LMFDB - Embedded newform 363.8.a.e.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [363,8,Mod(1,363)] 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("363.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(363, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	363.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:	$113.395764251$
Analytic rank:	$1$
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	$\chi$	$=$	363.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.40077 q^{2} -27.0000 q^{3} -122.236 q^{4} -505.769 q^{5} -64.8207 q^{6} -1401.07 q^{7} -600.759 q^{8} +729.000 q^{9} -1214.23 q^{10} +3300.38 q^{12} +4339.66 q^{13} -3363.65 q^{14} +13655.8 q^{15} +14204.0 q^{16} -28538.2 q^{17} +1750.16 q^{18} +5370.67 q^{19} +61823.3 q^{20} +37829.0 q^{21} -85439.7 q^{23} +16220.5 q^{24} +177677. q^{25} +10418.5 q^{26} -19683.0 q^{27} +171262. q^{28} -5942.59 q^{29} +32784.3 q^{30} +284424. q^{31} +110998. q^{32} -68513.6 q^{34} +708619. q^{35} -89110.3 q^{36} +98242.9 q^{37} +12893.7 q^{38} -117171. q^{39} +303845. q^{40} -96064.1 q^{41} +90818.6 q^{42} +198016. q^{43} -368706. q^{45} -205121. q^{46} +217235. q^{47} -383507. q^{48} +1.13946e6 q^{49} +426561. q^{50} +770532. q^{51} -530464. q^{52} +844220. q^{53} -47254.3 q^{54} +841708. q^{56} -145008. q^{57} -14266.8 q^{58} -1.14817e6 q^{59} -1.66923e6 q^{60} -3.20736e6 q^{61} +682837. q^{62} -1.02138e6 q^{63} -1.55163e6 q^{64} -2.19486e6 q^{65} +2.09841e6 q^{67} +3.48841e6 q^{68} +2.30687e6 q^{69} +1.70123e6 q^{70} +641493. q^{71} -437953. q^{72} +3.25736e6 q^{73} +235858. q^{74} -4.79728e6 q^{75} -656491. q^{76} -281300. q^{78} -7.68697e6 q^{79} -7.18392e6 q^{80} +531441. q^{81} -230627. q^{82} -4.06514e6 q^{83} -4.62408e6 q^{84} +1.44337e7 q^{85} +475389. q^{86} +160450. q^{87} +1.64848e6 q^{89} -885176. q^{90} -6.08018e6 q^{91} +1.04438e7 q^{92} -7.67946e6 q^{93} +521531. q^{94} -2.71632e6 q^{95} -2.99693e6 q^{96} -3.61970e6 q^{97} +2.73559e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{2} - 81 q^{3} - 15 q^{4} - 444 q^{5} + 243 q^{6} - 1614 q^{7} - 3153 q^{8} + 2187 q^{9} - 2880 q^{10} + 405 q^{12} - 20772 q^{13} + 36258 q^{14} + 11988 q^{15} + 12225 q^{16} + 14538 q^{17} - 6561 q^{18}+ \cdots - 24377397 q^{98}+O(q^{100}) $ 3 * q - 9 * q^2 - 81 * q^3 - 15 * q^4 - 444 * q^5 + 243 * q^6 - 1614 * q^7 - 3153 * q^8 + 2187 * q^9 - 2880 * q^10 + 405 * q^12 - 20772 * q^13 + 36258 * q^14 + 11988 * q^15 + 12225 * q^16 + 14538 * q^17 - 6561 * q^18 - 24492 * q^19 + 80112 * q^20 + 43578 * q^21 + 35094 * q^23 + 85131 * q^24 + 29121 * q^25 + 203832 * q^26 - 59049 * q^27 - 278034 * q^28 + 179862 * q^29 + 77760 * q^30 + 288888 * q^31 + 519567 * q^32 - 491586 * q^34 + 532872 * q^35 - 10935 * q^36 + 107562 * q^37 - 686328 * q^38 + 560844 * q^39 + 237360 * q^40 + 135198 * q^41 - 978966 * q^42 - 193536 * q^43 - 323676 * q^45 - 16422 * q^46 - 591486 * q^47 - 330075 * q^48 + 4461159 * q^49 + 1192245 * q^50 - 392526 * q^51 - 2449992 * q^52 + 79044 * q^53 + 177147 * q^54 + 752658 * q^56 + 661284 * q^57 + 2289930 * q^58 + 2532768 * q^59 - 2163024 * q^60 - 6678792 * q^61 + 2660808 * q^62 - 1176606 * q^63 - 3966303 * q^64 - 3191832 * q^65 + 7150356 * q^67 + 7821954 * q^68 - 947538 * q^69 + 4029120 * q^70 + 1390398 * q^71 - 2298537 * q^72 + 6429114 * q^73 - 2507478 * q^74 - 786267 * q^75 + 7654728 * q^76 - 5503464 * q^78 - 6873186 * q^79 - 7556016 * q^80 + 1594323 * q^81 - 1774590 * q^82 - 6505596 * q^83 + 7506918 * q^84 + 16546032 * q^85 - 6519468 * q^86 - 4856274 * q^87 - 8842962 * q^89 - 2099520 * q^90 + 3066648 * q^91 + 6921990 * q^92 - 7799976 * q^93 + 24038238 * q^94 + 190968 * q^95 - 14028309 * q^96 - 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	$0.466164\pi$
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	−122.236	−0.954971
$5$	−505.769	−1.80949	−0.904747	−	0.425950i	$-0.859940\pi$
$5$	−505.769	−1.80949	−0.904747	+	0.425950i	$0.859940\pi$
$6$	−64.8207	−0.122514
$7$	−1401.07	−1.54390	−0.771948	−	0.635686i	$-0.780717\pi$
$7$	−1401.07	−1.54390	−0.771948	+	0.635686i	$0.780717\pi$
$8$	−600.759	−0.414844
$9$	729.000	0.333333
$10$	−1214.23	−0.383974
$11$	0	0
$11$	0	0
$12$	3300.38	0.551353
$13$	4339.66	0.547840	0.273920	−	0.961752i	$-0.411680\pi$
$13$	4339.66	0.547840	0.273920	+	0.961752i	$0.411680\pi$
$14$	−3363.65	−0.327614
$15$	13655.8	1.04471
$16$	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	$0.748788\pi$
$18$	1750.16	0.0707333
$19$	5370.67	0.179635	0.0898176	−	0.995958i	$-0.471372\pi$
$19$	5370.67	0.179635	0.0898176	+	0.995958i	$0.471372\pi$
$20$	61823.3	1.72801
$21$	37829.0	0.891368
$22$	0	0
$23$	−85439.7	−1.46424	−0.732120	−	0.681175i	$-0.761469\pi$
$23$	−85439.7	−1.46424	−0.732120	+	0.681175i	$0.761469\pi$
$24$	16220.5	0.239511
$25$	177677.	2.27427
$26$	10418.5	0.116252
$27$	−19683.0	−0.192450
$28$	171262.	1.47438
$29$	−5942.59	−0.0452463	−0.0226231	−	0.999744i	$-0.507202\pi$
$29$	−5942.59	−0.0452463	−0.0226231	+	0.999744i	$0.507202\pi$
$30$	32784.3	0.221688
$31$	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
$36$	−89110.3	−0.318324
