LMFDB - Embedded newform 162.6.a.i.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [162,6,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, 6); 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, 6, names="a")

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

Newform invariants

comment:select newform

sage:traces = [3,-12,0,48,-54,0,132] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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			$6.81933$ 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)

$f(q)$	$=$	\(q-4.00000 q^{2} +16.0000 q^{4} +66.0868 q^{5} +114.190 q^{7} -64.0000 q^{8} -264.347 q^{10} +385.625 q^{11} +1032.57 q^{13} -456.762 q^{14} +256.000 q^{16} -959.020 q^{17} -464.576 q^{19} +1057.39 q^{20} -1542.50 q^{22} -2303.41 q^{23} +1242.47 q^{25} -4130.30 q^{26} +1827.05 q^{28} -7098.27 q^{29} +7763.06 q^{31} -1024.00 q^{32} +3836.08 q^{34} +7546.48 q^{35} +9317.57 q^{37} +1858.31 q^{38} -4229.56 q^{40} +13323.3 q^{41} -2112.58 q^{43} +6169.99 q^{44} +9213.62 q^{46} -2494.80 q^{47} -3767.56 q^{49} -4969.88 q^{50} +16521.2 q^{52} +10044.2 q^{53} +25484.7 q^{55} -7308.18 q^{56} +28393.1 q^{58} -5440.66 q^{59} +34188.8 q^{61} -31052.2 q^{62} +4096.00 q^{64} +68239.6 q^{65} +53585.2 q^{67} -15344.3 q^{68} -30185.9 q^{70} +970.010 q^{71} -72400.3 q^{73} -37270.3 q^{74} -7433.22 q^{76} +44034.6 q^{77} +32197.7 q^{79} +16918.2 q^{80} -53293.1 q^{82} +36093.2 q^{83} -63378.6 q^{85} +8450.34 q^{86} -24680.0 q^{88} -42622.2 q^{89} +117910. q^{91} -36854.5 q^{92} +9979.21 q^{94} -30702.4 q^{95} -43853.9 q^{97} +15070.2 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 12 q^{2} + 48 q^{4} - 54 q^{5} + 132 q^{7} - 192 q^{8} + 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}+ \cdots - 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

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
$5$	66.0868	1.18220	0.591099	−	0.806599i	$-0.298694\pi$
$5$	66.0868	1.18220	0.591099	+	0.806599i	$0.298694\pi$
$6$	0	0
$7$	114.190	0.880814	0.440407	−	0.897798i	$-0.354834\pi$
$7$	114.190	0.880814	0.440407	+	0.897798i	$0.354834\pi$
$8$	−64.0000	−0.353553
$9$	0	0
$10$	−264.347	−0.835940
$11$	385.625	0.960911	0.480455	−	0.877019i	$-0.340471\pi$
$11$	385.625	0.960911	0.480455	+	0.877019i	$0.340471\pi$
$12$	0	0
$13$	1032.57	1.69458	0.847292	−	0.531128i	$-0.178232\pi$
$13$	1032.57	1.69458	0.847292	+	0.531128i	$0.178232\pi$
$14$	−456.762	−0.622830
$15$	0	0
$16$	256.000	0.250000
$17$	−959.020	−0.804832	−0.402416	−	0.915457i	$-0.631829\pi$
$17$	−959.020	−0.804832	−0.402416	+	0.915457i	$0.631829\pi$
$18$	0	0
$19$	−464.576	−0.295239	−0.147619	−	0.989044i	$-0.547161\pi$
$19$	−464.576	−0.295239	−0.147619	+	0.989044i	$0.547161\pi$
$20$	1057.39	0.591099
$21$	0	0
$22$	−1542.50	−0.679466
$23$	−2303.41	−0.907927	−0.453963	−	0.891020i	$-0.649990\pi$
$23$	−2303.41	−0.907927	−0.453963	+	0.891020i	$0.649990\pi$
$24$	0	0
$25$	1242.47	0.397590
$26$	−4130.30	−1.19825
$27$	0	0
$28$	1827.05	0.440407
$29$	−7098.27	−1.56732	−0.783659	−	0.621191i	$-0.786649\pi$
$29$	−7098.27	−1.56732	−0.783659	+	0.621191i	$0.786649\pi$
$30$	0	0
$31$	7763.06	1.45087	0.725435	−	0.688291i	$-0.241639\pi$
$31$	7763.06	1.45087	0.725435	+	0.688291i	$0.241639\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	3836.08	0.569102
$35$	7546.48	1.04130
$36$	0	0
$37$	9317.57	1.11892	0.559459	−	0.828858i	$-0.311009\pi$
$37$	9317.57	1.11892	0.559459	+	0.828858i	$0.311009\pi$
$38$	1858.31	0.208765
$39$	0	0
$40$	−4229.56	−0.417970
$41$	13323.3	1.23780	0.618901	−	0.785469i	$-0.287578\pi$
$41$	13323.3	1.23780	0.618901	+	0.785469i	$0.287578\pi$
$42$	0	0
$43$	−2112.58	−0.174238	−0.0871190	−	0.996198i	$-0.527766\pi$
$43$	−2112.58	−0.174238	−0.0871190	+	0.996198i	$0.527766\pi$
$44$	6169.99	0.480455
$45$	0	0
$46$	9213.62	0.642001
$47$	−2494.80	−0.164737	−0.0823686	−	0.996602i	$-0.526248\pi$
$47$	−2494.80	−0.164737	−0.0823686	+	0.996602i	$0.526248\pi$
$48$	0	0
$49$	−3767.56	−0.224166
$50$	−4969.88	−0.281139
$51$	0	0
$52$	16521.2	0.847292
$53$	10044.2	0.491163	0.245582	−	0.969376i	$-0.421021\pi$
$53$	10044.2	0.491163	0.245582	+	0.969376i	$0.421021\pi$
$54$	0	0
$55$	25484.7	1.13599
$56$	−7308.18	−0.311415
$57$	0	0
$58$	28393.1	1.10826
$59$	−5440.66	−0.203480	−0.101740	−	0.994811i	$-0.532441\pi$
$59$	−5440.66	−0.203480	−0.101740	+	0.994811i	$0.532441\pi$
$60$	0	0
$61$	34188.8	1.17641	0.588206	−	0.808711i	$-0.299834\pi$
$61$	34188.8	1.17641	0.588206	+	0.808711i	$0.299834\pi$
$62$	−31052.2	−1.02592
$63$	0	0
$64$	4096.00	0.125000
$65$	68239.6	2.00333
$66$	0	0
$67$	53585.2	1.45834	0.729169	−	0.684334i	$-0.239907\pi$
$67$	53585.2	1.45834	0.729169	+	0.684334i	$0.239907\pi$
$68$	−15344.3	−0.402416
$69$	0	0
$70$	−30185.9	−0.736308
$71$	970.010	0.0228365	0.0114183	−	0.999935i	$-0.496365\pi$
$71$	970.010	0.0228365	0.0114183	+	0.999935i	$0.496365\pi$
$72$	0	0
$73$	−72400.3	−1.59013	−0.795066	−	0.606523i	$-0.792564\pi$
$73$	−72400.3	−1.59013	−0.795066	+	0.606523i	$0.792564\pi$
$74$	−37270.3	−0.791195
$75$	0	0
$76$	−7433.22	−0.147619
$77$	44034.6	0.846384
$78$	0	0
