LMFDB - Embedded newform 729.6.a.b.1.2

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 = [24,12] 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:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	\(q-9.81893 q^{2} +64.4115 q^{4} -77.7846 q^{5} +224.654 q^{7} -318.246 q^{8} +763.762 q^{10} -105.037 q^{11} -930.252 q^{13} -2205.86 q^{14} +1063.67 q^{16} -1594.51 q^{17} -979.171 q^{19} -5010.22 q^{20} +1031.35 q^{22} -1931.89 q^{23} +2925.45 q^{25} +9134.09 q^{26} +14470.3 q^{28} -2729.53 q^{29} +7368.21 q^{31} -260.228 q^{32} +15656.4 q^{34} -17474.6 q^{35} -12260.3 q^{37} +9614.42 q^{38} +24754.6 q^{40} +3873.68 q^{41} -14881.2 q^{43} -6765.58 q^{44} +18969.1 q^{46} +5291.61 q^{47} +33662.2 q^{49} -28724.8 q^{50} -59918.9 q^{52} -9959.46 q^{53} +8170.25 q^{55} -71495.1 q^{56} +26801.1 q^{58} +1647.45 q^{59} -45998.2 q^{61} -72348.0 q^{62} -31482.3 q^{64} +72359.3 q^{65} +30817.8 q^{67} -102705. q^{68} +171582. q^{70} +50609.8 q^{71} -37868.1 q^{73} +120383. q^{74} -63069.8 q^{76} -23596.9 q^{77} +18556.5 q^{79} -82737.1 q^{80} -38035.4 q^{82} +97148.8 q^{83} +124028. q^{85} +146118. q^{86} +33427.6 q^{88} -107778. q^{89} -208984. q^{91} -124436. q^{92} -51957.9 q^{94} +76164.4 q^{95} +3723.59 q^{97} -330527. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 12 q^{2} + 432 q^{4} + 33 q^{5} + 294 q^{7} + 843 q^{8} + 600 q^{10} + 30 q^{11} + 1014 q^{13} - 3720 q^{14} + 8448 q^{16} - 2709 q^{17} + 4332 q^{19} + 861 q^{20} - 2193 q^{22} + 4020 q^{23} + 15417 q^{25}+ \cdots + 944667 q^{98}+O(q^{100}) $ 24 * q + 12 * q^2 + 432 * q^4 + 33 * q^5 + 294 * q^7 + 843 * q^8 + 600 * q^10 + 30 * q^11 + 1014 * q^13 - 3720 * q^14 + 8448 * q^16 - 2709 * q^17 + 4332 * q^19 + 861 * q^20 - 2193 * q^22 + 4020 * q^23 + 15417 * q^25 + 17574 * q^26 + 4539 * q^28 - 15279 * q^29 + 17331 * q^31 + 43983 * q^32 + 20412 * q^34 + 2514 * q^35 + 11397 * q^37 + 78873 * q^38 + 52185 * q^40 - 31566 * q^41 + 34422 * q^43 - 63213 * q^44 + 62907 * q^46 + 507 * q^47 + 104106 * q^49 + 64929 * q^50 + 99237 * q^52 + 72297 * q^53 + 106611 * q^55 - 213900 * q^56 + 107733 * q^58 + 59226 * q^59 + 112452 * q^61 + 32100 * q^62 + 287115 * q^64 + 117795 * q^65 + 86055 * q^67 - 439146 * q^68 + 208623 * q^70 + 34992 * q^71 + 89580 * q^73 + 114360 * q^74 + 302031 * q^76 + 87639 * q^77 + 225600 * q^79 - 122331 * q^80 + 74526 * q^82 - 141234 * q^83 - 9900 * q^85 + 336804 * q^86 + 209895 * q^88 - 236637 * q^89 + 24354 * q^91 + 191454 * q^92 + 256080 * q^94 - 254139 * q^95 + 39363 * q^97 + 944667 * 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 $

$2$	−9.81893	−1.73576	−0.867879	−	0.496775i	$-0.834517\pi$
$2$	−9.81893	−1.73576	−0.867879	+	0.496775i	$0.834517\pi$
$3$	0	0
$3$	0	0
$4$	64.4115	2.01286
$5$	−77.7846	−1.39145	−0.695727	−	0.718307i	$-0.744918\pi$
$5$	−77.7846	−1.39145	−0.695727	+	0.718307i	$0.744918\pi$
$6$	0	0
$7$	224.654	1.73288	0.866439	−	0.499282i	$-0.166403\pi$
$7$	224.654	1.73288	0.866439	+	0.499282i	$0.166403\pi$
$8$	−318.246	−1.75808
$9$	0	0
$10$	763.762	2.41523
$11$	−105.037	−0.261734	−0.130867	−	0.991400i	$-0.541776\pi$
$11$	−105.037	−0.261734	−0.130867	+	0.991400i	$0.541776\pi$
$12$	0	0
$13$	−930.252	−1.52666	−0.763330	−	0.646009i	$-0.776437\pi$
$13$	−930.252	−1.52666	−0.763330	+	0.646009i	$0.776437\pi$
$14$	−2205.86	−3.00786
$15$	0	0
$16$	1063.67	1.03874
$17$	−1594.51	−1.33815	−0.669075	−	0.743195i	$-0.733309\pi$
$17$	−1594.51	−1.33815	−0.669075	+	0.743195i	$0.733309\pi$
$18$	0	0
$19$	−979.171	−0.622264	−0.311132	−	0.950367i	$-0.600708\pi$
$19$	−979.171	−0.622264	−0.311132	+	0.950367i	$0.600708\pi$
$20$	−5010.22	−2.80080
$21$	0	0
$22$	1031.35	0.454307
$23$	−1931.89	−0.761487	−0.380744	−	0.924681i	$-0.624332\pi$
$23$	−1931.89	−0.761487	−0.380744	+	0.924681i	$0.624332\pi$
$24$	0	0
$25$	2925.45	0.936143
$26$	9134.09	2.64991
$27$	0	0
$28$	14470.3	3.48804
$29$	−2729.53	−0.602689	−0.301345	−	0.953515i	$-0.597435\pi$
$29$	−2729.53	−0.602689	−0.301345	+	0.953515i	$0.597435\pi$
$30$	0	0
$31$	7368.21	1.37708	0.688538	−	0.725200i	$-0.258253\pi$
$31$	7368.21	1.37708	0.688538	+	0.725200i	$0.258253\pi$
$32$	−260.228	−0.0449241
$33$	0	0
$34$	15656.4	2.32271
$35$	−17474.6	−2.41122
$36$	0	0
$37$	−12260.3	−1.47230	−0.736152	−	0.676816i	$-0.763359\pi$
$37$	−12260.3	−1.47230	−0.736152	+	0.676816i	$0.763359\pi$
$38$	9614.42	1.08010
$39$	0	0
$40$	24754.6	2.44628
$41$	3873.68	0.359885	0.179943	−	0.983677i	$-0.442409\pi$
$41$	3873.68	0.359885	0.179943	+	0.983677i	$0.442409\pi$
$42$	0	0
$43$	−14881.2	−1.22735	−0.613673	−	0.789560i	$-0.710309\pi$
$43$	−14881.2	−1.22735	−0.613673	+	0.789560i	$0.710309\pi$
$44$	−6765.58	−0.526833
$45$	0	0
$46$	18969.1	1.32176
$47$	5291.61	0.349416	0.174708	−	0.984620i	$-0.444102\pi$
$47$	5291.61	0.349416	0.174708	+	0.984620i	$0.444102\pi$
$48$	0	0
$49$	33662.2	2.00287
$50$	−28724.8	−1.62492
$51$	0	0
$52$	−59918.9	−3.07295
$53$	−9959.46	−0.487019	−0.243510	−	0.969898i	$-0.578299\pi$
$53$	−9959.46	−0.487019	−0.243510	+	0.969898i	$0.578299\pi$
$54$	0	0
$55$	8170.25	0.364191
$56$	−71495.1	−3.04653
$57$	0	0
$58$	26801.1	1.04612
$59$	1647.45	0.0616146	0.0308073	−	0.999525i	$-0.490192\pi$
$59$	1647.45	0.0616146	0.0308073	+	0.999525i	$0.490192\pi$
$60$	0	0
