LMFDB - Embedded newform 669.6.a.d.1.13

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			1.13
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)

$f(q)$	$=$	\(q-6.68652 q^{2} +9.00000 q^{3} +12.7095 q^{4} -80.2857 q^{5} -60.1787 q^{6} -121.262 q^{7} +128.986 q^{8} +81.0000 q^{9} +536.832 q^{10} +450.028 q^{11} +114.386 q^{12} -784.551 q^{13} +810.818 q^{14} -722.572 q^{15} -1269.17 q^{16} +1613.01 q^{17} -541.608 q^{18} +2806.67 q^{19} -1020.39 q^{20} -1091.36 q^{21} -3009.12 q^{22} -3249.08 q^{23} +1160.88 q^{24} +3320.80 q^{25} +5245.91 q^{26} +729.000 q^{27} -1541.18 q^{28} -3966.14 q^{29} +4831.49 q^{30} -1046.29 q^{31} +4358.79 q^{32} +4050.25 q^{33} -10785.4 q^{34} +9735.58 q^{35} +1029.47 q^{36} -14717.4 q^{37} -18766.9 q^{38} -7060.96 q^{39} -10355.8 q^{40} +8520.30 q^{41} +7297.36 q^{42} -10128.7 q^{43} +5719.64 q^{44} -6503.15 q^{45} +21725.0 q^{46} -28238.9 q^{47} -11422.6 q^{48} -2102.60 q^{49} -22204.6 q^{50} +14517.1 q^{51} -9971.25 q^{52} -15332.0 q^{53} -4874.47 q^{54} -36130.8 q^{55} -15641.1 q^{56} +25260.0 q^{57} +26519.6 q^{58} -3042.30 q^{59} -9183.53 q^{60} +15619.3 q^{61} +6996.06 q^{62} -9822.20 q^{63} +11468.4 q^{64} +62988.2 q^{65} -27082.1 q^{66} +31949.0 q^{67} +20500.6 q^{68} -29241.7 q^{69} -65097.2 q^{70} +36132.8 q^{71} +10447.9 q^{72} -85110.6 q^{73} +98408.5 q^{74} +29887.2 q^{75} +35671.4 q^{76} -54571.2 q^{77} +47213.2 q^{78} +94544.5 q^{79} +101896. q^{80} +6561.00 q^{81} -56971.2 q^{82} -52560.5 q^{83} -13870.6 q^{84} -129502. q^{85} +67726.0 q^{86} -35695.2 q^{87} +58047.4 q^{88} -22746.1 q^{89} +43483.4 q^{90} +95135.9 q^{91} -41294.2 q^{92} -9416.64 q^{93} +188820. q^{94} -225336. q^{95} +39229.1 q^{96} +127494. q^{97} +14059.1 q^{98} +36452.3 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−6.68652	−1.18202	−0.591010	−	0.806664i	$-0.701271\pi$
$2$	−6.68652	−1.18202	−0.591010	+	0.806664i	$0.701271\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	12.7095	0.397172
$5$	−80.2857	−1.43620	−0.718098	−	0.695943i	$-0.754987\pi$
$5$	−80.2857	−1.43620	−0.718098	+	0.695943i	$0.754987\pi$
$6$	−60.1787	−0.682440
$7$	−121.262	−0.935359	−0.467680	−	0.883898i	$-0.654910\pi$
$7$	−121.262	−0.935359	−0.467680	+	0.883898i	$0.654910\pi$
$8$	128.986	0.712555
$9$	81.0000	0.333333
$10$	536.832	1.69761
$11$	450.028	1.12139	0.560697	−	0.828021i	$-0.310533\pi$
$11$	450.028	1.12139	0.560697	+	0.828021i	$0.310533\pi$
$12$	114.386	0.229308
$13$	−784.551	−1.28755	−0.643773	−	0.765217i	$-0.722632\pi$
$13$	−784.551	−1.28755	−0.643773	+	0.765217i	$0.722632\pi$
$14$	810.818	1.10561
$15$	−722.572	−0.829188
$16$	−1269.17	−1.23943
$17$	1613.01	1.35367	0.676837	−	0.736132i	$-0.263350\pi$
$17$	1613.01	1.35367	0.676837	+	0.736132i	$0.263350\pi$
$18$	−541.608	−0.394007
$19$	2806.67	1.78364	0.891821	−	0.452389i	$-0.149428\pi$
$19$	2806.67	1.78364	0.891821	+	0.452389i	$0.149428\pi$
$20$	−1020.39	−0.570417
$21$	−1091.36	−0.540030
$22$	−3009.12	−1.32551
$23$	−3249.08	−1.28068	−0.640340	−	0.768091i	$-0.721207\pi$
$23$	−3249.08	−1.28068	−0.640340	+	0.768091i	$0.721207\pi$
$24$	1160.88	0.411394
$25$	3320.80	1.06266
$26$	5245.91	1.52190
$27$	729.000	0.192450
$28$	−1541.18	−0.371499
$29$	−3966.14	−0.875735	−0.437867	−	0.899040i	$-0.644266\pi$
$29$	−3966.14	−0.875735	−0.437867	+	0.899040i	$0.644266\pi$
$30$	4831.49	0.980117
$31$	−1046.29	−0.195546	−0.0977730	−	0.995209i	$-0.531172\pi$
$31$	−1046.29	−0.195546	−0.0977730	+	0.995209i	$0.531172\pi$
$32$	4358.79	0.752473
$33$	4050.25	0.647437
$34$	−10785.4	−1.60007
$35$	9735.58	1.34336
$36$	1029.47	0.132391
$37$	−14717.4	−1.76737	−0.883687	−	0.468079i	$-0.844946\pi$
$37$	−14717.4	−1.76737	−0.883687	+	0.468079i	$0.844946\pi$
$38$	−18766.9	−2.10830
$39$	−7060.96	−0.743365
$40$	−10355.8	−1.02337
$41$	8520.30	0.791581	0.395791	−	0.918341i	$-0.370471\pi$
$41$	8520.30	0.791581	0.395791	+	0.918341i	$0.370471\pi$
$42$	7297.36	0.638326
$43$	−10128.7	−0.835380	−0.417690	−	0.908590i	$-0.637160\pi$
$43$	−10128.7	−0.835380	−0.417690	+	0.908590i	$0.637160\pi$
$44$	5719.64	0.445386
$45$	−6503.15	−0.478732
$46$	21725.0	1.51379
$47$	−28238.9	−1.86468	−0.932338	−	0.361589i	$-0.882234\pi$
$47$	−28238.9	−1.86468	−0.932338	+	0.361589i	$0.882234\pi$
$48$	−11422.6	−0.715583
$49$	−2102.60	−0.125103
$50$	−22204.6	−1.25608
$51$	14517.1	0.781545
$52$	−9971.25	−0.511377
$53$	−15332.0	−0.749736	−0.374868	−	0.927078i	$-0.622312\pi$
$53$	−15332.0	−0.749736	−0.374868	+	0.927078i	$0.622312\pi$
$54$	−4874.47	−0.227480
$55$	−36130.8	−1.61054
$56$	−15641.1	−0.666495
$57$	25260.0	1.02979
$58$	26519.6	1.03514
$59$	−3042.30	−0.113782	−0.0568908	−	0.998380i	$-0.518119\pi$
$59$	−3042.30	−0.113782	−0.0568908	+	0.998380i	$0.518119\pi$
$60$	−9183.53	−0.329330
$61$	15619.3	0.537448	0.268724	−	0.963217i	$-0.413398\pi$
$61$	15619.3	0.537448	0.268724	+	0.963217i	$0.413398\pi$
$62$	6996.06	0.231139
$63$	−9822.20	−0.311786
$64$	11468.4	0.349988
$65$	62988.2	1.84917
$66$	−27082.1	−0.765283
$67$	31949.0	0.869501	0.434750	−	0.900551i	$-0.356837\pi$
$67$	31949.0	0.869501	0.434750	+	0.900551i	$0.356837\pi$
$68$	20500.6	0.537642
$69$	−29241.7	−0.739401
$70$	−65097.2	−1.58788
$71$	36132.8	0.850659	0.425330	−	0.905039i	$-0.360158\pi$
$71$	36132.8	0.850659	0.425330	+	0.905039i	$0.360158\pi$
$72$	10447.9	0.237518
$73$	−85110.6	−1.86929	−0.934645	−	0.355582i	$-0.884283\pi$
$73$	−85110.6	−1.86929	−0.934645	+	0.355582i	$0.884283\pi$
$74$	98408.5	2.08907
$75$	29887.2	0.613525
$76$	35671.4	0.708413
$77$	−54571.2	−1.04891
$78$	47213.2	0.878672
$79$	94544.5	1.70439	0.852194	−	0.523226i	$-0.175272\pi$
$79$	94544.5	1.70439	0.852194	+	0.523226i	$0.175272\pi$
$80$	101896.	1.78006
$81$	6561.00	0.111111
$82$	−56971.2	−0.935665
