LMFDB - Embedded newform 432.8.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,8,Mod(1,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("432.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	432.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:	$134.950331009$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 54)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	432.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	120.000	0.429325	0.214663	−	0.976688i	$-0.431135\pi$
$5$	120.000	0.429325	0.214663	+	0.976688i	$0.431135\pi$
$6$	0	0
$7$	−377.000	−0.415430	−0.207715	−	0.978189i	$-0.566603\pi$
$7$	−377.000	−0.415430	−0.207715	+	0.978189i	$0.566603\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−600.000	−0.135918	−0.0679590	−	0.997688i	$-0.521649\pi$
$11$	−600.000	−0.135918	−0.0679590	+	0.997688i	$0.521649\pi$
$12$	0	0
$13$	5369.00	0.677785	0.338892	−	0.940825i	$-0.389948\pi$
$13$	5369.00	0.677785	0.338892	+	0.940825i	$0.389948\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	12168.0	0.600687	0.300343	−	0.953831i	$-0.402899\pi$
$17$	12168.0	0.600687	0.300343	+	0.953831i	$0.402899\pi$
$18$	0	0
$19$	−16211.0	−0.542216	−0.271108	−	0.962549i	$-0.587390\pi$
$19$	−16211.0	−0.542216	−0.271108	+	0.962549i	$0.587390\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−106392.	−1.82331	−0.911657	−	0.410952i	$-0.865197\pi$
$23$	−106392.	−1.82331	−0.911657	+	0.410952i	$0.865197\pi$
$24$	0	0
$25$	−63725.0	−0.815680
$26$	0	0
$27$	0	0
$28$	0	0
$29$	177216.	1.34930	0.674652	−	0.738136i	$-0.264294\pi$
$29$	177216.	1.34930	0.674652	+	0.738136i	$0.264294\pi$
$30$	0	0
$31$	268060.	1.61609	0.808046	−	0.589119i	$-0.200525\pi$
$31$	268060.	1.61609	0.808046	+	0.589119i	$0.200525\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−45240.0	−0.178355
$36$	0	0
$37$	114959.	0.373110	0.186555	−	0.982445i	$-0.440268\pi$
$37$	114959.	0.373110	0.186555	+	0.982445i	$0.440268\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−112128.	−0.254080	−0.127040	−	0.991898i	$-0.540548\pi$
$41$	−112128.	−0.254080	−0.127040	+	0.991898i	$0.540548\pi$
$42$	0	0
$43$	115048.	0.220668	0.110334	−	0.993895i	$-0.464808\pi$
$43$	115048.	0.220668	0.110334	+	0.993895i	$0.464808\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−561336.	−0.788643	−0.394321	−	0.918973i	$-0.629020\pi$
$47$	−561336.	−0.788643	−0.394321	+	0.918973i	$0.629020\pi$
$48$	0	0
$49$	−681414.	−0.827418
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.78776e6	−1.64947	−0.824734	−	0.565521i	$-0.808675\pi$
$53$	−1.78776e6	−1.64947	−0.824734	+	0.565521i	$0.808675\pi$
$54$	0	0
$55$	−72000.0	−0.0583530
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.78634e6	1.13236	0.566178	−	0.824283i	$-0.308421\pi$
$59$	1.78634e6	1.13236	0.566178	+	0.824283i	$0.308421\pi$
$60$	0	0
$61$	−1.30684e6	−0.737169	−0.368584	−	0.929594i	$-0.620157\pi$
$61$	−1.30684e6	−0.737169	−0.368584	+	0.929594i	$0.620157\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	644280.	0.290990
$66$	0	0
$67$	2.01382e6	0.818009	0.409005	−	0.912532i	$-0.365876\pi$
$67$	2.01382e6	0.818009	0.409005	+	0.912532i	$0.365876\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.06094e6	1.34655	0.673275	−	0.739392i	$-0.264887\pi$
$71$	4.06094e6	1.34655	0.673275	+	0.739392i	$0.264887\pi$
$72$	0	0
$73$	−3.85064e6	−1.15852	−0.579259	−	0.815144i	$-0.696658\pi$
$73$	−3.85064e6	−1.15852	−0.579259	+	0.815144i	$0.696658\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	226200.	0.0564644
$78$	0	0
$79$	−1.03723e6	−0.236690	−0.118345	−	0.992973i	$-0.537759\pi$
$79$	−1.03723e6	−0.236690	−0.118345	+	0.992973i	$0.537759\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−9.20357e6	−1.76678	−0.883391	−	0.468637i	$-0.844745\pi$
$83$	−9.20357e6	−1.76678	−0.883391	+	0.468637i	$0.844745\pi$
$84$	0	0
$85$	1.46016e6	0.257890
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.28930e6	−0.193861	−0.0969305	−	0.995291i	$-0.530902\pi$
$89$	−1.28930e6	−0.193861	−0.0969305	+	0.995291i	$0.530902\pi$
$90$	0	0
$91$	−2.02411e6	−0.281572
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.94532e6	−0.232787
$96$	0	0
$97$	8.55588e6	0.951840	0.475920	−	0.879489i	$-0.342115\pi$
$97$	8.55588e6	0.951840	0.475920	+	0.879489i	$0.342115\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.99006e7	1.92195	0.960974	−	0.276639i	$-0.0892208\pi$
$101$	1.99006e7	1.92195	0.960974	+	0.276639i	$0.0892208\pi$
