LMFDB - Embedded newform 81.5.b.a.80.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [81,5,Mod(80,81)] 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("81.80"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	81.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$8.37296700979$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.39400128.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 11x^{4} + 14x^{3} + 98x^{2} + 20x + 4 $ x^6 - x^5 + 11x^4 + 14x^3 + 98x^2 + 20x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{9} $
Twist minimal:	no (minimal twist has level 9)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			80.2
Root			$-1.28901 + 2.23263i$ of defining polynomial
Character	$\chi$	$=$	81.80
Dual form			81.5.b.a.80.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.46526i q^{2} -3.93857 q^{4} +16.0226i q^{5} +72.4837 q^{7} -53.8574i q^{8} +71.5451 q^{10} -96.1576i q^{11} +153.906 q^{13} -323.659i q^{14} -303.505 q^{16} -72.7905i q^{17} -190.660 q^{19} -63.1062i q^{20} -429.369 q^{22} -14.4552i q^{23} +368.276 q^{25} -687.231i q^{26} -285.482 q^{28} -716.262i q^{29} -302.568 q^{31} +493.509i q^{32} -325.029 q^{34} +1161.38i q^{35} +826.277 q^{37} +851.348i q^{38} +862.936 q^{40} +556.119i q^{41} +892.680 q^{43} +378.723i q^{44} -64.5464 q^{46} +3955.43i q^{47} +2852.89 q^{49} -1644.45i q^{50} -606.170 q^{52} +1966.96i q^{53} +1540.69 q^{55} -3903.79i q^{56} -3198.30 q^{58} +5414.94i q^{59} -1712.42 q^{61} +1351.04i q^{62} -2652.43 q^{64} +2465.97i q^{65} -4634.49 q^{67} +286.691i q^{68} +5185.86 q^{70} -6697.12i q^{71} -4823.86 q^{73} -3689.54i q^{74} +750.929 q^{76} -6969.86i q^{77} -5728.79 q^{79} -4862.94i q^{80} +2483.22 q^{82} -2832.23i q^{83} +1166.29 q^{85} -3986.05i q^{86} -5178.80 q^{88} +14277.7i q^{89} +11155.7 q^{91} +56.9330i q^{92} +17662.0 q^{94} -3054.87i q^{95} +7165.31 q^{97} -12738.9i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 30 q^{4} - 24 q^{7} - 36 q^{10} + 12 q^{13} - 30 q^{16} - 258 q^{19} + 738 q^{22} + 546 q^{25} + 1308 q^{28} - 2580 q^{31} - 1026 q^{34} + 12 q^{37} + 2628 q^{40} + 570 q^{43} - 5760 q^{46} + 3726 q^{49}+ \cdots + 57918 q^{97}+O(q^{100}) $ 6 * q - 30 * q^4 - 24 * q^7 - 36 * q^10 + 12 * q^13 - 30 * q^16 - 258 * q^19 + 738 * q^22 + 546 * q^25 + 1308 * q^28 - 2580 * q^31 - 1026 * q^34 + 12 * q^37 + 2628 * q^40 + 570 * q^43 - 5760 * q^46 + 3726 * q^49 + 480 * q^52 + 2016 * q^55 - 12924 * q^58 - 7260 * q^61 + 15450 * q^64 + 10110 * q^67 + 19368 * q^70 - 14622 * q^73 + 8094 * q^76 - 9528 * q^79 - 9702 * q^82 - 24732 * q^85 - 29574 * q^88 + 34836 * q^91 + 25416 * q^94 + 57918 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/81\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

Coefficient data

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

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

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

$2$	− 4.46526i	− 1.11632i	−0.829735	−	0.558158i	$-0.811508\pi$
$2$	− 4.46526i	− 1.11632i	0.829735	−	0.558158i	$-0.188492\pi$
$3$	0	0
$3$	0	0
$4$	−3.93857	−0.246161
$5$	16.0226i	0.640904i	0.947265	+	0.320452i	$0.103835\pi$
$5$	16.0226i	0.640904i	−0.947265	+	0.320452i	$0.896165\pi$
$6$	0	0
$7$	72.4837	1.47926	0.739630	−	0.673014i	$-0.235001\pi$
$7$	72.4837	1.47926	0.739630	+	0.673014i	$0.235001\pi$
$8$	− 53.8574i	− 0.841523i
$9$	0	0
$10$	71.5451	0.715451
$11$	− 96.1576i	− 0.794691i	−0.917669	−	0.397345i	$-0.869932\pi$
$11$	− 96.1576i	− 0.794691i	0.917669	−	0.397345i	$-0.130068\pi$
$12$	0	0
$13$	153.906	0.910686	0.455343	−	0.890316i	$-0.349517\pi$
$13$	153.906	0.910686	0.455343	+	0.890316i	$0.349517\pi$
$14$	− 323.659i	− 1.65132i
$15$	0	0
$16$	−303.505	−1.18557
$17$	− 72.7905i	− 0.251870i	−0.992038	−	0.125935i	$-0.959807\pi$
$17$	− 72.7905i	− 0.251870i	0.992038	−	0.125935i	$-0.0401931\pi$
$18$	0	0
$19$	−190.660	−0.528145	−0.264072	−	0.964503i	$-0.585066\pi$
$19$	−190.660	−0.528145	−0.264072	+	0.964503i	$0.585066\pi$
$20$	− 63.1062i	− 0.157765i
$21$	0	0
$22$	−429.369	−0.887126
$23$	− 14.4552i	− 0.0273256i	−0.999907	−	0.0136628i	$-0.995651\pi$
$23$	− 14.4552i	− 0.0273256i	0.999907	−	0.0136628i	$-0.00434914\pi$
$24$	0	0
$25$	368.276	0.589242
$26$	− 687.231i	− 1.01661i
$27$	0	0
$28$	−285.482	−0.364135
$29$	− 716.262i	− 0.851679i	−0.904799	−	0.425839i	$-0.859979\pi$
$29$	− 716.262i	− 0.851679i	0.904799	−	0.425839i	$-0.140021\pi$
$30$	0	0
$31$	−302.568	−0.314847	−0.157423	−	0.987531i	$-0.550319\pi$
$31$	−302.568	−0.314847	−0.157423	+	0.987531i	$0.550319\pi$
$32$	493.509i	0.481943i
$33$	0	0
$34$	−325.029	−0.281167
$35$	1161.38i	0.948063i
$36$	0	0
$37$	826.277	0.603562	0.301781	−	0.953377i	$-0.402419\pi$
$37$	826.277	0.603562	0.301781	+	0.953377i	$0.402419\pi$
$38$	851.348i	0.589576i
$39$	0	0
$40$	862.936	0.539335
$41$	556.119i	0.330826i	0.986224	+	0.165413i	$0.0528958\pi$
$41$	556.119i	0.330826i	−0.986224	+	0.165413i	$0.947104\pi$
$42$	0	0
$43$	892.680	0.482791	0.241395	−	0.970427i	$-0.422395\pi$
$43$	892.680	0.482791	0.241395	+	0.970427i	$0.422395\pi$
$44$	378.723i	0.195622i
$45$	0	0
$46$	−64.5464	−0.0305040
$47$	3955.43i	1.79060i	0.445465	+	0.895299i	$0.353038\pi$
$47$	3955.43i	1.79060i	−0.445465	+	0.895299i	$0.646962\pi$
$48$	0	0
$49$	2852.89	1.18821
$50$	− 1644.45i	− 0.657780i
$51$	0	0
$52$	−606.170	−0.224175
$53$	1966.96i	0.700234i	0.936706	+	0.350117i	$0.113858\pi$
