LMFDB - Embedded newform 538.6.a.d.1.14

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			1.14
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)

$f(q)$	$=$	\(q+4.00000 q^{2} -5.34471 q^{3} +16.0000 q^{4} -20.8011 q^{5} -21.3788 q^{6} +58.6538 q^{7} +64.0000 q^{8} -214.434 q^{9} -83.2043 q^{10} +437.510 q^{11} -85.5154 q^{12} +221.105 q^{13} +234.615 q^{14} +111.176 q^{15} +256.000 q^{16} +964.999 q^{17} -857.736 q^{18} +1195.72 q^{19} -332.817 q^{20} -313.487 q^{21} +1750.04 q^{22} -4646.53 q^{23} -342.061 q^{24} -2692.32 q^{25} +884.420 q^{26} +2444.85 q^{27} +938.460 q^{28} -8235.32 q^{29} +444.703 q^{30} +7270.22 q^{31} +1024.00 q^{32} -2338.37 q^{33} +3860.00 q^{34} -1220.06 q^{35} -3430.95 q^{36} -8028.54 q^{37} +4782.89 q^{38} -1181.74 q^{39} -1331.27 q^{40} +7530.45 q^{41} -1253.95 q^{42} +20669.1 q^{43} +7000.17 q^{44} +4460.46 q^{45} -18586.1 q^{46} +28906.8 q^{47} -1368.25 q^{48} -13366.7 q^{49} -10769.3 q^{50} -5157.64 q^{51} +3537.68 q^{52} +27887.3 q^{53} +9779.41 q^{54} -9100.68 q^{55} +3753.84 q^{56} -6390.79 q^{57} -32941.3 q^{58} +13306.5 q^{59} +1778.81 q^{60} +28984.5 q^{61} +29080.9 q^{62} -12577.4 q^{63} +4096.00 q^{64} -4599.22 q^{65} -9353.46 q^{66} +11998.6 q^{67} +15440.0 q^{68} +24834.4 q^{69} -4880.25 q^{70} -3848.58 q^{71} -13723.8 q^{72} -83655.9 q^{73} -32114.2 q^{74} +14389.6 q^{75} +19131.6 q^{76} +25661.6 q^{77} -4726.97 q^{78} -8037.48 q^{79} -5325.08 q^{80} +39040.4 q^{81} +30121.8 q^{82} +74025.9 q^{83} -5015.80 q^{84} -20073.0 q^{85} +82676.4 q^{86} +44015.4 q^{87} +28000.7 q^{88} +105411. q^{89} +17841.8 q^{90} +12968.6 q^{91} -74344.5 q^{92} -38857.2 q^{93} +115627. q^{94} -24872.3 q^{95} -5472.98 q^{96} +65941.4 q^{97} -53466.9 q^{98} -93817.1 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−5.34471	−0.342863	−0.171432	−	0.985196i	$-0.554839\pi$
$3$	−5.34471	−0.342863	−0.171432	+	0.985196i	$0.554839\pi$
$4$	16.0000	0.500000
$5$	−20.8011	−0.372101	−0.186050	−	0.982540i	$-0.559569\pi$
$5$	−20.8011	−0.372101	−0.186050	+	0.982540i	$0.559569\pi$
$6$	−21.3788	−0.242441
$7$	58.6538	0.452429	0.226215	−	0.974077i	$-0.427365\pi$
$7$	58.6538	0.452429	0.226215	+	0.974077i	$0.427365\pi$
$8$	64.0000	0.353553
$9$	−214.434	−0.882445
$10$	−83.2043	−0.263115
$11$	437.510	1.09020	0.545101	−	0.838371i	$-0.316491\pi$
$11$	437.510	1.09020	0.545101	+	0.838371i	$0.316491\pi$
$12$	−85.5154	−0.171432
$13$	221.105	0.362861	0.181430	−	0.983404i	$-0.441927\pi$
$13$	221.105	0.362861	0.181430	+	0.983404i	$0.441927\pi$
$14$	234.615	0.319916
$15$	111.176	0.127580
$16$	256.000	0.250000
$17$	964.999	0.809850	0.404925	−	0.914350i	$-0.367298\pi$
$17$	964.999	0.809850	0.404925	+	0.914350i	$0.367298\pi$
$18$	−857.736	−0.623983
$19$	1195.72	0.759882	0.379941	−	0.925011i	$-0.375944\pi$
$19$	1195.72	0.759882	0.379941	+	0.925011i	$0.375944\pi$
$20$	−332.817	−0.186050
$21$	−313.487	−0.155121
$22$	1750.04	0.770889
$23$	−4646.53	−1.83151	−0.915755	−	0.401737i	$-0.868406\pi$
$23$	−4646.53	−1.83151	−0.915755	+	0.401737i	$0.868406\pi$
$24$	−342.061	−0.121220
$25$	−2692.32	−0.861541
$26$	884.420	0.256581
$27$	2444.85	0.645421
$28$	938.460	0.226215
$29$	−8235.32	−1.81838	−0.909192	−	0.416377i	$-0.863299\pi$
$29$	−8235.32	−1.81838	−0.909192	+	0.416377i	$0.863299\pi$
$30$	444.703	0.0902125
$31$	7270.22	1.35876	0.679381	−	0.733786i	$-0.262248\pi$
$31$	7270.22	1.35876	0.679381	+	0.733786i	$0.262248\pi$
$32$	1024.00	0.176777
$33$	−2338.37	−0.373790
$34$	3860.00	0.572650
$35$	−1220.06	−0.168349
$36$	−3430.95	−0.441222
$37$	−8028.54	−0.964123	−0.482061	−	0.876137i	$-0.660112\pi$
$37$	−8028.54	−0.964123	−0.482061	+	0.876137i	$0.660112\pi$
$38$	4782.89	0.537318
$39$	−1181.74	−0.124412
$40$	−1331.27	−0.131558
$41$	7530.45	0.699618	0.349809	−	0.936821i	$-0.386246\pi$
$41$	7530.45	0.699618	0.349809	+	0.936821i	$0.386246\pi$
$42$	−1253.95	−0.109687
$43$	20669.1	1.70471	0.852355	−	0.522964i	$-0.175174\pi$
$43$	20669.1	1.70471	0.852355	+	0.522964i	$0.175174\pi$
$44$	7000.17	0.545101
$45$	4460.46	0.328359
$46$	−18586.1	−1.29507
$47$	28906.8	1.90877	0.954387	−	0.298571i	$-0.0965099\pi$
$47$	28906.8	1.90877	0.954387	+	0.298571i	$0.0965099\pi$
$48$	−1368.25	−0.0857158
$49$	−13366.7	−0.795308
$50$	−10769.3	−0.609201
$51$	−5157.64	−0.277668
$52$	3537.68	0.181430
$53$	27887.3	1.36370	0.681848	−	0.731494i	$-0.261177\pi$
$53$	27887.3	1.36370	0.681848	+	0.731494i	$0.261177\pi$
$54$	9779.41	0.456382
$55$	−9100.68	−0.405665
$56$	3753.84	0.159958
$57$	−6390.79	−0.260536
$58$	−32941.3	−1.28579
$59$	13306.5	0.497660	0.248830	−	0.968547i	$-0.419954\pi$
$59$	13306.5	0.497660	0.248830	+	0.968547i	$0.419954\pi$
$60$	1778.81	0.0637899
$61$	28984.5	0.997334	0.498667	−	0.866794i	$-0.333823\pi$
$61$	28984.5	0.997334	0.498667	+	0.866794i	$0.333823\pi$
$62$	29080.9	0.960789
$63$	−12577.4	−0.399244
$64$	4096.00	0.125000
$65$	−4599.22	−0.135021
$66$	−9353.46	−0.264309
$67$	11998.6	0.326545	0.163272	−	0.986581i	$-0.447795\pi$
$67$	11998.6	0.326545	0.163272	+	0.986581i	$0.447795\pi$
$68$	15440.0	0.404925
$69$	24834.4	0.627958
$70$	−4880.25	−0.119041
$71$	−3848.58	−0.0906056	−0.0453028	−	0.998973i	$-0.514425\pi$
$71$	−3848.58	−0.0906056	−0.0453028	+	0.998973i	$0.514425\pi$
$72$	−13723.8	−0.311991
$73$	−83655.9	−1.83734	−0.918670	−	0.395025i	$-0.870736\pi$
$73$	−83655.9	−1.83734	−0.918670	+	0.395025i	$0.870736\pi$
$74$	−32114.2	−0.681738
$75$	14389.6	0.295391
$76$	19131.6	0.379941
$77$	25661.6	0.493239
$78$	−4726.97	−0.0879723
$79$	−8037.48	−0.144895	−0.0724473	−	0.997372i	$-0.523081\pi$
$79$	−8037.48	−0.144895	−0.0724473	+	0.997372i	$0.523081\pi$
$80$	−5325.08	−0.0930252
$81$	39040.4	0.661153
$82$	30121.8	0.494705
$83$	74025.9	1.17947	0.589737	−	0.807595i	$-0.299231\pi$
