LMFDB - Embedded newform 162.4.a.d.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	$-0.631856\pi$
$5$	−9.00000	−0.804984	−0.402492	+	0.915423i	$0.631856\pi$
$6$	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	$-0.565907\pi$
$11$	−15.0000	−0.411152	−0.205576	+	0.978641i	$0.565907\pi$
$12$	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	$-0.596854\pi$
$17$	−42.0000	−0.599206	−0.299603	+	0.954064i	$0.596854\pi$
$18$	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	$-0.154895\pi$
$23$	195.000	1.76784	0.883920	+	0.467639i	$0.154895\pi$
$24$	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	$-0.384351\pi$
$29$	111.000	0.710765	0.355382	+	0.934721i	$0.384351\pi$
$30$	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	$-0.665598\pi$
$41$	−261.000	−0.994179	−0.497090	+	0.867699i	$0.665598\pi$
$42$	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	$-0.411434\pi$
$47$	177.000	0.549321	0.274661	+	0.961541i	$0.411434\pi$
$48$	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	$-0.452804\pi$
$53$	114.000	0.295455	0.147727	+	0.989028i	$0.452804\pi$
$54$	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	$-0.443870\pi$
$59$	159.000	0.350848	0.175424	+	0.984493i	$0.443870\pi$
$60$	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	$-0.458381\pi$
$71$	156.000	0.260758	0.130379	+	0.991464i	$0.458381\pi$
$72$	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	$-0.754080\pi$
$83$	−1083.00	−1.43223	−0.716113	+	0.697985i	$0.754080\pi$
$84$	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	$-0.715015\pi$
$89$	−1050.00	−1.25056	−0.625280	+	0.780401i	$0.715015\pi$
$90$	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	$-0.438946\pi$
$101$	387.000	0.381267	0.190633	+	0.981661i	$0.438946\pi$
$102$	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	$-0.501726\pi$
$107$	−12.0000	−0.0108419	−0.00542095	+	0.999985i	$0.501726\pi$
$108$	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	$-0.698185\pi$
$113$	−1401.00	−1.16633	−0.583164	+	0.812355i	$0.698185\pi$
$114$	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	$-0.332887\pi$
$131$	1503.00	1.00243	0.501213	+	0.865324i	$0.332887\pi$
$132$	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	$-0.811503\pi$
$137$	−2661.00	−1.65945	−0.829725	+	0.558173i	$0.811503\pi$
$138$	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	$-0.783624\pi$
$149$	−2829.00	−1.55544	−0.777721	+	0.628610i	$0.783624\pi$
$150$	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	$-0.550433\pi$
$167$	−681.000	−0.315553	−0.157777	+	0.987475i	$0.550433\pi$
$168$	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	$-0.838986\pi$
$173$	−3981.00	−1.74954	−0.874768	+	0.484541i	$0.838986\pi$
$174$	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	$-0.362592\pi$
$179$	2004.00	0.836793	0.418397	+	0.908264i	$0.362592\pi$
$180$	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	$-0.429422\pi$
$191$	1161.00	0.439827	0.219914	+	0.975519i	$0.429422\pi$
$192$	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	$-0.379365\pi$
$197$	2046.00	0.739957	0.369978	+	0.929040i	$0.379365\pi$
$198$	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	$-0.136034\pi$
$227$	6225.00	1.82012	0.910061	+	0.414474i	$0.136034\pi$
$228$	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	$-0.379255\pi$
$233$	2634.00	0.740597	0.370298	+	0.928913i	$0.379255\pi$
$234$	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	$-0.114713\pi$
$239$	6915.00	1.87152	0.935762	+	0.352633i	$0.114713\pi$
$240$	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	$-0.697299\pi$
$251$	−4620.00	−1.16180	−0.580900	+	0.813975i	$0.697299\pi$
$252$	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	$-0.366689\pi$
$257$	3351.00	0.813345	0.406672	+	0.913574i	$0.366689\pi$
$258$	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	$-0.522520\pi$
$263$	−603.000	−0.141379	−0.0706893	+	0.997498i	$0.522520\pi$
$264$	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	$-0.553277\pi$
$269$	−1470.00	−0.333188	−0.166594	+	0.986026i	$0.553277\pi$