$53$	1966.96i	0.700234i	−0.936706	+	0.350117i	$0.886142\pi$
$54$	0	0
$55$	1540.69	0.509320
$56$	− 3903.79i	− 1.24483i
$57$	0	0
$58$	−3198.30	−0.950742
$59$	5414.94i	1.55557i	0.628530	+	0.777785i	$0.283657\pi$
$59$	5414.94i	1.55557i	−0.628530	+	0.777785i	$0.716343\pi$
$60$	0	0
$61$	−1712.42	−0.460204	−0.230102	−	0.973166i	$-0.573906\pi$
$61$	−1712.42	−0.460204	−0.230102	+	0.973166i	$0.573906\pi$
$62$	1351.04i	0.351468i
$63$	0	0
$64$	−2652.43	−0.647565
$65$	2465.97i	0.583663i
$66$	0	0
$67$	−4634.49	−1.03241	−0.516205	−	0.856465i	$-0.672656\pi$
$67$	−4634.49	−1.03241	−0.516205	+	0.856465i	$0.672656\pi$
$68$	286.691i	0.0620006i
$69$	0	0
$70$	5185.86	1.05834
$71$	− 6697.12i	− 1.32853i	−0.747497	−	0.664265i	$-0.768744\pi$
$71$	− 6697.12i	− 1.32853i	0.747497	−	0.664265i	$-0.231256\pi$
$72$	0	0
$73$	−4823.86	−0.905208	−0.452604	−	0.891712i	$-0.649505\pi$
$73$	−4823.86	−0.905208	−0.452604	+	0.891712i	$0.649505\pi$
$74$	− 3689.54i	− 0.673766i
$75$	0	0
$76$	750.929	0.130008
$77$	− 6969.86i	− 1.17555i
$78$	0	0
$79$	−5728.79	−0.917929	−0.458964	−	0.888455i	$-0.651779\pi$
$79$	−5728.79	−0.917929	−0.458964	+	0.888455i	$0.651779\pi$
$80$	− 4862.94i	− 0.759834i
$81$	0	0
$82$	2483.22	0.369307
$83$	− 2832.23i	− 0.411124i	−0.978644	−	0.205562i	$-0.934098\pi$
$83$	− 2832.23i	− 0.411124i	0.978644	−	0.205562i	$-0.0659022\pi$
$84$	0	0
$85$	1166.29	0.161425
$86$	− 3986.05i	− 0.538947i
$87$	0	0
$88$	−5178.80	−0.668750
$89$	14277.7i	1.80251i	0.433290	+	0.901255i	$0.357353\pi$
$89$	14277.7i	1.80251i	−0.433290	+	0.901255i	$0.642647\pi$
$90$	0	0
$91$	11155.7	1.34714
$92$	56.9330i	0.00672649i
$93$	0	0
$94$	17662.0	1.99887
$95$	− 3054.87i	− 0.338490i
$96$	0	0
$97$	7165.31	0.761538	0.380769	−	0.924670i	$-0.375659\pi$
$97$	7165.31	0.761538	0.380769	+	0.924670i	$0.375659\pi$
$98$	− 12738.9i	− 1.32641i
$99$	0	0
$100$	−1450.48	−0.145048
$101$	1959.20i	0.192060i	0.995378	+	0.0960298i	$0.0306144\pi$
$101$	1959.20i	0.192060i	−0.995378	+	0.0960298i	$0.969386\pi$
$102$	0	0
$103$	−5155.48	−0.485953	−0.242976	−	0.970032i	$-0.578124\pi$
$103$	−5155.48	−0.485953	−0.242976	+	0.970032i	$0.578124\pi$
$104$	− 8288.98i	− 0.766363i
$105$	0	0
$106$	8782.98	0.781682
$107$	− 9117.08i	− 0.796321i	−0.917316	−	0.398161i	$-0.869649\pi$
$107$	− 9117.08i	− 0.796321i	0.917316	−	0.398161i	$-0.130351\pi$
$108$	0	0
$109$	−16161.1	−1.36024	−0.680122	−	0.733099i	$-0.738073\pi$
$109$	−16161.1	−1.36024	−0.680122	+	0.733099i	$0.738073\pi$
$110$	− 6879.60i	− 0.568562i
$111$	0	0
$112$	−21999.2	−1.75376
$113$	21099.2i	1.65238i	0.563393	+	0.826189i	$0.309496\pi$
$113$	21099.2i	1.65238i	−0.563393	+	0.826189i	$0.690504\pi$
$114$	0	0
$115$	231.611	0.0175131
$116$	2821.05i	0.209650i
$117$	0	0
$118$	24179.1	1.73651
$119$	− 5276.13i	− 0.372582i
$120$	0	0
$121$	5394.72	0.368467
$122$	7646.41i	0.513733i
$123$	0	0
$124$	1191.68	0.0775029
$125$	15914.9i	1.01855i
$126$	0	0
$127$	−20660.9	−1.28098	−0.640489	−	0.767968i	$-0.721268\pi$
$127$	−20660.9	−1.28098	−0.640489	+	0.767968i	$0.721268\pi$
$128$	19739.9i	1.20483i
$129$	0	0
$130$	11011.2	0.651552
$131$	3201.45i	0.186554i	0.995640	+	0.0932768i	$0.0297342\pi$
$131$	3201.45i	0.186554i	−0.995640	+	0.0932768i	$0.970266\pi$
$132$	0	0
$133$	−13819.8	−0.781263
$134$	20694.2i	1.15250i
$135$	0	0
$136$	−3920.31	−0.211955
$137$	3871.92i	0.206293i	0.994666	+	0.103147i	$0.0328911\pi$
$137$	3871.92i	0.206293i	−0.994666	+	0.103147i	$0.967109\pi$
$138$	0	0
$139$	−11679.2	−0.604484	−0.302242	−	0.953231i	$-0.597735\pi$
$139$	−11679.2	−0.604484	−0.302242	+	0.953231i	$0.597735\pi$
$140$	− 4574.17i	− 0.233376i
$141$	0	0
$142$	−29904.4	−1.48306
$143$	− 14799.2i	− 0.723714i
$144$	0	0
$145$	11476.4	0.545844
$146$	21539.8i	1.01050i
$147$	0	0
$148$	−3254.35	−0.148573
$149$	− 15091.8i	− 0.679778i	−0.940466	−	0.339889i	$-0.889610\pi$
$149$	− 15091.8i	− 0.679778i	0.940466	−	0.339889i	$-0.110390\pi$
$150$	0	0
$151$	30255.3	1.32693	0.663465	−	0.748207i	$-0.269085\pi$
$151$	30255.3	1.32693	0.663465	+	0.748207i	$0.269085\pi$
$152$	10268.5i	0.444446i
$153$	0	0
$154$	−31122.2	−1.31229
$155$	− 4847.92i	− 0.201787i
$156$	0	0
$157$	20622.7	0.836656	0.418328	−	0.908296i	$-0.362616\pi$
$157$	20622.7	0.836656	0.418328	+	0.908296i	$0.362616\pi$
$158$	25580.6i	1.02470i
$159$	0	0
$160$	−7907.31	−0.308879
$161$	− 1047.77i	− 0.0404216i
$162$	0	0
$163$	39790.7	1.49764	0.748818	−	0.662776i	$-0.230622\pi$
$163$	39790.7	1.49764	0.748818	+	0.662776i	$0.230622\pi$
$164$	− 2190.31i	− 0.0814364i
$165$	0	0
$166$	−12646.7	−0.458944
$167$	− 27450.7i	− 0.984283i	−0.870515	−	0.492141i	$-0.836214\pi$
$167$	− 27450.7i	− 0.984283i	0.870515	−	0.492141i	$-0.163786\pi$
$168$	0	0
$169$	−4873.95	−0.170651
$170$	− 5207.81i	− 0.180201i
$171$	0	0
$172$	−3515.89	−0.118844
$173$	− 10079.7i	− 0.336787i	−0.985720	−	0.168393i	$-0.946142\pi$
$173$	− 10079.7i	− 0.336787i	0.985720	−	0.168393i	$-0.0538578\pi$
$174$	0	0
$175$	26694.0	0.871642
$176$	29184.3i	0.942158i
$177$	0	0
$178$	63753.6	2.01217
$179$	− 1215.45i	− 0.0379343i	−0.999820	−	0.0189672i	$-0.993962\pi$
$179$	− 1215.45i	− 0.0379343i	0.999820	−	0.0189672i	$-0.00603779\pi$
$180$	0	0
$181$	28359.9	0.865661	0.432831	−	0.901475i	$-0.357515\pi$
$181$	28359.9	0.865661	0.432831	+	0.901475i	$0.357515\pi$
$182$	− 49813.0i	− 1.50383i
$183$	0	0
$184$	−778.522	−0.0229951
$185$	13239.1i	0.386826i
$186$	0	0
$187$	−6999.36	−0.200159
