LMFDB - Embedded newform 538.6.a.d.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [538,6,Mod(1,538)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(538, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("538.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 538 = 2 \cdot 269 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	538.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$86.2864950594$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	538.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.00000 q^{2} -7.35365 q^{3} +16.0000 q^{4} +78.1821 q^{5} -29.4146 q^{6} +209.235 q^{7} +64.0000 q^{8} -188.924 q^{9} +312.729 q^{10} +531.909 q^{11} -117.658 q^{12} +134.764 q^{13} +836.940 q^{14} -574.924 q^{15} +256.000 q^{16} +79.7966 q^{17} -755.695 q^{18} +2330.32 q^{19} +1250.91 q^{20} -1538.64 q^{21} +2127.63 q^{22} -281.583 q^{23} -470.634 q^{24} +2987.45 q^{25} +539.056 q^{26} +3176.22 q^{27} +3347.76 q^{28} +2779.11 q^{29} -2299.70 q^{30} -2442.38 q^{31} +1024.00 q^{32} -3911.47 q^{33} +319.186 q^{34} +16358.4 q^{35} -3022.78 q^{36} -10682.2 q^{37} +9321.29 q^{38} -991.008 q^{39} +5003.66 q^{40} -10311.9 q^{41} -6154.56 q^{42} +1275.58 q^{43} +8510.54 q^{44} -14770.5 q^{45} -1126.33 q^{46} -9613.12 q^{47} -1882.53 q^{48} +26972.3 q^{49} +11949.8 q^{50} -586.796 q^{51} +2156.23 q^{52} -10078.2 q^{53} +12704.9 q^{54} +41585.7 q^{55} +13391.0 q^{56} -17136.4 q^{57} +11116.4 q^{58} -19428.6 q^{59} -9198.78 q^{60} -39178.8 q^{61} -9769.52 q^{62} -39529.5 q^{63} +4096.00 q^{64} +10536.1 q^{65} -15645.9 q^{66} +31478.4 q^{67} +1276.75 q^{68} +2070.66 q^{69} +65433.8 q^{70} +48906.4 q^{71} -12091.1 q^{72} +940.198 q^{73} -42728.9 q^{74} -21968.6 q^{75} +37285.1 q^{76} +111294. q^{77} -3964.03 q^{78} -75460.1 q^{79} +20014.6 q^{80} +22551.7 q^{81} -41247.5 q^{82} -37665.2 q^{83} -24618.2 q^{84} +6238.67 q^{85} +5102.31 q^{86} -20436.6 q^{87} +34042.1 q^{88} -104482. q^{89} -59081.9 q^{90} +28197.4 q^{91} -4505.33 q^{92} +17960.4 q^{93} -38452.5 q^{94} +182190. q^{95} -7530.14 q^{96} -86520.0 q^{97} +107889. q^{98} -100490. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 128 q^{2} + 32 q^{3} + 512 q^{4} + 214 q^{5} + 128 q^{6} + 428 q^{7} + 2048 q^{8} + 3196 q^{9} + 856 q^{10} + 1357 q^{11} + 512 q^{12} + 1747 q^{13} + 1712 q^{14} + 2728 q^{15} + 8192 q^{16} + 3766 q^{17}+ \cdots + 338392 q^{99}+O(q^{100}) $ 32 * q + 128 * q^2 + 32 * q^3 + 512 * q^4 + 214 * q^5 + 128 * q^6 + 428 * q^7 + 2048 * q^8 + 3196 * q^9 + 856 * q^10 + 1357 * q^11 + 512 * q^12 + 1747 * q^13 + 1712 * q^14 + 2728 * q^15 + 8192 * q^16 + 3766 * q^17 + 12784 * q^18 + 1417 * q^19 + 3424 * q^20 + 9693 * q^21 + 5428 * q^22 + 11030 * q^23 + 2048 * q^24 + 30524 * q^25 + 6988 * q^26 + 11924 * q^27 + 6848 * q^28 + 21206 * q^29 + 10912 * q^30 + 18142 * q^31 + 32768 * q^32 + 27879 * q^33 + 15064 * q^34 + 14971 * q^35 + 51136 * q^36 + 24725 * q^37 + 5668 * q^38 + 25201 * q^39 + 13696 * q^40 + 38470 * q^41 + 38772 * q^42 + 18145 * q^43 + 21712 * q^44 + 54227 * q^45 + 44120 * q^46 + 50173 * q^47 + 8192 * q^48 + 114490 * q^49 + 122096 * q^50 + 28981 * q^51 + 27952 * q^52 + 109649 * q^53 + 47696 * q^54 + 53077 * q^55 + 27392 * q^56 + 90298 * q^57 + 84824 * q^58 + 78978 * q^59 + 43648 * q^60 + 105671 * q^61 + 72568 * q^62 + 138923 * q^63 + 131072 * q^64 + 149825 * q^65 + 111516 * q^66 + 135561 * q^67 + 60256 * q^68 + 281851 * q^69 + 59884 * q^70 + 306266 * q^71 + 204544 * q^72 + 253098 * q^73 + 98900 * q^74 + 440869 * q^75 + 22672 * q^76 + 501432 * q^77 + 100804 * q^78 + 331657 * q^79 + 54784 * q^80 + 613516 * q^81 + 153880 * q^82 + 421327 * q^83 + 155088 * q^84 + 526007 * q^85 + 72580 * q^86 + 617560 * q^87 + 86848 * q^88 + 392845 * q^89 + 216908 * q^90 + 306534 * q^91 + 176480 * q^92 + 471703 * q^93 + 200692 * q^94 + 337292 * q^95 + 32768 * q^96 + 451994 * q^97 + 457960 * q^98 + 338392 * 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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	−7.35365	−0.471737	−0.235868	−	0.971785i	$-0.575793\pi$
$3$	−7.35365	−0.471737	−0.235868	+	0.971785i	$0.575793\pi$
$4$	16.0000	0.500000
$5$	78.1821	1.39856	0.699282	−	0.714846i	$-0.253503\pi$
$5$	78.1821	1.39856	0.699282	+	0.714846i	$0.253503\pi$
$6$	−29.4146	−0.333568
$7$	209.235	1.61395	0.806973	−	0.590588i	$-0.201104\pi$
$7$	209.235	1.61395	0.806973	+	0.590588i	$0.201104\pi$
$8$	64.0000	0.353553
$9$	−188.924	−0.777464
$10$	312.729	0.988935
$11$	531.909	1.32543	0.662713	−	0.748874i	$-0.269405\pi$
$11$	531.909	1.32543	0.662713	+	0.748874i	$0.269405\pi$
$12$	−117.658	−0.235868
$13$	134.764	0.221165	0.110582	−	0.993867i	$-0.464728\pi$
$13$	134.764	0.221165	0.110582	+	0.993867i	$0.464728\pi$
$14$	836.940	1.14123
$15$	−574.924	−0.659754
$16$	256.000	0.250000
$17$	79.7966	0.0669672	0.0334836	−	0.999439i	$-0.489340\pi$
$17$	79.7966	0.0669672	0.0334836	+	0.999439i	$0.489340\pi$
$18$	−755.695	−0.549750
$19$	2330.32	1.48092	0.740461	−	0.672100i	$-0.234607\pi$
$19$	2330.32	1.48092	0.740461	+	0.672100i	$0.234607\pi$
$20$	1250.91	0.699282
$21$	−1538.64	−0.761358
$22$	2127.63	0.937217
$23$	−281.583	−0.110991	−0.0554954	−	0.998459i	$-0.517674\pi$
$23$	−281.583	−0.110991	−0.0554954	+	0.998459i	$0.517674\pi$
$24$	−470.634	−0.166784
$25$	2987.45	0.955983
$26$	539.056	0.156387
$27$	3176.22	0.838495
$28$	3347.76	0.806973
$29$	2779.11	0.613635	0.306817	−	0.951768i	$-0.400736\pi$
$29$	2779.11	0.613635	0.306817	+	0.951768i	$0.400736\pi$
$30$	−2299.70	−0.466517
$31$	−2442.38	−0.456467	−0.228233	−	0.973606i	$-0.573295\pi$
$31$	−2442.38	−0.456467	−0.228233	+	0.973606i	$0.573295\pi$
$32$	1024.00	0.176777
$33$	−3911.47	−0.625252
$34$	319.186	0.0473529
$35$	16358.4	2.25721
$36$	−3022.78	−0.388732
$37$	−10682.2	−1.28279	−0.641397	−	0.767209i	$-0.721645\pi$
$37$	−10682.2	−1.28279	−0.641397	+	0.767209i	$0.721645\pi$
$38$	9321.29	1.04717
$39$	−991.008	−0.104332
$40$	5003.66	0.494467
$41$	−10311.9	−0.958027	−0.479013	−	0.877808i	$-0.659005\pi$
$41$	−10311.9	−0.958027	−0.479013	+	0.877808i	$0.659005\pi$
$42$	−6154.56	−0.538361
$43$	1275.58	0.105205	0.0526024	−	0.998616i	$-0.483248\pi$
$43$	1275.58	0.105205	0.0526024	+	0.998616i	$0.483248\pi$