$37$	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
$39$	−117171.	−0.316296
$40$	303845.	0.750658
$41$	−96064.1	−0.217680	−0.108840	−	0.994059i	$-0.534714\pi$
$41$	−96064.1	−0.217680	−0.108840	+	0.994059i	$0.534714\pi$
$42$	90818.6	0.189148
$43$	198016.	0.379804	0.189902	−	0.981803i	$-0.439183\pi$
$43$	198016.	0.379804	0.189902	+	0.981803i	$0.439183\pi$
$44$	0	0
$45$	−368706.	−0.603165
$46$	−205121.	−0.310712
$47$	217235.	0.305202	0.152601	−	0.988288i	$-0.451235\pi$
$47$	217235.	0.305202	0.152601	+	0.988288i	$0.451235\pi$
$48$	−383507.	−0.500529
$49$	1.13946e6	1.38361
$50$	426561.	0.482599
$51$	770532.	0.813382
$52$	−530464.	−0.523172
$53$	844220.	0.778915	0.389458	−	0.921044i	$-0.372663\pi$
$53$	844220.	0.778915	0.389458	+	0.921044i	$0.372663\pi$
$54$	−47254.3	−0.0408379
$55$	0	0
$56$	841708.	0.640476
$57$	−145008.	−0.103712
$58$	−14266.8	−0.00960125
$59$	−1.14817e6	−0.727821	−0.363911	−	0.931434i	$-0.618559\pi$
$59$	−1.14817e6	−0.727821	−0.363911	+	0.931434i	$0.618559\pi$
$60$	−1.66923e6	−0.997670
$61$	−3.20736e6	−1.80923	−0.904615	−	0.426229i	$-0.859842\pi$
$61$	−3.20736e6	−1.80923	−0.904615	+	0.426229i	$0.859842\pi$
$62$	682837.	0.363870
$63$	−1.02138e6	−0.514632
$64$	−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
$69$	2.30687e6	0.845380
$70$	1.70123e6	0.592816
$71$	641493.	0.212710	0.106355	−	0.994328i	$-0.466082\pi$
$71$	641493.	0.212710	0.106355	+	0.994328i	$0.466082\pi$
$72$	−437953.	−0.138281
$73$	3.25736e6	0.980021	0.490011	−	0.871716i	$-0.336993\pi$
$73$	3.25736e6	0.980021	0.490011	+	0.871716i	$0.336993\pi$
$74$	235858.	0.0676613
$75$	−4.79728e6	−1.31305
$76$	−656491.	−0.171546
$77$	0	0
$78$	−281300.	−0.0671179
$79$	−7.68697e6	−1.75413	−0.877063	−	0.480376i	$-0.840500\pi$
$79$	−7.68697e6	−1.75413	−0.877063	+	0.480376i	$0.840500\pi$
$80$	−7.18392e6	−1.56872
$81$	531441.	0.111111
$82$	−230627.	−0.0461915
$83$	−4.06514e6	−0.780373	−0.390187	−	0.920736i	$-0.627590\pi$
$83$	−4.06514e6	−0.780373	−0.390187	+	0.920736i	$0.627590\pi$
$84$	−4.62408e6	−0.851231
$85$	1.44337e7	2.54925
$86$	475389.	0.0805944
$87$	160450.	0.0261230
$88$	0	0
$89$	1.64848e6	0.247867	0.123934	−	0.992290i	$-0.460449\pi$
$89$	1.64848e6	0.247867	0.123934	+	0.992290i	$0.460449\pi$
$90$	−885176.	−0.127991
$91$	−6.08018e6	−0.845808
$92$	1.04438e7	1.39831
$93$	−7.67946e6	−0.990012
$94$	521531.	0.0647638
$95$	−2.71632e6	−0.325049
$96$	−2.99693e6	−0.345723
$97$	−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	$0.278939\pi$
$102$	1.84987e6	0.172600
$103$	−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
$105$	−1.91327e7	−1.61293
$106$	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	$0.781371\pi$
$108$	2.40598e6	0.183784
$109$	−3.89843e6	−0.288335	−0.144167	−	0.989553i	$-0.546050\pi$
$109$	−3.89843e6	−0.288335	−0.144167	+	0.989553i	$0.546050\pi$
$110$	0	0
$111$	−2.65256e6	−0.184092
$112$	−1.99008e7	−1.33847
$113$	2.42731e6	0.158253	0.0791263	−	0.996865i	$-0.474787\pi$
$113$	2.42731e6	0.158253	0.0791263	+	0.996865i	$0.474787\pi$
$114$	−348131.	−0.0220077
$115$	4.32128e7	2.64953
$116$	726401.	0.0432089
$117$	3.16361e6	0.182613
$118$	−2.75649e6	−0.154444
$119$	3.99841e7	2.17507
$120$	−8.20382e6	−0.433393
$121$	0	0
$122$	−7.70013e6	−0.383918
$123$	2.59373e6	0.125677
$124$	−3.47670e7	−1.63754
$125$	−5.03504e7	−2.30578
$126$	−2.45210e6	−0.109205
$127$	2.46895e7	1.06955	0.534773	−	0.844996i	$-0.320397\pi$
$127$	2.46895e7	1.06955	0.534773	+	0.844996i	$0.320397\pi$
$128$	−1.79328e7	−0.755810
$129$	−5.34642e6	−0.219280
$130$	−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
$135$	9.95505e6	0.348237
$136$	1.71446e7	0.584441
$137$	1.15362e7	0.383301	0.191651	−	0.981463i	$-0.438616\pi$
$137$	1.15362e7	0.383301	0.191651	+	0.981463i	$0.438616\pi$
$138$	5.53826e6	0.179389
$139$	6.26452e7	1.97850	0.989249	−	0.146239i	$-0.0467169\pi$
$139$	6.26452e7	1.97850	0.989249	+	0.146239i	$0.0467169\pi$
$140$	−8.66190e7	−2.66787
$141$	−5.86535e6	−0.176208
$142$	1.54008e6	0.0451370
$143$	0	0
$144$	1.03547e7	0.288980
$145$	3.00558e6	0.0818729
$146$	7.82016e6	0.207960
$147$	−3.07655e7	−0.798829
$148$	−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	$0.673100\pi$
$150$	−1.15172e7	−0.278629
$151$	2.12508e7	0.502291	0.251145	−	0.967949i	$-0.419193\pi$
$151$	2.12508e7	0.502291	0.251145	+	0.967949i	$0.419193\pi$
$152$	−3.22648e6	−0.0745206
$153$	−2.08044e7	−0.469607
$154$	0	0
$155$	−1.43853e8	−3.10283
$156$	1.43225e7	0.302053
$157$	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
$159$	−2.27939e7	−0.449707
$160$	−5.61391e7	−1.08354
$161$	1.19707e8	2.26063
$162$	1.27587e6	0.0235778
$163$	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	$0.531609\pi$
$168$	−2.27261e7	−0.369779
$169$	−4.39159e7	−0.699871
$170$	3.46520e7	0.540950
$171$	3.91522e6	0.0598784
$172$	−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	$0.338761\pi$
$174$	385203.	0.00554328
$175$	−2.48939e8	−3.51123
$176$	0	0
$177$	3.10006e7	0.420208
$178$	3.95762e6	0.0525974
$179$	6.27322e7	0.817532	0.408766	−	0.912639i	$-0.365959\pi$
$179$	6.27322e7	0.817532	0.408766	+	0.912639i	$0.365959\pi$
