LMFDB - Embedded newform 729.6.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [729,6,Mod(1,729)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$116.919804644$
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			1.2
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)

$f(q)$	$=$	$q-10.3172 q^{2} +74.4455 q^{4} -13.8184 q^{5} -109.502 q^{7} -437.920 q^{8} +O(q^{10})$	\(q-10.3172 q^{2} +74.4455 q^{4} -13.8184 q^{5} -109.502 q^{7} -437.920 q^{8} +142.567 q^{10} +558.001 q^{11} -912.906 q^{13} +1129.75 q^{14} +2135.87 q^{16} -498.721 q^{17} +2862.57 q^{19} -1028.72 q^{20} -5757.03 q^{22} +387.431 q^{23} -2934.05 q^{25} +9418.67 q^{26} -8151.90 q^{28} -651.910 q^{29} -6066.06 q^{31} -8022.87 q^{32} +5145.42 q^{34} +1513.13 q^{35} -650.491 q^{37} -29533.8 q^{38} +6051.34 q^{40} +3590.03 q^{41} -14709.9 q^{43} +41540.7 q^{44} -3997.22 q^{46} +15325.9 q^{47} -4816.39 q^{49} +30271.3 q^{50} -67961.7 q^{52} +23094.4 q^{53} -7710.66 q^{55} +47953.0 q^{56} +6725.91 q^{58} -1588.02 q^{59} -3019.07 q^{61} +62585.0 q^{62} +14426.0 q^{64} +12614.9 q^{65} +6675.82 q^{67} -37127.5 q^{68} -15611.4 q^{70} +35684.6 q^{71} +46554.6 q^{73} +6711.27 q^{74} +213105. q^{76} -61102.0 q^{77} +14759.8 q^{79} -29514.3 q^{80} -37039.2 q^{82} -73279.7 q^{83} +6891.51 q^{85} +151765. q^{86} -244360. q^{88} -34020.8 q^{89} +99964.7 q^{91} +28842.5 q^{92} -158121. q^{94} -39556.0 q^{95} -13821.2 q^{97} +49691.9 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 42 q + 12 q^{2} + 624 q^{4} + 150 q^{5} + 573 q^{8}+O(q^{10}) $ 42 * q + 12 * q^2 + 624 * q^4 + 150 * q^5 + 573 * q^8	$ 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}+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

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−10.3172	−1.82385	−0.911924	−	0.410359i	$-0.865403\pi$
$2$	−10.3172	−1.82385	−0.911924	+	0.410359i	$0.865403\pi$
$3$	0	0
$3$	0	0
$4$	74.4455	2.32642
$5$	−13.8184	−0.247190	−0.123595	−	0.992333i	$-0.539442\pi$
$5$	−13.8184	−0.247190	−0.123595	+	0.992333i	$0.539442\pi$
$6$	0	0
$7$	−109.502	−0.844648	−0.422324	−	0.906445i	$-0.638785\pi$
$7$	−109.502	−0.844648	−0.422324	+	0.906445i	$0.638785\pi$
$8$	−437.920	−2.41919
$9$	0	0
$10$	142.567	0.450838
$11$	558.001	1.39044	0.695222	−	0.718795i	$-0.255306\pi$
$11$	558.001	1.39044	0.695222	+	0.718795i	$0.255306\pi$
$12$	0	0
$13$	−912.906	−1.49819	−0.749096	−	0.662461i	$-0.769512\pi$
$13$	−912.906	−1.49819	−0.749096	+	0.662461i	$0.769512\pi$
$14$	1129.75	1.54051
$15$	0	0
$16$	2135.87	2.08581
$17$	−498.721	−0.418538	−0.209269	−	0.977858i	$-0.567108\pi$
$17$	−498.721	−0.418538	−0.209269	+	0.977858i	$0.567108\pi$
$18$	0	0
$19$	2862.57	1.81916	0.909582	−	0.415523i	$-0.136402\pi$
$19$	2862.57	1.81916	0.909582	+	0.415523i	$0.136402\pi$
$20$	−1028.72	−0.575069
$21$	0	0
$22$	−5757.03	−2.53596
$23$	387.431	0.152713	0.0763563	−	0.997081i	$-0.475671\pi$
$23$	387.431	0.152713	0.0763563	+	0.997081i	$0.475671\pi$
$24$	0	0
$25$	−2934.05	−0.938897
$26$	9418.67	2.73248
$27$	0	0
$28$	−8151.90	−1.96501
$29$	−651.910	−0.143944	−0.0719719	−	0.997407i	$-0.522929\pi$
$29$	−651.910	−0.143944	−0.0719719	+	0.997407i	$0.522929\pi$
$30$	0	0
$31$	−6066.06	−1.13371	−0.566856	−	0.823817i	$-0.691840\pi$
$31$	−6066.06	−1.13371	−0.566856	+	0.823817i	$0.691840\pi$
$32$	−8022.87	−1.38502
$33$	0	0
$34$	5145.42	0.763350
$35$	1513.13	0.208789
$36$	0	0
$37$	−650.491	−0.0781154	−0.0390577	−	0.999237i	$-0.512436\pi$
$37$	−650.491	−0.0781154	−0.0390577	+	0.999237i	$0.512436\pi$
$38$	−29533.8	−3.31788
$39$	0	0
$40$	6051.34	0.598001
$41$	3590.03	0.333533	0.166766	−	0.985996i	$-0.446667\pi$
$41$	3590.03	0.333533	0.166766	+	0.985996i	$0.446667\pi$
$42$	0	0
$43$	−14709.9	−1.21321	−0.606607	−	0.795002i	$-0.707470\pi$
$43$	−14709.9	−1.21321	−0.606607	+	0.795002i	$0.707470\pi$
$44$	41540.7	3.23476
$45$	0	0
$46$	−3997.22	−0.278525
$47$	15325.9	1.01200	0.506002	−	0.862532i	$-0.331123\pi$
$47$	15325.9	1.01200	0.506002	+	0.862532i	$0.331123\pi$
$48$	0	0
$49$	−4816.39	−0.286571
$50$	30271.3	1.71241
$51$	0	0
$52$	−67961.7	−3.48543
$53$	23094.4	1.12932	0.564661	−	0.825323i	$-0.309007\pi$
$53$	23094.4	1.12932	0.564661	+	0.825323i	$0.309007\pi$
$54$	0	0
$55$	−7710.66	−0.343704
$56$	47953.0	2.04336
$57$	0	0
$58$	6725.91	0.262531
$59$	−1588.02	−0.0593918	−0.0296959	−	0.999559i	$-0.509454\pi$
$59$	−1588.02	−0.0593918	−0.0296959	+	0.999559i	$0.509454\pi$
$60$	0	0
$61$	−3019.07	−0.103884	−0.0519420	−	0.998650i	$-0.516541\pi$
$61$	−3019.07	−0.103884	−0.0519420	+	0.998650i	$0.516541\pi$
$62$	62585.0	2.06772
$63$	0	0
$64$	14426.0	0.440246
$65$	12614.9	0.370339
$66$	0	0
$67$	6675.82	0.181684	0.0908422	−	0.995865i	$-0.471044\pi$
$67$	6675.82	0.181684	0.0908422	+	0.995865i	$0.471044\pi$
$68$	−37127.5	−0.973696
$69$	0	0
$70$	−15611.4	−0.380799
$71$	35684.6	0.840107	0.420053	−	0.907499i	$-0.362011\pi$
$71$	35684.6	0.840107	0.420053	+	0.907499i	$0.362011\pi$
$72$	0	0
$73$	46554.6	1.02248	0.511240	−	0.859438i	$-0.329186\pi$
$73$	46554.6	1.02248	0.511240	+	0.859438i	$0.329186\pi$
$74$	6711.27	0.142471
$75$	0	0
$76$	213105.	4.23214
$77$	−61102.0	−1.17443
$78$	0	0
$79$	14759.8	0.266080	0.133040	−	0.991111i	$-0.457526\pi$
