LMFDB - Embedded newform 882.4.a.k.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [882,4,Mod(1,882)] 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("882.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(882, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	882.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,2,0,4,-7,0,0,8,0,-14,-35,0,66] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$52.0396846251$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	882.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +4.00000 q^{4} -7.00000 q^{5} +8.00000 q^{8} -14.0000 q^{10} -35.0000 q^{11} +66.0000 q^{13} +16.0000 q^{16} -59.0000 q^{17} +137.000 q^{19} -28.0000 q^{20} -70.0000 q^{22} +7.00000 q^{23} -76.0000 q^{25} +132.000 q^{26} -106.000 q^{29} +75.0000 q^{31} +32.0000 q^{32} -118.000 q^{34} +11.0000 q^{37} +274.000 q^{38} -56.0000 q^{40} +498.000 q^{41} +260.000 q^{43} -140.000 q^{44} +14.0000 q^{46} +171.000 q^{47} -152.000 q^{50} +264.000 q^{52} +417.000 q^{53} +245.000 q^{55} -212.000 q^{58} +17.0000 q^{59} +51.0000 q^{61} +150.000 q^{62} +64.0000 q^{64} -462.000 q^{65} +439.000 q^{67} -236.000 q^{68} +784.000 q^{71} +295.000 q^{73} +22.0000 q^{74} +548.000 q^{76} -495.000 q^{79} -112.000 q^{80} +996.000 q^{82} -932.000 q^{83} +413.000 q^{85} +520.000 q^{86} -280.000 q^{88} +873.000 q^{89} +28.0000 q^{92} +342.000 q^{94} -959.000 q^{95} -290.000 q^{97} +O(q^{100})$

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−7.00000	−0.626099	−0.313050	−	0.949737i	$-0.601351\pi$
$5$	−7.00000	−0.626099	−0.313050	+	0.949737i	$0.601351\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	0	0
$10$	−14.0000	−0.442719
$11$	−35.0000	−0.959354	−0.479677	−	0.877445i	$-0.659246\pi$
$11$	−35.0000	−0.959354	−0.479677	+	0.877445i	$0.659246\pi$
$12$	0	0
$13$	66.0000	1.40809	0.704043	−	0.710158i	$-0.251376\pi$
$13$	66.0000	1.40809	0.704043	+	0.710158i	$0.251376\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	−59.0000	−0.841741	−0.420871	−	0.907121i	$-0.638275\pi$
$17$	−59.0000	−0.841741	−0.420871	+	0.907121i	$0.638275\pi$
$18$	0	0
$19$	137.000	1.65421	0.827104	−	0.562049i	$-0.189987\pi$
$19$	137.000	1.65421	0.827104	+	0.562049i	$0.189987\pi$
$20$	−28.0000	−0.313050
$21$	0	0
$22$	−70.0000	−0.678366
$23$	7.00000	0.0634609	0.0317305	−	0.999496i	$-0.489898\pi$
$23$	7.00000	0.0634609	0.0317305	+	0.999496i	$0.489898\pi$
$24$	0	0
$25$	−76.0000	−0.608000
$26$	132.000	0.995667
$27$	0	0
$28$	0	0
$29$	−106.000	−0.678748	−0.339374	−	0.940651i	$-0.610215\pi$
$29$	−106.000	−0.678748	−0.339374	+	0.940651i	$0.610215\pi$
$30$	0	0
$31$	75.0000	0.434529	0.217264	−	0.976113i	$-0.430287\pi$
$31$	75.0000	0.434529	0.217264	+	0.976113i	$0.430287\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−118.000	−0.595201
$35$	0	0
$36$	0	0
$37$	11.0000	0.0488754	0.0244377	−	0.999701i	$-0.492220\pi$
$37$	11.0000	0.0488754	0.0244377	+	0.999701i	$0.492220\pi$
$38$	274.000	1.16970
$39$	0	0
$40$	−56.0000	−0.221359
$41$	498.000	1.89694	0.948470	−	0.316867i	$-0.102631\pi$
$41$	498.000	1.89694	0.948470	+	0.316867i	$0.102631\pi$
$42$	0	0
$43$	260.000	0.922084	0.461042	−	0.887378i	$-0.347476\pi$
$43$	260.000	0.922084	0.461042	+	0.887378i	$0.347476\pi$
$44$	−140.000	−0.479677
$45$	0	0
$46$	14.0000	0.0448736
$47$	171.000	0.530700	0.265350	−	0.964152i	$-0.414512\pi$
$47$	171.000	0.530700	0.265350	+	0.964152i	$0.414512\pi$
$48$	0	0
$49$	0	0
$50$	−152.000	−0.429921
$51$	0	0
$52$	264.000	0.704043
$53$	417.000	1.08074	0.540371	−	0.841427i	$-0.318284\pi$
$53$	417.000	1.08074	0.540371	+	0.841427i	$0.318284\pi$
$54$	0	0
$55$	245.000	0.600651
$56$	0	0
$57$	0	0
$58$	−212.000	−0.479948
$59$	17.0000	0.0375121	0.0187560	−	0.999824i	$-0.494029\pi$
$59$	17.0000	0.0375121	0.0187560	+	0.999824i	$0.494029\pi$
$60$	0	0
$61$	51.0000	0.107047	0.0535236	−	0.998567i	$-0.482955\pi$
$61$	51.0000	0.107047	0.0535236	+	0.998567i	$0.482955\pi$
$62$	150.000	0.307258
$63$	0	0
$64$	64.0000	0.125000
$65$	−462.000	−0.881601
$66$	0	0
$67$	439.000	0.800483	0.400242	−	0.916410i	$-0.368926\pi$
$67$	439.000	0.800483	0.400242	+	0.916410i	$0.368926\pi$
$68$	−236.000	−0.420871
$69$	0	0
$70$	0	0
$71$	784.000	1.31047	0.655237	−	0.755423i	$-0.272569\pi$
$71$	784.000	1.31047	0.655237	+	0.755423i	$0.272569\pi$
$72$	0	0
$73$	295.000	0.472974	0.236487	−	0.971635i	$-0.424004\pi$
$73$	295.000	0.472974	0.236487	+	0.971635i	$0.424004\pi$
$74$	22.0000	0.0345601
$75$	0	0
$76$	548.000	0.827104
$77$	0	0
$78$	0	0
$79$	−495.000	−0.704960	−0.352480	−	0.935819i	$-0.614662\pi$
$79$	−495.000	−0.704960	−0.352480	+	0.935819i	$0.614662\pi$
$80$	−112.000	−0.156525
$81$	0	0
$82$	996.000	1.34134
$83$	−932.000	−1.23253	−0.616267	−	0.787537i	$-0.711356\pi$
