LMFDB - Embedded newform 729.6.a.e.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [729,6,Mod(1,729)] 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("729.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(729, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 729 = 3^{6} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	729.a (trivial)

Newform invariants

comment:select newform

sage:traces = [42,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$116.919804644$
Analytic rank:	$0$
Dimension:	$42$
Twist minimal:	no (minimal twist has level 27)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	729.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-8.08006 q^{2} +33.2873 q^{4} -19.8788 q^{5} +75.2303 q^{7} -10.4014 q^{8} +160.622 q^{10} -188.727 q^{11} -837.401 q^{13} -607.865 q^{14} -981.149 q^{16} -930.189 q^{17} -2157.41 q^{19} -661.710 q^{20} +1524.92 q^{22} +2365.71 q^{23} -2729.83 q^{25} +6766.25 q^{26} +2504.21 q^{28} +101.657 q^{29} -5828.75 q^{31} +8260.59 q^{32} +7515.98 q^{34} -1495.49 q^{35} +10781.4 q^{37} +17432.0 q^{38} +206.767 q^{40} -14385.2 q^{41} +669.024 q^{43} -6282.21 q^{44} -19115.1 q^{46} +28444.1 q^{47} -11147.4 q^{49} +22057.2 q^{50} -27874.8 q^{52} -25335.8 q^{53} +3751.66 q^{55} -782.501 q^{56} -821.393 q^{58} -15341.6 q^{59} +8962.60 q^{61} +47096.7 q^{62} -35349.2 q^{64} +16646.5 q^{65} -61245.0 q^{67} -30963.5 q^{68} +12083.6 q^{70} -5104.28 q^{71} +35391.7 q^{73} -87114.1 q^{74} -71814.2 q^{76} -14198.0 q^{77} -55002.7 q^{79} +19504.0 q^{80} +116233. q^{82} +95577.4 q^{83} +18491.0 q^{85} -5405.75 q^{86} +1963.03 q^{88} +100742. q^{89} -62997.9 q^{91} +78748.2 q^{92} -229830. q^{94} +42886.6 q^{95} +93632.6 q^{97} +90071.6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 42 q + 12 q^{2} + 624 q^{4} + 150 q^{5} + 573 q^{8} + 3 q^{10} + 1452 q^{11} + 2256 q^{14} + 8448 q^{16} + 3465 q^{17} + 3 q^{19} + 4128 q^{20} + 96 q^{22} + 5019 q^{23} + 18750 q^{25} + 3903 q^{26} - 6 q^{28}+ \cdots + 463410 q^{98}+O(q^{100}) $ 42 * q + 12 * q^2 + 624 * q^4 + 150 * q^5 + 573 * q^8 + 3 * q^10 + 1452 * q^11 + 2256 * q^14 + 8448 * q^16 + 3465 * q^17 + 3 * q^19 + 4128 * q^20 + 96 * q^22 + 5019 * q^23 + 18750 * q^25 + 3903 * q^26 - 6 * q^28 + 13008 * q^29 + 24273 * q^32 + 35868 * q^35 + 3 * q^37 + 51801 * q^38 + 96 * q^40 + 55833 * q^41 + 110757 * q^44 + 3 * q^46 + 90129 * q^47 + 57624 * q^49 + 145362 * q^50 + 3072 * q^52 + 103203 * q^53 - 6 * q^55 + 227154 * q^56 - 192 * q^58 + 176856 * q^59 - 31851 * q^61 + 246066 * q^62 + 86019 * q^64 + 167160 * q^65 - 801 * q^67 + 374589 * q^68 + 9375 * q^70 + 279531 * q^71 + 27012 * q^73 + 413970 * q^74 + 96 * q^76 + 185190 * q^77 + 462057 * q^80 - 6 * q^82 + 295536 * q^83 + 319803 * q^86 + 3072 * q^88 + 154827 * q^89 + 91002 * q^91 + 330558 * q^92 + 96 * q^94 + 353244 * q^95 + 463410 * 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 $

$2$	−8.08006	−1.42837	−0.714183	−	0.699959i	$-0.753201\pi$
$2$	−8.08006	−1.42837	−0.714183	+	0.699959i	$0.753201\pi$
$3$	0	0
$3$	0	0
$4$	33.2873	1.04023
$5$	−19.8788	−0.355602	−0.177801	−	0.984066i	$-0.556898\pi$
$5$	−19.8788	−0.355602	−0.177801	+	0.984066i	$0.556898\pi$
$6$	0	0
$7$	75.2303	0.580293	0.290147	−	0.956982i	$-0.406296\pi$
$7$	75.2303	0.580293	0.290147	+	0.956982i	$0.406296\pi$
$8$	−10.4014	−0.0574602
$9$	0	0
$10$	160.622	0.507930
$11$	−188.727	−0.470275	−0.235138	−	0.971962i	$-0.575554\pi$
$11$	−188.727	−0.470275	−0.235138	+	0.971962i	$0.575554\pi$
$12$	0	0
$13$	−837.401	−1.37428	−0.687140	−	0.726525i	$-0.741134\pi$
$13$	−837.401	−1.37428	−0.687140	+	0.726525i	$0.741134\pi$
$14$	−607.865	−0.828871
$15$	0	0
$16$	−981.149	−0.958154
$17$	−930.189	−0.780637	−0.390318	−	0.920680i	$-0.627635\pi$
$17$	−930.189	−0.780637	−0.390318	+	0.920680i	$0.627635\pi$
$18$	0	0
$19$	−2157.41	−1.37103	−0.685517	−	0.728057i	$-0.740424\pi$
$19$	−2157.41	−1.37103	−0.685517	+	0.728057i	$0.740424\pi$
$20$	−661.710	−0.369907
$21$	0	0
$22$	1524.92	0.671725
$23$	2365.71	0.932487	0.466243	−	0.884656i	$-0.345607\pi$
$23$	2365.71	0.932487	0.466243	+	0.884656i	$0.345607\pi$
$24$	0	0
$25$	−2729.83	−0.873547
$26$	6766.25	1.96297
$27$	0	0
$28$	2504.21	0.603637
$29$	101.657	0.0224461	0.0112231	−	0.999937i	$-0.496428\pi$
$29$	101.657	0.0224461	0.0112231	+	0.999937i	$0.496428\pi$
$30$	0	0
$31$	−5828.75	−1.08936	−0.544680	−	0.838644i	$-0.683349\pi$
$31$	−5828.75	−1.08936	−0.544680	+	0.838644i	$0.683349\pi$
$32$	8260.59	1.42605
$33$	0	0
$34$	7515.98	1.11503
$35$	−1495.49	−0.206354
$36$	0	0
$37$	10781.4	1.29470	0.647351	−	0.762192i	$-0.275877\pi$
$37$	10781.4	1.29470	0.647351	+	0.762192i	$0.275877\pi$
$38$	17432.0	1.95834
$39$	0	0
$40$	206.767	0.0204330
$41$	−14385.2	−1.33646	−0.668231	−	0.743954i	$-0.732948\pi$
$41$	−14385.2	−1.33646	−0.668231	+	0.743954i	$0.732948\pi$
$42$	0	0
$43$	669.024	0.0551786	0.0275893	−	0.999619i	$-0.491217\pi$
$43$	669.024	0.0551786	0.0275893	+	0.999619i	$0.491217\pi$
$44$	−6282.21	−0.489194
$45$	0	0
$46$	−19115.1	−1.33193
$47$	28444.1	1.87822	0.939111	−	0.343615i	$-0.111651\pi$
$47$	28444.1	1.87822	0.939111	+	0.343615i	$0.111651\pi$
$48$	0	0
$49$	−11147.4	−0.663260
$50$	22057.2	1.24774
$51$	0	0
$52$	−27874.8	−1.42956
$53$	−25335.8	−1.23892	−0.619461	−	0.785027i	$-0.712649\pi$
$53$	−25335.8	−1.23892	−0.619461	+	0.785027i	$0.712649\pi$