$79$	32197.7	0.580440	0.290220	−	0.956960i	$-0.406271\pi$
$79$	32197.7	0.580440	0.290220	+	0.956960i	$0.406271\pi$
$80$	16918.2	0.295549
$81$	0	0
$82$	−53293.1	−0.875258
$83$	36093.2	0.575082	0.287541	−	0.957768i	$-0.407162\pi$
$83$	36093.2	0.575082	0.287541	+	0.957768i	$0.407162\pi$
$84$	0	0
$85$	−63378.6	−0.951470
$86$	8450.34	0.123205
$87$	0	0
$88$	−24680.0	−0.339733
$89$	−42622.2	−0.570375	−0.285188	−	0.958472i	$-0.592056\pi$
$89$	−42622.2	−0.570375	−0.285188	+	0.958472i	$0.592056\pi$
$90$	0	0
$91$	117910.	1.49261
$92$	−36854.5	−0.453963
$93$	0	0
$94$	9979.21	0.116487
$95$	−30702.4	−0.349030
$96$	0	0
$97$	−43853.9	−0.473237	−0.236618	−	0.971603i	$-0.576039\pi$
$97$	−43853.9	−0.473237	−0.236618	+	0.971603i	$0.576039\pi$
$98$	15070.2	0.158509
$99$	0	0
$100$	19879.5	0.198795
$101$	−24113.3	−0.235209	−0.117604	−	0.993061i	$-0.537522\pi$
$101$	−24113.3	−0.235209	−0.117604	+	0.993061i	$0.537522\pi$
$102$	0	0
$103$	−137221.	−1.27446	−0.637231	−	0.770673i	$-0.719920\pi$
$103$	−137221.	−1.27446	−0.637231	+	0.770673i	$0.719920\pi$
$104$	−66084.8	−0.599126
$105$	0	0
$106$	−40176.8	−0.347305
$107$	167979.	1.41839	0.709195	−	0.705012i	$-0.249059\pi$
$107$	167979.	1.41839	0.709195	+	0.705012i	$0.249059\pi$
$108$	0	0
$109$	−21224.0	−0.171104	−0.0855521	−	0.996334i	$-0.527265\pi$
$109$	−21224.0	−0.171104	−0.0855521	+	0.996334i	$0.527265\pi$
$110$	−101939.	−0.803263
$111$	0	0
$112$	29232.7	0.220204
$113$	28841.6	0.212482	0.106241	−	0.994340i	$-0.466118\pi$
$113$	28841.6	0.212482	0.106241	+	0.994340i	$0.466118\pi$
$114$	0	0
$115$	−152225.	−1.07335
$116$	−113572.	−0.783659
$117$	0	0
$118$	21762.6	0.143882
$119$	−109511.	−0.708908
$120$	0	0
$121$	−12344.7	−0.0766509
$122$	−136755.	−0.831849
$123$	0	0
$124$	124209.	0.725435
$125$	−124410.	−0.712167
$126$	0	0
$127$	−215062.	−1.18319	−0.591594	−	0.806236i	$-0.701501\pi$
$127$	−215062.	−1.18319	−0.591594	+	0.806236i	$0.701501\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	−272958.	−1.41657
$131$	189309.	0.963815	0.481908	−	0.876222i	$-0.339944\pi$
$131$	189309.	0.963815	0.481908	+	0.876222i	$0.339944\pi$
$132$	0	0
$133$	−53050.2	−0.260050
$134$	−214341.	−1.03120
$135$	0	0
$136$	61377.3	0.284551
$137$	3619.56	0.0164761	0.00823805	−	0.999966i	$-0.497378\pi$
$137$	3619.56	0.0164761	0.00823805	+	0.999966i	$0.497378\pi$
$138$	0	0
$139$	120880.	0.530662	0.265331	−	0.964157i	$-0.414519\pi$
$139$	120880.	0.530662	0.265331	+	0.964157i	$0.414519\pi$
$140$	120744.	0.520648
$141$	0	0
$142$	−3880.04	−0.0161479
$143$	398186.	1.62834
$144$	0	0
$145$	−469102.	−1.85288
$146$	289601.	1.12439
$147$	0	0
$148$	149081.	0.559459
$149$	−74816.3	−0.276077	−0.138039	−	0.990427i	$-0.544080\pi$
$149$	−74816.3	−0.276077	−0.138039	+	0.990427i	$0.544080\pi$
$150$	0	0
$151$	−359953.	−1.28470	−0.642352	−	0.766410i	$-0.722041\pi$
$151$	−359953.	−1.28470	−0.642352	+	0.766410i	$0.722041\pi$
$152$	29732.9	0.104383
$153$	0	0
$154$	−176138.	−0.598484
$155$	513036.	1.71521
$156$	0	0
$157$	47966.6	0.155307	0.0776533	−	0.996980i	$-0.475257\pi$
$157$	47966.6	0.155307	0.0776533	+	0.996980i	$0.475257\pi$
$158$	−128791.	−0.410433
$159$	0	0
$160$	−67672.9	−0.208985
$161$	−263027.	−0.799715
$162$	0	0
$163$	361063.	1.06442	0.532212	−	0.846611i	$-0.321361\pi$
$163$	361063.	1.06442	0.532212	+	0.846611i	$0.321361\pi$
$164$	213172.	0.618901
$165$	0	0
$166$	−144373.	−0.406644
$167$	−116466.	−0.323154	−0.161577	−	0.986860i	$-0.551658\pi$
$167$	−116466.	−0.323154	−0.161577	+	0.986860i	$0.551658\pi$
$168$	0	0
$169$	694917.	1.87161
$170$	253514.	0.672791
$171$	0	0
$172$	−33801.4	−0.0871190
$173$	−103526.	−0.262987	−0.131493	−	0.991317i	$-0.541977\pi$
$173$	−103526.	−0.262987	−0.131493	+	0.991317i	$0.541977\pi$
$174$	0	0
$175$	141878.	0.350203
$176$	98719.9	0.240228
$177$	0	0
$178$	170489.	0.403316
$179$	78941.2	0.184150	0.0920748	−	0.995752i	$-0.470650\pi$
$179$	78941.2	0.184150	0.0920748	+	0.995752i	$0.470650\pi$
$180$	0	0
$181$	586108.	1.32978	0.664892	−	0.746940i	$-0.268478\pi$
$181$	586108.	1.32978	0.664892	+	0.746940i	$0.268478\pi$
$182$	−471640.	−1.05544
$183$	0	0
$184$	147418.	0.321001
$185$	615769.	1.32278
$186$	0	0
$187$	−369822.	−0.773372
$188$	−39916.8	−0.0823686
$189$	0	0
$190$	122810.	0.246802
$191$	−772053.	−1.53131	−0.765656	−	0.643251i	$-0.777585\pi$
$191$	−772053.	−1.53131	−0.765656	+	0.643251i	$0.777585\pi$
$192$	0	0
$193$	−17439.3	−0.0337005	−0.0168503	−	0.999858i	$-0.505364\pi$
$193$	−17439.3	−0.0337005	−0.0168503	+	0.999858i	$0.505364\pi$
$194$	175416.	0.334629
$195$	0	0
$196$	−60280.9	−0.112083
$197$	−340729.	−0.625524	−0.312762	−	0.949832i	$-0.601254\pi$
$197$	−340729.	−0.625524	−0.312762	+	0.949832i	$0.601254\pi$
$198$	0	0
$199$	14239.7	0.0254899	0.0127450	−	0.999919i	$-0.495943\pi$
$199$	14239.7	0.0254899	0.0127450	+	0.999919i	$0.495943\pi$
$200$	−79518.1	−0.140569
$201$	0	0
$202$	96453.4	0.166318
$203$	−810554.	−1.38052
$204$	0	0
$205$	880493.	1.46333
$206$	548883.	0.901181
$207$	0	0
$208$	264339.	0.423646
$209$	−179152.	−0.283698
$210$	0	0
$211$	−48098.4	−0.0743745	−0.0371873	−	0.999308i	$-0.511840\pi$