$188$	− 15578.8i	− 0.440775i
$189$	0	0
$190$	−13640.8	−0.377862
$191$	9751.84i	0.267313i	0.991028	+	0.133656i	$0.0426719\pi$
$191$	9751.84i	0.267313i	−0.991028	+	0.133656i	$0.957328\pi$
$192$	0	0
$193$	−53402.8	−1.43367	−0.716836	−	0.697242i	$-0.754410\pi$
$193$	−53402.8	−1.43367	−0.716836	+	0.697242i	$0.754410\pi$
$194$	− 31995.0i	− 0.850117i
$195$	0	0
$196$	−11236.3	−0.292490
$197$	− 68537.7i	− 1.76603i	−0.469349	−	0.883013i	$-0.655511\pi$
$197$	− 68537.7i	− 1.76603i	0.469349	−	0.883013i	$-0.344489\pi$
$198$	0	0
$199$	−8237.42	−0.208010	−0.104005	−	0.994577i	$-0.533166\pi$
$199$	−8237.42	−0.208010	−0.104005	+	0.994577i	$0.533166\pi$
$200$	− 19834.4i	− 0.495860i
$201$	0	0
$202$	8748.34	0.214399
$203$	− 51917.3i	− 1.25985i
$204$	0	0
$205$	−8910.47	−0.212028
$206$	23020.6i	0.542477i
$207$	0	0
$208$	−46711.2	−1.07968
$209$	18333.4i	0.419712i
$210$	0	0
$211$	40787.4	0.916138	0.458069	−	0.888917i	$-0.348541\pi$
$211$	40787.4	0.916138	0.458069	+	0.888917i	$0.348541\pi$
$212$	− 7747.00i	− 0.172370i
$213$	0	0
$214$	−40710.2	−0.888946
$215$	14303.1i	0.309423i
$216$	0	0
$217$	−21931.2	−0.465740
$218$	72163.4i	1.51846i
$219$	0	0
$220$	−6068.13	−0.125375
$221$	− 11202.9i	− 0.229375i
$222$	0	0
$223$	−59919.6	−1.20492	−0.602461	−	0.798148i	$-0.705813\pi$
$223$	−59919.6	−1.20492	−0.602461	+	0.798148i	$0.705813\pi$
$224$	35771.4i	0.712918i
$225$	0	0
$226$	94213.5	1.84458
$227$	− 98810.4i	− 1.91757i	−0.284134	−	0.958785i	$-0.591706\pi$
$227$	− 98810.4i	− 1.91757i	0.284134	−	0.958785i	$-0.408294\pi$
$228$	0	0
$229$	−5317.94	−0.101408	−0.0507040	−	0.998714i	$-0.516147\pi$
$229$	−5317.94	−0.101408	−0.0507040	+	0.998714i	$0.516147\pi$
$230$	− 1034.20i	− 0.0195501i
$231$	0	0
$232$	−38576.0	−0.716707
$233$	18467.8i	0.340176i	0.985429	+	0.170088i	$0.0544052\pi$
$233$	18467.8i	0.340176i	−0.985429	+	0.170088i	$0.945595\pi$
$234$	0	0
$235$	−63376.3	−1.14760
$236$	− 21327.1i	− 0.382920i
$237$	0	0
$238$	−23559.3	−0.415919
$239$	25846.2i	0.452482i	0.974071	+	0.226241i	$0.0726437\pi$
$239$	25846.2i	0.452482i	−0.974071	+	0.226241i	$0.927356\pi$
$240$	0	0
$241$	83072.3	1.43028	0.715142	−	0.698979i	$-0.246362\pi$
$241$	83072.3	1.43028	0.715142	+	0.698979i	$0.246362\pi$
$242$	− 24088.9i	− 0.411325i
$243$	0	0
$244$	6744.49	0.113284
$245$	45710.7i	0.761527i
$246$	0	0
$247$	−29343.8	−0.480974
$248$	16295.5i	0.264951i
$249$	0	0
$250$	71064.1	1.13703
$251$	− 49051.6i	− 0.778585i	−0.921114	−	0.389292i	$-0.872720\pi$
$251$	− 49051.6i	− 0.778585i	0.921114	−	0.389292i	$-0.127280\pi$
$252$	0	0
$253$	−1389.98	−0.0217154
$254$	92256.3i	1.42998i
$255$	0	0
$256$	45705.2	0.697405
$257$	97105.7i	1.47021i	0.677955	+	0.735104i	$0.262867\pi$
$257$	97105.7i	1.47021i	−0.677955	+	0.735104i	$0.737133\pi$
$258$	0	0
$259$	59891.6	0.892825
$260$	− 9712.41i	− 0.143675i
$261$	0	0
$262$	14295.3	0.208253
$263$	− 41478.0i	− 0.599662i	−0.953992	−	0.299831i	$-0.903070\pi$
$263$	− 41478.0i	− 0.599662i	0.953992	−	0.299831i	$-0.0969303\pi$
$264$	0	0
$265$	−31515.8	−0.448783
$266$	61708.9i	0.872136i
$267$	0	0
$268$	18253.3	0.254139
$269$	115737.i	1.59944i	0.600370	+	0.799722i	$0.295020\pi$
$269$	115737.i	1.59944i	−0.600370	+	0.799722i	$0.704980\pi$
$270$	0	0
$271$	−5077.71	−0.0691400	−0.0345700	−	0.999402i	$-0.511006\pi$
$271$	−5077.71	−0.0691400	−0.0345700	+	0.999402i	$0.511006\pi$
$272$	22092.3i	0.298609i
$273$	0	0
$274$	17289.1	0.230289
$275$	− 35412.5i	− 0.468265i
$276$	0	0
$277$	−42385.8	−0.552409	−0.276205	−	0.961099i	$-0.589077\pi$
$277$	−42385.8	−0.552409	−0.276205	+	0.961099i	$0.589077\pi$
$278$	52150.9i	0.674795i
$279$	0	0
$280$	62548.8	0.797817
$281$	41456.6i	0.525027i	0.964928	+	0.262513i	$0.0845514\pi$
$281$	41456.6i	0.525027i	−0.964928	+	0.262513i	$0.915449\pi$
$282$	0	0
$283$	38388.0	0.479316	0.239658	−	0.970857i	$-0.422965\pi$
$283$	38388.0	0.479316	0.239658	+	0.970857i	$0.422965\pi$
$284$	26377.1i	0.327032i
$285$	0	0
$286$	−66082.4	−0.807893
$287$	40309.6i	0.489378i
$288$	0	0
$289$	78222.5	0.936561
$290$	− 51245.0i	− 0.609335i
$291$	0	0
$292$	18999.1	0.222827
$293$	− 12978.2i	− 0.151175i	−0.997139	−	0.0755876i	$-0.975917\pi$
$293$	− 12978.2i	− 0.151175i	0.997139	−	0.0755876i	$-0.0240833\pi$
$294$	0	0
$295$	−86761.4	−0.996971
$296$	− 44501.2i	− 0.507911i
$297$	0	0
$298$	−67388.7	−0.758847
$299$	− 2224.75i	− 0.0248850i
$300$	0	0
$301$	64704.8	0.714173
$302$	− 135098.i	− 1.48127i
$303$	0	0
$304$	57866.3	0.626150
$305$	− 27437.4i	− 0.294947i
$306$	0	0
$307$	−126105.	−1.33799	−0.668997	−	0.743265i	$-0.733276\pi$
$307$	−126105.	−1.33799	−0.668997	+	0.743265i	$0.733276\pi$
$308$	27451.3i	0.289375i
$309$	0	0
$310$	−21647.2	−0.225258
$311$	− 99803.5i	− 1.03187i	−0.856628	−	0.515935i	$-0.827445\pi$
$311$	− 99803.5i	− 1.03187i	0.856628	−	0.515935i	$-0.172555\pi$
$312$	0	0
$313$	−2225.35	−0.0227148	−0.0113574	−	0.999936i	$-0.503615\pi$
$313$	−2225.35	−0.0227148	−0.0113574	+	0.999936i	$0.503615\pi$
$314$	− 92085.9i	− 0.933972i
$315$	0	0
$316$	22563.3	0.225958
$317$	168069.i	1.67251i	0.548342	+	0.836254i	$0.315259\pi$
$317$	168069.i	1.67251i	−0.548342	+	0.836254i	$0.684741\pi$
$318$	0	0
$319$	−68874.0	−0.676821
$320$	− 42498.8i	− 0.415027i
$321$	0	0
$322$	−4678.56	−0.0451233
$323$	13878.3i	0.133024i
$324$	0	0
$325$	56679.9	0.536615
$326$	− 177676.i	− 1.67183i
$327$	0	0
$328$	29951.1	0.278398
$329$	286704.i	2.64876i
$330$	0	0
$331$	6758.86	0.0616904	0.0308452	−	0.999524i	$-0.490180\pi$