$54$	0	0
$55$	3751.66	0.167231
$56$	−782.501	−0.0333438
$57$	0	0
$58$	−821.393	−0.0320613
$59$	−15341.6	−0.573772	−0.286886	−	0.957965i	$-0.592620\pi$
$59$	−15341.6	−0.573772	−0.286886	+	0.957965i	$0.592620\pi$
$60$	0	0
$61$	8962.60	0.308397	0.154198	−	0.988040i	$-0.450721\pi$
$61$	8962.60	0.308397	0.154198	+	0.988040i	$0.450721\pi$
$62$	47096.7	1.55600
$63$	0	0
$64$	−35349.2	−1.07877
$65$	16646.5	0.488697
$66$	0	0
$67$	−61245.0	−1.66680	−0.833400	−	0.552671i	$-0.813609\pi$
$67$	−61245.0	−1.66680	−0.833400	+	0.552671i	$0.813609\pi$
$68$	−30963.5	−0.812040
$69$	0	0
$70$	12083.6	0.294748
$71$	−5104.28	−0.120168	−0.0600840	−	0.998193i	$-0.519137\pi$
$71$	−5104.28	−0.120168	−0.0600840	+	0.998193i	$0.519137\pi$
$72$	0	0
$73$	35391.7	0.777310	0.388655	−	0.921383i	$-0.372940\pi$
$73$	35391.7	0.777310	0.388655	+	0.921383i	$0.372940\pi$
$74$	−87114.1	−1.84931
$75$	0	0
$76$	−71814.2	−1.42619
$77$	−14198.0	−0.272898
$78$	0	0
$79$	−55002.7	−0.991554	−0.495777	−	0.868450i	$-0.665117\pi$
$79$	−55002.7	−0.991554	−0.495777	+	0.868450i	$0.665117\pi$
$80$	19504.0	0.340722
$81$	0	0
$82$	116233.	1.90896
$83$	95577.4	1.52286	0.761430	−	0.648247i	$-0.224498\pi$
$83$	95577.4	1.52286	0.761430	+	0.648247i	$0.224498\pi$
$84$	0	0
$85$	18491.0	0.277596
$86$	−5405.75	−0.0788151
$87$	0	0
$88$	1963.03	0.0270221
$89$	100742.	1.34814	0.674072	−	0.738666i	$-0.264544\pi$
$89$	100742.	1.34814	0.674072	+	0.738666i	$0.264544\pi$
$90$	0	0
$91$	−62997.9	−0.797485
$92$	78748.2	0.969999
$93$	0	0
$94$	−229830.	−2.68279
$95$	42886.6	0.487542
$96$	0	0
$97$	93632.6	1.01041	0.505205	−	0.862999i	$-0.331417\pi$
$97$	93632.6	1.01041	0.505205	+	0.862999i	$0.331417\pi$
$98$	90071.6	0.947377
$99$	0	0
$100$	−90868.8	−0.908688
$101$	−86230.6	−0.841120	−0.420560	−	0.907265i	$-0.638166\pi$
$101$	−86230.6	−0.841120	−0.420560	+	0.907265i	$0.638166\pi$
$102$	0	0
$103$	−119806.	−1.11272	−0.556361	−	0.830941i	$-0.687803\pi$
$103$	−119806.	−1.11272	−0.556361	+	0.830941i	$0.687803\pi$
$104$	8710.15	0.0789664
$105$	0	0
$106$	204714.	1.76963
$107$	−127097.	−1.07319	−0.536593	−	0.843841i	$-0.680289\pi$
$107$	−127097.	−1.07319	−0.536593	+	0.843841i	$0.680289\pi$
$108$	0	0
$109$	−112941.	−0.910511	−0.455256	−	0.890361i	$-0.650452\pi$
$109$	−112941.	−0.910511	−0.455256	+	0.890361i	$0.650452\pi$
$110$	−30313.6	−0.238867
$111$	0	0
$112$	−73812.2	−0.556010
$113$	93996.1	0.692490	0.346245	−	0.938144i	$-0.387457\pi$
$113$	93996.1	0.692490	0.346245	+	0.938144i	$0.387457\pi$
$114$	0	0
$115$	−47027.5	−0.331594
$116$	3383.88	0.0233491
$117$	0	0
$118$	123961.	0.819557
$119$	−69978.4	−0.452998
$120$	0	0
$121$	−125433.	−0.778841
$122$	−72418.3	−0.440503
$123$	0	0
$124$	−194023.	−1.13318
$125$	116387.	0.666238
$126$	0	0
$127$	46746.9	0.257184	0.128592	−	0.991698i	$-0.458954\pi$
$127$	46746.9	0.257184	0.128592	+	0.991698i	$0.458954\pi$
$128$	21284.8	0.114827
$129$	0	0
$130$	−134505.	−0.698038
$131$	−121299.	−0.617559	−0.308779	−	0.951134i	$-0.599920\pi$
$131$	−121299.	−0.617559	−0.308779	+	0.951134i	$0.599920\pi$
$132$	0	0
$133$	−162302.	−0.795601
$134$	494863.	2.38080
$135$	0	0
$136$	9675.27	0.0448555
$137$	−242183.	−1.10241	−0.551204	−	0.834370i	$-0.685831\pi$
$137$	−242183.	−1.10241	−0.551204	+	0.834370i	$0.685831\pi$
$138$	0	0
$139$	−338613.	−1.48651	−0.743254	−	0.669010i	$-0.766718\pi$
$139$	−338613.	−1.48651	−0.743254	+	0.669010i	$0.766718\pi$
$140$	−49780.7	−0.214655
$141$	0	0
$142$	41242.9	0.171644
$143$	158040.	0.646290
$144$	0	0
$145$	−2020.81	−0.00798189
$146$	−285967.	−1.11028
$147$	0	0
$148$	358883.	1.34679
$149$	76717.2	0.283092	0.141546	−	0.989932i	$-0.454793\pi$
$149$	76717.2	0.283092	0.141546	+	0.989932i	$0.454793\pi$
$150$	0	0
$151$	342014.	1.22068	0.610339	−	0.792140i	$-0.291033\pi$
$151$	342014.	1.22068	0.610339	+	0.792140i	$0.291033\pi$
$152$	22440.1	0.0787798
$153$	0	0
$154$	114721.	0.389798
$155$	115868.	0.387379
$156$	0	0
$157$	−44691.1	−0.144701	−0.0723506	−	0.997379i	$-0.523050\pi$
$157$	−44691.1	−0.144701	−0.0723506	+	0.997379i	$0.523050\pi$
$158$	444425.	1.41630
$159$	0	0
$160$	−164210.	−0.507108
$161$	177973.	0.541116
$162$	0	0
$163$	−102723.	−0.302831	−0.151415	−	0.988470i	$-0.548383\pi$
$163$	−102723.	−0.302831	−0.151415	+	0.988470i	$0.548383\pi$
$164$	−478845.	−1.39022
$165$	0	0
$166$	−772271.	−2.17520
$167$	51785.2	0.143686	0.0718430	−	0.997416i	$-0.477112\pi$
$167$	51785.2	0.143686	0.0718430	+	0.997416i	$0.477112\pi$
$168$	0	0
$169$	329947.	0.888644
$170$	−149408.	−0.396509
$171$	0	0
$172$	22270.0	0.0573983
$173$	−329392.	−0.836754	−0.418377	−	0.908274i	$-0.637401\pi$
$173$	−329392.	−0.836754	−0.418377	+	0.908274i	$0.637401\pi$
$174$	0	0
$175$	−205366.	−0.506914
$176$	185169.	0.450596
$177$	0	0
$178$	−814002.	−1.92564
$179$	325394.	0.759062	0.379531	−	0.925179i	$-0.376085\pi$
$179$	325394.	0.759062	0.379531	+	0.925179i	$0.376085\pi$
$180$	0	0
$181$	−226715.	−0.514379	−0.257189	−	0.966361i	$-0.582796\pi$
$181$	−226715.	−0.514379	−0.257189	+	0.966361i	$0.582796\pi$
$182$	509027.	1.13910
$183$	0	0
$184$	−24606.8	−0.0535809
$185$	−214321.	−0.460399
$186$	0	0
$187$	175552.	0.367114
$188$	946825.	1.95378
$189$	0	0
$190$	−346526.	−0.696389
$191$	409534.	0.812281	0.406140	−	0.913811i	$-0.366874\pi$
$191$	409534.	0.812281	0.406140	+	0.913811i	$0.366874\pi$
