LMFDB - Embedded newform 669.6.a.d.1.7

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			1.7
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)

$f(q)$	$=$	\(q-8.23985 q^{2} +9.00000 q^{3} +35.8952 q^{4} +71.4029 q^{5} -74.1587 q^{6} -149.227 q^{7} -32.0957 q^{8} +81.0000 q^{9} -588.350 q^{10} +441.170 q^{11} +323.057 q^{12} +590.317 q^{13} +1229.61 q^{14} +642.626 q^{15} -884.182 q^{16} +1948.51 q^{17} -667.428 q^{18} -165.391 q^{19} +2563.02 q^{20} -1343.04 q^{21} -3635.18 q^{22} +2313.91 q^{23} -288.862 q^{24} +1973.38 q^{25} -4864.12 q^{26} +729.000 q^{27} -5356.52 q^{28} +656.912 q^{29} -5295.15 q^{30} +280.478 q^{31} +8312.59 q^{32} +3970.53 q^{33} -16055.4 q^{34} -10655.2 q^{35} +2907.51 q^{36} -11990.1 q^{37} +1362.80 q^{38} +5312.85 q^{39} -2291.73 q^{40} +16067.0 q^{41} +11066.5 q^{42} +4384.92 q^{43} +15835.9 q^{44} +5783.64 q^{45} -19066.3 q^{46} +6219.53 q^{47} -7957.64 q^{48} +5461.59 q^{49} -16260.4 q^{50} +17536.6 q^{51} +21189.5 q^{52} +20557.7 q^{53} -6006.85 q^{54} +31500.9 q^{55} +4789.54 q^{56} -1488.52 q^{57} -5412.86 q^{58} +4007.23 q^{59} +23067.2 q^{60} -21726.3 q^{61} -2311.09 q^{62} -12087.4 q^{63} -40200.7 q^{64} +42150.3 q^{65} -32716.6 q^{66} +24741.3 q^{67} +69942.0 q^{68} +20825.2 q^{69} +87797.5 q^{70} +73404.2 q^{71} -2599.76 q^{72} -17255.2 q^{73} +98796.3 q^{74} +17760.4 q^{75} -5936.75 q^{76} -65834.4 q^{77} -43777.1 q^{78} -21699.4 q^{79} -63133.2 q^{80} +6561.00 q^{81} -132390. q^{82} -107758. q^{83} -48208.7 q^{84} +139129. q^{85} -36131.1 q^{86} +5912.21 q^{87} -14159.7 q^{88} -39299.9 q^{89} -47656.3 q^{90} -88091.0 q^{91} +83058.3 q^{92} +2524.30 q^{93} -51248.0 q^{94} -11809.4 q^{95} +74813.3 q^{96} +40453.4 q^{97} -45002.7 q^{98} +35734.8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−8.23985	−1.45661	−0.728307	−	0.685251i	$-0.759693\pi$
$2$	−8.23985	−1.45661	−0.728307	+	0.685251i	$0.759693\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	35.8952	1.12172
$5$	71.4029	1.27729	0.638647	−	0.769500i	$-0.279494\pi$
$5$	71.4029	1.27729	0.638647	+	0.769500i	$0.279494\pi$
$6$	−74.1587	−0.840977
$7$	−149.227	−1.15107	−0.575534	−	0.817778i	$-0.695206\pi$
$7$	−149.227	−1.15107	−0.575534	+	0.817778i	$0.695206\pi$
$8$	−32.0957	−0.177306
$9$	81.0000	0.333333
$10$	−588.350	−1.86053
$11$	441.170	1.09932	0.549661	−	0.835388i	$-0.314757\pi$
$11$	441.170	1.09932	0.549661	+	0.835388i	$0.314757\pi$
$12$	323.057	0.647628
$13$	590.317	0.968783	0.484392	−	0.874851i	$-0.339041\pi$
$13$	590.317	0.968783	0.484392	+	0.874851i	$0.339041\pi$
$14$	1229.61	1.67666
$15$	642.626	0.737446
$16$	−884.182	−0.863459
$17$	1948.51	1.63523	0.817616	−	0.575764i	$-0.195295\pi$
$17$	1948.51	1.63523	0.817616	+	0.575764i	$0.195295\pi$
$18$	−667.428	−0.485538
$19$	−165.391	−0.105106	−0.0525532	−	0.998618i	$-0.516736\pi$
$19$	−165.391	−0.105106	−0.0525532	+	0.998618i	$0.516736\pi$
$20$	2563.02	1.43277
$21$	−1343.04	−0.664570
$22$	−3635.18	−1.60129
$23$	2313.91	0.912068	0.456034	−	0.889962i	$-0.349269\pi$
$23$	2313.91	0.912068	0.456034	+	0.889962i	$0.349269\pi$
$24$	−288.862	−0.102367
$25$	1973.38	0.631481
$26$	−4864.12	−1.41114
$27$	729.000	0.192450
$28$	−5356.52	−1.29118
$29$	656.912	0.145048	0.0725241	−	0.997367i	$-0.476895\pi$
$29$	656.912	0.145048	0.0725241	+	0.997367i	$0.476895\pi$
$30$	−5295.15	−1.07417
$31$	280.478	0.0524196	0.0262098	−	0.999656i	$-0.491656\pi$
$31$	280.478	0.0524196	0.0262098	+	0.999656i	$0.491656\pi$
$32$	8312.59	1.43503
$33$	3970.53	0.634694
$34$	−16055.4	−2.38190
$35$	−10655.2	−1.47025
$36$	2907.51	0.373908
$37$	−11990.1	−1.43985	−0.719925	−	0.694052i	$-0.755824\pi$
$37$	−11990.1	−1.43985	−0.719925	+	0.694052i	$0.755824\pi$
$38$	1362.80	0.153099
$39$	5312.85	0.559327
$40$	−2291.73	−0.226471
$41$	16067.0	1.49271	0.746354	−	0.665549i	$-0.231803\pi$
$41$	16067.0	1.49271	0.746354	+	0.665549i	$0.231803\pi$
$42$	11066.5	0.968022
$43$	4384.92	0.361652	0.180826	−	0.983515i	$-0.442123\pi$
$43$	4384.92	0.361652	0.180826	+	0.983515i	$0.442123\pi$
$44$	15835.9	1.23314
$45$	5783.64	0.425765
$46$	−19066.3	−1.32853
$47$	6219.53	0.410689	0.205344	−	0.978690i	$-0.434169\pi$
$47$	6219.53	0.410689	0.205344	+	0.978690i	$0.434169\pi$
$48$	−7957.64	−0.498518
$49$	5461.59	0.324960
$50$	−16260.4	−0.919824
$51$	17536.6	0.944102
$52$	21189.5	1.08671
$53$	20557.7	1.00527	0.502636	−	0.864498i	$-0.332363\pi$
$53$	20557.7	1.00527	0.502636	+	0.864498i	$0.332363\pi$
$54$	−6006.85	−0.280326
$55$	31500.9	1.40416
$56$	4789.54	0.204091
$57$	−1488.52	−0.0606832
$58$	−5412.86	−0.211279
$59$	4007.23	0.149870	0.0749349	−	0.997188i	$-0.476125\pi$
$59$	4007.23	0.149870	0.0749349	+	0.997188i	$0.476125\pi$
$60$	23067.2	0.827212
$61$	−21726.3	−0.747585	−0.373793	−	0.927512i	$-0.621943\pi$
$61$	−21726.3	−0.747585	−0.373793	+	0.927512i	$0.621943\pi$
$62$	−2311.09	−0.0763552
$63$	−12087.4	−0.383690
$64$	−40200.7	−1.22683
$65$	42150.3	1.23742
$66$	−32716.6	−0.924504
$67$	24741.3	0.673341	0.336670	−	0.941623i	$-0.390699\pi$
$67$	24741.3	0.673341	0.336670	+	0.941623i	$0.390699\pi$
$68$	69942.0	1.83428
$69$	20825.2	0.526583
$70$	87797.5	2.14159
$71$	73404.2	1.72812	0.864062	−	0.503385i	$-0.167912\pi$
$71$	73404.2	1.72812	0.864062	+	0.503385i	$0.167912\pi$
$72$	−2599.76	−0.0591019
$73$	−17255.2	−0.378976	−0.189488	−	0.981883i	$-0.560683\pi$
$73$	−17255.2	−0.378976	−0.189488	+	0.981883i	$0.560683\pi$
$74$	98796.3	2.09730
$75$	17760.4	0.364586
$76$	−5936.75	−0.117900
$77$	−65834.4	−1.26539
$78$	−43777.1	−0.814724
$79$	−21699.4	−0.391184	−0.195592	−	0.980685i	$-0.562663\pi$
$79$	−21699.4	−0.391184	−0.195592	+	0.980685i	$0.562663\pi$
$80$	−63133.2	−1.10289
$81$	6561.00	0.111111
$82$	−132390.	−2.17430
