LMFDB - Embedded newform 162.6.a.j.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,6,Mod(1,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("162.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	162.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$25.9821788097$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.125628.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 63x + 159 $ x^3 - x^2 - 63*x + 159
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 18)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-8.54724$ of defining polynomial
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)

$f(q)$	$=$	$q+4.00000 q^{2} +16.0000 q^{4} +78.4839 q^{5} +221.131 q^{7} +64.0000 q^{8} +O(q^{10})$	\(q+4.00000 q^{2} +16.0000 q^{4} +78.4839 q^{5} +221.131 q^{7} +64.0000 q^{8} +313.936 q^{10} +230.289 q^{11} -771.586 q^{13} +884.524 q^{14} +256.000 q^{16} -769.880 q^{17} -383.367 q^{19} +1255.74 q^{20} +921.155 q^{22} +386.356 q^{23} +3034.73 q^{25} -3086.34 q^{26} +3538.10 q^{28} -789.767 q^{29} +3219.93 q^{31} +1024.00 q^{32} -3079.52 q^{34} +17355.2 q^{35} +2465.33 q^{37} -1533.47 q^{38} +5022.97 q^{40} -9243.46 q^{41} +10631.5 q^{43} +3684.62 q^{44} +1545.42 q^{46} +1952.06 q^{47} +32091.9 q^{49} +12138.9 q^{50} -12345.4 q^{52} -32589.2 q^{53} +18074.0 q^{55} +14152.4 q^{56} -3159.07 q^{58} -23832.1 q^{59} +37608.8 q^{61} +12879.7 q^{62} +4096.00 q^{64} -60557.1 q^{65} -23051.1 q^{67} -12318.1 q^{68} +69421.0 q^{70} -66050.4 q^{71} +65130.0 q^{73} +9861.33 q^{74} -6133.87 q^{76} +50924.0 q^{77} -70867.4 q^{79} +20091.9 q^{80} -36973.8 q^{82} +55287.0 q^{83} -60423.2 q^{85} +42526.0 q^{86} +14738.5 q^{88} +10598.6 q^{89} -170622. q^{91} +6181.69 q^{92} +7808.26 q^{94} -30088.1 q^{95} -82819.8 q^{97} +128368. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 12 q^{2} + 48 q^{4} + 54 q^{5} + 132 q^{7} + 192 q^{8}+O(q^{10}) $ 3 * q + 12 * q^2 + 48 * q^4 + 54 * q^5 + 132 * q^7 + 192 * q^8	$ 3 q + 12 q^{2} + 48 q^{4} + 54 q^{5} + 132 q^{7} + 192 q^{8} + 216 q^{10} + 315 q^{11} + 744 q^{13} + 528 q^{14} + 768 q^{16} + 1449 q^{17} + 1131 q^{19} + 864 q^{20} + 1260 q^{22} + 3168 q^{23} + 2883 q^{25} + 2976 q^{26} + 2112 q^{28} + 5148 q^{29} + 8610 q^{31} + 3072 q^{32} + 5796 q^{34} + 1350 q^{35} + 19968 q^{37} + 4524 q^{38} + 3456 q^{40} - 5049 q^{41} + 31389 q^{43} + 5040 q^{44} + 12672 q^{46} - 12924 q^{47} + 52857 q^{49} + 11532 q^{50} + 11904 q^{52} - 48024 q^{53} + 63126 q^{55} + 8448 q^{56} + 20592 q^{58} - 62955 q^{59} + 75966 q^{61} + 34440 q^{62} + 12288 q^{64} - 108702 q^{65} + 32991 q^{67} + 23184 q^{68} + 5400 q^{70} - 64836 q^{71} - 4233 q^{73} + 79872 q^{74} + 18096 q^{76} - 88740 q^{77} - 89202 q^{79} + 13824 q^{80} - 20196 q^{82} - 32634 q^{83} - 71388 q^{85} + 125556 q^{86} + 20160 q^{88} + 33066 q^{89} - 150918 q^{91} + 50688 q^{92} - 51696 q^{94} + 82944 q^{95} - 46245 q^{97} + 211428 q^{98}+O(q^{100}) $ 3 * q + 12 * q^2 + 48 * q^4 + 54 * q^5 + 132 * q^7 + 192 * q^8 + 216 * q^10 + 315 * q^11 + 744 * q^13 + 528 * q^14 + 768 * q^16 + 1449 * q^17 + 1131 * q^19 + 864 * q^20 + 1260 * q^22 + 3168 * q^23 + 2883 * q^25 + 2976 * q^26 + 2112 * q^28 + 5148 * q^29 + 8610 * q^31 + 3072 * q^32 + 5796 * q^34 + 1350 * q^35 + 19968 * q^37 + 4524 * q^38 + 3456 * q^40 - 5049 * q^41 + 31389 * q^43 + 5040 * q^44 + 12672 * q^46 - 12924 * q^47 + 52857 * q^49 + 11532 * q^50 + 11904 * q^52 - 48024 * q^53 + 63126 * q^55 + 8448 * q^56 + 20592 * q^58 - 62955 * q^59 + 75966 * q^61 + 34440 * q^62 + 12288 * q^64 - 108702 * q^65 + 32991 * q^67 + 23184 * q^68 + 5400 * q^70 - 64836 * q^71 - 4233 * q^73 + 79872 * q^74 + 18096 * q^76 - 88740 * q^77 - 89202 * q^79 + 13824 * q^80 - 20196 * q^82 - 32634 * q^83 - 71388 * q^85 + 125556 * q^86 + 20160 * q^88 + 33066 * q^89 - 150918 * q^91 + 50688 * q^92 - 51696 * q^94 + 82944 * q^95 - 46245 * q^97 + 211428 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	78.4839	1.40396	0.701982	−	0.712195i	$-0.252299\pi$
$5$	78.4839	1.40396	0.701982	+	0.712195i	$0.252299\pi$
$6$	0	0
$7$	221.131	1.70571	0.852854	−	0.522150i	$-0.174870\pi$
$7$	221.131	1.70571	0.852854	+	0.522150i	$0.174870\pi$
$8$	64.0000	0.353553
$9$	0	0
$10$	313.936	0.992752
$11$	230.289	0.573840	0.286920	−	0.957955i	$-0.407369\pi$
$11$	230.289	0.573840	0.286920	+	0.957955i	$0.407369\pi$
$12$	0	0
$13$	−771.586	−1.26627	−0.633134	−	0.774042i	$-0.718232\pi$
$13$	−771.586	−1.26627	−0.633134	+	0.774042i	$0.718232\pi$
$14$	884.524	1.20612
$15$	0	0
$16$	256.000	0.250000
$17$	−769.880	−0.646101	−0.323051	−	0.946382i	$-0.604708\pi$
$17$	−769.880	−0.646101	−0.323051	+	0.946382i	$0.604708\pi$
$18$	0	0
$19$	−383.367	−0.243630	−0.121815	−	0.992553i	$-0.538871\pi$
$19$	−383.367	−0.243630	−0.121815	+	0.992553i	$0.538871\pi$
$20$	1255.74	0.701982
$21$	0	0
$22$	921.155	0.405766
$23$	386.356	0.152289	0.0761444	−	0.997097i	$-0.475739\pi$
$23$	386.356	0.152289	0.0761444	+	0.997097i	$0.475739\pi$
$24$	0	0
$25$	3034.73	0.971114
$26$	−3086.34	−0.895387
$27$	0	0
$28$	3538.10	0.852854
$29$	−789.767	−0.174383	−0.0871914	−	0.996192i	$-0.527789\pi$
$29$	−789.767	−0.174383	−0.0871914	+	0.996192i	$0.527789\pi$
$30$	0	0
$31$	3219.93	0.601787	0.300893	−	0.953658i	$-0.402715\pi$
$31$	3219.93	0.601787	0.300893	+	0.953658i	$0.402715\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	−3079.52	−0.456863
$35$	17355.2	2.39475
$36$	0	0
$37$	2465.33	0.296054	0.148027	−	0.988983i	$-0.452708\pi$
$37$	2465.33	0.296054	0.148027	+	0.988983i	$0.452708\pi$
$38$	−1533.47	−0.172272
$39$	0	0
$40$	5022.97	0.496376
