LMFDB - Embedded newform 6038.2.a.d.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6038,2,Mod(1,6038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6038, 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("6038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6038 = 2 \cdot 3019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6038.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:	$48.2136727404$
Analytic rank:	$0$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	6038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.88111 q^{3} +1.00000 q^{4} +2.68970 q^{5} +2.88111 q^{6} +4.10918 q^{7} -1.00000 q^{8} +5.30078 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.88111 q^{3} +1.00000 q^{4} +2.68970 q^{5} +2.88111 q^{6} +4.10918 q^{7} -1.00000 q^{8} +5.30078 q^{9} -2.68970 q^{10} -5.04342 q^{11} -2.88111 q^{12} +3.90469 q^{13} -4.10918 q^{14} -7.74932 q^{15} +1.00000 q^{16} -4.33981 q^{17} -5.30078 q^{18} +1.79917 q^{19} +2.68970 q^{20} -11.8390 q^{21} +5.04342 q^{22} -3.73984 q^{23} +2.88111 q^{24} +2.23450 q^{25} -3.90469 q^{26} -6.62879 q^{27} +4.10918 q^{28} -7.72202 q^{29} +7.74932 q^{30} -7.85847 q^{31} -1.00000 q^{32} +14.5306 q^{33} +4.33981 q^{34} +11.0525 q^{35} +5.30078 q^{36} -0.274313 q^{37} -1.79917 q^{38} -11.2498 q^{39} -2.68970 q^{40} +4.62511 q^{41} +11.8390 q^{42} -12.9480 q^{43} -5.04342 q^{44} +14.2575 q^{45} +3.73984 q^{46} -6.67582 q^{47} -2.88111 q^{48} +9.88538 q^{49} -2.23450 q^{50} +12.5035 q^{51} +3.90469 q^{52} +5.51826 q^{53} +6.62879 q^{54} -13.5653 q^{55} -4.10918 q^{56} -5.18360 q^{57} +7.72202 q^{58} +11.2257 q^{59} -7.74932 q^{60} +11.1748 q^{61} +7.85847 q^{62} +21.7819 q^{63} +1.00000 q^{64} +10.5024 q^{65} -14.5306 q^{66} +0.208719 q^{67} -4.33981 q^{68} +10.7749 q^{69} -11.0525 q^{70} +0.250587 q^{71} -5.30078 q^{72} +16.4942 q^{73} +0.274313 q^{74} -6.43785 q^{75} +1.79917 q^{76} -20.7243 q^{77} +11.2498 q^{78} +13.1046 q^{79} +2.68970 q^{80} +3.19591 q^{81} -4.62511 q^{82} +8.82170 q^{83} -11.8390 q^{84} -11.6728 q^{85} +12.9480 q^{86} +22.2480 q^{87} +5.04342 q^{88} -9.48926 q^{89} -14.2575 q^{90} +16.0451 q^{91} -3.73984 q^{92} +22.6411 q^{93} +6.67582 q^{94} +4.83923 q^{95} +2.88111 q^{96} +3.40904 q^{97} -9.88538 q^{98} -26.7340 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9}+O(q^{10}) $ 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9} - 18 q^{10} - 4 q^{11} + 8 q^{12} + 40 q^{13} - 32 q^{14} + 6 q^{15} + 69 q^{16} + 6 q^{17} - 79 q^{18} + 13 q^{19} + 18 q^{20} + 23 q^{21} + 4 q^{22} + 11 q^{23} - 8 q^{24} + 111 q^{25} - 40 q^{26} + 32 q^{27} + 32 q^{28} + 23 q^{29} - 6 q^{30} + 30 q^{31} - 69 q^{32} + 37 q^{33} - 6 q^{34} - 12 q^{35} + 79 q^{36} + 81 q^{37} - 13 q^{38} - 8 q^{39} - 18 q^{40} - 13 q^{41} - 23 q^{42} + 42 q^{43} - 4 q^{44} + 89 q^{45} - 11 q^{46} + 35 q^{47} + 8 q^{48} + 115 q^{49} - 111 q^{50} - 21 q^{51} + 40 q^{52} + 41 q^{53} - 32 q^{54} + 28 q^{55} - 32 q^{56} + 39 q^{57} - 23 q^{58} - 24 q^{59} + 6 q^{60} + 69 q^{61} - 30 q^{62} + 79 q^{63} + 69 q^{64} + 15 q^{65} - 37 q^{66} + 87 q^{67} + 6 q^{68} + 51 q^{69} + 12 q^{70} - 22 q^{71} - 79 q^{72} + 104 q^{73} - 81 q^{74} + 39 q^{75} + 13 q^{76} + 47 q^{77} + 8 q^{78} + 16 q^{79} + 18 q^{80} + 105 q^{81} + 13 q^{82} + 30 q^{83} + 23 q^{84} + 74 q^{85} - 42 q^{86} + 47 q^{87} + 4 q^{88} - 4 q^{89} - 89 q^{90} + 70 q^{91} + 11 q^{92} + 104 q^{93} - 35 q^{94} - 36 q^{95} - 8 q^{96} + 158 q^{97} - 115 q^{98} + 34 q^{99}+O(q^{100}) \) 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9 - 18 * q^10 - 4 * q^11 + 8 * q^12 + 40 * q^13 - 32 * q^14 + 6 * q^15 + 69 * q^16 + 6 * q^17 - 79 * q^18 + 13 * q^19 + 18 * q^20 + 23 * q^21 + 4 * q^22 + 11 * q^23 - 8 * q^24 + 111 * q^25 - 40 * q^26 + 32 * q^27 + 32 * q^28 + 23 * q^29 - 6 * q^30 + 30 * q^31 - 69 * q^32 + 37 * q^33 - 6 * q^34 - 12 * q^35 + 79 * q^36 + 81 * q^37 - 13 * q^38 - 8 * q^39 - 18 * q^40 - 13 * q^41 - 23 * q^42 + 42 * q^43 - 4 * q^44 + 89 * q^45 - 11 * q^46 + 35 * q^47 + 8 * q^48 + 115 * q^49 - 111 * q^50 - 21 * q^51 + 40 * q^52 + 41 * q^53 - 32 * q^54 + 28 * q^55 - 32 * q^56 + 39 * q^57 - 23 * q^58 - 24 * q^59 + 6 * q^60 + 69 * q^61 - 30 * q^62 + 79 * q^63 + 69 * q^64 + 15 * q^65 - 37 * q^66 + 87 * q^67 + 6 * q^68 + 51 * q^69 + 12 * q^70 - 22 * q^71 - 79 * q^72 + 104 * q^73 - 81 * q^74 + 39 * q^75 + 13 * q^76 + 47 * q^77 + 8 * q^78 + 16 * q^79 + 18 * q^80 + 105 * q^81 + 13 * q^82 + 30 * q^83 + 23 * q^84 + 74 * q^85 - 42 * q^86 + 47 * q^87 + 4 * q^88 - 4 * q^89 - 89 * q^90 + 70 * q^91 + 11 * q^92 + 104 * q^93 - 35 * q^94 - 36 * q^95 - 8 * q^96 + 158 * q^97 - 115 * q^98 + 34 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.88111	−1.66341	−0.831704	−	0.555219i	$-0.812634\pi$
$3$	−2.88111	−1.66341	−0.831704	+	0.555219i	$0.812634\pi$
$4$	1.00000	0.500000
$5$	2.68970	1.20287	0.601436	−	0.798921i	$-0.294595\pi$
$5$	2.68970	1.20287	0.601436	+	0.798921i	$0.294595\pi$
$6$	2.88111	1.17621
$7$	4.10918	1.55312	0.776562	−	0.630040i	$-0.216962\pi$
$7$	4.10918	1.55312	0.776562	+	0.630040i	$0.216962\pi$
$8$	−1.00000	−0.353553
$9$	5.30078	1.76693
$10$	−2.68970	−0.850559
$11$	−5.04342	−1.52065	−0.760324	−	0.649544i	$-0.774960\pi$
$11$	−5.04342	−1.52065	−0.760324	+	0.649544i	$0.774960\pi$
$12$	−2.88111	−0.831704
$13$	3.90469	1.08297	0.541483	−	0.840712i	$-0.317863\pi$
$13$	3.90469	1.08297	0.541483	+	0.840712i	$0.317863\pi$
$14$	−4.10918	−1.09823
$15$	−7.74932	−2.00087
$16$	1.00000	0.250000
$17$	−4.33981	−1.05256	−0.526280	−	0.850312i	$-0.676413\pi$
$17$	−4.33981	−1.05256	−0.526280	+	0.850312i	$0.676413\pi$
$18$	−5.30078	−1.24941
$19$	1.79917	0.412758	0.206379	−	0.978472i	$-0.433832\pi$
$19$	1.79917	0.412758	0.206379	+	0.978472i	$0.433832\pi$
$20$	2.68970	0.601436
$21$	−11.8390	−2.58348
$22$	5.04342	1.07526
$23$	−3.73984	−0.779811	−0.389905	−	0.920855i	$-0.627492\pi$
$23$	−3.73984	−0.779811	−0.389905	+	0.920855i	$0.627492\pi$
$24$	2.88111	0.588104
$25$	2.23450	0.446901
$26$	−3.90469	−0.765772
$27$	−6.62879	−1.27571
$28$	4.10918	0.776562
$29$	−7.72202	−1.43394	−0.716972	−	0.697102i	$-0.754472\pi$
$29$	−7.72202	−1.43394	−0.716972	+	0.697102i	$0.754472\pi$
