LMFDB - Embedded newform 4027.2.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4027,2,Mod(1,4027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4027, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4027 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4027.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:	$32.1557568940$
Analytic rank:	$2$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	4027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034 q^{2} -3.23607 q^{3} -1.61803 q^{4} -1.00000 q^{5} -2.00000 q^{6} -2.38197 q^{7} -2.23607 q^{8} +7.47214 q^{9} +O(q^{10})$	\(q+0.618034 q^{2} -3.23607 q^{3} -1.61803 q^{4} -1.00000 q^{5} -2.00000 q^{6} -2.38197 q^{7} -2.23607 q^{8} +7.47214 q^{9} -0.618034 q^{10} -1.00000 q^{11} +5.23607 q^{12} -0.381966 q^{13} -1.47214 q^{14} +3.23607 q^{15} +1.85410 q^{16} +6.09017 q^{17} +4.61803 q^{18} -5.00000 q^{19} +1.61803 q^{20} +7.70820 q^{21} -0.618034 q^{22} -4.38197 q^{23} +7.23607 q^{24} -4.00000 q^{25} -0.236068 q^{26} -14.4721 q^{27} +3.85410 q^{28} -5.61803 q^{29} +2.00000 q^{30} -8.23607 q^{31} +5.61803 q^{32} +3.23607 q^{33} +3.76393 q^{34} +2.38197 q^{35} -12.0902 q^{36} -10.9443 q^{37} -3.09017 q^{38} +1.23607 q^{39} +2.23607 q^{40} -7.23607 q^{41} +4.76393 q^{42} -4.70820 q^{43} +1.61803 q^{44} -7.47214 q^{45} -2.70820 q^{46} -8.47214 q^{47} -6.00000 q^{48} -1.32624 q^{49} -2.47214 q^{50} -19.7082 q^{51} +0.618034 q^{52} +6.23607 q^{53} -8.94427 q^{54} +1.00000 q^{55} +5.32624 q^{56} +16.1803 q^{57} -3.47214 q^{58} -8.70820 q^{59} -5.23607 q^{60} +11.1803 q^{61} -5.09017 q^{62} -17.7984 q^{63} -0.236068 q^{64} +0.381966 q^{65} +2.00000 q^{66} +7.70820 q^{67} -9.85410 q^{68} +14.1803 q^{69} +1.47214 q^{70} -10.8541 q^{71} -16.7082 q^{72} -11.0902 q^{73} -6.76393 q^{74} +12.9443 q^{75} +8.09017 q^{76} +2.38197 q^{77} +0.763932 q^{78} +13.0000 q^{79} -1.85410 q^{80} +24.4164 q^{81} -4.47214 q^{82} +3.56231 q^{83} -12.4721 q^{84} -6.09017 q^{85} -2.90983 q^{86} +18.1803 q^{87} +2.23607 q^{88} -10.8541 q^{89} -4.61803 q^{90} +0.909830 q^{91} +7.09017 q^{92} +26.6525 q^{93} -5.23607 q^{94} +5.00000 q^{95} -18.1803 q^{96} +13.4721 q^{97} -0.819660 q^{98} -7.47214 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - 2 q^{3} - q^{4} - 2 q^{5} - 4 q^{6} - 7 q^{7} + 6 q^{9}+O(q^{10}) $ 2 * q - q^2 - 2 * q^3 - q^4 - 2 * q^5 - 4 * q^6 - 7 * q^7 + 6 * q^9	\( 2 q - q^{2} - 2 q^{3} - q^{4} - 2 q^{5} - 4 q^{6} - 7 q^{7} + 6 q^{9} + q^{10} - 2 q^{11} + 6 q^{12} - 3 q^{13} + 6 q^{14} + 2 q^{15} - 3 q^{16} + q^{17} + 7 q^{18} - 10 q^{19} + q^{20} + 2 q^{21} + q^{22} - 11 q^{23} + 10 q^{24} - 8 q^{25} + 4 q^{26} - 20 q^{27} + q^{28} - 9 q^{29} + 4 q^{30} - 12 q^{31} + 9 q^{32} + 2 q^{33} + 12 q^{34} + 7 q^{35} - 13 q^{36} - 4 q^{37} + 5 q^{38} - 2 q^{39} - 10 q^{41} + 14 q^{42} + 4 q^{43} + q^{44} - 6 q^{45} + 8 q^{46} - 8 q^{47} - 12 q^{48} + 13 q^{49} + 4 q^{50} - 26 q^{51} - q^{52} + 8 q^{53} + 2 q^{55} - 5 q^{56} + 10 q^{57} + 2 q^{58} - 4 q^{59} - 6 q^{60} + q^{62} - 11 q^{63} + 4 q^{64} + 3 q^{65} + 4 q^{66} + 2 q^{67} - 13 q^{68} + 6 q^{69} - 6 q^{70} - 15 q^{71} - 20 q^{72} - 11 q^{73} - 18 q^{74} + 8 q^{75} + 5 q^{76} + 7 q^{77} + 6 q^{78} + 26 q^{79} + 3 q^{80} + 22 q^{81} - 13 q^{83} - 16 q^{84} - q^{85} - 17 q^{86} + 14 q^{87} - 15 q^{89} - 7 q^{90} + 13 q^{91} + 3 q^{92} + 22 q^{93} - 6 q^{94} + 10 q^{95} - 14 q^{96} + 18 q^{97} - 24 q^{98} - 6 q^{99}+O(q^{100}) \) 2 * q - q^2 - 2 * q^3 - q^4 - 2 * q^5 - 4 * q^6 - 7 * q^7 + 6 * q^9 + q^10 - 2 * q^11 + 6 * q^12 - 3 * q^13 + 6 * q^14 + 2 * q^15 - 3 * q^16 + q^17 + 7 * q^18 - 10 * q^19 + q^20 + 2 * q^21 + q^22 - 11 * q^23 + 10 * q^24 - 8 * q^25 + 4 * q^26 - 20 * q^27 + q^28 - 9 * q^29 + 4 * q^30 - 12 * q^31 + 9 * q^32 + 2 * q^33 + 12 * q^34 + 7 * q^35 - 13 * q^36 - 4 * q^37 + 5 * q^38 - 2 * q^39 - 10 * q^41 + 14 * q^42 + 4 * q^43 + q^44 - 6 * q^45 + 8 * q^46 - 8 * q^47 - 12 * q^48 + 13 * q^49 + 4 * q^50 - 26 * q^51 - q^52 + 8 * q^53 + 2 * q^55 - 5 * q^56 + 10 * q^57 + 2 * q^58 - 4 * q^59 - 6 * q^60 + q^62 - 11 * q^63 + 4 * q^64 + 3 * q^65 + 4 * q^66 + 2 * q^67 - 13 * q^68 + 6 * q^69 - 6 * q^70 - 15 * q^71 - 20 * q^72 - 11 * q^73 - 18 * q^74 + 8 * q^75 + 5 * q^76 + 7 * q^77 + 6 * q^78 + 26 * q^79 + 3 * q^80 + 22 * q^81 - 13 * q^83 - 16 * q^84 - q^85 - 17 * q^86 + 14 * q^87 - 15 * q^89 - 7 * q^90 + 13 * q^91 + 3 * q^92 + 22 * q^93 - 6 * q^94 + 10 * q^95 - 14 * q^96 + 18 * q^97 - 24 * q^98 - 6 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0.618034	0.437016	0.218508	−	0.975835i	$-0.429881\pi$
$2$	0.618034	0.437016	0.218508	+	0.975835i	$0.429881\pi$
$3$	−3.23607	−1.86834	−0.934172	−	0.356822i	$-0.883860\pi$
$3$	−3.23607	−1.86834	−0.934172	+	0.356822i	$0.883860\pi$
$4$	−1.61803	−0.809017
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	−2.00000	−0.816497
$7$	−2.38197	−0.900299	−0.450149	−	0.892953i	$-0.648629\pi$
$7$	−2.38197	−0.900299	−0.450149	+	0.892953i	$0.648629\pi$
$8$	−2.23607	−0.790569
$9$	7.47214	2.49071
$10$	−0.618034	−0.195440
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	5.23607	1.51152
$13$	−0.381966	−0.105938	−0.0529692	−	0.998596i	$-0.516869\pi$
$13$	−0.381966	−0.105938	−0.0529692	+	0.998596i	$0.516869\pi$
$14$	−1.47214	−0.393445
$15$	3.23607	0.835549
$16$	1.85410	0.463525
$17$	6.09017	1.47708	0.738542	−	0.674208i	$-0.235515\pi$
$17$	6.09017	1.47708	0.738542	+	0.674208i	$0.235515\pi$
$18$	4.61803	1.08848
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	1.61803	0.361803
$21$	7.70820	1.68207
$22$	−0.618034	−0.131765
$23$	−4.38197	−0.913703	−0.456852	−	0.889543i	$-0.651023\pi$
$23$	−4.38197	−0.913703	−0.456852	+	0.889543i	$0.651023\pi$
$24$	7.23607	1.47706
$25$	−4.00000	−0.800000
$26$	−0.236068	−0.0462967
$27$	−14.4721	−2.78516
$28$	3.85410	0.728357
$29$	−5.61803	−1.04324	−0.521621	−	0.853177i	$-0.674673\pi$
$29$	−5.61803	−1.04324	−0.521621	+	0.853177i	$0.674673\pi$
$30$	2.00000	0.365148
$31$	−8.23607	−1.47924	−0.739621	−	0.673024i	$-0.764995\pi$
$31$	−8.23607	−1.47924	−0.739621	+	0.673024i	$0.764995\pi$
$32$	5.61803	0.993137
$33$	3.23607	0.563327