$192$	0	0
$193$	473119.	0.914275	0.457137	−	0.889396i	$-0.348875\pi$
$193$	473119.	0.914275	0.457137	+	0.889396i	$0.348875\pi$
$194$	−756556.	−1.44323
$195$	0	0
$196$	−371067.	−0.689941
$197$	372082.	0.683083	0.341542	−	0.939867i	$-0.389051\pi$
$197$	372082.	0.683083	0.341542	+	0.939867i	$0.389051\pi$
$198$	0	0
$199$	612667.	1.09671	0.548355	−	0.836246i	$-0.315254\pi$
$199$	612667.	1.09671	0.548355	+	0.836246i	$0.315254\pi$
$200$	28394.1	0.0501942
$201$	0	0
$202$	696748.	1.20143
$203$	7647.67	0.0130253
$204$	0	0
$205$	285960.	0.475249
$206$	968042.	1.58937
$207$	0	0
$208$	821615.	1.31677
$209$	407161.	0.644763
$210$	0	0
$211$	440687.	0.681435	0.340718	−	0.940166i	$-0.389330\pi$
$211$	440687.	0.681435	0.340718	+	0.940166i	$0.389330\pi$
$212$	−843359.	−1.28876
$213$	0	0
$214$	1.02695e6	1.53290
$215$	−13299.4	−0.0196216
$216$	0	0
$217$	−438499.	−0.632148
$218$	912569.	1.30054
$219$	0	0
$220$	124883.	0.173958
$221$	778941.	1.07281
$222$	0	0
$223$	−978312.	−1.31739	−0.658696	−	0.752409i	$-0.728892\pi$
$223$	−978312.	−1.31739	−0.658696	+	0.752409i	$0.728892\pi$
$224$	621446.	0.827530
$225$	0	0
$226$	−759494.	−0.989129
$227$	1.10134e6	1.41859	0.709294	−	0.704913i	$-0.249014\pi$
$227$	1.10134e6	1.41859	0.709294	+	0.704913i	$0.249014\pi$
$228$	0	0
$229$	−95039.8	−0.119761	−0.0598807	−	0.998206i	$-0.519072\pi$
$229$	−95039.8	−0.119761	−0.0598807	+	0.998206i	$0.519072\pi$
$230$	379985.	0.473638
$231$	0	0
$232$	−1057.37	−0.00128976
$233$	1.07694e6	1.29958	0.649790	−	0.760114i	$-0.274857\pi$
$233$	1.07694e6	1.29958	0.649790	+	0.760114i	$0.274857\pi$
$234$	0	0
$235$	−565433.	−0.667900
$236$	−510679.	−0.596854
$237$	0	0
$238$	565429.	0.647047
$239$	−1.25755e6	−1.42407	−0.712035	−	0.702144i	$-0.752226\pi$
$239$	−1.25755e6	−1.42407	−0.712035	+	0.702144i	$0.752226\pi$
$240$	0	0
$241$	646482.	0.716991	0.358495	−	0.933531i	$-0.383290\pi$
$241$	646482.	0.716991	0.358495	+	0.933531i	$0.383290\pi$
$242$	1.01351e6	1.11247
$243$	0	0
$244$	298341.	0.320803
$245$	221597.	0.235857
$246$	0	0
$247$	1.80661e6	1.88418
$248$	60627.2	0.0625948
$249$	0	0
$250$	−940413.	−0.951631
$251$	46905.0	0.0469932	0.0234966	−	0.999724i	$-0.492520\pi$
$251$	46905.0	0.0469932	0.0234966	+	0.999724i	$0.492520\pi$
$252$	0	0
$253$	−446474.	−0.438526
$254$	−377718.	−0.367353
$255$	0	0
$256$	959192.	0.914757
$257$	609152.	0.575298	0.287649	−	0.957736i	$-0.407126\pi$
$257$	609152.	0.575298	0.287649	+	0.957736i	$0.407126\pi$
$258$	0	0
$259$	811086.	0.751307
$260$	554117.	0.508356
$261$	0	0
$262$	980101.	0.882100
$263$	−817175.	−0.728493	−0.364247	−	0.931303i	$-0.618673\pi$
$263$	−817175.	−0.728493	−0.364247	+	0.931303i	$0.618673\pi$
$264$	0	0
$265$	503644.	0.440564
$266$	1.31141e6	1.13641
$267$	0	0
$268$	−2.03868e6	−1.73385
$269$	−1.06427e6	−0.896753	−0.448376	−	0.893845i	$-0.647998\pi$
$269$	−1.06427e6	−0.896753	−0.448376	+	0.893845i	$0.647998\pi$
$270$	0	0
$271$	1.32722e6	1.09779	0.548894	−	0.835892i	$-0.315049\pi$
$271$	1.32722e6	1.09779	0.548894	+	0.835892i	$0.315049\pi$
$272$	912654.	0.747970
$273$	0	0
$274$	1.95685e6	1.57464
$275$	515193.	0.410808
$276$	0	0
$277$	1.86419e6	1.45979	0.729895	−	0.683559i	$-0.239569\pi$
$277$	1.86419e6	1.45979	0.729895	+	0.683559i	$0.239569\pi$
$278$	2.73601e6	2.12328
$279$	0	0
$280$	15555.1	0.0118571
$281$	453541.	0.342650	0.171325	−	0.985215i	$-0.445195\pi$
$281$	453541.	0.342650	0.171325	+	0.985215i	$0.445195\pi$
$282$	0	0
$283$	157416.	0.116838	0.0584188	−	0.998292i	$-0.481394\pi$
$283$	157416.	0.116838	0.0584188	+	0.998292i	$0.481394\pi$
$284$	−169908.	−0.125002
$285$	0	0
$286$	−1.27697e6	−0.923138
$287$	−1.08220e6	−0.775540
$288$	0	0
$289$	−554605.	−0.390607
$290$	16328.3	0.0114011
$291$	0	0
$292$	1.17809e6	0.808579
$293$	2.82928e6	1.92534	0.962670	−	0.270679i	$-0.0872481\pi$
$293$	2.82928e6	1.92534	0.962670	+	0.270679i	$0.0872481\pi$
$294$	0	0
$295$	304971.	0.204035
$296$	−112141.	−0.0743938
$297$	0	0
$298$	−619879.	−0.404358
$299$	−1.98105e6	−1.28150
$300$	0	0
$301$	50330.8	0.0320197
$302$	−2.76349e6	−1.74358
$303$	0	0
$304$	2.11674e6	1.31366
$305$	−178166.	−0.109667
$306$	0	0
$307$	−124495.	−0.0753885	−0.0376943	−	0.999289i	$-0.512001\pi$
$307$	−124495.	−0.0753885	−0.0376943	+	0.999289i	$0.512001\pi$
$308$	−472613.	−0.283876
$309$	0	0
$310$	−936224.	−0.553319
$311$	−2.01606e6	−1.18196	−0.590979	−	0.806687i	$-0.701259\pi$
$311$	−2.01606e6	−1.18196	−0.590979	+	0.806687i	$0.701259\pi$
$312$	0	0
$313$	2.35273e6	1.35741	0.678704	−	0.734412i	$-0.262542\pi$
$313$	2.35273e6	1.35741	0.678704	+	0.734412i	$0.262542\pi$
$314$	361106.	0.206686
$315$	0	0
$316$	−1.83089e6	−1.03144
$317$	560966.	0.313537	0.156768	−	0.987635i	$-0.449892\pi$
$317$	560966.	0.313537	0.156768	+	0.987635i	$0.449892\pi$
$318$	0	0
$319$	−19185.4	−0.0105559
$320$	702699.	0.383614
$321$	0	0
$322$	−1.43803e6	−0.772911
$323$	2.00680e6	1.07028
$324$	0	0
$325$	2.28597e6	1.20050
$326$	830010.	0.432553
$327$	0	0
$328$	149626.	0.0767933
$329$	2.13985e6	1.08992
$330$	0	0
$331$	1.88017e6	0.943250	0.471625	−	0.881799i	$-0.343668\pi$
$331$	1.88017e6	0.943250	0.471625	+	0.881799i	$0.343668\pi$
$332$	3.18151e6	1.58412
$333$	0	0
$334$	−418428.	−0.205236
$335$	1.21747e6	0.592718
$336$	0	0
$337$	3.99555e6	1.91647	0.958234	−	0.285987i	$-0.0923214\pi$
$337$	3.99555e6	1.91647	0.958234	+	0.285987i	$0.0923214\pi$