$41$	−9243.46	−0.858766	−0.429383	−	0.903122i	$-0.641269\pi$
$41$	−9243.46	−0.858766	−0.429383	+	0.903122i	$0.641269\pi$
$42$	0	0
$43$	10631.5	0.876847	0.438424	−	0.898769i	$-0.355537\pi$
$43$	10631.5	0.876847	0.438424	+	0.898769i	$0.355537\pi$
$44$	3684.62	0.286920
$45$	0	0
$46$	1545.42	0.107684
$47$	1952.06	0.128899	0.0644495	−	0.997921i	$-0.479471\pi$
$47$	1952.06	0.128899	0.0644495	+	0.997921i	$0.479471\pi$
$48$	0	0
$49$	32091.9	1.90944
$50$	12138.9	0.686681
$51$	0	0
$52$	−12345.4	−0.633134
$53$	−32589.2	−1.59362	−0.796809	−	0.604231i	$-0.793480\pi$
$53$	−32589.2	−1.59362	−0.796809	+	0.604231i	$0.793480\pi$
$54$	0	0
$55$	18074.0	0.805651
$56$	14152.4	0.603059
$57$	0	0
$58$	−3159.07	−0.123307
$59$	−23832.1	−0.891317	−0.445659	−	0.895203i	$-0.647030\pi$
$59$	−23832.1	−0.891317	−0.445659	+	0.895203i	$0.647030\pi$
$60$	0	0
$61$	37608.8	1.29409	0.647046	−	0.762451i	$-0.276004\pi$
$61$	37608.8	1.29409	0.647046	+	0.762451i	$0.276004\pi$
$62$	12879.7	0.425528
$63$	0	0
$64$	4096.00	0.125000
$65$	−60557.1	−1.77779
$66$	0	0
$67$	−23051.1	−0.627343	−0.313672	−	0.949531i	$-0.601559\pi$
$67$	−23051.1	−0.627343	−0.313672	+	0.949531i	$0.601559\pi$
$68$	−12318.1	−0.323051
$69$	0	0
$70$	69421.0	1.69335
$71$	−66050.4	−1.55500	−0.777498	−	0.628885i	$-0.783511\pi$
$71$	−66050.4	−1.55500	−0.777498	+	0.628885i	$0.783511\pi$
$72$	0	0
$73$	65130.0	1.43045	0.715227	−	0.698893i	$-0.246324\pi$
$73$	65130.0	1.43045	0.715227	+	0.698893i	$0.246324\pi$
$74$	9861.33	0.209342
$75$	0	0
$76$	−6133.87	−0.121815
$77$	50924.0	0.978804
$78$	0	0
$79$	−70867.4	−1.27755	−0.638776	−	0.769392i	$-0.720559\pi$
$79$	−70867.4	−1.27755	−0.638776	+	0.769392i	$0.720559\pi$
$80$	20091.9	0.350991
$81$	0	0
$82$	−36973.8	−0.607239
$83$	55287.0	0.880903	0.440451	−	0.897776i	$-0.354818\pi$
$83$	55287.0	0.880903	0.440451	+	0.897776i	$0.354818\pi$
$84$	0	0
$85$	−60423.2	−0.907103
$86$	42526.0	0.620024
$87$	0	0
$88$	14738.5	0.202883
$89$	10598.6	0.141831	0.0709156	−	0.997482i	$-0.477408\pi$
$89$	10598.6	0.141831	0.0709156	+	0.997482i	$0.477408\pi$
$90$	0	0
$91$	−170622.	−2.15988
$92$	6181.69	0.0761444
$93$	0	0
$94$	7808.26	0.0911453
$95$	−30088.1	−0.342047
$96$	0	0
$97$	−82819.8	−0.893727	−0.446864	−	0.894602i	$-0.647459\pi$
$97$	−82819.8	−0.893727	−0.446864	+	0.894602i	$0.647459\pi$
$98$	128368.	1.35018
$99$	0	0
$100$	48555.7	0.485557
$101$	21208.2	0.206872	0.103436	−	0.994636i	$-0.467016\pi$
$101$	21208.2	0.206872	0.103436	+	0.994636i	$0.467016\pi$
$102$	0	0
$103$	131755.	1.22370	0.611848	−	0.790976i	$-0.290427\pi$
$103$	131755.	1.22370	0.611848	+	0.790976i	$0.290427\pi$
$104$	−49381.5	−0.447693
$105$	0	0
$106$	−130357.	−1.12686
$107$	54796.5	0.462693	0.231347	−	0.972871i	$-0.425687\pi$
$107$	54796.5	0.462693	0.231347	+	0.972871i	$0.425687\pi$
$108$	0	0
$109$	160570.	1.29448	0.647242	−	0.762284i	$-0.275922\pi$
$109$	160570.	1.29448	0.647242	+	0.762284i	$0.275922\pi$
$110$	72295.9	0.569681
$111$	0	0
$112$	56609.6	0.426427
$113$	−66435.6	−0.489446	−0.244723	−	0.969593i	$-0.578697\pi$
$113$	−66435.6	−0.489446	−0.244723	+	0.969593i	$0.578697\pi$
$114$	0	0
$115$	30322.7	0.213808
$116$	−12636.3	−0.0871914
$117$	0	0
$118$	−95328.4	−0.630256
$119$	−170244.	−1.10206
$120$	0	0
$121$	−108018.	−0.670708
$122$	150435.	0.915062
$123$	0	0
$124$	51518.9	0.300893
$125$	−7084.72	−0.0405553
$126$	0	0
$127$	−299603.	−1.64830	−0.824152	−	0.566368i	$-0.808348\pi$
$127$	−299603.	−1.64830	−0.824152	+	0.566368i	$0.808348\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	−242228.	−1.25709
$131$	−332448.	−1.69257	−0.846283	−	0.532733i	$-0.821165\pi$
$131$	−332448.	−1.69257	−0.846283	+	0.532733i	$0.821165\pi$
$132$	0	0
$133$	−84774.3	−0.415561
$134$	−92204.5	−0.443599
$135$	0	0
$136$	−49272.3	−0.228431
$137$	−187203.	−0.852139	−0.426070	−	0.904690i	$-0.640102\pi$
$137$	−187203.	−0.852139	−0.426070	+	0.904690i	$0.640102\pi$
$138$	0	0
$139$	−210569.	−0.924396	−0.462198	−	0.886777i	$-0.652939\pi$
$139$	−210569.	−0.924396	−0.462198	+	0.886777i	$0.652939\pi$
$140$	277684.	1.19738
$141$	0	0
$142$	−264201.	−1.09955
$143$	−177687.	−0.726635
$144$	0	0
$145$	−61984.0	−0.244827
$146$	260520.	1.01148
$147$	0	0
$148$	39445.3	0.148027
$149$	485744.	1.79243	0.896215	−	0.443620i	$-0.146306\pi$
$149$	485744.	1.79243	0.896215	+	0.443620i	$0.146306\pi$
$150$	0	0
$151$	210210.	0.750260	0.375130	−	0.926972i	$-0.377598\pi$
$151$	210210.	0.750260	0.375130	+	0.926972i	$0.377598\pi$
$152$	−24535.5	−0.0861361
$153$	0	0
$154$	203696.	0.692119
$155$	252713.	0.844887
$156$	0	0
$157$	−392247.	−1.27002	−0.635009	−	0.772504i	$-0.719004\pi$
$157$	−392247.	−1.27002	−0.635009	+	0.772504i	$0.719004\pi$
$158$	−283470.	−0.903366
$159$	0	0
$160$	80367.6	0.248188
$161$	85435.3	0.259760
$162$	0	0
$163$	190208.	0.560738	0.280369	−	0.959892i	$-0.409543\pi$
$163$	190208.	0.560738	0.280369	+	0.959892i	$0.409543\pi$
$164$	−147895.	−0.429383
$165$	0	0
$166$	221148.	0.622892
$167$	−401239.	−1.11330	−0.556650	−	0.830747i	$-0.687913\pi$
$167$	−401239.	−1.11330	−0.556650	+	0.830747i	$0.687913\pi$
$168$	0	0
$169$	224051.	0.603435
$170$	−241693.	−0.641418
$171$	0	0
$172$	170104.	0.438424
$173$	94783.7	0.240779	0.120389	−	0.992727i	$-0.461586\pi$
$173$	94783.7	0.240779	0.120389	+	0.992727i	$0.461586\pi$
$174$	0	0
$175$	671073.	1.65644
$176$	58953.9	0.143460
$177$	0	0
$178$	42394.3	0.100290
$179$	−218816.	−0.510442	−0.255221	−	0.966883i	$-0.582148\pi$
$179$	−218816.	−0.510442	−0.255221	+	0.966883i	$0.582148\pi$
$180$	0	0
$181$	−344513.	−0.781645	−0.390823	−	0.920466i	$-0.627809\pi$
$181$	−344513.	−0.781645	−0.390823	+	0.920466i	$0.627809\pi$
