LMFDB - Embedded newform 15.18.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [15,18,Mod(1,15)] 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("15.1"); S:= CuspForms(chi, 18); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(15, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 18, names="a")

Level:	$ N $	$=$	$ 15 = 3 \cdot 5 $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	15.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-356] 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:	$27.4833131017$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{849}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 212 $ x^2 - x - 212
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$15.0688$ of defining polynomial
Character	$\chi$	$=$	15.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-702.477 q^{2} +6561.00 q^{3} +362402. q^{4} +390625. q^{5} -4.60895e6 q^{6} -2.67481e6 q^{7} -1.62504e8 q^{8} +4.30467e7 q^{9} -2.74405e8 q^{10} -9.86334e8 q^{11} +2.37772e9 q^{12} +2.81488e9 q^{13} +1.87899e9 q^{14} +2.56289e9 q^{15} +6.66544e10 q^{16} +1.28501e10 q^{17} -3.02393e10 q^{18} -1.22560e11 q^{19} +1.41563e11 q^{20} -1.75494e10 q^{21} +6.92877e11 q^{22} +1.14646e11 q^{23} -1.06619e12 q^{24} +1.52588e11 q^{25} -1.97739e12 q^{26} +2.82430e11 q^{27} -9.69355e11 q^{28} -4.90027e12 q^{29} -1.80037e12 q^{30} +7.64406e12 q^{31} -2.55235e13 q^{32} -6.47134e12 q^{33} -9.02688e12 q^{34} -1.04485e12 q^{35} +1.56002e13 q^{36} +3.18983e12 q^{37} +8.60956e13 q^{38} +1.84684e13 q^{39} -6.34781e13 q^{40} -5.53137e12 q^{41} +1.23281e13 q^{42} +2.42435e13 q^{43} -3.57449e14 q^{44} +1.68151e13 q^{45} -8.05365e13 q^{46} -5.82161e13 q^{47} +4.37320e14 q^{48} -2.25476e14 q^{49} -1.07189e14 q^{50} +8.43094e13 q^{51} +1.02012e15 q^{52} -6.45746e14 q^{53} -1.98400e14 q^{54} -3.85287e14 q^{55} +4.34667e14 q^{56} -8.04117e14 q^{57} +3.44233e15 q^{58} -1.04433e12 q^{59} +9.28796e14 q^{60} -1.56088e15 q^{61} -5.36978e15 q^{62} -1.15142e14 q^{63} +9.19314e15 q^{64} +1.09956e15 q^{65} +4.54597e15 q^{66} +1.31151e15 q^{67} +4.65689e15 q^{68} +7.52195e14 q^{69} +7.33981e14 q^{70} -7.09230e15 q^{71} -6.99526e15 q^{72} -1.17619e16 q^{73} -2.24078e15 q^{74} +1.00113e15 q^{75} -4.44160e16 q^{76} +2.63826e15 q^{77} -1.29737e16 q^{78} -7.14674e15 q^{79} +2.60369e16 q^{80} +1.85302e15 q^{81} +3.88566e15 q^{82} +9.04707e15 q^{83} -6.35994e15 q^{84} +5.01956e15 q^{85} -1.70305e16 q^{86} -3.21507e16 q^{87} +1.60283e17 q^{88} -1.16786e16 q^{89} -1.18122e16 q^{90} -7.52927e15 q^{91} +4.15481e16 q^{92} +5.01527e16 q^{93} +4.08955e16 q^{94} -4.78750e16 q^{95} -1.67460e17 q^{96} -8.32538e16 q^{97} +1.58392e17 q^{98} -4.24585e16 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 356 q^{2} + 13122 q^{3} + 351376 q^{4} + 781250 q^{5} - 2335716 q^{6} - 20754552 q^{7} - 211737408 q^{8} + 86093442 q^{9} - 139062500 q^{10} - 1131629912 q^{11} + 2305377936 q^{12} - 446672524 q^{13}+ \cdots - 48\!\cdots\!52 q^{99}+O(q^{100}) $ 2 * q - 356 * q^2 + 13122 * q^3 + 351376 * q^4 + 781250 * q^5 - 2335716 * q^6 - 20754552 * q^7 - 211737408 * q^8 + 86093442 * q^9 - 139062500 * q^10 - 1131629912 * q^11 + 2305377936 * q^12 - 446672524 * q^13 - 4385222016 * q^14 + 5125781250 * q^15 + 51041317120 * q^16 + 20662021036 * q^17 - 15324632676 * q^18 - 194376127216 * q^19 + 137256250000 * q^20 - 136170615672 * q^21 + 642535396112 * q^22 + 281815042584 * q^23 - 1389209133888 * q^24 + 305175781250 * q^25 - 3107442504536 * q^26 + 564859072962 * q^27 - 770012242944 * q^28 - 4737062625892 * q^29 - 912389062500 * q^30 + 2256407074872 * q^31 - 24479953196032 * q^32 - 7424623852632 * q^33 - 6320225301928 * q^34 - 8107246875000 * q^35 + 15125584638096 * q^36 - 33686114979452 * q^37 + 61213036677904 * q^38 - 2930618429964 * q^39 - 82709925000000 * q^40 - 13051545859228 * q^41 - 28771441646976 * q^42 - 33149929851400 * q^43 - 355847272731712 * q^44 + 33630250781250 * q^45 - 22616426690688 * q^46 + 84299585768440 * q^47 + 334882081624320 * q^48 - 131229302667470 * q^49 - 54321289062500 * q^50 + 135563520017196 * q^51 + 1056079273719136 * q^52 + 155836696419964 * q^53 - 100544914987236 * q^54 - 442042934375000 * q^55 + 1324797309480960 * q^56 - 1275301770664176 * q^57 + 3498878281127368 * q^58 + 963392423116456 * q^59 + 900538256250000 * q^60 - 2858793214972660 * q^61 - 7236477385534944 * q^62 - 893415409423992 * q^63 + 11601155857485824 * q^64 - 174481454687500 * q^65 + 4215674733890832 * q^66 + 6027859197776136 * q^67 + 4570758410249888 * q^68 + 1848988494393624 * q^69 - 1712977350000000 * q^70 + 2206536655060304 * q^71 - 9114601127439168 * q^72 - 18626002136017276 * q^73 - 15017443813698616 * q^74 + 2002258300781250 * q^75 - 43624167666234944 * q^76 + 5265163905377376 * q^77 - 20387930272260696 * q^78 + 4997891097934440 * q^79 + 19938014500000000 * q^80 + 3706040377703682 * q^81 + 1280087848985176 * q^82 - 23351553482511288 * q^83 - 5052050325955584 * q^84 + 8071101967187500 * q^85 - 36916031436093872 * q^86 - 31079867888477412 * q^87 + 167436503858533632 * q^88 - 60857257652216796 * q^89 - 5986184639062500 * q^90 + 51438794903148912 * q^91 + 39704913484919808 * q^92 + 14804286818235192 * q^93 + 90273897275494496 * q^94 - 75928174693750000 * q^95 - 160612972919165952 * q^96 + 42869862659782756 * q^97 + 191045888549711804 * q^98 - 48712957097118552 * q^99

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 $

$2$	−702.477	−1.94034	−0.970168	−	0.242432i	$-0.922055\pi$
$2$	−702.477	−1.94034	−0.970168	+	0.242432i	$0.922055\pi$
$3$	6561.00	0.577350
$3$	6561.00	0.577350
$4$	362402.	2.76491
$5$	390625.	0.447214
$5$	390625.	0.447214
$6$	−4.60895e6	−1.12025
$7$	−2.67481e6	−0.175372	−0.0876858	−	0.996148i	$-0.527947\pi$
$7$	−2.67481e6	−0.175372	−0.0876858	+	0.996148i	$0.527947\pi$
$8$	−1.62504e8	−3.42451
$9$	4.30467e7	0.333333
$10$	−2.74405e8	−0.867745
$11$	−9.86334e8	−1.38735	−0.693675	−	0.720288i	$-0.744010\pi$
$11$	−9.86334e8	−1.38735	−0.693675	+	0.720288i	$0.744010\pi$
$12$	2.37772e9	1.59632
$13$	2.81488e9	0.957065	0.478533	−	0.878070i	$-0.341169\pi$
$13$	2.81488e9	0.957065	0.478533	+	0.878070i	$0.341169\pi$
$14$	1.87899e9	0.340280
$15$	2.56289e9	0.258199
$16$	6.66544e10	3.87980
$17$	1.28501e10	0.446776	0.223388	−	0.974730i	$-0.428288\pi$
$17$	1.28501e10	0.446776	0.223388	+	0.974730i	$0.428288\pi$
$18$	−3.02393e10	−0.646779
$19$	−1.22560e11	−1.65555	−0.827777	−	0.561057i	$-0.810395\pi$
$19$	−1.22560e11	−1.65555	−0.827777	+	0.561057i	$0.810395\pi$
$20$	1.41563e11	1.23650
$21$	−1.75494e10	−0.101251
$22$	6.92877e11	2.69193
$23$	1.14646e11	0.305263	0.152631	−	0.988283i	$-0.451225\pi$
$23$	1.14646e11	0.305263	0.152631	+	0.988283i	$0.451225\pi$
$24$	−1.06619e12	−1.97714
$25$	1.52588e11	0.200000
$26$	−1.97739e12	−1.85703
$27$	2.82430e11	0.192450
