LMFDB - Embedded newform 528.8.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [528,8,Mod(1,528)]

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

lf = mfeigenbasis(mf)

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, 8, names="a")

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

chi := DirichletCharacter("528.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 528 = 2^{4} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	528.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:	$164.939293456$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 66)
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)

$f(q)$

$=$

$q+27.0000 q^{3} +286.000 q^{7} +729.000 q^{9} +O(q^{10})$

$q+27.0000 q^{3} +286.000 q^{7} +729.000 q^{9} +1331.00 q^{11} -7432.00 q^{13} -9282.00 q^{17} +15496.0 q^{19} +7722.00 q^{21} +63882.0 q^{23} -78125.0 q^{25} +19683.0 q^{27} -102486. q^{29} +24232.0 q^{31} +35937.0 q^{33} -80890.0 q^{37} -200664. q^{39} +27426.0 q^{41} -218564. q^{43} -447378. q^{47} -741747. q^{49} -250614. q^{51} +438600. q^{53} +418392. q^{57} +207480. q^{59} +111428. q^{61} +208494. q^{63} -1.00329e6 q^{67} +1.72481e6 q^{69} +483474. q^{71} +5.76387e6 q^{73} -2.10938e6 q^{75} +380666. q^{77} -7.11897e6 q^{79} +531441. q^{81} -2.82572e6 q^{83} -2.76712e6 q^{87} -4.29845e6 q^{89} -2.12555e6 q^{91} +654264. q^{93} +2.25616e6 q^{97} +970299. q^{99} +O(q^{100})$

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
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	286.000	0.315154	0.157577	−	0.987507i	$-0.449632\pi$
$7$	286.000	0.315154	0.157577	+	0.987507i	$0.449632\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	1331.00	0.301511
$11$	1331.00	0.301511
$12$	0	0
$13$	−7432.00	−0.938218	−0.469109	−	0.883140i	$-0.655425\pi$
$13$	−7432.00	−0.938218	−0.469109	+	0.883140i	$0.655425\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−9282.00	−0.458216	−0.229108	−	0.973401i	$-0.573581\pi$
$17$	−9282.00	−0.458216	−0.229108	+	0.973401i	$0.573581\pi$
$18$	0	0
$19$	15496.0	0.518301	0.259150	−	0.965837i	$-0.416557\pi$
$19$	15496.0	0.518301	0.259150	+	0.965837i	$0.416557\pi$
$20$	0	0
$21$	7722.00	0.181954
$22$	0	0
$23$	63882.0	1.09479	0.547395	−	0.836874i	$-0.315619\pi$
$23$	63882.0	1.09479	0.547395	+	0.836874i	$0.315619\pi$
$24$	0	0
$25$	−78125.0	−1.00000
$26$	0	0
$27$	19683.0	0.192450
$28$	0	0
$29$	−102486.	−0.780318	−0.390159	−	0.920748i	$-0.627580\pi$
$29$	−102486.	−0.780318	−0.390159	+	0.920748i	$0.627580\pi$
$30$	0	0
$31$	24232.0	0.146091	0.0730455	−	0.997329i	$-0.476728\pi$
$31$	24232.0	0.146091	0.0730455	+	0.997329i	$0.476728\pi$
$32$	0	0
$33$	35937.0	0.174078
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−80890.0	−0.262536	−0.131268	−	0.991347i	$-0.541905\pi$
$37$	−80890.0	−0.262536	−0.131268	+	0.991347i	$0.541905\pi$
$38$	0	0
$39$	−200664.	−0.541681
$40$	0	0
$41$	27426.0	0.0621468	0.0310734	−	0.999517i	$-0.490107\pi$
$41$	27426.0	0.0621468	0.0310734	+	0.999517i	$0.490107\pi$
$42$	0	0
$43$	−218564.	−0.419217	−0.209609	−	0.977785i	$-0.567219\pi$
$43$	−218564.	−0.419217	−0.209609	+	0.977785i	$0.567219\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−447378.	−0.628539	−0.314269	−	0.949334i	$-0.601760\pi$
$47$	−447378.	−0.628539	−0.314269	+	0.949334i	$0.601760\pi$
$48$	0	0
$49$	−741747.	−0.900678
$50$	0	0
$51$	−250614.	−0.264551
$52$	0	0
$53$	438600.	0.404672	0.202336	−	0.979316i	$-0.435147\pi$
$53$	438600.	0.404672	0.202336	+	0.979316i	$0.435147\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	418392.	0.299241
$58$	0	0
$59$	207480.	0.131521	0.0657604	−	0.997835i	$-0.479053\pi$
$59$	207480.	0.131521	0.0657604	+	0.997835i	$0.479053\pi$
$60$	0	0
$61$	111428.	0.0628550	0.0314275	−	0.999506i	$-0.489995\pi$
$61$	111428.	0.0628550	0.0314275	+	0.999506i	$0.489995\pi$
$62$	0	0
$63$	208494.	0.105051
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−1.00329e6	−0.407536	−0.203768	−	0.979019i	$-0.565319\pi$
$67$	−1.00329e6	−0.407536	−0.203768	+	0.979019i	$0.565319\pi$
$68$	0	0
$69$	1.72481e6	0.632078
$70$	0	0
$71$	483474.	0.160313	0.0801565	−	0.996782i	$-0.474458\pi$
$71$	483474.	0.160313	0.0801565	+	0.996782i	$0.474458\pi$
$72$	0	0
$73$	5.76387e6	1.73414	0.867069	−	0.498187i	$-0.166001\pi$
$73$	5.76387e6	1.73414	0.867069	+	0.498187i	$0.166001\pi$
$74$	0	0
$75$	−2.10938e6	−0.577350
$76$	0	0
$77$	380666.	0.0950225
$78$	0	0
$79$	−7.11897e6	−1.62451	−0.812254	−	0.583303i	$-0.801760\pi$