$83$	−932.000	−1.23253	−0.616267	+	0.787537i	$0.711356\pi$
$84$	0	0
$85$	413.000	0.527013
$86$	520.000	0.652012
$87$	0	0
$88$	−280.000	−0.339183
$89$	873.000	1.03975	0.519875	−	0.854242i	$-0.325978\pi$
$89$	873.000	1.03975	0.519875	+	0.854242i	$0.325978\pi$
$90$	0	0
$91$	0	0
$92$	28.0000	0.0317305
$93$	0	0
$94$	342.000	0.375262
$95$	−959.000	−1.03570
$96$	0	0
$97$	−290.000	−0.303557	−0.151779	−	0.988415i	$-0.548500\pi$
$97$	−290.000	−0.303557	−0.151779	+	0.988415i	$0.548500\pi$
$98$	0	0
$99$	0	0
$100$	−304.000	−0.304000
$101$	1085.00	1.06893	0.534463	−	0.845192i	$-0.320514\pi$
$101$	1085.00	1.06893	0.534463	+	0.845192i	$0.320514\pi$
$102$	0	0
$103$	1553.00	1.48565	0.742823	−	0.669487i	$-0.233486\pi$
$103$	1553.00	1.48565	0.742823	+	0.669487i	$0.233486\pi$
$104$	528.000	0.497833
$105$	0	0
$106$	834.000	0.764200
$107$	−129.000	−0.116550	−0.0582752	−	0.998301i	$-0.518560\pi$
$107$	−129.000	−0.116550	−0.0582752	+	0.998301i	$0.518560\pi$
$108$	0	0
$109$	−965.000	−0.847984	−0.423992	−	0.905666i	$-0.639372\pi$
$109$	−965.000	−0.847984	−0.423992	+	0.905666i	$0.639372\pi$
$110$	490.000	0.424724
$111$	0	0
$112$	0	0
$113$	50.0000	0.0416248	0.0208124	−	0.999783i	$-0.493375\pi$
$113$	50.0000	0.0416248	0.0208124	+	0.999783i	$0.493375\pi$
$114$	0	0
$115$	−49.0000	−0.0397328
$116$	−424.000	−0.339374
$117$	0	0
$118$	34.0000	0.0265250
$119$	0	0
$120$	0	0
$121$	−106.000	−0.0796394
$122$	102.000	0.0756938
$123$	0	0
$124$	300.000	0.217264
$125$	1407.00	1.00677
$126$	0	0
$127$	936.000	0.653989	0.326994	−	0.945026i	$-0.393964\pi$
$127$	936.000	0.653989	0.326994	+	0.945026i	$0.393964\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	−924.000	−0.623386
$131$	755.000	0.503547	0.251773	−	0.967786i	$-0.418986\pi$
$131$	755.000	0.503547	0.251773	+	0.967786i	$0.418986\pi$
$132$	0	0
$133$	0	0
$134$	878.000	0.566027
$135$	0	0
$136$	−472.000	−0.297600
$137$	2357.00	1.46987	0.734935	−	0.678138i	$-0.237213\pi$
$137$	2357.00	1.46987	0.734935	+	0.678138i	$0.237213\pi$
$138$	0	0
$139$	28.0000	0.0170858	0.00854291	−	0.999964i	$-0.497281\pi$
$139$	28.0000	0.0170858	0.00854291	+	0.999964i	$0.497281\pi$
$140$	0	0
$141$	0	0
$142$	1568.00	0.926645
$143$	−2310.00	−1.35085
$144$	0	0
$145$	742.000	0.424964
$146$	590.000	0.334443
$147$	0	0
$148$	44.0000	0.0244377
$149$	−2295.00	−1.26184	−0.630919	−	0.775849i	$-0.717322\pi$
$149$	−2295.00	−1.26184	−0.630919	+	0.775849i	$0.717322\pi$
$150$	0	0
$151$	−1109.00	−0.597676	−0.298838	−	0.954304i	$-0.596599\pi$
$151$	−1109.00	−0.597676	−0.298838	+	0.954304i	$0.596599\pi$
$152$	1096.00	0.584851
$153$	0	0
$154$	0	0
$155$	−525.000	−0.272058
$156$	0	0
$157$	1559.00	0.792495	0.396248	−	0.918144i	$-0.370312\pi$
$157$	1559.00	0.792495	0.396248	+	0.918144i	$0.370312\pi$
$158$	−990.000	−0.498482
$159$	0	0
$160$	−224.000	−0.110680
$161$	0	0
$162$	0	0
$163$	−2251.00	−1.08167	−0.540834	−	0.841129i	$-0.681891\pi$
$163$	−2251.00	−1.08167	−0.540834	+	0.841129i	$0.681891\pi$
$164$	1992.00	0.948470
$165$	0	0
$166$	−1864.00	−0.871533
$167$	−2788.00	−1.29187	−0.645934	−	0.763393i	$-0.723532\pi$
$167$	−2788.00	−1.29187	−0.645934	+	0.763393i	$0.723532\pi$
$168$	0	0
$169$	2159.00	0.982704
$170$	826.000	0.372655
$171$	0	0
$172$	1040.00	0.461042
$173$	−1579.00	−0.693926	−0.346963	−	0.937879i	$-0.612787\pi$
$173$	−1579.00	−0.693926	−0.346963	+	0.937879i	$0.612787\pi$
$174$	0	0
$175$	0	0
$176$	−560.000	−0.239839
$177$	0	0
$178$	1746.00	0.735215
$179$	−2451.00	−1.02344	−0.511722	−	0.859151i	$-0.670992\pi$
$179$	−2451.00	−1.02344	−0.511722	+	0.859151i	$0.670992\pi$
$180$	0	0
$181$	−1170.00	−0.480472	−0.240236	−	0.970715i	$-0.577225\pi$
$181$	−1170.00	−0.480472	−0.240236	+	0.970715i	$0.577225\pi$
$182$	0	0
$183$	0	0
$184$	56.0000	0.0224368
$185$	−77.0000	−0.0306008
$186$	0	0
$187$	2065.00	0.807528
$188$	684.000	0.265350
$189$	0	0
$190$	−1918.00	−0.732349
$191$	1275.00	0.483014	0.241507	−	0.970399i	$-0.422358\pi$
$191$	1275.00	0.483014	0.241507	+	0.970399i	$0.422358\pi$
$192$	0	0
$193$	35.0000	0.0130537	0.00652683	−	0.999979i	$-0.497922\pi$
$193$	35.0000	0.0130537	0.00652683	+	0.999979i	$0.497922\pi$
$194$	−580.000	−0.214647
$195$	0	0
$196$	0	0
$197$	2734.00	0.988779	0.494389	−	0.869241i	$-0.335392\pi$
$197$	2734.00	0.988779	0.494389	+	0.869241i	$0.335392\pi$
$198$	0	0
$199$	2243.00	0.799005	0.399503	−	0.916732i	$-0.369183\pi$
$199$	2243.00	0.799005	0.399503	+	0.916732i	$0.369183\pi$
$200$	−608.000	−0.214960
$201$	0	0
$202$	2170.00	0.755845
$203$	0	0
$204$	0	0
$205$	−3486.00	−1.18767
$206$	3106.00	1.05051
$207$	0	0
$208$	1056.00	0.352021
$209$	−4795.00	−1.58697
$210$	0	0
$211$	1172.00	0.382388	0.191194	−	0.981552i	$-0.438764\pi$
$211$	1172.00	0.382388	0.191194	+	0.981552i	$0.438764\pi$