$338$	−2.66599e6	−1.26931
$339$	0	0
$340$	615516.	0.288763
$341$	1.10004e6	0.512299
$342$	0	0
$343$	−2.10302e6	−0.965178
$344$	−6958.79	−0.00317057
$345$	0	0
$346$	2.66151e6	1.19519
$347$	−1.34610e6	−0.600142	−0.300071	−	0.953917i	$-0.597010\pi$
$347$	−1.34610e6	−0.600142	−0.300071	+	0.953917i	$0.597010\pi$
$348$	0	0
$349$	−2.63720e6	−1.15899	−0.579495	−	0.814975i	$-0.696750\pi$
$349$	−2.63720e6	−1.15899	−0.579495	+	0.814975i	$0.696750\pi$
$350$	1.65937e6	0.724058
$351$	0	0
$352$	−1.55900e6	−0.670638
$353$	−4.49432e6	−1.91967	−0.959836	−	0.280560i	$-0.909480\pi$
$353$	−4.49432e6	−1.91967	−0.959836	+	0.280560i	$0.909480\pi$
$354$	0	0
$355$	101467.	0.0427320
$356$	3.35343e6	1.40238
$357$	0	0
$358$	−2.62920e6	−1.08422
$359$	−1.05112e6	−0.430444	−0.215222	−	0.976565i	$-0.569048\pi$
$359$	−1.05112e6	−0.430444	−0.215222	+	0.976565i	$0.569048\pi$
$360$	0	0
$361$	2.17830e6	0.879732
$362$	1.83187e6	0.734721
$363$	0	0
$364$	−2.09703e6	−0.829566
$365$	−703543.	−0.276413
$366$	0	0
$367$	−1.71297e6	−0.663871	−0.331935	−	0.943302i	$-0.607702\pi$
$367$	−1.71297e6	−0.663871	−0.331935	+	0.943302i	$0.607702\pi$
$368$	−2.32112e6	−0.893466
$369$	0	0
$370$	1.73172e6	0.657618
$371$	−1.90602e6	−0.718939
$372$	0	0
$373$	−4.79985e6	−1.78631	−0.893153	−	0.449753i	$-0.851512\pi$
$373$	−4.79985e6	−1.78631	−0.893153	+	0.449753i	$0.851512\pi$
$374$	−1.41847e6	−0.524373
$375$	0	0
$376$	−295858.	−0.107923
$377$	−85127.5	−0.0308473
$378$	0	0
$379$	1.68242e6	0.601640	0.300820	−	0.953681i	$-0.402740\pi$
$379$	1.68242e6	0.601640	0.300820	+	0.953681i	$0.402740\pi$
$380$	1.42758e6	0.507155
$381$	0	0
$382$	−3.30905e6	−1.16023
$383$	−488528.	−0.170174	−0.0850869	−	0.996374i	$-0.527117\pi$
$383$	−488528.	−0.170174	−0.0850869	+	0.996374i	$0.527117\pi$
$384$	0	0
$385$	282238.	0.0970430
$386$	−3.82282e6	−1.30592
$387$	0	0
$388$	3.11677e6	1.05106
$389$	−2.21813e6	−0.743211	−0.371606	−	0.928391i	$-0.621193\pi$
$389$	−2.21813e6	−0.743211	−0.371606	+	0.928391i	$0.621193\pi$
$390$	0	0
$391$	−2.20056e6	−0.727933
$392$	115949.	0.0381110
$393$	0	0
$394$	−3.00645e6	−0.975692
$395$	1.09339e6	0.352599
$396$	0	0
$397$	−1.14498e6	−0.364605	−0.182303	−	0.983242i	$-0.558355\pi$
$397$	−1.14498e6	−0.364605	−0.182303	+	0.983242i	$0.558355\pi$
$398$	−4.95038e6	−1.56650
$399$	0	0
$400$	2.67838e6	0.836992
$401$	2.35583e6	0.731614	0.365807	−	0.930691i	$-0.380793\pi$
$401$	2.35583e6	0.731614	0.365807	+	0.930691i	$0.380793\pi$
$402$	0	0
$403$	4.88100e6	1.49709
$404$	−2.87038e6	−0.874956
$405$	0	0
$406$	−61793.6	−0.0186049
$407$	−2.03474e6	−0.608867
$408$	0	0
$409$	4.22519e6	1.24893	0.624465	−	0.781053i	$-0.285317\pi$
$409$	4.22519e6	1.24893	0.624465	+	0.781053i	$0.285317\pi$
$410$	−2.31057e6	−0.678829
$411$	0	0
$412$	−3.98803e6	−1.15748
$413$	−1.15415e6	−0.332956
$414$	0	0
$415$	−1.89996e6	−0.541533
$416$	−6.91742e6	−1.95980
$417$	0	0
$418$	−3.28988e6	−0.920958
$419$	1.97909e6	0.550720	0.275360	−	0.961341i	$-0.411203\pi$
$419$	1.97909e6	0.550720	0.275360	+	0.961341i	$0.411203\pi$
$420$	0	0
$421$	2.09756e6	0.576780	0.288390	−	0.957513i	$-0.406880\pi$
$421$	2.09756e6	0.576780	0.288390	+	0.957513i	$0.406880\pi$
$422$	−3.56078e6	−0.973338
$423$	0	0
$424$	263528.	0.0711887
$425$	2.53926e6	0.681923
$426$	0	0
$427$	674259.	0.178961
$428$	−4.23071e6	−1.11636
$429$	0	0
$430$	107460.	0.0280268
$431$	−4.31651e6	−1.11928	−0.559641	−	0.828735i	$-0.689061\pi$
$431$	−4.31651e6	−1.11928	−0.559641	+	0.828735i	$0.689061\pi$
$432$	0	0
$433$	−4.01768e6	−1.02981	−0.514903	−	0.857249i	$-0.672172\pi$
$433$	−4.01768e6	−1.02981	−0.514903	+	0.857249i	$0.672172\pi$
$434$	3.54309e6	0.902939
$435$	0	0
$436$	−3.75950e6	−0.947139
$437$	−5.10381e6	−1.27847
$438$	0	0
$439$	5.11191e6	1.26597	0.632983	−	0.774166i	$-0.281830\pi$
$439$	5.11191e6	1.26597	0.632983	+	0.774166i	$0.281830\pi$
$440$	−39022.5	−0.00960912
$441$	0	0
$442$	−6.29389e6	−1.53237
$443$	2.47932e6	0.600237	0.300119	−	0.953902i	$-0.402974\pi$
$443$	2.47932e6	0.600237	0.300119	+	0.953902i	$0.402974\pi$
$444$	0	0
$445$	−2.00263e6	−0.479403
$446$	7.90482e6	1.88172
$447$	0	0
$448$	−2.65933e6	−0.626004
$449$	3.76384e6	0.881080	0.440540	−	0.897733i	$-0.354787\pi$
$449$	3.76384e6	0.881080	0.440540	+	0.897733i	$0.354787\pi$
$450$	0	0
$451$	2.71488e6	0.628505
$452$	3.12888e6	0.720348
$453$	0	0
$454$	−8.89888e6	−2.02626
$455$	1.25232e6	0.283588
$456$	0	0
$457$	4.58365e6	1.02665	0.513323	−	0.858195i	$-0.328414\pi$
$457$	4.58365e6	1.02665	0.513323	+	0.858195i	$0.328414\pi$
$458$	767927.	0.171063
$459$	0	0
$460$	−1.56542e6	−0.344934
$461$	7.02513e6	1.53958	0.769790	−	0.638298i	$-0.220361\pi$
$461$	7.02513e6	1.53958	0.769790	+	0.638298i	$0.220361\pi$
$462$	0	0
$463$	−6.97639e6	−1.51244	−0.756220	−	0.654317i	$-0.772956\pi$
$463$	−6.97639e6	−1.51244	−0.756220	+	0.654317i	$0.772956\pi$
$464$	−99740.5	−0.0215068
$465$	0	0
$466$	−8.70176e6	−1.85628
$467$	622520.	0.132087	0.0660437	−	0.997817i	$-0.478962\pi$
$467$	622520.	0.132087	0.0660437	+	0.997817i	$0.478962\pi$
$468$	0	0
$469$	−4.60748e6	−0.967233
$470$	4.56873e6	0.954005
$471$	0	0
$472$	159574.	0.0329691
$473$	−126263.	−0.0259491
$474$	0	0
$475$	5.88936e6	1.19766
$476$	−2.32939e6	−0.471221
$477$	0	0
$478$	1.01611e7	2.03409
$479$	2.84674e6	0.566903	0.283452	−	0.958987i	$-0.408520\pi$
$479$	2.84674e6	0.566903	0.283452	+	0.958987i	$0.408520\pi$