$44$	8510.54	0.662713
$45$	−14770.5	−1.08733
$46$	−1126.33	−0.0784824
$47$	−9613.12	−0.634775	−0.317387	−	0.948296i	$-0.602806\pi$
$47$	−9613.12	−0.634775	−0.317387	+	0.948296i	$0.602806\pi$
$48$	−1882.53	−0.117934
$49$	26972.3	1.60482
$50$	11949.8	0.675982
$51$	−586.796	−0.0315909
$52$	2156.23	0.110582
$53$	−10078.2	−0.492828	−0.246414	−	0.969165i	$-0.579252\pi$
$53$	−10078.2	−0.492828	−0.246414	+	0.969165i	$0.579252\pi$
$54$	12704.9	0.592906
$55$	41585.7	1.85369
$56$	13391.0	0.570616
$57$	−17136.4	−0.698605
$58$	11116.4	0.433905
$59$	−19428.6	−0.726627	−0.363314	−	0.931667i	$-0.618355\pi$
$59$	−19428.6	−0.726627	−0.363314	+	0.931667i	$0.618355\pi$
$60$	−9198.78	−0.329877
$61$	−39178.8	−1.34811	−0.674056	−	0.738680i	$-0.735449\pi$
$61$	−39178.8	−1.34811	−0.674056	+	0.738680i	$0.735449\pi$
$62$	−9769.52	−0.322771
$63$	−39529.5	−1.25479
$64$	4096.00	0.125000
$65$	10536.1	0.309313
$66$	−15645.9	−0.442120
$67$	31478.4	0.856694	0.428347	−	0.903614i	$-0.359096\pi$
$67$	31478.4	0.856694	0.428347	+	0.903614i	$0.359096\pi$
$68$	1276.75	0.0334836
$69$	2070.66	0.0523585
$70$	65433.8	1.59609
$71$	48906.4	1.15138	0.575692	−	0.817667i	$-0.304733\pi$
$71$	48906.4	1.15138	0.575692	+	0.817667i	$0.304733\pi$
$72$	−12091.1	−0.274875
$73$	940.198	0.0206496	0.0103248	−	0.999947i	$-0.496713\pi$
$73$	940.198	0.0206496	0.0103248	+	0.999947i	$0.496713\pi$
$74$	−42728.9	−0.907073
$75$	−21968.6	−0.450972
$76$	37285.1	0.740461
$77$	111294.	2.13917
$78$	−3964.03	−0.0737735
$79$	−75460.1	−1.36035	−0.680174	−	0.733051i	$-0.738096\pi$
$79$	−75460.1	−1.36035	−0.680174	+	0.733051i	$0.738096\pi$
$80$	20014.6	0.349641
$81$	22551.7	0.381915
$82$	−41247.5	−0.677427
$83$	−37665.2	−0.600130	−0.300065	−	0.953919i	$-0.597008\pi$
$83$	−37665.2	−0.600130	−0.300065	+	0.953919i	$0.597008\pi$
$84$	−24618.2	−0.380679
$85$	6238.67	0.0936579
$86$	5102.31	0.0743910
$87$	−20436.6	−0.289474
$88$	34042.1	0.468609
$89$	−104482.	−1.39819	−0.699094	−	0.715030i	$-0.746413\pi$
$89$	−104482.	−1.39819	−0.699094	+	0.715030i	$0.746413\pi$
$90$	−59081.9	−0.768861
$91$	28197.4	0.356948
$92$	−4505.33	−0.0554954
$93$	17960.4	0.215332
$94$	−38452.5	−0.448854
$95$	182190.	2.07116
$96$	−7530.14	−0.0833921
$97$	−86520.0	−0.933657	−0.466828	−	0.884348i	$-0.654603\pi$
$97$	−86520.0	−0.933657	−0.466828	+	0.884348i	$0.654603\pi$
$98$	107889.	1.13478
$99$	−100490.	−1.03047
$100$	47799.2	0.477992
$101$	165927.	1.61850	0.809252	−	0.587462i	$-0.199873\pi$
$101$	165927.	1.61850	0.809252	+	0.587462i	$0.199873\pi$
$102$	−2347.18	−0.0223381
$103$	192819.	1.79084	0.895420	−	0.445222i	$-0.146875\pi$
$103$	192819.	1.79084	0.895420	+	0.445222i	$0.146875\pi$
$104$	8624.90	0.0781935
$105$	−120294.	−1.06481
$106$	−40313.0	−0.348482
$107$	79542.2	0.671643	0.335821	−	0.941926i	$-0.390986\pi$
$107$	79542.2	0.671643	0.335821	+	0.941926i	$0.390986\pi$
$108$	50819.5	0.419248
$109$	−52730.5	−0.425104	−0.212552	−	0.977150i	$-0.568178\pi$
$109$	−52730.5	−0.425104	−0.212552	+	0.977150i	$0.568178\pi$
$110$	166343.	1.31076
$111$	78553.3	0.605141
$112$	53564.2	0.403487
$113$	59186.6	0.436041	0.218021	−	0.975944i	$-0.430040\pi$
$113$	59186.6	0.436041	0.218021	+	0.975944i	$0.430040\pi$
$114$	−68545.5	−0.493988
$115$	−22014.8	−0.155228
$116$	44465.7	0.306817
$117$	−25460.2	−0.171948
$118$	−77714.5	−0.513803
$119$	16696.2	0.108081
$120$	−36795.1	−0.233258
$121$	121876.	0.756752
$122$	−156715.	−0.953260
$123$	75829.8	0.451936
$124$	−39078.1	−0.228233
$125$	−10754.1	−0.0615603
$126$	−158118.	−0.887268
$127$	165139.	0.908531	0.454266	−	0.890866i	$-0.349902\pi$
$127$	165139.	0.908531	0.454266	+	0.890866i	$0.349902\pi$
$128$	16384.0	0.0883883
$129$	−9380.14	−0.0496290
$130$	42144.6	0.218717
$131$	384523.	1.95769	0.978846	−	0.204597i	$-0.0655884\pi$
$131$	384523.	1.95769	0.978846	+	0.204597i	$0.0655884\pi$
$132$	−62583.5	−0.312626
$133$	487585.	2.39013
$134$	125914.	0.605774
$135$	248323.	1.17269
$136$	5106.98	0.0236765
$137$	−357282.	−1.62633	−0.813166	−	0.582032i	$-0.802258\pi$
$137$	−357282.	−1.62633	−0.813166	+	0.582032i	$0.802258\pi$
$138$	8282.66	0.0370230
$139$	−22344.9	−0.0980940	−0.0490470	−	0.998796i	$-0.515618\pi$
$139$	−22344.9	−0.0980940	−0.0490470	+	0.998796i	$0.515618\pi$
$140$	261735.	1.12860
$141$	70691.5	0.299447
$142$	195626.	0.814151
$143$	71682.2	0.293137
$144$	−48364.5	−0.194366
$145$	217276.	0.858208
$146$	3760.79	0.0146015
$147$	−198345.	−0.757055
$148$	−170915.	−0.641397
$149$	174005.	0.642089	0.321044	−	0.947064i	$-0.395966\pi$
$149$	174005.	0.642089	0.321044	+	0.947064i	$0.395966\pi$
$150$	−87874.6	−0.318886
$151$	91029.4	0.324892	0.162446	−	0.986717i	$-0.448062\pi$
$151$	91029.4	0.324892	0.162446	+	0.986717i	$0.448062\pi$
$152$	149141.	0.523585
$153$	−15075.5	−0.0520646
$154$	445175.	1.51262
$155$	−190951.	−0.638398
$156$	−15856.1	−0.0521658
$157$	−30654.0	−0.0992517	−0.0496258	−	0.998768i	$-0.515803\pi$
$157$	−30654.0	−0.0992517	−0.0496258	+	0.998768i	$0.515803\pi$
$158$	−301840.	−0.961911
$159$	74111.8	0.232485
$160$	80058.5	0.247234
$161$	−58917.1	−0.179133
$162$	90206.9	0.270055
$163$	285164.	0.840670	0.420335	−	0.907369i	$-0.361913\pi$
$163$	285164.	0.840670	0.420335	+	0.907369i	$0.361913\pi$
$164$	−164990.	−0.479013
$165$	−305807.	−0.874455
$166$	−150661.	−0.424356
$167$	75380.3	0.209154	0.104577	−	0.994517i	$-0.466651\pi$
$167$	75380.3	0.209154	0.104577	+	0.994517i	$0.466651\pi$
$168$	−98473.0	−0.269181
$169$	−353132.	−0.951086
$170$	24954.7	0.0662262
$171$	−440253.	−1.15136
$172$	20409.2	0.0526024
$173$	261395.	0.664022	0.332011	−	0.943275i	$-0.392273\pi$
$173$	261395.	0.664022	0.332011	+	0.943275i	$0.392273\pi$
$174$	−81746.3	−0.204689
$175$	625079.	1.54291
$176$	136169.	0.331356
$177$	142871.	0.342777
$178$	−417927.	−0.988668
$179$	21388.2	0.0498933	0.0249467	−	0.999689i	$-0.492058\pi$
$179$	21388.2	0.0498933	0.0249467	+	0.999689i	$0.492058\pi$
$180$	−236328.	−0.543667
$181$	606702.	1.37651	0.688255	−	0.725469i	$-0.258377\pi$
$181$	606702.	1.37651	0.688255	+	0.725469i	$0.258377\pi$
$182$	112789.	0.252400
$183$	288107.	0.635954
$184$	−18021.3	−0.0392412
$185$	−835159.	−1.79407
$186$	71841.7	0.152263
$187$	42444.5	0.0887600
$188$	−153810.	−0.317387