$83$	−107758.	−1.71694	−0.858470	−	0.512863i	$-0.828585\pi$
$83$	−107758.	−1.71694	−0.858470	+	0.512863i	$0.828585\pi$
$84$	−48208.7	−0.745464
$85$	139129.	2.08867
$86$	−36131.1	−0.526788
$87$	5912.21	0.0837436
$88$	−14159.7	−0.194916
$89$	−39299.9	−0.525916	−0.262958	−	0.964807i	$-0.584698\pi$
$89$	−39299.9	−0.525916	−0.262958	+	0.964807i	$0.584698\pi$
$90$	−47656.3	−0.620175
$91$	−88091.0	−1.11514
$92$	83058.3	1.02309
$93$	2524.30	0.0302645
$94$	−51248.0	−0.598215
$95$	−11809.4	−0.134252
$96$	74813.3	0.828516
$97$	40453.4	0.436541	0.218271	−	0.975888i	$-0.429958\pi$
$97$	40453.4	0.436541	0.218271	+	0.975888i	$0.429958\pi$
$98$	−45002.7	−0.473341
$99$	35734.8	0.366441
$100$	70834.8	0.708348
$101$	−77503.5	−0.755994	−0.377997	−	0.925807i	$-0.623387\pi$
$101$	−77503.5	−0.755994	−0.377997	+	0.925807i	$0.623387\pi$
$102$	−144499.	−1.37519
$103$	89471.5	0.830982	0.415491	−	0.909597i	$-0.363610\pi$
$103$	89471.5	0.830982	0.415491	+	0.909597i	$0.363610\pi$
$104$	−18946.7	−0.171771
$105$	−95897.0	−0.848852
$106$	−169392.	−1.46429
$107$	−167249.	−1.41223	−0.706113	−	0.708099i	$-0.749553\pi$
$107$	−167249.	−1.41223	−0.706113	+	0.708099i	$0.749553\pi$
$108$	26167.6	0.215876
$109$	−246430.	−1.98668	−0.993340	−	0.115222i	$-0.963242\pi$
$109$	−246430.	−1.98668	−0.993340	+	0.115222i	$0.963242\pi$
$110$	−259563.	−2.04532
$111$	−107911.	−0.831297
$112$	131943.	0.993900
$113$	13820.8	0.101821	0.0509104	−	0.998703i	$-0.483788\pi$
$113$	13820.8	0.101821	0.0509104	+	0.998703i	$0.483788\pi$
$114$	12265.2	0.0883919
$115$	165220.	1.16498
$116$	23580.0	0.162704
$117$	47815.6	0.322928
$118$	−33019.0	−0.218303
$119$	−290769.	−1.88226
$120$	−20625.6	−0.130753
$121$	33580.4	0.208508
$122$	179021.	1.08894
$123$	144603.	0.861815
$124$	10067.8	0.0588004
$125$	−82229.1	−0.470707
$126$	99598.1	0.558888
$127$	70949.1	0.390335	0.195168	−	0.980770i	$-0.437475\pi$
$127$	70949.1	0.390335	0.195168	+	0.980770i	$0.437475\pi$
$128$	65245.1	0.351984
$129$	39464.3	0.208800
$130$	−347313.	−1.80245
$131$	206316.	1.05040	0.525201	−	0.850978i	$-0.323990\pi$
$131$	206316.	1.05040	0.525201	+	0.850978i	$0.323990\pi$
$132$	142523.	0.711951
$133$	24680.8	0.120985
$134$	−203864.	−0.980798
$135$	52052.7	0.245815
$136$	−62538.7	−0.289936
$137$	−58053.3	−0.264256	−0.132128	−	0.991233i	$-0.542181\pi$
$137$	−58053.3	−0.264256	−0.132128	+	0.991233i	$0.542181\pi$
$138$	−171597.	−0.767028
$139$	214235.	0.940489	0.470245	−	0.882536i	$-0.344166\pi$
$139$	214235.	0.940489	0.470245	+	0.882536i	$0.344166\pi$
$140$	−382471.	−1.64922
$141$	55975.8	0.237111
$142$	−604840.	−2.51721
$143$	260430.	1.06500
$144$	−71618.7	−0.287820
$145$	46905.5	0.185269
$146$	142180.	0.552022
$147$	49154.4	0.187615
$148$	−430385.	−1.61511
$149$	195759.	0.722363	0.361182	−	0.932496i	$-0.382373\pi$
$149$	195759.	0.722363	0.361182	+	0.932496i	$0.382373\pi$
$150$	−146343.	−0.531061
$151$	−410473.	−1.46502	−0.732508	−	0.680759i	$-0.761650\pi$
$151$	−410473.	−1.46502	−0.732508	+	0.680759i	$0.761650\pi$
$152$	5308.36	0.0186359
$153$	157829.	0.545077
$154$	542466.	1.84319
$155$	20026.9	0.0669553
$156$	190706.	0.627411
$157$	−294794.	−0.954487	−0.477243	−	0.878771i	$-0.658364\pi$
$157$	−294794.	−0.954487	−0.477243	+	0.878771i	$0.658364\pi$
$158$	178800.	0.569803
$159$	185019.	0.580395
$160$	593543.	1.83296
$161$	−345297.	−1.04985
$162$	−54061.7	−0.161846
$163$	656002.	1.93391	0.966955	−	0.254948i	$-0.0820583\pi$
$163$	656002.	1.93391	0.966955	+	0.254948i	$0.0820583\pi$
$164$	576727.	1.67441
$165$	283508.	0.810691
$166$	887912.	2.50092
$167$	275449.	0.764276	0.382138	−	0.924105i	$-0.375188\pi$
$167$	275449.	0.764276	0.382138	+	0.924105i	$0.375188\pi$
$168$	43105.9	0.117832
$169$	−22819.3	−0.0614589
$170$	−1.14640e6	−3.04239
$171$	−13396.7	−0.0350354
$172$	157398.	0.405674
$173$	368098.	0.935080	0.467540	−	0.883972i	$-0.345140\pi$
$173$	368098.	0.935080	0.467540	+	0.883972i	$0.345140\pi$
$174$	−48715.8	−0.121982
$175$	−294481.	−0.726878
$176$	−390075.	−0.949219
$177$	36065.1	0.0865274
$178$	323825.	0.766057
$179$	−567554.	−1.32396	−0.661980	−	0.749522i	$-0.730284\pi$
$179$	−567554.	−1.32396	−0.661980	+	0.749522i	$0.730284\pi$
$180$	207605.	0.477591
$181$	150491.	0.341440	0.170720	−	0.985320i	$-0.445391\pi$
$181$	150491.	0.341440	0.170720	+	0.985320i	$0.445391\pi$
$182$	725857.	1.62432
$183$	−195536.	−0.431619
$184$	−74266.7	−0.161715
$185$	−856125.	−1.83911
$186$	−20799.9	−0.0440837
$187$	859623.	1.79765
$188$	223251.	0.460680
$189$	−108786.	−0.221523
$190$	97307.9	0.195553
$191$	−636828.	−1.26310	−0.631552	−	0.775334i	$-0.717582\pi$
$191$	−636828.	−1.26310	−0.631552	+	0.775334i	$0.717582\pi$
$192$	−361806.	−0.708310
$193$	−758636.	−1.46602	−0.733011	−	0.680217i	$-0.761885\pi$
$193$	−758636.	−1.46602	−0.733011	+	0.680217i	$0.761885\pi$
$194$	−333330.	−0.635872
$195$	379353.	0.714426
$196$	196045.	0.364515
$197$	439872.	0.807533	0.403767	−	0.914862i	$-0.367701\pi$
$197$	439872.	0.807533	0.403767	+	0.914862i	$0.367701\pi$
$198$	−294450.	−0.533762
$199$	−27237.0	−0.0487559	−0.0243779	−	0.999703i	$-0.507761\pi$
$199$	−27237.0	−0.0487559	−0.0243779	+	0.999703i	$0.507761\pi$
$200$	−63337.1	−0.111965
$201$	222671.	0.388754
$202$	638618.	1.10119
$203$	−98028.8	−0.166961
$204$	629478.	1.05902
$205$	1.14723e6	1.90663
$206$	−737232.	−1.21042
$207$	187427.	0.304023
$208$	−521947.	−0.836504
$209$	−72965.8	−0.115546
$210$	790177.	1.23645
$211$	1.01336e6	1.56696	0.783479	−	0.621419i	$-0.213443\pi$
$211$	1.01336e6	1.56696	0.783479	+	0.621419i	$0.213443\pi$
$212$	737921.	1.12764
$213$	660638.	0.997733
$214$	1.37811e6	2.05707
$215$	313096.	0.461936
$216$	−23397.8	−0.0341225
$217$	−41854.7	−0.0603386
$218$	2.03055e6	2.89383
$219$	−155297.	−0.218802
$220$	1.13073e6	1.57508