$61$	−45998.2	−1.58277	−0.791383	−	0.611321i	$-0.790638\pi$
$61$	−45998.2	−1.58277	−0.791383	+	0.611321i	$0.790638\pi$
$62$	−72348.0	−2.39027
$63$	0	0
$64$	−31482.3	−0.960762
$65$	72359.3	2.12428
$66$	0	0
$67$	30817.8	0.838716	0.419358	−	0.907821i	$-0.362255\pi$
$67$	30817.8	0.838716	0.419358	+	0.907821i	$0.362255\pi$
$68$	−102705.	−2.69351
$69$	0	0
$70$	171582.	4.18530
$71$	50609.8	1.19149	0.595743	−	0.803175i	$-0.296858\pi$
$71$	50609.8	1.19149	0.595743	+	0.803175i	$0.296858\pi$
$72$	0	0
$73$	−37868.1	−0.831700	−0.415850	−	0.909433i	$-0.636516\pi$
$73$	−37868.1	−0.831700	−0.415850	+	0.909433i	$0.636516\pi$
$74$	120383.	2.55557
$75$	0	0
$76$	−63069.8	−1.25253
$77$	−23596.9	−0.453553
$78$	0	0
$79$	18556.5	0.334525	0.167263	−	0.985912i	$-0.446507\pi$
$79$	18556.5	0.334525	0.167263	+	0.985912i	$0.446507\pi$
$80$	−82737.1	−1.44536
$81$	0	0
$82$	−38035.4	−0.624674
$83$	97148.8	1.54790	0.773949	−	0.633249i	$-0.218279\pi$
$83$	97148.8	1.54790	0.773949	+	0.633249i	$0.218279\pi$
$84$	0	0
$85$	124028.	1.86197
$86$	146118.	2.13038
$87$	0	0
$88$	33427.6	0.460149
$89$	−107778.	−1.44230	−0.721151	−	0.692778i	$-0.756387\pi$
$89$	−107778.	−1.44230	−0.721151	+	0.692778i	$0.756387\pi$
$90$	0	0
$91$	−208984.	−2.64552
$92$	−124436.	−1.53277
$93$	0	0
$94$	−51957.9	−0.606502
$95$	76164.4	0.865851
$96$	0	0
$97$	3723.59	0.0401820	0.0200910	−	0.999798i	$-0.493604\pi$
$97$	3723.59	0.0401820	0.0200910	+	0.999798i	$0.493604\pi$
$98$	−330527.	−3.47650
$99$	0	0
$100$	188432.	1.88432
$101$	−170223.	−1.66040	−0.830202	−	0.557463i	$-0.811775\pi$
$101$	−170223.	−1.66040	−0.830202	+	0.557463i	$0.811775\pi$
$102$	0	0
$103$	−146836.	−1.36377	−0.681883	−	0.731462i	$-0.738838\pi$
$103$	−146836.	−1.36377	−0.681883	+	0.731462i	$0.738838\pi$
$104$	296049.	2.68399
$105$	0	0
$106$	97791.3	0.845348
$107$	−122455.	−1.03399	−0.516996	−	0.855988i	$-0.672950\pi$
$107$	−122455.	−1.03399	−0.516996	+	0.855988i	$0.672950\pi$
$108$	0	0
$109$	86313.5	0.695845	0.347922	−	0.937523i	$-0.386887\pi$
$109$	86313.5	0.695845	0.347922	+	0.937523i	$0.386887\pi$
$110$	−80223.2	−0.632147
$111$	0	0
$112$	238957.	1.80001
$113$	−121204.	−0.892935	−0.446467	−	0.894800i	$-0.647318\pi$
$113$	−121204.	−0.892935	−0.446467	+	0.894800i	$0.647318\pi$
$114$	0	0
$115$	150271.	1.05957
$116$	−175813.	−1.21313
$117$	0	0
$118$	−16176.2	−0.106948
$119$	−358212.	−2.31885
$120$	0	0
$121$	−150018.	−0.931495
$122$	451654.	2.74730
$123$	0	0
$124$	474597.	2.77186
$125$	15522.3	0.0888547
$126$	0	0
$127$	−19675.3	−0.108246	−0.0541231	−	0.998534i	$-0.517236\pi$
$127$	−19675.3	−0.108246	−0.0541231	+	0.998534i	$0.517236\pi$
$128$	317449.	1.71258
$129$	0	0
$130$	−710491.	−3.68723
$131$	141007.	0.717896	0.358948	−	0.933358i	$-0.383136\pi$
$131$	141007.	0.717896	0.358948	+	0.933358i	$0.383136\pi$
$132$	0	0
$133$	−219974.	−1.07831
$134$	−302598.	−1.45581
$135$	0	0
$136$	507446.	2.35257
$137$	−32119.4	−0.146206	−0.0731031	−	0.997324i	$-0.523290\pi$
$137$	−32119.4	−0.146206	−0.0731031	+	0.997324i	$0.523290\pi$
$138$	0	0
$139$	276266.	1.21281	0.606403	−	0.795158i	$-0.292612\pi$
$139$	276266.	1.21281	0.606403	+	0.795158i	$0.292612\pi$
$140$	−1.12556e6	−4.85344
$141$	0	0
$142$	−496934.	−2.06813
$143$	97710.8	0.399579
$144$	0	0
$145$	212316.	0.838614
$146$	371825.	1.44363
$147$	0	0
$148$	−789705.	−2.96354
$149$	26146.0	0.0964804	0.0482402	−	0.998836i	$-0.484639\pi$
$149$	26146.0	0.0964804	0.0482402	+	0.998836i	$0.484639\pi$
$150$	0	0
$151$	−12850.1	−0.0458631	−0.0229315	−	0.999737i	$-0.507300\pi$
$151$	−12850.1	−0.0458631	−0.0229315	+	0.999737i	$0.507300\pi$
$152$	311617.	1.09399
$153$	0	0
$154$	231696.	0.787259
$155$	−573133.	−1.91614
$156$	0	0
$157$	88963.9	0.288048	0.144024	−	0.989574i	$-0.453996\pi$
$157$	88963.9	0.288048	0.144024	+	0.989574i	$0.453996\pi$
$158$	−182205.	−0.580655
$159$	0	0
$160$	20241.8	0.0625098
$161$	−434006.	−1.31957
$162$	0	0
$163$	−198942.	−0.586485	−0.293243	−	0.956038i	$-0.594734\pi$
$163$	−198942.	−0.586485	−0.293243	+	0.956038i	$0.594734\pi$
$164$	249509.	0.724398
$165$	0	0
$166$	−953897.	−2.68678
$167$	−82697.2	−0.229456	−0.114728	−	0.993397i	$-0.536600\pi$
$167$	−82697.2	−0.229456	−0.114728	+	0.993397i	$0.536600\pi$
$168$	0	0
$169$	494076.	1.33069
$170$	−1.21783e6	−3.23194
$171$	0	0
$172$	−958521.	−2.47047
$173$	−251666.	−0.639307	−0.319653	−	0.947535i	$-0.603566\pi$
$173$	−251666.	−0.639307	−0.319653	+	0.947535i	$0.603566\pi$
$174$	0	0
$175$	657212.	1.62222
$176$	−111725.	−0.271874
$177$	0	0
$178$	1.05827e6	2.50349
$179$	−387612.	−0.904199	−0.452100	−	0.891967i	$-0.649325\pi$
$179$	−387612.	−0.904199	−0.452100	+	0.891967i	$0.649325\pi$
$180$	0	0
$181$	268627.	0.609472	0.304736	−	0.952437i	$-0.401432\pi$
$181$	268627.	0.609472	0.304736	+	0.952437i	$0.401432\pi$
$182$	2.05200e6	4.59198
$183$	0	0
$184$	614816.	1.33875
$185$	953665.	2.04864
$186$	0	0
$187$	167482.	0.350239
$188$	340840.	0.703325
$189$	0	0
$190$	−747854.	−1.50291
$191$	−51024.6	−0.101204	−0.0506018	−	0.998719i	$-0.516114\pi$
$191$	−51024.6	−0.101204	−0.0506018	+	0.998719i	$0.516114\pi$
$192$	0	0
$193$	−355230.	−0.686462	−0.343231	−	0.939251i	$-0.611521\pi$
$193$	−355230.	−0.686462	−0.343231	+	0.939251i	$0.611521\pi$
$194$	−36561.6	−0.0697463
$195$	0	0
$196$	2.16823e6	4.03149