$180$	4.50692e7	0.576005
$181$	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
$183$	8.65988e7	1.04456
$184$	5.13287e7	0.607432
$185$	−4.96882e7	−0.576969
$186$	−1.84366e7	−0.210080
$187$	0	0
$188$	−2.65540e7	−0.291459
$189$	2.75773e7	0.297123
$190$	−6.52125e6	−0.0689752
$191$	1.79024e8	1.85906	0.929531	−	0.368743i	$-0.120212\pi$
$191$	1.79024e8	1.85906	0.929531	+	0.368743i	$0.120212\pi$
$192$	4.18940e7	0.427167
$193$	−1.83687e8	−1.83919	−0.919596	−	0.392865i	$-0.871484\pi$
$193$	−1.83687e8	−1.83919	−0.919596	+	0.392865i	$0.871484\pi$
$194$	−8.69006e6	−0.0854509
$195$	5.92613e7	0.572335
$196$	−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
$201$	−5.66570e7	−0.492116
$202$	3.18184e7	0.271611
$203$	8.32601e6	0.0698555
$204$	−9.41869e7	−0.776757
$205$	4.85862e7	0.393890
$206$	−537001.	−0.00427996
$207$	−6.22856e7	−0.488080
$208$	6.16404e7	0.474945
$209$	0	0
$210$	−4.59332e7	−0.342262
$211$	7.45606e7	0.546413	0.273206	−	0.961955i	$-0.411916\pi$
$211$	7.45606e7	0.546413	0.273206	+	0.961955i	$0.411916\pi$
$212$	−1.03194e8	−0.743842
$213$	−1.73203e7	−0.122808
$214$	−4.70483e7	−0.328168
$215$	−1.00150e8	−0.687253
$216$	1.18247e7	0.0798369
$217$	−3.98500e8	−2.64739
$218$	−9.35921e6	−0.0611845
$219$	−8.79487e7	−0.565815
$220$	0	0
$221$	−1.23846e8	−0.771808
$222$	−6.36818e6	−0.0390643
$223$	−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
$225$	1.29527e8	0.758089
$226$	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	$0.421108\pi$
$228$	1.77253e7	0.0990423
$229$	−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	$0.477750\pi$
$234$	7.59509e6	0.0387505
$235$	−1.09871e8	−0.552261
$236$	1.40348e8	0.695048
$237$	2.07548e8	1.01274
$238$	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	$0.660229\pi$
$240$	1.93966e8	0.905704
$241$	2.42169e8	1.11445	0.557224	−	0.830363i	$-0.311867\pi$
$241$	2.42169e8	1.11445	0.557224	+	0.830363i	$0.311867\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	3.92056e8	1.72776
$245$	−5.76306e8	−2.50364
$246$	6.22694e6	0.0266687
$247$	2.33069e7	0.0984113
$248$	−1.70871e8	−0.711355
$249$	1.09759e8	0.450549
$250$	−1.20880e8	−0.489286
$251$	1.95655e8	0.780968	0.390484	−	0.920610i	$-0.372308\pi$
$251$	1.95655e8	0.780968	0.390484	+	0.920610i	$0.372308\pi$
$252$	1.24850e8	0.491459
$253$	0	0
$254$	5.92738e7	0.226957
$255$	−3.89711e8	−1.47181
$256$	1.55556e8	0.579491
$257$	3.55159e8	1.30514	0.652570	−	0.757728i	$-0.273691\pi$
$257$	3.55159e8	1.30514	0.652570	+	0.757728i	$0.273691\pi$
$258$	−1.28355e7	−0.0465312
$259$	−1.37646e8	−0.492281
$260$	2.68292e8	0.946676
$261$	−4.33215e6	−0.0150821
$262$	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
$267$	−4.45091e7	−0.143106
$268$	−2.56502e8	−0.813989
$269$	3.77104e8	1.18121	0.590606	−	0.806960i	$-0.298889\pi$
$269$	3.77104e8	1.18121	0.590606	+	0.806960i	$0.298889\pi$
$270$	2.38997e7	0.0738959
$271$	−2.18482e8	−0.666842	−0.333421	−	0.942778i	$-0.608203\pi$
$271$	−2.18482e8	−0.666842	−0.333421	+	0.942778i	$0.608203\pi$
$272$	−4.05356e8	−1.22136
$273$	1.64165e8	0.488327
$274$	2.76957e7	0.0813365
$275$	0	0
$276$	−2.81984e8	−0.807313
$277$	1.41112e8	0.398920	0.199460	−	0.979906i	$-0.436081\pi$
$277$	1.41112e8	0.398920	0.199460	+	0.979906i	$0.436081\pi$
$278$	1.50396e8	0.419837
$279$	2.07345e8	0.571583
$280$	−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	$0.238924\pi$
$282$	−1.40813e7	−0.0373914
$283$	−1.36059e8	−0.356841	−0.178421	−	0.983954i	$-0.557099\pi$
$283$	−1.36059e8	−0.356841	−0.178421	+	0.983954i	$0.557099\pi$
$284$	−7.84138e7	−0.203132
$285$	7.33406e7	0.187667
$286$	0	0
$287$	1.34593e8	0.336074
$288$	8.09172e7	0.199603
$289$	4.04091e8	0.984773
$290$	7.21569e6	0.0173734
$291$	9.77319e7	0.232494
$292$	−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	$0.934752\pi$
$294$	−7.38609e7	−0.169511
$295$	5.80709e8	1.31699
$296$	−5.90203e7	−0.132276
$297$	0	0
$298$	−1.00313e8	−0.219584
$299$	−3.70779e8	−0.802170
$300$	5.86402e8	1.25392
$301$	−2.77434e8	−0.586378
$302$	5.10181e7	0.106586
$303$	−3.57842e8	−0.738997
$304$	7.62849e7	0.155733
$305$	1.62219e9	3.27379
$306$	−4.99464e7	−0.0996504
$307$	1.21154e8	0.238976	0.119488	−	0.992836i	$-0.461875\pi$
$307$	1.21154e8	0.238976	0.119488	+	0.992836i	$0.461875\pi$
$308$	0	0
$309$	6.03933e6	0.0116449
$310$	−3.45357e8	−0.658420
$311$	−4.86049e8	−0.916259	−0.458130	−	0.888885i	$-0.651480\pi$
$311$	−4.86049e8	−0.916259	−0.458130	+	0.888885i	$0.651480\pi$
$312$	7.03914e7	0.131214
$313$	−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
$315$	5.16584e8	0.931223
$316$	9.39627e8	1.67514
$317$	−6.16413e8	−1.08684	−0.543418	−	0.839462i	$-0.682870\pi$
$317$	−6.16413e8	−1.08684	−0.543418	+	0.839462i	$0.682870\pi$
$318$	−5.47229e7	−0.0954277
$319$	0	0
$320$	7.84765e8	1.33880
$321$	5.29124e8	0.892874
$322$	2.87389e8	0.479706
$323$	−1.53269e8	−0.253074
$324$	−6.49614e7	−0.106108
$325$	7.71058e8	1.24594
$326$	1.86554e8	0.298224
$327$	1.05258e8	0.166470
$328$	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
$333$	7.16191e7	0.106286
$334$	−2.86504e7	−0.0420745
$335$	−1.06131e9	−1.54236
$336$	5.37322e8	0.772764