$34$	3.76393	0.645509
$35$	2.38197	0.402626
$36$	−12.0902	−2.01503
$37$	−10.9443	−1.79923	−0.899614	−	0.436687i	$-0.856152\pi$
$37$	−10.9443	−1.79923	−0.899614	+	0.436687i	$0.856152\pi$
$38$	−3.09017	−0.501292
$39$	1.23607	0.197929
$40$	2.23607	0.353553
$41$	−7.23607	−1.13008	−0.565042	−	0.825062i	$-0.691140\pi$
$41$	−7.23607	−1.13008	−0.565042	+	0.825062i	$0.691140\pi$
$42$	4.76393	0.735091
$43$	−4.70820	−0.717994	−0.358997	−	0.933339i	$-0.616881\pi$
$43$	−4.70820	−0.717994	−0.358997	+	0.933339i	$0.616881\pi$
$44$	1.61803	0.243928
$45$	−7.47214	−1.11388
$46$	−2.70820	−0.399303
$47$	−8.47214	−1.23579	−0.617894	−	0.786261i	$-0.712014\pi$
$47$	−8.47214	−1.23579	−0.617894	+	0.786261i	$0.712014\pi$
$48$	−6.00000	−0.866025
$49$	−1.32624	−0.189463
$50$	−2.47214	−0.349613
$51$	−19.7082	−2.75970
$52$	0.618034	0.0857059
$53$	6.23607	0.856590	0.428295	−	0.903639i	$-0.359114\pi$
$53$	6.23607	0.856590	0.428295	+	0.903639i	$0.359114\pi$
$54$	−8.94427	−1.21716
$55$	1.00000	0.134840
$56$	5.32624	0.711748
$57$	16.1803	2.14314
$58$	−3.47214	−0.455914
$59$	−8.70820	−1.13371	−0.566856	−	0.823817i	$-0.691840\pi$
$59$	−8.70820	−1.13371	−0.566856	+	0.823817i	$0.691840\pi$
$60$	−5.23607	−0.675973
$61$	11.1803	1.43150	0.715748	−	0.698359i	$-0.246086\pi$
$61$	11.1803	1.43150	0.715748	+	0.698359i	$0.246086\pi$
$62$	−5.09017	−0.646452
$63$	−17.7984	−2.24238
$64$	−0.236068	−0.0295085
$65$	0.381966	0.0473771
$66$	2.00000	0.246183
$67$	7.70820	0.941707	0.470853	−	0.882211i	$-0.343946\pi$
$67$	7.70820	0.941707	0.470853	+	0.882211i	$0.343946\pi$
$68$	−9.85410	−1.19499
$69$	14.1803	1.70711
$70$	1.47214	0.175954
$71$	−10.8541	−1.28814	−0.644072	−	0.764964i	$-0.722756\pi$
$71$	−10.8541	−1.28814	−0.644072	+	0.764964i	$0.722756\pi$
$72$	−16.7082	−1.96908
$73$	−11.0902	−1.29801	−0.649003	−	0.760786i	$-0.724814\pi$
$73$	−11.0902	−1.29801	−0.649003	+	0.760786i	$0.724814\pi$
$74$	−6.76393	−0.786291
$75$	12.9443	1.49468
$76$	8.09017	0.928006
$77$	2.38197	0.271450
$78$	0.763932	0.0864983
$79$	13.0000	1.46261	0.731307	−	0.682048i	$-0.238911\pi$
$79$	13.0000	1.46261	0.731307	+	0.682048i	$0.238911\pi$
$80$	−1.85410	−0.207295
$81$	24.4164	2.71293
$82$	−4.47214	−0.493865
$83$	3.56231	0.391014	0.195507	−	0.980702i	$-0.437365\pi$
$83$	3.56231	0.391014	0.195507	+	0.980702i	$0.437365\pi$
$84$	−12.4721	−1.36082
$85$	−6.09017	−0.660572
$86$	−2.90983	−0.313775
$87$	18.1803	1.94914
$88$	2.23607	0.238366
$89$	−10.8541	−1.15053	−0.575266	−	0.817966i	$-0.695102\pi$
$89$	−10.8541	−1.15053	−0.575266	+	0.817966i	$0.695102\pi$
$90$	−4.61803	−0.486784
$91$	0.909830	0.0953761
$92$	7.09017	0.739201
$93$	26.6525	2.76373
$94$	−5.23607	−0.540059
$95$	5.00000	0.512989
$96$	−18.1803	−1.85552
$97$	13.4721	1.36789	0.683944	−	0.729534i	$-0.260263\pi$
$97$	13.4721	1.36789	0.683944	+	0.729534i	$0.260263\pi$
$98$	−0.819660	−0.0827982
$99$	−7.47214	−0.750978
$100$	6.47214	0.647214
$101$	−5.85410	−0.582505	−0.291252	−	0.956646i	$-0.594072\pi$
$101$	−5.85410	−0.582505	−0.291252	+	0.956646i	$0.594072\pi$
$102$	−12.1803	−1.20603
$103$	0.326238	0.0321452	0.0160726	−	0.999871i	$-0.494884\pi$
$103$	0.326238	0.0321452	0.0160726	+	0.999871i	$0.494884\pi$
$104$	0.854102	0.0837516
$105$	−7.70820	−0.752244
$106$	3.85410	0.374343
$107$	−9.32624	−0.901601	−0.450801	−	0.892625i	$-0.648861\pi$
$107$	−9.32624	−0.901601	−0.450801	+	0.892625i	$0.648861\pi$
$108$	23.4164	2.25324
$109$	5.47214	0.524136	0.262068	−	0.965049i	$-0.415596\pi$
$109$	5.47214	0.524136	0.262068	+	0.965049i	$0.415596\pi$
$110$	0.618034	0.0589272
$111$	35.4164	3.36158
$112$	−4.41641	−0.417311
$113$	−6.23607	−0.586640	−0.293320	−	0.956014i	$-0.594760\pi$
$113$	−6.23607	−0.586640	−0.293320	+	0.956014i	$0.594760\pi$
$114$	10.0000	0.936586
$115$	4.38197	0.408620
$116$	9.09017	0.844001
$117$	−2.85410	−0.263862
$118$	−5.38197	−0.495450
$119$	−14.5066	−1.32982
$120$	−7.23607	−0.660560
$121$	−10.0000	−0.909091
$122$	6.90983	0.625587
$123$	23.4164	2.11139
$124$	13.3262	1.19673
$125$	9.00000	0.804984
$126$	−11.0000	−0.979958
$127$	−15.2361	−1.35198	−0.675991	−	0.736910i	$-0.736284\pi$
$127$	−15.2361	−1.35198	−0.675991	+	0.736910i	$0.736284\pi$
$128$	−11.3820	−1.00603
$129$	15.2361	1.34146
$130$	0.236068	0.0207045
$131$	4.79837	0.419236	0.209618	−	0.977783i	$-0.432778\pi$
$131$	4.79837	0.419236	0.209618	+	0.977783i	$0.432778\pi$
$132$	−5.23607	−0.455741
$133$	11.9098	1.03271
$134$	4.76393	0.411541
$135$	14.4721	1.24556
$136$	−13.6180	−1.16774
$137$	−11.6180	−0.992596	−0.496298	−	0.868152i	$-0.665308\pi$
$137$	−11.6180	−0.992596	−0.496298	+	0.868152i	$0.665308\pi$
$138$	8.76393	0.746035
$139$	15.7639	1.33708	0.668540	−	0.743677i	$-0.266920\pi$
$139$	15.7639	1.33708	0.668540	+	0.743677i	$0.266920\pi$
$140$	−3.85410	−0.325731
$141$	27.4164	2.30888
$142$	−6.70820	−0.562940
$143$	0.381966	0.0319416
$144$	13.8541	1.15451
$145$	5.61803	0.466552
$146$	−6.85410	−0.567250
$147$	4.29180	0.353981
$148$	17.7082	1.45561
$149$	−3.09017	−0.253157	−0.126578	−	0.991957i	$-0.540399\pi$
$149$	−3.09017	−0.253157	−0.126578	+	0.991957i	$0.540399\pi$
$150$	8.00000	0.653197
$151$	−6.70820	−0.545906	−0.272953	−	0.962027i	$-0.588000\pi$
$151$	−6.70820	−0.545906	−0.272953	+	0.962027i	$0.588000\pi$
$152$	11.1803	0.906845
$153$	45.5066	3.67899
$154$	1.47214	0.118628
$155$	8.23607	0.661537
$156$	−2.00000	−0.160128
$157$	0.854102	0.0681648	0.0340824	−	0.999419i	$-0.489149\pi$
$157$	0.854102	0.0681648	0.0340824	+	0.999419i	$0.489149\pi$
$158$	8.03444	0.639186
$159$	−20.1803	−1.60041
$160$	−5.61803	−0.444145
$161$	10.4377	0.822606
$162$	15.0902	1.18560
$163$	−17.0902	−1.33861	−0.669303	−	0.742990i	$-0.733407\pi$
$163$	−17.0902	−1.33861	−0.669303	+	0.742990i	$0.733407\pi$
$164$	11.7082	0.914257
$165$	−3.23607	−0.251928
$166$	2.20163	0.170879
$167$	−15.7082	−1.21554	−0.607769	−	0.794114i	$-0.707935\pi$
$167$	−15.7082	−1.21554	−0.607769	+	0.794114i	$0.707935\pi$
$168$	−17.2361	−1.32979
$169$	−12.8541	−0.988777
$170$	−3.76393	−0.288680
$171$	−37.3607	−2.85704
$172$	7.61803	0.580870
$173$	7.52786	0.572333	0.286166	−	0.958180i	$-0.407619\pi$
$173$	7.52786	0.572333	0.286166	+	0.958180i	$0.407619\pi$
