LMFDB - Embedded newform 99.8.a.e.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [99,8,Mod(1,99)] 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("99.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(99, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 99 = 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	99.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-9] 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:	$30.9261175229$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.115512.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 70x - 194 $ x^3 - x^2 - 70*x - 194
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$9.97132$ of defining polynomial
Character	$\chi$	$=$	99.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.40077 q^{2} -122.236 q^{4} +505.769 q^{5} +1401.07 q^{7} -600.759 q^{8} +1214.23 q^{10} +1331.00 q^{11} -4339.66 q^{13} +3363.65 q^{14} +14204.0 q^{16} -28538.2 q^{17} -5370.67 q^{19} -61823.3 q^{20} +3195.42 q^{22} +85439.7 q^{23} +177677. q^{25} -10418.5 q^{26} -171262. q^{28} -5942.59 q^{29} +284424. q^{31} +110998. q^{32} -68513.6 q^{34} +708619. q^{35} +98242.9 q^{37} -12893.7 q^{38} -303845. q^{40} -96064.1 q^{41} -198016. q^{43} -162697. q^{44} +205121. q^{46} -217235. q^{47} +1.13946e6 q^{49} +426561. q^{50} +530464. q^{52} -844220. q^{53} +673178. q^{55} -841708. q^{56} -14266.8 q^{58} +1.14817e6 q^{59} +3.20736e6 q^{61} +682837. q^{62} -1.55163e6 q^{64} -2.19486e6 q^{65} +2.09841e6 q^{67} +3.48841e6 q^{68} +1.70123e6 q^{70} -641493. q^{71} -3.25736e6 q^{73} +235858. q^{74} +656491. q^{76} +1.86483e6 q^{77} +7.68697e6 q^{79} +7.18392e6 q^{80} -230627. q^{82} -4.06514e6 q^{83} -1.44337e7 q^{85} -475389. q^{86} -799610. q^{88} -1.64848e6 q^{89} -6.08018e6 q^{91} -1.04438e7 q^{92} -521531. q^{94} -2.71632e6 q^{95} -3.61970e6 q^{97} +2.73559e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{2} - 15 q^{4} + 444 q^{5} + 1614 q^{7} - 3153 q^{8} + 2880 q^{10} + 3993 q^{11} + 20772 q^{13} - 36258 q^{14} + 12225 q^{16} + 14538 q^{17} + 24492 q^{19} - 80112 q^{20} - 11979 q^{22} - 35094 q^{23}+ \cdots - 24377397 q^{98}+O(q^{100}) $ 3 * q - 9 * q^2 - 15 * q^4 + 444 * q^5 + 1614 * q^7 - 3153 * q^8 + 2880 * q^10 + 3993 * q^11 + 20772 * q^13 - 36258 * q^14 + 12225 * q^16 + 14538 * q^17 + 24492 * q^19 - 80112 * q^20 - 11979 * q^22 - 35094 * q^23 + 29121 * q^25 - 203832 * q^26 + 278034 * q^28 + 179862 * q^29 + 288888 * q^31 + 519567 * q^32 - 491586 * q^34 + 532872 * q^35 + 107562 * q^37 + 686328 * q^38 - 237360 * q^40 + 135198 * q^41 + 193536 * q^43 - 19965 * q^44 + 16422 * q^46 + 591486 * q^47 + 4461159 * q^49 + 1192245 * q^50 + 2449992 * q^52 - 79044 * q^53 + 590964 * q^55 - 752658 * q^56 + 2289930 * q^58 - 2532768 * q^59 + 6678792 * q^61 + 2660808 * q^62 - 3966303 * q^64 - 3191832 * q^65 + 7150356 * q^67 + 7821954 * q^68 + 4029120 * q^70 - 1390398 * q^71 - 6429114 * q^73 - 2507478 * q^74 - 7654728 * q^76 + 2148234 * q^77 + 6873186 * q^79 + 7556016 * q^80 - 1774590 * q^82 - 6505596 * q^83 - 16546032 * q^85 + 6519468 * q^86 - 4196643 * q^88 + 8842962 * q^89 + 3066648 * q^91 - 6921990 * q^92 - 24038238 * q^94 + 190968 * q^95 - 1764774 * q^97 - 24377397 * 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 $

$2$	2.40077	0.212200	0.106100	−	0.994355i	$-0.466164\pi$
$2$	2.40077	0.212200	0.106100	+	0.994355i	$0.466164\pi$
$3$	0	0
$3$	0	0
$4$	−122.236	−0.954971
$5$	505.769	1.80949	0.904747	−	0.425950i	$-0.140060\pi$
$5$	505.769	1.80949	0.904747	+	0.425950i	$0.140060\pi$
$6$	0	0
$7$	1401.07	1.54390	0.771948	−	0.635686i	$-0.219283\pi$
$7$	1401.07	1.54390	0.771948	+	0.635686i	$0.219283\pi$
$8$	−600.759	−0.414844
$9$	0	0
$10$	1214.23	0.383974
$11$	1331.00	0.301511
$11$	1331.00	0.301511
$12$	0	0
$13$	−4339.66	−0.547840	−0.273920	−	0.961752i	$-0.588320\pi$
$13$	−4339.66	−0.547840	−0.273920	+	0.961752i	$0.588320\pi$
$14$	3363.65	0.327614
$15$	0	0
$16$	14204.0	0.866941
$17$	−28538.2	−1.40882	−0.704410	−	0.709793i	$-0.748788\pi$
$17$	−28538.2	−1.40882	−0.704410	+	0.709793i	$0.748788\pi$
$18$	0	0
$19$	−5370.67	−0.179635	−0.0898176	−	0.995958i	$-0.528628\pi$
$19$	−5370.67	−0.179635	−0.0898176	+	0.995958i	$0.528628\pi$
$20$	−61823.3	−1.72801
$21$	0	0
$22$	3195.42	0.0639806
$23$	85439.7	1.46424	0.732120	−	0.681175i	$-0.238531\pi$
$23$	85439.7	1.46424	0.732120	+	0.681175i	$0.238531\pi$
$24$	0	0
$25$	177677.	2.27427
$26$	−10418.5	−0.116252
$27$	0	0
$28$	−171262.	−1.47438
$29$	−5942.59	−0.0452463	−0.0226231	−	0.999744i	$-0.507202\pi$
$29$	−5942.59	−0.0452463	−0.0226231	+	0.999744i	$0.507202\pi$
$30$	0	0
$31$	284424.	1.71475	0.857375	−	0.514692i	$-0.172094\pi$
$31$	284424.	1.71475	0.857375	+	0.514692i	$0.172094\pi$
$32$	110998.	0.598809
$33$	0	0
$34$	−68513.6	−0.298951
$35$	708619.	2.79367
$36$	0	0
$37$	98242.9	0.318857	0.159428	−	0.987210i	$-0.449035\pi$
$37$	98242.9	0.318857	0.159428	+	0.987210i	$0.449035\pi$
$38$	−12893.7	−0.0381185
$39$	0	0
$40$	−303845.	−0.750658
$41$	−96064.1	−0.217680	−0.108840	−	0.994059i	$-0.534714\pi$
$41$	−96064.1	−0.217680	−0.108840	+	0.994059i	$0.534714\pi$
$42$	0	0
$43$	−198016.	−0.379804	−0.189902	−	0.981803i	$-0.560817\pi$
$43$	−198016.	−0.379804	−0.189902	+	0.981803i	$0.560817\pi$
$44$	−162697.	−0.287935
$45$	0	0
$46$	205121.	0.310712
$47$	−217235.	−0.305202	−0.152601	−	0.988288i	$-0.548765\pi$
$47$	−217235.	−0.305202	−0.152601	+	0.988288i	$0.548765\pi$
$48$	0	0
$49$	1.13946e6	1.38361
