LMFDB - Embedded newform 8002.2.a.f.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8002,2,Mod(1,8002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8002, 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("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8002 = 2 \cdot 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8002.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:	$63.8962916974$
Analytic rank:	$1$
Dimension:	$89$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.39329 q^{3} +1.00000 q^{4} -1.79652 q^{5} +2.39329 q^{6} -1.61915 q^{7} -1.00000 q^{8} +2.72784 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.39329 q^{3} +1.00000 q^{4} -1.79652 q^{5} +2.39329 q^{6} -1.61915 q^{7} -1.00000 q^{8} +2.72784 q^{9} +1.79652 q^{10} -4.17733 q^{11} -2.39329 q^{12} +6.76076 q^{13} +1.61915 q^{14} +4.29960 q^{15} +1.00000 q^{16} +7.19075 q^{17} -2.72784 q^{18} -3.88233 q^{19} -1.79652 q^{20} +3.87510 q^{21} +4.17733 q^{22} -5.61047 q^{23} +2.39329 q^{24} -1.77250 q^{25} -6.76076 q^{26} +0.651362 q^{27} -1.61915 q^{28} -1.09741 q^{29} -4.29960 q^{30} +0.183100 q^{31} -1.00000 q^{32} +9.99756 q^{33} -7.19075 q^{34} +2.90884 q^{35} +2.72784 q^{36} -1.69725 q^{37} +3.88233 q^{38} -16.1805 q^{39} +1.79652 q^{40} -5.90059 q^{41} -3.87510 q^{42} -5.56149 q^{43} -4.17733 q^{44} -4.90063 q^{45} +5.61047 q^{46} -7.12695 q^{47} -2.39329 q^{48} -4.37835 q^{49} +1.77250 q^{50} -17.2095 q^{51} +6.76076 q^{52} +10.9848 q^{53} -0.651362 q^{54} +7.50467 q^{55} +1.61915 q^{56} +9.29153 q^{57} +1.09741 q^{58} +0.0652261 q^{59} +4.29960 q^{60} -0.578412 q^{61} -0.183100 q^{62} -4.41678 q^{63} +1.00000 q^{64} -12.1459 q^{65} -9.99756 q^{66} +1.29552 q^{67} +7.19075 q^{68} +13.4275 q^{69} -2.90884 q^{70} +1.26608 q^{71} -2.72784 q^{72} +11.1755 q^{73} +1.69725 q^{74} +4.24211 q^{75} -3.88233 q^{76} +6.76372 q^{77} +16.1805 q^{78} +2.95337 q^{79} -1.79652 q^{80} -9.74241 q^{81} +5.90059 q^{82} -0.273935 q^{83} +3.87510 q^{84} -12.9184 q^{85} +5.56149 q^{86} +2.62642 q^{87} +4.17733 q^{88} +12.0823 q^{89} +4.90063 q^{90} -10.9467 q^{91} -5.61047 q^{92} -0.438212 q^{93} +7.12695 q^{94} +6.97469 q^{95} +2.39329 q^{96} +11.8296 q^{97} +4.37835 q^{98} -11.3951 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9}+O(q^{10}) $ 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9} + 18 q^{10} - 26 q^{11} - 12 q^{12} + 2 q^{13} + 27 q^{14} - 21 q^{15} + 89 q^{16} - 60 q^{17} - 95 q^{18} + q^{19} - 18 q^{20} - 6 q^{21} + 26 q^{22} - 45 q^{23} + 12 q^{24} + 107 q^{25} - 2 q^{26} - 45 q^{27} - 27 q^{28} - 18 q^{29} + 21 q^{30} - 38 q^{31} - 89 q^{32} - 29 q^{33} + 60 q^{34} - 47 q^{35} + 95 q^{36} - 15 q^{37} - q^{38} - 38 q^{39} + 18 q^{40} - 50 q^{41} + 6 q^{42} - 15 q^{43} - 26 q^{44} - 35 q^{45} + 45 q^{46} - 121 q^{47} - 12 q^{48} + 132 q^{49} - 107 q^{50} + 6 q^{51} + 2 q^{52} - 46 q^{53} + 45 q^{54} - 37 q^{55} + 27 q^{56} - 42 q^{57} + 18 q^{58} - 34 q^{59} - 21 q^{60} + 41 q^{61} + 38 q^{62} - 131 q^{63} + 89 q^{64} - 57 q^{65} + 29 q^{66} - 11 q^{67} - 60 q^{68} + 15 q^{69} + 47 q^{70} - 66 q^{71} - 95 q^{72} - 47 q^{73} + 15 q^{74} - 46 q^{75} + q^{76} - 106 q^{77} + 38 q^{78} - 51 q^{79} - 18 q^{80} + 113 q^{81} + 50 q^{82} - 141 q^{83} - 6 q^{84} - 7 q^{85} + 15 q^{86} - 110 q^{87} + 26 q^{88} - 30 q^{89} + 35 q^{90} + 37 q^{91} - 45 q^{92} - 44 q^{93} + 121 q^{94} - 98 q^{95} + 12 q^{96} + 3 q^{97} - 132 q^{98} - 71 q^{99}+O(q^{100}) \) 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9 + 18 * q^10 - 26 * q^11 - 12 * q^12 + 2 * q^13 + 27 * q^14 - 21 * q^15 + 89 * q^16 - 60 * q^17 - 95 * q^18 + q^19 - 18 * q^20 - 6 * q^21 + 26 * q^22 - 45 * q^23 + 12 * q^24 + 107 * q^25 - 2 * q^26 - 45 * q^27 - 27 * q^28 - 18 * q^29 + 21 * q^30 - 38 * q^31 - 89 * q^32 - 29 * q^33 + 60 * q^34 - 47 * q^35 + 95 * q^36 - 15 * q^37 - q^38 - 38 * q^39 + 18 * q^40 - 50 * q^41 + 6 * q^42 - 15 * q^43 - 26 * q^44 - 35 * q^45 + 45 * q^46 - 121 * q^47 - 12 * q^48 + 132 * q^49 - 107 * q^50 + 6 * q^51 + 2 * q^52 - 46 * q^53 + 45 * q^54 - 37 * q^55 + 27 * q^56 - 42 * q^57 + 18 * q^58 - 34 * q^59 - 21 * q^60 + 41 * q^61 + 38 * q^62 - 131 * q^63 + 89 * q^64 - 57 * q^65 + 29 * q^66 - 11 * q^67 - 60 * q^68 + 15 * q^69 + 47 * q^70 - 66 * q^71 - 95 * q^72 - 47 * q^73 + 15 * q^74 - 46 * q^75 + q^76 - 106 * q^77 + 38 * q^78 - 51 * q^79 - 18 * q^80 + 113 * q^81 + 50 * q^82 - 141 * q^83 - 6 * q^84 - 7 * q^85 + 15 * q^86 - 110 * q^87 + 26 * q^88 - 30 * q^89 + 35 * q^90 + 37 * q^91 - 45 * q^92 - 44 * q^93 + 121 * q^94 - 98 * q^95 + 12 * q^96 + 3 * q^97 - 132 * q^98 - 71 * 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
$3$	−2.39329	−1.38177	−0.690883	−	0.722966i	$-0.742778\pi$
$3$	−2.39329	−1.38177	−0.690883	+	0.722966i	$0.742778\pi$
$4$	1.00000	0.500000
$5$	−1.79652	−0.803430	−0.401715	−	0.915765i	$-0.631586\pi$
$5$	−1.79652	−0.803430	−0.401715	+	0.915765i	$0.631586\pi$
$6$	2.39329	0.977057
$7$	−1.61915	−0.611981	−0.305991	−	0.952035i	$-0.598988\pi$
$7$	−1.61915	−0.611981	−0.305991	+	0.952035i	$0.598988\pi$
$8$	−1.00000	−0.353553
$9$	2.72784	0.909279
$10$	1.79652	0.568111
$11$	−4.17733	−1.25951	−0.629756	−	0.776793i	$-0.716845\pi$
$11$	−4.17733	−1.25951	−0.629756	+	0.776793i	$0.716845\pi$
$12$	−2.39329	−0.690883
$13$	6.76076	1.87510	0.937549	−	0.347855i	$-0.113090\pi$
$13$	6.76076	1.87510	0.937549	+	0.347855i	$0.113090\pi$
$14$	1.61915	0.432736
$15$	4.29960	1.11015
$16$	1.00000	0.250000
$17$	7.19075	1.74401	0.872006	−	0.489495i	$-0.162819\pi$
$17$	7.19075	1.74401	0.872006	+	0.489495i	$0.162819\pi$
$18$	−2.72784	−0.642958
$19$	−3.88233	−0.890667	−0.445333	−	0.895365i	$-0.646915\pi$
$19$	−3.88233	−0.890667	−0.445333	+	0.895365i	$0.646915\pi$
$20$	−1.79652	−0.401715
$21$	3.87510	0.845615
$22$	4.17733	0.890609
$23$	−5.61047	−1.16986	−0.584932	−	0.811082i	$-0.698879\pi$
$23$	−5.61047	−1.16986	−0.584932	+	0.811082i	$0.698879\pi$
$24$	2.39329	0.488528
$25$	−1.77250	−0.354500
$26$	−6.76076	−1.32589
$27$	0.651362	0.125355
$28$	−1.61915	−0.305991
$29$	−1.09741	−0.203784	−0.101892	−	0.994795i	$-0.532490\pi$
$29$	−1.09741	−0.203784	−0.101892	+	0.994795i	$0.532490\pi$
$30$	−4.29960	−0.784997
$31$	0.183100	0.0328858	0.0164429	−	0.999865i	$-0.494766\pi$