$197$	661385.	1.21420	0.607098	−	0.794627i	$-0.292333\pi$
$197$	661385.	1.21420	0.607098	+	0.794627i	$0.292333\pi$
$198$	0	0
$199$	−136835.	−0.244943	−0.122471	−	0.992472i	$-0.539082\pi$
$199$	−136835.	−0.244943	−0.122471	+	0.992472i	$0.539082\pi$
$200$	−931011.	−1.64581
$201$	0	0
$202$	1.67140e6	2.88206
$203$	−613199.	−1.04439
$204$	0	0
$205$	−301313.	−0.500763
$206$	1.44177e6	2.36717
$207$	0	0
$208$	−989481.	−1.58580
$209$	102849.	0.162868
$210$	0	0
$211$	−413951.	−0.640093	−0.320047	−	0.947402i	$-0.603699\pi$
$211$	−413951.	−0.640093	−0.320047	+	0.947402i	$0.603699\pi$
$212$	−641503.	−0.980301
$213$	0	0
$214$	1.20238e6	1.79476
$215$	1.15753e6	1.70780
$216$	0	0
$217$	1.65529e6	2.38630
$218$	−847506.	−1.20782
$219$	0	0
$220$	526258.	0.733064
$221$	1.48330e6	2.04290
$222$	0	0
$223$	−390155.	−0.525382	−0.262691	−	0.964880i	$-0.584610\pi$
$223$	−390155.	−0.525382	−0.262691	+	0.964880i	$0.584610\pi$
$224$	−58461.2	−0.0778480
$225$	0	0
$226$	1.19009e6	1.54992
$227$	461470.	0.594400	0.297200	−	0.954815i	$-0.403947\pi$
$227$	461470.	0.594400	0.297200	+	0.954815i	$0.403947\pi$
$228$	0	0
$229$	74595.7	0.0939994	0.0469997	−	0.998895i	$-0.485034\pi$
$229$	74595.7	0.0939994	0.0469997	+	0.998895i	$0.485034\pi$
$230$	−1.47550e6	−1.83917
$231$	0	0
$232$	868663.	1.05957
$233$	167417.	0.202028	0.101014	−	0.994885i	$-0.467791\pi$
$233$	167417.	0.202028	0.101014	+	0.994885i	$0.467791\pi$
$234$	0	0
$235$	−411606.	−0.486196
$236$	106115.	0.124021
$237$	0	0
$238$	3.51726e6	4.02497
$239$	−706366.	−0.799899	−0.399949	−	0.916537i	$-0.630972\pi$
$239$	−706366.	−0.799899	−0.399949	+	0.916537i	$0.630972\pi$
$240$	0	0
$241$	1.44753e6	1.60541	0.802704	−	0.596377i	$-0.203394\pi$
$241$	1.44753e6	1.60541	0.802704	+	0.596377i	$0.203394\pi$
$242$	1.47302e6	1.61685
$243$	0	0
$244$	−2.96281e6	−3.18588
$245$	−2.61840e6	−2.78690
$246$	0	0
$247$	910876.	0.949986
$248$	−2.34490e6	−2.42101
$249$	0	0
$250$	−152412.	−0.154230
$251$	−1.58229e6	−1.58526	−0.792631	−	0.609701i	$-0.791289\pi$
$251$	−1.58229e6	−1.58526	−0.792631	+	0.609701i	$0.791289\pi$
$252$	0	0
$253$	202920.	0.199307
$254$	193191.	0.187889
$255$	0	0
$256$	−2.10958e6	−2.01186
$257$	947115.	0.894478	0.447239	−	0.894414i	$-0.352407\pi$
$257$	947115.	0.894478	0.447239	+	0.894414i	$0.352407\pi$
$258$	0	0
$259$	−2.75433e6	−2.55133
$260$	4.66077e6	4.27587
$261$	0	0
$262$	−1.38454e6	−1.24609
$263$	1.58841e6	1.41604	0.708018	−	0.706194i	$-0.249590\pi$
$263$	1.58841e6	1.41604	0.708018	+	0.706194i	$0.249590\pi$
$264$	0	0
$265$	774693.	0.677665
$266$	2.15991e6	1.87168
$267$	0	0
$268$	1.98502e6	1.68822
$269$	−211624.	−0.178313	−0.0891567	−	0.996018i	$-0.528417\pi$
$269$	−211624.	−0.178313	−0.0891567	+	0.996018i	$0.528417\pi$
$270$	0	0
$271$	1.00424e6	0.830642	0.415321	−	0.909675i	$-0.363669\pi$
$271$	1.00424e6	0.830642	0.415321	+	0.909675i	$0.363669\pi$
$272$	−1.69603e6	−1.38999
$273$	0	0
$274$	315378.	0.253779
$275$	−307280.	−0.245020
$276$	0	0
$277$	1.71934e6	1.34636	0.673182	−	0.739477i	$-0.264927\pi$
$277$	1.71934e6	1.34636	0.673182	+	0.739477i	$0.264927\pi$
$278$	−2.71264e6	−2.10514
$279$	0	0
$280$	5.56122e6	4.23911
$281$	1.94339e6	1.46823	0.734117	−	0.679024i	$-0.237597\pi$
$281$	1.94339e6	1.46823	0.734117	+	0.679024i	$0.237597\pi$
$282$	0	0
$283$	−988317.	−0.733551	−0.366775	−	0.930310i	$-0.619538\pi$
$283$	−988317.	−0.733551	−0.366775	+	0.930310i	$0.619538\pi$
$284$	3.25985e6	2.39829
$285$	0	0
$286$	−959416.	−0.693573
$287$	870236.	0.623637
$288$	0	0
$289$	1.12260e6	0.790645
$290$	−2.08471e6	−1.45563
$291$	0	0
$292$	−2.43914e6	−1.67409
$293$	886046.	0.602958	0.301479	−	0.953473i	$-0.402520\pi$
$293$	886046.	0.602958	0.301479	+	0.953473i	$0.402520\pi$
$294$	0	0
$295$	−128147.	−0.0857338
$296$	3.90180e6	2.58843
$297$	0	0
$298$	−256726.	−0.167467
$299$	1.79714e6	1.16253
$300$	0	0
$301$	−3.34312e6	−2.12684
$302$	126174.	0.0796073
$303$	0	0
$304$	−1.04151e6	−0.646370
$305$	3.57795e6	2.20234
$306$	0	0
$307$	−1.12119e6	−0.678945	−0.339473	−	0.940616i	$-0.610249\pi$
$307$	−1.12119e6	−0.678945	−0.339473	+	0.940616i	$0.610249\pi$
$308$	−1.51991e6	−0.912938
$309$	0	0
$310$	5.62756e6	3.32595
$311$	3.30770e6	1.93921	0.969606	−	0.244672i	$-0.0786801\pi$
$311$	3.30770e6	1.93921	0.969606	+	0.244672i	$0.0786801\pi$
$312$	0	0
$313$	−1.72639e6	−0.996041	−0.498021	−	0.867165i	$-0.665940\pi$
$313$	−1.72639e6	−0.996041	−0.498021	+	0.867165i	$0.665940\pi$
$314$	−873531.	−0.499982
$315$	0	0
$316$	1.19525e6	0.673352
$317$	671534.	0.375336	0.187668	−	0.982233i	$-0.439907\pi$
$317$	671534.	0.375336	0.187668	+	0.982233i	$0.439907\pi$
$318$	0	0
$319$	286702.	0.157744
$320$	2.44884e6	1.33686
$321$	0	0
$322$	4.26147e6	2.29045
$323$	1.56130e6	0.832682
$324$	0	0
$325$	−2.72140e6	−1.42917
$326$	1.95340e6	1.01800
$327$	0	0
$328$	−1.23278e6	−0.632706
$329$	1.18878e6	0.605496
$330$	0	0
$331$	−336611.	−0.168872	−0.0844361	−	0.996429i	$-0.526909\pi$
$331$	−336611.	−0.168872	−0.0844361	+	0.996429i	$0.526909\pi$
$332$	6.25749e6	3.11570
$333$	0	0
$334$	811998.	0.398280
$335$	−2.39715e6	−1.16703
$336$	0	0
$337$	1.37938e6	0.661620	0.330810	−	0.943697i	$-0.392678\pi$
$337$	1.37938e6	0.661620	0.330810	+	0.943697i	$0.392678\pi$
$338$	−4.85130e6	−2.30976
$339$	0	0
$340$	7.98884e6	3.74789