$79$	14759.8	0.266080	0.133040	+	0.991111i	$0.457526\pi$
$80$	−29514.3	−0.515593
$81$	0	0
$82$	−37039.2	−0.608313
$83$	−73279.7	−1.16758	−0.583792	−	0.811903i	$-0.698432\pi$
$83$	−73279.7	−1.16758	−0.583792	+	0.811903i	$0.698432\pi$
$84$	0	0
$85$	6891.51	0.103459
$86$	151765.	2.21272
$87$	0	0
$88$	−244360.	−3.36375
$89$	−34020.8	−0.455271	−0.227636	−	0.973746i	$-0.573099\pi$
$89$	−34020.8	−0.455271	−0.227636	+	0.973746i	$0.573099\pi$
$90$	0	0
$91$	99964.7	1.26544
$92$	28842.5	0.355274
$93$	0	0
$94$	−158121.	−1.84574
$95$	−39556.0	−0.449680
$96$	0	0
$97$	−13821.2	−0.149148	−0.0745740	−	0.997215i	$-0.523760\pi$
$97$	−13821.2	−0.149148	−0.0745740	+	0.997215i	$0.523760\pi$
$98$	49691.9	0.522661
$99$	0	0
$100$	−218427.	−2.18427
$101$	−57259.6	−0.558528	−0.279264	−	0.960214i	$-0.590091\pi$
$101$	−57259.6	−0.558528	−0.279264	+	0.960214i	$0.590091\pi$
$102$	0	0
$103$	−187180.	−1.73847	−0.869234	−	0.494400i	$-0.835388\pi$
$103$	−187180.	−1.73847	−0.869234	+	0.494400i	$0.835388\pi$
$104$	399780.	3.62441
$105$	0	0
$106$	−238271.	−2.05971
$107$	−200880.	−1.69620	−0.848100	−	0.529837i	$-0.822253\pi$
$107$	−200880.	−1.69620	−0.848100	+	0.529837i	$0.822253\pi$
$108$	0	0
$109$	12021.0	0.0969112	0.0484556	−	0.998825i	$-0.484570\pi$
$109$	12021.0	0.0969112	0.0484556	+	0.998825i	$0.484570\pi$
$110$	79552.8	0.626865
$111$	0	0
$112$	−233882.	−1.76178
$113$	−91658.2	−0.675267	−0.337633	−	0.941278i	$-0.609626\pi$
$113$	−91658.2	−0.675267	−0.337633	+	0.941278i	$0.609626\pi$
$114$	0	0
$115$	−5353.67	−0.0377491
$116$	−48531.8	−0.334874
$117$	0	0
$118$	16384.0	0.108322
$119$	54610.7	0.353517
$120$	0	0
$121$	150314.	0.933333
$122$	31148.5	0.189469
$123$	0	0
$124$	−451591.	−2.63749
$125$	83726.2	0.479277
$126$	0	0
$127$	40462.2	0.222608	0.111304	−	0.993786i	$-0.464497\pi$
$127$	40462.2	0.222608	0.111304	+	0.993786i	$0.464497\pi$
$128$	107896.	0.582075
$129$	0	0
$130$	−130151.	−0.675442
$131$	−170720.	−0.869175	−0.434588	−	0.900630i	$-0.643106\pi$
$131$	−170720.	−0.869175	−0.434588	+	0.900630i	$0.643106\pi$
$132$	0	0
$133$	−313456.	−1.53655
$134$	−68876.0	−0.331365
$135$	0	0
$136$	218400.	1.01252
$137$	−145149.	−0.660713	−0.330356	−	0.943856i	$-0.607169\pi$
$137$	−145149.	−0.660713	−0.330356	+	0.943856i	$0.607169\pi$
$138$	0	0
$139$	11770.0	0.0516700	0.0258350	−	0.999666i	$-0.491776\pi$
$139$	11770.0	0.0516700	0.0258350	+	0.999666i	$0.491776\pi$
$140$	112646.	0.485731
$141$	0	0
$142$	−368166.	−1.53223
$143$	−509403.	−2.08315
$144$	0	0
$145$	9008.34	0.0355815
$146$	−480315.	−1.86485
$147$	0	0
$148$	−48426.1	−0.181729
$149$	488903.	1.80409	0.902043	−	0.431647i	$-0.142067\pi$
$149$	488903.	1.80409	0.902043	+	0.431647i	$0.142067\pi$
$150$	0	0
$151$	−263361.	−0.939959	−0.469979	−	0.882677i	$-0.655739\pi$
$151$	−263361.	−0.939959	−0.469979	+	0.882677i	$0.655739\pi$
$152$	−1.25358e6	−4.40091
$153$	0	0
$154$	630404.	2.14199
$155$	83823.1	0.280243
$156$	0	0
$157$	−76462.0	−0.247569	−0.123785	−	0.992309i	$-0.539503\pi$
$157$	−76462.0	−0.247569	−0.123785	+	0.992309i	$0.539503\pi$
$158$	−152280.	−0.485290
$159$	0	0
$160$	110863.	0.342363
$161$	−42424.4	−0.128988
$162$	0	0
$163$	−466161.	−1.37425	−0.687126	−	0.726538i	$-0.741128\pi$
$163$	−466161.	−1.37425	−0.687126	+	0.726538i	$0.741128\pi$
$164$	267261.	0.775938
$165$	0	0
$166$	756044.	2.12950
$167$	368817.	1.02334	0.511670	−	0.859182i	$-0.329027\pi$
$167$	368817.	1.02334	0.511670	+	0.859182i	$0.329027\pi$
$168$	0	0
$169$	462104.	1.24458
$170$	−71101.3	−0.188693
$171$	0	0
$172$	−1.09508e6	−2.82245
$173$	204375.	0.519175	0.259587	−	0.965720i	$-0.416413\pi$
$173$	204375.	0.519175	0.259587	+	0.965720i	$0.416413\pi$
$174$	0	0
$175$	321284.	0.793037
$176$	1.19182e6	2.90021
$177$	0	0
$178$	351001.	0.830345
$179$	−274980.	−0.641458	−0.320729	−	0.947171i	$-0.603928\pi$
$179$	−274980.	−0.641458	−0.320729	+	0.947171i	$0.603928\pi$
$180$	0	0
$181$	373053.	0.846396	0.423198	−	0.906037i	$-0.360907\pi$
$181$	373053.	0.846396	0.423198	+	0.906037i	$0.360907\pi$
$182$	−1.03136e6	−2.30798
$183$	0	0
$184$	−169664.	−0.369441
$185$	8988.72	0.0193094
$186$	0	0
$187$	−278287.	−0.581954
$188$	1.14095e6	2.35435
$189$	0	0
$190$	408109.	0.820148
$191$	−348840.	−0.691899	−0.345949	−	0.938253i	$-0.612443\pi$
$191$	−348840.	−0.691899	−0.345949	+	0.938253i	$0.612443\pi$
$192$	0	0
$193$	545457.	1.05407	0.527033	−	0.849845i	$-0.323305\pi$
$193$	545457.	1.05407	0.527033	+	0.849845i	$0.323305\pi$
$194$	142597.	0.272023
$195$	0	0
$196$	−358558.	−0.666684
$197$	735997.	1.35117	0.675586	−	0.737281i	$-0.263891\pi$
$197$	735997.	1.35117	0.675586	+	0.737281i	$0.263891\pi$
$198$	0	0
$199$	296024.	0.529900	0.264950	−	0.964262i	$-0.414645\pi$
$199$	296024.	0.529900	0.264950	+	0.964262i	$0.414645\pi$
$200$	1.28488e6	2.27137
$201$	0	0
$202$	590762.	1.01867
$203$	71385.2	0.121582
$204$	0	0
$205$	−49608.4	−0.0824461
$206$	1.93118e6	3.17070
$207$	0	0
$208$	−1.94985e6	−3.12495
$209$	1.59732e6	2.52945
$210$	0	0
$211$	205386.	0.317588	0.158794	−	0.987312i	$-0.449239\pi$
$211$	205386.	0.317588	0.158794	+	0.987312i	$0.449239\pi$
$212$	1.71928e6	2.62728
$213$	0	0
$214$	2.07253e6	3.09361