$31$	0.183100	0.0328858	0.0164429	+	0.999865i	$0.494766\pi$
$32$	−1.00000	−0.176777
$33$	9.99756	1.74035
$34$	−7.19075	−1.23320
$35$	2.90884	0.491684
$36$	2.72784	0.454640
$37$	−1.69725	−0.279025	−0.139513	−	0.990220i	$-0.544554\pi$
$37$	−1.69725	−0.279025	−0.139513	+	0.990220i	$0.544554\pi$
$38$	3.88233	0.629796
$39$	−16.1805	−2.59095
$40$	1.79652	0.284055
$41$	−5.90059	−0.921517	−0.460759	−	0.887525i	$-0.652423\pi$
$41$	−5.90059	−0.921517	−0.460759	+	0.887525i	$0.652423\pi$
$42$	−3.87510	−0.597940
$43$	−5.56149	−0.848120	−0.424060	−	0.905634i	$-0.639395\pi$
$43$	−5.56149	−0.848120	−0.424060	+	0.905634i	$0.639395\pi$
$44$	−4.17733	−0.629756
$45$	−4.90063	−0.730543
$46$	5.61047	0.827219
$47$	−7.12695	−1.03957	−0.519786	−	0.854297i	$-0.673988\pi$
$47$	−7.12695	−1.03957	−0.519786	+	0.854297i	$0.673988\pi$
$48$	−2.39329	−0.345442
$49$	−4.37835	−0.625479
$50$	1.77250	0.250669
$51$	−17.2095	−2.40982
$52$	6.76076	0.937549
$53$	10.9848	1.50888	0.754440	−	0.656369i	$-0.227908\pi$
$53$	10.9848	1.50888	0.754440	+	0.656369i	$0.227908\pi$
$54$	−0.651362	−0.0886391
$55$	7.50467	1.01193
$56$	1.61915	0.216368
$57$	9.29153	1.23069
$58$	1.09741	0.144097
$59$	0.0652261	0.00849172	0.00424586	−	0.999991i	$-0.498648\pi$
$59$	0.0652261	0.00849172	0.00424586	+	0.999991i	$0.498648\pi$
$60$	4.29960	0.555077
$61$	−0.578412	−0.0740580	−0.0370290	−	0.999314i	$-0.511789\pi$
$61$	−0.578412	−0.0740580	−0.0370290	+	0.999314i	$0.511789\pi$
$62$	−0.183100	−0.0232538
$63$	−4.41678	−0.556462
$64$	1.00000	0.125000
$65$	−12.1459	−1.50651
$66$	−9.99756	−1.23061
$67$	1.29552	0.158273	0.0791366	−	0.996864i	$-0.474784\pi$
$67$	1.29552	0.158273	0.0791366	+	0.996864i	$0.474784\pi$
$68$	7.19075	0.872006
$69$	13.4275	1.61648
$70$	−2.90884	−0.347673
$71$	1.26608	0.150256	0.0751279	−	0.997174i	$-0.476063\pi$
$71$	1.26608	0.150256	0.0751279	+	0.997174i	$0.476063\pi$
$72$	−2.72784	−0.321479
$73$	11.1755	1.30799	0.653996	−	0.756498i	$-0.273091\pi$
$73$	11.1755	1.30799	0.653996	+	0.756498i	$0.273091\pi$
$74$	1.69725	0.197301
$75$	4.24211	0.489836
$76$	−3.88233	−0.445333
$77$	6.76372	0.770797
$78$	16.1805	1.83208
$79$	2.95337	0.332280	0.166140	−	0.986102i	$-0.446870\pi$
$79$	2.95337	0.332280	0.166140	+	0.986102i	$0.446870\pi$
$80$	−1.79652	−0.200858
$81$	−9.74241	−1.08249
$82$	5.90059	0.651611
$83$	−0.273935	−0.0300682	−0.0150341	−	0.999887i	$-0.504786\pi$
$83$	−0.273935	−0.0300682	−0.0150341	+	0.999887i	$0.504786\pi$
$84$	3.87510	0.422808
$85$	−12.9184	−1.40119
$86$	5.56149	0.599711
$87$	2.62642	0.281582
$88$	4.17733	0.445305
$89$	12.0823	1.28072	0.640359	−	0.768075i	$-0.278785\pi$
$89$	12.0823	1.28072	0.640359	+	0.768075i	$0.278785\pi$
$90$	4.90063	0.516572
$91$	−10.9467	−1.14752
$92$	−5.61047	−0.584932
$93$	−0.438212	−0.0454405
$94$	7.12695	0.735088
$95$	6.97469	0.715588
$96$	2.39329	0.244264
$97$	11.8296	1.20112	0.600559	−	0.799580i	$-0.294945\pi$
$97$	11.8296	1.20112	0.600559	+	0.799580i	$0.294945\pi$
$98$	4.37835	0.442281
$99$	−11.3951	−1.14525
$100$	−1.77250	−0.177250
$101$	15.7004	1.56225	0.781123	−	0.624377i	$-0.214647\pi$
$101$	15.7004	1.56225	0.781123	+	0.624377i	$0.214647\pi$
$102$	17.2095	1.70400
$103$	13.6441	1.34439	0.672196	−	0.740373i	$-0.265351\pi$
$103$	13.6441	1.34439	0.672196	+	0.740373i	$0.265351\pi$
$104$	−6.76076	−0.662947
$105$	−6.96170	−0.679393
$106$	−10.9848	−1.06694
$107$	8.54142	0.825730	0.412865	−	0.910792i	$-0.364528\pi$
$107$	8.54142	0.825730	0.412865	+	0.910792i	$0.364528\pi$
$108$	0.651362	0.0626773
$109$	−1.73730	−0.166403	−0.0832015	−	0.996533i	$-0.526515\pi$
$109$	−1.73730	−0.166403	−0.0832015	+	0.996533i	$0.526515\pi$
$110$	−7.50467	−0.715542
$111$	4.06200	0.385548
$112$	−1.61915	−0.152995
$113$	−17.1155	−1.61009	−0.805045	−	0.593213i	$-0.797859\pi$
$113$	−17.1155	−1.61009	−0.805045	+	0.593213i	$0.797859\pi$
$114$	−9.29153	−0.870232
$115$	10.0794	0.939905
$116$	−1.09741	−0.101892
$117$	18.4423	1.70499
$118$	−0.0652261	−0.00600455
$119$	−11.6429	−1.06730
$120$	−4.29960	−0.392498
$121$	6.45006	0.586369
$122$	0.578412	0.0523669
$123$	14.1218	1.27332
$124$	0.183100	0.0164429
$125$	12.1670	1.08825
$126$	4.41678	0.393478
$127$	3.32014	0.294615	0.147308	−	0.989091i	$-0.452939\pi$
$127$	3.32014	0.294615	0.147308	+	0.989091i	$0.452939\pi$
$128$	−1.00000	−0.0883883
$129$	13.3103	1.17190
$130$	12.1459	1.06526
$131$	−0.0662424	−0.00578762	−0.00289381	−	0.999996i	$-0.500921\pi$
$131$	−0.0662424	−0.00578762	−0.00289381	+	0.999996i	$0.500921\pi$
$132$	9.99756	0.870176
$133$	6.28607	0.545071
$134$	−1.29552	−0.111916
$135$	−1.17019	−0.100714
$136$	−7.19075	−0.616601
$137$	6.41317	0.547914	0.273957	−	0.961742i	$-0.411667\pi$
$137$	6.41317	0.547914	0.273957	+	0.961742i	$0.411667\pi$
$138$	−13.4275	−1.14302
$139$	6.50622	0.551851	0.275925	−	0.961179i	$-0.411016\pi$
$139$	6.50622	0.551851	0.275925	+	0.961179i	$0.411016\pi$
$140$	2.90884	0.245842
$141$	17.0568	1.43645
$142$	−1.26608	−0.106247
$143$	−28.2419	−2.36171
$144$	2.72784	0.227320
$145$	1.97152	0.163726
$146$	−11.1755	−0.924890
$147$	10.4787	0.864266
$148$	−1.69725	−0.139513
$149$	−8.06844	−0.660992	−0.330496	−	0.943807i	$-0.607216\pi$
$149$	−8.06844	−0.660992	−0.330496	+	0.943807i	$0.607216\pi$
$150$	−4.24211	−0.346366
$151$	11.4788	0.934133	0.467066	−	0.884222i	$-0.345311\pi$
$151$	11.4788	0.934133	0.467066	+	0.884222i	$0.345311\pi$
$152$	3.88233	0.314898
$153$	19.6152	1.58579
$154$	−6.76372	−0.545036
$155$	−0.328944	−0.0264214
$156$	−16.1805	−1.29547
$157$	13.0283	1.03977	0.519887	−	0.854235i	$-0.325974\pi$
$157$	13.0283	1.03977	0.519887	+	0.854235i	$0.325974\pi$
$158$	−2.95337	−0.234958
$159$	−26.2898	−2.08492
$160$	1.79652	0.142028
$161$	9.08420	0.715935
$162$	9.74241	0.765436
$163$	−5.12514	−0.401432	−0.200716	−	0.979649i	$-0.564327\pi$
$163$	−5.12514	−0.401432	−0.200716	+	0.979649i	$0.564327\pi$
$164$	−5.90059	−0.460759
$165$	−17.9609	−1.39825
$166$	0.273935	0.0212614
$167$	−9.63461	−0.745549	−0.372774	−	0.927922i	$-0.621593\pi$
$167$	−9.63461	−0.745549	−0.372774	+	0.927922i	$0.621593\pi$
$168$	−3.87510	−0.298970
$169$	32.7079	2.51599
$170$	12.9184	0.990792
$171$	−10.5904	−0.809865
$172$	−5.56149	−0.424060
$173$	−2.43200	−0.184902	−0.0924509	−	0.995717i	$-0.529470\pi$