$331$	6758.86	0.0616904	0.0308452	+	0.999524i	$0.490180\pi$
$332$	11154.9i	0.101202i
$333$	0	0
$334$	−122574.	−1.09877
$335$	− 74256.6i	− 0.661676i
$336$	0	0
$337$	−222093.	−1.95557	−0.977787	−	0.209600i	$-0.932784\pi$
$337$	−222093.	−1.95557	−0.977787	+	0.209600i	$0.932784\pi$
$338$	21763.5i	0.190500i
$339$	0	0
$340$	−4593.53	−0.0397364
$341$	29094.2i	0.250206i
$342$	0	0
$343$	32754.4	0.278408
$344$	− 48077.5i	− 0.406280i
$345$	0	0
$346$	−45008.4	−0.375960
$347$	− 162548.i	− 1.34996i	−0.737835	−	0.674981i	$-0.764152\pi$
$347$	− 162548.i	− 1.34996i	0.737835	−	0.674981i	$-0.235848\pi$
$348$	0	0
$349$	−72696.9	−0.596850	−0.298425	−	0.954433i	$-0.596461\pi$
$349$	−72696.9	−0.596850	−0.298425	+	0.954433i	$0.596461\pi$
$350$	− 119196.i	− 0.973027i
$351$	0	0
$352$	47454.7	0.382995
$353$	− 128629.i	− 1.03226i	−0.856509	−	0.516131i	$-0.827372\pi$
$353$	− 128629.i	− 1.03226i	0.856509	−	0.516131i	$-0.172628\pi$
$354$	0	0
$355$	107305.	0.851461
$356$	− 56233.6i	− 0.443707i
$357$	0	0
$358$	−5427.32	−0.0423467
$359$	− 95302.8i	− 0.739463i	−0.929139	−	0.369732i	$-0.879450\pi$
$359$	− 95302.8i	− 0.739463i	0.929139	−	0.369732i	$-0.120550\pi$
$360$	0	0
$361$	−93969.7	−0.721063
$362$	− 126635.i	− 0.966351i
$363$	0	0
$364$	−43937.4	−0.331613
$365$	− 77290.7i	− 0.580152i
$366$	0	0
$367$	104146.	0.773233	0.386616	−	0.922241i	$-0.373644\pi$
$367$	104146.	0.773233	0.386616	+	0.922241i	$0.373644\pi$
$368$	4387.23i	0.0323963i
$369$	0	0
$370$	59116.1	0.431820
$371$	142572.i	1.03583i
$372$	0	0
$373$	245831.	1.76693	0.883463	−	0.468501i	$-0.155206\pi$
$373$	245831.	1.76693	0.883463	+	0.468501i	$0.155206\pi$
$374$	31254.0i	0.223441i
$375$	0	0
$376$	213029.	1.50683
$377$	− 110237.i	− 0.775612i
$378$	0	0
$379$	200830.	1.39814	0.699070	−	0.715053i	$-0.253597\pi$
$379$	200830.	1.39814	0.699070	+	0.715053i	$0.253597\pi$
$380$	12031.8i	0.0833230i
$381$	0	0
$382$	43544.5	0.298406
$383$	135338.i	0.922615i	0.887240	+	0.461308i	$0.152620\pi$
$383$	135338.i	0.922615i	−0.887240	+	0.461308i	$0.847380\pi$
$384$	0	0
$385$	111675.	0.753417
$386$	238458.i	1.60043i
$387$	0	0
$388$	−28221.1	−0.187461
$389$	202177.i	1.33608i	0.744126	+	0.668039i	$0.232866\pi$
$389$	202177.i	1.33608i	−0.744126	+	0.668039i	$0.767134\pi$
$390$	0	0
$391$	−1052.20	−0.00688251
$392$	− 153649.i	− 0.999904i
$393$	0	0
$394$	−306039.	−1.97144
$395$	− 91790.2i	− 0.588304i
$396$	0	0
$397$	102384.	0.649608	0.324804	−	0.945781i	$-0.394702\pi$
$397$	102384.	0.649608	0.324804	+	0.945781i	$0.394702\pi$
$398$	36782.2i	0.232205i
$399$	0	0
$400$	−111774.	−0.698585
$401$	− 90097.2i	− 0.560302i	−0.959956	−	0.280151i	$-0.909615\pi$
$401$	− 90097.2i	− 0.560302i	0.959956	−	0.280151i	$-0.0903846\pi$
$402$	0	0
$403$	−46567.0	−0.286727
$404$	− 7716.45i	− 0.0472775i
$405$	0	0
$406$	−231824.	−1.40639
$407$	− 79452.8i	− 0.479645i
$408$	0	0
$409$	149722.	0.895033	0.447517	−	0.894276i	$-0.352309\pi$
$409$	149722.	0.895033	0.447517	+	0.894276i	$0.352309\pi$
$410$	39787.6i	0.236690i
$411$	0	0
$412$	20305.2	0.119623
$413$	392495.i	2.30109i
$414$	0	0
$415$	45379.7	0.263491
$416$	75954.1i	0.438899i
$417$	0	0
$418$	81863.6	0.468531
$419$	126351.i	0.719697i	0.933011	+	0.359848i	$0.117172\pi$
$419$	126351.i	0.719697i	−0.933011	+	0.359848i	$0.882828\pi$
$420$	0	0
$421$	−215289.	−1.21467	−0.607335	−	0.794446i	$-0.707762\pi$
$421$	−215289.	−1.21467	−0.607335	+	0.794446i	$0.707762\pi$
$422$	− 182126.i	− 1.02270i
$423$	0	0
$424$	105935.	0.589263
$425$	− 26807.0i	− 0.148413i
$426$	0	0
$427$	−124123.	−0.680762
$428$	35908.3i	0.196023i
$429$	0	0
$430$	63866.9	0.345413
$431$	121972.i	0.656607i	0.944572	+	0.328304i	$0.106477\pi$
$431$	121972.i	0.656607i	−0.944572	+	0.328304i	$0.893523\pi$
$432$	0	0
$433$	−152419.	−0.812951	−0.406475	−	0.913662i	$-0.633242\pi$
$433$	−152419.	−0.812951	−0.406475	+	0.913662i	$0.633242\pi$
$434$	97928.7i	0.519913i
$435$	0	0
$436$	63651.5	0.334839
$437$	2756.04i	0.0144319i
$438$	0	0
$439$	231128.	1.19929	0.599643	−	0.800267i	$-0.295309\pi$
$439$	231128.	1.19929	0.599643	+	0.800267i	$0.295309\pi$
$440$	− 82977.9i	− 0.428605i
$441$	0	0
$442$	−50023.9	−0.256055
$443$	20185.7i	0.102858i	0.998677	+	0.0514289i	$0.0163775\pi$
$443$	20185.7i	0.102858i	−0.998677	+	0.0514289i	$0.983622\pi$
$444$	0	0
$445$	−228766.	−1.15524
$446$	267557.i	1.34507i
$447$	0	0
$448$	−192258.	−0.957917
$449$	92962.8i	0.461123i	0.973058	+	0.230561i	$0.0740562\pi$
$449$	92962.8i	0.461123i	−0.973058	+	0.230561i	$0.925944\pi$
$450$	0	0
$451$	53475.0	0.262905
$452$	− 83100.7i	− 0.406750i
$453$	0	0
$454$	−441215.	−2.14061
$455$	178743.i	0.863388i
$456$	0	0
$457$	220887.	1.05764	0.528819	−	0.848735i	$-0.322635\pi$
$457$	220887.	1.05764	0.528819	+	0.848735i	$0.322635\pi$
$458$	23746.0i	0.113203i
$459$	0	0
$460$	−912.215	−0.00431103
$461$	− 106195.i	− 0.499690i	−0.968286	−	0.249845i	$-0.919620\pi$
$461$	− 106195.i	− 0.499690i	0.968286	−	0.249845i	$-0.0803797\pi$
$462$	0	0
$463$	−68179.9	−0.318049	−0.159025	−	0.987275i	$-0.550835\pi$
$463$	−68179.9	−0.318049	−0.159025	+	0.987275i	$0.550835\pi$
$464$	217389.i	1.00972i
$465$	0	0
$466$	82463.6	0.379744
$467$	− 224350.i	− 1.02871i	−0.857578	−	0.514353i	$-0.828032\pi$
$467$	− 224350.i	− 1.02871i	0.857578	−	0.514353i	$-0.171968\pi$
$468$	0	0
$469$	−335925.	−1.52720
$470$	282992.i	1.28109i
$471$	0	0
$472$	291635.	1.30905
$473$	− 85838.0i	− 0.383669i