$480$	0	0
$481$	−9.02834e6	−1.77928
$482$	−5.22361e6	−1.02412
$483$	0	0
$484$	−4.17533e6	−0.810172
$485$	−1.86130e6	−0.359304
$486$	0	0
$487$	3.37660e6	0.645146	0.322573	−	0.946545i	$-0.395452\pi$
$487$	3.37660e6	0.645146	0.322573	+	0.946545i	$0.395452\pi$
$488$	−93223.7	−0.0177205
$489$	0	0
$490$	−1.79051e6	−0.336889
$491$	6.92796e6	1.29689	0.648443	−	0.761263i	$-0.275421\pi$
$491$	6.92796e6	1.29689	0.648443	+	0.761263i	$0.275421\pi$
$492$	0	0
$493$	−94560.0	−0.0175223
$494$	−1.45975e7	−2.69130
$495$	0	0
$496$	5.71888e6	1.04377
$497$	−383996.	−0.0697326
$498$	0	0
$499$	−2.56810e6	−0.461700	−0.230850	−	0.972989i	$-0.574151\pi$
$499$	−2.56810e6	−0.461700	−0.230850	+	0.972989i	$0.574151\pi$
$500$	3.87421e6	0.693039
$501$	0	0
$502$	−378995.	−0.0671234
$503$	−2.64801e6	−0.466659	−0.233330	−	0.972398i	$-0.574962\pi$
$503$	−2.64801e6	−0.466659	−0.233330	+	0.972398i	$0.574962\pi$
$504$	0	0
$505$	1.71416e6	0.299104
$506$	3.60754e6	0.626375
$507$	0	0
$508$	1.55608e6	0.267530
$509$	−1.32297e6	−0.226337	−0.113169	−	0.993576i	$-0.536100\pi$
$509$	−1.32297e6	−0.226337	−0.113169	+	0.993576i	$0.536100\pi$
$510$	0	0
$511$	2.66253e6	0.451068
$512$	−8.43144e6	−1.42143
$513$	0	0
$514$	−4.92198e6	−0.821736
$515$	2.38160e6	0.395686
$516$	0	0
$517$	−5.36816e6	−0.883281
$518$	−6.55362e6	−1.07314
$519$	0	0
$520$	−173147.	−0.0280806
$521$	3.30481e6	0.533399	0.266699	−	0.963780i	$-0.414067\pi$
$521$	3.30481e6	0.533399	0.266699	+	0.963780i	$0.414067\pi$
$522$	0	0
$523$	7.01628e6	1.12164	0.560819	−	0.827939i	$-0.310487\pi$
$523$	7.01628e6	1.12164	0.560819	+	0.827939i	$0.310487\pi$
$524$	−4.03771e6	−0.642402
$525$	0	0
$526$	6.60282e6	1.04055
$527$	5.42184e6	0.850394
$528$	0	0
$529$	−839739.	−0.130468
$530$	−4.06947e6	−0.629286
$531$	0	0
$532$	−5.40260e6	−0.827607
$533$	1.20462e7	1.83667
$534$	0	0
$535$	2.52653e6	0.381628
$536$	637034.	0.0957746
$537$	0	0
$538$	8.59939e6	1.28089
$539$	2.10382e6	0.311915
$540$	0	0
$541$	−5.60437e6	−0.823254	−0.411627	−	0.911352i	$-0.635039\pi$
$541$	−5.60437e6	−0.823254	−0.411627	+	0.911352i	$0.635039\pi$
$542$	−1.07240e7	−1.56804
$543$	0	0
$544$	−7.68391e6	−1.11323
$545$	2.24513e6	0.323780
$546$	0	0
$547$	−1.37033e6	−0.195819	−0.0979096	−	0.995195i	$-0.531216\pi$
$547$	−1.37033e6	−0.195819	−0.0979096	+	0.995195i	$0.531216\pi$
$548$	−8.06162e6	−1.14676
$549$	0	0
$550$	−4.16279e6	−0.586784
$551$	−219315.	−0.0307744
$552$	0	0
$553$	−4.13787e6	−0.575392
$554$	−1.50627e7	−2.08511
$555$	0	0
$556$	−1.12715e7	−1.54631
$557$	−2.14503e6	−0.292951	−0.146476	−	0.989214i	$-0.546793\pi$
$557$	−2.14503e6	−0.292951	−0.146476	+	0.989214i	$0.546793\pi$
$558$	0	0
$559$	−560241.	−0.0758308
$560$	1.46729e6	0.197719
$561$	0	0
$562$	−3.66464e6	−0.489430
$563$	1.29272e7	1.71883	0.859415	−	0.511279i	$-0.170828\pi$
$563$	1.29272e7	1.71883	0.859415	+	0.511279i	$0.170828\pi$
$564$	0	0
$565$	−1.86853e6	−0.246251
$566$	−1.27193e6	−0.166887
$567$	0	0
$568$	53091.7	0.00690487
$569$	−1.89376e6	−0.245214	−0.122607	−	0.992455i	$-0.539125\pi$
$569$	−1.89376e6	−0.245214	−0.122607	+	0.992455i	$0.539125\pi$
$570$	0	0
$571$	−5.54404e6	−0.711600	−0.355800	−	0.934562i	$-0.615792\pi$
$571$	−5.54404e6	−0.711600	−0.355800	+	0.934562i	$0.615792\pi$
$572$	5.26073e6	0.672289
$573$	0	0
$574$	8.74426e6	1.10775
$575$	−6.45801e6	−0.814571
$576$	0	0
$577$	7.90126e6	0.988000	0.494000	−	0.869462i	$-0.335534\pi$
$577$	7.90126e6	0.988000	0.494000	+	0.869462i	$0.335534\pi$
$578$	4.48124e6	0.557929
$579$	0	0
$580$	−67267.4	−0.00830299
$581$	7.19032e6	0.883706
$582$	0	0
$583$	4.78154e6	0.582635
$584$	−368123.	−0.0446644
$585$	0	0
$586$	−2.28608e7	−2.75009
$587$	−1.54646e7	−1.85243	−0.926216	−	0.376993i	$-0.876958\pi$
$587$	−1.54646e7	−1.85243	−0.926216	+	0.376993i	$0.876958\pi$
$588$	0	0
$589$	1.25750e7	1.49355
$590$	−2.46419e6	−0.291436
$591$	0	0
$592$	−1.05781e7	−1.24052
$593$	6.33686e6	0.740009	0.370005	−	0.929030i	$-0.379356\pi$
$593$	6.33686e6	0.740009	0.370005	+	0.929030i	$0.379356\pi$
$594$	0	0
$595$	1.39108e6	0.161087
$596$	2.55371e6	0.294480
$597$	0	0
$598$	1.60070e7	1.83045
$599$	7.43389e6	0.846543	0.423272	−	0.906003i	$-0.360882\pi$
$599$	7.43389e6	0.846543	0.423272	+	0.906003i	$0.360882\pi$
$600$	0	0
$601$	1.11862e7	1.26327	0.631636	−	0.775265i	$-0.282384\pi$
$601$	1.11862e7	1.26327	0.631636	+	0.775265i	$0.282384\pi$
$602$	−406676.	−0.0457359
$603$	0	0
$604$	1.13847e7	1.26978
$605$	2.49346e6	0.276958
$606$	0	0
$607$	−226968.	−0.0250031	−0.0125015	−	0.999922i	$-0.503979\pi$
$607$	−226968.	−0.0250031	−0.0125015	+	0.999922i	$0.503979\pi$
$608$	−1.78214e7	−1.95517
$609$	0	0
$610$	1.43959e6	0.156644
$611$	−2.38191e7	−2.58120
$612$	0	0
$613$	9.81295e6	1.05475	0.527373	−	0.849634i	$-0.323177\pi$
$613$	9.81295e6	1.05475	0.527373	+	0.849634i	$0.323177\pi$
$614$	1.00593e6	0.107682
$615$	0	0
$616$	147679.	0.0156808
$617$	−1.02365e7	−1.08253	−0.541263	−	0.840853i	$-0.682054\pi$
$617$	−1.02365e7	−1.08253	−0.541263	+	0.840853i	$0.682054\pi$
$618$	0	0
$619$	2.70809e6	0.284077	0.142038	−	0.989861i	$-0.454634\pi$
$619$	2.70809e6	0.284077	0.142038	+	0.989861i	$0.454634\pi$
$620$	3.85695e6	0.402962
$621$	0	0
$622$	1.62899e7	1.68827
$623$	7.57886e6	0.782319
$624$	0	0
$625$	6.21710e6	0.636631
$626$	−1.90101e7	−1.93887
$627$	0	0
$628$	−1.48765e6	−0.150522
$629$	−1.00287e7	−1.01069