$50$	426561.	0.482599
$51$	0	0
$52$	530464.	0.523172
$53$	−844220.	−0.778915	−0.389458	−	0.921044i	$-0.627337\pi$
$53$	−844220.	−0.778915	−0.389458	+	0.921044i	$0.627337\pi$
$54$	0	0
$55$	673178.	0.545583
$56$	−841708.	−0.640476
$57$	0	0
$58$	−14266.8	−0.00960125
$59$	1.14817e6	0.727821	0.363911	−	0.931434i	$-0.381441\pi$
$59$	1.14817e6	0.727821	0.363911	+	0.931434i	$0.381441\pi$
$60$	0	0
$61$	3.20736e6	1.80923	0.904615	−	0.426229i	$-0.140158\pi$
$61$	3.20736e6	1.80923	0.904615	+	0.426229i	$0.140158\pi$
$62$	682837.	0.363870
$63$	0	0
$64$	−1.55163e6	−0.739874
$65$	−2.19486e6	−0.991313
$66$	0	0
$67$	2.09841e6	0.852370	0.426185	−	0.904636i	$-0.359857\pi$
$67$	2.09841e6	0.852370	0.426185	+	0.904636i	$0.359857\pi$
$68$	3.48841e6	1.34538
$69$	0	0
$70$	1.70123e6	0.592816
$71$	−641493.	−0.212710	−0.106355	−	0.994328i	$-0.533918\pi$
$71$	−641493.	−0.212710	−0.106355	+	0.994328i	$0.533918\pi$
$72$	0	0
$73$	−3.25736e6	−0.980021	−0.490011	−	0.871716i	$-0.663007\pi$
$73$	−3.25736e6	−0.980021	−0.490011	+	0.871716i	$0.663007\pi$
$74$	235858.	0.0676613
$75$	0	0
$76$	656491.	0.171546
$77$	1.86483e6	0.465502
$78$	0	0
$79$	7.68697e6	1.75413	0.877063	−	0.480376i	$-0.159500\pi$
$79$	7.68697e6	1.75413	0.877063	+	0.480376i	$0.159500\pi$
$80$	7.18392e6	1.56872
$81$	0	0
$82$	−230627.	−0.0461915
$83$	−4.06514e6	−0.780373	−0.390187	−	0.920736i	$-0.627590\pi$
$83$	−4.06514e6	−0.780373	−0.390187	+	0.920736i	$0.627590\pi$
$84$	0	0
$85$	−1.44337e7	−2.54925
$86$	−475389.	−0.0805944
$87$	0	0
$88$	−799610.	−0.125080
$89$	−1.64848e6	−0.247867	−0.123934	−	0.992290i	$-0.539551\pi$
$89$	−1.64848e6	−0.247867	−0.123934	+	0.992290i	$0.539551\pi$
$90$	0	0
$91$	−6.08018e6	−0.845808
$92$	−1.04438e7	−1.39831
$93$	0	0
$94$	−521531.	−0.0647638
$95$	−2.71632e6	−0.325049
$96$	0	0
$97$	−3.61970e6	−0.402691	−0.201345	−	0.979520i	$-0.564531\pi$
$97$	−3.61970e6	−0.402691	−0.201345	+	0.979520i	$0.564531\pi$
$98$	2.73559e6	0.293602
$99$	0	0
$100$	−2.17186e7	−2.17186
$101$	1.32534e7	1.27998	0.639990	−	0.768383i	$-0.278939\pi$
$101$	1.32534e7	1.27998	0.639990	+	0.768383i	$0.278939\pi$
$102$	0	0
$103$	−223679.	−0.0201695	−0.0100847	−	0.999949i	$-0.503210\pi$
$103$	−223679.	−0.0201695	−0.0100847	+	0.999949i	$0.503210\pi$
$104$	2.60709e6	0.227268
$105$	0	0
$106$	−2.02678e6	−0.165286
$107$	−1.95972e7	−1.54650	−0.773252	−	0.634099i	$-0.781371\pi$
$107$	−1.95972e7	−1.54650	−0.773252	+	0.634099i	$0.781371\pi$
$108$	0	0
$109$	3.89843e6	0.288335	0.144167	−	0.989553i	$-0.453950\pi$
$109$	3.89843e6	0.288335	0.144167	+	0.989553i	$0.453950\pi$
$110$	1.61614e6	0.115773
$111$	0	0
$112$	1.99008e7	1.33847
$113$	−2.42731e6	−0.158253	−0.0791263	−	0.996865i	$-0.525213\pi$
$113$	−2.42731e6	−0.158253	−0.0791263	+	0.996865i	$0.525213\pi$
$114$	0	0
$115$	4.32128e7	2.64953
$116$	726401.	0.0432089
$117$	0	0
$118$	2.75649e6	0.154444
$119$	−3.99841e7	−2.17507
$120$	0	0
$121$	1.77156e6	0.0909091
$122$	7.70013e6	0.383918
$123$	0	0
$124$	−3.47670e7	−1.63754
$125$	5.03504e7	2.30578
$126$	0	0
$127$	−2.46895e7	−1.06955	−0.534773	−	0.844996i	$-0.679603\pi$
$127$	−2.46895e7	−1.06955	−0.534773	+	0.844996i	$0.679603\pi$
$128$	−1.79328e7	−0.755810
$129$	0	0
$130$	−5.26936e6	−0.210357
$131$	1.29109e7	0.501774	0.250887	−	0.968016i	$-0.419278\pi$
$131$	1.29109e7	0.501774	0.250887	+	0.968016i	$0.419278\pi$
$132$	0	0
$133$	−7.52471e6	−0.277338
$134$	5.03779e6	0.180873
$135$	0	0
$136$	1.71446e7	0.584441
$137$	−1.15362e7	−0.383301	−0.191651	−	0.981463i	$-0.561384\pi$
$137$	−1.15362e7	−0.383301	−0.191651	+	0.981463i	$0.561384\pi$
$138$	0	0
$139$	−6.26452e7	−1.97850	−0.989249	−	0.146239i	$-0.953283\pi$
$139$	−6.26452e7	−1.97850	−0.989249	+	0.146239i	$0.953283\pi$
$140$	−8.66190e7	−2.66787
$141$	0	0
$142$	−1.54008e6	−0.0451370
$143$	−5.77609e6	−0.165180
$144$	0	0
$145$	−3.00558e6	−0.0818729
$146$	−7.82016e6	−0.207960
$147$	0	0
$148$	−1.20089e7	−0.304499
$149$	−4.17839e7	−1.03480	−0.517400	−	0.855743i	$-0.673100\pi$
$149$	−4.17839e7	−1.03480	−0.517400	+	0.855743i	$0.673100\pi$
$150$	0	0
$151$	−2.12508e7	−0.502291	−0.251145	−	0.967949i	$-0.580807\pi$
$151$	−2.12508e7	−0.502291	−0.251145	+	0.967949i	$0.580807\pi$
$152$	3.22648e6	0.0745206
$153$	0	0
$154$	4.47702e6	0.0987794
$155$	1.43853e8	3.10283
$156$	0	0
$157$	5.27515e6	0.108789	0.0543946	−	0.998520i	$-0.482677\pi$
$157$	5.27515e6	0.108789	0.0543946	+	0.998520i	$0.482677\pi$
$158$	1.84546e7	0.372225
$159$	0	0
$160$	5.61391e7	1.08354
$161$	1.19707e8	2.26063
$162$	0	0
$163$	7.77059e7	1.40539	0.702695	−	0.711491i	$-0.251980\pi$
$163$	7.77059e7	1.40539	0.702695	+	0.711491i	$0.251980\pi$
$164$	1.17425e7	0.207878
$165$	0	0
$166$	−9.75946e6	−0.165595
$167$	−1.19339e7	−0.198278	−0.0991389	−	0.995074i	$-0.531609\pi$
$167$	−1.19339e7	−0.198278	−0.0991389	+	0.995074i	$0.531609\pi$
$168$	0	0
$169$	−4.39159e7	−0.699871
$170$	−3.46520e7	−0.540950
$171$	0	0
$172$	2.42047e7	0.362702
$173$	6.60810e7	0.970321	0.485160	−	0.874425i	$-0.338761\pi$
$173$	6.60810e7	0.970321	0.485160	+	0.874425i	$0.338761\pi$
$174$	0	0
$175$	2.48939e8	3.51123
$176$	1.89055e7	0.261393
$177$	0	0
$178$	−3.95762e6	−0.0525974
$179$	−6.27322e7	−0.817532	−0.408766	−	0.912639i	$-0.634041\pi$
$179$	−6.27322e7	−0.817532	−0.408766	+	0.912639i	$0.634041\pi$
$180$	0	0
$181$	7.92117e7	0.992920	0.496460	−	0.868059i	$-0.334633\pi$
$181$	7.92117e7	0.992920	0.496460	+	0.868059i	$0.334633\pi$
$182$	−1.45971e7	−0.179480
$183$	0	0
$184$	−5.13287e7	−0.607432
$185$	4.96882e7	0.576969
$186$	0	0