$28$	−9.69355e11	−0.484886
$29$	−4.90027e12	−1.81902	−0.909510	−	0.415682i	$-0.863543\pi$
$29$	−4.90027e12	−1.81902	−0.909510	+	0.415682i	$0.863543\pi$
$30$	−1.80037e12	−0.500993
$31$	7.64406e12	1.60972	0.804859	−	0.593466i	$-0.202241\pi$
$31$	7.64406e12	1.60972	0.804859	+	0.593466i	$0.202241\pi$
$32$	−2.55235e13	−4.10361
$33$	−6.47134e12	−0.800987
$34$	−9.02688e12	−0.866896
$35$	−1.04485e12	−0.0784286
$36$	1.56002e13	0.921635
$37$	3.18983e12	0.149298	0.0746488	−	0.997210i	$-0.476216\pi$
$37$	3.18983e12	0.149298	0.0746488	+	0.997210i	$0.476216\pi$
$38$	8.60956e13	3.21233
$39$	1.84684e13	0.552562
$40$	−6.34781e13	−1.53149
$41$	−5.53137e12	−0.108186	−0.0540928	−	0.998536i	$-0.517227\pi$
$41$	−5.53137e12	−0.108186	−0.0540928	+	0.998536i	$0.517227\pi$
$42$	1.23281e13	0.196461
$43$	2.42435e13	0.316311	0.158155	−	0.987414i	$-0.449445\pi$
$43$	2.42435e13	0.316311	0.158155	+	0.987414i	$0.449445\pi$
$44$	−3.57449e14	−3.83589
$45$	1.68151e13	0.149071
$46$	−8.05365e13	−0.592313
$47$	−5.82161e13	−0.356625	−0.178312	−	0.983974i	$-0.557064\pi$
$47$	−5.82161e13	−0.356625	−0.178312	+	0.983974i	$0.557064\pi$
$48$	4.37320e14	2.24000
$49$	−2.25476e14	−0.969245
$50$	−1.07189e14	−0.388067
$51$	8.43094e13	0.257946
$52$	1.02012e15	2.64620
$53$	−6.45746e14	−1.42468	−0.712339	−	0.701835i	$-0.752364\pi$
$53$	−6.45746e14	−1.42468	−0.712339	+	0.701835i	$0.752364\pi$
$54$	−1.98400e14	−0.373418
$55$	−3.85287e14	−0.620442
$56$	4.34667e14	0.600562
$57$	−8.04117e14	−0.955835
$58$	3.44233e15	3.52951
$59$	−1.04433e12	−0.000925970 0	−0.000462985	−	1.00000i	$-0.500147\pi$
$59$	−1.04433e12	−0.000925970 0	−0.000462985		1.00000i	$0.500147\pi$
$60$	9.28796e14	0.713896
$61$	−1.56088e15	−1.04247	−0.521237	−	0.853412i	$-0.674529\pi$
$61$	−1.56088e15	−1.04247	−0.521237	+	0.853412i	$0.674529\pi$
$62$	−5.36978e15	−3.12340
$63$	−1.15142e14	−0.0584572
$64$	9.19314e15	4.08258
$65$	1.09956e15	0.428013
$66$	4.54597e15	1.55419
$67$	1.31151e15	0.394581	0.197290	−	0.980345i	$-0.436786\pi$
$67$	1.31151e15	0.394581	0.197290	+	0.980345i	$0.436786\pi$
$68$	4.65689e15	1.23529
$69$	7.52195e14	0.176244
$70$	7.33981e14	0.152178
$71$	−7.09230e15	−1.30344	−0.651720	−	0.758459i	$-0.725952\pi$
$71$	−7.09230e15	−1.30344	−0.651720	+	0.758459i	$0.725952\pi$
$72$	−6.99526e15	−1.14150
$73$	−1.17619e16	−1.70700	−0.853501	−	0.521091i	$-0.825525\pi$
$73$	−1.17619e16	−1.70700	−0.853501	+	0.521091i	$0.825525\pi$
$74$	−2.24078e15	−0.289688
$75$	1.00113e15	0.115470
$76$	−4.44160e16	−4.57745
$77$	2.63826e15	0.243302
$78$	−1.29737e16	−1.07216
$79$	−7.14674e15	−0.530003	−0.265001	−	0.964248i	$-0.585372\pi$
$79$	−7.14674e15	−0.530003	−0.265001	+	0.964248i	$0.585372\pi$
$80$	2.60369e16	1.73510
$81$	1.85302e15	0.111111
$82$	3.88566e15	0.209917
$83$	9.04707e15	0.440904	0.220452	−	0.975398i	$-0.429247\pi$
$83$	9.04707e15	0.440904	0.220452	+	0.975398i	$0.429247\pi$
$84$	−6.35994e15	−0.279949
$85$	5.01956e15	0.199804
$86$	−1.70305e16	−0.613749
$87$	−3.21507e16	−1.05021
$88$	1.60283e17	4.75100
$89$	−1.16786e16	−0.314467	−0.157233	−	0.987561i	$-0.550257\pi$
$89$	−1.16786e16	−0.314467	−0.157233	+	0.987561i	$0.550257\pi$
$90$	−1.18122e16	−0.289248
$91$	−7.52927e15	−0.167842
$92$	4.15481e16	0.844023
$93$	5.01527e16	0.929371
$94$	4.08955e16	0.691972
$95$	−4.78750e16	−0.740386
$96$	−1.67460e17	−2.36922
$97$	−8.32538e16	−1.07856	−0.539280	−	0.842126i	$-0.681304\pi$
$97$	−8.32538e16	−1.07856	−0.539280	+	0.842126i	$0.681304\pi$
$98$	1.58392e17	1.88066
$99$	−4.24585e16	−0.462450
$100$	5.52981e16	0.552981
$101$	−1.36210e16	−0.125164	−0.0625818	−	0.998040i	$-0.519933\pi$
$101$	−1.36210e16	−0.125164	−0.0625818	+	0.998040i	$0.519933\pi$
$102$	−5.92254e16	−0.500503
$103$	1.19030e17	0.925847	0.462923	−	0.886398i	$-0.346801\pi$
$103$	1.19030e17	0.925847	0.462923	+	0.886398i	$0.346801\pi$
$104$	−4.57429e17	−3.27748
$105$	−6.85524e15	−0.0452808
$106$	4.53622e17	2.76436
$107$	3.05375e17	1.71819	0.859096	−	0.511814i	$-0.171026\pi$
$107$	3.05375e17	1.71819	0.859096	+	0.511814i	$0.171026\pi$
$108$	1.02353e17	0.532106
$109$	−2.94297e17	−1.41469	−0.707344	−	0.706869i	$-0.750107\pi$
$109$	−2.94297e17	−1.41469	−0.707344	+	0.706869i	$0.750107\pi$
$110$	2.70655e17	1.20387
$111$	2.09285e16	0.0861970
$112$	−1.78288e17	−0.680407
$113$	−7.25034e16	−0.256562	−0.128281	−	0.991738i	$-0.540946\pi$
$113$	−7.25034e16	−0.256562	−0.128281	+	0.991738i	$0.540946\pi$
$114$	5.64873e17	1.85464
$115$	4.47838e16	0.136518
$116$	−1.77587e18	−5.02942
$117$	1.21171e17	0.319022
$118$	7.33620e14	0.00179669
$119$	−3.43715e16	−0.0783519
$120$	−4.16480e17	−0.884205
$121$	4.67408e17	0.924742
$122$	1.09648e18	2.02275
$123$	−3.62913e16	−0.0624610
$124$	2.77022e18	4.45072
$125$	5.96046e16	0.0894427
$126$	8.08844e16	0.113427
$127$	−1.00107e18	−1.31261	−0.656304	−	0.754497i	$-0.727881\pi$
$127$	−1.00107e18	−1.31261	−0.656304	+	0.754497i	$0.727881\pi$
$128$	−3.11255e18	−3.81797
$129$	1.59062e17	0.182622
$130$	−7.72418e17	−0.830489
$131$	6.36150e17	0.640846	0.320423	−	0.947275i	$-0.396175\pi$
$131$	6.36150e17	0.640846	0.320423	+	0.947275i	$0.396175\pi$
$132$	−2.34522e18	−2.21465
$133$	3.27825e17	0.290337
$134$	−9.21306e17	−0.765620
$135$	1.10324e17	0.0860663
$136$	−2.08819e18	−1.52999
$137$	2.75383e17	0.189588	0.0947942	−	0.995497i	$-0.469781\pi$
$137$	2.75383e17	0.189588	0.0947942	+	0.995497i	$0.469781\pi$
$138$	−5.28400e17	−0.341972
$139$	3.61048e17	0.219755	0.109878	−	0.993945i	$-0.464954\pi$
$139$	3.61048e17	0.219755	0.109878	+	0.993945i	$0.464954\pi$
$140$	−3.78654e17	−0.216848
$141$	−3.81956e17	−0.205897
$142$	4.98218e18	2.52911
$143$	−2.77641e18	−1.32779
$144$	2.86926e18	1.29327
$145$	−1.91417e18	−0.813490
$146$	8.26248e18	3.31216
$147$	−1.47935e18	−0.559594
$148$	1.15600e18	0.412794
$149$	−4.02824e18	−1.35841	−0.679207	−	0.733947i	$-0.737676\pi$
$149$	−4.02824e18	−1.35841	−0.679207	+	0.733947i	$0.737676\pi$
$150$	−7.03270e17	−0.224051
$151$	−3.02464e18	−0.910689	−0.455345	−	0.890315i	$-0.650484\pi$
$151$	−3.02464e18	−0.910689	−0.455345	+	0.890315i	$0.650484\pi$
$152$	1.99165e19	5.66947
$153$	5.53154e17	0.148925
$154$	−1.85331e18	−0.472088
$155$	2.98596e18	0.719888
$156$	6.69300e18	1.52778
$157$	−3.80909e18	−0.823520	−0.411760	−	0.911292i	$-0.635086\pi$
$157$	−3.80909e18	−0.823520	−0.411760	+	0.911292i	$0.635086\pi$
$158$	5.02042e18	1.02838
$159$	−4.23674e18	−0.822539
$160$	−9.97012e18	−1.83519
$161$	−3.06657e17	−0.0535345
$162$	−1.30170e18	−0.215593
$163$	−7.01276e18	−1.10229	−0.551143	−	0.834411i	$-0.685808\pi$
$163$	−7.01276e18	−1.10229	−0.551143	+	0.834411i	$0.685808\pi$
$164$	−2.00458e18	−0.299123
$165$	−2.52787e18	−0.358212
$166$	−6.35535e18	−0.855502
$167$	−5.20936e18	−0.666337	−0.333169	−	0.942867i	$-0.608118\pi$
$167$	−5.20936e18	−0.666337	−0.333169	+	0.942867i	$0.608118\pi$
$168$	2.85185e18	0.346735