$83$	−52560.5	−0.837461	−0.418730	−	0.908111i	$-0.637525\pi$
$83$	−52560.5	−0.837461	−0.418730	+	0.908111i	$0.637525\pi$
$84$	−13870.6	−0.214485
$85$	−129502.	−1.94414
$86$	67726.0	0.987437
$87$	−35695.2	−0.505606
$88$	58047.4	0.799054
$89$	−22746.1	−0.304391	−0.152196	−	0.988350i	$-0.548634\pi$
$89$	−22746.1	−0.304391	−0.152196	+	0.988350i	$0.548634\pi$
$90$	43483.4	0.565871
$91$	95135.9	1.20432
$92$	−41294.2	−0.508651
$93$	−9416.64	−0.112899
$94$	188820.	2.20408
$95$	−225336.	−2.56166
$96$	39229.1	0.434440
$97$	127494.	1.37582	0.687908	−	0.725798i	$-0.258529\pi$
$97$	127494.	1.37582	0.687908	+	0.725798i	$0.258529\pi$
$98$	14059.1	0.147874
$99$	36452.3	0.373798
$100$	42205.8	0.422058
$101$	2971.48	0.0289848	0.0144924	−	0.999895i	$-0.495387\pi$
$101$	2971.48	0.0289848	0.0144924	+	0.999895i	$0.495387\pi$
$102$	−97068.7	−0.923802
$103$	49608.3	0.460745	0.230373	−	0.973102i	$-0.426005\pi$
$103$	49608.3	0.460745	0.230373	+	0.973102i	$0.426005\pi$
$104$	−101196.	−0.917447
$105$	87620.3	0.775588
$106$	102517.	0.886203
$107$	47209.4	0.398629	0.199315	−	0.979936i	$-0.436128\pi$
$107$	47209.4	0.398629	0.199315	+	0.979936i	$0.436128\pi$
$108$	9265.23	0.0764358
$109$	126133.	1.01686	0.508430	−	0.861103i	$-0.330226\pi$
$109$	126133.	1.01686	0.508430	+	0.861103i	$0.330226\pi$
$110$	241590.	1.90369
$111$	−132457.	−1.02039
$112$	153902.	1.15931
$113$	−160538.	−1.18272	−0.591360	−	0.806408i	$-0.701409\pi$
$113$	−160538.	−1.18272	−0.591360	+	0.806408i	$0.701409\pi$
$114$	−168902.	−1.21723
$115$	260855.	1.83931
$116$	−50407.6	−0.347818
$117$	−63548.6	−0.429182
$118$	20342.4	0.134492
$119$	−195596.	−1.26617
$120$	−93201.8	−0.590842
$121$	41474.4	0.257523
$122$	−104439.	−0.635275
$123$	76682.7	0.457020
$124$	−13297.9	−0.0776655
$125$	−15720.0	−0.0899864
$126$	65676.3	0.368538
$127$	−25751.3	−0.141674	−0.0708369	−	0.997488i	$-0.522567\pi$
$127$	−25751.3	−0.141674	−0.0708369	+	0.997488i	$0.522567\pi$
$128$	−216165.	−1.16617
$129$	−91158.7	−0.482307
$130$	−421172.	−2.18575
$131$	−194300.	−0.989222	−0.494611	−	0.869115i	$-0.664689\pi$
$131$	−194300.	−0.989222	−0.494611	+	0.869115i	$0.664689\pi$
$132$	51476.7	0.257144
$133$	−340342.	−1.66835
$134$	−213627.	−1.02777
$135$	−58528.3	−0.276396
$136$	208056.	0.964567
$137$	−237560.	−1.08137	−0.540683	−	0.841226i	$-0.681834\pi$
$137$	−237560.	−1.08137	−0.540683	+	0.841226i	$0.681834\pi$
$138$	195525.	0.873987
$139$	−374846.	−1.64557	−0.822784	−	0.568354i	$-0.807580\pi$
$139$	−374846.	−1.64557	−0.822784	+	0.568354i	$0.807580\pi$
$140$	123735.	0.533545
$141$	−254150.	−1.07657
$142$	−241603.	−1.00550
$143$	−353070.	−1.44384
$144$	−102803.	−0.413142
$145$	318424.	1.25773
$146$	569094.	2.20954
$147$	−18923.4	−0.0722282
$148$	−187052.	−0.701952
$149$	−192839.	−0.711588	−0.355794	−	0.934564i	$-0.615790\pi$
$149$	−192839.	−0.711588	−0.355794	+	0.934564i	$0.615790\pi$
$150$	−199841.	−0.725199
$151$	267986.	0.956467	0.478233	−	0.878233i	$-0.341277\pi$
$151$	267986.	0.956467	0.478233	+	0.878233i	$0.341277\pi$
$152$	362022.	1.27094
$153$	130654.	0.451225
$154$	364891.	1.23983
$155$	84002.4	0.280842
$156$	−89741.3	−0.295244
$157$	383958.	1.24318	0.621591	−	0.783342i	$-0.286487\pi$
$157$	383958.	1.24318	0.621591	+	0.783342i	$0.286487\pi$
$158$	−632173.	−2.01462
$159$	−137988.	−0.432860
$160$	−349948.	−1.08070
$161$	393989.	1.19790
$162$	−43870.2	−0.131336
$163$	−144259.	−0.425279	−0.212639	−	0.977131i	$-0.568206\pi$
$163$	−144259.	−0.425279	−0.212639	+	0.977131i	$0.568206\pi$
$164$	108289.	0.314394
$165$	−325178.	−0.929845
$166$	351447.	0.989896
$167$	−483341.	−1.34110	−0.670552	−	0.741863i	$-0.733943\pi$
$167$	−483341.	−1.34110	−0.670552	+	0.741863i	$0.733943\pi$
$168$	−140770.	−0.384801
$169$	244227.	0.657773
$170$	865915.	2.29801
$171$	227340.	0.594547
$172$	−128731.	−0.331790
$173$	635228.	1.61367	0.806834	−	0.590778i	$-0.201179\pi$
$173$	635228.	1.61367	0.806834	+	0.590778i	$0.201179\pi$
$174$	238677.	0.597636
$175$	−402686.	−0.993965
$176$	−571163.	−1.38988
$177$	−27380.7	−0.0656918
$178$	152092.	0.359797
$179$	199421.	0.465198	0.232599	−	0.972573i	$-0.425277\pi$
$179$	199421.	0.465198	0.232599	+	0.972573i	$0.425277\pi$
$180$	−82651.8	−0.190139
$181$	−159176.	−0.361145	−0.180573	−	0.983562i	$-0.557795\pi$
$181$	−159176.	−0.361145	−0.180573	+	0.983562i	$0.557795\pi$
$182$	−636128.	−1.42353
$183$	140574.	0.310296
$184$	−419086.	−0.912555
$185$	1.18160e6	2.53829
$186$	62964.5	0.133448
$187$	725899.	1.51800
$188$	−358903.	−0.740597
$189$	−88399.8	−0.180010
$190$	1.50671e6	3.02793
$191$	782991.	1.55301	0.776503	−	0.630113i	$-0.216991\pi$
$191$	782991.	1.55301	0.776503	+	0.630113i	$0.216991\pi$
$192$	103216.	0.202066
$193$	578398.	1.11772	0.558861	−	0.829261i	$-0.311239\pi$
$193$	578398.	1.11772	0.558861	+	0.829261i	$0.311239\pi$
$194$	−852491.	−1.62624
$195$	566894.	1.06762
$196$	−26723.1	−0.0496874
$197$	177170.	0.325256	0.162628	−	0.986687i	$-0.448003\pi$
$197$	177170.	0.325256	0.162628	+	0.986687i	$0.448003\pi$
$198$	−243739.	−0.441837
$199$	364773.	0.652965	0.326482	−	0.945203i	$-0.394137\pi$
$199$	364773.	0.652965	0.326482	+	0.945203i	$0.394137\pi$
$200$	428337.	0.757201
$201$	287541.	0.502007
$202$	−19868.9	−0.0342606
$203$	480940.	0.819127
$204$	184505.	0.310408
$205$	−684059.	−1.13686
$206$	−331707.	−0.544610
$207$	−263175.	−0.426893
$208$	995730.	1.59582
$209$	1.26308e6	2.00016
$210$	−585874.	−0.916761
$211$	−76016.9	−0.117545	−0.0587725	−	0.998271i	$-0.518719\pi$
$211$	−76016.9	−0.117545	−0.0587725	+	0.998271i	$0.518719\pi$
$212$	−194862.	−0.297774
$213$	325195.	0.491128
$214$	−315666.	−0.471188
$215$	813193.	1.19977
$216$	94030.9	0.137131
$217$	126875.	0.182906
$218$	−843388.	−1.20195
