LMFDB - Embedded newform 669.6.a.d.1.19

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [669,6,Mod(1,669)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(669, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("669.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 669 = 3 \cdot 223 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	669.a (trivial)

Newform invariants

comment:select newform

sage:traces = [51] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	yes
Analytic conductor:	$107.296775455$
Analytic rank:	$0$
Dimension:	$51$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	669.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.13600 q^{2} +9.00000 q^{3} -22.1655 q^{4} -69.7481 q^{5} -28.2240 q^{6} +144.343 q^{7} +169.863 q^{8} +81.0000 q^{9} +218.730 q^{10} +9.52941 q^{11} -199.489 q^{12} +902.753 q^{13} -452.658 q^{14} -627.733 q^{15} +176.605 q^{16} +174.036 q^{17} -254.016 q^{18} -673.507 q^{19} +1546.00 q^{20} +1299.08 q^{21} -29.8842 q^{22} -1007.02 q^{23} +1528.77 q^{24} +1739.79 q^{25} -2831.04 q^{26} +729.000 q^{27} -3199.42 q^{28} +6850.62 q^{29} +1968.57 q^{30} +2964.67 q^{31} -5989.45 q^{32} +85.7647 q^{33} -545.778 q^{34} -10067.6 q^{35} -1795.41 q^{36} +5887.00 q^{37} +2112.12 q^{38} +8124.78 q^{39} -11847.6 q^{40} +1157.45 q^{41} -4073.93 q^{42} +4856.73 q^{43} -211.224 q^{44} -5649.59 q^{45} +3158.00 q^{46} -22608.8 q^{47} +1589.45 q^{48} +4027.78 q^{49} -5455.99 q^{50} +1566.33 q^{51} -20010.0 q^{52} -20693.3 q^{53} -2286.14 q^{54} -664.658 q^{55} +24518.5 q^{56} -6061.56 q^{57} -21483.6 q^{58} -48135.9 q^{59} +13914.0 q^{60} -14504.1 q^{61} -9297.22 q^{62} +11691.7 q^{63} +13131.6 q^{64} -62965.3 q^{65} -268.958 q^{66} +16748.7 q^{67} -3857.60 q^{68} -9063.14 q^{69} +31572.1 q^{70} +67759.3 q^{71} +13758.9 q^{72} -15219.7 q^{73} -18461.6 q^{74} +15658.1 q^{75} +14928.6 q^{76} +1375.50 q^{77} -25479.3 q^{78} +66285.0 q^{79} -12317.9 q^{80} +6561.00 q^{81} -3629.77 q^{82} -94589.5 q^{83} -28794.8 q^{84} -12138.7 q^{85} -15230.7 q^{86} +61655.6 q^{87} +1618.69 q^{88} +117203. q^{89} +17717.1 q^{90} +130306. q^{91} +22321.0 q^{92} +26682.1 q^{93} +70901.1 q^{94} +46975.8 q^{95} -53905.1 q^{96} -93697.5 q^{97} -12631.1 q^{98} +771.882 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 51 q + 18 q^{2} + 459 q^{3} + 940 q^{4} + 97 q^{5} + 162 q^{6} + 794 q^{7} + 411 q^{8} + 4131 q^{9} + 1256 q^{10} + 2349 q^{11} + 8460 q^{12} + 2954 q^{13} + 1816 q^{14} + 873 q^{15} + 18580 q^{16} + 1567 q^{17}+ \cdots + 190269 q^{99}+O(q^{100}) $ 51 * q + 18 * q^2 + 459 * q^3 + 940 * q^4 + 97 * q^5 + 162 * q^6 + 794 * q^7 + 411 * q^8 + 4131 * q^9 + 1256 * q^10 + 2349 * q^11 + 8460 * q^12 + 2954 * q^13 + 1816 * q^14 + 873 * q^15 + 18580 * q^16 + 1567 * q^17 + 1458 * q^18 + 12864 * q^19 + 3994 * q^20 + 7146 * q^21 + 5748 * q^22 + 6061 * q^23 + 3699 * q^24 + 52170 * q^25 + 14726 * q^26 + 37179 * q^27 + 40542 * q^28 + 4105 * q^29 + 11304 * q^30 + 27770 * q^31 + 3060 * q^32 + 21141 * q^33 + 37991 * q^34 + 26512 * q^35 + 76140 * q^36 + 38060 * q^37 + 12624 * q^38 + 26586 * q^39 + 45349 * q^40 + 54340 * q^41 + 16344 * q^42 + 66199 * q^43 + 52675 * q^44 + 7857 * q^45 + 63342 * q^46 + 63205 * q^47 + 167220 * q^48 + 205487 * q^49 + 48002 * q^50 + 14103 * q^51 + 105206 * q^52 + 11517 * q^53 + 13122 * q^54 + 121327 * q^55 + 153279 * q^56 + 115776 * q^57 + 145784 * q^58 + 219984 * q^59 + 35946 * q^60 + 163444 * q^61 + 415346 * q^62 + 64314 * q^63 + 558819 * q^64 + 191452 * q^65 + 51732 * q^66 + 247277 * q^67 + 310695 * q^68 + 54549 * q^69 + 366374 * q^70 + 238881 * q^71 + 33291 * q^72 + 380013 * q^73 + 355558 * q^74 + 469530 * q^75 + 513873 * q^76 + 91288 * q^77 + 132534 * q^78 + 369295 * q^79 + 319302 * q^80 + 334611 * q^81 + 93313 * q^82 + 310346 * q^83 + 364878 * q^84 + 336089 * q^85 + 72078 * q^86 + 36945 * q^87 + 480438 * q^88 + 325708 * q^89 + 101736 * q^90 + 555778 * q^91 + 39057 * q^92 + 249930 * q^93 + 297564 * q^94 - 67658 * q^95 + 27540 * q^96 + 418485 * q^97 - 75973 * q^98 + 190269 * 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 $

$2$	−3.13600	−0.554372	−0.277186	−	0.960816i	$-0.589402\pi$
$2$	−3.13600	−0.554372	−0.277186	+	0.960816i	$0.589402\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−22.1655	−0.692672
$5$	−69.7481	−1.24769	−0.623846	−	0.781548i	$-0.714431\pi$
$5$	−69.7481	−1.24769	−0.623846	+	0.781548i	$0.714431\pi$
$6$	−28.2240	−0.320067
$7$	144.343	1.11340	0.556698	−	0.830715i	$-0.312068\pi$
$7$	144.343	1.11340	0.556698	+	0.830715i	$0.312068\pi$
$8$	169.863	0.938370
$9$	81.0000	0.333333
$10$	218.730	0.691685
$11$	9.52941	0.0237457	0.0118728	−	0.999930i	$-0.496221\pi$
$11$	9.52941	0.0237457	0.0118728	+	0.999930i	$0.496221\pi$
$12$	−199.489	−0.399914
$13$	902.753	1.48153	0.740765	−	0.671764i	$-0.234463\pi$
$13$	902.753	1.48153	0.740765	+	0.671764i	$0.234463\pi$
$14$	−452.658	−0.617235
$15$	−627.733	−0.720355
$16$	176.605	0.172466
$17$	174.036	0.146055	0.0730277	−	0.997330i	$-0.476734\pi$
$17$	174.036	0.146055	0.0730277	+	0.997330i	$0.476734\pi$
$18$	−254.016	−0.184791
$19$	−673.507	−0.428014	−0.214007	−	0.976832i	$-0.568652\pi$
$19$	−673.507	−0.428014	−0.214007	+	0.976832i	$0.568652\pi$
$20$	1546.00	0.864241
$21$	1299.08	0.642819
$22$	−29.8842	−0.0131639
$23$	−1007.02	−0.396933	−0.198466	−	0.980108i	$-0.563596\pi$
$23$	−1007.02	−0.396933	−0.198466	+	0.980108i	$0.563596\pi$
$24$	1528.77	0.541768
$25$	1739.79	0.556734
$26$	−2831.04	−0.821319
$27$	729.000	0.192450
$28$	−3199.42	−0.771217
$29$	6850.62	1.51264	0.756319	−	0.654203i	$-0.226996\pi$
$29$	6850.62	1.51264	0.756319	+	0.654203i	$0.226996\pi$
$30$	1968.57	0.399345
$31$	2964.67	0.554080	0.277040	−	0.960858i	$-0.410647\pi$
$31$	2964.67	0.554080	0.277040	+	0.960858i	$0.410647\pi$
$32$	−5989.45	−1.03398
$33$	85.7647	0.0137096
$34$	−545.778	−0.0809690
$35$	−10067.6	−1.38917
$36$	−1795.41	−0.230891
$37$	5887.00	0.706951	0.353476	−	0.935444i	$-0.385000\pi$
$37$	5887.00	0.706951	0.353476	+	0.935444i	$0.385000\pi$
$38$	2112.12	0.237279
$39$	8124.78	0.855362
$40$	−11847.6	−1.17080
$41$	1157.45	0.107534	0.0537668	−	0.998554i	$-0.482877\pi$
$41$	1157.45	0.107534	0.0537668	+	0.998554i	$0.482877\pi$