$189$	664576.	1.35329
$190$	728758.	1.46453
$191$	−777092.	−1.54131	−0.770654	−	0.637254i	$-0.780070\pi$
$191$	−777092.	−1.54131	−0.770654	+	0.637254i	$0.780070\pi$
$192$	−30120.5	−0.0589671
$193$	−501121.	−0.968389	−0.484194	−	0.874960i	$-0.660887\pi$
$193$	−501121.	−0.968389	−0.484194	+	0.874960i	$0.660887\pi$
$194$	−346080.	−0.660195
$195$	−77479.1	−0.145914
$196$	431556.	0.802412
$197$	150209.	0.275760	0.137880	−	0.990449i	$-0.455971\pi$
$197$	150209.	0.275760	0.137880	+	0.990449i	$0.455971\pi$
$198$	−401961.	−0.728653
$199$	−107859.	−0.193074	−0.0965369	−	0.995329i	$-0.530777\pi$
$199$	−107859.	−0.193074	−0.0965369	+	0.995329i	$0.530777\pi$
$200$	191197.	0.337991
$201$	−231481.	−0.404134
$202$	663708.	1.14445
$203$	581486.	0.990374
$204$	−9388.74	−0.0157954
$205$	−806204.	−1.33986
$206$	771276.	1.26632
$207$	53197.8	0.0862915
$208$	34499.6	0.0552912
$209$	1.23952e6	1.96285
$210$	−481177.	−0.752933
$211$	745813.	1.15325	0.576626	−	0.817008i	$-0.304369\pi$
$211$	745813.	1.15325	0.576626	+	0.817008i	$0.304369\pi$
$212$	−161252.	−0.246414
$213$	−359641.	−0.543150
$214$	318169.	0.474923
$215$	99727.3	0.147136
$216$	203278.	0.296453
$217$	−511032.	−0.736713
$218$	−210922.	−0.300594
$219$	−6913.88	−0.00974119
$220$	665372.	0.926847
$221$	10753.7	0.0148108
$222$	314213.	0.427900
$223$	−813088.	−1.09490	−0.547451	−	0.836838i	$-0.684402\pi$
$223$	−813088.	−1.09490	−0.547451	+	0.836838i	$0.684402\pi$
$224$	214257.	0.285308
$225$	−564400.	−0.743243
$226$	236747.	0.308328
$227$	−92029.3	−0.118539	−0.0592696	−	0.998242i	$-0.518877\pi$
$227$	−92029.3	−0.118539	−0.0592696	+	0.998242i	$0.518877\pi$
$228$	−274182.	−0.349302
$229$	403346.	0.508263	0.254132	−	0.967170i	$-0.418210\pi$
$229$	403346.	0.508263	0.254132	+	0.967170i	$0.418210\pi$
$230$	−88059.1	−0.109763
$231$	−818416.	−1.00912
$232$	177863.	0.216953
$233$	−775722.	−0.936087	−0.468043	−	0.883705i	$-0.655041\pi$
$233$	−775722.	−0.936087	−0.468043	+	0.883705i	$0.655041\pi$
$234$	−101841.	−0.121585
$235$	−751574.	−0.887774
$236$	−310858.	−0.363314
$237$	554907.	0.641726
$238$	66784.9	0.0764251
$239$	−1.66853e6	−1.88946	−0.944731	−	0.327845i	$-0.893677\pi$
$239$	−1.66853e6	−1.88946	−0.944731	+	0.327845i	$0.893677\pi$
$240$	−147181.	−0.164939
$241$	−758105.	−0.840788	−0.420394	−	0.907342i	$-0.638108\pi$
$241$	−758105.	−0.840788	−0.420394	+	0.907342i	$0.638108\pi$
$242$	487503.	0.535105
$243$	−937658.	−1.01866
$244$	−626860.	−0.674056
$245$	2.10875e6	2.24445
$246$	303319.	0.319567
$247$	314044.	0.327528
$248$	−156312.	−0.161385
$249$	276977.	0.283104
$250$	−43016.6	−0.0435297
$251$	39153.1	0.0392267	0.0196134	−	0.999808i	$-0.493756\pi$
$251$	39153.1	0.0392267	0.0196134	+	0.999808i	$0.493756\pi$
$252$	−632472.	−0.627393
$253$	−149777.	−0.147110
$254$	660555.	0.642428
$255$	−45877.0	−0.0441819
$256$	65536.0	0.0625000
$257$	952241.	0.899320	0.449660	−	0.893200i	$-0.351545\pi$
$257$	952241.	0.899320	0.449660	+	0.893200i	$0.351545\pi$
$258$	−37520.6	−0.0350930
$259$	−2.23509e6	−2.07036
$260$	168578.	0.154657
$261$	−525039.	−0.477079
$262$	1.53809e6	1.38430
$263$	−434832.	−0.387644	−0.193822	−	0.981037i	$-0.562088\pi$
$263$	−434832.	−0.387644	−0.193822	+	0.981037i	$0.562088\pi$
$264$	−250334.	−0.221060
$265$	−787938.	−0.689251
$266$	1.95034e6	1.69008
$267$	768322.	0.659577
$268$	503654.	0.428347
$269$	−72361.0	−0.0609711
$269$	−72361.0	−0.0609711
$270$	993294.	0.829217
$271$	−1.88741e6	−1.56114	−0.780571	−	0.625067i	$-0.785072\pi$
$271$	−1.88741e6	−1.56114	−0.780571	+	0.625067i	$0.785072\pi$
$272$	20427.9	0.0167418
$273$	−207354.	−0.168386
$274$	−1.42913e6	−1.14999
$275$	1.58905e6	1.26708
$276$	33130.6	0.0261792
$277$	−785236.	−0.614895	−0.307447	−	0.951565i	$-0.599475\pi$
$277$	−785236.	−0.614895	−0.307447	+	0.951565i	$0.599475\pi$
$278$	−89379.8	−0.0693629
$279$	461424.	0.354887
$280$	1.04694e6	0.798044
$281$	263295.	0.198919	0.0994597	−	0.995042i	$-0.468289\pi$
$281$	263295.	0.198919	0.0994597	+	0.995042i	$0.468289\pi$
$282$	282766.	0.211741
$283$	−897989.	−0.666507	−0.333254	−	0.942837i	$-0.608147\pi$
$283$	−897989.	−0.666507	−0.333254	+	0.942837i	$0.608147\pi$
$284$	782503.	0.575692
$285$	−1.33976e6	−0.977044
$286$	286729.	0.207279
$287$	−2.15760e6	−1.54620
$288$	−193458.	−0.137438
$289$	−1.41349e6	−0.995515
$290$	869106.	0.606845
$291$	636238.	0.440440
$292$	15043.2	0.0103248
$293$	1.33088e6	0.905669	0.452835	−	0.891595i	$-0.350413\pi$
$293$	1.33088e6	0.905669	0.452835	+	0.891595i	$0.350413\pi$
$294$	−793379.	−0.535318
$295$	−1.51897e6	−1.01624
$296$	−683662.	−0.453536
$297$	1.68946e6	1.11136
$298$	696018.	0.454025
$299$	−37947.3	−0.0245473
$300$	−351498.	−0.225486
$301$	266895.	0.169795
$302$	364118.	0.229733
$303$	−1.22017e6	−0.763508
$304$	596562.	0.370230
$305$	−3.06308e6	−1.88542
$306$	−60301.9	−0.0368152
$307$	1.78875e6	1.08319	0.541593	−	0.840641i	$-0.317822\pi$
$307$	1.78875e6	1.08319	0.541593	+	0.840641i	$0.317822\pi$
$308$	1.78070e6	1.06958
$309$	−1.41792e6	−0.844805
$310$	−763802.	−0.451416
$311$	624672.	0.366227	0.183114	−	0.983092i	$-0.441382\pi$
$311$	624672.	0.366227	0.183114	+	0.983092i	$0.441382\pi$
$312$	−63424.5	−0.0368868
$313$	3.23634e6	1.86721	0.933605	−	0.358304i	$-0.116645\pi$
$313$	3.23634e6	1.86721	0.933605	+	0.358304i	$0.116645\pi$
$314$	−122616.	−0.0701815
$315$	−3.09050e6	−1.75490
$316$	−1.20736e6	−0.680174
$317$	328023.	0.183340	0.0916698	−	0.995789i	$-0.470780\pi$
$317$	328023.	0.183340	0.0916698	+	0.995789i	$0.470780\pi$
$318$	296447.	0.164392
$319$	1.47823e6	0.813327
$320$	320234.	0.174821
$321$	−584926.	−0.316839
$322$	−235668.	−0.126666
$323$	185952.	0.0991731
$324$	360827.	0.190958
$325$	402601.	0.211430
$326$	1.14066e6	0.594444
$327$	387762.	0.200537
$328$	−659959.	−0.338714
$329$	−2.01140e6	−1.02449
$330$	−1.22323e6	−0.618333
$331$	−205936.	−0.103315	−0.0516573	−	0.998665i	$-0.516450\pi$
$331$	−205936.	−0.103315	−0.0516573	+	0.998665i	$0.516450\pi$
$332$	−602644.	−0.300065
$333$	2.01813e6	0.997327
$334$	301521.	0.147894
$335$	2.46105e6	1.19814
$336$	−393892.	−0.190340
$337$	−249388.	−0.119619	−0.0598095	−	0.998210i	$-0.519049\pi$
$337$	−249388.	−0.119619	−0.0598095	+	0.998210i	$0.519049\pi$