$337$	−3.37346e8	−0.480143	−0.240071	−	0.970755i	$-0.577171\pi$
$337$	−3.37346e8	−0.480143	−0.240071	+	0.970755i	$0.577171\pi$
$338$	−1.05432e8	−0.148512
$339$	−6.55374e7	−0.0913672
$340$	−1.76433e9	−2.43446
$341$	0	0
$342$	9.39953e6	0.0127062
$343$	−4.42629e8	−0.592257
$344$	−1.18960e8	−0.157560
$345$	−1.16674e9	−1.52971
$346$	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	$0.688395\pi$
$348$	−1.96128e7	−0.0249467
$349$	−1.33122e8	−0.167634	−0.0838170	−	0.996481i	$-0.526711\pi$
$349$	−1.33122e8	−0.167634	−0.0838170	+	0.996481i	$0.526711\pi$
$350$	−5.97644e8	−0.745082
$351$	−8.54175e7	−0.105432
$352$	0	0
$353$	4.51506e8	0.546326	0.273163	−	0.961968i	$-0.411930\pi$
$353$	4.51506e8	0.546326	0.273163	+	0.961968i	$0.411930\pi$
$354$	7.44253e7	0.0891680
$355$	−3.24447e8	−0.384897
$356$	−2.01505e8	−0.236706
$357$	−1.07957e9	−1.25578
$358$	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	$0.624256\pi$
$360$	2.21503e8	0.250219
$361$	−8.65028e8	−0.967731
$362$	1.90169e8	0.210697
$363$	0	0
$364$	7.43219e8	0.807722
$365$	−1.64747e9	−1.77334
$366$	2.07904e8	0.221655
$367$	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
$369$	−7.00307e7	−0.0725598
$370$	−1.19290e8	−0.122433
$371$	−1.18281e9	−1.20256
$372$	9.38709e8	0.945433
$373$	1.24452e9	1.24172	0.620858	−	0.783923i	$-0.286784\pi$
$373$	1.24452e9	1.24172	0.620858	+	0.783923i	$0.286784\pi$
$374$	0	0
$375$	1.35946e9	1.33124
$376$	−1.30506e8	−0.126611
$377$	−2.57888e7	−0.0247877
$378$	6.62067e7	0.0630494
$379$	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
$381$	−6.66617e8	−0.617503
$382$	4.29794e8	0.394493
$383$	−2.71920e8	−0.247312	−0.123656	−	0.992325i	$-0.539462\pi$
$383$	−2.71920e8	−0.247312	−0.123656	+	0.992325i	$0.539462\pi$
$384$	4.84185e8	0.436367
$385$	0	0
$386$	−4.40989e8	−0.390276
$387$	1.44353e8	0.126601
$388$	4.42459e8	0.384558
$389$	1.19397e9	1.02842	0.514211	−	0.857664i	$-0.328085\pi$
$389$	1.19397e9	1.02842	0.514211	+	0.857664i	$0.328085\pi$
$390$	1.42273e8	0.121449
$391$	2.43830e9	2.06285
$392$	−6.84543e8	−0.573984
$393$	−3.48595e8	−0.289700
$394$	−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
$399$	2.03167e8	0.160121
$400$	2.52372e9	1.97166
$401$	3.81045e7	0.0295101	0.0147551	−	0.999891i	$-0.495303\pi$
$401$	3.81045e7	0.0295101	0.0147551	+	0.999891i	$0.495303\pi$
$402$	−1.36020e8	−0.104427
$403$	1.23431e9	0.939409
$404$	−1.62005e9	−1.22234
$405$	−2.68786e8	−0.201055
$406$	1.99888e7	0.0148233
$407$	0	0
$408$	−4.62904e8	−0.337427
$409$	1.10488e9	0.798515	0.399258	−	0.916839i	$-0.369268\pi$
$409$	1.10488e9	0.798515	0.399258	+	0.916839i	$0.369268\pi$
$410$	1.16644e8	0.0835833
$411$	−3.11477e8	−0.221299
$412$	2.73417e7	0.0192613
$413$	1.60867e9	1.12368
$414$	−1.49533e8	−0.103571
$415$	2.05602e9	1.41208
$416$	4.81692e8	0.328052
$417$	−1.69142e9	−1.14229
$418$	0	0
$419$	−4.54418e8	−0.301791	−0.150896	−	0.988550i	$-0.548216\pi$
$419$	−4.54418e8	−0.301791	−0.150896	+	0.988550i	$0.548216\pi$
$420$	2.33871e9	1.54030
$421$	−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
$423$	1.58364e8	0.101734
$424$	−5.07173e8	−0.323129
$425$	−5.07059e9	−3.20403
$426$	−4.15820e7	−0.0260599
$427$	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	$0.442820\pi$
$432$	−2.79577e8	−0.166843
$433$	−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
$435$	−8.11506e7	−0.0472693
$436$	4.76529e8	0.275351
$437$	−4.58869e8	−0.263029
$438$	−2.11144e8	−0.120066
$439$	1.35965e9	0.767011	0.383506	−	0.923539i	$-0.374717\pi$
$439$	1.35965e9	0.767011	0.383506	+	0.923539i	$0.374717\pi$
$440$	0	0
$441$	8.30669e8	0.461204
$442$	−2.97326e8	−0.163778
$443$	1.86938e9	1.02161	0.510805	−	0.859696i	$-0.329347\pi$
$443$	1.86938e9	1.02161	0.510805	+	0.859696i	$0.329347\pi$
$444$	3.24239e8	0.175802
$445$	−8.33752e8	−0.448515
$446$	−4.30914e7	−0.0229995
$447$	1.12816e9	0.597443
$448$	2.17395e9	1.14229
$449$	−1.94847e9	−1.01586	−0.507928	−	0.861400i	$-0.669588\pi$
$449$	−1.94847e9	−1.01586	−0.507928	+	0.861400i	$0.669588\pi$
$450$	3.10963e8	0.160866
$451$	0	0
$452$	−2.96706e8	−0.151127
$453$	−5.73771e8	−0.289998
$454$	2.07585e8	0.104112
$455$	3.07517e9	1.53048
$456$	8.71150e7	0.0430245
$457$	2.15115e9	1.05430	0.527150	−	0.849772i	$-0.323261\pi$
$457$	2.15115e9	1.05430	0.527150	+	0.849772i	$0.323261\pi$
$458$	−2.80865e8	−0.136605
$459$	5.61718e8	0.271127
$460$	−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
$465$	3.88403e9	1.79142
$466$	6.47531e7	0.0296422
$467$	−2.10774e8	−0.0957653	−0.0478827	−	0.998853i	$-0.515247\pi$
$467$	−2.10774e8	−0.0957653	−0.0478827	+	0.998853i	$0.515247\pi$
$468$	−3.86708e8	−0.174391
$469$	−2.94003e9	−1.31597
$470$	−2.63774e8	−0.117190
$471$	−1.42429e8	−0.0628095
$472$	6.89774e8	0.301933
$473$	0	0
$474$	4.98275e8	0.214904
$475$	9.54246e8	0.408538
$476$	−4.88751e9	−2.07713
$477$	6.15436e8	0.259638
$478$	−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	$0.501010\pi$
$480$	1.51576e9	0.625583
$481$	4.26341e8	0.174682
$482$	5.81392e8	0.236485
$483$	−3.23210e9	−1.30518
$484$	0	0
$485$	1.83073e9	0.728666
$486$	−3.44484e7	−0.0136126