$215$	203266.	0.299895
$216$	0	0
$217$	664244.	0.957587
$218$	−124023.	−0.176751
$219$	0	0
$220$	−574024.	−0.799601
$221$	455285.	0.627051
$222$	0	0
$223$	929336.	1.25144	0.625721	−	0.780047i	$-0.284805\pi$
$223$	929336.	1.25144	0.625721	+	0.780047i	$0.284805\pi$
$224$	878518.	1.16985
$225$	0	0
$226$	945660.	1.23158
$227$	−51928.1	−0.0668864	−0.0334432	−	0.999441i	$-0.510647\pi$
$227$	−51928.1	−0.0668864	−0.0334432	+	0.999441i	$0.510647\pi$
$228$	0	0
$229$	−363808.	−0.458441	−0.229220	−	0.973375i	$-0.573618\pi$
$229$	−363808.	−0.458441	−0.229220	+	0.973375i	$0.573618\pi$
$230$	55235.1	0.0688487
$231$	0	0
$232$	285485.	0.348227
$233$	72237.5	0.0871712	0.0435856	−	0.999050i	$-0.486122\pi$
$233$	72237.5	0.0871712	0.0435856	+	0.999050i	$0.486122\pi$
$234$	0	0
$235$	−211780.	−0.250158
$236$	−118221.	−0.138170
$237$	0	0
$238$	−563432.	−0.644762
$239$	−965695.	−1.09357	−0.546783	−	0.837274i	$-0.684148\pi$
$239$	−965695.	−1.09357	−0.546783	+	0.837274i	$0.684148\pi$
$240$	0	0
$241$	613634.	0.680561	0.340280	−	0.940324i	$-0.389478\pi$
$241$	613634.	0.680561	0.340280	+	0.940324i	$0.389478\pi$
$242$	−1.55083e6	−1.70226
$243$	0	0
$244$	−224756.	−0.241678
$245$	66554.7	0.0708375
$246$	0	0
$247$	−2.61326e6	−2.72546
$248$	2.65645e6	2.74266
$249$	0	0
$250$	−863824.	−0.874128
$251$	996171.	0.998044	0.499022	−	0.866589i	$-0.333693\pi$
$251$	996171.	0.998044	0.499022	+	0.866589i	$0.333693\pi$
$252$	0	0
$253$	216187.	0.212338
$254$	−417458.	−0.406003
$255$	0	0
$256$	−1.57482e6	−1.50186
$257$	−1.69906e6	−1.60464	−0.802319	−	0.596896i	$-0.796401\pi$
$257$	−1.69906e6	−1.60464	−0.802319	+	0.596896i	$0.796401\pi$
$258$	0	0
$259$	71229.8	0.0659800
$260$	939120.	0.861564
$261$	0	0
$262$	1.76136e6	1.58524
$263$	−1.94185e6	−1.73112	−0.865560	−	0.500806i	$-0.833037\pi$
$263$	−1.94185e6	−1.73112	−0.865560	+	0.500806i	$0.833037\pi$
$264$	0	0
$265$	−319127.	−0.279158
$266$	3.23400e6	2.80244
$267$	0	0
$268$	496984.	0.422674
$269$	1.70606e6	1.43752	0.718758	−	0.695261i	$-0.244711\pi$
$269$	1.70606e6	1.43752	0.718758	+	0.695261i	$0.244711\pi$
$270$	0	0
$271$	1.65207e6	1.36648	0.683242	−	0.730192i	$-0.260569\pi$
$271$	1.65207e6	1.36648	0.683242	+	0.730192i	$0.260569\pi$
$272$	−1.06520e6	−0.872993
$273$	0	0
$274$	1.49754e6	1.20504
$275$	−1.63720e6	−1.30548
$276$	0	0
$277$	452293.	0.354177	0.177088	−	0.984195i	$-0.443332\pi$
$277$	452293.	0.354177	0.177088	+	0.984195i	$0.443332\pi$
$278$	−121434.	−0.0942382
$279$	0	0
$280$	−662632.	−0.505100
$281$	1.23906e6	0.936110	0.468055	−	0.883699i	$-0.344955\pi$
$281$	1.23906e6	0.936110	0.468055	+	0.883699i	$0.344955\pi$
$282$	0	0
$283$	−838139.	−0.622085	−0.311043	−	0.950396i	$-0.600678\pi$
$283$	−838139.	−0.622085	−0.311043	+	0.950396i	$0.600678\pi$
$284$	2.65655e6	1.95444
$285$	0	0
$286$	5.25563e6	3.79935
$287$	−393114.	−0.281718
$288$	0	0
$289$	−1.17113e6	−0.824826
$290$	−92941.2	−0.0648953
$291$	0	0
$292$	3.46578e6	2.37872
$293$	−1.81714e6	−1.23657	−0.618286	−	0.785953i	$-0.712173\pi$
$293$	−1.81714e6	−1.23657	−0.618286	+	0.785953i	$0.712173\pi$
$294$	0	0
$295$	21943.9	0.0146811
$296$	284863.	0.188976
$297$	0	0
$298$	−5.04413e6	−3.29038
$299$	−353688.	−0.228793
$300$	0	0
$301$	1.61075e6	1.02474
$302$	2.71716e6	1.71434
$303$	0	0
$304$	6.11409e6	3.79444
$305$	41718.7	0.0256792
$306$	0	0
$307$	1.75250e6	1.06124	0.530619	−	0.847610i	$-0.321959\pi$
$307$	1.75250e6	1.06124	0.530619	+	0.847610i	$0.321959\pi$
$308$	−4.54877e6	−2.73223
$309$	0	0
$310$	−864823.	−0.511120
$311$	3.31381e6	1.94280	0.971398	−	0.237457i	$-0.0763138\pi$
$311$	3.31381e6	1.94280	0.971398	+	0.237457i	$0.0763138\pi$
$312$	0	0
$313$	−517379.	−0.298502	−0.149251	−	0.988799i	$-0.547686\pi$
$313$	−517379.	−0.298502	−0.149251	+	0.988799i	$0.547686\pi$
$314$	788877.	0.451528
$315$	0	0
$316$	1.09880e6	0.619014
$317$	320377.	0.179066	0.0895331	−	0.995984i	$-0.471463\pi$
$317$	320377.	0.179066	0.0895331	+	0.995984i	$0.471463\pi$
$318$	0	0
$319$	−363767.	−0.200146
$320$	−199344.	−0.108825
$321$	0	0
$322$	437702.	0.235255
$323$	−1.42762e6	−0.761390
$324$	0	0
$325$	2.67851e6	1.40665
$326$	4.80949e6	2.50643
$327$	0	0
$328$	−1.57215e6	−0.806879
$329$	−1.67822e6	−0.854788
$330$	0	0
$331$	−1.09566e6	−0.549674	−0.274837	−	0.961491i	$-0.588624\pi$
$331$	−1.09566e6	−0.549674	−0.274837	+	0.961491i	$0.588624\pi$
$332$	−5.45534e6	−2.71629
$333$	0	0
$334$	−3.80518e6	−1.86642
$335$	−92248.9	−0.0449106
$336$	0	0
$337$	3.18258e6	1.52653	0.763265	−	0.646086i	$-0.223595\pi$
$337$	3.18258e6	1.52653	0.763265	+	0.646086i	$0.223595\pi$
$338$	−4.76764e6	−2.26993
$339$	0	0
$340$	513042.	0.240688
$341$	−3.38487e6	−1.57636
$342$	0	0
$343$	2.36780e6	1.08670
$344$	6.44175e6	2.93500
$345$	0	0
$346$	−2.10859e6	−0.946896
$347$	4.04481e6	1.80333	0.901664	−	0.432438i	$-0.142346\pi$
$347$	4.04481e6	1.80333	0.901664	+	0.432438i	$0.142346\pi$
$348$	0	0
$349$	−1.36149e6	−0.598344	−0.299172	−	0.954199i	$-0.596710\pi$
$349$	−1.36149e6	−0.598344	−0.299172	+	0.954199i	$0.596710\pi$
$350$	−3.31476e6	−1.44638
$351$	0	0
$352$	−4.47677e6	−1.92579
$353$	1.83945e6	0.785692	0.392846	−	0.919604i	$-0.371491\pi$
$353$	1.83945e6	0.785692	0.392846	+	0.919604i	$0.371491\pi$