$182$	−682486.	−1.52727
$183$	0	0
$184$	24726.8	0.0538422
$185$	193489.	0.415649
$186$	0	0
$187$	−177295.	−0.370759
$188$	31233.0	0.0644495
$189$	0	0
$190$	−120353.	−0.241864
$191$	−134953.	−0.267670	−0.133835	−	0.991004i	$-0.542729\pi$
$191$	−134953.	−0.267670	−0.133835	+	0.991004i	$0.542729\pi$
$192$	0	0
$193$	227351.	0.439343	0.219671	−	0.975574i	$-0.429502\pi$
$193$	227351.	0.439343	0.219671	+	0.975574i	$0.429502\pi$
$194$	−331279.	−0.631961
$195$	0	0
$196$	513471.	0.954720
$197$	430437.	0.790213	0.395107	−	0.918635i	$-0.370708\pi$
$197$	430437.	0.790213	0.395107	+	0.918635i	$0.370708\pi$
$198$	0	0
$199$	719704.	1.28831	0.644156	−	0.764894i	$-0.277209\pi$
$199$	719704.	1.28831	0.644156	+	0.764894i	$0.277209\pi$
$200$	194223.	0.343341
$201$	0	0
$202$	84833.0	0.146280
$203$	−174642.	−0.297446
$204$	0	0
$205$	−725463.	−1.20568
$206$	527019.	0.865283
$207$	0	0
$208$	−197526.	−0.316567
$209$	−88285.0	−0.139805
$210$	0	0
$211$	61400.1	0.0949430	0.0474715	−	0.998873i	$-0.484884\pi$
$211$	61400.1	0.0949430	0.0474715	+	0.998873i	$0.484884\pi$
$212$	−521427.	−0.796809
$213$	0	0
$214$	219186.	0.327174
$215$	834403.	1.23106
$216$	0	0
$217$	712027.	1.02647
$218$	642278.	0.915339
$219$	0	0
$220$	289183.	0.402825
$221$	594028.	0.818138
$222$	0	0
$223$	658027.	0.886097	0.443049	−	0.896498i	$-0.353897\pi$
$223$	658027.	0.886097	0.443049	+	0.896498i	$0.353897\pi$
$224$	226438.	0.301529
$225$	0	0
$226$	−265742.	−0.346091
$227$	−10134.3	−0.0130536	−0.00652679	−	0.999979i	$-0.502078\pi$
$227$	−10134.3	−0.0130536	−0.00652679	+	0.999979i	$0.502078\pi$
$228$	0	0
$229$	1.22304e6	1.54117	0.770584	−	0.637338i	$-0.219964\pi$
$229$	1.22304e6	1.54117	0.770584	+	0.637338i	$0.219964\pi$
$230$	121291.	0.151185
$231$	0	0
$232$	−50545.1	−0.0616537
$233$	−1.16513e6	−1.40599	−0.702996	−	0.711194i	$-0.748155\pi$
$233$	−1.16513e6	−1.40599	−0.702996	+	0.711194i	$0.748155\pi$
$234$	0	0
$235$	153206.	0.180969
$236$	−381314.	−0.445659
$237$	0	0
$238$	−680977.	−0.779274
$239$	1.31952e6	1.49424	0.747122	−	0.664687i	$-0.231435\pi$
$239$	1.31952e6	1.49424	0.747122	+	0.664687i	$0.231435\pi$
$240$	0	0
$241$	531809.	0.589812	0.294906	−	0.955526i	$-0.404712\pi$
$241$	531809.	0.589812	0.294906	+	0.955526i	$0.404712\pi$
$242$	−432072.	−0.474262
$243$	0	0
$244$	601741.	0.647046
$245$	2.51870e6	2.68078
$246$	0	0
$247$	295800.	0.308501
$248$	206076.	0.212764
$249$	0	0
$250$	−28338.9	−0.0286769
$251$	475.750	0.000476644 0	0.000238322	−	1.00000i	$-0.499924\pi$
$251$	475.750	0.000476644 0	0.000238322		1.00000i	$0.499924\pi$
$252$	0	0
$253$	88973.4	0.0873894
$254$	−1.19841e6	−1.16553
$255$	0	0
$256$	65536.0	0.0625000
$257$	−27206.6	−0.0256946	−0.0128473	−	0.999917i	$-0.504090\pi$
$257$	−27206.6	−0.0256946	−0.0128473	+	0.999917i	$0.504090\pi$
$258$	0	0
$259$	545161.	0.504982
$260$	−968913.	−0.888897
$261$	0	0
$262$	−1.32979e6	−1.19683
$263$	−139982.	−0.124791	−0.0623956	−	0.998051i	$-0.519874\pi$
$263$	−139982.	−0.124791	−0.0623956	+	0.998051i	$0.519874\pi$
$264$	0	0
$265$	−2.55773e6	−2.23738
$266$	−339097.	−0.293846
$267$	0	0
$268$	−368818.	−0.313672
$269$	1.63842e6	1.38053	0.690263	−	0.723558i	$-0.257495\pi$
$269$	1.63842e6	1.38053	0.690263	+	0.723558i	$0.257495\pi$
$270$	0	0
$271$	−280791.	−0.232252	−0.116126	−	0.993234i	$-0.537048\pi$
$271$	−280791.	−0.232252	−0.116126	+	0.993234i	$0.537048\pi$
$272$	−197089.	−0.161525
$273$	0	0
$274$	−748811.	−0.602554
$275$	698864.	0.557264
$276$	0	0
$277$	692808.	0.542517	0.271259	−	0.962506i	$-0.412560\pi$
$277$	692808.	0.542517	0.271259	+	0.962506i	$0.412560\pi$
$278$	−842277.	−0.653647
$279$	0	0
$280$	1.11074e6	0.846673
$281$	−2.10929e6	−1.59356	−0.796782	−	0.604266i	$-0.793466\pi$
$281$	−2.10929e6	−1.59356	−0.796782	+	0.604266i	$0.793466\pi$
$282$	0	0
$283$	−1.46269e6	−1.08564	−0.542821	−	0.839848i	$-0.682644\pi$
$283$	−1.46269e6	−1.08564	−0.542821	+	0.839848i	$0.682644\pi$
$284$	−1.05681e6	−0.777498
$285$	0	0
$286$	−710750.	−0.513809
$287$	−2.04402e6	−1.46480
$288$	0	0
$289$	−827142.	−0.582553
$290$	−247936.	−0.173119
$291$	0	0
$292$	1.04208e6	0.715227
$293$	−2.55491e6	−1.73863	−0.869313	−	0.494263i	$-0.835438\pi$
$293$	−2.55491e6	−1.73863	−0.869313	+	0.494263i	$0.835438\pi$
$294$	0	0
$295$	−1.87044e6	−1.25138
$296$	157781.	0.104671
$297$	0	0
$298$	1.94298e6	1.26744
$299$	−298107.	−0.192838
$300$	0	0
$301$	2.35096e6	1.49564
$302$	840841.	0.530514
$303$	0	0
$304$	−98141.9	−0.0609074
$305$	2.95169e6	1.81686
$306$	0	0
$307$	−2.14982e6	−1.30183	−0.650917	−	0.759149i	$-0.725615\pi$
$307$	−2.14982e6	−1.30183	−0.650917	+	0.759149i	$0.725615\pi$
$308$	814784.	0.489402
$309$	0	0
$310$	1.01085e6	0.597425
$311$	−367619.	−0.215525	−0.107762	−	0.994177i	$-0.534369\pi$
$311$	−367619.	−0.215525	−0.107762	+	0.994177i	$0.534369\pi$
$312$	0	0
$313$	182744.	0.105434	0.0527171	−	0.998609i	$-0.483212\pi$
$313$	182744.	0.105434	0.0527171	+	0.998609i	$0.483212\pi$
$314$	−1.56899e6	−0.898039
$315$	0	0
$316$	−1.13388e6	−0.638776
$317$	864964.	0.483448	0.241724	−	0.970345i	$-0.422287\pi$
$317$	864964.	0.483448	0.241724	+	0.970345i	$0.422287\pi$
$318$	0	0
$319$	−181874.	−0.100068
$320$	321470.	0.175495
$321$	0	0
$322$	341741.	0.183678
$323$	295146.	0.157409
$324$	0	0
$325$	−2.34155e6	−1.22969
$326$	760832.	0.396502
$327$	0	0
$328$	−591581.	−0.303620
$329$	431662.	0.219864
$330$	0	0
$331$	−2.99760e6	−1.50385	−0.751923	−	0.659251i	$-0.770873\pi$
$331$	−2.99760e6	−1.50385	−0.751923	+	0.659251i	$0.770873\pi$
$332$	884592.	0.440451
$333$	0	0
$334$	−1.60496e6	−0.787221
$335$	−1.80914e6	−0.880767
$336$	0	0
