LMFDB - Embedded newform 162.4.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [162,4,Mod(1,162)] 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("162.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	162.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-2,0,4,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$9.55830942093$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 18)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	162.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-2.00000 q^{2} +4.00000 q^{4} +9.00000 q^{5} -31.0000 q^{7} -8.00000 q^{8} -18.0000 q^{10} +15.0000 q^{11} -37.0000 q^{13} +62.0000 q^{14} +16.0000 q^{16} +42.0000 q^{17} -28.0000 q^{19} +36.0000 q^{20} -30.0000 q^{22} -195.000 q^{23} -44.0000 q^{25} +74.0000 q^{26} -124.000 q^{28} -111.000 q^{29} -205.000 q^{31} -32.0000 q^{32} -84.0000 q^{34} -279.000 q^{35} -166.000 q^{37} +56.0000 q^{38} -72.0000 q^{40} +261.000 q^{41} -43.0000 q^{43} +60.0000 q^{44} +390.000 q^{46} -177.000 q^{47} +618.000 q^{49} +88.0000 q^{50} -148.000 q^{52} -114.000 q^{53} +135.000 q^{55} +248.000 q^{56} +222.000 q^{58} -159.000 q^{59} +191.000 q^{61} +410.000 q^{62} +64.0000 q^{64} -333.000 q^{65} -421.000 q^{67} +168.000 q^{68} +558.000 q^{70} -156.000 q^{71} +182.000 q^{73} +332.000 q^{74} -112.000 q^{76} -465.000 q^{77} +1133.00 q^{79} +144.000 q^{80} -522.000 q^{82} +1083.00 q^{83} +378.000 q^{85} +86.0000 q^{86} -120.000 q^{88} +1050.00 q^{89} +1147.00 q^{91} -780.000 q^{92} +354.000 q^{94} -252.000 q^{95} -901.000 q^{97} -1236.00 q^{98} +O(q^{100})\)

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
$5$	9.00000	0.804984	0.402492	−	0.915423i	$-0.368144\pi$
$5$	9.00000	0.804984	0.402492	+	0.915423i	$0.368144\pi$
$6$	0	0
$7$	−31.0000	−1.67384	−0.836921	−	0.547323i	$-0.815647\pi$
$7$	−31.0000	−1.67384	−0.836921	+	0.547323i	$0.815647\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−18.0000	−0.569210
$11$	15.0000	0.411152	0.205576	−	0.978641i	$-0.434093\pi$
$11$	15.0000	0.411152	0.205576	+	0.978641i	$0.434093\pi$
$12$	0	0
$13$	−37.0000	−0.789381	−0.394691	−	0.918814i	$-0.629148\pi$
$13$	−37.0000	−0.789381	−0.394691	+	0.918814i	$0.629148\pi$
$14$	62.0000	1.18359
$15$	0	0
$16$	16.0000	0.250000
$17$	42.0000	0.599206	0.299603	−	0.954064i	$-0.403146\pi$
$17$	42.0000	0.599206	0.299603	+	0.954064i	$0.403146\pi$
$18$	0	0
$19$	−28.0000	−0.338086	−0.169043	−	0.985609i	$-0.554068\pi$
$19$	−28.0000	−0.338086	−0.169043	+	0.985609i	$0.554068\pi$
$20$	36.0000	0.402492
$21$	0	0
$22$	−30.0000	−0.290728
$23$	−195.000	−1.76784	−0.883920	−	0.467639i	$-0.845105\pi$
$23$	−195.000	−1.76784	−0.883920	+	0.467639i	$0.845105\pi$
$24$	0	0
$25$	−44.0000	−0.352000
$26$	74.0000	0.558177
$27$	0	0
$28$	−124.000	−0.836921
$29$	−111.000	−0.710765	−0.355382	−	0.934721i	$-0.615649\pi$
$29$	−111.000	−0.710765	−0.355382	+	0.934721i	$0.615649\pi$
$30$	0	0
$31$	−205.000	−1.18771	−0.593856	−	0.804571i	$-0.702395\pi$
$31$	−205.000	−1.18771	−0.593856	+	0.804571i	$0.702395\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−84.0000	−0.423702
$35$	−279.000	−1.34742
$36$	0	0
$37$	−166.000	−0.737574	−0.368787	−	0.929514i	$-0.620227\pi$
$37$	−166.000	−0.737574	−0.368787	+	0.929514i	$0.620227\pi$
$38$	56.0000	0.239063
$39$	0	0
$40$	−72.0000	−0.284605
$41$	261.000	0.994179	0.497090	−	0.867699i	$-0.334402\pi$
$41$	261.000	0.994179	0.497090	+	0.867699i	$0.334402\pi$
$42$	0	0
$43$	−43.0000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−43.0000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	60.0000	0.205576
$45$	0	0
$46$	390.000	1.25005
$47$	−177.000	−0.549321	−0.274661	−	0.961541i	$-0.588566\pi$
$47$	−177.000	−0.549321	−0.274661	+	0.961541i	$0.588566\pi$
$48$	0	0
$49$	618.000	1.80175
$50$	88.0000	0.248902
$51$	0	0
$52$	−148.000	−0.394691
$53$	−114.000	−0.295455	−0.147727	−	0.989028i	$-0.547196\pi$
$53$	−114.000	−0.295455	−0.147727	+	0.989028i	$0.547196\pi$
$54$	0	0
$55$	135.000	0.330971
$56$	248.000	0.591793
$57$	0	0
$58$	222.000	0.502587
$59$	−159.000	−0.350848	−0.175424	−	0.984493i	$-0.556130\pi$
$59$	−159.000	−0.350848	−0.175424	+	0.984493i	$0.556130\pi$
$60$	0	0
$61$	191.000	0.400902	0.200451	−	0.979704i	$-0.435759\pi$
$61$	191.000	0.400902	0.200451	+	0.979704i	$0.435759\pi$
$62$	410.000	0.839840
$63$	0	0
$64$	64.0000	0.125000
$65$	−333.000	−0.635439
$66$	0	0
$67$	−421.000	−0.767662	−0.383831	−	0.923403i	$-0.625395\pi$
$67$	−421.000	−0.767662	−0.383831	+	0.923403i	$0.625395\pi$
$68$	168.000	0.299603
$69$	0	0
$70$	558.000	0.952768
$71$	−156.000	−0.260758	−0.130379	−	0.991464i	$-0.541619\pi$
$71$	−156.000	−0.260758	−0.130379	+	0.991464i	$0.541619\pi$
$72$	0	0
$73$	182.000	0.291801	0.145901	−	0.989299i	$-0.453392\pi$
$73$	182.000	0.291801	0.145901	+	0.989299i	$0.453392\pi$