$187$	−3.79844e7	−0.424775
$188$	2.65540e7	0.291459
$189$	0	0
$190$	−6.52125e6	−0.0689752
$191$	−1.79024e8	−1.85906	−0.929531	−	0.368743i	$-0.879788\pi$
$191$	−1.79024e8	−1.85906	−0.929531	+	0.368743i	$0.879788\pi$
$192$	0	0
$193$	1.83687e8	1.83919	0.919596	−	0.392865i	$-0.128516\pi$
$193$	1.83687e8	1.83919	0.919596	+	0.392865i	$0.128516\pi$
$194$	−8.69006e6	−0.0854509
$195$	0	0
$196$	−1.39284e8	−1.32131
$197$	−2.45767e7	−0.229029	−0.114515	−	0.993422i	$-0.536531\pi$
$197$	−2.45767e7	−0.229029	−0.114515	+	0.993422i	$0.536531\pi$
$198$	0	0
$199$	−8.23253e7	−0.740538	−0.370269	−	0.928925i	$-0.620735\pi$
$199$	−8.23253e7	−0.740538	−0.370269	+	0.928925i	$0.620735\pi$
$200$	−1.06741e8	−0.943467
$201$	0	0
$202$	3.18184e7	0.271611
$203$	−8.32601e6	−0.0698555
$204$	0	0
$205$	−4.85862e7	−0.393890
$206$	−537001.	−0.00427996
$207$	0	0
$208$	−6.16404e7	−0.474945
$209$	−7.14837e6	−0.0541620
$210$	0	0
$211$	−7.45606e7	−0.546413	−0.273206	−	0.961955i	$-0.588084\pi$
$211$	−7.45606e7	−0.546413	−0.273206	+	0.961955i	$0.588084\pi$
$212$	1.03194e8	0.743842
$213$	0	0
$214$	−4.70483e7	−0.328168
$215$	−1.00150e8	−0.687253
$216$	0	0
$217$	3.98500e8	2.64739
$218$	9.35921e6	0.0611845
$219$	0	0
$220$	−8.22868e7	−0.521016
$221$	1.23846e8	0.771808
$222$	0	0
$223$	−1.79490e7	−0.108386	−0.0541930	−	0.998530i	$-0.517259\pi$
$223$	−1.79490e7	−0.108386	−0.0541930	+	0.998530i	$0.517259\pi$
$224$	1.55516e8	0.924499
$225$	0	0
$226$	−5.82741e6	−0.0335812
$227$	8.64663e7	0.490632	0.245316	−	0.969443i	$-0.421108\pi$
$227$	8.64663e7	0.490632	0.245316	+	0.969443i	$0.421108\pi$
$228$	0	0
$229$	−1.16990e8	−0.643759	−0.321879	−	0.946781i	$-0.604315\pi$
$229$	−1.16990e8	−0.643759	−0.321879	+	0.946781i	$0.604315\pi$
$230$	1.03744e8	0.562231
$231$	0	0
$232$	3.57007e6	0.0187702
$233$	2.69719e7	0.139690	0.0698450	−	0.997558i	$-0.477750\pi$
$233$	2.69719e7	0.139690	0.0698450	+	0.997558i	$0.477750\pi$
$234$	0	0
$235$	−1.09871e8	−0.552261
$236$	−1.40348e8	−0.695048
$237$	0	0
$238$	−9.59926e7	−0.461549
$239$	−2.03617e8	−0.964766	−0.482383	−	0.875960i	$-0.660229\pi$
$239$	−2.03617e8	−0.964766	−0.482383	+	0.875960i	$0.660229\pi$
$240$	0	0
$241$	−2.42169e8	−1.11445	−0.557224	−	0.830363i	$-0.688133\pi$
$241$	−2.42169e8	−1.11445	−0.557224	+	0.830363i	$0.688133\pi$
$242$	4.25310e6	0.0192909
$243$	0	0
$244$	−3.92056e8	−1.72776
$245$	5.76306e8	2.50364
$246$	0	0
$247$	2.33069e7	0.0984113
$248$	−1.70871e8	−0.711355
$249$	0	0
$250$	1.20880e8	0.489286
$251$	−1.95655e8	−0.780968	−0.390484	−	0.920610i	$-0.627692\pi$
$251$	−1.95655e8	−0.780968	−0.390484	+	0.920610i	$0.627692\pi$
$252$	0	0
$253$	1.13720e8	0.441485
$254$	−5.92738e7	−0.226957
$255$	0	0
$256$	1.55556e8	0.579491
$257$	−3.55159e8	−1.30514	−0.652570	−	0.757728i	$-0.726309\pi$
$257$	−3.55159e8	−1.30514	−0.652570	+	0.757728i	$0.726309\pi$
$258$	0	0
$259$	1.37646e8	0.492281
$260$	2.68292e8	0.946676
$261$	0	0
$262$	3.09962e7	0.106476
$263$	−2.80423e8	−0.950534	−0.475267	−	0.879842i	$-0.657649\pi$
$263$	−2.80423e8	−0.950534	−0.475267	+	0.879842i	$0.657649\pi$
$264$	0	0
$265$	−4.26980e8	−1.40944
$266$	−1.80651e7	−0.0588510
$267$	0	0
$268$	−2.56502e8	−0.813989
$269$	−3.77104e8	−1.18121	−0.590606	−	0.806960i	$-0.701111\pi$
$269$	−3.77104e8	−1.18121	−0.590606	+	0.806960i	$0.701111\pi$
$270$	0	0
$271$	2.18482e8	0.666842	0.333421	−	0.942778i	$-0.391797\pi$
$271$	2.18482e8	0.666842	0.333421	+	0.942778i	$0.391797\pi$
$272$	−4.05356e8	−1.22136
$273$	0	0
$274$	−2.76957e7	−0.0813365
$275$	2.36488e8	0.685717
$276$	0	0
$277$	−1.41112e8	−0.398920	−0.199460	−	0.979906i	$-0.563919\pi$
$277$	−1.41112e8	−0.398920	−0.199460	+	0.979906i	$0.563919\pi$
$278$	−1.50396e8	−0.419837
$279$	0	0
$280$	−4.25709e8	−1.15894
$281$	5.43983e8	1.46256	0.731279	−	0.682079i	$-0.238924\pi$
$281$	5.43983e8	1.46256	0.731279	+	0.682079i	$0.238924\pi$
$282$	0	0
$283$	1.36059e8	0.356841	0.178421	−	0.983954i	$-0.442901\pi$
$283$	1.36059e8	0.356841	0.178421	+	0.983954i	$0.442901\pi$
$284$	7.84138e7	0.203132
$285$	0	0
$286$	−1.38670e7	−0.0350512
$287$	−1.34593e8	−0.336074
$288$	0	0
$289$	4.04091e8	0.984773
$290$	−7.21569e6	−0.0173734
$291$	0	0
$292$	3.98167e8	0.935892
$293$	−8.43098e8	−1.95813	−0.979064	−	0.203551i	$-0.934752\pi$
$293$	−8.43098e8	−1.95813	−0.979064	+	0.203551i	$0.934752\pi$
$294$	0	0
$295$	5.80709e8	1.31699
$296$	−5.90203e7	−0.132276
$297$	0	0
$298$	−1.00313e8	−0.219584
$299$	−3.70779e8	−0.802170
$300$	0	0
$301$	−2.77434e8	−0.586378
$302$	−5.10181e7	−0.106586
$303$	0	0
$304$	−7.62849e7	−0.155733
$305$	1.62219e9	3.27379
$306$	0	0
$307$	−1.21154e8	−0.238976	−0.119488	−	0.992836i	$-0.538125\pi$
$307$	−1.21154e8	−0.238976	−0.119488	+	0.992836i	$0.538125\pi$
$308$	−2.27950e8	−0.444541
$309$	0	0
$310$	3.45357e8	0.658420
$311$	4.86049e8	0.916259	0.458130	−	0.888885i	$-0.348520\pi$
$311$	4.86049e8	0.916259	0.458130	+	0.888885i	$0.348520\pi$
$312$	0	0
$313$	−8.64030e7	−0.159266	−0.0796331	−	0.996824i	$-0.525375\pi$
$313$	−8.64030e7	−0.159266	−0.0796331	+	0.996824i	$0.525375\pi$
$314$	1.26644e7	0.0230850
$315$	0	0
$316$	−9.39627e8	−1.67514
$317$	6.16413e8	1.08684	0.543418	−	0.839462i	$-0.317130\pi$
$317$	6.16413e8	1.08684	0.543418	+	0.839462i	$0.317130\pi$
$318$	0	0
$319$	−7.90959e6	−0.0136423
$320$	−7.84765e8	−1.33880
$321$	0	0
$322$	2.87389e8	0.479706
$323$	1.53269e8	0.253074
$324$	0	0
$325$	−7.71058e8	−1.24594
$326$	1.86554e8	0.298224
$327$	0	0
$328$	5.77114e7	0.0903031
$329$	−3.04362e8	−0.471200
$330$	0	0