$487$	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
$489$	−2.09806e9	−0.811403
$490$	−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	$0.391335\pi$
$492$	−3.17048e8	−0.120018
$493$	1.69591e8	0.0637439
$494$	5.59544e7	0.0208829
$495$	0	0
$496$	4.03995e9	1.48659
$497$	−8.98779e8	−0.328402
$498$	2.63505e8	0.0956064
$499$	−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
$501$	3.22215e8	0.114476
$502$	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	$0.675607\pi$
$504$	6.13605e8	0.213492
$505$	−6.70317e9	−2.31612
$506$	0	0
$507$	1.18573e9	0.404071
$508$	−3.01796e9	−1.02139
$509$	−2.73347e9	−0.918760	−0.459380	−	0.888240i	$-0.651928\pi$
$509$	−2.73347e9	−0.918760	−0.459380	+	0.888240i	$0.651928\pi$
$510$	−9.35605e8	−0.312318
$511$	−4.56380e9	−1.51305
$512$	2.66885e9	0.878778
$513$	−1.05711e8	−0.0345708
$514$	8.52655e8	0.276951
$515$	1.13130e8	0.0364966
$516$	6.53527e8	0.209406
$517$	0	0
$518$	−3.30455e8	−0.104462
$519$	−1.78419e9	−0.560215
$520$	1.31858e9	0.411241
$521$	−3.96803e9	−1.22926	−0.614629	−	0.788816i	$-0.710694\pi$
$521$	−3.96803e9	−1.22926	−0.614629	+	0.788816i	$0.710694\pi$
$522$	−1.04005e7	−0.00320042
$523$	−1.14104e9	−0.348773	−0.174387	−	0.984677i	$-0.555794\pi$
$523$	−1.14104e9	−0.348773	−0.174387	+	0.984677i	$0.555794\pi$
$524$	−1.57819e9	−0.479180
$525$	6.72135e9	2.02721
$526$	−6.73229e8	−0.201703
$527$	−8.11696e9	−2.41577
$528$	0	0
$529$	3.89512e9	1.14400
$530$	−1.02508e9	−0.299083
$531$	−8.37017e8	−0.242607
$532$	9.19793e8	0.264850
$533$	−4.16885e8	−0.119254
$534$	−1.06856e8	−0.0303671
$535$	9.91165e9	2.79839
$536$	−1.26064e9	−0.353601
$537$	−1.69377e9	−0.472002
$538$	9.05338e8	0.250653
$539$	0	0
$540$	−1.21687e9	−0.332557
$541$	4.46759e9	1.21306	0.606531	−	0.795060i	$-0.292561\pi$
$541$	4.46759e9	1.21306	0.606531	+	0.795060i	$0.292561\pi$
$542$	−5.24524e8	−0.141504
$543$	−2.13872e9	−0.573263
$544$	−3.16767e9	−0.843614
$545$	1.97170e9	0.521740
$546$	3.94122e8	0.103623
$547$	−4.19205e9	−1.09514	−0.547572	−	0.836759i	$-0.684448\pi$
$547$	−4.19205e9	−1.09514	−0.547572	+	0.836759i	$0.684448\pi$
$548$	−1.41014e9	−0.366042
$549$	−2.33817e9	−0.603077
$550$	0	0
$551$	−3.19157e7	−0.00812782
$552$	−1.38587e9	−0.350701
$553$	1.07700e10	2.70819
$554$	3.38778e8	0.0846507
$555$	1.34158e9	0.333113
$556$	−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	$0.771865\pi$
$558$	4.97788e8	0.121290
$559$	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	$0.630478\pi$
$564$	7.16958e8	0.168274
$565$	−1.22766e9	−0.286357
$566$	−3.26646e8	−0.0757216
$567$	−7.44588e8	−0.171544
$568$	−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	$0.360336\pi$
$570$	1.76074e8	0.0398229
$571$	−3.95974e9	−0.890102	−0.445051	−	0.895505i	$-0.646814\pi$
$571$	−3.95974e9	−0.890102	−0.445051	+	0.895505i	$0.646814\pi$
$572$	0	0
$573$	−4.83364e9	−1.07333
$574$	3.23126e8	0.0713149
$575$	−1.51807e10	−3.33008
$576$	−1.13114e9	−0.246625
$577$	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
$579$	4.95954e9	1.06186
$580$	−3.67391e8	−0.0781862
$581$	5.69556e9	1.20481
$582$	2.34632e8	0.0493351
$583$	0	0
$584$	−1.95689e9	−0.406556
$585$	−1.60006e9	−0.330438
$586$	−2.02408e9	−0.415514
$587$	−2.24975e9	−0.459093	−0.229546	−	0.973298i	$-0.573724\pi$
$587$	−2.24975e9	−0.459093	−0.229546	+	0.973298i	$0.573724\pi$
$588$	3.76067e9	0.762859
$589$	1.52755e9	0.308029
$590$	1.39415e9	0.279465
$591$	6.63570e8	0.132230
$592$	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
$597$	2.22278e9	0.427550
$598$	−8.90155e8	−0.170220
$599$	4.67049e9	0.887909	0.443955	−	0.896049i	$-0.353575\pi$
$599$	4.67049e9	0.887909	0.443955	+	0.896049i	$0.353575\pi$
$600$	2.88201e9	0.544711
$601$	−5.37104e9	−1.00925	−0.504624	−	0.863339i	$-0.668369\pi$
$601$	−5.37104e9	−1.00925	−0.504624	+	0.863339i	$0.668369\pi$
$602$	−6.66055e8	−0.124429
$603$	1.52974e9	0.284123
$604$	−2.59762e9	−0.479673
$605$	0	0
$606$	−8.59096e8	−0.156815
$607$	8.70240e8	0.157935	0.0789675	−	0.996877i	$-0.474838\pi$
$607$	8.70240e8	0.157935	0.0789675	+	0.996877i	$0.474838\pi$
$608$	5.96132e8	0.107567
$609$	−2.24802e8	−0.0403311
$610$	3.89449e9	0.694698
$611$	9.42726e8	0.167202
$612$	2.54305e9	0.448461
$613$	−9.68587e9	−1.69835	−0.849175	−	0.528112i	$-0.822900\pi$
$613$	−9.68587e9	−1.69835	−0.849175	+	0.528112i	$0.822900\pi$
$614$	2.90864e8	0.0507107
$615$	−1.31183e9	−0.227412
$616$	0	0
$617$	−9.06146e9	−1.55310	−0.776551	−	0.630054i	$-0.783033\pi$
$617$	−9.06146e9	−1.55310	−0.776551	+	0.630054i	$0.783033\pi$
$618$	1.44990e7	0.00247104
$619$	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
$621$	1.68171e9	0.281793
$622$	−1.16689e9	−0.194430
$623$	−2.30965e9	−0.382681
$624$	−1.66429e9	−0.274210
$625$	1.15846e10	1.89803
$626$	−2.07434e8	−0.0337963
$627$	0	0
$628$	−6.44815e8	−0.103891
$629$	−2.80368e9	−0.449211
$630$	1.24020e9	0.197605
$631$	−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
$633$	−2.01314e9	−0.315472
$634$	−1.47986e9	−0.230626
$635$	−1.24872e10	−1.93534
$636$	2.78625e9	0.429457
$637$	4.94489e9	0.757999
$638$	0	0
$639$	4.67649e8	0.0709033
$640$	9.06984e9	1.36763
$641$	−3.15511e9	−0.473164	−0.236582	−	0.971612i	$-0.576027\pi$