$337$	718278.	0.344523	0.172261	−	0.985051i	$-0.444893\pi$
$337$	718278.	0.344523	0.172261	+	0.985051i	$0.444893\pi$
$338$	896205.	0.426693
$339$	0	0
$340$	−966771.	−0.453551
$341$	741514.	0.345329
$342$	0	0
$343$	3.37998e6	1.55124
$344$	680417.	0.310012
$345$	0	0
$346$	379135.	0.170256
$347$	−3.74926e6	−1.67156	−0.835779	−	0.549066i	$-0.814984\pi$
$347$	−3.74926e6	−1.67156	−0.835779	+	0.549066i	$0.814984\pi$
$348$	0	0
$349$	−2.33358e6	−1.02556	−0.512778	−	0.858521i	$-0.671384\pi$
$349$	−2.33358e6	−1.02556	−0.512778	+	0.858521i	$0.671384\pi$
$350$	2.68429e6	1.17128
$351$	0	0
$352$	235816.	0.101442
$353$	4.34569e6	1.85619	0.928094	−	0.372346i	$-0.121446\pi$
$353$	4.34569e6	1.85619	0.928094	+	0.372346i	$0.121446\pi$
$354$	0	0
$355$	−5.18389e6	−2.18316
$356$	169577.	0.0709156
$357$	0	0
$358$	−875264.	−0.360937
$359$	−2.71281e6	−1.11092	−0.555461	−	0.831542i	$-0.687458\pi$
$359$	−2.71281e6	−1.11092	−0.555461	+	0.831542i	$0.687458\pi$
$360$	0	0
$361$	−2.32913e6	−0.940645
$362$	−1.37805e6	−0.552707
$363$	0	0
$364$	−2.72994e6	−1.07994
$365$	5.11166e6	2.00830
$366$	0	0
$367$	3.91628e6	1.51778	0.758889	−	0.651220i	$-0.225742\pi$
$367$	3.91628e6	1.51778	0.758889	+	0.651220i	$0.225742\pi$
$368$	98907.1	0.0380722
$369$	0	0
$370$	773956.	0.293908
$371$	−7.20649e6	−2.71825
$372$	0	0
$373$	1.78064e6	0.662682	0.331341	−	0.943511i	$-0.392499\pi$
$373$	1.78064e6	0.662682	0.331341	+	0.943511i	$0.392499\pi$
$374$	−709178.	−0.262166
$375$	0	0
$376$	124932.	0.0455727
$377$	609373.	0.220815
$378$	0	0
$379$	−631740.	−0.225912	−0.112956	−	0.993600i	$-0.536032\pi$
$379$	−631740.	−0.225912	−0.112956	+	0.993600i	$0.536032\pi$
$380$	−481410.	−0.171024
$381$	0	0
$382$	−539812.	−0.189271
$383$	133606.	0.0465403	0.0232701	−	0.999729i	$-0.492592\pi$
$383$	133606.	0.0465403	0.0232701	+	0.999729i	$0.492592\pi$
$384$	0	0
$385$	3.99672e6	1.37420
$386$	909404.	0.310662
$387$	0	0
$388$	−1.32512e6	−0.446864
$389$	5.59469e6	1.87457	0.937286	−	0.348560i	$-0.113329\pi$
$389$	5.59469e6	1.87457	0.937286	+	0.348560i	$0.113329\pi$
$390$	0	0
$391$	−297448.	−0.0983940
$392$	2.05388e6	0.675089
$393$	0	0
$394$	1.72175e6	0.558765
$395$	−5.56195e6	−1.79364
$396$	0	0
$397$	4.19051e6	1.33441	0.667206	−	0.744873i	$-0.267490\pi$
$397$	4.19051e6	1.33441	0.667206	+	0.744873i	$0.267490\pi$
$398$	2.87881e6	0.910974
$399$	0	0
$400$	776891.	0.242778
$401$	−458218.	−0.142302	−0.0711511	−	0.997466i	$-0.522667\pi$
$401$	−458218.	−0.142302	−0.0711511	+	0.997466i	$0.522667\pi$
$402$	0	0
$403$	−2.48445e6	−0.762024
$404$	339332.	0.103436
$405$	0	0
$406$	−698568.	−0.210326
$407$	567738.	0.169888
$408$	0	0
$409$	16935.9	0.00500610	0.00250305	−	0.999997i	$-0.499203\pi$
$409$	16935.9	0.00500610	0.00250305	+	0.999997i	$0.499203\pi$
$410$	−2.90185e6	−0.852542
$411$	0	0
$412$	2.10808e6	0.611848
$413$	−5.27002e6	−1.52033
$414$	0	0
$415$	4.33914e6	1.23676
$416$	−790104.	−0.223847
$417$	0	0
$418$	−353140.	−0.0988567
$419$	4.49779e6	1.25160	0.625798	−	0.779985i	$-0.284774\pi$
$419$	4.49779e6	1.25160	0.625798	+	0.779985i	$0.284774\pi$
$420$	0	0
$421$	−434497.	−0.119476	−0.0597381	−	0.998214i	$-0.519027\pi$
$421$	−434497.	−0.119476	−0.0597381	+	0.998214i	$0.519027\pi$
$422$	245600.	0.0671348
$423$	0	0
$424$	−2.08571e6	−0.563429
$425$	−2.33638e6	−0.627438
$426$	0	0
$427$	8.31648e6	2.20734
$428$	876744.	0.231347
$429$	0	0
$430$	3.33761e6	0.870492
$431$	5.16700e6	1.33982	0.669908	−	0.742444i	$-0.266334\pi$
$431$	5.16700e6	1.33982	0.669908	+	0.742444i	$0.266334\pi$
$432$	0	0
$433$	3.95066e6	1.01263	0.506314	−	0.862349i	$-0.331008\pi$
$433$	3.95066e6	1.01263	0.506314	+	0.862349i	$0.331008\pi$
$434$	2.84811e6	0.725826
$435$	0	0
$436$	2.56911e6	0.647242
$437$	−148116.	−0.0371021
$438$	0	0
$439$	−487606.	−0.120756	−0.0603778	−	0.998176i	$-0.519231\pi$
$439$	−487606.	−0.120756	−0.0603778	+	0.998176i	$0.519231\pi$
$440$	1.15673e6	0.284840
$441$	0	0
$442$	2.37611e6	0.578511
$443$	−8055.04	−0.00195011	−0.000975053	−	1.00000i	$-0.500310\pi$
$443$	−8055.04	−0.00195011	−0.000975053		1.00000i	$0.500310\pi$
$444$	0	0
$445$	831817.	0.199126
$446$	2.63211e6	0.626565
$447$	0	0
$448$	905753.	0.213213
$449$	6.79644e6	1.59098	0.795492	−	0.605964i	$-0.207212\pi$
$449$	6.79644e6	1.59098	0.795492	+	0.605964i	$0.207212\pi$
$450$	0	0
$451$	−2.12866e6	−0.492794
$452$	−1.06297e6	−0.244723
$453$	0	0
$454$	−40537.3	−0.00923028
$455$	−1.33911e7	−3.03240
$456$	0	0
$457$	398117.	0.0891704	0.0445852	−	0.999006i	$-0.485803\pi$
$457$	398117.	0.0891704	0.0445852	+	0.999006i	$0.485803\pi$
$458$	4.89214e6	1.08977
$459$	0	0
$460$	485164.	0.106904
$461$	−1.24192e6	−0.272170	−0.136085	−	0.990697i	$-0.543452\pi$
$461$	−1.24192e6	−0.272170	−0.136085	+	0.990697i	$0.543452\pi$
$462$	0	0
$463$	−2.43414e6	−0.527706	−0.263853	−	0.964563i	$-0.584993\pi$
$463$	−2.43414e6	−0.527706	−0.263853	+	0.964563i	$0.584993\pi$
$464$	−202180.	−0.0435957
$465$	0	0
$466$	−4.66050e6	−0.994186
$467$	−1.08309e6	−0.229811	−0.114906	−	0.993376i	$-0.536657\pi$
$467$	−1.08309e6	−0.229811	−0.114906	+	0.993376i	$0.536657\pi$
$468$	0	0
$469$	−5.09732e6	−1.07006
$470$	612823.	0.127965
$471$	0	0
$472$	−1.52525e6	−0.315128
$473$	2.44832e6	0.503170
$474$	0	0
$475$	−1.16341e6	−0.236592
$476$	−2.72391e6	−0.551030
$477$	0	0
$478$	5.27808e6	1.05659
$479$	−3.33813e6	−0.664758	−0.332379	−	0.943146i	$-0.607851\pi$
$479$	−3.33813e6	−0.664758	−0.332379	+	0.943146i	$0.607851\pi$
$480$	0	0
$481$	−1.90221e6	−0.374884
$482$	2.12724e6	0.417060
$483$	0	0
$484$	−1.72829e6	−0.335354
$485$	−6.50003e6	−1.25476
$486$	0	0
$487$	−5.17210e6	−0.988198	−0.494099	−	0.869406i	$-0.664502\pi$