$169$	−7.26857e17	−0.0840256
$170$	−3.52613e18	−0.387688
$171$	−5.27581e18	−0.551851
$172$	8.78590e18	0.874570
$173$	4.55714e18	0.431818	0.215909	−	0.976413i	$-0.430729\pi$
$173$	4.55714e18	0.431818	0.215909	+	0.976413i	$0.430729\pi$
$174$	2.25851e19	2.03776
$175$	−4.08143e17	−0.0350743
$176$	−6.57436e19	−5.38264
$177$	−6.85187e15	−0.000534609 0
$178$	8.20393e18	0.610171
$179$	1.27111e19	0.901428	0.450714	−	0.892668i	$-0.351169\pi$
$179$	1.27111e19	0.901428	0.450714	+	0.892668i	$0.351169\pi$
$180$	6.09383e18	0.412168
$181$	1.67839e19	1.08299	0.541494	−	0.840704i	$-0.317859\pi$
$181$	1.67839e19	1.08299	0.541494	+	0.840704i	$0.317859\pi$
$182$	5.28914e18	0.325670
$183$	−1.02409e19	−0.601872
$184$	−1.86305e19	−1.04538
$185$	1.24603e18	0.0667679
$186$	−3.52311e19	−1.80329
$187$	−1.26745e19	−0.619835
$188$	−2.10976e19	−0.986034
$189$	−7.55445e17	−0.0337503
$190$	3.36311e19	1.43660
$191$	1.93607e19	0.790930	0.395465	−	0.918481i	$-0.370583\pi$
$191$	1.93607e19	0.790930	0.395465	+	0.918481i	$0.370583\pi$
$192$	6.03162e19	2.35708
$193$	5.41103e18	0.202322	0.101161	−	0.994870i	$-0.467744\pi$
$193$	5.41103e18	0.202322	0.101161	+	0.994870i	$0.467744\pi$
$194$	5.84839e19	2.09277
$195$	7.21423e18	0.247113
$196$	−8.17129e19	−2.67987
$197$	1.69857e19	0.533483	0.266741	−	0.963768i	$-0.414053\pi$
$197$	1.69857e19	0.533483	0.266741	+	0.963768i	$0.414053\pi$
$198$	2.98261e19	0.897309
$199$	−8.38703e18	−0.241745	−0.120872	−	0.992668i	$-0.538569\pi$
$199$	−8.38703e18	−0.241745	−0.120872	+	0.992668i	$0.538569\pi$
$200$	−2.47961e19	−0.684902
$201$	8.60482e18	0.227811
$202$	9.56845e18	0.242860
$203$	1.31073e19	0.319004
$204$	3.05539e19	0.713197
$205$	−2.16069e18	−0.0483821
$206$	−8.36156e19	−1.79645
$207$	4.93515e18	0.101754
$208$	1.87624e20	3.71322
$209$	1.20885e20	2.29683
$210$	4.81565e18	0.0878599
$211$	2.60946e19	0.457245	0.228623	−	0.973515i	$-0.426578\pi$
$211$	2.60946e19	0.457245	0.228623	+	0.973515i	$0.426578\pi$
$212$	−2.34020e20	−3.93910
$213$	−4.65326e19	−0.752542
$214$	−2.14519e20	−3.33387
$215$	9.47013e18	0.141459
$216$	−4.58959e19	−0.659048
$217$	−2.04464e19	−0.282299
$218$	2.06737e20	2.74497
$219$	−7.71700e19	−0.985539
$220$	−1.39629e20	−1.71546
$221$	3.61715e19	0.427594
$222$	−1.47018e19	−0.167251
$223$	1.19753e19	0.131128	0.0655641	−	0.997848i	$-0.479115\pi$
$223$	1.19753e19	0.131128	0.0655641	+	0.997848i	$0.479115\pi$
$224$	6.82705e19	0.719656
$225$	6.56841e18	0.0666667
$226$	5.09320e19	0.497816
$227$	1.24200e20	1.16924	0.584619	−	0.811308i	$-0.301244\pi$
$227$	1.24200e20	1.16924	0.584619	+	0.811308i	$0.301244\pi$
$228$	−2.91413e20	−2.64279
$229$	1.03099e20	0.900853	0.450426	−	0.892814i	$-0.351272\pi$
$229$	1.03099e20	0.900853	0.450426	+	0.892814i	$0.351272\pi$
$230$	−3.14596e19	−0.264890
$231$	1.73096e19	0.140470
$232$	7.96313e20	6.22925
$233$	1.13357e20	0.854919	0.427459	−	0.904035i	$-0.359409\pi$
$233$	1.13357e20	0.854919	0.427459	+	0.904035i	$0.359409\pi$
$234$	−8.51201e19	−0.619010
$235$	−2.27407e19	−0.159487
$236$	−3.78468e17	−0.00256022
$237$	−4.68898e19	−0.305997
$238$	2.41452e19	0.152029
$239$	−8.29065e19	−0.503740	−0.251870	−	0.967761i	$-0.581046\pi$
$239$	−8.29065e19	−0.503740	−0.251870	+	0.967761i	$0.581046\pi$
$240$	1.70828e20	1.00176
$241$	−1.62688e20	−0.920896	−0.460448	−	0.887687i	$-0.652311\pi$
$241$	−1.62688e20	−0.920896	−0.460448	+	0.887687i	$0.652311\pi$
$242$	−3.28343e20	−1.79431
$243$	1.21577e19	0.0641500
$244$	−5.65664e20	−2.88234
$245$	−8.80765e19	−0.433459
$246$	2.54938e19	0.121195
$247$	−3.44992e20	−1.58447
$248$	−1.24219e21	−5.51250
$249$	5.93578e19	0.254556
$250$	−4.18709e19	−0.173549
$251$	4.10409e20	1.64433	0.822167	−	0.569247i	$-0.192765\pi$
$251$	4.10409e20	1.64433	0.822167	+	0.569247i	$0.192765\pi$
$252$	−4.17276e19	−0.161629
$253$	−1.13080e20	−0.423507
$254$	7.03232e20	2.54690
$255$	3.29333e19	0.115357
$256$	9.81533e20	3.32556
$257$	−1.69582e20	−0.555839	−0.277920	−	0.960604i	$-0.589645\pi$
$257$	−1.69582e20	−0.555839	−0.277920	+	0.960604i	$0.589645\pi$
$258$	−1.11737e20	−0.354348
$259$	−8.53218e18	−0.0261826
$260$	3.98484e20	1.18341
$261$	−2.10941e20	−0.606340
$262$	−4.46881e20	−1.24346
$263$	−3.50624e20	−0.944534	−0.472267	−	0.881456i	$-0.656564\pi$
$263$	−3.50624e20	−0.944534	−0.472267	+	0.881456i	$0.656564\pi$
$264$	1.05162e21	2.74299
$265$	−2.52245e20	−0.637136
$266$	−2.30289e20	−0.563352
$267$	−7.66231e19	−0.181557
$268$	4.75294e20	1.09098
$269$	2.92843e20	0.651240	0.325620	−	0.945501i	$-0.394427\pi$
$269$	2.92843e20	0.651240	0.325620	+	0.945501i	$0.394427\pi$
$270$	−7.75001e19	−0.166998
$271$	5.64043e20	1.17780	0.588902	−	0.808204i	$-0.299560\pi$
$271$	5.64043e20	1.17780	0.588902	+	0.808204i	$0.299560\pi$
$272$	8.56515e20	1.73340
$273$	−4.93995e19	−0.0969037
$274$	−1.93450e20	−0.367865
$275$	−1.50503e20	−0.277470
$276$	2.72597e20	0.487297
$277$	7.92843e20	1.37439	0.687194	−	0.726474i	$-0.258842\pi$
$277$	7.92843e20	1.37439	0.687194	+	0.726474i	$0.258842\pi$
$278$	−2.53628e20	−0.426399
$279$	3.29052e20	0.536573
$280$	1.69792e20	0.268580
$281$	−4.07531e20	−0.625399	−0.312700	−	0.949852i	$-0.601233\pi$
$281$	−4.07531e20	−0.625399	−0.312700	+	0.949852i	$0.601233\pi$
$282$	2.68315e20	0.399510
$283$	9.05800e20	1.30872	0.654362	−	0.756182i	$-0.272937\pi$
$283$	9.05800e20	1.30872	0.654362	+	0.756182i	$0.272937\pi$
$284$	−2.57026e21	−3.60389
$285$	−3.14108e20	−0.427462
$286$	1.95037e21	2.57635
$287$	1.47953e19	0.0189727
$288$	−1.09870e21	−1.36787
$289$	−6.62116e20	−0.800391
$290$	1.34466e21	1.57845
$291$	−5.46228e20	−0.622707
$292$	−4.26254e21	−4.71970
$293$	1.22920e21	1.32205	0.661025	−	0.750364i	$-0.270122\pi$
$293$	1.22920e21	1.32205	0.661025	+	0.750364i	$0.270122\pi$
$294$	1.03921e21	1.08580
$295$	−4.07943e17	−0.000414106 0
$296$	−5.18359e20	−0.511271
$297$	−2.78570e20	−0.266996
$298$	2.82974e21	2.63578
$299$	3.22716e20	0.292157
$300$	3.62811e20	0.319264
$301$	−6.48468e19	−0.0554719
$302$	2.12474e21	1.76704
$303$	−8.93675e19	−0.0722632
$304$	−8.16918e21	−6.42322
$305$	−6.09717e20	−0.466208
$306$	−3.88578e20	−0.288965
$307$	−1.33769e21	−0.967565	−0.483783	−	0.875188i	$-0.660738\pi$
$307$	−1.33769e21	−0.967565	−0.483783	+	0.875188i	$0.660738\pi$
$308$	9.56108e20	0.672707
$309$	7.80954e20	0.534538
$310$	−2.09757e21	−1.39683
$311$	−8.26556e19	−0.0535561	−0.0267781	−	0.999641i	$-0.508525\pi$
$311$	−8.26556e19	−0.0535561	−0.0267781	+	0.999641i	$0.508525\pi$
$312$	−3.00119e21	−1.89226
$313$	−2.88314e20	−0.176905	−0.0884523	−	0.996080i	$-0.528192\pi$
$313$	−2.88314e20	−0.176905	−0.0884523	+	0.996080i	$0.528192\pi$
$314$	2.67580e21	1.59791
$315$	−4.49772e19	−0.0261429
$316$	−2.58999e21	−1.46541
$317$	−3.17162e21	−1.74694	−0.873468	−	0.486882i	$-0.838134\pi$
$317$	−3.17162e21	−1.74694	−0.873468	+	0.486882i	$0.838134\pi$
$318$	2.97621e21	1.59600