$173$	−2.43200	−0.184902	−0.0924509	+	0.995717i	$0.529470\pi$
$174$	−2.62642	−0.199108
$175$	2.86994	0.216947
$176$	−4.17733	−0.314878
$177$	−0.156105	−0.0117336
$178$	−12.0823	−0.905605
$179$	−17.2297	−1.28781	−0.643904	−	0.765106i	$-0.722686\pi$
$179$	−17.2297	−1.28781	−0.643904	+	0.765106i	$0.722686\pi$
$180$	−4.90063	−0.365271
$181$	1.61286	0.119883	0.0599415	−	0.998202i	$-0.480909\pi$
$181$	1.61286	0.119883	0.0599415	+	0.998202i	$0.480909\pi$
$182$	10.9467	0.811422
$183$	1.38431	0.102331
$184$	5.61047	0.413610
$185$	3.04914	0.224177
$186$	0.438212	0.0321313
$187$	−30.0381	−2.19660
$188$	−7.12695	−0.519786
$189$	−1.05465	−0.0767147
$190$	−6.97469	−0.505997
$191$	−5.82764	−0.421673	−0.210837	−	0.977521i	$-0.567619\pi$
$191$	−5.82764	−0.421673	−0.210837	+	0.977521i	$0.567619\pi$
$192$	−2.39329	−0.172721
$193$	−7.38076	−0.531279	−0.265639	−	0.964072i	$-0.585583\pi$
$193$	−7.38076	−0.531279	−0.265639	+	0.964072i	$0.585583\pi$
$194$	−11.8296	−0.849319
$195$	29.0686	2.08164
$196$	−4.37835	−0.312740
$197$	−6.33903	−0.451637	−0.225819	−	0.974169i	$-0.572506\pi$
$197$	−6.33903	−0.451637	−0.225819	+	0.974169i	$0.572506\pi$
$198$	11.3951	0.809813
$199$	8.22834	0.583292	0.291646	−	0.956526i	$-0.405797\pi$
$199$	8.22834	0.583292	0.291646	+	0.956526i	$0.405797\pi$
$200$	1.77250	0.125335
$201$	−3.10056	−0.218697
$202$	−15.7004	−1.10468
$203$	1.77687	0.124712
$204$	−17.2095	−1.20491
$205$	10.6006	0.740375
$206$	−13.6441	−0.950629
$207$	−15.3045	−1.06373
$208$	6.76076	0.468774
$209$	16.2177	1.12180
$210$	6.96170	0.480403
$211$	9.07906	0.625028	0.312514	−	0.949913i	$-0.398829\pi$
$211$	9.07906	0.625028	0.312514	+	0.949913i	$0.398829\pi$
$212$	10.9848	0.754440
$213$	−3.03009	−0.207619
$214$	−8.54142	−0.583879
$215$	9.99136	0.681405
$216$	−0.651362	−0.0443196
$217$	−0.296467	−0.0201255
$218$	1.73730	0.117665
$219$	−26.7462	−1.80734
$220$	7.50467	0.505965
$221$	48.6149	3.27019
$222$	−4.06200	−0.272624
$223$	21.6006	1.44648	0.723240	−	0.690597i	$-0.242652\pi$
$223$	21.6006	1.44648	0.723240	+	0.690597i	$0.242652\pi$
$224$	1.61915	0.108184
$225$	−4.83509	−0.322339
$226$	17.1155	1.13851
$227$	−11.9088	−0.790416	−0.395208	−	0.918592i	$-0.629328\pi$
$227$	−11.9088	−0.790416	−0.395208	+	0.918592i	$0.629328\pi$
$228$	9.29153	0.615347
$229$	−5.50831	−0.363999	−0.182000	−	0.983299i	$-0.558257\pi$
$229$	−5.50831	−0.363999	−0.182000	+	0.983299i	$0.558257\pi$
$230$	−10.0794	−0.664613
$231$	−16.1875	−1.06506
$232$	1.09741	0.0720485
$233$	−20.0246	−1.31185	−0.655927	−	0.754825i	$-0.727722\pi$
$233$	−20.0246	−1.31185	−0.655927	+	0.754825i	$0.727722\pi$
$234$	−18.4423	−1.20561
$235$	12.8037	0.835223
$236$	0.0652261	0.00424586
$237$	−7.06827	−0.459134
$238$	11.6429	0.754697
$239$	−13.5465	−0.876248	−0.438124	−	0.898915i	$-0.644357\pi$
$239$	−13.5465	−0.876248	−0.438124	+	0.898915i	$0.644357\pi$
$240$	4.29960	0.277538
$241$	−7.38671	−0.475820	−0.237910	−	0.971287i	$-0.576462\pi$
$241$	−7.38671	−0.475820	−0.237910	+	0.971287i	$0.576462\pi$
$242$	−6.45006	−0.414625
$243$	21.3623	1.37039
$244$	−0.578412	−0.0370290
$245$	7.86582	0.502529
$246$	−14.1218	−0.900375
$247$	−26.2475	−1.67009
$248$	−0.183100	−0.0116269
$249$	0.655605	0.0415473
$250$	−12.1670	−0.769506
$251$	−8.91881	−0.562951	−0.281475	−	0.959568i	$-0.590824\pi$
$251$	−8.91881	−0.562951	−0.281475	+	0.959568i	$0.590824\pi$
$252$	−4.41678	−0.278231
$253$	23.4368	1.47346
$254$	−3.32014	−0.208324
$255$	30.9174	1.93612
$256$	1.00000	0.0625000
$257$	12.6118	0.786704	0.393352	−	0.919388i	$-0.371315\pi$
$257$	12.6118	0.786704	0.393352	+	0.919388i	$0.371315\pi$
$258$	−13.3103	−0.828661
$259$	2.74809	0.170758
$260$	−12.1459	−0.753255
$261$	−2.99356	−0.185296
$262$	0.0662424	0.00409247
$263$	3.37365	0.208028	0.104014	−	0.994576i	$-0.466831\pi$
$263$	3.37365	0.208028	0.104014	+	0.994576i	$0.466831\pi$
$264$	−9.99756	−0.615307
$265$	−19.7345	−1.21228
$266$	−6.28607	−0.385423
$267$	−28.9164	−1.76965
$268$	1.29552	0.0791366
$269$	6.11287	0.372709	0.186354	−	0.982483i	$-0.440333\pi$
$269$	6.11287	0.372709	0.186354	+	0.982483i	$0.440333\pi$
$270$	1.17019	0.0712153
$271$	8.63836	0.524743	0.262371	−	0.964967i	$-0.415495\pi$
$271$	8.63836	0.524743	0.262371	+	0.964967i	$0.415495\pi$
$272$	7.19075	0.436003
$273$	26.1986	1.58561
$274$	−6.41317	−0.387434
$275$	7.40431	0.446497
$276$	13.4275	0.808240
$277$	−22.4446	−1.34856	−0.674282	−	0.738474i	$-0.735547\pi$
$277$	−22.4446	−1.34856	−0.674282	+	0.738474i	$0.735547\pi$
$278$	−6.50622	−0.390217
$279$	0.499468	0.0299024
$280$	−2.90884	−0.173837
$281$	−9.79819	−0.584511	−0.292255	−	0.956340i	$-0.594406\pi$
$281$	−9.79819	−0.584511	−0.292255	+	0.956340i	$0.594406\pi$
$282$	−17.0568	−1.01572
$283$	10.6557	0.633414	0.316707	−	0.948523i	$-0.397423\pi$
$283$	10.6557	0.633414	0.316707	+	0.948523i	$0.397423\pi$
$284$	1.26608	0.0751279
$285$	−16.6925	−0.988776
$286$	28.2419	1.66998
$287$	9.55394	0.563951
$288$	−2.72784	−0.160739
$289$	34.7068	2.04158
$290$	−1.97152	−0.115772
$291$	−28.3118	−1.65967
$292$	11.1755	0.653996
$293$	−6.82643	−0.398804	−0.199402	−	0.979918i	$-0.563900\pi$
$293$	−6.82643	−0.398804	−0.199402	+	0.979918i	$0.563900\pi$
$294$	−10.4787	−0.611129
$295$	−0.117180	−0.00682251
$296$	1.69725	0.0986504
$297$	−2.72095	−0.157886
$298$	8.06844	0.467392
$299$	−37.9311	−2.19361
$300$	4.24211	0.244918
$301$	9.00489	0.519033
$302$	−11.4788	−0.660532
$303$	−37.5756	−2.15866
$304$	−3.88233	−0.222667
$305$	1.03913	0.0595005
$306$	−19.6152	−1.12133
$307$	24.1286	1.37709	0.688547	−	0.725191i	$-0.258249\pi$
$307$	24.1286	1.37709	0.688547	+	0.725191i	$0.258249\pi$
$308$	6.76372	0.385399
$309$	−32.6543	−1.85764
$310$	0.328944	0.0186828
$311$	21.5829	1.22385	0.611927	−	0.790914i	$-0.290395\pi$
$311$	21.5829	1.22385	0.611927	+	0.790914i	$0.290395\pi$
$312$	16.1805	0.916038
$313$	−9.34367	−0.528135	−0.264068	−	0.964504i	$-0.585064\pi$
$313$	−9.34367	−0.528135	−0.264068	+	0.964504i	$0.585064\pi$
$314$	−13.0283	−0.735231
$315$	7.93485	0.447078
$316$	2.95337	0.166140
$317$	−20.4364	−1.14782	−0.573912	−	0.818917i	$-0.694575\pi$
$317$	−20.4364	−1.14782	−0.573912	+	0.818917i	$0.694575\pi$
$318$	26.2898	1.47426
$319$	4.58424	0.256668
$320$	−1.79652	−0.100429