$487$	−5.17210e6	−0.988198	−0.494099	+	0.869406i	$0.664502\pi$
$488$	2.40697e6	0.457531
$489$	0	0
$490$	1.00748e7	1.89560
$491$	607895.	0.113795	0.0568977	−	0.998380i	$-0.481879\pi$
$491$	607895.	0.113795	0.0568977	+	0.998380i	$0.481879\pi$
$492$	0	0
$493$	608025.	0.112669
$494$	1.18320e6	0.218143
$495$	0	0
$496$	824303.	0.150447
$497$	−1.46058e7	−2.65237
$498$	0	0
$499$	3.51290e6	0.631560	0.315780	−	0.948832i	$-0.397734\pi$
$499$	3.51290e6	0.631560	0.315780	+	0.948832i	$0.397734\pi$
$500$	−113355.	−0.0202776
$501$	0	0
$502$	1903.00	0.000337038 0
$503$	−8.23500e6	−1.45125	−0.725627	−	0.688088i	$-0.758450\pi$
$503$	−8.23500e6	−1.45125	−0.725627	+	0.688088i	$0.758450\pi$
$504$	0	0
$505$	1.66451e6	0.290440
$506$	355893.	0.0617936
$507$	0	0
$508$	−4.79365e6	−0.824152
$509$	−1.06058e7	−1.81447	−0.907236	−	0.420621i	$-0.861812\pi$
$509$	−1.06058e7	−1.81447	−0.907236	+	0.420621i	$0.861812\pi$
$510$	0	0
$511$	1.44023e7	2.43993
$512$	262144.	0.0441942
$513$	0	0
$514$	−108826.	−0.0181688
$515$	1.03406e7	1.71802
$516$	0	0
$517$	449538.	0.0739674
$518$	2.18065e6	0.357076
$519$	0	0
$520$	−3.87565e6	−0.628545
$521$	1.03709e7	1.67388	0.836939	−	0.547297i	$-0.184343\pi$
$521$	1.03709e7	1.67388	0.836939	+	0.547297i	$0.184343\pi$
$522$	0	0
$523$	8.86641e6	1.41740	0.708702	−	0.705508i	$-0.249281\pi$
$523$	8.86641e6	1.41740	0.708702	+	0.705508i	$0.249281\pi$
$524$	−5.31917e6	−0.846283
$525$	0	0
$526$	−559929.	−0.0882407
$527$	−2.47896e6	−0.388815
$528$	0	0
$529$	−6.28707e6	−0.976808
$530$	−1.02309e7	−1.58207
$531$	0	0
$532$	−1.35639e6	−0.207781
$533$	7.13212e6	1.08743
$534$	0	0
$535$	4.30064e6	0.649605
$536$	−1.47527e6	−0.221799
$537$	0	0
$538$	6.55368e6	0.976180
$539$	7.39041e6	1.09571
$540$	0	0
$541$	−1.22530e6	−0.179990	−0.0899949	−	0.995942i	$-0.528685\pi$
$541$	−1.22530e6	−0.179990	−0.0899949	+	0.995942i	$0.528685\pi$
$542$	−1.12316e6	−0.164227
$543$	0	0
$544$	−788357.	−0.114216
$545$	1.26021e7	1.81741
$546$	0	0
$547$	2.69010e6	0.384415	0.192207	−	0.981354i	$-0.438435\pi$
$547$	2.69010e6	0.384415	0.192207	+	0.981354i	$0.438435\pi$
$548$	−2.99524e6	−0.426070
$549$	0	0
$550$	2.79546e6	0.394045
$551$	302770.	0.0424849
$552$	0	0
$553$	−1.56710e7	−2.17913
$554$	2.77123e6	0.383618
$555$	0	0
$556$	−3.36911e6	−0.462198
$557$	−5.82722e6	−0.795835	−0.397918	−	0.917421i	$-0.630267\pi$
$557$	−5.82722e6	−0.795835	−0.397918	+	0.917421i	$0.630267\pi$
$558$	0	0
$559$	−8.20312e6	−1.11032
$560$	4.44294e6	0.598688
$561$	0	0
$562$	−8.43714e6	−1.12682
$563$	9.32234e6	1.23952	0.619760	−	0.784791i	$-0.287230\pi$
$563$	9.32234e6	1.23952	0.619760	+	0.784791i	$0.287230\pi$
$564$	0	0
$565$	−5.21413e6	−0.687165
$566$	−5.85076e6	−0.767665
$567$	0	0
$568$	−4.22722e6	−0.549774
$569$	1.07735e7	1.39500	0.697502	−	0.716583i	$-0.254295\pi$
$569$	1.07735e7	1.39500	0.697502	+	0.716583i	$0.254295\pi$
$570$	0	0
$571$	−3.33200e6	−0.427675	−0.213838	−	0.976869i	$-0.568596\pi$
$571$	−3.33200e6	−0.427675	−0.213838	+	0.976869i	$0.568596\pi$
$572$	−2.84300e6	−0.363318
$573$	0	0
$574$	−8.17606e6	−1.03577
$575$	1.17249e6	0.147890
$576$	0	0
$577$	3.94388e6	0.493156	0.246578	−	0.969123i	$-0.420694\pi$
$577$	3.94388e6	0.493156	0.246578	+	0.969123i	$0.420694\pi$
$578$	−3.30857e6	−0.411927
$579$	0	0
$580$	−991744.	−0.122414
$581$	1.22257e7	1.50256
$582$	0	0
$583$	−7.50493e6	−0.914482
$584$	4.16832e6	0.505742
$585$	0	0
$586$	−1.02196e7	−1.22939
$587$	−1.63457e6	−0.195798	−0.0978991	−	0.995196i	$-0.531212\pi$
$587$	−1.63457e6	−0.195798	−0.0978991	+	0.995196i	$0.531212\pi$
$588$	0	0
$589$	−1.23442e6	−0.146613
$590$	−7.48175e6	−0.884857
$591$	0	0
$592$	631125.	0.0740135
$593$	9.45304e6	1.10391	0.551957	−	0.833873i	$-0.313881\pi$
$593$	9.45304e6	1.10391	0.551957	+	0.833873i	$0.313881\pi$
$594$	0	0
$595$	−1.33614e7	−1.54725
$596$	7.77191e6	0.896215
$597$	0	0
$598$	−1.19243e6	−0.136357
$599$	−2.07256e6	−0.236015	−0.118008	−	0.993013i	$-0.537651\pi$
$599$	−2.07256e6	−0.236015	−0.118008	+	0.993013i	$0.537651\pi$
$600$	0	0
$601$	1.06103e7	1.19823	0.599115	−	0.800663i	$-0.295519\pi$
$601$	1.06103e7	1.19823	0.599115	+	0.800663i	$0.295519\pi$
$602$	9.40383e6	1.05758
$603$	0	0
$604$	3.36336e6	0.375130
$605$	−8.47769e6	−0.941649
$606$	0	0
$607$	7.35521e6	0.810258	0.405129	−	0.914259i	$-0.367227\pi$
$607$	7.35521e6	0.810258	0.405129	+	0.914259i	$0.367227\pi$
$608$	−392567.	−0.0430681
$609$	0	0
$610$	1.18068e7	1.28471
$611$	−1.50618e6	−0.163221
$612$	0	0
$613$	1.09586e7	1.17789	0.588946	−	0.808173i	$-0.299543\pi$
$613$	1.09586e7	1.17789	0.588946	+	0.808173i	$0.299543\pi$
$614$	−8.59927e6	−0.920535
$615$	0	0
$616$	3.25913e6	0.346059
$617$	3.77968e6	0.399707	0.199854	−	0.979826i	$-0.435953\pi$
$617$	3.77968e6	0.399707	0.199854	+	0.979826i	$0.435953\pi$
$618$	0	0
$619$	−492920.	−0.0517071	−0.0258535	−	0.999666i	$-0.508230\pi$
$619$	−492920.	−0.0517071	−0.0258535	+	0.999666i	$0.508230\pi$
$620$	4.04341e6	0.422443
$621$	0	0
$622$	−1.47048e6	−0.152399
$623$	2.34367e6	0.241923
$624$	0	0
$625$	−1.00396e7	−1.02805
$626$	730974.	0.0745532
$627$	0	0
$628$	−6.27595e6	−0.635009
$629$	−1.89801e6	−0.191281
$630$	0	0
$631$	−1.31134e6	−0.131112	−0.0655558	−	0.997849i	$-0.520882\pi$
$631$	−1.31134e6	−0.131112	−0.0655558	+	0.997849i	$0.520882\pi$
$632$	−4.53551e6	−0.451683
$633$	0	0
$634$	3.45986e6	0.341850
$635$	−2.35141e7	−2.31416
$636$	0	0
$637$	−2.47617e7	−2.41786
$638$	−727497.	−0.0707587
$639$	0	0
$640$	1.28588e6	0.124094
$641$	1.38461e6	0.133102	0.0665508	−	0.997783i	$-0.478801\pi$
$641$	1.38461e6	0.133102	0.0665508	+	0.997783i	$0.478801\pi$