$354$	0	0
$355$	−493103.	−0.207666
$356$	−2.53270e6	−1.05915
$357$	0	0
$358$	2.83703e6	1.16992
$359$	1.00752e6	0.412588	0.206294	−	0.978490i	$-0.433860\pi$
$359$	1.00752e6	0.412588	0.206294	+	0.978490i	$0.433860\pi$
$360$	0	0
$361$	5.71821e6	2.30936
$362$	−3.84887e6	−1.54370
$363$	0	0
$364$	7.44192e6	2.94396
$365$	−643308.	−0.252748
$366$	0	0
$367$	685537.	0.265684	0.132842	−	0.991137i	$-0.457590\pi$
$367$	685537.	0.265684	0.132842	+	0.991137i	$0.457590\pi$
$368$	827504.	0.318530
$369$	0	0
$370$	−92738.8	−0.0352174
$371$	−2.52888e6	−0.953879
$372$	0	0
$373$	4.05298e6	1.50835	0.754176	−	0.656672i	$-0.228037\pi$
$373$	4.05298e6	1.50835	0.754176	+	0.656672i	$0.228037\pi$
$374$	2.87115e6	1.06140
$375$	0	0
$376$	−6.71154e6	−2.44823
$377$	595133.	0.215655
$378$	0	0
$379$	2.74335e6	0.981034	0.490517	−	0.871432i	$-0.336808\pi$
$379$	2.74335e6	0.981034	0.490517	+	0.871432i	$0.336808\pi$
$380$	−2.94477e6	−1.04615
$381$	0	0
$382$	3.59906e6	1.26192
$383$	−138868.	−0.0483732	−0.0241866	−	0.999707i	$-0.507700\pi$
$383$	−138868.	−0.0483732	−0.0241866	+	0.999707i	$0.507700\pi$
$384$	0	0
$385$	844330.	0.290309
$386$	−5.62762e6	−1.92245
$387$	0	0
$388$	−1.02893e6	−0.346981
$389$	973937.	0.326330	0.163165	−	0.986599i	$-0.447830\pi$
$389$	973937.	0.326330	0.163165	+	0.986599i	$0.447830\pi$
$390$	0	0
$391$	−193220.	−0.0639161
$392$	2.10919e6	0.693269
$393$	0	0
$394$	−7.59346e6	−2.46433
$395$	−203956.	−0.0657725
$396$	0	0
$397$	−101183.	−0.0322206	−0.0161103	−	0.999870i	$-0.505128\pi$
$397$	−101183.	−0.0322206	−0.0161103	+	0.999870i	$0.505128\pi$
$398$	−3.05415e6	−0.966457
$399$	0	0
$400$	−6.26676e6	−1.95836
$401$	1.32819e6	0.412478	0.206239	−	0.978502i	$-0.433878\pi$
$401$	1.32819e6	0.412478	0.206239	+	0.978502i	$0.433878\pi$
$402$	0	0
$403$	5.53774e6	1.69852
$404$	−4.26272e6	−1.29937
$405$	0	0
$406$	−736499.	−0.221747
$407$	−362975.	−0.108615
$408$	0	0
$409$	−4.65745e6	−1.37670	−0.688350	−	0.725378i	$-0.741665\pi$
$409$	−4.65745e6	−1.37670	−0.688350	+	0.725378i	$0.741665\pi$
$410$	511821.	0.150369
$411$	0	0
$412$	−1.39347e7	−4.04441
$413$	173891.	0.0501651
$414$	0	0
$415$	1.01261e6	0.288616
$416$	7.32413e6	2.07502
$417$	0	0
$418$	−1.64799e7	−4.61332
$419$	4.06572e6	1.13136	0.565682	−	0.824624i	$-0.308613\pi$
$419$	4.06572e6	1.13136	0.565682	+	0.824624i	$0.308613\pi$
$420$	0	0
$421$	3.18645e6	0.876198	0.438099	−	0.898927i	$-0.355652\pi$
$421$	3.18645e6	0.876198	0.438099	+	0.898927i	$0.355652\pi$
$422$	−2.11901e6	−0.579232
$423$	0	0
$424$	−1.01135e7	−2.73204
$425$	1.46327e6	0.392964
$426$	0	0
$427$	330593.	0.0877454
$428$	−1.49546e7	−3.94607
$429$	0	0
$430$	−2.09715e6	−0.546963
$431$	2.47673e6	0.642221	0.321111	−	0.947042i	$-0.395944\pi$
$431$	2.47673e6	0.642221	0.321111	+	0.947042i	$0.395944\pi$
$432$	0	0
$433$	5.22487e6	1.33923	0.669616	−	0.742707i	$-0.266459\pi$
$433$	5.22487e6	1.33923	0.669616	+	0.742707i	$0.266459\pi$
$434$	−6.85316e6	−1.74649
$435$	0	0
$436$	894908.	0.225456
$437$	1.10905e6	0.277810
$438$	0	0
$439$	−5.01844e6	−1.24282	−0.621409	−	0.783486i	$-0.713440\pi$
$439$	−5.01844e6	−1.24282	−0.621409	+	0.783486i	$0.713440\pi$
$440$	3.37666e6	0.831486
$441$	0	0
$442$	−4.69729e6	−1.14365
$443$	7.88704e6	1.90943	0.954717	−	0.297516i	$-0.0961582\pi$
$443$	7.88704e6	1.90943	0.954717	+	0.297516i	$0.0961582\pi$
$444$	0	0
$445$	470113.	0.112539
$446$	−9.58819e6	−2.28244
$447$	0	0
$448$	−1.57967e6	−0.371853
$449$	−5.28784e6	−1.23783	−0.618917	−	0.785456i	$-0.712428\pi$
$449$	−5.28784e6	−1.23783	−0.618917	+	0.785456i	$0.712428\pi$
$450$	0	0
$451$	2.00324e6	0.463758
$452$	−6.82354e6	−1.57096
$453$	0	0
$454$	535755.	0.121991
$455$	−1.38135e6	−0.312806
$456$	0	0
$457$	−2.62778e6	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$457$	−2.62778e6	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$458$	3.75349e6	0.836126
$459$	0	0
$460$	−398556.	−0.0878204
$461$	2.16443e6	0.474341	0.237170	−	0.971468i	$-0.423780\pi$
$461$	2.16443e6	0.474341	0.237170	+	0.971468i	$0.423780\pi$
$462$	0	0
$463$	−1.99490e6	−0.432483	−0.216241	−	0.976340i	$-0.569380\pi$
$463$	−1.99490e6	−0.432483	−0.216241	+	0.976340i	$0.569380\pi$
$464$	−1.39240e6	−0.300240
$465$	0	0
$466$	−745292.	−0.158987
$467$	4.95215e6	1.05076	0.525378	−	0.850869i	$-0.323924\pi$
$467$	4.95215e6	1.05076	0.525378	+	0.850869i	$0.323924\pi$
$468$	0	0
$469$	−731013.	−0.153459
$470$	2.18498e6	0.456250
$471$	0	0
$472$	695426.	0.143680
$473$	−8.20812e6	−1.68691
$474$	0	0
$475$	−8.39893e6	−1.70801
$476$	4.06552e6	0.822430
$477$	0	0
$478$	9.96331e6	1.99450
$479$	4.74850e6	0.945622	0.472811	−	0.881164i	$-0.343239\pi$
$479$	4.74850e6	0.945622	0.472811	+	0.881164i	$0.343239\pi$
$480$	0	0
$481$	593837.	0.117032
$482$	−6.33101e6	−1.24124
$483$	0	0
$484$	1.11902e7	2.17133
$485$	190987.	0.0368680
$486$	0	0
$487$	−6.13362e6	−1.17191	−0.585956	−	0.810343i	$-0.699281\pi$
$487$	−6.13362e6	−1.17191	−0.585956	+	0.810343i	$0.699281\pi$
$488$	1.32211e6	0.251315
$489$	0	0
$490$	−686661.	−0.129197
$491$	179920.	0.0336804	0.0168402	−	0.999858i	$-0.494639\pi$
$491$	179920.	0.0336804	0.0168402	+	0.999858i	$0.494639\pi$
$492$	0	0
$493$	325121.	0.0602460
$494$	2.69616e7	4.97082