$83$	74025.9	1.17947	0.589737	+	0.807595i	$0.299231\pi$
$84$	−5015.80	−0.0775607
$85$	−20073.0	−0.301346
$86$	82676.4	1.20541
$87$	44015.4	0.623457
$88$	28000.7	0.385444
$89$	105411.	1.41063	0.705314	−	0.708895i	$-0.250806\pi$
$89$	105411.	1.41063	0.705314	+	0.708895i	$0.250806\pi$
$90$	17841.8	0.232185
$91$	12968.6	0.164169
$92$	−74344.5	−0.915755
$93$	−38857.2	−0.465870
$94$	115627.	1.34971
$95$	−24872.3	−0.282753
$96$	−5472.98	−0.0606102
$97$	65941.4	0.711588	0.355794	−	0.934564i	$-0.384210\pi$
$97$	65941.4	0.711588	0.355794	+	0.934564i	$0.384210\pi$
$98$	−53466.9	−0.562367
$99$	−93817.1	−0.962042
$100$	−43077.0	−0.430770
$101$	−38691.8	−0.377411	−0.188706	−	0.982034i	$-0.560429\pi$
$101$	−38691.8	−0.377411	−0.188706	+	0.982034i	$0.560429\pi$
$102$	−20630.6	−0.196341
$103$	−63265.7	−0.587591	−0.293796	−	0.955868i	$-0.594919\pi$
$103$	−63265.7	−0.587591	−0.293796	+	0.955868i	$0.594919\pi$
$104$	14150.7	0.128291
$105$	6520.87	0.0577208
$106$	111549.	0.964279
$107$	15469.0	0.130618	0.0653090	−	0.997865i	$-0.479197\pi$
$107$	15469.0	0.130618	0.0653090	+	0.997865i	$0.479197\pi$
$108$	39117.6	0.322711
$109$	216445.	1.74495	0.872473	−	0.488662i	$-0.162515\pi$
$109$	216445.	1.74495	0.872473	+	0.488662i	$0.162515\pi$
$110$	−36402.7	−0.286848
$111$	42910.2	0.330562
$112$	15015.4	0.113107
$113$	−26770.5	−0.197224	−0.0986121	−	0.995126i	$-0.531440\pi$
$113$	−26770.5	−0.197224	−0.0986121	+	0.995126i	$0.531440\pi$
$114$	−25563.2	−0.184227
$115$	96652.8	0.681506
$116$	−131765.	−0.909192
$117$	−47412.4	−0.320205
$118$	53225.9	0.351899
$119$	56600.8	0.366400
$120$	7115.25	0.0451063
$121$	30364.3	0.188538
$122$	115938.	0.705222
$123$	−40248.1	−0.239873
$124$	116323.	0.679381
$125$	121006.	0.692681
$126$	−50309.5	−0.282308
$127$	112879.	0.621017	0.310508	−	0.950571i	$-0.399501\pi$
$127$	112879.	0.621017	0.310508	+	0.950571i	$0.399501\pi$
$128$	16384.0	0.0883883
$129$	−110470.	−0.584482
$130$	−18396.9	−0.0954742
$131$	242147.	1.23282	0.616412	−	0.787424i	$-0.288586\pi$
$131$	242147.	1.23282	0.616412	+	0.787424i	$0.288586\pi$
$132$	−37413.9	−0.186895
$133$	70133.6	0.343793
$134$	47994.3	0.230902
$135$	−50855.6	−0.240162
$136$	61759.9	0.286325
$137$	−25756.3	−0.117242	−0.0586208	−	0.998280i	$-0.518670\pi$
$137$	−25756.3	−0.117242	−0.0586208	+	0.998280i	$0.518670\pi$
$138$	99337.5	0.444033
$139$	326208.	1.43205	0.716023	−	0.698077i	$-0.245961\pi$
$139$	326208.	1.43205	0.716023	+	0.698077i	$0.245961\pi$
$140$	−19521.0	−0.0841747
$141$	−154498.	−0.654449
$142$	−15394.3	−0.0640678
$143$	96735.7	0.395591
$144$	−54895.1	−0.220611
$145$	171304.	0.676622
$146$	−334624.	−1.29920
$147$	71441.3	0.272682
$148$	−128457.	−0.482061
$149$	−186110.	−0.686760	−0.343380	−	0.939197i	$-0.611572\pi$
$149$	−186110.	−0.686760	−0.343380	+	0.939197i	$0.611572\pi$
$150$	57558.6	0.208873
$151$	341810.	1.21995	0.609975	−	0.792421i	$-0.291180\pi$
$151$	341810.	1.21995	0.609975	+	0.792421i	$0.291180\pi$
$152$	76526.2	0.268659
$153$	−206929.	−0.714648
$154$	102647.	0.348773
$155$	−151228.	−0.505596
$156$	−18907.9	−0.0622058
$157$	−321628.	−1.04137	−0.520685	−	0.853749i	$-0.674323\pi$
$157$	−321628.	−1.04137	−0.520685	+	0.853749i	$0.674323\pi$
$158$	−32149.9	−0.102456
$159$	−149050.	−0.467561
$160$	−21300.3	−0.0657788
$161$	−272537.	−0.828629
$162$	156162.	0.467506
$163$	307709.	0.907132	0.453566	−	0.891223i	$-0.350152\pi$
$163$	307709.	0.907132	0.453566	+	0.891223i	$0.350152\pi$
$164$	120487.	0.349809
$165$	48640.5	0.139088
$166$	296104.	0.834014
$167$	685858.	1.90302	0.951510	−	0.307619i	$-0.0995323\pi$
$167$	685858.	1.90302	0.951510	+	0.307619i	$0.0995323\pi$
$168$	−20063.2	−0.0548437
$169$	−322406.	−0.868332
$170$	−80292.1	−0.213084
$171$	−256404.	−0.670554
$172$	330706.	0.852355
$173$	−73751.2	−0.187350	−0.0936750	−	0.995603i	$-0.529861\pi$
$173$	−73751.2	−0.187350	−0.0936750	+	0.995603i	$0.529861\pi$
$174$	176062.	0.440851
$175$	−157914.	−0.389786
$176$	112003.	0.272550
$177$	−71119.2	−0.170629
$178$	421645.	0.997464
$179$	−687827.	−1.60453	−0.802263	−	0.596971i	$-0.796371\pi$
$179$	−687827.	−1.60453	−0.802263	+	0.596971i	$0.796371\pi$
$180$	71367.3	0.164179
$181$	475160.	1.07806	0.539030	−	0.842286i	$-0.318791\pi$
$181$	475160.	1.07806	0.539030	+	0.842286i	$0.318791\pi$
$182$	51874.6	0.116085
$183$	−154914.	−0.341949
$184$	−297378.	−0.647536
$185$	167002.	0.358751
$186$	−155429.	−0.329419
$187$	422197.	0.882899
$188$	462508.	0.954387
$189$	143400.	0.292008
$190$	−99489.2	−0.199937
$191$	563398.	1.11746	0.558729	−	0.829350i	$-0.311289\pi$
$191$	563398.	1.11746	0.558729	+	0.829350i	$0.311289\pi$
$192$	−21891.9	−0.0428579
$193$	−986404.	−1.90617	−0.953085	−	0.302703i	$-0.902111\pi$
$193$	−986404.	−1.90617	−0.953085	+	0.302703i	$0.902111\pi$
$194$	263766.	0.503169
$195$	24581.5	0.0462937
$196$	−213868.	−0.397654
$197$	422267.	0.775214	0.387607	−	0.921825i	$-0.373302\pi$
$197$	422267.	0.775214	0.387607	+	0.921825i	$0.373302\pi$
$198$	−375268.	−0.680267
$199$	632544.	1.13229	0.566146	−	0.824305i	$-0.308434\pi$
$199$	632544.	1.13229	0.566146	+	0.824305i	$0.308434\pi$
$200$	−172308.	−0.304601
$201$	−64129.0	−0.111960
$202$	−154767.	−0.266870
$203$	−483033.	−0.822690
$204$	−82522.3	−0.138834
$205$	−156641.	−0.260329
$206$	−253063.	−0.415490
$207$	996375.	1.61621
$208$	56602.9	0.0907152
$209$	523141.	0.828424
$210$	26083.5	0.0408148
$211$	−939638.	−1.45296	−0.726482	−	0.687186i	$-0.758846\pi$
$211$	−939638.	−1.45296	−0.726482	+	0.687186i	$0.758846\pi$
$212$	446198.	0.681848
$213$	20569.6	0.0310653
$214$	61876.0	0.0923609
$215$	−429940.	−0.634324
$216$	156471.	0.228191
$217$	426426.	0.614744
$218$	865781.	1.23386
$219$	447117.	0.629957
$220$	−145611.	−0.202832