$42$	−4073.93	−0.356361
$43$	4856.73	0.400565	0.200282	−	0.979738i	$-0.435814\pi$
$43$	4856.73	0.400565	0.200282	+	0.979738i	$0.435814\pi$
$44$	−211.224	−0.0164479
$45$	−5649.59	−0.415897
$46$	3158.00	0.220048
$47$	−22608.8	−1.49291	−0.746453	−	0.665438i	$-0.768245\pi$
$47$	−22608.8	−1.49291	−0.746453	+	0.665438i	$0.768245\pi$
$48$	1589.45	0.0995733
$49$	4027.78	0.239649
$50$	−5455.99	−0.308638
$51$	1566.33	0.0843251
$52$	−20010.0	−1.02621
$53$	−20693.3	−1.01191	−0.505954	−	0.862561i	$-0.668859\pi$
$53$	−20693.3	−1.01191	−0.505954	+	0.862561i	$0.668859\pi$
$54$	−2286.14	−0.106689
$55$	−664.658	−0.0296273
$56$	24518.5	1.04478
$57$	−6061.56	−0.247114
$58$	−21483.6	−0.838564
$59$	−48135.9	−1.80028	−0.900139	−	0.435604i	$-0.856535\pi$
$59$	−48135.9	−1.80028	−0.900139	+	0.435604i	$0.856535\pi$
$60$	13914.0	0.498970
$61$	−14504.1	−0.499076	−0.249538	−	0.968365i	$-0.580279\pi$
$61$	−14504.1	−0.499076	−0.249538	+	0.968365i	$0.580279\pi$
$62$	−9297.22	−0.307166
$63$	11691.7	0.371132
$64$	13131.6	0.400744
$65$	−62965.3	−1.84849
$66$	−268.958	−0.00760020
$67$	16748.7	0.455822	0.227911	−	0.973682i	$-0.426811\pi$
$67$	16748.7	0.455822	0.227911	+	0.973682i	$0.426811\pi$
$68$	−3857.60	−0.101168
$69$	−9063.14	−0.229169
$70$	31572.1	0.770119
$71$	67759.3	1.59523	0.797614	−	0.603168i	$-0.206095\pi$
$71$	67759.3	1.59523	0.797614	+	0.603168i	$0.206095\pi$
$72$	13758.9	0.312790
$73$	−15219.7	−0.334272	−0.167136	−	0.985934i	$-0.553452\pi$
$73$	−15219.7	−0.334272	−0.167136	+	0.985934i	$0.553452\pi$
$74$	−18461.6	−0.391914
$75$	15658.1	0.321430
$76$	14928.6	0.296473
$77$	1375.50	0.0264383
$78$	−25479.3	−0.474189
$79$	66285.0	1.19494	0.597472	−	0.801890i	$-0.296172\pi$
$79$	66285.0	1.19494	0.597472	+	0.801890i	$0.296172\pi$
$80$	−12317.9	−0.215184
$81$	6561.00	0.111111
$82$	−3629.77	−0.0596136
$83$	−94589.5	−1.50712	−0.753560	−	0.657380i	$-0.771665\pi$
$83$	−94589.5	−1.50712	−0.753560	+	0.657380i	$0.771665\pi$
$84$	−28794.8	−0.445263
$85$	−12138.7	−0.182232
$86$	−15230.7	−0.222062
$87$	61655.6	0.873322
$88$	1618.69	0.0222822
$89$	117203.	1.56842	0.784211	−	0.620495i	$-0.213068\pi$
$89$	117203.	1.56842	0.784211	+	0.620495i	$0.213068\pi$
$90$	17717.1	0.230562
$91$	130306.	1.64953
$92$	22321.0	0.274944
$93$	26682.1	0.319898
$94$	70901.1	0.827625
$95$	46975.8	0.534029
$96$	−53905.1	−0.596969
$97$	−93697.5	−1.01111	−0.505555	−	0.862794i	$-0.668712\pi$
$97$	−93697.5	−1.01111	−0.505555	+	0.862794i	$0.668712\pi$
$98$	−12631.1	−0.132855
$99$	771.882	0.00791522
$100$	−38563.4	−0.385634
$101$	110449.	1.07736	0.538679	−	0.842511i	$-0.318924\pi$
$101$	110449.	1.07736	0.538679	+	0.842511i	$0.318924\pi$
$102$	−4912.00	−0.0467475
$103$	48959.2	0.454717	0.227358	−	0.973811i	$-0.426991\pi$
$103$	48959.2	0.454717	0.227358	+	0.973811i	$0.426991\pi$
$104$	153344.	1.39022
$105$	−90608.5	−0.802040
$106$	64894.3	0.560973
$107$	68185.2	0.575746	0.287873	−	0.957669i	$-0.407052\pi$
$107$	68185.2	0.575746	0.287873	+	0.957669i	$0.407052\pi$
$108$	−16158.6	−0.133305
$109$	22800.1	0.183811	0.0919054	−	0.995768i	$-0.470704\pi$
$109$	22800.1	0.183811	0.0919054	+	0.995768i	$0.470704\pi$
$110$	2084.37	0.0164245
$111$	52983.0	0.408159
$112$	25491.6	0.192023
$113$	−20268.9	−0.149326	−0.0746628	−	0.997209i	$-0.523788\pi$
$113$	−20268.9	−0.149326	−0.0746628	+	0.997209i	$0.523788\pi$
$114$	19009.1	0.136993
$115$	70237.4	0.495249
$116$	−151848.	−1.04776
$117$	73123.0	0.493844
$118$	150954.	0.998023
$119$	25120.9	0.162617
$120$	−106629.	−0.675959
$121$	−160960.	−0.999436
$122$	45484.9	0.276674
$123$	10417.1	0.0620845
$124$	−65713.4	−0.383796
$125$	96615.5	0.553059
$126$	−36665.3	−0.205745
$127$	214868.	1.18212	0.591062	−	0.806626i	$-0.298709\pi$
$127$	214868.	1.18212	0.591062	+	0.806626i	$0.298709\pi$
$128$	150482.	0.811819
$129$	43710.6	0.231266
$130$	197459.	1.02475
$131$	103896.	0.528956	0.264478	−	0.964392i	$-0.414800\pi$
$131$	103896.	0.528956	0.264478	+	0.964392i	$0.414800\pi$
$132$	−1901.02	−0.00949623
$133$	−97215.7	−0.476549
$134$	−52524.1	−0.252695
$135$	−50846.3	−0.240118
$136$	29562.3	0.137054
$137$	−314470.	−1.43146	−0.715728	−	0.698379i	$-0.753905\pi$
$137$	−314470.	−1.43146	−0.715728	+	0.698379i	$0.753905\pi$
$138$	28422.0	0.127045
$139$	120193.	0.527647	0.263824	−	0.964571i	$-0.415016\pi$
$139$	120193.	0.527647	0.263824	+	0.964571i	$0.415016\pi$
$140$	223154.	0.962241
$141$	−203479.	−0.861929
$142$	−212493.	−0.884350
$143$	8602.70	0.0351799
$144$	14305.0	0.0574886
$145$	−477818.	−1.88731
$146$	47729.1	0.185311
$147$	36250.0	0.138361
$148$	−130488.	−0.489685
$149$	383320.	1.41448	0.707238	−	0.706975i	$-0.249941\pi$
$149$	383320.	1.41448	0.707238	+	0.706975i	$0.249941\pi$
$150$	−49103.9	−0.178192
$151$	324607.	1.15855	0.579276	−	0.815131i	$-0.303335\pi$
$151$	324607.	1.15855	0.579276	+	0.815131i	$0.303335\pi$
$152$	−114404.	−0.401635
$153$	14096.9	0.0486851
$154$	−4313.57	−0.0146567
$155$	−206780.	−0.691321
$156$	−180090.	−0.592485
$157$	195639.	0.633442	0.316721	−	0.948519i	$-0.397418\pi$
$157$	195639.	0.633442	0.316721	+	0.948519i	$0.397418\pi$
$158$	−207870.	−0.662443
$159$	−186240.	−0.584225
$160$	417753.	1.29009
$161$	−145355.	−0.441943
$162$	−20575.3	−0.0615969
$163$	−117122.	−0.345279	−0.172640	−	0.984985i	$-0.555230\pi$
$163$	−117122.	−0.345279	−0.172640	+	0.984985i	$0.555230\pi$
$164$	−25655.5	−0.0744854
$165$	−5981.92	−0.0171053
$166$	296633.	0.835505
$167$	600937.	1.66739	0.833696	−	0.552223i	$-0.186220\pi$
$167$	600937.	1.66739	0.833696	+	0.552223i	$0.186220\pi$
$168$	220666.	0.603202
$169$	443670.	1.19493
$170$	38067.0	0.101024
$171$	−54554.0	−0.142671
$172$	−107652.	−0.277460
$173$	−91476.6	−0.232378	−0.116189	−	0.993227i	$-0.537068\pi$
$173$	−91476.6	−0.232378	−0.116189	+	0.993227i	$0.537068\pi$
$174$	−193352.	−0.484145
$175$	251126.	0.619865
$176$	1682.94	0.00409532
$177$	−433223.	−1.03939
$178$	−367548.	−0.869489
$179$	−331899.	−0.774235	−0.387117	−	0.922030i	$-0.626529\pi$
$179$	−331899.	−0.774235	−0.387117	+	0.922030i	$0.626529\pi$
$180$	125226.	0.288080
$181$	−245557.	−0.557129	−0.278565	−	0.960417i	$-0.589859\pi$
$181$	−245557.	−0.557129	−0.278565	+	0.960417i	$0.589859\pi$
$182$	−408639.	−0.914453
$183$	−130537.	−0.288142