$341$	−773934.	−0.360428
$342$	0	0
$343$	3.78658e6	1.73785
$344$	4.73589e6	2.15777
$345$	0	0
$346$	2.47109e6	1.10968
$347$	560232.	0.249772	0.124886	−	0.992171i	$-0.460143\pi$
$347$	560232.	0.249772	0.124886	+	0.992171i	$0.460143\pi$
$348$	0	0
$349$	227898.	0.100156	0.0500780	−	0.998745i	$-0.484053\pi$
$349$	227898.	0.100156	0.0500780	+	0.998745i	$0.484053\pi$
$350$	−6.45312e6	−2.81578
$351$	0	0
$352$	27333.6	0.0117582
$353$	3.01656e6	1.28847	0.644236	−	0.764827i	$-0.277176\pi$
$353$	3.01656e6	1.28847	0.644236	+	0.764827i	$0.277176\pi$
$354$	0	0
$355$	−3.93666e6	−1.65790
$356$	−6.94216e6	−2.90315
$357$	0	0
$358$	3.80593e6	1.56947
$359$	−4.42207e6	−1.81088	−0.905440	−	0.424474i	$-0.860459\pi$
$359$	−4.42207e6	−1.81088	−0.905440	+	0.424474i	$0.860459\pi$
$360$	0	0
$361$	−1.51732e6	−0.612788
$362$	−2.63763e6	−1.05790
$363$	0	0
$364$	−1.34610e7	−5.32505
$365$	2.94556e6	1.15727
$366$	0	0
$367$	140993.	0.0546426	0.0273213	−	0.999627i	$-0.491302\pi$
$367$	140993.	0.0546426	0.0273213	+	0.999627i	$0.491302\pi$
$368$	−2.05489e6	−0.790987
$369$	0	0
$370$	−9.36397e6	−3.55595
$371$	−2.23743e6	−0.843945
$372$	0	0
$373$	−4.61535e6	−1.71764	−0.858820	−	0.512277i	$-0.828802\pi$
$373$	−4.61535e6	−1.71764	−0.858820	+	0.512277i	$0.828802\pi$
$374$	−1.64450e6	−0.607931
$375$	0	0
$376$	−1.68403e6	−0.614300
$377$	2.53916e6	0.920102
$378$	0	0
$379$	−2.62222e6	−0.937716	−0.468858	−	0.883274i	$-0.655334\pi$
$379$	−2.62222e6	−0.937716	−0.468858	+	0.883274i	$0.655334\pi$
$380$	4.90586e6	1.74284
$381$	0	0
$382$	501007.	0.175665
$383$	−306222.	−0.106669	−0.0533346	−	0.998577i	$-0.516985\pi$
$383$	−306222.	−0.106669	−0.0533346	+	0.998577i	$0.516985\pi$
$384$	0	0
$385$	1.83548e6	0.631098
$386$	3.48798e6	1.19153
$387$	0	0
$388$	239842.	0.0808807
$389$	−371122.	−0.124349	−0.0621745	−	0.998065i	$-0.519804\pi$
$389$	−371122.	−0.124349	−0.0621745	+	0.998065i	$0.519804\pi$
$390$	0	0
$391$	3.08042e6	1.01898
$392$	−1.07129e7	−3.52120
$393$	0	0
$394$	−6.49410e6	−2.10755
$395$	−1.44341e6	−0.465477
$396$	0	0
$397$	266590.	0.0848921	0.0424461	−	0.999099i	$-0.486485\pi$
$397$	266590.	0.0848921	0.0424461	+	0.999099i	$0.486485\pi$
$398$	1.34358e6	0.425162
$399$	0	0
$400$	3.11171e6	0.972408
$401$	−3.84744e6	−1.19484	−0.597422	−	0.801927i	$-0.703808\pi$
$401$	−3.84744e6	−1.19484	−0.597422	+	0.801927i	$0.703808\pi$
$402$	0	0
$403$	−6.85429e6	−2.10233
$404$	−1.09643e7	−3.34216
$405$	0	0
$406$	6.02096e6	1.81280
$407$	1.28779e6	0.385352
$408$	0	0
$409$	−288628.	−0.0853161	−0.0426580	−	0.999090i	$-0.513583\pi$
$409$	−288628.	−0.0853161	−0.0426580	+	0.999090i	$0.513583\pi$
$410$	2.95857e6	0.869204
$411$	0	0
$412$	−9.45792e6	−2.74507
$413$	370106.	0.106771
$414$	0	0
$415$	−7.55668e6	−2.15383
$416$	242078.	0.0685839
$417$	0	0
$418$	−1.00987e6	−0.282699
$419$	5.11049e6	1.42209	0.711045	−	0.703146i	$-0.248222\pi$
$419$	5.11049e6	1.42209	0.711045	+	0.703146i	$0.248222\pi$
$420$	0	0
$421$	5.11020e6	1.40518	0.702591	−	0.711594i	$-0.252027\pi$
$421$	5.11020e6	1.40518	0.702591	+	0.711594i	$0.252027\pi$
$422$	4.06456e6	1.11105
$423$	0	0
$424$	3.16956e6	0.856217
$425$	−4.66465e6	−1.25270
$426$	0	0
$427$	−1.03337e7	−2.74274
$428$	−7.88751e6	−2.08128
$429$	0	0
$430$	−1.13657e7	−2.96432
$431$	−581904.	−0.150889	−0.0754446	−	0.997150i	$-0.524038\pi$
$431$	−581904.	−0.150889	−0.0754446	+	0.997150i	$0.524038\pi$
$432$	0	0
$433$	4.03750e6	1.03489	0.517443	−	0.855718i	$-0.326884\pi$
$433$	4.03750e6	1.03489	0.517443	+	0.855718i	$0.326884\pi$
$434$	−1.62532e7	−4.14205
$435$	0	0
$436$	5.55958e6	1.40064
$437$	1.89165e6	0.473846
$438$	0	0
$439$	3.61017e6	0.894060	0.447030	−	0.894519i	$-0.352482\pi$
$439$	3.61017e6	0.894060	0.447030	+	0.894519i	$0.352482\pi$
$440$	−2.60015e6	−0.640275
$441$	0	0
$442$	−1.45644e7	−3.54598
$443$	−109125.	−0.0264189	−0.0132094	−	0.999913i	$-0.504205\pi$
$443$	−109125.	−0.0264189	−0.0132094	+	0.999913i	$0.504205\pi$
$444$	0	0
$445$	8.38349e6	2.00690
$446$	3.83091e6	0.911936
$447$	0	0
$448$	−7.07260e6	−1.66488
$449$	1.29009e6	0.301998	0.150999	−	0.988534i	$-0.451751\pi$
$449$	1.29009e6	0.301998	0.150999	+	0.988534i	$0.451751\pi$
$450$	0	0
$451$	−406879.	−0.0941942
$452$	−7.80691e6	−1.79735
$453$	0	0
$454$	−4.53114e6	−1.03173
$455$	1.62558e7	3.68111
$456$	0	0
$457$	8.35403e6	1.87114	0.935568	−	0.353146i	$-0.114888\pi$
$457$	8.35403e6	1.87114	0.935568	+	0.353146i	$0.114888\pi$
$458$	−732450.	−0.163160
$459$	0	0
$460$	9.67919e6	2.13277
$461$	−5.54610e6	−1.21544	−0.607722	−	0.794150i	$-0.707917\pi$
$461$	−5.54610e6	−1.21544	−0.607722	+	0.794150i	$0.707917\pi$
$462$	0	0
$463$	8769.70	0.00190122	0.000950610	−	1.00000i	$-0.499697\pi$
$463$	8769.70	0.00190122	0.000950610		1.00000i	$0.499697\pi$
$464$	−2.90332e6	−0.626037
$465$	0	0
$466$	−1.64386e6	−0.350671
$467$	−6.14127e6	−1.30306	−0.651532	−	0.758621i	$-0.725873\pi$
$467$	−6.14127e6	−1.30306	−0.651532	+	0.758621i	$0.725873\pi$
$468$	0	0
$469$	6.92333e6	1.45339
$470$	4.04153e6	0.843919
$471$	0	0
$472$	−524296.	−0.108323
$473$	1.56308e6	0.321238
$474$	0	0
$475$	−2.86451e6	−0.582528
$476$	−2.30730e7	−4.66752
$477$	0	0
$478$	6.93576e6	1.38843
$479$	2.57553e6	0.512895	0.256448	−	0.966558i	$-0.417448\pi$
$479$	2.57553e6	0.512895	0.256448	+	0.966558i	$0.417448\pi$
$480$	0	0