$221$	1.15024e6	1.58419
$222$	889167.	1.21088
$223$	−49729.0	−0.0669650
$223$	−49729.0	−0.0669650
$224$	−1.24046e6	−1.65182
$225$	159844.	0.210494
$226$	−113881.	−0.148314
$227$	−165686.	−0.213413	−0.106706	−	0.994291i	$-0.534030\pi$
$227$	−165686.	−0.213413	−0.106706	+	0.994291i	$0.534030\pi$
$228$	−53430.8	−0.0680698
$229$	−215031.	−0.270964	−0.135482	−	0.990780i	$-0.543258\pi$
$229$	−215031.	−0.270964	−0.135482	+	0.990780i	$0.543258\pi$
$230$	−1.36139e6	−1.69693
$231$	−592510.	−0.730576
$232$	−21084.1	−0.0257179
$233$	805130.	0.971574	0.485787	−	0.874077i	$-0.338533\pi$
$233$	805130.	0.971574	0.485787	+	0.874077i	$0.338533\pi$
$234$	−393994.	−0.470381
$235$	444093.	0.524570
$236$	143840.	0.168113
$237$	−195295.	−0.225850
$238$	2.39589e6	2.74173
$239$	1.02310e6	1.15857	0.579285	−	0.815125i	$-0.303332\pi$
$239$	1.02310e6	1.15857	0.579285	+	0.815125i	$0.303332\pi$
$240$	−568198.	−0.636754
$241$	515136.	0.571320	0.285660	−	0.958331i	$-0.407787\pi$
$241$	515136.	0.571320	0.285660	+	0.958331i	$0.407787\pi$
$242$	−276698.	−0.303715
$243$	59049.0	0.0641500
$244$	−779869.	−0.838585
$245$	389974.	0.415069
$246$	−1.19151e6	−1.25533
$247$	−97633.3	−0.101825
$248$	−9002.14	−0.00929430
$249$	−969824.	−0.991276
$250$	677556.	0.685638
$251$	−833582.	−0.835149	−0.417575	−	0.908643i	$-0.637120\pi$
$251$	−833582.	−0.835149	−0.417575	+	0.908643i	$0.637120\pi$
$252$	−433878.	−0.430394
$253$	1.02083e6	1.00266
$254$	−584610.	−0.568568
$255$	1.25216e6	1.20590
$256$	748813.	0.714124
$257$	−96079.2	−0.0907395	−0.0453698	−	0.998970i	$-0.514447\pi$
$257$	−96079.2	−0.0907395	−0.0453698	+	0.998970i	$0.514447\pi$
$258$	−325180.	−0.304141
$259$	1.78924e6	1.65737
$260$	1.51299e6	1.38805
$261$	53209.9	0.0483494
$262$	−1.70002e6	−1.53003
$263$	574245.	0.511927	0.255964	−	0.966686i	$-0.417607\pi$
$263$	574245.	0.511927	0.255964	+	0.966686i	$0.417607\pi$
$264$	−127437.	−0.112535
$265$	1.46788e6	1.28403
$266$	−203366.	−0.176228
$267$	−353699.	−0.303638
$268$	888093.	0.755303
$269$	592517.	0.499253	0.249626	−	0.968342i	$-0.419692\pi$
$269$	592517.	0.499253	0.249626	+	0.968342i	$0.419692\pi$
$270$	−428907.	−0.358058
$271$	2.25720e6	1.86701	0.933507	−	0.358560i	$-0.116732\pi$
$271$	2.25720e6	1.86701	0.933507	+	0.358560i	$0.116732\pi$
$272$	−1.72283e6	−1.41196
$273$	−792819.	−0.643824
$274$	478351.	0.384919
$275$	870597.	0.694201
$276$	747525.	0.590681
$277$	−1.77986e6	−1.39375	−0.696877	−	0.717190i	$-0.745428\pi$
$277$	−1.77986e6	−1.39375	−0.696877	+	0.717190i	$0.745428\pi$
$278$	−1.76527e6	−1.36993
$279$	22718.7	0.0174732
$280$	341987.	0.260684
$281$	1.06974e6	0.808184	0.404092	−	0.914718i	$-0.367587\pi$
$281$	1.06974e6	0.808184	0.404092	+	0.914718i	$0.367587\pi$
$282$	−461232.	−0.345380
$283$	344480.	0.255681	0.127840	−	0.991795i	$-0.459195\pi$
$283$	344480.	0.255681	0.127840	+	0.991795i	$0.459195\pi$
$284$	2.63486e6	1.93848
$285$	−106285.	−0.0775103
$286$	−2.14591e6	−1.55130
$287$	−2.39762e6	−1.71821
$288$	673320.	0.478344
$289$	2.37682e6	1.67398
$290$	−386494.	−0.269866
$291$	364080.	0.252037
$292$	−619378.	−0.425107
$293$	235373.	0.160172	0.0800862	−	0.996788i	$-0.474480\pi$
$293$	235373.	0.160172	0.0800862	+	0.996788i	$0.474480\pi$
$294$	−405025.	−0.273283
$295$	286128.	0.191428
$296$	384830.	0.255293
$297$	321613.	0.211565
$298$	−1.61302e6	−1.05220
$299$	1.36594e6	0.883597
$300$	637513.	0.408965
$301$	−654348.	−0.416286
$302$	3.38224e6	2.13396
$303$	−697532.	−0.436473
$304$	146236.	0.0907550
$305$	−1.55132e6	−0.954886
$306$	−1.30049e6	−0.793967
$307$	−800842.	−0.484954	−0.242477	−	0.970157i	$-0.577960\pi$
$307$	−800842.	−0.484954	−0.242477	+	0.970157i	$0.577960\pi$
$308$	−2.36314e6	−1.41942
$309$	805244.	0.479768
$310$	−165019.	−0.0975281
$311$	204704.	0.120012	0.0600060	−	0.998198i	$-0.480888\pi$
$311$	204704.	0.120012	0.0600060	+	0.998198i	$0.480888\pi$
$312$	−170520.	−0.0991719
$313$	−552393.	−0.318704	−0.159352	−	0.987222i	$-0.550940\pi$
$313$	−552393.	−0.318704	−0.159352	+	0.987222i	$0.550940\pi$
$314$	2.42906e6	1.39032
$315$	−863073.	−0.490085
$316$	−778905.	−0.438800
$317$	162089.	0.0905951	0.0452976	−	0.998974i	$-0.485576\pi$
$317$	162089.	0.0905951	0.0452976	+	0.998974i	$0.485576\pi$
$318$	−1.52453e6	−0.845411
$319$	289810.	0.159455
$320$	−2.87045e6	−1.56702
$321$	−1.50524e6	−0.815349
$322$	2.84520e6	1.52923
$323$	−322266.	−0.171873
$324$	235508.	0.124636
$325$	1.16492e6	0.611768
$326$	−5.40536e6	−2.81696
$327$	−2.21787e6	−1.14701
$328$	−515682.	−0.264665
$329$	−928119.	−0.472731
$330$	−2.33606e6	−1.18086
$331$	1.80933e6	0.907709	0.453855	−	0.891076i	$-0.350049\pi$
$331$	1.80933e6	0.907709	0.453855	+	0.891076i	$0.350049\pi$
$332$	−3.86800e6	−1.92593
$333$	−971195.	−0.479950
$334$	−2.26966e6	−1.11325
$335$	1.76660e6	0.860055
$336$	1.18749e6	0.573829
$337$	−3.31073e6	−1.58799	−0.793997	−	0.607922i	$-0.792003\pi$
$337$	−3.31073e6	−1.58799	−0.793997	+	0.607922i	$0.792003\pi$
$338$	188027.	0.0895220
$339$	124387.	0.0587863
$340$	4.99406e6	2.34292
$341$	123738.	0.0576260
$342$	110387.	0.0510331
$343$	1.69304e6	0.777018
$344$	−140737.	−0.0641229
$345$	1.48698e6	0.672601
$346$	−3.03308e6	−1.36205
$347$	4.07902e6	1.81858	0.909289	−	0.416166i	$-0.136626\pi$
$347$	4.07902e6	1.81858	0.909289	+	0.416166i	$0.136626\pi$
$348$	212220.	0.0939373
$349$	−630834.	−0.277237	−0.138619	−	0.990346i	$-0.544266\pi$
$349$	−630834.	−0.277237	−0.138619	+	0.990346i	$0.544266\pi$
$350$	2.42648e6	1.05878
$351$	430341.	0.186442
$352$	3.66727e6	1.57756
$353$	2.29220e6	0.979073	0.489537	−	0.871983i	$-0.337166\pi$
$353$	2.29220e6	0.979073	0.489537	+	0.871983i	$0.337166\pi$
$354$	−297171.	−0.126037
$355$	5.24128e6	2.20732
$356$	−1.41068e6	−0.589933
$357$	−2.61692e6	−1.08673