$642$	0	0
$643$	7.94263e6	0.757595	0.378797	−	0.925480i	$-0.376338\pi$
$643$	7.94263e6	0.757595	0.378797	+	0.925480i	$0.376338\pi$
$644$	1.36696e6	0.129880
$645$	0	0
$646$	1.18058e6	0.111305
$647$	−2.30348e6	−0.216334	−0.108167	−	0.994133i	$-0.534498\pi$
$647$	−2.30348e6	−0.216334	−0.108167	+	0.994133i	$0.534498\pi$
$648$	0	0
$649$	−5.48826e6	−0.511474
$650$	−9.36622e6	−0.869522
$651$	0	0
$652$	3.04333e6	0.280369
$653$	−7.92492e6	−0.727298	−0.363649	−	0.931536i	$-0.618469\pi$
$653$	−7.92492e6	−0.727298	−0.363649	+	0.931536i	$0.618469\pi$
$654$	0	0
$655$	−2.60918e7	−2.37630
$656$	−2.36633e6	−0.214692
$657$	0	0
$658$	1.72665e6	0.155467
$659$	9.51790e6	0.853744	0.426872	−	0.904312i	$-0.359615\pi$
$659$	9.51790e6	0.853744	0.426872	+	0.904312i	$0.359615\pi$
$660$	0	0
$661$	1.28548e7	1.14436	0.572181	−	0.820128i	$-0.306098\pi$
$661$	1.28548e7	1.14436	0.572181	+	0.820128i	$0.306098\pi$
$662$	−1.19904e7	−1.06338
$663$	0	0
$664$	3.53837e6	0.311446
$665$	−6.65342e6	−0.583433
$666$	0	0
$667$	−305131.	−0.0265565
$668$	−6.41982e6	−0.556650
$669$	0	0
$670$	−7.23658e6	−0.622797
$671$	8.66089e6	0.742602
$672$	0	0
$673$	6.55099e6	0.557531	0.278766	−	0.960359i	$-0.410075\pi$
$673$	6.55099e6	0.557531	0.278766	+	0.960359i	$0.410075\pi$
$674$	2.87311e6	0.243614
$675$	0	0
$676$	3.58482e6	0.301718
$677$	9.16385e6	0.768433	0.384217	−	0.923243i	$-0.374472\pi$
$677$	9.16385e6	0.768433	0.384217	+	0.923243i	$0.374472\pi$
$678$	0	0
$679$	−1.83140e7	−1.52444
$680$	−3.86709e6	−0.320709
$681$	0	0
$682$	2.96606e6	0.244185
$683$	8.96737e6	0.735552	0.367776	−	0.929914i	$-0.380119\pi$
$683$	8.96737e6	0.735552	0.367776	+	0.929914i	$0.380119\pi$
$684$	0	0
$685$	−1.46924e7	−1.19637
$686$	1.35199e7	1.09689
$687$	0	0
$688$	2.72167e6	0.219212
$689$	2.51454e7	2.01795
$690$	0	0
$691$	1.37186e7	1.09299	0.546494	−	0.837463i	$-0.315962\pi$
$691$	1.37186e7	1.09299	0.546494	+	0.837463i	$0.315962\pi$
$692$	1.51654e6	0.120389
$693$	0	0
$694$	−1.49970e7	−1.18197
$695$	−1.65263e7	−1.29782
$696$	0	0
$697$	7.11635e6	0.554850
$698$	−9.33432e6	−0.725178
$699$	0	0
$700$	1.07372e7	0.828218
$701$	3.42100e6	0.262941	0.131470	−	0.991320i	$-0.458030\pi$
$701$	3.42100e6	0.262941	0.131470	+	0.991320i	$0.458030\pi$
$702$	0	0
$703$	−945126.	−0.0721276
$704$	943262.	0.0717300
$705$	0	0
$706$	1.73828e7	1.31252
$707$	4.68980e6	0.352863
$708$	0	0
$709$	1.32225e7	0.987865	0.493933	−	0.869500i	$-0.335559\pi$
$709$	1.32225e7	0.987865	0.493933	+	0.869500i	$0.335559\pi$
$710$	−2.07356e7	−1.54373
$711$	0	0
$712$	678308.	0.0501449
$713$	1.24404e6	0.0916454
$714$	0	0
$715$	−1.39456e7	−1.02017
$716$	−3.50106e6	−0.255221
$717$	0	0
$718$	−1.08513e7	−0.785541
$719$	−6.17969e6	−0.445804	−0.222902	−	0.974841i	$-0.571553\pi$
$719$	−6.17969e6	−0.445804	−0.222902	+	0.974841i	$0.571553\pi$
$720$	0	0
$721$	2.91351e7	2.08727
$722$	−9.31652e6	−0.665136
$723$	0	0
$724$	−5.51221e6	−0.390823
$725$	−2.39673e6	−0.169346
$726$	0	0
$727$	−9.66767e6	−0.678400	−0.339200	−	0.940714i	$-0.610156\pi$
$727$	−9.66767e6	−0.678400	−0.339200	+	0.940714i	$0.610156\pi$
$728$	−1.09198e7	−0.763634
$729$	0	0
$730$	2.04466e7	1.42009
$731$	−8.18499e6	−0.566532
$732$	0	0
$733$	2.27355e7	1.56295	0.781475	−	0.623937i	$-0.214468\pi$
$733$	2.27355e7	1.56295	0.781475	+	0.623937i	$0.214468\pi$
$734$	1.56651e7	1.07323
$735$	0	0
$736$	395628.	0.0269211
$737$	−5.30842e6	−0.359995
$738$	0	0
$739$	2.13860e7	1.44052	0.720260	−	0.693704i	$-0.244022\pi$
$739$	2.13860e7	1.44052	0.720260	+	0.693704i	$0.244022\pi$
$740$	3.09582e6	0.207825
$741$	0	0
$742$	−2.88260e7	−1.92209
$743$	8.61135e6	0.572268	0.286134	−	0.958190i	$-0.407630\pi$
$743$	8.61135e6	0.572268	0.286134	+	0.958190i	$0.407630\pi$
$744$	0	0
$745$	3.81231e7	2.51651
$746$	7.12257e6	0.468587
$747$	0	0
$748$	−2.83671e6	−0.185379
$749$	1.21172e7	0.789220
$750$	0	0
$751$	−7.25471e6	−0.469376	−0.234688	−	0.972071i	$-0.575407\pi$
$751$	−7.25471e6	−0.469376	−0.234688	+	0.972071i	$0.575407\pi$
$752$	499728.	0.0322247
$753$	0	0
$754$	2.43749e6	0.156140
$755$	1.64981e7	1.05334
$756$	0	0
$757$	−3.04305e6	−0.193005	−0.0965027	−	0.995333i	$-0.530766\pi$
$757$	−3.04305e6	−0.193005	−0.0965027	+	0.995333i	$0.530766\pi$
$758$	−2.52696e6	−0.159744
$759$	0	0
$760$	−1.92564e6	−0.120932
$761$	−2.67061e7	−1.67166	−0.835832	−	0.548986i	$-0.815014\pi$
$761$	−2.67061e7	−1.67166	−0.835832	+	0.548986i	$0.815014\pi$
$762$	0	0
$763$	3.55069e7	2.20801
$764$	−2.15925e6	−0.133835
$765$	0	0
$766$	534424.	0.0329089
$767$	1.83885e7	1.12865
$768$	0	0
$769$	4.90128e6	0.298878	0.149439	−	0.988771i	$-0.452253\pi$
$769$	4.90128e6	0.298878	0.149439	+	0.988771i	$0.452253\pi$
$770$	1.59869e7	0.971709
$771$	0	0
$772$	3.63761e6	0.219671
$773$	5.22612e6	0.314579	0.157290	−	0.987552i	$-0.449724\pi$
$773$	5.22612e6	0.314579	0.157290	+	0.987552i	$0.449724\pi$
$774$	0	0
$775$	9.77163e6	0.584404
$776$	−5.30047e6	−0.315980
$777$	0	0
$778$	2.23788e7	1.32552
$779$	3.54363e6	0.209221
$780$	0	0
$781$	−1.52106e7	−0.892319
$782$	−1.18979e6	−0.0695750
$783$	0	0
$784$	8.21554e6	0.477360
$785$	−3.07851e7	−1.78306
$786$	0	0
$787$	−544224.	−0.0313214	−0.0156607	−	0.999877i	$-0.504985\pi$
$787$	−544224.	−0.0313214	−0.0156607	+	0.999877i	$0.504985\pi$
$788$	6.88700e6	0.395107
$789$	0	0
$790$	−2.22478e7	−1.26829
$791$	−1.46910e7	−0.834852
$792$	0	0
$793$	−2.90184e7	−1.63867
$794$	1.67620e7	0.943572
$795$	0	0
$796$	1.15153e7	0.644156
$797$	−1.72097e7	−0.959683	−0.479842	−	0.877355i	$-0.659306\pi$
$797$	−1.72097e7	−0.959683	−0.479842	+	0.877355i	$0.659306\pi$
$798$	0	0
$799$	−1.50285e6	−0.0832818
$800$	3.10756e6	0.171670