$219$	−765996.	−1.07924
$220$	−459205.	−0.639662
$221$	−1.26549e6	−1.74292
$222$	885676.	1.20613
$223$	−49729.0	−0.0669650
$223$	−49729.0	−0.0669650
$224$	−528554.	−0.703832
$225$	268985.	0.354219
$226$	1.07344e6	1.39800
$227$	−1.01422e6	−1.30638	−0.653188	−	0.757196i	$-0.726569\pi$
$227$	−1.01422e6	−1.30638	−0.653188	+	0.757196i	$0.726569\pi$
$228$	321043.	0.409002
$229$	1.14663e6	1.44488	0.722442	−	0.691431i	$-0.243019\pi$
$229$	1.14663e6	1.44488	0.722442	+	0.691431i	$0.243019\pi$
$230$	−1.74421e6	−2.17410
$231$	−491141.	−0.605586
$232$	−511577.	−0.624009
$233$	508415.	0.613520	0.306760	−	0.951787i	$-0.400755\pi$
$233$	508415.	0.613520	0.306760	+	0.951787i	$0.400755\pi$
$234$	424919.	0.507302
$235$	2.26718e6	2.67804
$236$	−38666.2	−0.0451909
$237$	850901.	0.984029
$238$	1.30786e6	1.49664
$239$	743599.	0.842062	0.421031	−	0.907046i	$-0.361668\pi$
$239$	743599.	0.842062	0.421031	+	0.907046i	$0.361668\pi$
$240$	917068.	1.02772
$241$	−544832.	−0.604255	−0.302127	−	0.953268i	$-0.597697\pi$
$241$	−544832.	−0.604255	−0.302127	+	0.953268i	$0.597697\pi$
$242$	−277319.	−0.304398
$243$	59049.0	0.0641500
$244$	198513.	0.213459
$245$	168809.	0.179672
$246$	−512740.	−0.540207
$247$	−2.20198e6	−2.29652
$248$	−134957.	−0.139337
$249$	−473045.	−0.483508
$250$	105112.	0.106366
$251$	1.49465e6	1.49746	0.748729	−	0.662876i	$-0.230664\pi$
$251$	1.49465e6	1.49746	0.748729	+	0.662876i	$0.230664\pi$
$252$	−124835.	−0.123833
$253$	−1.46218e6	−1.43615
$254$	172186.	0.167461
$255$	−1.16551e6	−1.12245
$256$	1.07840e6	1.02844
$257$	−262600.	−0.248006	−0.124003	−	0.992282i	$-0.539573\pi$
$257$	−262600.	−0.248006	−0.124003	+	0.992282i	$0.539573\pi$
$258$	609534.	0.570097
$259$	1.78466e6	1.65313
$260$	800550.	0.734438
$261$	−321257.	−0.291912
$262$	1.29919e6	1.16928
$263$	64630.3	0.0576165	0.0288083	−	0.999585i	$-0.490829\pi$
$263$	64630.3	0.0576165	0.0288083	+	0.999585i	$0.490829\pi$
$264$	522427.	0.461334
$265$	1.23094e6	1.07677
$266$	2.27570e6	1.97202
$267$	−204715.	−0.175740
$268$	406056.	0.345342
$269$	1.48657e6	1.25258	0.626290	−	0.779590i	$-0.284572\pi$
$269$	1.48657e6	1.25258	0.626290	+	0.779590i	$0.284572\pi$
$270$	391351.	0.326706
$271$	779512.	0.644762	0.322381	−	0.946610i	$-0.395517\pi$
$271$	779512.	0.644762	0.322381	+	0.946610i	$0.395517\pi$
$272$	−2.04719e6	−1.67778
$273$	856223.	0.695313
$274$	1.58845e6	1.27820
$275$	1.49445e6	1.19166
$276$	−371648.	−0.293670
$277$	664160.	0.520084	0.260042	−	0.965597i	$-0.416264\pi$
$277$	664160.	0.520084	0.260042	+	0.965597i	$0.416264\pi$
$278$	2.50641e6	1.94509
$279$	−84749.7	−0.0651820
$280$	1.25576e6	0.957216
$281$	−2.03513e6	−1.53754	−0.768771	−	0.639525i	$-0.779131\pi$
$281$	−2.03513e6	−1.53754	−0.768771	+	0.639525i	$0.779131\pi$
$282$	1.69938e6	1.27253
$283$	630045.	0.467633	0.233817	−	0.972281i	$-0.424878\pi$
$283$	630045.	0.467633	0.233817	+	0.972281i	$0.424878\pi$
$284$	459230.	0.337858
$285$	−2.02802e6	−1.47897
$286$	2.36081e6	1.70665
$287$	−1.03319e6	−0.740413
$288$	353062.	0.250824
$289$	1.18194e6	0.832436
$290$	−2.12915e6	−1.48666
$291$	1.14745e6	0.794328
$292$	−1.08171e6	−0.742430
$293$	−871514.	−0.593069	−0.296535	−	0.955022i	$-0.595831\pi$
$293$	−871514.	−0.593069	−0.296535	+	0.955022i	$0.595831\pi$
$294$	126532.	0.0853752
$295$	244253.	0.163413
$296$	−1.89835e6	−1.25935
$297$	328071.	0.215812
$298$	1.28942e6	0.841112
$299$	2.54907e6	1.64893
$300$	379852.	0.243675
$301$	1.22823e6	0.781381
$302$	−1.79189e6	−1.13056
$303$	26743.4	0.0167344
$304$	−3.56215e6	−2.21069
$305$	−1.25401e6	−0.771880
$306$	−873618.	−0.533357
$307$	−2.85886e6	−1.73120	−0.865600	−	0.500737i	$-0.833062\pi$
$307$	−2.85886e6	−1.73120	−0.865600	+	0.500737i	$0.833062\pi$
$308$	−693573.	−0.416596
$309$	446474.	0.266011
$310$	−561683.	−0.331961
$311$	−1.98143e6	−1.16165	−0.580827	−	0.814027i	$-0.697271\pi$
$311$	−1.98143e6	−1.16165	−0.580827	+	0.814027i	$0.697271\pi$
$312$	−910766.	−0.529688
$313$	−1.79865e6	−1.03774	−0.518868	−	0.854854i	$-0.673646\pi$
$313$	−1.79865e6	−1.03774	−0.518868	+	0.854854i	$0.673646\pi$
$314$	−2.56734e6	−1.46947
$315$	788582.	0.447786
$316$	1.20161e6	0.676936
$317$	−905450.	−0.506077	−0.253038	−	0.967456i	$-0.581430\pi$
$317$	−905450.	−0.506077	−0.253038	+	0.967456i	$0.581430\pi$
$318$	922657.	0.511650
$319$	−1.78487e6	−0.982043
$320$	−920751.	−0.502652
$321$	424885.	0.230149
$322$	−2.63441e6	−1.41594
$323$	4.52719e6	2.41447
$324$	83387.1	0.0441303
$325$	−2.60534e6	−1.36822
$326$	964589.	0.502688
$327$	1.13519e6	0.587084
$328$	1.09900e6	0.564045
$329$	3.42430e6	1.74414
$330$	2.17431e6	1.09910
$331$	2.93067e6	1.47027	0.735134	−	0.677922i	$-0.237119\pi$
$331$	2.93067e6	1.47027	0.735134	+	0.677922i	$0.237119\pi$
$332$	−668019.	−0.332616
$333$	−1.19211e6	−0.589124
$334$	3.23187e6	1.58521
$335$	−2.56505e6	−1.24877
$336$	1.38512e6	0.669327
$337$	1.46579e6	0.703069	0.351535	−	0.936175i	$-0.385660\pi$
$337$	1.46579e6	0.703069	0.351535	+	0.936175i	$0.385660\pi$
$338$	−1.63303e6	−0.777501
$339$	−1.44484e6	−0.682843
$340$	−1.64590e6	−0.772159
$341$	−470861.	−0.219284
$342$	−1.52012e6	−0.702767
$343$	2.29301e6	1.05238
$344$	−1.30647e6	−0.595254
$345$	2.34769e6	1.06192
$346$	−4.24746e6	−1.90739
$347$	293204.	0.130721	0.0653606	−	0.997862i	$-0.479180\pi$
$347$	293204.	0.130721	0.0653606	+	0.997862i	$0.479180\pi$
$348$	−453669.	−0.200813
$349$	2.22309e6	0.976996	0.488498	−	0.872565i	$-0.337545\pi$
$349$	2.22309e6	0.976996	0.488498	+	0.872565i	$0.337545\pi$
$350$	2.69257e6	1.17489
$351$	−571937.	−0.247788
$352$	1.96158e6	0.843818
$353$	427242.	0.182489	0.0912446	−	0.995829i	$-0.470915\pi$
$353$	427242.	0.182489	0.0912446	+	0.995829i	$0.470915\pi$
$354$	183082.	0.0776491
$355$	−2.90095e6	−1.22171