$270$	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	$-0.147527\pi$
$281$	8427.00	1.78901	0.894507	+	0.447055i	$0.147527\pi$
$282$	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	$-0.699729\pi$
$293$	−5889.00	−1.17419	−0.587097	+	0.809516i	$0.699729\pi$
$294$	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	$-0.596812\pi$
$311$	−3285.00	−0.598956	−0.299478	+	0.954103i	$0.596812\pi$
$312$	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	$-0.434829\pi$
$317$	2295.00	0.406625	0.203312	+	0.979114i	$0.434829\pi$
$318$	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	$-0.402686\pi$
$347$	3891.00	0.601959	0.300980	+	0.953631i	$0.402686\pi$
$348$	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	$-0.383297\pi$
$353$	4755.00	0.716949	0.358475	+	0.933539i	$0.383297\pi$
$354$	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	$-0.390006\pi$
$359$	4608.00	0.677440	0.338720	+	0.940887i	$0.390006\pi$
$360$	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	$-0.201473\pi$
$383$	12087.0	1.61258	0.806288	+	0.591523i	$0.201473\pi$
$384$	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	$-0.687900\pi$
$389$	−8541.00	−1.11323	−0.556614	+	0.830771i	$0.687900\pi$
$390$	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	$-0.295985\pi$
$401$	9603.00	1.19589	0.597944	+	0.801538i	$0.295985\pi$
$402$	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	$-0.613210\pi$
$419$	−5973.00	−0.696420	−0.348210	+	0.937416i	$0.613210\pi$
$420$	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	$-0.474787\pi$
$431$	1416.00	0.158251	0.0791257	+	0.996865i	$0.474787\pi$
$432$	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	$-0.784820\pi$
$443$	−14547.0	−1.56016	−0.780078	+	0.625683i	$0.784820\pi$
$444$	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	$-0.555993\pi$
$449$	−3330.00	−0.350005	−0.175003	+	0.984568i	$0.555993\pi$
$450$	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	$-0.367033\pi$
$461$	8031.00	0.811369	0.405684	+	0.914013i	$0.367033\pi$
$462$	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	$-0.412807\pi$
$467$	5460.00	0.541025	0.270512	+	0.962716i	$0.412807\pi$
$468$	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	$-0.493487\pi$
$479$	429.000	0.0409217	0.0204609	+	0.999791i	$0.493487\pi$
$480$	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	$-0.735564\pi$
$491$	−14673.0	−1.34864	−0.674321	+	0.738438i	$0.735564\pi$
$492$	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	$-0.780078\pi$
$503$	−17388.0	−1.54134	−0.770669	+	0.637236i	$0.780078\pi$
$504$	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	$-0.552754\pi$
$509$	−3789.00	−0.329950	−0.164975	+	0.986298i	$0.552754\pi$
$510$	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	$-0.365022\pi$
$521$	9786.00	0.822903	0.411451	+	0.911432i	$0.365022\pi$
$522$	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	$-0.459868\pi$
$557$	3306.00	0.251490	0.125745	+	0.992063i	$0.459868\pi$
$558$	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	$-0.789654\pi$
$563$	−21093.0	−1.57898	−0.789488	+	0.613765i	$0.789654\pi$
$564$	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	$-0.484903\pi$
$569$	1287.00	0.0948222	0.0474111	+	0.998875i	$0.484903\pi$
$570$	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	$-0.716418\pi$
$587$	−17883.0	−1.25743	−0.628714	+	0.777637i	$0.716418\pi$
$588$	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	$-0.254703\pi$
$593$	20118.0	1.39317	0.696583	+	0.717476i	$0.254703\pi$
$594$	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	$-0.511564\pi$
$599$	−1065.00	−0.0726456	−0.0363228	+	0.999340i	$0.511564\pi$
$600$	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	$-0.606365\pi$
$617$	−10053.0	−0.655946	−0.327973	+	0.944687i	$0.606365\pi$
$618$	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	$-0.609816\pi$
$641$	−10977.0	−0.676389	−0.338195	+	0.941076i	$0.609816\pi$
$642$	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	$-0.172019\pi$