$212$	1668.00	0.540371
$213$	0	0
$214$	−258.000	−0.0824136
$215$	−1820.00	−0.577316
$216$	0	0
$217$	0	0
$218$	−1930.00	−0.599615
$219$	0	0
$220$	980.000	0.300325
$221$	−3894.00	−1.18524
$222$	0	0
$223$	2024.00	0.607790	0.303895	−	0.952706i	$-0.401713\pi$
$223$	2024.00	0.607790	0.303895	+	0.952706i	$0.401713\pi$
$224$	0	0
$225$	0	0
$226$	100.000	0.0294332
$227$	−2571.00	−0.751732	−0.375866	−	0.926674i	$-0.622655\pi$
$227$	−2571.00	−0.751732	−0.375866	+	0.926674i	$0.622655\pi$
$228$	0	0
$229$	895.000	0.258268	0.129134	−	0.991627i	$-0.458780\pi$
$229$	895.000	0.258268	0.129134	+	0.991627i	$0.458780\pi$
$230$	−98.0000	−0.0280953
$231$	0	0
$232$	−848.000	−0.239974
$233$	−1787.00	−0.502447	−0.251224	−	0.967929i	$-0.580833\pi$
$233$	−1787.00	−0.502447	−0.251224	+	0.967929i	$0.580833\pi$
$234$	0	0
$235$	−1197.00	−0.332271
$236$	68.0000	0.0187560
$237$	0	0
$238$	0	0
$239$	5100.00	1.38030	0.690150	−	0.723667i	$-0.257545\pi$
$239$	5100.00	1.38030	0.690150	+	0.723667i	$0.257545\pi$
$240$	0	0
$241$	−4177.00	−1.11645	−0.558225	−	0.829690i	$-0.688517\pi$
$241$	−4177.00	−1.11645	−0.558225	+	0.829690i	$0.688517\pi$
$242$	−212.000	−0.0563135
$243$	0	0
$244$	204.000	0.0535236
$245$	0	0
$246$	0	0
$247$	9042.00	2.32927
$248$	600.000	0.153629
$249$	0	0
$250$	2814.00	0.711892
$251$	4680.00	1.17689	0.588444	−	0.808538i	$-0.299741\pi$
$251$	4680.00	1.17689	0.588444	+	0.808538i	$0.299741\pi$
$252$	0	0
$253$	−245.000	−0.0608815
$254$	1872.00	0.462440
$255$	0	0
$256$	256.000	0.0625000
$257$	1749.00	0.424512	0.212256	−	0.977214i	$-0.431919\pi$
$257$	1749.00	0.424512	0.212256	+	0.977214i	$0.431919\pi$
$258$	0	0
$259$	0	0
$260$	−1848.00	−0.440800
$261$	0	0
$262$	1510.00	0.356061
$263$	4473.00	1.04873	0.524367	−	0.851492i	$-0.324302\pi$
$263$	4473.00	1.04873	0.524367	+	0.851492i	$0.324302\pi$
$264$	0	0
$265$	−2919.00	−0.676652
$266$	0	0
$267$	0	0
$268$	1756.00	0.400242
$269$	−1975.00	−0.447650	−0.223825	−	0.974629i	$-0.571854\pi$
$269$	−1975.00	−0.447650	−0.223825	+	0.974629i	$0.571854\pi$
$270$	0	0
$271$	−8439.00	−1.89163	−0.945817	−	0.324701i	$-0.894736\pi$
$271$	−8439.00	−1.89163	−0.945817	+	0.324701i	$0.894736\pi$
$272$	−944.000	−0.210435
$273$	0	0
$274$	4714.00	1.03935
$275$	2660.00	0.583287
$276$	0	0
$277$	527.000	0.114312	0.0571559	−	0.998365i	$-0.481797\pi$
$277$	527.000	0.114312	0.0571559	+	0.998365i	$0.481797\pi$
$278$	56.0000	0.0120815
$279$	0	0
$280$	0	0
$281$	202.000	0.0428837	0.0214418	−	0.999770i	$-0.493174\pi$
$281$	202.000	0.0428837	0.0214418	+	0.999770i	$0.493174\pi$
$282$	0	0
$283$	−7949.00	−1.66968	−0.834839	−	0.550494i	$-0.814439\pi$
$283$	−7949.00	−1.66968	−0.834839	+	0.550494i	$0.814439\pi$
$284$	3136.00	0.655237
$285$	0	0
$286$	−4620.00	−0.955197
$287$	0	0
$288$	0	0
$289$	−1432.00	−0.291472
$290$	1484.00	0.300495
$291$	0	0
$292$	1180.00	0.236487
$293$	−318.000	−0.0634053	−0.0317027	−	0.999497i	$-0.510093\pi$
$293$	−318.000	−0.0634053	−0.0317027	+	0.999497i	$0.510093\pi$
$294$	0	0
$295$	−119.000	−0.0234863
$296$	88.0000	0.0172801
$297$	0	0
$298$	−4590.00	−0.892254
$299$	462.000	0.0893584
$300$	0	0
$301$	0	0
$302$	−2218.00	−0.422621
$303$	0	0
$304$	2192.00	0.413552
$305$	−357.000	−0.0670222
$306$	0	0
$307$	−8132.00	−1.51178	−0.755892	−	0.654696i	$-0.772797\pi$
$307$	−8132.00	−1.51178	−0.755892	+	0.654696i	$0.772797\pi$
$308$	0	0
$309$	0	0
$310$	−1050.00	−0.192374
$311$	929.000	0.169385	0.0846925	−	0.996407i	$-0.473009\pi$
$311$	929.000	0.169385	0.0846925	+	0.996407i	$0.473009\pi$
$312$	0	0
$313$	−209.000	−0.0377424	−0.0188712	−	0.999822i	$-0.506007\pi$
$313$	−209.000	−0.0377424	−0.0188712	+	0.999822i	$0.506007\pi$
$314$	3118.00	0.560379
$315$	0	0
$316$	−1980.00	−0.352480
$317$	−7131.00	−1.26346	−0.631730	−	0.775188i	$-0.717655\pi$
$317$	−7131.00	−1.26346	−0.631730	+	0.775188i	$0.717655\pi$
$318$	0	0
$319$	3710.00	0.651160
$320$	−448.000	−0.0782624
$321$	0	0
$322$	0	0
$323$	−8083.00	−1.39242
$324$	0	0
$325$	−5016.00	−0.856116
$326$	−4502.00	−0.764855
$327$	0	0
$328$	3984.00	0.670670
$329$	0	0
$330$	0	0
$331$	−6571.00	−1.09116	−0.545581	−	0.838058i	$-0.683691\pi$
$331$	−6571.00	−1.09116	−0.545581	+	0.838058i	$0.683691\pi$
$332$	−3728.00	−0.616267
$333$	0	0
$334$	−5576.00	−0.913488
$335$	−3073.00	−0.501182
$336$	0	0
$337$	−11466.0	−1.85339	−0.926696	−	0.375813i	$-0.877364\pi$
$337$	−11466.0	−1.85339	−0.926696	+	0.375813i	$0.877364\pi$
$338$	4318.00	0.694876
$339$	0	0
$340$	1652.00	0.263507
$341$	−2625.00	−0.416867
$342$	0	0
$343$	0	0
$344$	2080.00	0.326006
$345$	0	0
$346$	−3158.00	−0.490680
$347$	9777.00	1.51256	0.756278	−	0.654251i	$-0.227016\pi$
$347$	9777.00	1.51256	0.756278	+	0.654251i	$0.227016\pi$
$348$	0	0
$349$	11914.0	1.82734	0.913670	−	0.406456i	$-0.133236\pi$