$211$	−48098.4	−0.0743745	−0.0371873	+	0.999308i	$0.511840\pi$
$212$	160707.	0.245582
$213$	0	0
$214$	−671916.	−1.00295
$215$	−139614.	−0.205984
$216$	0	0
$217$	886466.	1.27795
$218$	84896.0	0.120989
$219$	0	0
$220$	407755.	0.567993
$221$	−990259.	−1.36386
$222$	0	0
$223$	783977.	1.05570	0.527851	−	0.849337i	$-0.322998\pi$
$223$	783977.	1.05570	0.527851	+	0.849337i	$0.322998\pi$
$224$	−116931.	−0.155707
$225$	0	0
$226$	−115366.	−0.150248
$227$	555796.	0.715898	0.357949	−	0.933741i	$-0.383476\pi$
$227$	555796.	0.715898	0.357949	+	0.933741i	$0.383476\pi$
$228$	0	0
$229$	114692.	0.144526	0.0722630	−	0.997386i	$-0.476978\pi$
$229$	114692.	0.144526	0.0722630	+	0.997386i	$0.476978\pi$
$230$	608899.	0.758972
$231$	0	0
$232$	454289.	0.554131
$233$	111177.	0.134161	0.0670805	−	0.997748i	$-0.478632\pi$
$233$	111177.	0.134161	0.0670805	+	0.997748i	$0.478632\pi$
$234$	0	0
$235$	−164874.	−0.194752
$236$	−87050.6	−0.101740
$237$	0	0
$238$	438043.	0.501274
$239$	739047.	0.836907	0.418453	−	0.908238i	$-0.362572\pi$
$239$	739047.	0.836907	0.418453	+	0.908238i	$0.362572\pi$
$240$	0	0
$241$	−1.29036e6	−1.43109	−0.715547	−	0.698564i	$-0.753823\pi$
$241$	−1.29036e6	−1.43109	−0.715547	+	0.698564i	$0.753823\pi$
$242$	49378.8	0.0542004
$243$	0	0
$244$	547021.	0.588206
$245$	−248986.	−0.265008
$246$	0	0
$247$	−479710.	−0.500307
$248$	−496836.	−0.512960
$249$	0	0
$250$	497642.	0.503578
$251$	−1.29383e6	−1.29626	−0.648129	−	0.761530i	$-0.724448\pi$
$251$	−1.29383e6	−1.29626	−0.648129	+	0.761530i	$0.724448\pi$
$252$	0	0
$253$	−888250.	−0.872437
$254$	860247.	0.836640
$255$	0	0
$256$	65536.0	0.0625000
$257$	−624362.	−0.589663	−0.294831	−	0.955549i	$-0.595263\pi$
$257$	−624362.	−0.589663	−0.294831	+	0.955549i	$0.595263\pi$
$258$	0	0
$259$	1.06398e6	0.985559
$260$	1.09183e6	1.00167
$261$	0	0
$262$	−757237.	−0.681520
$263$	1.16499e6	1.03856	0.519281	−	0.854603i	$-0.326200\pi$
$263$	1.16499e6	1.03856	0.519281	+	0.854603i	$0.326200\pi$
$264$	0	0
$265$	663790.	0.580652
$266$	212201.	0.183883
$267$	0	0
$268$	857363.	0.729169
$269$	−1.58951e6	−1.33932	−0.669659	−	0.742669i	$-0.733560\pi$
$269$	−1.58951e6	−1.33932	−0.669659	+	0.742669i	$0.733560\pi$
$270$	0	0
$271$	−977878.	−0.808837	−0.404419	−	0.914574i	$-0.632526\pi$
$271$	−977878.	−0.808837	−0.404419	+	0.914574i	$0.632526\pi$
$272$	−245509.	−0.201208
$273$	0	0
$274$	−14478.2	−0.0116504
$275$	479127.	0.382049
$276$	0	0
$277$	−1.29049e6	−1.01054	−0.505271	−	0.862961i	$-0.668607\pi$
$277$	−1.29049e6	−1.01054	−0.505271	+	0.862961i	$0.668607\pi$
$278$	−483521.	−0.375235
$279$	0	0
$280$	−482975.	−0.368154
$281$	−1.66732e6	−1.25966	−0.629831	−	0.776732i	$-0.716876\pi$
$281$	−1.66732e6	−1.25966	−0.629831	+	0.776732i	$0.716876\pi$
$282$	0	0
$283$	1.12914e6	0.838070	0.419035	−	0.907970i	$-0.362368\pi$
$283$	1.12914e6	0.838070	0.419035	+	0.907970i	$0.362368\pi$
$284$	15520.2	0.0114183
$285$	0	0
$286$	−1.59274e6	−1.15141
$287$	1.52139e6	1.09027
$288$	0	0
$289$	−500138.	−0.352245
$290$	1.87641e6	1.31018
$291$	0	0
$292$	−1.15840e6	−0.795066
$293$	−2.05631e6	−1.39933	−0.699663	−	0.714473i	$-0.746666\pi$
$293$	−2.05631e6	−1.39933	−0.699663	+	0.714473i	$0.746666\pi$
$294$	0	0
$295$	−359556.	−0.240553
$296$	−596324.	−0.395597
$297$	0	0
$298$	299265.	0.195216
$299$	−2.37844e6	−1.53856
$300$	0	0
$301$	−241237.	−0.153471
$302$	1.43981e6	0.908423
$303$	0	0
$304$	−118932.	−0.0738097
$305$	2.25943e6	1.39075
$306$	0	0
$307$	1.52149e6	0.921346	0.460673	−	0.887570i	$-0.347608\pi$
$307$	1.52149e6	0.921346	0.460673	+	0.887570i	$0.347608\pi$
$308$	704554.	0.423192
$309$	0	0
$310$	−2.05214e6	−1.21284
$311$	2.70840e6	1.58786	0.793928	−	0.608011i	$-0.208032\pi$
$311$	2.70840e6	1.58786	0.793928	+	0.608011i	$0.208032\pi$
$312$	0	0
$313$	−651166.	−0.375691	−0.187845	−	0.982199i	$-0.560150\pi$
$313$	−651166.	−0.375691	−0.187845	+	0.982199i	$0.560150\pi$
$314$	−191866.	−0.109818
$315$	0	0
$316$	515164.	0.290220
$317$	116111.	0.0648969	0.0324484	−	0.999473i	$-0.489670\pi$
$317$	116111.	0.0648969	0.0324484	+	0.999473i	$0.489670\pi$
$318$	0	0
$319$	−2.73727e6	−1.50605
$320$	270692.	0.147775
$321$	0	0
$322$	1.05211e6	0.565484
$323$	445538.	0.237618
$324$	0	0
$325$	1.28294e6	0.673750
$326$	−1.44425e6	−0.752661
$327$	0	0
$328$	−852689.	−0.437629
$329$	−284882.	−0.145103
$330$	0	0
$331$	317657.	0.159363	0.0796816	−	0.996820i	$-0.474610\pi$
$331$	317657.	0.159363	0.0796816	+	0.996820i	$0.474610\pi$
$332$	577491.	0.287541
$333$	0	0
$334$	465866.	0.228504
$335$	3.54128e6	1.72404
$336$	0	0
$337$	430550.	0.206514	0.103257	−	0.994655i	$-0.467074\pi$
$337$	430550.	0.206514	0.103257	+	0.994655i	$0.467074\pi$
$338$	−2.77967e6	−1.32343
$339$	0	0
$340$	−1.01406e6	−0.475735
$341$	2.99363e6	1.39416
$342$	0	0
$343$	−2.34942e6	−1.07826
$344$	135205.	0.0616024
$345$	0	0
$346$	414103.	0.185960
$347$	4.14039e6	1.84594	0.922971	−	0.384870i	$-0.125754\pi$
$347$	4.14039e6	1.84594	0.922971	+	0.384870i	$0.125754\pi$
$348$	0	0
$349$	−3.46095e6	−1.52101	−0.760505	−	0.649332i	$-0.775049\pi$
$349$	−3.46095e6	−1.52101	−0.760505	+	0.649332i	$0.775049\pi$