$174$	11.2361	0.851804
$175$	9.52786	0.720239
$176$	−1.85410	−0.139758
$177$	28.1803	2.11816
$178$	−6.70820	−0.502801
$179$	8.79837	0.657621	0.328811	−	0.944396i	$-0.393352\pi$
$179$	8.79837	0.657621	0.328811	+	0.944396i	$0.393352\pi$
$180$	12.0902	0.901148
$181$	4.47214	0.332411	0.166206	−	0.986091i	$-0.446848\pi$
$181$	4.47214	0.332411	0.166206	+	0.986091i	$0.446848\pi$
$182$	0.562306	0.0416809
$183$	−36.1803	−2.67453
$184$	9.79837	0.722346
$185$	10.9443	0.804639
$186$	16.4721	1.20780
$187$	−6.09017	−0.445357
$188$	13.7082	0.999774
$189$	34.4721	2.50748
$190$	3.09017	0.224184
$191$	−18.4164	−1.33256	−0.666282	−	0.745700i	$-0.732115\pi$
$191$	−18.4164	−1.33256	−0.666282	+	0.745700i	$0.732115\pi$
$192$	0.763932	0.0551320
$193$	17.1803	1.23667	0.618334	−	0.785915i	$-0.287808\pi$
$193$	17.1803	1.23667	0.618334	+	0.785915i	$0.287808\pi$
$194$	8.32624	0.597789
$195$	−1.23607	−0.0885167
$196$	2.14590	0.153278
$197$	1.29180	0.0920367	0.0460183	−	0.998941i	$-0.485347\pi$
$197$	1.29180	0.0920367	0.0460183	+	0.998941i	$0.485347\pi$
$198$	−4.61803	−0.328189
$199$	−24.9443	−1.76825	−0.884126	−	0.467248i	$-0.845246\pi$
$199$	−24.9443	−1.76825	−0.884126	+	0.467248i	$0.845246\pi$
$200$	8.94427	0.632456
$201$	−24.9443	−1.75943
$202$	−3.61803	−0.254564
$203$	13.3820	0.939230
$204$	31.8885	2.23264
$205$	7.23607	0.505389
$206$	0.201626	0.0140480
$207$	−32.7426	−2.27577
$208$	−0.708204	−0.0491051
$209$	5.00000	0.345857
$210$	−4.76393	−0.328743
$211$	−8.05573	−0.554579	−0.277290	−	0.960786i	$-0.589436\pi$
$211$	−8.05573	−0.554579	−0.277290	+	0.960786i	$0.589436\pi$
$212$	−10.0902	−0.692996
$213$	35.1246	2.40670
$214$	−5.76393	−0.394014
$215$	4.70820	0.321097
$216$	32.3607	2.20187
$217$	19.6180	1.33176
$218$	3.38197	0.229056
$219$	35.8885	2.42512
$220$	−1.61803	−0.109088
$221$	−2.32624	−0.156480
$222$	21.8885	1.46906
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	−13.3820	−0.894120
$225$	−29.8885	−1.99257
$226$	−3.85410	−0.256371
$227$	−19.4164	−1.28871	−0.644356	−	0.764726i	$-0.722875\pi$
$227$	−19.4164	−1.28871	−0.644356	+	0.764726i	$0.722875\pi$
$228$	−26.1803	−1.73384
$229$	17.9787	1.18807	0.594033	−	0.804440i	$-0.297535\pi$
$229$	17.9787	1.18807	0.594033	+	0.804440i	$0.297535\pi$
$230$	2.70820	0.178574
$231$	−7.70820	−0.507163
$232$	12.5623	0.824756
$233$	−10.5623	−0.691960	−0.345980	−	0.938242i	$-0.612453\pi$
$233$	−10.5623	−0.691960	−0.345980	+	0.938242i	$0.612453\pi$
$234$	−1.76393	−0.115312
$235$	8.47214	0.552661
$236$	14.0902	0.917192
$237$	−42.0689	−2.73267
$238$	−8.96556	−0.581151
$239$	25.1246	1.62518	0.812588	−	0.582838i	$-0.198058\pi$
$239$	25.1246	1.62518	0.812588	+	0.582838i	$0.198058\pi$
$240$	6.00000	0.387298
$241$	−6.27051	−0.403919	−0.201960	−	0.979394i	$-0.564731\pi$
$241$	−6.27051	−0.403919	−0.201960	+	0.979394i	$0.564731\pi$
$242$	−6.18034	−0.397287
$243$	−35.5967	−2.28353
$244$	−18.0902	−1.15810
$245$	1.32624	0.0847302
$246$	14.4721	0.922710
$247$	1.90983	0.121520
$248$	18.4164	1.16944
$249$	−11.5279	−0.730549
$250$	5.56231	0.351791
$251$	6.76393	0.426936	0.213468	−	0.976950i	$-0.431524\pi$
$251$	6.76393	0.426936	0.213468	+	0.976950i	$0.431524\pi$
$252$	28.7984	1.81413
$253$	4.38197	0.275492
$254$	−9.41641	−0.590838
$255$	19.7082	1.23418
$256$	−6.56231	−0.410144
$257$	7.03444	0.438796	0.219398	−	0.975635i	$-0.429591\pi$
$257$	7.03444	0.438796	0.219398	+	0.975635i	$0.429591\pi$
$258$	9.41641	0.586240
$259$	26.0689	1.61984
$260$	−0.618034	−0.0383288
$261$	−41.9787	−2.59842
$262$	2.96556	0.183213
$263$	19.6180	1.20970	0.604850	−	0.796339i	$-0.293233\pi$
$263$	19.6180	1.20970	0.604850	+	0.796339i	$0.293233\pi$
$264$	−7.23607	−0.445349
$265$	−6.23607	−0.383079
$266$	7.36068	0.451312
$267$	35.1246	2.14959
$268$	−12.4721	−0.761857
$269$	1.76393	0.107549	0.0537744	−	0.998553i	$-0.482875\pi$
$269$	1.76393	0.107549	0.0537744	+	0.998553i	$0.482875\pi$
$270$	8.94427	0.544331
$271$	−18.3262	−1.11324	−0.556620	−	0.830767i	$-0.687902\pi$
$271$	−18.3262	−1.11324	−0.556620	+	0.830767i	$0.687902\pi$
$272$	11.2918	0.684666
$273$	−2.94427	−0.178195
$274$	−7.18034	−0.433780
$275$	4.00000	0.241209
$276$	−22.9443	−1.38108
$277$	−16.4721	−0.989715	−0.494857	−	0.868974i	$-0.664780\pi$
$277$	−16.4721	−0.989715	−0.494857	+	0.868974i	$0.664780\pi$
$278$	9.74265	0.584325
$279$	−61.5410	−3.68436
$280$	−5.32624	−0.318304
$281$	−20.2361	−1.20718	−0.603591	−	0.797294i	$-0.706264\pi$
$281$	−20.2361	−1.20718	−0.603591	+	0.797294i	$0.706264\pi$
$282$	16.9443	1.00902
$283$	4.52786	0.269154	0.134577	−	0.990903i	$-0.457033\pi$
$283$	4.52786	0.269154	0.134577	+	0.990903i	$0.457033\pi$
$284$	17.5623	1.04213
$285$	−16.1803	−0.958441
$286$	0.236068	0.0139590
$287$	17.2361	1.01741
$288$	41.9787	2.47362
$289$	20.0902	1.18177
$290$	3.47214	0.203891
$291$	−43.5967	−2.55569
$292$	17.9443	1.05011
$293$	9.32624	0.544845	0.272422	−	0.962178i	$-0.412175\pi$
$293$	9.32624	0.544845	0.272422	+	0.962178i	$0.412175\pi$
$294$	2.65248	0.154696
$295$	8.70820	0.507011
$296$	24.4721	1.42241
$297$	14.4721	0.839759
$298$	−1.90983	−0.110633
$299$	1.67376	0.0967962
$300$	−20.9443	−1.20922
$301$	11.2148	0.646409
$302$	−4.14590	−0.238570
$303$	18.9443	1.08832
$304$	−9.27051	−0.531700
$305$	−11.1803	−0.640184
$306$	28.1246	1.60778
$307$	22.4721	1.28255	0.641276	−	0.767310i	$-0.278405\pi$
$307$	22.4721	1.28255	0.641276	+	0.767310i	$0.278405\pi$
$308$	−3.85410	−0.219608
$309$	−1.05573	−0.0600583
$310$	5.09017	0.289102
$311$	8.09017	0.458751	0.229376	−	0.973338i	$-0.426332\pi$
$311$	8.09017	0.458751	0.229376	+	0.973338i	$0.426332\pi$
$312$	−2.76393	−0.156477
$313$	−29.6525	−1.67606	−0.838029	−	0.545626i	$-0.816292\pi$
$313$	−29.6525	−1.67606	−0.838029	+	0.545626i	$0.816292\pi$
$314$	0.527864	0.0297891
$315$	17.7984	1.00282
$316$	−21.0344	−1.18328
$317$	17.1246	0.961814	0.480907	−	0.876772i	$-0.340307\pi$
$317$	17.1246	0.961814	0.480907	+	0.876772i	$0.340307\pi$
$318$	−12.4721	−0.699403
$319$	5.61803	0.314550
$320$	0.236068	0.0131966
$321$	30.1803	1.68450
$322$	6.45085	0.359492
$323$	−30.4508	−1.69433
$324$	−39.5066	−2.19481
$325$	1.52786	0.0847506