$630$	0	0
$631$	−1.32666e7	−1.32644	−0.663219	−	0.748426i	$-0.730810\pi$
$631$	−1.32666e7	−1.32644	−0.663219	+	0.748426i	$0.730810\pi$
$632$	572106.	0.0569749
$633$	0	0
$634$	−4.53264e6	−0.447845
$635$	−929272.	−0.0914552
$636$	0	0
$637$	9.33485e6	0.911504
$638$	155019.	0.0150776
$639$	0	0
$640$	−423116.	−0.0408329
$641$	−1.64583e7	−1.58212	−0.791062	−	0.611736i	$-0.790472\pi$
$641$	−1.64583e7	−1.58212	−0.791062	+	0.611736i	$0.790472\pi$
$642$	0	0
$643$	8.89619e6	0.848549	0.424274	−	0.905534i	$-0.360529\pi$
$643$	8.89619e6	0.848549	0.424274	+	0.905534i	$0.360529\pi$
$644$	5.92425e6	0.562884
$645$	0	0
$646$	−1.62150e7	−1.52875
$647$	−8.09399e6	−0.760154	−0.380077	−	0.924955i	$-0.624103\pi$
$647$	−8.09399e6	−0.760154	−0.380077	+	0.924955i	$0.624103\pi$
$648$	0	0
$649$	2.89537e6	0.269831
$650$	−1.84707e7	−1.71475
$651$	0	0
$652$	−3.41938e6	−0.315013
$653$	9.10846e6	0.835914	0.417957	−	0.908467i	$-0.362746\pi$
$653$	9.10846e6	0.835914	0.417957	+	0.908467i	$0.362746\pi$
$654$	0	0
$655$	2.41127e6	0.219605
$656$	1.41140e7	1.28054
$657$	0	0
$658$	−1.72901e7	−1.55680
$659$	−9.01616e6	−0.808739	−0.404369	−	0.914596i	$-0.632509\pi$
$659$	−9.01616e6	−0.808739	−0.404369	+	0.914596i	$0.632509\pi$
$660$	0	0
$661$	1.02778e7	0.914944	0.457472	−	0.889224i	$-0.348755\pi$
$661$	1.02778e7	0.914944	0.457472	+	0.889224i	$0.348755\pi$
$662$	−1.51919e7	−1.34731
$663$	0	0
$664$	−994139.	−0.0875038
$665$	3.22637e6	0.282918
$666$	0	0
$667$	240491.	0.0209307
$668$	1.72379e6	0.149466
$669$	0	0
$670$	−9.83726e6	−0.846618
$671$	−1.69149e6	−0.145031
$672$	0	0
$673$	3.36561e6	0.286435	0.143217	−	0.989691i	$-0.454255\pi$
$673$	3.36561e6	0.286435	0.143217	+	0.989691i	$0.454255\pi$
$674$	−3.22842e7	−2.73742
$675$	0	0
$676$	1.09831e7	0.924392
$677$	7.88871e6	0.661507	0.330753	−	0.943717i	$-0.392697\pi$
$677$	7.88871e6	0.661507	0.330753	+	0.943717i	$0.392697\pi$
$678$	0	0
$679$	7.04400e6	0.586334
$680$	−192333.	−0.0159507
$681$	0	0
$682$	−8.88841e6	−0.731751
$683$	1.11301e7	0.912952	0.456476	−	0.889736i	$-0.349111\pi$
$683$	1.11301e7	0.912952	0.456476	+	0.889736i	$0.349111\pi$
$684$	0	0
$685$	4.81430e6	0.392019
$686$	1.69925e7	1.37863
$687$	0	0
$688$	−656412.	−0.0528695
$689$	2.12162e7	1.70263
$690$	0	0
$691$	−7.05092e6	−0.561760	−0.280880	−	0.959743i	$-0.590626\pi$
$691$	−7.05092e6	−0.561760	−0.280880	+	0.959743i	$0.590626\pi$
$692$	−1.09646e7	−0.870415
$693$	0	0
$694$	1.08766e7	0.857222
$695$	6.73122e6	0.528605
$696$	0	0
$697$	1.33810e7	1.04329
$698$	2.13087e7	1.65546
$699$	0	0
$700$	−6.83609e6	−0.527306
$701$	−7.27260e6	−0.558978	−0.279489	−	0.960149i	$-0.590165\pi$
$701$	−7.27260e6	−0.558978	−0.279489	+	0.960149i	$0.590165\pi$
$702$	0	0
$703$	−2.32598e7	−1.77508
$704$	6.67135e6	0.507320
$705$	0	0
$706$	3.63144e7	2.74199
$707$	−6.48715e6	−0.488096
$708$	0	0
$709$	−1.96896e7	−1.47103	−0.735516	−	0.677508i	$-0.763060\pi$
$709$	−1.96896e7	−1.47103	−0.735516	+	0.677508i	$0.763060\pi$
$710$	−819857.	−0.0610369
$711$	0	0
$712$	−1.04786e6	−0.0774646
$713$	−1.37892e7	−1.01581
$714$	0	0
$715$	−3.14164e6	−0.229822
$716$	1.08315e7	0.789598
$717$	0	0
$718$	8.49312e6	0.614831
$719$	7.95574e6	0.573929	0.286965	−	0.957941i	$-0.407354\pi$
$719$	7.95574e6	0.573929	0.286965	+	0.957941i	$0.407354\pi$
$720$	0	0
$721$	−9.01306e6	−0.645705
$722$	−1.76008e7	−1.25658
$723$	0	0
$724$	−7.54671e6	−0.535071
$725$	−277506.	−0.0196077
$726$	0	0
$727$	6.36487e6	0.446636	0.223318	−	0.974746i	$-0.428311\pi$
$727$	6.36487e6	0.446636	0.223318	+	0.974746i	$0.428311\pi$
$728$	655267.	0.0458236
$729$	0	0
$730$	5.68467e6	0.394819
$731$	−622318.	−0.0430744
$732$	0	0
$733$	2.56360e7	1.76234	0.881171	−	0.472797i	$-0.156756\pi$
$733$	2.56360e7	1.76234	0.881171	+	0.472797i	$0.156756\pi$
$734$	1.38409e7	0.948250
$735$	0	0
$736$	1.95422e7	1.32978
$737$	1.15586e7	0.783855
$738$	0	0
$739$	−6.62218e6	−0.446057	−0.223028	−	0.974812i	$-0.571594\pi$
$739$	−6.62218e6	−0.446057	−0.223028	+	0.974812i	$0.571594\pi$
$740$	−7.13415e6	−0.478920
$741$	0	0
$742$	1.54007e7	1.02691
$743$	359954.	0.0239207	0.0119604	−	0.999928i	$-0.496193\pi$
$743$	359954.	0.0239207	0.0119604	+	0.999928i	$0.496193\pi$
$744$	0	0
$745$	−1.52504e6	−0.100668
$746$	3.87831e7	2.55150
$747$	0	0
$748$	5.84364e6	0.381882
$749$	−9.56153e6	−0.622763
$750$	0	0
$751$	5.19498e6	0.336112	0.168056	−	0.985777i	$-0.446251\pi$
$751$	5.19498e6	0.336112	0.168056	+	0.985777i	$0.446251\pi$
$752$	−2.79079e7	−1.79963
$753$	0	0
$754$	687835.	0.0440612
$755$	−6.79881e6	−0.434076
$756$	0	0
$757$	6.85707e6	0.434909	0.217455	−	0.976070i	$-0.430225\pi$
$757$	6.85707e6	0.434909	0.217455	+	0.976070i	$0.430225\pi$
$758$	−1.35941e7	−0.859362
$759$	0	0
$760$	−446081.	−0.0280143
$761$	−1.13668e7	−0.711502	−0.355751	−	0.934581i	$-0.615775\pi$
$761$	−1.13668e7	−0.711502	−0.355751	+	0.934581i	$0.615775\pi$
$762$	0	0
$763$	−8.49658e6	−0.528364
$764$	1.36323e7	0.844957
$765$	0	0
$766$	3.94734e6	0.243070
$767$	1.28470e7	0.788524
$768$	0	0
$769$	4.13692e6	0.252268	0.126134	−	0.992013i	$-0.459743\pi$
$769$	4.13692e6	0.252268	0.126134	+	0.992013i	$0.459743\pi$
$770$	−2.28050e6	−0.138613
$771$	0	0
$772$	1.57488e7	0.951054
$773$	1.54856e6	0.0932136	0.0466068	−	0.998913i	$-0.485159\pi$
$773$	1.54856e6	0.0932136	0.0466068	+	0.998913i	$0.485159\pi$
$774$	0	0
$775$	1.59115e7	0.951607