$321$	−20.4421	−1.14097
$322$	−9.08420	−0.506243
$323$	−27.9168	−1.55333
$324$	−9.74241	−0.541245
$325$	−11.9834	−0.664722
$326$	5.12514	0.283855
$327$	4.15786	0.229930
$328$	5.90059	0.325806
$329$	11.5396	0.636198
$330$	17.9609	0.988712
$331$	−0.332119	−0.0182549	−0.00912747	−	0.999958i	$-0.502905\pi$
$331$	−0.332119	−0.0182549	−0.00912747	+	0.999958i	$0.502905\pi$
$332$	−0.273935	−0.0150341
$333$	−4.62981	−0.253712
$334$	9.63461	0.527182
$335$	−2.32744	−0.127161
$336$	3.87510	0.211404
$337$	−6.29543	−0.342934	−0.171467	−	0.985190i	$-0.554851\pi$
$337$	−6.29543	−0.342934	−0.171467	+	0.985190i	$0.554851\pi$
$338$	−32.7079	−1.77907
$339$	40.9624	2.22477
$340$	−12.9184	−0.700596
$341$	−0.764869	−0.0414200
$342$	10.5904	0.572661
$343$	18.4233	0.994763
$344$	5.56149	0.299856
$345$	−24.1228	−1.29873
$346$	2.43200	0.130745
$347$	−6.95375	−0.373297	−0.186648	−	0.982427i	$-0.559763\pi$
$347$	−6.95375	−0.373297	−0.186648	+	0.982427i	$0.559763\pi$
$348$	2.62642	0.140791
$349$	17.1365	0.917298	0.458649	−	0.888618i	$-0.348334\pi$
$349$	17.1365	0.917298	0.458649	+	0.888618i	$0.348334\pi$
$350$	−2.86994	−0.153405
$351$	4.40370	0.235052
$352$	4.17733	0.222652
$353$	−32.7865	−1.74505	−0.872525	−	0.488569i	$-0.837519\pi$
$353$	−32.7865	−1.74505	−0.872525	+	0.488569i	$0.837519\pi$
$354$	0.156105	0.00829689
$355$	−2.27454	−0.120720
$356$	12.0823	0.640359
$357$	27.8648	1.47476
$358$	17.2297	0.910618
$359$	21.7289	1.14681	0.573404	−	0.819273i	$-0.305623\pi$
$359$	21.7289	1.14681	0.573404	+	0.819273i	$0.305623\pi$
$360$	4.90063	0.258286
$361$	−3.92755	−0.206713
$362$	−1.61286	−0.0847700
$363$	−15.4369	−0.810225
$364$	−10.9467	−0.573762
$365$	−20.0770	−1.05088
$366$	−1.38431	−0.0723589
$367$	−6.31680	−0.329734	−0.164867	−	0.986316i	$-0.552720\pi$
$367$	−6.31680	−0.329734	−0.164867	+	0.986316i	$0.552720\pi$
$368$	−5.61047	−0.292466
$369$	−16.0959	−0.837917
$370$	−3.04914	−0.158517
$371$	−17.7861	−0.923406
$372$	−0.438212	−0.0227202
$373$	11.5385	0.597438	0.298719	−	0.954341i	$-0.403441\pi$
$373$	11.5385	0.597438	0.298719	+	0.954341i	$0.403441\pi$
$374$	30.0381	1.55323
$375$	−29.1191	−1.50370
$376$	7.12695	0.367544
$377$	−7.41932	−0.382114
$378$	1.05465	0.0542455
$379$	−0.637674	−0.0327551	−0.0163776	−	0.999866i	$-0.505213\pi$
$379$	−0.637674	−0.0327551	−0.0163776	+	0.999866i	$0.505213\pi$
$380$	6.97469	0.357794
$381$	−7.94607	−0.407089
$382$	5.82764	0.298168
$383$	−12.4585	−0.636599	−0.318300	−	0.947990i	$-0.603112\pi$
$383$	−12.4585	−0.636599	−0.318300	+	0.947990i	$0.603112\pi$
$384$	2.39329	0.122132
$385$	−12.1512	−0.619282
$386$	7.38076	0.375671
$387$	−15.1709	−0.771178
$388$	11.8296	0.600559
$389$	−4.99535	−0.253274	−0.126637	−	0.991949i	$-0.540418\pi$
$389$	−4.99535	−0.253274	−0.126637	+	0.991949i	$0.540418\pi$
$390$	−29.0686	−1.47195
$391$	−40.3435	−2.04026
$392$	4.37835	0.221140
$393$	0.158537	0.00799714
$394$	6.33903	0.319356
$395$	−5.30580	−0.266964
$396$	−11.3951	−0.572624
$397$	11.8161	0.593034	0.296517	−	0.955028i	$-0.404175\pi$
$397$	11.8161	0.593034	0.296517	+	0.955028i	$0.404175\pi$
$398$	−8.22834	−0.412449
$399$	−15.0444	−0.753161
$400$	−1.77250	−0.0886250
$401$	−9.23018	−0.460933	−0.230467	−	0.973080i	$-0.574025\pi$
$401$	−9.23018	−0.460933	−0.230467	+	0.973080i	$0.574025\pi$
$402$	3.10056	0.154642
$403$	1.23790	0.0616640
$404$	15.7004	0.781123
$405$	17.5025	0.869705
$406$	−1.77687	−0.0881846
$407$	7.08995	0.351436
$408$	17.2095	0.851999
$409$	4.10689	0.203073	0.101536	−	0.994832i	$-0.467624\pi$
$409$	4.10689	0.203073	0.101536	+	0.994832i	$0.467624\pi$
$410$	−10.6006	−0.523524
$411$	−15.3486	−0.757089
$412$	13.6441	0.672196
$413$	−0.105611	−0.00519677
$414$	15.3045	0.752173
$415$	0.492130	0.0241577
$416$	−6.76076	−0.331473
$417$	−15.5713	−0.762529
$418$	−16.2177	−0.793236
$419$	−2.20587	−0.107764	−0.0538819	−	0.998547i	$-0.517159\pi$
$419$	−2.20587	−0.107764	−0.0538819	+	0.998547i	$0.517159\pi$
$420$	−6.96170	−0.339696
$421$	−8.38322	−0.408573	−0.204287	−	0.978911i	$-0.565487\pi$
$421$	−8.38322	−0.408573	−0.204287	+	0.978911i	$0.565487\pi$
$422$	−9.07906	−0.441962
$423$	−19.4412	−0.945261
$424$	−10.9848	−0.533470
$425$	−12.7456	−0.618252
$426$	3.03009	0.146809
$427$	0.936535	0.0453221
$428$	8.54142	0.412865
$429$	67.5911	3.26333
$430$	−9.99136	−0.481826
$431$	−33.7752	−1.62689	−0.813446	−	0.581640i	$-0.802411\pi$
$431$	−33.7752	−1.62689	−0.813446	+	0.581640i	$0.802411\pi$
$432$	0.651362	0.0313387
$433$	−14.7229	−0.707536	−0.353768	−	0.935333i	$-0.615100\pi$
$433$	−14.7229	−0.707536	−0.353768	+	0.935333i	$0.615100\pi$
$434$	0.296467	0.0142309
$435$	−4.71843	−0.226231
$436$	−1.73730	−0.0832015
$437$	21.7817	1.04196
$438$	26.7462	1.27798
$439$	29.7620	1.42046	0.710231	−	0.703969i	$-0.248591\pi$
$439$	29.7620	1.42046	0.710231	+	0.703969i	$0.248591\pi$
$440$	−7.50467	−0.357771
$441$	−11.9434	−0.568735
$442$	−48.6149	−2.31237
$443$	32.5172	1.54494	0.772468	−	0.635053i	$-0.219022\pi$
$443$	32.5172	1.54494	0.772468	+	0.635053i	$0.219022\pi$
$444$	4.06200	0.192774
$445$	−21.7061	−1.02897
$446$	−21.6006	−1.02282
$447$	19.3101	0.913337
$448$	−1.61915	−0.0764976
$449$	4.94895	0.233555	0.116778	−	0.993158i	$-0.462743\pi$
$449$	4.94895	0.233555	0.116778	+	0.993158i	$0.462743\pi$
$450$	4.83509	0.227928
$451$	24.6487	1.16066
$452$	−17.1155	−0.805045
$453$	−27.4721	−1.29075
$454$	11.9088	0.558909
$455$	19.6660	0.921955
$456$	−9.29153	−0.435116
$457$	−36.6164	−1.71284	−0.856421	−	0.516277i	$-0.827317\pi$
$457$	−36.6164	−1.71284	−0.856421	+	0.516277i	$0.827317\pi$
$458$	5.50831	0.257386
$459$	4.68378	0.218620
$460$	10.0794	0.469952
$461$	6.28985	0.292947	0.146474	−	0.989215i	$-0.453208\pi$
$461$	6.28985	0.292947	0.146474	+	0.989215i	$0.453208\pi$
$462$	16.1875	0.753113
$463$	−40.0445	−1.86103	−0.930513	−	0.366260i	$-0.880638\pi$
$463$	−40.0445	−1.86103	−0.930513	+	0.366260i	$0.880638\pi$
$464$	−1.09741	−0.0509459
$465$	0.787259	0.0365082
$466$	20.0246	0.927620
$467$	3.68241	0.170401	0.0852007	−	0.996364i	$-0.472847\pi$
$467$	3.68241	0.170401	0.0852007	+	0.996364i	$0.472847\pi$
$468$	18.4423	0.852494
$469$	−2.09764	−0.0968602
$470$	−12.8037	−0.590592
$471$	−31.1806	−1.43672
$472$	−0.0652261	−0.00300228
$473$	23.2322	1.06822