$331$	3.23896e8	0.490916	0.245458	−	0.969407i	$-0.421062\pi$
$331$	3.23896e8	0.490916	0.245458	+	0.969407i	$0.421062\pi$
$332$	4.96908e8	0.745234
$333$	0	0
$334$	−2.86504e7	−0.0420745
$335$	1.06131e9	1.54236
$336$	0	0
$337$	3.37346e8	0.480143	0.240071	−	0.970755i	$-0.422829\pi$
$337$	3.37346e8	0.480143	0.240071	+	0.970755i	$0.422829\pi$
$338$	−1.05432e8	−0.148512
$339$	0	0
$340$	1.76433e9	2.43446
$341$	3.78569e8	0.517017
$342$	0	0
$343$	4.42629e8	0.592257
$344$	1.18960e8	0.157560
$345$	0	0
$346$	1.58645e8	0.205902
$347$	−8.68450e8	−1.11581	−0.557907	−	0.829904i	$-0.688395\pi$
$347$	−8.68450e8	−1.11581	−0.557907	+	0.829904i	$0.688395\pi$
$348$	0	0
$349$	1.33122e8	0.167634	0.0838170	−	0.996481i	$-0.473289\pi$
$349$	1.33122e8	0.167634	0.0838170	+	0.996481i	$0.473289\pi$
$350$	5.97644e8	0.745082
$351$	0	0
$352$	1.47738e8	0.180548
$353$	−4.51506e8	−0.546326	−0.273163	−	0.961968i	$-0.588070\pi$
$353$	−4.51506e8	−0.546326	−0.273163	+	0.961968i	$0.588070\pi$
$354$	0	0
$355$	−3.24447e8	−0.384897
$356$	2.01505e8	0.236706
$357$	0	0
$358$	−1.50605e8	−0.173480
$359$	−6.67176e8	−0.761044	−0.380522	−	0.924772i	$-0.624256\pi$
$359$	−6.67176e8	−0.761044	−0.380522	+	0.924772i	$0.624256\pi$
$360$	0	0
$361$	−8.65028e8	−0.967731
$362$	1.90169e8	0.210697
$363$	0	0
$364$	7.43219e8	0.807722
$365$	−1.64747e9	−1.77334
$366$	0	0
$367$	4.01307e7	0.0423785	0.0211892	−	0.999775i	$-0.493255\pi$
$367$	4.01307e7	0.0423785	0.0211892	+	0.999775i	$0.493255\pi$
$368$	1.21358e9	1.26941
$369$	0	0
$370$	1.19290e8	0.122433
$371$	−1.18281e9	−1.20256
$372$	0	0
$373$	−1.24452e9	−1.24172	−0.620858	−	0.783923i	$-0.713216\pi$
$373$	−1.24452e9	−1.24172	−0.620858	+	0.783923i	$0.713216\pi$
$374$	−9.11916e7	−0.0901372
$375$	0	0
$376$	1.30506e8	0.126611
$377$	2.57888e7	0.0247877
$378$	0	0
$379$	1.95357e9	1.84328	0.921640	−	0.388045i	$-0.126850\pi$
$379$	1.95357e9	1.84328	0.921640	+	0.388045i	$0.126850\pi$
$380$	3.32033e8	0.310412
$381$	0	0
$382$	−4.29794e8	−0.394493
$383$	2.71920e8	0.247312	0.123656	−	0.992325i	$-0.460538\pi$
$383$	2.71920e8	0.247312	0.123656	+	0.992325i	$0.460538\pi$
$384$	0	0
$385$	9.43172e8	0.842323
$386$	4.40989e8	0.390276
$387$	0	0
$388$	4.42459e8	0.384558
$389$	−1.19397e9	−1.02842	−0.514211	−	0.857664i	$-0.671915\pi$
$389$	−1.19397e9	−1.02842	−0.514211	+	0.857664i	$0.671915\pi$
$390$	0	0
$391$	−2.43830e9	−2.06285
$392$	−6.84543e8	−0.573984
$393$	0	0
$394$	−5.90029e7	−0.0486000
$395$	3.88783e9	3.17408
$396$	0	0
$397$	1.88808e9	1.51445	0.757224	−	0.653156i	$-0.226555\pi$
$397$	1.88808e9	1.51445	0.757224	+	0.653156i	$0.226555\pi$
$398$	−1.97644e8	−0.157142
$399$	0	0
$400$	2.52372e9	1.97166
$401$	−3.81045e7	−0.0295101	−0.0147551	−	0.999891i	$-0.504697\pi$
$401$	−3.81045e7	−0.0295101	−0.0147551	+	0.999891i	$0.504697\pi$
$402$	0	0
$403$	−1.23431e9	−0.939409
$404$	−1.62005e9	−1.22234
$405$	0	0
$406$	−1.99888e7	−0.0148233
$407$	1.30761e8	0.0961389
$408$	0	0
$409$	−1.10488e9	−0.798515	−0.399258	−	0.916839i	$-0.630732\pi$
$409$	−1.10488e9	−0.798515	−0.399258	+	0.916839i	$0.630732\pi$
$410$	−1.16644e8	−0.0835833
$411$	0	0
$412$	2.73417e7	0.0192613
$413$	1.60867e9	1.12368
$414$	0	0
$415$	−2.05602e9	−1.41208
$416$	−4.81692e8	−0.328052
$417$	0	0
$418$	−1.71616e7	−0.0114932
$419$	4.54418e8	0.301791	0.150896	−	0.988550i	$-0.451784\pi$
$419$	4.54418e8	0.301791	0.150896	+	0.988550i	$0.451784\pi$
$420$	0	0
$421$	−1.04677e9	−0.683697	−0.341849	−	0.939755i	$-0.611053\pi$
$421$	−1.04677e9	−0.683697	−0.341849	+	0.939755i	$0.611053\pi$
$422$	−1.79003e8	−0.115949
$423$	0	0
$424$	5.07173e8	0.323129
$425$	−5.07059e9	−3.20403
$426$	0	0
$427$	4.49375e9	2.79326
$428$	2.39549e9	1.47687
$429$	0	0
$430$	−2.40437e8	−0.145835
$431$	5.93961e8	0.357345	0.178672	−	0.983909i	$-0.442820\pi$
$431$	5.93961e8	0.357345	0.178672	+	0.983909i	$0.442820\pi$
$432$	0	0
$433$	−1.90455e9	−1.12742	−0.563708	−	0.825974i	$-0.690626\pi$
$433$	−1.90455e9	−1.12742	−0.563708	+	0.825974i	$0.690626\pi$
$434$	9.56704e8	0.561777
$435$	0	0
$436$	−4.76529e8	−0.275351
$437$	−4.58869e8	−0.263029
$438$	0	0
$439$	−1.35965e9	−0.767011	−0.383506	−	0.923539i	$-0.625283\pi$
$439$	−1.35965e9	−0.767011	−0.383506	+	0.923539i	$0.625283\pi$
$440$	−4.04418e8	−0.226332
$441$	0	0
$442$	2.97326e8	0.163778
$443$	−1.86938e9	−1.02161	−0.510805	−	0.859696i	$-0.670653\pi$
$443$	−1.86938e9	−1.02161	−0.510805	+	0.859696i	$0.670653\pi$
$444$	0	0
$445$	−8.33752e8	−0.448515
$446$	−4.30914e7	−0.0229995
$447$	0	0
$448$	−2.17395e9	−1.14229
$449$	1.94847e9	1.01586	0.507928	−	0.861400i	$-0.330412\pi$
$449$	1.94847e9	1.01586	0.507928	+	0.861400i	$0.330412\pi$
$450$	0	0
$451$	−1.27861e8	−0.0656328
$452$	2.96706e8	0.151127
$453$	0	0
$454$	2.07585e8	0.104112
$455$	−3.07517e9	−1.53048
$456$	0	0
$457$	−2.15115e9	−1.05430	−0.527150	−	0.849772i	$-0.676739\pi$
$457$	−2.15115e9	−1.05430	−0.527150	+	0.849772i	$0.676739\pi$
$458$	−2.80865e8	−0.136605
$459$	0	0
$460$	−5.28217e9	−2.53023
$461$	2.33979e9	1.11230	0.556152	−	0.831081i	$-0.312277\pi$
$461$	2.33979e9	1.11230	0.556152	+	0.831081i	$0.312277\pi$
$462$	0	0
$463$	−1.87930e9	−0.879961	−0.439981	−	0.898007i	$-0.645015\pi$
$463$	−1.87930e9	−0.879961	−0.439981	+	0.898007i	$0.645015\pi$
$464$	−8.44084e7	−0.0392259
$465$	0	0
$466$	6.47531e7	0.0296422
$467$	2.10774e8	0.0957653	0.0478827	−	0.998853i	$-0.484753\pi$
$467$	2.10774e8	0.0957653	0.0478827	+	0.998853i	$0.484753\pi$
$468$	0	0
$469$	2.94003e9	1.31597
$470$	−2.63774e8	−0.117190
$471$	0	0