$495$	0	0
$496$	−1.29563e7	−2.36471
$497$	−3.90752e6	−0.709594
$498$	0	0
$499$	1.31847e6	0.237039	0.118520	−	0.992952i	$-0.462185\pi$
$499$	1.31847e6	0.237039	0.118520	+	0.992952i	$0.462185\pi$
$500$	6.23304e6	1.11500
$501$	0	0
$502$	−1.02777e7	−1.82028
$503$	8.86305e6	1.56194	0.780968	−	0.624571i	$-0.214726\pi$
$503$	8.86305e6	1.56194	0.780968	+	0.624571i	$0.214726\pi$
$504$	0	0
$505$	791235.	0.138063
$506$	−2.23045e6	−0.387273
$507$	0	0
$508$	3.01223e6	0.517879
$509$	3.04610e6	0.521134	0.260567	−	0.965456i	$-0.416090\pi$
$509$	3.04610e6	0.521134	0.260567	+	0.965456i	$0.416090\pi$
$510$	0	0
$511$	−5.09780e6	−0.863636
$512$	1.27951e7	2.15709
$513$	0	0
$514$	1.75297e7	2.92661
$515$	2.58653e6	0.429733
$516$	0	0
$517$	8.55189e6	1.40714
$518$	−734895.	−0.120338
$519$	0	0
$520$	−5.52431e6	−0.895920
$521$	−1.00456e7	−1.62137	−0.810685	−	0.585483i	$-0.800905\pi$
$521$	−1.00456e7	−1.62137	−0.810685	+	0.585483i	$0.800905\pi$
$522$	0	0
$523$	9.83245e6	1.57184	0.785919	−	0.618330i	$-0.212190\pi$
$523$	9.83245e6	1.57184	0.785919	+	0.618330i	$0.212190\pi$
$524$	−1.27094e7	−2.02207
$525$	0	0
$526$	2.00346e7	3.15730
$527$	3.02527e6	0.474502
$528$	0	0
$529$	−6.28624e6	−0.976679
$530$	3.29251e6	0.509141
$531$	0	0
$532$	−2.33354e7	−3.57467
$533$	−3.27736e6	−0.499696
$534$	0	0
$535$	2.77583e6	0.419284
$536$	−2.92348e6	−0.439529
$537$	0	0
$538$	−1.76018e7	−2.62181
$539$	−2.68755e6	−0.398460
$540$	0	0
$541$	7.32261e6	1.07565	0.537827	−	0.843055i	$-0.319245\pi$
$541$	7.32261e6	1.07565	0.537827	+	0.843055i	$0.319245\pi$
$542$	−1.70448e7	−2.49226
$543$	0	0
$544$	4.00117e6	0.579683
$545$	−166110.	−0.0239555
$546$	0	0
$547$	3.30607e6	0.472436	0.236218	−	0.971700i	$-0.424092\pi$
$547$	3.30607e6	0.472436	0.236218	+	0.971700i	$0.424092\pi$
$548$	−1.08057e7	−1.53710
$549$	0	0
$550$	1.68914e7	2.38100
$551$	−1.86614e6	−0.261857
$552$	0	0
$553$	−1.61622e6	−0.224744
$554$	−4.66641e6	−0.645964
$555$	0	0
$556$	876221.	0.120206
$557$	1.13552e7	1.55080	0.775401	−	0.631469i	$-0.217548\pi$
$557$	1.13552e7	1.55080	0.775401	+	0.631469i	$0.217548\pi$
$558$	0	0
$559$	1.34287e7	1.81763
$560$	3.23186e6	0.435495
$561$	0	0
$562$	−1.27837e7	−1.70732
$563$	−2.89235e6	−0.384574	−0.192287	−	0.981339i	$-0.561590\pi$
$563$	−2.89235e6	−0.384574	−0.192287	+	0.981339i	$0.561590\pi$
$564$	0	0
$565$	1.26657e6	0.166920
$566$	8.64728e6	1.13459
$567$	0	0
$568$	−1.56270e7	−2.03238
$569$	−1.22363e6	−0.158441	−0.0792205	−	0.996857i	$-0.525243\pi$
$569$	−1.22363e6	−0.158441	−0.0792205	+	0.996857i	$0.525243\pi$
$570$	0	0
$571$	−684782.	−0.0878946	−0.0439473	−	0.999034i	$-0.513993\pi$
$571$	−684782.	−0.0878946	−0.0439473	+	0.999034i	$0.513993\pi$
$572$	−3.79227e7	−4.84629
$573$	0	0
$574$	4.05585e6	0.513810
$575$	−1.13674e6	−0.143381
$576$	0	0
$577$	−9.01720e6	−1.12754	−0.563770	−	0.825932i	$-0.690650\pi$
$577$	−9.01720e6	−1.12754	−0.563770	+	0.825932i	$0.690650\pi$
$578$	1.20829e7	1.50436
$579$	0	0
$580$	670630.	0.0827776
$581$	8.02424e6	0.986197
$582$	0	0
$583$	1.28867e7	1.57026
$584$	−2.03872e7	−2.47358
$585$	0	0
$586$	1.87479e7	2.25532
$587$	1.28173e7	1.53532	0.767662	−	0.640855i	$-0.221420\pi$
$587$	1.28173e7	1.53532	0.767662	+	0.640855i	$0.221420\pi$
$588$	0	0
$589$	−1.73645e7	−2.06241
$590$	−226400.	−0.0267761
$591$	0	0
$592$	−1.38937e6	−0.162934
$593$	−6.72932e6	−0.785841	−0.392920	−	0.919573i	$-0.628535\pi$
$593$	−6.72932e6	−0.785841	−0.392920	+	0.919573i	$0.628535\pi$
$594$	0	0
$595$	−754631.	−0.0873861
$596$	3.63966e7	4.19706
$597$	0	0
$598$	3.64909e6	0.417284
$599$	1.01547e6	0.115638	0.0578191	−	0.998327i	$-0.481585\pi$
$599$	1.01547e6	0.115638	0.0578191	+	0.998327i	$0.481585\pi$
$600$	0	0
$601$	−328831.	−0.0371353	−0.0185676	−	0.999828i	$-0.505911\pi$
$601$	−328831.	−0.0371353	−0.0185676	+	0.999828i	$0.505911\pi$
$602$	−1.66185e7	−1.86897
$603$	0	0
$604$	−1.96060e7	−2.18674
$605$	−2.07710e6	−0.230711
$606$	0	0
$607$	−2.40182e6	−0.264588	−0.132294	−	0.991211i	$-0.542234\pi$
$607$	−2.40182e6	−0.264588	−0.132294	+	0.991211i	$0.542234\pi$
$608$	−2.29660e7	−2.51957
$609$	0	0
$610$	−430421.	−0.0468349
$611$	−1.39911e7	−1.51618
$612$	0	0
$613$	2.69667e6	0.289852	0.144926	−	0.989443i	$-0.453706\pi$
$613$	2.69667e6	0.289852	0.144926	+	0.989443i	$0.453706\pi$
$614$	−1.80810e7	−1.93554
$615$	0	0
$616$	2.67578e7	2.84118
$617$	−9.38760e6	−0.992754	−0.496377	−	0.868107i	$-0.665337\pi$
$617$	−9.38760e6	−0.992754	−0.496377	+	0.868107i	$0.665337\pi$
$618$	0	0
$619$	−1.01003e7	−1.05952	−0.529759	−	0.848148i	$-0.677718\pi$
$619$	−1.01003e7	−1.05952	−0.529759	+	0.848148i	$0.677718\pi$
$620$	6.24025e6	0.651962
$621$	0	0
$622$	−3.41894e7	−3.54336
$623$	3.72534e6	0.384544
$624$	0	0
$625$	8.01195e6	0.820424
$626$	5.33792e6	0.544423
$627$	0	0
$628$	−5.69225e6	−0.575950
$629$	324413.	0.0326943
$630$	0	0
$631$	5.58510e6	0.558415	0.279208	−	0.960231i	$-0.409928\pi$
$631$	5.58510e6	0.558415	0.279208	+	0.960231i	$0.409928\pi$
$632$	−6.46361e6	−0.643698
$633$	0	0
$634$	−3.30541e6	−0.326589
$635$	−559122.	−0.0550265
$636$	0	0
$637$	4.39691e6	0.429338
$638$	3.75307e6	0.365035
$639$	0	0
$640$	−1.49094e6	−0.143883
$641$	7.43246e6	0.714476	0.357238	−	0.934013i	$-0.383719\pi$