$221$	213366.	0.293863
$222$	171641.	0.233743
$223$	242211.	0.326161	0.163081	−	0.986613i	$-0.447857\pi$
$223$	242211.	0.326161	0.163081	+	0.986613i	$0.447857\pi$
$224$	60061.5	0.0799790
$225$	577324.	0.760262
$226$	−107082.	−0.139459
$227$	1.20145e6	1.54754	0.773770	−	0.633466i	$-0.218368\pi$
$227$	1.20145e6	1.54754	0.773770	+	0.633466i	$0.218368\pi$
$228$	−102253.	−0.130268
$229$	−281461.	−0.354674	−0.177337	−	0.984150i	$-0.556748\pi$
$229$	−281461.	−0.354674	−0.177337	+	0.984150i	$0.556748\pi$
$230$	386611.	0.481898
$231$	−137154.	−0.169114
$232$	−527061.	−0.642896
$233$	−312879.	−0.377561	−0.188780	−	0.982019i	$-0.560453\pi$
$233$	−312879.	−0.377561	−0.188780	+	0.982019i	$0.560453\pi$
$234$	−189650.	−0.226419
$235$	−601292.	−0.710257
$236$	212903.	0.248830
$237$	42958.0	0.0496790
$238$	226403.	0.259084
$239$	−942502.	−1.06730	−0.533651	−	0.845705i	$-0.679181\pi$
$239$	−942502.	−1.06730	−0.533651	+	0.845705i	$0.679181\pi$
$240$	28461.0	0.0318949
$241$	−1.41608e6	−1.57052	−0.785262	−	0.619164i	$-0.787472\pi$
$241$	−1.41608e6	−1.57052	−0.785262	+	0.619164i	$0.787472\pi$
$242$	121457.	0.133317
$243$	−802759.	−0.872107
$244$	463751.	0.498667
$245$	278042.	0.295935
$246$	−160992.	−0.169616
$247$	264380.	0.275732
$248$	465294.	0.480395
$249$	−395647.	−0.404399
$250$	484026.	0.489800
$251$	−869964.	−0.871600	−0.435800	−	0.900044i	$-0.643534\pi$
$251$	−869964.	−0.871600	−0.435800	+	0.900044i	$0.643534\pi$
$252$	−201238.	−0.199622
$253$	−2.03291e6	−1.99671
$254$	451515.	0.439125
$255$	107284.	0.103320
$256$	65536.0	0.0625000
$257$	−367903.	−0.347456	−0.173728	−	0.984794i	$-0.555581\pi$
$257$	−367903.	−0.347456	−0.173728	+	0.984794i	$0.555581\pi$
$258$	−441881.	−0.413292
$259$	−470904.	−0.436197
$260$	−73587.5	−0.0675104
$261$	1.76593e6	1.60462
$262$	968589.	0.871738
$263$	−1.26334e6	−1.12624	−0.563120	−	0.826375i	$-0.690399\pi$
$263$	−1.26334e6	−1.12624	−0.563120	+	0.826375i	$0.690399\pi$
$264$	−149655.	−0.132155
$265$	−580087.	−0.507432
$266$	280534.	0.243098
$267$	−563393.	−0.483653
$268$	191977.	0.163272
$269$	−72361.0	−0.0609711
$269$	−72361.0	−0.0609711
$270$	−203422.	−0.169820
$271$	1.68321e6	1.39224	0.696121	−	0.717924i	$-0.254908\pi$
$271$	1.68321e6	1.39224	0.696121	+	0.717924i	$0.254908\pi$
$272$	247040.	0.202463
$273$	−69313.6	−0.0562875
$274$	−103025.	−0.0829023
$275$	−1.17792e6	−0.939253
$276$	397350.	0.313979
$277$	44585.7	0.0349137	0.0174569	−	0.999848i	$-0.494443\pi$
$277$	44585.7	0.0349137	0.0174569	+	0.999848i	$0.494443\pi$
$278$	1.30483e6	1.01261
$279$	−1.55898e6	−1.19903
$280$	−78083.9	−0.0595205
$281$	−1.27218e6	−0.961129	−0.480564	−	0.876959i	$-0.659568\pi$
$281$	−1.27218e6	−0.961129	−0.480564	+	0.876959i	$0.659568\pi$
$282$	−617993.	−0.462765
$283$	437086.	0.324415	0.162208	−	0.986757i	$-0.448139\pi$
$283$	437086.	0.324415	0.162208	+	0.986757i	$0.448139\pi$
$284$	−61577.3	−0.0453028
$285$	132935.	0.0969456
$286$	386943.	0.279725
$287$	441689.	0.316528
$288$	−219580.	−0.155996
$289$	−488634.	−0.344143
$290$	685214.	0.478444
$291$	−352438.	−0.243978
$292$	−1.33849e6	−0.918670
$293$	53466.0	0.0363838	0.0181919	−	0.999835i	$-0.494209\pi$
$293$	53466.0	0.0363838	0.0181919	+	0.999835i	$0.494209\pi$
$294$	285765.	0.192815
$295$	−276789.	−0.185180
$296$	−513827.	−0.340869
$297$	1.06965e6	0.703639
$298$	−744442.	−0.485613
$299$	−1.02737e6	−0.664583
$300$	230234.	0.147695
$301$	1.21232e6	0.771261
$302$	1.36724e6	0.862634
$303$	206796.	0.129401
$304$	306105.	0.189971
$305$	−602908.	−0.371109
$306$	−827715.	−0.505332
$307$	−964446.	−0.584026	−0.292013	−	0.956414i	$-0.594325\pi$
$307$	−964446.	−0.584026	−0.292013	+	0.956414i	$0.594325\pi$
$308$	410586.	0.246619
$309$	338137.	0.201464
$310$	−604913.	−0.357511
$311$	−1.69214e6	−0.992054	−0.496027	−	0.868307i	$-0.665208\pi$
$311$	−1.69214e6	−0.992054	−0.496027	+	0.868307i	$0.665208\pi$
$312$	−75631.5	−0.0439862
$313$	−91145.2	−0.0525864	−0.0262932	−	0.999654i	$-0.508370\pi$
$313$	−91145.2	−0.0525864	−0.0262932	+	0.999654i	$0.508370\pi$
$314$	−1.28651e6	−0.736359
$315$	261623.	0.148559
$316$	−128600.	−0.0724473
$317$	983171.	0.549517	0.274758	−	0.961513i	$-0.411402\pi$
$317$	983171.	0.549517	0.274758	+	0.961513i	$0.411402\pi$
$318$	−596199.	−0.330616
$319$	−3.60304e6	−1.98240
$320$	−85201.2	−0.0465126
$321$	−82677.4	−0.0447841
$322$	−1.09015e6	−0.585929
$323$	1.15387e6	0.615391
$324$	624647.	0.330577
$325$	−595284.	−0.312619
$326$	1.23083e6	0.641439
$327$	−1.15684e6	−0.598278
$328$	481949.	0.247352
$329$	1.69549e6	0.863586
$330$	194562.	0.0983498
$331$	3.05149e6	1.53088	0.765441	−	0.643506i	$-0.222521\pi$
$331$	3.05149e6	1.53088	0.765441	+	0.643506i	$0.222521\pi$
$332$	1.18441e6	0.589737
$333$	1.72159e6	0.850785
$334$	2.74343e6	1.34564
$335$	−249583.	−0.121508
$336$	−80252.8	−0.0387804
$337$	−708465.	−0.339816	−0.169908	−	0.985460i	$-0.554347\pi$
$337$	−708465.	−0.339816	−0.169908	+	0.985460i	$0.554347\pi$
$338$	−1.28962e6	−0.614003
$339$	143080.	0.0676209
$340$	−321168.	−0.150673
$341$	3.18080e6	1.48132
$342$	−1.02561e6	−0.474153
$343$	−1.76980e6	−0.812250
$344$	1.32282e6	0.602706
$345$	−516581.	−0.233664
$346$	−295005.	−0.132477
$347$	−769245.	−0.342958	−0.171479	−	0.985188i	$-0.554855\pi$
$347$	−769245.	−0.342958	−0.171479	+	0.985188i	$0.554855\pi$
$348$	704247.	0.311729
$349$	2.76692e6	1.21600	0.607999	−	0.793938i	$-0.291972\pi$
$349$	2.76692e6	1.21600	0.607999	+	0.793938i	$0.291972\pi$
$350$	−631658.	−0.275621
$351$	540569.	0.234198
$352$	448011.	0.192722
$353$	2.36767e6	1.01131	0.505654	−	0.862736i	$-0.331251\pi$
$353$	2.36767e6	1.01131	0.505654	+	0.862736i	$0.331251\pi$
$354$	−284477.	−0.120653
$355$	80054.7	0.0337144
$356$	1.68658e6	0.705314
$357$	−302515.	−0.125625