$184$	−171055.	−0.372469
$185$	−410607.	−0.882057
$186$	−83675.0	−0.177343
$187$	1658.46	0.00346818
$188$	501135.	1.03409
$189$	105226.	0.214273
$190$	−147316.	−0.296051
$191$	−829755.	−1.64576	−0.822880	−	0.568215i	$-0.807634\pi$
$191$	−829755.	−1.64576	−0.822880	+	0.568215i	$0.807634\pi$
$192$	118184.	0.231369
$193$	425647.	0.822539	0.411270	−	0.911514i	$-0.365085\pi$
$193$	425647.	0.822539	0.411270	+	0.911514i	$0.365085\pi$
$194$	293835.	0.560531
$195$	−566688.	−1.06723
$196$	−89277.7	−0.165998
$197$	−689790.	−1.26634	−0.633171	−	0.774012i	$-0.718247\pi$
$197$	−689790.	−1.26634	−0.633171	+	0.774012i	$0.718247\pi$
$198$	−2420.62	−0.00438798
$199$	844080.	1.51095	0.755477	−	0.655175i	$-0.227405\pi$
$199$	844080.	1.51095	0.755477	+	0.655175i	$0.227405\pi$
$200$	295527.	0.522422
$201$	150739.	0.263169
$202$	−346369.	−0.597257
$203$	988837.	1.68416
$204$	−34718.4	−0.0584096
$205$	−80730.1	−0.134169
$206$	−153536.	−0.252082
$207$	−81568.3	−0.132311
$208$	159431.	0.255514
$209$	−6418.12	−0.0101635
$210$	284148.	0.444628
$211$	−231711.	−0.358295	−0.179147	−	0.983822i	$-0.557334\pi$
$211$	−231711.	−0.358295	−0.179147	+	0.983822i	$0.557334\pi$
$212$	458678.	0.700920
$213$	609833.	0.921005
$214$	−213829.	−0.319177
$215$	−338747.	−0.499781
$216$	123830.	0.180589
$217$	427928.	0.616910
$218$	−71501.2	−0.101900
$219$	−136978.	−0.192992
$220$	14732.5	0.0205220
$221$	157112.	0.216386
$222$	−166155.	−0.226272
$223$	−49729.0	−0.0669650
$223$	−49729.0	−0.0669650
$224$	−864533.	−1.15123
$225$	140923.	0.185578
$226$	63563.3	0.0827819
$227$	−5254.64	−0.00676827	−0.00338414	−	0.999994i	$-0.501077\pi$
$227$	−5254.64	−0.00676827	−0.00338414	+	0.999994i	$0.501077\pi$
$228$	134357.	0.171169
$229$	1.12100e6	1.41260	0.706299	−	0.707914i	$-0.250364\pi$
$229$	1.12100e6	1.41260	0.706299	+	0.707914i	$0.250364\pi$
$230$	−220265.	−0.274552
$231$	12379.5	0.0152642
$232$	1.16367e6	1.41941
$233$	277717.	0.335129	0.167565	−	0.985861i	$-0.446410\pi$
$233$	277717.	0.335129	0.167565	+	0.985861i	$0.446410\pi$
$234$	−229314.	−0.273773
$235$	1.57692e6	1.86269
$236$	1.06696e6	1.24700
$237$	596565.	0.689901
$238$	−78779.0	−0.0901505
$239$	1.48294e6	1.67930	0.839652	−	0.543125i	$-0.182759\pi$
$239$	1.48294e6	1.67930	0.839652	+	0.543125i	$0.182759\pi$
$240$	−110861.	−0.124237
$241$	1.37457e6	1.52449	0.762246	−	0.647288i	$-0.224097\pi$
$241$	1.37457e6	1.52449	0.762246	+	0.647288i	$0.224097\pi$
$242$	504771.	0.554059
$243$	59049.0	0.0641500
$244$	321491.	0.345696
$245$	−280930.	−0.299008
$246$	−32668.0	−0.0344179
$247$	−608010.	−0.634116
$248$	503588.	0.519932
$249$	−851305.	−0.870136
$250$	−302986.	−0.306601
$251$	1.37332e6	1.37590	0.687952	−	0.725756i	$-0.258510\pi$
$251$	1.37332e6	1.37590	0.687952	+	0.725756i	$0.258510\pi$
$252$	−259153.	−0.257072
$253$	−9596.26	−0.00942542
$254$	−673827.	−0.655336
$255$	−109248.	−0.105212
$256$	−892121.	−0.850793
$257$	161764.	0.152774	0.0763868	−	0.997078i	$-0.475662\pi$
$257$	161764.	0.152774	0.0763868	+	0.997078i	$0.475662\pi$
$258$	−137076.	−0.128207
$259$	849744.	0.787116
$260$	1.39566e6	1.28040
$261$	554901.	0.504213
$262$	−325817.	−0.293238
$263$	−1.27505e6	−1.13668	−0.568340	−	0.822794i	$-0.692414\pi$
$263$	−1.27505e6	−1.13668	−0.568340	+	0.822794i	$0.692414\pi$
$264$	14568.2	0.0128646
$265$	1.44332e6	1.26255
$266$	304868.	0.264185
$267$	1.05482e6	0.905529
$268$	−371244.	−0.315735
$269$	1.03556e6	0.872556	0.436278	−	0.899812i	$-0.356296\pi$
$269$	1.03556e6	0.872556	0.436278	+	0.899812i	$0.356296\pi$
$270$	159454.	0.133115
$271$	−1.33765e6	−1.10642	−0.553211	−	0.833041i	$-0.686598\pi$
$271$	−1.33765e6	−1.10642	−0.553211	+	0.833041i	$0.686598\pi$
$272$	30735.7	0.0251896
$273$	1.17275e6	0.952356
$274$	986179.	0.793559
$275$	16579.2	0.0132200
$276$	200889.	0.158739
$277$	−1.10289e6	−0.863636	−0.431818	−	0.901961i	$-0.642128\pi$
$277$	−1.10289e6	−0.863636	−0.431818	+	0.901961i	$0.642128\pi$
$278$	−376927.	−0.292513
$279$	240138.	0.184693
$280$	−1.71012e6	−1.30356
$281$	70379.5	0.0531716	0.0265858	−	0.999647i	$-0.491536\pi$
$281$	70379.5	0.0531716	0.0265858	+	0.999647i	$0.491536\pi$
$282$	638110.	0.477829
$283$	2.56385e6	1.90295	0.951474	−	0.307729i	$-0.0995689\pi$
$283$	2.56385e6	1.90295	0.951474	+	0.307729i	$0.0995689\pi$
$284$	−1.50192e6	−1.10497
$285$	422782.	0.308322
$286$	−26978.1	−0.0195028
$287$	167070.	0.119727
$288$	−485146.	−0.344660
$289$	−1.38957e6	−0.978668
$290$	1.49844e6	1.04627
$291$	−843277.	−0.583765
$292$	337353.	0.231541
$293$	1.62471e6	1.10562	0.552810	−	0.833307i	$-0.313556\pi$
$293$	1.62471e6	1.10562	0.552810	+	0.833307i	$0.313556\pi$
$294$	−113680.	−0.0767036
$295$	3.35739e6	2.24619
$296$	999984.	0.663382
$297$	6946.94	0.00456985
$298$	−1.20209e6	−0.784146
$299$	−909087.	−0.588068
$300$	−347070.	−0.222646
$301$	701033.	0.445987
$302$	−1.01797e6	−0.642269
$303$	994045.	0.622013
$304$	−118945.	−0.0738178
$305$	1.01163e6	0.622693
$306$	−44208.0	−0.0269897
$307$	326857.	0.197930	0.0989649	−	0.995091i	$-0.468447\pi$
$307$	326857.	0.197930	0.0989649	+	0.995091i	$0.468447\pi$
$308$	−30488.6	−0.0183131
$309$	440632.	0.262531
$310$	648463.	0.383249
$311$	742171.	0.435114	0.217557	−	0.976048i	$-0.430191\pi$
$311$	742171.	0.435114	0.217557	+	0.976048i	$0.430191\pi$
$312$	1.38010e6	0.802646
$313$	−3.06504e6	−1.76838	−0.884188	−	0.467131i	$-0.845288\pi$
$313$	−3.06504e6	−1.76838	−0.884188	+	0.467131i	$0.845288\pi$
$314$	−613525.	−0.351162
$315$	−815477.	−0.463058
$316$	−1.46924e6	−0.827704
$317$	384285.	0.214786	0.107393	−	0.994217i	$-0.465750\pi$
$317$	384285.	0.214786	0.107393	+	0.994217i	$0.465750\pi$
$318$	584049.	0.323878
$319$	65282.4	0.0359186
$320$	−915901.	−0.500004
$321$	613667.	0.332407
$322$	455834.	0.245001
$323$	−117215.	−0.0625137
$324$	−145428.	−0.0769635
$325$	1.57060e6	0.824818
$326$	367295.	0.191413
$327$	205201.	0.106123
$328$	196609.	0.100906
$329$	−3.26341e6	−1.66219
$330$	18759.3	0.00948270
$331$	2.30112e6	1.15444	0.577218	−	0.816590i	$-0.304138\pi$
$331$	2.30112e6	1.15444	0.577218	+	0.816590i	$0.304138\pi$
$332$	2.09662e6	1.04394
$333$	476847.	0.235650
$334$	−1.88454e6	−0.924356
$335$	−1.16819e6	−0.568725
$336$	229425.	0.110864