$30$	7.74932	1.41483
$31$	−7.85847	−1.41142	−0.705711	−	0.708500i	$-0.749372\pi$
$31$	−7.85847	−1.41142	−0.705711	+	0.708500i	$0.749372\pi$
$32$	−1.00000	−0.176777
$33$	14.5306	2.52946
$34$	4.33981	0.744272
$35$	11.0525	1.86821
$36$	5.30078	0.883463
$37$	−0.274313	−0.0450967	−0.0225484	−	0.999746i	$-0.507178\pi$
$37$	−0.274313	−0.0450967	−0.0225484	+	0.999746i	$0.507178\pi$
$38$	−1.79917	−0.291864
$39$	−11.2498	−1.80141
$40$	−2.68970	−0.425279
$41$	4.62511	0.722321	0.361160	−	0.932504i	$-0.382381\pi$
$41$	4.62511	0.722321	0.361160	+	0.932504i	$0.382381\pi$
$42$	11.8390	1.82680
$43$	−12.9480	−1.97455	−0.987273	−	0.159033i	$-0.949162\pi$
$43$	−12.9480	−1.97455	−0.987273	+	0.159033i	$0.949162\pi$
$44$	−5.04342	−0.760324
$45$	14.2575	2.12539
$46$	3.73984	0.551410
$47$	−6.67582	−0.973769	−0.486884	−	0.873466i	$-0.661867\pi$
$47$	−6.67582	−0.973769	−0.486884	+	0.873466i	$0.661867\pi$
$48$	−2.88111	−0.415852
$49$	9.88538	1.41220
$50$	−2.23450	−0.316007
$51$	12.5035	1.75084
$52$	3.90469	0.541483
$53$	5.51826	0.757991	0.378995	−	0.925399i	$-0.376270\pi$
$53$	5.51826	0.757991	0.378995	+	0.925399i	$0.376270\pi$
$54$	6.62879	0.902064
$55$	−13.5653	−1.82914
$56$	−4.10918	−0.549113
$57$	−5.18360	−0.686585
$58$	7.72202	1.01395
$59$	11.2257	1.46146	0.730729	−	0.682668i	$-0.239180\pi$
$59$	11.2257	1.46146	0.730729	+	0.682668i	$0.239180\pi$
$60$	−7.74932	−1.00043
$61$	11.1748	1.43079	0.715394	−	0.698722i	$-0.246247\pi$
$61$	11.1748	1.43079	0.715394	+	0.698722i	$0.246247\pi$
$62$	7.85847	0.998026
$63$	21.7819	2.74426
$64$	1.00000	0.125000
$65$	10.5024	1.30267
$66$	−14.5306	−1.78860
$67$	0.208719	0.0254991	0.0127496	−	0.999919i	$-0.495942\pi$
$67$	0.208719	0.0254991	0.0127496	+	0.999919i	$0.495942\pi$
$68$	−4.33981	−0.526280
$69$	10.7749	1.29714
$70$	−11.0525	−1.32102
$71$	0.250587	0.0297392	0.0148696	−	0.999889i	$-0.495267\pi$
$71$	0.250587	0.0297392	0.0148696	+	0.999889i	$0.495267\pi$
$72$	−5.30078	−0.624703
$73$	16.4942	1.93050	0.965252	−	0.261319i	$-0.0841576\pi$
$73$	16.4942	1.93050	0.965252	+	0.261319i	$0.0841576\pi$
$74$	0.274313	0.0318882
$75$	−6.43785	−0.743378
$76$	1.79917	0.206379
$77$	−20.7243	−2.36176
$78$	11.2498	1.27379
$79$	13.1046	1.47439	0.737194	−	0.675681i	$-0.236150\pi$
$79$	13.1046	1.47439	0.737194	+	0.675681i	$0.236150\pi$
$80$	2.68970	0.300718
$81$	3.19591	0.355101
$82$	−4.62511	−0.510758
$83$	8.82170	0.968307	0.484154	−	0.874983i	$-0.339128\pi$
$83$	8.82170	0.968307	0.484154	+	0.874983i	$0.339128\pi$
$84$	−11.8390	−1.29174
$85$	−11.6728	−1.26609
$86$	12.9480	1.39622
$87$	22.2480	2.38523
$88$	5.04342	0.537630
$89$	−9.48926	−1.00586	−0.502930	−	0.864327i	$-0.667744\pi$
$89$	−9.48926	−1.00586	−0.502930	+	0.864327i	$0.667744\pi$
$90$	−14.2575	−1.50287
$91$	16.0451	1.68198
$92$	−3.73984	−0.389905
$93$	22.6411	2.34777
$94$	6.67582	0.688559
$95$	4.83923	0.496495
$96$	2.88111	0.294052
$97$	3.40904	0.346136	0.173068	−	0.984910i	$-0.444632\pi$
$97$	3.40904	0.346136	0.173068	+	0.984910i	$0.444632\pi$
$98$	−9.88538	−0.998574
$99$	−26.7340	−2.68687
$100$	2.23450	0.223450
$101$	−1.23109	−0.122498	−0.0612490	−	0.998123i	$-0.519508\pi$
$101$	−1.23109	−0.122498	−0.0612490	+	0.998123i	$0.519508\pi$
$102$	−12.5035	−1.23803
$103$	5.76837	0.568375	0.284187	−	0.958769i	$-0.408276\pi$
$103$	5.76837	0.568375	0.284187	+	0.958769i	$0.408276\pi$
$104$	−3.90469	−0.382886
$105$	−31.8434	−3.10760
$106$	−5.51826	−0.535981
$107$	10.6125	1.02595	0.512976	−	0.858403i	$-0.328543\pi$
$107$	10.6125	1.02595	0.512976	+	0.858403i	$0.328543\pi$
$108$	−6.62879	−0.637855
$109$	20.6955	1.98227	0.991135	−	0.132855i	$-0.0424146\pi$
$109$	20.6955	1.98227	0.991135	+	0.132855i	$0.0424146\pi$
$110$	13.5653	1.29340
$111$	0.790324	0.0750142
$112$	4.10918	0.388281
$113$	−2.51707	−0.236786	−0.118393	−	0.992967i	$-0.537774\pi$
$113$	−2.51707	−0.236786	−0.118393	+	0.992967i	$0.537774\pi$
$114$	5.18360	0.485489
$115$	−10.0591	−0.938013
$116$	−7.72202	−0.716972
$117$	20.6979	1.91352
$118$	−11.2257	−1.03341
$119$	−17.8331	−1.63476
$120$	7.74932	0.707413
$121$	14.4361	1.31237
$122$	−11.1748	−1.01172
$123$	−13.3254	−1.20151
$124$	−7.85847	−0.705711
$125$	−7.43836	−0.665307
$126$	−21.7819	−1.94048
$127$	6.58944	0.584719	0.292359	−	0.956309i	$-0.405560\pi$
$127$	6.58944	0.584719	0.292359	+	0.956309i	$0.405560\pi$
$128$	−1.00000	−0.0883883
$129$	37.3045	3.28448
$130$	−10.5024	−0.921126
$131$	5.33901	0.466472	0.233236	−	0.972420i	$-0.425069\pi$
$131$	5.33901	0.466472	0.233236	+	0.972420i	$0.425069\pi$
$132$	14.5306	1.26473
$133$	7.39312	0.641064
$134$	−0.208719	−0.0180306
$135$	−17.8295	−1.53452
$136$	4.33981	0.372136
$137$	9.85592	0.842048	0.421024	−	0.907049i	$-0.361671\pi$
$137$	9.85592	0.842048	0.421024	+	0.907049i	$0.361671\pi$
$138$	−10.7749	−0.917219
$139$	−7.44793	−0.631725	−0.315863	−	0.948805i	$-0.602294\pi$
$139$	−7.44793	−0.631725	−0.315863	+	0.948805i	$0.602294\pi$
$140$	11.0525	0.934105
$141$	19.2338	1.61977
$142$	−0.250587	−0.0210288
$143$	−19.6930	−1.64681
$144$	5.30078	0.441731
$145$	−20.7699	−1.72485
$146$	−16.4942	−1.36507
$147$	−28.4808	−2.34906
$148$	−0.274313	−0.0225484
$149$	16.6371	1.36296	0.681481	−	0.731836i	$-0.261336\pi$
$149$	16.6371	1.36296	0.681481	+	0.731836i	$0.261336\pi$
$150$	6.43785	0.525648
$151$	10.3859	0.845193	0.422596	−	0.906318i	$-0.361119\pi$
$151$	10.3859	0.845193	0.422596	+	0.906318i	$0.361119\pi$
$152$	−1.79917	−0.145932
$153$	−23.0044	−1.85979
$154$	20.7243	1.67001
$155$	−21.1369	−1.69776
$156$	−11.2498	−0.900706
$157$	−19.0267	−1.51850	−0.759250	−	0.650799i	$-0.774434\pi$
$157$	−19.0267	−1.51850	−0.759250	+	0.650799i	$0.774434\pi$
$158$	−13.1046	−1.04255
$159$	−15.8987	−1.26085
$160$	−2.68970	−0.212640
$161$	−15.3677	−1.21114
$162$	−3.19591	−0.251094
$163$	10.9323	0.856281	0.428141	−	0.903712i	$-0.359169\pi$
$163$	10.9323	0.856281	0.428141	+	0.903712i	$0.359169\pi$
$164$	4.62511	0.361160
$165$	39.0831	3.04261
$166$	−8.82170	−0.684697
$167$	−20.7742	−1.60756	−0.803780	−	0.594927i	$-0.797181\pi$
$167$	−20.7742	−1.60756	−0.803780	+	0.594927i	$0.797181\pi$
$168$	11.8390	0.913398
$169$	2.24658	0.172814
$170$	11.6728	0.895264
$171$	9.53700	0.729312
$172$	−12.9480	−0.987273