$358$	4.67656e6	1.92850
$359$	3.91592e6	1.60360	0.801802	−	0.597589i	$-0.203875\pi$
$359$	3.91592e6	1.60360	0.801802	+	0.597589i	$0.203875\pi$
$360$	−185630.	−0.0754905
$361$	−2.44874e6	−0.988953
$362$	−1.24003e6	−0.497347
$363$	302224.	0.120382
$364$	−3.16204e6	−1.25088
$365$	−1.23207e6	−0.484064
$366$	1.61119e6	0.628702
$367$	−2.51377e6	−0.974227	−0.487114	−	0.873339i	$-0.661950\pi$
$367$	−2.51377e6	−0.974227	−0.487114	+	0.873339i	$0.661950\pi$
$368$	−2.04592e6	−0.787533
$369$	1.30143e6	0.497569
$370$	7.05435e6	2.67888
$371$	−3.06775e6	−1.15714
$372$	90610.2	0.0339484
$373$	2.76456e6	1.02885	0.514427	−	0.857534i	$-0.328005\pi$
$373$	2.76456e6	1.02885	0.514427	+	0.857534i	$0.328005\pi$
$374$	−7.08317e6	−2.61848
$375$	−740062.	−0.271763
$376$	−199620.	−0.0728174
$377$	387786.	0.140520
$378$	896383.	0.322674
$379$	255046.	0.0912055	0.0456027	−	0.998960i	$-0.485479\pi$
$379$	255046.	0.0912055	0.0456027	+	0.998960i	$0.485479\pi$
$380$	−423902.	−0.150593
$381$	638542.	0.225360
$382$	5.24737e6	1.83985
$383$	2.13776e6	0.744665	0.372333	−	0.928099i	$-0.378558\pi$
$383$	2.13776e6	0.744665	0.372333	+	0.928099i	$0.378558\pi$
$384$	587206.	0.203218
$385$	−4.70077e6	−1.61628
$386$	6.25105e6	2.13543
$387$	355179.	0.120551
$388$	1.45208e6	0.489679
$389$	−1.53702e6	−0.514998	−0.257499	−	0.966279i	$-0.582898\pi$
$389$	−1.53702e6	−0.514998	−0.257499	+	0.966279i	$0.582898\pi$
$390$	−3.12581e6	−1.04064
$391$	4.50867e6	1.49144
$392$	−175294.	−0.0576171
$393$	1.85685e6	0.606450
$394$	−3.62448e6	−1.17626
$395$	−1.54940e6	−0.499657
$396$	1.28271e6	0.411045
$397$	−2.05751e6	−0.655188	−0.327594	−	0.944819i	$-0.606238\pi$
$397$	−2.05751e6	−0.655188	−0.327594	+	0.944819i	$0.606238\pi$
$398$	224429.	0.0710185
$399$	222127.	0.0698505
$400$	−1.74483e6	−0.545258
$401$	−2.96657e6	−0.921285	−0.460643	−	0.887586i	$-0.652381\pi$
$401$	−2.96657e6	−0.921285	−0.460643	+	0.887586i	$0.652381\pi$
$402$	−1.83478e6	−0.566264
$403$	165571.	0.0507833
$404$	−2.78200e6	−0.848017
$405$	468475.	0.141922
$406$	807743.	0.243197
$407$	−5.28966e6	−1.58286
$408$	−562849.	−0.167395
$409$	4.28668e6	1.26711	0.633553	−	0.773700i	$-0.281596\pi$
$409$	4.28668e6	1.26711	0.633553	+	0.773700i	$0.281596\pi$
$410$	−9.45301e6	−2.77722
$411$	−522480.	−0.152568
$412$	3.21160e6	0.932133
$413$	−597986.	−0.172511
$414$	−1.54437e6	−0.442844
$415$	−7.69426e6	−2.19304
$416$	4.90706e6	1.39023
$417$	1.92812e6	0.542992
$418$	601227.	0.168305
$419$	6.40941e6	1.78354	0.891770	−	0.452489i	$-0.149464\pi$
$419$	6.40941e6	1.78354	0.891770	+	0.452489i	$0.149464\pi$
$420$	−3.44224e6	−0.952178
$421$	4.98237e6	1.37003	0.685016	−	0.728528i	$-0.259795\pi$
$421$	4.98237e6	1.37003	0.685016	+	0.728528i	$0.259795\pi$
$422$	−8.34993e6	−2.28245
$423$	503782.	0.136896
$424$	−659813.	−0.178240
$425$	3.84514e6	1.03262
$426$	−5.44356e6	−1.45331
$427$	3.24214e6	0.860522
$428$	−6.00343e6	−1.58413
$429$	2.34387e6	0.614881
$430$	−2.57987e6	−0.672863
$431$	1.79826e6	0.466294	0.233147	−	0.972441i	$-0.425098\pi$
$431$	1.79826e6	0.466294	0.233147	+	0.972441i	$0.425098\pi$
$432$	−644568.	−0.166173
$433$	−4.04335e6	−1.03639	−0.518193	−	0.855264i	$-0.673395\pi$
$433$	−4.04335e6	−1.03639	−0.518193	+	0.855264i	$0.673395\pi$
$434$	344877.	0.0878901
$435$	422149.	0.106965
$436$	−8.84566e6	−2.22851
$437$	−382701.	−0.0958641
$438$	1.27962e6	0.318710
$439$	−6.75844e6	−1.67373	−0.836865	−	0.547409i	$-0.815614\pi$
$439$	−6.75844e6	−1.67373	−0.836865	+	0.547409i	$0.815614\pi$
$440$	−1.01104e6	−0.248965
$441$	442389.	0.108320
$442$	−9.47777e6	−2.30755
$443$	2.58421e6	0.625631	0.312816	−	0.949814i	$-0.398728\pi$
$443$	2.58421e6	0.625631	0.312816	+	0.949814i	$0.398728\pi$
$444$	−3.87347e6	−0.932487
$445$	−2.80613e6	−0.671750
$446$	409760.	0.0975421
$447$	1.76183e6	0.417057
$448$	5.99902e6	1.41216
$449$	1.61661e6	0.378434	0.189217	−	0.981935i	$-0.439405\pi$
$449$	1.61661e6	0.378434	0.189217	+	0.981935i	$0.439405\pi$
$450$	−1.31709e6	−0.306608
$451$	7.08828e6	1.64097
$452$	496100.	0.114215
$453$	−3.69426e6	−0.845827
$454$	1.36522e6	0.310860
$455$	−6.28995e6	−1.42436
$456$	47775.2	0.0107595
$457$	1.62866e6	0.364787	0.182394	−	0.983226i	$-0.441615\pi$
$457$	1.62866e6	0.364787	0.182394	+	0.983226i	$0.441615\pi$
$458$	1.77182e6	0.394690
$459$	1.42046e6	0.314701
$460$	5.93061e6	1.30679
$461$	−8.71180e6	−1.90922	−0.954609	−	0.297862i	$-0.903726\pi$
$461$	−8.71180e6	−1.90922	−0.954609	+	0.297862i	$0.903726\pi$
$462$	4.88219e6	1.06417
$463$	−6.31119e6	−1.36823	−0.684115	−	0.729374i	$-0.739811\pi$
$463$	−6.31119e6	−1.36823	−0.684115	+	0.729374i	$0.739811\pi$
$464$	−580830.	−0.125243
$465$	180242.	0.0386567
$466$	−6.63415e6	−1.41521
$467$	6.24110e6	1.32425	0.662123	−	0.749395i	$-0.269656\pi$
$467$	6.24110e6	1.32425	0.662123	+	0.749395i	$0.269656\pi$
$468$	1.71635e6	0.362236
$469$	−3.69206e6	−0.775062
$470$	−3.65926e6	−0.764097
$471$	−2.65315e6	−0.551073
$472$	−128615.	−0.0265728
$473$	1.93450e6	0.397572
$474$	1.60920e6	0.328976
$475$	−326380.	−0.0663727
$476$	−1.04372e7	−2.11138
$477$	1.66517e6	0.335091
$478$	−8.43017e6	−1.68759
$479$	274741.	0.0547122	0.0273561	−	0.999626i	$-0.491291\pi$
$479$	274741.	0.0547122	0.0273561	+	0.999626i	$0.491291\pi$
$480$	5.34189e6	1.05826
$481$	−7.07793e6	−1.39490
$482$	−4.24465e6	−0.832193
$483$	−3.10768e6	−0.606133
$484$	1.20537e6	0.233888
$485$	2.88849e6	0.557592
$486$	−486555.	−0.0934418
$487$	−6.09595e6	−1.16471	−0.582357	−	0.812933i	$-0.697869\pi$
$487$	−6.09595e6	−1.16471	−0.582357	+	0.812933i	$0.697869\pi$
$488$	697321.	0.132551
$489$	5.90402e6	1.11654
$490$	−3.21333e6	−0.604595
$491$	6.40956e6	1.19984	0.599922	−	0.800058i	$-0.295198\pi$
$491$	6.40956e6	1.19984	0.599922	+	0.800058i	$0.295198\pi$