$79$	−7.11897e6	−1.62451	−0.812254	+	0.583303i	$0.801760\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	−2.82572e6	−0.542446	−0.271223	−	0.962517i	$-0.587428\pi$
$83$	−2.82572e6	−0.542446	−0.271223	+	0.962517i	$0.587428\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.76712e6	−0.450517
$88$	0	0
$89$	−4.29845e6	−0.646320	−0.323160	−	0.946344i	$-0.604745\pi$
$89$	−4.29845e6	−0.646320	−0.323160	+	0.946344i	$0.604745\pi$
$90$	0	0
$91$	−2.12555e6	−0.295683
$92$	0	0
$93$	654264.	0.0843457
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.25616e6	0.250997	0.125498	−	0.992094i	$-0.459947\pi$
$97$	2.25616e6	0.250997	0.125498	+	0.992094i	$0.459947\pi$
$98$	0	0
$99$	970299.	0.100504
$100$	0	0
$101$	−1.32338e7	−1.27808	−0.639041	−	0.769172i	$-0.720669\pi$
$101$	−1.32338e7	−1.27808	−0.639041	+	0.769172i	$0.720669\pi$
$102$	0	0
$103$	−1.17830e7	−1.06249	−0.531245	−	0.847218i	$-0.678276\pi$
$103$	−1.17830e7	−1.06249	−0.531245	+	0.847218i	$0.678276\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.32269e7	1.83294	0.916470	−	0.400102i	$-0.131025\pi$
$107$	2.32269e7	1.83294	0.916470	+	0.400102i	$0.131025\pi$
$108$	0	0
$109$	−5.25350e6	−0.388558	−0.194279	−	0.980946i	$-0.562237\pi$
$109$	−5.25350e6	−0.388558	−0.194279	+	0.980946i	$0.562237\pi$
$110$	0	0
$111$	−2.18403e6	−0.151575
$112$	0	0
$113$	−1.67762e7	−1.09376	−0.546878	−	0.837212i	$-0.684184\pi$
$113$	−1.67762e7	−1.09376	−0.546878	+	0.837212i	$0.684184\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−5.41793e6	−0.312739
$118$	0	0
$119$	−2.65465e6	−0.144409
$120$	0	0
$121$	1.77156e6	0.0909091
$122$	0	0
$123$	740502.	0.0358805
$124$	0	0
$125$	0	0
$126$	0	0
$127$	7.50308e6	0.325032	0.162516	−	0.986706i	$-0.448039\pi$
$127$	7.50308e6	0.325032	0.162516	+	0.986706i	$0.448039\pi$
$128$	0	0
$129$	−5.90123e6	−0.242035
$130$	0	0
$131$	2.69784e7	1.04850	0.524248	−	0.851566i	$-0.324346\pi$
$131$	2.69784e7	1.04850	0.524248	+	0.851566i	$0.324346\pi$
$132$	0	0
$133$	4.43186e6	0.163345
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.26711e7	−1.08553	−0.542766	−	0.839884i	$-0.682623\pi$
$137$	−3.26711e7	−1.08553	−0.542766	+	0.839884i	$0.682623\pi$
$138$	0	0
$139$	−3.11583e7	−0.984062	−0.492031	−	0.870578i	$-0.663745\pi$
$139$	−3.11583e7	−0.984062	−0.492031	+	0.870578i	$0.663745\pi$
$140$	0	0
$141$	−1.20792e7	−0.362887
$142$	0	0
$143$	−9.89199e6	−0.282884
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−2.00272e7	−0.520007
$148$	0	0
$149$	−3.68970e7	−0.913775	−0.456887	−	0.889525i	$-0.651036\pi$
$149$	−3.68970e7	−0.913775	−0.456887	+	0.889525i	$0.651036\pi$
$150$	0	0
$151$	3.94619e7	0.932736	0.466368	−	0.884591i	$-0.345562\pi$
$151$	3.94619e7	0.932736	0.466368	+	0.884591i	$0.345562\pi$
$152$	0	0
$153$	−6.76658e6	−0.152739
$154$	0	0
$155$	0	0
$156$	0	0
$157$	5.38879e7	1.11133	0.555664	−	0.831407i	$-0.312464\pi$
$157$	5.38879e7	1.11133	0.555664	+	0.831407i	$0.312464\pi$
$158$	0	0
$159$	1.18422e7	0.233637
$160$	0	0
$161$	1.82703e7	0.345028
$162$	0	0
$163$	−407588.	−0.00737165	−0.00368583	−	0.999993i	$-0.501173\pi$
$163$	−407588.	−0.00737165	−0.00368583	+	0.999993i	$0.501173\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.16670e7	−0.526138	−0.263069	−	0.964777i	$-0.584735\pi$
$167$	−3.16670e7	−0.526138	−0.263069	+	0.964777i	$0.584735\pi$
$168$	0	0
$169$	−7.51389e6	−0.119746
$170$	0	0
$171$	1.12966e7	0.172767
$172$	0	0
$173$	1.45497e7	0.213644	0.106822	−	0.994278i	$-0.465932\pi$
$173$	1.45497e7	0.213644	0.106822	+	0.994278i	$0.465932\pi$
$174$	0	0
$175$	−2.23438e7	−0.315154
$176$	0	0
$177$	5.60196e6	0.0759335
$178$	0	0
$179$	−9.83688e7	−1.28195	−0.640976	−	0.767561i	$-0.721470\pi$
$179$	−9.83688e7	−1.28195	−0.640976	+	0.767561i	$0.721470\pi$
$180$	0	0
$181$	2.81986e6	0.0353470	0.0176735	−	0.999844i	$-0.494374\pi$
$181$	2.81986e6	0.0353470	0.0176735	+	0.999844i	$0.494374\pi$
$182$	0	0
$183$	3.00856e6	0.0362894
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.23543e7	−0.138157
$188$	0	0
$189$	5.62934e6	0.0606514
$190$	0	0
$191$	−2.90471e7	−0.301638	−0.150819	−	0.988561i	$-0.548191\pi$
$191$	−2.90471e7	−0.301638	−0.150819	+	0.988561i	$0.548191\pi$
$192$	0	0
$193$	−5.28725e7	−0.529395	−0.264697	−	0.964332i	$-0.585272\pi$
$193$	−5.28725e7	−0.529395	−0.264697	+	0.964332i	$0.585272\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.11321e7	−0.756068	−0.378034	−	0.925792i	$-0.623400\pi$
$197$	−8.11321e7	−0.756068	−0.378034	+	0.925792i	$0.623400\pi$
$198$	0	0
$199$	4.92590e7	0.443098	0.221549	−	0.975149i	$-0.428889\pi$
$199$	4.92590e7	0.443098	0.221549	+	0.975149i	$0.428889\pi$