$349$	11914.0	1.82734	0.913670	+	0.406456i	$0.133236\pi$
$350$	0	0
$351$	0	0
$352$	−1120.00	−0.169591
$353$	−9123.00	−1.37555	−0.687774	−	0.725925i	$-0.741412\pi$
$353$	−9123.00	−1.37555	−0.687774	+	0.725925i	$0.741412\pi$
$354$	0	0
$355$	−5488.00	−0.820487
$356$	3492.00	0.519875
$357$	0	0
$358$	−4902.00	−0.723684
$359$	−8149.00	−1.19802	−0.599008	−	0.800743i	$-0.704438\pi$
$359$	−8149.00	−1.19802	−0.599008	+	0.800743i	$0.704438\pi$
$360$	0	0
$361$	11910.0	1.73640
$362$	−2340.00	−0.339745
$363$	0	0
$364$	0	0
$365$	−2065.00	−0.296129
$366$	0	0
$367$	9671.00	1.37554	0.687769	−	0.725930i	$-0.258590\pi$
$367$	9671.00	1.37554	0.687769	+	0.725930i	$0.258590\pi$
$368$	112.000	0.0158652
$369$	0	0
$370$	−154.000	−0.0216381
$371$	0	0
$372$	0	0
$373$	−4109.00	−0.570391	−0.285196	−	0.958469i	$-0.592059\pi$
$373$	−4109.00	−0.570391	−0.285196	+	0.958469i	$0.592059\pi$
$374$	4130.00	0.571009
$375$	0	0
$376$	1368.00	0.187631
$377$	−6996.00	−0.955736
$378$	0	0
$379$	−3488.00	−0.472735	−0.236367	−	0.971664i	$-0.575957\pi$
$379$	−3488.00	−0.472735	−0.236367	+	0.971664i	$0.575957\pi$
$380$	−3836.00	−0.517849
$381$	0	0
$382$	2550.00	0.341543
$383$	−8717.00	−1.16297	−0.581485	−	0.813557i	$-0.697528\pi$
$383$	−8717.00	−1.16297	−0.581485	+	0.813557i	$0.697528\pi$
$384$	0	0
$385$	0	0
$386$	70.0000	0.00923033
$387$	0	0
$388$	−1160.00	−0.151779
$389$	−163.000	−0.0212453	−0.0106227	−	0.999944i	$-0.503381\pi$
$389$	−163.000	−0.0212453	−0.0106227	+	0.999944i	$0.503381\pi$
$390$	0	0
$391$	−413.000	−0.0534177
$392$	0	0
$393$	0	0
$394$	5468.00	0.699172
$395$	3465.00	0.441375
$396$	0	0
$397$	999.000	0.126293	0.0631466	−	0.998004i	$-0.479886\pi$
$397$	999.000	0.126293	0.0631466	+	0.998004i	$0.479886\pi$
$398$	4486.00	0.564982
$399$	0	0
$400$	−1216.00	−0.152000
$401$	14757.0	1.83773	0.918865	−	0.394573i	$-0.129107\pi$
$401$	14757.0	1.83773	0.918865	+	0.394573i	$0.129107\pi$
$402$	0	0
$403$	4950.00	0.611854
$404$	4340.00	0.534463
$405$	0	0
$406$	0	0
$407$	−385.000	−0.0468888
$408$	0	0
$409$	−133.000	−0.0160793	−0.00803964	−	0.999968i	$-0.502559\pi$
$409$	−133.000	−0.0160793	−0.00803964	+	0.999968i	$0.502559\pi$
$410$	−6972.00	−0.839811
$411$	0	0
$412$	6212.00	0.742823
$413$	0	0
$414$	0	0
$415$	6524.00	0.771688
$416$	2112.00	0.248917
$417$	0	0
$418$	−9590.00	−1.12216
$419$	6420.00	0.748538	0.374269	−	0.927320i	$-0.377894\pi$
$419$	6420.00	0.748538	0.374269	+	0.927320i	$0.377894\pi$
$420$	0	0
$421$	10266.0	1.18844	0.594221	−	0.804302i	$-0.297460\pi$
$421$	10266.0	1.18844	0.594221	+	0.804302i	$0.297460\pi$
$422$	2344.00	0.270389
$423$	0	0
$424$	3336.00	0.382100
$425$	4484.00	0.511779
$426$	0	0
$427$	0	0
$428$	−516.000	−0.0582752
$429$	0	0
$430$	−3640.00	−0.408224
$431$	15213.0	1.70020	0.850098	−	0.526625i	$-0.176543\pi$
$431$	15213.0	1.70020	0.850098	+	0.526625i	$0.176543\pi$
$432$	0	0
$433$	−1378.00	−0.152939	−0.0764693	−	0.997072i	$-0.524365\pi$
$433$	−1378.00	−0.152939	−0.0764693	+	0.997072i	$0.524365\pi$
$434$	0	0
$435$	0	0
$436$	−3860.00	−0.423992
$437$	959.000	0.104978
$438$	0	0
$439$	−2763.00	−0.300389	−0.150195	−	0.988656i	$-0.547990\pi$
$439$	−2763.00	−0.300389	−0.150195	+	0.988656i	$0.547990\pi$
$440$	1960.00	0.212362
$441$	0	0
$442$	−7788.00	−0.838094
$443$	−5849.00	−0.627301	−0.313651	−	0.949538i	$-0.601552\pi$
$443$	−5849.00	−0.627301	−0.313651	+	0.949538i	$0.601552\pi$
$444$	0	0
$445$	−6111.00	−0.650987
$446$	4048.00	0.429772
$447$	0	0
$448$	0	0
$449$	−4582.00	−0.481599	−0.240799	−	0.970575i	$-0.577410\pi$
$449$	−4582.00	−0.481599	−0.240799	+	0.970575i	$0.577410\pi$
$450$	0	0
$451$	−17430.0	−1.81984
$452$	200.000	0.0208124
$453$	0	0
$454$	−5142.00	−0.531555
$455$	0	0
$456$	0	0
$457$	11551.0	1.18235	0.591174	−	0.806544i	$-0.298665\pi$
$457$	11551.0	1.18235	0.591174	+	0.806544i	$0.298665\pi$
$458$	1790.00	0.182623
$459$	0	0
$460$	−196.000	−0.0198664
$461$	9494.00	0.959175	0.479587	−	0.877494i	$-0.340786\pi$
$461$	9494.00	0.959175	0.479587	+	0.877494i	$0.340786\pi$
$462$	0	0
$463$	−10160.0	−1.01982	−0.509908	−	0.860229i	$-0.670321\pi$
$463$	−10160.0	−1.01982	−0.509908	+	0.860229i	$0.670321\pi$
$464$	−1696.00	−0.169687
$465$	0	0
$466$	−3574.00	−0.355284
$467$	1307.00	0.129509	0.0647545	−	0.997901i	$-0.479374\pi$
$467$	1307.00	0.129509	0.0647545	+	0.997901i	$0.479374\pi$
$468$	0	0
$469$	0	0
$470$	−2394.00	−0.234951
$471$	0	0
$472$	136.000	0.0132625
$473$	−9100.00	−0.884606
$474$	0	0
$475$	−10412.0	−1.00576
$476$	0	0
$477$	0	0
$478$	10200.0	0.976019
$479$	−18287.0	−1.74437	−0.872186	−	0.489174i	$-0.837298\pi$
$479$	−18287.0	−1.74437	−0.872186	+	0.489174i	$0.837298\pi$
$480$	0	0
$481$	726.000	0.0688207
$482$	−8354.00	−0.789449
$483$	0	0
$484$	−424.000	−0.0398197