$801$	0	0
$802$	−1.83287e6	−0.100623
$803$	1.49987e7	0.820851
$804$	0	0
$805$	6.70530e6	0.364694
$806$	−9.93782e6	−0.538832
$807$	0	0
$808$	1.35733e6	0.0731402
$809$	−9.72387e6	−0.522357	−0.261179	−	0.965290i	$-0.584111\pi$
$809$	−9.72387e6	−0.522357	−0.261179	+	0.965290i	$0.584111\pi$
$810$	0	0
$811$	1.04900e6	0.0560045	0.0280023	−	0.999608i	$-0.491085\pi$
$811$	1.04900e6	0.0560045	0.0280023	+	0.999608i	$0.491085\pi$
$812$	−2.79427e6	−0.148723
$813$	0	0
$814$	2.27095e6	0.120129
$815$	1.49283e7	0.787256
$816$	0	0
$817$	−4.07577e6	−0.213626
$818$	67743.6	0.00353985
$819$	0	0
$820$	−1.16074e7	−0.602838
$821$	−5.72824e6	−0.296595	−0.148297	−	0.988943i	$-0.547379\pi$
$821$	−5.72824e6	−0.296595	−0.148297	+	0.988943i	$0.547379\pi$
$822$	0	0
$823$	3.20854e7	1.65123	0.825617	−	0.564232i	$-0.190828\pi$
$823$	3.20854e7	1.65123	0.825617	+	0.564232i	$0.190828\pi$
$824$	8.43230e6	0.432642
$825$	0	0
$826$	−2.10801e7	−1.07503
$827$	−5.90474e6	−0.300218	−0.150109	−	0.988669i	$-0.547963\pi$
$827$	−5.90474e6	−0.300218	−0.150109	+	0.988669i	$0.547963\pi$
$828$	0	0
$829$	−1.24236e6	−0.0627859	−0.0313929	−	0.999507i	$-0.509994\pi$
$829$	−1.24236e6	−0.0627859	−0.0313929	+	0.999507i	$0.509994\pi$
$830$	1.73566e7	0.874518
$831$	0	0
$832$	−3.16041e6	−0.158284
$833$	−2.47069e7	−1.23369
$834$	0	0
$835$	−3.14908e7	−1.56303
$836$	−1.41256e6	−0.0699023
$837$	0	0
$838$	1.79912e7	0.885012
$839$	−1.96891e7	−0.965651	−0.482825	−	0.875717i	$-0.660389\pi$
$839$	−1.96891e7	−0.965651	−0.482825	+	0.875717i	$0.660389\pi$
$840$	0	0
$841$	−1.98874e7	−0.969591
$842$	−1.73799e6	−0.0844824
$843$	0	0
$844$	982402.	0.0474715
$845$	1.75844e7	0.847201
$846$	0	0
$847$	−2.38862e7	−1.14403
$848$	−8.34284e6	−0.398405
$849$	0	0
$850$	−9.34551e6	−0.443666
$851$	952495.	0.0450857
$852$	0	0
$853$	2.86598e7	1.34865	0.674327	−	0.738433i	$-0.264434\pi$
$853$	2.86598e7	1.34865	0.674327	+	0.738433i	$0.264434\pi$
$854$	3.32659e7	1.56083
$855$	0	0
$856$	3.50698e6	0.163587
$857$	7.38539e6	0.343496	0.171748	−	0.985141i	$-0.445059\pi$
$857$	7.38539e6	0.343496	0.171748	+	0.985141i	$0.445059\pi$
$858$	0	0
$859$	7.68303e6	0.355263	0.177631	−	0.984097i	$-0.443157\pi$
$859$	7.68303e6	0.355263	0.177631	+	0.984097i	$0.443157\pi$
$860$	1.33504e7	0.615531
$861$	0	0
$862$	2.06680e7	0.947393
$863$	3.40976e7	1.55846	0.779232	−	0.626735i	$-0.215609\pi$
$863$	3.40976e7	1.55846	0.779232	+	0.626735i	$0.215609\pi$
$864$	0	0
$865$	7.43900e6	0.338045
$866$	1.58027e7	0.716037
$867$	0	0
$868$	1.13924e7	0.513236
$869$	−1.63200e7	−0.733111
$870$	0	0
$871$	1.77859e7	0.794385
$872$	1.02765e7	0.457670
$873$	0	0
$874$	−592464.	−0.0262351
$875$	−1.56665e6	−0.0691755
$876$	0	0
$877$	−1.26370e6	−0.0554810	−0.0277405	−	0.999615i	$-0.508831\pi$
$877$	−1.26370e6	−0.0554810	−0.0277405	+	0.999615i	$0.508831\pi$
$878$	−1.95042e6	−0.0853871
$879$	0	0
$880$	4.62694e6	0.201413
$881$	1.89395e7	0.822108	0.411054	−	0.911611i	$-0.365161\pi$
$881$	1.89395e7	0.822108	0.411054	+	0.911611i	$0.365161\pi$
$882$	0	0
$883$	3.16414e7	1.36570	0.682848	−	0.730560i	$-0.260741\pi$
$883$	3.16414e7	1.36570	0.682848	+	0.730560i	$0.260741\pi$
$884$	9.50445e6	0.409069
$885$	0	0
$886$	−32220.2	−0.00137893
$887$	2.78426e7	1.18823	0.594116	−	0.804379i	$-0.297502\pi$
$887$	2.78426e7	1.18823	0.594116	+	0.804379i	$0.297502\pi$
$888$	0	0
$889$	−6.62516e7	−2.81153
$890$	3.32727e6	0.140803
$891$	0	0
$892$	1.05284e7	0.443049
$893$	−748356.	−0.0314036
$894$	0	0
$895$	−1.71736e7	−0.716643
$896$	3.62301e6	0.150765
$897$	0	0
$898$	2.71858e7	1.12500
$899$	−2.54300e6	−0.104941
$900$	0	0
$901$	2.50898e7	1.02964
$902$	−8.51466e6	−0.348458
$903$	0	0
$904$	−4.25188e6	−0.173045
$905$	−2.70388e7	−1.09740
$906$	0	0
$907$	−8.97470e6	−0.362245	−0.181122	−	0.983461i	$-0.557973\pi$
$907$	−8.97470e6	−0.362245	−0.181122	+	0.983461i	$0.557973\pi$
$908$	−162149.	−0.00652679
$909$	0	0
$910$	−5.35642e7	−2.14423
$911$	−8.87730e6	−0.354393	−0.177196	−	0.984176i	$-0.556703\pi$
$911$	−8.87730e6	−0.354393	−0.177196	+	0.984176i	$0.556703\pi$
$912$	0	0
$913$	1.27320e7	0.505497
$914$	1.59247e6	0.0630530
$915$	0	0
$916$	1.95686e7	0.770584
$917$	−7.35146e7	−2.88702
$918$	0	0
$919$	−1.48072e7	−0.578343	−0.289172	−	0.957277i	$-0.593380\pi$
$919$	−1.48072e7	−0.578343	−0.289172	+	0.957277i	$0.593380\pi$
$920$	1.94065e6	0.0755925
$921$	0	0
$922$	−4.96768e6	−0.192454
$923$	5.09635e7	1.96904
$924$	0	0
$925$	7.48162e6	0.287502
$926$	−9.73654e6	−0.373145
$927$	0	0
$928$	−808721.	−0.0308268
$929$	−1.42606e7	−0.542123	−0.271062	−	0.962562i	$-0.587375\pi$
$929$	−1.42606e7	−0.542123	−0.271062	+	0.962562i	$0.587375\pi$
$930$	0	0
$931$	−1.23030e7	−0.465196
$932$	−1.86420e7	−0.702996
$933$	0	0
$934$	−4.33235e6	−0.162501
$935$	−1.39148e7	−0.520532
$936$	0	0
$937$	−3.54753e7	−1.32001	−0.660006	−	0.751261i	$-0.729446\pi$
$937$	−3.54753e7	−1.32001	−0.660006	+	0.751261i	$0.729446\pi$
$938$	−2.03893e7	−0.756650
$939$	0	0
$940$	2.45129e6	0.0904847
$941$	2.60015e7	0.957248	0.478624	−	0.878020i	$-0.341136\pi$
$941$	2.60015e7	0.957248	0.478624	+	0.878020i	$0.341136\pi$
$942$	0	0
$943$	−3.57126e6	−0.130780
$944$	−6.10102e6	−0.222829
$945$	0	0
$946$	9.79327e6	0.355795
$947$	2.23777e7	0.810852	0.405426	−	0.914128i	$-0.367123\pi$
$947$	2.23777e7	0.810852	0.405426	+	0.914128i	$0.367123\pi$
$948$	0	0
$949$	−5.02533e7	−1.81134
$950$	−4.65366e6	−0.167296
$951$	0	0
$952$	−1.08956e7	−0.389637
$953$	−4.62557e7	−1.64981	−0.824903	−	0.565275i	$-0.808770\pi$
$953$	−4.62557e7	−1.64981	−0.824903	+	0.565275i	$0.808770\pi$
$954$	0	0
$955$	−1.05916e7	−0.375798
$956$	2.11123e7	0.747122
$957$	0	0