$472$	−6.89774e8	−0.301933
$473$	−2.63559e8	−0.114515
$474$	0	0
$475$	−9.54246e8	−0.408538
$476$	4.88751e9	2.07713
$477$	0	0
$478$	−4.88838e8	−0.204723
$479$	−1.52690e7	−0.00634799	−0.00317399	−	0.999995i	$-0.501010\pi$
$479$	−1.52690e7	−0.00634799	−0.00317399	+	0.999995i	$0.501010\pi$
$480$	0	0
$481$	−4.26341e8	−0.174682
$482$	−5.81392e8	−0.236485
$483$	0	0
$484$	−2.16549e8	−0.0868156
$485$	−1.83073e9	−0.728666
$486$	0	0
$487$	3.35023e9	1.31439	0.657194	−	0.753721i	$-0.271743\pi$
$487$	3.35023e9	1.31439	0.657194	+	0.753721i	$0.271743\pi$
$488$	−1.92685e9	−0.750549
$489$	0	0
$490$	1.38358e9	0.531271
$491$	1.75625e9	0.669579	0.334790	−	0.942293i	$-0.391335\pi$
$491$	1.75625e9	0.669579	0.334790	+	0.942293i	$0.391335\pi$
$492$	0	0
$493$	1.69591e8	0.0637439
$494$	5.59544e7	0.0208829
$495$	0	0
$496$	4.03995e9	1.48659
$497$	−8.98779e8	−0.328402
$498$	0	0
$499$	−3.42934e9	−1.23555	−0.617773	−	0.786356i	$-0.711965\pi$
$499$	−3.42934e9	−1.23555	−0.617773	+	0.786356i	$0.711965\pi$
$500$	−6.15465e9	−2.20195
$501$	0	0
$502$	−4.69723e8	−0.165721
$503$	−2.99193e9	−1.04825	−0.524123	−	0.851643i	$-0.675607\pi$
$503$	−2.99193e9	−1.04825	−0.524123	+	0.851643i	$0.675607\pi$
$504$	0	0
$505$	6.70317e9	2.31612
$506$	2.73016e8	0.0936831
$507$	0	0
$508$	3.01796e9	1.02139
$509$	2.73347e9	0.918760	0.459380	−	0.888240i	$-0.348072\pi$
$509$	2.73347e9	0.918760	0.459380	+	0.888240i	$0.348072\pi$
$510$	0	0
$511$	−4.56380e9	−1.51305
$512$	2.66885e9	0.878778
$513$	0	0
$514$	−8.52655e8	−0.276951
$515$	−1.13130e8	−0.0364966
$516$	0	0
$517$	−2.89140e8	−0.0920219
$518$	3.30455e8	0.104462
$519$	0	0
$520$	1.31858e9	0.411241
$521$	3.96803e9	1.22926	0.614629	−	0.788816i	$-0.289306\pi$
$521$	3.96803e9	1.22926	0.614629	+	0.788816i	$0.289306\pi$
$522$	0	0
$523$	1.14104e9	0.348773	0.174387	−	0.984677i	$-0.444206\pi$
$523$	1.14104e9	0.348773	0.174387	+	0.984677i	$0.444206\pi$
$524$	−1.57819e9	−0.479180
$525$	0	0
$526$	−6.73229e8	−0.201703
$527$	−8.11696e9	−2.41577
$528$	0	0
$529$	3.89512e9	1.14400
$530$	−1.02508e9	−0.299083
$531$	0	0
$532$	9.19793e8	0.264850
$533$	4.16885e8	0.119254
$534$	0	0
$535$	−9.91165e9	−2.79839
$536$	−1.26064e9	−0.353601
$537$	0	0
$538$	−9.05338e8	−0.250653
$539$	1.51663e9	0.417175
$540$	0	0
$541$	−4.46759e9	−1.21306	−0.606531	−	0.795060i	$-0.707439\pi$
$541$	−4.46759e9	−1.21306	−0.606531	+	0.795060i	$0.707439\pi$
$542$	5.24524e8	0.141504
$543$	0	0
$544$	−3.16767e9	−0.843614
$545$	1.97170e9	0.521740
$546$	0	0
$547$	4.19205e9	1.09514	0.547572	−	0.836759i	$-0.315552\pi$
$547$	4.19205e9	1.09514	0.547572	+	0.836759i	$0.315552\pi$
$548$	1.41014e9	0.366042
$549$	0	0
$550$	5.67753e8	0.145509
$551$	3.19157e7	0.00812782
$552$	0	0
$553$	1.07700e10	2.70819
$554$	−3.38778e8	−0.0846507
$555$	0	0
$556$	7.65751e9	1.88941
$557$	−6.15006e9	−1.50795	−0.753973	−	0.656905i	$-0.771865\pi$
$557$	−6.15006e9	−1.50795	−0.753973	+	0.656905i	$0.771865\pi$
$558$	0	0
$559$	8.59320e8	0.208072
$560$	1.00652e10	2.42195
$561$	0	0
$562$	1.30598e9	0.310354
$563$	−3.37494e9	−0.797053	−0.398527	−	0.917157i	$-0.630478\pi$
$563$	−3.37494e9	−0.797053	−0.398527	+	0.917157i	$0.630478\pi$
$564$	0	0
$565$	−1.22766e9	−0.286357
$566$	3.26646e8	0.0757216
$567$	0	0
$568$	3.85383e8	0.0882416
$569$	3.73363e9	0.849646	0.424823	−	0.905276i	$-0.360336\pi$
$569$	3.73363e9	0.849646	0.424823	+	0.905276i	$0.360336\pi$
$570$	0	0
$571$	3.95974e9	0.890102	0.445051	−	0.895505i	$-0.353186\pi$
$571$	3.95974e9	0.890102	0.445051	+	0.895505i	$0.353186\pi$
$572$	7.06048e8	0.157742
$573$	0	0
$574$	−3.23126e8	−0.0713149
$575$	1.51807e10	3.33008
$576$	0	0
$577$	1.26740e9	0.274662	0.137331	−	0.990525i	$-0.456148\pi$
$577$	1.26740e9	0.274662	0.137331	+	0.990525i	$0.456148\pi$
$578$	9.70127e8	0.208969
$579$	0	0
$580$	3.67391e8	0.0781862
$581$	−5.69556e9	−1.20481
$582$	0	0
$583$	−1.12366e9	−0.234852
$584$	1.95689e9	0.406556
$585$	0	0
$586$	−2.02408e9	−0.415514
$587$	2.24975e9	0.459093	0.229546	−	0.973298i	$-0.426276\pi$
$587$	2.24975e9	0.459093	0.229546	+	0.973298i	$0.426276\pi$
$588$	0	0
$589$	−1.52755e9	−0.308029
$590$	1.39415e9	0.279465
$591$	0	0
$592$	1.39544e9	0.276430
$593$	5.95656e9	1.17302	0.586508	−	0.809943i	$-0.300502\pi$
$593$	5.95656e9	1.17302	0.586508	+	0.809943i	$0.300502\pi$
$594$	0	0
$595$	−2.02227e10	−3.93578
$596$	5.10751e9	0.988205
$597$	0	0
$598$	−8.90155e8	−0.170220
$599$	−4.67049e9	−0.887909	−0.443955	−	0.896049i	$-0.646425\pi$
$599$	−4.67049e9	−0.887909	−0.443955	+	0.896049i	$0.646425\pi$
$600$	0	0
$601$	5.37104e9	1.00925	0.504624	−	0.863339i	$-0.331631\pi$
$601$	5.37104e9	1.00925	0.504624	+	0.863339i	$0.331631\pi$
$602$	−6.66055e8	−0.124429
$603$	0	0
$604$	2.59762e9	0.479673
$605$	8.96000e8	0.164499
$606$	0	0
$607$	−8.70240e8	−0.157935	−0.0789675	−	0.996877i	$-0.525162\pi$
$607$	−8.70240e8	−0.157935	−0.0789675	+	0.996877i	$0.525162\pi$
$608$	−5.96132e8	−0.107567
$609$	0	0
$610$	3.89449e9	0.694698
$611$	9.42726e8	0.167202
$612$	0	0
$613$	9.68587e9	1.69835	0.849175	−	0.528112i	$-0.177100\pi$
$613$	9.68587e9	1.69835	0.849175	+	0.528112i	$0.177100\pi$
$614$	−2.90864e8	−0.0507107
$615$	0	0
$616$	−1.12031e9	−0.193111
$617$	9.06146e9	1.55310	0.776551	−	0.630054i	$-0.216967\pi$
$617$	9.06146e9	1.55310	0.776551	+	0.630054i	$0.216967\pi$
$618$	0	0
$619$	1.81062e9	0.306839	0.153419	−	0.988161i	$-0.450971\pi$
$619$	1.81062e9	0.306839	0.153419	+	0.988161i	$0.450971\pi$
$620$	−1.75841e10	−2.96311
$621$	0	0
$622$	1.16689e9	0.194430
$623$	−2.30965e9	−0.382681