$481$	1.14052e7	2.24771
$482$	−1.42132e7	−2.78660
$483$	0	0
$484$	−9.66289e6	−1.87497
$485$	−289638.	−0.0559114
$486$	0	0
$487$	2.42151e6	0.462662	0.231331	−	0.972875i	$-0.425692\pi$
$487$	2.42151e6	0.462662	0.231331	+	0.972875i	$0.425692\pi$
$488$	1.46388e7	2.78262
$489$	0	0
$490$	2.57099e7	4.83738
$491$	7.86505e6	1.47231	0.736153	−	0.676815i	$-0.236640\pi$
$491$	7.86505e6	1.47231	0.736153	+	0.676815i	$0.236640\pi$
$492$	0	0
$493$	4.35227e6	0.806489
$494$	−8.94383e6	−1.64895
$495$	0	0
$496$	7.83734e6	1.43042
$497$	1.13697e7	2.06470
$498$	0	0
$499$	9.14176e6	1.64353	0.821766	−	0.569825i	$-0.192989\pi$
$499$	9.14176e6	1.64353	0.821766	+	0.569825i	$0.192989\pi$
$500$	999812.	0.178852
$501$	0	0
$502$	1.55364e7	2.75163
$503$	−8.62863e6	−1.52062	−0.760312	−	0.649558i	$-0.774954\pi$
$503$	−8.62863e6	−1.52062	−0.760312	+	0.649558i	$0.774954\pi$
$504$	0	0
$505$	1.32407e7	2.31037
$506$	−1.99245e6	−0.345949
$507$	0	0
$508$	−1.26732e6	−0.217884
$509$	−4.61271e6	−0.789154	−0.394577	−	0.918863i	$-0.629109\pi$
$509$	−4.61271e6	−0.789154	−0.394577	+	0.918863i	$0.629109\pi$
$510$	0	0
$511$	−8.50721e6	−1.44123
$512$	1.05555e7	1.77952
$513$	0	0
$514$	−9.29966e6	−1.55260
$515$	1.14216e7	1.89762
$516$	0	0
$517$	−555814.	−0.0914541
$518$	2.70445e7	4.42848
$519$	0	0
$520$	−2.30281e7	−3.73464
$521$	300956.	0.0485745	0.0242873	−	0.999705i	$-0.492268\pi$
$521$	300956.	0.0485745	0.0242873	+	0.999705i	$0.492268\pi$
$522$	0	0
$523$	−4.56248e6	−0.729368	−0.364684	−	0.931131i	$-0.618823\pi$
$523$	−4.56248e6	−0.729368	−0.364684	+	0.931131i	$0.618823\pi$
$524$	9.08245e6	1.44502
$525$	0	0
$526$	−1.55965e7	−2.45790
$527$	−1.17487e7	−1.84273
$528$	0	0
$529$	−2.70415e6	−0.420137
$530$	−7.60666e6	−1.17626
$531$	0	0
$532$	−1.41689e7	−2.17048
$533$	−3.60350e6	−0.549422
$534$	0	0
$535$	9.52512e6	1.43875
$536$	−9.80765e6	−1.47453
$537$	0	0
$538$	2.07792e6	0.309509
$539$	−3.53577e6	−0.524219
$540$	0	0
$541$	−2.92926e6	−0.430294	−0.215147	−	0.976582i	$-0.569023\pi$
$541$	−2.92926e6	−0.430294	−0.215147	+	0.976582i	$0.569023\pi$
$542$	−9.86056e6	−1.44179
$543$	0	0
$544$	414936.	0.0601152
$545$	−6.71386e6	−0.968236
$546$	0	0
$547$	−8.87074e6	−1.26763	−0.633814	−	0.773486i	$-0.718511\pi$
$547$	−8.87074e6	−1.26763	−0.633814	+	0.773486i	$0.718511\pi$
$548$	−2.06886e6	−0.294292
$549$	0	0
$550$	3.01716e6	0.425296
$551$	2.67268e6	0.375032
$552$	0	0
$553$	4.16879e6	0.579692
$554$	−1.68821e7	−2.33696
$555$	0	0
$556$	1.77947e7	2.44120
$557$	4.45693e6	0.608693	0.304346	−	0.952561i	$-0.401562\pi$
$557$	4.45693e6	0.608693	0.304346	+	0.952561i	$0.401562\pi$
$558$	0	0
$559$	1.38433e7	1.87374
$560$	−1.85872e7	−2.50463
$561$	0	0
$562$	−1.90821e7	−2.54850
$563$	9.90939e6	1.31758	0.658788	−	0.752329i	$-0.271069\pi$
$563$	9.90939e6	1.31758	0.658788	+	0.752329i	$0.271069\pi$
$564$	0	0
$565$	9.42778e6	1.24248
$566$	9.70422e6	1.27327
$567$	0	0
$568$	−1.61064e7	−2.09472
$569$	8.63426e6	1.11801	0.559003	−	0.829165i	$-0.311184\pi$
$569$	8.63426e6	1.11801	0.559003	+	0.829165i	$0.311184\pi$
$570$	0	0
$571$	1.03013e7	1.32221	0.661107	−	0.750292i	$-0.270087\pi$
$571$	1.03013e7	1.32221	0.661107	+	0.750292i	$0.270087\pi$
$572$	6.29370e6	0.804296
$573$	0	0
$574$	−8.54478e6	−1.08248
$575$	−5.65164e6	−0.712861
$576$	0	0
$577$	−8.11115e6	−1.01425	−0.507123	−	0.861874i	$-0.669291\pi$
$577$	−8.11115e6	−1.01425	−0.507123	+	0.861874i	$0.669291\pi$
$578$	−1.10228e7	−1.37237
$579$	0	0
$580$	1.36756e7	1.68801
$581$	2.18248e7	2.68232
$582$	0	0
$583$	1.04611e6	0.127470
$584$	1.20514e7	1.46219
$585$	0	0
$586$	−8.70003e6	−1.04659
$587$	7.43685e6	0.890828	0.445414	−	0.895325i	$-0.353057\pi$
$587$	7.43685e6	0.890828	0.445414	+	0.895325i	$0.353057\pi$
$588$	0	0
$589$	−7.21474e6	−0.856904
$590$	1.25826e6	0.148813
$591$	0	0
$592$	−1.30409e7	−1.52934
$593$	−7.25150e6	−0.846820	−0.423410	−	0.905938i	$-0.639167\pi$
$593$	−7.25150e6	−0.846820	−0.423410	+	0.905938i	$0.639167\pi$
$594$	0	0
$595$	2.78634e7	3.22657
$596$	1.68410e6	0.194201
$597$	0	0
$598$	−1.76460e7	−2.01788
$599$	4.59027e6	0.522722	0.261361	−	0.965241i	$-0.415829\pi$
$599$	4.59027e6	0.522722	0.261361	+	0.965241i	$0.415829\pi$
$600$	0	0
$601$	1.35428e7	1.52941	0.764705	−	0.644381i	$-0.222885\pi$
$601$	1.35428e7	1.52941	0.764705	+	0.644381i	$0.222885\pi$
$602$	3.28259e7	3.69169
$603$	0	0
$604$	−827692.	−0.0923159
$605$	1.16691e7	1.29613
$606$	0	0
$607$	−5.21495e6	−0.574484	−0.287242	−	0.957858i	$-0.592738\pi$
$607$	−5.21495e6	−0.574484	−0.287242	+	0.957858i	$0.592738\pi$
$608$	254808.	0.0279547
$609$	0	0
$610$	−3.51317e7	−3.82274
$611$	−4.92253e6	−0.533440
$612$	0	0
$613$	4.36235e6	0.468888	0.234444	−	0.972130i	$-0.424673\pi$
$613$	4.36235e6	0.468888	0.234444	+	0.972130i	$0.424673\pi$
$614$	1.10089e7	1.17849
$615$	0	0
$616$	7.50962e6	0.797382
$617$	1.09030e7	1.15302	0.576508	−	0.817092i	$-0.304415\pi$
$617$	1.09030e7	1.15302	0.576508	+	0.817092i	$0.304415\pi$
$618$	0	0
$619$	−8.10259e6	−0.849958	−0.424979	−	0.905203i	$-0.639718\pi$
$619$	−8.10259e6	−0.849958	−0.424979	+	0.905203i	$0.639718\pi$
$620$	−3.69164e7	−3.85691
$621$	0	0
$622$	−3.24781e7	−3.36600
$623$	−2.42128e7	−2.49933
$624$	0	0
$625$	−1.03494e7	−1.05978
$626$	1.69513e7	1.72889
$627$	0	0
$628$	5.73029e6	0.579800
$629$	1.95492e7	1.97016