$492$	5.19055e6	0.966719
$493$	1.28000e6	0.237188
$494$	804484.	0.148320
$495$	2.55157e6	0.468052
$496$	−247993.	−0.0452622
$497$	−1.09539e7	−1.98919
$498$	7.99121e6	1.44391
$499$	−7.58918e6	−1.36441	−0.682203	−	0.731163i	$-0.738978\pi$
$499$	−7.58918e6	−1.36441	−0.682203	+	0.731163i	$0.738978\pi$
$500$	−2.95163e6	−0.528004
$501$	2.47904e6	0.441255
$502$	6.86860e6	1.21649
$503$	2.75453e6	0.485432	0.242716	−	0.970097i	$-0.421962\pi$
$503$	2.75453e6	0.485432	0.242716	+	0.970097i	$0.421962\pi$
$504$	387953.	0.0680303
$505$	−5.53398e6	−0.965626
$506$	−8.41149e6	−1.46048
$507$	−205373.	−0.0354833
$508$	2.54673e6	0.437849
$509$	−3.27074e6	−0.559566	−0.279783	−	0.960063i	$-0.590262\pi$
$509$	−3.27074e6	−0.559566	−0.279783	+	0.960063i	$0.590262\pi$
$510$	−1.03176e7	−1.75653
$511$	2.57493e6	0.436228
$512$	−8.25795e6	−1.39219
$513$	−120570.	−0.0202277
$514$	791679.	0.132172
$515$	6.38853e6	1.06141
$516$	1.41658e6	0.234216
$517$	2.74387e6	0.451479
$518$	−1.47430e7	−2.41414
$519$	3.31288e6	0.539868
$520$	−1.35285e6	−0.219402
$521$	3.91993e6	0.632680	0.316340	−	0.948646i	$-0.397546\pi$
$521$	3.91993e6	0.632680	0.316340	+	0.948646i	$0.397546\pi$
$522$	−438442.	−0.0704264
$523$	−7.50864e6	−1.20035	−0.600174	−	0.799870i	$-0.704902\pi$
$523$	−7.50864e6	−1.20035	−0.600174	+	0.799870i	$0.704902\pi$
$524$	7.40576e6	1.17826
$525$	−2.65033e6	−0.419663
$526$	−4.73170e6	−0.745680
$527$	546512.	0.0857183
$528$	−3.51067e6	−0.548032
$529$	−1.08215e6	−0.168131
$530$	−1.20951e7	−1.87034
$531$	324586.	0.0499566
$532$	885922.	0.135711
$533$	9.48461e6	1.44611
$534$	2.91443e6	0.442283
$535$	−1.19421e7	−1.80383
$536$	−794090.	−0.119387
$537$	−5.10799e6	−0.764389
$538$	−4.88226e6	−0.727218
$539$	2.40949e6	0.357235
$540$	1.86844e6	0.275737
$541$	1.05098e7	1.54384	0.771919	−	0.635721i	$-0.219297\pi$
$541$	1.05098e7	1.54384	0.771919	+	0.635721i	$0.219297\pi$
$542$	−1.85990e7	−2.71952
$543$	1.35442e6	0.197131
$544$	1.61971e7	2.34661
$545$	−1.75958e7	−2.53757
$546$	6.53271e6	0.937803
$547$	776514.	0.110964	0.0554818	−	0.998460i	$-0.482331\pi$
$547$	776514.	0.110964	0.0554818	+	0.998460i	$0.482331\pi$
$548$	−2.08383e6	−0.296423
$549$	−1.75983e6	−0.249195
$550$	−7.17359e6	−1.01118
$551$	−108648.	−0.0152455
$552$	−668401.	−0.0933661
$553$	3.23813e6	0.450279
$554$	1.46658e7	2.03016
$555$	−7.70513e6	−1.06181
$556$	7.69001e6	1.05497
$557$	−8.05840e6	−1.10055	−0.550277	−	0.834982i	$-0.685478\pi$
$557$	−8.05840e6	−1.10055	−0.550277	+	0.834982i	$0.685478\pi$
$558$	−187199.	−0.0254517
$559$	2.58849e6	0.350363
$560$	9.42115e6	1.26950
$561$	7.73661e6	1.03787
$562$	−8.81446e6	−1.17721
$563$	1.74718e6	0.232309	0.116154	−	0.993231i	$-0.462943\pi$
$563$	1.74718e6	0.232309	0.116154	+	0.993231i	$0.462943\pi$
$564$	2.00926e6	0.265973
$565$	986845.	0.130055
$566$	−2.83847e6	−0.372428
$567$	−979076.	−0.127897
$568$	−2.35596e6	−0.306406
$569$	−2.10807e6	−0.272963	−0.136482	−	0.990643i	$-0.543579\pi$
$569$	−2.10807e6	−0.272963	−0.136482	+	0.990643i	$0.543579\pi$
$570$	875772.	0.112903
$571$	1.08820e7	1.39675	0.698375	−	0.715733i	$-0.253907\pi$
$571$	1.08820e7	1.39675	0.698375	+	0.715733i	$0.253907\pi$
$572$	9.34819e6	1.19464
$573$	−5.73146e6	−0.729253
$574$	1.97561e7	2.50277
$575$	4.56623e6	0.575954
$576$	−3.25626e6	−0.408943
$577$	−1.16376e6	−0.145521	−0.0727604	−	0.997349i	$-0.523181\pi$
$577$	−1.16376e6	−0.145521	−0.0727604	+	0.997349i	$0.523181\pi$
$578$	−1.95846e7	−2.43835
$579$	−6.82772e6	−0.846408
$580$	1.68368e6	0.207821
$581$	1.60804e7	1.97632
$582$	−2.99997e6	−0.367121
$583$	9.06943e6	1.10512
$584$	553818.	0.0671946
$585$	3.41418e6	0.412474
$586$	−1.93944e6	−0.233309
$587$	1.45829e7	1.74682	0.873411	−	0.486984i	$-0.161903\pi$
$587$	1.45829e7	1.74682	0.873411	+	0.486984i	$0.161903\pi$
$588$	1.76440e6	0.210453
$589$	−46388.6	−0.00550963
$590$	−2.35765e6	−0.278837
$591$	3.95885e6	0.466230
$592$	1.06014e7	1.24325
$593$	8.69521e6	1.01541	0.507707	−	0.861530i	$-0.330493\pi$
$593$	8.69521e6	1.01541	0.507707	+	0.861530i	$0.330493\pi$
$594$	−2.65005e6	−0.308168
$595$	−2.07618e7	−2.40421
$596$	7.02680e6	0.810292
$597$	−245133.	−0.0281492
$598$	−1.12552e7	−1.28706
$599$	1.19426e7	1.35998	0.679990	−	0.733222i	$-0.261984\pi$
$599$	1.19426e7	1.35998	0.679990	+	0.733222i	$0.261984\pi$
$600$	−570034.	−0.0646431
$601$	118781.	0.0134141	0.00670706	−	0.999978i	$-0.497865\pi$
$601$	118781.	0.0134141	0.00670706	+	0.999978i	$0.497865\pi$
$602$	5.39173e6	0.606369
$603$	2.00404e6	0.224447
$604$	−1.47340e7	−1.64334
$605$	2.39774e6	0.266326
$606$	5.74756e6	0.635773
$607$	6.90113e6	0.760237	0.380118	−	0.924938i	$-0.375883\pi$
$607$	6.90113e6	0.760237	0.380118	+	0.924938i	$0.375883\pi$
$608$	−1.37483e6	−0.150831
$609$	−882260.	−0.0963947
$610$	1.27826e7	1.39090
$611$	3.67149e6	0.397868
$612$	5.66530e6	0.611427
$613$	2.93745e6	0.315732	0.157866	−	0.987461i	$-0.449539\pi$
$613$	2.93745e6	0.315732	0.157866	+	0.987461i	$0.449539\pi$
$614$	6.59882e6	0.706391
$615$	1.03251e7	1.10079
$616$	2.11300e6	0.224362
$617$	−1.11757e7	−1.18184	−0.590922	−	0.806729i	$-0.701236\pi$
$617$	−1.11757e7	−1.18184	−0.590922	+	0.806729i	$0.701236\pi$
$618$	−6.63509e6	−0.698836
$619$	5.61018e6	0.588505	0.294253	−	0.955728i	$-0.404929\pi$
$619$	5.61018e6	0.588505	0.294253	+	0.955728i	$0.404929\pi$
$620$	718870.	0.0751054
$621$	1.68684e6	0.175528
$622$	−1.68673e6	−0.174811
$623$	5.86459e6	0.605366
$624$	−4.69752e6	−0.482956
$625$	−1.20382e7	−1.23271
$626$	4.55164e6	0.464228
$627$	−656692.	−0.0667103
$628$	−1.05817e7	−1.07067
$629$	−2.33627e7	−2.35449
$630$	7.11159e6	0.713864
$631$	−1.15144e7	−1.15124	−0.575622	−	0.817716i	$-0.695240\pi$
$631$	−1.15144e7	−1.15124	−0.575622	+	0.817716i	$0.695240\pi$
$632$	696459.	0.0693590
$633$	9.12023e6	0.904683