$326$	−10.5623	−0.584992
$327$	−17.7082	−0.979266
$328$	16.1803	0.893410
$329$	20.1803	1.11258
$330$	−2.00000	−0.110096
$331$	−24.7639	−1.36115	−0.680574	−	0.732679i	$-0.738270\pi$
$331$	−24.7639	−1.36115	−0.680574	+	0.732679i	$0.738270\pi$
$332$	−5.76393	−0.316337
$333$	−81.7771	−4.48136
$334$	−9.70820	−0.531209
$335$	−7.70820	−0.421144
$336$	14.2918	0.779681
$337$	25.0344	1.36371	0.681856	−	0.731486i	$-0.261173\pi$
$337$	25.0344	1.36371	0.681856	+	0.731486i	$0.261173\pi$
$338$	−7.94427	−0.432111
$339$	20.1803	1.09605
$340$	9.85410	0.534414
$341$	8.23607	0.446008
$342$	−23.0902	−1.24857
$343$	19.8328	1.07087
$344$	10.5279	0.567624
$345$	−14.1803	−0.763444
$346$	4.65248	0.250119
$347$	16.0902	0.863766	0.431883	−	0.901930i	$-0.357849\pi$
$347$	16.0902	0.863766	0.431883	+	0.901930i	$0.357849\pi$
$348$	−29.4164	−1.57688
$349$	12.0557	0.645328	0.322664	−	0.946514i	$-0.395422\pi$
$349$	12.0557	0.645328	0.322664	+	0.946514i	$0.395422\pi$
$350$	5.88854	0.314756
$351$	5.52786	0.295056
$352$	−5.61803	−0.299442
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	17.4164	0.925672
$355$	10.8541	0.576076
$356$	17.5623	0.930800
$357$	46.9443	2.48455
$358$	5.43769	0.287391
$359$	26.2705	1.38650	0.693252	−	0.720695i	$-0.256177\pi$
$359$	26.2705	1.38650	0.693252	+	0.720695i	$0.256177\pi$
$360$	16.7082	0.880600
$361$	6.00000	0.315789
$362$	2.76393	0.145269
$363$	32.3607	1.69850
$364$	−1.47214	−0.0771609
$365$	11.0902	0.580486
$366$	−22.3607	−1.16881
$367$	21.0344	1.09799	0.548994	−	0.835826i	$-0.315011\pi$
$367$	21.0344	1.09799	0.548994	+	0.835826i	$0.315011\pi$
$368$	−8.12461	−0.423525
$369$	−54.0689	−2.81471
$370$	6.76393	0.351640
$371$	−14.8541	−0.771187
$372$	−43.1246	−2.23591
$373$	20.5066	1.06179	0.530895	−	0.847437i	$-0.321856\pi$
$373$	20.5066	1.06179	0.530895	+	0.847437i	$0.321856\pi$
$374$	−3.76393	−0.194628
$375$	−29.1246	−1.50399
$376$	18.9443	0.976976
$377$	2.14590	0.110519
$378$	21.3050	1.09581
$379$	30.5967	1.57165	0.785825	−	0.618449i	$-0.212239\pi$
$379$	30.5967	1.57165	0.785825	+	0.618449i	$0.212239\pi$
$380$	−8.09017	−0.415017
$381$	49.3050	2.52597
$382$	−11.3820	−0.582352
$383$	6.20163	0.316888	0.158444	−	0.987368i	$-0.449352\pi$
$383$	6.20163	0.316888	0.158444	+	0.987368i	$0.449352\pi$
$384$	36.8328	1.87962
$385$	−2.38197	−0.121396
$386$	10.6180	0.540444
$387$	−35.1803	−1.78832
$388$	−21.7984	−1.10664
$389$	−29.0689	−1.47385	−0.736925	−	0.675974i	$-0.763723\pi$
$389$	−29.0689	−1.47385	−0.736925	+	0.675974i	$0.763723\pi$
$390$	−0.763932	−0.0386832
$391$	−26.6869	−1.34962
$392$	2.96556	0.149783
$393$	−15.5279	−0.783277
$394$	0.798374	0.0402215
$395$	−13.0000	−0.654101
$396$	12.0902	0.607554
$397$	−24.5967	−1.23448	−0.617238	−	0.786777i	$-0.711748\pi$
$397$	−24.5967	−1.23448	−0.617238	+	0.786777i	$0.711748\pi$
$398$	−15.4164	−0.772755
$399$	−38.5410	−1.92946
$400$	−7.41641	−0.370820
$401$	−32.3262	−1.61430	−0.807148	−	0.590350i	$-0.798990\pi$
$401$	−32.3262	−1.61430	−0.807148	+	0.590350i	$0.798990\pi$
$402$	−15.4164	−0.768901
$403$	3.14590	0.156708
$404$	9.47214	0.471256
$405$	−24.4164	−1.21326
$406$	8.27051	0.410459
$407$	10.9443	0.542487
$408$	44.0689	2.18173
$409$	−7.34752	−0.363312	−0.181656	−	0.983362i	$-0.558146\pi$
$409$	−7.34752	−0.363312	−0.181656	+	0.983362i	$0.558146\pi$
$410$	4.47214	0.220863
$411$	37.5967	1.85451
$412$	−0.527864	−0.0260060
$413$	20.7426	1.02068
$414$	−20.2361	−0.994548
$415$	−3.56231	−0.174867
$416$	−2.14590	−0.105211
$417$	−51.0132	−2.49812
$418$	3.09017	0.151145
$419$	−16.2361	−0.793184	−0.396592	−	0.917995i	$-0.629807\pi$
$419$	−16.2361	−0.793184	−0.396592	+	0.917995i	$0.629807\pi$
$420$	12.4721	0.608578
$421$	16.8328	0.820381	0.410191	−	0.912000i	$-0.365462\pi$
$421$	16.8328	0.820381	0.410191	+	0.912000i	$0.365462\pi$
$422$	−4.97871	−0.242360
$423$	−63.3050	−3.07799
$424$	−13.9443	−0.677194
$425$	−24.3607	−1.18167
$426$	21.7082	1.05177
$427$	−26.6312	−1.28877
$428$	15.0902	0.729411
$429$	−1.23607	−0.0596779
$430$	2.90983	0.140324
$431$	−38.8541	−1.87154	−0.935768	−	0.352616i	$-0.885292\pi$
$431$	−38.8541	−1.87154	−0.935768	+	0.352616i	$0.885292\pi$
$432$	−26.8328	−1.29099
$433$	−1.67376	−0.0804359	−0.0402179	−	0.999191i	$-0.512805\pi$
$433$	−1.67376	−0.0804359	−0.0402179	+	0.999191i	$0.512805\pi$
$434$	12.1246	0.582000
$435$	−18.1803	−0.871681
$436$	−8.85410	−0.424035
$437$	21.9098	1.04809
$438$	22.1803	1.05982
$439$	8.88854	0.424227	0.212114	−	0.977245i	$-0.431965\pi$
$439$	8.88854	0.424227	0.212114	+	0.977245i	$0.431965\pi$
$440$	−2.23607	−0.106600
$441$	−9.90983	−0.471897
$442$	−1.43769	−0.0683841
$443$	35.1591	1.67046	0.835229	−	0.549903i	$-0.185335\pi$
$443$	35.1591	1.67046	0.835229	+	0.549903i	$0.185335\pi$
$444$	−57.3050	−2.71957
$445$	10.8541	0.514534
$446$	−2.47214	−0.117059
$447$	10.0000	0.472984
$448$	0.562306	0.0265665
$449$	−28.6525	−1.35219	−0.676097	−	0.736813i	$-0.736330\pi$
$449$	−28.6525	−1.35219	−0.676097	+	0.736813i	$0.736330\pi$
$450$	−18.4721	−0.870785
$451$	7.23607	0.340733
$452$	10.0902	0.474602
$453$	21.7082	1.01994
$454$	−12.0000	−0.563188
$455$	−0.909830	−0.0426535
$456$	−36.1803	−1.69430
$457$	13.0000	0.608114	0.304057	−	0.952654i	$-0.401659\pi$
$457$	13.0000	0.608114	0.304057	+	0.952654i	$0.401659\pi$
$458$	11.1115	0.519204
$459$	−88.1378	−4.11392
$460$	−7.09017	−0.330581
$461$	−11.3607	−0.529120	−0.264560	−	0.964369i	$-0.585227\pi$
$461$	−11.3607	−0.529120	−0.264560	+	0.964369i	$0.585227\pi$
$462$	−4.76393	−0.221638
$463$	−8.88854	−0.413086	−0.206543	−	0.978438i	$-0.566221\pi$
$463$	−8.88854	−0.413086	−0.206543	+	0.978438i	$0.566221\pi$
$464$	−10.4164	−0.483570
$465$	−26.6525	−1.23598
$466$	−6.52786	−0.302397
$467$	41.1246	1.90302	0.951510	−	0.307618i	$-0.0995318\pi$
$467$	41.1246	1.90302	0.951510	+	0.307618i	$0.0995318\pi$
$468$	4.61803	0.213469
$469$	−18.3607	−0.847817
$470$	5.23607	0.241522
$471$	−2.76393	−0.127355
$472$	19.4721	0.896278
$473$	4.70820	0.216483
$474$	−26.0000	−1.19422
$475$	20.0000	0.917663
$476$	23.4721	1.07584
$477$	46.5967	2.13352
$478$	15.5279	0.710228
$479$	−13.9787	−0.638704	−0.319352	−	0.947636i	$-0.603465\pi$