$358$	−2.75131e6	−1.13457
$359$	497978.	0.203927	0.101963	−	0.994788i	$-0.467488\pi$
$359$	497978.	0.203927	0.101963	+	0.994788i	$0.467488\pi$
$360$	285469.	0.116092
$361$	−1.04635e6	−0.422579
$362$	1.90064e6	0.762304
$363$	−162288.	−0.0646429
$364$	207498.	0.0820844
$365$	1.74013e6	0.683676
$366$	−619654.	−0.241795
$367$	1.08168e6	0.419211	0.209606	−	0.977786i	$-0.432782\pi$
$367$	1.08168e6	0.419211	0.209606	+	0.977786i	$0.432782\pi$
$368$	−1.18951e6	−0.457877
$369$	−1.61478e6	−0.617374
$370$	668009.	0.253675
$371$	1.63570e6	0.616976
$372$	−621715.	−0.232935
$373$	1.40354e6	0.522340	0.261170	−	0.965293i	$-0.415892\pi$
$373$	1.40354e6	0.522340	0.261170	+	0.965293i	$0.415892\pi$
$374$	1.68879e6	0.624304
$375$	−646744.	−0.237495
$376$	1.85003e6	0.674854
$377$	−1.82087e6	−0.659820
$378$	573599.	0.206481
$379$	3.50396e6	1.25303	0.626514	−	0.779410i	$-0.284481\pi$
$379$	3.50396e6	1.25303	0.626514	+	0.779410i	$0.284481\pi$
$380$	−397957.	−0.141376
$381$	−603305.	−0.212924
$382$	2.25359e6	0.790163
$383$	3.63459e6	1.26607	0.633035	−	0.774123i	$-0.281809\pi$
$383$	3.63459e6	1.26607	0.633035	+	0.774123i	$0.281809\pi$
$384$	−87567.7	−0.0303051
$385$	−533789.	−0.183535
$386$	−3.94562e6	−1.34787
$387$	−4.43216e6	−1.50431
$388$	1.05506e6	0.355794
$389$	−2.96417e6	−0.993182	−0.496591	−	0.867985i	$-0.665415\pi$
$389$	−2.96417e6	−0.993182	−0.496591	+	0.867985i	$0.665415\pi$
$390$	98326.0	0.0327346
$391$	−4.48390e6	−1.48325
$392$	−855471.	−0.281184
$393$	−1.29421e6	−0.422690
$394$	1.68907e6	0.548159
$395$	167188.	0.0539154
$396$	−1.50107e6	−0.481021
$397$	−411048.	−0.130893	−0.0654465	−	0.997856i	$-0.520847\pi$
$397$	−411048.	−0.130893	−0.0654465	+	0.997856i	$0.520847\pi$
$398$	2.53018e6	0.800651
$399$	−374844.	−0.117874
$400$	−689233.	−0.215385
$401$	2.22970e6	0.692445	0.346222	−	0.938153i	$-0.387464\pi$
$401$	2.22970e6	0.692445	0.346222	+	0.938153i	$0.387464\pi$
$402$	−256516.	−0.0791679
$403$	1.60748e6	0.493041
$404$	−619068.	−0.188706
$405$	−812083.	−0.246016
$406$	−1.93213e6	−0.581730
$407$	−3.51257e6	−1.05109
$408$	−330089.	−0.0981704
$409$	−4.98395e6	−1.47321	−0.736607	−	0.676321i	$-0.763573\pi$
$409$	−4.98395e6	−1.47321	−0.736607	+	0.676321i	$0.763573\pi$
$410$	−626565.	−0.184080
$411$	137660.	0.0401979
$412$	−1.01225e6	−0.293796
$413$	780474.	0.225156
$414$	3.98550e6	1.14283
$415$	−1.53982e6	−0.438884
$416$	226411.	0.0641453
$417$	−1.74348e6	−0.490996
$418$	2.09256e6	0.585785
$419$	2.54581e6	0.708421	0.354210	−	0.935166i	$-0.384750\pi$
$419$	2.54581e6	0.708421	0.354210	+	0.935166i	$0.384750\pi$
$420$	104334.	0.0288604
$421$	−3.41803e6	−0.939875	−0.469938	−	0.882700i	$-0.655724\pi$
$421$	−3.41803e6	−0.939875	−0.469938	+	0.882700i	$0.655724\pi$
$422$	−3.75855e6	−1.02740
$423$	−6.19859e6	−1.68439
$424$	1.78479e6	0.482139
$425$	−2.59808e6	−0.697719
$426$	82278.2	0.0219665
$427$	1.70005e6	0.451223
$428$	247504.	0.0653090
$429$	−517024.	−0.135634
$430$	−1.71976e6	−0.448535
$431$	−5.53488e6	−1.43521	−0.717604	−	0.696451i	$-0.754761\pi$
$431$	−5.53488e6	−1.43521	−0.717604	+	0.696451i	$0.754761\pi$
$432$	625882.	0.161355
$433$	−1.70429e6	−0.436842	−0.218421	−	0.975855i	$-0.570091\pi$
$433$	−1.70429e6	−0.436842	−0.218421	+	0.975855i	$0.570091\pi$
$434$	1.70570e6	0.434689
$435$	−915568.	−0.231989
$436$	3.46313e6	0.872473
$437$	−5.55596e6	−1.39173
$438$	1.78847e6	0.445447
$439$	−7.12110e6	−1.76354	−0.881770	−	0.471679i	$-0.843648\pi$
$439$	−7.12110e6	−1.76354	−0.881770	+	0.471679i	$0.843648\pi$
$440$	−582444.	−0.143424
$441$	2.86628e6	0.701815
$442$	853464.	0.207792
$443$	−1.80503e6	−0.436994	−0.218497	−	0.975838i	$-0.570115\pi$
$443$	−1.80503e6	−0.436994	−0.218497	+	0.975838i	$0.570115\pi$
$444$	686564.	0.165281
$445$	−2.19267e6	−0.524896
$446$	968846.	0.230631
$447$	994706.	0.235465
$448$	240246.	0.0565537
$449$	5.07667e6	1.18840	0.594200	−	0.804317i	$-0.297469\pi$
$449$	5.07667e6	1.18840	0.594200	+	0.804317i	$0.297469\pi$
$450$	2.30930e6	0.537587
$451$	3.29465e6	0.762724
$452$	−428328.	−0.0986121
$453$	−1.82687e6	−0.418276
$454$	4.80581e6	1.09428
$455$	−269762.	−0.0610874
$456$	−409011.	−0.0921133
$457$	−5.40205e6	−1.20995	−0.604976	−	0.796244i	$-0.706817\pi$
$457$	−5.40205e6	−1.20995	−0.604976	+	0.796244i	$0.706817\pi$
$458$	−1.12584e6	−0.250793
$459$	2.35928e6	0.522694
$460$	1.54645e6	0.340753
$461$	1.34662e6	0.295116	0.147558	−	0.989053i	$-0.452859\pi$
$461$	1.34662e6	0.295116	0.147558	+	0.989053i	$0.452859\pi$
$462$	−548616.	−0.119581
$463$	2.67519e6	0.579966	0.289983	−	0.957032i	$-0.406350\pi$
$463$	2.67519e6	0.579966	0.289983	+	0.957032i	$0.406350\pi$
$464$	−2.10824e6	−0.454596
$465$	808272.	0.173350
$466$	−1.25152e6	−0.266976
$467$	−2.93662e6	−0.623096	−0.311548	−	0.950230i	$-0.600848\pi$
$467$	−2.93662e6	−0.623096	−0.311548	+	0.950230i	$0.600848\pi$
$468$	−758599.	−0.160102
$469$	703762.	0.147739
$470$	−2.40517e6	−0.502227
$471$	1.71901e6	0.357047
$472$	851614.	0.175949
$473$	9.04294e6	1.85848
$474$	171832.	0.0351284
$475$	−3.21926e6	−0.654670
$476$	905613.	0.183200
$477$	−5.98000e6	−1.20339
$478$	−3.77001e6	−0.754697
$479$	3.23315e6	0.643854	0.321927	−	0.946764i	$-0.395669\pi$
$479$	3.23315e6	0.643854	0.321927	+	0.946764i	$0.395669\pi$
$480$	113844.	0.0225531
$481$	−1.77515e6	−0.349842
$482$	−5.66431e6	−1.11053
$483$	1.45663e6	0.284106
$484$	485828.	0.0942691
$485$	−1.37165e6	−0.264783
$486$	−3.21104e6	−0.616672
$487$	−2.29425e6	−0.438347	−0.219174	−	0.975686i	$-0.570336\pi$
$487$	−2.29425e6	−0.438347	−0.219174	+	0.975686i	$0.570336\pi$
$488$	1.85501e6	0.352611
$489$	−1.64461e6	−0.311022
$490$	1.11217e6	0.209257
$491$	−8.92130e6	−1.67003	−0.835016	−	0.550226i	$-0.814541\pi$
$491$	−8.92130e6	−1.67003	−0.835016	+	0.550226i	$0.814541\pi$