$338$	−1.41253e6	−0.672519
$339$	−435238.	−0.205697
$340$	99818.7	0.0468290
$341$	−1.29912e6	−0.605013
$342$	−1.76101e6	−0.814137
$343$	2.12693e6	0.976154
$344$	81636.9	0.0371955
$345$	161889.	0.0732267
$346$	1.04558e6	0.469535
$347$	2.49739e6	1.11343	0.556715	−	0.830703i	$-0.312061\pi$
$347$	2.49739e6	1.11343	0.556715	+	0.830703i	$0.312061\pi$
$348$	−326985.	−0.144737
$349$	3.34253e6	1.46896	0.734482	−	0.678628i	$-0.237425\pi$
$349$	3.34253e6	1.46896	0.734482	+	0.678628i	$0.237425\pi$
$350$	2.50031e6	1.09100
$351$	428040.	0.185446
$352$	544674.	0.234304
$353$	−1.57985e6	−0.674807	−0.337404	−	0.941360i	$-0.609549\pi$
$353$	−1.57985e6	−0.674807	−0.337404	+	0.941360i	$0.609549\pi$
$354$	571485.	0.242380
$355$	3.82361e6	1.61028
$356$	−1.67171e6	−0.699094
$357$	−122778.	−0.0509860
$358$	85552.9	0.0352799
$359$	1.59169e6	0.651812	0.325906	−	0.945402i	$-0.394331\pi$
$359$	1.59169e6	0.651812	0.325906	+	0.945402i	$0.394331\pi$
$360$	−945310.	−0.384431
$361$	2.95430e6	1.19313
$362$	2.42681e6	0.973339
$363$	−896231.	−0.356988
$364$	451158.	0.178474
$365$	73506.7	0.0288798
$366$	1.15243e6	0.449688
$367$	−2.74973e6	−1.06568	−0.532838	−	0.846217i	$-0.678874\pi$
$367$	−2.74973e6	−1.06568	−0.532838	+	0.846217i	$0.678874\pi$
$368$	−72085.3	−0.0277477
$369$	1.94816e6	0.744832
$370$	−3.34063e6	−1.26860
$371$	−2.10872e6	−0.795397
$372$	287367.	0.107666
$373$	5.01138e6	1.86503	0.932514	−	0.361133i	$-0.117610\pi$
$373$	5.01138e6	1.86503	0.932514	+	0.361133i	$0.117610\pi$
$374$	169778.	0.0627628
$375$	79082.2	0.0290403
$376$	−615240.	−0.224427
$377$	374524.	0.135714
$378$	2.65830e6	0.956918
$379$	−5.20063e6	−1.85976	−0.929882	−	0.367858i	$-0.880091\pi$
$379$	−5.20063e6	−1.85976	−0.929882	+	0.367858i	$0.880091\pi$
$380$	2.91503e6	1.03558
$381$	−1.21437e6	−0.428588
$382$	−3.10837e6	−1.08987
$383$	−4.09569e6	−1.42669	−0.713346	−	0.700812i	$-0.752821\pi$
$383$	−4.09569e6	−1.42669	−0.713346	+	0.700812i	$0.752821\pi$
$384$	−120482.	−0.0416960
$385$	8.70119e6	2.99176
$386$	−2.00449e6	−0.684754
$387$	−240987.	−0.0817930
$388$	−1.38432e6	−0.466828
$389$	2.36100e6	0.791084	0.395542	−	0.918448i	$-0.370557\pi$
$389$	2.36100e6	0.791084	0.395542	+	0.918448i	$0.370557\pi$
$390$	−309917.	−0.103177
$391$	−22469.4	−0.00743275
$392$	1.72623e6	0.567391
$393$	−2.82765e6	−0.923516
$394$	600838.	0.194992
$395$	−5.89963e6	−1.90253
$396$	−1.60784e6	−0.515235
$397$	−815197.	−0.259589	−0.129794	−	0.991541i	$-0.541432\pi$
$397$	−815197.	−0.259589	−0.129794	+	0.991541i	$0.541432\pi$
$398$	−431436.	−0.136524
$399$	−3.58553e6	−1.12751
$400$	764787.	0.238996
$401$	−242568.	−0.0753307	−0.0376654	−	0.999290i	$-0.511992\pi$
$401$	−242568.	−0.0753307	−0.0376654	+	0.999290i	$0.511992\pi$
$402$	−925924.	−0.285766
$403$	−329145.	−0.100954
$404$	2.65483e6	0.809252
$405$	1.76314e6	0.534133
$406$	2.32594e6	0.700300
$407$	−5.68196e6	−1.70025
$408$	−37555.0	−0.0111691
$409$	516130.	0.152564	0.0762818	−	0.997086i	$-0.475695\pi$
$409$	516130.	0.152564	0.0762818	+	0.997086i	$0.475695\pi$
$410$	−3.22481e6	−0.947426
$411$	2.62732e6	0.767201
$412$	3.08510e6	0.895420
$413$	−4.06515e6	−1.17274
$414$	212791.	0.0610173
$415$	−2.94475e6	−0.839321
$416$	137998.	0.0390968
$417$	164317.	0.0462745
$418$	4.95807e6	1.38794
$419$	−4.83912e6	−1.34658	−0.673288	−	0.739380i	$-0.735119\pi$
$419$	−4.83912e6	−1.34658	−0.673288	+	0.739380i	$0.735119\pi$
$420$	−1.92471e6	−0.532404
$421$	2.58319e6	0.710314	0.355157	−	0.934807i	$-0.384427\pi$
$421$	2.58319e6	0.710314	0.355157	+	0.934807i	$0.384427\pi$
$422$	2.98325e6	0.815472
$423$	1.81615e6	0.493515
$424$	−645007.	−0.174241
$425$	238388.	0.0640195
$426$	−1.43856e6	−0.384065
$427$	−8.19757e6	−2.17578
$428$	1.27268e6	0.335821
$429$	−527126.	−0.138284
$430$	398909.	0.104041
$431$	−5.55939e6	−1.44156	−0.720782	−	0.693162i	$-0.756217\pi$
$431$	−5.55939e6	−1.44156	−0.720782	+	0.693162i	$0.756217\pi$
$432$	813111.	0.209624
$433$	712137.	0.182534	0.0912670	−	0.995826i	$-0.470908\pi$
$433$	712137.	0.182534	0.0912670	+	0.995826i	$0.470908\pi$
$434$	−2.04413e6	−0.520935
$435$	−1.59777e6	−0.404848
$436$	−843688.	−0.212552
$437$	−656180.	−0.164369
$438$	−27655.5	−0.00688806
$439$	−62771.0	−0.0155452	−0.00777262	−	0.999970i	$-0.502474\pi$
$439$	−62771.0	−0.0155452	−0.00777262	+	0.999970i	$0.502474\pi$
$440$	2.66149e6	0.655379
$441$	−5.09571e6	−1.24769
$442$	43014.9	0.0104728
$443$	−6.93054e6	−1.67787	−0.838934	−	0.544233i	$-0.816821\pi$
$443$	−6.93054e6	−1.67787	−0.838934	+	0.544233i	$0.816821\pi$
$444$	1.25685e6	0.302571
$445$	−8.16861e6	−1.95546
$446$	−3.25235e6	−0.774213
$447$	−1.27957e6	−0.302897
$448$	857026.	0.201743
$449$	966104.	0.226156	0.113078	−	0.993586i	$-0.463929\pi$
$449$	966104.	0.226156	0.113078	+	0.993586i	$0.463929\pi$
$450$	−2.25760e6	−0.525552
$451$	−5.48497e6	−1.26979
$452$	946986.	0.218021
$453$	−669398.	−0.153264
$454$	−368117.	−0.0838198
$455$	2.20453e6	0.499215
$456$	−1.09673e6	−0.246994
$457$	−3.27739e6	−0.734069	−0.367035	−	0.930207i	$-0.619627\pi$
$457$	−3.27739e6	−0.734069	−0.367035	+	0.930207i	$0.619627\pi$
$458$	1.61338e6	0.359396
$459$	253451.	0.0561517
$460$	−352237.	−0.0776140
$461$	−5.30444e6	−1.16248	−0.581242	−	0.813730i	$-0.697433\pi$
$461$	−5.30444e6	−1.16248	−0.581242	+	0.813730i	$0.697433\pi$
$462$	−3.27366e6	−0.713558
$463$	−4.68712e6	−1.01614	−0.508070	−	0.861316i	$-0.669641\pi$
$463$	−4.68712e6	−1.01614	−0.508070	+	0.861316i	$0.669641\pi$
$464$	711451.	0.153409
$465$	1.40418e6	0.301156
$466$	−3.10289e6	−0.661913
$467$	−2.82859e6	−0.600175	−0.300087	−	0.953912i	$-0.597016\pi$
$467$	−2.82859e6	−0.600175	−0.300087	+	0.953912i	$0.597016\pi$
$468$	−407362.	−0.0859738
$469$	6.58638e6	1.38266
$470$	−3.00630e6	−0.627751
$471$	225419.	0.0468207
$472$	−1.24343e6	−0.256902
$473$	678490.	0.139441
$474$	2.21963e6	0.453769
$475$	6.96171e6	1.41574
$476$	267140.	0.0540407
$477$	1.90402e6	0.383156
$478$	−6.67411e6	−1.33605
$479$	−4.26906e6	−0.850146	−0.425073	−	0.905159i	$-0.639752\pi$
$479$	−4.26906e6	−0.850146	−0.425073	+	0.905159i	$0.639752\pi$
$480$	−588722.	−0.116629
$481$	−1.43958e6	−0.283709
$482$	−3.03242e6	−0.594527
$483$	433255.	0.0845038
$484$	1.95001e6	0.378376
$485$	−6.76432e6	−1.30578