$474$	7.06827	0.324657
$475$	6.88142	0.315741
$476$	−11.6429	−0.533651
$477$	29.9648	1.37199
$478$	13.5465	0.619601
$479$	29.7996	1.36158	0.680790	−	0.732479i	$-0.261637\pi$
$479$	29.7996	1.36158	0.680790	+	0.732479i	$0.261637\pi$
$480$	−4.29960	−0.196249
$481$	−11.4747	−0.523200
$482$	7.38671	0.336456
$483$	−21.7411	−0.989255
$484$	6.45006	0.293184
$485$	−21.2523	−0.965015
$486$	−21.3623	−0.969015
$487$	−28.4639	−1.28982	−0.644912	−	0.764257i	$-0.723106\pi$
$487$	−28.4639	−1.28982	−0.644912	+	0.764257i	$0.723106\pi$
$488$	0.578412	0.0261835
$489$	12.2660	0.554686
$490$	−7.86582	−0.355342
$491$	32.0151	1.44482	0.722410	−	0.691465i	$-0.243034\pi$
$491$	32.0151	1.44482	0.722410	+	0.691465i	$0.243034\pi$
$492$	14.1218	0.636661
$493$	−7.89119	−0.355401
$494$	26.2475	1.18093
$495$	20.4715	0.920127
$496$	0.183100	0.00822144
$497$	−2.04997	−0.0919538
$498$	−0.655605	−0.0293784
$499$	34.8962	1.56217	0.781083	−	0.624427i	$-0.214667\pi$
$499$	34.8962	1.56217	0.781083	+	0.624427i	$0.214667\pi$
$500$	12.1670	0.544123
$501$	23.0584	1.03017
$502$	8.91881	0.398066
$503$	7.99673	0.356557	0.178278	−	0.983980i	$-0.442947\pi$
$503$	7.99673	0.356557	0.178278	+	0.983980i	$0.442947\pi$
$504$	4.41678	0.196739
$505$	−28.2061	−1.25516
$506$	−23.4368	−1.04189
$507$	−78.2794	−3.47651
$508$	3.32014	0.147308
$509$	−21.4468	−0.950612	−0.475306	−	0.879820i	$-0.657663\pi$
$509$	−21.4468	−0.950612	−0.475306	+	0.879820i	$0.657663\pi$
$510$	−30.9174	−1.36904
$511$	−18.0948	−0.800467
$512$	−1.00000	−0.0441942
$513$	−2.52880	−0.111649
$514$	−12.6118	−0.556283
$515$	−24.5119	−1.08013
$516$	13.3103	0.585952
$517$	29.7716	1.30935
$518$	−2.74809	−0.120744
$519$	5.82049	0.255491
$520$	12.1459	0.532632
$521$	37.3774	1.63753	0.818767	−	0.574126i	$-0.194658\pi$
$521$	37.3774	1.63753	0.818767	+	0.574126i	$0.194658\pi$
$522$	2.99356	0.131024
$523$	−5.77735	−0.252626	−0.126313	−	0.991990i	$-0.540314\pi$
$523$	−5.77735	−0.252626	−0.126313	+	0.991990i	$0.540314\pi$
$524$	−0.0662424	−0.00289381
$525$	−6.86861	−0.299771
$526$	−3.37365	−0.147098
$527$	1.31663	0.0573532
$528$	9.99756	0.435088
$529$	8.47742	0.368583
$530$	19.7345	0.857211
$531$	0.177926	0.00772135
$532$	6.28607	0.272536
$533$	−39.8925	−1.72793
$534$	28.9164	1.25133
$535$	−15.3449	−0.663416
$536$	−1.29552	−0.0559580
$537$	41.2357	1.77945
$538$	−6.11287	−0.263545
$539$	18.2898	0.787798
$540$	−1.17019	−0.0503569
$541$	27.3589	1.17625	0.588126	−	0.808769i	$-0.299866\pi$
$541$	27.3589	1.17625	0.588126	+	0.808769i	$0.299866\pi$
$542$	−8.63836	−0.371049
$543$	−3.86004	−0.165650
$544$	−7.19075	−0.308301
$545$	3.12110	0.133693
$546$	−26.1986	−1.12120
$547$	−20.5486	−0.878597	−0.439298	−	0.898341i	$-0.644773\pi$
$547$	−20.5486	−0.878597	−0.439298	+	0.898341i	$0.644773\pi$
$548$	6.41317	0.273957
$549$	−1.57781	−0.0673395
$550$	−7.40431	−0.315721
$551$	4.26050	0.181503
$552$	−13.4275	−0.571512
$553$	−4.78195	−0.203349
$554$	22.4446	0.953579
$555$	−7.29748	−0.309761
$556$	6.50622	0.275925
$557$	−10.1144	−0.428562	−0.214281	−	0.976772i	$-0.568741\pi$
$557$	−10.1144	−0.428562	−0.214281	+	0.976772i	$0.568741\pi$
$558$	−0.499468	−0.0211442
$559$	−37.5999	−1.59031
$560$	2.90884	0.122921
$561$	71.8899	3.03519
$562$	9.79819	0.413312
$563$	36.7050	1.54693	0.773466	−	0.633838i	$-0.218521\pi$
$563$	36.7050	1.54693	0.773466	+	0.633838i	$0.218521\pi$
$564$	17.0568	0.718223
$565$	30.7484	1.29360
$566$	−10.6557	−0.447891
$567$	15.7744	0.662464
$568$	−1.26608	−0.0531235
$569$	−31.7692	−1.33183	−0.665916	−	0.746026i	$-0.731959\pi$
$569$	−31.7692	−1.33183	−0.665916	+	0.746026i	$0.731959\pi$
$570$	16.6925	0.699170
$571$	24.3687	1.01980	0.509899	−	0.860235i	$-0.329683\pi$
$571$	24.3687	1.01980	0.509899	+	0.860235i	$0.329683\pi$
$572$	−28.2419	−1.18085
$573$	13.9472	0.582654
$574$	−9.55394	−0.398774
$575$	9.94456	0.414717
$576$	2.72784	0.113660
$577$	34.5204	1.43710	0.718551	−	0.695474i	$-0.244805\pi$
$577$	34.5204	1.43710	0.718551	+	0.695474i	$0.244805\pi$
$578$	−34.7068	−1.44361
$579$	17.6643	0.734103
$580$	1.97152	0.0818630
$581$	0.443541	0.0184012
$582$	28.3118	1.17356
$583$	−45.8872	−1.90045
$584$	−11.1755	−0.462445
$585$	−33.1320	−1.36984
$586$	6.82643	0.281997
$587$	41.1431	1.69816	0.849079	−	0.528266i	$-0.177158\pi$
$587$	41.1431	1.69816	0.849079	+	0.528266i	$0.177158\pi$
$588$	10.4787	0.432133
$589$	−0.710855	−0.0292903
$590$	0.117180	0.00482424
$591$	15.1711	0.624058
$592$	−1.69725	−0.0697563
$593$	−30.3499	−1.24632	−0.623160	−	0.782094i	$-0.714152\pi$
$593$	−30.3499	−1.24632	−0.623160	+	0.782094i	$0.714152\pi$
$594$	2.72095	0.111642
$595$	20.9167	0.857503
$596$	−8.06844	−0.330496
$597$	−19.6928	−0.805973
$598$	37.9311	1.55112
$599$	−38.0468	−1.55455	−0.777275	−	0.629161i	$-0.783399\pi$
$599$	−38.0468	−1.55455	−0.777275	+	0.629161i	$0.783399\pi$
$600$	−4.24211	−0.173183
$601$	0.834478	0.0340391	0.0170195	−	0.999855i	$-0.494582\pi$
$601$	0.834478	0.0340391	0.0170195	+	0.999855i	$0.494582\pi$
$602$	−9.00489	−0.367012
$603$	3.53397	0.143915
$604$	11.4788	0.467066
$605$	−11.5877	−0.471107
$606$	37.5756	1.52640
$607$	−20.8263	−0.845315	−0.422658	−	0.906289i	$-0.638903\pi$
$607$	−20.8263	−0.845315	−0.422658	+	0.906289i	$0.638903\pi$
$608$	3.88233	0.157449
$609$	−4.25257	−0.172323
$610$	−1.03913	−0.0420732
$611$	−48.1836	−1.94930
$612$	19.6152	0.792897
$613$	37.5634	1.51717	0.758585	−	0.651574i	$-0.225891\pi$
$613$	37.5634	1.51717	0.758585	+	0.651574i	$0.225891\pi$
$614$	−24.1286	−0.973753
$615$	−25.3702	−1.02303
$616$	−6.76372	−0.272518
$617$	−41.7654	−1.68141	−0.840707	−	0.541491i	$-0.817860\pi$
$617$	−41.7654	−1.68141	−0.840707	+	0.541491i	$0.817860\pi$
$618$	32.6543	1.31355
$619$	6.84010	0.274927	0.137463	−	0.990507i	$-0.456105\pi$
$619$	6.84010	0.274927	0.137463	+	0.990507i	$0.456105\pi$
$620$	−0.328944	−0.0132107
$621$	−3.65445	−0.146648
$622$	−21.5829	−0.865395
$623$	−19.5630	−0.783776
$624$	−16.1805	−0.647737
$625$	−12.9957	−0.519830
$626$	9.34367	0.373448
$627$	−38.8138	−1.55007
$628$	13.0283	0.519887
$629$	−12.2045	−0.486624
$630$	−7.93485	−0.316132
$631$	15.5266	0.618105	0.309052	−	0.951045i	$-0.399988\pi$
$631$	15.5266	0.618105	0.309052	+	0.951045i	$0.399988\pi$
$632$	−2.95337	−0.117479
$633$	−21.7288	−0.863643