$474$	0	0
$475$	−70215.6	−0.311205
$476$	20780.4i	0.0917149i
$477$	0	0
$478$	115410.	0.505113
$479$	109169.i	0.475802i	0.971289	+	0.237901i	$0.0764594\pi$
$479$	109169.i	0.475802i	−0.971289	+	0.237901i	$0.923541\pi$
$480$	0	0
$481$	127169.	0.549656
$482$	− 370940.i	− 1.59665i
$483$	0	0
$484$	−21247.5	−0.0907021
$485$	114807.i	0.488073i
$486$	0	0
$487$	106309.	0.448240	0.224120	−	0.974562i	$-0.428049\pi$
$487$	106309.	0.448240	0.224120	+	0.974562i	$0.428049\pi$
$488$	92226.6i	0.387272i
$489$	0	0
$490$	204110.	0.850105
$491$	− 74446.6i	− 0.308803i	−0.988008	−	0.154402i	$-0.950655\pi$
$491$	− 74446.6i	− 0.308803i	0.988008	−	0.154402i	$-0.0493450\pi$
$492$	0	0
$493$	−52137.1	−0.214513
$494$	131028.i	0.536919i
$495$	0	0
$496$	91830.8	0.373271
$497$	− 485432.i	− 1.96524i
$498$	0	0
$499$	−111207.	−0.446612	−0.223306	−	0.974748i	$-0.571685\pi$
$499$	−111207.	−0.446612	−0.223306	+	0.974748i	$0.571685\pi$
$500$	− 62681.8i	− 0.250727i
$501$	0	0
$502$	−219028.	−0.869146
$503$	− 115897.i	− 0.458077i	−0.973417	−	0.229038i	$-0.926442\pi$
$503$	− 115897.i	− 0.458077i	0.973417	−	0.229038i	$-0.0735581\pi$
$504$	0	0
$505$	−31391.5	−0.123092
$506$	6206.63i	0.0242412i
$507$	0	0
$508$	81374.4	0.315326
$509$	− 196644.i	− 0.759007i	−0.925190	−	0.379503i	$-0.876095\pi$
$509$	− 196644.i	− 0.759007i	0.925190	−	0.379503i	$-0.123905\pi$
$510$	0	0
$511$	−349651.	−1.33904
$512$	111753.i	0.426305i
$513$	0	0
$514$	433603.	1.64122
$515$	− 82604.1i	− 0.311449i
$516$	0	0
$517$	380345.	1.42297
$518$	− 267432.i	− 0.996675i
$519$	0	0
$520$	132811.	0.491165
$521$	− 299306.i	− 1.10266i	−0.834289	−	0.551328i	$-0.814121\pi$
$521$	− 299306.i	− 1.10266i	0.834289	−	0.551328i	$-0.185879\pi$
$522$	0	0
$523$	−162892.	−0.595520	−0.297760	−	0.954641i	$-0.596240\pi$
$523$	−162892.	−0.595520	−0.297760	+	0.954641i	$0.596240\pi$
$524$	− 12609.1i	− 0.0459222i
$525$	0	0
$526$	−185210.	−0.669412
$527$	22024.1i	0.0793006i
$528$	0	0
$529$	279632.	0.999253
$530$	140726.i	0.500983i
$531$	0	0
$532$	54430.1	0.192316
$533$	85590.0i	0.301279i
$534$	0	0
$535$	146079.	0.510365
$536$	249602.i	0.868796i
$537$	0	0
$538$	516798.	1.78548
$539$	− 274327.i	− 0.944257i
$540$	0	0
$541$	−62918.7	−0.214974	−0.107487	−	0.994207i	$-0.534280\pi$
$541$	−62918.7	−0.214974	−0.107487	+	0.994207i	$0.534280\pi$
$542$	22673.3i	0.0771820i
$543$	0	0
$544$	35922.8	0.121387
$545$	− 258942.i	− 0.871786i
$546$	0	0
$547$	−14807.1	−0.0494876	−0.0247438	−	0.999694i	$-0.507877\pi$
$547$	−14807.1	−0.0494876	−0.0247438	+	0.999694i	$0.507877\pi$
$548$	− 15249.8i	− 0.0507813i
$549$	0	0
$550$	−158126.	−0.522732
$551$	136563.i	0.449810i
$552$	0	0
$553$	−415244.	−1.35785
$554$	189264.i	0.616663i
$555$	0	0
$556$	45999.5	0.148800
$557$	− 254392.i	− 0.819962i	−0.912094	−	0.409981i	$-0.865535\pi$
$557$	− 254392.i	− 0.819962i	0.912094	−	0.409981i	$-0.134465\pi$
$558$	0	0
$559$	137389.	0.439671
$560$	− 352484.i	− 1.12399i
$561$	0	0
$562$	185115.	0.586096
$563$	93307.5i	0.294374i	0.989109	+	0.147187i	$0.0470220\pi$
$563$	93307.5i	0.294374i	−0.989109	+	0.147187i	$0.952978\pi$
$564$	0	0
$565$	−338064.	−1.05902
$566$	− 171412.i	− 0.535068i
$567$	0	0
$568$	−360690.	−1.11799
$569$	158024.i	0.488089i	0.969764	+	0.244045i	$0.0784744\pi$
$569$	158024.i	0.488089i	−0.969764	+	0.244045i	$0.921526\pi$
$570$	0	0
$571$	−438375.	−1.34454	−0.672270	−	0.740306i	$-0.734681\pi$
$571$	−438375.	−1.34454	−0.672270	+	0.740306i	$0.734681\pi$
$572$	58287.8i	0.178150i
$573$	0	0
$574$	179993.	0.546300
$575$	− 5323.52i	− 0.0161014i
$576$	0	0
$577$	214770.	0.645094	0.322547	−	0.946554i	$-0.395461\pi$
$577$	214770.	0.645094	0.322547	+	0.946554i	$0.395461\pi$
$578$	− 349284.i	− 1.04550i
$579$	0	0
$580$	−45200.5	−0.134365
$581$	− 205291.i	− 0.608158i
$582$	0	0
$583$	189138.	0.556469
$584$	259801.i	0.761753i
$585$	0	0
$586$	−57951.3	−0.168759
$587$	− 656412.i	− 1.90502i	−0.304502	−	0.952512i	$-0.598490\pi$
$587$	− 656412.i	− 1.90502i	0.304502	−	0.952512i	$-0.401510\pi$
$588$	0	0
$589$	57687.6	0.166285
$590$	387413.i	1.11293i
$591$	0	0
$592$	−250779.	−0.715563
$593$	118369.i	0.336612i	0.985735	+	0.168306i	$0.0538296\pi$
$593$	118369.i	0.336612i	−0.985735	+	0.168306i	$0.946170\pi$
$594$	0	0
$595$	84537.3	0.238789
$596$	59440.0i	0.167335i
$597$	0	0
$598$	−9934.08	−0.0277796
$599$	− 354139.i	− 0.987006i	−0.869744	−	0.493503i	$-0.835716\pi$
$599$	− 354139.i	− 0.987006i	0.869744	−	0.493503i	$-0.164284\pi$
$600$	0	0
$601$	−595639.	−1.64905	−0.824526	−	0.565824i	$-0.808558\pi$
$601$	−595639.	−1.64905	−0.824526	+	0.565824i	$0.808558\pi$
$602$	− 288924.i	− 0.797242i
$603$	0	0
$604$	−119163.	−0.326638
$605$	86437.5i	0.236152i
$606$	0	0
$607$	9721.62	0.0263853	0.0131926	−	0.999913i	$-0.495801\pi$
$607$	9721.62	0.0263853	0.0131926	+	0.999913i	$0.495801\pi$
$608$	− 94092.6i	− 0.254536i
$609$	0	0
$610$	−122515.	−0.329254
$611$	608765.i	1.63067i
$612$	0	0
$613$	194450.	0.517473	0.258736	−	0.965948i	$-0.416694\pi$
$613$	194450.	0.517473	0.258736	+	0.965948i	$0.416694\pi$
$614$	563090.i	1.49362i
$615$	0	0
$616$	−375379.	−0.989255
$617$	487639.i	1.28094i	0.767984	+	0.640469i	$0.221260\pi$
$617$	487639.i	1.28094i	−0.767984	+	0.640469i	$0.778740\pi$
$618$	0	0
$619$	17256.5	0.0450371	0.0225185	−	0.999746i	$-0.492832\pi$
$619$	17256.5	0.0450371	0.0225185	+	0.999746i	$0.492832\pi$
$620$	19093.9i	0.0496719i
$621$	0	0
$622$	−445649.	−1.15189