$74$	332.000	0.521543
$75$	0	0
$76$	−112.000	−0.169043
$77$	−465.000	−0.688203
$78$	0	0
$79$	1133.00	1.61358	0.806788	−	0.590841i	$-0.201204\pi$
$79$	1133.00	1.61358	0.806788	+	0.590841i	$0.201204\pi$
$80$	144.000	0.201246
$81$	0	0
$82$	−522.000	−0.702991
$83$	1083.00	1.43223	0.716113	−	0.697985i	$-0.245920\pi$
$83$	1083.00	1.43223	0.716113	+	0.697985i	$0.245920\pi$
$84$	0	0
$85$	378.000	0.482351
$86$	86.0000	0.107833
$87$	0	0
$88$	−120.000	−0.145364
$89$	1050.00	1.25056	0.625280	−	0.780401i	$-0.284985\pi$
$89$	1050.00	1.25056	0.625280	+	0.780401i	$0.284985\pi$
$90$	0	0
$91$	1147.00	1.32130
$92$	−780.000	−0.883920
$93$	0	0
$94$	354.000	0.388429
$95$	−252.000	−0.272154
$96$	0	0
$97$	−901.000	−0.943121	−0.471560	−	0.881834i	$-0.656309\pi$
$97$	−901.000	−0.943121	−0.471560	+	0.881834i	$0.656309\pi$
$98$	−1236.00	−1.27403
$99$	0	0
$100$	−176.000	−0.176000
$101$	−387.000	−0.381267	−0.190633	−	0.981661i	$-0.561054\pi$
$101$	−387.000	−0.381267	−0.190633	+	0.981661i	$0.561054\pi$
$102$	0	0
$103$	551.000	0.527103	0.263552	−	0.964645i	$-0.415106\pi$
$103$	551.000	0.527103	0.263552	+	0.964645i	$0.415106\pi$
$104$	296.000	0.279088
$105$	0	0
$106$	228.000	0.208918
$107$	12.0000	0.0108419	0.00542095	−	0.999985i	$-0.498274\pi$
$107$	12.0000	0.0108419	0.00542095	+	0.999985i	$0.498274\pi$
$108$	0	0
$109$	−502.000	−0.441127	−0.220564	−	0.975373i	$-0.570790\pi$
$109$	−502.000	−0.441127	−0.220564	+	0.975373i	$0.570790\pi$
$110$	−270.000	−0.234032
$111$	0	0
$112$	−496.000	−0.418461
$113$	1401.00	1.16633	0.583164	−	0.812355i	$-0.301815\pi$
$113$	1401.00	1.16633	0.583164	+	0.812355i	$0.301815\pi$
$114$	0	0
$115$	−1755.00	−1.42308
$116$	−444.000	−0.355382
$117$	0	0
$118$	318.000	0.248087
$119$	−1302.00	−1.00298
$120$	0	0
$121$	−1106.00	−0.830954
$122$	−382.000	−0.283481
$123$	0	0
$124$	−820.000	−0.593856
$125$	−1521.00	−1.08834
$126$	0	0
$127$	−880.000	−0.614861	−0.307431	−	0.951571i	$-0.599469\pi$
$127$	−880.000	−0.614861	−0.307431	+	0.951571i	$0.599469\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	666.000	0.449324
$131$	−1503.00	−1.00243	−0.501213	−	0.865324i	$-0.667113\pi$
$131$	−1503.00	−1.00243	−0.501213	+	0.865324i	$0.667113\pi$
$132$	0	0
$133$	868.000	0.565903
$134$	842.000	0.542819
$135$	0	0
$136$	−336.000	−0.211851
$137$	2661.00	1.65945	0.829725	−	0.558173i	$-0.188497\pi$
$137$	2661.00	1.65945	0.829725	+	0.558173i	$0.188497\pi$
$138$	0	0
$139$	−121.000	−0.0738352	−0.0369176	−	0.999318i	$-0.511754\pi$
$139$	−121.000	−0.0738352	−0.0369176	+	0.999318i	$0.511754\pi$
$140$	−1116.00	−0.673709
$141$	0	0
$142$	312.000	0.184384
$143$	−555.000	−0.324555
$144$	0	0
$145$	−999.000	−0.572155
$146$	−364.000	−0.206335
$147$	0	0
$148$	−664.000	−0.368787
$149$	2829.00	1.55544	0.777721	−	0.628610i	$-0.216376\pi$
$149$	2829.00	1.55544	0.777721	+	0.628610i	$0.216376\pi$
$150$	0	0
$151$	461.000	0.248448	0.124224	−	0.992254i	$-0.460356\pi$
$151$	461.000	0.248448	0.124224	+	0.992254i	$0.460356\pi$
$152$	224.000	0.119532
$153$	0	0
$154$	930.000	0.486633
$155$	−1845.00	−0.956090
$156$	0	0
$157$	−2977.00	−1.51332	−0.756658	−	0.653811i	$-0.773169\pi$
$157$	−2977.00	−1.51332	−0.756658	+	0.653811i	$0.773169\pi$
$158$	−2266.00	−1.14097
$159$	0	0
$160$	−288.000	−0.142302
$161$	6045.00	2.95909
$162$	0	0
$163$	−3316.00	−1.59343	−0.796715	−	0.604355i	$-0.793431\pi$
$163$	−3316.00	−1.59343	−0.796715	+	0.604355i	$0.793431\pi$
$164$	1044.00	0.497090
$165$	0	0
$166$	−2166.00	−1.01274
$167$	681.000	0.315553	0.157777	−	0.987475i	$-0.449567\pi$
$167$	681.000	0.315553	0.157777	+	0.987475i	$0.449567\pi$
$168$	0	0
$169$	−828.000	−0.376878
$170$	−756.000	−0.341074
$171$	0	0
$172$	−172.000	−0.0762493
$173$	3981.00	1.74954	0.874768	−	0.484541i	$-0.161014\pi$
$173$	3981.00	1.74954	0.874768	+	0.484541i	$0.161014\pi$
$174$	0	0
$175$	1364.00	0.589193
$176$	240.000	0.102788
$177$	0	0
$178$	−2100.00	−0.884279
$179$	−2004.00	−0.836793	−0.418397	−	0.908264i	$-0.637408\pi$
$179$	−2004.00	−0.836793	−0.418397	+	0.908264i	$0.637408\pi$
$180$	0	0
$181$	1274.00	0.523181	0.261590	−	0.965179i	$-0.415753\pi$
$181$	1274.00	0.523181	0.261590	+	0.965179i	$0.415753\pi$
$182$	−2294.00	−0.934300
$183$	0	0
$184$	1560.00	0.625026
$185$	−1494.00	−0.593735
$186$	0	0
$187$	630.000	0.246365
$188$	−708.000	−0.274661
$189$	0	0
$190$	504.000	0.192442
$191$	−1161.00	−0.439827	−0.219914	−	0.975519i	$-0.570578\pi$
$191$	−1161.00	−0.439827	−0.219914	+	0.975519i	$0.570578\pi$
$192$	0	0
$193$	3611.00	1.34676	0.673382	−	0.739295i	$-0.264841\pi$
$193$	3611.00	1.34676	0.673382	+	0.739295i	$0.264841\pi$
$194$	1802.00	0.666887
$195$	0	0
$196$	2472.00	0.900875
$197$	−2046.00	−0.739957	−0.369978	−	0.929040i	$-0.620635\pi$
$197$	−2046.00	−0.739957	−0.369978	+	0.929040i	$0.620635\pi$
$198$	0	0