$319$	4.83331e21	2.52362
$320$	3.59107e21	1.82578
$321$	2.00357e21	0.991999
$322$	2.15420e20	0.103875
$323$	−1.57491e21	−0.739662
$324$	6.71538e20	0.307212
$325$	4.29517e20	0.191413
$326$	4.92630e21	2.13881
$327$	−1.93088e21	−0.816771
$328$	8.98868e20	0.370483
$329$	1.55717e20	0.0625419
$330$	1.77577e21	0.695053
$331$	−1.58984e21	−0.606478	−0.303239	−	0.952915i	$-0.598068\pi$
$331$	−1.58984e21	−0.606478	−0.303239	+	0.952915i	$0.598068\pi$
$332$	3.27867e21	1.21906
$333$	1.37312e20	0.0497659
$334$	3.65945e21	1.29292
$335$	5.12309e20	0.176462
$336$	−1.16975e21	−0.392833
$337$	−3.01551e20	−0.0987432	−0.0493716	−	0.998780i	$-0.515722\pi$
$337$	−3.01551e20	−0.0987432	−0.0493716	+	0.998780i	$0.515722\pi$
$338$	5.10600e20	0.163038
$339$	−4.75695e20	−0.148126
$340$	1.81910e21	0.552440
$341$	−7.53960e21	−2.23324
$342$	3.70613e21	1.07078
$343$	1.22535e21	0.345350
$344$	−3.93967e21	−1.08321
$345$	2.93826e20	0.0788185
$346$	−3.20129e21	−0.837873
$347$	−3.81902e21	−0.975331	−0.487665	−	0.873031i	$-0.662151\pi$
$347$	−3.81902e21	−0.975331	−0.487665	+	0.873031i	$0.662151\pi$
$348$	−1.16515e22	−2.90374
$349$	−3.13624e20	−0.0762769	−0.0381384	−	0.999272i	$-0.512143\pi$
$349$	−3.13624e20	−0.0762769	−0.0381384	+	0.999272i	$0.512143\pi$
$350$	2.86711e20	0.0680560
$351$	7.95006e20	0.184187
$352$	2.51747e22	5.69314
$353$	−4.65586e21	−1.02781	−0.513907	−	0.857846i	$-0.671802\pi$
$353$	−4.65586e21	−1.02781	−0.513907	+	0.857846i	$0.671802\pi$
$354$	4.81328e18	0.00103732
$355$	−2.77043e21	−0.582916
$356$	−4.23234e21	−0.869471
$357$	−2.25511e20	−0.0452365
$358$	−8.92923e21	−1.74907
$359$	3.57780e21	0.684405	0.342203	−	0.939626i	$-0.388827\pi$
$359$	3.57780e21	0.684405	0.342203	+	0.939626i	$0.388827\pi$
$360$	−2.73252e21	−0.510496
$361$	9.54059e21	1.74086
$362$	−1.17903e22	−2.10136
$363$	3.06667e21	0.533900
$364$	−2.72862e21	−0.464068
$365$	−4.59450e21	−0.763395
$366$	7.19400e21	1.16783
$367$	−5.72348e21	−0.907818	−0.453909	−	0.891048i	$-0.649971\pi$
$367$	−5.72348e21	−0.907818	−0.453909	+	0.891048i	$0.649971\pi$
$368$	7.64170e21	1.18436
$369$	−2.38107e20	−0.0360619
$370$	−8.75305e20	−0.129552
$371$	1.72725e21	0.249848
$372$	1.81754e22	2.56962
$373$	7.79326e21	1.07695	0.538473	−	0.842643i	$-0.319001\pi$
$373$	7.79326e21	1.07695	0.538473	+	0.842643i	$0.319001\pi$
$374$	8.90352e21	1.20269
$375$	3.91066e20	0.0516398
$376$	9.46034e21	1.22127
$377$	−1.37937e22	−1.74092
$378$	5.30683e20	0.0654869
$379$	4.41273e21	0.532444	0.266222	−	0.963912i	$-0.414225\pi$
$379$	4.41273e21	0.532444	0.266222	+	0.963912i	$0.414225\pi$
$380$	−1.73500e22	−2.04710
$381$	−6.56805e21	−0.757834
$382$	−1.36005e22	−1.53467
$383$	7.42094e21	0.818973	0.409486	−	0.912316i	$-0.365708\pi$
$383$	7.42094e21	0.818973	0.409486	+	0.912316i	$0.365708\pi$
$384$	−2.04215e22	−2.20430
$385$	1.03057e21	0.108808
$386$	−3.80112e21	−0.392572
$387$	1.04360e21	0.105437
$388$	−3.01713e22	−2.98212
$389$	4.44977e21	0.430295	0.215147	−	0.976582i	$-0.430977\pi$
$389$	4.44977e21	0.430295	0.215147	+	0.976582i	$0.430977\pi$
$390$	−5.06783e21	−0.479483
$391$	1.47322e21	0.136384
$392$	3.66407e22	3.31919
$393$	4.17378e21	0.369993
$394$	−1.19321e22	−1.03514
$395$	−2.79170e21	−0.237024
$396$	−1.53870e22	−1.27863
$397$	−4.94222e20	−0.0401979	−0.0200989	−	0.999798i	$-0.506398\pi$
$397$	−4.94222e20	−0.0401979	−0.0200989	+	0.999798i	$0.506398\pi$
$398$	5.89170e21	0.469066
$399$	2.15086e21	0.167626
$400$	1.01707e22	0.775960
$401$	2.49341e22	1.86237	0.931187	−	0.364542i	$-0.118775\pi$
$401$	2.49341e22	1.86237	0.931187	+	0.364542i	$0.118775\pi$
$402$	−6.04469e21	−0.442031
$403$	2.15171e22	1.54061
$404$	−4.93628e21	−0.346066
$405$	7.23836e20	0.0496904
$406$	−9.20757e21	−0.618976
$407$	−3.14624e21	−0.207128
$408$	−1.37006e22	−0.883340
$409$	7.82942e21	0.494404	0.247202	−	0.968964i	$-0.420489\pi$
$409$	7.82942e21	0.494404	0.247202	+	0.968964i	$0.420489\pi$
$410$	1.51783e21	0.0938775
$411$	1.80679e21	0.109459
$412$	4.31366e22	2.55988
$413$	2.79339e18	0.000162389 0
$414$	−3.46683e21	−0.197438
$415$	3.53401e21	0.197178
$416$	−7.18457e22	−3.92742
$417$	2.36884e21	0.126876
$418$	−8.49191e22	−4.45663
$419$	−1.37314e20	−0.00706150	−0.00353075	−	0.999994i	$-0.501124\pi$
$419$	−1.37314e20	−0.00706150	−0.00353075	+	0.999994i	$0.501124\pi$
$420$	−2.48435e21	−0.125197
$421$	−1.84158e21	−0.0909481	−0.0454740	−	0.998966i	$-0.514480\pi$
$421$	−1.84158e21	−0.0909481	−0.0454740	+	0.998966i	$0.514480\pi$
$422$	−1.83308e22	−0.887209
$423$	−2.50601e21	−0.118875
$424$	1.04936e23	4.87883
$425$	1.96077e21	0.0893552
$426$	3.26881e22	1.46018
$427$	4.17505e21	0.182820
$428$	1.10669e23	4.75064
$429$	−1.82161e22	−0.766597
$430$	−6.65255e21	−0.274477
$431$	−3.30639e21	−0.133751	−0.0668756	−	0.997761i	$-0.521303\pi$
$431$	−3.30639e21	−0.133751	−0.0668756	+	0.997761i	$0.521303\pi$
$432$	1.88252e22	0.746668
$433$	1.49902e22	0.582990	0.291495	−	0.956572i	$-0.405847\pi$
$433$	1.49902e22	0.582990	0.291495	+	0.956572i	$0.405847\pi$
$434$	1.43631e22	0.547755
$435$	−1.25589e22	−0.469669
$436$	−1.06654e23	−3.91148
$437$	−1.40511e22	−0.505379
$438$	5.42102e22	1.91228
$439$	5.27352e22	1.82453	0.912267	−	0.409597i	$-0.134331\pi$
$439$	5.27352e22	1.82453	0.912267	+	0.409597i	$0.134331\pi$
$440$	6.26106e22	2.12471
$441$	−9.70600e21	−0.323082
$442$	−2.54096e22	−0.829676
$443$	2.21332e22	0.708944	0.354472	−	0.935067i	$-0.384661\pi$
$443$	2.21332e22	0.708944	0.354472	+	0.935067i	$0.384661\pi$
$444$	7.58452e21	0.238327
$445$	−4.56194e21	−0.140634
$446$	−8.41239e21	−0.254433
$447$	−2.64293e22	−0.784280
$448$	−2.45899e22	−0.715968
$449$	1.01171e22	0.289041	0.144521	−	0.989502i	$-0.453836\pi$
$449$	1.01171e22	0.289041	0.144521	+	0.989502i	$0.453836\pi$
$450$	−4.61416e21	−0.129356
$451$	5.45578e21	0.150091
$452$	−2.62754e22	−0.709369
$453$	−1.98447e22	−0.525787
$454$	−8.72479e22	−2.26872
$455$	−2.94112e21	−0.0750613
$456$	1.30672e23	3.27327
$457$	−7.09706e22	−1.74498	−0.872490	−	0.488632i	$-0.837496\pi$
$457$	−7.09706e22	−1.74498	−0.872490	+	0.488632i	$0.837496\pi$
$458$	−7.24248e22	−1.74796
$459$	3.62924e21	0.0859821
$460$	1.62297e22	0.377459
$461$	1.09889e22	0.250899	0.125449	−	0.992100i	$-0.459963\pi$
$461$	1.09889e22	0.250899	0.125449	+	0.992100i	$0.459963\pi$
$462$	−1.21596e22	−0.272560
$463$	1.78028e22	0.391787	0.195893	−	0.980625i	$-0.437239\pi$
$463$	1.78028e22	0.391787	0.195893	+	0.980625i	$0.437239\pi$
$464$	−3.26625e23	−7.05743
$465$	1.95909e22	0.415628
$466$	−7.96310e22	−1.65883
$467$	7.82704e22	1.60105	0.800524	−	0.599301i	$-0.204555\pi$
$467$	7.82704e22	1.60105	0.800524	+	0.599301i	$0.204555\pi$
$468$	4.39127e22	0.882065
$469$	−3.50804e21	−0.0691983
$470$	1.59748e22	0.309459
$471$	−2.49914e22	−0.475459
$472$	1.69708e20	0.00317100
$473$	−2.39122e22	−0.438834
$474$	3.29390e22	0.593737
$475$	−1.87012e22	−0.331111