$350$	−567512.	−0.247631
$351$	0	0
$352$	−394880.	−0.169867
$353$	−2.64457e6	−1.12958	−0.564792	−	0.825233i	$-0.691044\pi$
$353$	−2.64457e6	−1.12958	−0.564792	+	0.825233i	$0.691044\pi$
$354$	0	0
$355$	64104.9	0.0269973
$356$	−681955.	−0.285188
$357$	0	0
$358$	−315765.	−0.130213
$359$	−2.13123e6	−0.872761	−0.436380	−	0.899762i	$-0.643740\pi$
$359$	−2.13123e6	−0.872761	−0.436380	+	0.899762i	$0.643740\pi$
$360$	0	0
$361$	−2.26027e6	−0.912834
$362$	−2.34443e6	−0.940299
$363$	0	0
$364$	1.88656e6	0.746307
$365$	−4.78471e6	−1.87985
$366$	0	0
$367$	−1.28007e6	−0.496099	−0.248049	−	0.968747i	$-0.579790\pi$
$367$	−1.28007e6	−0.496099	−0.248049	+	0.968747i	$0.579790\pi$
$368$	−589672.	−0.226982
$369$	0	0
$370$	−2.46307e6	−0.935348
$371$	1.14695e6	0.432624
$372$	0	0
$373$	−3.89466e6	−1.44943	−0.724716	−	0.689048i	$-0.758029\pi$
$373$	−3.89466e6	−1.44943	−0.724716	+	0.689048i	$0.758029\pi$
$374$	1.47929e6	0.546856
$375$	0	0
$376$	159667.	0.0582434
$377$	−7.32949e6	−2.65595
$378$	0	0
$379$	−2.83335e6	−1.01322	−0.506609	−	0.862176i	$-0.669101\pi$
$379$	−2.83335e6	−1.01322	−0.506609	+	0.862176i	$0.669101\pi$
$380$	−491238.	−0.174515
$381$	0	0
$382$	3.08821e6	1.08280
$383$	−2.36499e6	−0.823819	−0.411909	−	0.911225i	$-0.635138\pi$
$383$	−2.36499e6	−0.823819	−0.411909	+	0.911225i	$0.635138\pi$
$384$	0	0
$385$	2.91011e6	1.00059
$386$	69757.3	0.0238299
$387$	0	0
$388$	−701662.	−0.236618
$389$	1.74338e6	0.584143	0.292071	−	0.956397i	$-0.405656\pi$
$389$	1.74338e6	0.584143	0.292071	+	0.956397i	$0.405656\pi$
$390$	0	0
$391$	2.20901e6	0.730729
$392$	241124.	0.0792546
$393$	0	0
$394$	1.36292e6	0.442312
$395$	2.12785e6	0.686195
$396$	0	0
$397$	4.81109e6	1.53203	0.766015	−	0.642823i	$-0.222237\pi$
$397$	4.81109e6	1.53203	0.766015	+	0.642823i	$0.222237\pi$
$398$	−56958.9	−0.0180241
$399$	0	0
$400$	318072.	0.0993976
$401$	4.55407e6	1.41429	0.707145	−	0.707068i	$-0.249983\pi$
$401$	4.55407e6	1.41429	0.707145	+	0.707068i	$0.249983\pi$
$402$	0	0
$403$	8.01593e6	2.45862
$404$	−385813.	−0.117604
$405$	0	0
$406$	3.24221e6	0.976173
$407$	3.59308e6	1.07518
$408$	0	0
$409$	1.91283e6	0.565416	0.282708	−	0.959206i	$-0.408767\pi$
$409$	1.91283e6	0.565416	0.282708	+	0.959206i	$0.408767\pi$
$410$	−3.52197e6	−1.03473
$411$	0	0
$412$	−2.19553e6	−0.637231
$413$	−621271.	−0.179228
$414$	0	0
$415$	2.38528e6	0.679860
$416$	−1.05736e6	−0.299563
$417$	0	0
$418$	716608.	0.200605
$419$	−4.41395e6	−1.22826	−0.614132	−	0.789203i	$-0.710494\pi$
$419$	−4.41395e6	−1.22826	−0.614132	+	0.789203i	$0.710494\pi$
$420$	0	0
$421$	−1.74451e6	−0.479697	−0.239848	−	0.970810i	$-0.577098\pi$
$421$	−1.74451e6	−0.479697	−0.239848	+	0.970810i	$0.577098\pi$
$422$	192393.	0.0525907
$423$	0	0
$424$	−642829.	−0.173652
$425$	−1.19155e6	−0.319993
$426$	0	0
$427$	3.90403e6	1.03620
$428$	2.68766e6	0.709195
$429$	0	0
$430$	558456.	0.145652
$431$	2.21740e6	0.574976	0.287488	−	0.957784i	$-0.407180\pi$
$431$	2.21740e6	0.574976	0.287488	+	0.957784i	$0.407180\pi$
$432$	0	0
$433$	−3.63511e6	−0.931746	−0.465873	−	0.884851i	$-0.654260\pi$
$433$	−3.63511e6	−0.931746	−0.465873	+	0.884851i	$0.654260\pi$
$434$	−3.54587e6	−0.903645
$435$	0	0
$436$	−339584.	−0.0855521
$437$	1.07011e6	0.268055
$438$	0	0
$439$	−3.08827e6	−0.764810	−0.382405	−	0.923995i	$-0.624904\pi$
$439$	−3.08827e6	−0.764810	−0.382405	+	0.923995i	$0.624904\pi$
$440$	−1.63102e6	−0.401632
$441$	0	0
$442$	3.96104e6	0.964391
$443$	7.00097e6	1.69492	0.847459	−	0.530861i	$-0.178131\pi$
$443$	7.00097e6	1.69492	0.847459	+	0.530861i	$0.178131\pi$
$444$	0	0
$445$	−2.81677e6	−0.674296
$446$	−3.13591e6	−0.746493
$447$	0	0
$448$	467724.	0.110102
$449$	5.00196e6	1.17091	0.585456	−	0.810704i	$-0.300916\pi$
$449$	5.00196e6	1.17091	0.585456	+	0.810704i	$0.300916\pi$
$450$	0	0
$451$	5.13778e6	1.18942
$452$	461465.	0.106241
$453$	0	0
$454$	−2.22319e6	−0.506216
$455$	7.79230e6	1.76456
$456$	0	0
$457$	3.57162e6	0.799971	0.399985	−	0.916521i	$-0.369015\pi$
$457$	3.57162e6	0.799971	0.399985	+	0.916521i	$0.369015\pi$
$458$	−458770.	−0.102195
$459$	0	0
$460$	−2.43560e6	−0.536674
$461$	−5.50122e6	−1.20561	−0.602805	−	0.797888i	$-0.705950\pi$
$461$	−5.50122e6	−1.20561	−0.602805	+	0.797888i	$0.705950\pi$
$462$	0	0
$463$	−2.85416e6	−0.618765	−0.309382	−	0.950938i	$-0.600122\pi$
$463$	−2.85416e6	−0.618765	−0.309382	+	0.950938i	$0.600122\pi$
$464$	−1.81716e6	−0.391830
$465$	0	0
$466$	−444709.	−0.0948662
$467$	7.16552e6	1.52039	0.760196	−	0.649694i	$-0.225103\pi$
$467$	7.16552e6	1.52039	0.760196	+	0.649694i	$0.225103\pi$
$468$	0	0
$469$	6.11892e6	1.28452
$470$	659494.	0.137710
$471$	0	0
$472$	348202.	0.0719410
$473$	−814664.	−0.167427
$474$	0	0
$475$	−577222.	−0.117384
$476$	−1.75217e6	−0.354454
$477$	0	0
$478$	−2.95619e6	−0.591782
$479$	−3.28370e6	−0.653921	−0.326961	−	0.945038i	$-0.606024\pi$
$479$	−3.28370e6	−0.653921	−0.326961	+	0.945038i	$0.606024\pi$
$480$	0	0
$481$	9.62108e6	1.89610
$482$	5.16144e6	1.01194
$483$	0	0
$484$	−197515.	−0.0383255
$485$	−2.89816e6	−0.559459
$486$	0	0
$487$	2.18097e6	0.416704	0.208352	−	0.978054i	$-0.433190\pi$