$634$	20.4364	0.811635
$635$	−5.96472	−0.236703
$636$	−26.2898	−1.04246
$637$	−29.6010	−1.17283
$638$	−4.58424	−0.181492
$639$	3.45366	0.136625
$640$	1.79652	0.0710139
$641$	−0.466748	−0.0184354	−0.00921772	−	0.999958i	$-0.502934\pi$
$641$	−0.466748	−0.0184354	−0.00921772	+	0.999958i	$0.502934\pi$
$642$	20.4421	0.806785
$643$	−35.9831	−1.41903	−0.709517	−	0.704689i	$-0.751087\pi$
$643$	−35.9831	−1.41903	−0.709517	+	0.704689i	$0.751087\pi$
$644$	9.08420	0.357968
$645$	−23.9122	−0.941543
$646$	27.9168	1.09837
$647$	15.3544	0.603643	0.301821	−	0.953365i	$-0.402405\pi$
$647$	15.3544	0.603643	0.301821	+	0.953365i	$0.402405\pi$
$648$	9.74241	0.382718
$649$	−0.272471	−0.0106954
$650$	11.9834	0.470029
$651$	0.709531	0.0278087
$652$	−5.12514	−0.200716
$653$	−34.3706	−1.34503	−0.672513	−	0.740085i	$-0.734785\pi$
$653$	−34.3706	−1.34503	−0.672513	+	0.740085i	$0.734785\pi$
$654$	−4.15786	−0.162585
$655$	0.119006	0.00464995
$656$	−5.90059	−0.230379
$657$	30.4849	1.18933
$658$	−11.5396	−0.449860
$659$	−49.4173	−1.92502	−0.962512	−	0.271238i	$-0.912567\pi$
$659$	−49.4173	−1.92502	−0.962512	+	0.271238i	$0.912567\pi$
$660$	−17.9609	−0.699125
$661$	14.9311	0.580753	0.290376	−	0.956913i	$-0.406219\pi$
$661$	14.9311	0.580753	0.290376	+	0.956913i	$0.406219\pi$
$662$	0.332119	0.0129082
$663$	−116.350	−4.51864
$664$	0.273935	0.0106307
$665$	−11.2931	−0.437927
$666$	4.62981	0.179402
$667$	6.15699	0.238399
$668$	−9.63461	−0.372774
$669$	−51.6964	−1.99870
$670$	2.32744	0.0899167
$671$	2.41621	0.0932769
$672$	−3.87510	−0.149485
$673$	−24.2492	−0.934738	−0.467369	−	0.884062i	$-0.654798\pi$
$673$	−24.2492	−0.934738	−0.467369	+	0.884062i	$0.654798\pi$
$674$	6.29543	0.242491
$675$	−1.15454	−0.0444382
$676$	32.7079	1.25799
$677$	−45.2738	−1.74001	−0.870007	−	0.493038i	$-0.835886\pi$
$677$	−45.2738	−1.74001	−0.870007	+	0.493038i	$0.835886\pi$
$678$	−40.9624	−1.57315
$679$	−19.1540	−0.735062
$680$	12.9184	0.495396
$681$	28.5013	1.09217
$682$	0.764869	0.0292884
$683$	5.03657	0.192719	0.0963595	−	0.995347i	$-0.469280\pi$
$683$	5.03657	0.192719	0.0963595	+	0.995347i	$0.469280\pi$
$684$	−10.5904	−0.404932
$685$	−11.5214	−0.440210
$686$	−18.4233	−0.703403
$687$	13.1830	0.502962
$688$	−5.56149	−0.212030
$689$	74.2657	2.82930
$690$	24.1228	0.918340
$691$	−18.0696	−0.687400	−0.343700	−	0.939079i	$-0.611680\pi$
$691$	−18.0696	−0.687400	−0.343700	+	0.939079i	$0.611680\pi$
$692$	−2.43200	−0.0924509
$693$	18.4503	0.700870
$694$	6.95375	0.263961
$695$	−11.6886	−0.443373
$696$	−2.62642	−0.0995542
$697$	−42.4296	−1.60714
$698$	−17.1365	−0.648627
$699$	47.9246	1.81268
$700$	2.86994	0.108474
$701$	−0.319419	−0.0120643	−0.00603214	−	0.999982i	$-0.501920\pi$
$701$	−0.319419	−0.0120643	−0.00603214	+	0.999982i	$0.501920\pi$
$702$	−4.40370	−0.166207
$703$	6.58926	0.248519
$704$	−4.17733	−0.157439
$705$	−30.6430	−1.15408
$706$	32.7865	1.23394
$707$	−25.4213	−0.956065
$708$	−0.156105	−0.00586679
$709$	1.11047	0.0417047	0.0208524	−	0.999783i	$-0.493362\pi$
$709$	1.11047	0.0417047	0.0208524	+	0.999783i	$0.493362\pi$
$710$	2.27454	0.0853620
$711$	8.05632	0.302135
$712$	−12.0823	−0.452802
$713$	−1.02728	−0.0384719
$714$	−27.8648	−1.04281
$715$	50.7373	1.89747
$716$	−17.2297	−0.643904
$717$	32.4206	1.21077
$718$	−21.7289	−0.810915
$719$	3.30434	0.123231	0.0616155	−	0.998100i	$-0.480375\pi$
$719$	3.30434	0.123231	0.0616155	+	0.998100i	$0.480375\pi$
$720$	−4.90063	−0.182636
$721$	−22.0918	−0.822743
$722$	3.92755	0.146168
$723$	17.6785	0.657472
$724$	1.61286	0.0599415
$725$	1.94516	0.0722413
$726$	15.4369	0.572916
$727$	−18.5038	−0.686267	−0.343134	−	0.939287i	$-0.611488\pi$
$727$	−18.5038	−0.686267	−0.343134	+	0.939287i	$0.611488\pi$
$728$	10.9467	0.405711
$729$	−21.8990	−0.811075
$730$	20.0770	0.743085
$731$	−39.9913	−1.47913
$732$	1.38431	0.0511655
$733$	−50.5951	−1.86877	−0.934387	−	0.356260i	$-0.884052\pi$
$733$	−50.5951	−1.86877	−0.934387	+	0.356260i	$0.884052\pi$
$734$	6.31680	0.233157
$735$	−18.8252	−0.694378
$736$	5.61047	0.206805
$737$	−5.41182	−0.199347
$738$	16.0959	0.592497
$739$	−41.0603	−1.51043	−0.755213	−	0.655479i	$-0.772467\pi$
$739$	−41.0603	−1.51043	−0.755213	+	0.655479i	$0.772467\pi$
$740$	3.04914	0.112089
$741$	62.8178	2.30767
$742$	17.7861	0.652947
$743$	−17.0917	−0.627033	−0.313517	−	0.949583i	$-0.601507\pi$
$743$	−17.0917	−0.627033	−0.313517	+	0.949583i	$0.601507\pi$
$744$	0.438212	0.0160656
$745$	14.4951	0.531061
$746$	−11.5385	−0.422453
$747$	−0.747249	−0.0273404
$748$	−30.0381	−1.09830
$749$	−13.8298	−0.505331
$750$	29.1191	1.06328
$751$	−28.0069	−1.02199	−0.510993	−	0.859585i	$-0.670722\pi$
$751$	−28.0069	−1.02199	−0.510993	+	0.859585i	$0.670722\pi$
$752$	−7.12695	−0.259893
$753$	21.3453	0.777866
$754$	7.41932	0.270196
$755$	−20.6220	−0.750510
$756$	−1.05465	−0.0383573
$757$	21.2887	0.773751	0.386875	−	0.922132i	$-0.373554\pi$
$757$	21.2887	0.773751	0.386875	+	0.922132i	$0.373554\pi$
$758$	0.637674	0.0231614
$759$	−56.0910	−2.03598
$760$	−6.97469	−0.252999
$761$	−26.5896	−0.963874	−0.481937	−	0.876206i	$-0.660067\pi$
$761$	−26.5896	−0.963874	−0.481937	+	0.876206i	$0.660067\pi$
$762$	7.94607	0.287856
$763$	2.81295	0.101835
$764$	−5.82764	−0.210837
$765$	−35.2392	−1.27407
$766$	12.4585	0.450144
$767$	0.440978	0.0159228
$768$	−2.39329	−0.0863604
$769$	−16.3533	−0.589715	−0.294858	−	0.955541i	$-0.595272\pi$
$769$	−16.3533	−0.589715	−0.294858	+	0.955541i	$0.595272\pi$
$770$	12.1512	0.437898
$771$	−30.1837	−1.08704
$772$	−7.38076	−0.265639
$773$	−13.9209	−0.500701	−0.250351	−	0.968155i	$-0.580546\pi$
$773$	−13.9209	−0.500701	−0.250351	+	0.968155i	$0.580546\pi$
$774$	15.1709	0.545305
$775$	−0.324545	−0.0116580
$776$	−11.8296	−0.424660
$777$	−6.57699	−0.235948
$778$	4.99535	0.179092
$779$	22.9080	0.820765
$780$	29.0686	1.04082
$781$	−5.28882	−0.189249
$782$	40.3435	1.44268
$783$	−0.714811	−0.0255452
$784$	−4.37835	−0.156370
$785$	−23.4057	−0.835385
$786$	−0.158537	−0.00565483
$787$	−27.4495	−0.978469	−0.489234	−	0.872152i	$-0.662724\pi$
$787$	−27.4495	−0.978469	−0.489234	+	0.872152i	$0.662724\pi$
$788$	−6.33903	−0.225819
$789$	−8.07412	−0.287446
$790$	5.30580	0.188772
$791$	27.7126	0.985345
$792$	11.3951	0.404906
$793$	−3.91050	−0.138866