$337$	1.41010e6	0.676356	0.338178	−	0.941082i	$-0.390189\pi$
$337$	1.41010e6	0.676356	0.338178	+	0.941082i	$0.390189\pi$
$338$	−1.39135e6	−0.662437
$339$	−182420.	−0.0862131
$340$	269060.	0.126227
$341$	28251.6	0.0131570
$342$	171082.	0.0790930
$343$	−1.84459e6	−0.846571
$344$	824979.	0.375878
$345$	632137.	0.285932
$346$	286871.	0.128824
$347$	−1.26110e6	−0.562246	−0.281123	−	0.959672i	$-0.590707\pi$
$347$	−1.26110e6	−0.562246	−0.281123	+	0.959672i	$0.590707\pi$
$348$	−1.36663e6	−0.604926
$349$	−1.14627e6	−0.503761	−0.251880	−	0.967758i	$-0.581049\pi$
$349$	−1.14627e6	−0.503761	−0.251880	+	0.967758i	$0.581049\pi$
$350$	−787532.	−0.343636
$351$	658107.	0.285121
$352$	−57075.9	−0.0245525
$353$	−1.81472e6	−0.775128	−0.387564	−	0.921843i	$-0.626683\pi$
$353$	−1.81472e6	−0.775128	−0.387564	+	0.921843i	$0.626683\pi$
$354$	1.35859e6	0.576209
$355$	−4.72608e6	−1.99035
$356$	−2.59786e6	−1.08640
$357$	226088.	0.0938872
$358$	1.04083e6	0.429214
$359$	848971.	0.347662	0.173831	−	0.984776i	$-0.444385\pi$
$359$	848971.	0.347662	0.173831	+	0.984776i	$0.444385\pi$
$360$	−959657.	−0.390265
$361$	−2.02249e6	−0.816804
$362$	770067.	0.308857
$363$	−1.44864e6	−0.577025
$364$	−2.88829e6	−1.14258
$365$	1.06155e6	0.417068
$366$	409364.	0.159738
$367$	−664664.	−0.257595	−0.128797	−	0.991671i	$-0.541112\pi$
$367$	−664664.	−0.257595	−0.128797	+	0.991671i	$0.541112\pi$
$368$	−177844.	−0.0684573
$369$	93753.7	0.0358445
$370$	1.28766e6	0.488988
$371$	−2.98693e6	−1.12665
$372$	−591421.	−0.221584
$373$	2.27988e6	0.848475	0.424238	−	0.905551i	$-0.360542\pi$
$373$	2.27988e6	0.848475	0.424238	+	0.905551i	$0.360542\pi$
$374$	−5200.94	−0.00192266
$375$	869540.	0.319309
$376$	−3.84039e6	−1.40090
$377$	6.18442e6	2.24102
$378$	−329988.	−0.118787
$379$	966148.	0.345498	0.172749	−	0.984966i	$-0.444735\pi$
$379$	966148.	0.345498	0.172749	+	0.984966i	$0.444735\pi$
$380$	−1.04124e6	−0.369907
$381$	1.93381e6	0.682500
$382$	2.60211e6	0.912363
$383$	213363.	0.0743230	0.0371615	−	0.999309i	$-0.488168\pi$
$383$	213363.	0.0743230	0.0371615	+	0.999309i	$0.488168\pi$
$384$	1.35434e6	0.468704
$385$	−95938.4	−0.0329868
$386$	−1.33483e6	−0.455993
$387$	393395.	0.133522
$388$	2.07685e6	0.700367
$389$	−4.01270e6	−1.34451	−0.672253	−	0.740322i	$-0.734673\pi$
$389$	−4.01270e6	−1.34451	−0.672253	+	0.740322i	$0.734673\pi$
$390$	1.77713e6	0.591641
$391$	−175257.	−0.0579742
$392$	684171.	0.224879
$393$	935062.	0.305393
$394$	2.16318e6	0.702024
$395$	−4.62325e6	−1.49092
$396$	−17109.1	−0.00548265
$397$	4.63362e6	1.47552	0.737758	−	0.675065i	$-0.235884\pi$
$397$	4.63362e6	1.47552	0.737758	+	0.675065i	$0.235884\pi$
$398$	−2.64704e6	−0.837630
$399$	−874941.	−0.275135
$400$	307256.	0.0960176
$401$	−5.76455e6	−1.79021	−0.895106	−	0.445853i	$-0.852900\pi$
$401$	−5.76455e6	−1.79021	−0.895106	+	0.445853i	$0.852900\pi$
$402$	−472717.	−0.145893
$403$	2.67637e6	0.820887
$404$	−2.44817e6	−0.746255
$405$	−457617.	−0.138632
$406$	−3.10099e6	−0.933654
$407$	56099.6	0.0167870
$408$	266061.	0.0791282
$409$	3.56831e6	1.05476	0.527381	−	0.849629i	$-0.323174\pi$
$409$	3.56831e6	1.05476	0.527381	+	0.849629i	$0.323174\pi$
$410$	253170.	0.0743793
$411$	−2.83023e6	−0.826452
$412$	−1.08520e6	−0.314969
$413$	−6.94806e6	−2.00442
$414$	255798.	0.0733494
$415$	6.59743e6	1.88042
$416$	−5.40700e6	−1.53187
$417$	1.08174e6	0.304637
$418$	20127.2	0.00563434
$419$	−2.47704e6	−0.689284	−0.344642	−	0.938734i	$-0.612000\pi$
$419$	−2.47704e6	−0.689284	−0.344642	+	0.938734i	$0.612000\pi$
$420$	2.00838e6	0.555550
$421$	1.47218e6	0.404813	0.202407	−	0.979302i	$-0.435124\pi$
$421$	1.47218e6	0.404813	0.202407	+	0.979302i	$0.435124\pi$
$422$	726646.	0.198629
$423$	−1.83131e6	−0.497635
$424$	−3.51503e6	−0.949543
$425$	302787.	0.0813140
$426$	−1.91244e6	−0.510580
$427$	−2.09356e6	−0.555669
$428$	−1.51136e6	−0.398803
$429$	77424.3	0.0203111
$430$	1.06231e6	0.277065
$431$	−4.86714e6	−1.26206	−0.631031	−	0.775758i	$-0.717368\pi$
$431$	−4.86714e6	−1.26206	−0.631031	+	0.775758i	$0.717368\pi$
$432$	128745.	0.0331911
$433$	7.42681e6	1.90363	0.951815	−	0.306674i	$-0.0992161\pi$
$433$	7.42681e6	1.90363	0.951815	+	0.306674i	$0.0992161\pi$
$434$	−1.34198e6	−0.341998
$435$	−4.30036e6	−1.08964
$436$	−505376.	−0.127321
$437$	678232.	0.169893
$438$	429562.	0.106989
$439$	1.10468e6	0.273573	0.136787	−	0.990601i	$-0.456323\pi$
$439$	1.10468e6	0.273573	0.136787	+	0.990601i	$0.456323\pi$
$440$	−112901.	−0.0278013
$441$	326250.	0.0798829
$442$	−492703.	−0.119958
$443$	−4.24714e6	−1.02822	−0.514112	−	0.857723i	$-0.671878\pi$
$443$	−4.24714e6	−1.02822	−0.514112	+	0.857723i	$0.671878\pi$
$444$	−1.17439e6	−0.282720
$445$	−8.17466e6	−1.95691
$446$	155950.	0.0371235
$447$	3.44988e6	0.816649
$448$	1.89544e6	0.446186
$449$	2.48857e6	0.582552	0.291276	−	0.956639i	$-0.405920\pi$
$449$	2.48857e6	0.582552	0.291276	+	0.956639i	$0.405920\pi$
$450$	−441935.	−0.102879
$451$	11029.8	0.00255345
$452$	449270.	0.103434
$453$	2.92146e6	0.668891
$454$	16478.5	0.00375214
$455$	−9.08857e6	−2.05810
$456$	−1.02963e6	−0.231884
$457$	6.69515e6	1.49958	0.749790	−	0.661675i	$-0.230154\pi$
$457$	6.69515e6	1.49958	0.749790	+	0.661675i	$0.230154\pi$
$458$	−3.51547e6	−0.783104
$459$	126873.	0.0281084
$460$	−1.55685e6	−0.343045
$461$	−3.77198e6	−0.826642	−0.413321	−	0.910585i	$-0.635631\pi$
$461$	−3.77198e6	−0.826642	−0.413321	+	0.910585i	$0.635631\pi$
$462$	−38822.1	−0.00846202
$463$	−1.16882e6	−0.253394	−0.126697	−	0.991941i	$-0.540438\pi$
$463$	−1.16882e6	−0.253394	−0.126697	+	0.991941i	$0.540438\pi$
$464$	1.20986e6	0.260879
$465$	−1.86102e6	−0.399134
$466$	−870920.	−0.185786
$467$	283602.	0.0601752	0.0300876	−	0.999547i	$-0.490421\pi$
$467$	283602.	0.0601752	0.0300876	+	0.999547i	$0.490421\pi$
$468$	−1.62081e6	−0.342071
$469$	2.41756e6	0.507510
$470$	−4.94522e6	−1.03262
$471$	1.76075e6	0.365718
$472$	−8.17652e6	−1.68933
$473$	46281.8	0.00951167
$474$	−1.87083e6	−0.382462
$475$	−1.17176e6	−0.238290
$476$	−556816.	−0.112640
$477$	−1.67616e6	−0.337302
$478$	−4.65051e6	−0.930959
$479$	3.83084e6	0.762878	0.381439	−	0.924394i	$-0.375429\pi$
$479$	3.83084e6	0.762878	0.381439	+	0.924394i	$0.375429\pi$
$480$	3.75977e6	0.744833
$481$	5.31451e6	1.04737
$482$	−4.31066e6	−0.845135