$199$	2996.00	1.06724	0.533620	−	0.845724i	$-0.320831\pi$
$199$	2996.00	1.06724	0.533620	+	0.845724i	$0.320831\pi$
$200$	352.000	0.124451
$201$	0	0
$202$	774.000	0.269596
$203$	3441.00	1.18971
$204$	0	0
$205$	2349.00	0.800299
$206$	−1102.00	−0.372718
$207$	0	0
$208$	−592.000	−0.197345
$209$	−420.000	−0.139005
$210$	0	0
$211$	755.000	0.246333	0.123167	−	0.992386i	$-0.460695\pi$
$211$	755.000	0.246333	0.123167	+	0.992386i	$0.460695\pi$
$212$	−456.000	−0.147727
$213$	0	0
$214$	−24.0000	−0.00766638
$215$	−387.000	−0.122759
$216$	0	0
$217$	6355.00	1.98804
$218$	1004.00	0.311924
$219$	0	0
$220$	540.000	0.165485
$221$	−1554.00	−0.473002
$222$	0	0
$223$	−3463.00	−1.03991	−0.519954	−	0.854194i	$-0.674051\pi$
$223$	−3463.00	−1.03991	−0.519954	+	0.854194i	$0.674051\pi$
$224$	992.000	0.295896
$225$	0	0
$226$	−2802.00	−0.824718
$227$	−6225.00	−1.82012	−0.910061	−	0.414474i	$-0.863966\pi$
$227$	−6225.00	−1.82012	−0.910061	+	0.414474i	$0.863966\pi$
$228$	0	0
$229$	−1465.00	−0.422751	−0.211375	−	0.977405i	$-0.567794\pi$
$229$	−1465.00	−0.422751	−0.211375	+	0.977405i	$0.567794\pi$
$230$	3510.00	1.00627
$231$	0	0
$232$	888.000	0.251293
$233$	−2634.00	−0.740597	−0.370298	−	0.928913i	$-0.620745\pi$
$233$	−2634.00	−0.740597	−0.370298	+	0.928913i	$0.620745\pi$
$234$	0	0
$235$	−1593.00	−0.442195
$236$	−636.000	−0.175424
$237$	0	0
$238$	2604.00	0.709211
$239$	−6915.00	−1.87152	−0.935762	−	0.352633i	$-0.885287\pi$
$239$	−6915.00	−1.87152	−0.935762	+	0.352633i	$0.885287\pi$
$240$	0	0
$241$	−1489.00	−0.397987	−0.198994	−	0.980001i	$-0.563767\pi$
$241$	−1489.00	−0.397987	−0.198994	+	0.980001i	$0.563767\pi$
$242$	2212.00	0.587573
$243$	0	0
$244$	764.000	0.200451
$245$	5562.00	1.45038
$246$	0	0
$247$	1036.00	0.266879
$248$	1640.00	0.419920
$249$	0	0
$250$	3042.00	0.769572
$251$	4620.00	1.16180	0.580900	−	0.813975i	$-0.302701\pi$
$251$	4620.00	1.16180	0.580900	+	0.813975i	$0.302701\pi$
$252$	0	0
$253$	−2925.00	−0.726850
$254$	1760.00	0.434773
$255$	0	0
$256$	256.000	0.0625000
$257$	−3351.00	−0.813345	−0.406672	−	0.913574i	$-0.633311\pi$
$257$	−3351.00	−0.813345	−0.406672	+	0.913574i	$0.633311\pi$
$258$	0	0
$259$	5146.00	1.23458
$260$	−1332.00	−0.317720
$261$	0	0
$262$	3006.00	0.708822
$263$	603.000	0.141379	0.0706893	−	0.997498i	$-0.477480\pi$
$263$	603.000	0.141379	0.0706893	+	0.997498i	$0.477480\pi$
$264$	0	0
$265$	−1026.00	−0.237837
$266$	−1736.00	−0.400154
$267$	0	0
$268$	−1684.00	−0.383831
$269$	1470.00	0.333188	0.166594	−	0.986026i	$-0.446723\pi$
$269$	1470.00	0.333188	0.166594	+	0.986026i	$0.446723\pi$
$270$	0	0
$271$	2072.00	0.464447	0.232223	−	0.972662i	$-0.425400\pi$
$271$	2072.00	0.464447	0.232223	+	0.972662i	$0.425400\pi$
$272$	672.000	0.149801
$273$	0	0
$274$	−5322.00	−1.17341
$275$	−660.000	−0.144725
$276$	0	0
$277$	7139.00	1.54852	0.774262	−	0.632866i	$-0.218121\pi$
$277$	7139.00	1.54852	0.774262	+	0.632866i	$0.218121\pi$
$278$	242.000	0.0522093
$279$	0	0
$280$	2232.00	0.476384
$281$	−8427.00	−1.78901	−0.894507	−	0.447055i	$-0.852473\pi$
$281$	−8427.00	−1.78901	−0.894507	+	0.447055i	$0.852473\pi$
$282$	0	0
$283$	−457.000	−0.0959923	−0.0479962	−	0.998848i	$-0.515284\pi$
$283$	−457.000	−0.0959923	−0.0479962	+	0.998848i	$0.515284\pi$
$284$	−624.000	−0.130379
$285$	0	0
$286$	1110.00	0.229495
$287$	−8091.00	−1.66410
$288$	0	0
$289$	−3149.00	−0.640953
$290$	1998.00	0.404574
$291$	0	0
$292$	728.000	0.145901
$293$	5889.00	1.17419	0.587097	−	0.809516i	$-0.300271\pi$
$293$	5889.00	1.17419	0.587097	+	0.809516i	$0.300271\pi$
$294$	0	0
$295$	−1431.00	−0.282427
$296$	1328.00	0.260772
$297$	0	0
$298$	−5658.00	−1.09986
$299$	7215.00	1.39550
$300$	0	0
$301$	1333.00	0.255259
$302$	−922.000	−0.175679
$303$	0	0
$304$	−448.000	−0.0845216
$305$	1719.00	0.322720
$306$	0	0
$307$	−1204.00	−0.223830	−0.111915	−	0.993718i	$-0.535698\pi$
$307$	−1204.00	−0.223830	−0.111915	+	0.993718i	$0.535698\pi$
$308$	−1860.00	−0.344102
$309$	0	0
$310$	3690.00	0.676058
$311$	3285.00	0.598956	0.299478	−	0.954103i	$-0.403188\pi$
$311$	3285.00	0.598956	0.299478	+	0.954103i	$0.403188\pi$
$312$	0	0
$313$	−10057.0	−1.81615	−0.908075	−	0.418807i	$-0.862449\pi$
$313$	−10057.0	−1.81615	−0.908075	+	0.418807i	$0.862449\pi$
$314$	5954.00	1.07008
$315$	0	0
$316$	4532.00	0.806788
$317$	−2295.00	−0.406625	−0.203312	−	0.979114i	$-0.565171\pi$
$317$	−2295.00	−0.406625	−0.203312	+	0.979114i	$0.565171\pi$
$318$	0	0
$319$	−1665.00	−0.292232
$320$	576.000	0.100623
$321$	0	0
$322$	−12090.0	−2.09239
$323$	−1176.00	−0.202583
$324$	0	0
$325$	1628.00	0.277862
$326$	6632.00	1.12673
$327$	0	0
$328$	−2088.00	−0.351495
$329$	5487.00	0.919478
$330$	0	0
$331$	−6679.00	−1.10910	−0.554548	−	0.832151i	$-0.687109\pi$
$331$	−6679.00	−1.10910	−0.554548	+	0.832151i	$0.687109\pi$
$332$	4332.00	0.716113
$333$	0	0