$641$	7.43246e6	0.714476	0.357238	+	0.934013i	$0.383719\pi$
$642$	0	0
$643$	−5.89394e6	−0.562184	−0.281092	−	0.959681i	$-0.590697\pi$
$643$	−5.89394e6	−0.562184	−0.281092	+	0.959681i	$0.590697\pi$
$644$	−3.15830e6	−0.300081
$645$	0	0
$646$	1.47291e7	1.38866
$647$	1.45407e6	0.136560	0.0682800	−	0.997666i	$-0.478249\pi$
$647$	1.45407e6	0.136560	0.0682800	+	0.997666i	$0.478249\pi$
$648$	0	0
$649$	−886117.	−0.0825809
$650$	−2.76349e7	−2.56551
$651$	0	0
$652$	−3.47035e7	−3.19709
$653$	1.54930e7	1.42185	0.710925	−	0.703268i	$-0.248277\pi$
$653$	1.54930e7	1.42185	0.710925	+	0.703268i	$0.248277\pi$
$654$	0	0
$655$	2.35908e6	0.214852
$656$	7.66785e6	0.695687
$657$	0	0
$658$	1.73146e7	1.55900
$659$	1.29510e7	1.16169	0.580845	−	0.814014i	$-0.302722\pi$
$659$	1.29510e7	1.16169	0.580845	+	0.814014i	$0.302722\pi$
$660$	0	0
$661$	−1.29418e7	−1.15210	−0.576052	−	0.817413i	$-0.695408\pi$
$661$	−1.29418e7	−1.15210	−0.576052	+	0.817413i	$0.695408\pi$
$662$	1.13042e7	1.00252
$663$	0	0
$664$	3.20906e7	2.82461
$665$	4.33145e6	0.379821
$666$	0	0
$667$	−252570.	−0.0219820
$668$	2.74568e7	2.38072
$669$	0	0
$670$	951754.	0.0819102
$671$	−1.68465e6	−0.144445
$672$	0	0
$673$	1.24957e7	1.06347	0.531734	−	0.846911i	$-0.321541\pi$
$673$	1.24957e7	1.06347	0.531734	+	0.846911i	$0.321541\pi$
$674$	−3.28355e7	−2.78416
$675$	0	0
$676$	3.44016e7	2.89542
$677$	6.43805e6	0.539862	0.269931	−	0.962880i	$-0.412999\pi$
$677$	6.43805e6	0.539862	0.269931	+	0.962880i	$0.412999\pi$
$678$	0	0
$679$	1.51345e6	0.125977
$680$	−3.01793e6	−0.250286
$681$	0	0
$682$	3.49225e7	2.87504
$683$	−1.36221e7	−1.11736	−0.558679	−	0.829384i	$-0.688692\pi$
$683$	−1.36221e7	−1.11736	−0.558679	+	0.829384i	$0.688692\pi$
$684$	0	0
$685$	2.00572e6	0.163322
$686$	−2.44291e7	−1.98197
$687$	0	0
$688$	−3.14184e7	−2.53054
$689$	−2.10830e7	−1.69194
$690$	0	0
$691$	6.75023e6	0.537803	0.268902	−	0.963168i	$-0.413339\pi$
$691$	6.75023e6	0.537803	0.268902	+	0.963168i	$0.413339\pi$
$692$	1.52148e7	1.20782
$693$	0	0
$694$	−4.17313e7	−3.28900
$695$	−162642.	−0.0127723
$696$	0	0
$697$	−1.79042e6	−0.139596
$698$	1.40468e7	1.09129
$699$	0	0
$700$	2.39181e7	1.84494
$701$	−1.50466e7	−1.15649	−0.578247	−	0.815862i	$-0.696263\pi$
$701$	−1.50466e7	−1.15649	−0.578247	+	0.815862i	$0.696263\pi$
$702$	0	0
$703$	−1.86208e6	−0.142105
$704$	8.04972e6	0.612138
$705$	0	0
$706$	−1.89781e7	−1.43298
$707$	6.27003e6	0.471760
$708$	0	0
$709$	1.98697e7	1.48449	0.742243	−	0.670130i	$-0.233762\pi$
$709$	1.98697e7	1.48449	0.742243	+	0.670130i	$0.233762\pi$
$710$	5.08746e6	0.378752
$711$	0	0
$712$	1.48984e7	1.10139
$713$	−2.35018e6	−0.173132
$714$	0	0
$715$	7.03911e6	0.514935
$716$	−2.04710e7	−1.49230
$717$	0	0
$718$	−1.03948e7	−0.752498
$719$	8.31861e6	0.600107	0.300053	−	0.953922i	$-0.402995\pi$
$719$	8.31861e6	0.600107	0.300053	+	0.953922i	$0.402995\pi$
$720$	0	0
$721$	2.04965e7	1.46839
$722$	−5.89961e7	−4.21192
$723$	0	0
$724$	2.77721e7	1.96907
$725$	1.91274e6	0.135148
$726$	0	0
$727$	−5.32623e6	−0.373752	−0.186876	−	0.982383i	$-0.559836\pi$
$727$	−5.32623e6	−0.373752	−0.186876	+	0.982383i	$0.559836\pi$
$728$	−4.37766e7	−3.06135
$729$	0	0
$730$	6.63717e6	0.460973
$731$	7.33612e6	0.507777
$732$	0	0
$733$	2.13481e6	0.146757	0.0733784	−	0.997304i	$-0.476622\pi$
$733$	2.13481e6	0.146757	0.0733784	+	0.997304i	$0.476622\pi$
$734$	−7.07285e6	−0.484568
$735$	0	0
$736$	−3.10831e6	−0.211510
$737$	3.72511e6	0.252622
$738$	0	0
$739$	−1.07908e7	−0.726845	−0.363423	−	0.931624i	$-0.618392\pi$
$739$	−1.07908e7	−0.726845	−0.363423	+	0.931624i	$0.618392\pi$
$740$	669170.	0.0449218
$741$	0	0
$742$	2.60910e7	1.73973
$743$	2.77406e6	0.184350	0.0921750	−	0.995743i	$-0.470618\pi$
$743$	2.77406e6	0.184350	0.0921750	+	0.995743i	$0.470618\pi$
$744$	0	0
$745$	−6.75584e6	−0.445953
$746$	−4.18156e7	−2.75100
$747$	0	0
$748$	−2.07172e7	−1.35387
$749$	2.19967e7	1.43269
$750$	0	0
$751$	2.24300e7	1.45121	0.725605	−	0.688112i	$-0.241560\pi$
$751$	2.24300e7	1.45121	0.725605	+	0.688112i	$0.241560\pi$
$752$	3.27343e7	2.11085
$753$	0	0
$754$	−6.14013e6	−0.393323
$755$	3.63922e6	0.232349
$756$	0	0
$757$	−2.07779e6	−0.131784	−0.0658920	−	0.997827i	$-0.520989\pi$
$757$	−2.07779e6	−0.131784	−0.0658920	+	0.997827i	$0.520989\pi$
$758$	−2.83038e7	−1.78926
$759$	0	0
$760$	1.73224e7	1.08786
$761$	−1.52398e7	−0.953932	−0.476966	−	0.878922i	$-0.658263\pi$
$761$	−1.52398e7	−0.953932	−0.476966	+	0.878922i	$0.658263\pi$
$762$	0	0
$763$	−1.31632e6	−0.0818558
$764$	−2.59695e7	−1.60965
$765$	0	0
$766$	1.43273e6	0.0882253
$767$	1.44971e6	0.0889803
$768$	0	0
$769$	1.39007e6	0.0847656	0.0423828	−	0.999101i	$-0.486505\pi$
$769$	1.39007e6	0.0847656	0.0423828	+	0.999101i	$0.486505\pi$
$770$	−8.71116e6	−0.529480
$771$	0	0
$772$	4.06068e7	2.45220
$773$	1.89364e7	1.13985	0.569927	−	0.821695i	$-0.306971\pi$
$773$	1.89364e7	1.13985	0.569927	+	0.821695i	$0.306971\pi$
$774$	0	0
$775$	1.77981e7	1.06444
$776$	6.05260e6	0.360817
$777$	0	0
$778$	−1.00483e7	−0.595176
$779$	1.02767e7	0.606751
$780$	0	0
$781$	1.99120e7	1.16812
$782$	1.99350e6	0.116573
$783$	0	0
$784$	−1.02872e7	−0.597733
$785$	1.05658e6	0.0611967
$786$	0	0