$634$	−1.33559e6	−0.131962
$635$	5.06597e6	0.498573
$636$	6.64129e6	0.651043
$637$	3.22407e6	0.314815
$638$	−2.38800e6	−0.232264
$639$	5.94574e6	0.576042
$640$	4.65869e6	0.449587
$641$	−1.99471e6	−0.191750	−0.0958748	−	0.995393i	$-0.530565\pi$
$641$	−1.99471e6	−0.191750	−0.0958748	+	0.995393i	$0.530565\pi$
$642$	1.24030e7	1.18765
$643$	−1.33477e7	−1.27315	−0.636576	−	0.771214i	$-0.719650\pi$
$643$	−1.33477e7	−1.27315	−0.636576	+	0.771214i	$0.719650\pi$
$644$	−1.23945e7	−1.17765
$645$	2.81787e6	0.266699
$646$	2.65542e6	0.250353
$647$	−1.35684e7	−1.27429	−0.637143	−	0.770745i	$-0.719884\pi$
$647$	−1.35684e7	−1.27429	−0.637143	+	0.770745i	$0.719884\pi$
$648$	−210580.	−0.0197006
$649$	1.76787e6	0.164755
$650$	−9.59876e6	−0.891111
$651$	−376693.	−0.0348365
$652$	2.35473e7	2.16931
$653$	−8.27964e6	−0.759851	−0.379926	−	0.925017i	$-0.624050\pi$
$653$	−8.27964e6	−0.759851	−0.379926	+	0.925017i	$0.624050\pi$
$654$	1.82749e7	1.67075
$655$	1.47316e7	1.34167
$656$	−1.42061e7	−1.28889
$657$	−1.39767e6	−0.126325
$658$	7.64757e6	0.688587
$659$	8.88700e6	0.797153	0.398576	−	0.917135i	$-0.369504\pi$
$659$	8.88700e6	0.797153	0.398576	+	0.917135i	$0.369504\pi$
$660$	1.01766e7	0.909372
$661$	5.99404e6	0.533600	0.266800	−	0.963752i	$-0.414034\pi$
$661$	5.99404e6	0.533600	0.266800	+	0.963752i	$0.414034\pi$
$662$	−1.49086e7	−1.32218
$663$	1.03521e7	0.914630
$664$	3.45858e6	0.304423
$665$	1.76228e6	0.154533
$666$	8.00250e6	0.699101
$667$	1.52004e6	0.132294
$668$	9.88729e6	0.857307
$669$	−447561.	−0.0386622
$670$	−1.45565e7	−1.25277
$671$	−9.58499e6	−0.821837
$672$	−1.11641e7	−0.953679
$673$	9.42030e6	0.801728	0.400864	−	0.916138i	$-0.368710\pi$
$673$	9.42030e6	0.801728	0.400864	+	0.916138i	$0.368710\pi$
$674$	2.72799e7	2.31309
$675$	1.43859e6	0.121529
$676$	−819102.	−0.0689400
$677$	1.54362e7	1.29440	0.647200	−	0.762320i	$-0.275940\pi$
$677$	1.54362e7	1.29440	0.647200	+	0.762320i	$0.275940\pi$
$678$	−1.02493e6	−0.0856290
$679$	−6.03672e6	−0.502489
$680$	−4.46545e6	−0.370333
$681$	−1.49117e6	−0.123214
$682$	−1.01959e6	−0.0839389
$683$	−1.35048e7	−1.10774	−0.553870	−	0.832603i	$-0.686849\pi$
$683$	−1.35048e7	−1.10774	−0.553870	+	0.832603i	$0.686849\pi$
$684$	−480877.	−0.0393001
$685$	−4.14517e6	−0.337533
$686$	−1.39504e7	−1.13182
$687$	−1.93528e6	−0.156441
$688$	−3.87707e6	−0.312272
$689$	1.21355e7	0.973891
$690$	−1.22525e7	−0.979721
$691$	2.31713e7	1.84610	0.923051	−	0.384678i	$-0.125688\pi$
$691$	2.31713e7	1.84610	0.923051	+	0.384678i	$0.125688\pi$
$692$	1.32130e7	1.04890
$693$	−5.33259e6	−0.421798
$694$	−3.36105e7	−2.64897
$695$	1.52970e7	1.20128
$696$	−189757.	−0.0148482
$697$	3.13066e7	2.44092
$698$	5.19798e6	0.403828
$699$	7.24617e6	0.560939
$700$	−1.05704e7	−0.815357
$701$	−1.53562e6	−0.118029	−0.0590144	−	0.998257i	$-0.518796\pi$
$701$	−1.53562e6	−0.118029	−0.0590144	+	0.998257i	$0.518796\pi$
$702$	−3.54595e6	−0.271575
$703$	1.98305e6	0.151337
$704$	−1.77354e7	−1.34868
$705$	3.99683e6	0.302861
$706$	−1.88874e7	−1.42613
$707$	1.15656e7	0.870201
$708$	1.29456e6	0.0970599
$709$	−4.48906e6	−0.335382	−0.167691	−	0.985840i	$-0.553631\pi$
$709$	−4.48906e6	−0.335382	−0.167691	+	0.985840i	$0.553631\pi$
$710$	−4.31873e7	−3.21522
$711$	−1.75765e6	−0.130395
$712$	1.26136e6	0.0932479
$713$	649001.	0.0478103
$714$	2.15630e7	1.58294
$715$	1.85955e7	1.36032
$716$	−2.03725e7	−1.48512
$717$	9.20788e6	0.668901
$718$	−3.22666e7	−2.33583
$719$	3.19461e6	0.230460	0.115230	−	0.993339i	$-0.463240\pi$
$719$	3.19461e6	0.230460	0.115230	+	0.993339i	$0.463240\pi$
$720$	−5.11379e6	−0.367630
$721$	−1.33515e7	−0.956518
$722$	2.01773e7	1.44052
$723$	4.63623e6	0.329852
$724$	5.40191e6	0.383002
$725$	1.29634e6	0.0915953
$726$	−2.49028e6	−0.175350
$727$	−8.08345e6	−0.567232	−0.283616	−	0.958938i	$-0.591534\pi$
$727$	−8.08345e6	−0.567232	−0.283616	+	0.958938i	$0.591534\pi$
$728$	2.82735e6	0.197720
$729$	531441.	0.0370370
$730$	1.01521e7	0.705095
$731$	8.54405e6	0.591385
$732$	−7.01882e6	−0.484157
$733$	1.82682e7	1.25584	0.627921	−	0.778277i	$-0.283906\pi$
$733$	1.82682e7	1.25584	0.627921	+	0.778277i	$0.283906\pi$
$734$	2.07131e7	1.41907
$735$	3.50977e6	0.239640
$736$	1.92346e7	1.30885
$737$	1.09151e7	0.740218
$738$	−1.07236e7	−0.724766
$739$	−3.43808e6	−0.231582	−0.115791	−	0.993274i	$-0.536940\pi$
$739$	−3.43808e6	−0.231582	−0.115791	+	0.993274i	$0.536940\pi$
$740$	−3.07308e7	−2.06298
$741$	−878699.	−0.0587888
$742$	2.52778e7	1.68550
$743$	−7.59067e6	−0.504438	−0.252219	−	0.967670i	$-0.581160\pi$
$743$	−7.59067e6	−0.504438	−0.252219	+	0.967670i	$0.581160\pi$
$744$	−81019.2	−0.00536606
$745$	1.39778e7	0.922671
$746$	−2.27796e7	−1.49864
$747$	−8.72842e6	−0.572314
$748$	3.08563e7	2.01646
$749$	2.49580e7	1.62557
$750$	6.09800e6	0.395854
$751$	9.30430e6	0.601983	0.300991	−	0.953627i	$-0.402682\pi$
$751$	9.30430e6	0.601983	0.300991	+	0.953627i	$0.402682\pi$
$752$	−5.49919e6	−0.354613
$753$	−7.50224e6	−0.482174
$754$	−3.19530e6	−0.204684
$755$	−2.93090e7	−1.87126
$756$	−3.90490e6	−0.248488
$757$	2.91274e6	0.184741	0.0923703	−	0.995725i	$-0.470556\pi$
$757$	2.91274e6	0.184741	0.0923703	+	0.995725i	$0.470556\pi$
$758$	−2.10154e6	−0.132851
$759$	9.18747e6	0.578884
$760$	379032.	0.0238036
$761$	−1.33479e7	−0.835508	−0.417754	−	0.908560i	$-0.637183\pi$
$761$	−1.33479e7	−0.835508	−0.417754	+	0.908560i	$0.637183\pi$
$762$	−5.26149e6	−0.328263
$763$	3.67740e7	2.28681
$764$	−2.28591e7	−1.41685
$765$	1.12695e7	0.696224
$766$	−1.76148e7	−1.08469
$767$	2.36554e6	0.145191
$768$	6.73932e6	0.412299
$769$	−2.85066e7	−1.73832	−0.869160	−	0.494531i	$-0.835340\pi$
$769$	−2.85066e7	−1.73832	−0.869160	+	0.494531i	$0.835340\pi$
$770$	3.87336e7	2.35430
$771$	−864713.	−0.0523885