$630$	0	0
$631$	8.43939e6	0.843797	0.421898	−	0.906643i	$-0.361364\pi$
$631$	8.43939e6	0.843797	0.421898	+	0.906643i	$0.361364\pi$
$632$	−5.90554e6	−0.588122
$633$	0	0
$634$	−6.59375e6	−0.651492
$635$	1.53044e6	0.150620
$636$	0	0
$637$	−3.13143e7	−3.05770
$638$	−2.81511e6	−0.273806
$639$	0	0
$640$	−2.46927e7	−2.38297
$641$	−2.65870e6	−0.255579	−0.127789	−	0.991801i	$-0.540788\pi$
$641$	−2.65870e6	−0.255579	−0.127789	+	0.991801i	$0.540788\pi$
$642$	0	0
$643$	−2.40948e6	−0.229825	−0.114912	−	0.993376i	$-0.536659\pi$
$643$	−2.40948e6	−0.229825	−0.114912	+	0.993376i	$0.536659\pi$
$644$	−2.79549e7	−2.65610
$645$	0	0
$646$	−1.53303e7	−1.44534
$647$	1.28476e7	1.20660	0.603298	−	0.797516i	$-0.293853\pi$
$647$	1.28476e7	1.20660	0.603298	+	0.797516i	$0.293853\pi$
$648$	0	0
$649$	−173044.	−0.0161266
$650$	2.67213e7	2.48070
$651$	0	0
$652$	−1.28141e7	−1.18051
$653$	−6.74963e6	−0.619437	−0.309718	−	0.950828i	$-0.600235\pi$
$653$	−6.74963e6	−0.619437	−0.309718	+	0.950828i	$0.600235\pi$
$654$	0	0
$655$	−1.09682e7	−0.998919
$656$	4.12031e6	0.373827
$657$	0	0
$658$	−1.16725e7	−1.05099
$659$	−1.09503e7	−0.982229	−0.491115	−	0.871095i	$-0.663410\pi$
$659$	−1.09503e7	−0.982229	−0.491115	+	0.871095i	$0.663410\pi$
$660$	0	0
$661$	−9.77692e6	−0.870359	−0.435180	−	0.900344i	$-0.643315\pi$
$661$	−9.77692e6	−0.870359	−0.435180	+	0.900344i	$0.643315\pi$
$662$	3.30516e6	0.293121
$663$	0	0
$664$	−3.09172e7	−2.72132
$665$	1.71106e7	1.50041
$666$	0	0
$667$	5.27316e6	0.458940
$668$	−5.32665e6	−0.461863
$669$	0	0
$670$	2.35375e7	2.02569
$671$	4.83151e6	0.414264
$672$	0	0
$673$	1.07479e7	0.914715	0.457357	−	0.889283i	$-0.348796\pi$
$673$	1.07479e7	0.914715	0.457357	+	0.889283i	$0.348796\pi$
$674$	−1.35440e7	−1.14841
$675$	0	0
$676$	3.18242e7	2.67849
$677$	−1.87032e6	−0.156835	−0.0784177	−	0.996921i	$-0.524987\pi$
$677$	−1.87032e6	−0.156835	−0.0784177	+	0.996921i	$0.524987\pi$
$678$	0	0
$679$	836517.	0.0696306
$680$	−3.94715e7	−3.27349
$681$	0	0
$682$	7.59921e6	0.625615
$683$	5.71730e6	0.468963	0.234482	−	0.972121i	$-0.424661\pi$
$683$	5.71730e6	0.468963	0.234482	+	0.972121i	$0.424661\pi$
$684$	0	0
$685$	2.49839e6	0.203439
$686$	−3.71802e7	−3.01649
$687$	0	0
$688$	−1.58287e7	−1.27489
$689$	9.26481e6	0.743513
$690$	0	0
$691$	9.41471e6	0.750087	0.375044	−	0.927007i	$-0.377628\pi$
$691$	9.41471e6	0.750087	0.375044	+	0.927007i	$0.377628\pi$
$692$	−1.62102e7	−1.28683
$693$	0	0
$694$	−5.50088e6	−0.433544
$695$	−2.14893e7	−1.68756
$696$	0	0
$697$	−6.17662e6	−0.481580
$698$	−2.23772e6	−0.173847
$699$	0	0
$700$	4.23320e7	3.26530
$701$	1.87698e7	1.44266	0.721332	−	0.692589i	$-0.243530\pi$
$701$	1.87698e7	1.44266	0.721332	+	0.692589i	$0.243530\pi$
$702$	0	0
$703$	1.20050e7	0.916162
$704$	3.30680e6	0.251464
$705$	0	0
$706$	−2.96194e7	−2.23647
$707$	−3.82411e7	−2.87728
$708$	0	0
$709$	7.43878e6	0.555759	0.277879	−	0.960616i	$-0.410368\pi$
$709$	7.43878e6	0.555759	0.277879	+	0.960616i	$0.410368\pi$
$710$	3.86538e7	2.87771
$711$	0	0
$712$	3.43000e7	2.53568
$713$	−1.42346e7	−1.04863
$714$	0	0
$715$	−7.60040e6	−0.555995
$716$	−2.49666e7	−1.82002
$717$	0	0
$718$	4.34200e7	3.14325
$719$	−1.04144e6	−0.0751295	−0.0375648	−	0.999294i	$-0.511960\pi$
$719$	−1.04144e6	−0.0751295	−0.0375648	+	0.999294i	$0.511960\pi$
$720$	0	0
$721$	−3.29872e7	−2.36324
$722$	1.48985e7	1.06365
$723$	0	0
$724$	1.73027e7	1.22678
$725$	−7.98510e6	−0.564203
$726$	0	0
$727$	4.56479e6	0.320321	0.160160	−	0.987091i	$-0.448799\pi$
$727$	4.56479e6	0.320321	0.160160	+	0.987091i	$0.448799\pi$
$728$	6.65085e7	4.65102
$729$	0	0
$730$	−2.89222e7	−2.00874
$731$	2.37282e7	1.64237
$732$	0	0
$733$	−1.83949e7	−1.26456	−0.632278	−	0.774742i	$-0.717880\pi$
$733$	−1.83949e7	−1.26456	−0.632278	+	0.774742i	$0.717880\pi$
$734$	−1.38440e6	−0.0948464
$735$	0	0
$736$	502732.	0.0342091
$737$	−3.23701e6	−0.219520
$738$	0	0
$739$	1.71709e7	1.15660	0.578299	−	0.815825i	$-0.303717\pi$
$739$	1.71709e7	1.15660	0.578299	+	0.815825i	$0.303717\pi$
$740$	6.14269e7	4.12363
$741$	0	0
$742$	2.19692e7	1.46489
$743$	−1.78392e6	−0.118550	−0.0592752	−	0.998242i	$-0.518879\pi$
$743$	−1.78392e6	−0.118550	−0.0592752	+	0.998242i	$0.518879\pi$
$744$	0	0
$745$	−2.03375e6	−0.134248
$746$	4.53178e7	2.98141
$747$	0	0
$748$	1.07878e7	0.704982
$749$	−2.75100e7	−1.79178
$750$	0	0
$751$	−2.00358e7	−1.29630	−0.648152	−	0.761511i	$-0.724458\pi$
$751$	−2.00358e7	−1.29630	−0.648152	+	0.761511i	$0.724458\pi$
$752$	5.62852e6	0.362952
$753$	0	0
$754$	−2.49318e7	−1.59707
$755$	999538.	0.0638164
$756$	0	0
$757$	−2.54926e7	−1.61687	−0.808433	−	0.588589i	$-0.799684\pi$
$757$	−2.54926e7	−1.61687	−0.808433	+	0.588589i	$0.799684\pi$
$758$	2.57474e7	1.62765
$759$	0	0
$760$	−2.42390e7	−1.52223
$761$	1.84607e7	1.15554	0.577771	−	0.816199i	$-0.303923\pi$
$761$	1.84607e7	1.15554	0.577771	+	0.816199i	$0.303923\pi$
$762$	0	0
$763$	1.93906e7	1.20581
$764$	−3.28657e6	−0.203709
$765$	0	0
$766$	3.00677e6	0.185152
$767$	−1.53255e6	−0.0940645
$768$	0	0
$769$	1.55369e7	0.947432	0.473716	−	0.880678i	$-0.342912\pi$
$769$	1.55369e7	0.947432	0.473716	+	0.880678i	$0.342912\pi$
$770$	−1.80224e7	−1.09543
$771$	0	0
$772$	−2.28809e7	−1.38175
$773$	−9.75906e6	−0.587434	−0.293717	−	0.955892i	$-0.594892\pi$