$479$	−13.9787	−0.638704	−0.319352	+	0.947636i	$0.603465\pi$
$480$	18.1803	0.829815
$481$	4.18034	0.190607
$482$	−3.87539	−0.176519
$483$	−33.7771	−1.53691
$484$	16.1803	0.735470
$485$	−13.4721	−0.611738
$486$	−22.0000	−0.997940
$487$	−35.8885	−1.62627	−0.813133	−	0.582079i	$-0.802240\pi$
$487$	−35.8885	−1.62627	−0.813133	+	0.582079i	$0.802240\pi$
$488$	−25.0000	−1.13170
$489$	55.3050	2.50098
$490$	0.819660	0.0370285
$491$	−17.0000	−0.767199	−0.383600	−	0.923499i	$-0.625316\pi$
$491$	−17.0000	−0.767199	−0.383600	+	0.923499i	$0.625316\pi$
$492$	−37.8885	−1.70815
$493$	−34.2148	−1.54096
$494$	1.18034	0.0531060
$495$	7.47214	0.335848
$496$	−15.2705	−0.685666
$497$	25.8541	1.15971
$498$	−7.12461	−0.319261
$499$	28.6869	1.28420	0.642101	−	0.766620i	$-0.278063\pi$
$499$	28.6869	1.28420	0.642101	+	0.766620i	$0.278063\pi$
$500$	−14.5623	−0.651246
$501$	50.8328	2.27104
$502$	4.18034	0.186578
$503$	−5.27051	−0.235000	−0.117500	−	0.993073i	$-0.537488\pi$
$503$	−5.27051	−0.235000	−0.117500	+	0.993073i	$0.537488\pi$
$504$	39.7984	1.77276
$505$	5.85410	0.260504
$506$	2.70820	0.120394
$507$	41.5967	1.84738
$508$	24.6525	1.09378
$509$	40.6180	1.80036	0.900181	−	0.435515i	$-0.143434\pi$
$509$	40.6180	1.80036	0.900181	+	0.435515i	$0.143434\pi$
$510$	12.1803	0.539355
$511$	26.4164	1.16859
$512$	18.7082	0.826794
$513$	72.3607	3.19480
$514$	4.34752	0.191761
$515$	−0.326238	−0.0143758
$516$	−24.6525	−1.08526
$517$	8.47214	0.372604
$518$	16.1115	0.707897
$519$	−24.3607	−1.06932
$520$	−0.854102	−0.0374548
$521$	18.8541	0.826013	0.413007	−	0.910728i	$-0.364479\pi$
$521$	18.8541	0.826013	0.413007	+	0.910728i	$0.364479\pi$
$522$	−25.9443	−1.13555
$523$	−8.05573	−0.352252	−0.176126	−	0.984368i	$-0.556357\pi$
$523$	−8.05573	−0.352252	−0.176126	+	0.984368i	$0.556357\pi$
$524$	−7.76393	−0.339169
$525$	−30.8328	−1.34565
$526$	12.1246	0.528658
$527$	−50.1591	−2.18496
$528$	6.00000	0.261116
$529$	−3.79837	−0.165147
$530$	−3.85410	−0.167411
$531$	−65.0689	−2.82375
$532$	−19.2705	−0.835483
$533$	2.76393	0.119719
$534$	21.7082	0.939406
$535$	9.32624	0.403208
$536$	−17.2361	−0.744485
$537$	−28.4721	−1.22866
$538$	1.09017	0.0470006
$539$	1.32624	0.0571251
$540$	−23.4164	−1.00768
$541$	12.2016	0.524589	0.262294	−	0.964988i	$-0.415521\pi$
$541$	12.2016	0.524589	0.262294	+	0.964988i	$0.415521\pi$
$542$	−11.3262	−0.486504
$543$	−14.4721	−0.621059
$544$	34.2148	1.46695
$545$	−5.47214	−0.234401
$546$	−1.81966	−0.0778743
$547$	−23.2148	−0.992592	−0.496296	−	0.868153i	$-0.665307\pi$
$547$	−23.2148	−0.992592	−0.496296	+	0.868153i	$0.665307\pi$
$548$	18.7984	0.803027
$549$	83.5410	3.56544
$550$	2.47214	0.105412
$551$	28.0902	1.19668
$552$	−31.7082	−1.34959
$553$	−30.9656	−1.31679
$554$	−10.1803	−0.432521
$555$	−35.4164	−1.50334
$556$	−25.5066	−1.08172
$557$	−12.0000	−0.508456	−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000	−0.508456	−0.254228	+	0.967144i	$0.581821\pi$
$558$	−38.0344	−1.61013
$559$	1.79837	0.0760631
$560$	4.41641	0.186627
$561$	19.7082	0.832081
$562$	−12.5066	−0.527558
$563$	−2.67376	−0.112686	−0.0563428	−	0.998411i	$-0.517944\pi$
$563$	−2.67376	−0.112686	−0.0563428	+	0.998411i	$0.517944\pi$
$564$	−44.3607	−1.86792
$565$	6.23607	0.262353
$566$	2.79837	0.117624
$567$	−58.1591	−2.44245
$568$	24.2705	1.01837
$569$	−33.7426	−1.41457	−0.707283	−	0.706931i	$-0.750079\pi$
$569$	−33.7426	−1.41457	−0.707283	+	0.706931i	$0.750079\pi$
$570$	−10.0000	−0.418854
$571$	−30.3262	−1.26911	−0.634557	−	0.772876i	$-0.718817\pi$
$571$	−30.3262	−1.26911	−0.634557	+	0.772876i	$0.718817\pi$
$572$	−0.618034	−0.0258413
$573$	59.5967	2.48969
$574$	10.6525	0.444626
$575$	17.5279	0.730962
$576$	−1.76393	−0.0734972
$577$	8.00000	0.333044	0.166522	−	0.986038i	$-0.446746\pi$
$577$	8.00000	0.333044	0.166522	+	0.986038i	$0.446746\pi$
$578$	12.4164	0.516454
$579$	−55.5967	−2.31052
$580$	−9.09017	−0.377449
$581$	−8.48529	−0.352029
$582$	−26.9443	−1.11688
$583$	−6.23607	−0.258272
$584$	24.7984	1.02616
$585$	2.85410	0.118003
$586$	5.76393	0.238106
$587$	31.2705	1.29067	0.645336	−	0.763899i	$-0.276717\pi$
$587$	31.2705	1.29067	0.645336	+	0.763899i	$0.276717\pi$
$588$	−6.94427	−0.286377
$589$	41.1803	1.69681
$590$	5.38197	0.221572
$591$	−4.18034	−0.171956
$592$	−20.2918	−0.833988
$593$	19.1803	0.787642	0.393821	−	0.919187i	$-0.371153\pi$
$593$	19.1803	0.787642	0.393821	+	0.919187i	$0.371153\pi$
$594$	8.94427	0.366988
$595$	14.5066	0.594712
$596$	5.00000	0.204808
$597$	80.7214	3.30371
$598$	1.03444	0.0423015
$599$	31.1803	1.27399	0.636997	−	0.770866i	$-0.280176\pi$
$599$	31.1803	1.27399	0.636997	+	0.770866i	$0.280176\pi$
$600$	−28.9443	−1.18164
$601$	40.3607	1.64635	0.823173	−	0.567790i	$-0.192202\pi$
$601$	40.3607	1.64635	0.823173	+	0.567790i	$0.192202\pi$
$602$	6.93112	0.282491
$603$	57.5967	2.34552
$604$	10.8541	0.441647
$605$	10.0000	0.406558
$606$	11.7082	0.475613
$607$	17.0689	0.692805	0.346402	−	0.938086i	$-0.387403\pi$
$607$	17.0689	0.692805	0.346402	+	0.938086i	$0.387403\pi$
$608$	−28.0902	−1.13921
$609$	−43.3050	−1.75481
$610$	−6.90983	−0.279771
$611$	3.23607	0.130917
$612$	−73.6312	−2.97636
$613$	1.27051	0.0513154	0.0256577	−	0.999671i	$-0.491832\pi$
$613$	1.27051	0.0513154	0.0256577	+	0.999671i	$0.491832\pi$
$614$	13.8885	0.560496
$615$	−23.4164	−0.944241
$616$	−5.32624	−0.214600
$617$	29.2705	1.17839	0.589193	−	0.807992i	$-0.299446\pi$
$617$	29.2705	1.17839	0.589193	+	0.807992i	$0.299446\pi$
$618$	−0.652476	−0.0262464
$619$	−37.3820	−1.50251	−0.751254	−	0.660013i	$-0.770551\pi$
$619$	−37.3820	−1.50251	−0.751254	+	0.660013i	$0.770551\pi$
$620$	−13.3262	−0.535195
$621$	63.4164	2.54481
$622$	5.00000	0.200482
$623$	25.8541	1.03582
$624$	2.29180	0.0917453
$625$	11.0000	0.440000
$626$	−18.3262	−0.732464
$627$	−16.1803	−0.646181
$628$	−1.38197	−0.0551464
$629$	−66.6525	−2.65761
$630$	11.0000	0.438250
$631$	42.0902	1.67558	0.837792	−	0.545990i	$-0.183846\pi$
$631$	42.0902	1.67558	0.837792	+	0.545990i	$0.183846\pi$
$632$	−29.0689	−1.15630
$633$	26.0689	1.03615
$634$	10.5836	0.420328
$635$	15.2361	0.604625
$636$	32.6525	1.29475
$637$	0.506578	0.0200713