$334$	−1362.00	−0.223130
$335$	−3789.00	−0.617956
$336$	0	0
$337$	2183.00	0.352865	0.176433	−	0.984313i	$-0.443544\pi$
$337$	2183.00	0.352865	0.176433	+	0.984313i	$0.443544\pi$
$338$	1656.00	0.266493
$339$	0	0
$340$	1512.00	0.241176
$341$	−3075.00	−0.488330
$342$	0	0
$343$	−8525.00	−1.34200
$344$	344.000	0.0539164
$345$	0	0
$346$	−7962.00	−1.23711
$347$	−3891.00	−0.601959	−0.300980	−	0.953631i	$-0.597314\pi$
$347$	−3891.00	−0.601959	−0.300980	+	0.953631i	$0.597314\pi$
$348$	0	0
$349$	2795.00	0.428690	0.214345	−	0.976758i	$-0.431238\pi$
$349$	2795.00	0.428690	0.214345	+	0.976758i	$0.431238\pi$
$350$	−2728.00	−0.416622
$351$	0	0
$352$	−480.000	−0.0726821
$353$	−4755.00	−0.716949	−0.358475	−	0.933539i	$-0.616703\pi$
$353$	−4755.00	−0.716949	−0.358475	+	0.933539i	$0.616703\pi$
$354$	0	0
$355$	−1404.00	−0.209906
$356$	4200.00	0.625280
$357$	0	0
$358$	4008.00	0.591702
$359$	−4608.00	−0.677440	−0.338720	−	0.940887i	$-0.609994\pi$
$359$	−4608.00	−0.677440	−0.338720	+	0.940887i	$0.609994\pi$
$360$	0	0
$361$	−6075.00	−0.885698
$362$	−2548.00	−0.369944
$363$	0	0
$364$	4588.00	0.660650
$365$	1638.00	0.234895
$366$	0	0
$367$	3845.00	0.546887	0.273443	−	0.961888i	$-0.411837\pi$
$367$	3845.00	0.546887	0.273443	+	0.961888i	$0.411837\pi$
$368$	−3120.00	−0.441960
$369$	0	0
$370$	2988.00	0.419834
$371$	3534.00	0.494545
$372$	0	0
$373$	−8317.00	−1.15453	−0.577263	−	0.816559i	$-0.695879\pi$
$373$	−8317.00	−1.15453	−0.577263	+	0.816559i	$0.695879\pi$
$374$	−1260.00	−0.174206
$375$	0	0
$376$	1416.00	0.194214
$377$	4107.00	0.561064
$378$	0	0
$379$	12560.0	1.70228	0.851140	−	0.524939i	$-0.175912\pi$
$379$	12560.0	1.70228	0.851140	+	0.524939i	$0.175912\pi$
$380$	−1008.00	−0.136077
$381$	0	0
$382$	2322.00	0.311005
$383$	−12087.0	−1.61258	−0.806288	−	0.591523i	$-0.798527\pi$
$383$	−12087.0	−1.61258	−0.806288	+	0.591523i	$0.798527\pi$
$384$	0	0
$385$	−4185.00	−0.553993
$386$	−7222.00	−0.952306
$387$	0	0
$388$	−3604.00	−0.471560
$389$	8541.00	1.11323	0.556614	−	0.830771i	$-0.312100\pi$
$389$	8541.00	1.11323	0.556614	+	0.830771i	$0.312100\pi$
$390$	0	0
$391$	−8190.00	−1.05930
$392$	−4944.00	−0.637015
$393$	0	0
$394$	4092.00	0.523228
$395$	10197.0	1.29890
$396$	0	0
$397$	−13174.0	−1.66545	−0.832726	−	0.553686i	$-0.813221\pi$
$397$	−13174.0	−1.66545	−0.832726	+	0.553686i	$0.813221\pi$
$398$	−5992.00	−0.754653
$399$	0	0
$400$	−704.000	−0.0880000
$401$	−9603.00	−1.19589	−0.597944	−	0.801538i	$-0.704015\pi$
$401$	−9603.00	−1.19589	−0.597944	+	0.801538i	$0.704015\pi$
$402$	0	0
$403$	7585.00	0.937558
$404$	−1548.00	−0.190633
$405$	0	0
$406$	−6882.00	−0.841251
$407$	−2490.00	−0.303255
$408$	0	0
$409$	11471.0	1.38681	0.693404	−	0.720549i	$-0.256110\pi$
$409$	11471.0	1.38681	0.693404	+	0.720549i	$0.256110\pi$
$410$	−4698.00	−0.565897
$411$	0	0
$412$	2204.00	0.263552
$413$	4929.00	0.587264
$414$	0	0
$415$	9747.00	1.15292
$416$	1184.00	0.139544
$417$	0	0
$418$	840.000	0.0982913
$419$	5973.00	0.696420	0.348210	−	0.937416i	$-0.386790\pi$
$419$	5973.00	0.696420	0.348210	+	0.937416i	$0.386790\pi$
$420$	0	0
$421$	−8905.00	−1.03089	−0.515443	−	0.856924i	$-0.672373\pi$
$421$	−8905.00	−1.03089	−0.515443	+	0.856924i	$0.672373\pi$
$422$	−1510.00	−0.174184
$423$	0	0
$424$	912.000	0.104459
$425$	−1848.00	−0.210920
$426$	0	0
$427$	−5921.00	−0.671047
$428$	48.0000	0.00542095
$429$	0	0
$430$	774.000	0.0868037
$431$	−1416.00	−0.158251	−0.0791257	−	0.996865i	$-0.525213\pi$
$431$	−1416.00	−0.158251	−0.0791257	+	0.996865i	$0.525213\pi$
$432$	0	0
$433$	10766.0	1.19488	0.597438	−	0.801915i	$-0.296186\pi$
$433$	10766.0	1.19488	0.597438	+	0.801915i	$0.296186\pi$
$434$	−12710.0	−1.40576
$435$	0	0
$436$	−2008.00	−0.220564
$437$	5460.00	0.597682
$438$	0	0
$439$	4349.00	0.472817	0.236408	−	0.971654i	$-0.424030\pi$
$439$	4349.00	0.472817	0.236408	+	0.971654i	$0.424030\pi$
$440$	−1080.00	−0.117016
$441$	0	0
$442$	3108.00	0.334463
$443$	14547.0	1.56016	0.780078	−	0.625683i	$-0.215180\pi$
$443$	14547.0	1.56016	0.780078	+	0.625683i	$0.215180\pi$
$444$	0	0
$445$	9450.00	1.00668
$446$	6926.00	0.735326
$447$	0	0
$448$	−1984.00	−0.209230
$449$	3330.00	0.350005	0.175003	−	0.984568i	$-0.444007\pi$
$449$	3330.00	0.350005	0.175003	+	0.984568i	$0.444007\pi$
$450$	0	0
$451$	3915.00	0.408759
$452$	5604.00	0.583164
$453$	0	0
$454$	12450.0	1.28702
$455$	10323.0	1.06363
$456$	0	0
$457$	8147.00	0.833918	0.416959	−	0.908925i	$-0.363096\pi$
$457$	8147.00	0.833918	0.416959	+	0.908925i	$0.363096\pi$
$458$	2930.00	0.298930
$459$	0	0
$460$	−7020.00	−0.711542
$461$	−8031.00	−0.811369	−0.405684	−	0.914013i	$-0.632967\pi$
$461$	−8031.00	−0.811369	−0.405684	+	0.914013i	$0.632967\pi$
$462$	0	0
$463$	4283.00	0.429909	0.214955	−	0.976624i	$-0.431040\pi$
$463$	4283.00	0.429909	0.214955	+	0.976624i	$0.431040\pi$
$464$	−1776.00	−0.177691