$200$	0	0
$201$	−2.70889e7	−0.235291
$202$	0	0
$203$	−2.93110e7	−0.245920
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.65700e7	0.364930
$208$	0	0
$209$	2.06252e7	0.156274
$210$	0	0
$211$	−1.51736e8	−1.11199	−0.555994	−	0.831186i	$-0.687662\pi$
$211$	−1.51736e8	−1.11199	−0.555994	+	0.831186i	$0.687662\pi$
$212$	0	0
$213$	1.30538e7	0.0925568
$214$	0	0
$215$	0	0
$216$	0	0
$217$	6.93035e6	0.0460412
$218$	0	0
$219$	1.55624e8	1.00121
$220$	0	0
$221$	6.89838e7	0.429907
$222$	0	0
$223$	−1.94332e8	−1.17349	−0.586743	−	0.809774i	$-0.699590\pi$
$223$	−1.94332e8	−1.17349	−0.586743	+	0.809774i	$0.699590\pi$
$224$	0	0
$225$	−5.69531e7	−0.333333
$226$	0	0
$227$	−3.39809e7	−0.192817	−0.0964083	−	0.995342i	$-0.530735\pi$
$227$	−3.39809e7	−0.192817	−0.0964083	+	0.995342i	$0.530735\pi$
$228$	0	0
$229$	−2.25104e8	−1.23868	−0.619339	−	0.785124i	$-0.712599\pi$
$229$	−2.25104e8	−1.23868	−0.619339	+	0.785124i	$0.712599\pi$
$230$	0	0
$231$	1.02780e7	0.0548613
$232$	0	0
$233$	3.22683e8	1.67121	0.835603	−	0.549333i	$-0.185118\pi$
$233$	3.22683e8	1.67121	0.835603	+	0.549333i	$0.185118\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−1.92212e8	−0.937911
$238$	0	0
$239$	6.04520e7	0.286430	0.143215	−	0.989692i	$-0.454256\pi$
$239$	6.04520e7	0.286430	0.143215	+	0.989692i	$0.454256\pi$
$240$	0	0
$241$	−1.23561e8	−0.568620	−0.284310	−	0.958732i	$-0.591765\pi$
$241$	−1.23561e8	−0.568620	−0.284310	+	0.958732i	$0.591765\pi$
$242$	0	0
$243$	1.43489e7	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.15166e8	−0.486280
$248$	0	0
$249$	−7.62945e7	−0.313181
$250$	0	0
$251$	−4.22604e7	−0.168685	−0.0843423	−	0.996437i	$-0.526879\pi$
$251$	−4.22604e7	−0.168685	−0.0843423	+	0.996437i	$0.526879\pi$
$252$	0	0
$253$	8.50269e7	0.330092
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.70828e8	−0.627761	−0.313880	−	0.949463i	$-0.601629\pi$
$257$	−1.70828e8	−0.627761	−0.313880	+	0.949463i	$0.601629\pi$
$258$	0	0
$259$	−2.31345e7	−0.0827393
$260$	0	0
$261$	−7.47123e7	−0.260106
$262$	0	0
$263$	−6.84528e7	−0.232031	−0.116016	−	0.993247i	$-0.537012\pi$
$263$	−6.84528e7	−0.232031	−0.116016	+	0.993247i	$0.537012\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1.16058e8	−0.373153
$268$	0	0
$269$	−3.36792e8	−1.05494	−0.527471	−	0.849573i	$-0.676860\pi$
$269$	−3.36792e8	−1.05494	−0.527471	+	0.849573i	$0.676860\pi$
$270$	0	0
$271$	−3.85975e8	−1.17806	−0.589028	−	0.808112i	$-0.700489\pi$
$271$	−3.85975e8	−1.17806	−0.589028	+	0.808112i	$0.700489\pi$
$272$	0	0
$273$	−5.73899e7	−0.170713
$274$	0	0
$275$	−1.03984e8	−0.301511
$276$	0	0
$277$	3.62042e8	1.02348	0.511741	−	0.859140i	$-0.329001\pi$
$277$	3.62042e8	1.02348	0.511741	+	0.859140i	$0.329001\pi$
$278$	0	0
$279$	1.76651e7	0.0486970
$280$	0	0
$281$	3.80465e7	0.102292	0.0511462	−	0.998691i	$-0.483713\pi$
$281$	3.80465e7	0.102292	0.0511462	+	0.998691i	$0.483713\pi$
$282$	0	0
$283$	−3.02002e8	−0.792060	−0.396030	−	0.918238i	$-0.629612\pi$
$283$	−3.02002e8	−0.792060	−0.396030	+	0.918238i	$0.629612\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.84384e6	0.0195858
$288$	0	0
$289$	−3.24183e8	−0.790038
$290$	0	0
$291$	6.09163e7	0.144913
$292$	0	0
$293$	−6.70179e8	−1.55652	−0.778259	−	0.627944i	$-0.783897\pi$
$293$	−6.70179e8	−1.55652	−0.778259	+	0.627944i	$0.783897\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.61981e7	0.0580259
$298$	0	0
$299$	−4.74771e8	−1.02715
$300$	0	0
$301$	−6.25093e7	−0.132118
$302$	0	0
$303$	−3.57312e8	−0.737901
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−8.04368e6	−0.0158661	−0.00793306	−	0.999969i	$-0.502525\pi$
$307$	−8.04368e6	−0.0158661	−0.00793306	+	0.999969i	$0.502525\pi$
$308$	0	0
$309$	−3.18141e8	−0.613429
$310$	0	0
$311$	−6.09950e8	−1.14983	−0.574914	−	0.818214i	$-0.694965\pi$
$311$	−6.09950e8	−1.14983	−0.574914	+	0.818214i	$0.694965\pi$
$312$	0	0
$313$	1.23512e8	0.227668	0.113834	−	0.993500i	$-0.463687\pi$
$313$	1.23512e8	0.227668	0.113834	+	0.993500i	$0.463687\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.03290e8	−0.887382	−0.443691	−	0.896180i	$-0.646331\pi$
$317$	−5.03290e8	−0.887382	−0.443691	+	0.896180i	$0.646331\pi$
$318$	0	0
$319$	−1.36409e8	−0.235275
$320$	0	0
$321$	6.27127e8	1.05825
$322$	0	0
$323$	−1.43834e8	−0.237494
$324$	0	0
$325$	5.80625e8	0.938218
$326$	0	0
$327$	−1.41844e8	−0.224334
$328$	0	0
$329$	−1.27950e8	−0.198087
$330$	0	0
$331$	7.87410e8	1.19345	0.596723	−	0.802447i	$-0.296469\pi$
$331$	7.87410e8	1.19345	0.596723	+	0.802447i	$0.296469\pi$
$332$	0	0
$333$	−5.89688e7	−0.0875120