$356$	−289092.	−0.120896
$357$	−1.76037e6	−0.731025
$358$	−1.33343e6	−0.549873
$359$	−3.09615e6	−1.26790	−0.633951	−	0.773373i	$-0.718568\pi$
$359$	−3.09615e6	−1.26790	−0.633951	+	0.773373i	$0.718568\pi$
$360$	−838816.	−0.341123
$361$	5.40131e6	2.18138
$362$	1.06433e6	0.426881
$363$	373269.	0.148681
$364$	1.20913e6	0.478322
$365$	6.83317e6	2.68466
$366$	−939947.	−0.366776
$367$	525392.	0.203619	0.101809	−	0.994804i	$-0.467537\pi$
$367$	525392.	0.203619	0.101809	+	0.994804i	$0.467537\pi$
$368$	4.12364e6	1.58731
$369$	690145.	0.263860
$370$	−7.90080e6	−3.00031
$371$	1.85918e6	0.701272
$372$	−119681.	−0.0448402
$373$	338070.	0.125816	0.0629079	−	0.998019i	$-0.479963\pi$
$373$	338070.	0.125816	0.0629079	+	0.998019i	$0.479963\pi$
$374$	−4.85374e6	−1.79431
$375$	−141480.	−0.0519537
$376$	−3.64243e6	−1.32868
$377$	3.11163e6	1.12755
$378$	591087.	0.212775
$379$	1.88201e6	0.673015	0.336508	−	0.941681i	$-0.390754\pi$
$379$	1.88201e6	0.673015	0.336508	+	0.941681i	$0.390754\pi$
$380$	−2.86391e6	−1.01742
$381$	−231762.	−0.0817954
$382$	−5.23548e6	−1.83569
$383$	4.40319e6	1.53381	0.766904	−	0.641762i	$-0.221796\pi$
$383$	4.40319e6	1.53381	0.766904	+	0.641762i	$0.221796\pi$
$384$	−1.94548e6	−0.673286
$385$	4.38129e6	1.50643
$386$	−3.86747e6	−1.32117
$387$	−820428.	−0.278460
$388$	1.62039e6	0.546436
$389$	−340593.	−0.114120	−0.0570600	−	0.998371i	$-0.518173\pi$
$389$	−340593.	−0.114120	−0.0570600	+	0.998371i	$0.518173\pi$
$390$	−3.79055e6	−1.26194
$391$	−5.24079e6	−1.73362
$392$	−271207.	−0.0891426
$393$	−1.74870e6	−0.571127
$394$	−1.18465e6	−0.384459
$395$	−7.59058e6	−2.44783
$396$	463291.	0.148462
$397$	3.52628e6	1.12290	0.561450	−	0.827511i	$-0.310244\pi$
$397$	3.52628e6	1.12290	0.561450	+	0.827511i	$0.310244\pi$
$398$	−2.43906e6	−0.771818
$399$	−3.06308e6	−0.963220
$400$	−4.21467e6	−1.31708
$401$	4.48744e6	1.39360	0.696800	−	0.717266i	$-0.254607\pi$
$401$	4.48744e6	1.39360	0.696800	+	0.717266i	$0.254607\pi$
$402$	−1.92265e6	−0.593382
$403$	820870.	0.251774
$404$	37766.1	0.0115119
$405$	−526755.	−0.159577
$406$	−3.21582e6	−0.968224
$407$	−6.62327e6	−1.98192
$408$	1.87250e6	0.556893
$409$	−3.43818e6	−1.01629	−0.508147	−	0.861270i	$-0.669669\pi$
$409$	−3.43818e6	−1.01629	−0.508147	+	0.861270i	$0.669669\pi$
$410$	4.57397e6	1.34380
$411$	−2.13804e6	−0.624327
$412$	630497.	0.182995
$413$	368914.	0.106427
$414$	1.75973e6	0.504597
$415$	4.21986e6	1.20276
$416$	−3.41969e6	−0.968843
$417$	−3.37361e6	−0.950069
$418$	−8.44562e6	−2.36423
$419$	2.98158e6	0.829681	0.414841	−	0.909894i	$-0.363837\pi$
$419$	2.98158e6	0.829681	0.414841	+	0.909894i	$0.363837\pi$
$420$	1.11361e6	0.308042
$421$	−963070.	−0.264821	−0.132411	−	0.991195i	$-0.542272\pi$
$421$	−963070.	−0.264821	−0.132411	+	0.991195i	$0.542272\pi$
$422$	508288.	0.138941
$423$	−2.28735e6	−0.621558
$424$	−1.97761e6	−0.534228
$425$	5.35648e6	1.43849
$426$	−2.17442e6	−0.580524
$427$	−1.89402e6	−0.502707
$428$	600008.	0.158324
$429$	−3.17763e6	−0.833604
$430$	−5.43743e6	−1.41815
$431$	−7.15254e6	−1.85467	−0.927336	−	0.374229i	$-0.877908\pi$
$431$	−7.15254e6	−1.85467	−0.927336	+	0.374229i	$0.877908\pi$
$432$	−925227.	−0.238528
$433$	−180493.	−0.0462637	−0.0231318	−	0.999732i	$-0.507364\pi$
$433$	−180493.	−0.0462637	−0.0231318	+	0.999732i	$0.507364\pi$
$434$	−848353.	−0.216198
$435$	2.86582e6	0.726148
$436$	1.60308e6	0.403869
$437$	−9.11910e6	−2.28428
$438$	5.12184e6	1.27568
$439$	2.05955e6	0.510047	0.255023	−	0.966935i	$-0.417917\pi$
$439$	2.05955e6	0.510047	0.255023	+	0.966935i	$0.417917\pi$
$440$	−4.66038e6	−1.14760
$441$	−170311.	−0.0417010
$442$	8.46170e6	2.06016
$443$	4.25371e6	1.02981	0.514907	−	0.857246i	$-0.327827\pi$
$443$	4.25371e6	1.02981	0.514907	+	0.857246i	$0.327827\pi$
$444$	−1.68346e6	−0.405272
$445$	1.82619e6	0.437165
$446$	332514.	0.0791539
$447$	−1.73555e6	−0.410836
$448$	−1.39068e6	−0.327365
$449$	8.12326e6	1.90158	0.950790	−	0.309837i	$-0.100274\pi$
$449$	8.12326e6	1.90158	0.950790	+	0.309837i	$0.100274\pi$
$450$	−1.79857e6	−0.418694
$451$	3.83438e6	0.887674
$452$	−2.04036e6	−0.469743
$453$	2.41187e6	0.552216
$454$	6.78161e6	1.54416
$455$	−7.63806e6	−1.72964
$456$	3.25820e6	0.733779
$457$	6.26372e6	1.40295	0.701474	−	0.712695i	$-0.252525\pi$
$457$	6.26372e6	1.40295	0.701474	+	0.712695i	$0.252525\pi$
$458$	−7.66694e6	−1.70788
$459$	1.17588e6	0.260515
$460$	3.31534e6	0.730522
$461$	3.81500e6	0.836069	0.418035	−	0.908431i	$-0.362719\pi$
$461$	3.81500e6	0.836069	0.418035	+	0.908431i	$0.362719\pi$
$462$	3.28402e6	0.715815
$463$	−363787.	−0.0788670	−0.0394335	−	0.999222i	$-0.512555\pi$
$463$	−363787.	−0.0788670	−0.0394335	+	0.999222i	$0.512555\pi$
$464$	5.03371e6	1.08541
$465$	756022.	0.162144
$466$	−3.39953e6	−0.725193
$467$	−4.80791e6	−1.02015	−0.510075	−	0.860130i	$-0.670382\pi$
$467$	−4.80791e6	−1.02015	−0.510075	+	0.860130i	$0.670382\pi$
$468$	−807672.	−0.170459
$469$	−3.87419e6	−0.813296
$470$	−1.51595e7	−3.16549
$471$	3.45562e6	0.717751
$472$	−392415.	−0.0810756
$473$	−4.55822e6	−0.936790
$474$	−5.68956e6	−1.16314
$475$	9.32040e6	1.89540
$476$	−2.48593e6	−0.502889
$477$	−1.24189e6	−0.249912
$478$	−4.97209e6	−0.995335
$479$	536768.	0.106893	0.0534463	−	0.998571i	$-0.482979\pi$
$479$	536768.	0.106893	0.0534463	+	0.998571i	$0.482979\pi$
$480$	−3.14954e6	−0.623941
$481$	1.15466e7	2.27557
$482$	3.64303e6	0.714241
$483$	3.54590e6	0.691606
$484$	527119.	0.102281
$485$	−1.02359e7	−1.97594
$486$	−394832.	−0.0758266
$487$	7.41794e6	1.41730	0.708648	−	0.705562i	$-0.249305\pi$
$487$	7.41794e6	1.41730	0.708648	+	0.705562i	$0.249305\pi$
$488$	2.01467e6	0.382961
$489$	−1.29833e6	−0.245535
$490$	−1.12875e6	−0.212376
$491$	−7.95931e6	−1.48995	−0.744975	−	0.667093i	$-0.767538\pi$