$486$	−3.75063e6	−0.720301
$487$	5.60571e6	1.07105	0.535523	−	0.844521i	$-0.320115\pi$
$487$	5.60571e6	1.07105	0.535523	+	0.844521i	$0.320115\pi$
$488$	−2.50744e6	−0.476630
$489$	−2.09700e6	−0.396575
$490$	8.43500e6	1.58707
$491$	8.52761e6	1.59633	0.798167	−	0.602436i	$-0.205803\pi$
$491$	8.52761e6	1.59633	0.798167	+	0.602436i	$0.205803\pi$
$492$	1.21328e6	0.225968
$493$	221763.	0.0410934
$494$	1.25618e6	0.231597
$495$	−7.85654e6	−1.44118
$496$	−625250.	−0.114117
$497$	1.02329e7	1.85827
$498$	1.10791e6	0.200184
$499$	−47964.7	−0.00862324	−0.00431162	−	0.999991i	$-0.501372\pi$
$499$	−47964.7	−0.00862324	−0.00431162	+	0.999991i	$0.501372\pi$
$500$	−172066.	−0.0307801
$501$	−554320.	−0.0986658
$502$	156613.	0.0277375
$503$	−5.81901e6	−1.02549	−0.512743	−	0.858542i	$-0.671370\pi$
$503$	−5.81901e6	−1.02549	−0.512743	+	0.858542i	$0.671370\pi$
$504$	−2.52989e6	−0.443634
$505$	1.29725e7	2.26358
$506$	−599106.	−0.104023
$507$	2.59681e6	0.448662
$508$	2.64222e6	0.454266
$509$	−221005.	−0.0378102	−0.0189051	−	0.999821i	$-0.506018\pi$
$509$	−221005.	−0.0378102	−0.0189051	+	0.999821i	$0.506018\pi$
$510$	−183508.	−0.0312413
$511$	196722.	0.0333274
$512$	262144.	0.0441942
$513$	7.40161e6	1.24175
$514$	3.80896e6	0.635915
$515$	1.50750e7	2.50461
$516$	−150082.	−0.0248145
$517$	−5.11330e6	−0.841347
$518$	−8.94037e6	−1.46397
$519$	−1.92221e6	−0.313244
$520$	674313.	0.109359
$521$	1.04061e7	1.67955	0.839777	−	0.542932i	$-0.182686\pi$
$521$	1.04061e7	1.67955	0.839777	+	0.542932i	$0.182686\pi$
$522$	−2.10016e6	−0.337346
$523$	−5.80345e6	−0.927753	−0.463876	−	0.885900i	$-0.653542\pi$
$523$	−5.80345e6	−0.927753	−0.463876	+	0.885900i	$0.653542\pi$
$524$	6.15237e6	0.978846
$525$	−4.59661e6	−0.727846
$526$	−1.73933e6	−0.274105
$527$	−194894.	−0.0305683
$528$	−1.00134e6	−0.156313
$529$	−6.35705e6	−0.987681
$530$	−3.15175e6	−0.487374
$531$	3.67053e6	0.564927
$532$	7.80136e6	1.19506
$533$	−1.38967e6	−0.211882
$534$	3.07329e6	0.466391
$535$	6.21878e6	0.939336
$536$	2.01462e6	0.302887
$537$	−157282.	−0.0235365
$538$	−289444.	−0.0431131
$539$	1.43468e7	2.12707
$540$	3.97317e6	0.586345
$541$	−3.48884e6	−0.512493	−0.256246	−	0.966612i	$-0.582486\pi$
$541$	−3.48884e6	−0.512493	−0.256246	+	0.966612i	$0.582486\pi$
$542$	−7.54963e6	−1.10389
$543$	−4.46148e6	−0.649350
$544$	81711.7	0.0118382
$545$	−4.12258e6	−0.594536
$546$	−829414.	−0.119067
$547$	−6.63506e6	−0.948149	−0.474074	−	0.880485i	$-0.657217\pi$
$547$	−6.63506e6	−0.948149	−0.474074	+	0.880485i	$0.657217\pi$
$548$	−5.71651e6	−0.813166
$549$	7.40180e6	1.04811
$550$	6.35620e6	0.895964
$551$	6.47621e6	0.908745
$552$	132523.	0.0185115
$553$	−1.57889e7	−2.19553
$554$	−3.14094e6	−0.434796
$555$	6.14146e6	0.846329
$556$	−357519.	−0.0490470
$557$	−4.70063e6	−0.641975	−0.320988	−	0.947083i	$-0.604015\pi$
$557$	−4.70063e6	−0.641975	−0.320988	+	0.947083i	$0.604015\pi$
$558$	1.84570e6	0.250943
$559$	171902.	0.0232676
$560$	4.18776e6	0.564302
$561$	−312122.	−0.0418714
$562$	1.05318e6	0.140657
$563$	−3.03612e6	−0.403690	−0.201845	−	0.979417i	$-0.564694\pi$
$563$	−3.03612e6	−0.403690	−0.201845	+	0.979417i	$0.564694\pi$
$564$	1.13106e6	0.149723
$565$	4.62734e6	0.609832
$566$	−3.59196e6	−0.471292
$567$	4.71861e6	0.616391
$568$	3.13001e6	0.407076
$569$	1.06373e7	1.37737	0.688685	−	0.725061i	$-0.258188\pi$
$569$	1.06373e7	1.37737	0.688685	+	0.725061i	$0.258188\pi$
$570$	−5.35903e6	−0.690875
$571$	7.22715e6	0.927635	0.463817	−	0.885931i	$-0.346479\pi$
$571$	7.22715e6	0.927635	0.463817	+	0.885931i	$0.346479\pi$
$572$	1.14691e6	0.146569
$573$	5.71446e6	0.727091
$574$	−8.63041e6	−1.09333
$575$	−841215.	−0.106105
$576$	−773832.	−0.0971830
$577$	−1.46098e7	−1.82686	−0.913428	−	0.407000i	$-0.866575\pi$
$577$	−1.46098e7	−1.82686	−0.913428	+	0.407000i	$0.866575\pi$
$578$	−5.65396e6	−0.703936
$579$	3.68507e6	0.456825
$580$	3.47642e6	0.429104
$581$	−7.88089e6	−0.968578
$582$	2.54495e6	0.311438
$583$	−5.36070e6	−0.653206
$584$	60172.7	0.00730074
$585$	−1.99053e6	−0.240480
$586$	5.32352e6	0.640405
$587$	1.42905e7	1.71180	0.855898	−	0.517145i	$-0.173005\pi$
$587$	1.42905e7	1.71180	0.855898	+	0.517145i	$0.173005\pi$
$588$	−3.17351e6	−0.378527
$589$	−5.69153e6	−0.675991
$590$	−6.07588e6	−0.718587
$591$	−1.10459e6	−0.130086
$592$	−2.73465e6	−0.320699
$593$	−1.09071e7	−1.27371	−0.636855	−	0.770983i	$-0.719765\pi$
$593$	−1.09071e7	−1.27371	−0.636855	+	0.770983i	$0.719765\pi$
$594$	6.75783e6	0.785852
$595$	1.30535e6	0.151159
$596$	2.78407e6	0.321044
$597$	793157.	0.0910801
$598$	−151789.	−0.0173575
$599$	−8.01203e6	−0.912380	−0.456190	−	0.889882i	$-0.650786\pi$
$599$	−8.01203e6	−0.912380	−0.456190	+	0.889882i	$0.650786\pi$
$600$	−1.40599e6	−0.159443
$601$	7.55922e6	0.853671	0.426836	−	0.904329i	$-0.359628\pi$
$601$	7.55922e6	0.853671	0.426836	+	0.904329i	$0.359628\pi$
$602$	1.06758e6	0.120063
$603$	−5.94702e6	−0.666049
$604$	1.45647e6	0.162446
$605$	9.52850e6	1.05837
$606$	−4.88067e6	−0.539881
$607$	1.58096e7	1.74161	0.870803	−	0.491631i	$-0.163599\pi$
$607$	1.58096e7	1.74161	0.870803	+	0.491631i	$0.163599\pi$
$608$	2.38625e6	0.261792
$609$	−4.27604e6	−0.467196
$610$	−1.22523e7	−1.33320
$611$	−1.29550e6	−0.140390
$612$	−241208.	−0.0260323
$613$	2.82014e6	0.303123	0.151562	−	0.988448i	$-0.451570\pi$
$613$	2.82014e6	0.303123	0.151562	+	0.988448i	$0.451570\pi$
$614$	7.15498e6	0.765928
$615$	5.92854e6	0.632062
$616$	7.12281e6	0.756309
$617$	−1.25845e7	−1.33084	−0.665418	−	0.746471i	$-0.731747\pi$
$617$	−1.25845e7	−1.33084	−0.665418	+	0.746471i	$0.731747\pi$
$618$	−5.67169e6	−0.597368
$619$	3.84050e6	0.402866	0.201433	−	0.979502i	$-0.435440\pi$
$619$	3.84050e6	0.402866	0.201433	+	0.979502i	$0.435440\pi$
$620$	−3.05521e6	−0.319199
$621$	−894369.	−0.0930653
$622$	2.49869e6	0.258962
$623$	−2.18612e7	−2.25660
$624$	−253698.	−0.0260829
$625$	−1.01766e7	−1.04208
$626$	1.29454e7	1.32032
$627$	−9.11498e6	−0.925949
$628$	−490464.	−0.0496258
$629$	−852404.	−0.0859051
$630$	−1.23620e7	−1.24090
$631$	5.55681e6	0.555587	0.277793	−	0.960641i	$-0.410397\pi$
$631$	5.55681e6	0.555587	0.277793	+	0.960641i	$0.410397\pi$
$632$	−4.82945e6	−0.480955
$633$	−5.48445e6	−0.544031
$634$	1.31209e6	0.129641
$635$	1.29109e7	1.27064