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.57110e7	0.0223613	0.0111807	−	0.999937i	$-0.496441\pi$
$337$	1.57110e7	0.0223613	0.0111807	+	0.999937i	$0.496441\pi$
$338$	0	0
$339$	−4.52959e8	−0.631480
$340$	0	0
$341$	3.22528e7	0.0440481
$342$	0	0
$343$	−4.47673e8	−0.599006
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.84980e8	−0.237669	−0.118834	−	0.992914i	$-0.537916\pi$
$347$	−1.84980e8	−0.237669	−0.118834	+	0.992914i	$0.537916\pi$
$348$	0	0
$349$	−5.09794e8	−0.641956	−0.320978	−	0.947087i	$-0.604012\pi$
$349$	−5.09794e8	−0.641956	−0.320978	+	0.947087i	$0.604012\pi$
$350$	0	0
$351$	−1.46284e8	−0.180560
$352$	0	0
$353$	−1.24938e9	−1.51176	−0.755882	−	0.654708i	$-0.772791\pi$
$353$	−1.24938e9	−1.51176	−0.755882	+	0.654708i	$0.772791\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−7.16756e7	−0.0833744
$358$	0	0
$359$	1.08052e9	1.23255	0.616273	−	0.787532i	$-0.288642\pi$
$359$	1.08052e9	1.23255	0.616273	+	0.787532i	$0.288642\pi$
$360$	0	0
$361$	−6.53746e8	−0.731364
$362$	0	0
$363$	4.78321e7	0.0524864
$364$	0	0
$365$	0	0
$366$	0	0
$367$	6.18303e8	0.652935	0.326468	−	0.945208i	$-0.394142\pi$
$367$	6.18303e8	0.652935	0.326468	+	0.945208i	$0.394142\pi$
$368$	0	0
$369$	1.99936e7	0.0207156
$370$	0	0
$371$	1.25440e8	0.127534
$372$	0	0
$373$	−1.47762e9	−1.47429	−0.737143	−	0.675737i	$-0.763825\pi$
$373$	−1.47762e9	−1.47429	−0.737143	+	0.675737i	$0.763825\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.61676e8	0.732109
$378$	0	0
$379$	1.24874e9	1.17824	0.589120	−	0.808046i	$-0.299475\pi$
$379$	1.24874e9	1.17824	0.589120	+	0.808046i	$0.299475\pi$
$380$	0	0
$381$	2.02583e8	0.187658
$382$	0	0
$383$	8.56895e8	0.779349	0.389674	−	0.920953i	$-0.372588\pi$
$383$	8.56895e8	0.779349	0.389674	+	0.920953i	$0.372588\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.59333e8	−0.139739
$388$	0	0
$389$	−1.45082e9	−1.24966	−0.624828	−	0.780762i	$-0.714831\pi$
$389$	−1.45082e9	−1.24966	−0.624828	+	0.780762i	$0.714831\pi$
$390$	0	0
$391$	−5.92953e8	−0.501651
$392$	0	0
$393$	7.28417e8	0.605350
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−2.34266e9	−1.87907	−0.939536	−	0.342450i	$-0.888743\pi$
$397$	−2.34266e9	−1.87907	−0.939536	+	0.342450i	$0.888743\pi$
$398$	0	0
$399$	1.19660e8	0.0943071
$400$	0	0
$401$	−1.19862e9	−0.928273	−0.464136	−	0.885764i	$-0.653635\pi$
$401$	−1.19862e9	−0.928273	−0.464136	+	0.885764i	$0.653635\pi$
$402$	0	0
$403$	−1.80092e8	−0.137065
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.07665e8	−0.0791576
$408$	0	0
$409$	1.66474e9	1.20314	0.601568	−	0.798821i	$-0.294543\pi$
$409$	1.66474e9	1.20314	0.601568	+	0.798821i	$0.294543\pi$
$410$	0	0
$411$	−8.82121e8	−0.626732
$412$	0	0
$413$	5.93393e7	0.0414493
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−8.41275e8	−0.568149
$418$	0	0
$419$	2.75209e9	1.82774	0.913869	−	0.406009i	$-0.133080\pi$
$419$	2.75209e9	1.82774	0.913869	+	0.406009i	$0.133080\pi$
$420$	0	0
$421$	2.03898e9	1.33176	0.665881	−	0.746058i	$-0.268056\pi$
$421$	2.03898e9	1.33176	0.665881	+	0.746058i	$0.268056\pi$
$422$	0	0
$423$	−3.26139e8	−0.209513
$424$	0	0
$425$	7.25156e8	0.458216
$426$	0	0
$427$	3.18684e7	0.0198090
$428$	0	0
$429$	−2.67084e8	−0.163323
$430$	0	0
$431$	−2.80381e9	−1.68685	−0.843427	−	0.537243i	$-0.819466\pi$
$431$	−2.80381e9	−1.68685	−0.843427	+	0.537243i	$0.819466\pi$
$432$	0	0
$433$	1.73940e9	1.02966	0.514828	−	0.857293i	$-0.327856\pi$
$433$	1.73940e9	1.02966	0.514828	+	0.857293i	$0.327856\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9.89915e8	0.567431
$438$	0	0
$439$	−1.01004e9	−0.569787	−0.284893	−	0.958559i	$-0.591958\pi$
$439$	−1.01004e9	−0.569787	−0.284893	+	0.958559i	$0.591958\pi$
$440$	0	0
$441$	−5.40734e8	−0.300226
$442$	0	0
$443$	−2.79131e9	−1.52544	−0.762719	−	0.646730i	$-0.776136\pi$
$443$	−2.79131e9	−1.52544	−0.762719	+	0.646730i	$0.776136\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−9.96219e8	−0.527568
$448$	0	0
$449$	2.32051e9	1.20982	0.604911	−	0.796293i	$-0.293209\pi$
$449$	2.32051e9	1.20982	0.604911	+	0.796293i	$0.293209\pi$
$450$	0	0
$451$	3.65040e7	0.0187380
$452$	0	0
$453$	1.06547e9	0.538515
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.97318e9	0.967074	0.483537	−	0.875324i	$-0.339352\pi$
$457$	1.97318e9	0.967074	0.483537	+	0.875324i	$0.339352\pi$
$458$	0	0
$459$	−1.82698e8	−0.0881837
$460$	0	0
$461$	2.39194e9	1.13710	0.568548	−	0.822650i	$-0.307505\pi$
$461$	2.39194e9	1.13710	0.568548	+	0.822650i	$0.307505\pi$
$462$	0	0