$491$	−7.95931e6	−1.48995	−0.744975	+	0.667093i	$0.767538\pi$
$492$	974600.	0.181516
$493$	−6.39741e6	−1.18546
$494$	1.47236e7	2.71453
$495$	−2.92660e6	−0.536847
$496$	1.32793e6	0.242365
$497$	−4.38152e6	−0.795672
$498$	3.16302e6	0.571517
$499$	−3.15169e6	−0.566620	−0.283310	−	0.959028i	$-0.591433\pi$
$499$	−3.15169e6	−0.566620	−0.283310	+	0.959028i	$0.591433\pi$
$500$	−199793.	−0.0357401
$501$	−4.35007e6	−0.774286
$502$	−9.99399e6	−1.77003
$503$	6.47822e6	1.14166	0.570829	−	0.821069i	$-0.306622\pi$
$503$	6.47822e6	1.14166	0.570829	+	0.821069i	$0.306622\pi$
$504$	−1.26693e6	−0.222165
$505$	−238568.	−0.0416278
$506$	9.77687e6	1.69755
$507$	2.19804e6	0.379766
$508$	−327286.	−0.0562689
$509$	−4.09068e6	−0.699843	−0.349922	−	0.936779i	$-0.613792\pi$
$509$	−4.09068e6	−0.699843	−0.349922	+	0.936779i	$0.613792\pi$
$510$	7.79323e6	1.32676
$511$	1.03207e7	1.74846
$512$	−293470.	−0.0494754
$513$	2.04606e6	0.343262
$514$	1.75588e6	0.293148
$515$	−3.98284e6	−0.661720
$516$	−1.15858e6	−0.191559
$517$	−1.27083e7	−2.09103
$518$	−1.19332e7	−1.95403
$519$	5.71705e6	0.931651
$520$	8.12461e6	1.31763
$521$	1.18263e6	0.190878	0.0954391	−	0.995435i	$-0.469574\pi$
$521$	1.18263e6	0.190878	0.0954391	+	0.995435i	$0.469574\pi$
$522$	2.14809e6	0.345045
$523$	8.15650e6	1.30392	0.651958	−	0.758255i	$-0.273948\pi$
$523$	8.15650e6	1.30392	0.651958	+	0.758255i	$0.273948\pi$
$524$	−2.46945e6	−0.392891
$525$	−3.62417e6	−0.573866
$526$	−432152.	−0.0681039
$527$	−1.68768e6	−0.264706
$528$	−5.14047e6	−0.802450
$529$	4.12017e6	0.640142
$530$	−8.23069e6	−1.27276
$531$	−246426.	−0.0379272
$532$	−4.32558e6	−0.662621
$533$	−6.68461e6	−1.01920
$534$	1.36883e6	0.207729
$535$	−3.79024e6	−0.572509
$536$	4.12098e6	0.619567
$537$	1.79479e6	0.268582
$538$	−9.94000e6	−1.48058
$539$	−946231.	−0.140290
$540$	−743866.	−0.109777
$541$	3.54874e6	0.521292	0.260646	−	0.965435i	$-0.416065\pi$
$541$	3.54874e6	0.521292	0.260646	+	0.965435i	$0.416065\pi$
$542$	−5.21222e6	−0.762122
$543$	−1.43259e6	−0.208507
$544$	7.03076e6	1.01860
$545$	−1.01267e7	−1.46041
$546$	−5.72515e6	−0.821874
$547$	3.11608e6	0.445287	0.222643	−	0.974900i	$-0.428531\pi$
$547$	3.11608e6	0.445287	0.222643	+	0.974900i	$0.428531\pi$
$548$	−3.01928e6	−0.429489
$549$	1.26516e6	0.179149
$550$	−9.99269e6	−1.40856
$551$	−1.11316e7	−1.56200
$552$	−3.77178e6	−0.526864
$553$	−1.14646e7	−1.59422
$554$	−4.44092e6	−0.614750
$555$	1.06344e7	1.46548
$556$	−4.76411e6	−0.653574
$557$	3.90125e6	0.532802	0.266401	−	0.963862i	$-0.414165\pi$
$557$	3.90125e6	0.532802	0.266401	+	0.963862i	$0.414165\pi$
$558$	566680.	0.0770465
$559$	7.94651e6	1.07559
$560$	−1.23561e7	−1.66499
$561$	6.53309e6	0.876419
$562$	1.36079e7	1.81740
$563$	−8.61040e6	−1.14486	−0.572430	−	0.819954i	$-0.693999\pi$
$563$	−8.61040e6	−1.14486	−0.572430	+	0.819954i	$0.693999\pi$
$564$	−3.23012e6	−0.427584
$565$	1.28889e7	1.69862
$566$	−4.21281e6	−0.552752
$567$	−795598.	−0.103929
$568$	4.66063e6	0.606141
$569$	−1.07489e6	−0.139182	−0.0695909	−	0.997576i	$-0.522169\pi$
$569$	−1.07489e6	−0.139182	−0.0695909	+	0.997576i	$0.522169\pi$
$570$	1.35604e7	1.74818
$571$	−1.28247e7	−1.64610	−0.823049	−	0.567970i	$-0.807729\pi$
$571$	−1.28247e7	−1.64610	−0.823049	+	0.567970i	$0.807729\pi$
$572$	−4.48735e6	−0.573455
$573$	7.04692e6	0.896629
$574$	6.90842e6	0.875183
$575$	−1.07895e7	−1.36092
$576$	928942.	0.116663
$577$	5.18486e6	0.648332	0.324166	−	0.946000i	$-0.394916\pi$
$577$	5.18486e6	0.648332	0.324166	+	0.946000i	$0.394916\pi$
$578$	−7.90306e6	−0.983956
$579$	5.20558e6	0.645317
$580$	4.04702e6	0.499534
$581$	6.37358e6	0.783327
$582$	−7.67242e6	−0.938912
$583$	−6.89982e6	−0.840749
$584$	−1.09781e7	−1.33197
$585$	5.10205e6	0.616389
$586$	5.82739e6	0.701020
$587$	9.73801e6	1.16647	0.583237	−	0.812302i	$-0.301786\pi$
$587$	9.73801e6	1.16647	0.583237	+	0.812302i	$0.301786\pi$
$588$	−240508.	−0.0286870
$589$	−2.93660e6	−0.348784
$590$	−1.63320e6	−0.193157
$591$	1.59453e6	0.187787
$592$	1.86790e7	2.19053
$593$	1.14267e7	1.33439	0.667197	−	0.744881i	$-0.267494\pi$
$593$	1.14267e7	1.33439	0.667197	+	0.744881i	$0.267494\pi$
$594$	−2.19365e6	−0.255094
$595$	1.57036e7	1.81847
$596$	−2.45089e6	−0.282623
$597$	3.28296e6	0.376989
$598$	−1.70444e7	−1.94907
$599$	−1.23941e7	−1.41139	−0.705697	−	0.708514i	$-0.749366\pi$
$599$	−1.23941e7	−1.41139	−0.705697	+	0.708514i	$0.749366\pi$
$600$	3.85504e6	0.437170
$601$	−1.22935e7	−1.38831	−0.694157	−	0.719823i	$-0.744223\pi$
$601$	−1.22935e7	−1.38831	−0.694157	+	0.719823i	$0.744223\pi$
$602$	−8.21257e6	−0.923608
$603$	2.58787e6	0.289834
$604$	3.40597e6	0.379882
$605$	−3.32980e6	−0.369853
$606$	−178820.	−0.0197804
$607$	789049.	0.0869225	0.0434613	−	0.999055i	$-0.486161\pi$
$607$	789049.	0.0869225	0.0434613	+	0.999055i	$0.486161\pi$
$608$	1.22337e7	1.34214
$609$	4.32846e6	0.472923
$610$	8.38493e6	0.912378
$611$	2.21548e7	2.40085
$612$	1.66054e6	0.179214
$613$	−1.01565e7	−1.09167	−0.545837	−	0.837891i	$-0.683788\pi$
$613$	−1.01565e7	−1.09167	−0.545837	+	0.837891i	$0.683788\pi$
$614$	1.91158e7	2.04631
$615$	−6.15653e6	−0.656369
$616$	−7.03893e6	−0.747403
$617$	−1.50137e6	−0.158772	−0.0793862	−	0.996844i	$-0.525296\pi$
$617$	−1.50137e6	−0.158772	−0.0793862	+	0.996844i	$0.525296\pi$
$618$	−2.98536e6	−0.314431
$619$	−1.28761e7	−1.35069	−0.675347	−	0.737500i	$-0.736006\pi$
$619$	−1.28761e7	−1.35069	−0.675347	+	0.737500i	$0.736006\pi$
$620$	1.06763e6	0.111543
$621$	−2.36858e6	−0.246467
$622$	1.32488e7	1.37310
$623$	2.75823e6	0.284715
$624$	8.96157e6	0.921346
$625$	−9.11541e6	−0.933418
$626$	1.20267e7	1.22663
$627$	1.13677e7	1.15480
$628$	4.87992e6	0.493757
$629$	−2.37394e7	−2.39245
$630$	−5.27287e6	−0.529292
$631$	−1.49984e7	−1.49959	−0.749793	−	0.661672i	$-0.769847\pi$