$773$	−9.75906e6	−0.587434	−0.293717	+	0.955892i	$0.594892\pi$
$774$	0	0
$775$	2.15553e7	1.28914
$776$	−1.18502e6	−0.0706431
$777$	0	0
$778$	3.64402e6	0.215840
$779$	−3.79299e6	−0.223944
$780$	0	0
$781$	−5.31590e6	−0.311852
$782$	−3.02464e7	−1.76871
$783$	0	0
$784$	3.58055e7	2.08046
$785$	−6.92002e6	−0.400805
$786$	0	0
$787$	1.36290e7	0.784383	0.392191	−	0.919884i	$-0.371717\pi$
$787$	1.36290e7	0.784383	0.392191	+	0.919884i	$0.371717\pi$
$788$	4.26008e7	2.44401
$789$	0	0
$790$	1.41728e7	0.807955
$791$	−2.72288e7	−1.54735
$792$	0	0
$793$	4.27900e7	2.41635
$794$	−2.61763e6	−0.147352
$795$	0	0
$796$	−8.81375e6	−0.493035
$797$	1.33882e7	0.746578	0.373289	−	0.927715i	$-0.378230\pi$
$797$	1.33882e7	0.746578	0.373289	+	0.927715i	$0.378230\pi$
$798$	0	0
$799$	−8.43751e6	−0.467571
$800$	−761284.	−0.0420554
$801$	0	0
$802$	3.77778e7	2.07396
$803$	3.97755e6	0.217684
$804$	0	0
$805$	3.37590e7	1.83611
$806$	6.73019e7	3.64913
$807$	0	0
$808$	5.41727e7	2.91912
$809$	3.54909e7	1.90654	0.953270	−	0.302120i	$-0.0976944\pi$
$809$	3.54909e7	1.90654	0.953270	+	0.302120i	$0.0976944\pi$
$810$	0	0
$811$	−1.61925e7	−0.864495	−0.432247	−	0.901755i	$-0.642279\pi$
$811$	−1.61925e7	−0.864495	−0.432247	+	0.901755i	$0.642279\pi$
$812$	−3.94971e7	−2.10220
$813$	0	0
$814$	−1.26447e7	−0.668878
$815$	1.54746e7	0.816067
$816$	0	0
$817$	1.45713e7	0.763734
$818$	2.83402e6	0.148088
$819$	0	0
$820$	−1.94080e7	−1.00797
$821$	2.16190e7	1.11938	0.559689	−	0.828702i	$-0.310920\pi$
$821$	2.16190e7	1.11938	0.559689	+	0.828702i	$0.310920\pi$
$822$	0	0
$823$	1.14562e7	0.589578	0.294789	−	0.955562i	$-0.404751\pi$
$823$	1.14562e7	0.589578	0.294789	+	0.955562i	$0.404751\pi$
$824$	4.67300e7	2.39760
$825$	0	0
$826$	−3.63405e6	−0.185328
$827$	−3.51307e7	−1.78617	−0.893085	−	0.449887i	$-0.851464\pi$
$827$	−3.51307e7	−1.78617	−0.893085	+	0.449887i	$0.851464\pi$
$828$	0	0
$829$	−1.17530e7	−0.593968	−0.296984	−	0.954882i	$-0.595981\pi$
$829$	−1.17530e7	−0.593968	−0.296984	+	0.954882i	$0.595981\pi$
$830$	7.41985e7	3.73852
$831$	0	0
$832$	2.92864e7	1.46676
$833$	−5.36747e7	−2.68014
$834$	0	0
$835$	6.43257e6	0.319277
$836$	6.62466e6	0.327829
$837$	0	0
$838$	−5.01795e7	−2.46841
$839$	−2.81875e7	−1.38246	−0.691230	−	0.722635i	$-0.742931\pi$
$839$	−2.81875e7	−1.38246	−0.691230	+	0.722635i	$0.742931\pi$
$840$	0	0
$841$	−1.30608e7	−0.636766
$842$	−5.01767e7	−2.43906
$843$	0	0
$844$	−2.66632e7	−1.28842
$845$	−3.84315e7	−1.85160
$846$	0	0
$847$	−3.37021e7	−1.61417
$848$	−1.05936e7	−0.505886
$849$	0	0
$850$	4.58019e7	2.17438
$851$	2.36856e7	1.12114
$852$	0	0
$853$	2.14819e7	1.01088	0.505440	−	0.862862i	$-0.331330\pi$
$853$	2.14819e7	1.01088	0.505440	+	0.862862i	$0.331330\pi$
$854$	1.01466e8	4.76074
$855$	0	0
$856$	3.89708e7	1.81784
$857$	−1.80715e7	−0.840507	−0.420254	−	0.907407i	$-0.638059\pi$
$857$	−1.80715e7	−0.840507	−0.420254	+	0.907407i	$0.638059\pi$
$858$	0	0
$859$	−1.80796e7	−0.835999	−0.417999	−	0.908447i	$-0.637269\pi$
$859$	−1.80796e7	−0.835999	−0.417999	+	0.908447i	$0.637269\pi$
$860$	7.45582e7	3.43755
$861$	0	0
$862$	5.71368e6	0.261907
$863$	1.48203e7	0.677377	0.338688	−	0.940899i	$-0.390017\pi$
$863$	1.48203e7	0.677377	0.338688	+	0.940899i	$0.390017\pi$
$864$	0	0
$865$	1.95757e7	0.889566
$866$	−3.96439e7	−1.79631
$867$	0	0
$868$	1.06620e8	4.80329
$869$	−1.94912e6	−0.0875567
$870$	0	0
$871$	−2.86683e7	−1.28043
$872$	−2.74689e7	−1.22335
$873$	0	0
$874$	−1.85740e7	−0.822482
$875$	3.48713e6	0.153974
$876$	0	0
$877$	−3.52762e7	−1.54876	−0.774378	−	0.632723i	$-0.781937\pi$
$877$	−3.52762e7	−1.54876	−0.774378	+	0.632723i	$0.781937\pi$
$878$	−3.54481e7	−1.55187
$879$	0	0
$880$	8.69045e6	0.378299
$881$	2.90408e6	0.126057	0.0630287	−	0.998012i	$-0.479924\pi$
$881$	2.90408e6	0.126057	0.0630287	+	0.998012i	$0.479924\pi$
$882$	0	0
$883$	−2.16118e7	−0.932803	−0.466401	−	0.884573i	$-0.654450\pi$
$883$	−2.16118e7	−0.932803	−0.466401	+	0.884573i	$0.654450\pi$
$884$	9.55413e7	4.11207
$885$	0	0
$886$	1.07149e6	0.0458568
$887$	−2.88078e7	−1.22942	−0.614711	−	0.788752i	$-0.710728\pi$
$887$	−2.88078e7	−1.22942	−0.614711	+	0.788752i	$0.710728\pi$
$888$	0	0
$889$	−4.42013e6	−0.187578
$890$	−8.23170e7	−3.48349
$891$	0	0
$892$	−2.51305e7	−1.05752
$893$	−5.18139e6	−0.217429
$894$	0	0
$895$	3.01502e7	1.25815
$896$	7.13161e7	2.96769
$897$	0	0
$898$	−1.26673e7	−0.524196
$899$	−2.01118e7	−0.829949
$900$	0	0
$901$	1.58805e7	0.651705
$902$	3.99512e6	0.163498
$903$	0	0
$904$	3.85726e7	1.56985
$905$	−2.08951e7	−0.848052
$906$	0	0
$907$	−2.88533e6	−0.116460	−0.0582302	−	0.998303i	$-0.518546\pi$
$907$	−2.88533e6	−0.116460	−0.0582302	+	0.998303i	$0.518546\pi$
$908$	2.97239e7	1.19644
$909$	0	0
$910$	−1.59614e8	−6.38952
$911$	−1.43791e7	−0.574031	−0.287015	−	0.957926i	$-0.592663\pi$
$911$	−1.43791e7	−0.574031	−0.287015	+	0.957926i	$0.592663\pi$
$912$	0	0
$913$	−1.02042e7	−0.405137
$914$	−8.20276e7	−3.24784
$915$	0	0
$916$	4.80482e6	0.189207
$917$	3.16776e7	1.24403
$918$	0	0
$919$	4.00971e7	1.56612	0.783058	−	0.621949i	$-0.213659\pi$
$919$	4.00971e7	1.56612	0.783058	+	0.621949i	$0.213659\pi$
$920$	−4.78232e7	−1.86281
$921$	0	0
$922$	5.44567e7	2.10972
$923$	−4.70799e7	−1.81899
$924$	0	0
$925$	−3.58669e7	−1.37829
$926$	−86109.1	−0.00330006
$927$	0	0
$928$	710302.	0.0270753