$624$	0	0
$625$	1.15846e10	1.89803
$626$	−2.07434e8	−0.0337963
$627$	0	0
$628$	−6.44815e8	−0.103891
$629$	−2.80368e9	−0.449211
$630$	0	0
$631$	−6.77403e9	−1.07336	−0.536678	−	0.843787i	$-0.680321\pi$
$631$	−6.77403e9	−1.07336	−0.536678	+	0.843787i	$0.680321\pi$
$632$	−4.61802e9	−0.727689
$633$	0	0
$634$	1.47986e9	0.230626
$635$	−1.24872e10	−1.93534
$636$	0	0
$637$	−4.94489e9	−0.757999
$638$	−1.89891e7	−0.00289489
$639$	0	0
$640$	−9.06984e9	−1.36763
$641$	3.15511e9	0.473164	0.236582	−	0.971612i	$-0.423973\pi$
$641$	3.15511e9	0.473164	0.236582	+	0.971612i	$0.423973\pi$
$642$	0	0
$643$	−1.02056e10	−1.51391	−0.756953	−	0.653469i	$-0.773313\pi$
$643$	−1.02056e10	−1.51391	−0.756953	+	0.653469i	$0.773313\pi$
$644$	−1.46326e10	−2.15884
$645$	0	0
$646$	3.67964e8	0.0537021
$647$	5.60213e9	0.813183	0.406592	−	0.913610i	$-0.366717\pi$
$647$	5.60213e9	0.813183	0.406592	+	0.913610i	$0.366717\pi$
$648$	0	0
$649$	1.52822e9	0.219446
$650$	−1.85113e9	−0.264387
$651$	0	0
$652$	−9.49848e9	−1.34211
$653$	1.94634e9	0.273541	0.136771	−	0.990603i	$-0.456328\pi$
$653$	1.94634e9	0.273541	0.136771	+	0.990603i	$0.456328\pi$
$654$	0	0
$655$	6.52995e9	0.907957
$656$	−1.36449e9	−0.188715
$657$	0	0
$658$	−7.30703e8	−0.0999885
$659$	1.75943e9	0.239482	0.119741	−	0.992805i	$-0.461794\pi$
$659$	1.75943e9	0.239482	0.119741	+	0.992805i	$0.461794\pi$
$660$	0	0
$661$	1.11099e10	1.49625	0.748126	−	0.663557i	$-0.230954\pi$
$661$	1.11099e10	1.49625	0.748126	+	0.663557i	$0.230954\pi$
$662$	7.77599e8	0.104172
$663$	0	0
$664$	2.44217e9	0.323734
$665$	−3.80576e9	−0.501841
$666$	0	0
$667$	−5.07734e8	−0.0662515
$668$	1.45875e9	0.189350
$669$	0	0
$670$	2.54796e9	0.327288
$671$	4.26900e9	0.545503
$672$	0	0
$673$	8.91893e9	1.12787	0.563936	−	0.825818i	$-0.309286\pi$
$673$	8.91893e9	1.12787	0.563936	+	0.825818i	$0.309286\pi$
$674$	8.09888e8	0.101886
$675$	0	0
$676$	5.36811e9	0.668357
$677$	−4.65213e9	−0.576224	−0.288112	−	0.957597i	$-0.593028\pi$
$677$	−4.65213e9	−0.576224	−0.288112	+	0.957597i	$0.593028\pi$
$678$	0	0
$679$	−5.07147e9	−0.621712
$680$	8.67120e9	1.05754
$681$	0	0
$682$	9.08855e8	0.109711
$683$	1.30727e10	1.56998	0.784989	−	0.619510i	$-0.212669\pi$
$683$	1.30727e10	1.56998	0.784989	+	0.619510i	$0.212669\pi$
$684$	0	0
$685$	−5.83464e9	−0.693581
$686$	1.06265e9	0.125677
$687$	0	0
$688$	−2.81261e9	−0.329268
$689$	3.66363e9	0.426721
$690$	0	0
$691$	4.85256e9	0.559497	0.279749	−	0.960073i	$-0.409749\pi$
$691$	4.85256e9	0.559497	0.279749	+	0.960073i	$0.409749\pi$
$692$	−8.07750e9	−0.926628
$693$	0	0
$694$	−2.08495e9	−0.236775
$695$	−3.16840e10	−3.58008
$696$	0	0
$697$	2.74150e9	0.306671
$698$	3.19596e8	0.0355719
$699$	0	0
$700$	−3.04294e10	−3.35312
$701$	−2.91482e9	−0.319594	−0.159797	−	0.987150i	$-0.551084\pi$
$701$	−2.91482e9	−0.319594	−0.159797	+	0.987150i	$0.551084\pi$
$702$	0	0
$703$	−5.27631e8	−0.0572778
$704$	−2.06522e9	−0.223080
$705$	0	0
$706$	−1.08396e9	−0.115930
$707$	1.85690e10	1.97615
$708$	0	0
$709$	5.65291e9	0.595676	0.297838	−	0.954616i	$-0.403735\pi$
$709$	5.65291e9	0.595676	0.297838	+	0.954616i	$0.403735\pi$
$710$	−7.78922e8	−0.0816751
$711$	0	0
$712$	9.90341e8	0.102826
$713$	2.43011e10	2.51081
$714$	0	0
$715$	−2.92137e9	−0.298892
$716$	7.66815e9	0.780719
$717$	0	0
$718$	−1.60173e9	−0.161493
$719$	1.82027e9	0.182635	0.0913176	−	0.995822i	$-0.470892\pi$
$719$	1.82027e9	0.182635	0.0913176	+	0.995822i	$0.470892\pi$
$720$	0	0
$721$	−3.13391e8	−0.0311396
$722$	−2.07673e9	−0.205352
$723$	0	0
$724$	−9.68255e9	−0.948210
$725$	−1.05586e9	−0.102902
$726$	0	0
$727$	−3.58089e9	−0.345637	−0.172819	−	0.984954i	$-0.555287\pi$
$727$	−3.58089e9	−0.345637	−0.172819	+	0.984954i	$0.555287\pi$
$728$	3.65272e9	0.350879
$729$	0	0
$730$	−3.95519e9	−0.376303
$731$	5.65101e9	0.535076
$732$	0	0
$733$	−1.37367e10	−1.28830	−0.644152	−	0.764898i	$-0.722790\pi$
$733$	−1.37367e10	−1.28830	−0.644152	+	0.764898i	$0.722790\pi$
$734$	9.63444e7	0.00899270
$735$	0	0
$736$	9.48360e9	0.876801
$737$	2.79298e9	0.256999
$738$	0	0
$739$	−1.13883e10	−1.03802	−0.519009	−	0.854769i	$-0.673699\pi$
$739$	−1.13883e10	−1.03802	−0.519009	+	0.854769i	$0.673699\pi$
$740$	−6.07371e9	−0.550989
$741$	0	0
$742$	−2.83966e9	−0.255184
$743$	−2.15842e10	−1.93052	−0.965262	−	0.261285i	$-0.915854\pi$
$743$	−2.15842e10	−1.93052	−0.965262	+	0.261285i	$0.915854\pi$
$744$	0	0
$745$	−2.11330e10	−1.87247
$746$	−2.98781e9	−0.263492
$747$	0	0
$748$	4.64307e9	0.405648
$749$	−2.74571e10	−2.38764
$750$	0	0
$751$	3.24639e9	0.279680	0.139840	−	0.990174i	$-0.455341\pi$
$751$	3.24639e9	0.279680	0.139840	+	0.990174i	$0.455341\pi$
$752$	−3.08560e9	−0.264592
$753$	0	0
$754$	6.19129e7	0.00525995
$755$	−1.07480e10	−0.908892
$756$	0	0
$757$	−5.81302e9	−0.487042	−0.243521	−	0.969896i	$-0.578302\pi$
$757$	−5.81302e9	−0.487042	−0.243521	+	0.969896i	$0.578302\pi$
$758$	4.69006e9	0.391144
$759$	0	0
$760$	1.63185e9	0.134845
$761$	6.49747e9	0.534439	0.267219	−	0.963636i	$-0.413895\pi$
$761$	6.49747e9	0.534439	0.267219	+	0.963636i	$0.413895\pi$
$762$	0	0
$763$	5.46198e9	0.445158
$764$	2.18832e10	1.77535
$765$	0	0
$766$	6.52816e8	0.0524796
$767$	−4.98267e9	−0.398730
$768$	0	0
$769$	1.24166e10	0.984603	0.492301	−	0.870425i	$-0.336156\pi$
$769$	1.24166e10	0.984603	0.492301	+	0.870425i	$0.336156\pi$
$770$	2.26434e9	0.178741
$771$	0	0
$772$	−2.24532e10	−1.75638
$773$	−1.43991e10	−1.12126	−0.560631	−	0.828066i	$-0.689441\pi$
$773$	−1.43991e10	−1.12126	−0.560631	+	0.828066i	$0.689441\pi$
$774$	0	0