$794$	−11.8161	−0.419339
$795$	47.2304	1.67509
$796$	8.22834	0.291646
$797$	1.49879	0.0530899	0.0265450	−	0.999648i	$-0.491549\pi$
$797$	1.49879	0.0530899	0.0265450	+	0.999648i	$0.491549\pi$
$798$	15.0444	0.532565
$799$	−51.2480	−1.81303
$800$	1.77250	0.0626673
$801$	32.9585	1.16453
$802$	9.23018	0.325929
$803$	−46.6837	−1.64743
$804$	−3.10056	−0.109348
$805$	−16.3200	−0.575204
$806$	−1.23790	−0.0436030
$807$	−14.6299	−0.514996
$808$	−15.7004	−0.552338
$809$	−54.6251	−1.92052	−0.960258	−	0.279115i	$-0.909959\pi$
$809$	−54.6251	−1.92052	−0.960258	+	0.279115i	$0.909959\pi$
$810$	−17.5025	−0.614975
$811$	46.4138	1.62981	0.814905	−	0.579595i	$-0.196789\pi$
$811$	46.4138	1.62981	0.814905	+	0.579595i	$0.196789\pi$
$812$	1.77687	0.0623559
$813$	−20.6741	−0.725072
$814$	−7.08995	−0.248503
$815$	9.20745	0.322523
$816$	−17.2095	−0.602454
$817$	21.5915	0.755392
$818$	−4.10689	−0.143594
$819$	−29.8608	−1.04342
$820$	10.6006	0.370187
$821$	13.4511	0.469447	0.234724	−	0.972062i	$-0.424581\pi$
$821$	13.4511	0.469447	0.234724	+	0.972062i	$0.424581\pi$
$822$	15.3486	0.535343
$823$	4.22402	0.147240	0.0736200	−	0.997286i	$-0.476545\pi$
$823$	4.22402	0.147240	0.0736200	+	0.997286i	$0.476545\pi$
$824$	−13.6441	−0.475314
$825$	−17.7207	−0.616954
$826$	0.105611	0.00367467
$827$	10.6135	0.369068	0.184534	−	0.982826i	$-0.440922\pi$
$827$	10.6135	0.369068	0.184534	+	0.982826i	$0.440922\pi$
$828$	−15.3045	−0.531867
$829$	31.3436	1.08861	0.544304	−	0.838888i	$-0.316794\pi$
$829$	31.3436	1.08861	0.544304	+	0.838888i	$0.316794\pi$
$830$	−0.492130	−0.0170821
$831$	53.7164	1.86340
$832$	6.76076	0.234387
$833$	−31.4836	−1.09084
$834$	15.5713	0.539189
$835$	17.3088	0.598996
$836$	16.2177	0.560902
$837$	0.119265	0.00412238
$838$	2.20587	0.0762006
$839$	−47.2156	−1.63006	−0.815032	−	0.579415i	$-0.803281\pi$
$839$	−47.2156	−1.63006	−0.815032	+	0.579415i	$0.803281\pi$
$840$	6.96170	0.240202
$841$	−27.7957	−0.958472
$842$	8.38322	0.288905
$843$	23.4499	0.807658
$844$	9.07906	0.312514
$845$	−58.7605	−2.02142
$846$	19.4412	0.668401
$847$	−10.4436	−0.358847
$848$	10.9848	0.377220
$849$	−25.5021	−0.875230
$850$	12.7456	0.437170
$851$	9.52235	0.326422
$852$	−3.03009	−0.103809
$853$	−18.6558	−0.638764	−0.319382	−	0.947626i	$-0.603475\pi$
$853$	−18.6558	−0.638764	−0.319382	+	0.947626i	$0.603475\pi$
$854$	−0.936535	−0.0320476
$855$	19.0258	0.650670
$856$	−8.54142	−0.291940
$857$	40.2994	1.37660	0.688301	−	0.725425i	$-0.258357\pi$
$857$	40.2994	1.37660	0.688301	+	0.725425i	$0.258357\pi$
$858$	−67.5911	−2.30752
$859$	20.0340	0.683552	0.341776	−	0.939781i	$-0.388972\pi$
$859$	20.0340	0.683552	0.341776	+	0.939781i	$0.388972\pi$
$860$	9.99136	0.340703
$861$	−22.8653	−0.779249
$862$	33.7752	1.15039
$863$	−33.8646	−1.15277	−0.576383	−	0.817180i	$-0.695536\pi$
$863$	−33.8646	−1.15277	−0.576383	+	0.817180i	$0.695536\pi$
$864$	−0.651362	−0.0221598
$865$	4.36916	0.148556
$866$	14.7229	0.500304
$867$	−83.0635	−2.82098
$868$	−0.296467	−0.0100627
$869$	−12.3372	−0.418511
$870$	4.71843	0.159970
$871$	8.75871	0.296778
$872$	1.73730	0.0588323
$873$	32.2694	1.09215
$874$	−21.7817	−0.736777
$875$	−19.7001	−0.665986
$876$	−26.7462	−0.903670
$877$	−19.9604	−0.674015	−0.337008	−	0.941502i	$-0.609415\pi$
$877$	−19.9604	−0.674015	−0.337008	+	0.941502i	$0.609415\pi$
$878$	−29.7620	−1.00442
$879$	16.3376	0.551055
$880$	7.50467	0.252982
$881$	−20.5917	−0.693753	−0.346877	−	0.937911i	$-0.612758\pi$
$881$	−20.5917	−0.693753	−0.346877	+	0.937911i	$0.612758\pi$
$882$	11.9434	0.402157
$883$	31.7988	1.07011	0.535057	−	0.844816i	$-0.320290\pi$
$883$	31.7988	1.07011	0.535057	+	0.844816i	$0.320290\pi$
$884$	48.6149	1.63510
$885$	0.280447	0.00942711
$886$	−32.5172	−1.09244
$887$	−23.8611	−0.801176	−0.400588	−	0.916258i	$-0.631194\pi$
$887$	−23.8611	−0.801176	−0.400588	+	0.916258i	$0.631194\pi$
$888$	−4.06200	−0.136312
$889$	−5.37581	−0.180299
$890$	21.7061	0.727590
$891$	40.6972	1.36341
$892$	21.6006	0.723240
$893$	27.6691	0.925912
$894$	−19.3101	−0.645827
$895$	30.9536	1.03466
$896$	1.61915	0.0540920
$897$	90.7800	3.03106
$898$	−4.94895	−0.165149
$899$	−0.200936	−0.00670159
$900$	−4.83509	−0.161170
$901$	78.9890	2.63151
$902$	−24.6487	−0.820712
$903$	−21.5513	−0.717183
$904$	17.1155	0.569253
$905$	−2.89754	−0.0963176
$906$	27.4721	0.912701
$907$	18.8335	0.625356	0.312678	−	0.949859i	$-0.398774\pi$
$907$	18.8335	0.625356	0.312678	+	0.949859i	$0.398774\pi$
$908$	−11.9088	−0.395208
$909$	42.8281	1.42052
$910$	−19.6660	−0.651921
$911$	−28.1234	−0.931768	−0.465884	−	0.884846i	$-0.654264\pi$
$911$	−28.1234	−0.931768	−0.465884	+	0.884846i	$0.654264\pi$
$912$	9.29153	0.307673
$913$	1.14431	0.0378713
$914$	36.6164	1.21116
$915$	−2.48694	−0.0822158
$916$	−5.50831	−0.182000
$917$	0.107256	0.00354192
$918$	−4.68378	−0.154588
$919$	24.0765	0.794211	0.397106	−	0.917773i	$-0.370015\pi$
$919$	24.0765	0.794211	0.397106	+	0.917773i	$0.370015\pi$
$920$	−10.0794	−0.332306
$921$	−57.7469	−1.90282
$922$	−6.28985	−0.207145
$923$	8.55965	0.281744
$924$	−16.1875	−0.532531
$925$	3.00837	0.0989145
$926$	40.0445	1.31594
$927$	37.2189	1.22243
$928$	1.09741	0.0360242
$929$	−8.29456	−0.272136	−0.136068	−	0.990700i	$-0.543446\pi$
$929$	−8.29456	−0.272136	−0.136068	+	0.990700i	$0.543446\pi$
$930$	−0.787259	−0.0258152
$931$	16.9982	0.557093
$932$	−20.0246	−0.655927
$933$	−51.6541	−1.69108
$934$	−3.68241	−0.120492
$935$	53.9642	1.76482
$936$	−18.4423	−0.602804
$937$	−60.1922	−1.96639	−0.983196	−	0.182551i	$-0.941564\pi$
$937$	−60.1922	−1.96639	−0.983196	+	0.182551i	$0.941564\pi$
$938$	2.09764	0.0684905
$939$	22.3621	0.729760
$940$	12.8037	0.417612
$941$	1.70285	0.0555113	0.0277557	−	0.999615i	$-0.491164\pi$
$941$	1.70285	0.0555113	0.0277557	+	0.999615i	$0.491164\pi$
$942$	31.1806	1.01592
$943$	33.1051	1.07805
$944$	0.0652261	0.00212293
$945$	1.89471	0.0616349
$946$	−23.2322	−0.755343
$947$	10.2919	0.334442	0.167221	−	0.985919i	$-0.446521\pi$
$947$	10.2919	0.334442	0.167221	+	0.985919i	$0.446521\pi$
$948$	−7.06827	−0.229567
$949$	75.5548	2.45261
$950$	−6.88142	−0.223263
$951$	48.9103	1.58603
$952$	11.6429	0.377348
$953$	57.2579	1.85476	0.927382	−	0.374115i	$-0.122053\pi$
$953$	57.2579	1.85476	0.927382	+	0.374115i	$0.122053\pi$