$465$	0	0
$466$	5268.00	0.523681
$467$	−5460.00	−0.541025	−0.270512	−	0.962716i	$-0.587193\pi$
$467$	−5460.00	−0.541025	−0.270512	+	0.962716i	$0.587193\pi$
$468$	0	0
$469$	13051.0	1.28494
$470$	3186.00	0.312679
$471$	0	0
$472$	1272.00	0.124044
$473$	−645.000	−0.0627001
$474$	0	0
$475$	1232.00	0.119006
$476$	−5208.00	−0.501488
$477$	0	0
$478$	13830.0	1.32337
$479$	−429.000	−0.0409217	−0.0204609	−	0.999791i	$-0.506513\pi$
$479$	−429.000	−0.0409217	−0.0204609	+	0.999791i	$0.506513\pi$
$480$	0	0
$481$	6142.00	0.582227
$482$	2978.00	0.281419
$483$	0	0
$484$	−4424.00	−0.415477
$485$	−8109.00	−0.759197
$486$	0	0
$487$	−11296.0	−1.05107	−0.525535	−	0.850772i	$-0.676135\pi$
$487$	−11296.0	−1.05107	−0.525535	+	0.850772i	$0.676135\pi$
$488$	−1528.00	−0.141740
$489$	0	0
$490$	−11124.0	−1.02557
$491$	14673.0	1.34864	0.674321	−	0.738438i	$-0.264436\pi$
$491$	14673.0	1.34864	0.674321	+	0.738438i	$0.264436\pi$
$492$	0	0
$493$	−4662.00	−0.425894
$494$	−2072.00	−0.188712
$495$	0	0
$496$	−3280.00	−0.296928
$497$	4836.00	0.436467
$498$	0	0
$499$	13439.0	1.20564	0.602818	−	0.797879i	$-0.294045\pi$
$499$	13439.0	1.20564	0.602818	+	0.797879i	$0.294045\pi$
$500$	−6084.00	−0.544170
$501$	0	0
$502$	−9240.00	−0.821517
$503$	17388.0	1.54134	0.770669	−	0.637236i	$-0.219922\pi$
$503$	17388.0	1.54134	0.770669	+	0.637236i	$0.219922\pi$
$504$	0	0
$505$	−3483.00	−0.306914
$506$	5850.00	0.513961
$507$	0	0
$508$	−3520.00	−0.307431
$509$	3789.00	0.329950	0.164975	−	0.986298i	$-0.447246\pi$
$509$	3789.00	0.329950	0.164975	+	0.986298i	$0.447246\pi$
$510$	0	0
$511$	−5642.00	−0.488429
$512$	−512.000	−0.0441942
$513$	0	0
$514$	6702.00	0.575122
$515$	4959.00	0.424310
$516$	0	0
$517$	−2655.00	−0.225854
$518$	−10292.0	−0.872982
$519$	0	0
$520$	2664.00	0.224662
$521$	−9786.00	−0.822903	−0.411451	−	0.911432i	$-0.634978\pi$
$521$	−9786.00	−0.822903	−0.411451	+	0.911432i	$0.634978\pi$
$522$	0	0
$523$	−8008.00	−0.669532	−0.334766	−	0.942301i	$-0.608657\pi$
$523$	−8008.00	−0.669532	−0.334766	+	0.942301i	$0.608657\pi$
$524$	−6012.00	−0.501213
$525$	0	0
$526$	−1206.00	−0.0999698
$527$	−8610.00	−0.711684
$528$	0	0
$529$	25858.0	2.12526
$530$	2052.00	0.168176
$531$	0	0
$532$	3472.00	0.282952
$533$	−9657.00	−0.784786
$534$	0	0
$535$	108.000	0.00872756
$536$	3368.00	0.271409
$537$	0	0
$538$	−2940.00	−0.235599
$539$	9270.00	0.740793
$540$	0	0
$541$	−2938.00	−0.233483	−0.116742	−	0.993162i	$-0.537245\pi$
$541$	−2938.00	−0.233483	−0.116742	+	0.993162i	$0.537245\pi$
$542$	−4144.00	−0.328413
$543$	0	0
$544$	−1344.00	−0.105926
$545$	−4518.00	−0.355101
$546$	0	0
$547$	−10375.0	−0.810974	−0.405487	−	0.914101i	$-0.632898\pi$
$547$	−10375.0	−0.810974	−0.405487	+	0.914101i	$0.632898\pi$
$548$	10644.0	0.829725
$549$	0	0
$550$	1320.00	0.102336
$551$	3108.00	0.240300
$552$	0	0
$553$	−35123.0	−2.70087
$554$	−14278.0	−1.09497
$555$	0	0
$556$	−484.000	−0.0369176
$557$	−3306.00	−0.251490	−0.125745	−	0.992063i	$-0.540132\pi$
$557$	−3306.00	−0.251490	−0.125745	+	0.992063i	$0.540132\pi$
$558$	0	0
$559$	1591.00	0.120379
$560$	−4464.00	−0.336854
$561$	0	0
$562$	16854.0	1.26502
$563$	21093.0	1.57898	0.789488	−	0.613765i	$-0.210346\pi$
$563$	21093.0	1.57898	0.789488	+	0.613765i	$0.210346\pi$
$564$	0	0
$565$	12609.0	0.938875
$566$	914.000	0.0678768
$567$	0	0
$568$	1248.00	0.0921918
$569$	−1287.00	−0.0948222	−0.0474111	−	0.998875i	$-0.515097\pi$
$569$	−1287.00	−0.0948222	−0.0474111	+	0.998875i	$0.515097\pi$
$570$	0	0
$571$	15035.0	1.10192	0.550959	−	0.834532i	$-0.314262\pi$
$571$	15035.0	1.10192	0.550959	+	0.834532i	$0.314262\pi$
$572$	−2220.00	−0.162278
$573$	0	0
$574$	16182.0	1.17670
$575$	8580.00	0.622280
$576$	0	0
$577$	1190.00	0.0858585	0.0429292	−	0.999078i	$-0.486331\pi$
$577$	1190.00	0.0858585	0.0429292	+	0.999078i	$0.486331\pi$
$578$	6298.00	0.453222
$579$	0	0
$580$	−3996.00	−0.286077
$581$	−33573.0	−2.39732
$582$	0	0
$583$	−1710.00	−0.121477
$584$	−1456.00	−0.103167
$585$	0	0
$586$	−11778.0	−0.830281
$587$	17883.0	1.25743	0.628714	−	0.777637i	$-0.283582\pi$
$587$	17883.0	1.25743	0.628714	+	0.777637i	$0.283582\pi$
$588$	0	0
$589$	5740.00	0.401549
$590$	2862.00	0.199706
$591$	0	0
$592$	−2656.00	−0.184393
$593$	−20118.0	−1.39317	−0.696583	−	0.717476i	$-0.745297\pi$
$593$	−20118.0	−1.39317	−0.696583	+	0.717476i	$0.745297\pi$
$594$	0	0
$595$	−11718.0	−0.807380
$596$	11316.0	0.777721
$597$	0	0
$598$	−14430.0	−0.986767
$599$	1065.00	0.0726456	0.0363228	−	0.999340i	$-0.488436\pi$
$599$	1065.00	0.0726456	0.0363228	+	0.999340i	$0.488436\pi$
$600$	0	0
$601$	−20725.0	−1.40664	−0.703320	−	0.710874i	$-0.748300\pi$
$601$	−20725.0	−1.40664	−0.703320	+	0.710874i	$0.748300\pi$
$602$	−2666.00	−0.180495
$603$	0	0
$604$	1844.00	0.124224
$605$	−9954.00	−0.668905
$606$	0	0