$787$	2.30040e7	1.32394	0.661969	−	0.749532i	$-0.269721\pi$
$787$	2.30040e7	1.32394	0.661969	+	0.749532i	$0.269721\pi$
$788$	5.47916e7	3.14339
$789$	0	0
$790$	2.10427e6	0.119959
$791$	1.00367e7	0.570362
$792$	0	0
$793$	2.75613e6	0.155638
$794$	1.04393e6	0.0587654
$795$	0	0
$796$	2.20376e7	1.23277
$797$	−6.36761e6	−0.355084	−0.177542	−	0.984113i	$-0.556815\pi$
$797$	−6.36761e6	−0.355084	−0.177542	+	0.984113i	$0.556815\pi$
$798$	0	0
$799$	−7.64337e6	−0.423563
$800$	2.35395e7	1.30039
$801$	0	0
$802$	−1.37033e7	−0.752297
$803$	2.59775e7	1.42170
$804$	0	0
$805$	586236.	0.0318847
$806$	−5.71342e7	−3.09784
$807$	0	0
$808$	2.50752e7	1.35119
$809$	2.58080e7	1.38638	0.693190	−	0.720755i	$-0.256205\pi$
$809$	2.58080e7	1.38638	0.693190	+	0.720755i	$0.256205\pi$
$810$	0	0
$811$	6.93201e6	0.370090	0.185045	−	0.982730i	$-0.440757\pi$
$811$	6.93201e6	0.370090	0.185045	+	0.982730i	$0.440757\pi$
$812$	5.31431e6	0.282850
$813$	0	0
$814$	3.74490e6	0.198097
$815$	6.44158e6	0.339702
$816$	0	0
$817$	−4.21080e7	−2.20704
$818$	4.80520e7	2.51089
$819$	0	0
$820$	−3.69312e6	−0.191804
$821$	5.86257e6	0.303550	0.151775	−	0.988415i	$-0.451501\pi$
$821$	5.86257e6	0.303550	0.151775	+	0.988415i	$0.451501\pi$
$822$	0	0
$823$	−1.63833e7	−0.843142	−0.421571	−	0.906795i	$-0.638521\pi$
$823$	−1.63833e7	−0.843142	−0.421571	+	0.906795i	$0.638521\pi$
$824$	8.19700e7	4.20569
$825$	0	0
$826$	−1.79407e6	−0.0914935
$827$	−1.39265e7	−0.708075	−0.354037	−	0.935231i	$-0.615191\pi$
$827$	−1.39265e7	−0.708075	−0.354037	+	0.935231i	$0.615191\pi$
$828$	0	0
$829$	−7.22986e6	−0.365379	−0.182690	−	0.983171i	$-0.558480\pi$
$829$	−7.22986e6	−0.365379	−0.182690	+	0.983171i	$0.558480\pi$
$830$	−1.04473e7	−0.526391
$831$	0	0
$832$	−1.31696e7	−0.659574
$833$	2.40203e6	0.119941
$834$	0	0
$835$	−5.09645e6	−0.252960
$836$	1.18913e8	5.88456
$837$	0	0
$838$	−4.19470e7	−2.06343
$839$	2.27427e7	1.11542	0.557708	−	0.830037i	$-0.311681\pi$
$839$	2.27427e7	1.11542	0.557708	+	0.830037i	$0.311681\pi$
$840$	0	0
$841$	−2.00862e7	−0.979280
$842$	−3.28754e7	−1.59805
$843$	0	0
$844$	1.52900e7	0.738843
$845$	−6.38553e6	−0.307649
$846$	0	0
$847$	−1.64597e7	−0.788337
$848$	4.93268e7	2.35555
$849$	0	0
$850$	−1.50969e7	−0.716707
$851$	−252021.	−0.0119292
$852$	0	0
$853$	−5.45554e6	−0.256723	−0.128362	−	0.991727i	$-0.540972\pi$
$853$	−5.45554e6	−0.256723	−0.128362	+	0.991727i	$0.540972\pi$
$854$	−3.41081e6	−0.160034
$855$	0	0
$856$	8.79693e7	4.10343
$857$	−4.10626e7	−1.90983	−0.954914	−	0.296883i	$-0.904053\pi$
$857$	−4.10626e7	−1.90983	−0.954914	+	0.296883i	$0.904053\pi$
$858$	0	0
$859$	8.53171e6	0.394505	0.197253	−	0.980353i	$-0.436798\pi$
$859$	8.53171e6	0.394505	0.197253	+	0.980353i	$0.436798\pi$
$860$	1.51323e7	0.697682
$861$	0	0
$862$	−2.55530e7	−1.17131
$863$	−2.93589e6	−0.134188	−0.0670940	−	0.997747i	$-0.521373\pi$
$863$	−2.93589e6	−0.134188	−0.0670940	+	0.997747i	$0.521373\pi$
$864$	0	0
$865$	−2.82414e6	−0.128335
$866$	−5.39063e7	−2.44256
$867$	0	0
$868$	4.94499e7	2.22775
$869$	8.23598e6	0.369969
$870$	0	0
$871$	−6.09439e6	−0.272198
$872$	−5.26423e6	−0.234447
$873$	0	0
$874$	−1.14423e7	−0.506682
$875$	−9.16816e6	−0.404820
$876$	0	0
$877$	2.47763e7	1.08777	0.543885	−	0.839160i	$-0.316953\pi$
$877$	2.47763e7	1.08777	0.543885	+	0.839160i	$0.316953\pi$
$878$	5.17765e7	2.26671
$879$	0	0
$880$	−1.64690e7	−0.716903
$881$	2.25075e7	0.976984	0.488492	−	0.872568i	$-0.337547\pi$
$881$	2.25075e7	0.976984	0.488492	+	0.872568i	$0.337547\pi$
$882$	0	0
$883$	−3.56235e7	−1.53757	−0.768786	−	0.639506i	$-0.779139\pi$
$883$	−3.56235e7	−1.53757	−0.768786	+	0.639506i	$0.779139\pi$
$884$	3.38939e7	1.45878
$885$	0	0
$886$	−8.13725e7	−3.48252
$887$	1.26385e7	0.539369	0.269684	−	0.962949i	$-0.413081\pi$
$887$	1.26385e7	0.539369	0.269684	+	0.962949i	$0.413081\pi$
$888$	0	0
$889$	−4.43068e6	−0.188025
$890$	−4.85027e6	−0.205253
$891$	0	0
$892$	6.91849e7	2.91138
$893$	4.38716e7	1.84100
$894$	0	0
$895$	3.79977e6	0.158562
$896$	−1.18147e7	−0.491648
$897$	0	0
$898$	5.45559e7	2.25762
$899$	3.95453e6	0.163191
$900$	0	0
$901$	−1.15177e7	−0.472664
$902$	−2.06679e7	−0.845825
$903$	0	0
$904$	4.01390e7	1.63360
$905$	−5.15498e6	−0.209221
$906$	0	0
$907$	3.60859e7	1.45653	0.728264	−	0.685296i	$-0.240327\pi$
$907$	3.60859e7	1.45653	0.728264	+	0.685296i	$0.240327\pi$
$908$	−3.86581e6	−0.155606
$909$	0	0
$910$	1.42517e7	0.570510
$911$	−1.41298e7	−0.564078	−0.282039	−	0.959403i	$-0.591011\pi$
$911$	−1.41298e7	−0.564078	−0.282039	+	0.959403i	$0.591011\pi$
$912$	0	0
$913$	−4.08901e7	−1.62346
$914$	2.71114e7	1.07346
$915$	0	0
$916$	−2.70838e7	−1.06653
$917$	1.86942e7	0.734147
$918$	0	0
$919$	−4.34357e7	−1.69652	−0.848258	−	0.529584i	$-0.822348\pi$
$919$	−4.34357e7	−1.69652	−0.848258	+	0.529584i	$0.822348\pi$
$920$	2.34448e6	0.0913223
$921$	0	0
$922$	−2.23309e7	−0.865126
$923$	−3.25767e7	−1.25864
$924$	0	0
$925$	1.90857e6	0.0733423
$926$	2.05819e7	0.788783
$927$	0	0
$928$	5.23019e6	0.199365
$929$	3.19171e6	0.121334	0.0606672	−	0.998158i	$-0.480677\pi$
$929$	3.19171e6	0.121334	0.0606672	+	0.998158i	$0.480677\pi$
$930$	0	0
$931$	−1.37873e7	−0.521319
$932$	5.37776e6	0.202797
$933$	0	0
$934$	−5.10925e7	−1.91642
$935$	3.84547e6	0.143853