$102$	0	0
$103$	−6.99984e6	−0.631187	−0.315594	−	0.948894i	$-0.602204\pi$
$103$	−6.99984e6	−0.631187	−0.315594	+	0.948894i	$0.602204\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.48063e6	0.353587	0.176793	−	0.984248i	$-0.443428\pi$
$107$	4.48063e6	0.353587	0.176793	+	0.984248i	$0.443428\pi$
$108$	0	0
$109$	−1.47074e6	−0.108779	−0.0543893	−	0.998520i	$-0.517321\pi$
$109$	−1.47074e6	−0.108779	−0.0543893	+	0.998520i	$0.517321\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.47786e6	0.226745	0.113373	−	0.993553i	$-0.463835\pi$
$113$	3.47786e6	0.226745	0.113373	+	0.993553i	$0.463835\pi$
$114$	0	0
$115$	−1.27670e7	−0.782795
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.58734e6	−0.249543
$120$	0	0
$121$	−1.91272e7	−0.981526
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.70220e7	−0.779517
$126$	0	0
$127$	−2.10378e7	−0.911353	−0.455677	−	0.890145i	$-0.650603\pi$
$127$	−2.10378e7	−0.911353	−0.455677	+	0.890145i	$0.650603\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.06906e7	0.804125	0.402062	−	0.915612i	$-0.368294\pi$
$131$	2.06906e7	0.804125	0.402062	+	0.915612i	$0.368294\pi$
$132$	0	0
$133$	6.11155e6	0.225253
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.31521e7	−1.76603	−0.883016	−	0.469343i	$-0.844491\pi$
$137$	−5.31521e7	−1.76603	−0.883016	+	0.469343i	$0.844491\pi$
$138$	0	0
$139$	1.95009e7	0.615890	0.307945	−	0.951404i	$-0.400359\pi$
$139$	1.95009e7	0.615890	0.307945	+	0.951404i	$0.400359\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.22140e6	−0.0921231
$144$	0	0
$145$	2.12659e7	0.579290
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.51563e6	0.210894	0.105447	−	0.994425i	$-0.466373\pi$
$149$	8.51563e6	0.210894	0.105447	+	0.994425i	$0.466373\pi$
$150$	0	0
$151$	−6.37020e7	−1.50568	−0.752841	−	0.658202i	$-0.771317\pi$
$151$	−6.37020e7	−1.50568	−0.752841	+	0.658202i	$0.771317\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.21672e7	0.693829
$156$	0	0
$157$	−6.36342e7	−1.31233	−0.656163	−	0.754619i	$-0.727822\pi$
$157$	−6.36342e7	−1.31233	−0.656163	+	0.754619i	$0.727822\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.01098e7	0.757460
$162$	0	0
$163$	−6.87301e7	−1.24305	−0.621527	−	0.783392i	$-0.713488\pi$
$163$	−6.87301e7	−1.24305	−0.621527	+	0.783392i	$0.713488\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.29434e7	−0.215051	−0.107525	−	0.994202i	$-0.534293\pi$
$167$	−1.29434e7	−0.215051	−0.107525	+	0.994202i	$0.534293\pi$
$168$	0	0
$169$	−3.39224e7	−0.540608
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−4.22124e7	−0.619838	−0.309919	−	0.950763i	$-0.600302\pi$
$173$	−4.22124e7	−0.619838	−0.309919	+	0.950763i	$0.600302\pi$
$174$	0	0
$175$	2.40243e7	0.338858
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1.13640e8	−1.48096	−0.740482	−	0.672076i	$-0.765403\pi$
$179$	−1.13640e8	−1.48096	−0.740482	+	0.672076i	$0.765403\pi$
$180$	0	0
$181$	−1.27922e8	−1.60350	−0.801752	−	0.597658i	$-0.796098\pi$
$181$	−1.27922e8	−1.60350	−0.801752	+	0.597658i	$0.796098\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.37951e7	0.160185
$186$	0	0
$187$	−7.30080e6	−0.0816441
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.10840e7	0.218946	0.109473	−	0.993990i	$-0.465084\pi$
$191$	2.10840e7	0.218946	0.109473	+	0.993990i	$0.465084\pi$
$192$	0	0
$193$	−1.39821e8	−1.39998	−0.699990	−	0.714152i	$-0.746812\pi$
$193$	−1.39821e8	−1.39998	−0.699990	+	0.714152i	$0.746812\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.45966e8	1.36025	0.680127	−	0.733094i	$-0.261924\pi$
$197$	1.45966e8	1.36025	0.680127	+	0.733094i	$0.261924\pi$
$198$	0	0
$199$	5.11146e7	0.459790	0.229895	−	0.973215i	$-0.426162\pi$
$199$	5.11146e7	0.459790	0.229895	+	0.973215i	$0.426162\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−6.68104e7	−0.560542
$204$	0	0
$205$	−1.34554e7	−0.109083
$206$	0	0
$207$	0	0
$208$	0	0
$209$	9.72660e6	0.0736969
$210$	0	0
$211$	−9.80537e7	−0.718581	−0.359290	−	0.933226i	$-0.616981\pi$
$211$	−9.80537e7	−0.718581	−0.359290	+	0.933226i	$0.616981\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.38058e7	0.0947383
$216$	0	0
$217$	−1.01059e8	−0.671374
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.53300e7	0.407136
$222$	0	0
$223$	−2.16758e8	−1.30891	−0.654453	−	0.756103i	$-0.727101\pi$
$223$	−2.16758e8	−1.30891	−0.654453	+	0.756103i	$0.727101\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.39865e8	0.793629	0.396814	−	0.917899i	$-0.370116\pi$