$776$	−973910.	−0.0580583
$777$	0	0
$778$	1.79226e7	1.06158
$779$	3.10347e7	1.83233
$780$	0	0
$781$	963315.	0.0565120
$782$	1.77807e7	1.03975
$783$	0	0
$784$	1.09373e7	0.635505
$785$	888404.	0.0514560
$786$	0	0
$787$	−2.35939e7	−1.35788	−0.678941	−	0.734192i	$-0.737561\pi$
$787$	−2.35939e7	−1.35788	−0.678941	+	0.734192i	$0.737561\pi$
$788$	1.23856e7	0.710562
$789$	0	0
$790$	−8.83462e6	−0.503640
$791$	7.07135e6	0.401848
$792$	0	0
$793$	−7.50529e6	−0.423823
$794$	9.25153e6	0.520789
$795$	0	0
$796$	2.03940e7	1.14083
$797$	−1.02305e7	−0.570494	−0.285247	−	0.958454i	$-0.592076\pi$
$797$	−1.02305e7	−0.570494	−0.285247	+	0.958454i	$0.592076\pi$
$798$	0	0
$799$	−2.64583e7	−1.46621
$800$	−2.25500e7	−1.24573
$801$	0	0
$802$	−1.90352e7	−1.04501
$803$	−6.67937e6	−0.365550
$804$	0	0
$805$	−3.53789e6	−0.192422
$806$	−3.94388e7	−2.13838
$807$	0	0
$808$	896919.	0.0483309
$809$	2.67937e7	1.43933	0.719667	−	0.694319i	$-0.244294\pi$
$809$	2.67937e7	1.43933	0.719667	+	0.694319i	$0.244294\pi$
$810$	0	0
$811$	−3.77636e6	−0.201614	−0.100807	−	0.994906i	$-0.532142\pi$
$811$	−3.77636e6	−0.201614	−0.100807	+	0.994906i	$0.532142\pi$
$812$	254570.	0.0135493
$813$	0	0
$814$	1.64408e7	0.869684
$815$	2.04201e6	0.107687
$816$	0	0
$817$	−1.44336e6	−0.0756516
$818$	−3.41398e7	−1.78393
$819$	0	0
$820$	9.51884e6	0.494367
$821$	−2.57812e7	−1.33489	−0.667444	−	0.744660i	$-0.732612\pi$
$821$	−2.57812e7	−1.33489	−0.667444	+	0.744660i	$0.732612\pi$
$822$	0	0
$823$	−1.23986e7	−0.638077	−0.319038	−	0.947742i	$-0.603360\pi$
$823$	−1.23986e7	−0.638077	−0.319038	+	0.947742i	$0.603360\pi$
$824$	1.24615e6	0.0639372
$825$	0	0
$826$	9.32560e6	0.475583
$827$	1.27739e6	0.0649469	0.0324734	−	0.999473i	$-0.489662\pi$
$827$	1.27739e6	0.0649469	0.0324734	+	0.999473i	$0.489662\pi$
$828$	0	0
$829$	−1.57808e7	−0.797520	−0.398760	−	0.917055i	$-0.630559\pi$
$829$	−1.57808e7	−0.797520	−0.398760	+	0.917055i	$0.630559\pi$
$830$	1.53518e7	0.773506
$831$	0	0
$832$	2.96015e7	1.48253
$833$	1.03692e7	0.517765
$834$	0	0
$835$	−1.02943e6	−0.0510951
$836$	1.35533e7	0.670701
$837$	0	0
$838$	−1.59912e7	−0.786629
$839$	3.66378e7	1.79690	0.898452	−	0.439073i	$-0.144693\pi$
$839$	3.66378e7	1.79690	0.898452	+	0.439073i	$0.144693\pi$
$840$	0	0
$841$	−2.05008e7	−0.999496
$842$	−1.69484e7	−0.823853
$843$	0	0
$844$	1.46693e7	0.708848
$845$	−6.55895e6	−0.316004
$846$	0	0
$847$	−9.43637e6	−0.451956
$848$	2.48582e7	1.18708
$849$	0	0
$850$	−2.05174e7	−0.974035
$851$	2.55057e7	1.20729
$852$	0	0
$853$	6.09754e6	0.286934	0.143467	−	0.989655i	$-0.454175\pi$
$853$	6.09754e6	0.286934	0.143467	+	0.989655i	$0.454175\pi$
$854$	−5.44805e6	−0.255621
$855$	0	0
$856$	1.32199e6	0.0616655
$857$	3.38239e7	1.57315	0.786577	−	0.617493i	$-0.211851\pi$
$857$	3.38239e7	1.57315	0.786577	+	0.617493i	$0.211851\pi$
$858$	0	0
$859$	4.59291e6	0.212376	0.106188	−	0.994346i	$-0.466135\pi$
$859$	4.59291e6	0.212376	0.106188	+	0.994346i	$0.466135\pi$
$860$	−442700.	−0.0204110
$861$	0	0
$862$	3.48777e7	1.59874
$863$	4.00578e7	1.83088	0.915441	−	0.402451i	$-0.131842\pi$
$863$	4.00578e7	1.83088	0.915441	+	0.402451i	$0.131842\pi$
$864$	0	0
$865$	6.54791e6	0.297552
$866$	3.24630e7	1.47094
$867$	0	0
$868$	−1.45964e7	−0.657578
$869$	1.03805e7	0.466304
$870$	0	0
$871$	5.12866e7	2.29065
$872$	1.17474e6	0.0523181
$873$	0	0
$874$	4.12390e7	1.82612
$875$	8.75582e6	0.386613
$876$	0	0
$877$	2.80438e7	1.23123	0.615613	−	0.788048i	$-0.288908\pi$
$877$	2.80438e7	1.23123	0.615613	+	0.788048i	$0.288908\pi$
$878$	−4.13045e7	−1.80826
$879$	0	0
$880$	−3.68094e6	−0.160233
$881$	9.07614e6	0.393969	0.196984	−	0.980407i	$-0.436885\pi$
$881$	9.07614e6	0.393969	0.196984	+	0.980407i	$0.436885\pi$
$882$	0	0
$883$	2.73153e7	1.17897	0.589486	−	0.807779i	$-0.299330\pi$
$883$	2.73153e7	1.17897	0.589486	+	0.807779i	$0.299330\pi$
$884$	2.59288e7	1.11597
$885$	0	0
$886$	−2.00330e7	−0.857358
$887$	3.68421e7	1.57230	0.786151	−	0.618035i	$-0.212071\pi$
$887$	3.68421e7	1.57230	0.786151	+	0.618035i	$0.212071\pi$
$888$	0	0
$889$	3.51679e6	0.149242
$890$	1.61814e7	0.684763
$891$	0	0
$892$	−3.25654e7	−1.37039
$893$	−6.13654e7	−2.57510
$894$	0	0
$895$	−6.46844e6	−0.269924
$896$	1.60126e6	0.0666336
$897$	0	0
$898$	−3.04121e7	−1.25850
$899$	−592532.	−0.0244519
$900$	0	0
$901$	2.35670e7	0.967149
$902$	−2.19364e7	−0.897735
$903$	0	0
$904$	−977692.	−0.0397906
$905$	4.50681e6	0.182914
$906$	0	0
$907$	−3.88618e7	−1.56857	−0.784287	−	0.620398i	$-0.786971\pi$
$907$	−3.88618e7	−1.56857	−0.784287	+	0.620398i	$0.786971\pi$
$908$	3.66606e7	1.47565
$909$	0	0
$910$	−1.01188e7	−0.405067
$911$	3.05722e7	1.22048	0.610240	−	0.792217i	$-0.291073\pi$
$911$	3.05722e7	1.22048	0.610240	+	0.792217i	$0.291073\pi$
$912$	0	0
$913$	−1.80380e7	−0.716164
$914$	−3.70361e7	−1.46643
$915$	0	0
$916$	−3.16362e6	−0.124579
$917$	−9.12534e6	−0.358365
$918$	0	0
$919$	−4.08434e6	−0.159527	−0.0797633	−	0.996814i	$-0.525416\pi$
$919$	−4.08434e6	−0.159527	−0.0797633	+	0.996814i	$0.525416\pi$
$920$	489152.	0.0190535
$921$	0	0
$922$	−5.67634e7	−2.19908
$923$	4.27433e6	0.165144
$924$	0	0
$925$	−2.94314e7	−1.13098
$926$	5.63696e7	2.16032
$927$	0	0
$928$	839745.	0.0320094
$929$	2.37671e7	0.903517	0.451758	−	0.892140i	$-0.350797\pi$
$929$	2.37671e7	0.903517	0.451758	+	0.892140i	$0.350797\pi$
$930$	0	0
$931$	2.40495e7	0.909351