$487$	2.18097e6	0.416704	0.208352	+	0.978054i	$0.433190\pi$
$488$	−2.18808e6	−0.415925
$489$	0	0
$490$	995944.	0.187389
$491$	9.93949e6	1.86063	0.930315	−	0.366760i	$-0.119533\pi$
$491$	9.93949e6	1.86063	0.930315	+	0.366760i	$0.119533\pi$
$492$	0	0
$493$	6.80738e6	1.26143
$494$	1.91884e6	0.353770
$495$	0	0
$496$	1.98734e6	0.362718
$497$	110766.	0.0201148
$498$	0	0
$499$	−4.52772e6	−0.814007	−0.407004	−	0.913427i	$-0.633426\pi$
$499$	−4.52772e6	−0.814007	−0.407004	+	0.913427i	$0.633426\pi$
$500$	−1.99057e6	−0.356084
$501$	0	0
$502$	5.17530e6	0.916593
$503$	4.40282e6	0.775909	0.387955	−	0.921679i	$-0.373182\pi$
$503$	4.40282e6	0.775909	0.387955	+	0.921679i	$0.373182\pi$
$504$	0	0
$505$	−1.59357e6	−0.278063
$506$	3.55300e6	0.616906
$507$	0	0
$508$	−3.44099e6	−0.591594
$509$	−5.82125e6	−0.995915	−0.497957	−	0.867202i	$-0.665916\pi$
$509$	−5.82125e6	−0.995915	−0.497957	+	0.867202i	$0.665916\pi$
$510$	0	0
$511$	−8.26742e6	−1.40061
$512$	−262144.	−0.0441942
$513$	0	0
$514$	2.49745e6	0.416954
$515$	−9.06849e6	−1.50667
$516$	0	0
$517$	−962057.	−0.158298
$518$	−4.25591e6	−0.696896
$519$	0	0
$520$	−4.36733e6	−0.708285
$521$	−1.09649e7	−1.76975	−0.884875	−	0.465829i	$-0.845756\pi$
$521$	−1.09649e7	−1.76975	−0.884875	+	0.465829i	$0.845756\pi$
$522$	0	0
$523$	−3.83705e6	−0.613399	−0.306700	−	0.951806i	$-0.599225\pi$
$523$	−3.83705e6	−0.613399	−0.306700	+	0.951806i	$0.599225\pi$
$524$	3.02895e6	0.481908
$525$	0	0
$526$	−4.65996e6	−0.734374
$527$	−7.44493e6	−1.16771
$528$	0	0
$529$	−1.13066e6	−0.175669
$530$	−2.65516e6	−0.410583
$531$	0	0
$532$	−848803.	−0.130025
$533$	1.37573e7	2.09756
$534$	0	0
$535$	1.11012e7	1.67682
$536$	−3.42945e6	−0.515600
$537$	0	0
$538$	6.35805e6	0.947040
$539$	−1.45286e6	−0.215403
$540$	0	0
$541$	−4.88767e6	−0.717974	−0.358987	−	0.933343i	$-0.616878\pi$
$541$	−4.88767e6	−0.717974	−0.358987	+	0.933343i	$0.616878\pi$
$542$	3.91151e6	0.571934
$543$	0	0
$544$	982036.	0.142276
$545$	−1.40263e6	−0.202279
$546$	0	0
$547$	−6.93344e6	−0.990788	−0.495394	−	0.868668i	$-0.664976\pi$
$547$	−6.93344e6	−0.990788	−0.495394	+	0.868668i	$0.664976\pi$
$548$	57913.0	0.00823805
$549$	0	0
$550$	−1.91651e6	−0.270149
$551$	3.29769e6	0.462733
$552$	0	0
$553$	3.67667e6	0.511260
$554$	5.16195e6	0.714561
$555$	0	0
$556$	1.93408e6	0.265331
$557$	9.83771e6	1.34356	0.671778	−	0.740752i	$-0.265531\pi$
$557$	9.83771e6	1.34356	0.671778	+	0.740752i	$0.265531\pi$
$558$	0	0
$559$	−2.18140e6	−0.295261
$560$	1.93190e6	0.260324
$561$	0	0
$562$	6.66930e6	0.890716
$563$	−2.73217e6	−0.363277	−0.181638	−	0.983365i	$-0.558140\pi$
$563$	−2.73217e6	−0.363277	−0.181638	+	0.983365i	$0.558140\pi$
$564$	0	0
$565$	1.90605e6	0.251196
$566$	−4.51654e6	−0.592605
$567$	0	0
$568$	−62080.7	−0.00807394
$569$	−9.43997e6	−1.22233	−0.611167	−	0.791502i	$-0.709300\pi$
$569$	−9.43997e6	−1.22233	−0.611167	+	0.791502i	$0.709300\pi$
$570$	0	0
$571$	−1.26790e7	−1.62740	−0.813698	−	0.581288i	$-0.802549\pi$
$571$	−1.26790e7	−1.62740	−0.813698	+	0.581288i	$0.802549\pi$
$572$	6.37098e6	0.814172
$573$	0	0
$574$	−6.08556e6	−0.770940
$575$	−2.86191e6	−0.360983
$576$	0	0
$577$	−1.51657e7	−1.89637	−0.948187	−	0.317712i	$-0.897086\pi$
$577$	−1.51657e7	−1.89637	−0.948187	+	0.317712i	$0.897086\pi$
$578$	2.00055e6	0.249075
$579$	0	0
$580$	−7.50563e6	−0.926440
$581$	4.12149e6	0.506540
$582$	0	0
$583$	3.87329e6	0.471964
$584$	4.63362e6	0.562197
$585$	0	0
$586$	8.22522e6	0.989472
$587$	−1.82340e6	−0.218417	−0.109209	−	0.994019i	$-0.534832\pi$
$587$	−1.82340e6	−0.218417	−0.109209	+	0.994019i	$0.534832\pi$
$588$	0	0
$589$	−3.60653e6	−0.428353
$590$	1.43822e6	0.170097
$591$	0	0
$592$	2.38530e6	0.279730
$593$	−1.27957e7	−1.49427	−0.747133	−	0.664674i	$-0.768570\pi$
$593$	−1.27957e7	−1.49427	−0.747133	+	0.664674i	$0.768570\pi$
$594$	0	0
$595$	−7.23723e6	−0.838069
$596$	−1.19706e6	−0.138039
$597$	0	0
$598$	9.51375e6	1.08792
$599$	−8.76425e6	−0.998039	−0.499020	−	0.866591i	$-0.666306\pi$
$599$	−8.76425e6	−0.998039	−0.499020	+	0.866591i	$0.666306\pi$
$600$	0	0
$601$	1.66959e6	0.188548	0.0942741	−	0.995546i	$-0.469947\pi$
$601$	1.66959e6	0.188548	0.0942741	+	0.995546i	$0.469947\pi$
$602$	964947.	0.108521
$603$	0	0
$604$	−5.75924e6	−0.642352
$605$	−815822.	−0.0906165
$606$	0	0
$607$	−1.01707e7	−1.12042	−0.560208	−	0.828352i	$-0.689279\pi$
$607$	−1.01707e7	−1.12042	−0.560208	+	0.828352i	$0.689279\pi$
$608$	475726.	0.0521913
$609$	0	0
$610$	−9.03772e6	−0.983410
$611$	−2.57607e6	−0.279161
$612$	0	0
$613$	1.37829e7	1.48146	0.740730	−	0.671803i	$-0.234480\pi$
$613$	1.37829e7	1.48146	0.740730	+	0.671803i	$0.234480\pi$
$614$	−6.08595e6	−0.651490
$615$	0	0
$616$	−2.81822e6	−0.299242
$617$	1.21082e7	1.28046	0.640230	−	0.768184i	$-0.278839\pi$
$617$	1.21082e7	1.28046	0.640230	+	0.768184i	$0.278839\pi$
$618$	0	0
$619$	−1.69733e7	−1.78049	−0.890245	−	0.455481i	$-0.849467\pi$
$619$	−1.69733e7	−1.78049	−0.890245	+	0.455481i	$0.849467\pi$
$620$	8.20857e6	0.857607
$621$	0	0
$622$	−1.08336e7	−1.12278
$623$	−4.86704e6	−0.502395
$624$	0	0
$625$	−1.21046e7	−1.23951
$626$	2.60466e6	0.265654
$627$	0	0