$647$	28224.0	1.71499	0.857496	+	0.514490i	$0.172019\pi$
$648$	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	$-0.180197\pi$
$653$	28167.0	1.68799	0.843997	+	0.536348i	$0.180197\pi$
$654$	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	$-0.397210\pi$
$659$	10737.0	0.634680	0.317340	+	0.948312i	$0.397210\pi$
$660$	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	$-0.422892\pi$
$677$	8451.00	0.479761	0.239881	+	0.970802i	$0.422892\pi$
$678$	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	$-0.758185\pi$
$683$	−25884.0	−1.45011	−0.725054	+	0.688692i	$0.758185\pi$
$684$	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	$-0.490377\pi$
$701$	1122.00	0.0604527	0.0302264	+	0.999543i	$0.490377\pi$
$702$	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	$-0.466654\pi$
$719$	4032.00	0.209135	0.104568	+	0.994518i	$0.466654\pi$
$720$	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	$-0.487242\pi$
$743$	1623.00	0.0801374	0.0400687	+	0.999197i	$0.487242\pi$
$744$	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	$-0.464972\pi$
$761$	4611.00	0.219643	0.109822	+	0.993951i	$0.464972\pi$
$762$	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	$-0.801616\pi$
$773$	−34902.0	−1.62398	−0.811991	+	0.583670i	$0.801616\pi$
$774$	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	$-0.510166\pi$
$797$	−1437.00	−0.0638659	−0.0319330	+	0.999490i	$0.510166\pi$
$798$	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	$-0.311115\pi$
$809$	25734.0	1.11837	0.559184	+	0.829044i	$0.311115\pi$
$810$	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	$-0.102481\pi$
$821$	44631.0	1.89724	0.948619	+	0.316420i	$0.102481\pi$
$822$	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	$-0.601871\pi$
$827$	−14964.0	−0.629201	−0.314601	+	0.949224i	$0.601871\pi$
$828$	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	$-0.482328\pi$
$839$	2697.00	0.110978	0.0554891	+	0.998459i	$0.482328\pi$
$840$	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	$-0.518871\pi$
$857$	−2973.00	−0.118501	−0.0592507	+	0.998243i	$0.518871\pi$
$858$	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	$-0.491110\pi$
$863$	1416.00	0.0558531	0.0279265	+	0.999610i	$0.491110\pi$
$864$	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	$-0.305314\pi$
$881$	30030.0	1.14840	0.574198	+	0.818717i	$0.305314\pi$
$882$	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	$-0.495499\pi$
$887$	747.000	0.0282771	0.0141386	+	0.999900i	$0.495499\pi$
$888$	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	$-0.644545\pi$
$911$	−24123.0	−0.877311	−0.438656	+	0.898655i	$0.644545\pi$
$912$	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	$-0.392887\pi$
$929$	18699.0	0.660381	0.330191	+	0.943914i	$0.392887\pi$
$930$	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	$-0.434754\pi$
$941$	11751.0	0.407090	0.203545	+	0.979066i	$0.434754\pi$
$942$	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	$-0.646493\pi$
$947$	−25887.0	−0.888294	−0.444147	+	0.895954i	$0.646493\pi$
$948$	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	$-0.746380\pi$
$953$	−41130.0	−1.39804	−0.699020	+	0.715102i	$0.746380\pi$
$954$	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	$-0.678947\pi$
$971$	−32256.0	−1.06606	−0.533030	+	0.846096i	$0.678947\pi$
$972$	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	$-0.452844\pi$
$977$	9015.00	0.295205	0.147603	+	0.989047i	$0.452844\pi$
$978$	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	$-0.417033\pi$
$983$	15885.0	0.515415	0.257707	+	0.966223i	$0.417033\pi$
$984$	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.d.1.1		1
3.2	odd	2		162.4.a.a.1.1		1
4.3	odd	2		1296.4.a.b.1.1		1
9.2	odd	6		54.4.c.a.37.1		2
9.4	even	3		18.4.c.a.7.1	✓	2
9.5	odd	6		54.4.c.a.19.1		2
9.7	even	3		18.4.c.a.13.1	yes	2
12.11	even	2		1296.4.a.g.1.1		1
36.7	odd	6		144.4.i.a.49.1		2
36.11	even	6		432.4.i.a.145.1		2
36.23	even	6		432.4.i.a.289.1		2
36.31	odd	6		144.4.i.a.97.1		2

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