$485$	2030.00	0.190057
$486$	0	0
$487$	−14953.0	−1.39135	−0.695673	−	0.718359i	$-0.744894\pi$
$487$	−14953.0	−1.39135	−0.695673	+	0.718359i	$0.744894\pi$
$488$	408.000	0.0378469
$489$	0	0
$490$	0	0
$491$	−14352.0	−1.31914	−0.659569	−	0.751644i	$-0.729261\pi$
$491$	−14352.0	−1.31914	−0.659569	+	0.751644i	$0.729261\pi$
$492$	0	0
$493$	6254.00	0.571331
$494$	18084.0	1.64704
$495$	0	0
$496$	1200.00	0.108632
$497$	0	0
$498$	0	0
$499$	−5531.00	−0.496196	−0.248098	−	0.968735i	$-0.579805\pi$
$499$	−5531.00	−0.496196	−0.248098	+	0.968735i	$0.579805\pi$
$500$	5628.00	0.503384
$501$	0	0
$502$	9360.00	0.832186
$503$	−8400.00	−0.744607	−0.372304	−	0.928111i	$-0.621432\pi$
$503$	−8400.00	−0.744607	−0.372304	+	0.928111i	$0.621432\pi$
$504$	0	0
$505$	−7595.00	−0.669254
$506$	−490.000	−0.0430497
$507$	0	0
$508$	3744.00	0.326994
$509$	2385.00	0.207688	0.103844	−	0.994594i	$-0.466886\pi$
$509$	2385.00	0.207688	0.103844	+	0.994594i	$0.466886\pi$
$510$	0	0
$511$	0	0
$512$	512.000	0.0441942
$513$	0	0
$514$	3498.00	0.300175
$515$	−10871.0	−0.930162
$516$	0	0
$517$	−5985.00	−0.509130
$518$	0	0
$519$	0	0
$520$	−3696.00	−0.311693
$521$	9153.00	0.769674	0.384837	−	0.922985i	$-0.374258\pi$
$521$	9153.00	0.769674	0.384837	+	0.922985i	$0.374258\pi$
$522$	0	0
$523$	−13807.0	−1.15437	−0.577187	−	0.816612i	$-0.695850\pi$
$523$	−13807.0	−1.15437	−0.577187	+	0.816612i	$0.695850\pi$
$524$	3020.00	0.251773
$525$	0	0
$526$	8946.00	0.741567
$527$	−4425.00	−0.365761
$528$	0	0
$529$	−12118.0	−0.995973
$530$	−5838.00	−0.478465
$531$	0	0
$532$	0	0
$533$	32868.0	2.67105
$534$	0	0
$535$	903.000	0.0729721
$536$	3512.00	0.283014
$537$	0	0
$538$	−3950.00	−0.316536
$539$	0	0
$540$	0	0
$541$	8175.00	0.649669	0.324834	−	0.945771i	$-0.394691\pi$
$541$	8175.00	0.649669	0.324834	+	0.945771i	$0.394691\pi$
$542$	−16878.0	−1.33759
$543$	0	0
$544$	−1888.00	−0.148800
$545$	6755.00	0.530922
$546$	0	0
$547$	4656.00	0.363942	0.181971	−	0.983304i	$-0.441752\pi$
$547$	4656.00	0.363942	0.181971	+	0.983304i	$0.441752\pi$
$548$	9428.00	0.734935
$549$	0	0
$550$	5320.00	0.412446
$551$	−14522.0	−1.12279
$552$	0	0
$553$	0	0
$554$	1054.00	0.0808306
$555$	0	0
$556$	112.000	0.00854291
$557$	−7003.00	−0.532723	−0.266361	−	0.963873i	$-0.585821\pi$
$557$	−7003.00	−0.532723	−0.266361	+	0.963873i	$0.585821\pi$
$558$	0	0
$559$	17160.0	1.29837
$560$	0	0
$561$	0	0
$562$	404.000	0.0303233
$563$	19753.0	1.47867	0.739334	−	0.673339i	$-0.235141\pi$
$563$	19753.0	1.47867	0.739334	+	0.673339i	$0.235141\pi$
$564$	0	0
$565$	−350.000	−0.0260613
$566$	−15898.0	−1.18064
$567$	0	0
$568$	6272.00	0.463323
$569$	6897.00	0.508150	0.254075	−	0.967185i	$-0.418229\pi$
$569$	6897.00	0.508150	0.254075	+	0.967185i	$0.418229\pi$
$570$	0	0
$571$	24915.0	1.82603	0.913013	−	0.407932i	$-0.133750\pi$
$571$	24915.0	1.82603	0.913013	+	0.407932i	$0.133750\pi$
$572$	−9240.00	−0.675426
$573$	0	0
$574$	0	0
$575$	−532.000	−0.0385842
$576$	0	0
$577$	127.000	0.00916305	0.00458152	−	0.999990i	$-0.498542\pi$
$577$	127.000	0.00916305	0.00458152	+	0.999990i	$0.498542\pi$
$578$	−2864.00	−0.206102
$579$	0	0
$580$	2968.00	0.212482
$581$	0	0
$582$	0	0
$583$	−14595.0	−1.03681
$584$	2360.00	0.167222
$585$	0	0
$586$	−636.000	−0.0448343
$587$	−9044.00	−0.635921	−0.317961	−	0.948104i	$-0.602998\pi$
$587$	−9044.00	−0.635921	−0.317961	+	0.948104i	$0.602998\pi$
$588$	0	0
$589$	10275.0	0.718801
$590$	−238.000	−0.0166073
$591$	0	0
$592$	176.000	0.0122188
$593$	10701.0	0.741041	0.370521	−	0.928824i	$-0.379179\pi$
$593$	10701.0	0.741041	0.370521	+	0.928824i	$0.379179\pi$
$594$	0	0
$595$	0	0
$596$	−9180.00	−0.630919
$597$	0	0
$598$	924.000	0.0631859
$599$	−20799.0	−1.41874	−0.709369	−	0.704837i	$-0.751020\pi$
$599$	−20799.0	−1.41874	−0.709369	+	0.704837i	$0.751020\pi$
$600$	0	0
$601$	−1402.00	−0.0951560	−0.0475780	−	0.998868i	$-0.515150\pi$
$601$	−1402.00	−0.0951560	−0.0475780	+	0.998868i	$0.515150\pi$
$602$	0	0
$603$	0	0
$604$	−4436.00	−0.298838
$605$	742.000	0.0498621
$606$	0	0
$607$	6525.00	0.436312	0.218156	−	0.975914i	$-0.429996\pi$
$607$	6525.00	0.436312	0.218156	+	0.975914i	$0.429996\pi$
$608$	4384.00	0.292425
$609$	0	0
$610$	−714.000	−0.0473918
$611$	11286.0	0.747271
$612$	0	0
$613$	15051.0	0.991687	0.495844	−	0.868412i	$-0.334859\pi$
$613$	15051.0	0.991687	0.495844	+	0.868412i	$0.334859\pi$
$614$	−16264.0	−1.06899
$615$	0	0
$616$	0	0
$617$	−11150.0	−0.727524	−0.363762	−	0.931492i	$-0.618508\pi$
$617$	−11150.0	−0.727524	−0.363762	+	0.931492i	$0.618508\pi$
$618$	0	0
$619$	3415.00	0.221745	0.110873	−	0.993835i	$-0.464635\pi$
$619$	3415.00	0.221745	0.110873	+	0.993835i	$0.464635\pi$
$620$	−2100.00	−0.136029
$621$	0	0
$622$	1858.00	0.119773
$623$	0	0
$624$	0	0
$625$	−349.000	−0.0223360