$483$	−1.30820e6	−0.255156
$484$	3.56776e6	0.692281
$485$	6.53522e6	1.26155
$486$	−185178.	−0.0355630
$487$	−2.52648e6	−0.482719	−0.241359	−	0.970436i	$-0.577593\pi$
$487$	−2.52648e6	−0.482719	−0.241359	+	0.970436i	$0.577593\pi$
$488$	−2.46371e6	−0.468318
$489$	−1.05410e6	−0.199347
$490$	880996.	0.165762
$491$	−123430.	−0.0231055	−0.0115528	−	0.999933i	$-0.503677\pi$
$491$	−123430.	−0.0231055	−0.0115528	+	0.999933i	$0.503677\pi$
$492$	−230900.	−0.0430042
$493$	1.19226e6	0.220929
$494$	1.90672e6	0.351536
$495$	−53837.3	−0.00987575
$496$	523576.	0.0955599
$497$	9.78055e6	1.77612
$498$	2.66969e6	0.482379
$499$	5.91798e6	1.06395	0.531976	−	0.846759i	$-0.321450\pi$
$499$	5.91798e6	1.06395	0.531976	+	0.846759i	$0.321450\pi$
$500$	−2.14153e6	−0.383089
$501$	5.40843e6	0.962669
$502$	−4.30674e6	−0.762763
$503$	7.78155e6	1.37134	0.685672	−	0.727911i	$-0.259509\pi$
$503$	7.78155e6	1.37134	0.685672	+	0.727911i	$0.259509\pi$
$504$	1.98600e6	0.348259
$505$	−7.70363e6	−1.34421
$506$	30093.9	0.00522519
$507$	3.99303e6	0.689895
$508$	−4.76266e6	−0.818824
$509$	−5.37066e6	−0.918826	−0.459413	−	0.888223i	$-0.651940\pi$
$509$	−5.37066e6	−0.918826	−0.459413	+	0.888223i	$0.651940\pi$
$510$	342603.	0.0583264
$511$	−2.19686e6	−0.372177
$512$	−2.01773e6	−0.340163
$513$	−490986.	−0.0823713
$514$	−507291.	−0.0846934
$515$	−3.41481e6	−0.567346
$516$	−968866.	−0.160192
$517$	−215448.	−0.0354500
$518$	−2.66480e6	−0.436355
$519$	−823290.	−0.134163
$520$	−1.06955e7	−1.73457
$521$	839164.	0.135442	0.0677209	−	0.997704i	$-0.478427\pi$
$521$	839164.	0.135442	0.0677209	+	0.997704i	$0.478427\pi$
$522$	−1.74017e6	−0.279521
$523$	−3.46527e6	−0.553966	−0.276983	−	0.960875i	$-0.589335\pi$
$523$	−3.46527e6	−0.553966	−0.276983	+	0.960875i	$0.589335\pi$
$524$	−2.30290e6	−0.366393
$525$	2.26014e6	0.357879
$526$	3.99856e6	0.630143
$527$	515961.	0.0809264
$528$	15146.5	0.00236443
$529$	−5.42226e6	−0.842445
$530$	−4.52625e6	−0.699921
$531$	−3.89901e6	−0.600092
$532$	2.15483e6	0.330092
$533$	1.04489e6	0.159314
$534$	−3.30793e6	−0.502000
$535$	−4.75579e6	−0.718353
$536$	2.84499e6	0.427729
$537$	−2.98709e6	−0.447005
$538$	−3.24751e6	−0.483721
$539$	38382.3	0.00569062
$540$	1.12703e6	0.166323
$541$	9.68754e6	1.42305	0.711525	−	0.702661i	$-0.248005\pi$
$541$	9.68754e6	1.42305	0.711525	+	0.702661i	$0.248005\pi$
$542$	4.19488e6	0.613369
$543$	−2.21001e6	−0.321659
$544$	−1.04238e6	−0.151018
$545$	−1.59026e6	−0.229339
$546$	−3.67775e6	−0.527959
$547$	6.78906e6	0.970156	0.485078	−	0.874471i	$-0.338791\pi$
$547$	6.78906e6	0.970156	0.485078	+	0.874471i	$0.338791\pi$
$548$	6.97039e6	0.991529
$549$	−1.17483e6	−0.166359
$550$	−51992.4	−0.00732880
$551$	−4.61394e6	−0.647430
$552$	−1.53949e6	−0.215045
$553$	9.56774e6	1.33044
$554$	3.45865e6	0.478776
$555$	−3.69546e6	−0.509256
$556$	−2.66415e6	−0.365486
$557$	1.51688e6	0.207164	0.103582	−	0.994621i	$-0.466970\pi$
$557$	1.51688e6	0.207164	0.103582	+	0.994621i	$0.466970\pi$
$558$	−753075.	−0.102389
$559$	4.38443e6	0.593449
$560$	−1.77799e6	−0.239585
$561$	14926.2	0.00200236
$562$	−220710.	−0.0294769
$563$	−9.93096e6	−1.32044	−0.660222	−	0.751070i	$-0.729538\pi$
$563$	−9.93096e6	−1.32044	−0.660222	+	0.751070i	$0.729538\pi$
$564$	4.51021e6	0.597034
$565$	1.41372e6	0.186312
$566$	−8.04025e6	−1.05494
$567$	947032.	0.123711
$568$	1.15098e7	1.49691
$569$	−2.97756e6	−0.385549	−0.192774	−	0.981243i	$-0.561749\pi$
$569$	−2.97756e6	−0.385549	−0.192774	+	0.981243i	$0.561749\pi$
$570$	−1.32584e6	−0.170925
$571$	3.34731e6	0.429640	0.214820	−	0.976654i	$-0.431083\pi$
$571$	3.34731e6	0.429640	0.214820	+	0.976654i	$0.431083\pi$
$572$	−190683.	−0.0243681
$573$	−7.46780e6	−0.950180
$574$	−523931.	−0.0663735
$575$	−1.75200e6	−0.220986
$576$	1.06366e6	0.133581
$577$	1.07096e7	1.33916	0.669582	−	0.742738i	$-0.266473\pi$
$577$	1.07096e7	1.33916	0.669582	+	0.742738i	$0.266473\pi$
$578$	4.35769e6	0.542546
$579$	3.83083e6	0.474893
$580$	1.05911e7	1.30728
$581$	−1.36533e7	−1.67802
$582$	2.64452e6	0.323623
$583$	−197195.	−0.0240284
$584$	−2.58527e6	−0.313671
$585$	−5.10019e6	−0.616164
$586$	−5.09508e6	−0.612924
$587$	−7.80689e6	−0.935154	−0.467577	−	0.883952i	$-0.654873\pi$
$587$	−7.80689e6	−0.935154	−0.467577	+	0.883952i	$0.654873\pi$
$588$	−803499.	−0.0958390
$589$	−1.99673e6	−0.237154
$590$	−1.05288e7	−1.24522
$591$	−6.20811e6	−0.731123
$592$	1.03967e6	0.121925
$593$	8.45179e6	0.986988	0.493494	−	0.869749i	$-0.335719\pi$
$593$	8.45179e6	0.986988	0.493494	+	0.869749i	$0.335719\pi$
$594$	−21785.6	−0.00253340
$595$	−1.75213e6	−0.202896
$596$	−8.49648e6	−0.979768
$597$	7.59672e6	0.872350
$598$	2.85090e6	0.326008
$599$	1.39199e7	1.58515	0.792573	−	0.609777i	$-0.208741\pi$
$599$	1.39199e7	1.58515	0.792573	+	0.609777i	$0.208741\pi$
$600$	2.65974e6	0.301621
$601$	1.13558e7	1.28243	0.641213	−	0.767363i	$-0.278432\pi$
$601$	1.13558e7	1.28243	0.641213	+	0.767363i	$0.278432\pi$
$602$	−2.19844e6	−0.247243
$603$	1.35665e6	0.151941
$604$	−7.19508e6	−0.802497
$605$	1.12267e7	1.24699
$606$	−3.11733e6	−0.344826
$607$	−1.41617e7	−1.56007	−0.780034	−	0.625737i	$-0.784798\pi$
$607$	−1.41617e7	−1.56007	−0.780034	+	0.625737i	$0.784798\pi$
$608$	4.03394e6	0.442558
$609$	8.89953e6	0.972353
$610$	−3.17248e6	−0.345203
$611$	−2.04101e7	−2.21179
$612$	−312466.	−0.0337228
$613$	1.07943e7	1.16023	0.580113	−	0.814536i	$-0.303009\pi$
$613$	1.07943e7	1.16023	0.580113	+	0.814536i	$0.303009\pi$
$614$	−1.02502e6	−0.109727
$615$	−726571.	−0.0774623
$616$	233647.	0.0248089
$617$	−5.01677e6	−0.530531	−0.265266	−	0.964175i	$-0.585460\pi$
$617$	−5.01677e6	−0.530531	−0.265266	+	0.964175i	$0.585460\pi$
$618$	−1.38182e6	−0.145540
$619$	−411275.	−0.0431425	−0.0215712	−	0.999767i	$-0.506867\pi$
$619$	−411275.	−0.0431425	−0.0215712	+	0.999767i	$0.506867\pi$
$620$	4.58339e6	0.478858
$621$	−734115.	−0.0763897
$622$	−2.32745e6	−0.241215
$623$	1.69173e7	1.74627
$624$	1.43488e6	0.147521
$625$	−1.21756e7	−1.24678
$626$	9.61195e6	0.980338
$627$	−57763.1	−0.00586788
$628$	−4.33644e6	−0.438767
$629$	1.02455e6	0.103254
$630$	2.55734e6	0.256706
$631$	−3.44797e6	−0.344739	−0.172369	−	0.985032i	$-0.555142\pi$
$631$	−3.44797e6	−0.344739	−0.172369	+	0.985032i	$0.555142\pi$
$632$	1.12594e7	1.12130