$623$	1.03490e6i	2.66638i
$624$	0	0
$625$	−24825.0	−0.0635520
$626$	9936.78i	0.0253569i
$627$	0	0
$628$	−81224.1	−0.205952
$629$	− 60145.1i	− 0.152019i
$630$	0	0
$631$	−429836.	−1.07955	−0.539777	−	0.841808i	$-0.681491\pi$
$631$	−429836.	−1.07955	−0.539777	+	0.841808i	$0.681491\pi$
$632$	308538.i	0.772458i
$633$	0	0
$634$	750471.	1.86705
$635$	− 331041.i	− 0.820984i
$636$	0	0
$637$	439076.	1.08208
$638$	307540.i	0.755546i
$639$	0	0
$640$	−316285.	−0.772181
$641$	32308.0i	0.0786310i	0.999227	+	0.0393155i	$0.0125177\pi$
$641$	32308.0i	0.0786310i	−0.999227	+	0.0393155i	$0.987482\pi$
$642$	0	0
$643$	277048.	0.670091	0.335045	−	0.942202i	$-0.391248\pi$
$643$	277048.	0.670091	0.335045	+	0.942202i	$0.391248\pi$
$644$	4126.71i	0.00995022i
$645$	0	0
$646$	61970.1	0.148497
$647$	125452.i	0.299689i	0.988710	+	0.149844i	$0.0478773\pi$
$647$	125452.i	0.299689i	−0.988710	+	0.149844i	$0.952123\pi$
$648$	0	0
$649$	520687.	1.23620
$650$	− 253091.i	− 0.599031i
$651$	0	0
$652$	−156718.	−0.368659
$653$	283982.i	0.665985i	0.942930	+	0.332992i	$0.108058\pi$
$653$	283982.i	0.665985i	−0.942930	+	0.332992i	$0.891942\pi$
$654$	0	0
$655$	−51295.5	−0.119563
$656$	− 168785.i	− 0.392216i
$657$	0	0
$658$	1.28021e6	2.95685
$659$	− 359272.i	− 0.827280i	−0.910440	−	0.413640i	$-0.864257\pi$
$659$	− 359272.i	− 0.827280i	0.910440	−	0.413640i	$-0.135743\pi$
$660$	0	0
$661$	197804.	0.452723	0.226362	−	0.974043i	$-0.427317\pi$
$661$	197804.	0.452723	0.226362	+	0.974043i	$0.427317\pi$
$662$	− 30180.1i	− 0.0688659i
$663$	0	0
$664$	−152537.	−0.345970
$665$	− 221429.i	− 0.500715i
$666$	0	0
$667$	−10353.7	−0.0232726
$668$	108116.i	0.242292i
$669$	0	0
$670$	−331575.	−0.738639
$671$	164662.i	0.365720i
$672$	0	0
$673$	238072.	0.525628	0.262814	−	0.964847i	$-0.415349\pi$
$673$	238072.	0.525628	0.262814	+	0.964847i	$0.415349\pi$
$674$	991702.i	2.18304i
$675$	0	0
$676$	19196.4	0.0420075
$677$	− 250150.i	− 0.545787i	−0.962044	−	0.272894i	$-0.912019\pi$
$677$	− 250150.i	− 0.545787i	0.962044	−	0.272894i	$-0.0879807\pi$
$678$	0	0
$679$	519368.	1.12651
$680$	− 62813.6i	− 0.135843i
$681$	0	0
$682$	129913.	0.279309
$683$	− 267358.i	− 0.573129i	−0.958061	−	0.286564i	$-0.907487\pi$
$683$	− 267358.i	− 0.573129i	0.958061	−	0.286564i	$-0.0925132\pi$
$684$	0	0
$685$	−62038.2	−0.132214
$686$	− 146257.i	− 0.310791i
$687$	0	0
$688$	−270933.	−0.572380
$689$	302727.i	0.637694i
$690$	0	0
$691$	336686.	0.705129	0.352565	−	0.935787i	$-0.385310\pi$
$691$	336686.	0.705129	0.352565	+	0.935787i	$0.385310\pi$
$692$	39699.5i	0.0829036i
$693$	0	0
$694$	−725817.	−1.50698
$695$	− 187132.i	− 0.387417i
$696$	0	0
$697$	40480.2	0.0833253
$698$	324611.i	0.666273i
$699$	0	0
$700$	−105136.	−0.214564
$701$	− 635795.i	− 1.29384i	−0.762557	−	0.646921i	$-0.776056\pi$
$701$	− 635795.i	− 1.29384i	0.762557	−	0.646921i	$-0.223944\pi$
$702$	0	0
$703$	−157538.	−0.318768
$704$	255051.i	0.514614i
$705$	0	0
$706$	−574363.	−1.15233
$707$	142010.i	0.284106i
$708$	0	0
$709$	431026.	0.857455	0.428728	−	0.903434i	$-0.358962\pi$
$709$	431026.	0.857455	0.428728	+	0.903434i	$0.358962\pi$
$710$	− 479147.i	− 0.950499i
$711$	0	0
$712$	768959.	1.51685
$713$	4373.69i	0.00860337i
$714$	0	0
$715$	237122.	0.463831
$716$	4787.15i	0.00933793i
$717$	0	0
$718$	−425552.	−0.825475
$719$	798370.i	1.54435i	0.635409	+	0.772176i	$0.280832\pi$
$719$	798370.i	1.54435i	−0.635409	+	0.772176i	$0.719168\pi$
$720$	0	0
$721$	−373688.	−0.718850
$722$	419599.i	0.804934i
$723$	0	0
$724$	−111698.	−0.213092
$725$	− 263782.i	− 0.501845i
$726$	0	0
$727$	317113.	0.599992	0.299996	−	0.953940i	$-0.403015\pi$
$727$	317113.	0.599992	0.299996	+	0.953940i	$0.403015\pi$
$728$	− 600816.i	− 1.13365i
$729$	0	0
$730$	−345123.	−0.647632
$731$	− 64978.7i	− 0.121601i
$732$	0	0
$733$	−977843.	−1.81996	−0.909979	−	0.414654i	$-0.863902\pi$
$733$	−977843.	−1.81996	−0.909979	+	0.414654i	$0.863902\pi$
$734$	− 465039.i	− 0.863172i
$735$	0	0
$736$	7133.80	0.0131694
$737$	445641.i	0.820446i
$738$	0	0
$739$	−392565.	−0.718825	−0.359412	−	0.933179i	$-0.617023\pi$
$739$	−392565.	−0.718825	−0.359412	+	0.933179i	$0.617023\pi$
$740$	− 52143.2i	− 0.0952213i
$741$	0	0
$742$	636623.	1.15631
$743$	1.03860e6i	1.88135i	0.339314	+	0.940673i	$0.389805\pi$
$743$	1.03860e6i	1.88135i	−0.339314	+	0.940673i	$0.610195\pi$
$744$	0	0
$745$	241809.	0.435673
$746$	− 1.09770e6i	− 1.97245i
$747$	0	0
$748$	27567.5	0.0492713
$749$	− 660840.i	− 1.17797i
$750$	0	0
$751$	319565.	0.566604	0.283302	−	0.959031i	$-0.408570\pi$
$751$	319565.	0.566604	0.283302	+	0.959031i	$0.408570\pi$
$752$	− 1.20049e6i	− 2.12287i
$753$	0	0
$754$	−492237.	−0.865828
$755$	484769.i	0.850435i
$756$	0	0
$757$	500321.	0.873085	0.436543	−	0.899684i	$-0.356203\pi$
$757$	500321.	0.873085	0.436543	+	0.899684i	$0.356203\pi$
$758$	− 896760.i	− 1.56077i
$759$	0	0
$760$	−164528.	−0.284847
$761$	− 904875.i	− 1.56250i	−0.624220	−	0.781249i	$-0.714583\pi$
$761$	− 904875.i	− 1.56250i	0.624220	−	0.781249i	$-0.285417\pi$
$762$	0	0
$763$	−1.17141e6	−2.01215
$764$	− 38408.3i	− 0.0658019i
$765$	0	0
$766$	604318.	1.02993
$767$	833392.i	1.41664i
$768$	0	0
$769$	143377.	0.242453	0.121227	−	0.992625i	$-0.461317\pi$
$769$	143377.	0.242453	0.121227	+	0.992625i	$0.461317\pi$
$770$	− 498659.i	− 0.841051i
$771$	0	0
$772$	210331.	0.352914
$773$	− 710063.i	− 1.18833i	−0.804342	−	0.594166i	$-0.797482\pi$
$773$	− 710063.i	− 1.18833i	0.804342	−	0.594166i	$-0.202518\pi$