$227$	1.39865e8	0.793629	0.396814	+	0.917899i	$0.370116\pi$
$228$	0	0
$229$	−9.49569e7	−0.522519	−0.261260	−	0.965269i	$-0.584138\pi$
$229$	−9.49569e7	−0.522519	−0.261260	+	0.965269i	$0.584138\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.10948e8	−1.09252	−0.546261	−	0.837615i	$-0.683949\pi$
$233$	−2.10948e8	−1.09252	−0.546261	+	0.837615i	$0.683949\pi$
$234$	0	0
$235$	−6.73603e7	−0.338584
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.29152e8	0.611939	0.305969	−	0.952041i	$-0.401019\pi$
$239$	1.29152e8	0.611939	0.305969	+	0.952041i	$0.401019\pi$
$240$	0	0
$241$	−4.01868e7	−0.184937	−0.0924685	−	0.995716i	$-0.529476\pi$
$241$	−4.01868e7	−0.184937	−0.0924685	+	0.995716i	$0.529476\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−8.17697e7	−0.355231
$246$	0	0
$247$	−8.70369e7	−0.367506
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.68201e8	1.07054	0.535270	−	0.844681i	$-0.320210\pi$
$251$	2.68201e8	1.07054	0.535270	+	0.844681i	$0.320210\pi$
$252$	0	0
$253$	6.38352e7	0.247821
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.71864e8	−0.631566	−0.315783	−	0.948831i	$-0.602267\pi$
$257$	−1.71864e8	−0.631566	−0.315783	+	0.948831i	$0.602267\pi$
$258$	0	0
$259$	−4.33395e7	−0.155001
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2.02033e8	−0.684822	−0.342411	−	0.939550i	$-0.611243\pi$
$263$	−2.02033e8	−0.684822	−0.342411	+	0.939550i	$0.611243\pi$
$264$	0	0
$265$	−2.14531e8	−0.708158
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−9.73917e7	−0.305063	−0.152531	−	0.988299i	$-0.548743\pi$
$269$	−9.73917e7	−0.305063	−0.152531	+	0.988299i	$0.548743\pi$
$270$	0	0
$271$	−4.51945e8	−1.37941	−0.689705	−	0.724091i	$-0.742260\pi$
$271$	−4.51945e8	−1.37941	−0.689705	+	0.724091i	$0.742260\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.82350e7	0.110866
$276$	0	0
$277$	−2.09703e8	−0.592823	−0.296411	−	0.955060i	$-0.595790\pi$
$277$	−2.09703e8	−0.592823	−0.296411	+	0.955060i	$0.595790\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5.32204e8	1.43089	0.715444	−	0.698670i	$-0.246224\pi$
$281$	5.32204e8	1.43089	0.715444	+	0.698670i	$0.246224\pi$
$282$	0	0
$283$	−4.74761e8	−1.24515	−0.622576	−	0.782559i	$-0.713914\pi$
$283$	−4.74761e8	−1.24515	−0.622576	+	0.782559i	$0.713914\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.22723e7	0.105553
$288$	0	0
$289$	−2.62278e8	−0.639176
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2.32118e8	−0.539104	−0.269552	−	0.962986i	$-0.586876\pi$
$293$	−2.32118e8	−0.539104	−0.269552	+	0.962986i	$0.586876\pi$
$294$	0	0
$295$	2.14361e8	0.486149
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−5.71219e8	−1.23581
$300$	0	0
$301$	−4.33731e7	−0.0916722
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.56820e8	−0.316485
$306$	0	0
$307$	−7.53314e8	−1.48591	−0.742953	−	0.669343i	$-0.766576\pi$
$307$	−7.53314e8	−1.48591	−0.742953	+	0.669343i	$0.766576\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4.79778e8	−0.904439	−0.452219	−	0.891907i	$-0.649368\pi$
$311$	−4.79778e8	−0.904439	−0.452219	+	0.891907i	$0.649368\pi$
$312$	0	0
$313$	6.55525e8	1.20833	0.604163	−	0.796861i	$-0.293508\pi$
$313$	6.55525e8	1.20833	0.604163	+	0.796861i	$0.293508\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.93361e8	1.22251	0.611255	−	0.791434i	$-0.290665\pi$
$317$	6.93361e8	1.22251	0.611255	+	0.791434i	$0.290665\pi$
$318$	0	0
$319$	−1.06330e8	−0.183395
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.97255e8	−0.325702
$324$	0	0
$325$	−3.42140e8	−0.552855
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.11624e8	0.327626
$330$	0	0
$331$	1.84653e8	0.279872	0.139936	−	0.990161i	$-0.455310\pi$
$331$	1.84653e8	0.279872	0.139936	+	0.990161i	$0.455310\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.41658e8	0.351192
$336$	0	0
$337$	8.43883e8	1.20110	0.600548	−	0.799589i	$-0.294949\pi$
$337$	8.43883e8	1.20110	0.600548	+	0.799589i	$0.294949\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.60836e8	−0.219656
$342$	0	0
$343$	5.67369e8	0.759165
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.22250e8	−0.157071	−0.0785353	−	0.996911i	$-0.525024\pi$
$347$	−1.22250e8	−0.157071	−0.0785353	+	0.996911i	$0.525024\pi$
$348$	0	0
$349$	−1.05296e9	−1.32594	−0.662969	−	0.748646i	$-0.730704\pi$