$476$	−1.24563e22	−0.216636
$477$	−2.77973e22	−0.474893
$478$	5.82399e22	0.977425
$479$	−1.17366e23	−1.93504	−0.967522	−	0.252785i	$-0.918653\pi$
$479$	−1.17366e23	−1.93504	−0.967522	+	0.252785i	$0.918653\pi$
$480$	−6.54140e22	−1.05955
$481$	8.97899e21	0.142888
$482$	1.14285e23	1.78685
$483$	−2.01198e21	−0.0309081
$484$	1.69390e23	2.55683
$485$	−3.25210e22	−0.482347
$486$	−8.54048e21	−0.124473
$487$	4.40441e22	0.630800	0.315400	−	0.948959i	$-0.397861\pi$
$487$	4.40441e22	0.630800	0.315400	+	0.948959i	$0.397861\pi$
$488$	2.53648e23	3.56996
$489$	−4.60107e22	−0.636405
$490$	6.18717e22	0.841057
$491$	−9.19512e21	−0.122847	−0.0614235	−	0.998112i	$-0.519564\pi$
$491$	−9.19512e21	−0.122847	−0.0614235	+	0.998112i	$0.519564\pi$
$492$	−1.31520e22	−0.172699
$493$	−6.29689e22	−0.812695
$494$	2.42349e23	3.07441
$495$	−1.65853e22	−0.206814
$496$	5.09511e23	6.24539
$497$	1.89705e22	0.228586
$498$	−4.16975e22	−0.493924
$499$	−9.12259e22	−1.06234	−0.531170	−	0.847265i	$-0.678247\pi$
$499$	−9.12259e22	−1.06234	−0.531170	+	0.847265i	$0.678247\pi$
$500$	2.16008e22	0.247301
$501$	−3.41786e22	−0.384710
$502$	−2.88303e23	−3.19056
$503$	−4.20493e22	−0.457542	−0.228771	−	0.973480i	$-0.573471\pi$
$503$	−4.20493e22	−0.457542	−0.228771	+	0.973480i	$0.573471\pi$
$504$	1.87110e22	0.200187
$505$	−5.32071e21	−0.0559749
$506$	7.94359e22	0.821746
$507$	−4.76891e21	−0.0485122
$508$	−3.62791e23	−3.62924
$509$	3.12071e22	0.307010	0.153505	−	0.988148i	$-0.450944\pi$
$509$	3.12071e22	0.307010	0.153505	+	0.988148i	$0.450944\pi$
$510$	−2.31349e22	−0.223832
$511$	3.14609e22	0.299360
$512$	−2.81536e23	−2.63475
$513$	−3.46146e22	−0.318612
$514$	1.19128e23	1.07852
$515$	4.64960e22	0.414051
$516$	5.76443e22	0.504933
$517$	5.74206e22	0.494764
$518$	5.99366e21	0.0508030
$519$	2.98994e22	0.249310
$520$	−1.78683e23	−1.46573
$521$	−1.71736e23	−1.38593	−0.692965	−	0.720971i	$-0.743696\pi$
$521$	−1.71736e23	−1.38593	−0.692965	+	0.720971i	$0.743696\pi$
$522$	1.48181e23	1.17650
$523$	−3.54202e22	−0.276686	−0.138343	−	0.990384i	$-0.544178\pi$
$523$	−3.54202e22	−0.276686	−0.138343	+	0.990384i	$0.544178\pi$
$524$	2.30542e23	1.77188
$525$	−2.67783e21	−0.0202502
$526$	2.46305e23	1.83271
$527$	9.82268e22	0.719184
$528$	−4.31344e23	−3.10767
$529$	−1.27906e23	−0.906815
$530$	1.77196e23	1.23626
$531$	−4.49551e19	−0.000308657 0
$532$	1.18804e23	0.802755
$533$	−1.55701e22	−0.103541
$534$	5.38260e22	0.352283
$535$	1.19287e23	0.768399
$536$	−2.13126e23	−1.35125
$537$	8.33973e22	0.520440
$538$	−2.05716e23	−1.26363
$539$	2.22395e23	1.34468
$540$	3.99816e22	0.237965
$541$	−1.03474e23	−0.606257	−0.303128	−	0.952950i	$-0.598031\pi$
$541$	−1.03474e23	−0.606257	−0.303128	+	0.952950i	$0.598031\pi$
$542$	−3.96227e23	−2.28534
$543$	1.10119e23	0.625264
$544$	−3.27979e23	−1.83339
$545$	−1.14960e23	−0.632668
$546$	3.47020e22	0.188026
$547$	2.96616e23	1.58235	0.791176	−	0.611589i	$-0.209469\pi$
$547$	2.96616e23	1.58235	0.791176	+	0.611589i	$0.209469\pi$
$548$	9.97992e22	0.524194
$549$	−6.71906e22	−0.347491
$550$	1.05725e23	0.538385
$551$	6.00578e23	3.01149
$552$	−1.22235e23	−0.603548
$553$	1.91162e22	0.0929474
$554$	−5.56954e23	−2.66678
$555$	8.17518e21	0.0385485
$556$	1.30845e23	0.607603
$557$	−5.46640e22	−0.249996	−0.124998	−	0.992157i	$-0.539892\pi$
$557$	−5.46640e22	−0.249996	−0.124998	+	0.992157i	$0.539892\pi$
$558$	−2.31151e23	−1.04113
$559$	6.82427e22	0.302730
$560$	−6.96437e22	−0.304287
$561$	−8.31572e22	−0.357862
$562$	2.86281e23	1.21349
$563$	−1.18516e23	−0.494829	−0.247414	−	0.968910i	$-0.579581\pi$
$563$	−1.18516e23	−0.494829	−0.247414	+	0.968910i	$0.579581\pi$
$564$	−1.38422e23	−0.569287
$565$	−2.83217e22	−0.114738
$566$	−6.36304e23	−2.53936
$567$	−4.95647e21	−0.0194857
$568$	1.15253e24	4.46365
$569$	−4.39785e23	−1.67798	−0.838988	−	0.544149i	$-0.816853\pi$
$569$	−4.39785e23	−1.67798	−0.838988	+	0.544149i	$0.816853\pi$
$570$	2.20654e23	0.829421
$571$	−4.89721e22	−0.181360	−0.0906800	−	0.995880i	$-0.528904\pi$
$571$	−4.89721e22	−0.181360	−0.0906800	+	0.995880i	$0.528904\pi$
$572$	−1.00618e24	−3.67120
$573$	1.27026e23	0.456644
$574$	−1.03934e22	−0.0368134
$575$	1.74937e22	0.0610526
$576$	3.95735e23	1.36086
$577$	5.15333e23	1.74620	0.873099	−	0.487543i	$-0.162107\pi$
$577$	5.15333e23	1.74620	0.873099	+	0.487543i	$0.162107\pi$
$578$	4.65121e23	1.55303
$579$	3.55018e22	0.116811
$580$	−6.93698e23	−2.24922
$581$	−2.41992e22	−0.0773220
$582$	3.83713e23	1.20826
$583$	6.36922e23	1.97653
$584$	1.91136e24	5.84565
$585$	4.73326e22	0.142671
$586$	−8.63485e23	−2.56522
$587$	−3.59992e22	−0.105407	−0.0527035	−	0.998610i	$-0.516784\pi$
$587$	−3.59992e22	−0.105407	−0.0527035	+	0.998610i	$0.516784\pi$
$588$	−5.36118e23	−1.54722
$589$	−9.36857e23	−2.66498
$590$	2.86570e20	0.000803506 0
$591$	1.11443e23	0.308006
$592$	2.12616e23	0.579245
$593$	−3.67652e23	−0.987351	−0.493676	−	0.869646i	$-0.664347\pi$
$593$	−3.67652e23	−0.987351	−0.493676	+	0.869646i	$0.664347\pi$
$594$	1.95689e23	0.518062
$595$	−1.34264e22	−0.0350400
$596$	−1.45984e24	−3.75588
$597$	−5.50273e22	−0.139571
$598$	−2.26701e23	−0.566882
$599$	−6.97132e23	−1.71865	−0.859324	−	0.511432i	$-0.829115\pi$
$599$	−6.97132e23	−1.71865	−0.859324	+	0.511432i	$0.829115\pi$
$600$	−1.62687e23	−0.395429
$601$	3.44330e23	0.825166	0.412583	−	0.910920i	$-0.364627\pi$
$601$	3.44330e23	0.825166	0.412583	+	0.910920i	$0.364627\pi$
$602$	4.55534e22	0.107634
$603$	5.64562e22	0.131527
$604$	−1.09614e24	−2.51797
$605$	1.82581e23	0.413557
$606$	6.27786e22	0.140215
$607$	−3.08234e23	−0.678855	−0.339427	−	0.940632i	$-0.610233\pi$
$607$	−3.08234e23	−0.678855	−0.339427	+	0.940632i	$0.610233\pi$
$608$	3.12816e24	6.79374
$609$	8.59970e22	0.184177
$610$	4.28312e23	0.904601
$611$	−1.63872e23	−0.341313
$612$	2.00464e23	0.411765
$613$	−4.35176e23	−0.881558	−0.440779	−	0.897616i	$-0.645298\pi$
$613$	−4.35176e23	−0.881558	−0.440779	+	0.897616i	$0.645298\pi$
$614$	9.39699e23	1.87740
$615$	−1.41763e22	−0.0279334
$616$	−4.28727e23	−0.833190
$617$	6.04231e23	1.15819	0.579094	−	0.815261i	$-0.303406\pi$
$617$	6.04231e23	1.15819	0.579094	+	0.815261i	$0.303406\pi$
$618$	−5.48602e23	−1.03718
$619$	−3.33503e23	−0.621913	−0.310956	−	0.950424i	$-0.600649\pi$
$619$	−3.33503e23	−0.621913	−0.310956	+	0.950424i	$0.600649\pi$
$620$	1.08212e24	1.99042
$621$	3.23795e22	0.0587479
$622$	5.80637e22	0.103917
$623$	3.12379e22	0.0551485
$624$	1.23100e24	2.14383
$625$	2.32831e22	0.0400000
$626$	2.02534e23	0.343255
$627$	7.93128e23	1.32608
$628$	−1.38042e24	−2.27695
$629$	4.09896e22	0.0667026
$630$	3.15955e22	0.0507259
$631$	7.67831e23	1.21623	0.608116	−	0.793848i	$-0.291926\pi$
$631$	7.67831e23	1.21623	0.608116	+	0.793848i	$0.291926\pi$
$632$	1.16137e24	1.81500
$633$	1.71206e23	0.263991
$634$	2.22799e24	3.38964
$635$	−3.91045e23	−0.587016
$636$	−1.53540e24	−2.27424