$958$	−1.33525e7	−0.470055
$959$	−4.13963e7	−1.45350
$960$	0	0
$961$	−1.82612e7	−0.637852
$962$	−7.60886e6	−0.265083
$963$	0	0
$964$	8.50895e6	0.294906
$965$	1.78434e7	0.616821
$966$	0	0
$967$	−4.08625e7	−1.40527	−0.702634	−	0.711552i	$-0.747993\pi$
$967$	−4.08625e7	−1.40527	−0.702634	+	0.711552i	$0.747993\pi$
$968$	−6.91316e6	−0.237131
$969$	0	0
$970$	−2.60001e7	−0.887250
$971$	−3.69663e7	−1.25822	−0.629112	−	0.777315i	$-0.716581\pi$
$971$	−3.69663e7	−1.25822	−0.629112	+	0.777315i	$0.716581\pi$
$972$	0	0
$973$	−4.65634e7	−1.57675
$974$	−2.06884e7	−0.698762
$975$	0	0
$976$	9.62786e6	0.323523
$977$	−5.91657e7	−1.98305	−0.991525	−	0.129915i	$-0.958530\pi$
$977$	−5.91657e7	−1.98305	−0.991525	+	0.129915i	$0.958530\pi$
$978$	0	0
$979$	2.44073e6	0.0813885
$980$	4.02992e7	1.34039
$981$	0	0
$982$	2.43158e6	0.0804655
$983$	−1.82526e7	−0.602477	−0.301238	−	0.953549i	$-0.597400\pi$
$983$	−1.82526e7	−0.602477	−0.301238	+	0.953549i	$0.597400\pi$
$984$	0	0
$985$	3.37824e7	1.10943
$986$	2.43210e6	0.0796690
$987$	0	0
$988$	4.73280e6	0.154250
$989$	4.10755e6	0.133534
$990$	0	0
$991$	4.10661e7	1.32831	0.664154	−	0.747596i	$-0.268792\pi$
$991$	4.10661e7	1.32831	0.664154	+	0.747596i	$0.268792\pi$
$992$	3.29721e6	0.106382
$993$	0	0
$994$	−5.84231e7	−1.87551
$995$	5.64852e7	1.80874
$996$	0	0
$997$	1.87017e7	0.595859	0.297930	−	0.954588i	$-0.403704\pi$
$997$	1.87017e7	0.595859	0.297930	+	0.954588i	$0.403704\pi$
$998$	1.40516e7	0.446580
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.6.a.j.1.3		3
3.2	odd	2		162.6.a.i.1.1		3
9.2	odd	6		54.6.c.b.37.3		6
9.4	even	3		18.6.c.b.7.1	✓	6
9.5	odd	6		54.6.c.b.19.3		6
9.7	even	3		18.6.c.b.13.1	yes	6
36.7	odd	6		144.6.i.b.49.3		6
36.11	even	6		432.6.i.b.145.3		6
36.23	even	6		432.6.i.b.289.3		6
36.31	odd	6		144.6.i.b.97.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.6.c.b.7.1	✓	6	9.4	even	3
18.6.c.b.13.1	yes	6	9.7	even	3
54.6.c.b.19.3		6	9.5	odd	6
54.6.c.b.37.3		6	9.2	odd	6
144.6.i.b.49.3		6	36.7	odd	6
144.6.i.b.97.3		6	36.31	odd	6
162.6.a.i.1.1		3	3.2	odd	2
162.6.a.j.1.3		3	1.1	even	1	trivial
432.6.i.b.145.3		6	36.11	even	6
432.6.i.b.289.3		6	36.23	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	162.a (trivial)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} +16.0000 q^{4} +78.4839 q^{5} +221.131 q^{7} +64.0000 q^{8} +O(q^{10})\)	\(q+4.00000 q^{2} +16.0000 q^{4} +78.4839 q^{5} +221.131 q^{7} +64.0000 q^{8} +313.936 q^{10} +230.289 q^{11} -771.586 q^{13} +884.524 q^{14} +256.000 q^{16} -769.880 q^{17} -383.367 q^{19} +1255.74 q^{20} +921.155 q^{22} +386.356 q^{23} +3034.73 q^{25} -3086.34 q^{26} +3538.10 q^{28} -789.767 q^{29} +3219.93 q^{31} +1024.00 q^{32} -3079.52 q^{34} +17355.2 q^{35} +2465.33 q^{37} -1533.47 q^{38} +5022.97 q^{40} -9243.46 q^{41} +10631.5 q^{43} +3684.62 q^{44} +1545.42 q^{46} +1952.06 q^{47} +32091.9 q^{49} +12138.9 q^{50} -12345.4 q^{52} -32589.2 q^{53} +18074.0 q^{55} +14152.4 q^{56} -3159.07 q^{58} -23832.1 q^{59} +37608.8 q^{61} +12879.7 q^{62} +4096.00 q^{64} -60557.1 q^{65} -23051.1 q^{67} -12318.1 q^{68} +69421.0 q^{70} -66050.4 q^{71} +65130.0 q^{73} +9861.33 q^{74} -6133.87 q^{76} +50924.0 q^{77} -70867.4 q^{79} +20091.9 q^{80} -36973.8 q^{82} +55287.0 q^{83} -60423.2 q^{85} +42526.0 q^{86} +14738.5 q^{88} +10598.6 q^{89} -170622. q^{91} +6181.69 q^{92} +7808.26 q^{94} -30088.1 q^{95} -82819.8 q^{97} +128368. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 12 q^{2} + 48 q^{4} + 54 q^{5} + 132 q^{7} + 192 q^{8}+O(q^{10}) \) 3 * q + 12 * q^2 + 48 * q^4 + 54 * q^5 + 132 * q^7 + 192 * q^8	\( 3 q + 12 q^{2} + 48 q^{4} + 54 q^{5} + 132 q^{7} + 192 q^{8} + 216 q^{10} + 315 q^{11} + 744 q^{13} + 528 q^{14} + 768 q^{16} + 1449 q^{17} + 1131 q^{19} + 864 q^{20} + 1260 q^{22} + 3168 q^{23} + 2883 q^{25} + 2976 q^{26} + 2112 q^{28} + 5148 q^{29} + 8610 q^{31} + 3072 q^{32} + 5796 q^{34} + 1350 q^{35} + 19968 q^{37} + 4524 q^{38} + 3456 q^{40} - 5049 q^{41} + 31389 q^{43} + 5040 q^{44} + 12672 q^{46} - 12924 q^{47} + 52857 q^{49} + 11532 q^{50} + 11904 q^{52} - 48024 q^{53} + 63126 q^{55} + 8448 q^{56} + 20592 q^{58} - 62955 q^{59} + 75966 q^{61} + 34440 q^{62} + 12288 q^{64} - 108702 q^{65} + 32991 q^{67} + 23184 q^{68} + 5400 q^{70} - 64836 q^{71} - 4233 q^{73} + 79872 q^{74} + 18096 q^{76} - 88740 q^{77} - 89202 q^{79} + 13824 q^{80} - 20196 q^{82} - 32634 q^{83} - 71388 q^{85} + 125556 q^{86} + 20160 q^{88} + 33066 q^{89} - 150918 q^{91} + 50688 q^{92} - 51696 q^{94} + 82944 q^{95} - 46245 q^{97} + 211428 q^{98}+O(q^{100}) \) 3 * q + 12 * q^2 + 48 * q^4 + 54 * q^5 + 132 * q^7 + 192 * q^8 + 216 * q^10 + 315 * q^11 + 744 * q^13 + 528 * q^14 + 768 * q^16 + 1449 * q^17 + 1131 * q^19 + 864 * q^20 + 1260 * q^22 + 3168 * q^23 + 2883 * q^25 + 2976 * q^26 + 2112 * q^28 + 5148 * q^29 + 8610 * q^31 + 3072 * q^32 + 5796 * q^34 + 1350 * q^35 + 19968 * q^37 + 4524 * q^38 + 3456 * q^40 - 5049 * q^41 + 31389 * q^43 + 5040 * q^44 + 12672 * q^46 - 12924 * q^47 + 52857 * q^49 + 11532 * q^50 + 11904 * q^52 - 48024 * q^53 + 63126 * q^55 + 8448 * q^56 + 20592 * q^58 - 62955 * q^59 + 75966 * q^61 + 34440 * q^62 + 12288 * q^64 - 108702 * q^65 + 32991 * q^67 + 23184 * q^68 + 5400 * q^70 - 64836 * q^71 - 4233 * q^73 + 79872 * q^74 + 18096 * q^76 - 88740 * q^77 - 89202 * q^79 + 13824 * q^80 - 20196 * q^82 - 32634 * q^83 - 71388 * q^85 + 125556 * q^86 + 20160 * q^88 + 33066 * q^89 - 150918 * q^91 + 50688 * q^92 - 51696 * q^94 + 82944 * q^95 - 46245 * q^97 + 211428 * q^98

Introduction

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