$349$	−1.05296e9	−1.32594	−0.662969	+	0.748646i	$0.730704\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6.24584e8	−0.755752	−0.377876	−	0.925856i	$-0.623346\pi$
$353$	−6.24584e8	−0.755752	−0.377876	+	0.925856i	$0.623346\pi$
$354$	0	0
$355$	4.87313e8	0.578108
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7.95194e8	0.907074	0.453537	−	0.891237i	$-0.350162\pi$
$359$	7.95194e8	0.907074	0.453537	+	0.891237i	$0.350162\pi$
$360$	0	0
$361$	−6.31075e8	−0.706002
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4.62077e8	−0.497381
$366$	0	0
$367$	1.09445e9	1.15575	0.577876	−	0.816125i	$-0.303882\pi$
$367$	1.09445e9	1.15575	0.577876	+	0.816125i	$0.303882\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	6.73986e8	0.685239
$372$	0	0
$373$	8.82044e8	0.880054	0.440027	−	0.897985i	$-0.354969\pi$
$373$	8.82044e8	0.880054	0.440027	+	0.897985i	$0.354969\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.51473e8	0.914538
$378$	0	0
$379$	1.04251e9	0.983651	0.491826	−	0.870694i	$-0.336330\pi$
$379$	1.04251e9	0.983651	0.491826	+	0.870694i	$0.336330\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	8.36847e8	0.761115	0.380558	−	0.924757i	$-0.375732\pi$
$383$	8.36847e8	0.761115	0.380558	+	0.924757i	$0.375732\pi$
$384$	0	0
$385$	2.71440e7	0.0242416
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.47060e9	−1.26669	−0.633345	−	0.773870i	$-0.718319\pi$
$389$	−1.47060e9	−1.26669	−0.633345	+	0.773870i	$0.718319\pi$
$390$	0	0
$391$	−1.29458e9	−1.09524
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.24468e8	−0.101617
$396$	0	0
$397$	−1.26746e9	−1.01664	−0.508322	−	0.861167i	$-0.669734\pi$
$397$	−1.26746e9	−1.01664	−0.508322	+	0.861167i	$0.669734\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.27858e8	0.408801	0.204401	−	0.978887i	$-0.434476\pi$
$401$	5.27858e8	0.408801	0.204401	+	0.978887i	$0.434476\pi$
$402$	0	0
$403$	1.43921e9	1.09536
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6.89754e7	−0.0507124
$408$	0	0
$409$	−2.22791e8	−0.161015	−0.0805073	−	0.996754i	$-0.525654\pi$
$409$	−2.22791e8	−0.161015	−0.0805073	+	0.996754i	$0.525654\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−6.73452e8	−0.470415
$414$	0	0
$415$	−1.10443e9	−0.758524
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4.40581e8	0.292602	0.146301	−	0.989240i	$-0.453263\pi$
$419$	4.40581e8	0.292602	0.146301	+	0.989240i	$0.453263\pi$
$420$	0	0
$421$	2.18934e9	1.42997	0.714984	−	0.699140i	$-0.246434\pi$
$421$	2.18934e9	1.42997	0.714984	+	0.699140i	$0.246434\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−7.75406e8	−0.489968
$426$	0	0
$427$	4.92678e8	0.306242
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−3.03634e9	−1.82675	−0.913377	−	0.407114i	$-0.866535\pi$
$431$	−3.03634e9	−1.82675	−0.913377	+	0.407114i	$0.866535\pi$
$432$	0	0
$433$	2.21735e9	1.31258	0.656290	−	0.754509i	$-0.272125\pi$
$433$	2.21735e9	1.31258	0.656290	+	0.754509i	$0.272125\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.72472e9	0.988630
$438$	0	0
$439$	2.50094e9	1.41084	0.705419	−	0.708791i	$-0.250759\pi$
$439$	2.50094e9	1.41084	0.705419	+	0.708791i	$0.250759\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.40387e9	1.31370	0.656852	−	0.754019i	$-0.271887\pi$
$443$	2.40387e9	1.31370	0.656852	+	0.754019i	$0.271887\pi$
$444$	0	0
$445$	−1.54716e8	−0.0832294
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2.14935e9	1.12059	0.560294	−	0.828294i	$-0.310688\pi$
$449$	2.14935e9	1.12059	0.560294	+	0.828294i	$0.310688\pi$
$450$	0	0
$451$	6.72768e7	0.0345340
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.42894e8	−0.120886
$456$	0	0
$457$	2.30159e9	1.12803	0.564015	−	0.825765i	$-0.309256\pi$
$457$	2.30159e9	1.12803	0.564015	+	0.825765i	$0.309256\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.37928e9	0.655689	0.327844	−	0.944732i	$-0.393678\pi$
$461$	1.37928e9	0.655689	0.327844	+	0.944732i	$0.393678\pi$
$462$	0	0
$463$	1.72054e9	0.805620	0.402810	−	0.915284i	$-0.368033\pi$
$463$	1.72054e9	0.805620	0.402810	+	0.915284i	$0.368033\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.58750e9	−0.721280	−0.360640	−	0.932705i	$-0.617442\pi$
$467$	−1.58750e9	−0.721280	−0.360640	+	0.932705i	$0.617442\pi$
$468$	0	0
$469$	−7.59209e8	−0.339826
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−6.90288e7	−0.0299928
$474$	0	0
$475$	1.03305e9	0.442275
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1.75670e9	−0.730336	−0.365168	−	0.930942i	$-0.618988\pi$