$633$	−2.08540e6	−0.206862
$634$	−1.20512e6	−0.119071
$635$	−1.49866e7	−1.47493
$636$	4.12810e6	0.404676
$637$	3.63609e6	0.355047
$638$	−204726.	−0.0199123
$639$	5.48850e6	0.531743
$640$	−1.04958e7	−1.01290
$641$	−1.61676e7	−1.55418	−0.777089	−	0.629390i	$-0.783305\pi$
$641$	−1.61676e7	−1.55418	−0.777089	+	0.629390i	$0.783305\pi$
$642$	−1.92446e6	−0.184277
$643$	4.59621e6	0.438402	0.219201	−	0.975680i	$-0.429655\pi$
$643$	4.59621e6	0.438402	0.219201	+	0.975680i	$0.429655\pi$
$644$	3.22187e6	0.306121
$645$	−3.04873e6	−0.288549
$646$	367585.	0.0346559
$647$	1.67949e7	1.57731	0.788655	−	0.614836i	$-0.210778\pi$
$647$	1.67949e7	1.57731	0.788655	+	0.614836i	$0.210778\pi$
$648$	1.11447e6	0.104263
$649$	−458707.	−0.0427488
$650$	−4.92541e6	−0.457256
$651$	3.85136e6	0.356173
$652$	2.59607e6	0.239165
$653$	−6.02642e6	−0.553065	−0.276533	−	0.961005i	$-0.589185\pi$
$653$	−6.02642e6	−0.553065	−0.276533	+	0.961005i	$0.589185\pi$
$654$	−643511.	−0.0588317
$655$	−7.24653e6	−0.659974
$656$	204412.	0.0185459
$657$	−1.23280e6	−0.111424
$658$	1.02341e7	0.921474
$659$	7.67299e6	0.688258	0.344129	−	0.938922i	$-0.388174\pi$
$659$	7.67299e6	0.688258	0.344129	+	0.938922i	$0.388174\pi$
$660$	132592.	0.0118484
$661$	1.47810e7	1.31583	0.657914	−	0.753093i	$-0.271439\pi$
$661$	1.47810e7	1.31583	0.657914	+	0.753093i	$0.271439\pi$
$662$	−7.21633e6	−0.639987
$663$	1.41401e6	0.124930
$664$	−1.60673e7	−1.41424
$665$	6.78061e6	0.594586
$666$	−1.49539e6	−0.130638
$667$	−6.89869e6	−0.600416
$668$	−1.33201e7	−1.15496
$669$	−447561.	−0.0386622
$670$	3.66345e6	0.315285
$671$	−138216.	−0.0118509
$672$	−7.78080e6	−0.664662
$673$	1.91371e7	1.62869	0.814343	−	0.580383i	$-0.197097\pi$
$673$	1.91371e7	1.62869	0.814343	+	0.580383i	$0.197097\pi$
$674$	−4.42208e6	−0.374953
$675$	1.26831e6	0.107143
$676$	−9.83417e6	−0.827697
$677$	7.29229e6	0.611494	0.305747	−	0.952113i	$-0.401094\pi$
$677$	7.29229e6	0.611494	0.305747	+	0.952113i	$0.401094\pi$
$678$	572070.	0.0477941
$679$	−1.35245e7	−1.12577
$680$	−2.06192e6	−0.171001
$681$	−47291.7	−0.00390766
$682$	−88597.0	−0.00729387
$683$	1.77201e7	1.45350	0.726748	−	0.686904i	$-0.241031\pi$
$683$	1.77201e7	1.45350	0.726748	+	0.686904i	$0.241031\pi$
$684$	1.20922e6	0.0988244
$685$	2.19337e7	1.78602
$686$	5.78462e6	0.469315
$687$	1.00890e7	0.815564
$688$	857723.	0.0690838
$689$	−1.86810e7	−1.49917
$690$	−1.98238e6	−0.158513
$691$	−8.78404e6	−0.699841	−0.349920	−	0.936779i	$-0.613791\pi$
$691$	−8.78404e6	−0.699841	−0.349920	+	0.936779i	$0.613791\pi$
$692$	2.02762e6	0.160962
$693$	111415.	0.00881277
$694$	3.95482e6	0.311694
$695$	−8.38326e6	−0.658341
$696$	1.04730e7	0.819499
$697$	201439.	0.0157059
$698$	3.59471e6	0.279271
$699$	2.49945e6	0.193487
$700$	−5.56634e6	−0.429363
$701$	1.66519e7	1.27988	0.639938	−	0.768427i	$-0.278960\pi$
$701$	1.66519e7	1.27988	0.639938	+	0.768427i	$0.278960\pi$
$702$	−2.06382e6	−0.158063
$703$	−3.96493e6	−0.302585
$704$	125136.	0.00951592
$705$	1.41923e7	1.07542
$706$	5.69097e6	0.429709
$707$	1.59426e7	1.19952
$708$	9.60261e6	0.719956
$709$	435549.	0.0325403	0.0162701	−	0.999868i	$-0.494821\pi$
$709$	435549.	0.0325403	0.0162701	+	0.999868i	$0.494821\pi$
$710$	1.48210e7	1.10340
$711$	5.36908e6	0.398314
$712$	1.99084e7	1.47176
$713$	−2.98547e6	−0.219932
$714$	−709011.	−0.0520484
$715$	−600022.	−0.0438937
$716$	7.35670e6	0.536291
$717$	1.33465e7	0.969547
$718$	−2.66237e6	−0.192734
$719$	5.22384e6	0.376849	0.188425	−	0.982088i	$-0.439662\pi$
$719$	5.22384e6	0.376849	0.188425	+	0.982088i	$0.439662\pi$
$720$	−997747.	−0.0717281
$721$	7.06689e6	0.506279
$722$	6.34252e6	0.452813
$723$	1.23712e7	0.880166
$724$	5.44289e6	0.385908
$725$	1.19187e7	0.842137
$726$	4.54294e6	0.319886
$727$	−1.78524e7	−1.25274	−0.626370	−	0.779526i	$-0.715460\pi$
$727$	−1.78524e7	−1.25274	−0.626370	+	0.779526i	$0.715460\pi$
$728$	2.21341e7	1.54787
$729$	531441.	0.0370370
$730$	−3.32901e6	−0.231211
$731$	845247.	0.0585047
$732$	2.89342e6	0.199588
$733$	−9.53695e6	−0.655616	−0.327808	−	0.944744i	$-0.606310\pi$
$733$	−9.53695e6	−0.655616	−0.327808	+	0.944744i	$0.606310\pi$
$734$	2.08439e6	0.142803
$735$	−2.52837e6	−0.172632
$736$	6.03147e6	0.410420
$737$	159606.	0.0108238
$738$	−294012.	−0.0198712
$739$	−2.20032e7	−1.48209	−0.741044	−	0.671457i	$-0.765669\pi$
$739$	−2.20032e7	−1.48209	−0.741044	+	0.671457i	$0.765669\pi$
$740$	9.10130e6	0.610976
$741$	−5.47209e6	−0.366107
$742$	9.36701e6	0.624585
$743$	−2.44524e7	−1.62498	−0.812491	−	0.582973i	$-0.801889\pi$
$743$	−2.44524e7	−1.62498	−0.812491	+	0.582973i	$0.801889\pi$
$744$	4.53230e6	0.300183
$745$	−2.67358e7	−1.76483
$746$	−7.14970e6	−0.470371
$747$	−7.66175e6	−0.502373
$748$	−36760.7	−0.00240231
$749$	9.84203e6	0.641033
$750$	−2.72688e6	−0.177016
$751$	9.42910e6	0.610057	0.305028	−	0.952343i	$-0.401334\pi$
$751$	9.42910e6	0.610057	0.305028	+	0.952343i	$0.401334\pi$
$752$	−3.99282e6	−0.257475
$753$	1.23599e7	0.794379
$754$	−1.93944e7	−1.24236
$755$	−2.26407e7	−1.44552
$756$	−2.33238e6	−0.148421
$757$	8.37001e6	0.530867	0.265434	−	0.964129i	$-0.414485\pi$
$757$	8.37001e6	0.530867	0.265434	+	0.964129i	$0.414485\pi$
$758$	−3.02984e6	−0.191535
$759$	−86366.4	−0.00544177
$760$	7.97945e6	0.501117
$761$	1.59273e7	0.996969	0.498485	−	0.866899i	$-0.333890\pi$
$761$	1.59273e7	0.996969	0.498485	+	0.866899i	$0.333890\pi$
$762$	−6.06445e6	−0.378359
$763$	3.29103e6	0.204654
$764$	1.83919e7	1.13997
$765$	−983235.	−0.0607440
$766$	−669108.	−0.0412026
$767$	−4.34549e7	−2.66717
$768$	−8.02909e6	−0.491206
$769$	2.97274e6	0.181276	0.0906381	−	0.995884i	$-0.471109\pi$
$769$	2.97274e6	0.181276	0.0906381	+	0.995884i	$0.471109\pi$
$770$	300863.	0.0182870
$771$	1.45587e6	0.0882039
$772$	−9.43469e6	−0.569750
$773$	−1.13218e7	−0.681500	−0.340750	−	0.940154i	$-0.610681\pi$
$773$	−1.13218e7	−0.681500	−0.340750	+	0.940154i	$0.610681\pi$
$774$	−1.23369e6	−0.0740206
$775$	5.15792e6	0.308475
$776$	−1.59157e7	−0.948795
$777$	7.64770e6	0.454442
$778$	1.25838e7	0.745356
$779$	−779552.	−0.0460258
$780$	1.25609e7	0.739239
$781$	645706.	0.0378797
$782$	549607.	0.0321392
$783$	4.99411e6	0.291107
$784$	711326.	0.0413313
$785$	−1.36455e7	−0.790340
$786$	−2.93236e6	−0.169301