$636$	1.18579e6	0.116242
$637$	3.63489e6	0.354930
$638$	5.91292e6	0.575109
$639$	−9.23959e6	−0.895160
$640$	1.28094e6	0.123617
$641$	−1.23451e7	−1.18672	−0.593361	−	0.804936i	$-0.702199\pi$
$641$	−1.23451e7	−1.18672	−0.593361	+	0.804936i	$0.702199\pi$
$642$	−2.33970e6	−0.224039
$643$	1.31772e7	1.25689	0.628443	−	0.777856i	$-0.283693\pi$
$643$	1.31772e7	1.25689	0.628443	+	0.777856i	$0.283693\pi$
$644$	−942673.	−0.0895667
$645$	−733360.	−0.0694093
$646$	743807.	0.0701260
$647$	−5.15166e6	−0.483823	−0.241912	−	0.970298i	$-0.577774\pi$
$647$	−5.15166e6	−0.483823	−0.241912	+	0.970298i	$0.577774\pi$
$648$	1.44331e6	0.135027
$649$	−1.03342e7	−0.963090
$650$	1.61040e6	0.149503
$651$	3.75795e6	0.347535
$652$	4.56262e6	0.420335
$653$	1.47775e7	1.35618	0.678090	−	0.734979i	$-0.262808\pi$
$653$	1.47775e7	1.35618	0.678090	+	0.734979i	$0.262808\pi$
$654$	1.55105e6	0.141801
$655$	3.00629e7	2.73796
$656$	−2.63984e6	−0.239507
$657$	−177626.	−0.0160543
$658$	−8.04560e6	−0.724426
$659$	−4.44211e6	−0.398452	−0.199226	−	0.979954i	$-0.563843\pi$
$659$	−4.44211e6	−0.398452	−0.199226	+	0.979954i	$0.563843\pi$
$660$	−4.89291e6	−0.437228
$661$	1.05777e7	0.941646	0.470823	−	0.882228i	$-0.343957\pi$
$661$	1.05777e7	0.941646	0.470823	+	0.882228i	$0.343957\pi$
$662$	−823744.	−0.0730545
$663$	−79079.1	−0.00698679
$664$	−2.41058e6	−0.212178
$665$	3.81204e7	3.34275
$666$	8.07250e6	0.705217
$667$	−782550.	−0.0681079
$668$	1.20609e6	0.104577
$669$	5.97917e6	0.516506
$670$	9.84419e6	0.847214
$671$	−2.08395e7	−1.78682
$672$	−1.57557e6	−0.134590
$673$	6.91005e6	0.588090	0.294045	−	0.955792i	$-0.404999\pi$
$673$	6.91005e6	0.588090	0.294045	+	0.955792i	$0.404999\pi$
$674$	−997550.	−0.0845834
$675$	9.48878e6	0.801588
$676$	−5.65011e6	−0.475543
$677$	−1.38255e7	−1.15934	−0.579668	−	0.814853i	$-0.696818\pi$
$677$	−1.38255e7	−1.15934	−0.579668	+	0.814853i	$0.696818\pi$
$678$	−1.74095e6	−0.145450
$679$	−1.81030e7	−1.50687
$680$	399275.	0.0331131
$681$	676751.	0.0559193
$682$	−5.19649e6	−0.427809
$683$	9.27782e6	0.761017	0.380508	−	0.924777i	$-0.375749\pi$
$683$	9.27782e6	0.761017	0.380508	+	0.924777i	$0.375749\pi$
$684$	−7.04405e6	−0.575682
$685$	−2.79330e7	−2.27453
$686$	8.50772e6	0.690245
$687$	−2.96606e6	−0.239767
$688$	326548.	0.0263012
$689$	−1.35818e6	−0.108996
$690$	647556.	0.0517791
$691$	−1.05827e7	−0.843142	−0.421571	−	0.906795i	$-0.638521\pi$
$691$	−1.05827e7	−0.843142	−0.421571	+	0.906795i	$0.638521\pi$
$692$	4.18233e6	0.332011
$693$	−2.10261e7	−1.66313
$694$	9.98957e6	0.787314
$695$	−1.74698e6	−0.137191
$696$	−1.30794e6	−0.102345
$697$	−822852.	−0.0641563
$698$	1.33701e7	1.03871
$699$	5.70439e6	0.441587
$700$	1.00013e7	0.771453
$701$	4.94558e6	0.380122	0.190061	−	0.981772i	$-0.439131\pi$
$701$	4.94558e6	0.380122	0.190061	+	0.981772i	$0.439131\pi$
$702$	1.71216e6	0.131130
$703$	−2.48930e7	−1.89972
$704$	2.17870e6	0.165678
$705$	5.52681e6	0.418795
$706$	−6.31941e6	−0.477161
$707$	3.47177e7	2.61218
$708$	2.28594e6	0.171388
$709$	2.19381e7	1.63901	0.819507	−	0.573069i	$-0.194247\pi$
$709$	2.19381e7	1.63901	0.819507	+	0.573069i	$0.194247\pi$
$710$	1.52944e7	1.13864
$711$	1.42562e7	1.05762
$712$	−6.68683e6	−0.494334
$713$	687734.	0.0506636
$714$	−491113.	−0.0360525
$715$	5.60427e6	0.409972
$716$	342212.	0.0249467
$717$	1.22698e7	0.891329
$718$	6.36676e6	0.460900
$719$	2.39359e7	1.72674	0.863370	−	0.504572i	$-0.168350\pi$
$719$	2.39359e7	1.72674	0.863370	+	0.504572i	$0.168350\pi$
$720$	−3.78124e6	−0.271834
$721$	4.03445e7	2.89032
$722$	1.18172e7	0.843668
$723$	5.57484e6	0.396631
$724$	9.70724e6	0.688255
$725$	8.30243e6	0.586625
$726$	−3.58492e6	−0.252429
$727$	1.37310e7	0.963529	0.481764	−	0.876301i	$-0.339996\pi$
$727$	1.37310e7	0.963529	0.481764	+	0.876301i	$0.339996\pi$
$728$	1.80463e6	0.126200
$729$	1.41514e6	0.0986237
$730$	294027.	0.0204211
$731$	101787.	0.00704527
$732$	4.60971e6	0.317977
$733$	−8.31377e6	−0.571529	−0.285764	−	0.958300i	$-0.592247\pi$
$733$	−8.31377e6	−0.571529	−0.285764	+	0.958300i	$0.592247\pi$
$734$	−1.09989e7	−0.753546
$735$	−1.55070e7	−1.05879
$736$	−288341.	−0.0196206
$737$	1.67436e7	1.13548
$738$	7.79263e6	0.526675
$739$	−2.82089e6	−0.190010	−0.0950048	−	0.995477i	$-0.530287\pi$
$739$	−2.82089e6	−0.190010	−0.0950048	+	0.995477i	$0.530287\pi$
$740$	−1.33625e7	−0.897036
$741$	−2.30937e6	−0.154507
$742$	−8.43488e6	−0.562431
$743$	−1.52816e7	−1.01554	−0.507770	−	0.861493i	$-0.669530\pi$
$743$	−1.52816e7	−1.01554	−0.507770	+	0.861493i	$0.669530\pi$
$744$	1.14947e6	0.0761314
$745$	1.36041e7	0.898003
$746$	2.00455e7	1.31877
$747$	7.11586e6	0.466580
$748$	679112.	0.0443800
$749$	1.66430e7	1.08400
$750$	316329.	0.0205346
$751$	1.42000e7	0.918734	0.459367	−	0.888246i	$-0.348076\pi$
$751$	1.42000e7	0.918734	0.459367	+	0.888246i	$0.348076\pi$
$752$	−2.46096e6	−0.158694
$753$	−287918.	−0.0185047
$754$	1.49809e6	0.0959646
$755$	7.11687e6	0.454383
$756$	1.06332e7	0.676643
$757$	−3.14982e7	−1.99777	−0.998887	−	0.0471662i	$-0.984981\pi$
$757$	−3.14982e7	−1.99777	−0.998887	+	0.0471662i	$0.984981\pi$
$758$	−2.08025e7	−1.31505
$759$	1.10140e6	0.0693973
$760$	1.16601e7	0.732267
$761$	6.36011e6	0.398110	0.199055	−	0.979988i	$-0.436213\pi$
$761$	6.36011e6	0.398110	0.199055	+	0.979988i	$0.436213\pi$
$762$	−4.85749e6	−0.303057
$763$	−1.10331e7	−0.686096
$764$	−1.24335e7	−0.770654
$765$	−1.17863e6	−0.0728157
$766$	−1.63828e7	−1.00882
$767$	−2.61828e6	−0.160704
$768$	−481929.	−0.0294836
$769$	6.25574e6	0.381472	0.190736	−	0.981641i	$-0.438913\pi$
$769$	6.25574e6	0.381472	0.190736	+	0.981641i	$0.438913\pi$
$770$	3.48048e7	2.11549
$771$	−7.00245e6	−0.424242
$772$	−8.01794e6	−0.484194
$773$	5.73127e6	0.344986	0.172493	−	0.985011i	$-0.444818\pi$
$773$	5.73127e6	0.344986	0.172493	+	0.985011i	$0.444818\pi$
$774$	−963947.	−0.0578364
$775$	−7.29649e6	−0.436375
$776$	−5.53728e6	−0.330097
$777$	1.64361e7	0.976666
$778$	9.44401e6	0.559381
$779$	−2.40300e7	−1.41876
$780$	−1.23967e6	−0.0729572
$781$	2.60138e7	1.52607
$782$	−89877.5	−0.00525574
$783$	8.82704e6	0.514530
$784$	6.90490e6	0.401206
$785$	−2.39659e6	−0.138810
$786$	−1.13106e7	−0.653024
$787$	1.01636e7	0.584940	0.292470	−	0.956275i	$-0.405523\pi$