$774$	0	0
$775$	−111428.	−0.185521
$776$	− 385905.i	− 0.640851i
$777$	0	0
$778$	902772.	1.49148
$779$	− 106030.i	− 0.174724i
$780$	0	0
$781$	−643979.	−1.05577
$782$	4698.37i	0.00768305i
$783$	0	0
$784$	−865865.	−1.40870
$785$	330430.i	0.536216i
$786$	0	0
$787$	−601711.	−0.971490	−0.485745	−	0.874101i	$-0.661452\pi$
$787$	−601711.	−0.971490	−0.485745	+	0.874101i	$0.661452\pi$
$788$	269941.i	0.434726i
$789$	0	0
$790$	−409867.	−0.656733
$791$	1.52935e6i	2.44429i
$792$	0	0
$793$	−263552.	−0.419102
$794$	− 457172.i	− 0.725168i
$795$	0	0
$796$	32443.7	0.0512040
$797$	− 429811.i	− 0.676645i	−0.941030	−	0.338322i	$-0.890141\pi$
$797$	− 429811.i	− 0.676645i	0.941030	−	0.338322i	$-0.109859\pi$
$798$	0	0
$799$	287918.	0.450999
$800$	181748.i	0.283981i
$801$	0	0
$802$	−402308.	−0.625474
$803$	463850.i	0.719361i
$804$	0	0
$805$	16788.0	0.0259064
$806$	207934.i	0.320077i
$807$	0	0
$808$	105517.	0.161622
$809$	− 629639.i	− 0.962042i	−0.876709	−	0.481021i	$-0.840266\pi$
$809$	− 629639.i	− 0.962042i	0.876709	−	0.481021i	$-0.159734\pi$
$810$	0	0
$811$	577760.	0.878428	0.439214	−	0.898383i	$-0.355257\pi$
$811$	577760.	0.878428	0.439214	+	0.898383i	$0.355257\pi$
$812$	204480.i	0.310126i
$813$	0	0
$814$	−354778.	−0.535436
$815$	637550.i	0.959841i
$816$	0	0
$817$	−170199.	−0.254984
$818$	− 668548.i	− 0.999140i
$819$	0	0
$820$	35094.5	0.0521929
$821$	− 44719.6i	− 0.0663455i	−0.999450	−	0.0331727i	$-0.989439\pi$
$821$	− 44719.6i	− 0.0663455i	0.999450	−	0.0331727i	$-0.0105612\pi$
$822$	0	0
$823$	−292578.	−0.431958	−0.215979	−	0.976398i	$-0.569294\pi$
$823$	−292578.	−0.431958	−0.215979	+	0.976398i	$0.569294\pi$
$824$	277661.i	0.408940i
$825$	0	0
$826$	1.75259e6	2.56874
$827$	− 806491.i	− 1.17920i	−0.807694	−	0.589601i	$-0.799285\pi$
$827$	− 806491.i	− 1.17920i	0.807694	−	0.589601i	$-0.200715\pi$
$828$	0	0
$829$	204904.	0.298154	0.149077	−	0.988826i	$-0.452370\pi$
$829$	204904.	0.298154	0.149077	+	0.988826i	$0.452370\pi$
$830$	− 202632.i	− 0.294139i
$831$	0	0
$832$	−408224.	−0.589729
$833$	− 207663.i	− 0.299274i
$834$	0	0
$835$	439831.	0.630831
$836$	− 72207.5i	− 0.103317i
$837$	0	0
$838$	564189.	0.803409
$839$	− 771105.i	− 1.09544i	−0.836661	−	0.547721i	$-0.815495\pi$
$839$	− 771105.i	− 1.09544i	0.836661	−	0.547721i	$-0.184505\pi$
$840$	0	0
$841$	194250.	0.274643
$842$	961324.i	1.35596i
$843$	0	0
$844$	−160644.	−0.225517
$845$	− 78093.4i	− 0.109371i
$846$	0	0
$847$	391030.	0.545058
$848$	− 596981.i	− 0.830173i
$849$	0	0
$850$	−119700.	−0.165675
$851$	− 11944.0i	− 0.0164927i
$852$	0	0
$853$	1.08237e6	1.48757	0.743784	−	0.668420i	$-0.233029\pi$
$853$	1.08237e6	1.48757	0.743784	+	0.668420i	$0.233029\pi$
$854$	554240.i	0.759945i
$855$	0	0
$856$	−491023.	−0.670122
$857$	533345.i	0.726183i	0.931753	+	0.363092i	$0.118279\pi$
$857$	533345.i	0.726183i	−0.931753	+	0.363092i	$0.881721\pi$
$858$	0	0
$859$	1.13105e6	1.53283	0.766417	−	0.642343i	$-0.222038\pi$
$859$	1.13105e6	1.53283	0.766417	+	0.642343i	$0.222038\pi$
$860$	− 56333.6i	− 0.0761677i
$861$	0	0
$862$	544637.	0.732981
$863$	110025.i	0.147730i	0.997268	+	0.0738649i	$0.0235334\pi$
$863$	110025.i	0.147730i	−0.997268	+	0.0738649i	$0.976467\pi$
$864$	0	0
$865$	161503.	0.215848
$866$	680592.i	0.907510i
$867$	0	0
$868$	86377.7	0.114647
$869$	550867.i	0.729470i
$870$	0	0
$871$	−713275.	−0.940202
$872$	870394.i	1.14468i
$873$	0	0
$874$	12306.4	0.0161105
$875$	1.15357e6i	1.50670i
$876$	0	0
$877$	−408753.	−0.531450	−0.265725	−	0.964049i	$-0.585611\pi$
$877$	−408753.	−0.531450	−0.265725	+	0.964049i	$0.585611\pi$
$878$	− 1.03205e6i	− 1.33878i
$879$	0	0
$880$	−467608.	−0.603833
$881$	− 235809.i	− 0.303815i	−0.988395	−	0.151907i	$-0.951458\pi$
$881$	− 235809.i	− 0.303815i	0.988395	−	0.151907i	$-0.0485416\pi$
$882$	0	0
$883$	−273719.	−0.351061	−0.175531	−	0.984474i	$-0.556164\pi$
$883$	−273719.	−0.351061	−0.175531	+	0.984474i	$0.556164\pi$
$884$	44123.4i	0.0564631i
$885$	0	0
$886$	90134.6	0.114822
$887$	371896.i	0.472687i	0.971670	+	0.236344i	$0.0759491\pi$
$887$	371896.i	0.472687i	−0.971670	+	0.236344i	$0.924051\pi$
$888$	0	0
$889$	−1.49758e6	−1.89490
$890$	1.02150e6i	1.28961i
$891$	0	0
$892$	235997.	0.296604
$893$	− 754144.i	− 0.945695i
$894$	0	0
$895$	19474.7	0.0243122
$896$	1.43082e6i	1.78226i
$897$	0	0
$898$	415103.	0.514758
$899$	216718.i	0.268148i
$900$	0	0
$901$	143176.	0.176368
$902$	− 238780.i	− 0.293484i
$903$	0	0
$904$	1.13635e6	1.39051
$905$	454400.i	0.554806i
$906$	0	0
$907$	−143931.	−0.174960	−0.0874801	−	0.996166i	$-0.527881\pi$
$907$	−143931.	−0.174960	−0.0874801	+	0.996166i	$0.527881\pi$
$908$	389172.i	0.472030i
$909$	0	0
$910$	798134.	0.963814
$911$	891661.i	1.07439i	0.843457	+	0.537196i	$0.180517\pi$
$911$	891661.i	1.07439i	−0.843457	+	0.537196i	$0.819483\pi$
$912$	0	0
$913$	−272340.	−0.326716
$914$	− 986317.i	− 1.18066i
$915$	0	0
$916$	20945.1	0.0249627
$917$	232053.i	0.275961i
$918$	0	0
$919$	−582156.	−0.689300	−0.344650	−	0.938731i	$-0.612002\pi$
$919$	−582156.	−0.689300	−0.344650	+	0.938731i	$0.612002\pi$
$920$	− 12474.0i	− 0.0147377i
$921$	0	0
$922$	−474187.	−0.557812
$923$	− 1.03073e6i	− 1.20987i
$924$	0	0
$925$	304298.	0.355644
$926$	304441.i	0.355043i
$927$	0	0
$928$	353482.	0.410460
$929$	771371.i	0.893782i	0.894588	+	0.446891i	$0.147469\pi$
$929$	771371.i	0.893782i	−0.894588	+	0.446891i	$0.852531\pi$
$930$	0	0
$931$	−543932.	−0.627546
