150332
$936$	0	0
$937$	1.79853e7	0.669221	0.334611	−	0.942357i	$-0.391395\pi$
$937$	1.79853e7	0.669221	0.334611	+	0.942357i	$0.391395\pi$
$938$	0	0
$939$	1.18547e7	0.438761
$940$	0	0
$941$	−3.22586e7	−1.18760	−0.593802	−	0.804611i	$-0.702374\pi$
$941$	−3.22586e7	−1.18760	−0.593802	+	0.804611i	$0.702374\pi$
$942$	0	0
$943$	−1.30300e7	−0.477160
$944$	0	0
$945$	−4.96303e6	−0.180787
$946$	0	0
$947$	−4.41659e7	−1.60034	−0.800169	−	0.599774i	$-0.795257\pi$
$947$	−4.41659e7	−1.60034	−0.800169	+	0.599774i	$0.795257\pi$
$948$	0	0
$949$	−1.05880e7	−0.381635
$950$	0	0
$951$	2.02986e7	0.727804
$952$	0	0
$953$	1.87488e7	0.668714	0.334357	−	0.942446i	$-0.391481\pi$
$953$	1.87488e7	0.668714	0.334357	+	0.942446i	$0.391481\pi$
$954$	0	0
$955$	1.40580e7	0.498786
$956$	0	0
$957$	−5.06385e6	−0.178731
$958$	0	0
$959$	3.38716e7	1.18929
$960$	0	0
$961$	−3.30813e6	−0.115551
$962$	0	0
$963$	1.14706e7	0.398584
$964$	0	0
$965$	5.35596e6	0.185148
$966$	0	0
$967$	−1.08673e7	−0.373730	−0.186865	−	0.982386i	$-0.559833\pi$
$967$	−1.08673e7	−0.373730	−0.186865	+	0.982386i	$0.559833\pi$
$968$	0	0
$969$	1.40357e7	0.480202
$970$	0	0
$971$	4.79123e7	1.63079	0.815397	−	0.578902i	$-0.196519\pi$
$971$	4.79123e7	1.63079	0.815397	+	0.578902i	$0.196519\pi$
$972$	0	0
$973$	5.54882e7	1.87896
$974$	0	0
$975$	−5.21249e6	−0.175604
$976$	0	0
$977$	4.01385e7	1.34532	0.672658	−	0.739954i	$-0.265153\pi$
$977$	4.01385e7	1.34532	0.672658	+	0.739954i	$0.265153\pi$
$978$	0	0
$979$	−1.60410e7	−0.534902
$980$	0	0
$981$	−1.94246e7	−0.644436
$982$	0	0
$983$	−3.22682e6	−0.106510	−0.0532551	−	0.998581i	$-0.516960\pi$
$983$	−3.22682e6	−0.106510	−0.0532551	+	0.998581i	$0.516960\pi$
$984$	0	0
$985$	−1.13961e7	−0.374254
$986$	0	0
$987$	3.29377e7	1.07622
$988$	0	0
$989$	−2.10589e7	−0.684614
$990$	0	0
$991$	5.95345e6	0.192568	0.0962841	−	0.995354i	$-0.469304\pi$
$991$	5.95345e6	0.192568	0.0962841	+	0.995354i	$0.469304\pi$
$992$	0	0
$993$	2.85364e7	0.918387
$994$	0	0
$995$	2.36146e7	0.756175
$996$	0	0
$997$	3.20783e7	1.02205	0.511027	−	0.859565i	$-0.329265\pi$
$997$	3.20783e7	1.02205	0.511027	+	0.859565i	$0.329265\pi$
$998$	0	0
$999$	5.91802e6	0.187613

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	528.6.a.i.1.1		1
4.3	odd	2		33.6.a.a.1.1	✓	1
12.11	even	2		99.6.a.b.1.1		1
20.19	odd	2		825.6.a.b.1.1		1
44.43	even	2		363.6.a.c.1.1		1
132.131	odd	2		1089.6.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.a.a.1.1	✓	1	4.3	odd	2
99.6.a.b.1.1		1	12.11	even	2
363.6.a.c.1.1		1	44.43	even	2
528.6.a.i.1.1		1	1.1	even	1	trivial
825.6.a.b.1.1		1	20.19	odd	2
1089.6.a.d.1.1		1	132.131	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	528.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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