$787$	−1.66830e7	−0.960149	−0.480074	−	0.877228i	$-0.659390\pi$
$787$	−1.66830e7	−0.960149	−0.480074	+	0.877228i	$0.659390\pi$
$788$	1.52895e7	0.877159
$789$	−1.14755e7	−0.656262
$790$	1.44985e7	0.826525
$791$	−2.92567e6	−0.166258
$792$	131114.	0.00742740
$793$	−1.30936e7	−0.739396
$794$	−1.45310e7	−0.817985
$795$	1.29899e7	0.728932
$796$	−1.87095e7	−1.04660
$797$	1.65718e7	0.924113	0.462056	−	0.886851i	$-0.347112\pi$
$797$	1.65718e7	0.924113	0.462056	+	0.886851i	$0.347112\pi$
$798$	2.74382e6	0.152527
$799$	−3.93475e6	−0.218047
$800$	−1.04204e7	−0.575652
$801$	9.49342e6	0.522807
$802$	1.80776e7	0.992444
$803$	−145035.	−0.00793751
$804$	−3.34120e6	−0.182290
$805$	1.01382e7	0.551408
$806$	−8.39309e6	−0.455076
$807$	9.32002e6	0.503771
$808$	1.87613e7	1.01096
$809$	−7.20315e6	−0.386947	−0.193473	−	0.981106i	$-0.561975\pi$
$809$	−7.20315e6	−0.386947	−0.193473	+	0.981106i	$0.561975\pi$
$810$	1.43509e6	0.0768539
$811$	−3.68576e7	−1.96777	−0.983886	−	0.178794i	$-0.942780\pi$
$811$	−3.68576e7	−1.96777	−0.983886	+	0.178794i	$0.942780\pi$
$812$	−2.19181e7	−1.16657
$813$	−1.20389e7	−0.638793
$814$	−175928.	−0.00930626
$815$	8.16905e6	0.430802
$816$	276621.	0.0145432
$817$	−3.27104e6	−0.171447
$818$	−1.11902e7	−0.584730
$819$	1.05548e7	0.549843
$820$	1.78942e6	0.0929348
$821$	−4.26516e6	−0.220840	−0.110420	−	0.993885i	$-0.535220\pi$
$821$	−4.26516e6	−0.220840	−0.110420	+	0.993885i	$0.535220\pi$
$822$	8.87561e6	0.458162
$823$	5.98989e6	0.308261	0.154131	−	0.988050i	$-0.450742\pi$
$823$	5.98989e6	0.308261	0.154131	+	0.988050i	$0.450742\pi$
$824$	8.31635e6	0.426692
$825$	149213.	0.00763258
$826$	2.17891e7	1.11119
$827$	−3.85955e6	−0.196233	−0.0981167	−	0.995175i	$-0.531282\pi$
$827$	−3.85955e6	−0.196233	−0.0981167	+	0.995175i	$0.531282\pi$
$828$	1.80800e6	0.0916480
$829$	119831.	0.00605594	0.00302797	−	0.999995i	$-0.499036\pi$
$829$	119831.	0.00605594	0.00302797	+	0.999995i	$0.499036\pi$
$830$	−2.06896e7	−1.04245
$831$	−9.92597e6	−0.498621
$832$	1.18546e7	0.593714
$833$	700980.	0.0350020
$834$	−3.39234e6	−0.168882
$835$	−4.19142e7	−2.08039
$836$	142261.	0.00703995
$837$	2.16125e6	0.106633
$838$	7.76801e6	0.382120
$839$	2.93998e7	1.44191	0.720957	−	0.692979i	$-0.243702\pi$
$839$	2.93998e7	1.44191	0.720957	+	0.692979i	$0.243702\pi$
$840$	−1.53910e7	−0.752610
$841$	2.64199e7	1.28808
$842$	−4.61675e6	−0.224417
$843$	633415.	0.0306987
$844$	5.13599e6	0.248181
$845$	−3.09451e7	−1.49091
$846$	5.74299e6	0.275875
$847$	−2.32334e7	−1.11277
$848$	−3.65455e6	−0.174520
$849$	2.30747e7	1.09867
$850$	−949541.	−0.0450782
$851$	−5.92830e6	−0.280612
$852$	−1.35173e7	−0.637954
$853$	−4.07050e7	−1.91547	−0.957735	−	0.287653i	$-0.907125\pi$
$853$	−4.07050e7	−1.91547	−0.957735	+	0.287653i	$0.907125\pi$
$854$	6.56541e6	0.308047
$855$	3.80504e6	0.178010
$856$	1.15822e7	0.540263
$857$	−2.00135e7	−0.930833	−0.465416	−	0.885092i	$-0.654095\pi$
$857$	−2.00135e7	−0.930833	−0.465416	+	0.885092i	$0.654095\pi$
$858$	−242803.	−0.0112599
$859$	5.86249e6	0.271081	0.135541	−	0.990772i	$-0.456723\pi$
$859$	5.86249e6	0.271081	0.135541	+	0.990772i	$0.456723\pi$
$860$	7.50851e6	0.346184
$861$	1.50363e6	0.0691246
$862$	1.52633e7	0.699651
$863$	7.97755e6	0.364622	0.182311	−	0.983241i	$-0.441642\pi$
$863$	7.97755e6	0.364622	0.182311	+	0.983241i	$0.441642\pi$
$864$	−4.36631e6	−0.198990
$865$	6.38032e6	0.289936
$866$	−2.32905e7	−1.05532
$867$	−1.25061e7	−0.565034
$868$	−9.48525e6	−0.427316
$869$	631657.	0.0283747
$870$	1.34859e7	0.604064
$871$	1.51200e7	0.675314
$872$	3.87290e6	0.172482
$873$	−7.58949e6	−0.337037
$874$	−2.12694e6	−0.0941837
$875$	1.39457e7	0.615774
$876$	3.03618e6	0.133680
$877$	−1.15892e7	−0.508810	−0.254405	−	0.967098i	$-0.581880\pi$
$877$	−1.15892e7	−0.508810	−0.254405	+	0.967098i	$0.581880\pi$
$878$	−3.46427e6	−0.151661
$879$	1.46224e7	0.638330
$880$	−117382.	−0.00510969
$881$	6.39499e6	0.277588	0.138794	−	0.990321i	$-0.455677\pi$
$881$	6.39499e6	0.277588	0.138794	+	0.990321i	$0.455677\pi$
$882$	−1.02312e6	−0.0442849
$883$	1.76219e7	0.760590	0.380295	−	0.924865i	$-0.375822\pi$
$883$	1.76219e7	0.760590	0.380295	+	0.924865i	$0.375822\pi$
$884$	−3.48246e6	−0.149884
$885$	3.02165e7	1.29684
$886$	1.33190e7	0.570018
$887$	−1.58521e6	−0.0676515	−0.0338258	−	0.999428i	$-0.510769\pi$
$887$	−1.58521e6	−0.0676515	−0.0338258	+	0.999428i	$0.510769\pi$
$888$	8.99985e6	0.383004
$889$	3.10146e7	1.31617
$890$	2.56358e7	1.08485
$891$	62522.4	0.00263841
$892$	1.10227e6	0.0463847
$893$	1.52272e7	0.638984
$894$	−1.08188e7	−0.452727
$895$	2.31493e7	0.966006
$896$	2.17209e7	0.903875
$897$	−8.18178e6	−0.339521
$898$	−7.80417e6	−0.322950
$899$	2.03099e7	0.838123
$900$	−3.12363e6	−0.128545
$901$	−3.60139e6	−0.147795
$902$	−34589.6	−0.00141556
$903$	6.30929e6	0.257491
$904$	−3.44294e6	−0.140123
$905$	1.71271e7	0.695125
$906$	−9.16172e6	−0.370814
$907$	2.66378e7	1.07518	0.537589	−	0.843207i	$-0.319335\pi$
$907$	2.66378e7	1.07518	0.537589	+	0.843207i	$0.319335\pi$
$908$	116472.	0.00468819
$909$	8.94640e6	0.359119
$910$	2.85018e7	1.14095
$911$	3.92288e7	1.56606	0.783032	−	0.621981i	$-0.213672\pi$
$911$	3.92288e7	1.56606	0.783032	+	0.621981i	$0.213672\pi$
$912$	−1.07050e6	−0.0426187
$913$	−901382.	−0.0357875
$914$	−2.09960e7	−0.831326
$915$	9.10470e6	0.359512
$916$	−2.48476e7	−0.978466
$917$	1.49966e7	0.588937
$918$	−397872.	−0.0155825
$919$	3.90436e7	1.52497	0.762484	−	0.647007i	$-0.223979\pi$
$919$	3.90436e7	1.52497	0.762484	+	0.647007i	$0.223979\pi$
$920$	1.19307e7	0.464727
$921$	2.94171e6	0.114275
$922$	1.18289e7	0.458267
$923$	6.11699e7	2.36338
$924$	−274398.	−0.0105731
$925$	1.02422e7	0.393584
$926$	3.66543e6	0.140474
$927$	3.96569e6	0.151572
$928$	−4.10315e7	−1.56404
$929$	−2.13297e7	−0.810858	−0.405429	−	0.914127i	$-0.632878\pi$
$929$	−2.13297e7	−0.810858	−0.405429	+	0.914127i	$0.632878\pi$
$930$	5.83617e6	0.221269
$931$	−2.71274e6	−0.102573
$932$	−6.15573e6	−0.232134
$933$	6.67954e6	0.251213
$934$	−889378.	−0.0333595
$935$	−115675.	−0.00432722
$936$	1.24209e7	0.463408
$937$	−3.37564e7	−1.25605	−0.628025	−	0.778193i	$-0.716137\pi$
$937$	−3.37564e7	−1.25605	−0.628025	+	0.778193i	$0.716137\pi$
$938$	−7.58146e6	−0.281349
$939$	−2.75853e7	−1.02097
$940$	−3.49532e7	−1.29023