$492$	−643969.	−0.119937
$493$	−7.94708e6	−1.47262
$494$	1.05752e6	0.194972
$495$	1.95150e6	0.357977
$496$	1.86118e6	0.339690
$497$	−225734.	−0.0409926
$498$	−1.58259e6	−0.285953
$499$	−6.96412e6	−1.25203	−0.626016	−	0.779811i	$-0.715315\pi$
$499$	−6.96412e6	−1.25203	−0.626016	+	0.779811i	$0.715315\pi$
$500$	1.93610e6	0.346341
$501$	−3.66571e6	−0.652476
$502$	−3.47986e6	−0.616314
$503$	−1.82337e6	−0.321333	−0.160666	−	0.987009i	$-0.551364\pi$
$503$	−1.82337e6	−0.321333	−0.160666	+	0.987009i	$0.551364\pi$
$504$	−804951.	−0.141154
$505$	804830.	0.140435
$506$	−8.13162e6	−1.41189
$507$	1.72316e6	0.297719
$508$	1.80606e6	0.310508
$509$	551249.	0.0943090	0.0471545	−	0.998888i	$-0.484985\pi$
$509$	551249.	0.0943090	0.0471545	+	0.998888i	$0.484985\pi$
$510$	429138.	0.0730586
$511$	−4.90674e6	−0.831267
$512$	262144.	0.0441942
$513$	2.92336e6	0.490444
$514$	−1.47161e6	−0.245689
$515$	1.31600e6	0.218643
$516$	−1.76753e6	−0.292241
$517$	1.26470e7	2.08095
$518$	−1.88362e6	−0.308438
$519$	394179.	0.0642355
$520$	−294350.	−0.0477371
$521$	−6.58558e6	−1.06292	−0.531459	−	0.847084i	$-0.678356\pi$
$521$	−6.58558e6	−1.06292	−0.531459	+	0.847084i	$0.678356\pi$
$522$	7.06373e6	1.13464
$523$	4.56933e6	0.730463	0.365232	−	0.930917i	$-0.380990\pi$
$523$	4.56933e6	0.730463	0.365232	+	0.930917i	$0.380990\pi$
$524$	3.87435e6	0.616412
$525$	844007.	0.133643
$526$	−5.05336e6	−0.796372
$527$	7.01575e6	1.10039
$528$	−598622.	−0.0934475
$529$	1.51539e7	2.35443
$530$	−2.32035e6	−0.358809
$531$	−2.85336e6	−0.439157
$532$	1.12214e6	0.171897
$533$	1.66502e6	0.253864
$534$	−2.25357e6	−0.341994
$535$	−321772.	−0.0486031
$536$	767909.	0.115451
$537$	3.67624e6	0.550133
$538$	−289444.	−0.0431131
$539$	−5.84808e6	−0.867045
$540$	−813689.	−0.120081
$541$	1.19493e7	1.75529	0.877647	−	0.479308i	$-0.159112\pi$
$541$	1.19493e7	1.75529	0.877647	+	0.479308i	$0.159112\pi$
$542$	6.73284e6	0.984464
$543$	−2.53959e6	−0.369628
$544$	988159.	0.143163
$545$	−4.50230e6	−0.649296
$546$	−277254.	−0.0398013
$547$	−1.00960e7	−1.44272	−0.721359	−	0.692561i	$-0.756482\pi$
$547$	−1.00960e7	−1.44272	−0.721359	+	0.692561i	$0.756482\pi$
$548$	−412101.	−0.0586208
$549$	−6.21525e6	−0.880092
$550$	−4.71166e6	−0.664152
$551$	−9.84716e6	−1.38176
$552$	1.58940e6	0.222017
$553$	−471428.	−0.0655546
$554$	178343.	0.0246877
$555$	−892579.	−0.123003
$556$	5.21932e6	0.716023
$557$	9.43144e6	1.28807	0.644036	−	0.764995i	$-0.277259\pi$
$557$	9.43144e6	1.28807	0.644036	+	0.764995i	$0.277259\pi$
$558$	−6.23593e6	−0.847844
$559$	4.57004e6	0.618572
$560$	−312336.	−0.0420873
$561$	−2.25652e6	−0.302714
$562$	−5.08871e6	−0.679621
$563$	−9.12331e6	−1.21306	−0.606529	−	0.795062i	$-0.707439\pi$
$563$	−9.12331e6	−1.21306	−0.606529	+	0.795062i	$0.707439\pi$
$564$	−2.47197e6	−0.327224
$565$	556855.	0.0733873
$566$	1.74834e6	0.229396
$567$	2.28987e6	0.299125
$568$	−246309.	−0.0320339
$569$	8.40861e6	1.08879	0.544394	−	0.838830i	$-0.316760\pi$
$569$	8.40861e6	1.08879	0.544394	+	0.838830i	$0.316760\pi$
$570$	531741.	0.0685509
$571$	−511522.	−0.0656559	−0.0328279	−	0.999461i	$-0.510451\pi$
$571$	−511522.	−0.0656559	−0.0328279	+	0.999461i	$0.510451\pi$
$572$	1.54777e6	0.197796
$573$	−3.01120e6	−0.383136
$574$	1.76676e6	0.223819
$575$	1.25099e7	1.57792
$576$	−878322.	−0.110306
$577$	−1.17483e7	−1.46905	−0.734523	−	0.678583i	$-0.762594\pi$
$577$	−1.17483e7	−1.46905	−0.734523	+	0.678583i	$0.762594\pi$
$578$	−1.95453e6	−0.243346
$579$	5.27204e6	0.653556
$580$	2.74086e6	0.338311
$581$	4.34190e6	0.533629
$582$	−1.40975e6	−0.172518
$583$	1.22010e7	1.48670
$584$	−5.35398e6	−0.649598
$585$	986230.	0.119148
$586$	213864.	0.0257272
$587$	−4.34766e6	−0.520787	−0.260394	−	0.965503i	$-0.583852\pi$
$587$	−4.34766e6	−0.520787	−0.260394	+	0.965503i	$0.583852\pi$
$588$	1.14306e6	0.136341
$589$	8.69316e6	1.03250
$590$	−1.10716e6	−0.130942
$591$	−2.25690e6	−0.265793
$592$	−2.05531e6	−0.241031
$593$	−4.76066e6	−0.555944	−0.277972	−	0.960589i	$-0.589662\pi$
$593$	−4.76066e6	−0.555944	−0.277972	+	0.960589i	$0.589662\pi$
$594$	4.27859e6	0.497548
$595$	−1.17736e6	−0.136338
$596$	−2.97777e6	−0.343380
$597$	−3.38077e6	−0.388221
$598$	−4.10948e6	−0.469931
$599$	−1.10166e7	−1.25452	−0.627262	−	0.778808i	$-0.715825\pi$
$599$	−1.10166e7	−1.25452	−0.627262	+	0.778808i	$0.715825\pi$
$600$	920937.	0.104436
$601$	2.19510e6	0.247895	0.123948	−	0.992289i	$-0.460445\pi$
$601$	2.19510e6	0.247895	0.123948	+	0.992289i	$0.460445\pi$
$602$	4.84928e6	0.545364
$603$	−2.57291e6	−0.288158
$604$	5.46895e6	0.609975
$605$	−631610.	−0.0701553
$606$	827185.	0.0915000
$607$	5.96464e6	0.657072	0.328536	−	0.944492i	$-0.393445\pi$
$607$	5.96464e6	0.657072	0.328536	+	0.944492i	$0.393445\pi$
$608$	1.22442e6	0.134329
$609$	2.58167e6	0.282070
$610$	−2.41163e6	−0.262414
$611$	6.39143e6	0.692620
$612$	−3.31086e6	−0.357324
$613$	−1.35062e7	−1.45171	−0.725857	−	0.687845i	$-0.758557\pi$
$613$	−1.35062e7	−1.45171	−0.725857	+	0.687845i	$0.758557\pi$
$614$	−3.85779e6	−0.412969
$615$	837203.	0.0892571
$616$	1.64234e6	0.174386
$617$	8.32419e6	0.880297	0.440148	−	0.897925i	$-0.354926\pi$
$617$	8.32419e6	0.880297	0.440148	+	0.897925i	$0.354926\pi$
$618$	1.35255e6	0.142456
$619$	−209491.	−0.0219754	−0.0109877	−	0.999940i	$-0.503498\pi$
$619$	−209491.	−0.0219754	−0.0109877	+	0.999940i	$0.503498\pi$
$620$	−2.41965e6	−0.252798
$621$	−1.13601e7	−1.18210
$622$	−6.76856e6	−0.701488
$623$	6.18277e6	0.638209
$624$	−302526.	−0.0311029
$625$	5.89642e6	0.603794
$626$	−364581.	−0.0371842
$627$	−2.79604e6	−0.284036
$628$	−5.14605e6	−0.520685
$629$	−7.74754e6	−0.780795
$630$	1.04649e6	0.105047
$631$	2.74375e6	0.274329	0.137164	−	0.990548i	$-0.456201\pi$
$631$	2.74375e6	0.274329	0.137164	+	0.990548i	$0.456201\pi$