$631$	−1.49984e7	−1.49959	−0.749793	+	0.661672i	$0.769847\pi$
$632$	1.21949e7	1.21447
$633$	−684152.	−0.0678646
$634$	6.05431e6	0.598193
$635$	2.06746e6	0.203471
$636$	−1.75376e6	−0.171920
$637$	1.64960e6	0.161076
$638$	1.19346e7	1.16080
$639$	2.92676e6	0.283553
$640$	1.73550e7	1.67484
$641$	5.01419e6	0.482009	0.241005	−	0.970524i	$-0.422523\pi$
$641$	5.01419e6	0.482009	0.241005	+	0.970524i	$0.422523\pi$
$642$	−2.84100e6	−0.272040
$643$	−8.12806e6	−0.775282	−0.387641	−	0.921811i	$-0.626710\pi$
$643$	−8.12806e6	−0.775282	−0.387641	+	0.921811i	$0.626710\pi$
$644$	5.00741e6	0.475771
$645$	7.31874e6	0.692687
$646$	−3.02711e7	−2.85395
$647$	−1.99771e7	−1.87617	−0.938083	−	0.346411i	$-0.887400\pi$
$647$	−1.99771e7	−1.87617	−0.938083	+	0.346411i	$0.887400\pi$
$648$	846278.	0.0791727
$649$	−1.36912e6	−0.127594
$650$	1.74206e7	1.61726
$651$	1.14188e6	0.105601
$652$	−1.83346e6	−0.168909
$653$	1.33861e7	1.22849	0.614245	−	0.789115i	$-0.289461\pi$
$653$	1.33861e7	1.22849	0.614245	+	0.789115i	$0.289461\pi$
$654$	−7.59049e6	−0.693946
$655$	1.55995e7	1.42072
$656$	−1.08137e7	−0.981107
$657$	−6.89396e6	−0.623097
$658$	−2.28966e7	−2.06161
$659$	−206812.	−0.0185508	−0.00927539	−	0.999957i	$-0.502952\pi$
$659$	−206812.	−0.0185508	−0.00927539	+	0.999957i	$0.502952\pi$
$660$	−4.13285e6	−0.369309
$661$	3.39735e6	0.302438	0.151219	−	0.988500i	$-0.451680\pi$
$661$	3.39735e6	0.302438	0.151219	+	0.988500i	$0.451680\pi$
$662$	−1.95960e7	−1.73789
$663$	−1.13894e7	−1.00627
$664$	−6.77958e6	−0.596737
$665$	2.73246e7	2.39607
$666$	7.97109e6	0.696357
$667$	1.28863e7	1.12154
$668$	−6.14302e6	−0.532649
$669$	−447561.	−0.0386622
$670$	1.71512e7	1.47608
$671$	7.02912e6	0.602691
$672$	−4.75698e6	−0.406358
$673$	1.34509e7	1.14476	0.572379	−	0.819989i	$-0.306021\pi$
$673$	1.34509e7	1.14476	0.572379	+	0.819989i	$0.306021\pi$
$674$	−9.80106e6	−0.831042
$675$	2.42086e6	0.204508
$676$	3.10400e6	0.261249
$677$	3.65208e6	0.306245	0.153122	−	0.988207i	$-0.451067\pi$
$677$	3.65208e6	0.306245	0.153122	+	0.988207i	$0.451067\pi$
$678$	9.66096e6	0.807135
$679$	−1.54601e7	−1.28688
$680$	−1.67039e7	−1.38531
$681$	−9.12799e6	−0.754236
$682$	3.14842e6	0.259198
$683$	−3.61457e6	−0.296486	−0.148243	−	0.988951i	$-0.547362\pi$
$683$	−3.61457e6	−0.296486	−0.148243	+	0.988951i	$0.547362\pi$
$684$	2.88939e6	0.236138
$685$	1.90727e7	1.55305
$686$	−1.53323e7	−1.24393
$687$	1.03196e7	0.834205
$688$	1.28551e7	1.03539
$689$	1.20287e7	0.965319
$690$	−1.56979e7	−1.25522
$691$	3.28020e6	0.261339	0.130670	−	0.991426i	$-0.458287\pi$
$691$	3.28020e6	0.261339	0.130670	+	0.991426i	$0.458287\pi$
$692$	8.07343e6	0.640904
$693$	−4.42027e6	−0.349635
$694$	−1.96051e6	−0.154515
$695$	3.00948e7	2.36336
$696$	−4.60419e6	−0.360272
$697$	1.37433e7	1.07154
$698$	−1.48647e7	−1.15483
$699$	4.57574e6	0.354216
$700$	−5.11794e6	−0.394775
$701$	−1.13695e7	−0.873867	−0.436934	−	0.899494i	$-0.643936\pi$
$701$	−1.13695e7	−0.873867	−0.436934	+	0.899494i	$0.643936\pi$
$702$	3.82427e6	0.292891
$703$	−4.13071e7	−3.15236
$704$	5.16111e6	0.392475
$705$	2.04046e7	1.54617
$706$	−2.85676e6	−0.215706
$707$	−360327.	−0.0271112
$708$	−347995.	−0.0260910
$709$	−1.96004e6	−0.146436	−0.0732182	−	0.997316i	$-0.523327\pi$
$709$	−1.96004e6	−0.146436	−0.0732182	+	0.997316i	$0.523327\pi$
$710$	1.93972e7	1.44409
$711$	7.65810e6	0.568129
$712$	−2.93393e6	−0.216895
$713$	3.39949e6	0.250432
$714$	1.17707e7	0.864087
$715$	2.83465e7	2.07364
$716$	2.53454e6	0.184764
$717$	6.69239e6	0.486165
$718$	2.07025e7	1.49869
$719$	−1.01752e7	−0.734041	−0.367021	−	0.930213i	$-0.619622\pi$
$719$	−1.01752e7	−0.734041	−0.367021	+	0.930213i	$0.619622\pi$
$720$	8.25361e6	0.593353
$721$	−6.01558e6	−0.430963
$722$	−3.61159e7	−2.57843
$723$	−4.90349e6	−0.348867
$724$	−2.02305e6	−0.143437
$725$	−1.31707e7	−0.930605
$726$	−2.49587e6	−0.175744
$727$	2.64867e7	1.85863	0.929313	−	0.369293i	$-0.120400\pi$
$727$	2.64867e7	1.85863	0.929313	+	0.369293i	$0.120400\pi$
$728$	1.22712e7	0.858142
$729$	531441.	0.0370370
$730$	−4.56901e7	−3.17333
$731$	−1.63377e7	−1.13083
$732$	1.78662e6	0.123241
$733$	6.11724e6	0.420529	0.210264	−	0.977645i	$-0.432567\pi$
$733$	6.11724e6	0.420529	0.210264	+	0.977645i	$0.432567\pi$
$734$	−3.51304e6	−0.240682
$735$	1.51928e6	0.103734
$736$	−1.41620e7	−0.963677
$737$	1.43779e7	0.975053
$738$	−4.61466e6	−0.311888
$739$	2.08754e7	1.40612	0.703062	−	0.711128i	$-0.251816\pi$
$739$	2.08754e7	1.40612	0.703062	+	0.711128i	$0.251816\pi$
$740$	1.50176e7	1.00814
$741$	−1.98178e7	−1.32590
$742$	−1.24314e7	−0.828918
$743$	1.92753e7	1.28094	0.640471	−	0.767983i	$-0.278739\pi$
$743$	1.92753e7	1.28094	0.640471	+	0.767983i	$0.278739\pi$
$744$	−1.21462e6	−0.0804464
$745$	1.54822e7	1.02198
$746$	−2.26051e6	−0.148717
$747$	−4.25740e6	−0.279154
$748$	9.22583e6	0.602908
$749$	−5.72469e6	−0.372861
$750$	946007.	0.0614103
$751$	2.30681e6	0.149249	0.0746247	−	0.997212i	$-0.476224\pi$
$751$	2.30681e6	0.149249	0.0746247	+	0.997212i	$0.476224\pi$
$752$	3.58400e7	2.31113
$753$	1.34518e7	0.864558
$754$	−2.08060e7	−1.33279
$755$	−2.15155e7	−1.37367
$756$	−1.12352e6	−0.0714950
$757$	−614210.	−0.0389562	−0.0194781	−	0.999810i	$-0.506200\pi$
$757$	−614210.	−0.0389562	−0.0194781	+	0.999810i	$0.506200\pi$
$758$	−1.25841e7	−0.795518
$759$	−1.31596e7	−0.829160
$760$	−2.90652e7	−1.82532
$761$	2.50209e6	0.156618	0.0783090	−	0.996929i	$-0.475048\pi$
$761$	2.50209e6	0.156618	0.0783090	+	0.996929i	$0.475048\pi$
$762$	1.54968e6	0.0966839
$763$	−1.52951e7	−0.951130
$764$	9.95143e6	0.616811
$765$	−1.04896e7	−0.648047
$766$	−2.94420e7	−1.81299
$767$	2.38684e6	0.146499
$768$	9.70561e6	0.593772
$769$	−2.24026e6	−0.136610	−0.0683052	−	0.997664i	$-0.521759\pi$