$787$	1.01636e7	0.584940	0.292470	+	0.956275i	$0.405523\pi$
$788$	2.40335e6	0.137880
$789$	3.19761e6	0.182866
$790$	−2.35985e7	−1.34529
$791$	1.23839e7	0.703747
$792$	−6.43137e6	−0.364326
$793$	−5.27989e6	−0.298155
$794$	−3.26079e6	−0.183557
$795$	5.79422e6	0.325145
$796$	−1.72574e6	−0.0965369
$797$	−6.91836e6	−0.385796	−0.192898	−	0.981219i	$-0.561789\pi$
$797$	−6.91836e6	−0.385796	−0.192898	+	0.981219i	$0.561789\pi$
$798$	−1.43421e7	−0.797271
$799$	−767094.	−0.0425091
$800$	3.05915e6	0.168996
$801$	1.97391e7	1.08704
$802$	−970271.	−0.0532669
$803$	500099.	0.0273695
$804$	−3.70370e6	−0.202067
$805$	−4.60626e6	−0.250530
$806$	−1.31658e6	−0.0713855
$807$	532117.	0.0287623
$808$	1.06193e7	0.572227
$809$	1.05785e7	0.568269	0.284134	−	0.958784i	$-0.408294\pi$
$809$	1.05785e7	0.568269	0.284134	+	0.958784i	$0.408294\pi$
$810$	7.05256e6	0.377689
$811$	1.51357e7	0.808073	0.404037	−	0.914743i	$-0.367607\pi$
$811$	1.51357e7	0.808073	0.404037	+	0.914743i	$0.367607\pi$
$812$	9.30378e6	0.495187
$813$	1.38793e7	0.736448
$814$	−2.27278e7	−1.20226
$815$	2.22947e7	1.17573
$816$	−150220.	−0.00789772
$817$	2.97250e6	0.155800
$818$	2.06452e6	0.107879
$819$	−5.32715e6	−0.277514
$820$	−1.28993e7	−0.669931
$821$	8.18292e6	0.423692	0.211846	−	0.977303i	$-0.432052\pi$
$821$	8.18292e6	0.423692	0.211846	+	0.977303i	$0.432052\pi$
$822$	1.05093e7	0.542493
$823$	−1.27321e7	−0.655242	−0.327621	−	0.944809i	$-0.606247\pi$
$823$	−1.27321e7	−0.655242	−0.327621	+	0.944809i	$0.606247\pi$
$824$	1.23404e7	0.633158
$825$	−1.16853e7	−0.597730
$826$	−1.62606e7	−0.829251
$827$	1.47732e7	0.751123	0.375562	−	0.926797i	$-0.377450\pi$
$827$	1.47732e7	0.751123	0.375562	+	0.926797i	$0.377450\pi$
$828$	851165.	0.0431457
$829$	−3.08227e7	−1.55770	−0.778850	−	0.627210i	$-0.784197\pi$
$829$	−3.08227e7	−1.55770	−0.778850	+	0.627210i	$0.784197\pi$
$830$	−1.17790e7	−0.593490
$831$	5.77435e6	0.290069
$832$	551994.	0.0276456
$833$	2.15230e6	0.107471
$834$	657268.	0.0327210
$835$	5.89340e6	0.292516
$836$	1.98323e7	0.981425
$837$	−7.75753e6	−0.382745
$838$	−1.93565e7	−0.952174
$839$	−8.38326e6	−0.411157	−0.205579	−	0.978641i	$-0.565908\pi$
$839$	−8.38326e6	−0.411157	−0.205579	+	0.978641i	$0.565908\pi$
$840$	−7.69883e6	−0.376467
$841$	−1.27877e7	−0.623452
$842$	1.03327e7	0.502268
$843$	−1.93618e6	−0.0938376
$844$	1.19330e7	0.576626
$845$	−2.76086e7	−1.33016
$846$	7.26459e6	0.348968
$847$	2.55007e7	1.22136
$848$	−2.58003e6	−0.123207
$849$	6.60350e6	0.314416
$850$	953552.	0.0452686
$851$	3.00793e6	0.142378
$852$	−5.75425e6	−0.271575
$853$	−2.62065e7	−1.23321	−0.616604	−	0.787274i	$-0.711492\pi$
$853$	−2.62065e7	−1.23321	−0.616604	+	0.787274i	$0.711492\pi$
$854$	−3.27903e7	−1.53851
$855$	−3.44200e7	−1.61026
$856$	5.09070e6	0.237462
$857$	2.91581e7	1.35615	0.678074	−	0.734994i	$-0.262815\pi$
$857$	2.91581e7	1.35615	0.678074	+	0.734994i	$0.262815\pi$
$858$	−2.10850e6	−0.0977813
$859$	−750586.	−0.0347070	−0.0173535	−	0.999849i	$-0.505524\pi$
$859$	−750586.	−0.0347070	−0.0173535	+	0.999849i	$0.505524\pi$
$860$	1.59564e6	0.0735678
$861$	1.58663e7	0.729401
$862$	−2.22376e7	−1.01934
$863$	2.08397e7	0.952500	0.476250	−	0.879310i	$-0.341996\pi$
$863$	2.08397e7	0.952500	0.476250	+	0.879310i	$0.341996\pi$
$864$	3.25245e6	0.148226
$865$	2.04364e7	0.928678
$866$	2.84855e6	0.129071
$867$	1.03943e7	0.469621
$868$	−8.17651e6	−0.368357
$869$	−4.01379e7	−1.80304
$870$	−6.39110e6	−0.286271
$871$	4.24216e6	0.189470
$872$	−3.37475e6	−0.150297
$873$	1.63457e7	0.725885
$874$	−2.62472e6	−0.116226
$875$	−2.25014e6	−0.0993550
$876$	−110622.	−0.00487059
$877$	2.06088e7	0.904801	0.452400	−	0.891815i	$-0.350568\pi$
$877$	2.06088e7	0.904801	0.452400	+	0.891815i	$0.350568\pi$
$878$	−251084.	−0.0109922
$879$	−9.78682e6	−0.427238
$880$	1.06460e7	0.463423
$881$	−7.52049e6	−0.326442	−0.163221	−	0.986589i	$-0.552188\pi$
$881$	−7.52049e6	−0.326442	−0.163221	+	0.986589i	$0.552188\pi$
$882$	−2.03828e7	−0.882252
$883$	−2.85717e7	−1.23320	−0.616602	−	0.787275i	$-0.711491\pi$
$883$	−2.85717e7	−1.23320	−0.616602	+	0.787275i	$0.711491\pi$
$884$	172059.	0.00740539
$885$	1.11700e7	0.479396
$886$	−2.77222e7	−1.18643
$887$	−3.33766e7	−1.42440	−0.712202	−	0.701975i	$-0.752302\pi$
$887$	−3.33766e7	−1.42440	−0.712202	+	0.701975i	$0.752302\pi$
$888$	5.02741e6	0.213950
$889$	3.45528e7	1.46632
$890$	−3.26744e7	−1.38272
$891$	1.19954e7	0.506200
$892$	−1.30094e7	−0.547451
$893$	−2.24017e7	−0.940051
$894$	−5.11828e6	−0.214180
$895$	1.67218e6	0.0697790
$896$	3.42811e6	0.142654
$897$	279051.	0.0115798
$898$	3.86441e6	0.159916
$899$	−6.78763e6	−0.280104
$900$	−9.03040e6	−0.371621
$901$	−804209.	−0.0330033
$902$	−2.19399e7	−0.897879
$903$	−1.96265e6	−0.0800985
$904$	3.78794e6	0.154164
$905$	4.74333e7	1.92514
$906$	−2.67759e6	−0.108374
$907$	1.60743e7	0.648803	0.324401	−	0.945920i	$-0.394837\pi$
$907$	1.60743e7	0.648803	0.324401	+	0.945920i	$0.394837\pi$
$908$	−1.47247e6	−0.0592696
$909$	−3.13476e7	−1.25833
$910$	8.81812e6	0.352998
$911$	6.00724e6	0.239817	0.119908	−	0.992785i	$-0.461740\pi$
$911$	6.00724e6	0.239817	0.119908	+	0.992785i	$0.461740\pi$
$912$	−4.38691e6	−0.174651
$913$	−2.00345e7	−0.795428
$914$	−1.31095e7	−0.519065
$915$	2.25248e7	0.889423
$916$	6.45353e6	0.254132
$917$	8.04557e7	3.15961
$918$	1.01380e6	0.0397052
$919$	1.61954e7	0.632563	0.316282	−	0.948665i	$-0.397566\pi$
$919$	1.61954e7	0.632563	0.316282	+	0.948665i	$0.397566\pi$
$920$	−1.40895e6	−0.0548814
$921$	−1.31538e7	−0.510978
$922$	−2.12178e7	−0.822001
$923$	6.59083e6	0.254646
$924$	−1.30947e7	−0.504562
$925$	−3.19126e7	−1.22633
$926$	−1.87485e7	−0.718520
$927$	−3.64281e7	−1.39231
$928$	2.84580e6	0.108476
$929$	1.34026e7	0.509508	0.254754	−	0.967006i	$-0.418005\pi$
$929$	1.34026e7	0.509508	0.254754	+	0.967006i	$0.418005\pi$
$930$	5.61673e6	0.212949
$931$	6.28541e7	2.37662
$932$	−1.24115e7	−0.468043
$933$	−4.59362e6	−0.172763
$934$	−1.13144e7	−0.424388
$935$	3.31840e6	0.124137
$936$	−1.62945e6	−0.0607927
$937$	2.14662e7	0.798741	0.399371	−	0.916790i	$-0.369229\pi$
$937$	2.14662e7	0.798741	0.399371	+	0.916790i	$0.369229\pi$
$938$	2.63455e7	0.977687
$939$	−2.37989e7	−0.880832
$940$	−1.20252e7	−0.443887
$941$	1.80652e7	0.665072	0.332536	−	0.943091i	$-0.392096\pi$