$173$	14.5270	1.10447	0.552235	−	0.833688i	$-0.313775\pi$
$173$	14.5270	1.10447	0.552235	+	0.833688i	$0.313775\pi$
$174$	−22.2480	−1.68661
$175$	9.18199	0.694093
$176$	−5.04342	−0.380162
$177$	−32.3424	−2.43100
$178$	9.48926	0.711250
$179$	−20.1995	−1.50978	−0.754892	−	0.655849i	$-0.772311\pi$
$179$	−20.1995	−1.50978	−0.754892	+	0.655849i	$0.772311\pi$
$180$	14.2575	1.06269
$181$	−14.4387	−1.07322	−0.536611	−	0.843830i	$-0.680296\pi$
$181$	−14.4387	−1.07322	−0.536611	+	0.843830i	$0.680296\pi$
$182$	−16.0451	−1.18934
$183$	−32.1958	−2.37998
$184$	3.73984	0.275705
$185$	−0.737820	−0.0542456
$186$	−22.6411	−1.66012
$187$	21.8875	1.60057
$188$	−6.67582	−0.486884
$189$	−27.2389	−1.98134
$190$	−4.83923	−0.351075
$191$	20.7454	1.50109	0.750544	−	0.660821i	$-0.229792\pi$
$191$	20.7454	1.50109	0.750544	+	0.660821i	$0.229792\pi$
$192$	−2.88111	−0.207926
$193$	20.6964	1.48976	0.744880	−	0.667198i	$-0.232506\pi$
$193$	20.6964	1.48976	0.744880	+	0.667198i	$0.232506\pi$
$194$	−3.40904	−0.244755
$195$	−30.2587	−2.16687
$196$	9.88538	0.706098
$197$	19.7708	1.40861	0.704305	−	0.709898i	$-0.251259\pi$
$197$	19.7708	1.40861	0.704305	+	0.709898i	$0.251259\pi$
$198$	26.7340	1.89990
$199$	18.2542	1.29401	0.647004	−	0.762486i	$-0.276022\pi$
$199$	18.2542	1.29401	0.647004	+	0.762486i	$0.276022\pi$
$200$	−2.23450	−0.158003
$201$	−0.601343	−0.0424155
$202$	1.23109	0.0866192
$203$	−31.7312	−2.22709
$204$	12.5035	0.875418
$205$	12.4402	0.868859
$206$	−5.76837	−0.401901
$207$	−19.8241	−1.37787
$208$	3.90469	0.270741
$209$	−9.07396	−0.627659
$210$	31.8434	2.19740
$211$	8.32833	0.573346	0.286673	−	0.958029i	$-0.407451\pi$
$211$	8.32833	0.573346	0.286673	+	0.958029i	$0.407451\pi$
$212$	5.51826	0.378995
$213$	−0.721967	−0.0494684
$214$	−10.6125	−0.725458
$215$	−34.8262	−2.37513
$216$	6.62879	0.451032
$217$	−32.2919	−2.19211
$218$	−20.6955	−1.40168
$219$	−47.5217	−3.21122
$220$	−13.5653	−0.914572
$221$	−16.9456	−1.13989
$222$	−0.790324	−0.0530431
$223$	−8.78349	−0.588186	−0.294093	−	0.955777i	$-0.595018\pi$
$223$	−8.78349	−0.588186	−0.294093	+	0.955777i	$0.595018\pi$
$224$	−4.10918	−0.274556
$225$	11.8446	0.789641
$226$	2.51707	0.167433
$227$	−19.1395	−1.27033	−0.635167	−	0.772375i	$-0.719069\pi$
$227$	−19.1395	−1.27033	−0.635167	+	0.772375i	$0.719069\pi$
$228$	−5.18360	−0.343292
$229$	0.951510	0.0628776	0.0314388	−	0.999506i	$-0.489991\pi$
$229$	0.951510	0.0628776	0.0314388	+	0.999506i	$0.489991\pi$
$230$	10.0591	0.663275
$231$	59.7090	3.92856
$232$	7.72202	0.506975
$233$	21.4003	1.40198	0.700991	−	0.713170i	$-0.252741\pi$
$233$	21.4003	1.40198	0.700991	+	0.713170i	$0.252741\pi$
$234$	−20.6979	−1.35306
$235$	−17.9560	−1.17132
$236$	11.2257	0.730729
$237$	−37.7559	−2.45251
$238$	17.8331	1.15595
$239$	20.2624	1.31066	0.655332	−	0.755341i	$-0.272529\pi$
$239$	20.2624	1.31066	0.655332	+	0.755341i	$0.272529\pi$
$240$	−7.74932	−0.500217
$241$	8.23835	0.530679	0.265339	−	0.964155i	$-0.414516\pi$
$241$	8.23835	0.530679	0.265339	+	0.964155i	$0.414516\pi$
$242$	−14.4361	−0.927985
$243$	10.6786	0.685032
$244$	11.1748	0.715394
$245$	26.5887	1.69869
$246$	13.3254	0.849598
$247$	7.02519	0.447002
$248$	7.85847	0.499013
$249$	−25.4163	−1.61069
$250$	7.43836	0.470443
$251$	28.1402	1.77619	0.888097	−	0.459657i	$-0.152028\pi$
$251$	28.1402	1.77619	0.888097	+	0.459657i	$0.152028\pi$
$252$	21.7819	1.37213
$253$	18.8616	1.18582
$254$	−6.58944	−0.413458
$255$	33.6306	2.10603
$256$	1.00000	0.0625000
$257$	−14.3300	−0.893879	−0.446939	−	0.894564i	$-0.647486\pi$
$257$	−14.3300	−0.893879	−0.446939	+	0.894564i	$0.647486\pi$
$258$	−37.3045	−2.32248
$259$	−1.12720	−0.0700408
$260$	10.5024	0.651334
$261$	−40.9327	−2.53367
$262$	−5.33901	−0.329845
$263$	2.39980	0.147978	0.0739889	−	0.997259i	$-0.476427\pi$
$263$	2.39980	0.147978	0.0739889	+	0.997259i	$0.476427\pi$
$264$	−14.5306	−0.894298
$265$	14.8425	0.911766
$266$	−7.39312	−0.453301
$267$	27.3396	1.67315
$268$	0.208719	0.0127496
$269$	−2.39283	−0.145893	−0.0729466	−	0.997336i	$-0.523240\pi$
$269$	−2.39283	−0.145893	−0.0729466	+	0.997336i	$0.523240\pi$
$270$	17.8295	1.08507
$271$	2.08975	0.126943	0.0634715	−	0.997984i	$-0.479783\pi$
$271$	2.08975	0.126943	0.0634715	+	0.997984i	$0.479783\pi$
$272$	−4.33981	−0.263140
$273$	−46.2276	−2.79782
$274$	−9.85592	−0.595418
$275$	−11.2695	−0.679579
$276$	10.7749	0.648572
$277$	−1.14421	−0.0687491	−0.0343745	−	0.999409i	$-0.510944\pi$
$277$	−1.14421	−0.0687491	−0.0343745	+	0.999409i	$0.510944\pi$
$278$	7.44793	0.446697
$279$	−41.6560	−2.49388
$280$	−11.0525	−0.660512
$281$	−23.6244	−1.40931	−0.704656	−	0.709549i	$-0.748899\pi$
$281$	−23.6244	−1.40931	−0.704656	+	0.709549i	$0.748899\pi$
$282$	−19.2338	−1.14535
$283$	−4.76859	−0.283463	−0.141732	−	0.989905i	$-0.545267\pi$
$283$	−4.76859	−0.283463	−0.141732	+	0.989905i	$0.545267\pi$
$284$	0.250587	0.0148696
$285$	−13.9423	−0.825873
$286$	19.6930	1.16447
$287$	19.0054	1.12185
$288$	−5.30078	−0.312351
$289$	1.83398	0.107881
$290$	20.7699	1.21965
$291$	−9.82182	−0.575765
$292$	16.4942	0.965252
$293$	25.3972	1.48372	0.741861	−	0.670554i	$-0.233944\pi$
$293$	25.3972	1.48372	0.741861	+	0.670554i	$0.233944\pi$
$294$	28.4808	1.66104
$295$	30.1937	1.75795
$296$	0.274313	0.0159441
$297$	33.4317	1.93991
$298$	−16.6371	−0.963760
$299$	−14.6029	−0.844508
$300$	−6.43785	−0.371689
$301$	−53.2056	−3.06672
$302$	−10.3859	−0.597641
$303$	3.54690	0.203764
$304$	1.79917	0.103189
$305$	30.0569	1.72105
$306$	23.0044	1.31507
$307$	34.1402	1.94848	0.974242	−	0.225505i	$-0.0724031\pi$
$307$	34.1402	1.94848	0.974242	+	0.225505i	$0.0724031\pi$
$308$	−20.7243	−1.18088
$309$	−16.6193	−0.945439
$310$	21.1369	1.20050
$311$	0.301451	0.0170937	0.00854687	−	0.999963i	$-0.497279\pi$
$311$	0.301451	0.0170937	0.00854687	+	0.999963i	$0.497279\pi$
$312$	11.2498	0.636896
$313$	−20.9581	−1.18462	−0.592311	−	0.805709i	$-0.701784\pi$
$313$	−20.9581	−1.18462	−0.592311	+	0.805709i	$0.701784\pi$
$314$	19.0267	1.07374
$315$	58.5867	3.30099
$316$	13.1046	0.737194
$317$	11.6252	0.652939	0.326469	−	0.945208i	$-0.394141\pi$
$317$	11.6252	0.652939	0.326469	+	0.945208i	$0.394141\pi$
$318$	15.8987	0.891554
$319$	38.9454	2.18052