$769$	−2.24026e6	−0.136610	−0.0683052	+	0.997664i	$0.521759\pi$
$770$	−2.92956e7	−1.78063
$771$	−2.36340e6	−0.143186
$772$	7.35116e6	0.443928
$773$	2.46858e7	1.48593	0.742966	−	0.669330i	$-0.233419\pi$
$773$	2.46858e7	1.48593	0.742966	+	0.669330i	$0.233419\pi$
$774$	5.48581e6	0.329146
$775$	−3.47453e6	−0.207798
$776$	1.64450e7	0.980344
$777$	1.60620e7	0.954434
$778$	2.27738e6	0.134892
$779$	2.39137e7	1.41190
$780$	7.20495e6	0.424028
$781$	1.62608e7	0.953924
$782$	3.50427e7	2.04918
$783$	−2.89131e6	−0.168535
$784$	2.66857e6	0.155056
$785$	−3.08264e7	−1.78545
$786$	1.16927e7	0.675084
$787$	−3.53044e6	−0.203185	−0.101593	−	0.994826i	$-0.532394\pi$
$787$	−3.53044e6	−0.203185	−0.101593	+	0.994826i	$0.532394\pi$
$788$	2.25175e6	0.129183
$789$	581673.	0.0332649
$790$	5.07545e7	2.89339
$791$	1.94671e7	1.10627
$792$	4.70184e6	0.266351
$793$	−1.22541e7	−0.691989
$794$	−2.35786e7	−1.32729
$795$	1.10784e7	0.621672
$796$	4.63608e6	0.259340
$797$	1.87821e7	1.04737	0.523683	−	0.851913i	$-0.324557\pi$
$797$	1.87821e7	1.04737	0.523683	+	0.851913i	$0.324557\pi$
$798$	2.04813e7	1.13855
$799$	−4.55496e7	−2.52416
$800$	1.44747e7	0.799620
$801$	−1.84243e6	−0.101464
$802$	−3.00054e7	−1.64726
$803$	−3.83022e7	−2.09621
$804$	3.65450e6	0.199383
$805$	−3.16317e7	−1.72041
$806$	−5.48876e6	−0.297602
$807$	1.33792e7	0.723178
$808$	383280.	0.0206532
$809$	−1.94958e7	−1.04730	−0.523649	−	0.851934i	$-0.675430\pi$
$809$	−1.94958e7	−1.04730	−0.523649	+	0.851934i	$0.675430\pi$
$810$	3.52215e6	0.188624
$811$	−2.12098e7	−1.13236	−0.566180	−	0.824282i	$-0.691579\pi$
$811$	−2.12098e7	−1.13236	−0.566180	+	0.824282i	$0.691579\pi$
$812$	6.11252e6	0.325334
$813$	7.01560e6	0.372253
$814$	4.42866e7	2.34267
$815$	1.15819e7	0.610783
$816$	−1.84247e7	−0.968667
$817$	−2.84281e7	−1.49002
$818$	2.29894e7	1.20128
$819$	7.70601e6	0.401439
$820$	−8.69406e6	−0.451531
$821$	497764.	0.0257730	0.0128865	−	0.999917i	$-0.495898\pi$
$821$	497764.	0.0257730	0.0128865	+	0.999917i	$0.495898\pi$
$822$	1.42961e7	0.737967
$823$	824502.	0.0424319	0.0212159	−	0.999775i	$-0.493246\pi$
$823$	824502.	0.0424319	0.0212159	+	0.999775i	$0.493246\pi$
$824$	6.39878e6	0.328306
$825$	1.34501e7	0.688003
$826$	−2.46675e6	−0.125799
$827$	3.01215e6	0.153149	0.0765743	−	0.997064i	$-0.475602\pi$
$827$	3.01215e6	0.153149	0.0765743	+	0.997064i	$0.475602\pi$
$828$	−3.34483e6	−0.169550
$829$	7.21018e6	0.364384	0.182192	−	0.983263i	$-0.441681\pi$
$829$	7.21018e6	0.364384	0.182192	+	0.983263i	$0.441681\pi$
$830$	−2.82162e7	−1.42168
$831$	5.97744e6	0.300271
$832$	−8.99756e6	−0.450626
$833$	−3.39152e6	−0.169349
$834$	2.25577e7	1.12300
$835$	3.88054e7	1.92609
$836$	1.60531e7	0.794410
$837$	−762748.	−0.0376329
$838$	−1.99364e7	−0.980700
$839$	8.61276e6	0.422413	0.211207	−	0.977441i	$-0.432261\pi$
$839$	8.61276e6	0.422413	0.211207	+	0.977441i	$0.432261\pi$
$840$	1.13018e7	0.552649
$841$	−4.78092e6	−0.233089
$842$	6.43959e6	0.313024
$843$	−1.83162e7	−0.887700
$844$	−966138.	−0.0466856
$845$	−1.96079e7	−0.944691
$846$	1.52944e7	0.734695
$847$	−5.02925e6	−0.240877
$848$	1.94589e7	0.929242
$849$	5.67040e6	0.269988
$850$	−3.58162e7	−1.70033
$851$	4.78182e7	2.26344
$852$	4.13307e6	0.195063
$853$	3.35348e7	1.57806	0.789029	−	0.614356i	$-0.210584\pi$
$853$	3.35348e7	1.57806	0.789029	+	0.614356i	$0.210584\pi$
$854$	1.26644e7	0.594210
$855$	−1.82522e7	−0.853886
$856$	6.08936e6	0.284045
$857$	2.07569e7	0.965407	0.482703	−	0.875784i	$-0.339655\pi$
$857$	2.07569e7	0.965407	0.482703	+	0.875784i	$0.339655\pi$
$858$	2.12473e7	0.985337
$859$	1.77155e7	0.819165	0.409583	−	0.912273i	$-0.365674\pi$
$859$	1.77155e7	0.819165	0.409583	+	0.912273i	$0.365674\pi$
$860$	1.03353e7	0.476515
$861$	−9.29868e6	−0.427478
$862$	4.78256e7	2.19226
$863$	3.00503e7	1.37348	0.686739	−	0.726904i	$-0.259042\pi$
$863$	3.00503e7	1.37348	0.686739	+	0.726904i	$0.259042\pi$
$864$	3.17756e6	0.144813
$865$	−5.09997e7	−2.31754
$866$	1.20687e6	0.0546846
$867$	1.06375e7	0.480607
$868$	1.61252e6	0.0726451
$869$	4.25477e7	1.91129
$870$	−1.91623e7	−0.858322
$871$	−2.50656e7	−1.11952
$872$	1.62694e7	0.724568
$873$	1.03270e7	0.458605
$874$	6.09750e7	2.70006
$875$	1.90623e6	0.0841696
$876$	−9.73543e6	−0.428642
$877$	4.27729e7	1.87789	0.938943	−	0.344072i	$-0.111806\pi$
$877$	4.27729e7	1.87789	0.938943	+	0.344072i	$0.111806\pi$
$878$	−1.37712e7	−0.602886
$879$	−7.84363e6	−0.342409
$880$	4.58563e7	1.99615
$881$	1.15515e7	0.501415	0.250708	−	0.968063i	$-0.419337\pi$
$881$	1.15515e7	0.501415	0.250708	+	0.968063i	$0.419337\pi$
$882$	1.13879e6	0.0492914
$883$	1.51036e6	0.0651895	0.0325948	−	0.999469i	$-0.489623\pi$
$883$	1.51036e6	0.0651895	0.0325948	+	0.999469i	$0.489623\pi$
$884$	−1.60837e7	−0.692239
$885$	2.19828e6	0.0943463
$886$	−2.84425e7	−1.21726
$887$	−4.32216e7	−1.84456	−0.922278	−	0.386527i	$-0.873675\pi$
$887$	−4.32216e7	−1.84456	−0.922278	+	0.386527i	$0.873675\pi$
$888$	−1.70851e7	−0.727086
$889$	3.12264e6	0.132516
$890$	−1.22108e7	−0.516738
$891$	2.95263e6	0.124599
$892$	−632031.	−0.0265966
$893$	−7.92573e7	−3.32591
$894$	1.16048e7	0.485616
$895$	−1.60106e7	−0.668115
$896$	2.62125e7	1.09078
$897$	2.29416e7	0.952013
$898$	−5.43163e7	−2.24771
$899$	4.14974e6	0.171246
$900$	3.41867e6	0.140686
$901$	−2.47306e7	−1.01490
$902$	−2.56386e7	−1.04925
$903$	1.10541e7	0.451130
$904$	−2.07072e7	−0.842752
$905$	1.27796e7	0.518675
$906$	−1.61270e7	−0.652731
$907$	−4.28946e7	−1.73135	−0.865673	−	0.500609i	$-0.833109\pi$
$907$	−4.28946e7	−1.73135	−0.865673	+	0.500609i	$0.833109\pi$
$908$	−1.28903e7	−0.518856
$909$	240690.	0.00966159
$910$	5.10720e7	2.04446
$911$	−2.42343e7	−0.967462	−0.483731	−	0.875217i	$-0.660719\pi$
$911$	−2.42343e7	−0.967462	−0.483731	+	0.875217i	$0.660719\pi$