$626$	−418.000	−0.0266879
$627$	0	0
$628$	6236.00	0.396248
$629$	−649.000	−0.0411404
$630$	0	0
$631$	−21184.0	−1.33648	−0.668242	−	0.743944i	$-0.732953\pi$
$631$	−21184.0	−1.33648	−0.668242	+	0.743944i	$0.732953\pi$
$632$	−3960.00	−0.249241
$633$	0	0
$634$	−14262.0	−0.893401
$635$	−6552.00	−0.409462
$636$	0	0
$637$	0	0
$638$	7420.00	0.460440
$639$	0	0
$640$	−896.000	−0.0553399
$641$	10705.0	0.659629	0.329814	−	0.944046i	$-0.393014\pi$
$641$	10705.0	0.659629	0.329814	+	0.944046i	$0.393014\pi$
$642$	0	0
$643$	6860.00	0.420734	0.210367	−	0.977622i	$-0.432534\pi$
$643$	6860.00	0.420734	0.210367	+	0.977622i	$0.432534\pi$
$644$	0	0
$645$	0	0
$646$	−16166.0	−0.984586
$647$	−14463.0	−0.878824	−0.439412	−	0.898286i	$-0.644813\pi$
$647$	−14463.0	−0.878824	−0.439412	+	0.898286i	$0.644813\pi$
$648$	0	0
$649$	−595.000	−0.0359874
$650$	−10032.0	−0.605365
$651$	0	0
$652$	−9004.00	−0.540834
$653$	−5979.00	−0.358310	−0.179155	−	0.983821i	$-0.557336\pi$
$653$	−5979.00	−0.358310	−0.179155	+	0.983821i	$0.557336\pi$
$654$	0	0
$655$	−5285.00	−0.315270
$656$	7968.00	0.474235
$657$	0	0
$658$	0	0
$659$	6940.00	0.410234	0.205117	−	0.978737i	$-0.434243\pi$
$659$	6940.00	0.410234	0.205117	+	0.978737i	$0.434243\pi$
$660$	0	0
$661$	13399.0	0.788443	0.394221	−	0.919015i	$-0.371014\pi$
$661$	13399.0	0.788443	0.394221	+	0.919015i	$0.371014\pi$
$662$	−13142.0	−0.771568
$663$	0	0
$664$	−7456.00	−0.435766
$665$	0	0
$666$	0	0
$667$	−742.000	−0.0430740
$668$	−11152.0	−0.645934
$669$	0	0
$670$	−6146.00	−0.354389
$671$	−1785.00	−0.102696
$672$	0	0
$673$	29510.0	1.69023	0.845117	−	0.534582i	$-0.179531\pi$
$673$	29510.0	1.69023	0.845117	+	0.534582i	$0.179531\pi$
$674$	−22932.0	−1.31055
$675$	0	0
$676$	8636.00	0.491352
$677$	26001.0	1.47607	0.738035	−	0.674762i	$-0.235754\pi$
$677$	26001.0	1.47607	0.738035	+	0.674762i	$0.235754\pi$
$678$	0	0
$679$	0	0
$680$	3304.00	0.186327
$681$	0	0
$682$	−5250.00	−0.294770
$683$	8805.00	0.493285	0.246643	−	0.969106i	$-0.420673\pi$
$683$	8805.00	0.493285	0.246643	+	0.969106i	$0.420673\pi$
$684$	0	0
$685$	−16499.0	−0.920284
$686$	0	0
$687$	0	0
$688$	4160.00	0.230521
$689$	27522.0	1.52178
$690$	0	0
$691$	28685.0	1.57920	0.789601	−	0.613620i	$-0.210287\pi$
$691$	28685.0	1.57920	0.789601	+	0.613620i	$0.210287\pi$
$692$	−6316.00	−0.346963
$693$	0	0
$694$	19554.0	1.06954
$695$	−196.000	−0.0106974
$696$	0	0
$697$	−29382.0	−1.59673
$698$	23828.0	1.29212
$699$	0	0
$700$	0	0
$701$	3146.00	0.169505	0.0847523	−	0.996402i	$-0.472990\pi$
$701$	3146.00	0.169505	0.0847523	+	0.996402i	$0.472990\pi$
$702$	0	0
$703$	1507.00	0.0808500
$704$	−2240.00	−0.119919
$705$	0	0
$706$	−18246.0	−0.972659
$707$	0	0
$708$	0	0
$709$	1259.00	0.0666893	0.0333447	−	0.999444i	$-0.489384\pi$
$709$	1259.00	0.0666893	0.0333447	+	0.999444i	$0.489384\pi$
$710$	−10976.0	−0.580172
$711$	0	0
$712$	6984.00	0.367607
$713$	525.000	0.0275756
$714$	0	0
$715$	16170.0	0.845767
$716$	−9804.00	−0.511722
$717$	0	0
$718$	−16298.0	−0.847125
$719$	−16425.0	−0.851946	−0.425973	−	0.904736i	$-0.640068\pi$
$719$	−16425.0	−0.851946	−0.425973	+	0.904736i	$0.640068\pi$
$720$	0	0
$721$	0	0
$722$	23820.0	1.22782
$723$	0	0
$724$	−4680.00	−0.240236
$725$	8056.00	0.412679
$726$	0	0
$727$	−6032.00	−0.307723	−0.153861	−	0.988092i	$-0.549171\pi$
$727$	−6032.00	−0.307723	−0.153861	+	0.988092i	$0.549171\pi$
$728$	0	0
$729$	0	0
$730$	−4130.00	−0.209395
$731$	−15340.0	−0.776156
$732$	0	0
$733$	15243.0	0.768094	0.384047	−	0.923314i	$-0.374530\pi$
$733$	15243.0	0.768094	0.384047	+	0.923314i	$0.374530\pi$
$734$	19342.0	0.972652
$735$	0	0
$736$	224.000	0.0112184
$737$	−15365.0	−0.767947
$738$	0	0
$739$	−10053.0	−0.500414	−0.250207	−	0.968192i	$-0.580499\pi$
$739$	−10053.0	−0.500414	−0.250207	+	0.968192i	$0.580499\pi$
$740$	−308.000	−0.0153004
$741$	0	0
$742$	0	0
$743$	−24384.0	−1.20399	−0.601993	−	0.798501i	$-0.705627\pi$
$743$	−24384.0	−1.20399	−0.601993	+	0.798501i	$0.705627\pi$
$744$	0	0
$745$	16065.0	0.790035
$746$	−8218.00	−0.403328
$747$	0	0
$748$	8260.00	0.403764
$749$	0	0
$750$	0	0
$751$	11589.0	0.563101	0.281550	−	0.959546i	$-0.409151\pi$
$751$	11589.0	0.563101	0.281550	+	0.959546i	$0.409151\pi$
$752$	2736.00	0.132675
$753$	0	0
$754$	−13992.0	−0.675807
$755$	7763.00	0.374205
$756$	0	0
$757$	14562.0	0.699161	0.349581	−	0.936906i	$-0.386324\pi$
$757$	14562.0	0.699161	0.349581	+	0.936906i	$0.386324\pi$
$758$	−6976.00	−0.334274
$759$	0	0
$760$	−7672.00	−0.366175
$761$	22765.0	1.08440	0.542201	−	0.840249i	$-0.317591\pi$
$761$	22765.0	1.08440	0.542201	+	0.840249i	$0.317591\pi$
$762$	0	0
$763$	0	0
$764$	5100.00	0.241507
$765$	0	0
$766$	−17434.0	−0.822345
$767$	1122.00	0.0528202
$768$	0	0
$769$	3766.00	0.176600	0.0883000	−	0.996094i	$-0.471857\pi$