$954$	−29.9648	−0.970146
$955$	10.4695	0.338785
$956$	−13.5465	−0.438124
$957$	−10.9714	−0.354655
$958$	−29.7996	−0.962782
$959$	−10.3839	−0.335313
$960$	4.29960	0.138769
$961$	−30.9665	−0.998919
$962$	11.4747	0.369958
$963$	23.2996	0.750819
$964$	−7.38671	−0.237910
$965$	13.2597	0.426845
$966$	21.7411	0.699509
$967$	40.7037	1.30894	0.654471	−	0.756087i	$-0.272891\pi$
$967$	40.7037	1.30894	0.654471	+	0.756087i	$0.272891\pi$
$968$	−6.45006	−0.207313
$969$	66.8130	2.14634
$970$	21.2523	0.682369
$971$	−18.4838	−0.593175	−0.296587	−	0.955006i	$-0.595849\pi$
$971$	−18.4838	−0.593175	−0.296587	+	0.955006i	$0.595849\pi$
$972$	21.3623	0.685197
$973$	−10.5345	−0.337722
$974$	28.4639	0.912043
$975$	28.6799	0.918490
$976$	−0.578412	−0.0185145
$977$	26.4247	0.845401	0.422700	−	0.906270i	$-0.361082\pi$
$977$	26.4247	0.845401	0.422700	+	0.906270i	$0.361082\pi$
$978$	−12.2660	−0.392222
$979$	−50.4716	−1.61308
$980$	7.86582	0.251264
$981$	−4.73907	−0.151307
$982$	−32.0151	−1.02164
$983$	49.9561	1.59335	0.796676	−	0.604406i	$-0.206590\pi$
$983$	49.9561	1.59335	0.796676	+	0.604406i	$0.206590\pi$
$984$	−14.1218	−0.450187
$985$	11.3882	0.362859
$986$	7.89119	0.251307
$987$	−27.6176	−0.879078
$988$	−26.2475	−0.835043
$989$	31.2026	0.992185
$990$	−20.4715	−0.650628
$991$	13.1779	0.418611	0.209305	−	0.977850i	$-0.432880\pi$
$991$	13.1779	0.418611	0.209305	+	0.977850i	$0.432880\pi$
$992$	−0.183100	−0.00581344
$993$	0.794858	0.0252241
$994$	2.04997	0.0650211
$995$	−14.7824	−0.468634
$996$	0.655605	0.0207736
$997$	−20.5793	−0.651754	−0.325877	−	0.945412i	$-0.605659\pi$
$997$	−20.5793	−0.651754	−0.325877	+	0.945412i	$0.605659\pi$
$998$	−34.8962	−1.10462
$999$	−1.10552	−0.0349771

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.f.1.16	✓	89

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.f.1.16	✓	89	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:	\( N \)	\(=\)	\( 8002 = 2 \cdot 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8002.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.39329 q^{3} +1.00000 q^{4} -1.79652 q^{5} +2.39329 q^{6} -1.61915 q^{7} -1.00000 q^{8} +2.72784 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.39329 q^{3} +1.00000 q^{4} -1.79652 q^{5} +2.39329 q^{6} -1.61915 q^{7} -1.00000 q^{8} +2.72784 q^{9} +1.79652 q^{10} -4.17733 q^{11} -2.39329 q^{12} +6.76076 q^{13} +1.61915 q^{14} +4.29960 q^{15} +1.00000 q^{16} +7.19075 q^{17} -2.72784 q^{18} -3.88233 q^{19} -1.79652 q^{20} +3.87510 q^{21} +4.17733 q^{22} -5.61047 q^{23} +2.39329 q^{24} -1.77250 q^{25} -6.76076 q^{26} +0.651362 q^{27} -1.61915 q^{28} -1.09741 q^{29} -4.29960 q^{30} +0.183100 q^{31} -1.00000 q^{32} +9.99756 q^{33} -7.19075 q^{34} +2.90884 q^{35} +2.72784 q^{36} -1.69725 q^{37} +3.88233 q^{38} -16.1805 q^{39} +1.79652 q^{40} -5.90059 q^{41} -3.87510 q^{42} -5.56149 q^{43} -4.17733 q^{44} -4.90063 q^{45} +5.61047 q^{46} -7.12695 q^{47} -2.39329 q^{48} -4.37835 q^{49} +1.77250 q^{50} -17.2095 q^{51} +6.76076 q^{52} +10.9848 q^{53} -0.651362 q^{54} +7.50467 q^{55} +1.61915 q^{56} +9.29153 q^{57} +1.09741 q^{58} +0.0652261 q^{59} +4.29960 q^{60} -0.578412 q^{61} -0.183100 q^{62} -4.41678 q^{63} +1.00000 q^{64} -12.1459 q^{65} -9.99756 q^{66} +1.29552 q^{67} +7.19075 q^{68} +13.4275 q^{69} -2.90884 q^{70} +1.26608 q^{71} -2.72784 q^{72} +11.1755 q^{73} +1.69725 q^{74} +4.24211 q^{75} -3.88233 q^{76} +6.76372 q^{77} +16.1805 q^{78} +2.95337 q^{79} -1.79652 q^{80} -9.74241 q^{81} +5.90059 q^{82} -0.273935 q^{83} +3.87510 q^{84} -12.9184 q^{85} +5.56149 q^{86} +2.62642 q^{87} +4.17733 q^{88} +12.0823 q^{89} +4.90063 q^{90} -10.9467 q^{91} -5.61047 q^{92} -0.438212 q^{93} +7.12695 q^{94} +6.97469 q^{95} +2.39329 q^{96} +11.8296 q^{97} +4.37835 q^{98} -11.3951 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9}+O(q^{10}) \) 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9} + 18 q^{10} - 26 q^{11} - 12 q^{12} + 2 q^{13} + 27 q^{14} - 21 q^{15} + 89 q^{16} - 60 q^{17} - 95 q^{18} + q^{19} - 18 q^{20} - 6 q^{21} + 26 q^{22} - 45 q^{23} + 12 q^{24} + 107 q^{25} - 2 q^{26} - 45 q^{27} - 27 q^{28} - 18 q^{29} + 21 q^{30} - 38 q^{31} - 89 q^{32} - 29 q^{33} + 60 q^{34} - 47 q^{35} + 95 q^{36} - 15 q^{37} - q^{38} - 38 q^{39} + 18 q^{40} - 50 q^{41} + 6 q^{42} - 15 q^{43} - 26 q^{44} - 35 q^{45} + 45 q^{46} - 121 q^{47} - 12 q^{48} + 132 q^{49} - 107 q^{50} + 6 q^{51} + 2 q^{52} - 46 q^{53} + 45 q^{54} - 37 q^{55} + 27 q^{56} - 42 q^{57} + 18 q^{58} - 34 q^{59} - 21 q^{60} + 41 q^{61} + 38 q^{62} - 131 q^{63} + 89 q^{64} - 57 q^{65} + 29 q^{66} - 11 q^{67} - 60 q^{68} + 15 q^{69} + 47 q^{70} - 66 q^{71} - 95 q^{72} - 47 q^{73} + 15 q^{74} - 46 q^{75} + q^{76} - 106 q^{77} + 38 q^{78} - 51 q^{79} - 18 q^{80} + 113 q^{81} + 50 q^{82} - 141 q^{83} - 6 q^{84} - 7 q^{85} + 15 q^{86} - 110 q^{87} + 26 q^{88} - 30 q^{89} + 35 q^{90} + 37 q^{91} - 45 q^{92} - 44 q^{93} + 121 q^{94} - 98 q^{95} + 12 q^{96} + 3 q^{97} - 132 q^{98} - 71 q^{99}+O(q^{100}) \) 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9 + 18 * q^10 - 26 * q^11 - 12 * q^12 + 2 * q^13 + 27 * q^14 - 21 * q^15 + 89 * q^16 - 60 * q^17 - 95 * q^18 + q^19 - 18 * q^20 - 6 * q^21 + 26 * q^22 - 45 * q^23 + 12 * q^24 + 107 * q^25 - 2 * q^26 - 45 * q^27 - 27 * q^28 - 18 * q^29 + 21 * q^30 - 38 * q^31 - 89 * q^32 - 29 * q^33 + 60 * q^34 - 47 * q^35 + 95 * q^36 - 15 * q^37 - q^38 - 38 * q^39 + 18 * q^40 - 50 * q^41 + 6 * q^42 - 15 * q^43 - 26 * q^44 - 35 * q^45 + 45 * q^46 - 121 * q^47 - 12 * q^48 + 132 * q^49 - 107 * q^50 + 6 * q^51 + 2 * q^52 - 46 * q^53 + 45 * q^54 - 37 * q^55 + 27 * q^56 - 42 * q^57 + 18 * q^58 - 34 * q^59 - 21 * q^60 + 41 * q^61 + 38 * q^62 - 131 * q^63 + 89 * q^64 - 57 * q^65 + 29 * q^66 - 11 * q^67 - 60 * q^68 + 15 * q^69 + 47 * q^70 - 66 * q^71 - 95 * q^72 - 47 * q^73 + 15 * q^74 - 46 * q^75 + q^76 - 106 * q^77 + 38 * q^78 - 51 * q^79 - 18 * q^80 + 113 * q^81 + 50 * q^82 - 141 * q^83 - 6 * q^84 - 7 * q^85 + 15 * q^86 - 110 * q^87 + 26 * q^88 - 30 * q^89 + 35 * q^90 + 37 * q^91 - 45 * q^92 - 44 * q^93 + 121 * q^94 - 98 * q^95 + 12 * q^96 + 3 * q^97 - 132 * q^98 - 71 * q^99

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