$320$	2.68970	0.150359
$321$	−30.5759	−1.70658
$322$	15.3677	0.856408
$323$	−7.80806	−0.434452
$324$	3.19591	0.177551
$325$	8.72504	0.483978
$326$	−10.9323	−0.605482
$327$	−59.6260	−3.29733
$328$	−4.62511	−0.255379
$329$	−27.4322	−1.51238
$330$	−39.0831	−2.15145
$331$	1.82900	0.100531	0.0502654	−	0.998736i	$-0.483993\pi$
$331$	1.82900	0.100531	0.0502654	+	0.998736i	$0.483993\pi$
$332$	8.82170	0.484154
$333$	−1.45407	−0.0796826
$334$	20.7742	1.13672
$335$	0.561393	0.0306722
$336$	−11.8390	−0.645870
$337$	23.9701	1.30573	0.652867	−	0.757472i	$-0.273566\pi$
$337$	23.9701	1.30573	0.652867	+	0.757472i	$0.273566\pi$
$338$	−2.24658	−0.122198
$339$	7.25196	0.393872
$340$	−11.6728	−0.633047
$341$	39.6335	2.14628
$342$	−9.53700	−0.515702
$343$	11.8565	0.640193
$344$	12.9480	0.698108
$345$	28.9812	1.56030
$346$	−14.5270	−0.780978
$347$	25.5354	1.37081	0.685407	−	0.728160i	$-0.259624\pi$
$347$	25.5354	1.37081	0.685407	+	0.728160i	$0.259624\pi$
$348$	22.2480	1.19262
$349$	23.3790	1.25145	0.625726	−	0.780043i	$-0.284803\pi$
$349$	23.3790	1.25145	0.625726	+	0.780043i	$0.284803\pi$
$350$	−9.18199	−0.490798
$351$	−25.8833	−1.38155
$352$	5.04342	0.268815
$353$	1.61447	0.0859297	0.0429649	−	0.999077i	$-0.486320\pi$
$353$	1.61447	0.0859297	0.0429649	+	0.999077i	$0.486320\pi$
$354$	32.3424	1.71898
$355$	0.674004	0.0357724
$356$	−9.48926	−0.502930
$357$	51.3790	2.71927
$358$	20.1995	1.06758
$359$	−20.8967	−1.10289	−0.551443	−	0.834213i	$-0.685922\pi$
$359$	−20.8967	−1.10289	−0.551443	+	0.834213i	$0.685922\pi$
$360$	−14.2575	−0.751437
$361$	−15.7630	−0.829631
$362$	14.4387	0.758882
$363$	−41.5918	−2.18300
$364$	16.0451	0.840990
$365$	44.3646	2.32215
$366$	32.1958	1.68290
$367$	19.4217	1.01381	0.506903	−	0.862003i	$-0.330790\pi$
$367$	19.4217	1.01381	0.506903	+	0.862003i	$0.330790\pi$
$368$	−3.73984	−0.194953
$369$	24.5167	1.27629
$370$	0.737820	0.0383574
$371$	22.6755	1.17725
$372$	22.6411	1.17389
$373$	8.22448	0.425847	0.212924	−	0.977069i	$-0.431701\pi$
$373$	8.22448	0.425847	0.212924	+	0.977069i	$0.431701\pi$
$374$	−21.8875	−1.13178
$375$	21.4307	1.10668
$376$	6.67582	0.344279
$377$	−30.1521	−1.55291
$378$	27.2389	1.40102
$379$	−29.3085	−1.50548	−0.752739	−	0.658319i	$-0.771268\pi$
$379$	−29.3085	−1.50548	−0.752739	+	0.658319i	$0.771268\pi$
$380$	4.83923	0.248247
$381$	−18.9849	−0.972625
$382$	−20.7454	−1.06143
$383$	−15.3420	−0.783941	−0.391971	−	0.919978i	$-0.628206\pi$
$383$	−15.3420	−0.783941	−0.391971	+	0.919978i	$0.628206\pi$
$384$	2.88111	0.147026
$385$	−55.7423	−2.84089
$386$	−20.6964	−1.05342
$387$	−68.6343	−3.48888
$388$	3.40904	0.173068
$389$	12.2686	0.622045	0.311023	−	0.950403i	$-0.399329\pi$
$389$	12.2686	0.622045	0.311023	+	0.950403i	$0.399329\pi$
$390$	30.2587	1.53221
$391$	16.2302	0.820797
$392$	−9.88538	−0.499287
$393$	−15.3823	−0.775933
$394$	−19.7708	−0.996037
$395$	35.2476	1.77350
$396$	−26.7340	−1.34344
$397$	29.3681	1.47395	0.736973	−	0.675923i	$-0.236255\pi$
$397$	29.3681	1.47395	0.736973	+	0.675923i	$0.236255\pi$
$398$	−18.2542	−0.915002
$399$	−21.3004	−1.06635
$400$	2.23450	0.111725
$401$	9.78948	0.488863	0.244432	−	0.969667i	$-0.421399\pi$
$401$	9.78948	0.488863	0.244432	+	0.969667i	$0.421399\pi$
$402$	0.601343	0.0299923
$403$	−30.6848	−1.52852
$404$	−1.23109	−0.0612490
$405$	8.59605	0.427141
$406$	31.7312	1.57479
$407$	1.38347	0.0685762
$408$	−12.5035	−0.619014
$409$	−11.5338	−0.570309	−0.285154	−	0.958482i	$-0.592045\pi$
$409$	−11.5338	−0.570309	−0.285154	+	0.958482i	$0.592045\pi$
$410$	−12.4402	−0.614376
$411$	−28.3960	−1.40067
$412$	5.76837	0.284187
$413$	46.1283	2.26983
$414$	19.8241	0.974300
$415$	23.7278	1.16475
$416$	−3.90469	−0.191443
$417$	21.4583	1.05082
$418$	9.07396	0.443822
$419$	−21.7515	−1.06263	−0.531316	−	0.847174i	$-0.678302\pi$
$419$	−21.7515	−1.06263	−0.531316	+	0.847174i	$0.678302\pi$
$420$	−31.8434	−1.55380
$421$	15.8528	0.772618	0.386309	−	0.922369i	$-0.373750\pi$
$421$	15.8528	0.772618	0.386309	+	0.922369i	$0.373750\pi$
$422$	−8.32833	−0.405417
$423$	−35.3871	−1.72058
$424$	−5.51826	−0.267990
$425$	−9.69733	−0.470390
$426$	0.721967	0.0349794
$427$	45.9193	2.22219
$428$	10.6125	0.512976
$429$	56.7375	2.73931
$430$	34.8262	1.67947
$431$	−30.3080	−1.45989	−0.729944	−	0.683507i	$-0.760454\pi$
$431$	−30.3080	−1.45989	−0.729944	+	0.683507i	$0.760454\pi$
$432$	−6.62879	−0.318928
$433$	−7.11826	−0.342082	−0.171041	−	0.985264i	$-0.554713\pi$
$433$	−7.11826	−0.342082	−0.171041	+	0.985264i	$0.554713\pi$
$434$	32.2919	1.55006
$435$	59.8404	2.86913
$436$	20.6955	0.991135
$437$	−6.72861	−0.321873
$438$	47.5217	2.27067
$439$	−30.5801	−1.45951	−0.729754	−	0.683709i	$-0.760366\pi$
$439$	−30.5801	−1.45951	−0.729754	+	0.683709i	$0.760366\pi$
$440$	13.5653	0.646700
$441$	52.4002	2.49525
$442$	16.9456	0.806020
$443$	22.6452	1.07591	0.537954	−	0.842974i	$-0.319198\pi$
$443$	22.6452	1.07591	0.537954	+	0.842974i	$0.319198\pi$
$444$	0.790324	0.0375071
$445$	−25.5233	−1.20992
$446$	8.78349	0.415910
$447$	−47.9332	−2.26716
$448$	4.10918	0.194141
$449$	−31.7113	−1.49655	−0.748275	−	0.663389i	$-0.769118\pi$
$449$	−31.7113	−1.49655	−0.748275	+	0.663389i	$0.769118\pi$
$450$	−11.8446	−0.558360
$451$	−23.3263	−1.09839
$452$	−2.51707	−0.118393
$453$	−29.9229	−1.40590
$454$	19.1395	0.898262
$455$	43.1565	2.02321
$456$	5.18360	0.242744
$457$	22.6067	1.05750	0.528749	−	0.848779i	$-0.322661\pi$
$457$	22.6067	1.05750	0.528749	+	0.848779i	$0.322661\pi$
$458$	−0.951510	−0.0444611
$459$	28.7677	1.34276
$460$	−10.0591	−0.469006
$461$	6.22957	0.290140	0.145070	−	0.989421i	$-0.453659\pi$
$461$	6.22957	0.290140	0.145070	+	0.989421i	$0.453659\pi$
$462$	−59.7090	−2.77791
$463$	2.88429	0.134044	0.0670221	−	0.997751i	$-0.478650\pi$
$463$	2.88429	0.134044	0.0670221	+	0.997751i	$0.478650\pi$
$464$	−7.72202	−0.358486
$465$	60.8978	2.82407
$466$	−21.4003	−0.991351
$467$	−32.9579	−1.52511	−0.762555	−	0.646924i	$-0.776055\pi$
$467$	−32.9579	−1.52511	−0.762555	+	0.646924i	$0.776055\pi$
$468$	20.6979	0.956760
$469$	0.857666	0.0396033
$470$	17.9560	0.828248
$471$	54.8181	2.52588
$472$	−11.2257	−0.516703
$473$	65.3020	3.00259
$474$	37.7559	1.73419
$475$	4.02025	0.184462