$772$	−2.72314e7	−1.64447
$773$	−6.15622e6	−0.370566	−0.185283	−	0.982685i	$-0.559320\pi$
$773$	−6.15622e6	−0.370566	−0.185283	+	0.982685i	$0.559320\pi$
$774$	−2.92662e6	−0.175596
$775$	553489.	0.0331020
$776$	−1.29838e6	−0.0774012
$777$	1.61031e7	0.956880
$778$	1.26648e7	0.750154
$779$	−2.65734e6	−0.156893
$780$	1.36169e7	0.801389
$781$	3.23838e7	1.89976
$782$	−3.71508e7	−2.17246
$783$	478889.	0.0279145
$784$	−4.82904e6	−0.280589
$785$	−2.10492e7	−1.21916
$786$	−1.53001e7	−0.883363
$787$	−5.46039e6	−0.314258	−0.157129	−	0.987578i	$-0.550224\pi$
$787$	−5.46039e6	−0.314258	−0.157129	+	0.987578i	$0.550224\pi$
$788$	1.57893e7	0.905830
$789$	5.16821e6	0.295561
$790$	1.27669e7	0.727807
$791$	−2.06243e6	−0.117203
$792$	−1.14694e6	−0.0649720
$793$	−1.28254e7	−0.724248
$794$	1.69536e7	0.954356
$795$	1.32109e7	0.741335
$796$	−977678.	−0.0546906
$797$	−1.47544e7	−0.822765	−0.411383	−	0.911463i	$-0.634954\pi$
$797$	−1.47544e7	−0.822765	−0.411383	+	0.911463i	$0.634954\pi$
$798$	−1.83030e6	−0.101745
$799$	1.21188e7	0.671571
$800$	1.64039e7	0.906196
$801$	−3.18329e6	−0.175305
$802$	2.44441e7	1.34196
$803$	−7.61247e6	−0.416617
$804$	7.99283e6	0.436074
$805$	−2.46553e7	−1.34097
$806$	−1.36428e6	−0.0739716
$807$	5.33265e6	0.288244
$808$	2.48753e6	0.134042
$809$	2.45767e7	1.32024	0.660120	−	0.751160i	$-0.270505\pi$
$809$	2.45767e7	1.32024	0.660120	+	0.751160i	$0.270505\pi$
$810$	−3.86016e6	−0.206725
$811$	−825944.	−0.0440959	−0.0220480	−	0.999757i	$-0.507019\pi$
$811$	−825944.	−0.0440959	−0.0220480	+	0.999757i	$0.507019\pi$
$812$	−3.51876e6	−0.187284
$813$	2.03148e7	1.07792
$814$	4.35860e7	2.30561
$815$	4.68405e7	2.47017
$816$	−1.55055e7	−0.815193
$817$	−725229.	−0.0380119
$818$	−3.53216e7	−1.84568
$819$	−7.13537e6	−0.371712
$820$	4.11800e7	2.13871
$821$	6.25802e6	0.324025	0.162013	−	0.986789i	$-0.448201\pi$
$821$	6.25802e6	0.324025	0.162013	+	0.986789i	$0.448201\pi$
$822$	4.30515e6	0.222233
$823$	−1.67942e7	−0.864291	−0.432146	−	0.901804i	$-0.642243\pi$
$823$	−1.67942e7	−0.864291	−0.432146	+	0.901804i	$0.642243\pi$
$824$	−2.87165e6	−0.147338
$825$	7.83537e6	0.400797
$826$	4.92731e6	0.251281
$827$	−1.10649e7	−0.562578	−0.281289	−	0.959623i	$-0.590762\pi$
$827$	−1.10649e7	−0.562578	−0.281289	+	0.959623i	$0.590762\pi$
$828$	6.72772e6	0.341030
$829$	−3.30122e7	−1.66835	−0.834176	−	0.551499i	$-0.814056\pi$
$829$	−3.30122e7	−1.66835	−0.834176	+	0.551499i	$0.814056\pi$
$830$	6.33995e7	3.19441
$831$	−1.60187e7	−0.804684
$832$	−2.37312e7	−1.18853
$833$	1.06420e7	0.531384
$834$	−1.58874e7	−0.790929
$835$	1.96679e7	0.976205
$836$	−2.61912e6	−0.129610
$837$	204468.	0.0100882
$838$	−5.28126e7	−2.59793
$839$	2.73255e7	1.34018	0.670090	−	0.742280i	$-0.266256\pi$
$839$	2.73255e7	1.34018	0.670090	+	0.742280i	$0.266256\pi$
$840$	3.07788e6	0.150506
$841$	−2.00796e7	−0.978961
$842$	−4.10540e7	−1.99561
$843$	9.62762e6	0.466605
$844$	3.63747e7	1.75769
$845$	−1.62936e6	−0.0785012
$846$	−4.15109e6	−0.199405
$847$	−5.01109e6	−0.240007
$848$	−1.81767e7	−0.868012
$849$	3.10032e6	0.147617
$850$	−3.16834e7	−1.50413
$851$	−2.77439e7	−1.31324
$852$	2.37137e7	1.11918
$853$	−2.04074e7	−0.960316	−0.480158	−	0.877182i	$-0.659421\pi$
$853$	−2.04074e7	−0.960316	−0.480158	+	0.877182i	$0.659421\pi$
$854$	−2.67148e7	−1.25345
$855$	−956564.	−0.0447506
$856$	5.36798e6	0.250395
$857$	−1.33604e7	−0.621393	−0.310696	−	0.950509i	$-0.600562\pi$
$857$	−1.33604e7	−0.621393	−0.310696	+	0.950509i	$0.600562\pi$
$858$	−1.93132e7	−0.895644
$859$	1.13841e7	0.526400	0.263200	−	0.964741i	$-0.415222\pi$
$859$	1.13841e7	0.526400	0.263200	+	0.964741i	$0.415222\pi$
$860$	1.12387e7	0.518165
$861$	−2.15786e7	−0.992009
$862$	−1.48174e7	−0.679211
$863$	5.23427e6	0.239237	0.119619	−	0.992820i	$-0.461833\pi$
$863$	5.23427e6	0.239237	0.119619	+	0.992820i	$0.461833\pi$
$864$	6.05988e6	0.276172
$865$	2.62833e7	1.19437
$866$	3.33166e7	1.50962
$867$	2.13914e7	0.966475
$868$	−1.50238e6	−0.0676833
$869$	−9.57315e6	−0.430036
$870$	−3.47845e6	−0.155807
$871$	1.46052e7	0.652321
$872$	7.90936e6	0.352249
$873$	3.27672e6	0.145514
$874$	3.15340e6	0.139637
$875$	1.22708e7	0.541816
$876$	−5.57440e6	−0.245436
$877$	−4.34612e7	−1.90811	−0.954054	−	0.299636i	$-0.903135\pi$
$877$	−4.34612e7	−1.90811	−0.954054	+	0.299636i	$0.903135\pi$
$878$	5.56886e7	2.43798
$879$	2.11836e6	0.0924755
$880$	−2.78525e7	−1.21243
$881$	4.45487e6	0.193373	0.0966864	−	0.995315i	$-0.469176\pi$
$881$	4.45487e6	0.193373	0.0966864	+	0.995315i	$0.469176\pi$
$882$	−3.64522e6	−0.157780
$883$	2.04007e7	0.880529	0.440264	−	0.897868i	$-0.354885\pi$
$883$	2.04007e7	0.880529	0.440264	+	0.897868i	$0.354885\pi$
$884$	4.12879e7	1.77702
$885$	2.57515e6	0.110521
$886$	−2.12935e7	−0.911303
$887$	−1.97489e7	−0.842816	−0.421408	−	0.906871i	$-0.638464\pi$
$887$	−1.97489e7	−0.842816	−0.421408	+	0.906871i	$0.638464\pi$
$888$	3.46347e6	0.147394
$889$	−1.05875e7	−0.449303
$890$	2.31221e7	0.978480
$891$	2.89452e6	0.122147
$892$	−1.78503e6	−0.0751162
$893$	−1.02866e6	−0.0431660
$894$	−1.45172e7	−0.607490
$895$	−4.05250e7	−1.69109
$896$	−9.73631e6	−0.405158
$897$	1.22935e7	0.510145
$898$	−1.33206e7	−0.551232
$899$	184249.	0.00760338
$900$	5.73762e6	0.236116
$901$	4.00567e7	1.64385
$902$	−5.84064e7	−2.39025
$903$	−5.88913e6	−0.240343
$904$	−443589.	−0.0180534
$905$	1.07455e7	0.436120
$906$	3.04401e7	1.23204
$907$	−2.14610e7	−0.866226	−0.433113	−	0.901340i	$-0.642585\pi$
$907$	−2.14610e7	−0.866226	−0.433113	+	0.901340i	$0.642585\pi$
$908$	−5.94731e6	−0.239390
$909$	−6.27779e6	−0.251998
$910$	5.18283e7	2.07474
$911$	−3.70251e7	−1.47809	−0.739044	−	0.673658i	$-0.764722\pi$
$911$	−3.70251e7	−1.47809	−0.739044	+	0.673658i	$0.764722\pi$
$912$	1.31612e6	0.0523974
$913$	−4.75398e7	−1.88747