$463$	2.38336e9	1.11598	0.557991	−	0.829847i	$-0.311573\pi$
$463$	2.38336e9	1.11598	0.557991	+	0.829847i	$0.311573\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−2.15610e9	−0.979627	−0.489813	−	0.871827i	$-0.662935\pi$
$467$	−2.15610e9	−0.979627	−0.489813	+	0.871827i	$0.662935\pi$
$468$	0	0
$469$	−2.86942e8	−0.128436
$470$	0	0
$471$	1.45497e9	0.641626
$472$	0	0
$473$	−2.90909e8	−0.126399
$474$	0	0
$475$	−1.21062e9	−0.518301
$476$	0	0
$477$	3.19739e8	0.134891
$478$	0	0
$479$	−1.77087e8	−0.0736229	−0.0368114	−	0.999322i	$-0.511720\pi$
$479$	−1.77087e8	−0.0736229	−0.0368114	+	0.999322i	$0.511720\pi$
$480$	0	0
$481$	6.01174e8	0.246316
$482$	0	0
$483$	4.93297e8	0.199202
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1.85468e9	0.727640	0.363820	−	0.931469i	$-0.381472\pi$
$487$	1.85468e9	0.727640	0.363820	+	0.931469i	$0.381472\pi$
$488$	0	0
$489$	−1.10049e7	−0.00425602
$490$	0	0
$491$	1.13886e9	0.434197	0.217099	−	0.976150i	$-0.430341\pi$
$491$	1.13886e9	0.434197	0.217099	+	0.976150i	$0.430341\pi$
$492$	0	0
$493$	9.51275e8	0.357554
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.38274e8	0.0505233
$498$	0	0
$499$	4.67880e9	1.68571	0.842855	−	0.538141i	$-0.180873\pi$
$499$	4.67880e9	1.68571	0.842855	+	0.538141i	$0.180873\pi$
$500$	0	0
$501$	−8.55009e8	−0.303766
$502$	0	0
$503$	−1.04743e9	−0.366977	−0.183488	−	0.983022i	$-0.558739\pi$
$503$	−1.04743e9	−0.366977	−0.183488	+	0.983022i	$0.558739\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−2.02875e8	−0.0691355
$508$	0	0
$509$	4.44149e9	1.49285	0.746426	−	0.665468i	$-0.231768\pi$
$509$	4.44149e9	1.49285	0.746426	+	0.665468i	$0.231768\pi$
$510$	0	0
$511$	1.64847e9	0.546521
$512$	0	0
$513$	3.05008e8	0.0997471
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−5.95460e8	−0.189512
$518$	0	0
$519$	3.92841e8	0.123348
$520$	0	0
$521$	4.39144e8	0.136043	0.0680213	−	0.997684i	$-0.478331\pi$
$521$	4.39144e8	0.136043	0.0680213	+	0.997684i	$0.478331\pi$
$522$	0	0
$523$	2.96971e9	0.907732	0.453866	−	0.891070i	$-0.350045\pi$
$523$	2.96971e9	0.907732	0.453866	+	0.891070i	$0.350045\pi$
$524$	0	0
$525$	−6.03281e8	−0.181954
$526$	0	0
$527$	−2.24921e8	−0.0669412
$528$	0	0
$529$	6.76084e8	0.198567
$530$	0	0
$531$	1.51253e8	0.0438402
$532$	0	0
$533$	−2.03830e8	−0.0583073
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−2.65596e9	−0.740135
$538$	0	0
$539$	−9.87265e8	−0.271565
$540$	0	0
$541$	1.79271e9	0.486766	0.243383	−	0.969930i	$-0.421743\pi$
$541$	1.79271e9	0.486766	0.243383	+	0.969930i	$0.421743\pi$
$542$	0	0
$543$	7.61362e7	0.0204076
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−2.25811e8	−0.0589915	−0.0294957	−	0.999565i	$-0.509390\pi$
$547$	−2.25811e8	−0.0589915	−0.0294957	+	0.999565i	$0.509390\pi$
$548$	0	0
$549$	8.12310e7	0.0209517
$550$	0	0
$551$	−1.58812e9	−0.404439
$552$	0	0
$553$	−2.03602e9	−0.511971
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3.64087e8	−0.0892714	−0.0446357	−	0.999003i	$-0.514213\pi$
$557$	−3.64087e8	−0.0892714	−0.0446357	+	0.999003i	$0.514213\pi$
$558$	0	0
$559$	1.62437e9	0.393317
$560$	0	0
$561$	−3.33567e8	−0.0797652
$562$	0	0
$563$	3.79244e9	0.895652	0.447826	−	0.894121i	$-0.352198\pi$
$563$	3.79244e9	0.895652	0.447826	+	0.894121i	$0.352198\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.51992e8	0.0350171
$568$	0	0
$569$	−3.58750e9	−0.816393	−0.408196	−	0.912894i	$-0.633842\pi$
$569$	−3.58750e9	−0.816393	−0.408196	+	0.912894i	$0.633842\pi$
$570$	0	0
$571$	8.16196e9	1.83471	0.917356	−	0.398068i	$-0.130319\pi$
$571$	8.16196e9	1.83471	0.917356	+	0.398068i	$0.130319\pi$
$572$	0	0
$573$	−7.84272e8	−0.174151
$574$	0	0
$575$	−4.99078e9	−1.09479
$576$	0	0
$577$	−6.01839e9	−1.30426	−0.652131	−	0.758106i	$-0.726125\pi$
$577$	−6.01839e9	−1.30426	−0.652131	+	0.758106i	$0.726125\pi$
$578$	0	0
$579$	−1.42756e9	−0.305646
$580$	0	0
$581$	−8.08157e8	−0.170954
$582$	0	0
$583$	5.83777e8	0.122013
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6.19124e9	1.26341	0.631704	−	0.775209i	$-0.282356\pi$
$587$	6.19124e9	1.26341	0.631704	+	0.775209i	$0.282356\pi$
$588$	0	0
$589$	3.75499e8	0.0757191
$590$	0	0
$591$	−2.19057e9	−0.436516
$592$	0	0
$593$	3.07325e9	0.605210	0.302605	−	0.953116i	$-0.402144\pi$
$593$	3.07325e9	0.605210	0.302605	+	0.953116i	$0.402144\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.32999e9	0.255823
$598$	0	0
$599$	5.29706e9	1.00703	0.503513	−	0.863988i	$-0.332041\pi$
$599$	5.29706e9	1.00703	0.503513	+	0.863988i	$0.332041\pi$
$600$	0	0
$601$	7.29749e9	1.37124	0.685619	−	0.727961i	$-0.259532\pi$