$476$	−17.8331	−0.817378
$477$	29.2511	1.33931
$478$	−20.2624	−0.926780
$479$	22.5413	1.02994	0.514970	−	0.857208i	$-0.327803\pi$
$479$	22.5413	1.02994	0.514970	+	0.857208i	$0.327803\pi$
$480$	7.74932	0.353707
$481$	−1.07110	−0.0488382
$482$	−8.23835	−0.375247
$483$	44.2760	2.01463
$484$	14.4361	0.656184
$485$	9.16931	0.416357
$486$	−10.6786	−0.484391
$487$	29.1421	1.32056	0.660278	−	0.751021i	$-0.270438\pi$
$487$	29.1421	1.32056	0.660278	+	0.751021i	$0.270438\pi$
$488$	−11.1748	−0.505860
$489$	−31.4970	−1.42435
$490$	−26.5887	−1.20116
$491$	−12.0970	−0.545931	−0.272965	−	0.962024i	$-0.588004\pi$
$491$	−12.0970	−0.545931	−0.272965	+	0.962024i	$0.588004\pi$
$492$	−13.3254	−0.600757
$493$	33.5121	1.50931
$494$	−7.02519	−0.316078
$495$	−71.9066	−3.23196
$496$	−7.85847	−0.352856
$497$	1.02971	0.0461886
$498$	25.4163	1.13893
$499$	−38.8508	−1.73920	−0.869601	−	0.493755i	$-0.835624\pi$
$499$	−38.8508	−1.73920	−0.869601	+	0.493755i	$0.835624\pi$
$500$	−7.43836	−0.332654
$501$	59.8528	2.67403
$502$	−28.1402	−1.25596
$503$	26.8996	1.19940	0.599698	−	0.800227i	$-0.295288\pi$
$503$	26.8996	1.19940	0.599698	+	0.800227i	$0.295288\pi$
$504$	−21.7819	−0.970241
$505$	−3.31127	−0.147349
$506$	−18.8616	−0.838500
$507$	−6.47263	−0.287460
$508$	6.58944	0.292359
$509$	5.15193	0.228355	0.114178	−	0.993460i	$-0.463577\pi$
$509$	5.15193	0.228355	0.114178	+	0.993460i	$0.463577\pi$
$510$	−33.6306	−1.48919
$511$	67.7778	2.99831
$512$	−1.00000	−0.0441942
$513$	−11.9263	−0.526560
$514$	14.3300	0.632068
$515$	15.5152	0.683682
$516$	37.3045	1.64224
$517$	33.6690	1.48076
$518$	1.12720	0.0495263
$519$	−41.8540	−1.83718
$520$	−10.5024	−0.460563
$521$	28.7445	1.25932	0.629659	−	0.776872i	$-0.283195\pi$
$521$	28.7445	1.25932	0.629659	+	0.776872i	$0.283195\pi$
$522$	40.9327	1.79158
$523$	−16.0192	−0.700469	−0.350234	−	0.936662i	$-0.613898\pi$
$523$	−16.0192	−0.700469	−0.350234	+	0.936662i	$0.613898\pi$
$524$	5.33901	0.233236
$525$	−26.4543	−1.15456
$526$	−2.39980	−0.104636
$527$	34.1043	1.48561
$528$	14.5306	0.632364
$529$	−9.01358	−0.391895
$530$	−14.8425	−0.644716
$531$	59.5048	2.58229
$532$	7.39312	0.320532
$533$	18.0596	0.782248
$534$	−27.3396	−1.18310
$535$	28.5446	1.23409
$536$	−0.208719	−0.00901531
$537$	58.1970	2.51139
$538$	2.39283	0.103162
$539$	−49.8561	−2.14745
$540$	−17.8295	−0.767258
$541$	−37.6038	−1.61671	−0.808357	−	0.588692i	$-0.799643\pi$
$541$	−37.6038	−1.61671	−0.808357	+	0.588692i	$0.799643\pi$
$542$	−2.08975	−0.0897623
$543$	41.5995	1.78521
$544$	4.33981	0.186068
$545$	55.6648	2.38442
$546$	46.2276	1.97836
$547$	24.9123	1.06517	0.532586	−	0.846376i	$-0.321220\pi$
$547$	24.9123	1.06517	0.532586	+	0.846376i	$0.321220\pi$
$548$	9.85592	0.421024
$549$	59.2352	2.52810
$550$	11.2695	0.480535
$551$	−13.8932	−0.591871
$552$	−10.7749	−0.458610
$553$	53.8494	2.28991
$554$	1.14421	0.0486129
$555$	2.12574	0.0902325
$556$	−7.44793	−0.315863
$557$	40.6572	1.72270	0.861350	−	0.508012i	$-0.169619\pi$
$557$	40.6572	1.72270	0.861350	+	0.508012i	$0.169619\pi$
$558$	41.6560	1.76344
$559$	−50.5578	−2.13837
$560$	11.0525	0.467053
$561$	−63.0602	−2.66240
$562$	23.6244	0.996534
$563$	15.0162	0.632856	0.316428	−	0.948617i	$-0.397516\pi$
$563$	15.0162	0.632856	0.316428	+	0.948617i	$0.397516\pi$
$564$	19.2338	0.809887
$565$	−6.77018	−0.284824
$566$	4.76859	0.200439
$567$	13.1326	0.551517
$568$	−0.250587	−0.0105144
$569$	16.2553	0.681458	0.340729	−	0.940162i	$-0.389326\pi$
$569$	16.2553	0.681458	0.340729	+	0.940162i	$0.389326\pi$
$570$	13.9423	0.583981
$571$	21.7806	0.911488	0.455744	−	0.890111i	$-0.349373\pi$
$571$	21.7806	0.911488	0.455744	+	0.890111i	$0.349373\pi$
$572$	−19.6930	−0.823404
$573$	−59.7698	−2.49692
$574$	−19.0054	−0.793271
$575$	−8.35669	−0.348498
$576$	5.30078	0.220866
$577$	−6.94457	−0.289106	−0.144553	−	0.989497i	$-0.546174\pi$
$577$	−6.94457	−0.289106	−0.144553	+	0.989497i	$0.546174\pi$
$578$	−1.83398	−0.0762835
$579$	−59.6286	−2.47808
$580$	−20.7699	−0.862425
$581$	36.2500	1.50390
$582$	9.82182	0.407127
$583$	−27.8309	−1.15264
$584$	−16.4942	−0.682537
$585$	55.6711	2.30172
$586$	−25.3972	−1.04915
$587$	7.11890	0.293829	0.146914	−	0.989149i	$-0.453066\pi$
$587$	7.11890	0.293829	0.146914	+	0.989149i	$0.453066\pi$
$588$	−28.4808	−1.17453
$589$	−14.1387	−0.582576
$590$	−30.1937	−1.24306
$591$	−56.9617	−2.34309
$592$	−0.274313	−0.0112742
$593$	−19.0603	−0.782714	−0.391357	−	0.920239i	$-0.627994\pi$
$593$	−19.0603	−0.782714	−0.391357	+	0.920239i	$0.627994\pi$
$594$	−33.4317	−1.37172
$595$	−47.9657	−1.96640
$596$	16.6371	0.681481
$597$	−52.5924	−2.15246
$598$	14.6029	0.597157
$599$	−23.7935	−0.972175	−0.486087	−	0.873910i	$-0.661576\pi$
$599$	−23.7935	−0.972175	−0.486087	+	0.873910i	$0.661576\pi$
$600$	6.43785	0.262824
$601$	−17.7908	−0.725701	−0.362850	−	0.931847i	$-0.618196\pi$
$601$	−17.7908	−0.725701	−0.362850	+	0.931847i	$0.618196\pi$
$602$	53.2056	2.16850
$603$	1.10638	0.0450551
$604$	10.3859	0.422596
$605$	38.8287	1.57861
$606$	−3.54690	−0.144083
$607$	−42.5191	−1.72580	−0.862899	−	0.505377i	$-0.831354\pi$
$607$	−42.5191	−1.72580	−0.862899	+	0.505377i	$0.831354\pi$
$608$	−1.79917	−0.0729660
$609$	91.4209	3.70456
$610$	−30.0569	−1.21697
$611$	−26.0670	−1.05456
$612$	−23.0044	−0.929897
$613$	13.9374	0.562926	0.281463	−	0.959572i	$-0.409180\pi$
$613$	13.9374	0.562926	0.281463	+	0.959572i	$0.409180\pi$
$614$	−34.1402	−1.37779
$615$	−35.8415	−1.44527
$616$	20.7243	0.835007
$617$	35.0312	1.41030	0.705152	−	0.709056i	$-0.250879\pi$
$617$	35.0312	1.41030	0.705152	+	0.709056i	$0.250879\pi$
$618$	16.6193	0.668526
$619$	−10.6595	−0.428440	−0.214220	−	0.976785i	$-0.568721\pi$
$619$	−10.6595	−0.428440	−0.214220	+	0.976785i	$0.568721\pi$
$620$	−21.1369	−0.848880
$621$	24.7906	0.994813
$622$	−0.301451	−0.0120871
$623$	−38.9931	−1.56222
$624$	−11.2498	−0.450353
$625$	−31.1795	−1.24718
$626$	20.9581	0.837655
$627$	26.1431	1.04405
$628$	−19.0267	−0.759250
$629$	1.19047	0.0474670
$630$	−58.5867	−2.33415
$631$	−34.6237	−1.37835	−0.689174	−	0.724596i	$-0.742026\pi$
$631$	−34.6237	−1.37835	−0.689174	+	0.724596i	$0.742026\pi$
$632$	−13.1046	−0.521275
$633$	−23.9948	−0.953708
$634$	−11.6252	−0.461697