$479$	−1.75670e9	−0.730336	−0.365168	+	0.930942i	$0.618988\pi$
$480$	0	0
$481$	6.17215e8	0.252888
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.02671e9	0.408649
$486$	0	0
$487$	−1.52004e9	−0.596353	−0.298177	−	0.954511i	$-0.596378\pi$
$487$	−1.52004e9	−0.596353	−0.298177	+	0.954511i	$0.596378\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3.64824e9	−1.39091	−0.695453	−	0.718572i	$-0.744796\pi$
$491$	−3.64824e9	−1.39091	−0.695453	+	0.718572i	$0.744796\pi$
$492$	0	0
$493$	2.15636e9	0.810509
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.53098e9	−0.559398
$498$	0	0
$499$	−4.83503e8	−0.174200	−0.0870998	−	0.996200i	$-0.527760\pi$
$499$	−4.83503e8	−0.174200	−0.0870998	+	0.996200i	$0.527760\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	4.20389e9	1.47287	0.736434	−	0.676510i	$-0.236508\pi$
$503$	4.20389e9	1.47287	0.736434	+	0.676510i	$0.236508\pi$
$504$	0	0
$505$	2.38807e9	0.825140
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−2.67771e9	−0.900018	−0.450009	−	0.893024i	$-0.648579\pi$
$509$	−2.67771e9	−0.900018	−0.450009	+	0.893024i	$0.648579\pi$
$510$	0	0
$511$	1.45169e9	0.481284
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8.39981e8	−0.270984
$516$	0	0
$517$	3.36802e8	0.107191
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−4.83951e9	−1.49923	−0.749616	−	0.661873i	$-0.769762\pi$
$521$	−4.83951e9	−1.49923	−0.749616	+	0.661873i	$0.769762\pi$
$522$	0	0
$523$	−6.47019e8	−0.197770	−0.0988851	−	0.995099i	$-0.531528\pi$
$523$	−6.47019e8	−0.197770	−0.0988851	+	0.995099i	$0.531528\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.26175e9	0.970765
$528$	0	0
$529$	7.91443e9	2.32448
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−6.02015e8	−0.172212
$534$	0	0
$535$	5.37676e8	0.151804
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.08848e8	0.112461
$540$	0	0
$541$	−3.47237e8	−0.0942835	−0.0471418	−	0.998888i	$-0.515011\pi$
$541$	−3.47237e8	−0.0942835	−0.0471418	+	0.998888i	$0.515011\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.76489e8	−0.0467014
$546$	0	0
$547$	2.88628e9	0.754021	0.377010	−	0.926209i	$-0.376952\pi$
$547$	2.88628e9	0.754021	0.377010	+	0.926209i	$0.376952\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.87285e9	−0.731614
$552$	0	0
$553$	3.91036e8	0.0983284
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−8.74890e8	−0.214516	−0.107258	−	0.994231i	$-0.534207\pi$
$557$	−8.74890e8	−0.214516	−0.107258	+	0.994231i	$0.534207\pi$
$558$	0	0
$559$	6.17693e8	0.149565
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8.07444e9	1.90692	0.953462	−	0.301515i	$-0.0974922\pi$
$563$	8.07444e9	1.90692	0.953462	+	0.301515i	$0.0974922\pi$
$564$	0	0
$565$	4.17344e8	0.0973474
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5.50834e9	1.25351	0.626755	−	0.779216i	$-0.284383\pi$
$569$	5.50834e9	1.25351	0.626755	+	0.779216i	$0.284383\pi$
$570$	0	0
$571$	2.81462e9	0.632693	0.316347	−	0.948644i	$-0.397544\pi$
$571$	2.81462e9	0.632693	0.316347	+	0.948644i	$0.397544\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6.77983e9	1.48724
$576$	0	0
$577$	−4.74376e9	−1.02803	−0.514017	−	0.857780i	$-0.671843\pi$
$577$	−4.74376e9	−1.02803	−0.514017	+	0.857780i	$0.671843\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	3.46975e9	0.733975
$582$	0	0
$583$	1.07266e9	0.224192
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−4.92039e9	−1.00408	−0.502038	−	0.864846i	$-0.667416\pi$
$587$	−4.92039e9	−1.00408	−0.502038	+	0.864846i	$0.667416\pi$
$588$	0	0
$589$	−4.34552e9	−0.876271
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1.96899e9	−0.387750	−0.193875	−	0.981026i	$-0.562106\pi$
$593$	−1.96899e9	−0.387750	−0.193875	+	0.981026i	$0.562106\pi$
$594$	0	0
$595$	−5.50480e8	−0.107135
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−8.28842e9	−1.57572	−0.787858	−	0.615857i	$-0.788810\pi$
$599$	−8.28842e9	−1.57572	−0.787858	+	0.615857i	$0.788810\pi$
$600$	0	0
$601$	7.19345e9	1.35169	0.675844	−	0.737045i	$-0.263779\pi$
$601$	7.19345e9	1.35169	0.675844	+	0.737045i	$0.263779\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2.29526e9	−0.421394
$606$	0	0
$607$	5.35656e9	0.972132	0.486066	−	0.873922i	$-0.338431\pi$
$607$	5.35656e9	0.972132	0.486066	+	0.873922i	$0.338431\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3.01381e9	−0.534530
$612$	0	0