$607$	−15745.0	−1.05283	−0.526417	−	0.850227i	$-0.676465\pi$
$607$	−15745.0	−1.05283	−0.526417	+	0.850227i	$0.676465\pi$
$608$	896.000	0.0597658
$609$	0	0
$610$	−3438.00	−0.228198
$611$	6549.00	0.433624
$612$	0	0
$613$	5042.00	0.332210	0.166105	−	0.986108i	$-0.446881\pi$
$613$	5042.00	0.332210	0.166105	+	0.986108i	$0.446881\pi$
$614$	2408.00	0.158272
$615$	0	0
$616$	3720.00	0.243317
$617$	10053.0	0.655946	0.327973	−	0.944687i	$-0.393635\pi$
$617$	10053.0	0.655946	0.327973	+	0.944687i	$0.393635\pi$
$618$	0	0
$619$	−5983.00	−0.388493	−0.194246	−	0.980953i	$-0.562226\pi$
$619$	−5983.00	−0.388493	−0.194246	+	0.980953i	$0.562226\pi$
$620$	−7380.00	−0.478045
$621$	0	0
$622$	−6570.00	−0.423526
$623$	−32550.0	−2.09324
$624$	0	0
$625$	−8189.00	−0.524096
$626$	20114.0	1.28421
$627$	0	0
$628$	−11908.0	−0.756658
$629$	−6972.00	−0.441958
$630$	0	0
$631$	−19696.0	−1.24261	−0.621304	−	0.783570i	$-0.713397\pi$
$631$	−19696.0	−1.24261	−0.621304	+	0.783570i	$0.713397\pi$
$632$	−9064.00	−0.570485
$633$	0	0
$634$	4590.00	0.287527
$635$	−7920.00	−0.494954
$636$	0	0
$637$	−22866.0	−1.42227
$638$	3330.00	0.206639
$639$	0	0
$640$	−1152.00	−0.0711512
$641$	10977.0	0.676389	0.338195	−	0.941076i	$-0.390184\pi$
$641$	10977.0	0.676389	0.338195	+	0.941076i	$0.390184\pi$
$642$	0	0
$643$	−15829.0	−0.970816	−0.485408	−	0.874288i	$-0.661329\pi$
$643$	−15829.0	−0.970816	−0.485408	+	0.874288i	$0.661329\pi$
$644$	24180.0	1.47954
$645$	0	0
$646$	2352.00	0.143248
$647$	−28224.0	−1.71499	−0.857496	−	0.514490i	$-0.827981\pi$
$647$	−28224.0	−1.71499	−0.857496	+	0.514490i	$0.827981\pi$
$648$	0	0
$649$	−2385.00	−0.144252
$650$	−3256.00	−0.196478
$651$	0	0
$652$	−13264.0	−0.796715
$653$	−28167.0	−1.68799	−0.843997	−	0.536348i	$-0.819803\pi$
$653$	−28167.0	−1.68799	−0.843997	+	0.536348i	$0.819803\pi$
$654$	0	0
$655$	−13527.0	−0.806937
$656$	4176.00	0.248545
$657$	0	0
$658$	−10974.0	−0.650169
$659$	−10737.0	−0.634680	−0.317340	−	0.948312i	$-0.602790\pi$
$659$	−10737.0	−0.634680	−0.317340	+	0.948312i	$0.602790\pi$
$660$	0	0
$661$	10127.0	0.595907	0.297954	−	0.954580i	$-0.403696\pi$
$661$	10127.0	0.595907	0.297954	+	0.954580i	$0.403696\pi$
$662$	13358.0	0.784250
$663$	0	0
$664$	−8664.00	−0.506368
$665$	7812.00	0.455543
$666$	0	0
$667$	21645.0	1.25652
$668$	2724.00	0.157777
$669$	0	0
$670$	7578.00	0.436961
$671$	2865.00	0.164832
$672$	0	0
$673$	251.000	0.0143764	0.00718822	−	0.999974i	$-0.497712\pi$
$673$	251.000	0.0143764	0.00718822	+	0.999974i	$0.497712\pi$
$674$	−4366.00	−0.249513
$675$	0	0
$676$	−3312.00	−0.188439
$677$	−8451.00	−0.479761	−0.239881	−	0.970802i	$-0.577108\pi$
$677$	−8451.00	−0.479761	−0.239881	+	0.970802i	$0.577108\pi$
$678$	0	0
$679$	27931.0	1.57864
$680$	−3024.00	−0.170537
$681$	0	0
$682$	6150.00	0.345302
$683$	25884.0	1.45011	0.725054	−	0.688692i	$-0.241815\pi$
$683$	25884.0	1.45011	0.725054	+	0.688692i	$0.241815\pi$
$684$	0	0
$685$	23949.0	1.33583
$686$	17050.0	0.948939
$687$	0	0
$688$	−688.000	−0.0381246
$689$	4218.00	0.233226
$690$	0	0
$691$	6365.00	0.350414	0.175207	−	0.984532i	$-0.443941\pi$
$691$	6365.00	0.350414	0.175207	+	0.984532i	$0.443941\pi$
$692$	15924.0	0.874768
$693$	0	0
$694$	7782.00	0.425649
$695$	−1089.00	−0.0594362
$696$	0	0
$697$	10962.0	0.595718
$698$	−5590.00	−0.303130
$699$	0	0
$700$	5456.00	0.294596
$701$	−1122.00	−0.0604527	−0.0302264	−	0.999543i	$-0.509623\pi$
$701$	−1122.00	−0.0604527	−0.0302264	+	0.999543i	$0.509623\pi$
$702$	0	0
$703$	4648.00	0.249364
$704$	960.000	0.0513940
$705$	0	0
$706$	9510.00	0.506960
$707$	11997.0	0.638181
$708$	0	0
$709$	4283.00	0.226871	0.113435	−	0.993545i	$-0.463814\pi$
$709$	4283.00	0.226871	0.113435	+	0.993545i	$0.463814\pi$
$710$	2808.00	0.148426
$711$	0	0
$712$	−8400.00	−0.442139
$713$	39975.0	2.09969
$714$	0	0
$715$	−4995.00	−0.261262
$716$	−8016.00	−0.418397
$717$	0	0
$718$	9216.00	0.479022
$719$	−4032.00	−0.209135	−0.104568	−	0.994518i	$-0.533346\pi$
$719$	−4032.00	−0.209135	−0.104568	+	0.994518i	$0.533346\pi$
$720$	0	0
$721$	−17081.0	−0.882288
$722$	12150.0	0.626283
$723$	0	0
$724$	5096.00	0.261590
$725$	4884.00	0.250189
$726$	0	0
$727$	24005.0	1.22462	0.612308	−	0.790619i	$-0.290241\pi$
$727$	24005.0	1.22462	0.612308	+	0.790619i	$0.290241\pi$
$728$	−9176.00	−0.467150
$729$	0	0
$730$	−3276.00	−0.166096
$731$	−1806.00	−0.0913780
$732$	0	0
$733$	−37501.0	−1.88967	−0.944837	−	0.327541i	$-0.893780\pi$
$733$	−37501.0	−1.88967	−0.944837	+	0.327541i	$0.893780\pi$
$734$	−7690.00	−0.386707
$735$	0	0
$736$	6240.00	0.312513
$737$	−6315.00	−0.315626
$738$	0	0
$739$	−880.000	−0.0438042	−0.0219021	−	0.999760i	$-0.506972\pi$
$739$	−880.000	−0.0438042	−0.0219021	+	0.999760i	$0.506972\pi$
$740$	−5976.00	−0.296868
$741$	0	0
$742$	−7068.00	−0.349696
$743$	−1623.00	−0.0801374	−0.0400687	−	0.999197i	$-0.512758\pi$