$632$	−514399.	−0.0512280
$633$	5.02209e6	0.498168
$634$	3.93268e6	0.388567
$635$	−2.34800e6	−0.231081
$636$	−2.38480e6	−0.233781
$637$	−2.95545e6	−0.288586
$638$	−1.44122e7	−1.40177
$639$	825267.	0.0799544
$640$	−340805.	−0.0328894
$641$	1.64669e7	1.58295	0.791476	−	0.611200i	$-0.209313\pi$
$641$	1.64669e7	1.58295	0.791476	+	0.611200i	$0.209313\pi$
$642$	−330710.	−0.0316672
$643$	1.65608e7	1.57963	0.789813	−	0.613347i	$-0.210177\pi$
$643$	1.65608e7	1.57963	0.789813	+	0.613347i	$0.210177\pi$
$644$	−4.36058e6	−0.414314
$645$	2.29790e6	0.217486
$646$	4.61548e6	0.435147
$647$	−6.94675e6	−0.652410	−0.326205	−	0.945299i	$-0.605770\pi$
$647$	−6.94675e6	−0.652410	−0.326205	+	0.945299i	$0.605770\pi$
$648$	2.49859e6	0.233753
$649$	5.82172e6	0.542549
$650$	−2.38114e6	−0.221055
$651$	−2.27912e6	−0.210773
$652$	4.92334e6	0.453566
$653$	6.91120e6	0.634264	0.317132	−	0.948381i	$-0.397280\pi$
$653$	6.91120e6	0.634264	0.317132	+	0.948381i	$0.397280\pi$
$654$	−4.62735e6	−0.423047
$655$	−5.03692e6	−0.458735
$656$	1.92779e6	0.174905
$657$	1.79387e7	1.62135
$658$	6.78196e6	0.610647
$659$	8.97537e6	0.805080	0.402540	−	0.915402i	$-0.368127\pi$
$659$	8.97537e6	0.805080	0.402540	+	0.915402i	$0.368127\pi$
$660$	778248.	0.0695438
$661$	1.07922e7	0.960743	0.480372	−	0.877065i	$-0.340502\pi$
$661$	1.07922e7	0.960743	0.480372	+	0.877065i	$0.340502\pi$
$662$	1.22060e7	1.08250
$663$	−1.14038e6	−0.100755
$664$	4.73766e6	0.417007
$665$	−1.45885e6	−0.127926
$666$	6.88637e6	0.601596
$667$	3.82657e7	3.33039
$668$	1.09737e7	0.951510
$669$	−1.29455e6	−0.111829
$670$	−998334.	−0.0859189
$671$	1.26810e7	1.08729
$672$	−321011.	−0.0274219
$673$	−1.36372e7	−1.16061	−0.580306	−	0.814399i	$-0.697067\pi$
$673$	−1.36372e7	−1.16061	−0.580306	+	0.814399i	$0.697067\pi$
$674$	−2.83386e6	−0.240286
$675$	−6.58231e6	−0.556057
$676$	−5.15849e6	−0.434166
$677$	4.41430e6	0.370160	0.185080	−	0.982723i	$-0.440746\pi$
$677$	4.41430e6	0.370160	0.185080	+	0.982723i	$0.440746\pi$
$678$	572322.	0.0478152
$679$	3.86771e6	0.321943
$680$	−1.28467e6	−0.106542
$681$	−6.42142e6	−0.530595
$682$	1.27232e7	1.04745
$683$	1.69116e7	1.38718	0.693590	−	0.720370i	$-0.256028\pi$
$683$	1.69116e7	1.38718	0.693590	+	0.720370i	$0.256028\pi$
$684$	−4.10246e6	−0.335277
$685$	535758.	0.0436257
$686$	−7.07921e6	−0.574347
$687$	1.50433e6	0.121605
$688$	5.29129e6	0.426177
$689$	6.16603e6	0.494832
$690$	−2.06633e6	−0.165225
$691$	1.14843e7	0.914972	0.457486	−	0.889217i	$-0.348750\pi$
$691$	1.14843e7	0.914972	0.457486	+	0.889217i	$0.348750\pi$
$692$	−1.18002e6	−0.0936750
$693$	−5.50273e6	−0.435256
$694$	−3.07698e6	−0.242508
$695$	−6.78547e6	−0.532866
$696$	2.81699e6	0.220425
$697$	7.26687e6	0.566586
$698$	1.10677e7	0.859840
$699$	1.67225e6	0.129452
$700$	−2.52663e6	−0.194893
$701$	3.62485e6	0.278609	0.139305	−	0.990250i	$-0.455513\pi$
$701$	3.62485e6	0.278609	0.139305	+	0.990250i	$0.455513\pi$
$702$	2.16228e6	0.165603
$703$	−9.59991e6	−0.732620
$704$	1.79204e6	0.136275
$705$	3.21373e6	0.243521
$706$	9.47067e6	0.715103
$707$	−2.26942e6	−0.170752
$708$	−1.13791e6	−0.0853147
$709$	−1.34016e7	−1.00125	−0.500624	−	0.865665i	$-0.666896\pi$
$709$	−1.34016e7	−1.00125	−0.500624	+	0.865665i	$0.666896\pi$
$710$	320219.	0.0238397
$711$	1.72351e6	0.127861
$712$	6.74633e6	0.498732
$713$	−3.37813e7	−2.48858
$714$	−1.21006e6	−0.0888304
$715$	−2.01221e6	−0.147200
$716$	−1.10052e7	−0.802263
$717$	5.03740e6	0.365939
$718$	1.99191e6	0.144198
$719$	1.85603e7	1.33895	0.669473	−	0.742837i	$-0.266520\pi$
$719$	1.85603e7	1.33895	0.669473	+	0.742837i	$0.266520\pi$
$720$	1.14188e6	0.0820896
$721$	−3.71077e6	−0.265844
$722$	−4.18539e6	−0.298808
$723$	7.56853e6	0.538475
$724$	7.60256e6	0.539030
$725$	2.21721e7	1.56661
$726$	−649153.	−0.0457094
$727$	−2.19304e7	−1.53890	−0.769449	−	0.638708i	$-0.779469\pi$
$727$	−2.19304e7	−1.53890	−0.769449	+	0.638708i	$0.779469\pi$
$728$	829993.	0.0580425
$729$	−5.19631e6	−0.362140
$730$	6.96053e6	0.483432
$731$	1.99457e7	1.38056
$732$	−2.47862e6	−0.170975
$733$	−779053.	−0.0535559	−0.0267779	−	0.999641i	$-0.508525\pi$
$733$	−779053.	−0.0535559	−0.0267779	+	0.999641i	$0.508525\pi$
$734$	4.32671e6	0.296427
$735$	−1.48606e6	−0.101465
$736$	−4.75805e6	−0.323768
$737$	5.24950e6	0.356000
$738$	−6.45914e6	−0.436550
$739$	−2.05306e7	−1.38290	−0.691448	−	0.722426i	$-0.743027\pi$
$739$	−2.05306e7	−1.38290	−0.691448	+	0.722426i	$0.743027\pi$
$740$	2.67204e6	0.179375
$741$	−1.41304e6	−0.0945382
$742$	6.54279e6	0.436268
$743$	−2.79721e6	−0.185889	−0.0929443	−	0.995671i	$-0.529628\pi$
$743$	−2.79721e6	−0.185889	−0.0929443	+	0.995671i	$0.529628\pi$
$744$	−2.48686e6	−0.164710
$745$	3.87130e6	0.255544
$746$	5.61416e6	0.369350
$747$	−1.58737e7	−1.04082
$748$	6.75515e6	0.441450
$749$	907316.	0.0590954
$750$	−2.58698e6	−0.167934
$751$	6.59049e6	0.426401	0.213200	−	0.977008i	$-0.431611\pi$
$751$	6.59049e6	0.426401	0.213200	+	0.977008i	$0.431611\pi$
$752$	7.40013e6	0.477194
$753$	4.64971e6	0.298840
$754$	−7.28348e6	−0.466563
$755$	−7.11001e6	−0.453944
$756$	2.29440e6	0.146004
$757$	2.13801e7	1.35603	0.678015	−	0.735048i	$-0.262840\pi$
$757$	2.13801e7	1.35603	0.678015	+	0.735048i	$0.262840\pi$
$758$	1.40158e7	0.886025
$759$	1.08653e7	0.684600
$760$	−1.59183e6	−0.0999683
$761$	2.25310e7	1.41032	0.705162	−	0.709046i	$-0.250874\pi$
$761$	2.25310e7	1.41032	0.705162	+	0.709046i	$0.250874\pi$
$762$	−2.41322e6	−0.150560
$763$	1.26953e7	0.789465
$764$	9.01436e6	0.558729
$765$	4.30434e6	0.265921
$766$	1.45383e7	0.895247
$767$	2.94213e6	0.180581
$768$	−350271.	−0.0214290
$769$	2.53420e7	1.54534	0.772672	−	0.634805i	$-0.218920\pi$
$769$	2.53420e7	1.54534	0.772672	+	0.634805i	$0.218920\pi$
$770$	−2.13516e6	−0.129779