$637$	−6.34688e23	−0.927631
$638$	−3.39529e24	−4.89667
$639$	−3.05300e23	−0.434480
$640$	−1.21584e24	−1.70745
$641$	−1.34739e24	−1.86724	−0.933620	−	0.358264i	$-0.883369\pi$
$641$	−1.34739e24	−1.86724	−0.933620	+	0.358264i	$0.883369\pi$
$642$	−1.40746e24	−1.92481
$643$	1.07291e24	1.44800	0.723999	−	0.689801i	$-0.242302\pi$
$643$	1.07291e24	1.44800	0.723999	+	0.689801i	$0.242302\pi$
$644$	−1.11133e23	−0.148018
$645$	6.21335e22	0.0816711
$646$	1.10634e24	1.43519
$647$	2.56767e23	0.328740	0.164370	−	0.986399i	$-0.447441\pi$
$647$	2.56767e23	0.328740	0.164370	+	0.986399i	$0.447441\pi$
$648$	−3.01123e23	−0.380501
$649$	1.03006e21	0.00128465
$650$	−3.01726e23	−0.371406
$651$	−1.34149e23	−0.162985
$652$	−2.54144e24	−3.04772
$653$	9.55573e23	1.13110	0.565551	−	0.824713i	$-0.308664\pi$
$653$	9.55573e23	1.13110	0.565551	+	0.824713i	$0.308664\pi$
$654$	1.35640e24	1.58481
$655$	2.48496e23	0.286595
$656$	−3.68690e23	−0.419739
$657$	−5.06312e23	−0.569001
$658$	−1.09388e23	−0.121352
$659$	−4.48249e23	−0.490901	−0.245450	−	0.969409i	$-0.578936\pi$
$659$	−4.48249e23	−0.490901	−0.245450	+	0.969409i	$0.578936\pi$
$660$	−9.16103e23	−0.990424
$661$	3.31071e23	0.353353	0.176677	−	0.984269i	$-0.443465\pi$
$661$	3.31071e23	0.353353	0.176677	+	0.984269i	$0.443465\pi$
$662$	1.11682e24	1.17677
$663$	2.37321e23	0.246872
$664$	−1.47018e24	−1.50988
$665$	1.28057e23	0.129843
$666$	−9.64583e22	−0.0965625
$667$	−5.61799e23	−0.555279
$668$	−1.88788e24	−1.84236
$669$	7.85701e22	0.0757069
$670$	−3.59885e23	−0.342396
$671$	1.53955e24	1.44628
$672$	4.47923e23	0.415494
$673$	−9.61434e23	−0.880626	−0.440313	−	0.897844i	$-0.645132\pi$
$673$	−9.61434e23	−0.880626	−0.440313	+	0.897844i	$0.645132\pi$
$674$	2.11833e23	0.191595
$675$	4.30953e22	0.0384900
$676$	−2.63414e23	−0.232323
$677$	1.08092e24	0.941432	0.470716	−	0.882285i	$-0.343996\pi$
$677$	1.08092e24	0.941432	0.470716	+	0.882285i	$0.343996\pi$
$678$	3.34165e23	0.287414
$679$	2.22688e23	0.189149
$680$	−8.15698e23	−0.684232
$681$	8.14879e23	0.675060
$682$	5.29640e24	4.33325
$683$	4.56949e23	0.369226	0.184613	−	0.982811i	$-0.440897\pi$
$683$	4.56949e23	0.369226	0.184613	+	0.982811i	$0.440897\pi$
$684$	−1.91196e24	−1.52582
$685$	1.07571e23	0.0847865
$686$	−8.60778e23	−0.670095
$687$	6.76434e23	0.520107
$688$	1.61594e24	1.22722
$689$	−1.81770e24	−1.36351
$690$	−2.06406e23	−0.152934
$691$	−1.38418e24	−1.01305	−0.506523	−	0.862226i	$-0.669070\pi$
$691$	−1.38418e24	−1.01305	−0.506523	+	0.862226i	$0.669070\pi$
$692$	1.65152e24	1.19394
$693$	1.13568e23	0.0811007
$694$	2.68278e24	1.89247
$695$	1.41035e23	0.0982776
$696$	5.22461e24	3.59646
$697$	−7.10785e22	−0.0483348
$698$	2.20314e23	0.148003
$699$	7.43739e23	0.493588
$700$	−1.47912e23	−0.0969772
$701$	9.18978e22	0.0595254	0.0297627	−	0.999557i	$-0.490525\pi$
$701$	9.18978e22	0.0595254	0.0297627	+	0.999557i	$0.490525\pi$
$702$	−5.58473e23	−0.357385
$703$	−3.90946e23	−0.247170
$704$	−9.06751e24	−5.66396
$705$	−1.49202e23	−0.0920801
$706$	3.27063e24	1.99431
$707$	3.64336e22	0.0219501
$708$	−2.48313e21	−0.00147814
$709$	−1.54021e24	−0.905913	−0.452956	−	0.891533i	$-0.649631\pi$
$709$	−1.54021e24	−0.905913	−0.452956	+	0.891533i	$0.649631\pi$
$710$	1.94616e24	1.13105
$711$	−3.07644e23	−0.176668
$712$	1.89781e24	1.07690
$713$	8.76365e23	0.491387
$714$	1.58417e23	0.0877740
$715$	−1.08454e24	−0.593804
$716$	4.60651e24	2.49236
$717$	−5.43949e23	−0.290834
$718$	−2.51332e24	−1.32798
$719$	3.23522e24	1.68931	0.844654	−	0.535313i	$-0.179806\pi$
$719$	3.23522e24	1.68931	0.844654	+	0.535313i	$0.179806\pi$
$720$	1.12080e24	0.578366
$721$	−3.18382e23	−0.162367
$722$	−6.70204e24	−3.37786
$723$	−1.06740e24	−0.531679
$724$	6.08250e24	2.99436
$725$	−7.47722e23	−0.363804
$726$	−2.15426e24	−1.03595
$727$	−2.33512e24	−1.10986	−0.554929	−	0.831898i	$-0.687255\pi$
$727$	−2.33512e24	−1.10986	−0.554929	+	0.831898i	$0.687255\pi$
$728$	1.22354e24	0.574777
$729$	7.97664e22	0.0370370
$730$	3.22753e24	1.48124
$731$	3.11531e23	0.141320
$732$	−3.71132e24	−1.66412
$733$	1.81465e24	0.804282	0.402141	−	0.915578i	$-0.368266\pi$
$733$	1.81465e24	0.804282	0.402141	+	0.915578i	$0.368266\pi$
$734$	4.02061e24	1.76147
$735$	−5.77870e23	−0.250258
$736$	−2.92618e24	−1.25268
$737$	−1.29359e24	−0.547422
$738$	1.67265e23	0.0699722
$739$	−1.21901e24	−0.504114	−0.252057	−	0.967712i	$-0.581107\pi$
$739$	−1.21901e24	−0.504114	−0.252057	+	0.967712i	$0.581107\pi$
$740$	4.51562e23	0.184607
$741$	−2.26349e24	−0.914797
$742$	−1.21335e24	−0.484790
$743$	4.70556e23	0.185869	0.0929344	−	0.995672i	$-0.470375\pi$
$743$	4.70556e23	0.185869	0.0929344	+	0.995672i	$0.470375\pi$
$744$	−8.15001e24	−3.18264
$745$	−1.57353e24	−0.607501
$746$	−5.47458e24	−2.08964
$747$	3.89447e23	0.146968
$748$	−4.59325e24	−1.71379
$749$	−8.16820e23	−0.301322
$750$	−2.74715e23	−0.100199
$751$	−3.20027e24	−1.15411	−0.577056	−	0.816705i	$-0.695798\pi$
$751$	−3.20027e24	−1.15411	−0.577056	+	0.816705i	$0.695798\pi$
$752$	−3.88036e24	−1.38363
$753$	2.69269e24	0.949356
$754$	9.68975e24	3.37797
$755$	−1.18150e24	−0.407273
$756$	−2.73775e23	−0.0933164
$757$	−5.33142e24	−1.79692	−0.898458	−	0.439060i	$-0.855312\pi$
$757$	−5.33142e24	−1.79692	−0.898458	+	0.439060i	$0.855312\pi$
$758$	−3.09984e24	−1.03312
$759$	−7.41916e23	−0.244512
$760$	7.77988e24	2.53546
$761$	3.19775e24	1.03057	0.515283	−	0.857020i	$-0.327687\pi$
$761$	3.19775e24	1.03057	0.515283	+	0.857020i	$0.327687\pi$
$762$	4.61390e24	1.47045
$763$	7.87188e23	0.248096
$764$	7.01636e24	2.18685
$765$	2.16076e23	0.0666014
$766$	−5.21304e24	−1.58908
$767$	−2.93967e21	−0.000886214 0
$768$	6.43984e24	1.92001
$769$	−3.24975e24	−0.958245	−0.479122	−	0.877748i	$-0.659045\pi$
$769$	−3.24975e24	−0.958245	−0.479122	+	0.877748i	$0.659045\pi$
$770$	−7.23950e23	−0.211124
$771$	−1.11263e24	−0.320914
$772$	1.96097e24	0.559401
$773$	4.35391e24	1.22844	0.614220	−	0.789135i	$-0.289471\pi$
$773$	4.35391e24	1.22844	0.614220	+	0.789135i	$0.289471\pi$
$774$	−7.33108e23	−0.204583
$775$	1.16639e24	0.321944
$776$	1.35291e25	3.69354
$777$	−5.59797e22	−0.0151165
$778$	−3.12586e24	−0.834917
$779$	6.77925e23	0.179107
$780$	2.61445e24	0.683245
$781$	6.99538e24	1.80833
$782$	−1.03490e24	−0.264631
$783$	−1.38398e24	−0.350070
$784$	−1.50290e25	−3.76048
$785$	−1.48792e24	−0.368289
$786$	−2.93199e24	−0.717910
$787$	1.67282e24	0.405194	0.202597	−	0.979262i	$-0.435062\pi$
$787$	1.67282e24	0.405194	0.202597	+	0.979262i	$0.435062\pi$
$788$	6.15565e24	1.47503
$789$	−2.30044e24	−0.545327
$790$	1.96110e24	0.459907
$791$	1.93933e23	0.0449936
$792$	6.89966e24	1.58367
$793$	−4.39368e24	−0.997715
$794$	3.47180e23	0.0779974
$795$	−1.65498e24	−0.367850
$796$	−3.03948e24	−0.668402
$797$	9.07285e23	0.197400	0.0987002	−	0.995117i	$-0.468532\pi$
$797$	9.07285e23	0.197400	0.0987002	+	0.995117i	$0.468532\pi$
$798$	−1.51093e24	−0.325251