$932$	3.58485e7	1.35186
$933$	0	0
$934$	−5.03000e6	−0.188669
$935$	−3.48975e6	−0.130547
$936$	0	0
$937$	−1.24958e7	−0.464958	−0.232479	−	0.972601i	$-0.574684\pi$
$937$	−1.24958e7	−0.464958	−0.232479	+	0.972601i	$0.574684\pi$
$938$	3.72287e7	1.38156
$939$	0	0
$940$	−1.88217e7	−0.694768
$941$	−6.64744e6	−0.244726	−0.122363	−	0.992485i	$-0.539047\pi$
$941$	−6.64744e6	−0.244726	−0.122363	+	0.992485i	$0.539047\pi$
$942$	0	0
$943$	−3.40313e7	−1.24623
$944$	1.50524e7	0.549762
$945$	0	0
$946$	1.02021e6	0.0370648
$947$	−4.32942e6	−0.156875	−0.0784377	−	0.996919i	$-0.524993\pi$
$947$	−4.32942e6	−0.156875	−0.0784377	+	0.996919i	$0.524993\pi$
$948$	0	0
$949$	−2.96370e7	−1.06824
$950$	−4.75864e7	−1.71070
$951$	0	0
$952$	727873.	0.0260294
$953$	−4.06536e7	−1.44999	−0.724997	−	0.688752i	$-0.758159\pi$
$953$	−4.06536e7	−1.44999	−0.724997	+	0.688752i	$0.758159\pi$
$954$	0	0
$955$	−8.14102e6	−0.288849
$956$	−4.18605e7	−1.48136
$957$	0	0
$958$	−2.30018e7	−0.809745
$959$	−1.82195e7	−0.639720
$960$	0	0
$961$	5.34522e6	0.186705
$962$	7.29495e7	2.54147
$963$	0	0
$964$	2.15196e7	0.745834
$965$	−9.40502e6	−0.325118
$966$	0	0
$967$	−2.92765e7	−1.00682	−0.503411	−	0.864047i	$-0.667922\pi$
$967$	−2.92765e7	−1.00682	−0.503411	+	0.864047i	$0.667922\pi$
$968$	1.30468e6	0.0447523
$969$	0	0
$970$	1.50394e7	0.513217
$971$	−2.51690e6	−0.0856677	−0.0428338	−	0.999082i	$-0.513639\pi$
$971$	−2.51690e6	−0.0856677	−0.0428338	+	0.999082i	$0.513639\pi$
$972$	0	0
$973$	−2.54740e7	−0.862610
$974$	−2.72831e7	−0.921504
$975$	0	0
$976$	−8.79365e6	−0.295491
$977$	2.91375e7	0.976599	0.488300	−	0.872676i	$-0.337617\pi$
$977$	2.91375e7	0.976599	0.488300	+	0.872676i	$0.337617\pi$
$978$	0	0
$979$	−1.90128e7	−0.633999
$980$	7.37635e6	0.245345
$981$	0	0
$982$	−5.59783e7	−1.85243
$983$	4.98825e6	0.164651	0.0823254	−	0.996605i	$-0.473765\pi$
$983$	4.98825e6	0.164651	0.0823254	+	0.996605i	$0.473765\pi$
$984$	0	0
$985$	−7.39654e6	−0.242906
$986$	764050.	0.0250282
$987$	0	0
$988$	6.01373e7	1.95998
$989$	1.58272e6	0.0514533
$990$	0	0
$991$	1.19267e7	0.385778	0.192889	−	0.981221i	$-0.438214\pi$
$991$	1.19267e7	0.385778	0.192889	+	0.981221i	$0.438214\pi$
$992$	−4.81489e7	−1.55349
$993$	0	0
$994$	3.10271e6	0.0996037
$995$	−1.21791e7	−0.389993
$996$	0	0
$997$	−2.01817e6	−0.0643014	−0.0321507	−	0.999483i	$-0.510236\pi$
$997$	−2.01817e6	−0.0643014	−0.0321507	+	0.999483i	$0.510236\pi$
$998$	2.07504e7	0.659477
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.6.a.e.1.8		42
3.2	odd	2		729.6.a.c.1.35		42
27.5	odd	18		27.6.e.a.25.12	yes	84
27.11	odd	18		27.6.e.a.13.12	✓	84
27.16	even	9		81.6.e.a.10.3		84
27.22	even	9		81.6.e.a.73.3		84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.6.e.a.13.12	✓	84	27.11	odd	18
27.6.e.a.25.12	yes	84	27.5	odd	18
81.6.e.a.10.3		84	27.16	even	9
81.6.e.a.73.3		84	27.22	even	9
729.6.a.c.1.35		42	3.2	odd	2
729.6.a.e.1.8		42	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 729 = 3^{6} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	729.a (trivial)

\(f(q)\)	\(=\)	\(q-8.08006 q^{2} +33.2873 q^{4} -19.8788 q^{5} +75.2303 q^{7} -10.4014 q^{8} +160.622 q^{10} -188.727 q^{11} -837.401 q^{13} -607.865 q^{14} -981.149 q^{16} -930.189 q^{17} -2157.41 q^{19} -661.710 q^{20} +1524.92 q^{22} +2365.71 q^{23} -2729.83 q^{25} +6766.25 q^{26} +2504.21 q^{28} +101.657 q^{29} -5828.75 q^{31} +8260.59 q^{32} +7515.98 q^{34} -1495.49 q^{35} +10781.4 q^{37} +17432.0 q^{38} +206.767 q^{40} -14385.2 q^{41} +669.024 q^{43} -6282.21 q^{44} -19115.1 q^{46} +28444.1 q^{47} -11147.4 q^{49} +22057.2 q^{50} -27874.8 q^{52} -25335.8 q^{53} +3751.66 q^{55} -782.501 q^{56} -821.393 q^{58} -15341.6 q^{59} +8962.60 q^{61} +47096.7 q^{62} -35349.2 q^{64} +16646.5 q^{65} -61245.0 q^{67} -30963.5 q^{68} +12083.6 q^{70} -5104.28 q^{71} +35391.7 q^{73} -87114.1 q^{74} -71814.2 q^{76} -14198.0 q^{77} -55002.7 q^{79} +19504.0 q^{80} +116233. q^{82} +95577.4 q^{83} +18491.0 q^{85} -5405.75 q^{86} +1963.03 q^{88} +100742. q^{89} -62997.9 q^{91} +78748.2 q^{92} -229830. q^{94} +42886.6 q^{95} +93632.6 q^{97} +90071.6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q + 12 q^{2} + 624 q^{4} + 150 q^{5} + 573 q^{8} + 3 q^{10} + 1452 q^{11} + 2256 q^{14} + 8448 q^{16} + 3465 q^{17} + 3 q^{19} + 4128 q^{20} + 96 q^{22} + 5019 q^{23} + 18750 q^{25} + 3903 q^{26} - 6 q^{28}+ \cdots + 463410 q^{98}+O(q^{100}) \) 42 * q + 12 * q^2 + 624 * q^4 + 150 * q^5 + 573 * q^8 + 3 * q^10 + 1452 * q^11 + 2256 * q^14 + 8448 * q^16 + 3465 * q^17 + 3 * q^19 + 4128 * q^20 + 96 * q^22 + 5019 * q^23 + 18750 * q^25 + 3903 * q^26 - 6 * q^28 + 13008 * q^29 + 24273 * q^32 + 35868 * q^35 + 3 * q^37 + 51801 * q^38 + 96 * q^40 + 55833 * q^41 + 110757 * q^44 + 3 * q^46 + 90129 * q^47 + 57624 * q^49 + 145362 * q^50 + 3072 * q^52 + 103203 * q^53 - 6 * q^55 + 227154 * q^56 - 192 * q^58 + 176856 * q^59 - 31851 * q^61 + 246066 * q^62 + 86019 * q^64 + 167160 * q^65 - 801 * q^67 + 374589 * q^68 + 9375 * q^70 + 279531 * q^71 + 27012 * q^73 + 413970 * q^74 + 96 * q^76 + 185190 * q^77 + 462057 * q^80 - 6 * q^82 + 295536 * q^83 + 319803 * q^86 + 3072 * q^88 + 154827 * q^89 + 91002 * q^91 + 330558 * q^92 + 96 * q^94 + 353244 * q^95 + 463410 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

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