$769$	3766.00	0.176600	0.0883000	+	0.996094i	$0.471857\pi$
$770$	0	0
$771$	0	0
$772$	140.000	0.00652683
$773$	26861.0	1.24984	0.624918	−	0.780691i	$-0.285132\pi$
$773$	26861.0	1.24984	0.624918	+	0.780691i	$0.285132\pi$
$774$	0	0
$775$	−5700.00	−0.264194
$776$	−2320.00	−0.107324
$777$	0	0
$778$	−326.000	−0.0150227
$779$	68226.0	3.13793
$780$	0	0
$781$	−27440.0	−1.25721
$782$	−826.000	−0.0377720
$783$	0	0
$784$	0	0
$785$	−10913.0	−0.496180
$786$	0	0
$787$	−2097.00	−0.0949809	−0.0474905	−	0.998872i	$-0.515122\pi$
$787$	−2097.00	−0.0949809	−0.0474905	+	0.998872i	$0.515122\pi$
$788$	10936.0	0.494389
$789$	0	0
$790$	6930.00	0.312099
$791$	0	0
$792$	0	0
$793$	3366.00	0.150732
$794$	1998.00	0.0893027
$795$	0	0
$796$	8972.00	0.399503
$797$	35334.0	1.57038	0.785191	−	0.619254i	$-0.212565\pi$
$797$	35334.0	1.57038	0.785191	+	0.619254i	$0.212565\pi$
$798$	0	0
$799$	−10089.0	−0.446712
$800$	−2432.00	−0.107480
$801$	0	0
$802$	29514.0	1.29947
$803$	−10325.0	−0.453750
$804$	0	0
$805$	0	0
$806$	9900.00	0.432646
$807$	0	0
$808$	8680.00	0.377922
$809$	−42535.0	−1.84852	−0.924259	−	0.381766i	$-0.875316\pi$
$809$	−42535.0	−1.84852	−0.924259	+	0.381766i	$0.875316\pi$
$810$	0	0
$811$	30676.0	1.32821	0.664106	−	0.747638i	$-0.268812\pi$
$811$	30676.0	1.32821	0.664106	+	0.747638i	$0.268812\pi$
$812$	0	0
$813$	0	0
$814$	−770.000	−0.0331554
$815$	15757.0	0.677231
$816$	0	0
$817$	35620.0	1.52532
$818$	−266.000	−0.0113698
$819$	0	0
$820$	−13944.0	−0.593836
$821$	−37343.0	−1.58743	−0.793715	−	0.608290i	$-0.791856\pi$
$821$	−37343.0	−1.58743	−0.793715	+	0.608290i	$0.791856\pi$
$822$	0	0
$823$	2815.00	0.119228	0.0596141	−	0.998222i	$-0.481013\pi$
$823$	2815.00	0.119228	0.0596141	+	0.998222i	$0.481013\pi$
$824$	12424.0	0.525256
$825$	0	0
$826$	0	0
$827$	9276.00	0.390034	0.195017	−	0.980800i	$-0.437524\pi$
$827$	9276.00	0.390034	0.195017	+	0.980800i	$0.437524\pi$
$828$	0	0
$829$	18571.0	0.778043	0.389021	−	0.921229i	$-0.372813\pi$
$829$	18571.0	0.778043	0.389021	+	0.921229i	$0.372813\pi$
$830$	13048.0	0.545666
$831$	0	0
$832$	4224.00	0.176011
$833$	0	0
$834$	0	0
$835$	19516.0	0.808837
$836$	−19180.0	−0.793486
$837$	0	0
$838$	12840.0	0.529296
$839$	−29048.0	−1.19529	−0.597645	−	0.801761i	$-0.703897\pi$
$839$	−29048.0	−1.19529	−0.597645	+	0.801761i	$0.703897\pi$
$840$	0	0
$841$	−13153.0	−0.539301
$842$	20532.0	0.840356
$843$	0	0
$844$	4688.00	0.191194
$845$	−15113.0	−0.615270
$846$	0	0
$847$	0	0
$848$	6672.00	0.270186
$849$	0	0
$850$	8968.00	0.361882
$851$	77.0000	0.00310168
$852$	0	0
$853$	32090.0	1.28809	0.644045	−	0.764988i	$-0.277255\pi$
$853$	32090.0	1.28809	0.644045	+	0.764988i	$0.277255\pi$
$854$	0	0
$855$	0	0
$856$	−1032.00	−0.0412068
$857$	24537.0	0.978026	0.489013	−	0.872277i	$-0.337357\pi$
$857$	24537.0	0.978026	0.489013	+	0.872277i	$0.337357\pi$
$858$	0	0
$859$	20825.0	0.827171	0.413585	−	0.910465i	$-0.364276\pi$
$859$	20825.0	0.827171	0.413585	+	0.910465i	$0.364276\pi$
$860$	−7280.00	−0.288658
$861$	0	0
$862$	30426.0	1.20222
$863$	22847.0	0.901183	0.450591	−	0.892730i	$-0.351213\pi$
$863$	22847.0	0.901183	0.450591	+	0.892730i	$0.351213\pi$
$864$	0	0
$865$	11053.0	0.434466
$866$	−2756.00	−0.108144
$867$	0	0
$868$	0	0
$869$	17325.0	0.676307
$870$	0	0
$871$	28974.0	1.12715
$872$	−7720.00	−0.299808
$873$	0	0
$874$	1918.00	0.0742303
$875$	0	0
$876$	0	0
$877$	−42737.0	−1.64553	−0.822763	−	0.568385i	$-0.807568\pi$
$877$	−42737.0	−1.64553	−0.822763	+	0.568385i	$0.807568\pi$
$878$	−5526.00	−0.212407
$879$	0	0
$880$	3920.00	0.150163
$881$	−6162.00	−0.235645	−0.117822	−	0.993035i	$-0.537591\pi$
$881$	−6162.00	−0.235645	−0.117822	+	0.993035i	$0.537591\pi$
$882$	0	0
$883$	7748.00	0.295290	0.147645	−	0.989040i	$-0.452831\pi$
$883$	7748.00	0.295290	0.147645	+	0.989040i	$0.452831\pi$
$884$	−15576.0	−0.592622
$885$	0	0
$886$	−11698.0	−0.443569
$887$	25923.0	0.981296	0.490648	−	0.871358i	$-0.336760\pi$
$887$	25923.0	0.981296	0.490648	+	0.871358i	$0.336760\pi$
$888$	0	0
$889$	0	0
$890$	−12222.0	−0.460317
$891$	0	0
$892$	8096.00	0.303895
$893$	23427.0	0.877889
$894$	0	0
$895$	17157.0	0.640777
$896$	0	0
$897$	0	0
$898$	−9164.00	−0.340542
$899$	−7950.00	−0.294936
$900$	0	0
$901$	−24603.0	−0.909706
$902$	−34860.0	−1.28682
$903$	0	0
$904$	400.000	0.0147166
$905$	8190.00	0.300823
$906$	0	0
$907$	31935.0	1.16911	0.584556	−	0.811353i	$-0.301269\pi$
$907$	31935.0	1.16911	0.584556	+	0.811353i	$0.301269\pi$
$908$	−10284.0	−0.375866
$909$	0	0
$910$	0	0
$911$	−3408.00	−0.123943	−0.0619715	−	0.998078i	$-0.519739\pi$
$911$	−3408.00	−0.123943	−0.0619715	+	0.998078i	$0.519739\pi$
$912$	0	0
$913$	32620.0	1.18244
$914$	23102.0	0.836046
$915$	0	0
$916$	3580.00	0.129134
$917$	0	0
$918$	0	0