$912$	−3.20594e7	−1.27634
$913$	−2.36537e7	−0.939123
$914$	−4.18825e7	−1.65831
$915$	−1.12861e7	−0.445645
$916$	1.45731e7	0.573868
$917$	2.35611e7	0.925278
$918$	−7.86256e6	−0.307934
$919$	4.11713e6	0.160807	0.0804037	−	0.996762i	$-0.474379\pi$
$919$	4.11713e6	0.160807	0.0804037	+	0.996762i	$0.474379\pi$
$920$	3.36467e7	1.31061
$921$	−2.57297e7	−0.999508
$922$	−2.55091e7	−0.988251
$923$	−2.83480e7	−1.09526
$924$	−6.24216e6	−0.240522
$925$	−4.88737e7	−1.87811
$926$	2.43247e6	0.0932224
$927$	4.01827e6	0.153582
$928$	−1.72875e7	−0.658966
$929$	4.19272e7	1.59388	0.796942	−	0.604055i	$-0.206449\pi$
$929$	4.19272e7	1.59388	0.796942	+	0.604055i	$0.206449\pi$
$930$	−5.05515e6	−0.191658
$931$	−5.90132e6	−0.223139
$932$	6.46171e6	0.243673
$933$	−1.78328e7	−0.670681
$934$	3.21482e7	1.20584
$935$	−5.82794e7	−2.18015
$936$	−8.19689e6	−0.305816
$937$	4.60244e7	1.71253	0.856267	−	0.516533i	$-0.172778\pi$
$937$	4.60244e7	1.71253	0.856267	+	0.516533i	$0.172778\pi$
$938$	2.59048e7	0.961332
$939$	−1.61879e7	−0.599137
$940$	2.88148e7	1.06364
$941$	−2.11098e7	−0.777158	−0.388579	−	0.921415i	$-0.627034\pi$
$941$	−2.11098e7	−0.777158	−0.388579	+	0.921415i	$0.627034\pi$
$942$	−2.31061e7	−0.848397
$943$	−2.76831e7	−1.01376
$944$	3.86120e6	0.141024
$945$	7.09724e6	0.258529
$946$	3.04786e7	1.10730
$947$	2.78097e7	1.00768	0.503838	−	0.863798i	$-0.331921\pi$
$947$	2.78097e7	1.00768	0.503838	+	0.863798i	$0.331921\pi$
$948$	1.08145e7	0.390829
$949$	6.67736e7	2.40680
$950$	−6.23210e7	−2.24040
$951$	−8.14905e6	−0.292184
$952$	−2.52292e7	−0.902217
$953$	7.10863e6	0.253544	0.126772	−	0.991932i	$-0.459538\pi$
$953$	7.10863e6	0.253544	0.126772	+	0.991932i	$0.459538\pi$
$954$	8.30392e6	0.295401
$955$	−6.28630e7	−2.23042
$956$	9.45078e6	0.334444
$957$	−1.60639e7	−0.566983
$958$	−3.58911e6	−0.126349
$959$	2.88070e7	1.01147
$960$	−8.28676e6	−0.290206
$961$	−2.75344e7	−0.961762
$962$	−7.72064e7	−2.68977
$963$	3.82396e6	0.132876
$964$	−6.92455e6	−0.239993
$965$	−4.64371e7	−1.60527
$966$	−2.37097e7	−0.817492
$967$	4.36683e7	1.50176	0.750880	−	0.660439i	$-0.229630\pi$
$967$	4.36683e7	1.50176	0.750880	+	0.660439i	$0.229630\pi$
$968$	5.34962e6	0.183499
$969$	4.07447e7	1.39400
$970$	6.84428e7	2.33560
$971$	−2.82197e6	−0.0960514	−0.0480257	−	0.998846i	$-0.515293\pi$
$971$	−2.82197e6	−0.0960514	−0.0480257	+	0.998846i	$0.515293\pi$
$972$	750484.	0.0254786
$973$	4.54544e7	1.53920
$974$	−4.96002e7	−1.67527
$975$	−2.34480e7	−0.789941
$976$	−1.98236e7	−0.666127
$977$	−3.69112e7	−1.23715	−0.618574	−	0.785727i	$-0.712289\pi$
$977$	−3.69112e7	−1.23715	−0.618574	+	0.785727i	$0.712289\pi$
$978$	8.68130e6	0.290227
$979$	−1.02364e7	−0.341342
$980$	2.14548e6	0.0713608
$981$	1.02167e7	0.338953
$982$	5.32200e7	1.76115
$983$	−4.13069e7	−1.36345	−0.681724	−	0.731609i	$-0.738770\pi$
$983$	−4.13069e7	−1.36345	−0.681724	+	0.731609i	$0.738770\pi$
$984$	9.89101e6	0.325651
$985$	−1.42242e7	−0.467131
$986$	4.27764e7	1.40124
$987$	3.08187e7	1.00698
$988$	−2.79860e7	−0.912114
$989$	3.29091e7	1.06986
$990$	1.95688e7	0.634564
$991$	1.18442e7	0.383107	0.191553	−	0.981482i	$-0.438647\pi$
$991$	1.18442e7	0.383107	0.191553	+	0.981482i	$0.438647\pi$
$992$	−4.56057e6	−0.147143
$993$	2.63760e7	0.848860
$994$	2.92971e7	0.940501
$995$	−2.92861e7	−0.937785
$996$	−6.01217e6	−0.192036
$997$	1.66243e7	0.529670	0.264835	−	0.964294i	$-0.414682\pi$
$997$	1.66243e7	0.529670	0.264835	+	0.964294i	$0.414682\pi$
$998$	2.10738e7	0.669757
$999$	−1.07290e7	−0.340131

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	669.6.a.d.1.13	✓	51

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
669.6.a.d.1.13	✓	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)

\(f(q)\)	\(=\)	\(q-6.68652 q^{2} +9.00000 q^{3} +12.7095 q^{4} -80.2857 q^{5} -60.1787 q^{6} -121.262 q^{7} +128.986 q^{8} +81.0000 q^{9} +536.832 q^{10} +450.028 q^{11} +114.386 q^{12} -784.551 q^{13} +810.818 q^{14} -722.572 q^{15} -1269.17 q^{16} +1613.01 q^{17} -541.608 q^{18} +2806.67 q^{19} -1020.39 q^{20} -1091.36 q^{21} -3009.12 q^{22} -3249.08 q^{23} +1160.88 q^{24} +3320.80 q^{25} +5245.91 q^{26} +729.000 q^{27} -1541.18 q^{28} -3966.14 q^{29} +4831.49 q^{30} -1046.29 q^{31} +4358.79 q^{32} +4050.25 q^{33} -10785.4 q^{34} +9735.58 q^{35} +1029.47 q^{36} -14717.4 q^{37} -18766.9 q^{38} -7060.96 q^{39} -10355.8 q^{40} +8520.30 q^{41} +7297.36 q^{42} -10128.7 q^{43} +5719.64 q^{44} -6503.15 q^{45} +21725.0 q^{46} -28238.9 q^{47} -11422.6 q^{48} -2102.60 q^{49} -22204.6 q^{50} +14517.1 q^{51} -9971.25 q^{52} -15332.0 q^{53} -4874.47 q^{54} -36130.8 q^{55} -15641.1 q^{56} +25260.0 q^{57} +26519.6 q^{58} -3042.30 q^{59} -9183.53 q^{60} +15619.3 q^{61} +6996.06 q^{62} -9822.20 q^{63} +11468.4 q^{64} +62988.2 q^{65} -27082.1 q^{66} +31949.0 q^{67} +20500.6 q^{68} -29241.7 q^{69} -65097.2 q^{70} +36132.8 q^{71} +10447.9 q^{72} -85110.6 q^{73} +98408.5 q^{74} +29887.2 q^{75} +35671.4 q^{76} -54571.2 q^{77} +47213.2 q^{78} +94544.5 q^{79} +101896. q^{80} +6561.00 q^{81} -56971.2 q^{82} -52560.5 q^{83} -13870.6 q^{84} -129502. q^{85} +67726.0 q^{86} -35695.2 q^{87} +58047.4 q^{88} -22746.1 q^{89} +43483.4 q^{90} +95135.9 q^{91} -41294.2 q^{92} -9416.64 q^{93} +188820. q^{94} -225336. q^{95} +39229.1 q^{96} +127494. q^{97} +14059.1 q^{98} +36452.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 51 q + 18 q^{2} + 459 q^{3} + 940 q^{4} + 97 q^{5} + 162 q^{6} + 794 q^{7} + 411 q^{8} + 4131 q^{9} + 1256 q^{10} + 2349 q^{11} + 8460 q^{12} + 2954 q^{13} + 1816 q^{14} + 873 q^{15} + 18580 q^{16} + 1567 q^{17}+ \cdots + 190269 q^{99}+O(q^{100}) \) 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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