$929$	−3.70883e7	−1.40993	−0.704965	−	0.709242i	$-0.749037\pi$
$929$	−3.70883e7	−1.40993	−0.704965	+	0.709242i	$0.749037\pi$
$930$	0	0
$931$	−3.29611e7	−1.24631
$932$	1.07836e7	0.406653
$933$	0	0
$934$	6.03007e7	2.26180
$935$	−1.30275e7	−0.487342
$936$	0	0
$937$	9.91089e6	0.368777	0.184388	−	0.982853i	$-0.440970\pi$
$937$	9.91089e6	0.368777	0.184388	+	0.982853i	$0.440970\pi$
$938$	−6.79797e7	−2.52274
$939$	0	0
$940$	−2.65121e7	−0.978644
$941$	3.56335e7	1.31185	0.655925	−	0.754826i	$-0.272279\pi$
$941$	3.56335e7	1.31185	0.655925	+	0.754826i	$0.272279\pi$
$942$	0	0
$943$	−7.48352e6	−0.274048
$944$	1.75235e6	0.0640015
$945$	0	0
$946$	−1.53477e7	−0.557592
$947$	1.44861e7	0.524899	0.262450	−	0.964946i	$-0.415470\pi$
$947$	1.44861e7	0.524899	0.262450	+	0.964946i	$0.415470\pi$
$948$	0	0
$949$	3.52269e7	1.26972
$950$	2.81265e7	1.01113
$951$	0	0
$952$	1.14000e8	4.07672
$953$	1.98924e7	0.709505	0.354753	−	0.934960i	$-0.384565\pi$
$953$	1.98924e7	0.709505	0.354753	+	0.934960i	$0.384565\pi$
$954$	0	0
$955$	3.96893e6	0.140820
$956$	−4.54981e7	−1.61008
$957$	0	0
$958$	−2.52890e7	−0.890262
$959$	−7.21573e6	−0.253358
$960$	0	0
$961$	2.56614e7	0.896337
$962$	−1.11987e8	−3.90148
$963$	0	0
$964$	9.32377e7	3.23146
$965$	2.76315e7	0.955180
$966$	0	0
$967$	3.02252e7	1.03945	0.519724	−	0.854334i	$-0.326035\pi$
$967$	3.02252e7	1.03945	0.519724	+	0.854334i	$0.326035\pi$
$968$	4.77427e7	1.63764
$969$	0	0
$970$	2.84393e6	0.0970488
$971$	−1.51348e7	−0.515144	−0.257572	−	0.966259i	$-0.582922\pi$
$971$	−1.51348e7	−0.515144	−0.257572	+	0.966259i	$0.582922\pi$
$972$	0	0
$973$	6.20642e7	2.10164
$974$	−2.37766e7	−0.803069
$975$	0	0
$976$	−4.89269e7	−1.64408
$977$	−1.33828e7	−0.448548	−0.224274	−	0.974526i	$-0.572001\pi$
$977$	−1.33828e7	−0.448548	−0.224274	+	0.974526i	$0.572001\pi$
$978$	0	0
$979$	1.13207e7	0.377500
$980$	−1.68655e8	−5.60963
$981$	0	0
$982$	−7.72264e7	−2.55557
$983$	4.91967e7	1.62387	0.811937	−	0.583746i	$-0.198414\pi$
$983$	4.91967e7	1.62387	0.811937	+	0.583746i	$0.198414\pi$
$984$	0	0
$985$	−5.14456e7	−1.68950
$986$	−4.27346e7	−1.39987
$987$	0	0
$988$	5.86709e7	1.91219
$989$	2.87489e7	0.934609
$990$	0	0
$991$	5.27043e7	1.70476	0.852378	−	0.522927i	$-0.175160\pi$
$991$	5.27043e7	1.70476	0.852378	+	0.522927i	$0.175160\pi$
$992$	−1.91742e6	−0.0618639
$993$	0	0
$994$	−1.11638e8	−3.58382
$995$	1.06437e7	0.340827
$996$	0	0
$997$	4.95166e7	1.57766	0.788829	−	0.614612i	$-0.210688\pi$
$997$	4.95166e7	1.57766	0.788829	+	0.614612i	$0.210688\pi$
$998$	−8.97623e7	−2.85278
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.6.a.b.1.2	yes	24
3.2	odd	2		729.6.a.a.1.23	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
729.6.a.a.1.23	✓	24	3.2	odd	2
729.6.a.b.1.2	yes	24	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)

\(f(q)\)	\(=\)	\(q-9.81893 q^{2} +64.4115 q^{4} -77.7846 q^{5} +224.654 q^{7} -318.246 q^{8} +763.762 q^{10} -105.037 q^{11} -930.252 q^{13} -2205.86 q^{14} +1063.67 q^{16} -1594.51 q^{17} -979.171 q^{19} -5010.22 q^{20} +1031.35 q^{22} -1931.89 q^{23} +2925.45 q^{25} +9134.09 q^{26} +14470.3 q^{28} -2729.53 q^{29} +7368.21 q^{31} -260.228 q^{32} +15656.4 q^{34} -17474.6 q^{35} -12260.3 q^{37} +9614.42 q^{38} +24754.6 q^{40} +3873.68 q^{41} -14881.2 q^{43} -6765.58 q^{44} +18969.1 q^{46} +5291.61 q^{47} +33662.2 q^{49} -28724.8 q^{50} -59918.9 q^{52} -9959.46 q^{53} +8170.25 q^{55} -71495.1 q^{56} +26801.1 q^{58} +1647.45 q^{59} -45998.2 q^{61} -72348.0 q^{62} -31482.3 q^{64} +72359.3 q^{65} +30817.8 q^{67} -102705. q^{68} +171582. q^{70} +50609.8 q^{71} -37868.1 q^{73} +120383. q^{74} -63069.8 q^{76} -23596.9 q^{77} +18556.5 q^{79} -82737.1 q^{80} -38035.4 q^{82} +97148.8 q^{83} +124028. q^{85} +146118. q^{86} +33427.6 q^{88} -107778. q^{89} -208984. q^{91} -124436. q^{92} -51957.9 q^{94} +76164.4 q^{95} +3723.59 q^{97} -330527. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 12 q^{2} + 432 q^{4} + 33 q^{5} + 294 q^{7} + 843 q^{8} + 600 q^{10} + 30 q^{11} + 1014 q^{13} - 3720 q^{14} + 8448 q^{16} - 2709 q^{17} + 4332 q^{19} + 861 q^{20} - 2193 q^{22} + 4020 q^{23} + 15417 q^{25}+ \cdots + 944667 q^{98}+O(q^{100}) \) 24 * q + 12 * q^2 + 432 * q^4 + 33 * q^5 + 294 * q^7 + 843 * q^8 + 600 * q^10 + 30 * q^11 + 1014 * q^13 - 3720 * q^14 + 8448 * q^16 - 2709 * q^17 + 4332 * q^19 + 861 * q^20 - 2193 * q^22 + 4020 * q^23 + 15417 * q^25 + 17574 * q^26 + 4539 * q^28 - 15279 * q^29 + 17331 * q^31 + 43983 * q^32 + 20412 * q^34 + 2514 * q^35 + 11397 * q^37 + 78873 * q^38 + 52185 * q^40 - 31566 * q^41 + 34422 * q^43 - 63213 * q^44 + 62907 * q^46 + 507 * q^47 + 104106 * q^49 + 64929 * q^50 + 99237 * q^52 + 72297 * q^53 + 106611 * q^55 - 213900 * q^56 + 107733 * q^58 + 59226 * q^59 + 112452 * q^61 + 32100 * q^62 + 287115 * q^64 + 117795 * q^65 + 86055 * q^67 - 439146 * q^68 + 208623 * q^70 + 34992 * q^71 + 89580 * q^73 + 114360 * q^74 + 302031 * q^76 + 87639 * q^77 + 225600 * q^79 - 122331 * q^80 + 74526 * q^82 - 141234 * q^83 - 9900 * q^85 + 336804 * q^86 + 209895 * q^88 - 236637 * q^89 + 24354 * q^91 + 191454 * q^92 + 256080 * q^94 - 254139 * q^95 + 39363 * q^97 + 944667 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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