$932$	− 72736.8i	− 0.0837380i
$933$	0	0
$934$	−1.00178e6	−1.14836
$935$	− 112148.i	− 0.128283i
$936$	0	0
$937$	−1.46097e6	−1.66403	−0.832015	−	0.554753i	$-0.812813\pi$
$937$	−1.46097e6	−1.66403	−0.832015	+	0.554753i	$0.812813\pi$
$938$	1.49999e6i	1.70484i
$939$	0	0
$940$	249612.	0.282494
$941$	169346.i	0.191248i	0.995418	+	0.0956240i	$0.0304846\pi$
$941$	169346.i	0.191248i	−0.995418	+	0.0956240i	$0.969515\pi$
$942$	0	0
$943$	8038.83	0.00904002
$944$	− 1.64346e6i	− 1.84423i
$945$	0	0
$946$	−383289.	−0.428296
$947$	− 593888.i	− 0.662224i	−0.943591	−	0.331112i	$-0.892576\pi$
$947$	− 593888.i	− 0.662224i	0.943591	−	0.331112i	$-0.107424\pi$
$948$	0	0
$949$	−742420.	−0.824361
$950$	313531.i	0.347403i
$951$	0	0
$952$	−284159.	−0.313536
$953$	1.25477e6i	1.38159i	0.723050	+	0.690796i	$0.242740\pi$
$953$	1.25477e6i	1.38159i	−0.723050	+	0.690796i	$0.757260\pi$
$954$	0	0
$955$	−156250.	−0.171322
$956$	− 101797.i	− 0.111383i
$957$	0	0
$958$	487466.	0.531145
$959$	280651.i	0.305161i
$960$	0	0
$961$	−831974.	−0.900872
$962$	− 567843.i	− 0.613590i
$963$	0	0
$964$	−327186.	−0.352080
$965$	− 855652.i	− 0.918846i
$966$	0	0
$967$	−1.05786e6	−1.13129	−0.565645	−	0.824649i	$-0.691373\pi$
$967$	−1.05786e6	−1.13129	−0.565645	+	0.824649i	$0.691373\pi$
$968$	− 290546.i	− 0.310073i
$969$	0	0
$970$	512643.	0.544843
$971$	− 775745.i	− 0.822774i	−0.911461	−	0.411387i	$-0.865044\pi$
$971$	− 775745.i	− 0.822774i	0.911461	−	0.411387i	$-0.134956\pi$
$972$	0	0
$973$	−846555.	−0.894189
$974$	− 474696.i	− 0.500377i
$975$	0	0
$976$	519728.	0.545603
$977$	− 496516.i	− 0.520168i	−0.965586	−	0.260084i	$-0.916250\pi$
$977$	− 496516.i	− 0.520168i	0.965586	−	0.260084i	$-0.0837503\pi$
$978$	0	0
$979$	1.37291e6	1.43244
$980$	− 180035.i	− 0.187458i
$981$	0	0
$982$	−332424.	−0.344722
$983$	− 300964.i	− 0.311464i	−0.987799	−	0.155732i	$-0.950226\pi$
$983$	− 300964.i	− 0.311464i	0.987799	−	0.155732i	$-0.0497736\pi$
$984$	0	0
$985$	1.09815e6	1.13185
$986$	232806.i	0.239464i
$987$	0	0
$988$	115572.	0.118397
$989$	− 12903.9i	− 0.0131925i
$990$	0	0
$991$	1.70094e6	1.73197	0.865987	−	0.500067i	$-0.166691\pi$
$991$	1.70094e6	1.73197	0.865987	+	0.500067i	$0.166691\pi$
$992$	− 149320.i	− 0.151738i
$993$	0	0
$994$	−2.16758e6	−2.19383
$995$	− 131985.i	− 0.133315i
$996$	0	0
$997$	−625837.	−0.629609	−0.314805	−	0.949156i	$-0.601939\pi$
$997$	−625837.	−0.629609	−0.314805	+	0.949156i	$0.601939\pi$
$998$	496568.i	0.498560i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.5.b.a.80.2		6
3.2	odd	2	inner	81.5.b.a.80.5		6
4.3	odd	2		1296.5.e.c.161.5		6
9.2	odd	6		9.5.d.a.5.3	yes	6
9.4	even	3		9.5.d.a.2.3	✓	6
9.5	odd	6		27.5.d.a.8.1		6
9.7	even	3		27.5.d.a.17.1		6
12.11	even	2		1296.5.e.c.161.2		6
36.7	odd	6		432.5.q.a.17.1		6
36.11	even	6		144.5.q.a.113.3		6
36.23	even	6		432.5.q.a.305.1		6
36.31	odd	6		144.5.q.a.65.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.5.d.a.2.3	✓	6	9.4	even	3
9.5.d.a.5.3	yes	6	9.2	odd	6
27.5.d.a.8.1		6	9.5	odd	6
27.5.d.a.17.1		6	9.7	even	3
81.5.b.a.80.2		6	1.1	even	1	trivial
81.5.b.a.80.5		6	3.2	odd	2	inner
144.5.q.a.65.3		6	36.31	odd	6
144.5.q.a.113.3		6	36.11	even	6
432.5.q.a.17.1		6	36.7	odd	6
432.5.q.a.305.1		6	36.23	even	6
1296.5.e.c.161.2		6	12.11	even	2
1296.5.e.c.161.5		6	4.3	odd	2

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:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	81.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.46526i q^{2} -3.93857 q^{4} +16.0226i q^{5} +72.4837 q^{7} -53.8574i q^{8} +71.5451 q^{10} -96.1576i q^{11} +153.906 q^{13} -323.659i q^{14} -303.505 q^{16} -72.7905i q^{17} -190.660 q^{19} -63.1062i q^{20} -429.369 q^{22} -14.4552i q^{23} +368.276 q^{25} -687.231i q^{26} -285.482 q^{28} -716.262i q^{29} -302.568 q^{31} +493.509i q^{32} -325.029 q^{34} +1161.38i q^{35} +826.277 q^{37} +851.348i q^{38} +862.936 q^{40} +556.119i q^{41} +892.680 q^{43} +378.723i q^{44} -64.5464 q^{46} +3955.43i q^{47} +2852.89 q^{49} -1644.45i q^{50} -606.170 q^{52} +1966.96i q^{53} +1540.69 q^{55} -3903.79i q^{56} -3198.30 q^{58} +5414.94i q^{59} -1712.42 q^{61} +1351.04i q^{62} -2652.43 q^{64} +2465.97i q^{65} -4634.49 q^{67} +286.691i q^{68} +5185.86 q^{70} -6697.12i q^{71} -4823.86 q^{73} -3689.54i q^{74} +750.929 q^{76} -6969.86i q^{77} -5728.79 q^{79} -4862.94i q^{80} +2483.22 q^{82} -2832.23i q^{83} +1166.29 q^{85} -3986.05i q^{86} -5178.80 q^{88} +14277.7i q^{89} +11155.7 q^{91} +56.9330i q^{92} +17662.0 q^{94} -3054.87i q^{95} +7165.31 q^{97} -12738.9i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 30 q^{4} - 24 q^{7} - 36 q^{10} + 12 q^{13} - 30 q^{16} - 258 q^{19} + 738 q^{22} + 546 q^{25} + 1308 q^{28} - 2580 q^{31} - 1026 q^{34} + 12 q^{37} + 2628 q^{40} + 570 q^{43} - 5760 q^{46} + 3726 q^{49}+ \cdots + 57918 q^{97}+O(q^{100}) \) 6 * q - 30 * q^4 - 24 * q^7 - 36 * q^10 + 12 * q^13 - 30 * q^16 - 258 * q^19 + 738 * q^22 + 546 * q^25 + 1308 * q^28 - 2580 * q^31 - 1026 * q^34 + 12 * q^37 + 2628 * q^40 + 570 * q^43 - 5760 * q^46 + 3726 * q^49 + 480 * q^52 + 2016 * q^55 - 12924 * q^58 - 7260 * q^61 + 15450 * q^64 + 10110 * q^67 + 19368 * q^70 - 14622 * q^73 + 8094 * q^76 - 9528 * q^79 - 9702 * q^82 - 24732 * q^85 - 29574 * q^88 + 34836 * q^91 + 25416 * q^94 + 57918 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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