$635$	17.7236	0.703342
$636$	−15.8987	−0.630424
$637$	38.5993	1.52936
$638$	−38.9454	−1.54186
$639$	1.32830	0.0525469
$640$	−2.68970	−0.106320
$641$	−0.964740	−0.0381049	−0.0190525	−	0.999818i	$-0.506065\pi$
$641$	−0.964740	−0.0381049	−0.0190525	+	0.999818i	$0.506065\pi$
$642$	30.5759	1.20673
$643$	−26.8572	−1.05914	−0.529572	−	0.848265i	$-0.677648\pi$
$643$	−26.8572	−1.05914	−0.529572	+	0.848265i	$0.677648\pi$
$644$	−15.3677	−0.605572
$645$	100.338	3.95080
$646$	7.80806	0.307204
$647$	1.04268	0.0409918	0.0204959	−	0.999790i	$-0.493475\pi$
$647$	1.04268	0.0409918	0.0204959	+	0.999790i	$0.493475\pi$
$648$	−3.19591	−0.125547
$649$	−56.6157	−2.22236
$650$	−8.72504	−0.342224
$651$	93.0363	3.64638
$652$	10.9323	0.428141
$653$	−15.0694	−0.589712	−0.294856	−	0.955542i	$-0.595272\pi$
$653$	−15.0694	−0.589712	−0.294856	+	0.955542i	$0.595272\pi$
$654$	59.6260	2.33156
$655$	14.3604	0.561106
$656$	4.62511	0.180580
$657$	87.4323	3.41106
$658$	27.4322	1.06942
$659$	22.3615	0.871081	0.435541	−	0.900169i	$-0.356557\pi$
$659$	22.3615	0.871081	0.435541	+	0.900169i	$0.356557\pi$
$660$	39.0831	1.52131
$661$	−12.8123	−0.498342	−0.249171	−	0.968459i	$-0.580158\pi$
$661$	−12.8123	−0.498342	−0.249171	+	0.968459i	$0.580158\pi$
$662$	−1.82900	−0.0710861
$663$	48.8221	1.89609
$664$	−8.82170	−0.342348
$665$	19.8853	0.771118
$666$	1.45407	0.0563441
$667$	28.8791	1.11820
$668$	−20.7742	−0.803780
$669$	25.3062	0.978393
$670$	−0.561393	−0.0216885
$671$	−56.3592	−2.17572
$672$	11.8390	0.456699
$673$	−41.1640	−1.58676	−0.793378	−	0.608729i	$-0.791680\pi$
$673$	−41.1640	−1.58676	−0.793378	+	0.608729i	$0.791680\pi$
$674$	−23.9701	−0.923294
$675$	−14.8121	−0.570116
$676$	2.24658	0.0864068
$677$	−20.9677	−0.805856	−0.402928	−	0.915232i	$-0.632008\pi$
$677$	−20.9677	−0.805856	−0.402928	+	0.915232i	$0.632008\pi$
$678$	−7.25196	−0.278510
$679$	14.0084	0.537592
$680$	11.6728	0.447632
$681$	55.1430	2.11308
$682$	−39.6335	−1.51765
$683$	16.7110	0.639428	0.319714	−	0.947514i	$-0.396413\pi$
$683$	16.7110	0.639428	0.319714	+	0.947514i	$0.396413\pi$
$684$	9.53700	0.364656
$685$	26.5095	1.01288
$686$	−11.8565	−0.452685
$687$	−2.74140	−0.104591
$688$	−12.9480	−0.493637
$689$	21.5471	0.820878
$690$	−28.9812	−1.10330
$691$	37.2339	1.41644	0.708221	−	0.705990i	$-0.249498\pi$
$691$	37.2339	1.41644	0.708221	+	0.705990i	$0.249498\pi$
$692$	14.5270	0.552235
$693$	−109.855	−4.17305
$694$	−25.5354	−0.969312
$695$	−20.0327	−0.759885
$696$	−22.2480	−0.843307
$697$	−20.0721	−0.760285
$698$	−23.3790	−0.884909
$699$	−61.6566	−2.33207
$700$	9.18199	0.347046
$701$	−4.34689	−0.164180	−0.0820899	−	0.996625i	$-0.526159\pi$
$701$	−4.34689	−0.164180	−0.0820899	+	0.996625i	$0.526159\pi$
$702$	25.8833	0.976904
$703$	−0.493535	−0.0186140
$704$	−5.04342	−0.190081
$705$	51.7331	1.94838
$706$	−1.61447	−0.0607615
$707$	−5.05877	−0.190255
$708$	−32.3424	−1.21550
$709$	−30.1519	−1.13238	−0.566190	−	0.824275i	$-0.691583\pi$
$709$	−30.1519	−1.13238	−0.566190	+	0.824275i	$0.691583\pi$
$710$	−0.674004	−0.0252949
$711$	69.4648	2.60513
$712$	9.48926	0.355625
$713$	29.3894	1.10064
$714$	−51.3790	−1.92281
$715$	−52.9682	−1.98090
$716$	−20.1995	−0.754892
$717$	−58.3781	−2.18017
$718$	20.8967	0.779858
$719$	−11.6630	−0.434958	−0.217479	−	0.976065i	$-0.569783\pi$
$719$	−11.6630	−0.434958	−0.217479	+	0.976065i	$0.569783\pi$
$720$	14.2575	0.531346
$721$	23.7033	0.882757
$722$	15.7630	0.586638
$723$	−23.7356	−0.882736
$724$	−14.4387	−0.536611
$725$	−17.2549	−0.640830
$726$	41.5918	1.54362
$727$	14.4208	0.534838	0.267419	−	0.963580i	$-0.413829\pi$
$727$	14.4208	0.534838	0.267419	+	0.963580i	$0.413829\pi$
$728$	−16.0451	−0.594670
$729$	−40.3539	−1.49459
$730$	−44.3646	−1.64201
$731$	56.1918	2.07833
$732$	−32.1958	−1.18999
$733$	3.78736	0.139889	0.0699446	−	0.997551i	$-0.477718\pi$
$733$	3.78736	0.139889	0.0699446	+	0.997551i	$0.477718\pi$
$734$	−19.4217	−0.716869
$735$	−76.6050	−2.82562
$736$	3.73984	0.137852
$737$	−1.05266	−0.0387752
$738$	−24.5167	−0.902471
$739$	−17.5684	−0.646263	−0.323131	−	0.946354i	$-0.604736\pi$
$739$	−17.5684	−0.646263	−0.323131	+	0.946354i	$0.604736\pi$
$740$	−0.737820	−0.0271228
$741$	−20.2403	−0.743547
$742$	−22.6755	−0.832445
$743$	9.15941	0.336026	0.168013	−	0.985785i	$-0.446265\pi$
$743$	9.15941	0.336026	0.168013	+	0.985785i	$0.446265\pi$
$744$	−22.6411	−0.830062
$745$	44.7488	1.63947
$746$	−8.22448	−0.301120
$747$	46.7619	1.71093
$748$	21.8875	0.800286
$749$	43.6089	1.59343
$750$	−21.4307	−0.782539
$751$	−9.94125	−0.362761	−0.181381	−	0.983413i	$-0.558057\pi$
$751$	−9.94125	−0.362761	−0.181381	+	0.983413i	$0.558057\pi$
$752$	−6.67582	−0.243442
$753$	−81.0749	−2.95453
$754$	30.1521	1.09807
$755$	27.9350	1.01666
$756$	−27.2389	−0.990669
$757$	−10.0616	−0.365697	−0.182848	−	0.983141i	$-0.558532\pi$
$757$	−10.0616	−0.365697	−0.182848	+	0.983141i	$0.558532\pi$
$758$	29.3085	1.06453
$759$	−54.3422	−1.97250
$760$	−4.83923	−0.175537
$761$	45.6676	1.65545	0.827724	−	0.561135i	$-0.189635\pi$
$761$	45.6676	1.65545	0.827724	+	0.561135i	$0.189635\pi$
$762$	18.9849	0.687750
$763$	85.0416	3.07871
$764$	20.7454	0.750544
$765$	−61.8750	−2.23709
$766$	15.3420	0.554330
$767$	43.8327	1.58271
$768$	−2.88111	−0.103963
$769$	−18.1217	−0.653485	−0.326743	−	0.945113i	$-0.605951\pi$
$769$	−18.1217	−0.653485	−0.326743	+	0.945113i	$0.605951\pi$
$770$	55.7423	2.00881
$771$	41.2862	1.48688
$772$	20.6964	0.744880
$773$	−29.9694	−1.07793	−0.538963	−	0.842330i	$-0.681184\pi$
$773$	−29.9694	−1.07793	−0.538963	+	0.842330i	$0.681184\pi$
$774$	68.6343	2.46701
$775$	−17.5598	−0.630766
$776$	−3.40904	−0.122377
$777$	3.24759	0.116506
$778$	−12.2686	−0.439852
$779$	8.32135	0.298143
$780$	−30.2587	−1.08343
$781$	−1.26381	−0.0452228
$782$	−16.2302	−0.580391
$783$	51.1876	1.82930
$784$	9.88538	0.353049
$785$	−51.1763	−1.82656
$786$	15.3823	0.548667
$787$	−39.4427	−1.40598	−0.702990	−	0.711199i	$-0.748152\pi$
$787$	−39.4427	−1.40598	−0.702990	+	0.711199i	$0.748152\pi$
$788$	19.7708	0.704305
$789$	−6.91407	−0.246147
$790$	−35.2476	−1.25405
$791$	−10.3431	−0.367759
$792$	26.7340	0.949952
$793$	43.6341	1.54949
$794$	−29.3681	−1.04224
$795$	−42.7628	−1.51664