$941$	1.80652e7	0.665072	0.332536	+	0.943091i	$0.392096\pi$
$942$	901675.	0.0331072
$943$	2.90365e6	0.106332
$944$	−4.97372e6	−0.181657
$945$	5.19579e7	1.89266
$946$	2.71396e6	0.0985997
$947$	−3.47075e7	−1.25762	−0.628808	−	0.777561i	$-0.716457\pi$
$947$	−3.47075e7	−1.25762	−0.628808	+	0.777561i	$0.716457\pi$
$948$	8.87852e6	0.320863
$949$	126705.	0.00456697
$950$	2.78469e7	1.00108
$951$	−2.41217e6	−0.0864880
$952$	1.06856e6	0.0382126
$953$	1.44089e7	0.513922	0.256961	−	0.966422i	$-0.417279\pi$
$953$	1.44089e7	0.513922	0.256961	+	0.966422i	$0.417279\pi$
$954$	7.61608e6	0.270932
$955$	−6.07547e7	−2.15562
$956$	−2.66964e7	−0.944731
$957$	−1.08704e7	−0.383676
$958$	−1.70762e7	−0.601144
$959$	−7.47558e7	−2.62481
$960$	−2.35489e6	−0.0824693
$961$	−2.26639e7	−0.791638
$962$	−5.75832e6	−0.200612
$963$	−1.50274e7	−0.522178
$964$	−1.21297e7	−0.420394
$965$	−3.91787e7	−1.35435
$966$	1.73302e6	0.0597532
$967$	−5.59960e7	−1.92571	−0.962855	−	0.270019i	$-0.912970\pi$
$967$	−5.59960e7	−1.92571	−0.962855	+	0.270019i	$0.912970\pi$
$968$	7.80004e6	0.267552
$969$	−1.36742e6	−0.0467836
$970$	−2.70573e7	−0.923325
$971$	−1.73124e6	−0.0589264	−0.0294632	−	0.999566i	$-0.509380\pi$
$971$	−1.73124e6	−0.0589264	−0.0294632	+	0.999566i	$0.509380\pi$
$972$	−1.50025e7	−0.509329
$973$	−4.67535e6	−0.158318
$974$	2.24228e7	0.757344
$975$	−2.96058e6	−0.0997392
$976$	−1.00298e7	−0.337028
$977$	−2.65121e7	−0.888602	−0.444301	−	0.895878i	$-0.646548\pi$
$977$	−2.65121e7	−0.888602	−0.444301	+	0.895878i	$0.646548\pi$
$978$	−8.38798e6	−0.280421
$979$	−5.55747e7	−1.85319
$980$	3.37400e7	1.12223
$981$	9.96205e6	0.330504
$982$	3.41105e7	1.12878
$983$	−3.12916e7	−1.03287	−0.516433	−	0.856328i	$-0.672740\pi$
$983$	−3.12916e7	−1.03287	−0.516433	+	0.856328i	$0.672740\pi$
$984$	4.85311e6	0.159784
$985$	1.17437e7	0.385669
$986$	887052.	0.0290574
$987$	1.47911e7	0.483291
$988$	5.02470e6	0.163764
$989$	−359181.	−0.0116768
$990$	−3.14262e7	−1.01907
$991$	−7.58503e6	−0.245343	−0.122671	−	0.992447i	$-0.539146\pi$
$991$	−7.58503e6	−0.245343	−0.122671	+	0.992447i	$0.539146\pi$
$992$	−2.50100e6	−0.0806927
$993$	1.51438e6	0.0487373
$994$	4.09317e7	1.31400
$995$	−8.43264e6	−0.270026
$996$	4.43163e6	0.141552
$997$	−4.59581e7	−1.46428	−0.732139	−	0.681155i	$-0.761478\pi$
$997$	−4.59581e7	−1.46428	−0.732139	+	0.681155i	$0.761478\pi$
$998$	−191859.	−0.00609755
$999$	−3.39290e7	−1.07562

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	538.6.a.d.1.11	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.6.a.d.1.11	✓	32	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 538 = 2 \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	538.a (trivial)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} -7.35365 q^{3} +16.0000 q^{4} +78.1821 q^{5} -29.4146 q^{6} +209.235 q^{7} +64.0000 q^{8} -188.924 q^{9} +312.729 q^{10} +531.909 q^{11} -117.658 q^{12} +134.764 q^{13} +836.940 q^{14} -574.924 q^{15} +256.000 q^{16} +79.7966 q^{17} -755.695 q^{18} +2330.32 q^{19} +1250.91 q^{20} -1538.64 q^{21} +2127.63 q^{22} -281.583 q^{23} -470.634 q^{24} +2987.45 q^{25} +539.056 q^{26} +3176.22 q^{27} +3347.76 q^{28} +2779.11 q^{29} -2299.70 q^{30} -2442.38 q^{31} +1024.00 q^{32} -3911.47 q^{33} +319.186 q^{34} +16358.4 q^{35} -3022.78 q^{36} -10682.2 q^{37} +9321.29 q^{38} -991.008 q^{39} +5003.66 q^{40} -10311.9 q^{41} -6154.56 q^{42} +1275.58 q^{43} +8510.54 q^{44} -14770.5 q^{45} -1126.33 q^{46} -9613.12 q^{47} -1882.53 q^{48} +26972.3 q^{49} +11949.8 q^{50} -586.796 q^{51} +2156.23 q^{52} -10078.2 q^{53} +12704.9 q^{54} +41585.7 q^{55} +13391.0 q^{56} -17136.4 q^{57} +11116.4 q^{58} -19428.6 q^{59} -9198.78 q^{60} -39178.8 q^{61} -9769.52 q^{62} -39529.5 q^{63} +4096.00 q^{64} +10536.1 q^{65} -15645.9 q^{66} +31478.4 q^{67} +1276.75 q^{68} +2070.66 q^{69} +65433.8 q^{70} +48906.4 q^{71} -12091.1 q^{72} +940.198 q^{73} -42728.9 q^{74} -21968.6 q^{75} +37285.1 q^{76} +111294. q^{77} -3964.03 q^{78} -75460.1 q^{79} +20014.6 q^{80} +22551.7 q^{81} -41247.5 q^{82} -37665.2 q^{83} -24618.2 q^{84} +6238.67 q^{85} +5102.31 q^{86} -20436.6 q^{87} +34042.1 q^{88} -104482. q^{89} -59081.9 q^{90} +28197.4 q^{91} -4505.33 q^{92} +17960.4 q^{93} -38452.5 q^{94} +182190. q^{95} -7530.14 q^{96} -86520.0 q^{97} +107889. q^{98} -100490. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 128 q^{2} + 32 q^{3} + 512 q^{4} + 214 q^{5} + 128 q^{6} + 428 q^{7} + 2048 q^{8} + 3196 q^{9} + 856 q^{10} + 1357 q^{11} + 512 q^{12} + 1747 q^{13} + 1712 q^{14} + 2728 q^{15} + 8192 q^{16} + 3766 q^{17}+ \cdots + 338392 q^{99}+O(q^{100}) \) 32 * q + 128 * q^2 + 32 * q^3 + 512 * q^4 + 214 * q^5 + 128 * q^6 + 428 * q^7 + 2048 * q^8 + 3196 * q^9 + 856 * q^10 + 1357 * q^11 + 512 * q^12 + 1747 * q^13 + 1712 * q^14 + 2728 * q^15 + 8192 * q^16 + 3766 * q^17 + 12784 * q^18 + 1417 * q^19 + 3424 * q^20 + 9693 * q^21 + 5428 * q^22 + 11030 * q^23 + 2048 * q^24 + 30524 * q^25 + 6988 * q^26 + 11924 * q^27 + 6848 * q^28 + 21206 * q^29 + 10912 * q^30 + 18142 * q^31 + 32768 * q^32 + 27879 * q^33 + 15064 * q^34 + 14971 * q^35 + 51136 * q^36 + 24725 * q^37 + 5668 * q^38 + 25201 * q^39 + 13696 * q^40 + 38470 * q^41 + 38772 * q^42 + 18145 * q^43 + 21712 * q^44 + 54227 * q^45 + 44120 * q^46 + 50173 * q^47 + 8192 * q^48 + 114490 * q^49 + 122096 * q^50 + 28981 * q^51 + 27952 * q^52 + 109649 * q^53 + 47696 * q^54 + 53077 * q^55 + 27392 * q^56 + 90298 * q^57 + 84824 * q^58 + 78978 * q^59 + 43648 * q^60 + 105671 * q^61 + 72568 * q^62 + 138923 * q^63 + 131072 * q^64 + 149825 * q^65 + 111516 * q^66 + 135561 * q^67 + 60256 * q^68 + 281851 * q^69 + 59884 * q^70 + 306266 * q^71 + 204544 * q^72 + 253098 * q^73 + 98900 * q^74 + 440869 * q^75 + 22672 * q^76 + 501432 * q^77 + 100804 * q^78 + 331657 * q^79 + 54784 * q^80 + 613516 * q^81 + 153880 * q^82 + 421327 * q^83 + 155088 * q^84 + 526007 * q^85 + 72580 * q^86 + 617560 * q^87 + 86848 * q^88 + 392845 * q^89 + 216908 * q^90 + 306534 * q^91 + 176480 * q^92 + 471703 * q^93 + 200692 * q^94 + 337292 * q^95 + 32768 * q^96 + 451994 * q^97 + 457960 * q^98 + 338392 * q^99

Introduction

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