$799$	−7.48082e23	−0.159331
$800$	−3.89458e24	−0.820721
$801$	−5.02724e23	−0.104822
$802$	−1.75156e25	−3.61363
$803$	1.16012e25	2.36821
$804$	3.11840e24	0.629877
$805$	−1.19788e23	−0.0239413
$806$	−1.51153e25	−2.98929
$807$	1.92135e24	0.375994
$808$	2.21347e24	0.428624
$809$	−9.92795e23	−0.190238	−0.0951189	−	0.995466i	$-0.530323\pi$
$809$	−9.92795e23	−0.190238	−0.0951189	+	0.995466i	$0.530323\pi$
$810$	−5.08478e23	−0.0964161
$811$	5.63576e24	1.05749	0.528744	−	0.848781i	$-0.322663\pi$
$811$	5.63576e24	1.05749	0.528744	+	0.848781i	$0.322663\pi$
$812$	4.75011e24	0.882017
$813$	3.70068e24	0.680006
$814$	2.21016e24	0.401898
$815$	−2.73936e24	−0.492957
$816$	5.61959e24	1.00078
$817$	−2.97129e24	−0.523670
$818$	−5.49999e24	−0.959310
$819$	−3.24110e23	−0.0559474
$820$	−7.83038e23	−0.133772
$821$	−9.18781e24	−1.55344	−0.776721	−	0.629845i	$-0.783118\pi$
$821$	−9.18781e24	−1.55344	−0.776721	+	0.629845i	$0.783118\pi$
$822$	−1.26923e24	−0.212387
$823$	6.83078e24	1.13128	0.565642	−	0.824651i	$-0.308628\pi$
$823$	6.83078e24	1.13128	0.565642	+	0.824651i	$0.308628\pi$
$824$	−1.93428e25	−3.17057
$825$	−9.87448e23	−0.160197
$826$	−1.96229e21	−0.000315089 0
$827$	−9.93808e24	−1.57945	−0.789724	−	0.613462i	$-0.789776\pi$
$827$	−9.93808e24	−1.57945	−0.789724	+	0.613462i	$0.789776\pi$
$828$	1.78851e24	0.281341
$829$	1.00372e25	1.56278	0.781391	−	0.624042i	$-0.214511\pi$
$829$	1.00372e25	1.56278	0.781391	+	0.624042i	$0.214511\pi$
$830$	−2.48256e24	−0.382592
$831$	5.20185e24	0.793503
$832$	2.58776e25	3.90729
$833$	−2.89738e24	−0.433035
$834$	−1.66405e24	−0.246182
$835$	−2.03491e24	−0.297995
$836$	4.38090e25	6.35053
$837$	2.15891e24	0.309790
$838$	9.64602e22	0.0137017
$839$	1.16269e25	1.63489	0.817446	−	0.576005i	$-0.195389\pi$
$839$	1.16269e25	1.63489	0.817446	+	0.576005i	$0.195389\pi$
$840$	1.11400e24	0.155065
$841$	1.67555e25	2.30883
$842$	1.29367e24	0.176470
$843$	−2.67381e24	−0.361074
$844$	9.45671e24	1.26424
$845$	−2.83928e23	−0.0375774
$846$	1.76042e24	0.230657
$847$	−1.25023e24	−0.162174
$848$	−4.30419e25	−5.52747
$849$	5.94296e24	0.755592
$850$	−1.37739e24	−0.173379
$851$	3.65703e23	0.0455750
$852$	−1.68635e25	−2.08071
$853$	1.09678e23	0.0133984	0.00669922	−	0.999978i	$-0.497868\pi$
$853$	1.09678e23	0.0133984	0.00669922	+	0.999978i	$0.497868\pi$
$854$	−2.93287e24	−0.354733
$855$	−2.06086e24	−0.246795
$856$	−4.96246e25	−5.88397
$857$	1.52818e25	1.79406	0.897030	−	0.441970i	$-0.145720\pi$
$857$	1.52818e25	1.79406	0.897030	+	0.441970i	$0.145720\pi$
$858$	1.27964e25	1.48746
$859$	−1.62639e25	−1.87190	−0.935951	−	0.352130i	$-0.885457\pi$
$859$	−1.62639e25	−1.87190	−0.935951	+	0.352130i	$0.885457\pi$
$860$	3.43199e24	0.391119
$861$	9.70723e22	0.0109539
$862$	2.32266e24	0.259522
$863$	−8.64833e24	−0.956842	−0.478421	−	0.878130i	$-0.658791\pi$
$863$	−8.64833e24	−0.956842	−0.478421	+	0.878130i	$0.658791\pi$
$864$	−7.20859e24	−0.789739
$865$	1.78013e24	0.193115
$866$	−1.05303e25	−1.13120
$867$	−4.34414e24	−0.462106
$868$	−7.40981e24	−0.780530
$869$	7.04908e24	0.735299
$870$	8.82231e24	0.911316
$871$	3.69175e24	0.377640
$872$	4.78244e25	4.84462
$873$	−3.58381e24	−0.359520
$874$	9.87056e24	0.980606
$875$	−1.59431e23	−0.0156857
$876$	−2.79665e25	−2.72492
$877$	1.02051e25	0.984740	0.492370	−	0.870386i	$-0.336131\pi$
$877$	1.02051e25	0.984740	0.492370	+	0.870386i	$0.336131\pi$
$878$	−3.70452e25	−3.54021
$879$	8.06479e24	0.763286
$880$	−2.56811e25	−2.40719
$881$	−9.18419e23	−0.0852601	−0.0426301	−	0.999091i	$-0.513574\pi$
$881$	−9.18419e23	−0.0852601	−0.0426301	+	0.999091i	$0.513574\pi$
$882$	6.81824e24	0.626887
$883$	8.44090e24	0.768640	0.384320	−	0.923200i	$-0.374436\pi$
$883$	8.44090e24	0.768640	0.384320	+	0.923200i	$0.374436\pi$
$884$	1.31086e25	1.18226
$885$	−2.67651e21	−0.000239084 0
$886$	−1.55480e25	−1.37559
$887$	−1.20407e25	−1.05512	−0.527558	−	0.849519i	$-0.676892\pi$
$887$	−1.20407e25	−1.05512	−0.527558	+	0.849519i	$0.676892\pi$
$888$	−3.40096e24	−0.295183
$889$	2.67768e24	0.230194
$890$	3.20466e24	0.272877
$891$	−1.82770e24	−0.154150
$892$	4.33988e24	0.362557
$893$	7.13498e24	0.590412
$894$	1.85660e25	1.52177
$895$	4.96526e24	0.403131
$896$	8.32548e24	0.669563
$897$	2.11734e24	0.168677
$898$	−7.10700e24	−0.560838
$899$	−3.74580e25	−2.92811
$900$	2.38040e24	0.184327
$901$	−8.29789e24	−0.636512
$902$	−3.83256e24	−0.291228
$903$	−4.25460e23	−0.0320267
$904$	1.17821e25	0.878599
$905$	6.55620e24	0.484327
$906$	1.39404e25	1.02020
$907$	−2.51467e25	−1.82314	−0.911568	−	0.411150i	$-0.865127\pi$
$907$	−2.51467e25	−1.82314	−0.911568	+	0.411150i	$0.865127\pi$
$908$	4.50104e25	3.23284
$909$	−5.86340e23	−0.0417212
$910$	2.06607e24	0.145644
$911$	−8.57548e24	−0.598897	−0.299448	−	0.954112i	$-0.596803\pi$
$911$	−8.57548e24	−0.598897	−0.299448	+	0.954112i	$0.596803\pi$
$912$	−5.35980e25	−3.70845
$913$	−8.92343e24	−0.611688
$914$	4.98552e25	3.38585
$915$	−4.00036e24	−0.269165
$916$	3.73633e25	2.49077
$917$	−1.70158e24	−0.112386
$918$	−2.54946e24	−0.166834
$919$	1.26569e25	0.820626	0.410313	−	0.911945i	$-0.365419\pi$
$919$	1.26569e25	0.820626	0.410313	+	0.911945i	$0.365419\pi$
$920$	−7.27753e24	−0.467507
$921$	−8.77661e24	−0.558624
$922$	−7.71948e24	−0.486828
$923$	−1.99640e25	−1.24748
$924$	6.27303e24	0.388388
$925$	4.86729e23	0.0298595
$926$	−1.25060e25	−0.760198
$927$	5.12384e24	0.308616
$928$	1.25072e26	7.46454
$929$	−1.61298e25	−0.953881	−0.476941	−	0.878936i	$-0.658254\pi$
$929$	−1.61298e25	−0.953881	−0.476941	+	0.878936i	$0.658254\pi$
$930$	−1.37622e25	−0.806457
$931$	2.76343e25	1.60464
$932$	4.10810e25	2.36377
$933$	−5.42304e23	−0.0309206
$934$	−5.49832e25	−3.10657
$935$	−4.95097e24	−0.277199
$936$	−1.96908e25	−1.09249
$937$	2.27726e24	0.125206	0.0626032	−	0.998038i	$-0.480060\pi$
$937$	2.27726e24	0.125206	0.0626032	+	0.998038i	$0.480060\pi$
$938$	2.46432e24	0.134268
$939$	−1.89163e24	−0.102136
$940$	−8.24126e24	−0.440968
$941$	−1.42919e25	−0.757839	−0.378919	−	0.925430i	$-0.623704\pi$
$941$	−1.42919e25	−0.757839	−0.378919	+	0.925430i	$0.623704\pi$
$942$	1.75559e25	0.922551
$943$	−6.34151e23	−0.0330251
$944$	−6.96095e22	−0.00359258
$945$	−2.95096e23	−0.0150936
$946$	1.67978e25	0.851486
$947$	−1.82863e24	−0.0918653	−0.0459326	−	0.998945i	$-0.514626\pi$
$947$	−1.82863e24	−0.0918653	−0.0459326	+	0.998945i	$0.514626\pi$
$948$	−1.69929e25	−0.846053
$949$	−3.31084e25	−1.63371
$950$	1.31372e25	0.642467
$951$	−2.08090e25	−1.00859
$952$	5.58550e24	0.268317
$953$	1.66599e25	0.793201	0.396601	−	0.917991i	$-0.370190\pi$
$953$	1.66599e25	0.793201	0.396601	+	0.917991i	$0.370190\pi$
$954$	1.95269e25	0.921452
$955$	7.56278e24	0.353715
$956$	−3.00455e25	−1.39279
$957$	3.17113e25	1.45701
$958$	8.24470e25	3.75464
$959$	−7.36596e23	−0.0332484
$960$	2.35610e25	1.05412
$961$	3.58816e25	1.59119
$962$	−6.30753e24	−0.277250