$601$	7.29749e9	1.37124	0.685619	+	0.727961i	$0.259532\pi$
$602$	0	0
$603$	−7.31400e8	−0.135845
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−4.46311e9	−0.809986	−0.404993	−	0.914320i	$-0.632726\pi$
$607$	−4.46311e9	−0.809986	−0.404993	+	0.914320i	$0.632726\pi$
$608$	0	0
$609$	−7.91397e8	−0.141982
$610$	0	0
$611$	3.32491e9	0.589707
$612$	0	0
$613$	3.35886e8	0.0588953	0.0294477	−	0.999566i	$-0.490625\pi$
$613$	3.35886e8	0.0588953	0.0294477	+	0.999566i	$0.490625\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.23466e9	0.897203	0.448601	−	0.893732i	$-0.351922\pi$
$617$	5.23466e9	0.897203	0.448601	+	0.893732i	$0.351922\pi$
$618$	0	0
$619$	6.44510e9	1.09223	0.546113	−	0.837712i	$-0.316107\pi$
$619$	6.44510e9	1.09223	0.546113	+	0.837712i	$0.316107\pi$
$620$	0	0
$621$	1.25739e9	0.210693
$622$	0	0
$623$	−1.22936e9	−0.203690
$624$	0	0
$625$	6.10352e9	1.00000
$626$	0	0
$627$	5.56880e8	0.0902246
$628$	0	0
$629$	7.50821e8	0.120298
$630$	0	0
$631$	5.93275e9	0.940055	0.470027	−	0.882652i	$-0.344244\pi$
$631$	5.93275e9	0.940055	0.470027	+	0.882652i	$0.344244\pi$
$632$	0	0
$633$	−4.09687e9	−0.642007
$634$	0	0
$635$	0	0
$636$	0	0
$637$	5.51266e9	0.845033
$638$	0	0
$639$	3.52453e8	0.0534377
$640$	0	0
$641$	−5.32589e9	−0.798710	−0.399355	−	0.916796i	$-0.630766\pi$
$641$	−5.32589e9	−0.798710	−0.399355	+	0.916796i	$0.630766\pi$
$642$	0	0
$643$	2.36887e9	0.351401	0.175700	−	0.984444i	$-0.443781\pi$
$643$	2.36887e9	0.351401	0.175700	+	0.984444i	$0.443781\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−4.18245e9	−0.607108	−0.303554	−	0.952814i	$-0.598173\pi$
$647$	−4.18245e9	−0.607108	−0.303554	+	0.952814i	$0.598173\pi$
$648$	0	0
$649$	2.76156e8	0.0396550
$650$	0	0
$651$	1.87120e8	0.0265819
$652$	0	0
$653$	−9.28180e9	−1.30448	−0.652238	−	0.758014i	$-0.726170\pi$
$653$	−9.28180e9	−1.30448	−0.652238	+	0.758014i	$0.726170\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4.20186e9	0.578046
$658$	0	0
$659$	−3.76855e9	−0.512950	−0.256475	−	0.966551i	$-0.582561\pi$
$659$	−3.76855e9	−0.512950	−0.256475	+	0.966551i	$0.582561\pi$
$660$	0	0
$661$	8.82310e9	1.18827	0.594136	−	0.804364i	$-0.297494\pi$
$661$	8.82310e9	1.18827	0.594136	+	0.804364i	$0.297494\pi$
$662$	0	0
$663$	1.86256e9	0.248207
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6.54701e9	−0.854285
$668$	0	0
$669$	−5.24697e9	−0.677512
$670$	0	0
$671$	1.48311e8	0.0189515
$672$	0	0
$673$	1.06625e10	1.34836	0.674178	−	0.738569i	$-0.264498\pi$
$673$	1.06625e10	1.34836	0.674178	+	0.738569i	$0.264498\pi$
$674$	0	0
$675$	−1.53773e9	−0.192450
$676$	0	0
$677$	−5.83935e9	−0.723277	−0.361638	−	0.932318i	$-0.617782\pi$
$677$	−5.83935e9	−0.723277	−0.361638	+	0.932318i	$0.617782\pi$
$678$	0	0
$679$	6.45261e8	0.0791027
$680$	0	0
$681$	−9.17485e8	−0.111323
$682$	0	0
$683$	7.34084e9	0.881603	0.440801	−	0.897605i	$-0.354694\pi$
$683$	7.34084e9	0.881603	0.440801	+	0.897605i	$0.354694\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−6.07780e9	−0.715151
$688$	0	0
$689$	−3.25968e9	−0.379671
$690$	0	0
$691$	−1.33998e10	−1.54499	−0.772496	−	0.635019i	$-0.780992\pi$
$691$	−1.33998e10	−1.54499	−0.772496	+	0.635019i	$0.780992\pi$
$692$	0	0
$693$	2.77506e8	0.0316742
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−2.54568e8	−0.0284767
$698$	0	0
$699$	8.71244e9	0.964872
$700$	0	0
$701$	−9.65086e9	−1.05816	−0.529082	−	0.848571i	$-0.677464\pi$
$701$	−9.65086e9	−1.05816	−0.529082	+	0.848571i	$0.677464\pi$
$702$	0	0
$703$	−1.25347e9	−0.136073
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.78486e9	−0.402793
$708$	0	0
$709$	4.15520e9	0.437855	0.218927	−	0.975741i	$-0.429744\pi$
$709$	4.15520e9	0.437855	0.218927	+	0.975741i	$0.429744\pi$
$710$	0	0
$711$	−5.18973e9	−0.541503
$712$	0	0
$713$	1.54799e9	0.159939
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.63220e9	0.165370
$718$	0	0
$719$	3.58869e9	0.360069	0.180034	−	0.983660i	$-0.442379\pi$
$719$	3.58869e9	0.360069	0.180034	+	0.983660i	$0.442379\pi$
$720$	0	0
$721$	−3.36993e9	−0.334848
$722$	0	0
$723$	−3.33615e9	−0.328293
$724$	0	0
$725$	8.00672e9	0.780318
$726$	0	0
$727$	−1.88213e10	−1.81668	−0.908342	−	0.418227i	$-0.862652\pi$
$727$	−1.88213e10	−1.81668	−0.908342	+	0.418227i	$0.862652\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	2.02871e9	0.192092
$732$	0	0
$733$	1.51519e10	1.42103	0.710515	−	0.703682i	$-0.248462\pi$
$733$	1.51519e10	1.42103	0.710515	+	0.703682i	$0.248462\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.33538e9	−0.122877
$738$	0	0