$771$	1.96633e6	0.119130
$772$	−1.57825e7	−0.953085
$773$	−2.09446e7	−1.26074	−0.630368	−	0.776296i	$-0.717096\pi$
$773$	−2.09446e7	−1.26074	−0.630368	+	0.776296i	$0.717096\pi$
$774$	−1.77286e7	−1.06371
$775$	−1.95737e7	−1.17063
$776$	4.22025e6	0.251584
$777$	2.51685e6	0.149556
$778$	−1.18567e7	−0.702286
$779$	9.00432e6	0.531627
$780$	393304.	0.0231469
$781$	−1.68379e6	−0.0987783
$782$	−1.79356e7	−1.04881
$783$	−2.01341e7	−1.17362
$784$	−3.42188e6	−0.198827
$785$	6.69021e6	0.387495
$786$	−5.17683e6	−0.298887
$787$	−2.21871e7	−1.27692	−0.638460	−	0.769655i	$-0.720428\pi$
$787$	−2.21871e7	−1.27692	−0.638460	+	0.769655i	$0.720428\pi$
$788$	6.75628e6	0.387607
$789$	6.75219e6	0.386147
$790$	668753.	0.0381240
$791$	−1.57019e6	−0.0892300
$792$	−6.00430e6	−0.340133
$793$	6.40861e6	0.361893
$794$	−1.64419e6	−0.0925554
$795$	3.10040e6	0.173980
$796$	1.01207e7	0.566146
$797$	−1.28661e6	−0.0717466	−0.0358733	−	0.999356i	$-0.511421\pi$
$797$	−1.28661e6	−0.0717466	−0.0358733	+	0.999356i	$0.511421\pi$
$798$	−1.49938e6	−0.0833495
$799$	2.78950e7	1.54582
$800$	−2.75693e6	−0.152300
$801$	−2.26038e7	−1.24480
$802$	8.91879e6	0.489632
$803$	−3.66003e7	−2.00307
$804$	−1.02606e6	−0.0559801
$805$	5.66905e6	0.308334
$806$	6.42993e6	0.348633
$807$	386749.	0.0209047
$808$	−2.47627e6	−0.133435
$809$	2.09774e7	1.12689	0.563443	−	0.826155i	$-0.309476\pi$
$809$	2.09774e7	1.12689	0.563443	+	0.826155i	$0.309476\pi$
$810$	−3.24833e6	−0.173959
$811$	1.50740e7	0.804781	0.402391	−	0.915468i	$-0.368179\pi$
$811$	1.50740e7	0.804781	0.402391	+	0.915468i	$0.368179\pi$
$812$	−7.72852e6	−0.411345
$813$	−8.99627e6	−0.477349
$814$	−1.40503e7	−0.743231
$815$	−6.40067e6	−0.337545
$816$	−1.32036e6	−0.0694170
$817$	2.47145e7	1.29538
$818$	−1.99358e7	−1.04172
$819$	−2.78092e6	−0.144870
$820$	−2.50626e6	−0.130164
$821$	2.21213e7	1.14539	0.572695	−	0.819769i	$-0.305898\pi$
$821$	2.21213e7	1.14539	0.572695	+	0.819769i	$0.305898\pi$
$822$	550640.	0.0284242
$823$	2.43064e6	0.125090	0.0625449	−	0.998042i	$-0.480078\pi$
$823$	2.43064e6	0.125090	0.0625449	+	0.998042i	$0.480078\pi$
$824$	−4.04901e6	−0.207745
$825$	6.29562e6	0.322035
$826$	3.12190e6	0.159209
$827$	−2.47959e7	−1.26071	−0.630356	−	0.776306i	$-0.717091\pi$
$827$	−2.47959e7	−1.26071	−0.630356	+	0.776306i	$0.717091\pi$
$828$	1.59420e7	0.808103
$829$	−1.31507e7	−0.664603	−0.332302	−	0.943173i	$-0.607825\pi$
$829$	−1.31507e7	−0.664603	−0.332302	+	0.943173i	$0.607825\pi$
$830$	−6.15927e6	−0.310338
$831$	−238298.	−0.0119706
$832$	905646.	0.0453576
$833$	−1.28989e7	−0.644080
$834$	−6.97394e6	−0.347187
$835$	−1.42666e7	−0.708115
$836$	8.37025e6	0.414212
$837$	1.77746e7	0.876974
$838$	1.01832e7	0.500929
$839$	−2.46911e7	−1.21097	−0.605487	−	0.795855i	$-0.707022\pi$
$839$	−2.46911e7	−1.21097	−0.605487	+	0.795855i	$0.707022\pi$
$840$	417336.	0.0204074
$841$	4.73094e7	2.30652
$842$	−1.36721e7	−0.664592
$843$	6.79942e6	0.329536
$844$	−1.50342e7	−0.726482
$845$	6.70638e6	0.323107
$846$	−2.47944e7	−1.19104
$847$	1.78098e6	0.0853002
$848$	7.13916e6	0.340924
$849$	−2.33610e6	−0.111230
$850$	−1.03923e7	−0.493362
$851$	3.73049e7	1.76580
$852$	329113.	0.0155327
$853$	−1.80848e7	−0.851022	−0.425511	−	0.904953i	$-0.639906\pi$
$853$	−1.80848e7	−0.851022	−0.425511	+	0.904953i	$0.639906\pi$
$854$	6.80019e6	0.319063
$855$	5.33347e6	0.249514
$856$	990017.	0.0461804
$857$	1.02834e7	0.478281	0.239141	−	0.970985i	$-0.423134\pi$
$857$	1.02834e7	0.478281	0.239141	+	0.970985i	$0.423134\pi$
$858$	−2.06810e6	−0.0959075
$859$	−9.38414e6	−0.433922	−0.216961	−	0.976180i	$-0.569614\pi$
$859$	−9.38414e6	−0.433922	−0.216961	+	0.976180i	$0.569614\pi$
$860$	−6.87903e6	−0.317162
$861$	−2.36070e6	−0.108526
$862$	−2.21395e7	−1.01485
$863$	6.41955e6	0.293412	0.146706	−	0.989180i	$-0.453133\pi$
$863$	6.41955e6	0.293412	0.146706	+	0.989180i	$0.453133\pi$
$864$	2.50353e6	0.114095
$865$	1.53410e6	0.0697131
$866$	−6.81717e6	−0.308894
$867$	2.61161e6	0.117994
$868$	6.82281e6	0.307372
$869$	−3.51648e6	−0.157964
$870$	−3.66227e6	−0.164041
$871$	2.65295e6	0.118490
$872$	1.38525e7	0.616932
$873$	−1.41401e7	−0.627937
$874$	−2.22238e7	−0.984103
$875$	7.09748e6	0.313389
$876$	7.15387e6	0.314978
$877$	−7.90128e6	−0.346895	−0.173448	−	0.984843i	$-0.555491\pi$
$877$	−7.90128e6	−0.346895	−0.173448	+	0.984843i	$0.555491\pi$
$878$	−2.84844e7	−1.24701
$879$	−285760.	−0.0124747
$880$	−2.32978e6	−0.101416
$881$	−4.99225e6	−0.216699	−0.108349	−	0.994113i	$-0.534557\pi$
$881$	−4.99225e6	−0.216699	−0.108349	+	0.994113i	$0.534557\pi$
$882$	1.14651e7	0.496258
$883$	5.81865e6	0.251143	0.125571	−	0.992085i	$-0.459924\pi$
$883$	5.81865e6	0.251143	0.125571	+	0.992085i	$0.459924\pi$
$884$	3.41386e6	0.146931
$885$	1.47936e6	0.0634914
$886$	−7.22012e6	−0.309001
$887$	1.65122e7	0.704685	0.352342	−	0.935871i	$-0.385385\pi$
$887$	1.65122e7	0.704685	0.352342	+	0.935871i	$0.385385\pi$
$888$	2.74625e6	0.116871
$889$	6.62077e6	0.280966
$890$	−8.77068e6	−0.371157
$891$	1.70806e7	0.720790
$892$	3.87538e6	0.163081
$893$	3.45645e7	1.45044
$894$	3.97882e6	0.166499
$895$	1.43075e7	0.597045
$896$	960983.	0.0399895
$897$	5.49100e6	0.227861
$898$	2.03067e7	0.840326
$899$	−5.98726e7	−2.47075
$900$	9.23719e6	0.380131
$901$	2.69113e7	1.10439
$902$	1.31786e7	0.539328
$903$	−6.47950e6	−0.264437
$904$	−1.71331e6	−0.0697293
$905$	−9.88384e6	−0.401147
$906$	−7.30749e6	−0.295766
$907$	−4.30350e7	−1.73701	−0.868507	−	0.495677i	$-0.834920\pi$
$907$	−4.30350e7	−1.73701	−0.868507	+	0.495677i	$0.834920\pi$
$908$	1.92232e7	0.773770
$909$	8.29683e6	0.333045
$910$	−1.07905e6	−0.0431953
$911$	−2.78893e6	−0.111337	−0.0556687	−	0.998449i	$-0.517729\pi$
$911$	−2.78893e6	−0.111337	−0.0556687	+	0.998449i	$0.517729\pi$