$638$	3.47214	0.137463
$639$	−81.1033	−3.20840
$640$	11.3820	0.449912
$641$	−20.5410	−0.811321	−0.405661	−	0.914024i	$-0.632959\pi$
$641$	−20.5410	−0.811321	−0.405661	+	0.914024i	$0.632959\pi$
$642$	18.6525	0.736155
$643$	45.1803	1.78174	0.890869	−	0.454260i	$-0.150096\pi$
$643$	45.1803	1.78174	0.890869	+	0.454260i	$0.150096\pi$
$644$	−16.8885	−0.665502
$645$	−15.2361	−0.599920
$646$	−18.8197	−0.740450
$647$	−2.65248	−0.104280	−0.0521398	−	0.998640i	$-0.516604\pi$
$647$	−2.65248	−0.104280	−0.0521398	+	0.998640i	$0.516604\pi$
$648$	−54.5967	−2.14476
$649$	8.70820	0.341827
$650$	0.944272	0.0370374
$651$	−63.4853	−2.48818
$652$	27.6525	1.08295
$653$	−9.61803	−0.376383	−0.188191	−	0.982132i	$-0.560263\pi$
$653$	−9.61803	−0.376383	−0.188191	+	0.982132i	$0.560263\pi$
$654$	−10.9443	−0.427955
$655$	−4.79837	−0.187488
$656$	−13.4164	−0.523823
$657$	−82.8673	−3.23296
$658$	12.4721	0.486214
$659$	−10.0000	−0.389545	−0.194772	−	0.980848i	$-0.562397\pi$
$659$	−10.0000	−0.389545	−0.194772	+	0.980848i	$0.562397\pi$
$660$	5.23607	0.203814
$661$	−33.5623	−1.30542	−0.652711	−	0.757607i	$-0.726368\pi$
$661$	−33.5623	−1.30542	−0.652711	+	0.757607i	$0.726368\pi$
$662$	−15.3050	−0.594844
$663$	7.52786	0.292358
$664$	−7.96556	−0.309124
$665$	−11.9098	−0.461843
$666$	−50.5410	−1.95842
$667$	24.6180	0.953214
$668$	25.4164	0.983390
$669$	12.9443	0.500454
$670$	−4.76393	−0.184047
$671$	−11.1803	−0.431612
$672$	43.3050	1.67052
$673$	−12.7984	−0.493341	−0.246671	−	0.969099i	$-0.579337\pi$
$673$	−12.7984	−0.493341	−0.246671	+	0.969099i	$0.579337\pi$
$674$	15.4721	0.595964
$675$	57.8885	2.22813
$676$	20.7984	0.799937
$677$	4.36068	0.167595	0.0837973	−	0.996483i	$-0.473295\pi$
$677$	4.36068	0.167595	0.0837973	+	0.996483i	$0.473295\pi$
$678$	12.4721	0.478989
$679$	−32.0902	−1.23151
$680$	13.6180	0.522228
$681$	62.8328	2.40776
$682$	5.09017	0.194913
$683$	−11.2361	−0.429936	−0.214968	−	0.976621i	$-0.568965\pi$
$683$	−11.2361	−0.429936	−0.214968	+	0.976621i	$0.568965\pi$
$684$	60.4508	2.31140
$685$	11.6180	0.443902
$686$	12.2574	0.467988
$687$	−58.1803	−2.21972
$688$	−8.72949	−0.332809
$689$	−2.38197	−0.0907457
$690$	−8.76393	−0.333637
$691$	−28.0557	−1.06729	−0.533645	−	0.845709i	$-0.679178\pi$
$691$	−28.0557	−1.06729	−0.533645	+	0.845709i	$0.679178\pi$
$692$	−12.1803	−0.463027
$693$	17.7984	0.676104
$694$	9.94427	0.377479
$695$	−15.7639	−0.597960
$696$	−40.6525	−1.54093
$697$	−44.0689	−1.66923
$698$	7.45085	0.282019
$699$	34.1803	1.29282
$700$	−15.4164	−0.582685
$701$	−14.4377	−0.545304	−0.272652	−	0.962113i	$-0.587901\pi$
$701$	−14.4377	−0.545304	−0.272652	+	0.962113i	$0.587901\pi$
$702$	3.41641	0.128944
$703$	54.7214	2.06386
$704$	0.236068	0.00889715
$705$	−27.4164	−1.03256
$706$	3.70820	0.139560
$707$	13.9443	0.524428
$708$	−45.5967	−1.71363
$709$	−17.8197	−0.669231	−0.334616	−	0.942355i	$-0.608607\pi$
$709$	−17.8197	−0.669231	−0.334616	+	0.942355i	$0.608607\pi$
$710$	6.70820	0.251754
$711$	97.1378	3.64295
$712$	24.2705	0.909576
$713$	36.0902	1.35159
$714$	29.0132	1.08579
$715$	−0.381966	−0.0142847
$716$	−14.2361	−0.532027
$717$	−81.3050	−3.03639
$718$	16.2361	0.605925
$719$	19.6738	0.733708	0.366854	−	0.930279i	$-0.380435\pi$
$719$	19.6738	0.733708	0.366854	+	0.930279i	$0.380435\pi$
$720$	−13.8541	−0.516312
$721$	−0.777088	−0.0289403
$722$	3.70820	0.138005
$723$	20.2918	0.754660
$724$	−7.23607	−0.268926
$725$	22.4721	0.834594
$726$	20.0000	0.742270
$727$	−2.00000	−0.0741759	−0.0370879	−	0.999312i	$-0.511808\pi$
$727$	−2.00000	−0.0741759	−0.0370879	+	0.999312i	$0.511808\pi$
$728$	−2.03444	−0.0754014
$729$	41.9443	1.55349
$730$	6.85410	0.253682
$731$	−28.6738	−1.06054
$732$	58.5410	2.16374
$733$	−36.2918	−1.34047	−0.670234	−	0.742150i	$-0.733806\pi$
$733$	−36.2918	−1.34047	−0.670234	+	0.742150i	$0.733806\pi$
$734$	13.0000	0.479839
$735$	−4.29180	−0.158305
$736$	−24.6180	−0.907433
$737$	−7.70820	−0.283935
$738$	−33.4164	−1.23007
$739$	−33.9443	−1.24866	−0.624330	−	0.781161i	$-0.714628\pi$
$739$	−33.9443	−1.24866	−0.624330	+	0.781161i	$0.714628\pi$
$740$	−17.7082	−0.650967
$741$	−6.18034	−0.227040
$742$	−9.18034	−0.337021
$743$	40.2492	1.47660	0.738300	−	0.674472i	$-0.235629\pi$
$743$	40.2492	1.47660	0.738300	+	0.674472i	$0.235629\pi$
$744$	−59.5967	−2.18492
$745$	3.09017	0.113215
$746$	12.6738	0.464019
$747$	26.6180	0.973903
$748$	9.85410	0.360302
$749$	22.2148	0.811710
$750$	−18.0000	−0.657267
$751$	−47.6312	−1.73809	−0.869043	−	0.494736i	$-0.835265\pi$
$751$	−47.6312	−1.73809	−0.869043	+	0.494736i	$0.835265\pi$
$752$	−15.7082	−0.572819
$753$	−21.8885	−0.797663
$754$	1.32624	0.0482987
$755$	6.70820	0.244137
$756$	−55.7771	−2.02859
$757$	32.8885	1.19535	0.597677	−	0.801737i	$-0.296090\pi$
$757$	32.8885	1.19535	0.597677	+	0.801737i	$0.296090\pi$
$758$	18.9098	0.686836
$759$	−14.1803	−0.514714
$760$	−11.1803	−0.405554
$761$	−39.2705	−1.42355	−0.711777	−	0.702405i	$-0.752110\pi$
$761$	−39.2705	−1.42355	−0.711777	+	0.702405i	$0.752110\pi$
$762$	30.4721	1.10389
$763$	−13.0344	−0.471878
$764$	29.7984	1.07807
$765$	−45.5066	−1.64529
$766$	3.83282	0.138485
$767$	3.32624	0.120103
$768$	21.2361	0.766291
$769$	2.29180	0.0826443	0.0413221	−	0.999146i	$-0.486843\pi$
$769$	2.29180	0.0826443	0.0413221	+	0.999146i	$0.486843\pi$
$770$	−1.47214	−0.0530521
$771$	−22.7639	−0.819823
$772$	−27.7984	−1.00049
$773$	−9.03444	−0.324946	−0.162473	−	0.986713i	$-0.551947\pi$
$773$	−9.03444	−0.324946	−0.162473	+	0.986713i	$0.551947\pi$
$774$	−21.7426	−0.781523
$775$	32.9443	1.18339
$776$	−30.1246	−1.08141
$777$	−84.3607	−3.02642
$778$	−17.9656	−0.644096
$779$	36.1803	1.29630
$780$	2.00000	0.0716115
$781$	10.8541	0.388390
$782$	−16.4934	−0.589804
$783$	81.3050	2.90560
$784$	−2.45898	−0.0878207
$785$	−0.854102	−0.0304842
$786$	−9.59675	−0.342305
$787$	−34.8673	−1.24288	−0.621442	−	0.783460i	$-0.713453\pi$
$787$	−34.8673	−1.24288	−0.621442	+	0.783460i	$0.713453\pi$
$788$	−2.09017	−0.0744592
$789$	−63.4853	−2.26014
$790$	−8.03444	−0.285853
$791$	14.8541	0.528151
$792$	16.7082	0.593700
$793$	−4.27051	−0.151650
$794$	−15.2016	−0.539486
$795$	20.1803	0.715723