$796$	18.2542	0.647004
$797$	−20.5624	−0.728357	−0.364178	−	0.931329i	$-0.618650\pi$
$797$	−20.5624	−0.728357	−0.364178	+	0.931329i	$0.618650\pi$
$798$	21.3004	0.754024
$799$	28.9718	1.02495
$800$	−2.23450	−0.0790017
$801$	−50.3004	−1.77728
$802$	−9.78948	−0.345678
$803$	−83.1873	−2.93562
$804$	−0.601343	−0.0212077
$805$	−41.3345	−1.45685
$806$	30.6848	1.08083
$807$	6.89399	0.242680
$808$	1.23109	0.0433096
$809$	−13.8231	−0.485993	−0.242996	−	0.970027i	$-0.578130\pi$
$809$	−13.8231	−0.485993	−0.242996	+	0.970027i	$0.578130\pi$
$810$	−8.59605	−0.302035
$811$	−36.9514	−1.29754	−0.648770	−	0.760985i	$-0.724716\pi$
$811$	−36.9514	−1.29754	−0.648770	+	0.760985i	$0.724716\pi$
$812$	−31.7312	−1.11355
$813$	−6.02079	−0.211158
$814$	−1.38347	−0.0484907
$815$	29.4046	1.03000
$816$	12.5035	0.437709
$817$	−23.2956	−0.815010
$818$	11.5338	0.403269
$819$	85.0513	2.97193
$820$	12.4402	0.434430
$821$	15.2384	0.531825	0.265913	−	0.963997i	$-0.414327\pi$
$821$	15.2384	0.531825	0.265913	+	0.963997i	$0.414327\pi$
$822$	28.3960	0.990423
$823$	−43.6971	−1.52318	−0.761592	−	0.648057i	$-0.775582\pi$
$823$	−43.6971	−1.52318	−0.761592	+	0.648057i	$0.775582\pi$
$824$	−5.76837	−0.200951
$825$	32.4687	1.13042
$826$	−46.1283	−1.60501
$827$	−31.4358	−1.09313	−0.546565	−	0.837416i	$-0.684065\pi$
$827$	−31.4358	−1.09313	−0.546565	+	0.837416i	$0.684065\pi$
$828$	−19.8241	−0.688934
$829$	43.2819	1.50324	0.751622	−	0.659595i	$-0.229272\pi$
$829$	43.2819	1.50324	0.751622	+	0.659595i	$0.229272\pi$
$830$	−23.7278	−0.823602
$831$	3.29660	0.114358
$832$	3.90469	0.135371
$833$	−42.9007	−1.48642
$834$	−21.4583	−0.743040
$835$	−55.8766	−1.93369
$836$	−9.07396	−0.313830
$837$	52.0921	1.80057
$838$	21.7515	0.751394
$839$	2.87787	0.0993550	0.0496775	−	0.998765i	$-0.484181\pi$
$839$	2.87787	0.0993550	0.0496775	+	0.998765i	$0.484181\pi$
$840$	31.8434	1.09870
$841$	30.6296	1.05619
$842$	−15.8528	−0.546324
$843$	68.0643	2.34426
$844$	8.32833	0.286673
$845$	6.04263	0.207873
$846$	35.3871	1.21663
$847$	59.3204	2.03827
$848$	5.51826	0.189498
$849$	13.7388	0.471515
$850$	9.69733	0.332616
$851$	1.02589	0.0351669
$852$	−0.721967	−0.0247342
$853$	33.4109	1.14397	0.571983	−	0.820265i	$-0.306174\pi$
$853$	33.4109	1.14397	0.571983	+	0.820265i	$0.306174\pi$
$854$	−45.9193	−1.57133
$855$	25.6517	0.877270
$856$	−10.6125	−0.362729
$857$	−28.6782	−0.979629	−0.489814	−	0.871827i	$-0.662935\pi$
$857$	−28.6782	−0.979629	−0.489814	+	0.871827i	$0.662935\pi$
$858$	−56.7375	−1.93699
$859$	51.7442	1.76549	0.882745	−	0.469852i	$-0.155693\pi$
$859$	51.7442	1.76549	0.882745	+	0.469852i	$0.155693\pi$
$860$	−34.8262	−1.18756
$861$	−54.7566	−1.86610
$862$	30.3080	1.03230
$863$	13.0645	0.444722	0.222361	−	0.974964i	$-0.428624\pi$
$863$	13.0645	0.444722	0.222361	+	0.974964i	$0.428624\pi$
$864$	6.62879	0.225516
$865$	39.0734	1.32854
$866$	7.11826	0.241888
$867$	−5.28389	−0.179450
$868$	−32.2919	−1.09606
$869$	−66.0922	−2.24202
$870$	−59.8404	−2.02878
$871$	0.814984	0.0276147
$872$	−20.6955	−0.700839
$873$	18.0706	0.611596
$874$	6.72861	0.227599
$875$	−30.5656	−1.03331
$876$	−47.5217	−1.60561
$877$	−57.2024	−1.93159	−0.965793	−	0.259313i	$-0.916504\pi$
$877$	−57.2024	−1.93159	−0.965793	+	0.259313i	$0.916504\pi$
$878$	30.5801	1.03203
$879$	−73.1721	−2.46803
$880$	−13.5653	−0.457286
$881$	16.3191	0.549805	0.274903	−	0.961472i	$-0.411354\pi$
$881$	16.3191	0.549805	0.274903	+	0.961472i	$0.411354\pi$
$882$	−52.4002	−1.76441
$883$	−5.21725	−0.175574	−0.0877872	−	0.996139i	$-0.527980\pi$
$883$	−5.21725	−0.175574	−0.0877872	+	0.996139i	$0.527980\pi$
$884$	−16.9456	−0.569943
$885$	−86.9914	−2.92418
$886$	−22.6452	−0.760781
$887$	−15.8556	−0.532379	−0.266189	−	0.963921i	$-0.585765\pi$
$887$	−15.8556	−0.532379	−0.266189	+	0.963921i	$0.585765\pi$
$888$	−0.790324	−0.0265215
$889$	27.0772	0.908141
$890$	25.5233	0.855542
$891$	−16.1183	−0.539984
$892$	−8.78349	−0.294093
$893$	−12.0109	−0.401931
$894$	47.9332	1.60313
$895$	−54.3308	−1.81608
$896$	−4.10918	−0.137278
$897$	42.0725	1.40476
$898$	31.7113	1.05822
$899$	60.6832	2.02390
$900$	11.8446	0.394820
$901$	−23.9482	−0.797830
$902$	23.3263	0.776682
$903$	153.291	5.10120
$904$	2.51707	0.0837166
$905$	−38.8359	−1.29095
$906$	29.9229	0.994122
$907$	49.7159	1.65079	0.825395	−	0.564556i	$-0.190953\pi$
$907$	49.7159	1.65079	0.825395	+	0.564556i	$0.190953\pi$
$908$	−19.1395	−0.635167
$909$	−6.52573	−0.216445
$910$	−43.1565	−1.43062
$911$	−58.1651	−1.92710	−0.963548	−	0.267537i	$-0.913790\pi$
$911$	−58.1651	−1.92710	−0.963548	+	0.267537i	$0.913790\pi$
$912$	−5.18360	−0.171646
$913$	−44.4915	−1.47245
$914$	−22.6067	−0.747763
$915$	−86.5972	−2.86281
$916$	0.951510	0.0314388
$917$	21.9390	0.724489
$918$	−28.7677	−0.949475
$919$	34.5221	1.13878	0.569389	−	0.822068i	$-0.307180\pi$
$919$	34.5221	1.13878	0.569389	+	0.822068i	$0.307180\pi$
$920$	10.0591	0.331638
$921$	−98.3616	−3.24112
$922$	−6.22957	−0.205160
$923$	0.978463	0.0322065
$924$	59.7090	1.96428
$925$	−0.612953	−0.0201538
$926$	−2.88429	−0.0947836
$927$	30.5769	1.00428
$928$	7.72202	0.253488
$929$	−5.43302	−0.178252	−0.0891259	−	0.996020i	$-0.528407\pi$
$929$	−5.43302	−0.178252	−0.0891259	+	0.996020i	$0.528407\pi$
$930$	−60.8978	−1.99692
$931$	17.7855	0.582895
$932$	21.4003	0.700991
$933$	−0.868513	−0.0284339
$934$	32.9579	1.07842
$935$	58.8708	1.92528
$936$	−20.6979	−0.676531
$937$	20.0024	0.653450	0.326725	−	0.945119i	$-0.394055\pi$
$937$	20.0024	0.653450	0.326725	+	0.945119i	$0.394055\pi$
$938$	−0.857666	−0.0280038
$939$	60.3826	1.97051
$940$	−17.9560	−0.585660
$941$	27.3104	0.890294	0.445147	−	0.895457i	$-0.353151\pi$
$941$	27.3104	0.890294	0.445147	+	0.895457i	$0.353151\pi$
$942$	−54.8181	−1.78607
$943$	−17.2972	−0.563273
$944$	11.2257	0.365365
$945$	−73.2645	−2.38330
$946$	−65.3020	−2.12315
$947$	21.9496	0.713265	0.356632	−	0.934245i	$-0.383925\pi$
$947$	21.9496	0.713265	0.356632	+	0.934245i	$0.383925\pi$
$948$	−37.7559	−1.22625
$949$	64.4048	2.09067
$950$	−4.02025	−0.130434
$951$	−33.4936	−1.08610
$952$	17.8331	0.577974
$953$	38.9344	1.26121	0.630604	−	0.776104i	$-0.282807\pi$
$953$	38.9344	1.26121	0.630604	+	0.776104i	$0.282807\pi$
$954$	−29.2511	−0.947038