$641$	−3.15511e9	−0.473164	−0.236582	+	0.971612i	$0.576027\pi$
$642$	1.27030e9	0.189468
$643$	−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
$645$	2.70405e9	0.396786
$646$	−3.67964e8	−0.0537021
$647$	−5.60213e9	−0.813183	−0.406592	−	0.913610i	$-0.633283\pi$
$647$	−5.60213e9	−0.813183	−0.406592	+	0.913610i	$0.633283\pi$
$648$	−3.19268e8	−0.0460938
$649$	0	0
$650$	1.85113e9	0.264387
$651$	1.07595e10	1.52847
$652$	−9.49848e9	−1.34211
$653$	−1.94634e9	−0.273541	−0.136771	−	0.990603i	$-0.543672\pi$
$653$	−1.94634e9	−0.273541	−0.136771	+	0.990603i	$0.543672\pi$
$654$	2.52699e8	0.0353249
$655$	−6.52995e9	−0.907957
$656$	−1.36449e9	−0.188715
$657$	2.37461e9	0.326674
$658$	−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
$663$	3.34384e9	0.445604
$664$	2.44217e9	0.323734
$665$	3.80576e9	0.501841
$666$	1.71941e8	0.0225538
$667$	5.07734e8	0.0662515
$668$	1.45875e9	0.189350
$669$	4.84623e8	0.0625767
$670$	−2.54796e9	−0.327288
$671$	0	0
$672$	4.19893e9	0.533760
$673$	−8.91893e9	−1.12787	−0.563936	−	0.825818i	$-0.690714\pi$
$673$	−8.91893e9	−1.12787	−0.563936	+	0.825818i	$0.690714\pi$
$674$	−8.09888e8	−0.101886
$675$	−3.49722e9	−0.437683
$676$	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	$0.593028\pi$
$678$	−1.57340e8	−0.0193881
$679$	5.07147e9	0.621712
$680$	−8.67120e9	−1.05754
$681$	−2.33459e9	−0.283267
$682$	0	0
$683$	−1.30727e10	−1.56998	−0.784989	−	0.619510i	$-0.787331\pi$
$683$	−1.30727e10	−1.56998	−0.784989	+	0.619510i	$0.787331\pi$
$684$	−4.78582e8	−0.0571821
$685$	−5.83464e9	−0.693581
$686$	−1.06265e9	−0.125677
$687$	3.15872e9	0.371674
$688$	2.81261e9	0.329268
$689$	3.66363e9	0.426721
$690$	−2.80108e9	−0.324604
$691$	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
$696$	−9.63918e7	−0.0108370
$697$	2.74150e9	0.306671
$698$	−3.19596e8	−0.0355719
$699$	−7.28240e8	−0.0806500
$700$	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	$0.551084\pi$
$702$	−2.05068e8	−0.0223726
$703$	5.27631e8	0.0572778
$704$	0	0
$705$	2.96651e9	0.318848
$706$	1.08396e9	0.115930
$707$	−1.85690e10	−1.97615
$708$	−3.78940e9	−0.401286
$709$	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
$711$	−5.60380e9	−0.584708
$712$	−9.90341e8	−0.102826
$713$	−2.43011e10	−2.51081
$714$	−2.59180e9	−0.266476
$715$	0	0
$716$	−7.66815e9	−0.780719
$717$	5.49767e9	0.557008
$718$	−1.60173e9	−0.161493
$719$	−1.82027e9	−0.182635	−0.0913176	−	0.995822i	$-0.529108\pi$
$719$	−1.82027e9	−0.182635	−0.0913176	+	0.995822i	$0.529108\pi$
$720$	−5.23708e9	−0.522908
$721$	3.13391e8	0.0311396
$722$	−2.07673e9	−0.205352
$723$	−6.53857e9	−0.643426
$724$	−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
$729$	3.87420e8	0.0370370
$730$	−3.95519e9	−0.376303
$731$	−5.65101e9	−0.535076
$732$	−1.05855e10	−0.997524
$733$	1.37367e10	1.28830	0.644152	−	0.764898i	$-0.277210\pi$
$733$	1.37367e10	1.28830	0.644152	+	0.764898i	$0.277210\pi$
$734$	9.63444e7	0.00899270
$735$	1.55603e10	1.44548
$736$	−9.48360e9	−0.876801
$737$	0	0
$738$	−1.68127e8	−0.0153972
$739$	1.13883e10	1.03802	0.519009	−	0.854769i	$-0.326301\pi$
$739$	1.13883e10	1.03802	0.519009	+	0.854769i	$0.326301\pi$
$740$	6.07371e9	0.550989
$741$	−6.29286e8	−0.0568178
$742$	−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	$0.915854\pi$
$744$	4.61350e9	0.410701
$745$	2.11330e10	1.87247
$746$	2.98781e9	0.263492
$747$	−2.96349e9	−0.260124
$748$	0	0
$749$	2.74571e10	2.38764
$750$	3.26375e9	0.282489
$751$	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
$753$	−5.28269e9	−0.450892
$754$	−6.19129e7	−0.00525995
$755$	−1.07480e10	−0.908892
$756$	−3.37095e9	−0.283744
$757$	−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	$0.413895\pi$
$762$	−1.60039e9	−0.131034
$763$	5.46198e9	0.445158
$764$	−2.18832e10	−1.77535
$765$	1.05222e10	0.849750
$766$	−6.52816e8	−0.0524796
$767$	−4.98267e9	−0.398730
$768$	−4.20001e9	−0.334570
$769$	−1.24166e10	−0.984603	−0.492301	−	0.870425i	$-0.663844\pi$
$769$	−1.24166e10	−0.984603	−0.492301	+	0.870425i	$0.663844\pi$
$770$	0	0
$771$	−9.58930e9	−0.753523
$772$	2.24532e10	1.75638
$773$	1.43991e10	1.12126	0.560631	−	0.828066i	$-0.310559\pi$
$773$	1.43991e10	1.12126	0.560631	+	0.828066i	$0.310559\pi$
$774$	3.46559e8	0.0268648
$775$	5.05357e10	3.89980
$776$	2.17457e9	0.167054
$777$	3.71643e9	0.284219
$778$	2.86645e9	0.218231
$779$	−5.15929e8	−0.0391029
$780$	−7.24389e9	−0.546564
$781$	0	0
$782$	5.85378e9	0.437737
$783$	1.16968e8	0.00870765
$784$	1.61849e10	1.19951
$785$	−2.66801e9	−0.196853
$786$	−8.36896e8	−0.0614742
$787$	1.42293e10	1.04057	0.520287	−	0.853991i	$-0.325825\pi$
$787$	1.42293e10	1.04057	0.520287	+	0.853991i	$0.325825\pi$
$788$	3.00416e9	0.218717
$789$	7.57141e9	0.548791
$790$	9.33378e9	0.673539
$791$	−3.40084e9	−0.244325
$792$	0	0
$793$	−1.39189e10	−0.991169
$794$	4.53285e9	0.321365
$795$	1.15285e10	0.813742
$796$	1.00631e10	0.707193
$797$	−1.15055e10	−0.805010	−0.402505	−	0.915418i	$-0.631860\pi$
$797$	−1.15055e10	−0.805010	−0.402505	+	0.915418i	$0.631860\pi$
$798$	4.87757e8	0.0339777
$799$	−6.19950e9	−0.429975
$800$	1.97217e10	1.36185