$613$	−1.01444e10	−1.77874	−0.889371	−	0.457187i	$-0.848857\pi$
$613$	−1.01444e10	−1.77874	−0.889371	+	0.457187i	$0.848857\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−2.40428e9	−0.412085	−0.206043	−	0.978543i	$-0.566059\pi$
$617$	−2.40428e9	−0.412085	−0.206043	+	0.978543i	$0.566059\pi$
$618$	0	0
$619$	2.75619e9	0.467081	0.233541	−	0.972347i	$-0.424969\pi$
$619$	2.75619e9	0.467081	0.233541	+	0.972347i	$0.424969\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.86068e8	0.0805357
$624$	0	0
$625$	2.93588e9	0.481014
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.39882e9	0.224122
$630$	0	0
$631$	−3.92975e9	−0.622676	−0.311338	−	0.950299i	$-0.600777\pi$
$631$	−3.92975e9	−0.622676	−0.311338	+	0.950299i	$0.600777\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2.52453e9	−0.391267
$636$	0	0
$637$	−3.65851e9	−0.560811
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.21986e10	1.82939	0.914696	−	0.404143i	$-0.132430\pi$
$641$	1.21986e10	1.82939	0.914696	+	0.404143i	$0.132430\pi$
$642$	0	0
$643$	−4.64582e8	−0.0689166	−0.0344583	−	0.999406i	$-0.510971\pi$
$643$	−4.64582e8	−0.0689166	−0.0344583	+	0.999406i	$0.510971\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.98889e9	−1.01448	−0.507239	−	0.861805i	$-0.669334\pi$
$647$	−6.98889e9	−1.01448	−0.507239	+	0.861805i	$0.669334\pi$
$648$	0	0
$649$	−1.07181e9	−0.153908
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3.44988e9	0.484850	0.242425	−	0.970170i	$-0.422057\pi$
$653$	3.44988e9	0.484850	0.242425	+	0.970170i	$0.422057\pi$
$654$	0	0
$655$	2.48287e9	0.345231
$656$	0	0
$657$	0	0
$658$	0	0
$659$	9.87453e9	1.34406	0.672029	−	0.740525i	$-0.265423\pi$
$659$	9.87453e9	1.34406	0.672029	+	0.740525i	$0.265423\pi$
$660$	0	0
$661$	−1.22825e10	−1.65417	−0.827087	−	0.562075i	$-0.810003\pi$
$661$	−1.22825e10	−1.65417	−0.827087	+	0.562075i	$0.810003\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	7.33386e8	0.0967067
$666$	0	0
$667$	−1.88544e10	−2.46021
$668$	0	0
$669$	0	0
$670$	0	0
$671$	7.84102e8	0.100194
$672$	0	0
$673$	9.21233e9	1.16498	0.582488	−	0.812840i	$-0.302079\pi$
$673$	9.21233e9	1.16498	0.582488	+	0.812840i	$0.302079\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.74226e9	0.463525	0.231763	−	0.972772i	$-0.425551\pi$
$677$	3.74226e9	0.463525	0.231763	+	0.972772i	$0.425551\pi$
$678$	0	0
$679$	−3.22557e9	−0.395423
$680$	0	0
$681$	0	0
$682$	0	0
$683$	4.17620e9	0.501544	0.250772	−	0.968046i	$-0.419316\pi$
$683$	4.17620e9	0.501544	0.250772	+	0.968046i	$0.419316\pi$
$684$	0	0
$685$	−6.37825e9	−0.758202
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−9.59848e9	−1.11798
$690$	0	0
$691$	6.94731e9	0.801020	0.400510	−	0.916292i	$-0.368833\pi$
$691$	6.94731e9	0.801020	0.400510	+	0.916292i	$0.368833\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.34011e9	0.264417
$696$	0	0
$697$	−1.36437e9	−0.152622
$698$	0	0
$699$	0	0
$700$	0	0
$701$	9.64715e9	1.05776	0.528879	−	0.848697i	$-0.322613\pi$
$701$	9.64715e9	1.05776	0.528879	+	0.848697i	$0.322613\pi$
$702$	0	0
$703$	−1.86360e9	−0.202306
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−7.50253e9	−0.798435
$708$	0	0
$709$	7.72407e9	0.813925	0.406962	−	0.913445i	$-0.366588\pi$
$709$	7.72407e9	0.813925	0.406962	+	0.913445i	$0.366588\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.85194e10	−2.94664
$714$	0	0
$715$	−3.86568e8	−0.0395508
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.11572e9	−0.111945	−0.0559725	−	0.998432i	$-0.517826\pi$
$719$	−1.11572e9	−0.111945	−0.0559725	+	0.998432i	$0.517826\pi$
$720$	0	0
$721$	2.63894e9	0.262214
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.12931e10	−1.10060
$726$	0	0
$727$	7.85026e9	0.757728	0.378864	−	0.925452i	$-0.376315\pi$
$727$	7.85026e9	0.757728	0.378864	+	0.925452i	$0.376315\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.39990e9	0.132552
$732$	0	0
$733$	−1.43652e10	−1.34725	−0.673625	−	0.739073i	$-0.735264\pi$
$733$	−1.43652e10	−1.34725	−0.673625	+	0.739073i	$0.735264\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.20829e9	−0.111182
$738$	0	0
$739$	−5.13869e9	−0.468379	−0.234189	−	0.972191i	$-0.575244\pi$
$739$	−5.13869e9	−0.468379	−0.234189	+	0.972191i	$0.575244\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.12182e10	1.00338	0.501688	−	0.865049i	$-0.332713\pi$
$743$	1.12182e10	1.00338	0.501688	+	0.865049i	$0.332713\pi$
$744$	0	0
$745$	1.02188e9	0.0905422