$628$	767466.	0.0776533
$629$	−8.93574e6	−0.900541
$630$	0	0
$631$	7.90092e6	0.789958	0.394979	−	0.918690i	$-0.370752\pi$
$631$	7.90092e6	0.789958	0.394979	+	0.918690i	$0.370752\pi$
$632$	−2.06065e6	−0.205217
$633$	0	0
$634$	−464443.	−0.0458890
$635$	−1.42127e7	−1.39876
$636$	0	0
$637$	−3.89028e6	−0.379868
$638$	1.09491e7	1.06494
$639$	0	0
$640$	−1.08277e6	−0.104492
$641$	−1.06761e7	−1.02629	−0.513144	−	0.858302i	$-0.671519\pi$
$641$	−1.06761e7	−1.02629	−0.513144	+	0.858302i	$0.671519\pi$
$642$	0	0
$643$	9.37769e6	0.894476	0.447238	−	0.894415i	$-0.352408\pi$
$643$	9.37769e6	0.894476	0.447238	+	0.894415i	$0.352408\pi$
$644$	−4.20843e6	−0.399858
$645$	0	0
$646$	−1.78215e6	−0.168021
$647$	9.44013e6	0.886579	0.443289	−	0.896379i	$-0.353811\pi$
$647$	9.44013e6	0.886579	0.443289	+	0.896379i	$0.353811\pi$
$648$	0	0
$649$	−2.09805e6	−0.195526
$650$	−5.13177e6	−0.476413
$651$	0	0
$652$	5.77701e6	0.532212
$653$	−1.48891e7	−1.36642	−0.683212	−	0.730220i	$-0.739418\pi$
$653$	−1.48891e7	−1.36642	−0.683212	+	0.730220i	$0.739418\pi$
$654$	0	0
$655$	1.25109e7	1.13942
$656$	3.41076e6	0.309451
$657$	0	0
$658$	1.13953e6	0.102603
$659$	7.64781e6	0.685999	0.343000	−	0.939336i	$-0.388557\pi$
$659$	7.64781e6	0.685999	0.343000	+	0.939336i	$0.388557\pi$
$660$	0	0
$661$	−5.00539e6	−0.445588	−0.222794	−	0.974866i	$-0.571518\pi$
$661$	−5.00539e6	−0.445588	−0.222794	+	0.974866i	$0.571518\pi$
$662$	−1.27063e6	−0.112687
$663$	0	0
$664$	−2.30996e6	−0.203322
$665$	−3.50592e6	−0.307431
$666$	0	0
$667$	1.63502e7	1.42301
$668$	−1.86346e6	−0.161577
$669$	0	0
$670$	−1.41651e7	−1.21908
$671$	1.31840e7	1.13043
$672$	0	0
$673$	2.14440e6	0.182502	0.0912511	−	0.995828i	$-0.470913\pi$
$673$	2.14440e6	0.182502	0.0912511	+	0.995828i	$0.470913\pi$
$674$	−1.72220e6	−0.146027
$675$	0	0
$676$	1.11187e7	0.935807
$677$	9.50926e6	0.797398	0.398699	−	0.917082i	$-0.369462\pi$
$677$	9.50926e6	0.797398	0.398699	+	0.917082i	$0.369462\pi$
$678$	0	0
$679$	−5.00769e6	−0.416834
$680$	4.05623e6	0.336396
$681$	0	0
$682$	−1.19745e7	−0.985817
$683$	5.47266e6	0.448897	0.224449	−	0.974486i	$-0.427942\pi$
$683$	5.47266e6	0.448897	0.224449	+	0.974486i	$0.427942\pi$
$684$	0	0
$685$	239205.	0.0194780
$686$	9.39767e6	0.762447
$687$	0	0
$688$	−540822.	−0.0435595
$689$	1.03714e7	0.832317
$690$	0	0
$691$	6.88167e6	0.548276	0.274138	−	0.961690i	$-0.411608\pi$
$691$	6.88167e6	0.548276	0.274138	+	0.961690i	$0.411608\pi$
$692$	−1.65641e6	−0.131493
$693$	0	0
$694$	−1.65616e7	−1.30528
$695$	7.98859e6	0.627348
$696$	0	0
$697$	−1.27773e7	−0.996223
$698$	1.38438e7	1.07552
$699$	0	0
$700$	2.27005e6	0.175102
$701$	−5.19615e6	−0.399380	−0.199690	−	0.979859i	$-0.563994\pi$
$701$	−5.19615e6	−0.399380	−0.199690	+	0.979859i	$0.563994\pi$
$702$	0	0
$703$	−4.32872e6	−0.330348
$704$	1.57952e6	0.120114
$705$	0	0
$706$	1.05783e7	0.798737
$707$	−2.75351e6	−0.207175
$708$	0	0
$709$	574821.	0.0429454	0.0214727	−	0.999769i	$-0.493164\pi$
$709$	574821.	0.0429454	0.0214727	+	0.999769i	$0.493164\pi$
$710$	−256420.	−0.0190900
$711$	0	0
$712$	2.72782e6	0.201658
$713$	−1.78815e7	−1.31728
$714$	0	0
$715$	2.63149e7	1.92502
$716$	1.26306e6	0.0920748
$717$	0	0
$718$	8.52494e6	0.617135
$719$	400906.	0.0289215	0.0144607	−	0.999895i	$-0.495397\pi$
$719$	400906.	0.0289215	0.0144607	+	0.999895i	$0.495397\pi$
$720$	0	0
$721$	−1.56693e7	−1.12256
$722$	9.04107e6	0.645471
$723$	0	0
$724$	9.37772e6	0.664892
$725$	−8.81938e6	−0.623151
$726$	0	0
$727$	1.05431e7	0.739830	0.369915	−	0.929066i	$-0.379387\pi$
$727$	1.05431e7	0.739830	0.369915	+	0.929066i	$0.379387\pi$
$728$	−7.54624e6	−0.527719
$729$	0	0
$730$	1.91388e7	1.32925
$731$	2.02601e6	0.140232
$732$	0	0
$733$	1.39374e7	0.958122	0.479061	−	0.877782i	$-0.340977\pi$
$733$	1.39374e7	0.958122	0.479061	+	0.877782i	$0.340977\pi$
$734$	5.12028e6	0.350795
$735$	0	0
$736$	2.35869e6	0.160500
$737$	2.06638e7	1.40133
$738$	0	0
$739$	−9.88034e6	−0.665519	−0.332760	−	0.943012i	$-0.607980\pi$
$739$	−9.88034e6	−0.665519	−0.332760	+	0.943012i	$0.607980\pi$
$740$	9.85230e6	0.661391
$741$	0	0
$742$	−4.58781e6	−0.305911
$743$	9.33390e6	0.620285	0.310142	−	0.950690i	$-0.399623\pi$
$743$	9.33390e6	0.620285	0.310142	+	0.950690i	$0.399623\pi$
$744$	0	0
$745$	−4.94438e6	−0.326378
$746$	1.55787e7	1.02490
$747$	0	0
$748$	−5.91715e6	−0.386686
$749$	1.91816e7	1.24934
$750$	0	0
$751$	2.87522e7	1.86025	0.930124	−	0.367245i	$-0.119699\pi$
$751$	2.87522e7	1.86025	0.930124	+	0.367245i	$0.119699\pi$
$752$	−638669.	−0.0411843
$753$	0	0
$754$	2.93179e7	1.87804
$755$	−2.37881e7	−1.51877
$756$	0	0
$757$	−4.94587e6	−0.313691	−0.156846	−	0.987623i	$-0.550133\pi$
$757$	−4.94587e6	−0.313691	−0.156846	+	0.987623i	$0.550133\pi$
$758$	1.13334e7	0.716453
$759$	0	0
$760$	1.96495e6	0.123401
$761$	−9.13658e6	−0.571902	−0.285951	−	0.958244i	$-0.592310\pi$
$761$	−9.13658e6	−0.571902	−0.285951	+	0.958244i	$0.592310\pi$
$762$	0	0
$763$	−2.42358e6	−0.150711
$764$	−1.23528e7	−0.765656
$765$	0	0
$766$	9.45995e6	0.582528
$767$	−5.61789e6	−0.344814
$768$	0	0
$769$	1.65244e7	1.00765	0.503826	−	0.863805i	$-0.331925\pi$