$941$	3.35271e6	0.123430	0.0617151	−	0.998094i	$-0.480343\pi$
$941$	3.35271e6	0.123430	0.0617151	+	0.998094i	$0.480343\pi$
$942$	−5.52172e6	−0.202744
$943$	−1.16557e6	−0.0426835
$944$	−8.50105e6	−0.310486
$945$	−7.33929e6	−0.267347
$946$	−145140.	−0.00527300
$947$	−1.36141e7	−0.493302	−0.246651	−	0.969104i	$-0.579330\pi$
$947$	−1.36141e7	−0.493302	−0.246651	+	0.969104i	$0.579330\pi$
$948$	−1.32232e7	−0.477875
$949$	−1.37397e7	−0.495234
$950$	3.67465e6	0.132101
$951$	3.45857e6	0.124007
$952$	4.26711e6	0.152595
$953$	2.09892e6	0.0748623	0.0374312	−	0.999299i	$-0.488083\pi$
$953$	2.09892e6	0.0748623	0.0374312	+	0.999299i	$0.488083\pi$
$954$	5.25644e6	0.186991
$955$	5.78738e7	2.05340
$956$	−3.28701e7	−1.16321
$957$	587542.	0.0207376
$958$	−1.20135e7	−0.422918
$959$	−4.53914e7	−1.59378
$960$	−8.24311e6	−0.288678
$961$	−1.98399e7	−0.692995
$962$	−1.66663e7	−0.580633
$963$	5.52300e6	0.191915
$964$	−3.04681e7	−1.05597
$965$	−2.96881e7	−1.02628
$966$	4.10251e6	0.141451
$967$	6.82598e6	0.234746	0.117373	−	0.993088i	$-0.462553\pi$
$967$	6.82598e6	0.234746	0.117373	+	0.993088i	$0.462553\pi$
$968$	−2.73412e7	−0.937841
$969$	−1.05493e6	−0.0360923
$970$	−2.04944e7	−0.699370
$971$	1.19216e7	0.405776	0.202888	−	0.979202i	$-0.434967\pi$
$971$	1.19216e7	0.405776	0.202888	+	0.979202i	$0.434967\pi$
$972$	−1.30885e6	−0.0444349
$973$	1.73490e7	0.587480
$974$	7.92306e6	0.267606
$975$	1.41354e7	0.476209
$976$	−2.56150e6	−0.0860736
$977$	2.42970e7	0.814359	0.407180	−	0.913348i	$-0.366512\pi$
$977$	2.42970e7	0.814359	0.407180	+	0.913348i	$0.366512\pi$
$978$	3.30566e6	0.110512
$979$	1.11687e6	0.0372432
$980$	6.22695e6	0.207114
$981$	1.84681e6	0.0612703
$982$	387076.	0.0128091
$983$	−382326.	−0.0126197	−0.00630987	−	0.999980i	$-0.502009\pi$
$983$	−382326.	−0.0126197	−0.00630987	+	0.999980i	$0.502009\pi$
$984$	1.76948e6	0.0582582
$985$	4.81115e7	1.58000
$986$	−3.73892e6	−0.122477
$987$	−2.93707e7	−0.959668
$988$	1.34768e7	0.439234
$989$	−4.89080e6	−0.158997
$990$	168834.	0.00547484
$991$	−1.01407e6	−0.0328008	−0.0164004	−	0.999866i	$-0.505221\pi$
$991$	−1.01407e6	−0.0328008	−0.0164004	+	0.999866i	$0.505221\pi$
$992$	−1.77568e7	−0.572908
$993$	2.07101e7	0.666514
$994$	−3.06718e7	−0.984631
$995$	−5.88730e7	−1.88520
$996$	1.88696e7	0.602718
$997$	2.16742e7	0.690566	0.345283	−	0.938499i	$-0.387783\pi$
$997$	2.16742e7	0.690566	0.345283	+	0.938499i	$0.387783\pi$
$998$	−1.85588e7	−0.589825
$999$	4.29162e6	0.136053

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	669.6.a.d.1.19	✓	51

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
669.6.a.d.1.19	✓	51	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 669 = 3 \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	669.a (trivial)

\(f(q)\)	\(=\)	\(q-3.13600 q^{2} +9.00000 q^{3} -22.1655 q^{4} -69.7481 q^{5} -28.2240 q^{6} +144.343 q^{7} +169.863 q^{8} +81.0000 q^{9} +218.730 q^{10} +9.52941 q^{11} -199.489 q^{12} +902.753 q^{13} -452.658 q^{14} -627.733 q^{15} +176.605 q^{16} +174.036 q^{17} -254.016 q^{18} -673.507 q^{19} +1546.00 q^{20} +1299.08 q^{21} -29.8842 q^{22} -1007.02 q^{23} +1528.77 q^{24} +1739.79 q^{25} -2831.04 q^{26} +729.000 q^{27} -3199.42 q^{28} +6850.62 q^{29} +1968.57 q^{30} +2964.67 q^{31} -5989.45 q^{32} +85.7647 q^{33} -545.778 q^{34} -10067.6 q^{35} -1795.41 q^{36} +5887.00 q^{37} +2112.12 q^{38} +8124.78 q^{39} -11847.6 q^{40} +1157.45 q^{41} -4073.93 q^{42} +4856.73 q^{43} -211.224 q^{44} -5649.59 q^{45} +3158.00 q^{46} -22608.8 q^{47} +1589.45 q^{48} +4027.78 q^{49} -5455.99 q^{50} +1566.33 q^{51} -20010.0 q^{52} -20693.3 q^{53} -2286.14 q^{54} -664.658 q^{55} +24518.5 q^{56} -6061.56 q^{57} -21483.6 q^{58} -48135.9 q^{59} +13914.0 q^{60} -14504.1 q^{61} -9297.22 q^{62} +11691.7 q^{63} +13131.6 q^{64} -62965.3 q^{65} -268.958 q^{66} +16748.7 q^{67} -3857.60 q^{68} -9063.14 q^{69} +31572.1 q^{70} +67759.3 q^{71} +13758.9 q^{72} -15219.7 q^{73} -18461.6 q^{74} +15658.1 q^{75} +14928.6 q^{76} +1375.50 q^{77} -25479.3 q^{78} +66285.0 q^{79} -12317.9 q^{80} +6561.00 q^{81} -3629.77 q^{82} -94589.5 q^{83} -28794.8 q^{84} -12138.7 q^{85} -15230.7 q^{86} +61655.6 q^{87} +1618.69 q^{88} +117203. q^{89} +17717.1 q^{90} +130306. q^{91} +22321.0 q^{92} +26682.1 q^{93} +70901.1 q^{94} +46975.8 q^{95} -53905.1 q^{96} -93697.5 q^{97} -12631.1 q^{98} +771.882 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 51 q + 18 q^{2} + 459 q^{3} + 940 q^{4} + 97 q^{5} + 162 q^{6} + 794 q^{7} + 411 q^{8} + 4131 q^{9} + 1256 q^{10} + 2349 q^{11} + 8460 q^{12} + 2954 q^{13} + 1816 q^{14} + 873 q^{15} + 18580 q^{16} + 1567 q^{17}+ \cdots + 190269 q^{99}+O(q^{100}) \) 51 * q + 18 * q^2 + 459 * q^3 + 940 * q^4 + 97 * q^5 + 162 * q^6 + 794 * q^7 + 411 * q^8 + 4131 * q^9 + 1256 * q^10 + 2349 * q^11 + 8460 * q^12 + 2954 * q^13 + 1816 * q^14 + 873 * q^15 + 18580 * q^16 + 1567 * q^17 + 1458 * q^18 + 12864 * q^19 + 3994 * q^20 + 7146 * q^21 + 5748 * q^22 + 6061 * q^23 + 3699 * q^24 + 52170 * q^25 + 14726 * q^26 + 37179 * q^27 + 40542 * q^28 + 4105 * q^29 + 11304 * q^30 + 27770 * q^31 + 3060 * q^32 + 21141 * q^33 + 37991 * q^34 + 26512 * q^35 + 76140 * q^36 + 38060 * q^37 + 12624 * q^38 + 26586 * q^39 + 45349 * q^40 + 54340 * q^41 + 16344 * q^42 + 66199 * q^43 + 52675 * q^44 + 7857 * q^45 + 63342 * q^46 + 63205 * q^47 + 167220 * q^48 + 205487 * q^49 + 48002 * q^50 + 14103 * q^51 + 105206 * q^52 + 11517 * q^53 + 13122 * q^54 + 121327 * q^55 + 153279 * q^56 + 115776 * q^57 + 145784 * q^58 + 219984 * q^59 + 35946 * q^60 + 163444 * q^61 + 415346 * q^62 + 64314 * q^63 + 558819 * q^64 + 191452 * q^65 + 51732 * q^66 + 247277 * q^67 + 310695 * q^68 + 54549 * q^69 + 366374 * q^70 + 238881 * q^71 + 33291 * q^72 + 380013 * q^73 + 355558 * q^74 + 469530 * q^75 + 513873 * q^76 + 91288 * q^77 + 132534 * q^78 + 369295 * q^79 + 319302 * q^80 + 334611 * q^81 + 93313 * q^82 + 310346 * q^83 + 364878 * q^84 + 336089 * q^85 + 72078 * q^86 + 36945 * q^87 + 480438 * q^88 + 325708 * q^89 + 101736 * q^90 + 555778 * q^91 + 39057 * q^92 + 249930 * q^93 + 297564 * q^94 - 67658 * q^95 + 27540 * q^96 + 418485 * q^97 - 75973 * q^98 + 190269 * q^99

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