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.68920e9	−0.146891
$750$	0	0
$751$	−2.16341e10	−1.86380	−0.931899	−	0.362717i	$-0.881849\pi$
$751$	−2.16341e10	−1.86380	−0.931899	+	0.362717i	$0.881849\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−7.64424e9	−0.646427
$756$	0	0
$757$	−1.67963e10	−1.40727	−0.703635	−	0.710561i	$-0.748441\pi$
$757$	−1.67963e10	−1.40727	−0.703635	+	0.710561i	$0.748441\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.41213e10	1.98406	0.992030	−	0.126002i	$-0.0402146\pi$
$761$	2.41213e10	1.98406	0.992030	+	0.126002i	$0.0402146\pi$
$762$	0	0
$763$	5.54470e8	0.0451900
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.59088e9	0.767494
$768$	0	0
$769$	1.66703e10	1.32191	0.660954	−	0.750426i	$-0.270152\pi$
$769$	1.66703e10	1.32191	0.660954	+	0.750426i	$0.270152\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1.34400e10	−1.04658	−0.523288	−	0.852156i	$-0.675295\pi$
$773$	−1.34400e10	−1.04658	−0.523288	+	0.852156i	$0.675295\pi$
$774$	0	0
$775$	−1.70821e10	−1.31821
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.81771e9	0.137766
$780$	0	0
$781$	−2.43657e9	−0.183020
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−7.63611e9	−0.563415
$786$	0	0
$787$	−1.88162e10	−1.37601	−0.688004	−	0.725707i	$-0.741513\pi$
$787$	−1.88162e10	−1.37601	−0.688004	+	0.725707i	$0.741513\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.31115e9	−0.0941968
$792$	0	0
$793$	−7.01641e9	−0.499642
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.78944e9	0.405072	0.202536	−	0.979275i	$-0.435082\pi$
$797$	5.78944e9	0.405072	0.202536	+	0.979275i	$0.435082\pi$
$798$	0	0
$799$	−6.83034e9	−0.473727
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.31038e9	0.157463
$804$	0	0
$805$	4.81317e9	0.325197
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.81286e10	1.86779	0.933897	−	0.357543i	$-0.116385\pi$
$809$	2.81286e10	1.86779	0.933897	+	0.357543i	$0.116385\pi$
$810$	0	0
$811$	−6.63925e9	−0.437065	−0.218532	−	0.975830i	$-0.570127\pi$
$811$	−6.63925e9	−0.437065	−0.218532	+	0.975830i	$0.570127\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−8.24761e9	−0.533674
$816$	0	0
$817$	−1.86504e9	−0.119650
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5.35199e9	0.337532	0.168766	−	0.985656i	$-0.446022\pi$
$821$	5.35199e9	0.337532	0.168766	+	0.985656i	$0.446022\pi$
$822$	0	0
$823$	−1.10559e9	−0.0691343	−0.0345672	−	0.999402i	$-0.511005\pi$
$823$	−1.10559e9	−0.0691343	−0.0345672	+	0.999402i	$0.511005\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−9.56978e9	−0.588346	−0.294173	−	0.955752i	$-0.595044\pi$
$827$	−9.56978e9	−0.588346	−0.294173	+	0.955752i	$0.595044\pi$
$828$	0	0
$829$	−2.36632e10	−1.44256	−0.721278	−	0.692646i	$-0.756445\pi$
$829$	−2.36632e10	−1.44256	−0.721278	+	0.692646i	$0.756445\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−8.29145e9	−0.497019
$834$	0	0
$835$	−1.55321e9	−0.0923267
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−2.07240e10	−1.21145	−0.605726	−	0.795673i	$-0.707117\pi$
$839$	−2.07240e10	−1.21145	−0.605726	+	0.795673i	$0.707117\pi$
$840$	0	0
$841$	1.41556e10	0.820622
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−4.07068e9	−0.232097
$846$	0	0
$847$	7.21094e9	0.407756
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1.22307e10	−0.680297
$852$	0	0
$853$	−1.96254e10	−1.08267	−0.541336	−	0.840806i	$-0.682081\pi$
$853$	−1.96254e10	−1.08267	−0.541336	+	0.840806i	$0.682081\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.34086e10	1.27040	0.635202	−	0.772346i	$-0.280917\pi$
$857$	2.34086e10	1.27040	0.635202	+	0.772346i	$0.280917\pi$
$858$	0	0
$859$	−6.37986e9	−0.343428	−0.171714	−	0.985147i	$-0.554930\pi$
$859$	−6.37986e9	−0.343428	−0.171714	+	0.985147i	$0.554930\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.63727e9	−0.0867127	−0.0433564	−	0.999060i	$-0.513805\pi$
$863$	−1.63727e9	−0.0867127	−0.0433564	+	0.999060i	$0.513805\pi$
$864$	0	0
$865$	−5.06548e9	−0.266112
$866$	0	0
$867$	0	0
$868$	0	0
$869$	6.22339e8	0.0321705
$870$	0	0
$871$	1.08122e10	0.554434
$872$	0	0
$873$	0	0
$874$	0	0
$875$	6.41729e9	0.323835
$876$	0	0
$877$	−3.31776e10	−1.66091	−0.830455	−	0.557085i	$-0.811920\pi$
$877$	−3.31776e10	−1.66091	−0.830455	+	0.557085i	$0.811920\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−5.88175e9	−0.289795	−0.144897	−	0.989447i	$-0.546285\pi$