$955$	55.7991	1.80562
$956$	20.2624	0.655332
$957$	−112.206	−3.62710
$958$	−22.5413	−0.728277
$959$	40.4998	1.30781
$960$	−7.74932	−0.250108
$961$	30.7555	0.992113
$962$	1.07110	0.0345338
$963$	56.2547	1.81278
$964$	8.23835	0.265339
$965$	55.6672	1.79199
$966$	−44.2760	−1.42456
$967$	23.2461	0.747543	0.373772	−	0.927521i	$-0.378064\pi$
$967$	23.2461	0.747543	0.373772	+	0.927521i	$0.378064\pi$
$968$	−14.4361	−0.463992
$969$	22.4959	0.722671
$970$	−9.16931	−0.294409
$971$	−11.7159	−0.375980	−0.187990	−	0.982171i	$-0.560197\pi$
$971$	−11.7159	−0.375980	−0.187990	+	0.982171i	$0.560197\pi$
$972$	10.6786	0.342516
$973$	−30.6049	−0.981148
$974$	−29.1421	−0.933774
$975$	−25.1378	−0.805053
$976$	11.1748	0.357697
$977$	−10.6515	−0.340771	−0.170386	−	0.985377i	$-0.554501\pi$
$977$	−10.6515	−0.340771	−0.170386	+	0.985377i	$0.554501\pi$
$978$	31.4970	1.00716
$979$	47.8583	1.52956
$980$	26.5887	0.849346
$981$	109.702	3.50253
$982$	12.0970	0.386032
$983$	−12.7341	−0.406156	−0.203078	−	0.979163i	$-0.565094\pi$
$983$	−12.7341	−0.406156	−0.203078	+	0.979163i	$0.565094\pi$
$984$	13.3254	0.424799
$985$	53.1775	1.69438
$986$	−33.5121	−1.06724
$987$	79.0350	2.51571
$988$	7.02519	0.223501
$989$	48.4234	1.53977
$990$	71.9066	2.28534
$991$	−44.1522	−1.40254	−0.701270	−	0.712896i	$-0.747383\pi$
$991$	−44.1522	−1.40254	−0.701270	+	0.712896i	$0.747383\pi$
$992$	7.85847	0.249507
$993$	−5.26954	−0.167224
$994$	−1.02971	−0.0326603
$995$	49.0985	1.55653
$996$	−25.4163	−0.805345
$997$	46.5435	1.47405	0.737024	−	0.675867i	$-0.236231\pi$
$997$	46.5435	1.47405	0.737024	+	0.675867i	$0.236231\pi$
$998$	38.8508	1.22980
$999$	1.81836	0.0575304

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6038.2.a.d.1.6	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6038.2.a.d.1.6	✓	69	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:	\( N \)	\(=\)	\( 6038 = 2 \cdot 3019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6038.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.88111 q^{3} +1.00000 q^{4} +2.68970 q^{5} +2.88111 q^{6} +4.10918 q^{7} -1.00000 q^{8} +5.30078 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.88111 q^{3} +1.00000 q^{4} +2.68970 q^{5} +2.88111 q^{6} +4.10918 q^{7} -1.00000 q^{8} +5.30078 q^{9} -2.68970 q^{10} -5.04342 q^{11} -2.88111 q^{12} +3.90469 q^{13} -4.10918 q^{14} -7.74932 q^{15} +1.00000 q^{16} -4.33981 q^{17} -5.30078 q^{18} +1.79917 q^{19} +2.68970 q^{20} -11.8390 q^{21} +5.04342 q^{22} -3.73984 q^{23} +2.88111 q^{24} +2.23450 q^{25} -3.90469 q^{26} -6.62879 q^{27} +4.10918 q^{28} -7.72202 q^{29} +7.74932 q^{30} -7.85847 q^{31} -1.00000 q^{32} +14.5306 q^{33} +4.33981 q^{34} +11.0525 q^{35} +5.30078 q^{36} -0.274313 q^{37} -1.79917 q^{38} -11.2498 q^{39} -2.68970 q^{40} +4.62511 q^{41} +11.8390 q^{42} -12.9480 q^{43} -5.04342 q^{44} +14.2575 q^{45} +3.73984 q^{46} -6.67582 q^{47} -2.88111 q^{48} +9.88538 q^{49} -2.23450 q^{50} +12.5035 q^{51} +3.90469 q^{52} +5.51826 q^{53} +6.62879 q^{54} -13.5653 q^{55} -4.10918 q^{56} -5.18360 q^{57} +7.72202 q^{58} +11.2257 q^{59} -7.74932 q^{60} +11.1748 q^{61} +7.85847 q^{62} +21.7819 q^{63} +1.00000 q^{64} +10.5024 q^{65} -14.5306 q^{66} +0.208719 q^{67} -4.33981 q^{68} +10.7749 q^{69} -11.0525 q^{70} +0.250587 q^{71} -5.30078 q^{72} +16.4942 q^{73} +0.274313 q^{74} -6.43785 q^{75} +1.79917 q^{76} -20.7243 q^{77} +11.2498 q^{78} +13.1046 q^{79} +2.68970 q^{80} +3.19591 q^{81} -4.62511 q^{82} +8.82170 q^{83} -11.8390 q^{84} -11.6728 q^{85} +12.9480 q^{86} +22.2480 q^{87} +5.04342 q^{88} -9.48926 q^{89} -14.2575 q^{90} +16.0451 q^{91} -3.73984 q^{92} +22.6411 q^{93} +6.67582 q^{94} +4.83923 q^{95} +2.88111 q^{96} +3.40904 q^{97} -9.88538 q^{98} -26.7340 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9}+O(q^{10}) \) 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9} - 18 q^{10} - 4 q^{11} + 8 q^{12} + 40 q^{13} - 32 q^{14} + 6 q^{15} + 69 q^{16} + 6 q^{17} - 79 q^{18} + 13 q^{19} + 18 q^{20} + 23 q^{21} + 4 q^{22} + 11 q^{23} - 8 q^{24} + 111 q^{25} - 40 q^{26} + 32 q^{27} + 32 q^{28} + 23 q^{29} - 6 q^{30} + 30 q^{31} - 69 q^{32} + 37 q^{33} - 6 q^{34} - 12 q^{35} + 79 q^{36} + 81 q^{37} - 13 q^{38} - 8 q^{39} - 18 q^{40} - 13 q^{41} - 23 q^{42} + 42 q^{43} - 4 q^{44} + 89 q^{45} - 11 q^{46} + 35 q^{47} + 8 q^{48} + 115 q^{49} - 111 q^{50} - 21 q^{51} + 40 q^{52} + 41 q^{53} - 32 q^{54} + 28 q^{55} - 32 q^{56} + 39 q^{57} - 23 q^{58} - 24 q^{59} + 6 q^{60} + 69 q^{61} - 30 q^{62} + 79 q^{63} + 69 q^{64} + 15 q^{65} - 37 q^{66} + 87 q^{67} + 6 q^{68} + 51 q^{69} + 12 q^{70} - 22 q^{71} - 79 q^{72} + 104 q^{73} - 81 q^{74} + 39 q^{75} + 13 q^{76} + 47 q^{77} + 8 q^{78} + 16 q^{79} + 18 q^{80} + 105 q^{81} + 13 q^{82} + 30 q^{83} + 23 q^{84} + 74 q^{85} - 42 q^{86} + 47 q^{87} + 4 q^{88} - 4 q^{89} - 89 q^{90} + 70 q^{91} + 11 q^{92} + 104 q^{93} - 35 q^{94} - 36 q^{95} - 8 q^{96} + 158 q^{97} - 115 q^{98} + 34 q^{99}+O(q^{100}) \) 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9 - 18 * q^10 - 4 * q^11 + 8 * q^12 + 40 * q^13 - 32 * q^14 + 6 * q^15 + 69 * q^16 + 6 * q^17 - 79 * q^18 + 13 * q^19 + 18 * q^20 + 23 * q^21 + 4 * q^22 + 11 * q^23 - 8 * q^24 + 111 * q^25 - 40 * q^26 + 32 * q^27 + 32 * q^28 + 23 * q^29 - 6 * q^30 + 30 * q^31 - 69 * q^32 + 37 * q^33 - 6 * q^34 - 12 * q^35 + 79 * q^36 + 81 * q^37 - 13 * q^38 - 8 * q^39 - 18 * q^40 - 13 * q^41 - 23 * q^42 + 42 * q^43 - 4 * q^44 + 89 * q^45 - 11 * q^46 + 35 * q^47 + 8 * q^48 + 115 * q^49 - 111 * q^50 - 21 * q^51 + 40 * q^52 + 41 * q^53 - 32 * q^54 + 28 * q^55 - 32 * q^56 + 39 * q^57 - 23 * q^58 - 24 * q^59 + 6 * q^60 + 69 * q^61 - 30 * q^62 + 79 * q^63 + 69 * q^64 + 15 * q^65 - 37 * q^66 + 87 * q^67 + 6 * q^68 + 51 * q^69 + 12 * q^70 - 22 * q^71 - 79 * q^72 + 104 * q^73 - 81 * q^74 + 39 * q^75 + 13 * q^76 + 47 * q^77 + 8 * q^78 + 16 * q^79 + 18 * q^80 + 105 * q^81 + 13 * q^82 + 30 * q^83 + 23 * q^84 + 74 * q^85 - 42 * q^86 + 47 * q^87 + 4 * q^88 - 4 * q^89 - 89 * q^90 + 70 * q^91 + 11 * q^92 + 104 * q^93 - 35 * q^94 - 36 * q^95 - 8 * q^96 + 158 * q^97 - 115 * q^98 + 34 * q^99

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