$775$	5.05357e10	3.89980
$776$	2.17457e9	0.167054
$777$	0	0
$778$	−2.86645e9	−0.218231
$779$	5.15929e8	0.0391029
$780$	0	0
$781$	−8.53827e8	−0.0641345
$782$	−5.85378e9	−0.437737
$783$	0	0
$784$	1.61849e10	1.19951
$785$	2.66801e9	0.196853
$786$	0	0
$787$	−1.42293e10	−1.04057	−0.520287	−	0.853991i	$-0.674175\pi$
$787$	−1.42293e10	−1.04057	−0.520287	+	0.853991i	$0.674175\pi$
$788$	3.00416e9	0.218717
$789$	0	0
$790$	9.33378e9	0.673539
$791$	−3.40084e9	−0.244325
$792$	0	0
$793$	−1.39189e10	−0.991169
$794$	4.53285e9	0.321365
$795$	0	0
$796$	1.00631e10	0.707193
$797$	1.15055e10	0.805010	0.402505	−	0.915418i	$-0.368140\pi$
$797$	1.15055e10	0.805010	0.402505	+	0.915418i	$0.368140\pi$
$798$	0	0
$799$	6.19950e9	0.429975
$800$	1.97217e10	1.36185
$801$	0	0
$802$	−9.14800e7	−0.00626204
$803$	−4.33554e9	−0.295488
$804$	0	0
$805$	6.05443e10	4.09060
$806$	−2.96328e9	−0.199342
$807$	0	0
$808$	−7.96211e9	−0.530993
$809$	9.95848e9	0.661262	0.330631	−	0.943760i	$-0.392739\pi$
$809$	9.95848e9	0.661262	0.330631	+	0.943760i	$0.392739\pi$
$810$	0	0
$811$	−9.67643e9	−0.637004	−0.318502	−	0.947922i	$-0.603180\pi$
$811$	−9.67643e9	−0.637004	−0.318502	+	0.947922i	$0.603180\pi$
$812$	1.01774e9	0.0667100
$813$	0	0
$814$	3.13927e8	0.0204006
$815$	3.93012e10	2.54305
$816$	0	0
$817$	1.06348e9	0.0682262
$818$	−2.65256e9	−0.169445
$819$	0	0
$820$	5.93900e9	0.376153
$821$	−1.72563e10	−1.08829	−0.544146	−	0.838990i	$-0.683146\pi$
$821$	−1.72563e10	−1.08829	−0.544146	+	0.838990i	$0.683146\pi$
$822$	0	0
$823$	2.71904e10	1.70026	0.850131	−	0.526572i	$-0.176523\pi$
$823$	2.71904e10	1.70026	0.850131	+	0.526572i	$0.176523\pi$
$824$	1.34377e8	0.00836720
$825$	0	0
$826$	3.86205e9	0.238445
$827$	1.12774e10	0.693329	0.346664	−	0.937989i	$-0.387314\pi$
$827$	1.12774e10	0.693329	0.346664	+	0.937989i	$0.387314\pi$
$828$	0	0
$829$	−6.11311e9	−0.372667	−0.186334	−	0.982487i	$-0.559661\pi$
$829$	−6.11311e9	−0.372667	−0.186334	+	0.982487i	$0.559661\pi$
$830$	−4.93603e9	−0.299643
$831$	0	0
$832$	6.73354e9	0.405333
$833$	−3.25183e10	−1.94926
$834$	0	0
$835$	−6.03578e9	−0.358782
$836$	8.73790e8	0.0517232
$837$	0	0
$838$	1.09095e9	0.0640400
$839$	7.39055e8	0.0432026	0.0216013	−	0.999767i	$-0.493124\pi$
$839$	7.39055e8	0.0432026	0.0216013	+	0.999767i	$0.493124\pi$
$840$	0	0
$841$	−1.72146e10	−0.997953
$842$	−2.51305e9	−0.145080
$843$	0	0
$844$	9.11401e9	0.521809
$845$	−2.22113e10	−1.26641
$846$	0	0
$847$	2.48209e9	0.140354
$848$	−1.19913e10	−0.675274
$849$	0	0
$850$	−1.21733e10	−0.679895
$851$	8.39385e9	0.466883
$852$	0	0
$853$	−2.19811e10	−1.21263	−0.606314	−	0.795225i	$-0.707352\pi$
$853$	−2.19811e10	−1.21263	−0.606314	+	0.795225i	$0.707352\pi$
$854$	1.07885e10	0.592730
$855$	0	0
$856$	1.17732e10	0.641558
$857$	2.15503e10	1.16955	0.584777	−	0.811194i	$-0.301182\pi$
$857$	2.15503e10	1.16955	0.584777	+	0.811194i	$0.301182\pi$
$858$	0	0
$859$	1.15575e10	0.622137	0.311069	−	0.950387i	$-0.399313\pi$
$859$	1.15575e10	0.622137	0.311069	+	0.950387i	$0.399313\pi$
$860$	1.22420e10	0.656307
$861$	0	0
$862$	1.42596e9	0.0758285
$863$	−2.47775e10	−1.31226	−0.656130	−	0.754648i	$-0.727808\pi$
$863$	−2.47775e10	−1.31226	−0.656130	+	0.754648i	$0.727808\pi$
$864$	0	0
$865$	3.34217e10	1.75579
$866$	−4.57237e9	−0.239237
$867$	0	0
$868$	−4.87111e10	−2.52819
$869$	1.02314e10	0.528889
$870$	0	0
$871$	−9.10638e9	−0.466963
$872$	−2.34201e9	−0.119614
$873$	0	0
$874$	−1.10164e9	−0.0558147
$875$	7.05446e10	3.55988
$876$	0	0
$877$	1.37586e10	0.688771	0.344386	−	0.938828i	$-0.388087\pi$
$877$	1.37586e10	0.688771	0.344386	+	0.938828i	$0.388087\pi$
$878$	−3.26420e9	−0.162760
$879$	0	0
$880$	9.56180e9	0.472988
$881$	−3.21022e10	−1.58168	−0.790841	−	0.612022i	$-0.790356\pi$
$881$	−3.21022e10	−1.58168	−0.790841	+	0.612022i	$0.790356\pi$
$882$	0	0
$883$	3.40295e8	0.0166339	0.00831694	−	0.999965i	$-0.497353\pi$
$883$	3.40295e8	0.0166339	0.00831694	+	0.999965i	$0.497353\pi$
$884$	−1.51385e10	−0.737055
$885$	0	0
$886$	−4.48795e9	−0.216786
$887$	1.22982e10	0.591711	0.295856	−	0.955233i	$-0.404395\pi$
$887$	1.22982e10	0.591711	0.295856	+	0.955233i	$0.404395\pi$
$888$	0	0
$889$	−3.45918e10	−1.65127
$890$	−2.00164e9	−0.0951747
$891$	0	0
$892$	2.19402e9	0.103506
$893$	1.16670e9	0.0548250
$894$	0	0
$895$	−3.17280e10	−1.47932
$896$	−2.51252e10	−1.16689
$897$	0	0
$898$	4.67782e9	0.215564
$899$	−1.69022e9	−0.0775861
$900$	0	0
$901$	2.40925e10	1.09735
$902$	−3.06965e8	−0.0139273
$903$	0	0
$904$	1.45823e9	0.0656502
$905$	4.00628e10	1.79668
$906$	0	0
$907$	−2.08631e10	−0.928438	−0.464219	−	0.885720i	$-0.653665\pi$
$907$	−2.08631e10	−0.928438	−0.464219	+	0.885720i	$0.653665\pi$
$908$	−1.05693e10	−0.468540
$909$	0	0
$910$	−7.38276e9	−0.324768
$911$	−2.52053e10	−1.10453	−0.552266	−	0.833668i	$-0.686237\pi$
$911$	−2.52053e10	−1.10453	−0.552266	+	0.833668i	$0.686237\pi$
$912$	0	0
$913$	−5.41070e9	−0.235291
$914$	−5.16441e9	−0.223722
$915$	0	0
$916$	1.43004e10	0.614771
$917$	1.80892e10	0.774687
$918$	0	0
$919$	4.38722e9	0.186460	0.0932299	−	0.995645i	$-0.470281\pi$
$919$	4.38722e9	0.186460	0.0932299	+	0.995645i	$0.470281\pi$
$920$	−2.59605e10	−1.09914
$921$	0	0
$922$	5.61728e9	0.236031
$923$	2.78386e9	0.116531
$924$	0	0
$925$	1.74555e10	0.725165
$926$	−4.51177e9	−0.186728
$927$	0	0
$928$	−6.59613e8	−0.0270939
$929$	4.04223e10	1.65412	0.827058	−	0.562117i	$-0.190013\pi$
$929$	4.04223e10	1.65412	0.827058	+	0.562117i	$0.190013\pi$
$930$	0	0
$931$	−6.11969e9	−0.248545