$739$	1.56672e10	1.42802	0.714011	−	0.700134i	$-0.246876\pi$
$739$	1.56672e10	1.42802	0.714011	+	0.700134i	$0.246876\pi$
$740$	0	0
$741$	−3.10949e9	−0.280754
$742$	0	0
$743$	1.36080e10	1.21712	0.608560	−	0.793508i	$-0.291747\pi$
$743$	1.36080e10	1.21712	0.608560	+	0.793508i	$0.291747\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−2.05995e9	−0.180815
$748$	0	0
$749$	6.64290e9	0.577659
$750$	0	0
$751$	1.58059e10	1.36170	0.680849	−	0.732424i	$-0.261611\pi$
$751$	1.58059e10	1.36170	0.680849	+	0.732424i	$0.261611\pi$
$752$	0	0
$753$	−1.14103e9	−0.0973901
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.62525e10	1.36171	0.680855	−	0.732418i	$-0.261608\pi$
$757$	1.62525e10	1.36171	0.680855	+	0.732418i	$0.261608\pi$
$758$	0	0
$759$	2.29573e9	0.190579
$760$	0	0
$761$	−1.41695e10	−1.16549	−0.582745	−	0.812655i	$-0.698021\pi$
$761$	−1.41695e10	−1.16549	−0.582745	+	0.812655i	$0.698021\pi$
$762$	0	0
$763$	−1.50250e9	−0.122456
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.54199e9	−0.123395
$768$	0	0
$769$	1.53701e10	1.21881	0.609403	−	0.792861i	$-0.291409\pi$
$769$	1.53701e10	1.21881	0.609403	+	0.792861i	$0.291409\pi$
$770$	0	0
$771$	−4.61237e9	−0.362438
$772$	0	0
$773$	−6.56178e9	−0.510968	−0.255484	−	0.966813i	$-0.582235\pi$
$773$	−6.56178e9	−0.510968	−0.255484	+	0.966813i	$0.582235\pi$
$774$	0	0
$775$	−1.89312e9	−0.146091
$776$	0	0
$777$	−6.24633e8	−0.0477695
$778$	0	0
$779$	4.24993e8	0.0322108
$780$	0	0
$781$	6.43504e8	0.0483362
$782$	0	0
$783$	−2.01723e9	−0.150172
$784$	0	0
$785$	0	0
$786$	0	0
$787$	8.27689e9	0.605279	0.302639	−	0.953105i	$-0.402132\pi$
$787$	8.27689e9	0.605279	0.302639	+	0.953105i	$0.402132\pi$
$788$	0	0
$789$	−1.84823e9	−0.133963
$790$	0	0
$791$	−4.79801e9	−0.344701
$792$	0	0
$793$	−8.28133e8	−0.0589717
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.77680e10	1.24318	0.621589	−	0.783344i	$-0.286487\pi$
$797$	1.77680e10	1.24318	0.621589	+	0.783344i	$0.286487\pi$
$798$	0	0
$799$	4.15256e9	0.288007
$800$	0	0
$801$	−3.13357e9	−0.215440
$802$	0	0
$803$	7.67171e9	0.522863
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−9.09337e9	−0.609071
$808$	0	0
$809$	−2.94411e10	−1.95494	−0.977472	−	0.211065i	$-0.932307\pi$
$809$	−2.94411e10	−1.95494	−0.977472	+	0.211065i	$0.932307\pi$
$810$	0	0
$811$	2.34340e10	1.54267	0.771336	−	0.636428i	$-0.219589\pi$
$811$	2.34340e10	1.54267	0.771336	+	0.636428i	$0.219589\pi$
$812$	0	0
$813$	−1.04213e10	−0.680151
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−3.38687e9	−0.217281
$818$	0	0
$819$	−1.54953e9	−0.0985611
$820$	0	0
$821$	−4.73345e9	−0.298523	−0.149261	−	0.988798i	$-0.547690\pi$
$821$	−4.73345e9	−0.298523	−0.149261	+	0.988798i	$0.547690\pi$
$822$	0	0
$823$	−2.15606e10	−1.34822	−0.674111	−	0.738630i	$-0.735473\pi$
$823$	−2.15606e10	−1.34822	−0.674111	+	0.738630i	$0.735473\pi$
$824$	0	0
$825$	−2.80758e9	−0.174078
$826$	0	0
$827$	−4.49887e9	−0.276588	−0.138294	−	0.990391i	$-0.544162\pi$
$827$	−4.49887e9	−0.276588	−0.138294	+	0.990391i	$0.544162\pi$
$828$	0	0
$829$	−2.37595e10	−1.44843	−0.724214	−	0.689576i	$-0.757797\pi$
$829$	−2.37595e10	−1.44843	−0.724214	+	0.689576i	$0.757797\pi$
$830$	0	0
$831$	9.77515e9	0.590908
$832$	0	0
$833$	6.88490e9	0.412705
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4.76958e8	0.0281152
$838$	0	0
$839$	−1.95950e10	−1.14545	−0.572727	−	0.819746i	$-0.694115\pi$
$839$	−1.95950e10	−1.14545	−0.572727	+	0.819746i	$0.694115\pi$
$840$	0	0
$841$	−6.74650e9	−0.391104
$842$	0	0
$843$	1.02726e9	0.0590585
$844$	0	0
$845$	0	0
$846$	0	0
$847$	5.06666e8	0.0286504
$848$	0	0
$849$	−8.15407e9	−0.457296
$850$	0	0
$851$	−5.16741e9	−0.287422
$852$	0	0
$853$	3.45578e10	1.90644	0.953222	−	0.302271i	$-0.0977447\pi$
$853$	3.45578e10	1.90644	0.953222	+	0.302271i	$0.0977447\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.15573e10	1.71264	0.856321	−	0.516444i	$-0.172744\pi$
$857$	3.15573e10	1.71264	0.856321	+	0.516444i	$0.172744\pi$
$858$	0	0
$859$	2.59041e10	1.39442	0.697209	−	0.716868i	$-0.254425\pi$
$859$	2.59041e10	1.39442	0.697209	+	0.716868i	$0.254425\pi$
$860$	0	0
$861$	2.11784e8	0.0113079
$862$	0	0
$863$	−3.70854e10	−1.96411	−0.982055	−	0.188596i	$-0.939606\pi$
$863$	−3.70854e10	−1.96411	−0.982055	+	0.188596i	$0.939606\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−8.75295e9	−0.456129
$868$	0	0
$869$	−9.47534e9	−0.489808
$870$	0	0
$871$	7.45647e9	0.382357
$872$	0	0
$873$	1.64474e9	0.0836656
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−3.23264e9	−0.161830	−0.0809149	−	0.996721i	$-0.525784\pi$