$801$	1.20174e9	0.0826225
$802$	9.14800e7	0.00626204
$803$	0	0
$804$	6.92555e9	0.469957
$805$	−6.05443e10	−4.09060
$806$	2.96328e9	0.199342
$807$	−1.01818e10	−0.681973
$808$	−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	$0.392739\pi$
$810$	−6.45293e8	−0.0426638
$811$	9.67643e9	0.637004	0.318502	−	0.947922i	$-0.396820\pi$
$811$	9.67643e9	0.637004	0.318502	+	0.947922i	$0.396820\pi$
$812$	−1.01774e9	−0.0667100
$813$	5.89901e9	0.385001
$814$	0	0
$815$	−3.93012e10	−2.54305
$816$	1.09446e10	0.705155
$817$	1.06348e9	0.0682262
$818$	2.65256e9	0.169445
$819$	−4.43245e9	−0.281936
$820$	−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	$0.683146\pi$
$822$	−7.47784e8	−0.0469596
$823$	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	$0.387314\pi$
$828$	7.61356e9	0.466103
$829$	−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
$831$	−3.81003e9	−0.230316
$832$	−6.73354e9	−0.405333
$833$	−3.25183e10	−1.94926
$834$	−4.06070e9	−0.242393
$835$	6.03578e9	0.358782
$836$	0	0
$837$	−5.59833e9	−0.330004
$838$	−1.09095e9	−0.0640400
$839$	−7.39055e8	−0.0432026	−0.0216013	−	0.999767i	$-0.506876\pi$
$839$	−7.39055e8	−0.0432026	−0.0216013	+	0.999767i	$0.506876\pi$
$840$	1.14942e10	0.669113
$841$	−1.72146e10	−0.997953
$842$	−2.51305e9	−0.145080
$843$	−1.46875e10	−0.844408
$844$	−9.11401e9	−0.521809
$845$	2.22113e10	1.26641
$846$	3.80196e8	0.0215879
$847$	0	0
$848$	1.19913e10	0.675274
$849$	3.67359e9	0.206022
$850$	−1.21733e10	−0.679895
$851$	−8.39385e9	−0.466883
$852$	2.11717e9	0.117278
$853$	2.19811e10	1.21263	0.606314	−	0.795225i	$-0.292648\pi$
$853$	2.19811e10	1.21263	0.606314	+	0.795225i	$0.292648\pi$
$854$	1.07885e10	0.592730
$855$	−1.98020e9	−0.108350
$856$	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
$861$	−3.63401e9	−0.194033
$862$	1.42596e9	0.0758285
$863$	2.47775e10	1.31226	0.656130	−	0.754648i	$-0.272192\pi$
$863$	2.47775e10	1.31226	0.656130	+	0.754648i	$0.272192\pi$
$864$	−2.18476e9	−0.115241
$865$	−3.34217e10	−1.75579
$866$	−4.57237e9	−0.239237
$867$	−1.09104e10	−0.568559
$868$	4.87111e10	2.52819
$869$	0	0
$870$	−1.94824e8	−0.0100305
$871$	9.10638e9	0.466963
$872$	2.34201e9	0.119614
$873$	−2.63876e9	−0.134230
$874$	−1.10164e9	−0.0558147
$875$	7.05446e10	3.55988
$876$	1.07505e10	0.540338
$877$	−1.37586e10	−0.688771	−0.344386	−	0.938828i	$-0.611913\pi$
$877$	−1.37586e10	−0.688771	−0.344386	+	0.938828i	$0.611913\pi$
$878$	3.26420e9	0.162760
$879$	2.27636e10	1.13053
$880$	0	0
$881$	3.21022e10	1.58168	0.790841	−	0.612022i	$-0.209644\pi$
$881$	3.21022e10	1.58168	0.790841	+	0.612022i	$0.209644\pi$
$882$	1.99424e9	0.0978674
$883$	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
$885$	−1.56792e10	−0.760363
$886$	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	$0.404395\pi$
$888$	1.59355e9	0.0763695
$889$	−3.45918e10	−1.65127
$890$	−2.00164e9	−0.0951747
$891$	0	0
$892$	2.19402e9	0.103506
$893$	1.16670e9	0.0548250
$894$	2.70846e9	0.126777
$895$	−3.17280e10	−1.47932
$896$	2.51252e10	1.16689
$897$	1.00110e10	0.463133
$898$	−4.67782e9	−0.215564
$899$	−1.69022e9	−0.0775861
$900$	−1.58329e10	−0.723953
$901$	−2.40925e10	−1.09735
$902$	0	0
$903$	7.49073e9	0.338545
$904$	−1.45823e9	−0.0656502
$905$	−4.00628e10	−1.79668
$906$	−1.37749e9	−0.0615375
$907$	−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
$909$	9.66174e9	0.426660
$910$	7.38276e9	0.324768
$911$	2.52053e10	1.10453	0.552266	−	0.833668i	$-0.313763\pi$
$911$	2.52053e10	1.10453	0.552266	+	0.833668i	$0.313763\pi$
$912$	−2.05969e9	−0.0899126
$913$	0	0
$914$	5.16441e9	0.223722
$915$	−4.37990e10	−1.89012
$916$	1.43004e10	0.614771
$917$	−1.80892e10	−0.774687
$918$	1.34855e9	0.0575332
$919$	−4.38722e9	−0.186460	−0.0932299	−	0.995645i	$-0.529719\pi$
$919$	−4.38722e9	−0.186460	−0.0932299	+	0.995645i	$0.529719\pi$
$920$	−2.59605e10	−1.09914
$921$	−3.27117e9	−0.137973
$922$	5.61728e9	0.236031
$923$	2.78386e9	0.116531
$924$	0	0
$925$	1.74555e10	0.725165
$926$	−4.51177e9	−0.186728
$927$	−1.63062e8	−0.00672316
$928$	−6.59613e8	−0.0270939
$929$	−4.04223e10	−1.65412	−0.827058	−	0.562117i	$-0.809987\pi$
$929$	−4.04223e10	−1.65412	−0.827058	+	0.562117i	$0.809987\pi$
$930$	9.32465e9	0.380139
$931$	6.11969e9	0.248545
$932$	−3.29694e9	−0.133400
$933$	1.31233e10	0.529003
$934$	−5.06019e8	−0.0203214
$935$	0	0
$936$	−1.90057e9	−0.0757562
$937$	4.80084e9	0.190647	0.0953233	−	0.995446i	$-0.469612\pi$
$937$	4.80084e9	0.190647	0.0953233	+	0.995446i	$0.469612\pi$
$938$	−7.05831e9	−0.279249
$939$	2.33288e9	0.0919524
$940$	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	$0.445362\pi$
$942$	−3.41939e8	−0.0133282
$943$	8.20769e9	0.318735
$944$	−1.63086e10	−0.630978
$945$	−1.39478e10	−0.537642
$946$	0	0
$947$	1.14448e10	0.437909	0.218955	−	0.975735i	$-0.429735\pi$
$947$	1.14448e10	0.437909	0.218955	+	0.975735i	$0.429735\pi$
$948$	−2.53699e10	−0.967142
$949$	1.41358e10	0.536895
$950$	2.29092e9	0.0866917
$951$	1.66431e10	0.627485
$952$	−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	$0.546847\pi$
$954$	1.47752e9	0.0550952
$955$	−9.05447e10	−3.36396
$956$	2.48894e10	0.921324
$957$	0	0