$796$	40.3607	1.43055
$797$	12.1246	0.429476	0.214738	−	0.976672i	$-0.431110\pi$
$797$	12.1246	0.429476	0.214738	+	0.976672i	$0.431110\pi$
$798$	−23.8197	−0.843207
$799$	−51.5967	−1.82536
$800$	−22.4721	−0.794510
$801$	−81.1033	−2.86565
$802$	−19.9787	−0.705473
$803$	11.0902	0.391364
$804$	40.3607	1.42341
$805$	−10.4377	−0.367880
$806$	1.94427	0.0684841
$807$	−5.70820	−0.200938
$808$	13.0902	0.460511
$809$	−19.9787	−0.702414	−0.351207	−	0.936298i	$-0.614229\pi$
$809$	−19.9787	−0.702414	−0.351207	+	0.936298i	$0.614229\pi$
$810$	−15.0902	−0.530215
$811$	−39.2705	−1.37897	−0.689487	−	0.724298i	$-0.742164\pi$
$811$	−39.2705	−1.37897	−0.689487	+	0.724298i	$0.742164\pi$
$812$	−21.6525	−0.759853
$813$	59.3050	2.07992
$814$	6.76393	0.237076
$815$	17.0902	0.598643
$816$	−36.5410	−1.27919
$817$	23.5410	0.823596
$818$	−4.54102	−0.158773
$819$	6.79837	0.237554
$820$	−11.7082	−0.408868
$821$	−23.4721	−0.819183	−0.409592	−	0.912269i	$-0.634329\pi$
$821$	−23.4721	−0.819183	−0.409592	+	0.912269i	$0.634329\pi$
$822$	23.2361	0.810451
$823$	−51.6525	−1.80049	−0.900246	−	0.435381i	$-0.856613\pi$
$823$	−51.6525	−1.80049	−0.900246	+	0.435381i	$0.856613\pi$
$824$	−0.729490	−0.0254130
$825$	−12.9443	−0.450662
$826$	12.8197	0.446053
$827$	−19.2918	−0.670841	−0.335421	−	0.942068i	$-0.608878\pi$
$827$	−19.2918	−0.670841	−0.335421	+	0.942068i	$0.608878\pi$
$828$	52.9787	1.84114
$829$	−16.8541	−0.585367	−0.292684	−	0.956209i	$-0.594548\pi$
$829$	−16.8541	−0.585367	−0.292684	+	0.956209i	$0.594548\pi$
$830$	−2.20163	−0.0764196
$831$	53.3050	1.84913
$832$	0.0901699	0.00312608
$833$	−8.07701	−0.279852
$834$	−31.5279	−1.09172
$835$	15.7082	0.543605
$836$	−8.09017	−0.279804
$837$	119.193	4.11993
$838$	−10.0344	−0.346634
$839$	−18.3475	−0.633427	−0.316713	−	0.948521i	$-0.602579\pi$
$839$	−18.3475	−0.633427	−0.316713	+	0.948521i	$0.602579\pi$
$840$	17.2361	0.594701
$841$	2.56231	0.0883554
$842$	10.4033	0.358520
$843$	65.4853	2.25543
$844$	13.0344	0.448664
$845$	12.8541	0.442195
$846$	−39.1246	−1.34513
$847$	23.8197	0.818453
$848$	11.5623	0.397051
$849$	−14.6525	−0.502872
$850$	−15.0557	−0.516407
$851$	47.9574	1.64396
$852$	−56.8328	−1.94706
$853$	−28.7082	−0.982950	−0.491475	−	0.870892i	$-0.663542\pi$
$853$	−28.7082	−0.982950	−0.491475	+	0.870892i	$0.663542\pi$
$854$	−16.4590	−0.563215
$855$	37.3607	1.27771
$856$	20.8541	0.712779
$857$	−11.0689	−0.378106	−0.189053	−	0.981967i	$-0.560542\pi$
$857$	−11.0689	−0.378106	−0.189053	+	0.981967i	$0.560542\pi$
$858$	−0.763932	−0.0260802
$859$	−5.88854	−0.200915	−0.100457	−	0.994941i	$-0.532031\pi$
$859$	−5.88854	−0.200915	−0.100457	+	0.994941i	$0.532031\pi$
$860$	−7.61803	−0.259773
$861$	−55.7771	−1.90088
$862$	−24.0132	−0.817891
$863$	50.1935	1.70861	0.854303	−	0.519775i	$-0.173984\pi$
$863$	50.1935	1.70861	0.854303	+	0.519775i	$0.173984\pi$
$864$	−81.3050	−2.76605
$865$	−7.52786	−0.255955
$866$	−1.03444	−0.0351518
$867$	−65.0132	−2.20796
$868$	−31.7426	−1.07742
$869$	−13.0000	−0.440995
$870$	−11.2361	−0.380938
$871$	−2.94427	−0.0997628
$872$	−12.2361	−0.414366
$873$	100.666	3.40702
$874$	13.5410	0.458032
$875$	−21.4377	−0.724726
$876$	−58.0689	−1.96197
$877$	−21.0689	−0.711446	−0.355723	−	0.934591i	$-0.615765\pi$
$877$	−21.0689	−0.711446	−0.355723	+	0.934591i	$0.615765\pi$
$878$	5.49342	0.185394
$879$	−30.1803	−1.01796
$880$	1.85410	0.0625018
$881$	−11.6738	−0.393299	−0.196650	−	0.980474i	$-0.563006\pi$
$881$	−11.6738	−0.393299	−0.196650	+	0.980474i	$0.563006\pi$
$882$	−6.12461	−0.206226
$883$	38.4853	1.29513	0.647567	−	0.762009i	$-0.275787\pi$
$883$	38.4853	1.29513	0.647567	+	0.762009i	$0.275787\pi$
$884$	3.76393	0.126595
$885$	−28.1803	−0.947272
$886$	21.7295	0.730016
$887$	−44.4508	−1.49251	−0.746257	−	0.665658i	$-0.768151\pi$
$887$	−44.4508	−1.49251	−0.746257	+	0.665658i	$0.768151\pi$
$888$	−79.1935	−2.65756
$889$	36.2918	1.21719
$890$	6.70820	0.224860
$891$	−24.4164	−0.817980
$892$	6.47214	0.216703
$893$	42.3607	1.41755
$894$	6.18034	0.206701
$895$	−8.79837	−0.294097
$896$	27.1115	0.905730
$897$	−5.41641	−0.180849
$898$	−17.7082	−0.590930
$899$	46.2705	1.54321
$900$	48.3607	1.61202
$901$	37.9787	1.26525
$902$	4.47214	0.148906
$903$	−36.2918	−1.20772
$904$	13.9443	0.463780
$905$	−4.47214	−0.148659
$906$	13.4164	0.445730
$907$	13.5066	0.448479	0.224239	−	0.974534i	$-0.428010\pi$
$907$	13.5066	0.448479	0.224239	+	0.974534i	$0.428010\pi$
$908$	31.4164	1.04259
$909$	−43.7426	−1.45085
$910$	−0.562306	−0.0186403
$911$	−50.8328	−1.68417	−0.842083	−	0.539348i	$-0.818671\pi$
$911$	−50.8328	−1.68417	−0.842083	+	0.539348i	$0.818671\pi$
$912$	30.0000	0.993399
$913$	−3.56231	−0.117895
$914$	8.03444	0.265756
$915$	36.1803	1.19609
$916$	−29.0902	−0.961166
$917$	−11.4296	−0.377437
$918$	−54.4721	−1.79785
$919$	−9.38197	−0.309483	−0.154741	−	0.987955i	$-0.549454\pi$
$919$	−9.38197	−0.309483	−0.154741	+	0.987955i	$0.549454\pi$
$920$	−9.79837	−0.323043
$921$	−72.7214	−2.39625
$922$	−7.02129	−0.231234
$923$	4.14590	0.136464
$924$	12.4721	0.410303
$925$	43.7771	1.43938
$926$	−5.49342	−0.180525
$927$	2.43769	0.0800644
$928$	−31.5623	−1.03608
$929$	−5.65248	−0.185452	−0.0927259	−	0.995692i	$-0.529558\pi$
$929$	−5.65248	−0.185452	−0.0927259	+	0.995692i	$0.529558\pi$
$930$	−16.4721	−0.540143
$931$	6.63119	0.217328
$932$	17.0902	0.559807
$933$	−26.1803	−0.857106
$934$	25.4164	0.831650
$935$	6.09017	0.199170
$936$	6.38197	0.208601
$937$	−45.1033	−1.47346	−0.736731	−	0.676186i	$-0.763631\pi$
$937$	−45.1033	−1.47346	−0.736731	+	0.676186i	$0.763631\pi$
$938$	−11.3475	−0.370510
$939$	95.9574	3.13145
$940$	−13.7082	−0.447112
$941$	7.72949	0.251974	0.125987	−	0.992032i	$-0.459790\pi$
$941$	7.72949	0.251974	0.125987	+	0.992032i	$0.459790\pi$
$942$	−1.70820	−0.0556563
$943$	31.7082	1.03256
$944$	−16.1459	−0.525504
$945$	−34.4721	−1.12138
$946$	2.90983	0.0946067
$947$	−2.70820	−0.0880048	−0.0440024	−	0.999031i	$-0.514011\pi$
$947$	−2.70820	−0.0880048	−0.0440024	+	0.999031i	$0.514011\pi$
$948$	68.0689	2.21077
$949$	4.23607	0.137509
$950$	12.3607	0.401033
$951$	−55.4164	−1.79700
$952$	32.4377	1.05131
$953$	46.4508	1.50469	0.752345	−	0.658769i	$-0.228923\pi$