$963$	1.31454e25	0.572731
$964$	−5.89584e25	−2.54619
$965$	2.11368e24	0.0904810
$966$	1.41337e24	0.0599722
$967$	−3.59794e25	−1.51331	−0.756657	−	0.653812i	$-0.773169\pi$
$967$	−3.59794e25	−1.51331	−0.756657	+	0.653812i	$0.773169\pi$
$968$	−7.59556e25	−3.16679
$969$	−1.03330e25	−0.427044
$970$	2.28453e25	0.935915
$971$	−2.68946e25	−1.09220	−0.546099	−	0.837721i	$-0.683888\pi$
$971$	−2.68946e25	−1.09220	−0.546099	+	0.837721i	$0.683888\pi$
$972$	4.40596e24	0.177369
$973$	−9.65735e23	−0.0385389
$974$	−3.09400e25	−1.22396
$975$	2.81806e24	0.110512
$976$	−1.04039e26	−4.04459
$977$	1.26185e25	0.486301	0.243151	−	0.969989i	$-0.421819\pi$
$977$	1.26185e25	0.486301	0.243151	+	0.969989i	$0.421819\pi$
$978$	3.23215e25	1.23484
$979$	1.15190e25	0.436276
$980$	−3.19191e25	−1.19847
$981$	−1.26685e25	−0.471563
$982$	6.45936e24	0.238365
$983$	−2.72481e25	−0.996852	−0.498426	−	0.866932i	$-0.666088\pi$
$983$	−2.72481e25	−0.996852	−0.498426	+	0.866932i	$0.666088\pi$
$984$	5.89747e24	0.213898
$985$	6.63504e24	0.238581
$986$	4.42342e25	1.57690
$987$	1.02166e24	0.0361086
$988$	−1.25026e26	−4.38092
$989$	2.77944e24	0.0965580
$990$	1.16508e25	0.401289
$991$	−2.34443e25	−0.800593	−0.400296	−	0.916386i	$-0.631093\pi$
$991$	−2.34443e25	−0.800593	−0.400296	+	0.916386i	$0.631093\pi$
$992$	−1.95103e26	−6.60565
$993$	−1.04309e25	−0.350150
$994$	−1.33264e25	−0.443535
$995$	−3.27619e24	−0.108112
$996$	2.15114e25	0.703823
$997$	5.52750e25	1.79316	0.896582	−	0.442877i	$-0.146042\pi$
$997$	5.52750e25	1.79316	0.896582	+	0.442877i	$0.146042\pi$
$998$	6.40841e25	2.06130
$999$	9.00902e23	0.0287323

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.18.a.a.1.1	✓	2
3.2	odd	2		45.18.a.b.1.2		2
5.2	odd	4		75.18.b.b.49.1		4
5.3	odd	4		75.18.b.b.49.4		4
5.4	even	2		75.18.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.18.a.a.1.1	✓	2	1.1	even	1	trivial
45.18.a.b.1.2		2	3.2	odd	2
75.18.a.c.1.2		2	5.4	even	2
75.18.b.b.49.1		4	5.2	odd	4
75.18.b.b.49.4		4	5.3	odd	4

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:	\( N \)	\(=\)	\( 15 = 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	15.a (trivial)

\(f(q)\)	\(=\)	\(q-702.477 q^{2} +6561.00 q^{3} +362402. q^{4} +390625. q^{5} -4.60895e6 q^{6} -2.67481e6 q^{7} -1.62504e8 q^{8} +4.30467e7 q^{9} -2.74405e8 q^{10} -9.86334e8 q^{11} +2.37772e9 q^{12} +2.81488e9 q^{13} +1.87899e9 q^{14} +2.56289e9 q^{15} +6.66544e10 q^{16} +1.28501e10 q^{17} -3.02393e10 q^{18} -1.22560e11 q^{19} +1.41563e11 q^{20} -1.75494e10 q^{21} +6.92877e11 q^{22} +1.14646e11 q^{23} -1.06619e12 q^{24} +1.52588e11 q^{25} -1.97739e12 q^{26} +2.82430e11 q^{27} -9.69355e11 q^{28} -4.90027e12 q^{29} -1.80037e12 q^{30} +7.64406e12 q^{31} -2.55235e13 q^{32} -6.47134e12 q^{33} -9.02688e12 q^{34} -1.04485e12 q^{35} +1.56002e13 q^{36} +3.18983e12 q^{37} +8.60956e13 q^{38} +1.84684e13 q^{39} -6.34781e13 q^{40} -5.53137e12 q^{41} +1.23281e13 q^{42} +2.42435e13 q^{43} -3.57449e14 q^{44} +1.68151e13 q^{45} -8.05365e13 q^{46} -5.82161e13 q^{47} +4.37320e14 q^{48} -2.25476e14 q^{49} -1.07189e14 q^{50} +8.43094e13 q^{51} +1.02012e15 q^{52} -6.45746e14 q^{53} -1.98400e14 q^{54} -3.85287e14 q^{55} +4.34667e14 q^{56} -8.04117e14 q^{57} +3.44233e15 q^{58} -1.04433e12 q^{59} +9.28796e14 q^{60} -1.56088e15 q^{61} -5.36978e15 q^{62} -1.15142e14 q^{63} +9.19314e15 q^{64} +1.09956e15 q^{65} +4.54597e15 q^{66} +1.31151e15 q^{67} +4.65689e15 q^{68} +7.52195e14 q^{69} +7.33981e14 q^{70} -7.09230e15 q^{71} -6.99526e15 q^{72} -1.17619e16 q^{73} -2.24078e15 q^{74} +1.00113e15 q^{75} -4.44160e16 q^{76} +2.63826e15 q^{77} -1.29737e16 q^{78} -7.14674e15 q^{79} +2.60369e16 q^{80} +1.85302e15 q^{81} +3.88566e15 q^{82} +9.04707e15 q^{83} -6.35994e15 q^{84} +5.01956e15 q^{85} -1.70305e16 q^{86} -3.21507e16 q^{87} +1.60283e17 q^{88} -1.16786e16 q^{89} -1.18122e16 q^{90} -7.52927e15 q^{91} +4.15481e16 q^{92} +5.01527e16 q^{93} +4.08955e16 q^{94} -4.78750e16 q^{95} -1.67460e17 q^{96} -8.32538e16 q^{97} +1.58392e17 q^{98} -4.24585e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 356 q^{2} + 13122 q^{3} + 351376 q^{4} + 781250 q^{5} - 2335716 q^{6} - 20754552 q^{7} - 211737408 q^{8} + 86093442 q^{9} - 139062500 q^{10} - 1131629912 q^{11} + 2305377936 q^{12} - 446672524 q^{13}+ \cdots - 48\!\cdots\!52 q^{99}+O(q^{100}) \) 2 * q - 356 * q^2 + 13122 * q^3 + 351376 * q^4 + 781250 * q^5 - 2335716 * q^6 - 20754552 * q^7 - 211737408 * q^8 + 86093442 * q^9 - 139062500 * q^10 - 1131629912 * q^11 + 2305377936 * q^12 - 446672524 * q^13 - 4385222016 * q^14 + 5125781250 * q^15 + 51041317120 * q^16 + 20662021036 * q^17 - 15324632676 * q^18 - 194376127216 * q^19 + 137256250000 * q^20 - 136170615672 * q^21 + 642535396112 * q^22 + 281815042584 * q^23 - 1389209133888 * q^24 + 305175781250 * q^25 - 3107442504536 * q^26 + 564859072962 * q^27 - 770012242944 * q^28 - 4737062625892 * q^29 - 912389062500 * q^30 + 2256407074872 * q^31 - 24479953196032 * q^32 - 7424623852632 * q^33 - 6320225301928 * q^34 - 8107246875000 * q^35 + 15125584638096 * q^36 - 33686114979452 * q^37 + 61213036677904 * q^38 - 2930618429964 * q^39 - 82709925000000 * q^40 - 13051545859228 * q^41 - 28771441646976 * q^42 - 33149929851400 * q^43 - 355847272731712 * q^44 + 33630250781250 * q^45 - 22616426690688 * q^46 + 84299585768440 * q^47 + 334882081624320 * q^48 - 131229302667470 * q^49 - 54321289062500 * q^50 + 135563520017196 * q^51 + 1056079273719136 * q^52 + 155836696419964 * q^53 - 100544914987236 * q^54 - 442042934375000 * q^55 + 1324797309480960 * q^56 - 1275301770664176 * q^57 + 3498878281127368 * q^58 + 963392423116456 * q^59 + 900538256250000 * q^60 - 2858793214972660 * q^61 - 7236477385534944 * q^62 - 893415409423992 * q^63 + 11601155857485824 * q^64 - 174481454687500 * q^65 + 4215674733890832 * q^66 + 6027859197776136 * q^67 + 4570758410249888 * q^68 + 1848988494393624 * q^69 - 1712977350000000 * q^70 + 2206536655060304 * q^71 - 9114601127439168 * q^72 - 18626002136017276 * q^73 - 15017443813698616 * q^74 + 2002258300781250 * q^75 - 43624167666234944 * q^76 + 5265163905377376 * q^77 - 20387930272260696 * q^78 + 4997891097934440 * q^79 + 19938014500000000 * q^80 + 3706040377703682 * q^81 + 1280087848985176 * q^82 - 23351553482511288 * q^83 - 5052050325955584 * q^84 + 8071101967187500 * q^85 - 36916031436093872 * q^86 - 31079867888477412 * q^87 + 167436503858533632 * q^88 - 60857257652216796 * q^89 - 5986184639062500 * q^90 + 51438794903148912 * q^91 + 39704913484919808 * q^92 + 14804286818235192 * q^93 + 90273897275494496 * q^94 - 75928174693750000 * q^95 - 160612972919165952 * q^96 + 42869862659782756 * q^97 + 191045888549711804 * q^98 - 48712957097118552 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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