$769$	1.65244e7	1.00765	0.503826	+	0.863805i	$0.331925\pi$
$770$	−1.16404e7	−0.707526
$771$	0	0
$772$	−279029.	−0.0168503
$773$	−1.01763e7	−0.612550	−0.306275	−	0.951943i	$-0.599083\pi$
$773$	−1.01763e7	−0.612550	−0.306275	+	0.951943i	$0.599083\pi$
$774$	0	0
$775$	9.64536e6	0.576852
$776$	2.80665e6	0.167315
$777$	0	0
$778$	−6.97353e6	−0.413051
$779$	−6.18968e6	−0.365447
$780$	0	0
$781$	374060.	0.0219439
$782$	−8.83605e6	−0.516703
$783$	0	0
$784$	−964495.	−0.0560415
$785$	3.16996e6	0.183603
$786$	0	0
$787$	−1.87635e7	−1.07988	−0.539941	−	0.841703i	$-0.681553\pi$
$787$	−1.87635e7	−1.07988	−0.539941	+	0.841703i	$0.681553\pi$
$788$	−5.45167e6	−0.312762
$789$	0	0
$790$	−8.51138e6	−0.485213
$791$	3.29343e6	0.187158
$792$	0	0
$793$	3.53025e7	1.99353
$794$	−1.92444e7	−1.08331
$795$	0	0
$796$	227835.	0.0127450
$797$	2.47965e6	0.138275	0.0691376	−	0.997607i	$-0.477975\pi$
$797$	2.47965e6	0.138275	0.0691376	+	0.997607i	$0.477975\pi$
$798$	0	0
$799$	2.39257e6	0.132586
$800$	−1.27229e6	−0.0702847
$801$	0	0
$802$	−1.82163e7	−1.00005
$803$	−2.79193e7	−1.52797
$804$	0	0
$805$	−1.73826e7	−0.945421
$806$	−3.20637e7	−1.73851
$807$	0	0
$808$	1.54325e6	0.0831589
$809$	2.50880e7	1.34770	0.673852	−	0.738867i	$-0.264639\pi$
$809$	2.50880e7	1.34770	0.673852	+	0.738867i	$0.264639\pi$
$810$	0	0
$811$	−4.28465e6	−0.228751	−0.114376	−	0.993438i	$-0.536487\pi$
$811$	−4.28465e6	−0.228751	−0.114376	+	0.993438i	$0.536487\pi$
$812$	−1.29689e7	−0.690258
$813$	0	0
$814$	−1.43723e7	−0.760267
$815$	2.38615e7	1.25836
$816$	0	0
$817$	981457.	0.0514418
$818$	−7.65132e6	−0.399809
$819$	0	0
$820$	1.40879e7	0.731663
$821$	−2.03127e7	−1.05174	−0.525871	−	0.850564i	$-0.676261\pi$
$821$	−2.03127e7	−1.05174	−0.525871	+	0.850564i	$0.676261\pi$
$822$	0	0
$823$	−1.31663e7	−0.677583	−0.338792	−	0.940861i	$-0.610018\pi$
$823$	−1.31663e7	−0.677583	−0.338792	+	0.940861i	$0.610018\pi$
$824$	8.78213e6	0.450590
$825$	0	0
$826$	2.48508e6	0.126733
$827$	−2.82859e7	−1.43816	−0.719079	−	0.694928i	$-0.755436\pi$
$827$	−2.82859e7	−1.43816	−0.719079	+	0.694928i	$0.755436\pi$
$828$	0	0
$829$	1.36608e7	0.690381	0.345190	−	0.938533i	$-0.387814\pi$
$829$	1.36608e7	0.690381	0.345190	+	0.938533i	$0.387814\pi$
$830$	−9.54113e6	−0.480734
$831$	0	0
$832$	4.22942e6	0.211823
$833$	3.61316e6	0.180416
$834$	0	0
$835$	−7.69690e6	−0.382032
$836$	−2.86643e6	−0.141849
$837$	0	0
$838$	1.76558e7	0.868514
$839$	151283.	0.00741969	0.00370984	−	0.999993i	$-0.498819\pi$
$839$	151283.	0.00741969	0.00370984	+	0.999993i	$0.498819\pi$
$840$	0	0
$841$	2.98742e7	1.45649
$842$	6.97802e6	0.339197
$843$	0	0
$844$	−769574.	−0.0371873
$845$	4.59249e7	2.21262
$846$	0	0
$847$	−1.40965e6	−0.0675152
$848$	2.57132e6	0.122791
$849$	0	0
$850$	4.76621e6	0.226270
$851$	−2.14621e7	−1.01590
$852$	0	0
$853$	−2.77662e7	−1.30660	−0.653301	−	0.757098i	$-0.726616\pi$
$853$	−2.77662e7	−1.30660	−0.653301	+	0.757098i	$0.726616\pi$
$854$	−1.56161e7	−0.732705
$855$	0	0
$856$	−1.07507e7	−0.501477
$857$	3.52272e7	1.63842	0.819212	−	0.573491i	$-0.194411\pi$
$857$	3.52272e7	1.63842	0.819212	+	0.573491i	$0.194411\pi$
$858$	0	0
$859$	−4.04681e7	−1.87124	−0.935622	−	0.353005i	$-0.885160\pi$
$859$	−4.04681e7	−1.87124	−0.935622	+	0.353005i	$0.885160\pi$
$860$	−2.23382e6	−0.102992
$861$	0	0
$862$	−8.86958e6	−0.406570
$863$	1.96865e7	0.899789	0.449895	−	0.893082i	$-0.351462\pi$
$863$	1.96865e7	0.899789	0.449895	+	0.893082i	$0.351462\pi$
$864$	0	0
$865$	−6.84170e6	−0.310902
$866$	1.45404e7	0.658844
$867$	0	0
$868$	1.41835e7	0.638974
$869$	1.24162e7	0.557751
$870$	0	0
$871$	5.53307e7	2.47127
$872$	1.35834e6	0.0604945
$873$	0	0
$874$	−4.28043e6	−0.189544
$875$	−1.42065e7	−0.627287
$876$	0	0
$877$	2.62442e7	1.15222	0.576109	−	0.817373i	$-0.304570\pi$
$877$	2.62442e7	1.15222	0.576109	+	0.817373i	$0.304570\pi$
$878$	1.23531e7	0.540802
$879$	0	0
$880$	6.52408e6	0.283996
$881$	1.29423e7	0.561786	0.280893	−	0.959739i	$-0.409369\pi$
$881$	1.29423e7	0.561786	0.280893	+	0.959739i	$0.409369\pi$
$882$	0	0
$883$	3.39097e6	0.146360	0.0731800	−	0.997319i	$-0.476685\pi$
$883$	3.39097e6	0.146360	0.0731800	+	0.997319i	$0.476685\pi$
$884$	−1.58442e7	−0.681928
$885$	0	0
$886$	−2.80039e7	−1.19849
$887$	−7.67591e6	−0.327583	−0.163791	−	0.986495i	$-0.552372\pi$
$887$	−7.67591e6	−0.327583	−0.163791	+	0.986495i	$0.552372\pi$
$888$	0	0
$889$	−2.45580e7	−1.04217
$890$	1.12671e7	0.476799
$891$	0	0
$892$	1.25436e7	0.527851
$893$	1.15903e6	0.0486368
$894$	0	0
$895$	5.21697e6	0.217701
$896$	−1.87090e6	−0.0778537
$897$	0	0
$898$	−2.00078e7	−0.827959
$899$	−5.51042e7	−2.27398
$900$	0	0
$901$	−9.63260e6	−0.395304
$902$	−2.05511e7	−0.841045
$903$	0	0
$904$	−1.84586e6	−0.0751239
$905$	3.87340e7	1.57207
$906$	0	0
$907$	1.08887e7	0.439499	0.219750	−	0.975556i	$-0.429476\pi$
$907$	1.08887e7	0.439499	0.219750	+	0.975556i	$0.429476\pi$
$908$	8.89274e6	0.357949
$909$	0	0
$910$	−3.11692e7	−1.24773
$911$	180478.	0.00720492	0.00360246	−	0.999994i	$-0.498853\pi$
$911$	180478.	0.00720492	0.00360246	+	0.999994i	$0.498853\pi$
$912$	0	0
$913$	1.39184e7	0.552602
$914$	−1.42865e7	−0.565665