$914$	−1.34199e7	−0.531354
$915$	−1.39619e7	−0.551304
$916$	−7.71856e6	−0.303947
$917$	−3.07879e7	−1.20908
$918$	−1.17044e7	−0.458397
$919$	−3.50073e7	−1.36732	−0.683660	−	0.729801i	$-0.739613\pi$
$919$	−3.50073e7	−1.36732	−0.683660	+	0.729801i	$0.739613\pi$
$920$	−5.30286e6	−0.206557
$921$	−7.20758e6	−0.279988
$922$	7.17839e7	2.78099
$923$	4.33317e7	1.67418
$924$	−2.12682e7	−0.819505
$925$	−2.36609e7	−0.909238
$926$	5.20033e7	1.99298
$927$	7.24719e6	0.276994
$928$	5.46064e6	0.208149
$929$	9.91282e6	0.376841	0.188420	−	0.982088i	$-0.439663\pi$
$929$	9.91282e6	0.376841	0.188420	+	0.982088i	$0.439663\pi$
$930$	−1.48517e6	−0.0563078
$931$	−903301.	−0.0341553
$932$	2.89003e7	1.08984
$933$	1.84233e6	0.0692889
$934$	−5.14257e7	−1.92892
$935$	6.13796e7	2.29612
$936$	−1.53468e6	−0.0572569
$937$	−7.06990e6	−0.263066	−0.131533	−	0.991312i	$-0.541990\pi$
$937$	−7.06990e6	−0.263066	−0.131533	+	0.991312i	$0.541990\pi$
$938$	3.04220e7	1.12897
$939$	−4.97154e6	−0.184004
$940$	1.59408e7	0.588424
$941$	−1.38641e7	−0.510410	−0.255205	−	0.966887i	$-0.582143\pi$
$941$	−1.38641e7	−0.510410	−0.255205	+	0.966887i	$0.582143\pi$
$942$	2.18616e7	0.802701
$943$	3.71776e7	1.36145
$944$	−3.54312e6	−0.129406
$945$	−7.76766e6	−0.282951
$946$	−1.59400e7	−0.579109
$947$	−2.83757e7	−1.02819	−0.514093	−	0.857735i	$-0.671871\pi$
$947$	−2.83757e7	−1.02819	−0.514093	+	0.857735i	$0.671871\pi$
$948$	−7.01014e6	−0.253341
$949$	−1.01860e7	−0.367146
$950$	2.68932e6	0.0966794
$951$	1.45880e6	0.0523051
$952$	9.33245e6	0.333736
$953$	−1.11592e7	−0.398016	−0.199008	−	0.979998i	$-0.563772\pi$
$953$	−1.11592e7	−0.398016	−0.199008	+	0.979998i	$0.563772\pi$
$954$	−1.37208e7	−0.488098
$955$	−4.54714e7	−1.61336
$956$	3.67243e7	1.29960
$957$	2.60829e6	0.0920612
$958$	−2.26382e6	−0.0796946
$959$	8.66310e6	0.304177
$960$	−2.58340e7	−0.904720
$961$	−2.85505e7	−0.997252
$962$	5.83211e7	2.03183
$963$	−1.35472e7	−0.470742
$964$	1.84909e7	0.640864
$965$	−5.41688e7	−1.87254
$966$	2.56068e7	0.882902
$967$	−4.02717e7	−1.38495	−0.692474	−	0.721443i	$-0.743479\pi$
$967$	−4.02717e7	−1.38495	−0.692474	+	0.721443i	$0.743479\pi$
$968$	−1.07779e6	−0.0369696
$969$	−2.90039e6	−0.0992311
$970$	−2.38007e7	−0.812196
$971$	−4.23365e7	−1.44101	−0.720505	−	0.693449i	$-0.756090\pi$
$971$	−4.23365e7	−1.44101	−0.720505	+	0.693449i	$0.756090\pi$
$972$	2.11957e6	0.0719587
$973$	−3.19696e7	−1.08257
$974$	5.02297e7	1.69654
$975$	1.04843e7	0.353205
$976$	1.92100e7	0.645509
$977$	1.55990e7	0.522830	0.261415	−	0.965226i	$-0.415811\pi$
$977$	1.55990e7	0.522830	0.261415	+	0.965226i	$0.415811\pi$
$978$	−4.86482e7	−1.62637
$979$	−1.73380e7	−0.578151
$980$	1.39982e7	0.465593
$981$	−1.99609e7	−0.662227
$982$	−5.28139e7	−1.74771
$983$	−1.97648e7	−0.652393	−0.326197	−	0.945302i	$-0.605767\pi$
$983$	−1.97648e7	−0.652393	−0.326197	+	0.945302i	$0.605767\pi$
$984$	−4.64114e6	−0.152805
$985$	3.14081e7	1.03146
$986$	−1.05470e7	−0.345491
$987$	−8.35307e6	−0.272931
$988$	−3.50456e6	−0.114220
$989$	1.01463e7	0.329851
$990$	−2.10246e7	−0.681772
$991$	−4.33057e7	−1.40075	−0.700375	−	0.713775i	$-0.746984\pi$
$991$	−4.33057e7	−1.40075	−0.700375	+	0.713775i	$0.746984\pi$
$992$	2.33150e6	0.0752238
$993$	1.62839e7	0.524066
$994$	9.02582e7	2.89748
$995$	−1.94480e6	−0.0622756
$996$	−3.48120e7	−1.11194
$997$	−5.21220e7	−1.66067	−0.830334	−	0.557266i	$-0.811850\pi$
$997$	−5.21220e7	−1.66067	−0.830334	+	0.557266i	$0.811850\pi$
$998$	6.25337e7	1.98741
$999$	−8.74075e6	−0.277099

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	669.6.a.d.1.7	✓	51

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
669.6.a.d.1.7	✓	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)

\(f(q)\)	\(=\)	\(q-8.23985 q^{2} +9.00000 q^{3} +35.8952 q^{4} +71.4029 q^{5} -74.1587 q^{6} -149.227 q^{7} -32.0957 q^{8} +81.0000 q^{9} -588.350 q^{10} +441.170 q^{11} +323.057 q^{12} +590.317 q^{13} +1229.61 q^{14} +642.626 q^{15} -884.182 q^{16} +1948.51 q^{17} -667.428 q^{18} -165.391 q^{19} +2563.02 q^{20} -1343.04 q^{21} -3635.18 q^{22} +2313.91 q^{23} -288.862 q^{24} +1973.38 q^{25} -4864.12 q^{26} +729.000 q^{27} -5356.52 q^{28} +656.912 q^{29} -5295.15 q^{30} +280.478 q^{31} +8312.59 q^{32} +3970.53 q^{33} -16055.4 q^{34} -10655.2 q^{35} +2907.51 q^{36} -11990.1 q^{37} +1362.80 q^{38} +5312.85 q^{39} -2291.73 q^{40} +16067.0 q^{41} +11066.5 q^{42} +4384.92 q^{43} +15835.9 q^{44} +5783.64 q^{45} -19066.3 q^{46} +6219.53 q^{47} -7957.64 q^{48} +5461.59 q^{49} -16260.4 q^{50} +17536.6 q^{51} +21189.5 q^{52} +20557.7 q^{53} -6006.85 q^{54} +31500.9 q^{55} +4789.54 q^{56} -1488.52 q^{57} -5412.86 q^{58} +4007.23 q^{59} +23067.2 q^{60} -21726.3 q^{61} -2311.09 q^{62} -12087.4 q^{63} -40200.7 q^{64} +42150.3 q^{65} -32716.6 q^{66} +24741.3 q^{67} +69942.0 q^{68} +20825.2 q^{69} +87797.5 q^{70} +73404.2 q^{71} -2599.76 q^{72} -17255.2 q^{73} +98796.3 q^{74} +17760.4 q^{75} -5936.75 q^{76} -65834.4 q^{77} -43777.1 q^{78} -21699.4 q^{79} -63133.2 q^{80} +6561.00 q^{81} -132390. q^{82} -107758. q^{83} -48208.7 q^{84} +139129. q^{85} -36131.1 q^{86} +5912.21 q^{87} -14159.7 q^{88} -39299.9 q^{89} -47656.3 q^{90} -88091.0 q^{91} +83058.3 q^{92} +2524.30 q^{93} -51248.0 q^{94} -11809.4 q^{95} +74813.3 q^{96} +40453.4 q^{97} -45002.7 q^{98} +35734.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 51 q + 18 q^{2} + 459 q^{3} + 940 q^{4} + 97 q^{5} + 162 q^{6} + 794 q^{7} + 411 q^{8} + 4131 q^{9} + 1256 q^{10} + 2349 q^{11} + 8460 q^{12} + 2954 q^{13} + 1816 q^{14} + 873 q^{15} + 18580 q^{16} + 1567 q^{17}+ \cdots + 190269 q^{99}+O(q^{100}) \) 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

Introduction

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