$958$	−3.66573e7	−0.00134704
$959$	−1.61630e10	−0.591777
$960$	−2.11887e10	−0.772955
$961$	5.33846e10	1.94037
$962$	1.02354e9	0.0370676
$963$	−1.42864e10	−0.515501
$964$	−2.96019e10	−1.06426
$965$	9.29030e10	3.32801
$966$	−7.75951e9	−0.276958
$967$	−2.69859e10	−0.959718	−0.479859	−	0.877346i	$-0.659312\pi$
$967$	−2.69859e10	−0.959718	−0.479859	+	0.877346i	$0.659312\pi$
$968$	0	0
$969$	4.13827e9	0.146112
$970$	4.39516e9	0.154623
$971$	5.49167e10	1.92503	0.962514	−	0.271232i	$-0.0874311\pi$
$971$	5.49167e10	1.92503	0.962514	+	0.271232i	$0.0874311\pi$
$972$	1.75396e9	0.0612614
$973$	−8.77705e10	−3.05459
$974$	8.04313e9	0.278913
$975$	−2.08186e10	−0.719341
$976$	−4.55573e10	−1.56850
$977$	2.25460e10	0.773460	0.386730	−	0.922193i	$-0.373604\pi$
$977$	2.25460e10	0.773460	0.386730	+	0.922193i	$0.373604\pi$
$978$	−5.03695e9	−0.172179
$979$	0	0
$980$	7.04455e10	2.39090
$981$	−2.84195e9	−0.0961115
$982$	4.21635e9	0.142085
$983$	3.93701e10	1.32199	0.660996	−	0.750389i	$-0.270134\pi$
$983$	3.93701e10	1.32199	0.660996	+	0.750389i	$0.270134\pi$
$984$	−1.55821e9	−0.0521365
$985$	1.24301e10	0.414427
$986$	4.07148e8	0.0135264
$987$	8.21778e9	0.272047
$988$	−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
$993$	−8.74519e9	−0.283431
$994$	−2.15776e9	−0.0696868
$995$	4.16376e10	1.34000
$996$	−1.34165e10	−0.430261
$997$	−1.96960e10	−0.629426	−0.314713	−	0.949187i	$-0.601908\pi$
$997$	−1.96960e10	−0.629426	−0.314713	+	0.949187i	$0.601908\pi$
$998$	−8.23306e9	−0.262183
$999$	−1.93372e9	−0.0613640

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.a.d.1.2	✓	3	11.10	odd	2
99.8.a.e.1.2		3	33.32	even	2
363.8.a.e.1.2		3	1.1	even	1	trivial
528.8.a.o.1.1		3	44.43	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:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	363.a (trivial)

\(f(q)\)	\(=\)	\(q+2.40077 q^{2} -27.0000 q^{3} -122.236 q^{4} -505.769 q^{5} -64.8207 q^{6} -1401.07 q^{7} -600.759 q^{8} +729.000 q^{9} -1214.23 q^{10} +3300.38 q^{12} +4339.66 q^{13} -3363.65 q^{14} +13655.8 q^{15} +14204.0 q^{16} -28538.2 q^{17} +1750.16 q^{18} +5370.67 q^{19} +61823.3 q^{20} +37829.0 q^{21} -85439.7 q^{23} +16220.5 q^{24} +177677. q^{25} +10418.5 q^{26} -19683.0 q^{27} +171262. q^{28} -5942.59 q^{29} +32784.3 q^{30} +284424. q^{31} +110998. q^{32} -68513.6 q^{34} +708619. q^{35} -89110.3 q^{36} +98242.9 q^{37} +12893.7 q^{38} -117171. q^{39} +303845. q^{40} -96064.1 q^{41} +90818.6 q^{42} +198016. q^{43} -368706. q^{45} -205121. q^{46} +217235. q^{47} -383507. q^{48} +1.13946e6 q^{49} +426561. q^{50} +770532. q^{51} -530464. q^{52} +844220. q^{53} -47254.3 q^{54} +841708. q^{56} -145008. q^{57} -14266.8 q^{58} -1.14817e6 q^{59} -1.66923e6 q^{60} -3.20736e6 q^{61} +682837. q^{62} -1.02138e6 q^{63} -1.55163e6 q^{64} -2.19486e6 q^{65} +2.09841e6 q^{67} +3.48841e6 q^{68} +2.30687e6 q^{69} +1.70123e6 q^{70} +641493. q^{71} -437953. q^{72} +3.25736e6 q^{73} +235858. q^{74} -4.79728e6 q^{75} -656491. q^{76} -281300. q^{78} -7.68697e6 q^{79} -7.18392e6 q^{80} +531441. q^{81} -230627. q^{82} -4.06514e6 q^{83} -4.62408e6 q^{84} +1.44337e7 q^{85} +475389. q^{86} +160450. q^{87} +1.64848e6 q^{89} -885176. q^{90} -6.08018e6 q^{91} +1.04438e7 q^{92} -7.67946e6 q^{93} +521531. q^{94} -2.71632e6 q^{95} -2.99693e6 q^{96} -3.61970e6 q^{97} +2.73559e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{2} - 81 q^{3} - 15 q^{4} - 444 q^{5} + 243 q^{6} - 1614 q^{7} - 3153 q^{8} + 2187 q^{9} - 2880 q^{10} + 405 q^{12} - 20772 q^{13} + 36258 q^{14} + 11988 q^{15} + 12225 q^{16} + 14538 q^{17} - 6561 q^{18}+ \cdots - 24377397 q^{98}+O(q^{100}) \) 3 * q - 9 * q^2 - 81 * q^3 - 15 * q^4 - 444 * q^5 + 243 * q^6 - 1614 * q^7 - 3153 * q^8 + 2187 * q^9 - 2880 * q^10 + 405 * q^12 - 20772 * q^13 + 36258 * q^14 + 11988 * q^15 + 12225 * q^16 + 14538 * q^17 - 6561 * q^18 - 24492 * q^19 + 80112 * q^20 + 43578 * q^21 + 35094 * q^23 + 85131 * q^24 + 29121 * q^25 + 203832 * q^26 - 59049 * q^27 - 278034 * q^28 + 179862 * q^29 + 77760 * q^30 + 288888 * q^31 + 519567 * q^32 - 491586 * q^34 + 532872 * q^35 - 10935 * q^36 + 107562 * q^37 - 686328 * q^38 + 560844 * q^39 + 237360 * q^40 + 135198 * q^41 - 978966 * q^42 - 193536 * q^43 - 323676 * q^45 - 16422 * q^46 - 591486 * q^47 - 330075 * q^48 + 4461159 * q^49 + 1192245 * q^50 - 392526 * q^51 - 2449992 * q^52 + 79044 * q^53 + 177147 * q^54 + 752658 * q^56 + 661284 * q^57 + 2289930 * q^58 + 2532768 * q^59 - 2163024 * q^60 - 6678792 * q^61 + 2660808 * q^62 - 1176606 * q^63 - 3966303 * q^64 - 3191832 * q^65 + 7150356 * q^67 + 7821954 * q^68 - 947538 * q^69 + 4029120 * q^70 + 1390398 * q^71 - 2298537 * q^72 + 6429114 * q^73 - 2507478 * q^74 - 786267 * q^75 + 7654728 * q^76 - 5503464 * q^78 - 6873186 * q^79 - 7556016 * q^80 + 1594323 * q^81 - 1774590 * q^82 - 6505596 * q^83 + 7506918 * q^84 + 16546032 * q^85 - 6519468 * q^86 - 4856274 * q^87 - 8842962 * q^89 - 2099520 * q^90 + 3066648 * q^91 + 6921990 * q^92 - 7799976 * q^93 + 24038238 * q^94 + 190968 * q^95 - 14028309 * q^96 - 1764774 * q^97 - 24377397 * q^98

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