$919$	13909.0	0.499255	0.249628	−	0.968342i	$-0.419692\pi$
$919$	13909.0	0.499255	0.249628	+	0.968342i	$0.419692\pi$
$920$	−392.000	−0.0140477
$921$	0	0
$922$	18988.0	0.678239
$923$	51744.0	1.84526
$924$	0	0
$925$	−836.000	−0.0297162
$926$	−20320.0	−0.721119
$927$	0	0
$928$	−3392.00	−0.119987
$929$	24537.0	0.866559	0.433279	−	0.901260i	$-0.357356\pi$
$929$	24537.0	0.866559	0.433279	+	0.901260i	$0.357356\pi$
$930$	0	0
$931$	0	0
$932$	−7148.00	−0.251224
$933$	0	0
$934$	2614.00	0.0915768
$935$	−14455.0	−0.505593
$936$	0	0
$937$	−32758.0	−1.14211	−0.571055	−	0.820912i	$-0.693466\pi$
$937$	−32758.0	−1.14211	−0.571055	+	0.820912i	$0.693466\pi$
$938$	0	0
$939$	0	0
$940$	−4788.00	−0.166135
$941$	38561.0	1.33587	0.667934	−	0.744220i	$-0.267179\pi$
$941$	38561.0	1.33587	0.667934	+	0.744220i	$0.267179\pi$
$942$	0	0
$943$	3486.00	0.120382
$944$	272.000	0.00937801
$945$	0	0
$946$	−18200.0	−0.625511
$947$	−39661.0	−1.36094	−0.680470	−	0.732776i	$-0.738224\pi$
$947$	−39661.0	−1.36094	−0.680470	+	0.732776i	$0.738224\pi$
$948$	0	0
$949$	19470.0	0.665988
$950$	−20824.0	−0.711179
$951$	0	0
$952$	0	0
$953$	46618.0	1.58458	0.792290	−	0.610144i	$-0.208889\pi$
$953$	46618.0	1.58458	0.792290	+	0.610144i	$0.208889\pi$
$954$	0	0
$955$	−8925.00	−0.302415
$956$	20400.0	0.690150
$957$	0	0
$958$	−36574.0	−1.23346
$959$	0	0
$960$	0	0
$961$	−24166.0	−0.811185
$962$	1452.00	0.0486636
$963$	0	0
$964$	−16708.0	−0.558225
$965$	−245.000	−0.00817288
$966$	0	0
$967$	14816.0	0.492710	0.246355	−	0.969180i	$-0.420767\pi$
$967$	14816.0	0.492710	0.246355	+	0.969180i	$0.420767\pi$
$968$	−848.000	−0.0281568
$969$	0	0
$970$	4060.00	0.134390
$971$	16875.0	0.557718	0.278859	−	0.960332i	$-0.410044\pi$
$971$	16875.0	0.557718	0.278859	+	0.960332i	$0.410044\pi$
$972$	0	0
$973$	0	0
$974$	−29906.0	−0.983830
$975$	0	0
$976$	816.000	0.0267618
$977$	15837.0	0.518598	0.259299	−	0.965797i	$-0.416508\pi$
$977$	15837.0	0.518598	0.259299	+	0.965797i	$0.416508\pi$
$978$	0	0
$979$	−30555.0	−0.997489
$980$	0	0
$981$	0	0
$982$	−28704.0	−0.932771
$983$	−9915.00	−0.321708	−0.160854	−	0.986978i	$-0.551425\pi$
$983$	−9915.00	−0.321708	−0.160854	+	0.986978i	$0.551425\pi$
$984$	0	0
$985$	−19138.0	−0.619073
$986$	12508.0	0.403992
$987$	0	0
$988$	36168.0	1.16463
$989$	1820.00	0.0585163
$990$	0	0
$991$	−43681.0	−1.40017	−0.700087	−	0.714057i	$-0.746856\pi$
$991$	−43681.0	−1.40017	−0.700087	+	0.714057i	$0.746856\pi$
$992$	2400.00	0.0768146
$993$	0	0
$994$	0	0
$995$	−15701.0	−0.500256
$996$	0	0
$997$	−47113.0	−1.49657	−0.748287	−	0.663375i	$-0.769123\pi$
$997$	−47113.0	−1.49657	−0.748287	+	0.663375i	$0.769123\pi$
$998$	−11062.0	−0.350863
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.4.a.k.1.1		1
3.2	odd	2		98.4.a.b.1.1		1
7.2	even	3		126.4.g.c.109.1		2
7.3	odd	6		882.4.g.d.667.1		2
7.4	even	3		126.4.g.c.37.1		2
7.5	odd	6		882.4.g.d.361.1		2
7.6	odd	2		882.4.a.p.1.1		1
12.11	even	2		784.4.a.l.1.1		1
15.14	odd	2		2450.4.a.bh.1.1		1
21.2	odd	6		14.4.c.b.11.1	yes	2
21.5	even	6		98.4.c.e.67.1		2
21.11	odd	6		14.4.c.b.9.1	✓	2
21.17	even	6		98.4.c.e.79.1		2
21.20	even	2		98.4.a.c.1.1		1
84.11	even	6		112.4.i.b.65.1		2
84.23	even	6		112.4.i.b.81.1		2
84.83	odd	2		784.4.a.j.1.1		1
105.2	even	12		350.4.j.d.249.1		4
105.23	even	12		350.4.j.d.249.2		4
105.32	even	12		350.4.j.d.149.2		4
105.44	odd	6		350.4.e.b.151.1		2
105.53	even	12		350.4.j.d.149.1		4
105.74	odd	6		350.4.e.b.51.1		2
105.104	even	2		2450.4.a.bf.1.1		1
168.11	even	6		448.4.i.d.65.1		2
168.53	odd	6		448.4.i.c.65.1		2
168.107	even	6		448.4.i.d.193.1		2
168.149	odd	6		448.4.i.c.193.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.4.c.b.9.1	✓	2	21.11	odd	6
14.4.c.b.11.1	yes	2	21.2	odd	6
98.4.a.b.1.1		1	3.2	odd	2
98.4.a.c.1.1		1	21.20	even	2
98.4.c.e.67.1		2	21.5	even	6
98.4.c.e.79.1		2	21.17	even	6
112.4.i.b.65.1		2	84.11	even	6
112.4.i.b.81.1		2	84.23	even	6
126.4.g.c.37.1		2	7.4	even	3
126.4.g.c.109.1		2	7.2	even	3
350.4.e.b.51.1		2	105.74	odd	6
350.4.e.b.151.1		2	105.44	odd	6
350.4.j.d.149.1		4	105.53	even	12
350.4.j.d.149.2		4	105.32	even	12
350.4.j.d.249.1		4	105.2	even	12
350.4.j.d.249.2		4	105.23	even	12
448.4.i.c.65.1		2	168.53	odd	6
448.4.i.c.193.1		2	168.149	odd	6
448.4.i.d.65.1		2	168.11	even	6
448.4.i.d.193.1		2	168.107	even	6
784.4.a.j.1.1		1	84.83	odd	2
784.4.a.l.1.1		1	12.11	even	2
882.4.a.k.1.1		1	1.1	even	1	trivial
882.4.a.p.1.1		1	7.6	odd	2
882.4.g.d.361.1		2	7.5	odd	6
882.4.g.d.667.1		2	7.3	odd	6
2450.4.a.bf.1.1		1	105.104	even	2
2450.4.a.bh.1.1		1	15.14	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:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	882.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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