$915$	0	0
$916$	1.83508e6	0.0722630
$917$	2.16173e7	0.848942
$918$	0	0
$919$	−2.24831e7	−0.878148	−0.439074	−	0.898451i	$-0.644693\pi$
$919$	−2.24831e7	−0.878148	−0.439074	+	0.898451i	$0.644693\pi$
$920$	9.74239e6	0.379486
$921$	0	0
$922$	2.20049e7	0.852495
$923$	1.00161e6	0.0386984
$924$	0	0
$925$	1.15768e7	0.444871
$926$	1.14166e7	0.437533
$927$	0	0
$928$	7.26862e6	0.277065
$929$	4.13240e7	1.57095	0.785477	−	0.618891i	$-0.212418\pi$
$929$	4.13240e7	1.57095	0.785477	+	0.618891i	$0.212418\pi$
$930$	0	0
$931$	1.75032e6	0.0661824
$932$	1.77884e6	0.0670805
$933$	0	0
$934$	−2.86621e7	−1.07508
$935$	−2.44403e7	−0.914278
$936$	0	0
$937$	2.56470e7	0.954305	0.477153	−	0.878820i	$-0.341669\pi$
$937$	2.56470e7	0.954305	0.477153	+	0.878820i	$0.341669\pi$
$938$	−2.44757e7	−0.908296
$939$	0	0
$940$	−2.63798e6	−0.0973759
$941$	−1.79976e7	−0.662583	−0.331292	−	0.943528i	$-0.607484\pi$
$941$	−1.79976e7	−0.662583	−0.331292	+	0.943528i	$0.607484\pi$
$942$	0	0
$943$	−3.06889e7	−1.12383
$944$	−1.39281e6	−0.0508700
$945$	0	0
$946$	3.25866e6	0.118389
$947$	4.50796e7	1.63345	0.816723	−	0.577030i	$-0.195789\pi$
$947$	4.50796e7	1.63345	0.816723	+	0.577030i	$0.195789\pi$
$948$	0	0
$949$	−7.47587e7	−2.69461
$950$	2.30889e6	0.0830030
$951$	0	0
$952$	7.00869e6	0.250637
$953$	−2.21374e7	−0.789577	−0.394788	−	0.918772i	$-0.629182\pi$
$953$	−2.21374e7	−0.789577	−0.394788	+	0.918772i	$0.629182\pi$
$954$	0	0
$955$	−5.10225e7	−1.81031
$956$	1.18247e7	0.418453
$957$	0	0
$958$	1.31348e7	0.462392
$959$	413319.	0.0145124
$960$	0	0
$961$	3.16359e7	1.10502
$962$	−3.84843e7	−1.34075
$963$	0	0
$964$	−2.06458e7	−0.715547
$965$	−1.15251e6	−0.0398407
$966$	0	0
$967$	−1.96323e7	−0.675156	−0.337578	−	0.941297i	$-0.609608\pi$
$967$	−1.96323e7	−0.675156	−0.337578	+	0.941297i	$0.609608\pi$
$968$	790061.	0.0271002
$969$	0	0
$970$	1.15927e7	0.395598
$971$	3.46539e7	1.17952	0.589758	−	0.807580i	$-0.299223\pi$
$971$	3.46539e7	1.17952	0.589758	+	0.807580i	$0.299223\pi$
$972$	0	0
$973$	1.38034e7	0.467415
$974$	−8.72389e6	−0.294654
$975$	0	0
$976$	8.75234e6	0.294103
$977$	−2.68013e7	−0.898296	−0.449148	−	0.893457i	$-0.648272\pi$
$977$	−2.68013e7	−0.898296	−0.449148	+	0.893457i	$0.648272\pi$
$978$	0	0
$979$	−1.64362e7	−0.548080
$980$	−3.98377e6	−0.132504
$981$	0	0
$982$	−3.97579e7	−1.31566
$983$	2.38080e7	0.785849	0.392924	−	0.919571i	$-0.371463\pi$
$983$	2.38080e7	0.785849	0.392924	+	0.919571i	$0.371463\pi$
$984$	0	0
$985$	−2.25177e7	−0.739492
$986$	−2.72295e7	−0.891965
$987$	0	0
$988$	−7.67536e6	−0.250153
$989$	4.86614e6	0.158195
$990$	0	0
$991$	−2.37156e7	−0.767096	−0.383548	−	0.923521i	$-0.625298\pi$
$991$	−2.37156e7	−0.767096	−0.383548	+	0.923521i	$0.625298\pi$
$992$	−7.94937e6	−0.256480
$993$	0	0
$994$	−443063.	−0.0142233
$995$	941058.	0.0301341
$996$	0	0
$997$	−4.08340e7	−1.30102	−0.650511	−	0.759497i	$-0.725445\pi$
$997$	−4.08340e7	−1.30102	−0.650511	+	0.759497i	$0.725445\pi$
$998$	1.81109e7	0.575590
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.6.a.i.1.3		3
3.2	odd	2		162.6.a.j.1.1		3
9.2	odd	6		18.6.c.b.13.3	yes	6
9.4	even	3		54.6.c.b.19.1		6
9.5	odd	6		18.6.c.b.7.3	✓	6
9.7	even	3		54.6.c.b.37.1		6
36.7	odd	6		432.6.i.b.145.1		6
36.11	even	6		144.6.i.b.49.1		6
36.23	even	6		144.6.i.b.97.1		6
36.31	odd	6		432.6.i.b.289.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.6.c.b.7.3	✓	6	9.5	odd	6
18.6.c.b.13.3	yes	6	9.2	odd	6
54.6.c.b.19.1		6	9.4	even	3
54.6.c.b.37.1		6	9.7	even	3
144.6.i.b.49.1		6	36.11	even	6
144.6.i.b.97.1		6	36.23	even	6
162.6.a.i.1.3		3	1.1	even	1	trivial
162.6.a.j.1.1		3	3.2	odd	2
432.6.i.b.145.1		6	36.7	odd	6
432.6.i.b.289.1		6	36.31	odd	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +16.0000 q^{4} +66.0868 q^{5} +114.190 q^{7} -64.0000 q^{8} -264.347 q^{10} +385.625 q^{11} +1032.57 q^{13} -456.762 q^{14} +256.000 q^{16} -959.020 q^{17} -464.576 q^{19} +1057.39 q^{20} -1542.50 q^{22} -2303.41 q^{23} +1242.47 q^{25} -4130.30 q^{26} +1827.05 q^{28} -7098.27 q^{29} +7763.06 q^{31} -1024.00 q^{32} +3836.08 q^{34} +7546.48 q^{35} +9317.57 q^{37} +1858.31 q^{38} -4229.56 q^{40} +13323.3 q^{41} -2112.58 q^{43} +6169.99 q^{44} +9213.62 q^{46} -2494.80 q^{47} -3767.56 q^{49} -4969.88 q^{50} +16521.2 q^{52} +10044.2 q^{53} +25484.7 q^{55} -7308.18 q^{56} +28393.1 q^{58} -5440.66 q^{59} +34188.8 q^{61} -31052.2 q^{62} +4096.00 q^{64} +68239.6 q^{65} +53585.2 q^{67} -15344.3 q^{68} -30185.9 q^{70} +970.010 q^{71} -72400.3 q^{73} -37270.3 q^{74} -7433.22 q^{76} +44034.6 q^{77} +32197.7 q^{79} +16918.2 q^{80} -53293.1 q^{82} +36093.2 q^{83} -63378.6 q^{85} +8450.34 q^{86} -24680.0 q^{88} -42622.2 q^{89} +117910. q^{91} -36854.5 q^{92} +9979.21 q^{94} -30702.4 q^{95} -43853.9 q^{97} +15070.2 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 12 q^{2} + 48 q^{4} - 54 q^{5} + 132 q^{7} - 192 q^{8} + 216 q^{10} - 315 q^{11} + 744 q^{13} - 528 q^{14} + 768 q^{16} - 1449 q^{17} + 1131 q^{19} - 864 q^{20} + 1260 q^{22} - 3168 q^{23} + 2883 q^{25}+ \cdots - 211428 q^{98}+O(q^{100}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more