$953$	46.4508	1.50469	0.752345	+	0.658769i	$0.228923\pi$
$954$	28.7984	0.932382
$955$	18.4164	0.595941
$956$	−40.6525	−1.31480
$957$	−18.1803	−0.587687
$958$	−8.63932	−0.279124
$959$	27.6738	0.893632
$960$	−0.763932	−0.0246558
$961$	36.8328	1.18816
$962$	2.58359	0.0832984
$963$	−69.6869	−2.24563
$964$	10.1459	0.326777
$965$	−17.1803	−0.553055
$966$	−20.8754	−0.671655
$967$	30.6525	0.985717	0.492859	−	0.870109i	$-0.335952\pi$
$967$	30.6525	0.985717	0.492859	+	0.870109i	$0.335952\pi$
$968$	22.3607	0.718699
$969$	98.5410	3.16559
$970$	−8.32624	−0.267339
$971$	−35.2361	−1.13078	−0.565390	−	0.824824i	$-0.691274\pi$
$971$	−35.2361	−1.13078	−0.565390	+	0.824824i	$0.691274\pi$
$972$	57.5967	1.84742
$973$	−37.5492	−1.20377
$974$	−22.1803	−0.710704
$975$	−4.94427	−0.158343
$976$	20.7295	0.663535
$977$	5.09017	0.162849	0.0814245	−	0.996680i	$-0.474053\pi$
$977$	5.09017	0.162849	0.0814245	+	0.996680i	$0.474053\pi$
$978$	34.1803	1.09297
$979$	10.8541	0.346899
$980$	−2.14590	−0.0685482
$981$	40.8885	1.30547
$982$	−10.5066	−0.335278
$983$	21.1803	0.675548	0.337774	−	0.941227i	$-0.390326\pi$
$983$	21.1803	0.675548	0.337774	+	0.941227i	$0.390326\pi$
$984$	−52.3607	−1.66920
$985$	−1.29180	−0.0411600
$986$	−21.1459	−0.673423
$987$	−65.3050	−2.07868
$988$	−3.09017	−0.0983114
$989$	20.6312	0.656034
$990$	4.61803	0.146771
$991$	6.41641	0.203824	0.101912	−	0.994793i	$-0.467504\pi$
$991$	6.41641	0.203824	0.101912	+	0.994793i	$0.467504\pi$
$992$	−46.2705	−1.46909
$993$	80.1378	2.54310
$994$	15.9787	0.506814
$995$	24.9443	0.790787
$996$	18.6525	0.591026
$997$	12.9787	0.411040	0.205520	−	0.978653i	$-0.434111\pi$
$997$	12.9787	0.411040	0.205520	+	0.978653i	$0.434111\pi$
$998$	17.7295	0.561217
$999$	158.387	5.01114

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4027.2.a.a.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4027.2.a.a.1.2	✓	2	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:	\( N \)	\(=\)	\( 4027 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4027.a (trivial)

\(f(q)\)	\(=\)	\(q+0.618034 q^{2} -3.23607 q^{3} -1.61803 q^{4} -1.00000 q^{5} -2.00000 q^{6} -2.38197 q^{7} -2.23607 q^{8} +7.47214 q^{9} +O(q^{10})\)	\(q+0.618034 q^{2} -3.23607 q^{3} -1.61803 q^{4} -1.00000 q^{5} -2.00000 q^{6} -2.38197 q^{7} -2.23607 q^{8} +7.47214 q^{9} -0.618034 q^{10} -1.00000 q^{11} +5.23607 q^{12} -0.381966 q^{13} -1.47214 q^{14} +3.23607 q^{15} +1.85410 q^{16} +6.09017 q^{17} +4.61803 q^{18} -5.00000 q^{19} +1.61803 q^{20} +7.70820 q^{21} -0.618034 q^{22} -4.38197 q^{23} +7.23607 q^{24} -4.00000 q^{25} -0.236068 q^{26} -14.4721 q^{27} +3.85410 q^{28} -5.61803 q^{29} +2.00000 q^{30} -8.23607 q^{31} +5.61803 q^{32} +3.23607 q^{33} +3.76393 q^{34} +2.38197 q^{35} -12.0902 q^{36} -10.9443 q^{37} -3.09017 q^{38} +1.23607 q^{39} +2.23607 q^{40} -7.23607 q^{41} +4.76393 q^{42} -4.70820 q^{43} +1.61803 q^{44} -7.47214 q^{45} -2.70820 q^{46} -8.47214 q^{47} -6.00000 q^{48} -1.32624 q^{49} -2.47214 q^{50} -19.7082 q^{51} +0.618034 q^{52} +6.23607 q^{53} -8.94427 q^{54} +1.00000 q^{55} +5.32624 q^{56} +16.1803 q^{57} -3.47214 q^{58} -8.70820 q^{59} -5.23607 q^{60} +11.1803 q^{61} -5.09017 q^{62} -17.7984 q^{63} -0.236068 q^{64} +0.381966 q^{65} +2.00000 q^{66} +7.70820 q^{67} -9.85410 q^{68} +14.1803 q^{69} +1.47214 q^{70} -10.8541 q^{71} -16.7082 q^{72} -11.0902 q^{73} -6.76393 q^{74} +12.9443 q^{75} +8.09017 q^{76} +2.38197 q^{77} +0.763932 q^{78} +13.0000 q^{79} -1.85410 q^{80} +24.4164 q^{81} -4.47214 q^{82} +3.56231 q^{83} -12.4721 q^{84} -6.09017 q^{85} -2.90983 q^{86} +18.1803 q^{87} +2.23607 q^{88} -10.8541 q^{89} -4.61803 q^{90} +0.909830 q^{91} +7.09017 q^{92} +26.6525 q^{93} -5.23607 q^{94} +5.00000 q^{95} -18.1803 q^{96} +13.4721 q^{97} -0.819660 q^{98} -7.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 2 q^{3} - q^{4} - 2 q^{5} - 4 q^{6} - 7 q^{7} + 6 q^{9}+O(q^{10}) \) 2 * q - q^2 - 2 * q^3 - q^4 - 2 * q^5 - 4 * q^6 - 7 * q^7 + 6 * q^9	\( 2 q - q^{2} - 2 q^{3} - q^{4} - 2 q^{5} - 4 q^{6} - 7 q^{7} + 6 q^{9} + q^{10} - 2 q^{11} + 6 q^{12} - 3 q^{13} + 6 q^{14} + 2 q^{15} - 3 q^{16} + q^{17} + 7 q^{18} - 10 q^{19} + q^{20} + 2 q^{21} + q^{22} - 11 q^{23} + 10 q^{24} - 8 q^{25} + 4 q^{26} - 20 q^{27} + q^{28} - 9 q^{29} + 4 q^{30} - 12 q^{31} + 9 q^{32} + 2 q^{33} + 12 q^{34} + 7 q^{35} - 13 q^{36} - 4 q^{37} + 5 q^{38} - 2 q^{39} - 10 q^{41} + 14 q^{42} + 4 q^{43} + q^{44} - 6 q^{45} + 8 q^{46} - 8 q^{47} - 12 q^{48} + 13 q^{49} + 4 q^{50} - 26 q^{51} - q^{52} + 8 q^{53} + 2 q^{55} - 5 q^{56} + 10 q^{57} + 2 q^{58} - 4 q^{59} - 6 q^{60} + q^{62} - 11 q^{63} + 4 q^{64} + 3 q^{65} + 4 q^{66} + 2 q^{67} - 13 q^{68} + 6 q^{69} - 6 q^{70} - 15 q^{71} - 20 q^{72} - 11 q^{73} - 18 q^{74} + 8 q^{75} + 5 q^{76} + 7 q^{77} + 6 q^{78} + 26 q^{79} + 3 q^{80} + 22 q^{81} - 13 q^{83} - 16 q^{84} - q^{85} - 17 q^{86} + 14 q^{87} - 15 q^{89} - 7 q^{90} + 13 q^{91} + 3 q^{92} + 22 q^{93} - 6 q^{94} + 10 q^{95} - 14 q^{96} + 18 q^{97} - 24 q^{98} - 6 q^{99}+O(q^{100}) \) 2 * q - q^2 - 2 * q^3 - q^4 - 2 * q^5 - 4 * q^6 - 7 * q^7 + 6 * q^9 + q^10 - 2 * q^11 + 6 * q^12 - 3 * q^13 + 6 * q^14 + 2 * q^15 - 3 * q^16 + q^17 + 7 * q^18 - 10 * q^19 + q^20 + 2 * q^21 + q^22 - 11 * q^23 + 10 * q^24 - 8 * q^25 + 4 * q^26 - 20 * q^27 + q^28 - 9 * q^29 + 4 * q^30 - 12 * q^31 + 9 * q^32 + 2 * q^33 + 12 * q^34 + 7 * q^35 - 13 * q^36 - 4 * q^37 + 5 * q^38 - 2 * q^39 - 10 * q^41 + 14 * q^42 + 4 * q^43 + q^44 - 6 * q^45 + 8 * q^46 - 8 * q^47 - 12 * q^48 + 13 * q^49 + 4 * q^50 - 26 * q^51 - q^52 + 8 * q^53 + 2 * q^55 - 5 * q^56 + 10 * q^57 + 2 * q^58 - 4 * q^59 - 6 * q^60 + q^62 - 11 * q^63 + 4 * q^64 + 3 * q^65 + 4 * q^66 + 2 * q^67 - 13 * q^68 + 6 * q^69 - 6 * q^70 - 15 * q^71 - 20 * q^72 - 11 * q^73 - 18 * q^74 + 8 * q^75 + 5 * q^76 + 7 * q^77 + 6 * q^78 + 26 * q^79 + 3 * q^80 + 22 * q^81 - 13 * q^83 - 16 * q^84 - q^85 - 17 * q^86 + 14 * q^87 - 15 * q^89 - 7 * q^90 + 13 * q^91 + 3 * q^92 + 22 * q^93 - 6 * q^94 + 10 * q^95 - 14 * q^96 + 18 * q^97 - 24 * q^98 - 6 * q^99

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