$881$	−5.88175e9	−0.289795	−0.144897	+	0.989447i	$0.546285\pi$
$882$	0	0
$883$	−5.84882e9	−0.285894	−0.142947	−	0.989730i	$-0.545658\pi$
$883$	−5.84882e9	−0.285894	−0.142947	+	0.989730i	$0.545658\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	7.81095e9	0.375813	0.187906	−	0.982187i	$-0.439830\pi$
$887$	7.81095e9	0.375813	0.187906	+	0.982187i	$0.439830\pi$
$888$	0	0
$889$	7.93124e9	0.378604
$890$	0	0
$891$	0	0
$892$	0	0
$893$	9.09982e9	0.427615
$894$	0	0
$895$	−1.36368e10	−0.635815
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.75045e10	2.18060
$900$	0	0
$901$	−2.17535e10	−0.990813
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.53506e10	−0.688424
$906$	0	0
$907$	4.12701e10	1.83658	0.918289	−	0.395910i	$-0.129571\pi$
$907$	4.12701e10	1.83658	0.918289	+	0.395910i	$0.129571\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−6.66422e9	−0.292035	−0.146017	−	0.989282i	$-0.546646\pi$
$911$	−6.66422e9	−0.292035	−0.146017	+	0.989282i	$0.546646\pi$
$912$	0	0
$913$	5.52214e9	0.240137
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−7.80035e9	−0.334058
$918$	0	0
$919$	−5.45018e9	−0.231636	−0.115818	−	0.993270i	$-0.536949\pi$
$919$	−5.45018e9	−0.231636	−0.115818	+	0.993270i	$0.536949\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.18032e10	0.912671
$924$	0	0
$925$	−7.32576e9	−0.304338
$926$	0	0
$927$	0	0
$928$	0	0
$929$	7.78356e9	0.318510	0.159255	−	0.987237i	$-0.449091\pi$
$929$	7.78356e9	0.318510	0.159255	+	0.987237i	$0.449091\pi$
$930$	0	0
$931$	1.10464e10	0.448639
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−8.76096e8	−0.0350519
$936$	0	0
$937$	4.59669e10	1.82539	0.912697	−	0.408638i	$-0.133996\pi$
$937$	4.59669e10	1.82539	0.912697	+	0.408638i	$0.133996\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1.63152e10	−0.638305	−0.319152	−	0.947703i	$-0.603398\pi$
$941$	−1.63152e10	−0.638305	−0.319152	+	0.947703i	$0.603398\pi$
$942$	0	0
$943$	1.19295e10	0.463268
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.13934e10	−1.20119	−0.600597	−	0.799552i	$-0.705071\pi$
$947$	−3.13934e10	−1.20119	−0.600597	+	0.799552i	$0.705071\pi$
$948$	0	0
$949$	−2.06741e10	−0.785226
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−4.82350e10	−1.80525	−0.902625	−	0.430428i	$-0.858363\pi$
$953$	−4.82350e10	−1.80525	−0.902625	+	0.430428i	$0.858363\pi$
$954$	0	0
$955$	2.53008e9	0.0939989
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.00383e10	0.733663
$960$	0	0
$961$	4.43435e10	1.61175
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.67785e10	−0.601047
$966$	0	0
$967$	−3.22238e9	−0.114600	−0.0572999	−	0.998357i	$-0.518249\pi$
$967$	−3.22238e9	−0.114600	−0.0572999	+	0.998357i	$0.518249\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−2.25397e10	−0.790096	−0.395048	−	0.918660i	$-0.629272\pi$
$971$	−2.25397e10	−0.790096	−0.395048	+	0.918660i	$0.629272\pi$
$972$	0	0
$973$	−7.35184e9	−0.255859
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.31576e10	1.48056	0.740280	−	0.672298i	$-0.234693\pi$
$977$	4.31576e10	1.48056	0.740280	+	0.672298i	$0.234693\pi$
$978$	0	0
$979$	7.73582e8	0.0263492
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.28659e10	0.767806	0.383903	−	0.923373i	$-0.374580\pi$
$983$	2.28659e10	0.767806	0.383903	+	0.923373i	$0.374580\pi$
$984$	0	0
$985$	1.75159e10	0.583991
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.22402e10	−0.402347
$990$	0	0
$991$	−1.13727e10	−0.371198	−0.185599	−	0.982626i	$-0.559423\pi$
$991$	−1.13727e10	−0.371198	−0.185599	+	0.982626i	$0.559423\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	6.13376e9	0.197399
$996$	0	0
$997$	−4.31535e10	−1.37906	−0.689530	−	0.724257i	$-0.742183\pi$
$997$	−4.31535e10	−1.37906	−0.689530	+	0.724257i	$0.742183\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.8.a.g.1.1		1
3.2	odd	2		432.8.a.b.1.1		1
4.3	odd	2		54.8.a.f.1.1	yes	1
12.11	even	2		54.8.a.a.1.1	✓	1
36.7	odd	6		162.8.c.b.109.1		2
36.11	even	6		162.8.c.k.109.1		2
36.23	even	6		162.8.c.k.55.1		2
36.31	odd	6		162.8.c.b.55.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.8.a.a.1.1	✓	1	12.11	even	2
54.8.a.f.1.1	yes	1	4.3	odd	2
162.8.c.b.55.1		2	36.31	odd	6
162.8.c.b.109.1		2	36.7	odd	6
162.8.c.k.55.1		2	36.23	even	6
162.8.c.k.109.1		2	36.11	even	6
432.8.a.b.1.1		1	3.2	odd	2
432.8.a.g.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	432.a (trivial)

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