$912$	−1.63604e6	−0.0651339
$913$	3.23871e7	1.28586
$914$	−2.16082e7	−0.855565
$915$	3.22237e6	0.127240
$916$	−4.50338e6	−0.177337
$917$	1.42028e7	0.557766
$918$	9.43712e6	0.369601
$919$	−3.29514e7	−1.28702	−0.643509	−	0.765438i	$-0.722522\pi$
$919$	−3.29514e7	−1.28702	−0.643509	+	0.765438i	$0.722522\pi$
$920$	6.18578e6	0.240949
$921$	5.15469e6	0.200241
$922$	5.38648e6	0.208679
$923$	−850941.	−0.0328772
$924$	−2.19446e6	−0.0845568
$925$	2.16154e7	0.830631
$926$	1.07008e7	0.410098
$927$	1.35663e7	0.518517
$928$	−8.43297e6	−0.321448
$929$	−8.55328e6	−0.325157	−0.162579	−	0.986696i	$-0.551981\pi$
$929$	−8.55328e6	−0.325157	−0.162579	+	0.986696i	$0.551981\pi$
$930$	3.23309e6	0.122577
$931$	−1.59829e7	−0.604340
$932$	−5.00607e6	−0.188780
$933$	9.04400e6	0.340139
$934$	−1.17465e7	−0.440596
$935$	−8.78215e6	−0.328528
$936$	−3.03440e6	−0.113209
$937$	−3.22959e7	−1.20171	−0.600854	−	0.799359i	$-0.705173\pi$
$937$	−3.22959e7	−1.20171	−0.600854	+	0.799359i	$0.705173\pi$
$938$	2.81505e6	0.104467
$939$	487145.	0.0180299
$940$	−9.62067e6	−0.355128
$941$	−2.88955e7	−1.06379	−0.531896	−	0.846810i	$-0.678520\pi$
$941$	−2.88955e7	−1.06379	−0.531896	+	0.846810i	$0.678520\pi$
$942$	6.87604e6	0.252471
$943$	−3.49905e7	−1.28136
$944$	3.40646e6	0.124415
$945$	−2.98287e6	−0.108656
$946$	3.61718e7	1.31414
$947$	4.43561e7	1.60723	0.803616	−	0.595148i	$-0.202906\pi$
$947$	4.43561e7	1.60723	0.803616	+	0.595148i	$0.202906\pi$
$948$	687328.	0.0248395
$949$	−1.84967e7	−0.666699
$950$	−1.28770e7	−0.462921
$951$	−5.25476e6	−0.188409
$952$	3.62245e6	0.129542
$953$	−1.95319e7	−0.696647	−0.348324	−	0.937374i	$-0.613249\pi$
$953$	−1.95319e7	−0.696647	−0.348324	+	0.937374i	$0.613249\pi$
$954$	−2.39200e7	−0.850922
$955$	−1.17193e7	−0.415807
$956$	−1.50800e7	−0.533651
$957$	1.92572e7	0.679694
$958$	1.29326e7	0.455274
$959$	−1.51070e6	−0.0530435
$960$	455376.	0.0159475
$961$	2.42269e7	0.846233
$962$	−7.10060e6	−0.247376
$963$	−3.31708e6	−0.115263
$964$	−2.26572e7	−0.785262
$965$	2.05183e7	0.709288
$966$	5.82652e6	0.200894
$967$	−3.76088e7	−1.29337	−0.646686	−	0.762756i	$-0.723846\pi$
$967$	−3.76088e7	−1.29337	−0.646686	+	0.762756i	$0.723846\pi$
$968$	1.94331e6	0.0666583
$969$	−6.16711e6	−0.210995
$970$	−5.48661e6	−0.187230
$971$	5.93287e6	0.201937	0.100969	−	0.994890i	$-0.467806\pi$
$971$	5.93287e6	0.201937	0.100969	+	0.994890i	$0.467806\pi$
$972$	−1.28441e7	−0.436053
$973$	1.91333e7	0.647900
$974$	−9.17700e6	−0.309958
$975$	3.18162e6	0.107186
$976$	7.42002e6	0.249333
$977$	1.21591e7	0.407536	0.203768	−	0.979019i	$-0.434681\pi$
$977$	1.21591e7	0.407536	0.203768	+	0.979019i	$0.434681\pi$
$978$	−6.57845e6	−0.219926
$979$	4.61186e7	1.53787
$980$	4.44868e6	0.147967
$981$	−4.64133e7	−1.53982
$982$	−3.56852e7	−1.18089
$983$	1.40827e7	0.464838	0.232419	−	0.972616i	$-0.425336\pi$
$983$	1.40827e7	0.464838	0.232419	+	0.972616i	$0.425336\pi$
$984$	−2.57588e6	−0.0848081
$985$	−8.78361e6	−0.288458
$986$	−3.17883e7	−1.04130
$987$	−9.06190e6	−0.296092
$988$	4.23008e6	0.137866
$989$	−9.60396e7	−3.12219
$990$	7.80599e6	0.253128
$991$	2.48259e7	0.803010	0.401505	−	0.915857i	$-0.368487\pi$
$991$	2.48259e7	0.803010	0.401505	+	0.915857i	$0.368487\pi$
$992$	7.44470e6	0.240197
$993$	−1.63093e7	−0.524884
$994$	−902935.	−0.0289862
$995$	−1.31576e7	−0.421327
$996$	−6.33035e6	−0.202199
$997$	5.19041e7	1.65373	0.826863	−	0.562404i	$-0.190123\pi$
$997$	5.19041e7	1.65373	0.826863	+	0.562404i	$0.190123\pi$
$998$	−2.78565e7	−0.885320
$999$	−1.96286e7	−0.622265

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	538.6.a.d.1.14	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.6.a.d.1.14	✓	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)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} -5.34471 q^{3} +16.0000 q^{4} -20.8011 q^{5} -21.3788 q^{6} +58.6538 q^{7} +64.0000 q^{8} -214.434 q^{9} -83.2043 q^{10} +437.510 q^{11} -85.5154 q^{12} +221.105 q^{13} +234.615 q^{14} +111.176 q^{15} +256.000 q^{16} +964.999 q^{17} -857.736 q^{18} +1195.72 q^{19} -332.817 q^{20} -313.487 q^{21} +1750.04 q^{22} -4646.53 q^{23} -342.061 q^{24} -2692.32 q^{25} +884.420 q^{26} +2444.85 q^{27} +938.460 q^{28} -8235.32 q^{29} +444.703 q^{30} +7270.22 q^{31} +1024.00 q^{32} -2338.37 q^{33} +3860.00 q^{34} -1220.06 q^{35} -3430.95 q^{36} -8028.54 q^{37} +4782.89 q^{38} -1181.74 q^{39} -1331.27 q^{40} +7530.45 q^{41} -1253.95 q^{42} +20669.1 q^{43} +7000.17 q^{44} +4460.46 q^{45} -18586.1 q^{46} +28906.8 q^{47} -1368.25 q^{48} -13366.7 q^{49} -10769.3 q^{50} -5157.64 q^{51} +3537.68 q^{52} +27887.3 q^{53} +9779.41 q^{54} -9100.68 q^{55} +3753.84 q^{56} -6390.79 q^{57} -32941.3 q^{58} +13306.5 q^{59} +1778.81 q^{60} +28984.5 q^{61} +29080.9 q^{62} -12577.4 q^{63} +4096.00 q^{64} -4599.22 q^{65} -9353.46 q^{66} +11998.6 q^{67} +15440.0 q^{68} +24834.4 q^{69} -4880.25 q^{70} -3848.58 q^{71} -13723.8 q^{72} -83655.9 q^{73} -32114.2 q^{74} +14389.6 q^{75} +19131.6 q^{76} +25661.6 q^{77} -4726.97 q^{78} -8037.48 q^{79} -5325.08 q^{80} +39040.4 q^{81} +30121.8 q^{82} +74025.9 q^{83} -5015.80 q^{84} -20073.0 q^{85} +82676.4 q^{86} +44015.4 q^{87} +28000.7 q^{88} +105411. q^{89} +17841.8 q^{90} +12968.6 q^{91} -74344.5 q^{92} -38857.2 q^{93} +115627. q^{94} -24872.3 q^{95} -5472.98 q^{96} +65941.4 q^{97} -53466.9 q^{98} -93817.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 128 q^{2} + 32 q^{3} + 512 q^{4} + 214 q^{5} + 128 q^{6} + 428 q^{7} + 2048 q^{8} + 3196 q^{9} + 856 q^{10} + 1357 q^{11} + 512 q^{12} + 1747 q^{13} + 1712 q^{14} + 2728 q^{15} + 8192 q^{16} + 3766 q^{17}+ \cdots + 338392 q^{99}+O(q^{100}) \) 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

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