$743$	−1623.00	−0.0801374	−0.0400687	+	0.999197i	$0.512758\pi$
$744$	0	0
$745$	25461.0	1.25211
$746$	16634.0	0.816373
$747$	0	0
$748$	2520.00	0.123182
$749$	−372.000	−0.0181476
$750$	0	0
$751$	−6889.00	−0.334731	−0.167366	−	0.985895i	$-0.553526\pi$
$751$	−6889.00	−0.334731	−0.167366	+	0.985895i	$0.553526\pi$
$752$	−2832.00	−0.137330
$753$	0	0
$754$	−8214.00	−0.396732
$755$	4149.00	0.199997
$756$	0	0
$757$	−12850.0	−0.616963	−0.308482	−	0.951230i	$-0.599821\pi$
$757$	−12850.0	−0.616963	−0.308482	+	0.951230i	$0.599821\pi$
$758$	−25120.0	−1.20369
$759$	0	0
$760$	2016.00	0.0962211
$761$	−4611.00	−0.219643	−0.109822	−	0.993951i	$-0.535028\pi$
$761$	−4611.00	−0.219643	−0.109822	+	0.993951i	$0.535028\pi$
$762$	0	0
$763$	15562.0	0.738378
$764$	−4644.00	−0.219914
$765$	0	0
$766$	24174.0	1.14026
$767$	5883.00	0.276953
$768$	0	0
$769$	−2305.00	−0.108089	−0.0540445	−	0.998539i	$-0.517211\pi$
$769$	−2305.00	−0.108089	−0.0540445	+	0.998539i	$0.517211\pi$
$770$	8370.00	0.391732
$771$	0	0
$772$	14444.0	0.673382
$773$	34902.0	1.62398	0.811991	−	0.583670i	$-0.198384\pi$
$773$	34902.0	1.62398	0.811991	+	0.583670i	$0.198384\pi$
$774$	0	0
$775$	9020.00	0.418075
$776$	7208.00	0.333443
$777$	0	0
$778$	−17082.0	−0.787171
$779$	−7308.00	−0.336118
$780$	0	0
$781$	−2340.00	−0.107211
$782$	16380.0	0.749038
$783$	0	0
$784$	9888.00	0.450437
$785$	−26793.0	−1.21820
$786$	0	0
$787$	26255.0	1.18919	0.594593	−	0.804027i	$-0.297313\pi$
$787$	26255.0	1.18919	0.594593	+	0.804027i	$0.297313\pi$
$788$	−8184.00	−0.369978
$789$	0	0
$790$	−20394.0	−0.918463
$791$	−43431.0	−1.95225
$792$	0	0
$793$	−7067.00	−0.316465
$794$	26348.0	1.17765
$795$	0	0
$796$	11984.0	0.533620
$797$	1437.00	0.0638659	0.0319330	−	0.999490i	$-0.489834\pi$
$797$	1437.00	0.0638659	0.0319330	+	0.999490i	$0.489834\pi$
$798$	0	0
$799$	−7434.00	−0.329156
$800$	1408.00	0.0622254
$801$	0	0
$802$	19206.0	0.845620
$803$	2730.00	0.119975
$804$	0	0
$805$	54405.0	2.38202
$806$	−15170.0	−0.662953
$807$	0	0
$808$	3096.00	0.134798
$809$	−25734.0	−1.11837	−0.559184	−	0.829044i	$-0.688885\pi$
$809$	−25734.0	−1.11837	−0.559184	+	0.829044i	$0.688885\pi$
$810$	0	0
$811$	−3220.00	−0.139420	−0.0697099	−	0.997567i	$-0.522207\pi$
$811$	−3220.00	−0.139420	−0.0697099	+	0.997567i	$0.522207\pi$
$812$	13764.0	0.594854
$813$	0	0
$814$	4980.00	0.214434
$815$	−29844.0	−1.28269
$816$	0	0
$817$	1204.00	0.0515577
$818$	−22942.0	−0.980621
$819$	0	0
$820$	9396.00	0.400149
$821$	−44631.0	−1.89724	−0.948619	−	0.316420i	$-0.897519\pi$
$821$	−44631.0	−1.89724	−0.948619	+	0.316420i	$0.897519\pi$
$822$	0	0
$823$	24947.0	1.05662	0.528310	−	0.849052i	$-0.322826\pi$
$823$	24947.0	1.05662	0.528310	+	0.849052i	$0.322826\pi$
$824$	−4408.00	−0.186359
$825$	0	0
$826$	−9858.00	−0.415259
$827$	14964.0	0.629201	0.314601	−	0.949224i	$-0.398129\pi$
$827$	14964.0	0.629201	0.314601	+	0.949224i	$0.398129\pi$
$828$	0	0
$829$	−7462.00	−0.312625	−0.156312	−	0.987708i	$-0.549961\pi$
$829$	−7462.00	−0.312625	−0.156312	+	0.987708i	$0.549961\pi$
$830$	−19494.0	−0.815237
$831$	0	0
$832$	−2368.00	−0.0986726
$833$	25956.0	1.07962
$834$	0	0
$835$	6129.00	0.254015
$836$	−1680.00	−0.0695024
$837$	0	0
$838$	−11946.0	−0.492444
$839$	−2697.00	−0.110978	−0.0554891	−	0.998459i	$-0.517672\pi$
$839$	−2697.00	−0.110978	−0.0554891	+	0.998459i	$0.517672\pi$
$840$	0	0
$841$	−12068.0	−0.494813
$842$	17810.0	0.728947
$843$	0	0
$844$	3020.00	0.123167
$845$	−7452.00	−0.303381
$846$	0	0
$847$	34286.0	1.39089
$848$	−1824.00	−0.0738637
$849$	0	0
$850$	3696.00	0.149143
$851$	32370.0	1.30391
$852$	0	0
$853$	−39625.0	−1.59054	−0.795272	−	0.606253i	$-0.792672\pi$
$853$	−39625.0	−1.59054	−0.795272	+	0.606253i	$0.792672\pi$
$854$	11842.0	0.474502
$855$	0	0
$856$	−96.0000	−0.00383319
$857$	2973.00	0.118501	0.0592507	−	0.998243i	$-0.481129\pi$
$857$	2973.00	0.118501	0.0592507	+	0.998243i	$0.481129\pi$
$858$	0	0
$859$	−45229.0	−1.79650	−0.898250	−	0.439485i	$-0.855161\pi$
$859$	−45229.0	−1.79650	−0.898250	+	0.439485i	$0.855161\pi$
$860$	−1548.00	−0.0613795
$861$	0	0
$862$	2832.00	0.111901
$863$	−1416.00	−0.0558531	−0.0279265	−	0.999610i	$-0.508890\pi$
$863$	−1416.00	−0.0558531	−0.0279265	+	0.999610i	$0.508890\pi$
$864$	0	0
$865$	35829.0	1.40835
$866$	−21532.0	−0.844904
$867$	0	0
$868$	25420.0	0.994022
$869$	16995.0	0.663424
$870$	0	0
$871$	15577.0	0.605978
$872$	4016.00	0.155962
$873$	0	0
$874$	−10920.0	−0.422625
$875$	47151.0	1.82171
$876$	0	0
$877$	17555.0	0.675930	0.337965	−	0.941159i	$-0.390262\pi$
$877$	17555.0	0.675930	0.337965	+	0.941159i	$0.390262\pi$
$878$	−8698.00	−0.334332
$879$	0	0
$880$	2160.00	0.0827427
$881$	−30030.0	−1.14840	−0.574198	−	0.818717i	$-0.694686\pi$
$881$	−30030.0	−1.14840	−0.574198	+	0.818717i	$0.694686\pi$
$882$	0	0
$883$	−35740.0	−1.36211	−0.681057	−	0.732230i	$-0.738479\pi$