$932$	−3.29694e9	−0.133400
$933$	0	0
$934$	5.06019e8	0.0203214
$935$	−1.92113e10	−0.768628
$936$	0	0
$937$	−4.80084e9	−0.190647	−0.0953233	−	0.995446i	$-0.530388\pi$
$937$	−4.80084e9	−0.190647	−0.0953233	+	0.995446i	$0.530388\pi$
$938$	7.05831e9	0.279249
$939$	0	0
$940$	1.34302e10	0.527394
$941$	8.73176e9	0.341616	0.170808	−	0.985304i	$-0.445362\pi$
$941$	8.73176e9	0.341616	0.170808	+	0.985304i	$0.445362\pi$
$942$	0	0
$943$	−8.20769e9	−0.318735
$944$	1.63086e10	0.630978
$945$	0	0
$946$	−6.32743e8	−0.0243001
$947$	−1.14448e10	−0.437909	−0.218955	−	0.975735i	$-0.570265\pi$
$947$	−1.14448e10	−0.437909	−0.218955	+	0.975735i	$0.570265\pi$
$948$	0	0
$949$	1.41358e10	0.536895
$950$	−2.29092e9	−0.0866917
$951$	0	0
$952$	2.40208e10	0.902316
$953$	−7.83640e9	−0.293286	−0.146643	−	0.989189i	$-0.546847\pi$
$953$	−7.83640e9	−0.293286	−0.146643	+	0.989189i	$0.546847\pi$
$954$	0	0
$955$	−9.05447e10	−3.36396
$956$	2.48894e10	0.921324
$957$	0	0
$958$	−3.66573e7	−0.00134704
$959$	−1.61630e10	−0.591777
$960$	0	0
$961$	5.33846e10	1.94037
$962$	−1.02354e9	−0.0370676
$963$	0	0
$964$	2.96019e10	1.06426
$965$	9.29030e10	3.32801
$966$	0	0
$967$	2.69859e10	0.959718	0.479859	−	0.877346i	$-0.340688\pi$
$967$	2.69859e10	0.959718	0.479859	+	0.877346i	$0.340688\pi$
$968$	−1.06428e9	−0.0377131
$969$	0	0
$970$	−4.39516e9	−0.154623
$971$	−5.49167e10	−1.92503	−0.962514	−	0.271232i	$-0.912569\pi$
$971$	−5.49167e10	−1.92503	−0.962514	+	0.271232i	$0.912569\pi$
$972$	0	0
$973$	−8.77705e10	−3.05459
$974$	8.04313e9	0.278913
$975$	0	0
$976$	4.55573e10	1.56850
$977$	−2.25460e10	−0.773460	−0.386730	−	0.922193i	$-0.626396\pi$
$977$	−2.25460e10	−0.773460	−0.386730	+	0.922193i	$0.626396\pi$
$978$	0	0
$979$	−2.19413e9	−0.0747349
$980$	−7.04455e10	−2.39090
$981$	0	0
$982$	4.21635e9	0.142085
$983$	−3.93701e10	−1.32199	−0.660996	−	0.750389i	$-0.729866\pi$
$983$	−3.93701e10	−1.32199	−0.660996	+	0.750389i	$0.729866\pi$
$984$	0	0
$985$	−1.24301e10	−0.414427
$986$	4.07148e8	0.0135264
$987$	0	0
$988$	−2.84895e9	−0.0939800
$989$	−1.69184e10	−0.556125
$990$	0	0
$991$	5.49438e10	1.79333	0.896666	−	0.442708i	$-0.145982\pi$
$991$	5.49438e10	1.79333	0.896666	+	0.442708i	$0.145982\pi$
$992$	3.15704e10	1.02681
$993$	0	0
$994$	−2.15776e9	−0.0696868
$995$	−4.16376e10	−1.34000
$996$	0	0
$997$	1.96960e10	0.629426	0.314713	−	0.949187i	$-0.398092\pi$
$997$	1.96960e10	0.629426	0.314713	+	0.949187i	$0.398092\pi$
$998$	−8.23306e9	−0.262183
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	99.8.a.e.1.2		3
3.2	odd	2		33.8.a.d.1.2	✓	3
12.11	even	2		528.8.a.o.1.1		3
33.32	even	2		363.8.a.e.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.a.d.1.2	✓	3	3.2	odd	2
99.8.a.e.1.2		3	1.1	even	1	trivial
363.8.a.e.1.2		3	33.32	even	2
528.8.a.o.1.1		3	12.11	even	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:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	99.a (trivial)

\(f(q)\)	\(=\)	\(q+2.40077 q^{2} -122.236 q^{4} +505.769 q^{5} +1401.07 q^{7} -600.759 q^{8} +1214.23 q^{10} +1331.00 q^{11} -4339.66 q^{13} +3363.65 q^{14} +14204.0 q^{16} -28538.2 q^{17} -5370.67 q^{19} -61823.3 q^{20} +3195.42 q^{22} +85439.7 q^{23} +177677. q^{25} -10418.5 q^{26} -171262. q^{28} -5942.59 q^{29} +284424. q^{31} +110998. q^{32} -68513.6 q^{34} +708619. q^{35} +98242.9 q^{37} -12893.7 q^{38} -303845. q^{40} -96064.1 q^{41} -198016. q^{43} -162697. q^{44} +205121. q^{46} -217235. q^{47} +1.13946e6 q^{49} +426561. q^{50} +530464. q^{52} -844220. q^{53} +673178. q^{55} -841708. q^{56} -14266.8 q^{58} +1.14817e6 q^{59} +3.20736e6 q^{61} +682837. q^{62} -1.55163e6 q^{64} -2.19486e6 q^{65} +2.09841e6 q^{67} +3.48841e6 q^{68} +1.70123e6 q^{70} -641493. q^{71} -3.25736e6 q^{73} +235858. q^{74} +656491. q^{76} +1.86483e6 q^{77} +7.68697e6 q^{79} +7.18392e6 q^{80} -230627. q^{82} -4.06514e6 q^{83} -1.44337e7 q^{85} -475389. q^{86} -799610. q^{88} -1.64848e6 q^{89} -6.08018e6 q^{91} -1.04438e7 q^{92} -521531. q^{94} -2.71632e6 q^{95} -3.61970e6 q^{97} +2.73559e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{2} - 15 q^{4} + 444 q^{5} + 1614 q^{7} - 3153 q^{8} + 2880 q^{10} + 3993 q^{11} + 20772 q^{13} - 36258 q^{14} + 12225 q^{16} + 14538 q^{17} + 24492 q^{19} - 80112 q^{20} - 11979 q^{22} - 35094 q^{23}+ \cdots - 24377397 q^{98}+O(q^{100}) \) 3 * q - 9 * q^2 - 15 * q^4 + 444 * q^5 + 1614 * q^7 - 3153 * q^8 + 2880 * q^10 + 3993 * q^11 + 20772 * q^13 - 36258 * q^14 + 12225 * q^16 + 14538 * q^17 + 24492 * q^19 - 80112 * q^20 - 11979 * q^22 - 35094 * q^23 + 29121 * q^25 - 203832 * q^26 + 278034 * q^28 + 179862 * q^29 + 288888 * q^31 + 519567 * q^32 - 491586 * q^34 + 532872 * q^35 + 107562 * q^37 + 686328 * q^38 - 237360 * q^40 + 135198 * q^41 + 193536 * q^43 - 19965 * q^44 + 16422 * q^46 + 591486 * q^47 + 4461159 * q^49 + 1192245 * q^50 + 2449992 * q^52 - 79044 * q^53 + 590964 * q^55 - 752658 * q^56 + 2289930 * q^58 - 2532768 * q^59 + 6678792 * q^61 + 2660808 * q^62 - 3966303 * q^64 - 3191832 * q^65 + 7150356 * q^67 + 7821954 * q^68 + 4029120 * q^70 - 1390398 * q^71 - 6429114 * q^73 - 2507478 * q^74 - 7654728 * q^76 + 2148234 * q^77 + 6873186 * q^79 + 7556016 * q^80 - 1774590 * q^82 - 6505596 * q^83 - 16546032 * q^85 + 6519468 * q^86 - 4196643 * q^88 + 8842962 * q^89 + 3066648 * q^91 - 6921990 * q^92 - 24038238 * q^94 + 190968 * q^95 - 1764774 * q^97 - 24377397 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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