$936$	0	0
$937$	−3.03647e7	−1.12985	−0.564923	−	0.825144i	$-0.691094\pi$
$937$	−3.03647e7	−1.12985	−0.564923	+	0.825144i	$0.691094\pi$
$938$	7.54204e6	0.279886
$939$	0	0
$940$	−1.57660e7	−0.581973
$941$	3.85401e7	1.41886	0.709429	−	0.704777i	$-0.248953\pi$
$941$	3.85401e7	1.41886	0.709429	+	0.704777i	$0.248953\pi$
$942$	0	0
$943$	1.39089e6	0.0509347
$944$	−3.39181e6	−0.123880
$945$	0	0
$946$	8.46852e7	3.07666
$947$	−4.03020e7	−1.46033	−0.730167	−	0.683269i	$-0.760558\pi$
$947$	−4.03020e7	−1.46033	−0.730167	+	0.683269i	$0.760558\pi$
$948$	0	0
$949$	−4.25000e7	−1.53187
$950$	8.66538e7	3.11515
$951$	0	0
$952$	−2.39151e7	−0.855226
$953$	−5.89261e6	−0.210172	−0.105086	−	0.994463i	$-0.533512\pi$
$953$	−5.89261e6	−0.210172	−0.105086	+	0.994463i	$0.533512\pi$
$954$	0	0
$955$	4.82039e6	0.171031
$956$	−7.18916e7	−2.54410
$957$	0	0
$958$	−4.89914e7	−1.72467
$959$	1.58941e7	0.558069
$960$	0	0
$961$	8.16794e6	0.285301
$962$	−6.12676e6	−0.213449
$963$	0	0
$964$	4.56823e7	1.58327
$965$	−7.53733e6	−0.260555
$966$	0	0
$967$	3.46156e7	1.19044	0.595218	−	0.803564i	$-0.297066\pi$
$967$	3.46156e7	1.19044	0.595218	+	0.803564i	$0.297066\pi$
$968$	−6.58256e7	−2.25791
$969$	0	0
$970$	−1.97046e6	−0.0672415
$971$	−3.52792e7	−1.20080	−0.600400	−	0.799700i	$-0.704992\pi$
$971$	−3.52792e7	−1.20080	−0.600400	+	0.799700i	$0.704992\pi$
$972$	0	0
$973$	−1.28883e6	−0.0436429
$974$	6.32821e7	2.13739
$975$	0	0
$976$	−6.44836e6	−0.216683
$977$	−1.26965e7	−0.425548	−0.212774	−	0.977101i	$-0.568250\pi$
$977$	−1.26965e7	−0.425548	−0.212774	+	0.977101i	$0.568250\pi$
$978$	0	0
$979$	−1.89837e7	−0.633029
$980$	4.95469e6	0.164798
$981$	0	0
$982$	−1.85628e6	−0.0614278
$983$	5.32245e7	1.75682	0.878412	−	0.477905i	$-0.158604\pi$
$983$	5.32245e7	1.75682	0.878412	+	0.477905i	$0.158604\pi$
$984$	0	0
$985$	−1.01703e7	−0.333997
$986$	−3.35435e6	−0.109879
$987$	0	0
$988$	−1.94545e8	−6.34057
$989$	−5.69906e6	−0.185273
$990$	0	0
$991$	−5.02997e7	−1.62698	−0.813489	−	0.581581i	$-0.802435\pi$
$991$	−5.02997e7	−1.62698	−0.813489	+	0.581581i	$0.802435\pi$
$992$	4.86672e7	1.57021
$993$	0	0
$994$	4.03148e7	1.29419
$995$	−4.09057e6	−0.130986
$996$	0	0
$997$	5.36507e7	1.70938	0.854688	−	0.519141i	$-0.173748\pi$
$997$	5.36507e7	1.70938	0.854688	+	0.519141i	$0.173748\pi$
$998$	−1.36030e7	−0.432323
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.6.a.e.1.2		42
3.2	odd	2		729.6.a.c.1.41		42
27.2	odd	18		27.6.e.a.4.1	✓	84
27.13	even	9		81.6.e.a.19.14		84
27.14	odd	18		27.6.e.a.7.1	yes	84
27.25	even	9		81.6.e.a.64.14		84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.6.e.a.4.1	✓	84	27.2	odd	18
27.6.e.a.7.1	yes	84	27.14	odd	18
81.6.e.a.19.14		84	27.13	even	9
81.6.e.a.64.14		84	27.25	even	9
729.6.a.c.1.41		42	3.2	odd	2
729.6.a.e.1.2		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-10.3172 q^{2} +74.4455 q^{4} -13.8184 q^{5} -109.502 q^{7} -437.920 q^{8} +O(q^{10})\)	\(q-10.3172 q^{2} +74.4455 q^{4} -13.8184 q^{5} -109.502 q^{7} -437.920 q^{8} +142.567 q^{10} +558.001 q^{11} -912.906 q^{13} +1129.75 q^{14} +2135.87 q^{16} -498.721 q^{17} +2862.57 q^{19} -1028.72 q^{20} -5757.03 q^{22} +387.431 q^{23} -2934.05 q^{25} +9418.67 q^{26} -8151.90 q^{28} -651.910 q^{29} -6066.06 q^{31} -8022.87 q^{32} +5145.42 q^{34} +1513.13 q^{35} -650.491 q^{37} -29533.8 q^{38} +6051.34 q^{40} +3590.03 q^{41} -14709.9 q^{43} +41540.7 q^{44} -3997.22 q^{46} +15325.9 q^{47} -4816.39 q^{49} +30271.3 q^{50} -67961.7 q^{52} +23094.4 q^{53} -7710.66 q^{55} +47953.0 q^{56} +6725.91 q^{58} -1588.02 q^{59} -3019.07 q^{61} +62585.0 q^{62} +14426.0 q^{64} +12614.9 q^{65} +6675.82 q^{67} -37127.5 q^{68} -15611.4 q^{70} +35684.6 q^{71} +46554.6 q^{73} +6711.27 q^{74} +213105. q^{76} -61102.0 q^{77} +14759.8 q^{79} -29514.3 q^{80} -37039.2 q^{82} -73279.7 q^{83} +6891.51 q^{85} +151765. q^{86} -244360. q^{88} -34020.8 q^{89} +99964.7 q^{91} +28842.5 q^{92} -158121. q^{94} -39556.0 q^{95} -13821.2 q^{97} +49691.9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q + 12 q^{2} + 624 q^{4} + 150 q^{5} + 573 q^{8}+O(q^{10}) \) 42 * q + 12 * q^2 + 624 * q^4 + 150 * q^5 + 573 * q^8	\( 42 q + 12 q^{2} + 624 q^{4} + 150 q^{5} + 573 q^{8} + 3 q^{10} + 1452 q^{11} + 2256 q^{14} + 8448 q^{16} + 3465 q^{17} + 3 q^{19} + 4128 q^{20} + 96 q^{22} + 5019 q^{23} + 18750 q^{25} + 3903 q^{26} - 6 q^{28} + 13008 q^{29} + 24273 q^{32} + 35868 q^{35} + 3 q^{37} + 51801 q^{38} + 96 q^{40} + 55833 q^{41} + 110757 q^{44} + 3 q^{46} + 90129 q^{47} + 57624 q^{49} + 145362 q^{50} + 3072 q^{52} + 103203 q^{53} - 6 q^{55} + 227154 q^{56} - 192 q^{58} + 176856 q^{59} - 31851 q^{61} + 246066 q^{62} + 86019 q^{64} + 167160 q^{65} - 801 q^{67} + 374589 q^{68} + 9375 q^{70} + 279531 q^{71} + 27012 q^{73} + 413970 q^{74} + 96 q^{76} + 185190 q^{77} + 462057 q^{80} - 6 q^{82} + 295536 q^{83} + 319803 q^{86} + 3072 q^{88} + 154827 q^{89} + 91002 q^{91} + 330558 q^{92} + 96 q^{94} + 353244 q^{95} + 463410 q^{98}+O(q^{100}) \) 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

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Database

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