$883$	−35740.0	−1.36211	−0.681057	+	0.732230i	$0.738479\pi$
$884$	−6216.00	−0.236501
$885$	0	0
$886$	−29094.0	−1.10320
$887$	−747.000	−0.0282771	−0.0141386	−	0.999900i	$-0.504501\pi$
$887$	−747.000	−0.0282771	−0.0141386	+	0.999900i	$0.504501\pi$
$888$	0	0
$889$	27280.0	1.02918
$890$	−18900.0	−0.711831
$891$	0	0
$892$	−13852.0	−0.519954
$893$	4956.00	0.185718
$894$	0	0
$895$	−18036.0	−0.673606
$896$	3968.00	0.147948
$897$	0	0
$898$	−6660.00	−0.247491
$899$	22755.0	0.844184
$900$	0	0
$901$	−4788.00	−0.177038
$902$	−7830.00	−0.289036
$903$	0	0
$904$	−11208.0	−0.412359
$905$	11466.0	0.421152
$906$	0	0
$907$	−20539.0	−0.751914	−0.375957	−	0.926637i	$-0.622686\pi$
$907$	−20539.0	−0.751914	−0.375957	+	0.926637i	$0.622686\pi$
$908$	−24900.0	−0.910061
$909$	0	0
$910$	−20646.0	−0.752097
$911$	24123.0	0.877311	0.438656	−	0.898655i	$-0.355455\pi$
$911$	24123.0	0.877311	0.438656	+	0.898655i	$0.355455\pi$
$912$	0	0
$913$	16245.0	0.588862
$914$	−16294.0	−0.589669
$915$	0	0
$916$	−5860.00	−0.211375
$917$	46593.0	1.67790
$918$	0	0
$919$	23312.0	0.836770	0.418385	−	0.908270i	$-0.362596\pi$
$919$	23312.0	0.836770	0.418385	+	0.908270i	$0.362596\pi$
$920$	14040.0	0.503136
$921$	0	0
$922$	16062.0	0.573724
$923$	5772.00	0.205837
$924$	0	0
$925$	7304.00	0.259626
$926$	−8566.00	−0.303992
$927$	0	0
$928$	3552.00	0.125647
$929$	−18699.0	−0.660381	−0.330191	−	0.943914i	$-0.607113\pi$
$929$	−18699.0	−0.660381	−0.330191	+	0.943914i	$0.607113\pi$
$930$	0	0
$931$	−17304.0	−0.609147
$932$	−10536.0	−0.370298
$933$	0	0
$934$	10920.0	0.382562
$935$	5670.00	0.198320
$936$	0	0
$937$	−31234.0	−1.08898	−0.544488	−	0.838769i	$-0.683276\pi$
$937$	−31234.0	−1.08898	−0.544488	+	0.838769i	$0.683276\pi$
$938$	−26102.0	−0.908593
$939$	0	0
$940$	−6372.00	−0.221098
$941$	−11751.0	−0.407090	−0.203545	−	0.979066i	$-0.565246\pi$
$941$	−11751.0	−0.407090	−0.203545	+	0.979066i	$0.565246\pi$
$942$	0	0
$943$	−50895.0	−1.75755
$944$	−2544.00	−0.0877120
$945$	0	0
$946$	1290.00	0.0443356
$947$	25887.0	0.888294	0.444147	−	0.895954i	$-0.353507\pi$
$947$	25887.0	0.888294	0.444147	+	0.895954i	$0.353507\pi$
$948$	0	0
$949$	−6734.00	−0.230342
$950$	−2464.00	−0.0841502
$951$	0	0
$952$	10416.0	0.354606
$953$	41130.0	1.39804	0.699020	−	0.715102i	$-0.253620\pi$
$953$	41130.0	1.39804	0.699020	+	0.715102i	$0.253620\pi$
$954$	0	0
$955$	−10449.0	−0.354054
$956$	−27660.0	−0.935762
$957$	0	0
$958$	858.000	0.0289360
$959$	−82491.0	−2.77766
$960$	0	0
$961$	12234.0	0.410661
$962$	−12284.0	−0.411697
$963$	0	0
$964$	−5956.00	−0.198994
$965$	32499.0	1.08412
$966$	0	0
$967$	−28645.0	−0.952597	−0.476298	−	0.879284i	$-0.658022\pi$
$967$	−28645.0	−0.952597	−0.476298	+	0.879284i	$0.658022\pi$
$968$	8848.00	0.293787
$969$	0	0
$970$	16218.0	0.536834
$971$	32256.0	1.06606	0.533030	−	0.846096i	$-0.321053\pi$
$971$	32256.0	1.06606	0.533030	+	0.846096i	$0.321053\pi$
$972$	0	0
$973$	3751.00	0.123588
$974$	22592.0	0.743218
$975$	0	0
$976$	3056.00	0.100226
$977$	−9015.00	−0.295205	−0.147603	−	0.989047i	$-0.547156\pi$
$977$	−9015.00	−0.295205	−0.147603	+	0.989047i	$0.547156\pi$
$978$	0	0
$979$	15750.0	0.514170
$980$	22248.0	0.725190
$981$	0	0
$982$	−29346.0	−0.953634
$983$	−15885.0	−0.515415	−0.257707	−	0.966223i	$-0.582967\pi$
$983$	−15885.0	−0.515415	−0.257707	+	0.966223i	$0.582967\pi$
$984$	0	0
$985$	−18414.0	−0.595654
$986$	9324.00	0.301153
$987$	0	0
$988$	4144.00	0.133439
$989$	8385.00	0.269593
$990$	0	0
$991$	34904.0	1.11883	0.559416	−	0.828887i	$-0.311026\pi$
$991$	34904.0	1.11883	0.559416	+	0.828887i	$0.311026\pi$
$992$	6560.00	0.209960
$993$	0	0
$994$	−9672.00	−0.308629
$995$	26964.0	0.859112
$996$	0	0
$997$	−35341.0	−1.12263	−0.561314	−	0.827603i	$-0.689704\pi$
$997$	−35341.0	−1.12263	−0.561314	+	0.827603i	$0.689704\pi$
$998$	−26878.0	−0.852513
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.4.a.a.1.1		1
3.2	odd	2		162.4.a.d.1.1		1
4.3	odd	2		1296.4.a.g.1.1		1
9.2	odd	6		18.4.c.a.13.1	yes	2
9.4	even	3		54.4.c.a.19.1		2
9.5	odd	6		18.4.c.a.7.1	✓	2
9.7	even	3		54.4.c.a.37.1		2
12.11	even	2		1296.4.a.b.1.1		1
36.7	odd	6		432.4.i.a.145.1		2
36.11	even	6		144.4.i.a.49.1		2
36.23	even	6		144.4.i.a.97.1		2
36.31	odd	6		432.4.i.a.289.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.4.c.a.7.1	✓	2	9.5	odd	6
18.4.c.a.13.1	yes	2	9.2	odd	6
54.4.c.a.19.1		2	9.4	even	3
54.4.c.a.37.1		2	9.7	even	3
144.4.i.a.49.1		2	36.11	even	6
144.4.i.a.97.1		2	36.23	even	6
162.4.a.a.1.1		1	1.1	even	1	trivial
162.4.a.d.1.1		1	3.2	odd	2
432.4.i.a.145.1		2	36.7	odd	6
432.4.i.a.289.1		2	36.31	odd	6
1296.4.a.b.1.1		1	12.11	even	2
1296.4.a.g.1.1		1	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	162.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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