$877$	−3.23264e9	−0.161830	−0.0809149	+	0.996721i	$0.525784\pi$
$878$	0	0
$879$	−1.80948e10	−0.898656
$880$	0	0
$881$	−2.77886e9	−0.136915	−0.0684575	−	0.997654i	$-0.521808\pi$
$881$	−2.77886e9	−0.136915	−0.0684575	+	0.997654i	$0.521808\pi$
$882$	0	0
$883$	3.56842e10	1.74427	0.872136	−	0.489264i	$-0.162735\pi$
$883$	3.56842e10	1.74427	0.872136	+	0.489264i	$0.162735\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.39173e10	−1.15075	−0.575374	−	0.817890i	$-0.695143\pi$
$887$	−2.39173e10	−1.15075	−0.575374	+	0.817890i	$0.695143\pi$
$888$	0	0
$889$	2.14588e9	0.102435
$890$	0	0
$891$	7.07348e8	0.0335013
$892$	0	0
$893$	−6.93257e9	−0.325772
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.28188e10	−0.593027
$898$	0	0
$899$	−2.48344e9	−0.113997
$900$	0	0
$901$	−4.07109e9	−0.185427
$902$	0	0
$903$	−1.68775e9	−0.0762784
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1.01893e10	−0.453439	−0.226720	−	0.973960i	$-0.572800\pi$
$907$	−1.01893e10	−0.453439	−0.226720	+	0.973960i	$0.572800\pi$
$908$	0	0
$909$	−9.64742e9	−0.426028
$910$	0	0
$911$	−3.17483e10	−1.39125	−0.695627	−	0.718403i	$-0.744874\pi$
$911$	−3.17483e10	−1.39125	−0.695627	+	0.718403i	$0.744874\pi$
$912$	0	0
$913$	−3.76104e9	−0.163554
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.71583e9	0.330438
$918$	0	0
$919$	−9.55588e9	−0.406131	−0.203066	−	0.979165i	$-0.565090\pi$
$919$	−9.55588e9	−0.406131	−0.203066	+	0.979165i	$0.565090\pi$
$920$	0	0
$921$	−2.17179e8	−0.00916030
$922$	0	0
$923$	−3.59318e9	−0.150409
$924$	0	0
$925$	6.31953e9	0.262536
$926$	0	0
$927$	−8.58980e9	−0.354164
$928$	0	0
$929$	−6.44204e9	−0.263614	−0.131807	−	0.991275i	$-0.542078\pi$
$929$	−6.44204e9	−0.263614	−0.131807	+	0.991275i	$0.542078\pi$
$930$	0	0
$931$	−1.14941e10	−0.466822
$932$	0	0
$933$	−1.64687e10	−0.663854
$934$	0	0
$935$	0	0
$936$	0	0
$937$	8.57994e9	0.340718	0.170359	−	0.985382i	$-0.445507\pi$
$937$	8.57994e9	0.340718	0.170359	+	0.985382i	$0.445507\pi$
$938$	0	0
$939$	3.33481e9	0.131444
$940$	0	0
$941$	−7.49911e9	−0.293390	−0.146695	−	0.989182i	$-0.546864\pi$
$941$	−7.49911e9	−0.293390	−0.146695	+	0.989182i	$0.546864\pi$
$942$	0	0
$943$	1.75203e9	0.0680378
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.01400e10	−0.387983	−0.193991	−	0.981003i	$-0.562143\pi$
$947$	−1.01400e10	−0.387983	−0.193991	+	0.981003i	$0.562143\pi$
$948$	0	0
$949$	−4.28371e10	−1.62700
$950$	0	0
$951$	−1.35888e10	−0.512330
$952$	0	0
$953$	−2.15349e10	−0.805969	−0.402984	−	0.915207i	$-0.632027\pi$
$953$	−2.15349e10	−0.805969	−0.402984	+	0.915207i	$0.632027\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−3.68304e9	−0.135836
$958$	0	0
$959$	−9.34395e9	−0.342110
$960$	0	0
$961$	−2.69254e10	−0.978657
$962$	0	0
$963$	1.69324e10	0.610980
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−9.96532e9	−0.354404	−0.177202	−	0.984175i	$-0.556705\pi$
$967$	−9.96532e9	−0.354404	−0.177202	+	0.984175i	$0.556705\pi$
$968$	0	0
$969$	−3.88351e9	−0.137117
$970$	0	0
$971$	2.31708e10	0.812222	0.406111	−	0.913824i	$-0.366885\pi$
$971$	2.31708e10	0.812222	0.406111	+	0.913824i	$0.366885\pi$
$972$	0	0
$973$	−8.91129e9	−0.310131
$974$	0	0
$975$	1.56769e10	0.541681
$976$	0	0
$977$	−1.55053e10	−0.531923	−0.265962	−	0.963984i	$-0.585689\pi$
$977$	−1.55053e10	−0.531923	−0.265962	+	0.963984i	$0.585689\pi$
$978$	0	0
$979$	−5.72124e9	−0.194873
$980$	0	0
$981$	−3.82980e9	−0.129519
$982$	0	0
$983$	−7.39916e9	−0.248454	−0.124227	−	0.992254i	$-0.539645\pi$
$983$	−7.39916e9	−0.248454	−0.124227	+	0.992254i	$0.539645\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−3.45465e9	−0.114365
$988$	0	0
$989$	−1.39623e10	−0.458955
$990$	0	0
$991$	2.21601e10	0.723293	0.361646	−	0.932315i	$-0.382215\pi$
$991$	2.21601e10	0.723293	0.361646	+	0.932315i	$0.382215\pi$
$992$	0	0
$993$	2.12601e10	0.689037
$994$	0	0
$995$	0	0
$996$	0	0
$997$	3.52224e10	1.12561	0.562803	−	0.826591i	$-0.309723\pi$
$997$	3.52224e10	1.12561	0.562803	+	0.826591i	$0.309723\pi$
$998$	0	0
$999$	−1.59216e9	−0.0505251

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	528.8.a.c.1.1		1
4.3	odd	2		66.8.a.b.1.1	✓	1
12.11	even	2		198.8.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
66.8.a.b.1.1	✓	1	4.3	odd	2
198.8.a.a.1.1		1	12.11	even	2
528.8.a.c.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:	\( N \)	\(=\)	\( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	528.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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